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STUDIESANDEXERCISES

IN


F O RMAL L OG I C

INCLUDINGAGENERALISATIONOFLOGICALPROCESSESINTHEIRAPPLICATIONTOCOMPLEXINFERENCES

BY


JOHNNEVILLEKEYNES,M.A.,Sc.D.

UNIVERSITYLECTURERINMORALSCIENCEANDFORMERLYFELLOWOFPEMBROKECOLLEGEINTHEUNIVERSITYOFCAMBRIDGE

FOURTHEDITIONRE-WRITTENANDENLARGED

ᲅᲢᲡᲗᲢᲡ

MACMILLANANDCO.,LIMITED

NEWYORK:THEMACMILLANCOMPANY

1906

[TheRightofTranslationandReproductionisreserved]

FirstEdition(Crown8vo.)printed1884.

SecondEdition(Crown8vo.)1887.

ThirdEdition(Demy8vo.)1894.

FourthEdition(Demy8vo.)1906.


PREFACETOTHEFOURTHEDITION.

INthiseditionmanyofthesectionshavebeenre-writtenandagooddealofnew matter has been introduced. The following are some of the moreimportantmodifications.

InPart Ianewdefinitionof“connotativename”isproposed, in thehopethat some misunderstanding may thereby be avoided; and the treatment ofnegativenameshasbeenrevised.

InPart II theproblemof the import of judgments andpropositions in itsvarious aspects is dealt with in much more detail than before, and greaterimportance is attached to distinctionsofmodality.Partly in consequenceofthis, the treatment of conditional and hypothetical propositions has beenmodified. Ihavepartially re-written thechapteron theexistential importofpropositions in order to meet some recent criticisms and to explain mypositionmoreclearly.ManyotherminorchangesinPartIIhavebeenmade.

Amongst the changes in Part III are a more systematic treatment of theprocess of the indirect reduction of syllogisms, and the introduction of achapteronthecharacteristicsofinference.

Anappendixonthefundamentallawsofthoughthasbeenadded;andthetreatmentofcomplexpropositionswhichpreviouslyconstitutedPartIVofthebookhasnowbeenplacedinanappendix.

ThereaderofthiseditionwillperceivemyindebtednesstoSigwart’sLogic.I have receivedvaluablehelp fromProfessor J.S.Mackenzie and frommyson,MrJ.M.Keynes;andIcannotexpresstoostronglythedebtIoncemoreowetoMrW.E.Johnson,whobyhiscriticismshasenabledmetoimprovemyexpositioninmanypartsofthebook,andalsotoavoidsomeerrors.

J.N.KEYNES.


6,HARVEYROAD, CAMBRIDGE, 4September1906.

vi


PREFACETOTHEFIRSTEDITION.1

1 Withsomeomissions.

IN addition toa somewhatdetailedexpositionofcertainportionsofwhatmaybecalled thebook-workofformal logic, thefollowingpagescontainanumberofproblemsworkedoutindetailandunsolvedproblems,bymeansofwhichthestudentmaytesthiscommandoverlogicalprocesses.

In theexpositoryportionsofParts I, II,andIII,dealingrespectivelywithterms, propositions, and syllogisms, the traditional lines are in the mainfollowed, though with certain modifications; e.g., in the systematisation ofimmediate inferences, and in several points of detail in connexionwith thesyllogism. For purposes of illustration Euler’s diagrams are employed to agreaterextentthanisusualinEnglishmanuals.

In Part IV, which contains a generalisation of logical processes in theirapplicationtocomplexinferences,asomewhatnewdepartureistaken.SofarasIamaware thispartconstitutes thefirstsystematicattempt thathasbeenmade to deal with formal reasonings of the most complicated characterwithout the aidofmathematicalorother symbolsofoperation, andwithoutabandoning the ordinary non-equational or predicative form of proposition.ThisattempthasonthewholemetwithgreatersuccessthanIhadanticipated;andIbelievethatthemethodsformulatedwillbefoundtobebothaseasyandaseffectiveasthesymbolicalmethodsofBooleandhisfollowers.Thebookconcludes with a general and sure method of solution of what ProfessorJevonscalledtheinverseproblem,andwhichhehimselfseemedtoregardassolubleonlybyaseriesofguesses.

ThewritersonlogictowhomIhavebeenchieflyindebtedareDeMorgan,Jevons,andVenn.ToMrVennIampeculiarlyindebted,notmerelybyreasonofhispublishedwritings,viiespeciallyhisSymbolicLogic,butalsoformost


valuablesuggestionsandcriticismswhilethisbookwasinprogress.Iamgladtohave thisopportunityofexpressing tohimmy thanks for theungrudginghelphehasaffordedme.IamalsoundergreatobligationtoMissMartinofNewnhamCollege,and toMrCaldecottofSt John’sCollege, forcriticismswhichIhavefoundextremelyhelpful.

CAMBRIDGE,19January1884.

viii


PREFACETOTHESECONDEDITION.

THISeditionhasbeencarefullyrevised,andnumeroussectionshavebeenalmostentirelyre-written.

In addition to the introduction of some brief prefatory sections, thefollowingareamong themore importantmodifications. InPart Ianattempthasbeenmade todifferentiate themeaningsof the three termsconnotation,intension,comprehension,withthehopethatsuchdifferentiationofmeaningmayhelptoremoveanambiguitywhichisthesourceofmuchofthecurrentcontroversy on the subject of connotation. In Part II a distinction betweenconditionalandhypotheticalpropositionsisadoptedforwhichIamindebtedto Mr W. E. Johnson; and the treatment of the existential import ofpropositionshasbeenbothexpandedandsystematised. InPart IVparticularpropositions,whichinthefirsteditionwerepracticallyneglected,aretreatedindetail;and,whilethenumberofmereexerciseshasbeendiminished,manypoints of theory have received considerable development. Throughout thebooktheunansweredexercisesarenowseparatedfromtheexpositorymatterandplacedtogetherattheendoftheseveralchaptersinwhichtheyoccur.Anindexhasbeenadded.

Ihavetothankseveralfriendsandcorrespondents,amongstwhomImustespeciallymentionMrHenryLaurieoftheUniversityofMelbourneandMrW.E.JohnsonofKing’sCollege,Cambridge,forsuggestionsandcriticismsfromwhichIhavederivedthegreatestassistance.MrJohnsonhaskindlyreadthe proof sheets throughout; and I am particularly indebted to him for thegenerousmannerinwhichhehasplacedatmydisposalnotonlyhistimebutalsotheresultsofhisownworkonvariouspointsofformallogic.

CAMBRIDGE,22June1887.


ix


PREFACETOTHETHIRDEDITION.

THIS edition has been in great part re-written and the book is againconsiderablyenlarged.

InPart I themutual relations between the extension and the intension ofnamesare examined fromanewpointofview, and thedistinctionbetweenreal and verbal propositions is treated more fully than in the two earliereditions.InPartIImoreattentionispaidtotablesofequivalentpropositions,certaindevelopmentsofEuler’sandLambert’sdiagramsare introduced, theinterpretationofpropositionsinextensionandintensionisdiscussedinmoredetail,andabriefexplanationisgivenofthenatureoflogicalequations.Thechapters on the existential import of propositions and on conditional,hypothetical, and disjunctive (or, as I now prefer to call them, alternative)propositionshavealsobeenexpanded,and thepositionwhich I takeon thevariousquestionsraisedinthesechaptersisIhopemoreclearlyexplained.InParts III and IV there is less absolutely new matter, but the minormodificationsarenumerous.Anappendixisaddedcontainingabriefaccountofthedoctrineofdivision.

In the preface to earlier editions I was glad to have the opportunity ofacknowledgingmyindebtednesstoProfessorCaldecott,toMrW.E.Johnson,to Professor Henry Laurie, to Dr Venn, and to Mrs Ward. In the presentedition my indebtedness to Mr Johnson is again very great. Many newdevelopmentsareduetohissuggestion,andineveryimportantdiscussioninthebookIhavebeenmostmateriallyhelpedbyhiscriticismandadvice.

CAMBRIDGE,25July1894.


TABLEOFCONTENTS.


INTRODUCTION.SECTION PAGE

1. TheGeneralCharacterofLogic 12. FormalLogic 13. LogicandLanguage 34. LogicandPsychology 55. TheUtilityofLogic 6

PARTI.

TERMS.

CHAPTERI.

THELOGICOFTERMS.

6. TheThreePartsofLogicalDoctrine 87. NamesandConcepts 108. TheLogicofTerms 119. GeneralandSingularNames 1110. ProperNames 1311. CollectiveNames 1412. ConcreteandAbstractNames 1613. CanAbstractNamesbesubdividedintoGeneraland

Singular?19

14,15. Exercises 21

CHAPTERII.

EXTENSIONANDINTENSION.

16. TheExtensionandtheIntensionofNames 2217. Connotation,SubjectiveIntension,andComprehension. 2318. Sigwart’sdistinctionbetweenEmpirical,Metaphysical, 27


andLogicalConceptsxii

19. ConnotationandEtymology 2820. FixityofConnotation 2821. ExtensionandDenotation 2922. DependenceofExtensionandIntensionupononeanother 3123. InverseVariationofExtensionandIntension 3524. ConnotativeNames 4025. Arepropernamesconnotative? 41

26to30. Exercises

47

CHAPTERIII.

REAL,VERBAL,ANDFORMALPROPOSITIONS.

31. Real,Verbal,andFormalPropositions 4932. NatureoftheAnalysisinvolvedinAnalyticPropositions 53

33to37. Exercises

56

CHAPTERIV.

NEGATIVENAMESANDRELATIVENAMES.

38. PositiveandNegativeNames 5739. IndefiniteCharacterofNegativeNames 5940. ContradictoryTerms 6141. ContraryTerms 6242. RelativeNames 63

43to45. Exercises

65

PARTII.


PROPOSITIONS.

CHAPTERI.

IMPORTOFJUDGMENTSANDPROPOSITIONS.

46. JudgmentsandPropositions 6647. TheAbstractCharacterofLogic 6848. NatureoftheEnquiryintotheImportofPropositions 7049. TheObjectiveReferenceinJudgments 7450. TheUniversalityofJudgments 7651. TheNecessityofJudgments 7752. Exercise 78

xiii

CHAPTERII.

KINDSOFJUDGMENTSANDPROPOSITIONS.

53. TheClassificationofJudgments 7954. Kant’sClassificationofJudgments 8155. SimpleJudgmentsandCompoundJudgments 8256. TheModalityofJudgments 8457. ModalityinrelationtoSimpleJudgments 8558. SubjectiveDistinctionsofModality 8659. ObjectiveDistinctionsofModality 8760. ModalityinrelationtoCompoundJudgments 9061. TheQuantityandtheQualityofPropositions 9162. ThetraditionalSchemeofPropositions 9263. TheDistributionofTermsinaProposition 9564. TheDistinctionbetweenSubjectandPredicateinthe

traditionalSchemeofPropositions96

65. UniversalPropositions 9766. ParticularPropositions 10067. SingularPropositions 102


68. PlurativePropositionsandNumericallyDefinitePropositions

103

69. IndefinitePropositions 10570. MultipleQuantification 10571. InfiniteorLimitativePropositions 106

72to78. Exercises

107

CHAPTERIII.

THEOPPOSITIONOFPROPOSITIONS.

79. TheSquareofOpposition 10980. ContradictoryOpposition 11181. ContraryOpposition 11482. TheOppositionofSingularPropositions 11583. TheOppositionofModalPropositions 11684. ExtensionoftheDoctrineofOpposition 11785. TheNatureofSignificantDenial 119

86to95. Exercises.

124

xiv

CHAPTERIV.

IMMEDIATEINFERENCES.

96. TheConversionofCategoricalPropositions 12697. SimpleConversionandConversionperaccidens. 12898. InconvertibilityofParticularNegativePropositions 13099. LegitimacyofConversion 130100. TableofPropositionsconnectinganytwoterms 132101. TheObversionofCategoricalPropositions 133102. TheContrapositionofCategoricalPropositions 134103. TheInversionofCategoricalPropositions 137104. TheValidityofInversion 139


105. SummaryofResults 140106. TableofPropositionsconnectinganytwotermsandtheir

contradictories141

107. MutualRelationsofthenon-equivalentPropositionsconnectinganytwotermsandtheircontradictories

142

108. TheEliminationofNegativeTerms 144109. OtherImmediateInferences 147110. Reductionofimmediateinferencestothemediateform 151

111to124. Exercises

153

CHAPTERV.

THEDIAGRAMMATICREPRESENTATIONOFPROPOSITIONS.

125. TheuseofDiagramsinLogic 156126. Euler’sDiagrams 157127. Lambert’sDiagrams 163128. DrVenn’sDiagrams 166129. Expressionofthepossiblerelationsbetweenanytwo

classesbymeansofthepropositionalformsA,E,I,O168

130. Euler’sdiagramsandtheclass-relationsbetweenS,not-S,P,not-P

170

131. Lambert’sdiagramsandtheclass-relationsbetweenS,not-S,P,not-P

174

132to134. Exercises

176

xv

CHAPTERVI.

PROPOSITIONSINEXTENSIONANDININTENSION.

135. FourfoldImplicationofPropositionsinConnotationand

Denotation177

(1)Subjectindenotation,predicateinconnotation 179(2)Subjectindenotation,predicateindenotation 181


(3)Subjectinconnotation,predicateinconnotation 184(4)Subjectinconnotation,predicateindenotation 186

136. TheReadingofPropositionsinComprehension 187

CHAPTERVII.

LOGICALEQUATIONSANDTHEQUANTIFICATIONOFTHEPREDICATE.

137. TheemploymentofthesymbolofEqualityinLogic 189138. TypesofLogicalEquations 191139. TheexpressionofPropositionsasEquations 194140. Theeightpropositionalformsresultingfromtheexplicit

QuantificationofthePredicate195

141. SirWilliamHamilton’sfundamentalPostulateofLogic 195142. AdvantagesclaimedfortheQuantificationofthe

Predicate196

143. ObjectionsurgedagainsttheQuantificationofthePredicate

197

144. ThemeaningtobeattachedtothewordsomeintheeightpropositionalformsrecognisedbySirWilliamHamilton

199

145. Theuseofsomeinthesenseofsomeonly 202146. TheinterpretationoftheeightHamiltonianformsof

proposition,somebeingusedinitsordinarylogicalsense

203

147. ThepropositionsUandY 204148. Thepropositionη 206149. Thepropositionω 206150. SixfoldScheduleofPropositionsobtainedbyrecognising

Yandη,inadditiontoA,E,I,O207

151,152. Exercises

209

CHAPTERVIII.

THEEXISTENTIALIMPORTOFCATEGORICALPROPOSITIONS.


153. ExistenceandtheUniverseofDiscourse 210154. FormalLogicandtheExistentialImportofPropositions 215155. TheExistentialFormulationofPropositions 218156. VariousSuppositionsconcerningtheExistentialImportof

CategoricalPropositions218

xvi

157. ReductionofthetraditionalformsofpropositiontotheformofExistentialPropositions

221

158. ImmediateInferencesandtheExistentialImportofPropositions

223

159. TheDoctrineofOppositionandtheExistentialImportofPropositions

227

160. TheOppositionofModalPropositionsconsideredinconnexionwiththeirExistentialImport

231

161. Jevons’sCriterionofConsistency 232162. TheExistentialImportofthePropositionsincludedinthe

TraditionalSchedule234

163. TheExistentialImportofModalPropositions 244

164to172. Exercises

245

CHAPTERIX.

CONDITIONALANDHYPOTHETICALPROPOSITIONS.

173. ThedistinctionbetweenConditionalPropositionsand

HypotheticalPropositions249

174. TheImportofConditionalPropositions 252175. ConditionalPropositionsandCategoricalPropositions 253176. TheOppositionofConditionalPropositions 256177. ImmediateInferencesfromConditionalPropositions 259178. TheImportofHypotheticalPropositions 261179. TheOppositionofHypotheticalPropositions 264180. ImmediateInferencesfromHypotheticalPropositions 268181. HypotheticalPropositionsandCategoricalPropositions 270182. AllegedReciprocalCharacterofConditionaland 270


HypotheticalJudgments183to188. Exercises

273

CHAPTERX.

DISJUNCTIVE(ORALTERNATIVE)PROPOSITIONS.

189. ThetermsDisjunctiveandAlternativeasappliedto

Propositions275

190. TwotypesofAlternativePropositions 276191. TheImportofDisjunctive(Alternative)Propositions 277192. SchemeofAssertoricandModalPropositions 282

193. TheRelationofDisjunctive(Alternative)PropositionstoConditionalsandHypotheticals

282

194to196. Exercises

284

xvii

PARTIII.

SYLLOGISMS.

CHAPTERI.

THERULESOFTHESYLLOGISM.

197. TheTermsoftheSyllogism 285198. ThePropositionsoftheSyllogism 287199. TheRulesoftheSyllogism 287200. CorollariesfromtheRulesoftheSyllogism 289201. RestatementoftheRulesoftheSyllogism 291202. DependenceoftheRulesoftheSyllogismuponone

another291

203. StatementoftheindependentRulesoftheSyllogism 293


204. ProofoftheRuleofQuality 294205. Twonegativepremissesmayyieldavalidconclusion;but

notsyllogistically295

206. OtherapparentexceptionstotheRulesoftheSyllogism 297207. Syllogismswithtwosingularpremisses 298208. Chargeofincompletenessbroughtagainsttheordinary

syllogisticconclusion300

209. TheconnexionbetweentheDictumdeomnietnulloandtheordinaryRulesoftheSyllogism

301

210to242. Exercises

302

CHAPTERII.

THEFIGURESANDMOODSOFTHESYLLOGISM.

243. FigureandMood 309244. TheSpecialRulesoftheFigures;andtheDetermination

oftheLegitimateMoodsineachFigure309

245. WeakenedConclusionsandSubalternMoods 313246. StrengthenedSyllogisms 314247. Thepeculiaritiesandusesofeachofthefourfiguresof

thesyllogism315

248to255. Exercises

317

xviii

CHAPTERIII.

THEREDUCTIONOFSYLLOGISMS.

256. TheProblemofReduction 318257. IndirectReduction 318258. ThemnemoniclinesBarbara,Celarent,&c. 319259. ThedirectreductionofBarocoandBocardo 323260. ExtensionoftheDoctrineofReduction 324


261. IsReductionanessentialpartoftheDoctrineoftheSyllogism?

325

262. TheFourthFigure 328263. IndirectMoods 329264. FurtherdiscussionoftheprocessofIndirectReduction 331265. TheAntilogism 332266. EquivalenceoftheMoodsofthefirstthreeFiguresshewn

bytheMethodofIndirectReduction333

267. TheMoodsofFigure4intheirrelationtooneanother 334268. EquivalenceoftheSpecialRulesoftheFirstThree

Figures335

269. SchemeoftheValidMoodsofFigure1 336270. SchemeoftheValidMoodsofFigure2 336271. SchemeoftheValidMoodsofFigure3 337272. DictumforFigure4 338

273to287. Exercises

339

CHAPTERIV.

THEDIAGRAMMATICREPRESENTATIONOFSYLLOGISMS.

288. Euler’sdiagramsandsyllogisticreasonings 341289. Lambert’sdiagramsandsyllogisticreasonings 344290. DrVenn’sdiagramsandsyllogisticreasonings 345

291to300. Exercises

347

CHAPTERV.

CONDITIONALANDHYPOTHETICALSYLLOGISMS.

301. TheConditionalSyllogism,theHypotheticalSyllogism,

andtheHypothetico-CategoricalSyllogism348

302. DistinctionsofMoodandFigureinthecaseofConditionalandHypotheticalSyllogisms

349


303. FallaciesinHypotheticalSyllogisms 350304. TheReductionofConditionalandHypothetical

Syllogisms351

xix

305. TheMoodsoftheMixedHypotheticalSyllogism 352306. FallaciesinMixedHypotheticalSyllogisms 353307. TheReductionofMixedHypotheticalSyllogisms 354

308. Isthereasoningcontainedinthemixedhypotheticalsyllogismmediateorimmediate?

354

309to315. Exercises

358

CHAPTERVI.

DISJUNCTIVESYLLOGISMS.

316. TheDisjunctiveSyllogism 359317. Themodusponendotollens 361318. TheDilemma 363


319to

321.Exercises

366

CHAPTERVII.

IRREGULARANDCOMPOUNDSYLLOGISMS.

322. TheEnthymeme 367323. ThePolysyllogismandtheEpicheirema 368324. TheSorites 370325. TheSpecialRulesoftheSorites 372326. ThepossibilityofaSoritesinaFigureotherthantheFirst 373327. Ultra-totalDistributionoftheMiddleTerm 376328. TheQuantificationofthePredicateandtheSyllogism 378329. TableofvalidmoodsresultingfromtherecognitionofYand

ηinadditiontoA,E,I,O381

330. FormalInferencesnotreducibletoordinarySyllogisms 384331to

341.Exercises

388

CHAPTERVIII.

PROBLEMSONTHESYLLOGISM.

342. Bearingoftheexistentialinterpretationofpropositionsupon

thevalidityofsyllogisticreasonings390

343. Connexionbetweenthetruthandfalsityofpremissesandconclusioninavalidsyllogism

394

344. Argumentsfromthetruthofonepremissandthefalsityoftheotherpremissinavalidsyllogism,orfromthefalsityofonepremisstothetruthoftheconclusion,orfromthetruthofonepremisstothefalsityoftheconclusion

396

345. NumericalMoodsoftheSyllogism 400346 403


to375.

Exercises

xx

CHAPTERIX.

THECHARACTERISTICSOFINFERENCE.

376. TheNatureofLogicalInference 413377. TheParadoxofInference 414378. Thenatureofthedifferencethattheremustbebetween

premissesandconclusioninaninference415

379. TheDirectImportandtheImplicationsofaProposition 420380. SyllogismsandImmediateInferences 423381. ThechargeofpetitioprincipiibroughtagainstSyllogistic

Reasoning424

CHAPTERX.

EXAMPLESOFARGUMENTSANDFALLACIES.

382to

408.Exercises

431

APPENDIXA.

THEDOCTRINEOFDIVISION.

409. LogicalDivision 441410. PhysicalDivision,MetaphysicalDivision,andVerbal

Division442

411. RulesofLogicalDivision 443412. DivisionbyDichotomy 445413. TheplaceoftheDoctrineofDivisioninLogic 446


APPENDIXB.

THEFUNDAMENTALLAWSOFTHOUGHT.

414. TheThreeLawsofThought 450415. TheLawofIdentity 451416. TheLawofContradiction 454417. TheSophismof“TheLiar” 457418. TheLawofExcludedMiddle 458419. Groundsonwhichtheabsoluteuniversalityandnecessityof

thelawofexcludedmiddlehavebeendenied460

420. AretheLawsofThoughtalsoLawsofThings? 463421. MutualRelationsofthethreeLawsofThought 464422. TheLawsofThoughtinrelationtoImmediateInferences 464423. TheLawsofThoughtandFormalMediateInferences 466

xxi

APPENDIXC.

AGENERALISATIONOFLOGICALPROCESSESINTHEIRAPPLICATIONTOCOMPLEXPROPOSITIONS.

CHAPTERI.

THECOMBINATIONOFTERMS.

424. ComplexTerms 468425. OrderofCombinationinComplexTerms 469426. TheOppositionofComplexTerms 470427. DualityofFormalEquivalencesinthecaseofComplex

Terms472

428. LawsofDistribution 472


429. LawsofTautology 473430. LawsofDevelopmentandReduction 474431. LawsofAbsorption 475432. LawsofExclusionandInclusion 475433. SummaryofFormalEquivalencesofComplexTerms 475434. TheConjunctiveCombinationofAlternativeTerms 478435to

439.Exercises

477

CHAPTERII.

COMPLEXPROPOSITIONSANDCOMPOUNDPROPOSITIONS.

440. ComplexPropositions 478441. TheOppositionofComplexPropositions 478442. CompoundPropositions 478443. TheOppositionofCompoundPropositions 480444. FormalEquivalencesofCompoundPropositions 480445. TheSimplificationofComplexPropositions 481446. TheResolutionofUniversalComplexPropositionsinto

EquivalentCompoundPropositions483

447. TheResolutionofParticularComplexPropositionsintoEquivalentCompoundPropositions

484

448. TheOmissionofTermsfromComplexPropositions 485449. TheIntroductionofTermsintoComplexPropositions 485450. InterpretationofAnomalousForms 486451to

453.Exercises

487

xxii

CHAPTERIII.

IMMEDIATEINFERENCESFROMCOMPLEXPROPOSITIONS.

454. TheObversionofComplexPropositions 488


455. TheConversionofComplexPropositions 489456. TheContrapositionofComplexPropositions 490457. SummaryoftheresultsobtainablebyObversion,Conversion,

andContraposition493

458to

473.Exercises

494

CHAPTERIV.

THECOMBINATIONOFCOMPLEXPROPOSITIONS.

474. TheProblemofcombiningComplexPropositions 498475. TheConjunctiveCombinationofUniversalAffirmatives 498476. TheConjunctiveCombinationofUniversalNegatives 499477. TheConjunctiveCombinationofUniversalswithParticulars

ofthesameQuality500

478. TheConjunctiveCombinationofAffirmativeswithNegatives

501

479. TheConjunctiveCombinationofParticularswithParticulars 501480. TheAlternativeCombinationofUniversalPropositions 502481. TheAlternativeCombinationofParticularPropositions 502482. TheAlternativeCombinationofParticularswithUniversals 502483to

486.Exercises

503

CHAPTERV.

INFERENCESFROMCOMBINATIONSOFCOMPLEXPROPOSITIONS.

487. Conditionsunderwhichauniversalpropositionaffords

informationinregardtoanygiventerm504

488. Informationjointlyaffordedbyaseriesofuniversalpropositionswithregardtoanygiventerm

506

489. TheProblemofElimination 508


490. EliminationfromUniversalAffirmatives 509491. EliminationfromUniversalNegatives 510492. EliminationfromParticularAffirmatives 511493. EliminationfromParticularNegatives 511494. Orderofprocedureintheprocessofelimination 511495to

533.Exercises

512

xxiii

CHAPTERVI.

THEINVERSEPROBLEM.

534. NatureoftheInverseProblem 525535. AGeneralSolutionoftheInverseProblem 527536. AnotherMethodofSolutionoftheInverseProblem 530537. AThirdMethodofSolutionoftheInverseProblem 531538. MrJohnson’sNotationfortheSolutionofLogicalProblems 533539. TheInverseProblemandSchröder’sLawofReciprocal

Equivalences534

540to

550.Exercises

535

INDEX 539


xxiv

REFERENCE LIST OF INITIAL LETTERS SHEWING THEAUTHORSHIP OR SOURCE OF QUESTIONS ANDPROBLEMS.

B=ProfessorJ.I.Beare,TrinityCollege,Dublin;C=UniversityofCambridge;J=MrW.E.Johnson,King’sCollege,Cambridge;K=DrJ.N.Keynes,PembrokeCollege,Cambridge;L=UniversityofLondon;M=UniversityofMelbourne;N=ProfessorJ.S.Nicholson,UniversityofEdinburgh;O=UniversityofOxford;O’S=MrC.A.O’Sullivan,TrinityCollege,Dublin;R=thelateProfessorG.CroomRobertson;RR=MrR.A.P.Rogers,TrinityCollege,Dublin;T=DrF.A.Tarleton,TrinityCollege,Dublin;V=DrJ.Venn,GonvilleandCaiusCollege,Cambridge;W=ProfessorJ.Ward,TrinityCollege,Cambridge.

Note.AfewproblemshavebeenselectedfromthepublishedwritingsofBoole,DeMorgan,Jevons,Solly,Venn,andWhately,fromthePortRoyalLogic,andfromtheJohnsHopkinsStudiesinLogic.Inthesecasesthesourceoftheproblemisappendedinfull.


S T UD I E S A NDEX E R C I S E S I N F O RMAL

LOG I C .


INTRODUCTION.

1.TheGeneralCharacterofLogic.—Logicmaybedefinedasthesciencewhich investigates the general principles of valid thought. Its object is todiscuss the characteristics of judgments, regarded not as psychologicalphenomenabutasexpressingourknowledgeandbeliefs;and,inparticular,itseeks to determine the conditions under which we are justified in passingfromgivenjudgmentstootherjudgmentsthatfollowfromthem.

Asthusdefined,logichasinviewanideal;it isconcernedfundamentallywithhowweoughttothink,andonlyindirectlyandasameanstoanendwithhowwe actually think. It may accordingly be described as a normative orregulative science. This character it possesses in common with ethics andaesthetics. These three branches of knowledge—all of them based onpsychology—formauniquetrio,tobedistinguishedfrompositivesciencesontheonehand,andfrompracticalartsontheother.Itmaybesaidroughlythatthey are concerned with the ideal in the domains of thought, action, andfeelingrespectively.Logicseekstodeterminethegeneralprinciplesofvalidthought,ethics thegeneralprinciplesofrightconduct,aesthetics thegeneralprinciplesofcorrecttaste.

2. Formal Logic.—As regards the scope of logic, one of the principalquestions ordinarily raised is whether the science is formal or material,subjectiveorobjective,concernedwith2thoughtsorwiththings.Itisusualtosaythat logicis formal, insofaras it isconcernedmerelywiththeformofthought, that is, with our manner of thinking irrespective of the particularobjects aboutwhichwe are thinking; and that it ismaterial, in so far as itregardsasfundamentaltheobjectivereferenceofourthought,andrecognisesasof essential importance thedifferencesexisting in theobjects themselvesaboutwhichwethink.

Logic is certainly formal, or at any rate non-material, in the sense that itcannot guarantee the actual objective or material truth of any particular


conclusions.Moreoveranyvalidreasoningwhatsoevermustconformtosomedefinite type, or—in other words—must be reducible to some determinateform;andoneofthemainobjectsoflogicisbyabstractiontodiscoverwhatarethevarioustypesorformstowhichallvalidreasoningmaybereduced.

But, on the other hand, it is essential that logic should recognise anobjectivereferenceineveryjudgment,thatis,areferenceoutsidethestateofmind which constitutes the judgment itself: apart from this, as we shallendeavourtoshewinmoredetaillateron,thetruenatureofjudgmentcannotbe understood. It is,moreover, possible for logic to examine and formulatecertain general conditions which must be satisfied if our thoughts andjudgmentsaretohaveobjectivevalidity;andthesciencemayrecogniseanddiscuss certaingeneralpresuppositions relating to externalnaturewhichareinvolvedinpassingfromtheparticularfactsofobservationtogenerallaws.

Logicfullytreatedhasthenbothaformalandamaterialside.Thequestionmayindeedberaisedwhetherthedistinctionbetweenformandmatterisnotarelative, rather than an absolute, distinction. All sciences are in a senseformal, since theyabstract to someextent from thematterof thought.Thusphysics abstracts in themain from the chemical properties of bodies,whilegeometry abstracts also from their physical properties, considering theirfigure only. In this way we becomemore andmore formal as we becomemore andmore general; and logicmaybe said to bemore abstract,more 3general, more formal, than any other science, except perhaps puremathematics.

It is tobeadded that,within thedomainof logic itself, theanswer to thequestionwhethertwogivenpropositionshaveorhavenotthesameformmaydependupon theparticular systemofpropositions inconnexionwithwhichtheyareconsidered.Thus,ifwecarryouranalysisnofurtherthanisusualinordinary formal logic, the two propositions,Every angle in a semi-circle isequaltoarightangle,Anytwosidesofatrianglearetogethergreaterthanthe thirdside,maybeconsidered tobe identical in form.Each isuniversal,andeachisaffirmative;theydifferonlyinmatter.Butitwillbefoundthatinthe logicof relatives, towhichfurther referencewillsubsequentlybemade,thetwopropositions(oneexpressinganequalityandtheotheraninequality)


mayberegardedasdifferinginformaswellasinmatter;and,moreover,thatthe difference between them in form is capable of being symbolicallyexpressed.

The difficulty of assigning a distinctive scope to formal logic parexcellenceisincreasedbythefactthatcertainproblemsfallingnaturallyintothedomainofmateriallogic—forexample,theinductivemethods—admituptoacertainpointofapurelyformaltreatment.

Itisnotpossiblethentodrawahardandfastlineandtosaythatacertaindeterminateportionoflogicisformal,andthattherestisnotformal.Wemustcontentourselveswiththestatementthatwhenwespeakofformallogicinadistinctivesensewemeanthemostabstractpartsofthescience,inwhichnopresuppositionsaremade relating toexternalnature, and inwhich—beyondthe recognition of the necessary objective reference contained in alljudgments—thereisanabstractionfromthematterofthought.Becausetheyaresoabstract,theproblemsofformallogicasthusconceivedadmitusuallyof symbolic treatment; and it iswith problems admitting of such treatmentthatweshallmoreparticularlyconcernourselvesinthefollowingpages.

3. Logic and Language.—Some logicians, in their treatment of theproblemsofformallogic,endeavourtoabstractnot4merelyfromthematterofthoughtbutalsofromthelanguagewhichistheinstrumentofthought.Thismethodoftreatmentisnotadoptedinthefollowingpages.Inordertojustifythe adoption of the alternativemethod, it is not necessary tomaintain thatthought is altogether impossible without language. It is enough that allthought-processesofanydegreeofcomplexityareasamatteroffactcarriedonbytheaidoflanguage,andthatthought-productsarenormallyexpressedinlanguage.Thatlanguageisinthissensetheuniversalinstrumentofthoughtwillnotbedenied;anditseemsafaircorollarythattheprinciplesbywhichvalid thought is regulated, and more especially the application of theseprinciples to the criticism of thought-products, cannot be adequatelydiscussed,unlessaccountistakenofthewayinwhichthisinstrumentactuallyperformsitsfunctions.

Languageisfullofambiguities,anditisimpossibletoproceedfarwiththeproblemswithwhichlogicisconcerneduntilapreciseinterpretationhasbeen


placed upon certain forms of words as representing thought. In ordinarydiscourse,totakeasimpleexample,thewordsomemayormaynotbeusedinasenseinwhichitisexclusiveofall ;itmaybeunderstoodtomeannot-allaswell asnot-none, or its fullmeaningmay be taken to benot-none. Thelogicianmustdecideinwhichofthesesensesthewordistobeunderstoodinany given scheme of propositional forms.Now, if thoughtwere consideredexclusivelyinitself,suchaquestionasthiscouldnotarise;ithastodowiththeexpressionof thought in language.The fact that suchquestionsdoariseandcannothelparisingshews thatactually toeliminateall considerationoflanguagefromlogicisanimpossibility.Anotinfrequentresultofattemptingto rise above mere considerations of language is needless prolixity anddogmatisminregardtowhatarereallyverbalquestions,thoughtheyarenotrecognisedassuch.

The method of treating logic here advocated is sometimes callednominalist, and the opposed method conceptualist. A word or two ofexplanation is, however, desirable in order that this use of terms may notprove misleading. Nominalism and conceptualism usually denote certaindoctrines concerning the 5 nature of general notions. Nominalism isunderstoodtoinvolvetheassertionthatgeneralitybelongstolanguagealoneand that there is nothing general in thought. But a so-called nominalisttreatment of logic does not involve this. It involves no more than a clearrecognitionof the importanceof languageas the instrumentof thought;andthis isacircumstanceuponwhichmodernadvocatesofconceptualismhavethemselvesinsisted.

Itisperhapsnecessarytoaddthatontheviewheretakenlogicinnowaybecomesamerebranchofgrammar,nordoesitceasetohaveaplaceamongstthe mental sciences. Whatever may be the aid derived from language, itremainstruethatthevalidityofformalreasoningsdependsultimatelyonlawsofthought.Formallogicis,therefore,stillconcernedprimarilywiththought,andonlysecondarilywithlanguageastheinstrumentofthought.

In our subsequent discussion of the relation of terms toconcepts, and ofpropositionstojudgments,weshall return toaconsiderationof thequestionraisedinthissection.2


2 Seesections7and46.

4. Logic and Psychology.—Since processes of reasoning are mentalprocessesdependingupontheconstitutionofourminds, theyfallwithinthecognizance of psychology as well as of logic. But laws of reasoning areregarded from different points of view by these two sciences. Psychologydeals with such laws in the sense of uniformities, that is, as laws inaccordancewithwhichmenare foundby experiencenormally to think andreason.Logic,on theotherhand,dealswith lawsof reasoningas regulativeandauthoritative,asaffordingcriteriaby theaidofwhichvalidand invalidreasoningsmaybediscriminated,andasdetermining theformal relations inwhichdifferentproductsofthoughtstandtooneanother.

Looking at the relations between logic and psychology from a slightlydifferent standpoint, we observe that while the latter is concernedwith theactual, the former is concerned with the ideal. Logic does not, likepsychology,treatofallthewaysinwhichmenactuallyreachconclusions,orof all the various modes in which, through the association of ideas orotherwise, one belief actually generates another. It is concerned with 6reasonings only as regards their cogency, and with the dependence of onejudgmentuponanotheronlyinsofarasitisadependenceinrespectofproof.

There are various other ways in which the contrast between the twosciences may be expressed. We may, for example, say that psychology isconcerned with thought-processes, logic with thought-products; or thatpsychology is concerned with the origin of our beliefs, logic with theirvalidity.

Logichasthusauniquecharacterofitsown,andisnotamerebranchofpsychology.Psychologicalandlogicaldiscussionsarenodoubtapttooverlapone another at certain points, in connexion, for example, with theories ofconceptionandjudgment.Inthefollowingpages,however,thepsychologicalsideoflogiciscomparativelylittletouchedupon.Themetaphysicalquestionsalsotowhichlogictendstogiveriseareasfaraspossibleavoided.

5.TheUtilityofLogic.—Wehaveseenthatlogichasinviewanidealandtreats of what ought to be. Its object is, however, to investigate generalprinciples, and it puts forward no claim to be a practical art. Its utility is


accordinglynottobemeasuredbyanydirecthelpthatitmayaffordtowardsthe attainment of particular scientific truths. No doubt the procedure in allsciences is subject to the general principles formulated by logic; but, indetails, the weighing of evidence will often be better performed by thejudgmentoftheexpertthanbyanyformalorsystematicobservanceoflogicalrules.

Itisimportanttobearinmindthat,inthestudyoflogic,ourimmediateaimisthescientificinvestigationofgeneralprinciplesrecognisedasauthoritativeinrelation to thought-products,not theformulationofasystemofrulesandprecepts. It may be said that the art of dealing with particular concretearguments, with the object of determining their validity, is related to thescience of logic in the sameway as the art of casuistry (that is, the art ofdecidingwhatitisrighttodoinparticularconcretecircumstances)isrelatedtothescienceofethics.Moreover,justasintheartofcasuistrywemeetwithproblemswhich are elusive and difficult to decide because in the concretethey cannot be brought exactly under the abstract 7 formulae of ethicalscience, so in the art of detecting fallacies wemeet with arguments whichcannoteasilybebroughtundertheabstractformulaeoflogicalscience.Asitwould be a mistake to subordinate ethics to the treatment of casuisticalquestions, so it would be a mistake to mould the science of logic withconstant reference to concrete arguments which, either because of theambiguityofthetermsemployed,orbecauseoftheuncertainbearingofthecontext inwhich theyoccur,eludeanyattempt to reduce them toa form towhichgeneralprinciplesaredirectlyapplicable.

Whereinthenconsiststheutilityoflogic?Inanswertothisquestion,itmaybe observed primarily that if logic determines truly the principles of validthought,thenitsstudyisofvaluesimplyinthatitaddstoourknowledge.Tojustifythestudyoflogicitis,asManselhasobserved,sufficienttoshewthatwhat it teaches is true, and that by its aidwe advance in the knowledgeofourselvesandofourcapacities.

Tothisitmustbeadded(inqualificationofwhathasbeensaidpreviously)that,whilelogicisnottoberegardedasanartofattainingtruth,itstilldoespossess utility as propaedeutic to other studies and independently of the


addition that itmakes toourknowledge.Fallaciousargumentscannodoubtusually be recognised as such by an acute intellect apart from any logicalstudy;and,aswehaveseen,itisnottheprimaryfunctionoflogictodealwithparticularconcretearguments.Atthesametime,itisonlybytheaidoflogicthatwecananalyseareasoning,explainpreciselywhyafallaciousargumentisfaulty,andgivethefallacyaname.Inotherwords,whilelogicisnottobeidentifiedwith thecriticismofparticularconcretearguments, suchcriticismwhensystematicallyundertakenmustbebasedonlogic.

Greater, however, than the indirect value of logic in its subsequentapplicationtotheexaminationofparticularreasoningsisitsvalueasageneralintellectual discipline. The study of logic cultivates the power of abstractthought; and it is not too much to say that, when undertaken withthoroughness,itaffordsauniquementaltraining.


PARTI.

TERMS.


CHAPTERI.

THELOGICOFTERMS.

6.TheThreePartsofLogicalDoctrine—Ithasbeenusualtodividelogicaldoctrine into three parts, dealingwith terms (or concepts), propositions (orjudgments), and reasonings respectively; and itwill be convenient to adoptthisarrangementinthepresenttreatise.Atthesametime,wemayinpassingtouchuponcertainobjections thathavebeen raised to thismodeof treatingthesubject.

MrBosanquettreatsoflogicintwoparts,notinthree,givingnoseparatediscussionofnames(orconcepts).Hismaingroundfortakingupthispositionis that “the name or concept has no reality in living language or livingthought, except when referred to its place in a proposition or judgment”(Essentials of Logic, p. 87). He urges that “we ought not to think ofpropositionsasbuiltupbyputtingwordsornamestogether,butofwordsornamesasdistinguishedthoughnotseparableelementsinpropositions.”Thereis undoubted force in this argument, and attention should be called to thepointsraisedbyMrBosanquet,eventhoughwemaynotbe ledtoquite thesameconclusion.

Logic is essentially concernedwith truth and falsity as characteristics ofthought, and truth and falsity are embodied in judgments and in judgmentsonly.Hence the judgment9 (or theproposition as expressing the judgment)mayberegardedasfundamentallythelogicalunit.Itwould,moreover,nowbegenerallyagreedthattheconceptisnotbyitselfacompletementalstate,butisrealisedonlyasoccurringinacontext.Correspondinglythenamedoesnotbyitselfexpressanymentalstate.Ifamerenameispronounceditleavesusinastateofexpectancy,exceptinsofarasitistheabbreviatedexpressionofaproposition,asitmaybewhenspokeninanswertoaquestionorwhenthespecialcircumstancesormannerofitsutteranceconnectitwithacontextthatgivesitpredicativeforce.


At the same time, in ideal analysis the developed judgment yields theconcept as at any rate a distinguishable element of which it is composed,whilethepropositionsimilarlyyieldstheterm;andinorderthattheimportofjudgmentsandpropositionsmaybeproperlyunderstoodsomediscussionofconceptsandtermsisnecessary.

Thequestionas to theproperorderof treatment remains. Indealingwiththisquestionweneednottroubleourselveswiththeenquiryastowhethertheconcept or the judgment has psychological priority, that is to say, as towhether in the first instance theprocessof forming judgments requires thatconceptsshouldhavebeenalreadyformed,orwhetherontheotherhandtheprocessofformingconceptionsitselfinvolvestheformationofjudgments,orwhetherthetwoprocessesgoonparipassu.It isenoughthatthedevelopedjudgmentandtheproposition,asweareconcernedwiththeminlogic,yieldrespectively the concept, and the term as elements out of which they areconstituted.

Weshallthengiveaseparatediscussionofterms,andshallenteruponthispartofthesubjectbeforediscussingpropositions.Butindoingthisweshallendeavourconstantly tobear inmind that theproposition is the true logicalunit, and that the logical import of terms cannot be properly understoodexceptwithreferencetotheiremploymentinpropositions.3

3 In this connexion attentionmaybe called toMill’swell knowndictum that“namesarenamesofthings,notofourideas,”Apartfromitscontext,theforceofthis antithesis may easily be misunderstood. It is clear that every name that isemployedinanintelligiblesensemusthavesomementalequivalent,mustcallupsomeideaorothertoourminds,andmustthereforeinthissensebethenameofanidea.Itisnot,however,Mill’sintentiontodenythis.Nor,ontheotherhand,doeshe intend to assert that things actually exist corresponding to all the names weemploy. His dictum really has reference to predication. What he means is thatwhenanynameappearsas thesubjectofaproposition,anassertion ismadenotaboutthecorrespondingidea,butaboutsomethingwhichisdistinctbothfromthename and the idea, though both are related to it. He is in fact affirming theobjective reference that is essential to the conception of truth or falsity. Thediscussion may, therefore, be said to be properly part of the discussion of theimportofpropositionsratherthanofnames,anditwouldcertainlybelesspuzzlingifitwereintroducedinthatconnexion.Ourspecialobject,however,inreferringtothematterhereisnottocriticiseMill,buttoillustratethedifficultyofdiscussingnames logically apart from the use that may be made of them for purposes ofpredication.


107.NamesandConcepts.—Wehaveintheprecedingsectionspokenmoreorlessindiscriminatelyofnames(orterms)andofconcepts,andthishasbeenintentional. We have already expressed our disagreement with those whowould exclude from logic all consideration of language. Our judgmentscannothavecertaintyanduniversalvalidityunlesstheideaswhichenterintothemare fixedanddetermined;and,apart from theaid thatwederive fromlanguage,ourideascannotbethusfixedanddetermined.

It is, therefore, amistake to treat of concepts to the exclusion of names.But,ontheotherhand,wemustnotforgetthatthelogicianisconcernedwithnamesonlyasrepresentiveofideas.Hisrealaimistotreatofideas,thoughhemaythinkitwisertodosonotdirectly,butindirectlybyconsideringthenamesbywhichideasarerepresented.Forthisreasonitiswell,nowandthenatanyrate,toreferexplicitlytotheconcept.

Theso-calledconceptualistschooloflogiciansareaptintheirtreatmentofthe first part of logical doctrine to discuss problems of a markedlypsychologicalcharacter,as, forexample, themodeofformationofconceptsandthecontroversybetweenconceptualismandnominalism.Apart,however,from the fact that the conceptualist logiciansdonotdraw so clear a lineofdistinctionasdothenominalistsbetweenlogicandpsychology,thedifferencebetween the two schools is to a large extent 11 a mere difference ofphraseology.Practicallythesamepoints,forexample,areraisedwhetherwediscuss the extension and intension of concepts or the denotation andconnotationofnames.At the same time, itmustbe said that theattempt todeal with the intension of concepts to the entire exclusion of anyconsideration of the connotation of names appears to be responsible for agooddealofconfusion.

8.TheLogicofTerms.—Attentionhasalreadybeencalledtotherelationofdependence that exists between the logic of terms and the logic ofpropositions. Itwill be found thatwe cannot in general fully determine thelogical characteristics of a given name without explicit reference to itsemployment as a constituent of a proposition. We cannot again properlydiscuss or understand the import of so-called negative names withoutreferencetonegativejudgments.


It must be added that in dealing with distinctions between names, it isparticularlydifficultforthelogicianwhofollowsatallonthetraditionallinestoavoiddiscussingproblems thatbelongmoreappropriately topsychology,metaphysics, or grammar; and to some of the questionswhich arise itmayhardlybepossible togiveacompletely satisfactoryanswer from thepurelylogical point of view. This remark applies especially to the distinctionbetween abstract and concrete terms, a distinction, moreover, which is oflittle further logical utility or significance. It is introduced in the followingpages in accordance with custom; but adequately to discriminate betweenthingsandtheirattributesisthefunctionofmetaphysicsratherthanoflogic.Theportionof the logicof terms (or concepts) towhichby far thegreatestimportance attaches is thatwhich is concernedwith thedistinctionbetweenextensionandintension.

9.General and Singular Names.—A general name is a name which isactuallyorpotentiallypredicable in the same senseof eachof an indefinitenumberofunits;asingularorindividualnameisanamewhichisunderstoodin theparticular circumstances inwhich it is employed todenote someonedeterminateunitonly.

Thenatureandlogical importanceof thisdistinctionmay12beillustratedbyconsideringnamesas thesubjectsofpropositions.Ageneralnameis thenameofadivisibleclass,andpredicationispossibleinrespectofthewholeorapartoftheclass;asingularnameisthenameofaunitindivisible.Hencewe may take as the test or criterion of a general name, the possibility ofprefixingallorsometoitwithanymeaning.

Thus,primeministerofEnglandisageneralname,sinceitisapplicabletomorethanoneindividual,andstatementsmaybemadewhicharetrueofallprimeministersofEnglandoronlyofsome.ThenameGod issingulartoamonotheistasthenameoftheDeity,generaltoapolytheist,orasthenameofanyobjectofworship.Universeisgeneralinsofaraswedistinguishdifferentkindsofuniverses,e.g.,thematerialuniverse,theterrestrialuniverse,&c.;itissingularifwemeanthetotalityofallthings.Space isgeneral ifwemeananyportionofspace,singularifwemeanspaceasawhole.Waterisgeneral.Professor Bain takes a different view here; he says, “Names of material—


earth,atone,salt,mercury,water,flame—aresingular.Theyeachdenotetheentirecollectionofone speciesofmaterial” (Logic,Deduction, pp. 48, 49).Butwhenwepredicateanythingofthesetermsitisgenerallyofanyportion(or of some particular portion) of the material in question, and not of theentirecollectionofitconsideredasoneaggregate ;thus,ifwesay,“Wateriscomposed of oxygen and hydrogen,” we mean any and every particle ofwater, and the name has all the distinctive characters of the general name.Again,wecandistinguishthiswaterfromthatwater,andwecansay,“somewaterisnotfittodrink”;butthewordsomecannot,aswehaveseenabove,beattachedtoareallysingularname.Similarlywithregardtotheothertermsin question. It is also to be observed that we distinguish between differentkindsofstone,salt,&c.4

4 Terms of the kind here under discussion are called by Jevons substantialterms. (SeePrinciplesofScience,2,§4.)Theirpeculiarity is that,althoughtheyare concrete, the things denoted by them possess a peculiar homogeneity oruniformityofstructure;alsowedonotasaruleusetheindefinitearticlewiththemaswedowithothergeneralnames.

Anameistoberegardedasgeneralifitispotentially13predicableofmorethan one object, although as amatter of fact it happens that it can be trulyaffirmedofonlyone,e.g.,anEnglish sovereign six timesmarried.Areallysingularnameisnotevenpotentiallyapplicabletomorethanoneindividual;e.g.,thelastof theMohicans, theeldestsonofKingEdward theFirst.Thismaybedifferentlyexpressedbysayingthatareallysingularnameimpliesinitssignificationtheuniquenessof thecorrespondingobject.Wemaytakeasexamplesthesummumbonum,thecentreofgravityofthematerialuniverse.It is not easy to find such names except in caseswhere uniqueness resultsfrom some explicit or implicit limitation in time or space or from somerelation to anobject denotedby apropername.Even in such a case as thecentreofgravityof thematerialuniversesomelimitationin timeappears tobe necessary, for the centre of gravity of the universe may be differentlysituatedatdifferentperiods.

Anygeneralnamemaybetransformedintoasingularnamebymeansofanindividualisingprefix,suchasademonstrativepronoun(e.g.,thisbook),orbythe use of the definite article,which usually indicates a restriction to some


one determinate person or thing (e.g., the Queen, the pole star). Suchrestriction by means of the definite article may sometimes need to beinterpreted by the context, e.g., the garden, the river ; in other cases somelimitationofplaceortimeorcircumstanceisintroducedwhichunequivocallydefines the individual reference, e.g., the first man, the present LordChancellor,theauthorofParadiseLost.

On the other hand, propositions with singular names as subjects maysometimesadmitofsubdivisionintouniversalandparticular.Thisisthecasewhen,withreferencetodifferenttimesordifferentconditions,adistinctionismadeorimpliedinregardtothemannerofexistence,actualorpotential,oftheobjectdenotedbythename:forexample,“Homersometimesnods,”“Thepresent Pope always dwells in the Vatican,” “This country is sometimessubjecttoearthquakes.”5

5 Comparesections70and82.

10. Proper Names.—A proper name is a name assigned as a mark todistinguishanindividualpersonorthingfromothers,14withoutimplyinginits signification thepossessionby the individual inquestionof any specificattributes.Suchnamesaregiventoobjectswhichpossess interest inrespectoftheirindividualityandindependentlyoftheirspecificnature.Forthemostpart they are confined to persons and places; but they are also given todomestic animals, and sometimes to inanimate objects to which affection-value is attached, as, for example, by children to their dolls. Proper namesform a sub-class of singular names, being distinguished from the singularnames ofwhich exampleswere given in the preceding section in that theydenoteindividualobjectswithoutatthesametimenecessarilyconveyinganyinformationastoparticularpropertiesbelongingtothoseobjects.6

6 Proper names are farther discussed in section 25 in connexion with theconnotationofnames.

Manypropernames,e.g.,John,Victoria,areasamatteroffactassignedtomorethanoneindividual;buttheyarenotthereforegeneralnames,sinceoneachparticularoccasionoftheiruse,withtheexceptionnotedbelow,thereisan understood reference to some one determinate individual only. There is,moreover, no implication that different individuals who may happen to be


calledbythesamepropernamehavethisnameassignedtothemonaccountofpropertieswhichtheypossess incommon.7Theexceptionabovereferredtooccurswhenwespeakoftheclasscomposedofthosewhobearthename,andwhoareconstitutedadistinct classby thiscommon featurealone:e.g.,“AllVictorias arehonoured in theirname,”“SomeJohnsarenotofAnglo-Saxon origin, but are negroes.” The subjects of such propositions as thesemust,however,beregardedaselliptical;writtenoutmorefully,theybecomeallpersonscalledVictoria,someindividualsnamedJohn.

7 ProfessorBainbringsoutthisdistinctioninhisdefinitionofageneralname:“Ageneralnameisapplicabletoanumberofthingsinvirtueoftheirbeingsimilar,orhavingsomethingincommon.”

11.Collective Names.—A collective name is one which is applied to agroupofsimilarthingsregardedasconstitutingasinglewhole;e.g.,regiment,nation, army. Anon-collective name, e.g., stone, may also be the name ofsomethingwhich is15 composedofanumberofpreciselysimilarparts,butthisisnotinthesamewaypresenttothemindintheuseofthename.8

8 Tocollectivenameasabovedefinedthereisnodistinctiveantitheticalterminordinary use. The antithesis between the collective and the distributive use ofnamesarises,asweshallsee,inconnexionwithpredicationonly.

Acollectivenamemaybesingularorgeneral.Itisthenameofagrouporcollectionofthings,andsofarasitiscapableofbeingcorrectlyaffirmedinthesamesenseofonlyonesuchgroup,itissingular;e.g.,the29thregimentoffoot,theEnglishnation,theBodleianLibrary,Butifitiscapableofbeingcorrectlyaffirmedinthesamesenseofeachofseveralsuchgroupsitistoberegardedasgeneral;e.g.,regiment,nation,library.9

9 It is pointed out byDrVenn that certain proper namesmaybe regarded ascollective,thoughsuchnamesarenotcommon.“Oneinstanceofthemisexhibitedinthecaseofgeographicalgroups.Forinstance,theSeychelles,andthePyrenees,are distinctly, in their present usage, proper names, denoting respectively twogroups of things. They simply denote these groups, and give us no informationwhateveraboutanyoftheircharacteristics”(EmpiricalLogic,p.172).

Somelogiciansimplyanantithesisbetweencollectiveandgeneralnames,eitherregardingcollectivesasasub-classofsingulars,orelserecognisingathreefold division into singular, collective, and general. There is, properlyspeaking,nosuchantithesis;andboththeabovealternativesmustberegarded


asmisleading,ifnotactuallyerroneous;for,aswehavejustseen,theclassofcollectivenamesoverlapseachoftheotherclasses.

Thecorrectandreallyimportantlogicalantithesisisbetweenthecollectiveand thedistributiveuseofnames.Acollectivenamesuchasnation, or anyname in the plural number, is the name of a collection or group of similarthings.Thesewemayregardasonewhole,andsomethingmaybepredicatedof them that is trueof themonly as awhole; in this case thename is usedcollectively.Ontheotherhand,thegroupmayberegardedasaseriesofunits,and something may be predicated of these which is true of them takenindividually;inthiscasethenameisuseddistributively.10

10 It is held byDrVenn (Empirical Logic, p. 170) that substantial terms arealwaysusedcollectivelywhentheyappearassubjectsofgeneralpropositions.If,however,we takesuchapropositionas“Oil is lighter thanwater” it seemsclearthatthesubjectisusednotcollectively,butdistributively;fortheassertionismadeof each and every portion of oil, whereas if we used the term collectively ourassertionwouldapplyonlytoall theportionstakentogether.Thesameisclearlytrue in other instances; for example, in the propositions, “Water is composed ofoxygenandhydrogen,”“Icemeltswhenthetemperaturerisesabove32°Fahr.”

16 The above distinctionmay be illustrated by the propositions, “All theanglesofatriangleareequaltotworightangles,”“Alltheanglesofatrianglearelessthantworightangles.”Inthefirstcasethepredicationistrueonlyofthe angles all taken together,while in the second it is true only of each ofthemtakenseparately;inthefirstcase,therefore,thetermisusedcollectively,in the second distributively. Compare again the propositions, “The peoplefilledthechurch,”“Thepeopleallfellontheirknees.”11

11 Wheninanargumentwepassfromthecollectivetothedistributiveuseofaterm,orviceversâ,wehavewhat is technicallycalleda fallacyofdivisionorofcomposition as the case may be. The following are examples: The people whoattended Great St Mary’s contributed more than those who attended Little StMary’s,therefore,A(whoattendedtheformer)gavemorethanB(whoattendedthelatter);Alltheanglesofatrianglearelessthantworightangles,thereforeA,B,andC,whicharealltheanglesofatriangle,aretogetherlessthantworightangles.Thepointoftheoldriddle,“Whydowhitesheepeatmorethanblack?”consistsintheunexpecteduseoftermscollectivelyinsteadofdistributively.

12.ConcreteandAbstractNames.—Thedistinctionbetweenconcreteandabstract names, as ordinarily recognised,may bemost briefly expressed bysayingthataconcretenameisthenameofathing,whilstanabstractnameis


thenameofanattribute.Thequestion,however,atoncearisesastowhatismeantbyathingasdistinguishedfromanattribute ;andtheonlyanswertobegiven is that by a thing we mean whatever is regarded as possessingattributes.Itwouldappear,therefore,thatourdefinitionsmaybemademoreexplicit by saying that a concrete name is the name of anything which isregarded as possessing attributes, i.e., as a subject of attributes ; while anabstract name is the nameof anythingwhich is regarded as an attribute ofsomethingelse,i.e.,asanattributeofsubjects.12

12 Thedistinctionissometimesexpressedbysayingthatanabstractnameisthenameofanattribute,aconcretenamethenameofasubstance.Ifbysubstanceismerelymeantwhateverpossesses attributes, then thisdistinction is equivalent tothatgiven in the text;but if,aswouldordinarilybe thecase,a fullermeaning isgiventotheterm,thenthedivisionofnamesintoabstractandconcreteisnolongeran exhaustive one. Take such names as astronomy, proposition, triangle: thesenames certainly do not denote attributes; but, on the other hand, it seemsparadoxical toregardthemasnamesofsubstances.Onthewhole, therefore, it isbesttoavoidthetermsubstanceinthisconnexion.

17Thisdistinctionisinmostcaseseasyofapplication;forexample,planetriangleisthenameofallfiguresthatpossesstheattributeofbeingboundedby three straight lines, and is a concretename; triangularity is thenameofthisdistinctiveattributeoftriangles,andisanabstractname.Similarly,man,living being, generous are concretes; humanity, life, generosity are thecorrespondingabstracts.13

13 Itwill be observed that, according to the above definitions, a name is notcalledabstract,simplybecausethecorrespondingideaistheresultofabstraction,i.e.,attendingtosomequalitiesofathingorclassofthingstotheexclusionasfaraspossibleofothers. In this senseallgeneralnames, suchasman, living being,&c.,wouldbeabstract.

Abstract and concrete names usually go in pairs as in the aboveillustrations.Aconcretegeneralnameisthenameofaclassofthingsgroupedtogether in virtue of some quality or set of qualitieswhich they possess incommon;thenamegiventothequalityorqualitiesthemselvesapartfromtheindividuals towhich they belong is the corresponding abstract.14 Using thetermsconnoteanddenoteintheirtechnicalsenses,asdefinedinthefollowingchapter, an abstract name denotes the qualities which are connoted by thecorresponding concrete name. This relation between concretes and the


corresponding abstracts is the one point in connexion with abstract andconcretenamesthatisofreallogicalimportance,anditmaybeobservedthatitdoesnot in itselfgiverise to thesomewhatfruitlesssubtletieswithwhichthedistinctionisapttobe18associated.Forwhentwonamesaregivenwhichare thus related, there will never be any difficulty in determiningwhich isconcreteandwhichisabstractinrelationtotheother.


14 Thus, in the case of every general concrete name there is or may beconstructedacorrespondingabstract.Butthisisnottrueofpropernamesorothersingular names regarded strictly as such.Wemay indeed have such abstracts asCaesarism and Bismarckism. These names, however, do not denote all thedifferentiating attributes of Caesar and Bismarck respectively, but only certainqualitiessupposedtobespeciallycharacteristicoftheseindividuals.Informingtheabove abstracts we generalise, and contemplate a certain type of character andconductthatmaypossiblybecommontoawholeclass.Comparepage45.

Butwhilst the distinction is absolute and unmistakeablewhen names arethusgiveninpairs, theapplicationofourdefinitionsisbynomeansalwayseasywhenweconsidernamesinthemselvesandnotinthisdefiniterelationtoother names. We shall find indeed that if we adopt the definitions givenabove, then the division of names into abstract and concrete is not anexclusiveonein thesense thateverynamecanonceandforallbeassignedexclusivelytooneorotherofthetwocategories.

We are at any rate driven to this if we once admit that attributes maythemselves be the subjects of attributes, and it is difficult to see how thisadmission can be avoided. If, for example, we say that “unpunctuality isirritating,”weascribetheattributeofbeingirritatingtounpunctuality,whichisitselfanattribute.Unpunctuality, therefore,althoughprimarilyanabstractname,canalsobeused insuchaway that it is,according toourdefinition,concrete.

Similarlywhenweconsiderthatanattributemayappearindifferentformsor in different degrees,wemust regard it as somethingwhich can itself bemodified by the addition of a further attribute; as, for example, when wedistinguish physical courage frommoral courage, or thewhiteness of snowfrom the whiteness of smoke, or when we observe that the beauty of adiamonddiffersinitscharacteristicsfromthebeautyofalandscape.

Hence, if the definitions under discussion are adopted, we arrive at theconclusion that while some names are concrete and never anything butconcrete,nameswhichareprimarily formedasabstractsandcontinue tobeusedassuchareaptalsotobeusedasconcretes,thatistosay,theyarenamesofattributeswhichcanthemselvesberegardedaspossessingattributes.Theyare abstract names when viewed in one relation, concrete when viewed in


another.1515 Theuseofthesametermasbothabstractandconcreteinthemannerabove

describedmustbedistinguishedfromthenotunfrequentcaseofquiteanotherkindinwhichanameoriginallyabstractchangesitsmeaningandcomestobeusedinthesenseofthecorrespondingconcrete;as,forexample,whenwetalkoftheDeitymeaning thereby God, not the qualities of God. Compare Jevons, ElementaryLessonsinLogic,pp.21,22.

19Itmustbeadmittedthatthisresultisparadoxical.Asyieldingadivisionofnamesthatisnon-exclusive,itisalsounscientific.Therearetwowaysofavoidingthisdifficulty.

In the first place,wemay furthermodify our definitions and say that anabstractnameisthenameofanythingwhichcanberegardedasanattributeofsomethingelse(whetheritisorisnotitselfasubjectofattributes),whileaconcretenameisthenameofthatwhichcannotberegardedasanattributeofsomething else. This distinction is simple and easy of application, it is inaccordancewithpopularusage,anditsatisfiestheconditionthatthemembersof adivision shallbemutuallyexclusive.But itmaybedoubtedwhether ithasanylogicalvalue.

Asecondwayofavoidingthedifficultyistogiveupforlogicalpurposesthedistinctionbetweenconcreteandabstractnames,andtosubstituteforitadistinction between the concrete and the abstract use of names. A name isthen used in a concrete sense when the thing called by the name iscontemplated as a subject of attributes, and in an abstract sense when thethingcalledbythenameiscontemplatedasanattributeofsubjects.Itfollowsfromwhat has been already said that some names can be used as concreteonly,whileotherscanbeusedeitherasabstractorasconcrete.Thissolutionissatisfactoryfromthelogicalpointofview,sincelogicisconcernednotwithnamesassuch,butwiththeuseofnamesinpropositions.Itmaybeaddedthatas logicians we have very little to do with the abstract use of names, Aconsideration of the import of propositions will shew that when a nameappearseitherasthesubjectorasthepredicateofanon-verbalpropositionitsuseisalwaysconcrete.

13.CanAbstractNamesbesubdivided intoGeneralandSingular?—Thequestionwhetheranyabstractnamescanbeconsideredgeneralhasgivenrise


to much difference of opinion amongst logicians. On the one hand, it isargued that all 20 abstract names must necessarily be singular, since anattributeconsideredpurelyassuchandapartfromitsconcretemanifestationsis one and indivisible, and cannot admit of numerical distinction.16 On theother hand, it is urged that some abstracts must certainly be consideredgeneralsincetheyarenamesofattributesofwhichtherearevariouskindsorsubdivisions; and in confirmation of this view it is pointed out that wefrequentlywriteabstractsinthepluralnumber,aswhenwesay,“Rednessandyellownessarecolours,”“Patienceandmeeknessarevirtues.”17

16 ThisrepresentstheviewtakenbyJevons.SeePrinciplesofScience,2,§3.17 CompareMill,Logic,i.2,§4.

The solution of the question really depends upon our use of the termabstract.

If we adopt the definition given in the last paragraph but one of theprecedingsection, and includeunder abstractnames thenamesof attributeswhicharethemselvesthesubjectsofattributes,theselatterattributespossiblyvaryingindifferentinstances,thentherecanbenodoubtthatsomeabstractsaregeneral; for theyare thenamesofaclassof thingswhich,whilehavingsomethingincommon,arealsodistinguishableinterse.

So far, however, as the question is raised in regard to the abstract (asdistinguishedfromtheconcrete)useofnamesinthemannerindicatedinthelastparagraphoftheprecedingsection,weareledtotheconclusionthatitisonlywhen names are used in a concrete sense that they can be consideredgeneral. For it is clear that the name of an attribute can be described asgeneralonlyinsofarastheattributeisregardedasexhibitingcharacteristicswhichvaryindifferentinstances,onlyinsofar,thatistosay,asitisitselfasubjectofattributes;andwhentheattributeissoregarded,thenameisusedinaconcrete,notanabstract,sense.

Takethepropositions,“Somecoloursarepainfullyvivid,”“Allyellowsareagreeable,”“Somecourage is the resultof ignorance,”“Somecruelty is theresultoffear,”“Allcrueltyisdetestable.”Thesubjectsofthesepropositionsare certainly 21 general. According to the definition given in the lastparagraphbutoneoftheprecedingsectiontheyarealsoabstract.If,however,


in place of distinguishing between abstract and concrete names per se, wedistinguishbetweentheabstractandtheconcreteuseofnamesasproposedinthelastparagraphoftheprecedingsection,thenthetermsinquestionareallusedinaconcrete,notanabstract,sense.

EXERCISES.

14. Discuss Mill’s statement that “names are names of things, not of our ideas,” with specialreferencetothefollowingnames:dodo,mermaid,chimaera,toothache,jealousy,idea.[C.]

15.Discussthelogicalcharacteristicsofadjectives.[K.]


CHAPTERII.

EXTENSIONANDINTENSION.

16.TheExtensionandtheIntensionofNames.18—Everyconcretegeneralname is the name of a real or imaginary class of objectswhich possess incommoncertainattributes;andthereare,therefore,twoaspectsunderwhichitmaybe regarded.Wemayconsider thename(i) in relation to theobjectswhich are called by it; or (ii) in relation to the qualities belonging to thoseobjects.Itisdesirabletohavetermsbywhichtorefertothisbroaddistinctionwithoutregardtofurtherrefinementsofmeaning;andthetermsextensionandintensionwill accordingly be employed to express in themost generalwaythesetwoaspectsofnamesrespectively.19

18 Wemay speak also of the extension and the intension of concepts. In thediscussion, however, of questions concerning extension and intension, it isessential to recognise the part played by language as the instrument of thought.Hence it seemsbetter to start fromnames rather than fromconcepts.Neglect toconsider names explicitly in this connexion has been responsible for muchconfusion.

19 It is usual to employ the terms comprehension and connotation as simplysynonymous with intension, and denotation as synonymous with extension. Weshall,however,presentlyfind itconvenient todifferentiate themeaningsof theseterms.The force of the termsextension and intension in themost general sensemightperhapsalsobeexpressedbythepairoftermsapplicationandimplication.

Theextensionofanamethenconsistsofobjectsofwhichthenamecanbepredicated;itsintensionconsistsofpropertieswhichcanbepredicatedofit.For example,by theextensionofplane trianglewemeana certainclassofgeometrical figures, and by its intension certain qualities belonging to suchfigures. 23 Similarly, by the extension ofman is meant a certain class ofmaterial objects, and by its intension the qualities of rationality, animality,&c.,belongingtotheseobjects.

17. Connotation, Subjective Intension, and Comprehension.—The termintension has been used in the preceding section to express in the most


generalway that aspect of general names underwhichwe consider not theobjects called by the names but the qualities belonging to those objects.Takinganygeneralname,however,thereareatleastthreedifferentpointsofview fromwhich the qualities of the corresponding classmay be regarded;anditistoawantofdiscriminationbetweenthesepointsofviewthatwemayattribute many of the controversies and misunderstandings to which theproblemoftheconnotationofnameshasgivenrise.

(1)There are those qualitieswhich are essential to the class in the sensethatthenameimpliestheminitsdefinition.Wereanyofthissetofqualitiesabsent the namewould not be applicable; and any individual thing lackingthem would accordingly not be regarded as a member of the class. Thestandpointheretakenmaybesaidtobeconventional,sinceweareconcernedwiththesetofcharacteristicswhicharesupposedtohavebeenconventionallyagreeduponasdeterminingtheapplicationofthename.

(2)Therearethosequalitieswhichinthemindofanygivenindividualareassociatedwith thename in suchaway that theyarenormallycalledup inideawhenthenameisused.Thesequalitieswillincludethemarksbywhichthe individual in question usually recognises or identifies an object asbelonging to the class. Theymay not exhaust the essential qualities of theclassinthesenseindicatedintheprecedingparagraph,butontheotherhandtheywill probably include some that are not essential to it. The standpointheretakenissubjectiveandrelative.Evenwhenthereisagreementastotheactual meaning of a name, the qualities that we naturally think of inconnexionwith itmay vary both from individual to individual, and, in thecaseofanygivenindividual,fromtimetotime.

Wemayconsiderasaspecialcaseunderthisheadthe24completegroupofattributesknownatanygiventimetobelongtotheclass.Alltheseattributescanbecalledupinideabyanypersonwhoseknowledgeoftheclassisfullyuptodate;andthisgroupmay,therefore,beregardedasconstitutingthemostscientificformofintensionfromthesubjectivepointofview.

(3)Thereisthesum-totalofqualitiesactuallypossessedincommonbyallmembersoftheclass.Thesewillincludeall thequalitiesincludedunderthetwoprecedingheads,20 andusuallymanyothers inaddition.The standpoint


heretakenisobjective.2120 Itishereassumed,asregardsthequalitiesmentallyassociatedwiththename,

thatourknowledgeoftheclass,sofarasitextends,iscorrect.21 Whentheobjectivestandpointistaken,thereisanimpliedreferencetosome

particular universe of discourse, within which the class denoted by the name issupposed to be included. The force of this remark will be made clearer at asubsequentstage.

Inseekingtogiveaprecisemeaningtoconnotation,wemaystartfromtheaboveclassification.Itsuggeststhreedistinctsensesinwhichthetermmightpossiblybeused,andasamatteroffactall threeof thesesenseshavebeenselected by different logicians, sometimes without any clear recognition ofdivergencefromtheusageofotherwriters. It isdesirable thatweshouldbequiteclearinourownmindsinwhichsenseweintendtoemploytheterm.

(i)AccordingtoMill’susage,whichisthatadoptedinthefollowingpages,theconventionalstandpoint is takenwhenwespeakof theconnotation of aname.Onthisview,wedonotmeanby theconnotationofaclass-nameallthequalitiespossessedincommonbytheclass;nordowenecessarilymeanthose particular qualitieswhichmay bementally associatedwith the name;butwemean just thosequalitiesonaccountof thepossessionofwhichanyindividual isplaced in theclassandcalledby thename.Inotherwords,weinclude in theconnotationofaclass-nameonly thoseattributesuponwhichtheclassificationisbased,andintheabsenceofanyofwhichthenamewouldnotberegardedasapplicable.Forexample,althoughallequilateraltrianglesareequiangular,equiangularityisnotonthisviewincludedintheconnotationof equilateral 25 triangle, since it is not a property upon which theclassification of triangles into equilateral and non-equilateral is based;although all kangaroosmay happen to beAustralian kangaroos, this is notpart of what is necessarily implied by the use of the name, for an animalsubsequently found in the interiorofNewGuinea,butotherwisepossessingallthepropertiesofkangaroos,wouldnothavethenamekangaroodeniedtoit;althoughallruminantanimalsarecloven-hoofed,wecannotregardcloven-hoofedaspartofthemeaningofruminant,and(asMillobserves)ifananimalwere to be discoveredwhich chewed the cud, but had its feet undivided, itwouldcertainlystillbecalledruminant.


(ii)Somewriterswhoregardpropernamesasconnotativeappeartoincludeintheconnotationofanameall thoseattributeswhichthenamesuggeststothe mind, whether or not they are actually implied by it. And it is to beobserved in this connexion that a name may in the mind of any givenindividual be closely associated with properties which even the sameindividualwouldinnowayregardasimpliedinthemeaningofthename,as,forinstance,“Trinityundergraduate”withabluegown.Thisinterpretationofconnotation is, therefore, clearly to be distinguished from that given in theprecedingparagraph.

We may further distinguish the view, apparently taken by some writers,accordingtowhichtheconnotationofaclass-nameatanygiventimewouldincludeallthepropertiesknownatthattimetobelongtotheclass.

(iii)Otherwritersusetheterminstillanothersenseandwouldincludeinthe connotation of a class-name all the properties, known and unknown,whicharepossessedincommonbyallmembersoftheclass.Thus,MrE.C.Beneckewrites,—“Justastheword‘man’denoteseverycreature,orclassofcreatures,havingtheattributesofhumanity,whetherweknowhimornot,sodoes thewordproperly connote thewhole of theproperties common to theclass, whether we know them or not. Many of the facts, known tophysiologists and anatomists about the constitution of man’s brain, forexample,arenotinvolvedinmostmen’sideaofthebrain;thepossession26ofabrainpreciselysoconstituteddoesnot,therefore,formanypartoftheirmeaning of the word ‘man.’ Yet surely this is properly connoted by theword….Wehave thus thedenotationof the concretenameon theone sideand its connotation on the other, occupying perfectly analogous positions.Given the connotation,—the denotation is all the objects that possess thewholeofthepropertiessoconnoted.Giventhedenotation,—theconnotationis the whole of the properties possessed in common by all the objects sodenoted”(Mind, 1881,p. 532). Jevonsuses the term in the same sense. “Atermtakeninintent(connotation)hasforitsmeaningthewholeinfiniteseriesofqualitiesandcircumstanceswhichathingpossesses.Ofthesequalitiesorcircumstancessomemaybeknownandformthedescriptionordefinitionofthemeaning;theinfiniteremainderareunknown”(PureLogic,p.6).22


22 Bain appears to use the term in an intermediate sense, including in theconnotationofaclass-namenotall theattributescommontotheclassbutall theindependentattributes,thatis,allthatcannotbederivedorinferredfromothers.

Whilerejectingtheuseof the termconnotation inanybut thefirstof theabovementionedsenses,weshallfinditconvenienttohavedistinctivetermswhich can be used with the other meanings that have been indicated. Thethree terms connotation, intension, and comprehension are commonlyemployed almost synonymously, and there will certainly be a gain inendeavouringtodifferentiatetheirmeanings.Intension,asalreadysuggested,may be used to indicate in the most general way what may be called theimplicational aspect of names; the complex terms conventional intension,subjective intension, and objective intension will then explain themselves.Connotation may be used as equivalent to conventional intension ; andcomprehension as equivalent to objective intension. Subjective intension islessimportantfromthelogicalstandpoint,andweneednotseektoinventasingletermtobeusedasitsequivalent.23

23 For anyonewho isgiven themeaningof anamebutknowsnothingof theobjects denoted by the name, subjective intension coincides with connotation.Were the ideal of knowledge to be reached, subjective intensionwould coincidewithcomprehension.

27 Conventional intension or connotation will then include only thoseattributeswhichconstitutethemeaningofaname;24subjectiveintensionwillinclude those that are mentally associated with it, whether or not they areactuallysignifiedbyit;objectiveintensionorcomprehensionwillincludealltheattributespossessed incommonbyallmembersof theclassdenotedbythe name.Wemight perhaps speakmore strictly of the connotation of thename itself, the subjective intension of the notion which is the mentalequivalentofthename,andthecomprehensionoftheclasswhichisdenotedbythename.25

24 It is tobeobserved that in speakingof the connotationof anamewemayhaveinvieweitherthesignificationthatthenamebearsincommonacceptation,orsome specialmeaning assigned to it by explicit definition for some scientific orotherspecificpurpose.

25 Thedistinctionsofmeaningindicatedinthissectionwillbefoundessentialfor clearness of view in discussing certain questions to whichwe shall pass onimmediately; in particular, the questions whether connotation and denotation


necessarilyvaryinversely,andwhetherpropernamesareconnotative.

18. Sigwart’s distinction between Empirical, Metaphysical, and LogicalConcepts.—Sigwart observes that in speaking of concepts we ought todistinguish between three meanings of the word. These three meanings of“concept”hedescribesasfollows.26

26 Logic, I. p. 245. This and future references to Sigwart are to the EnglishtranslationofhisworkbyMrsBosanquet.

(1)By a conceptmaybemeant a natural psychological production,—thegeneralideawhichhasbeendevelopedinthenaturalcourseofthought.Suchideas are different for different people, and are continually in process offormation;evenfor the individualhimself theychange,so thataworddoesnot always keep the same meaning even for the same person. Strictlyspeaking,itisonlybyafictionwhichneglectsindividualpeculiaritiesthatwecan speak of the concepts corresponding to the terms used in ordinarylanguage.

(2)Incontrastwiththisempiricalmeaningtheconceptmaybeviewedasan ideal; it is then the mark at which we aim in our endeavour to attainknowledge,forweseektofindinitanadequatecopyoftheessenceofthings.28

(3) Between these two meanings of the word, which may be called theempiricalandthemetaphysical,thereliesthelogical.Thishasitsorigininthelogicaldemandforcertaintyanduniversalvalidityinourjudgments.Allthatis required is thatour ideas shouldbeabsolutely fixedanddetermined,andthatallwhomakeuseofthesamesystemofdenotationshouldhavethesameideas.

Thisthreefolddistinctionmaybeusefullycomparedwiththatdrawnintheprecedingsection.Sigwartisapproachingthequestionfromadifferentpointof view, but it will be observed that his three “meanings of concept”correspond broadly with subjective intension, objective intension, andconventionalintensionrespectively.

Itmay be added thatMrBosanquet’s distinction (Logic, I. pp. 41 to 46)between the objective reference of a name (its logical meaning) and itscontentfortheindividualmind(thepsychicalidea)appearstosomeextentto


correspondtothedistinctionbetweenconnotationandsubjectiveintension.

19.ConnotationandEtymology.—Theconnotationofanamemustnotbeconfusedwithitsetymology.Indealingwithnamesfromtheetymologicalorhistorical point of viewwe consider the circumstances inwhich theywerefirstimposedandthereasonsfortheiradoption;alsothesuccessivechanges,if any, in theirmeaning thathave subsequentlyoccurred. Inmakingprecisetheconnotationtobeattachedtoanamewemaybehelpedbyconsideringitsetymology. But we must clearly distinguish between the two; in finallydeciding upon the connotation to be assigned to a name for any particularscientificpurpose,wemayindeedfinditnecessarytodepartnotmerelyfromitsoriginalmeaning,butalsofromitscurrentmeaningineverydaydiscourse.

20.FixityofConnotation.—Ithasbeenalreadypointedoutthatsubjectiveintensionisvariable.Agivennamewillalmostcertainlycallupinthemindsofdifferentpersonsdifferentideas;andeveninthecaseofthesamepersonitwillprobablydosoatdifferenttimes.Thequestionmayberaisedhowfarthesameistrueofconnotation.Ithasbeenimpliedintheprecedingsectionthatthescientificuseofanamemaydiffer29fromitsuseineverydaydiscourse;andtherecanbenodoubtthatasamatteroffactdifferentpeoplemaybythesamenameintendtosignifydifferentthings,thatistosay,theywouldincludedifferent attributes in the connotation of the name. It is, moreover, notunfrequentlythecasethatsomeofusmaybeunabletosaypreciselywhatisthemeaningthatweourselvesattachtothewordsweuse.

Atthesametimeacleardistinctionoughttobedrawnbetweensubjectiveintensionandconnotationinrespectof theirvariability.Subjectiveintensionisnecessarilyvariable; it canneverbeotherwise.Connotation,on theotherhand,isonlyvariablebyaccident;andinsofarasthereisvariationlanguagefailsofitspurpose.“Identicalreference,”asMrBosanquetputsit,“istherootandessenceofthesystemofsignswhichwecalllanguage”(Logic,I.p.16).It isonlybysomeconventionalagreementwhichshallmakelanguagefixedthatscientificdiscussionscanbesatisfactorilycarriedon;andtherewouldbeno variation in the connotation of names in the case of an ideal languageproperly employed. In dealing with reasonings from the point of view oflogicaldoctrine, it is, therefore,nounreasonableassumptiontomakethatin


any given argument the connotation of the names employed is fixed anddefinite; in other words, that every name employed is either used in itsordinary sense and that this is precisely determined, or else that, the namebeingusedwitha specialmeaning, suchmeaning isadhered toconsistentlyandwithoutequivocation.

21.Extension and Denotation.—The terms extension and denotation areusually employed as synonymous, but there will be some advantage indrawinga certaindistinctionbetween them.We shall find thatwhennamesare regardedas the subjectsofpropositions there is an implied reference tosomeuniverseofdiscourse,whichmaybemoreorlesslimited.Forexample,we shouldnaturallyunderstand suchpropositionsasallmenaremortal,nomenareperfect,torefertoallmenwhohaveactuallyexistedontheearth,orarenowexisting,orwillexisthereafter,butweshouldnotunderstandthemtorefer to all fictitious persons or all beings possessing the essentialcharacteristics of men whom we are able to conceive or imagine. 30 Themeaningofuniverseofdiscoursewillbefurtherillustratedsubsequently.Theonlyreasonforintroducingtheconceptionatthispointisthatweproposetousethetermdenotationorobjectiveextensionratherthanthetermextensionsimplywhenthereisanexplicitorimplicitlimitationtotheobjectsactuallytobefoundinsomerestricteduniverse.Bythesubjectiveextensionofageneralname,ontheotherhand,weshallunderstandthewholerangeofobjectsrealorimaginarytowhichthenamecanbecorrectlyapplied,theonlylimitationbeingthatoflogicalconceivability.Everyname,therefore,whichcanbeusedin an intelligible sense will have a positive subjective extension, but itsdenotation in a universewhich is in someway restrictedby time, place, orcircumstancemaybezero.27

27 The value of the above distinction may be illustrated by reference to thedivergenceofviewindicatedinthefollowingquotationfromMrMonck,whousesthe terms denotation and extension as synonymous: “It is a matter of accidentwhetherageneralnamewillhaveanyextension(ordenotation)ornot.Unicorn,griffin,anddragon aregeneralnamesbecause theyhaveameaning,andwecansupposeanotherworldinwhichsuchbeingsexist;butthetermshavenoextension,because there are no such animals in thisworld. Some logicians speak of thesetermsashavinganextension,becausewecansupposeindividualscorrespondingtothem. In this way every general termwould have an extensionwhichmight beeitherrealorimaginary.Itis,however,moreconvenienttousethewordextension


forarealextension(past,present,orfuture)only”(IntroductiontoLogic,p.10).Itshouldbeadded,inordertopreventpossiblemisapprehension,thatbyuniverseofdiscourse,asusedinthetext,wedonotnecessarilymeanthematerialuniverse;wemay, for example,mean theuniverseof fairy-land,orofheraldry, and in suchacase,unicorn,griffin,anddragonmayhavedenotation (in our special sense), aswellassubjectiveextension,greater thanzero.What is theparticularuniverseofreferenceinanygivenpropositionwillgenerallybedeterminedbythecontext.Forlogical purposeswemayassume that it is conventionallyunderstoodandagreedupon,andthat it remains thesamethroughout thecourseofanygivenargument.AsDrVennremarks,“Wemightincludeamongsttheassumptionsoflogicthatthespeakerandhearershouldbeinagreement,notonlyastothemeaningofthewordstheyuse,butalsoastotheconventionallimitationsunderwhichtheyapplytheminthecircumstancesofthecase”(EmpiricalLogic,p.180).

Inthesensehereindicated,denotationisincertainrespectsthecorrelativeofcomprehension rather thanofconnotation. For in speaking of denotationweare,asinthecaseofcomprehension,takinganobjectivestandpoint;andthereis,moreover,inthecaseofcomprehension,asinthatofdenotation,a31tacit reference to some particular universe of discourse. Since, however,denotationisgenerallyspeakingdeterminedbyconnotation,thereisoneveryimportantrespectinwhichconnotationanddenotationarestillcorrelatives.

22.DependenceofExtensionand Intensionupononeanother.28—Takingany class-name X, let us first suppose that there has been a conventionalagreementtouseitwhereveracertainselectedsetofpropertiesP1,P2,…Pm,are present. This set of propertieswill constitute the connotation ofX, andwill, with reference to a given universe of discourse,29 determine thedenotationofthename,say,Q1,Q2,…Qy ; thatis,Q1,Q2,…Qy,areall theindividualspossessingincommonthepropertiesP1,P2,…Pm.

28 Thissectionmaybeomittedonafirstreading.29 Itwill be assumed in the remainder of this section thatwe are throughout

speakingwithreferencetoagivenuniverseofdiscourse.

These properties may not, and almost certainly will not, exhaust thepropertiescommontoQ1,Q2,…Qy.LetallthecommonpropertiesbeP1,P2,…Px ;theywillincludeP1,P2,…Pm,andinallprobabilitymorebesides,andwillconstitutethecomprehensionoftheclass-name.

NowitwillalwaysbepossibleinoneormorewaystomakeoutofQ1,Q2,…Qy,aselectionQ1,Q2,…Qn,whichshallbepreciselytypicalofthewhole


class;30 thatistosay,Q1,Q2,…Qnwillpossess incommonthoseattributesand only those attributes (namely, P1, P2, … Px) which are possessed incommonbyQ1,Q2,…Qy.31Q1,Q2,…Qnmaybecalledtheexemplificationor32extensivedefinition ofX. The reason for selecting the name extensivedefinition will appear in a moment. It will sometimes be convenientcorrespondingly to speak of the connotation of a name as its intensivedefinition.

30 ItmaychancetobenecessarytomakeQ1,Q2,…QncoincidewithQ1,Q2,…Qy. But thismust be regarded as the limiting case; usually a smaller number ofindividualswillbesufficient.

31 Mr Johnsonpointsout tome that inpursuing this lineof argument certainrestrictions of a somewhat subtle kind are necessary in regard to what may becalled our “universe of attributes.” The “universe of objects”which is what wemeanbythe“universeofdiscourse,”impliesindividualityofobjectandlimitationofrangeofobjects ;andifwearetoworkoutathoroughgoingreciprocitybetweenattributesandobjects,wemustrecogniseinour“universeofattributes”restrictionsanalogousto theabove,namely,simplicityofattributeand limitation of range ofattributes.By“simplicityofattribute”ismeantthattheuniverseofattributesmustnotcontainanyattributewhichisalogicalfunctionofanyotherattributeorsetofattributes.Thus,ifA,Baretwoattributesrecognisedinouruniverse,wemustnotadmitsuchattributesasX(=AandB),orY (=AorB),orZ (=not-A).Wemayindeed have a negatively defined attribute, but it must not be the formalcontradictory of another or formally involve the contradictory of another. Thefollowing example will shew the necessity of this restriction. LetP1,P2, P3, beselectedastypicalofthewholeclassP1,P2,P3P4,P5,P6;andletA1beanattributepossessedbyP1alone,A2anattributepossessedbyP2alone,andsoon.ThenifwerecogniseA1orA2orA3asadistinctattribute,itisatonceclearthatP1,P2,P3willno longer be typical of thewhole class; and the same result follows ifnot-A4 isrecognisedasadistinctattribute.Similarly,withouttherestrictioninquestionanyselection (short of the whole) would necessarily fail to be typical of the wholeclass.Asaconcreteexample,supposethatweselectfromtheclassofprofessionalmen a set of examples that have in common no attribute except those that arecommontothewholeclass.Itmayturnoutthatourexamplesareallbarristersordoctors, but none of them solicitors.Now ifwe recognise as a distinct attributebeing“eitherabarristeroradoctor,”ourselectedgroupwilltherebyhaveanextracommonattributenotpossessedbyeveryprofessionalman.Thesameresultwillfollow ifwe recognise the attribute “non-solicitor.”Notmuch need be added asregards the necessity of some limitation in the range of attributes which arerecognised. Themere fact of our having selected a certain group would indeedconstituteanadditionalattribute,whichwouldatoncecausetheselectiontofailinitspurpose,unless thiswereexcludedas inessential.Similarly, suchattributesasposition in space or in time&c.must in general be regarded as inessential. For


example,Imightdrawonasheetofpaperanumberoftrianglessufficientlytypicalof the whole class of triangles, but for this it would be necessary to reject asinessentialthecommonpropertywhichtheywouldpossessofallbeingdrawnonaparticularsheetofpaper.

Wehavethen,withreferencetoX, (1) Connotation:P1…Pm ; (2) Denotation:Q1…Qn…Qy ; (3) Comprehension:P1…Pm…Px ; (4) Exemplification:Q1…Qn.

Ofthese,eithertheconnotationortheexemplificationwillsufficetomarkoutorcompletelyidentifytheclass,althoughtheydonotexhausteitherallitscommon properties or all the individuals contained in it. In other words,whether we start from the connotation or from the exemplification, thedenotationandthecomprehensionwillbethesame.32

32 It will be observed that connotation and exemplification are distinguishedfromcomprehensionanddenotationinthattheyareselective,whilethelatterpairareexhaustive.Inmakingourselectionouraimwillusuallybetofindtheminimumlistwhichwillsufficeforourpurpose.

33 For a concrete illustration of the above, the termmetalmay be taken.Fromthechemicalpointofviewametalmaybedefinedasanelementwhichcan replace hydrogen in an acid and thus form a salt. This then is theconnotation of the name. Its denotation consists of the complete list ofelementsfulfillingtheaboveconditionnowknowntochemists,andpossiblyof others not yet discovered.33 The members of the whole class thusconstituted are, however, found to possess other properties in commonbesidesthosecontainedinthedefinitionofthename,forexample,fusibility,the characteristic lustre termed metallic, a high degree of opacity, and thepropertyofbeinggoodconductorsofheatandelectricity.Thecompletelistofthesepropertiesformsthecomprehensionofthename.Nowachemistwouldnodoubtbeable from the fulldenotationofmetal tomakea selectionofalimited number of metals which would be precisely typical of the wholeclass;34 that is to say, his selected listwould possess in commononly suchproperties as are common to the whole class. This selected class wouldconstitutetheexemplificationofthename.


33 Itisnecessarytodistinguishbetweentheknownextensionofatermanditsfulldenotation,justaswedistinguishbetweentheknownintensionofatermanditsfullcomprehension.

34 He would take metals as different from one another as possible, such asaluminium,antimony,copper,gold,iron,mercury,sodium,zinc.

Wehavesofarassumedthat(1)connotationorintensivedefinitionhasfirstbeen arbitrarily fixed, and that this has successively determined (2)denotation, (3) comprehension, and—with a certain range of choice—(4)exemplification.Butitisclearthattheoreticallywemightstartbyarbitrarilyfixing (i) the exemplification or extensive definition ; and that this wouldsuccessivelydetermine(ii)comprehension, (iii)denotation,and then—againwithacertainrangeofchoice35—(iv)connotation.

35 Itisordinarilysaidthat“ofthedenotationandconnotationofatermonemay,bothcannot,bearbitrary,”andthisisbroadlytrue.Itispossible,however,tomakethestatementrathermoreexact.Giveneitherintensiveorextensivedefinition,thenbothdenotationandcomprehensionare,withreferencetoanyassigneduniverseofdiscourse, absolutely fixed.But different intensive definitions, and also differentextensive definitions, may sometimes yield the same results; and it is thereforepossiblethat,everythingelsebeinggiven,connotationorexemplificationmaystillbewithin certain limits indeterminate. For example, given the class of parallelstraightlines, theconnotationmaybedetermined in twoor threedifferentways;or, given the class of equilateral equiangular triangles, we may select asconnotation either having three equal sides or having three equal angles.Again,given the connotation ofmetal, itwould no doubt be possible to select inmorewaysthanonealimitednumberofmetalsnotpossessingincommonanyattributeswhicharenotalsopossessedbytheremainingmembersoftheclass.

34Itisinterestingfromatheoreticalpointofviewtonotethepossibilityofthis second order of procedure; and this order may, moreover, be said torepresentwhat actually occurs—at any rate in the first instance—in certaincases,as,forexample,inthecaseofnaturalgroupsintheanimal,vegetable,and mineral kingdoms. Men form classes out of vaguely recognisedresemblanceslongbeforetheyareabletogiveanintensivedefinitionoftheclass-name, and in such a case if they are asked to explain their use of thename, their reply will be to enumerate typical examples of the class. Thiswouldnodoubtordinarilybedoneinanunscientificmanner,butitwouldbepossibletoworkitoutscientifically.Theextensivedefinitionofanamewilltaketheform:XisthenameoftheclassofwhichQ1,Q2,…Qnare typical.


Thisprimitiveformofdefinitionmayalsobecalleddefinitionbytype.3636 Itisnotofcoursemeantthatwhenwestartfromanextensivedefinition,we

areclassingthingstogetheratrandomwithoutanyguidingprincipleofselection.Nodoubtweshallbeguidedbyaresemblancebetweentheobjectswhichweplacein the same class, and in this sense intension may be said always to have thepriority. But the resemblance may be unanalysed, so that we may be far morefamiliarwiththeapplicationoftheclass-namethanwithitsimplication;andevenwhen a connotation has been assigned to the name, it may be extensivelycontrolled,andconstantlysubjecttomodification,justbecausewearemuchmoreconcernedtokeepthedenotationfixedthantheconnotation.

In this connexion the names of simple feelings which are incapable ofanalysis may be specially considered. For the names of ultimate elements,thereis,saysSigwart,37nodefinition;wemustassumethateveryoneattachesthesamemeaningtothem.Tosuchnameswemayindeedbeabletoassignaproximategenus,aswhenwesay“red isacolour”;butwe35 cannot addaspecificdifference.Itis,however,onlyanintensivedefinitionthatiswantingin these cases; and the deficiency is supplied by means of an extensivedefinition.Thewayinwhichwemakecleartoothersouruseofsuchatermas“red”isbypointingoutorotherwiseindicatingvariousobjectswhichgiverise inus to the feeling.Thus“red” is thecolourof fieldpoppies,hipsandhaws,ordinarysealing-wax,bricksmadefromcertainkindsofclay,&c.Thisisnothingmoreorlessthananextensivedefinitionasabovedefined.

37 Logic,I.p.289.

Inthecaseofmostnames,however,whereformaldefinitionisattempted,itismoreusual,aswellasreallysimpler,tostartfromanintensivedefinition,and this in general correspondswith the ultimate procedure of science. Forlogical purposes, it is accordingly best to assume this order of procedure,unlessanexplicitstatementismadetothecontrary.38

38 It isworthnoticingthat inpracticeanintensivedefinitionisoftenfollowedby an enumeration of typical examples,which, ifwell selected,may themselvesalmostamounttoanextensivedefinition.Inthiscase,wemaybesaidtohavethetwokindsofdefinitionsupplementingoneanother.

23. Inverse Variation of Extension and Intension.39—In general, asintension is increased or diminished, extension is diminished or increasedaccordingly, and vice versâ. If, for example, rational is added to theconnotation of animal, the denotation is diminished, since all non-rational


animals are now excluded, whereas theywere previously included. On theotherhand,ifthedenotationofanimalistobeextendedsoastoincludethevegetablekingdom,itcanonlybebyomittingsensitivefromtheconnotation.Hencethefollowinglawhasbeenformulated:“Inaseriesofcommontermsstandingtooneanotherinarelationofsubordination40theextensionandtheintensionvaryinversely.” Is this lawtobeaccepted?Itmustbeobservedatthe outset that the notion of inverse variation is at any rate not to beinterpreted in any strictmathematical or numerical sense. It is certainlynottrue that whenever the number of 36 attributes included in the intension isaltered in any manner, then the number of individuals included in theextension will be altered in some assigned numerical proportion. If, forexample, to the connotation of a given name different single attributes areadded, the denotationwill be affected in very different degrees in differentcases. Thus, the addition of resident to the connotation ofmember of theSenateof theUniversityofCambridgewill reduce its denotation in amuchgreaterdegreethantheadditionofnon-resident.Thereisinshortnoregularlawof variation.The statementmust not then be understood tomeanmorethan that any increase or diminution of the intension of a name willnecessarilybeaccompaniedbysomediminutionorincreaseoftheextensionasthecasemaybe,andviceversâ.41Wewilldiscusstheallegedlawinthisform, considering, first, connotation and denotation, exemplification andcomprehension;and,secondly,denotationandcomprehension.42

39 Thissectionmaybeomittedonafirstreading.40 AsintheTreeofPorphyry:Substance,CorporealSubstance(Body),Animate

Body(LivingBeing),SensitiveLivingBeing(Animal),RationalAnimal(Man).Inthis series of terms the intension is at each step increased, and the extensiondiminished.

41 Ithasbeensaidthatwhiletheextensionofatermiscapableofquantitativemeasurement, the same is not equally true of intension. “The parts of extensionmay be counted, but it is inept to count the parts in intension. For they are notexternaltoeachother,andtheyformawholesuchascannotbedividedintounitsexcept by themost arbitrary dilaceration.And if itwere so divided, all its partswouldvaryinvalue,andtherewouldbenoreasontoexpectthattenofthem(thatis, ten attributes) should have twice the amount or value of five” (Bosanquet,Logic,I.p.59).Thereissomeforceinthis,anditisdecisiveagainstinterpretinginversevariationinthepresentconnexioninanystrictnumericalsense.But,atthesame time,noerror is committedandnodifficultyof interpretationarises, ifwe


content ourselves with speakingmerely of the enlargement or restriction of theintensionofaterm.Therecanbenodoubtthatintensionisincreasedwhenwepassfromanimaltoman,orfrommantonegro;oragainwhenwepassfromtriangletoisoscelestriangle,orfromisoscelestriangletoright-angledisoscelestriangle.

42 Thediscussionispurposelymadeasformalandexactaspossible.Ifindeedthedoctrineofinversevariationcannotbetreatedwithprecision,itisbetternottoattempttodealwithitatall.

A.(1)Letconnotationbesupposedarbitrarilyfixed,andusedtodeterminedenotation in some assigned universe of discourse.Then itwill not be truethatconnotationanddenotationwillnecessarilyvary inversely.Forsupposetheconnotationofaname, i.e., theattributessignifiedbyit, tobea,b,c. Itmay happen that in fact wherever the attributes a and b are present, theattributesc andd are alsopresent. 37 In this case, if c is dropped from theconnotation, or d added to it, the denotation of the name will remainunaffected.Wehaveconcreteexamplesofthis,ifwesupposeequiangularityadded to theconnotationofequilateral triangle,orcloven-hoofed to that ofruminant,orhavingjawsopeningupanddowntothatofvertebrate,orifwesuppose invalid dropped from the connotation of invalid syllogism withundistributedmiddle.It isclear,however,thatifanyalteration indenotationtakesplacewhenconnotationisaltered,itmustnecessarilybeintheoppositedirection. Some individuals possessing the attributes a andb may lack theattributecortheattributed ;butnoindividualspossessingtheattributesa,b,c,ora,b,c,dcanfailtopossesstheattributesa,b,ora,b,c.Forexample,ifto the connotation ofmetal we add fusible, it makes no difference to thedenotation;butifweaddhavinggreatweight,weexcludepotassium,sodium,&c.

Thelawofvariationofdenotationwithconnotationmaythenbestatedasfollows:—Iftheconnotationofatermisarbitrarilyenlargedorrestricted,thedenotationinanassigneduniverseofdiscoursewilleitherremainunalteredorwillchangeintheoppositedirection.43

43 Since reference is here made to the actual denotation of a term in someassigned universe of discourse, the above law may be said to turn partly onmaterial, andnotonpurely formal,considerations. It should, therefore,beaddedthat although an alteration in the connotation of a termwill not always alter itsactual denotation in an assigned universe of discourse, it will always affectpotentiallyitssubjectiveextension.If,forexample,theconnotationofatermXis


a,b,c, andweaddd ; then the (real or imaginary) class ofX’s that are notd isnecessarily excluded from, while it was previously included in, the subjectiveextensionofthetermX.Hence,iftheconnotationofatermisarbitrarilyenlargedor restricted, the subjective extension will be potentially restricted or enlargedaccordingly.Cf.Jevons,PrinciplesofScience,30,§13.

(2)Letexemplificationbesupposedarbitrarilyfixed,andusedtodeterminecomprehension.Itisunnecessarytoshewindetailthatacorrespondinglawofvariationofcomprehensionwithexemplificationwillholdgood,namely:—Iftheexemplification(extensivedefinition)ofa termisarbitrarilyenlargedorrestricted,thecomprehensioninanassigneduniverseofdiscoursewilleitherremainunalteredorwillchangeintheoppositedirection.38

B.Wemaynowconsidertherelationbetweenthecomprehensionandthedenotationofaterm.LetP1,P2,…Pxbe the totalityofattributespossessedbytheclassX,andQ1,Q2,…QythetotalityofobjectsincludedintheclassX.Both these groups are objectively, not arbitrarily,44 determined; and therelation between them is reciprocal. P1, P2, … Px are the only attributespossessedincommonbytheobjectsQ1,Q2,…Qy ;andQ1,Q2,…QyaretheonlyobjectspossessingalltheattributesP1,P2,…Px.

44 What may be arbitrary is the intensive definition (P1, P2, … Pm) or theextensivedefinition(Q1,Q2,…Qn)whichdeterminesthemboth.

Wecannotsupposeanydirectarbitraryalterationeither incomprehensionor in denotation. We can, however, establish the following law of inversevariation,namely, thatanyarbitraryalteration in either intensivedefinitionorextensivedefinitionwhichresults inanalterationofeitherdenotationorcomprehensionwillalsoresultinanalterationintheoppositedirectionoftheother.

LetX andY be two termswhich are so related that the definition (eitherintensiveorextensive,asthecasemaybe)ofYincludesallthatisincludedinthe definition of X and more besides. We have to shew that either thedenotations and comprehensions of X and Y will be identical or if thedenotation of one includes more than the denotation of the other then itscomprehensionwillincludeless,andviceversâ.

(a)LetXandYbedeterminedbyconnotationorintensivedefinition.Thus,letXbedeterminedbythesetofpropertiesP1…PmandYbythesetP1…


Pm+1,whichincludestheadditionalpropertyPm+1. ThenPm+1eitherdoesordoesnotalwaysaccompanyP1…Pm. If the former,noobject included in thedenotationofX is excluded fromthatofY,sothatthedenotationsofXandYarethesame;anditfollowsthatthecomprehensionsofXandYarealsothesame. If the latter, then the denotation ofY is less than that ofX by all thoseobjectsthatpossessP1…PmwithoutalsopossessingPm+1.Atthesametime,thecomprehensionofYincludesat39leastPm+1 inadditiontothepropertiesincludedinthecomprehensionofX.

(b)LetXandYbedeterminedbyexemplificationorextensivedefinition.Thus,letXbedeterminedbythesetofexamplesQ1…Qn,andYbythesetQ1…Qn+1whichincludestheadditionalobjectQn+1. ThenQn+1eitherdoesordoesnotpossessallthepropertiescommontoQ1

…Qn. Iftheformer,nopropertyincludedinthecomprehensionofXisexcludedfrom thatofY, so that thecomprehensionsofX andY are the same; and itfollowsthatthedenotationsofXandYarealsothesame. Ifthelatter,thenthecomprehensionofYislessthanthatofXbyallthosepropertiesthatbelongtoQ1…QnwithoutalsobelongingtoQn+1.Atthesametime, the denotation of Y includes at leastQn+1 in addition to the objectsincludedinthedenotationofX.

All cases have nowbeen considered, and it has been shewn that the lawaboveformulatedholdsgooduniversally.Thislawandthetwolawsgivenonpage37musttogetherbesubstitutedforthelawofinverserelationbetweenextension and intension in its usual form if full precision of statement isdesired.

It should be observed that in speakingof variations in comprehensionordenotation,noreferenceisintendedtochangesinthingsorinourknowledgeof them. The variation is always supposed to have originated in somearbitraryalterationintheintensiveorextensivedefinitionofagiventerm,orin passing from the consideration of one term to that of another with adifferent extensive or intensive definition. Thus fresh things may bediscoveredtobelongtoaclass,andthecomprehensionoftheclass-namemay


not therebybeaffected.But in thiscase thedenotationhasnot itselfvaried;only our knowledge of it has varied. Or we may discover fresh attributespreviouslyoverlooked;inwhichcasesimilarremarkswillapply.Again,newthingsmaybebroughtintoexistencewhichcomeunderthedenotationofthename, and still its comprehensionmay remain unchanged.Or possibly newqualities may be developed by 40 the whole of the class. In these cases,however, there isnoarbitrary alteration in theapplicationor implicationofthename,andhencenorealexceptiontowhathasbeenlaiddownabove.

24.ConnotativeNames.—Mill’suseofthewordconnotative,whichisthatgenerally adopted in modern works on logic, is as follows: “A non-connotativetermisonewhichsignifiesasubjectonly,oranattributeonly.Aconnotative term is one which denotes a subject, and implies an attribute”(Logic, I.2,§5).According to thisdefinition,aconnotativenamemustnotonlypossessextension,butmustalsohaveaconventionalintensionassignedtoit.

Mill considers that the following kinds of names are connotative in theabovesense:—(1)Allconcretegeneralnames.(2)Somesingularnames.Forexample, city is a general name, and as such no one would deny it to beconnotative. Now if we say the largest city in the world, we haveindividualised the name, but it does not thereby cease to be connotative.Proper names are, however, according toMill, non-connotative, since theymerelydenoteasubjectanddonotimplyanyattributes.Tothispoint,whichisasubjectofcontroversy,weshallreturninthefollowingsection.(3)Whileadmitting that most abstract names are non-connotative, since they merelysignify an attribute and do not denote a subject, Mill maintains that someabstractsmayjustlybe“consideredasconnotative;forattributesthemselvesmay have attributes ascribed to them; and awordwhich denotes attributesmayconnoteanattributeofthoseattributes”(Logic,I.2,§5).

ThewordingofMill’sdefinitionisunfortunateandisprobablyresponsibleforagooddealof thecontroversy thathascentred round thequestionas towhethercertainclassesofnamesareorarenotconnotative.

All names that we are able to use in an intelligible sense must havesubjectiveintensionforus.Forwemustknowtowhatobjectsorwhatkinds


of objects the names are applicable, and we cannot but associate somepropertieswiththeseobjectsandthereforewiththenames.

Moreover all names that have denotation in any given 41 universe ofdiscourse must have comprehension also; for no object can exist withoutpossessingpropertiesofsomekind.

Ifthenanynamecanproperlybedescribedasnon-connotative,itcannotbeinthesensethatithasnosubjectiveintensionornocomprehension.Thisisatleast obscuredwhenMill speaksof non-connotativenames as not implyinganyattributes;andifmisunderstandingistobeavoided,hisdefinitionsmustbeamended,soastomakeitquiteclearthatinanon-connotativenameitisconnotation only that is lacking, and not either subjective intension orcomprehension.

A connotative name may be defined as a name whose application isdeterminedbyconnotationorintensivedefinition,thatis,byaconventionallyassigned attribute or set of attributes. A non-connotative name is anexemplificative name, a name whose application is determined byexemplificationorextensivedefinitioninthesenseexplainedinsection22;inotherwords,itisanamewhoseapplicationisdeterminedbypointingoutorindicating,bymeansofadescriptionorotherwise,theparticularindividual(ifthenameissingular),ortypicalindividuals(ifthenameisgeneral),towhichthenameisattached.

Ifitisallowedthattheapplicationofanynamescanbedeterminedinthelatterway,asdistinguishedfromtheformer,thenitmustbeallowedthatsomenamesarenon-connotative.

25. Are proper names connotative?—To this question absolutelycontradictory answers are given by ordinarily clear thinkers as beingobviouslycorrect.Tosomeextent,however,thedivergenceismerelyverbal,the terms “connotation” and “connotative name” being used in differentsenses.

It isnecessaryat theoutset toguardagainstamisconceptionwhichquiteobscures the real point at issue. Thus, with reference toMill, Jevons says,“Logicianshaveerroneouslyasserted,as it seems tome, thatsingular terms


aredevoidofmeaningin intension, thefactbeingthat theyexceedallothertermsinthatkindofmeaning”(PrinciplesofScience,2,§2,withareferencetoMillinafoot-note).ButMilldistinctlystatesthatsomesingularnamesareconnotative,e.g.,the42sun,45thefirstemperorofRome(Logic,I.2,§5).Wemay certainly narrow down the extension of a term till it becomesindividualised without destroying its connotation; “the present Professor ofPure Mathematics in University College, London” is a singular term—itsextensioncannotbefurtherdiminished—butitiscertainlyconnotative.

45 The question has been asked onwhat grounds the sun can be regarded asconnotative, while John is considered non-connotative; compare T. H. Green,PhilosophicalWorks, ii. p.204.Theanswer is thatsun is ageneralnamewithadefinitesignificationwhichdeterminesitsapplication,andthatitdoesnotloseitsconnotationwhenindividualisedbytheprefixthe ;whileJohn,ontheotherhand,isanamegiventoanobjectmerelyasamarkforpurposesoffuturereference,andwithout signifying the possession by that object of any conventionally selectedattributes.

Itmustthenbeunderstoodthatonlyoneclassofsingularnames,namely,propernames,areaffirmedtobenon-connotative;andthatnomoreismeantby this than that their application is not determined by a conventionallyassigned set of attributes.46 The ground may be further cleared by ourexplicitlyrecognisingthat,althoughpropernameshavenoconnotation,theynevertheless have both subjective intension and comprehension. Anindividualobjectcanberecognisedonly through itsattributes;andapropername when understood by me to be a mark of a certain individualundoubtedly suggests to my mind certain qualities.47 The qualities thussuggestedbythenameconstituteitssubjectiveintension.Thecomprehensionof the name will include a good deal more than its subjective intension,namely,43thewholeofthepropertiesthatbelongtotheindividualdenoted.

46 Thetreatmentofthequestionadoptedinthisworkhasbeencriticisedontheground that it is question-begging, since in section 10 proper names have reallybeendefinedasnon-connotative.Thiscriticismcannot,however,bepressedunlessit is at the same timemaintained that the definition given in section10 yields adenotationdifferentfromthatordinarilyunderstoodtobelongtopropernames.

47 Apropernamemayhavesuggestiveforceevenforthosewhoarenotactuallyacquaintedwith the person or thing denoted by it. ThusWilliamStanley Jevonsmaysuggestanyorallofthefollowingtoonewhoneverheardthenamebefore:anorganisedbeing,ahumanbeing,amale,anAnglo-Saxon,havingsomerelative


named Stanley, having parents named Jevons. But at the same time, the namecannotbesaidnecessarily tosignifyanyof these things, in thesensethat if theywere wanting it would be misapplied. Consider, for example, such a name asVictoriaNyanza.Somefurtherremarksbearingonthispointwillbefoundlateroninthissection.

Itwillbefoundthatmostwriterswhoregardpropernamesaspossessingconnotationreallymeantherebyeithersubjectiveintensionorcomprehension.Thus Jevons puts his case as follows:—“Any proper name such as JohnSmith,isalmostwithoutmeaninguntilweknowtheJohnSmithinquestion.It is true that thenamealoneconnotes the fact thathe isaTeuton,and isamale;but,sosoonasweknowtheexactindividualitdenotesthenamesurelyimplies,also,thepeculiarfeatures,form,andcharacter,ofthatindividual.Infact, as it is only by the peculiar qualities, features, or circumstances of athing, thatwecaneverrecogniseit,nonamecouldhaveanyfixedmeaningunlesswe attached to it,mentally at least, such a definition of the kind ofthing denoted by it, that we should know whether any given thing wasdenotedbyitornot.IfthenameJohnSmithdoesnotsuggesttomymindthequalities of John Smith, how shall I know him when I meet him? For hecertainlydoesnotbearhisnamewrittenuponhisbrow”(ElementaryLessonsin Logic, p. 43). A wrong criterion of connotation in Mill’s sense is heretaken.TheconnotationofanameisnotthequalityorqualitiesbywhichIorany one else may happen to recognise the class which it denotes. Forexample,ImayrecogniseanEnglishmanabroadbythecutofhisclothes,oraFrenchmanbyhispronunciation,oraproctorbyhisbands,orabarristerbyhiswig;butIdonotmeananyofthesethingsbythesenames,nordothey(inMill’s sense) form any part of the connotation of the names.Compare twosuch names asHenry Montagu Butler and the Master of Trinity College,Cambridge.Atthepresenttimetheydenotethesameperson;butthenamesare not equivalent,—the one is given to a certain individual as a mark todistinguish him from others, and has no further signification; the other isgiven because of the performance of certain functions, on the cessation ofwhichthenamewouldceasetoapply.Surelythereisadistinctionhere,andonewhichitisimportantthatweshouldnotoverlook.

Itmay indeed fairly be said thatmany, if notmost, proper 44 names dosignifysomething,inthesensethattheywerechoseninthefirstinstancefora


specialreason.Forexample,Strongi’th’arm,Smith,Jungfrau.Butsuchnamesevenifinacertainsenseconnotativewhenfirstimposedsoonceasetobeso,since their subsequent application to thepersonsor thingsdesignated isnotdependent on the continuance of the attributewith reference towhich theywereoriginallygiven.AsMillputsit,thenameoncegivenisindependentofthe reason. In other words, we ought carefully to distinguish between theconnotation of a name and itshistory. Thus, amanmay in his youth havebeenstrong,butweshouldnotcontinuetocallingstronginhisdotage;whilstthenameStrongi’th’armoncegivenwouldnotbetakenfromhim.Again,thenameSmithmayinthefirstinstancehavebeengivenbecauseamanpliedacertain handicraft, but he would still be called by the same name if hechangedhistrade,andhisdescendantscontinuetobecalledSmithwhatevertheiroccupationsmaybe.48

48 Itcannot,however,besaidthatthenamenecessarilyimpliesancestorsofthesamename.AsDrVennremarks,“hewhochangeshisfamilynamemaygrosslydeceivegenealogists,buthedoesnottellafalsehood”(EmpiricalLogic,p.185).

Ithasbeenarguedthatpropernamesmustbeconnotativebecausetheuseofapropernameconveysmoreinformationthantheuseofageneralname.“Few persons,” says Mr Benecke,49 “will deny that if I say the principalspeakerwasMrGladstone,Iamgivingnotlessbutmoreinformationthanif,insteadofMrGladstone,IsayeitheramemberofParliament,oraneminentman,orastatesman,oraLiberalleader.ItwillbeadmittedthatthepredicateMr Gladstone tells us all that is told us by all these other connotativepredicatesputtogether,andmore;and,ifso,Icannotseehowitcanbedeniedthat it also connotesmore.” It is clear, however, that the information givenwhen a thing is called by any name depends not on the connotation of thename, but on its intension for theperson addressed.To anyonewhoknowsthat Mr Gladstone was Prime Minister in 1892 the same information isafforded whether a speaker is referred to as Mr Gladstone or as PrimeMinisterof45GreatBritainand Ireland in 1892.But it certainly cannot bemaintainedthattheconnotationofthesetwonamesisidentical.

49 InapaperontheConnotationofProperNamesreadbeforetheAristotelianSociety.

In criticismof theposition that the applicationof a propername such as


Gladstone is determined by some attribute or set of attributes, we maynaturallyask,whatattributeorsetofattributes?Theanswercannotbethattheconnotation consists of the complete group of attributes possessed by theindividualdesignated;foritisabsurdtorequireanysuchenumerationasthisinordertodeterminetheapplicationofthename.Itis,however,impossibletoselect some particular attributes of the individual in question, and point tothemasagroup thatwouldbeacceptedasconstituting thedefinitionof thename;andifit issaidthattheapplicationofthenameisdeterminedbyanysetofattributes thatwill suffice for identification, thecase isgivenup.Forthisamountstoidentifyingtheindividualbyadescription(thatis,practicallyby exemplification), not by a particular set of attributes conventionallyattachedtothenameassuch.Thetruthisthatnoonewouldeverproposetogive an intensivedefinition of a proper name.All names, however, that areconnotativemustnecessarilyadmitofintensivedefinition.50

50 Mr Bosanquet arrives at the conclusion that “a proper name has aconnotation, but not a fixed general connotation. It is attached to a uniqueindividual, and connotes whatever may be involved in his identity, or isinstrumentalinbringingitbeforethemind”(EssentialsofLogic,p.93).SofarasIcan understand this statement, it amounts to saying that proper names havecomprehension and subjective intension, but not connotation, in the senses inwhichIhavedefinedtheseterms.

Proper names of course become connotative when they are used todesignateacertaintypeofperson;forexample,aDiogenes,aThomas,aDonQuixote,aPaulPry,aBenedick,aSocrates.But,whensoused,suchnameshavereallyceasedtobepropernamesatall;theyhavecometopossessallthecharacteristicsofgeneralnames.51

51 CompareGray’slines,—

“SomevillageHampden,that,withdauntlessbreast,Thelittletyrantofhisfieldswithstood,SomemuteingloriousMiltonheremayrest,SomeCromwellguiltlessofhiscountry’sblood.”

Attentionmaybecalledtoaclassofsingularnames,suchas46MissSmith,Captain Jones, President Roosevelt, the Lake of Lucerne, the Falls ofNiagara, which may be said to be partially but only partially connotative.Their peculiarity is that they are partly made up of elements that have a


general and permanent signification, and that consequently some change intheobjectdenotedmightrenderthemnolongerapplicable,as,forexample,ifCaptainJonesreceivedpromotionandweremadeamajor;while,atthesametime, such connotation as they possess is by itself insufficient to determinecompletely their application. Itmaybe said that their application is limited,but not determined, by reference to specific assignable attributes. Theyoccupy an intermediate position, therefore, between connotative singularnames,suchasthefirstman,andstrictlypropernames.

WemayinthisconnexiontouchuponJevons’sargumentthatsuchanameas“JohnSmith”connotesatanyrate“Teuton”and“male.”Thisisnotstrictlythecase,since“JohnSmith”mightbeadahlia,oraracehorse,oranegro,orthepseudonymofawoman,asinthecaseofGeorgeEliot.Innoneofthesecasescouldthenamebesaidtobemisappliedasitwouldbeifadahliaorahorsewerecalledaman,oranegroaTeuton,orawomanamale.Atthesametime, it cannotbedenied that certainpropernamesare inpractice somuchlimited to certain classes of objects, that some incongruitywould be felt iftheywereappliedtoobjectsbelongingtoanyotherclass.Itis,forexample,unlikelythataparentwoulddeliberatelyhavehisdaughterchristened“JohnRichard.”Sofarasthisisthecase,thenamesinquestionmaybesaidtobepartially connotative in the same way as the names referred to in theprecedingparagraph,thoughtoalessextent;thatistosay,theirapplicationislimited,thoughnotdetermined,byreferencetospecificattributes.Weshouldhaveastillclearercaseofa similarkind if the right tobearacertainnamecarriedwithitspecificlegalorsocialprivileges.52

52 CompareBosanquet,Logic,i.p.53.

The position has been taken that every proper name is at least partiallyconnotativeinasmuchasitnecessarilyimpliesindividualityandthepropertyofbeingcalledbythenameinquestion.Ifwerefertoanythingbyanynamewhatsoever, it 47 must at any rate have the quality of being called by thatname.IfwecallamanJohnwhenhereallypassesbythenameofJames,wemakeamistake;weattribute tohimaqualitywhichhedoesnotpossess,—that of passing by the name of John. This argument, although it does notappear to establish the conclusion that proper names are in any degree


connotative,neverthelesscallsattentiontoadistinctivepeculiarityofpropernamesthatisworthyofnotice.Thedenotationofconnotativenamesmay,andusuallydoes,varyfromtimetotime;andthisistrueofconnotativesingularnamesaswellasofgeneralnames.Butitisclearlyessentialinthecaseofapropernamethat(inanygivenuse)thenameshallbeconsistentlyaffixedtothesameindividualobject.Itis,however,onethingtosaythattheidentityoftheobjectcalledbythenamewiththattowhichthenamehaspreviouslybeenassigned is a condition essential to the correct use of a proper name, andanother thing to say that this is connoted by a proper name. If indeed byconnotationwemeantheattributesbyreasonof thepossessionofwhichbyanyobject thename isapplicable to thatobject, it seemsacaseofὕστερονπρότερον to include in the connotation the property of being called by thename.

EXERCISES.

26.Aresuchconceptsas“equilateraltriangle”and“equiangulartriangle”identicalordifferent?[K.] [Thisquestionshouldbeconsideredwithreferencetothediscussioninsections17and18.]

27.LetX1,X2,X3,X4,andX5constitutethewholeofacertainuniverseofdiscourse:alsoleta,b,c,d,e,fexhaustthepropertiesofX1;a,b,c,d,e,g,thoseofX2;b,c,d,f,g,thoseofX3;a,b,d,e,f,thoseofX4;anda,c,e,f,gthoseofX5. (i)Given that, under these conditions, a term has the connotationa,b, find its denotation and itscomprehension,anddetermineanexemplificationthatwouldyieldthesameresult. (ii) Given that, under the same conditions, a term has the exemplification X4, X5, find itscomprehensionanditsdenotation,anddetermineaconnotationthatwouldyieldthesameresult.[K.]

4828.Onwhatgroundsmayitbeheldthatnamesmaypossess(a)denotationwithoutconnotation,(b)connotationwithoutdenotation? Giveillustrationsshewingthatthedenotationofatermofwhichtheconnotationisknownmustberegarded as relative to the proposition inwhich it is used as subject and to the context inwhich thepropositionoccurs.[J.]

29.Whatdoyouconsidertobethequestionreallyatissuewhenitisaskedwhetherpropernamesareconnotative? Enquirewhetherthefollowingnamesarerespectivelyconnotativeornon-connotative:Caesar,Czar,Lord Beaconsfield, the highest mountain in Europe, Mont Blanc, the Weisshorn, Greenland, theClaimant,thepolestar,Homer,aDanielcometojudgment.[K.]

30.Bringoutanyspecialpointsthatariseinthediscussionoftheextensionalandintensionalaspects


ofthefollowingtermsrespectively:theRosaceae,equilateraltriangle,colour,giant.[C.]


CHAPTERIII.

REAL,VERBAL,ANDFORMALPROPOSITIONS.

31. Real (Synthetic), Verbal (Analytic or Synonymous), and FormalPropositions.—(1) A real proposition is one which gives information ofsomethingmorethanthemeaningorapplicationofthetermwhichconstitutesits subject; aswhen a proposition predicates of a connotative subject someattribute not included in its connotation, or when a connotative term ispredicatedofanon-connotativesubject.Forexample,Allbodieshaveweight,Theanglesofany triangleare together equal to two rightangles,Negativepropositionsdistributetheirpredicates,Wordsworthisagreatpoet.

Realpropositionsarealsodescribedassynthetic,ampliative,accidental.

(2)Averbalproposition isonewhichgives informationonly in regard tothemeaningorapplicationofthetermwhichconstitutesitssubject.53


53 Althoughverbalpropositionsmaybedistinguishedfromrealpropositionsinaccordance with the above definitions, it may be argued that every verbalpropositionimpliesarealpropositionofacertainsortbehindit.Forthequestionastowhatmeaning isattached toagiven terminordinarydiscourse,orbyagivenindividual,isaquestionofmatteroffact,andastatementrespectingitmaybetrueorfalse.Thus,Xmeansabc isaverbalproposition;butsuchpropositionsasThemeaning commonly attached to the termX is abc,Themeaning attached in thisworktothetermXisabc,ThemeaningwithwhichitwouldbemostconvenienttoemploythetermXisabc,arereal.Lookedatfromthispointofviewthedistinctionbetweenverbalandrealpropositionsmayperhapsbethoughttobearathersubtleone.Itremainstrue,however,thatthepropositionXmeansabcisverbalrelativelytoitssubjectX.OutofthegivenmaterialwecannotbyanymanipulationobtainarealpredicationaboutX, thatis,aboutthethingsignifiedbythetermX,butonlyaboutthemeaningof the termX.The realproposition involvedcan thusonlybeobtainedbysubstitutingfortheoriginalsubjectanothersubject.

50Twoclassesofverbalpropositionsaretobedistinguished,whichmaybecalledrespectivelyanalyticandsynonymous.Intheformerthepredicategivesapartialorcompleteanalysisof theconnotationof thesubject;e.g.,Bodiesareextended,Anequilateraltriangleisatrianglehavingthreeequalsides,Anegativepropositionhasanegativecopula.54Definitionsare includedunderthis division of verbal propositions; and the importance of definitions is sogreat,thatitisclearlyerroneoustospeakofverbalpropositionsasbeinginallcases trivial. Ingeneral theyare trivial only in so far as their truenature ismisunderstood;when,forexample,peoplewastetimeinpretendingtoprovewhat has been already assumed in the meaning assigned to the termsemployed.55

54 Sincewedonotherereallyadvancebeyondananalysisofthesubject-notion,Dr Bain describes the verbal proposition as the “notion under the guise of theproposition.”Hence theappropriatenessof treatingverbalpropositionsunder thegeneralheadofTerms.

55 Byaverbaldisputeismeantadisputethatturnsonthemeaningofwords.DrVennobservesthatpurelyverbaldisputesareveryrare,since“adifferentusageofwords almost necessarily entails different convictions as to facts” (EmpiricalLogic,p.296).This is trueand important; itought indeedalways tobeborne inmindthattheproblemofscientificdefinitionisnotamerequestionofwords,butaquestion of things. At the same time, disputes which are partly verbal areexceedinglycommon, and it is alsoverycommon for their true character in thisrespect tobeunrecognised.When this is thecase, thecontroversy ismore likelythannottobefruitless.Thequestionswhetherpropernamesareconnotative,andwhethereverysyllogisminvolvesapetitioprincipii,maybetakenasexamples.We


certainly go a long way towards the solution of these questions by clearlydifferentiating between different meanings which may be attached to the termsemployed.

Besides propositions giving a more or less complete analysis of theconnotationofnames,thefollowing—whichwemayspeakofassynonymouspropositions—are to be included under the head of verbal propositions: (a)wherethesubjectandpredicatearebothpropernames,e.g.,TullyisCicero ;(b)where they are dictionary synonyms,e.g.,Wealth is riches,A story is atale,Charityislove.Inthesecasesinformationisgivenonlyinregardtotheapplication or meaning of the terms which appear as the subjects of thepropositions.

Analytic propositions are also described as explicative and as essential.Verynearlythesamedistinction,therefore,as51thatbetweenverbalandrealpropositions is expressed by the pairs of terms—analytic and synthetic,explicative and ampliative, essential and accidental. These terms do not,however,coverquitethesamegroundasverbalandreal,sincetheyleaveoutofaccountsynonymouspropositions,whichcannot,forexample,beproperlydescribedaseitheranalyticorsynthetic.56

56 Thus,Manselcallsattentionto“aclassofpropositionswhicharenot,inthestrict sense of theword, analytical,viz., those inwhich the predicate is a singletermsynonymouswiththesubject”(Mansel’sAldrich,p.170).

The distinction between real and verbal propositions as above givenassumes that the use of terms is fixed by their connotation and that thisconnotationisdeterminate.57Whetheranygivenpropositionisasamatteroffactverbalorrealwilldependonthemeaningattachedtothetermswhichitcontains;and it isclear that logiccannot laydownanyrulefordeterminingunderwhichcategoryanygivenpropositionshouldbeplaced.58 Still,whilewecannotwithcertaintydistinguisharealpropositionbyitsform,itmaybeobserved that theattachmentofa signofquantity, suchasall,every, some,&c., to the subject of a proposition may in general be regarded as anindicationthatintheviewofthepersonlayingdownthe52propositionafactisbeingstatedandnotmerelya termexplained.Verbalpropositions,on theother hand, are usually unquantified or indesignate (see section 69). Forexample, in order to give a partially correct idea of themeaning of such a


name as square, we should not say “all squares are four-sided figures,” or“everysquareisafour-sidedfigure,”but“asquareisafour-sidedfigure.”59

57 Wecan,however,adaptthedistinctiontothecaseinwhichtheuseoftermsisfixed by extensive definition. We may say that whilst a proposition (expressedaffirmatively and with a copula of inclusion) is intensively verbal when theconnotationofthepredicateisapartorthewholeoftheconnotationofthesubject,itisextensivelyverbalwhenthesubjecttakeninextensionisapartorthewholeoftheextensivedefinitionofthepredicate.Thus,iftheuseofthetermmetalisfixedby an extensive definition, that is to say, by the enumeration of certain typicalmetals,ofwhichwemaysupposeirontobeone,thenitisaverbalpropositiontosaythatironisametal.If,however,tinisnotincludedamongstthetypicalmetals,thenitisarealpropositiontosaythattinisametal.

58 It does not follow from this that the distinction between verbal and realpropositionsisofnologicalimportance.Althoughthelogiciancannotquâlogiciandetermineindoubtfulcasestowhichcategoryagivenpropositionbelongs,hecanpointoutwhataretheconditionsuponwhichthisdepends,andhecanshewthatinanydiscussionorargumentnoprogressispossibleuntilitisclearlyunderstoodbyallwhoaretakingpartwhetherthepropositionslaiddownaretobeinterpretedasbeingrealormerelyverbal.Torefertoananalogouscase,itwillnotbesaidthatthe distinction between truth and falsity is of no logical importance because thelogiciancannotquâlogiciandeterminewhetheragivenpropositionistrueorfalse.

59 Itshouldbeaddedthatwemayformallydistinguishafulldefinitionfromarealpropositionbyconnectingthesubjectandthepredicatebytheword“means”insteadoftheword“is.”

(3)Therearepropositionsusuallyclassedasverbalwhichoughtrather tobeplaced inaclassby themselves,namely, thosewhicharevalidwhatevermaybethemeaningofthetermsinvolved;e.g.,AllAisA,NoAisnot-A,AllZiseitherBornot-B,IfallAisBthennonot-BisA,IfallAisBandallBisC then all A is C. These may be called formal propositions, since theirvalidityisdeterminedbytheirbareform.60

60 Propositionswhichareinappearancepurelytautologoushavesometimesanepigrammaticforceandareusedforrhetoricalpurposes,e.g.,Aman’saman(fora’that). In such cases, however, there is usually an implication which gives thepropositionthecharacterofarealproposition;thus,intheaboveinstancethetrueforce of the proposition is thatEveryman is as such entitled to respect. “In theproposition, Children are children, the subject-term means only the agecharacteristicofchildhood; thepredicate-term theothercharacteristicswhichareconnectedwithit.Bytheproposition,Wariswar,wemeantosaythatwhenonceastate of warfare has arisen, we need not be surprised that all the consequencesusuallyconnectedwithitappearalso.Thusthepredicateaddsnewdeterminationstothemeaninginwhichthesubjectwasfirsttaken”(Sigwart,Logic,I.p.86).


Formalpropositionsare theonlypropositionswhosevalidity isexaminedandguaranteedbylogicitselfirrespectiveofothersourcesofknowledge,andmany of the results reached in formal logic may be summed up in suchpropositions; foranyformallyvalidreasoningcanbeexpressedbyaformalhypotheticalpropositionasinthelasttwooftheexamplesgivenabove.

A formal proposition as here defined must not be confused with aproposition expressed in symbols.A formal proposition need not indeed beexpressedinsymbolsatall.Thus,thepropositionAnanimalisananimalisaformal proposition; 53 All S is P is not. Strictly speaking, a symbolicexpression,suchasAllSisP,istoberegardedasapropositionalform,ratherthanasapropositionperse.Foritcannotbedescribedasinitselfeithertrueor false.Whatweare largelyconcernedwith in logic are relationsbetweenpropositional forms; because these involve corresponding relations betweenallpropositionsfallingintotheformsinquestion.

Wehavethenthreeclassesofpropositions—formal,verbal,andreal—thevalidityorinvalidityofwhichisdeterminedrespectivelybytheirbareform,bythemeremeaningorapplicationofthetermsinvolved,byquestionsoffactconcerningthethingsdenotedbytheseterms.61

61 Realpropositionsaredividedintotrueandfalseaccordingastheydoordonotaccuratelycorrespondwithfacts.Byverbalandformalpropositionsweusuallymean propositions which from the point of view taken are valid. A propositionwhichfromeitherofthesepointsofviewisinvalidisspokenofasacontradictioninterms.Properlyspeakingweoughttodistinguishbetweenaverbalcontradictionin terms and a formal contradiction in terms, the contradiction depending in thefirst caseupon the forceof the termsemployedand in the secondcaseupon themere form of the proposition; e.g.,Somemen are not animals,A is not-A.Anypurely formal fallacymaybe said to resolve itself into a formal contradiction interms. It should be added that a mere term, if it is complex, may involve acontradictioninterms;e.g.,RomanCatholic (if theseparate termsare interpretedliterally),Anot-A.

32.NatureoftheAnalysisinvolvedinAnalyticPropositions.—Confusionisnotunfrequentlyintroducedintodiscussionsrelatingtoanalyticpropositionsbyawantofagreementastothenatureoftheanalysisinvolved.Ifidentified,as above,with a division of the verbal proposition, an analytic propositiongivesananalysis,partialorcomplete,oftheconnotationofthesubject-term.Somewriters,however,appear tohaveinviewananalysisof thesubjective


intensionofthesubject-term.Thereisofcoursenothingabsolutelyincorrectin this interpretation, if consistently adhered to,but itmakes thedistinctionbetween analytic and synthetic propositions logically valueless and for allpracticalpurposesnugatory.“Bothintensionandextension,”saysMrBradley,“are relative toourknowledge.And theperceptionof this truth is fatal toawell-known Kantian distinction. A judgment is not fixed as ‘synthetic’ or‘analytic’: its character varies with the knowledge 54 possessed by variouspersonsandatdifferenttimes.Ifthemeaningofawordwereconfinedtothatattribute or group of attributes with which it set out, we could distinguishthosejudgmentswhichassertwithinthewholeonepartofitscontentsfromthose which add an element from outside; and the distinction thus madewould remain valid for ever. But in actual practice the meaning itself isenlargedby synthesis.What is added to-day is implied to-morrow.Wemayeven say that a synthetic judgment, so soon as it is made, is at onceanalytic.”62

62 Principles of Logic, p. 172. Professor Veitch expresses himself somewhatsimilarly.“Logicallyalljudgmentsareanalytic,forjudgmentisanassertionbytheperson judging of what he knows of the subject spoken of. To the personaddressed, real or imaginary, the judgmentmaycontain apredicatenew—anewknowledge. But the person making the judgment speaks analytically, andanalyticallyonly;forhesetsforthapartofwhatheknowsbelongstothesubjectspokenof.Infact,itisimpossibleanyonecanjudgeotherwise.Wemustjudgebyour real or supposed knowledge of the thing already in themind” (Institutes ofLogic,p.237).

Ifbyintensionismeantsubjectiveintension,andbyananalyticjudgmentone which analyses the intension of the subject, the above statements areunimpeachable. It is indeed so obviously true that in this sense syntheticjudgmentsareonlyanalyticjudgmentsinthemaking,thattodwelluponthedistinction itself at any lengthwouldbe onlywaste of time. It is, however,misleading to identify subjective intension with meaning ;63 and this isespeciallythecaseinthepresentconnexion,sinceitmaybemaintainedwitha certain degree of plausibility that some synthetic judgments are onlyanalytic judgments in the making, even when by an analytic judgment ismeantonewhichanalysestheconnotationofthesubject.Forundoubtedlytheconnotationofnamesisnotinpracticeunalterablyfixed.Asourknowledgeprogresses, many of our 55 definitions are modified, and hence a form of


wordswhichissyntheticatoneperiodmaybecomeanalyticatanother.63 ComparethefollowingcriticismofMill’sdistinctionbetweenrealandverbal

propositions:“Ifeveryproposition ismerelyverbalwhichassertssomethingofathing under a name that already presupposes what is about to be asserted, theneverystatementbyascientificmanisforhimmerelyverbal”(T.H.Green,Works,ii.p.233).Thiscriticismseemstoloseitsforceifwebearinmindthedistinctionbetweenconnotationandsubjectiveintension.

But,inthefirstplace,itisveryfarindeedfrombeingauniversalrulethatnewly-discovered properties of a class are taken ultimately into theconnotation or intensive definition of the class-name. Dr Bain (Logic,Deduction,pp.69to73)seemstoimplythecontrary;buthisdoctrineonthispointisnotdefensibleonthegroundeitheroflogicalexpediencyorofactualpractice. As to logical expediency, it is a generally recognised principle ofdefinition that we ought to aim at including in a definition the minimumnumber of properties necessary for identification rather than themaximumwhichitispossibletoinclude.64Andastowhatactuallyoccurs,itiseasytofind cases where we are able to say with confidence that certain commonpropertiesofaclassneverwillasamatteroffactbeincludedinthedefinitionoftheclass-name;forexample,equiangularitywillneverbeincludedin thedefinitionofequilateraltriangle,orhavingclovenhoofs in thedefinitionofruminantanimal.

64 Ifweincludeinthedefinitionofaclass-nameallthecommonpropertiesoftheclass,howarewetomakeanyuniversalstatementoffactabouttheclassatall?GiventhatthepropertyPbelongstothewholeoftheclassS,thenbyhypothesisPbecomespartofthemeaningofS,andthepropositionAllSisPmerelymakesthisverbalstatement,andisnoassertionofanymatteroffactatall.Weare,therefore,involvedinakindofviciouscircle.

In the second place, even when freshly discovered properties of thingscomeultimatelytobeincludedintheconnotationoftheirnames,theprocessis at any rate gradual, and it would, therefore, be incorrect to say—in thesense in which we are now using the terms—that a synthetic judgmentbecomes in theveryprocessof its formationanalytic.On theotherhand, itmayreasonablybeassumedthatinanygivendiscussionthemeaningofourtermsisfixed,andthedistinctionbetweenanalyticandsyntheticpropositionsthenbecomeshighlysignificantandimportant.Itmaybeaddedthatwhenanamechangesitsmeaning,anypropositioninwhichitoccursdoesnotstrictly


speakingremainthesamepropositionasbefore.Weought56rathertosaythatthesameformofwordsnowexpressesadifferentproposition.65

65 This point is brought out byMrMonck in the admirable discussionof theabovequestioncontainedinhisIntroductiontoLogic,pp.130to134.

EXERCISES.

33. State which of the following propositions you consider real, and which verbal, giving yourreasonsineachcase:

(i) Allpropernamesaresingular;(ii) Asyllogismcontainsthreeandonlythreeterms;(iii) Menarevertebrates;(iv) Allisnotgoldthatglitters;(v) Thedodoisanextinctbird;(vi) Logicisthescienceofreasoning;(vii) Twoandtwoarefour;(viii) Allequilateraltrianglesareequiangular;(ix) Betweenanytwopointsone,andonlyone,straightlinecanbedrawn;(x) Anytwosidesofatrianglearetogethergreaterthanthethirdside.

[C.]

34. Enquirewhether the following propositions are real or verbal: (a)Homerwrote the Iliad, (b)MiltonwroteParadiseLost.[C.]

35.Howwouldyoucharacteriseapropositionwhichisformally inferredfromtheconjunctionofaverbalpropositionwitharealmaterialproposition?Explainyourviewbytheaidofanillustration.[J.]

36.Ifallxisy,andsomexisz,andpisthenameofthosez’swhicharex ;isitaverbalpropositiontosaythatallpisy?[V.]

37. Is it possible tomake any termwhatever the subject (a) of a verbal proposition, (b) of a realproposition?[J.]


CHAPTERIV.

NEGATIVENAMESANDRELATIVENAMES.

38.Positive andNegativeNames.—A pair of names of the formsA andnot-Aarecommonlydescribedaspositiveandnegativerespectively.Thetrueimportof thenegativenamenot-A, including the questionwhether it reallyhasanysignificationatall,has,however,givenrisetomuchdiscussion.

Strictlyspeakingneitheraffirmationnornegationhasanymeaningexceptinreferencetojudgmentsorpropositions.Aconceptoratermcannotbeitselfeitheraffirmedordenied.IfIaffirm,itmustbeajudgmentorapropositionthatIaffirm;ifIdeny,itmustbeajudgmentorapropositionthatIdeny.

Starting from this position, Sigwart is led to the conclusion that, “takenliterally, the formula not-A, where A denotes any idea, has no meaningwhatever”(Logic,I.p.134).Apartfromthefactthatthemereabsenceofanideaisnotitselfanidea,not-AcannotbeinterpretedtomeantheabsenceofAin thought; for,on thecontrary, it implies thepresenceofA in thought.Wecannot,forinstance,thinkofnot-whiteexceptbythinkingofwhite.Noragaincanweinterpretnot-AasdenotingwhateverdoesnotnecessarilyaccompanyA in thought. For, if so,A and not-A would not as a rule be exclusive orincompatible.Forexample,square,solid,donotnecessarilyaccompanywhitein thought; but there is no opposition between these ideas and the idea ofwhite. In order to interpretnot-A as a real negationwemust, saysSigwart,tacitly introduce a judgment or rather a series of judgments, 58meaningbynot-A“whateverisnotA,”thatis,everythingwhatsoeverofwhichAmustbedenied. “Imust review in thought all possible things in order to denyAofthem,andthesewouldbethepositiveobjectsdenotedbynot-A.Buteveniftherewereanyuseinthis,itwouldbeanimpossibletask”(p.135).

Whilst agreeingwithmuch that Sigwart says in this connexion, I cannotaltogether accept his conclusion. We shall return to the question from the


morecontroversialpointofviewinthefollowingsection.Inthemeantimewemayindicate theresult towhichSigwart’sgeneralargumentreallyseemstoleadus.

We must agree that not-A cannot be regarded as representing anyindependent concept; that is to say,we cannot form any idea ofnot-A thatnegates the notion A. It is, therefore, true that, taken literally (that is, asrepresenting an ideawhich is the pure negationof the ideaA), the formulanot-A isunintelligible.Regardingnot-A,however,asequivalent towhateveris not A, we may say that its justification and explanation is to be foundprimarilybyreferencetotheextensionofthename.ThethinkingofanythingasA involves itsbeingdistinguished from thatwhich isnotA.Thuson theextensivesideeveryconceptdividestheuniversewithreferencetowhichitisthought (whatever that may be) into two mutually exclusive subdivisions,namely, a portion of whichA can be predicated and a portion of whichAcannotbepredicated.ThesewedesignateAandnot-A respectively.While itmaybesaidthatAandnot-A involve intensivelyonlyoneconcept, theyareextensivelymutuallyexclusive.

Confiningourselvestoconnotativenames,wemayexpressthedistinctionbetweenpositiveandnegativenames somewhatdifferentlyby saying that apositive name implies the presence in the things called by the name of acertainspecifiedattributeorsetofattributes,whileanegativenameimpliestheabsenceofoneorotherofcertainspecifiedattributes.Anegativename,therefore,has itsdenotationdetermined indirectly.Theclassdenotedby thepositivename isdeterminedpositively, and then thenegativenamedenoteswhatisleft.

5939.IndefiniteCharacterofNegativeNames.—Infiniteandindefinitearedesignations that have been applied to negative nameswhen interpreted insuchawayasnottoinvolverestrictiontoalimiteduniverseofdiscourse.Forwithout such restriction (explicit or implicit) a negativename, for example,not-white,mustbeunderstoodtodenotethewholeinfiniteorindefiniteclassofthingsofwhichwhitecannot trulybeaffirmed, includingsuchentitiesasvirtue,adream,time,asoliloquy,NewGuinea,theSevenAgesofMan.

Many logicians hold that no significant term can be really infinite or


indefiniteinthisway.66Theysaythatifatermlikenot-whiteistohaveanymeaningatall, itmustbeunderstoodasdenoting,notall thingswhatsoeverexceptwhite things, but only things that are black, red, green, yellow, etc.,that is, all coloured things except such as are white. In other words, theuniverse of discourse which any pair of contradictory terms A and not-AbetweenthemexhaustisconsideredtobenecessarilylimitedtotheproximategenusofwhichAisaspecies;as,forexample,inthecaseofwhiteandnot-white,theuniverseofcolour.

66 ThisisattherootofSigwart’sfinaldifficultywithregardtonegativenames,asindicatedintheprecedingsection.Lateronhepointsoutthatindivisionwearejustified in including negative characteristics of the form not-A in a concept,althoughwecannot regardnot-A itself as an independent concept.Thuswemaydividetheconceptorganicbeingintofeelingandnot-feeling,aspecificdifferencebeinghereconstitutedbytheabsenceofacharacteristicwhichiscompatiblewiththeremainingcharacteristics,butisnotnecessarilyconnectedwiththem(Logic,I.p.278).ComparealsoLotze,Logic,§40.

It is doubtless the case that we seldom or never make use of negativenames except with reference to some proximate genus. For instance, inspeakingofnon-votersweareprobably referring to the inhabitantsof sometownorlocalitywhomwesubdivideintothosewhohavevotesandthosewhohave not. In a similar way we ordinarily deny red only of things that arecoloured,squarenessonlyofthingsthathavesomefigure,etc.,sothatthereisan implicit limitationofsphere. Itmaybegrantedfurther thatapropositioncontaining a negative name interpreted as infinite can have little or nopractical value. But it does not follow that some limitation 60 of sphere isnecessary inorder thatanegative termmayhavemeaning.Theargumentisused that it is an utterly impossible feat to hold together in any one idea achaoticmassofthemostdifferentthings.Buttheanswertothisargumentisthatwedonotprofesstoholdtogetherthethingsdenotedbyanegativenamebyreferencetoanypositiveelementswhichtheymayhaveincommon:theyareheld together simplyby the fact that theyall lack someoneorotherofcertaindeterminateelements.Inotherwords,theargumentonlyshewsthatanegativenamehasnopositiveconceptcorrespondingtoit.67Itmaybeaddedthat if this argument had force, itwould apply also to the subdivision of agenuswith reference to the presence or absence of a certain quality. Ifwe


divide coloured objects into red and not-red, we may say equally that wecannotholdtogethercolouredobjectsotherthanredbyanypositiveelementthat they have in common: the fact that they are all coloured is obviouslyinsufficientforthepurpose.

67 Foragoodstatementofthecounter-argument,compareMrsLaddFranklininMind,January,1892,pp.130,1.

A somewhat different argument is implied by Sigwartwhen he says, “IfA=mortal,wherewilljustice,virtue,law,order,distancefindaplace?Theyareneithermortalbeings,noryetnot-mortalbeings,fortheyarenotbeingsatall.” The answer seems clear. They are not-(mortal beings), and thereforenot-A.Asarule,itisneedlesstoexcludeexplicitlyfromaspecieswhatdoesnotevenbelongtosomehighergenus.Butthefactoftheexclusionremains.

Granting then that inpracticewe rarely, if ever, employanegativenameexcept with reference to some proximate genus, we nevertheless hold thatnot-Aisperfectlyintelligiblewhatevertheuniverseofdiscoursemaybeandhowever wide it may be. For it denotes in that universe whatever is notdenoted by the corresponding positive name.Moreover in formal processeswe should be unnecessarily hampered if not allowed to pass unreservedlyfromXisnotAtoXisnot-A.68

68 Writerswhotaketheviewwhichweareherecriticisingmustinconsistencydenytheuniversalvalidityoftheprocessofimmediateinferencecalledobversion.ThusLotze, rightlyonhisownview,willnotallowus topass fromspirit isnotmattertospiritisnot-matter ;infactherejectsaltogethertheformofjudgmentSisnot-P(Logic,§40).Somewriters,whofollowLotzeonthegeneralquestionhereraised,appeartogoagooddealfurtherthanhedoes,notmerelydisallowingsuchapropositionasvirtueisnot-bluebutalsosuchapropositionasvirtueisnotblue,onthegroundthatifwesay“virtueisnotblue,”thereisnorealpredication,sincethenotionofcolourisabsolutelyforeigntoanunextendedandabstractconceptsuchas“virtue.”Lotze,however,expresslydrawsadistinctionbetweenthetwoformsSisnon-QandSisnotQ,andtellsusthat“everythingwhichitiswishedtosecureby the affirmative predicatenon-Q is secured by the intelligible negation ofQ”(Logic,§72;cf.§40).OnthemoreextremeviewitiswrongtosaythatVirtueiseitherblueoritisnotblue ;butLotzehimselfdoesnotthusdenytheuniversalityofthelawofexcludedmiddle.

61 From this point of view attention may be called to the difference inordinaryusebetweensuchformsasunholy,immoral,discourteousandsuchforms asnon-holy,non-moral,non-courteous. The lattermay be used with


reference to any universe of discourse, however extensive. But not so theformer; in their case there is undoubtedly a restriction to some universe ofdiscoursethatismoreorlesslimitedinitsrange.Wecan,forexample,speakofatableasnon-moral,althoughwecannotspeakofitasimmoral.Awantofrecognition of this distinctionmay be partly responsible for the denial thatanytermscanproperlybedescribedasinfiniteorindefinite.69

69 It shouldbe added that in theordinaryuseof language thenegativeprefixdoesnot alwaysmakea termnegativeasheredefined.Thus, asMillpointsout,“the word unpleasant, notwithstanding its negative form, does not connote themereabsenceofpleasantness,buta lessdegreeofwhat is signifiedby thewordpainful,which,itishardlynecessarytosay,ispositive.”Ontheotherhand,somenamespositiveinformmay,withreferencetoalimiteduniverseofdiscourse,benegative in force;e.g.,alien, foreign.Another example is the termTuranian, asemployed in the science of language.This termhas been used to denote groupslyingoutsidetheAryanandSemiticgroups,butnotdistinguishedbyanypositivecharacteristicswhichtheypossessincommon.

40.ContradictoryTerms.—Apositivenameandthecorrespondingnegativearespokenofascontradictory.Wemaydefinecontradictorytermsasapairof terms so related that between them they exhaust the entire universe towhich reference is made, whilst in that universe there is no individual ofwhichbothcanbeaffirmedatthesametime.Itisdesirabletorepeatherethatcontradictioncanexistprimarilybetween62 judgmentsorpropositionsonly,so that as applied to terms or ideas the notion of contradiction must beinterpreted with reference to predication. A and not-A are spoken of ascontradictory because they cannot without contradiction be predicatedtogetherof thesamesubject.Thusit is intheirexclusivecharacter that theyare termed contradictory; as between them exhausting the universe ofdiscoursetheymightratherbecalledcomplementary.70

70 Dr Venn (Empirical Logic, p. 191) distinguishes between formalcontradictoriesandmaterialcontradictories,accordingastherelationinwhichthepairoftermsstandtooneanotherisorisnotapparentfromtheirmereform.ThusA andnot-A are formal contradictories; so are human and non-human. Materialcontradictories,ontheotherhand,arenotconstructed“fortheexpresspurposeofindicatingtheirmutualrelation.”Noformalcontradiction,forexample,isapparentbetweenBritishandForeign,orbetweenBritishandAlien ; andyet“within theirrangeof appropriate application—which in the latter case includes persons only,and in the formercase isextended toproduceofmostkinds—these twopairsofterms fulfil tolerably well the conditions of mutual exclusion and collective


exhaustion.”

41.ContraryTerms.—Twotermsareusuallyspokenofascontrary71tooneanother when they denote things which can be regarded as standing atopposite ends of some definite scale in the universe to which reference ismade; e.g., first and last, black andwhite,wise and foolish, pleasant andpainful.72Contrariesdifferfromcontradictoriesinthattheyadmitofamean,andthereforedonotbetweenthemexhausttheentireuniverseofdiscourse.Itfollowsthat,althoughtwocontrariescannotbothbetrueofthesamethingatthesametime, theymaybothbefalse.Thus,acolourmaybeneitherblacknor 63 white, but blue; a feeling may be neither pleasant nor painful, butindifferent.

71 De Morgan uses the terms contrary and contradictory as equivalent, hisdefinitionofthemcorrespondingtothatgivenintheprecedingsection.

72 Ithasbeenalreadypointedoutthatthenegativeprefixdoesnotalwaysmakea term really negative in force. Thus pleasant and unpleasant are notcontradictories,fortheyadmitofamean;whenwesaythatanythingisunpleasant,we intend somethingmore than themere denial that it ispleasant. It should beadded that a pair of terms of this kind may also fail to be contraries as abovedefined, sincewhile admitting of amean theymay at the same time not denoteextremes.Unpleasant, for example, denotes only that which is mildly painful:unlessintendedironically,itwouldbeamisuseoftermstospeakofthetorturesoftheInquisitionasmerelyunpleasant.CompareCarvethRead,Logic,p.49.

Itwillbeobservedthatnoteverytermhasacontraryasabovedefined,forthe thing denoted by a term may not be capable of being regarded asrepresentingtheextremeinanydefinitescale.Thusbluecanhardlybesaidtohave a contrary in the universe of colour, or indifferent in the universe offeeling.

Bysomewriters,thetermcontraryisusedinawidersensethantheabove,contrarietybeingidentifiedwithsimpleincompatibility(ameanbetweenthetwoincompatiblesbeingpossible);thus,blueandyellowequallywithblack,wouldinthissensebecalledcontrariesofwhite.73Otherwritersusethetermrepugnant to express the mere relation of incompatibility; thus red, blue,yellowareinthissenserepugnanttooneanother.74

73 There ismuch to be said in favour of thiswider use of the termcontrary.Comparethediscussionofcontrarypropositionsinsection81.

74 So longasweareconfined tosimple terms the relationsofcontrarietyand


repugnancycannotbeexpressedformallyor inmeresymbols.But it isotherwisewhenwepassontotheconsiderationofcomplexterms.Thus,whileXYandnot-Xor not-Y are formal contradictories, XY and X not-Y may be said to be formalrepugnants, XY and not-X not-Y formal contraries (in the narrower of the twosensesindicatedabove).

42.RelativeNames.—Anameissaidtoberelative,when,overandabovethe object that it denotes, it implies in its signification another object, towhich in explaining itsmeaning referencemust bemade.Thenameof thisother object is called the correlative of the first. Non-relative names aresometimescalledabsolute.

Jevonsconsidersthatincertainrespectsallnamesarerelative.“Thefactisthateverythingmustreallyhaverelationstosomethingelse,thewatertotheelements of which it is composed, the gas to the coal from which it ismanufactured,thetreetothesoilinwhichitisrooted“(ElementaryLessonsinLogic,p.26).Again,bythelawofrelativity,consciousnessispossibleonlyin circumstances of change. We cannot think of any object except asdistinguishedfromsomethingelse.Everyterm,therefore,impliesitsnegativeasanobject64ofthought.Takethetermman.Itisanambiguousterm,andinmanyofitsmeaningsisclearlyrelative,—forexample,asopposedtomaster,toofficer,towife.Ifinanysenseitisabsoluteitiswhenopposedtonot-man;buteven in thiscase itmaybesaid toberelative tonot-man.Toavoid thisdifficulty, Jevons remarks, “Logicians have been content to consider asrelative terms those only which imply some peculiar and striking kind ofrelationarisingfrompositionintimeorspace,fromconnexionofcauseandeffect,&c.;anditisinthisspecialsense,therefore,thatthestudentmustusethedistinction.”

A more satisfactory solution of the difficulty may be found by callingattention to the distinction already drawn between the point of view ofconnotation (which has to do with the signification of names) and thesubjective and objective points of view respectively. From the subjectivepointofviewallnotionsarerelativebythelawofrelativityabovereferredto.Again, from the objective point of view all things, at any rate in thephenomenalworld,arerelativeinthesensethattheycouldnotexistwithouttheexistenceofsomethingelse;e.g.,manwithoutoxygen,oratreewithout


soil.Butwhenwe say that aname is relative,wedonotmean thatwhat itdenotescannotexistorbethoughtaboutwithoutsomethingelsealsoexistingor being thought about; wemean that its signification cannot be explainedwithout reference to something elsewhich is called by a correlative name,e.g., husband, parent. It cannot be said that in this sense all names arerelative.

Thefactorfactsconstitutingthegroundofbothcorrelativenamesiscalledthe fundamentumrelationis. For example, in the caseof partner, the fact ofpartnership; in the case of husband andwife, the factswhich constitute themarriage tie; in the case of ruler and subject, the controlwhich the formerexercisesoverthelatter.

Sometimes the relation which each correlative bears to the other is thesame; forexample, in thecaseofpartner,where thecorrelativename is thesamenameoveragain.Sometimesitisnotthesame;forexample,fatherandson,slave-ownerandslave.65

The consideration of relative names is not of importance except inconnexionwiththelogicofrelatives,towhichfurtherreferencewillbemadesubsequently.

EXERCISES.


43.Giveoneexampleofeachofthefollowing,—(i)acollectivegeneralname,(ii)asingularabstractname,(iii)aconnotativesingularname,(iv)aconnotativeabstractname.Addreasonsjustifyingyourexampleineachcase.[K.]

44.Discussthelogicalcharacteristicsofthefollowingnames:—beauty,fault,MrsGrundy,immortal,nobility,slave,sovereign,theTimes,truth,ungenerous.[K.]

[Indiscussingthecharacterofanynameitisnecessaryfirstofalltodeterminewhetheritisunivocal,thatis,usedinonedefinitesenseonly,orequivocal(orambiguous), thatis,usedinmoresensesthanone.Inthelattercase,itslogicalcharacteristicsmayvaryaccordingtothesenseinwhichitisused.]

45. It has been maintained that the doctrine of terms is extra-logical. Justify or controvert thisposition.[J.]


PARTII.

PROPOSITIONS.


CHAPTERI.

IMPORTOFJUDGMENTSANDPROPOSITIONS.

46. Judgments and Propositions.—In passing to the next division of oursubjectweareconfronted,firstofall,withaquestionwhichispartly,butnotentirely,aquestionofphraseology.Shallwespeakofthejudgmentoroftheproposition? The usage of logicians differs widely. Some treat almostexclusivelyofjudgments;othersalmostexclusivelyofpropositions.Itwillbefoundthatfor themostpart theformerarethosewhotendtoemphasisethepsychologicalorthemetaphysicalaspectsoflogic,whilethelatterarethosewhoaremoreinclinedtodevelopthesymbolicorthematerialaspects.

To a certain extent it is a matter of little importance which of thealternativesisostensivelyadopted.Thosewhodealwithjudgmentsfromthelogicalstandpointmustwhenpressedadmitthattheycandealwiththemonlyas expressed in language, and all their illustrations necessarily consist ofjudgments expressed in language.But a judgment expressed in language ispreciselywhatismeantbyaproposition.Henceintreatingofjudgmentsitisimpossiblenottotreatalsoofpropositions.67

Ontheotherhand,so faraswe treatofpropositions in logic,we treatofthemnotasgrammaticalsentences,butasassertions,asverbalexpressionsofjudgments. The logical proposition is the proposition as understood; and apropositionasunderstoodisajudgment.Henceintreatingofpropositionsinlogicwenecessarilytreatalsoofjudgments.

Inalargedegree,then,theproblemdoesresolveitselfintoamerelyverbalquestion.At thesame time, reasonsandcounter-reasonsmaybeadduced infavouroftheonealternativeandinfavouroftheother.

On the one side, it is said that the use of the term proposition tends toconfuse the sentence as a grammatical combination of words with thepropositionasapprehendedandintellectuallyaffirmed;anditisurgedthatin


treatingofpropositionsthelogiciantendstobecomeameregrammarian.

Ontheotherside, it issubmittedthat the logicianisprimarilyconcerned,not with the process of judgment, the discussion of which belongs to thesphereofpsychology,butwithjudgmentasaproduct,andmoreoverthatheisconcernedwiththisproductonlyinsofarasitassumesafixedanddefiniteform,which it cannot dountil it receivesverbal expression; and it is urgedthat ifweconcentrateour attentionon judgmentswithout explicit regard totheir expression in language, our treatment tends to become toopsychological.

Ithasbeen saidabove that logicallywecandealwith judgmentsonlyasexpressedinpropositions;andnodoubtalljudgmentscanwithmoreorlesseffortbesoexpressed.Butasamatteroffactweconstantlyjudgeinavaguesort ofwaywithout the precision that is necessary even in loosemodes ofexpression, and we find that to give expression to our judgments maysometimesrequireveryconsiderableeffort.Itmustberememberedthatlogichasinviewanideal.Itsobjectistodeterminetheconditionstowhichvalidjudgmentsmustconform,anditisconcernedwiththecharacteristicsofactualjudgments only in subordination to this end. From this point of view it isspecially important thatwe shoulddealwith judgments in theonly form inwhich it is possible for them to attain precision; and this considerationappearstobeconclusiveinfavourofour68treatingexplicitlyofpropositionsinsomepartatanyrateofalogicalcourse.

Nodoubt in dealingwith propositionswehave to raise certain questionsthat relate to the usage of language. Unfortunately the same propositionalform may be understood as expressing very different judgments. It isthereforerequisitethatinanyscientifictreatmentoflogicweshoulddiscusstheinterpretationofthepropositionalformsthatwerecognise.Thisproblemisakin to theproblemofdefinitionwhichhas tobefacedsooneror later ineveryscience;and,asisalsotrueofadefinition,thesolutioninanyparticularcaseislargelyofthenatureofaconvention.Butthisdoesnotdetractfromitsimportanceasconducingtoclearnessofthought.

Thequestionof the interpretationofpropositional forms isasamatteroffactonethatcannotbealtogetheravoidedonanytreatmentoflogic;anditis


of importance to recogniseexplicitly that indiscussing thisquestionwearenot dealing with judgments pure and simple.Words are like mathematicalsymbols,andthemeaningofagivenformofwordsisnotsomethinginherenteitherinthewordsthemselvesorinthethoughtsthattheymayrepresent,butisdependentonaconventionestablishedbythosewhoemploythewords.Inthe force of a given judgment, however, there can be nothing that isdependentonconvention.Thisdistinctionisnotalwaysrememberedbythosewhoconfine their attentionmainly to judgments, and theyareconsequentlysometimes led to express themselves with an appearance of dogmatism onquestionsthatdonotreallyadmitofdogmatictreatment.

But while in certain aspects of logical enquiry it is requisite to dealexplicitlywithpropositions, itmust never be forgotten that as logiciansweare concernedwith propositions only as the expressions of judgments; andtherearenumerousoccasionswhenwehavetogobehindpropositionalformsandaskwhat are the fundamental characteristicsof the judgments that theyexpress.

47.The Abstract Character of Logic.—Reference has been made in thepreceding section to the necessity for logical purposes of making ourjudgmentsprecise.Foronlyif they69areprecise is itpossible todeterminewith accuracy what are their logical implications considered eitherindividuallyorinconjunctionwithoneanother.Ithasalsobeenpointedoutthat we can make our judgments precise only by expressing them inpropositionalforms,theinterpretationofwhichhasbeenagreedupon.

But this is notwithout its disadvantages. Sometimes the full force of anactual judgment hardly admits of being expressed in words, and even theforce of a proposition as understood may not be found exclusively in thewordsofwhichitcomposed,butmaydependpartlyonthecontextinwhichitisplaced.Hencetheisolatedpropositionmustfrequentlyberegardedasinasenseanabstraction,leavingbehinditsomeportionoftheactualjudgmentforwhichitstands.

Thisisindeedmuchlesstrueofthepropositionsofsciencethanofthoseofeveryday life; and themore fully a statement is independent of context themore fully may it be regarded as fulfilling its purpose from the scientific


standpoint. Still the abstract character of logicmust be frankly recognised.“Justasthoughtisabstractinitsdealingswithreality,sologicisabstractinitsdealingswithordinarythought.”75

75 Hobhouse,TheoryofKnowledge,p.7.

Thattheyareinsomedegreeabstractionsistruenotonlyofpropositions,but also of inferences, aswe have to dealwith them in logic.Much of thereasoningofeverydaylifedoesnotadmitofexpressionintheformofdefinitepremisses and conclusions such as would satisfy the canons of logic. Thegroundsuponwhichourconclusionsarebasedareoftensocomplex,andtheinfluencewhichsomeofthemexertuponourbeliefsissosubtleanddelicate,that they cannotbe completely set forth.Thiswill be realised at once if anattemptismadetoapplytherulesoflogictoanyordinaryinference;andanexplanation isherein foundwhy the illustrationsgiven in logical text-booksfrequentlyappearsoartificialandunreal.

It must be admitted that the abstract character of logic detracts to someextentfromitsutilityasanart,thoughtheextentofthisdrawbackmayeasilybeexaggerated.Regardedasascience,however, thevalue,oflogicremainsunimpaired.70Other sciencesbesides logichave toproceedbyabstractionsandseparationsthatdonotfullycorrespondtothecomplexitiesofnature;andthis often becomes the more true the higher the stage that the science hasreached. Its necessary abstractness does not prevent logic from analysingsuccessfully the characteristics of the developed judgment or fromdetermining the principles of valid reasoning. If we were to seek to treatlogicalproblemswithoutabstractionweshouldbeindangerofdestroyingthescientificcharacteroflogicwithoutachievinganyvaluableresultevenfromthe purely utilitarian point of view. It is of little value to criticise receivedsystemswithoutprovidinganynewconstructivesystemintheirplace.

48.Nature of the Enquiry into the Import of Propositions.—Under thegeneralheadoftheimportofpropositionsitisusualtoincludeproblemsthatarereallyverydifferentincharacter.76

76 CompareMrW.E.JohnsoninMind,April,1895,p.242.

(1) There is, in the first place, the fundamental problem or series ofproblems as to what are the essential characteristics of judgments, and


therefore of propositions as expressing judgments. The discussion ofquestions of this character must be based directly on psychological orphilosophical considerations, and in the solutions nothing arbitrary orconventionalcanfindaplace.

Underthisheadaretobeincludedsuchproblemsasthefollowing:Doalljudgments contain a reference to reality? In what sense, if any, can alljudgments claim topossessuniversalityor necessity?What is thenatureofsignificantdenial?Aredistinctionsofmodalitysubjectiveorobjective?

(2) In the interpretation of propositional forms we have an enquiry of averydifferentcharacter,anenquirywhichrelatesdistinctivelytopropositions,andnottojudgmentsconsideredapartfromtheirexpression.Theproblemisindeed to determine what is the precise judgment that a given propositionshall be understood to express; and, in consequence of the uncertainty andambiguity of ordinary language, the solution of the problem includes anoptionalorselectiveelement.

71Asasimpleillustrationofthekindofproblemthatweherehaveinview,wemaynotethatinthetraditionalschemeofpropositions,AllSisP,NoSisP,SomeSisP,SomeSisnotP,thesignsofquantityhavetobeinterpreted.The existential and modal import of these propositions is also partly aquestionofinterpretation.

In connexion with the interpretation of propositions, the distinctionbetween meaning and implication has to be considered. What we do ininterpreting propositions is to assign to them a meaning; and when themeaninghasoncebeenfixed,theimplicationsaredeterminedinaccordancewithlogicalprinciples.

The dividing line between meaning and implication is not in practicealwayseasy todraw,andsomewritersseek to ignore itby includingwithinthescopeofmeaningalltheimplicationsofaproposition.Butthisisafatalerror. The assignment of meaning is within certain limits arbitrary andselective.Butifelementanecessarilyinvolveselementb,thenahavingbeenassigned as part of the meaning of a given propositional form, it is noquestion of meaning as to whether the form in question does or does not


imply b, and there is nothing arbitrary or selective in the solution of thisquestion.

Sometimestheelementsaandbmutuallyinvolveoneanother.Itmaythenbeaquestionofinterpretationwhetherashallbeincludedinmeaning,bthusbecoming an implication, or whether b shall be included in meaning, abecominganimplication.

Afailuretorecognisewhatisreallythepointatissueinacaselikethishassometimes caused discussions to take a wrong turn. Thus the question israisedwhether the importof thepropositionAllS isP is that theclassS isincluded in the class P, or that the set of attributes S is invariablyaccompaniedbythesetofattributesP ;andtheseareregardedasantagonistictheories.Iftheimplicationsofapropositionareregardedaspartofitsimport,then the propositionmaybe said to import both these things.But if by theimportofapropositionwe intend tosignify itsmeaningonly, thenwemayadoptaninterpretationthatwillmakeeitherofthem(butnotboth)partofitsimport, or our interpretation may be such 72 that the proposition importsneitherofthem.Thequestionhereraisedisdealtwithinmoredetaillateron.

(3) A third problem, distinct from both those described above, arises inconnexionwiththeexpressionofjudgmentsinpropositionalform.

In ordinary discourse we meet with an infinite variety of forms ofstatement. To recognise and deal separately with all these forms in ourtreatment of logical problems would, however, be impracticable.We have,therefore,insomeatanyrateofourdiscussions,tolimitourselvestoacertainnumberofselectedforms;andinsuchdiscussionswehavetoassumethatthejudgmentswithwhichwe are dealing are at the outset expressed in one orotheroracombinationoftheseselectedforms.

This reductionof a statement to somecanonical formhasbeencalledbyMrJohnsonitsformulation.

A given statement, since it involves many different relations whichmutually implicateoneanother,maybeformulated inanumberofdifferentways; and it is needless to say that there is no one scheme of formulatingpropositionsthatweareboundtoaccepttotheexclusionofothers.Different


schemes are useful for different purposes, and several schedules ofpropositions(forexample,equationalandexistentialschedules)willpresentlybe considered in addition to the traditional fourfold schedule. It should beaddedthatagivenschememayprofesstocoverpartonlyofthefield.Thusthe traditional schedule (All S is P, etc.) professes to be a scheme forcategorical judgments only, and (as traditionally interpreted) for assertoricjudgmentsonly.

Withreferencetothereductionofastatementtoaforminwhichitbelongstoagivenscheduletwopointscallfornotice.

(a)Thereisdangerlestsomepartoftheforceoftheoriginalstatementmaybelost.

To a certain extent this is inevitable, especially if the original statementcontainssuggestionorinnuendoinadditiontowhatitdefinitelyaffirms;andthismust be taken in connexionwithwhat has already been said about theabstract character of logic. If, however, there is any substantial loss of 73import,theschemestandscondemnedsofarasitprofessestobeacompleteschemeofformulation.Itmay,aswehaveseen,notprofesstobeacompletescheme,butonlytoformulatestatementsfallingwithinacertaincategory,forexample,assertoricstatementsorcategoricalstatements.

Itistobeaddedthatastatementwhichdoesnotadmitofbeingtranslatedinto any one of the simple forms included in a given schememay still becapable of being expressed by a conjunctive or disjunctive combination ofsuch simple forms. Thus, if the statement Some S is P is made with anemphasisonsome,implyingnotall,thenthestatementcannotbeexpressedinany one of the forms of the traditional schedule of propositions, but it isequivalenttoSomeSisPandsomeSisnotP.

(b)Inthereductionofastatementtoaforminwhichitbelongstoagivenschedule theremaybe involvedwhatmustbeadmitted tobe inference. As,forinstance,ifstatementsaregivenintheordinarypredicativeformandhavetobeexpressedinanequationalscheme.

Itmayperhapsbeurged that this is legitimate,simplyon theground thatoneofthepostulatesoflogicisthatwebeallowedtosubstituteforanygiven


formofwordsthetechnicalform(andinanequationalsystemthiswillbeanequation)whichisequivalenttoit.Havewenot,however,inrealityaviciouscircleifaprocesswhichinvolvesinferenceistoberegardedasapostulateoflogic?

The difficulty here raised is a serious one only if we suppose ourselvesrigidlylimitedinlogictoasingleschemeofformulation;andthesolutionisto be found in our not confining ourselves to any one scheme, but in ourrecognisingseveralandinvestigatingthelogicalrelationsbetweenthem.Wecanthenrefusetoregardanysubstitutionofonesetofwordsforanotheraspre-logical except in so far as it consistsofamerelyverbal transformation:andourpostulatewillmerelybethatwearefreetomakeverbalchangesasweplease;itwillnotbyitselfauthoriseanychangeofaninferentialcharacter.Forachangeofthiskind,appealmustbemadetologicalprinciples.

74Wehave then in thissectiondistinguishedbetweenthreeproblemsanyor all of which may be involved in discussions concerning the import ofpropositions.Wehave (1) the discussion of the essential nature of judgments and of thefundamentaldistinctionsbetweenjudgments; (2) theinterpretationofpropositionalforms; (3) thediscussionandcomparisonoflogicalschedulesorschemesof propositions, drawn upwith a view to the expression of judgments in alimitednumberofpropositionalforms.

These problems are inter-related and do not admit of being discussed incompleteisolation.Itisclear,forinstance,thatthedrawingupofascheduleof propositions needs to be supplemented by the exact interpretation of thedifferentformswhichitisproposedtorecognise;andbothinthedrawingupofthescheduleandintheinterpretationweshallbeguidedandcontrolledbyaconsiderationoffundamentaldistinctionsbetweenjudgments.

The problems are, however, in themselves distinct; and somemisunderstandingmaybeavoidedifwecanmakeitclearwhatistheactualproblemthatwearediscussingatanygivenpoint.

In particular, it is important to recognise that in the formulation and


interpretationofpropositionsthereisanarbitraryandselectiveelementwhichis absent from themore fundamental problem. Systems of formulation andinterpretation, therefore, if only theyare intelligible and self-consistent, canhardly be condemned as radically wrong, though they may be rejected asinconvenient or unsuitable. When, however, we are dealing with thefundamental importof judgments, thequestionsraiseddobecomequestionsofabsoluterightorwrong.

It should be added that in the present treatise, since it is concernedwithlogic in itsmore formalaspects,questionsof interpretationand formulationoccupy a position of greater relative importance than they would in atreatmentofthesciencemorefullydevelopedonthephilosophicalside.

49.TheObjectiveReferenceinJudgments.—Ajudgmentcanbeformedorunderstood only through the occurrence of certain psychical events in theminds of those who form or 75 understand it; and in this sense it may beincluded amongst subjective states. It is, however, distinguished from allothersubjectivestatesbythefactthatitclaimstobetrue.

This claim to be true implies an objective reference. For a merelysubjectivestateisnot,assuch,eithertrueorfalse;itissimplyanoccurrence.Thus,thedistinctionbetweentruthandfalsityisinapplicabletoanemotionora volition. An emotionmay be pleasurable or painful; it may be strong orweak;itmayormaynotimpeltoaction;butwecannotdescribeitastrueorfalse.

Andthesameappliestoajudgmentregardedasnomorethanasubjectiveconnexion of ideas. The claim to truth necessarily involvesmore than this,namely, a reference to something external to the psychical occurrenceinvolvedintheformationofthejudgment.Everyjudgmentimplies,therefore,on the part of the judgingmind, the recognition of an objective system ofrealityofsomesort.Thevaliditythatisclaimedforjudgmentisanobjectivevalidity.

Theword“objective”isalwaysadangerouswordtouse,andsomefurtherexplanationmaybegivenofthemeaningtobeattachedtoithere.Whenwesay that a judgment refers to an objective system, we mean a system that


subsistsindependentlyoftheactofjudgmentitself,andthatisnotdependenton the passing fancy of the person who forms the judgment. An objectivesystemofrealityinthissensemay,however,includesubjectivestates,thatis,states of consciousness. A body of psychological doctrine consists ofjudgments relating to states ofmind. But such judgments have an externalreference (that is, external to the judgments themselves) just asmuch as abody of judgments relating to material phenomena. Indeed the doctrine ofjudgment here laid down is not inconsistent with the theory of subjectiveidealismthatresolvesallphenomenaintostatesofconsciousness.

Evenwhena judgment relates topurely fictitiousobjects there is still anexternalreference,—inthiscase,totheworldofconvention.

Theparticularaspectorportionofthetotalsystemofrealityreferredtoinanyjudgmentmaysometimesbe76convenientlyspokenofastheuniverseofdiscourse. The limits, if any, intended to be placed upon the universe ofdiscourse inanygivenpropositionareusuallynotexplicitlystated;but theymust be considered to be implicit in the judgmentwhich the proposition ismeant to express, and to be capable of being themselves expressed shouldtherebeanydangerofmisunderstanding.Atthesametime,itisonlyfairtoadd that attempts to define the universe of discourse are likely to raisemetaphysicaldifficultiesastotheultimatenatureofreality.Whatisofmainimportance from the logical standpoint is the recognition that there is areference to some system of reality which is to be distinguished from theuncontrolled course of our own ideas. And so far as a distinction can bedrawnbetween different systems of reality, there is need of the assumptionthat,whenwecombinejudgmentsorviewthemintheirmutualrelations,theuniverseofdiscourseisthesamethroughout.

50.TheUniversality of Judgments.—The fundamental characteristic thenof judgments is theirobjectivereference, theirclaimtoobjectivevalidity. Itfollowsthatalljudgmentsclaimuniversality, that is tosay, theyclaimtobeacknowledgedastruenotforagivenpersononly,orforalimitednumberofpersons, but for everyone; and again, not for a given time only, or for alimitedtime,butforalltime.Inotherwords,theimportofajudgmentisnotmerelytoexpresssomeconnexionof ideas inmyownmind;but toexpress


something thatclaims tobe true.And truth isnot relative to the individual,norisitwhenfullysetforthlimitedbyconsiderationsoftime.

Weshallhave subsequently todealwith theordinarydistinctionbetweenuniversal and particular propositions; but it will be clear that the claim touniversalitywhichwearenowconsideringisonethatmustbemadeonbehalfof so-called particular, as well as of so-called universal, propositions. Thejudgmentthatsomemenaresixfeetinheightclaimsuniversalacceptancejustasmuchasthejudgmentthatallmenaremortal.

Somejudgmentsagaincontainanexplicitorimplicitreferencetotime.Butthisisreallypartofthejudgment.As77soonasthejudgmentisfullystateditbecomes independent of time. It may perhaps be said that the judgmentFranceisunderBourbonrulewastruetwocenturiesago,butisnottruenow.But the judgment as it stands,without context, is incompletely stated.ThatFrance is (orwas) underBourbon rule in the year 1906A.D. is for all timefalse;thatFranceis(orwas)underBourbonruleintheyear1706A.D.isforalltimetrue.

In regard to the nature and significance of the reference to time injudgments, Mr Bosanquet draws a useful distinction between the time ofpredicationandthetimeinpredication.77Bythetimeofpredicationismeantthe time atwhich some thinking beingmakes the judgment; and this in nowayaffectsthetruthofthejudgment.But,asSigwartpointsout,everythingwhichexistsasaparticularthingoccupiesadefinitepositionintime.Henceall judgments relating toparticular things, includingsingular judgmentsandso-callednarrative judgments, relate to somedefinite time, past, present, orfuture,with reference towhichalone thestatementsmadearevalid.This isthetimeinpredication,andthereferencetoitmustberegardedasanintrinsicpartofthejudgmentitself,althoughitisnotalwaysexplicitlymentioned.

77 Logic,I.p.215.CompareSigwart,Logic,§15.

Itwillbeseen that the recognitionof theuniversalityofall judgments inthe sense here indicated is but the recognition in another aspect of theirobjectivecharacter.

51. The Necessity of Judgments.—A further characteristic that has been


ascribedtoalljudgments,whenconsideredinrelationtothejudgingmind,isnecessity.Thistooisconnectedwiththeclaimtoobjectivevalidity.Whenwejudge, we are not free to judge as we will. No doubt by controlling theintellectualinfluencestowhichwesubjectourselveswemayindirectlyandinthe long run modify within certain limits our beliefs. This is a questionbelongingtopsychologyintowhichweneednotnowenter.Butatanygivenmomentthejudgmentsweformaredeterminedbyourmentalhistoryandthecircumstancesinwhichweareplaced.Weareboundtojudgeaswedojudge;sofaraswefeelaquestiontobean78openoneourjudgmentregardingitissuspended. It must be granted that we not unfrequently make statementswhichdonotbetraythedoubtswhichasamatteroffactwefeelwithregardtothepointatissue;butsuchstatementsdonotrepresentourrealjudgments.Thepropositionsweutteraretheexpressionsofpossiblejudgments,butnotofourjudgments.

Inanydiscussionofthemodalityofjudgments,othersensesinwhichtheterm“necessary”maybeappliedtojudgmentshavetobeconsidered.Inhereaffirming necessity as a characteristic of all judgments, we are merelydeclaringover again in another aspect their objective character.Themerelysubjective sequence of ideas in our minds is more or less under our owncontrol.Atanyratewecanatwillbringgivenideastogetherinourmind.Butajudgmentismorethanarelationbetweenideas.Itclaimstobetrueofsomesystemofreality;andhenceitisnotsomuchdeterminedbyus,asforusbythe knowledge which we have come to possess or think we have come topossessaboutthatsystemofreality.

EXERCISE.

52. “Whatisoncetrueisalwaystrue.” “Whatistrueto-daymaybefalseto-morrow.” Examinethesestatements.[L.]


CHAPTERII.

KINDSOFJUDGMENTSANDPROPOSITIONS.

53.TheClassificationofJudgments.—Itiscustomaryforlogicianstooffera classification of judgments or propositions. There is, however, so muchvariationintheobjectstheyhaveinviewindrawinguptheirclassifications,thatveryoftentheirresultsarenotreallycomparable.

(1) Our object in classifying propositions may, in the first place, be toproduceaworkingschemefor theformulationof judgments.Anillustrationofthisisaffordedbythetraditionalschemeofpropositions(AllSisP,NoSisP, etc.),orby theHamiltonianschemebasedupon thequantificationof thepredicate. A classification of this kind is essentially formal. The differentpropositional forms that are recognised must receive clearly definedinterpretations;andtheresultingscheme,ifitisworthanythingatall,willbeorderlyandcompact.Ontheotherhand,itisnotlikelytobecomprehensiveorexhaustive;formanynaturalmodesofjudgmentwillnotfindaplaceinit,atanyrateuntiltheyhavebeenexpressedinamodified,thoughasnearlyaspossibleequivalent,form.

There are many ways of formulating judgments, each of which has itsspecialmeritsandisfromsomeparticularpointofviewspeciallyappropriate.Wemust,however,giveuptheideathatanyoneofthesewayscanholdthefield as a fundamental and essentially suitable classification of judgmentslookedatfromthepsychologicalpointofview.

(2)Fromthepsychologicalstandpointourendeavourmustbetogiveratherwhatmaybecalledanaturalhistory80classificationofjudgments.Primitivetypesofjudgment,whichinalogicalschemeofformulationarenotlikelytofind a place at all,will nowbe regarded as of equal importancewithmoredeveloped and scientific types. Our object may indeed be (as with MrBosanquet)tosketchthedevelopmentofjudgmentsfromthemostprimitive


typestothosewhichgiveexpressiontotheidealofknowledge.

Inaclassificationofthiskindthedividinglinesarenotsoclearandsharplydefinedasinaschemeframedforthelogicalformulationofjudgments.Thedifferent types,moreover, do not stand out inmarked distinction from oneanother,anditisdifficulttoarrangethedifferentclassesinduesubordination,and with complete avoidance of cross divisions. The underlying plan isindeed apt to be obscured by details, so that thewhole discussion tends tobecomesomewhatcumbrous.

(3)AclassificationofpropositionsofstillanotherkindisgivenbyMillinthelaterpartofhischapterontheImportofPropositions.Theconclusionatwhich he arrives is that every proposition affirms, or denies, either simpleexistence, or else some sequence, coexistence, causation, or resemblance.This classification is certainly not a formal one; it is not a scheme for thelogicalformulationofjudgments.Nor,ontheotherhand,canitberegardedasapsychologicalclassificationof typesof judgment,designed to illustratethe nature and growth of thought. Mill’s point of view is objective andmaterial.Inoneplacehedescribeshisschemeasaclassificationofmattersoffact,ofallthingsthatcanbebelieved;andthemainusethathesubsequentlymakesofit isinconnexionwiththeenquiryastothemethodsofproofthatareappropriateaccordingtothenatureofthematteroffactthatisasserted.

Inthepagesthatfollowvariousschemesforformulatingjudgmentswillbeconsidered.Forreasonsalreadystated,however,noschemeof thiskindcanbe regarded as constituting an exhaustive classification of judgments. Thetraditionalscheme,forexample,isludicrouslyunsatisfactoryandincompleteifputforwardasaffordingsuchaclassification.

Weshallnot attempt togivewhathasbeen spokenof aboveasanaturalhistory classification of judgments. The really 81 important distinctionsinvolvedinsuchaclassificationcanberaisedindependently,andthegeneralplanofthisworkistodwellprincipallyonthemoreformalaspectsoflogic.Itmay be added that even from a broader point of view the problem of theevolutionofthoughtishardlytoberegardedasprimarilyalogicalproblem.

Again,suchaclassificationasMill’sinvolvesmaterialconsiderationsthat


areoutsidethescopeofthistreatise.

Without, however, professing to give any complete scheme ofclassification, we shall endeavour to touch upon the most fundamentaldifferencesthatmayexistbetweenjudgments.

54. Kant’s Classification of Judgments.—Kant classified judgmentsaccording to four different principles (Quantity, Quality, Relation, andModality)eachyieldingthreesubdivisions,asfollows:

(1) Quantity. (i) Singular ThisSisP.(ii) Particular SomeSisP.(iii) Universal AllSisP.

(2) Quality. (i) Affirmative AllSisP.(ii) Negative NoSisP.(iii) Infinite AllSisnot-P.

(3) Relation. (i) Categorical SisP.(ii) Hypothetical IfSisPthenQisR.(iii) Disjunctive EitherSisPorQisR.

(4) Modality. (i) Problematic SmaybeP.(ii) Assertoric SisP.(iii) Apodeictic SmustbeP.

Thisarrangementisopentocriticismfromseveralpointsofview;anditssymmetry,althoughattractive,isnotreallydefensible.Atthesametimeithasthe great merit of making prominent what really are the fundamentaldistinctionsbetweenjudgments.

The first distinction that we shall consider is that between simple andcompoundjudgments(replacingKant’sdistinctionaccordingtorelation).Weshallthenconsiderinturndistinctionsofmodality,ofquantity,andofquality.82

55. Simple Judgments and Compound Judgments.—Under the head ofrelation, Kant gave the well-known threefold division of judgments intocategorical,wheretheaffirmationordenialisabsolute(SisP);hypothetical(orconditional),wheretheaffirmationordenialismadeunderacondition(IfAisBthenSisP);anddisjunctive,where theaffirmationordenial ismade


withanalternative(EitherSisPorQisR).

These threekindsof judgmentcannot,however,properlybeco-ordinatedasonanequalitywithoneanotherinathreefolddivision.Forthecategoricaljudgmentappearsasanelementinboththeothers,andhencethedistinctionbetween the categorical, on the one hand, and the hypothetical and thedisjunctive,ontheother,appearstobeonadifferentlevelfromthatbetweenthetwolatter.Moreover,thehypotheticalandthedisjunctivedonotexhaustthemodes inwhich categorical judgmentsmay be combined so as to formfurtherjudgments.Itis,therefore,betternottostartfromtheabovethreefolddivision,butfromatwofold,namely,intosimpleandcompound.

Acompoundjudgmentmaybedefinedasajudgmentintothecompositionofwhichotherjudgmentsenteraselements.78Therearethreeprincipalwaysin which judgments may be combined, and in each case the denial of thevalidityofthecombinationyieldsafurtherformofjudgment,sothattherearesixkindsofcompoundjudgmentstobeconsidered.

78 Thedistinctionhereimpliedhasbeencriticisedonthegroundthat(a)iftheso-calledelementsare really judgments, thecombinationof themyieldsnofreshjudgment; while (b) if the combination is really an independent judgment, theelements intowhich it can be analysed are not themselves judgments. Itwill beseenthat(a)isintendedtoapplytoconjunctivesyntheses,and(b)tohypotheticalsanddisjunctives.Weshallconsidertheargumentundertheseheadsseverally.

(1)Wemayaffirmtwoormoresimplejudgmentstogether.Thus,giventhatPandQstandseparatelyforjudgments,wemayaffirm“PandQ.”

Ithasbeenheldthatasynthesisoftwoindependentjudgmentsinthiswaydoes not really yield any fresh judgment distinct from the two judgmentsthemselves.79Inasensethisistrue.Anyonemay,however,bechallengedforholding two 83 judgments together on grounds which would have noapplication to either taken separately. Hence it is convenient to regard thecombination as constituting a distinct logical whole, which demands somekindofseparatetreatment;andonthisgroundthedescriptionof“PandQ”asacompoundjudgmentmaybejustified.

79 CompareSigwart,Logic,i.p.214.

Thesynthesis involved isconjunctive.HencePandQmaybe spokenof


more distinctively as a conjunctive judgment. Its denial yields “Not both PandQ” and this form is more truly disjunctive than the form (P or Q) towhichthatdesignationismorecommonlyapplied.

(2)Without committing ourselves to the affirmation of eitherP orQ wemayholdthemtobesorelatedthatthetruthoftheformerinvolvesthatofthelatter.Thisyieldsthehypotheticaljudgment,“IfPthenQ.”

Ithasbeenheldthat toregardthisasacombinationof judgments,andtospeakofitasinthissenseacompoundjudgment,ismisleading,sincePandQ are here not judgments at all, that is to say, they are not at themomentintended as statements. Neither P nor Q is affirmed to be true. What isaffirmedtobetrueisacertainrelationbetweenthem.80


ItiscertainlythecasethatwhenIjudge“IfPthenQ,”Pneednotbemyjudgment,norneedQ ;myobjectmayevenbetoestablishthefalsityofPonthe ground of the known falsity ofQ. A more impersonal view, however,beingtaken,PandQaresuppositions,thatis,possiblejudgments,sothattheyhavemeaningasjudgments;andIfPthenQmayfairlybesaidtoexpressarelationbetweenjudgmentsinthesenseofitsforcebeingthattheacceptanceofPasatruejudgmentinvolvestheacceptanceofQasatruejudgmentalso.Thedescriptionofthehypotheticaljudgmentascompoundappearsthereforeto be in this sense justified. Such a judgment as If P then Q cannot beinterpreted except on the supposition that P and Q taken separately havemeaningasjudgments.

Aswegetacompoundjudgmentwhenwedeclaretwojudgmentstobesorelated that if one is accepted the othermust be accepted also, soweget acompound judgment when 84 we deny that this relation subsists betweenthem. Thus in addition to the judgment “If P thenQ,” we have its denial,namely, “If P then not necessarily Q.”81 The best mode of describing thisformofpropositionwillbeconsideredinasubsequentchapter.

81 IngivingthisasthecontradictoryofIfPthenQ,weareassumingaparticulardoctrineoftheimportofthehypotheticaljudgment.Thequestionwillbediscussedmorefullylateron.

(3)Wehaveanotherformofcompoundjudgmentwhenweaffirmthatone


orotheroftwogivenjudgmentsistrue.Thisformofjudgment,“PorQ,”isusuallycalleddisjunctive, thoughalternativewouldbeabettername. Ithasbeen already pointed out thatNot both P and Q is the more distinctivelydisjunctiveform.

ItmaybedeniedthatPorQisacompoundjudgmentonthesamegroundsasthoseonwhichthisisdeniedofIfPthenQ.Since,however,thepointsatissuearepracticallythesameasbefore,thediscussionneednotberepeated.

The denial of “PorQ” yields “Neither P norQ.” Thismay be called aremotivejudgmentifadistinctivenameiswantedforit.

Itshouldbeaddedthatnotallformsofpropositionwhichwouldordinarilybe described as hypothetical or disjunctive are really the expressions ofcompoundjudgmentsasabovedescribed.ThustheformsIfanySisPitisQ(Ifatriangleisisoscelestheanglesatitsbaseareequal).EverySiseitherPorQ(Everybloodvesseliseitheraveinoranartery),donot—liketheformsIfPistrueQistrue(If there isarighteousGodthewickedwillnotescapetheirjustpunishment),EitherPorQistrue(Eitherfreewill isafactorthesense of obligation is an illusion)—express any relation between twoindependent judgments or propositions. This point will be developedsubsequentlyinadistinctionthatwillbedrawnbetweenthetruehypothetical(IfPistrueQistrue)andtheconditional(IfanySisPitisQ).

56.TheModalityofJudgments.—Verydifferent accountsof themodalityofjudgmentsorpropositionsaregivenbydifferentwriters,andtheproblemstowhichdistinctionsofmodalitygive85riseareasarulenoteasyofsolution.At thesame timesuchdistinctionsareofa fundamentalcharacter, and theyare apt to present themselves in a disguised form, thus obscuring manyquestionsthatatfirstsightappeartohavenoconnexionwithmodalityatall.It is a drawback to have to deal with so difficult a problem nearly at thecommencementofourtreatmentofjudgments,andthespaceatourdisposalwillnotadmitofourdealingwithitingreatdetail.Moreover,itcanhardlybehoped that the solution offeredwill be accepted by all readers. Still a briefconsideration of modal distinctions at this stage will help to make somesubsequentdiscussionseasier.


Themainpointatissueiswhetherdistinctionsofmodalityaresubjectiveorobjective. Inattempting todecide thisquestion itwillbeconvenient todealseparatelywithsimplejudgmentsandcompoundjudgments.

57.Modality in relation to Simple Judgments.—TheAristoteliandoctrineofmodals,whichwas also the scholastic doctrine, gave a fourfold divisioninto(a)necessary,(b)contingent,(c)possible,and(d)impossible,accordingasapropositionexpresses(a)thatwhichisnecessaryandunchangeable,andwhichcannotthereforebeotherwise;or(b) thatwhichhappenstobeatanygiven time, butmight have been otherwise; or (c) thatwhich is not at anygiventime,butmaybeatsomeothertime;or(d)thatwhichcannotbe.Thepoint of view here taken is objective, not subjective; that is to say, thedistinctionsindicateddependuponmaterialconsiderations,anddonotrelateto the varying degrees of belief with which different propositions areaccepted.82


82 The consideration of modality as above conceived has sometimes beenregardedasextra-logicalonthegroundthatnecessity,contingency,possibility,andimpossibilitydependuponmattersoffactwithwhichthelogicianassuchhasnoconcern.Butitalsodependsuponmattersoffactwhetheranygivenpredicatecanrightlybepredicatedaffirmativelyornegatively,universallyorparticularly,ofanygiven subject. Distinctions of quality and quantity can nevertheless be formallyexpressed,andifdistinctionsofmodalitycanalsobeformallyexpressed,thereisnoinitialreasonwhytheyshouldnotberecognisedbythelogician,eventhoughheisnotcompetenttodeterminethevalidityofanygivenmodal.Insofar,however,as the modality of a proposition is something that does not admit of formalexpression, so thatpropositionsof thesameformmayhaveadifferentmodality,then the argument that thedoctrineofmodals is extra-logical ismoreworthyofconsideration.

86Kant’sdoctrineofmodalityisdistinguishedfromthescholasticdoctrineinthatthepointofviewtakenissubjective,notobjective,accordingtooneofthesensesinwhichKantusestheseterms.Kantdividesjudgmentsaccordingto modality into (a) apodeictic judgments—S must be P, (b) assertoricjudgments—S is P, and (c) problematic judgments—S may be P ; and thedistinctions between these three classes have come to be interpreted asdepending upon the character of the belief with which the judgments areaccepted.

Thedistinctionbetweenthesetwodoctrinesisfundamental;for,asSigwartputs it,83 the statement that a judgment is possible or necessary is not thesameasthestatementthatitispossibleornecessaryforapredicatetobelongto a subject. The former (which is the Kantian doctrine) refers to thesubjective possibility or necessity of judgment; the latter (which is theAristoteliandoctrine)referstotheobjectivepossibilityornecessityofwhatisstatedinthejudgment.

83 Logic,i.p.176.

58. Subjective Distinctions of Modality.—We must reject the view thatsubjective distinctions of modality can be drawn in relation to simplejudgments.84Foralljudgments,aswehaveseen,possessthecharacteristicofnecessity, and hence this characteristic cannot be made the distinguishingmarkofaparticularclassofjudgments,theapodeictic.

84 What follows in this section is basedmainly onSigwart’s treatment of thesubject(Logic,§31).


We may touch on two ways in which it has been attempted to draw adistinction, from the subjective point of view, between assertoric andapodeicticjudgments.

The assertoric judgment has been regarded as expressing what has onlysubjectivevalidity,thatis,whatcanbeaffirmedtobetrueonlyforthepersonforming the judgment, while the apodeictic judgment expresses what hasuniversalvalidityandcanbeaffirmedtobetrueforeveryone.

This again conflicts with the general doctrine of judgment already laiddown.Weholdthateveryjudgmentclaimstobetrue,andthattruthcannotberelativetotheindividual.Theassertoricjudgment,therefore,asthusdefinedis no true 87 judgment at all, and we find that all judgments are reallyapodeictic.

Another suggested ground of distinction is that between immediateknowledge and knowledge that is based on inference, the former beingexpressedbytheassertoricjudgment,andthelatterbytheapodeictic.

There is no doubt that we often say a thing is so and sowhen this is amatterofdirectperception,whilewesayitmustbesoandsowhenwecannototherwiseaccountforcertainperceivedfacts.Thus,ifIhavebeenoutintherain,Isayithasrained ;if,withouthavingobservedanyrainfall,Inoticethattheroadsandroofsarewet,Isayitmusthaverained.

It is obvious, however, that this distinction is quite inconsistentwith theascriptionofanysuperiorcertaintytotheapodeicticjudgment.Forthatwhichwe know mediately must always be based on that which we knowimmediately;and,sinceintheprocessofinferenceerrormaybecommitted,itfollows that that whichwe knowmediatelymust have inferior certainty tothat of which we have immediate knowledge. Accordingly in ordinarydiscourse thestatement thatanythingmustbe so and sowouldgenerallybeunderstoodasexpressingacertaindegreeofdoubt.

We cannot then justify the recognition of the apodeictic judgment asexpressingahigherdegreeofcertaintythanthemerelyassertoric.

On the other hand, the so-called problematic judgment, interpreted asexpressingmere uncertainty,85 cannot be regarded as in itself expressing a


judgment at all. It may imply a judgment in regard to the validity ofargumentsbroughtforwardinsupportorindisproofofagiventhesis;anditimpliesalsoajudgmentastothestateofmindofthepersonwhoisinastateofuncertainty;butitisinitselfameresuspensionofjudgment.

85 Theproblematicjudgmentasinterpretedinthefollowingsectiondoesmorethan expressmere uncertainty.The formof propositionSmay beP is no doubtambiguous.

59.ObjectiveDistinctionsofModality.—Wehavenexttoconsiderwhether,having regard not to the judgment as a 88 subjective product, but to theobjective fact expressed in a judgment, any valid distinction can be drawnbetween thenecessary, theactual (orcontingent), and thepossible ; andouranswermustbeintheaffirmative,providedthatwearepreparedtoadmittheconceptionoftheoperationoflaw.

Thus the judgment Planets move in elliptic orbits is in this sense ajudgment of necessity. It expresses something which we regard as themanifestation of a law, and it has an indefinitelywide application. For webelieveittoholdgoodnotonlyoftheplanetswithwhichweareacquainted,but also of other planets (if such there be) which have not yet beendiscovered.

Nowtakethejudgment,AllthekingswhoruledinFranceintheeighteenthcenturywerenamedLouis.This isastatementof fact,butclearly isnot theexpression of any law. The proposition relates to a limited number ofindividuals who happened to have the same name given to them; but werecognise that their names might have been different, and that their beingkingsofFrancewasnotdependentontheirpossessingthenameinquestion.Thisthenwemaycallajudgmentofactuality.

Wehaveajudgmentofpossibilitywhenwemakesuchastatementasthataseedlingrosemaybeproduceddifferentincolourfromanyroseswithwhichweareat present acquainted,meaning that there is nothing in the inherentnatureof roses (or in the laws regulating theproductionof roses) to renderthisimpossible.

Wehave thena judgmentofnecessity (anapodeictic judgment)when theintention is to give expression to some law relating to the class of objects


denotedby thesubject-term;wehavea judgmentofactuality (anassertoricjudgment) when the intention is to state a fact, as distinguished from theaffirmation or denial of a law; we have a judgment of possibility (aproblematicjudgment)whentheintentionistodenytheoperationofanylawrenderingsomecomplexofpropertiesimpossible.86

86 Thecaseofapropositionwhichmayberegardedasexpressingaparticularinstanceof theoperationofa lawneedstobespeciallyconsidered.Granting,forinstance, that the proposition Every triangle has its angles equal to two rightanglesisapodeictic,arewetodescribethepropositionThistrianglehasitsanglesequaltotworightanglesasapodeicticorasassertoric?Therightanswerseemstobethat,asthusbarelystated,thepropositionmaybemerelyassertoric;foritmaydo no more than express a fact that has been ascertained by measurement. If,however, the proposition is interpreted asmeaningThis figure, being a triangle,hasitsanglesequaltotworightangles,thenitisapodeictic.

89Ishallnotattempttogivehereanyadequatephilosophicanalysisoftheconceptionofobjectivenecessity.Itmustsuffice tosaythatweallhavetheconception of the operation of law, and that for our present purpose thevalidityofthisconceptionisassumed.

With regard to this treatment of modality the objection may perhaps beraised that,whatever theirvalue in themselves, thedistinctions involvedarenot of a kindwithwhich formal logic has any concern. It is true that, in asense, judgments of necessity are the peculiar concern of inductive, asdistinguished from formal, logic. The main function of inductive logic isindeed todeterminehowapodeictic judgments (as abovedefined) are tobeestablished on the basis of individual observations; for what we mean byinductionistheprocessofpassingfromparticularstothelawsbywhichtheyare governed. Granting this, however, there are also many problems, withwhich logic in itsmore formalaspectshas todeal, in the solutionofwhichsome recognition of the distinctions under discussion is desirable, if notessential.

Butitwillbesaidthatthedistinctionscannotbeappliedformally:that,forexample, given a proposition in the bare formS isP, or given an ordinaryuniversal affirmative propositionAll S is P, it cannot be determined, apartfrom thematter of the proposition,whether it is apodeictic (in the sense inwhichthattermisusedinthissection)ormerelyassertoric.Thisistrueifwe


are limited to the traditional schedule of propositions. But it is to beremembered that the formulation and the interpretation of propositions arewithincertainlimitsunderourowncontrol,andthatitiswithinourpowersoto interpret propositional forms for logical purposes as to bring outdistinctionsthatarenotmadeclearinordinarydiscourseorinthetraditionallogic. Thus, the form S as such is P might be used for giving formalexpression to the apodeictic judgment, S is P being interpreted as merelyassertoric.

90 Another solution, however, and one that may be made to yield asymmetrical scheme, is to utilise the conditional (as distinguished from thetruehypothetical,87)proposition, and todifferentiate it from thecategorical,byinterpretingitasmodal,88whilethecategoricalremainsmerelyassertoric.

87 Seesection173.88 Here and elsewhere in speaking of a proposition as modal (in

contradistinction toassertoric)wemeanaproposition that is either apodeicticorproblematic.

Thus,weshouldhave,— IfanythingisSitisP,—apodeictic; AllSisP,—assertoric; IfanythingisSitmaybeP,—problematic.89

89 Itwillbeobserved that in thisscheme(leavingononeside thequestionofexistential import) the categorical proposition All S is P is inferable from theconditionalIfanythingisSitisP,butnotviceversâ.

It is of course not pretended that the differentiation here proposed isadoptedintheordinaryuseofthepropositionalformsinquestion;weshall,for example, have presently to point out that in the customary usage ofcategoricals theuniversalaffirmativehas frequentlyanapodeictic force.Weshallreturntoadiscussionofthesuggestedschemelateron.

60.ModalityinrelationtoCompoundJudgments.—Wemaynowconsidertheapplicationofdistinctionsofmodalitytocompoundjudgments,thatis,tojudgmentswhichexpressa relation inwhichsimple judgmentsstandone toanother.It isonethingtosaythatasamatteroffact twojudgmentsarenotbothtrue; it isanother thingtosaythat twojudgmentsaresorelatedtooneanotherthattheycannotbothbetrue.Wemaydescribetheonestatementas


assertoric, the other as apodeictic. An apodeictic judgment thus conceivedexpressesa relationofgroundandconsequence;anobligation, therefore, toaffirm the truth of a certain proposition when the truth of a certain otherpropositionor combinationof propositions is admitted.Theobligationmaysometimesdependupontheassistanceofcertainotherpropositionswhichareleftunexpressed.90

90 Inanapodeicticcompoundjudgment,thenecessitymay(atanyrateincertaincases)bedescribedassubjective.Thisissointhecaseofaformalhypothetical;as,for example, in the proposition If all S is P then all not-P is not-S, or in thepropositionIfallSisMandallMisPthenallSisP.

91 In section 55 a threefold classification of compound judgments wasgiven; the distinction now under consideration points, however, to a morefundamentaltwofoldclassification.Fromthispointofviewaschememaybesuggested inwhichconjunctives (PandQ) and so-calleddisjunctives (P orQ)wouldberegardedasassertoric,whilehypotheticals(IfPthenQ)wouldberegardedasmodal.Theenquiryastohowfarthisisinaccordancewiththeordinary usage of the propositional forms in question must be deferred. Itmay,however,bedesirabletopointoutatoncethat,ifthisschemeisadopted,certain ordinarily recognised logical relations are not valid. For thehypotheticalIfPthenQisordinarilyregardedasequivalenttothedisjunctiveEithernot-PorQ,andthisasequivalenttothedenialoftheconjunctiveBothPandnot-Q.If,however,theconjunctive(and,therefore,itsdenial)andalsothedisjunctivearemerelyassertoric,whilethehypotheticalisapodeictic,itisclearthat thisequivalencenolongerholdsgood.Thedisjunctivecanindeedstill be inferred from the hypothetical, but not the hypothetical from thedisjunctive.Thisresultwillbeconsideredfurtheratalaterstage.

So far we have spoken only of the apodeictic form, If P then Q. Thecorrespondingproblematicformis,IfPthenpossiblyQ ;forexample,IfallSisPitisstillpossiblethatsomePisnotS.ThisdeniestheobligationtoadmitthatallPisSwhenithasbeenadmittedthatallSisP. It is tobeobservedthatinanytreatmentofmodality,theapodeicticandtheproblematicinvolveone another, since the one form is always required to express thecontradictoryoftheother.

61. The Quantity and the Quality of Propositions.—Propositions are


commonlydividedintouniversalandparticular,accordingasthepredicationismadeofthewholeorofapartofthesubject.Thisdivisionofpropositionsissaidtobeaccordingtotheirquantity.

Kantaddedathirdsubdivision,namely,singular ;andotherlogicianshaveaddedafourth,namely,indefinite.Undertheheadofquantitytherehavealsoto be considered what are called plurative and numerically definitepropositions; and the possibility of multiple quantification has to berecognised.The 92 questionmay also be raisedwhether there are not somepropositions,e.g.,hypotheticalpropositions,whichdonotadmitofdivisionaccording to quantity at all. The discussion of the various points hereindicatedmay,however,convenientlybedeferreduntilthetraditionalschemeof categorical propositions, which is based on the definitive division intouniversalandparticular,hasbeenbrieflytouchedupon.

Anotherprimarydivisionofpropositions is intoaffirmativeandnegative,accordingasthepredicateisaffirmedordeniedofthesubject.Thisdivisionofpropositionsissaidtobeaccordingtotheirquality.

Here,again,Kantaddedathirdsubdivision,namely,infinite.Thisthreefolddivision and the more fundamental question as to the true significance oflogicaldenial,willalsobedeferreduntilsomeaccounthasbeengivenofthetraditionalschemeofpropositions.

62. The traditional Scheme of Propositions.—The traditional scheme offormulatingpropositionsisintendedprimarilyforcategoricals,anditisbasedondistinctionsofquantityandqualityonly,distinctionsofmodalitynotbeingtakenintoaccount.For thepurposesof the traditionalschemethefollowinganalysisofthecategoricalpropositionmaybegiven.

Acategoricalpropositionconsistsoftwoterms(whicharerespectivelythesubjectandthepredicate),unitedbyacopula,andusuallyprecededbyasignofquantity.Itthuscontainsfourelements,twoofwhich—thesubjectandthepredicate—constituteitsmatter,whiletheremainingtwo—thecopulaandthesignofquantity—constituteitsform.91

91 The logical analysis of a proposition must be distinguished from itsgrammatical analysis.Grammatically only two elements are recognised, namely,thesubjectandthepredicate.Logicallywefurtheranalysethegrammaticalsubject


into quantity and logical subject, and the grammatical predicate into copula andlogicalpredicate.

The subject is that term about which affirmation or denial ismade. Thepredicateisthattermwhichisaffirmedordeniedofthesubject.

When propositions are brought into one of the forms recognised in thetraditional scheme the subjectprecedes thepredicate. Inordinarydiscourse,however,thisorderissometimes93invertedforthesakeofliteraryeffect,forexample,intheproposition—Sweetaretheusesofadversity.

Thesignofquantityattachedtothesubjectindicatestheextenttowhichtheindividuals denoted by the subject-term are referred to. Thus, in theproposition All S is P the sign of quantity is all, and the affirmation isunderstoodtobemadeofeachandeveryindividualdenotedbythetermS.

Thecopulaisthelinkofconnexionbetweenthesubjectandthepredicate,andindicateswhetherthelatterisaffirmedordeniedoftheformer.

Thedifferent elementsof theproposition asheredistinguished arebynomeansalwaysseparatelyexpressedinthepropositionsofordinarydiscourse;but by analysis and expansion they may be made to appear without anychange of meaning. Some grammatical change of form is, therefore, oftennecessary before propositions can be dealt with in the traditional scheme.Thusinsuchapropositionas“Allthatlovevirtueloveangling,”thecopulaisnotseparatelyexpressed.Thepropositionmay,however,bewritten—

signofquantity subject copula predicate

All loversofvirtue are loversofangling ;

and in this form the four different elements are made distinct. The olderlogicians distinguished between propositions secundi adjacentis andpropositionstertiiadjacentis. Intheformer, thecopulaandthepredicatearenot separated, e.g., Theman runs, All that love virtue love angling; in thelatter,theyaremadedistinct,e.g.,Themanisrunning,Allloversofvirtueareloversofangling.

Thetraditionalschemeofpropositionsisobtainedbyacombinationofthe


division(accordingtoquantity)intouniversalandparticular,andthedivision(accordingtoquality)intoaffirmativeandnegative.Thiscombinationyieldsfourfundamentalformsofpropositionasfollows:— (1)theuniversalaffirmative—AllSisP(orEverySisP,orAnySisP,orAllS’sareP’s)—usuallydenotedbythesymbolA;94 (2)theparticularaffirmative—SomeSisP(orSomeS’sareP’s)—usuallydenotedbythesymbolI; (3)theuniversalnegative—NoSisP(orNoS’sareP’s)—usuallydenotedbythesymbolE; (4)theparticularnegative—SomeSisnotP(orNotallSisP,orSomeS’sarenotP’s,orNotallS’sareP’s)—usuallydenotedbythesymbolO.

These symbolsA, I,E,O, are taken from the Latin words affirmo andnego, theaffirmative symbolsbeing the first twovowelsof the former, andthenegativesymbolsthetwovowelsofthelatter.

Besides these symbols, itwill sometimes be found convenient to use thefollowing,—

SaP=AllSisP ;

SiP=SomeSisP ;

SeP=NoSisP ;

SoP=SomeSisnotP.

Theseformsareusefulwhenitisdesiredthatthesymbolwhichisusedtodenote the proposition as a whole should also indicate what symbols havebeenchosenforthesubjectandthepredicaterespectively.Thus,

MaP=AllMisP ;

PoQ=SomePisnotQ.

Itwillfurtherbefoundconvenientsometimestodenotenot-SbySʹ,not-PbyPʹ,andsoon.Thusweshallhave

SʹaPʹ=Allnot-Sisnot-P ;

PiQʹ=SomePisnot-Q.

ItisbetternottowritetheuniversalnegativeintheformAllSisnotP ;92


forthisformisambiguousandwouldusuallybeinterpretedasbeingmerelyparticular,thenotbeingtakentoqualifytheall,sothatwehaveAllSisnotP=Not-all S is P. Thus, “All that glitters is not gold” is intended for anOproposition,andisequivalentto“Somethingsthatglitterarenotgold.”

92 SimilarremarksapplytotheformEverySisnotP.

95Thetraditionalschemeofformulationissomewhatlimitedinitsscope,andfrommorepointsofviewthanoneitisopentocriticism.Ithas,however,the merit of simplicity, and it has met with wide acceptation. For thesereasonsitisasaruleconvenienttoadoptitasabasisofdiscussion,thoughitisalsonotinfrequentlynecessarytolookbeyondit.

63. The Distribution of Terms in a Proposition.—A term is said to bedistributedwhenreference ismade toall the individualsdenotedby it; it issaidtobeundistributedwhentheyareonlyreferredtopartially,thatis,wheninformationisgivenwithregardtoaportionoftheclassdenotedbytheterm,but we are left in ignorance with regard to the remainder of the class. Itfollows immediately from this definition that the subject is distributed in auniversal, and undistributed in a particular,93 proposition. It can further beshewnthat thepredicate isdistributed inanegative,andundistributed inanaffirmativeproposition.Thus,ifIsayAllSisP,IidentifyeverymemberoftheclassSwithsomememberoftheclassP,andIthereforeimplythatatanyratesomePisS,butImakenoimplicationwithregardtothewholeofP.Itisleft an open question whether there is or is not anyP outside the class S.SimilarlyifIsaySomeSisP.ButifIsayNoSisP,inexcludingthewholeofSfromP,IamalsoexcludingthewholeofPfromS,andthereforePaswellas S is distributed. Again, if I say Some S is not P, although I make anassertionwithregardtoapartonlyofS,IexcludethispartfromthewholeofP, and therefore thewhole ofP from it. In this case, then, the predicate isdistributed,althoughthesubjectisnot.94

93 Somebeingusedinthesenseofsome,itmaybeall.Ifbysomeweunderstandsome, but not all, then we are not really left in ignorance with regard to theremainderoftheclasswhichformsthesubjectofourproposition.

94 Hencewemaysaythatthequantityofaproposition,sofarasitspredicateisconcerned,isdeterminedbyitsquality.Theaboveresults,however,nolongerholdgood if we explicitly quantify the predicate as in Hamilton’s doctrine of the


quantification of the predicate. According to this doctrine, the predicate of anaffirmativepropositionissometimesexpresslydistributed,whilethepredicateofanegative proposition is sometimes given undistributed. For example, such formsareintroducedasSomeSisallP,NoSissomeP.Thisdoctrinewillbediscussedinchapter7.

96Summingupourresults,wefindthat Adistributesitssubjectonly, Idistributesneitheritssubjectnoritspredicate, Edistributesbothitssubjectanditspredicate, Odistributesitspredicateonly.

64. The Distinction between Subject and Predicate in the traditionalScheme of Propositions.—The nature of the distinction ordinarily drawnbetween thesubjectand thepredicateofapropositionmaybeexpressedbysaying that thesubject is thatofwhichsomething isaffirmedordenied, thepredicatethatwhichisaffirmedordeniedofthesubject;orwemaysaythatthe subject is that which we regard as the determined or qualified notion,whilethepredicateis thatwhichweregardasthedeterminingorqualifyingnotion.

Itfollowsthatthesubjectmustbegivenfirstinidea,sincewecannotassertapredicateuntilwehavesomethingaboutwhichtoassertit.Canit,however,be said thatbecause the subject logically comes first inorderof thought, itmustnecessarilydosoinorderofstatement,thesubjectalwaysprecedingthecopula, and the predicate always following it? In other words, can weconsidertheorderofthetermsinapropositiontosufficeasacriterion?Ifthesubject and the predicate are pure synonyms95 or if the proposition ispracticallyreducedtoanequation,asinthedoctrineofthequantificationofthepredicate,itisdifficulttoseewhatothercriterioncanbetaken;oritmayratherbesaidthatinthesecasesthedistinctionbetweensubjectandpredicatelosesallimportance.Thetwoareplacedonanequality,andnothingisleftbywhichtodistinguishthemexcepttheorderinwhichtheyarestated.ThisviewisindicatedbyProfessorBaynesinhisEssayontheNewAnalyticofLogicalForms. In such a proposition, for example, as “Great is Diana of theEphesians,” he would call “great” the subject, reading the proposition,“(Some)greatis(all)DianaoftheEphesians.”


95 For illustrations of this point, and on the general question raised in thissection,compareVenn,EmpiricalLogic,pp.208to214.

Withreferencetothetraditionalschemeofpropositions,however,itcannotbe said that the order of terms is always a 97 sufficient criterion. In theproposition just quoted, “Diana of the Ephesians” would generally beacceptedasthesubject.Whatfurthercriterionthencanbegiven?InthecaseofEandIpropositions(propositions,aswillbeshewn,whichcanbesimplyconverted) we must appeal to the context or to the question to which theproposition is ananswer. Ifone termclearlyconveys information regardingtheotherterm,itisthepredicate.Itwillbeshewnalsothatitismoreusualforthe subject tobe read inextensionand thepredicate in intension.96 If theseconsiderationsarenotdecisive,thentheorderofthetermsmustsuffice.InthecaseofAandOpropositions(propositions,aswillbeshewn,whichcannotbesimplyconverted)afurthercriterionmaybeadded.Fromtherulesrelatingto the distribution of terms in a proposition it follows that in affirmativepropositionsthedistributedterm(ifeither termisdistributed) is thesubject;whilst in negative propositions, if only one term is distributed, it is thepredicate.ItisdoubtfuliftheinversionoftermseveroccursinthecaseofanOproposition;butinApropositionsitisnotinfrequent.Applyingtheaboveconsiderations to such a proposition as “Workers of miracles were theApostles,”itisclearthatthelattertermisdistributedwhiletheformerisnot;thelattertermis,therefore,thesubject.Sinceasingulartermisequivalenttoa distributed term, it follows further as a corollary that in an affirmativepropositionifoneandonlyonetermissingularitisthesubject.Thisdecidessuchacaseas“GreatisDianaoftheEphesians.”

96 The subject is often a substantive and the predicate an adjective.Comparesection135.

65. Universal Propositions.—In discussing the import of the universalpropositionAll S is P, attentionmust first be called to a certain ambiguityresultingfromthefactthatthewordallmaybeusedeitherdistributivelyorcollectively.Intheproposition,Alltheanglesofatrianglearelessthantworightangles,itisuseddistributively,thepredicateapplyingtoeachandeveryangle of a triangle taken separately. In the proposition.All the angles of atriangle are equal to two right angles, it is used collectively, the predicate


applying toall the98angles taken together,andnot toeachseparately.ThisambiguityattachestothesymbolicformAllSisP,butnottotheformAllS’sareP’s.Ambiguitymayalsobeavoidedbyusingeveryinsteadofall,asthesignof quantity. In any case the ambiguity is not of a dangerous character,anditmaybeassumedthatallistobeinterpreteddistributively,unlessbythecontextorinsomeotherwayanindicationisgiventothecontrary.

Amore important distinctionbetweenpropositions expressed in the formAll S is P remains to be considered. For such propositionsmay bemerelyassertoric or they may be apodeictic, in the sense given to these terms insection59.

Itwillbeconvenientheretocommencewithathreefolddistinction.

(1)ThepropositionAllSisPmay,inthefirstplace,makeapredicationofalimitednumberofparticularobjectswhichadmitofbeingenumerated:e.g.,Allthebooksonthatshelfarenovels,Allmysonsareinthearmy,Allthemeninthisyear’selevenwereatpublicschools.Apropositionofthiskindmaybecalled distinctively an enumerative universal. It is clear that such apropositioncannotclaimtobeapodeictic.

(2)The propositionAll S isPmay, in the second place, expresswhat isusuallydescribedasanempiricallaworuniformity:e.g.,Alllionsaretawny,Allscarletflowersarewithoutsweetscent,Allvioletsarewhiteoryelloworhaveatingeofblueinthem.Manypropositionsrelatingtotheuseofdrugs,tothesuccessionofcertainkindsofweathertocertainappearancesofsky,andsoon, fall into this class.Apropositionof this kind expresses a uniformitywhichhasbeen found toholdgoodwithin the rangeofour experience,butwhichweshouldhesitatetoextendmuchbeyondthatrangeeitherinspaceorintime.Thepredicationwhichitmakesisnotlimitedtoadefinitenumberofobjectswhichcanbeenumerated,butatthesametimeitcannotberegardedas expressing a necessary relation between subject and predicate. Such apropositionis,therefore,assertoric,notapodeictic.

(3)ThepropositionAllSisPmay,inthethirdplace,expressalawinthestrict sense, that is to say, a uniformity 99 that we believe to hold gooduniversally and unconditionally: e.g., All equilateral triangles are


equiangular,Allbodieshaveweight,Allarsenicispoisonous.Apropositionof this kind is to be regarded as expressing a necessary relation betweensubjectandpredicate,anditis,therefore,apodeictic.

Propositions falling under the first two of the above categories may bedescribed as empirically universal, and those falling under the third asunconditionallyuniversal.97

97 IhaveborrowedthesetermsfromSigwart,Logic,§27;butIcannotbesurethatmyusageof themcorresponds exactlywithhis. In section27he appears toincludeunder empiricallyuniversal judgmentsonly such judgments asbelong tothe first of the three classes distinguished from one another above.At the sametime,hisdescriptionoftheunconditionallyuniversaljudgmentappliestothethirdclassonly:suchajudgment,hesays,expressesanecessaryconnexionbetweenthepredicatePandthesubjectS ;itmeans,IfanythingisS itmustalsobeP.Anditseemsclearfromhissubsequenttreatment(in§96)ofjudgmentsbelongingtothesecondclassthathedoesnotregardthemasunconditionallyuniversal.

Lotze (Logic, § 68) indicates the distinction we are discussing by the termsuniversalandgeneral.Butagainthereseemssomeuncertaintyastowhichtermhewould apply to judgments belonging to our second class. In the universaljudgment, he says,we havemerely a summation ofwhat is found to be true ineveryindividualinstanceofthesubject;inthegeneraljudgmentthepredicationisofthewholeofanindefiniteclass,includingbothexaminedandunexaminedcases.From this it would appear that the universal judgment corresponds to (1) only,while thegeneral judgment includesboth(2)and(3).Lotze,however,continues,“Theuniversaljudgmentisonlyacollectionofmanysingularjudgments,thesumofwhosesubjectsdoesasamatteroffactfillupthewholeextentoftheuniversalconcept;… theuniversal proposition,Allmenaremortal, leaves it still an openquestionwhether,strictlyspeaking,theymightnotallliveforever,andwhetheritis not merely a remarkable concatenation of circumstances, different in everydifferent case, which finally results in the fact that no one remains alive. Thegeneraljudgment,ontheotherhand,Manismortal,assertsbyitsformthatitliesin the character of mankind that mortality is inseparable from everyone whopartakesinit.”Theillustrationheregivenseemstoimplythatajudgmentmayberegardedasuniversal,thoughitrelatestoaclassofobjects,notallofwhichcanbeenumerated.

Ifthisdistinctionisregardedmerelyasadistinctionbetweendifferentwaysin which judgments may be obtained (for example, by enumeration orempiricalgeneralisationontheonehand,orbyabstractreasoningortheaidoftheprincipleofcausalityontheotherhand),withoutanyrealdifferenceofcontent, it becomes merely genetic and can hardly be retained as a 100distinctionbetweenjudgmentsconsideredinandbythemselves.Ifweareso


toretainit, itmustbeasadistinctionbetweenthemerelyassertoricandtheapodeicticinthesensealreadyexplained.Inordertobeabletodealwithitasaformaldistinction,wemustfurtherbepreparedtoassigndistinctiveformsof expression to the two kinds of universal judgments respectively. Lotzeappears toregardtheformsAllSisPandS isPassufficientlyserving thispurpose.Butthisishardlyborneoutbythecurrentusageoftheseforms.AlltheS’sarePmightservefortheenumerativeuniversalandSassuchisPforthe unconditionally universal. These forms do not, however, fit into anygenerally recognised schedules; andour secondclassofuniversalwouldbeleft out. Another solution, which has been already indicated in section 59,wouldbe touse thecategorical formfor theempiricallyuniversal judgmentonly, adopting the conditional form for the unconditionally universaljudgment.

Themost importantoutcomeof theabovediscussionis thatapropositionordinarily expressed in the form All S is P may be either assertoric orapodeictic. Itwillbe found that thisdistinctionhasan importantbearingonseveralquestionssubsequentlytoberaised.

66.ParticularPropositions.—In dealingwith particular propositions it isnecessarytoassignaprecisesignificationtothesignofquantitysome.

Initsordinaryuse,thewordsomeisalwaysunderstoodtobeexclusiveofnone, but in its relation to all there is ambiguity. For it is sometimesinterpreted as excluding all as well as none, while sometimes it is notregarded as carrying this further implication. The word may, therefore, bedefined in twoconflictingsenses: first,asequivalentsimply tooneat least,that is,as thepurecontradictoryofnone, andhenceas coveringeverycase(including all) which is inconsistent with none ; secondly, as any quantityintermediatebetweennoneandallandhencecarryingwithittheimplicationnot all as well as not none. In ordinary speech the latter of these twomeaningsisprobablythemoreusual.98Ithas,however,beencustomarywith101logiciansininterpretingthetraditionalschemetoadopttheothermeaning,sothatSomeSisPisnotinconsistentwithAllSisP.Usingthewordinthissense,ifwewanttoexpressSome,butnotall,SisP,wemustmakeuseoftwopropositions—SomeSisP,SomeSisnotP.Theparticularpropositionas


thusinterpretedisindefinite,thoughwithacertainlimit;thatis,itisindefiniteinsofarthatitmayapplytoanynumberfromasingleoneuptoall,butontheotherhanditisdefiniteinsofarasitexcludesnone.Weshallhenceforthinterpretsomeinthisindefinitesenseunlessanexplicitindicationisgiventothecontrary.


98 Wemight indeed go further and say that in ordinary speech some usuallymeans considerably less than all, so that it becomes still more limited in itssignification.Incommonlanguage,asisremarkedbyDeMorgan,“some usuallymeansarathersmallfractionofthewhole;alargerfractionwouldbeexpressedbyagoodmany ;andsomewhatmorethanhalfbymost ;whileastilllargerproportionwouldbeagreatmajorityornearlyall”(FormalLogic,p.58).

Mr Bosanquet regards the particular proposition as unscientific, on theground that it always depends either upon imperfect description or uponincomplete enumeration.99 I may, for instance, know that all S’s of someparticulardescriptionareP,butnotcaringornot troubling todefine themIcontentmyselfwithsayingSomeSisP,forexample,Sometruthisbetterkeptto oneself.100 Contrasted with this, we have the particular proposition ofincomplete enumeration where our ground for asserting it is simply theobservationofindividualinstancesinwhichthepropositionisfoundtoholdgood.

99 EssentialsofLogic,pp.116,117.100 It is implied that a proposition of this kind might be expanded into the

propositionAll S that isA isP, that is,AllAS isP.Mr Bosanquet gives, as anexample,Someengines candraga trainatamileaminute fora longdistance.“This does notmean a certain number of engines, though of course there are acertainnumber.Itmeanscertainenginesofaparticularmake,notspecifiedinthejudgment.”

It is true that the particular proposition is not in itself ofmuch scientificimportance;anditsindefinitecharacternaturallylimitsitspracticalutility.Itseems, however, hardly correct to describe it as unscientific, since—aswillsubsequently be shewn in more detail—it may be regarded as possessingdistinctivefunctions.Twosuchfunctionsmaybedistinguished, though theyareoften implicated theone in theother. In thefirstplace, theutilityof theparticularpropositionoftendepends102ratheronwhatitdeniesthanonwhatit affirms, and the proposition that it denies is not indefinite. One of theprincipal functions of the particular affirmative is to deny the universalnegative,andof theparticularnegative todeny theuniversalaffirmative. Inthesecondplace,thedistinctivepurposeoftheparticularpropositionmaybetoaffirmexistence;andthis isprobablyasarule thecasewithpropositionswhicharedescribedasresultingfromincompletedescription.If,forexample,we say that “some engines can drag a train at a mile a minute for a long


distance,” our object is primarily to affirm that thereare such engines; andthiswouldnotbesoclearlyexpressedin theuniversalpropositionofwhichtheparticularissaidtobetheincompleteandimperfectexpression.

Therelationoftheparticularproposition,SomeSisP, to theproblematicproposition,SmaybeP,willbeconsideredsubsequently.

67. Singular Propositions.—By a singular or individual proposition ismeant a proposition inwhich the affirmation or denial ismade of a singleindividualonly:forexample,Brutusisanhonourableman ;MuchAdoaboutNothingisaplayofShakespeare’s ;Myboatisontheshore.

Singularpropositionsmayberegardedasformingasub-classofuniversals,sinceineverysingularpropositiontheaffirmationordenialisofthewholeofthesubject.101More definitely, the singular propositionmay be said to fallintoline,asarule,withtheenumerativeuniversalproposition.

101 It is argued by Father Clarke that singulars ought to be included underparticulars,onthegroundthatwhenapredicateisassertedofonememberonlyofaclass,itisassertedofaportiononlyoftheclass.“NowifIsay,ThisHottentotisa great rascal, my assertion has reference to a smaller portion of the HottentotnationthanthepropositionSomeHottentotsaregreatrascals.Thesameisthecaseevenifthesubjectbeapropername.LondonisalargecitymustnecessarilybeamorerestrictedpropositionthanSomecitiesarelargecities ;andifthelattershouldbereckonedunderparticulars,muchmoretheformer”(Logic,p.274).Thisviewfails to recognise thatwhat is reallycharacteristicof theparticularproposition isnot its restricted character—since the particular is not inconsistent with theuniversal—butitsindefinitecharacter.

Hamilton distinguishes between universal and singular propositions, thepredicationbeing in the former caseof awholeundivided, and in the lattercase of a unit indivisible. The 103 distinction here indicated is sometimesuseful; but it can with advantage be expressed somewhat differently. Asingularpropositionmaygenerallywithout riskof confusionbedenotedbyone of the symbols A or E; and in syllogistic inferences a singular mayordinarily be treated as equivalent to a universal proposition. The use ofindependent symbols for singular propositions (affirmative and negative)wouldintroduceconsiderableadditionalcomplexityintothetreatmentofthesyllogism;andforthisreasonitseemsdesirableasaruletoincludesingularsunder universals. Universal propositions may, however, be divided into


generalandsingular,andtherewillthenbetermswherebytocallattentiontothedistinctionwheneveritmaybenecessaryorusefultodoso.

There is also a certain class of propositions which, while singular,inasmuchastheyrelatebuttoasingleindividual,possessalsotheindefinitecharacterwhichbelongstotheparticularproposition:forexample,Acertainman had two sons ;A great statesmanwas present ;AnEnglish officer waskilled.Havingtwosuchpropositionsin thesamediscoursewecannot,apartfromthecontext,besurethatthesameindividualisreferredtoinbothcases.Carrying thedistinction indicated in theprecedingparagrapha little further,wehaveafourfolddivisionofpropositions:—generaldefinite,“AllS isP”;general indefinite,“SomeS isP”;singulardefinite, “ThisS isP”; singularindefinite,“AcertainSisP.”Thisclassificationadmitsofourworkingwiththeordinarytwofolddistinctionintouniversalandparticular—or,asitishereexpressed, definite and indefinite—wherever this is adequate, as in thetraditional doctrine of the syllogism;while at the same time it introduces afurtherdistinctionwhichmayincertainconnexionsbeofimportance.

68.PlurativePropositionsandNumericallyDefinitePropositions.—Othersignsofquantitybesidesallandsomearesometimesrecognisedbylogicians.Thus,propositionsof theformsMostS’sareP’s,FewS’sareP’s,arecalledplurativepropositions.Mostmaybeinterpretedasequivalenttoatleastonemore than half. Few has a negative force; and Few S’s are P’s may beregarded as equivalent to Most S’s are not 104 P’s.102 Formal logicians(excepting De Morgan and Hamilton) have not as a rule recognised theseadditionalsignsofquantity;andit is truethat inmanylogicalcombinationstheycannotbe regardedasyieldingmore thanparticularpropositions,MostS’sareP’sbeingreducedtoSomeS’sareP’s,andFewS’sareP’stoSomeS’sare not P’s. Sometimes, however, we are able to make use of the extraknowledgegivenus;e.g., fromMostM’sareP’s,MostM’sareS’s,we caninferSomeS’sareP’s,althoughfromSomeM’sareP’s,SomeM’sareS’s,wecaninfernothing.

102 Withperhapsthefurtherimplication“althoughsomeS’sareP’s”;thus,FewS’sareP’sisgivenbyKantasanexampleoftheexponibleproposition(thatis,aproposition which, though not compound in form, can nevertheless be resolvedintoaconjunctionoftwoormoresimplerpropositions,whichareindependentof


one another), on the ground that it contains both an affirmation and a negation,thoughoneoftheminaconcealedway.Itshouldbeaddedthatafewhasnot thesamesignificationas few, butmust be regarded as affirmative, and generally, assimplyequivalenttosome ;e.g.,AfewS’sareP’s=SomeS’sareP’s.Sometimes,however,itmeansasmallnumber,andinthiscasethepropositionisperhapsbestregarded as singular, the subject being collective. Thus, “a few peasantssuccessfully defended the citadel” may be rendered “a small band of peasantssuccessfully defended the citadel,” rather than “some peasants successfullydefendedthecitadel,”sincethestressisintendedtobelaidatleastasmuchonthepaucity of their numbers as on the fact that they were peasants. Whilst thepropositioninterpretedinthiswayissingular,notgeneral,itissingularindefinite,notsingulardefinite;forwhatsmallbandisalludedtoisleftindeterminate.

Numericallydefinitepropositionsarethoseinwhichapredicationismadeofsomedefiniteproportionofaclass;e.g.,Two-thirdsofSareP.Acertainambiguitymay lurk in numerically definite propositions; e.g., in the abovepropositionisitmeantthatexactlytwo-thirdsofSneithermorenorlessareP,so thatwearealsogiven implicitlyone-thirdofSarenotP,or is itmerelymeant that at least two-thirds of S but perhaps more are P? In ordinarydiscoursewe should no doubtmean sometimes the one and sometimes theother.Ifwearetofixourinterpretation,itwillprobablybebesttoadoptthefirstalternative,onthegroundthatiffiguresareintroducedatallweshouldaimatbeingquitedeterminate.103Somesuchwords105asatleastcanthenbeusedwhenit isnotprofessedtostatemorethantheminimumproportionofS’sthatareP’s.

103 De Morgan remarks that “a perfectly definite particular, as to quantity,wouldexpresshowmanyX’sare inexistence,howmanyY’s,andhowmanyoftheX’sareorarenotY’s;asin70ofthe100X’sareamongthe200Y’s”(FormalLogic, p. 58). He contrasts the definite particular with the indefinite particularwhichisoftheformSomeX’sareY’s.ItwillbenoticedthatDeMorgan’sdefiniteparticular, as here defined, is still more explicit than the numerically definiteproposition,asdefinedinthetext.

69. IndefinitePropositions.—According to quantity, propositions have bysome logicians been divided into (1)Universal, (2) Particular, (3) Singular,(4)Indefinite.Singularpropositionshavealreadybeendiscussed.

By an indefinite proposition is meant one “in which the quantity is notexplicitly declared by one of the designatory termsall, every, some,many,&c.”;e.g.,SisP,Cretansareliars.WemayperhapssaywithHamilton,thatindesignate would be a better term to employ. At any rate the so-called


indefinitepropositionisnottheexpressionofadistinctformofjudgment.Itisa formofpropositionwhich is the imperfect expressionof a judgment.Forreasons already stated, the particular has more claim to be regarded as anindefinitejudgment.

Whenapropositionisgivenintheindesignateform,wecangenerallytellfromour knowledgeof the subject-matter or from the contextwhether it ismeant to be universal or particular. Probably in the majority of casesindesignate propositions are intended to be understood as universals, e.g.,“Cometsaresubject to the lawofgravitation”;but ifwearereally indoubtwithregardtothequantityoftheproposition,itmustlogicallyberegardedasparticular.104

104 In the Port Royal Logic a distinction is drawn between metaphysicaluniversalityandmoraluniversality.“Wecallmetaphysicaluniversalitythatwhichisperfectandwithoutexception;andmoraluniversalitythatwhichadmitsofsomeexception,sinceinmoralthingsitissufficientthatthingsaregenerallysuch”(PortRoyalLogic, ProfessorBaynes’s translation, p. 150).The following are given asexamples of moral universals: All women love to talk ; All young people areinconstant ;Alloldpeoplepraisepasttimes.Indesignatepropositionsmayalmostwithout exceptionbe regarded as universals eithermetaphysical ormoral.But itseems clear that moral universals have in reality no valid claim to be calleduniversalsatall.Logicallytheyoughtnottobetreatedasmorethanparticulars,oratanyratepluratives.

70.MultipleQuantification.—Theapplicationofapredicatetoasubjectissometimes limited with reference to times or conditions, and this may betreatedasyieldingasecondaryquantificationoftheproposition;forexample,All men are 106 sometimes unhappy, In some countries all foreigners areunpopular.Thisdifferentiationmaybecarriedfurthersoastoyieldtripleoranyhigherorderofquantification.Thus,wehavetriplequantificationintheproposition,Inallcountriesallforeignersaresometimesunpopular.105

105 For a further development of the notion ofmultiple quantification seeMrJohnson’sarticlesonTheLogicalCalculusinMind,1892.

In this way a proposition with a singular term for subject may, withreference to some secondary quantification, be classified as universal orparticular as the case may be; for example,Gladstone is always eloquent,Browningissometimesobscure.


71. Infinite or LimitativePropositions.—In place of the ordinary twofolddivisionofpropositionsinrespectofquality,Kantgaveathreefolddivision,recognising a class of infinite (or limitative) judgments, which are neitheraffirmative nor negative. Thus, S is P being affirmative, and S is not Pnegative,S is not-P is spoken of as infinite or limitative.106 It is, however,difficulttojustifytheseparaterecognitionofthisthirdclass,whetherwetakethe purely formal stand-point, or have regard to the real content of thepropositions. From the formal stand-point we might substitute some othersymbol,sayQ,fornot-P,andfromthispointofviewSomeSisnot-Pmustberegarded as simply affirmative. On the other hand, Some S is not-P isequivalentinmeaningtoSomeS isnotP,and(assumingP tobeapositiveterm) these two propositions must, having regard to their real content, beequallynegativeinforce.

106 An infinite judgment, in the sense inwhich the term is hereused,maybedescribed as the affirmative predication of a negative. Some writers, however,includeunderpropositionesinfinitae thosewhosesubject,aswellasthosewhosepredicate, is negative. Thus Father Clarke defines propositiones infinitae aspropositionsinwhich“thesubjectorpredicateisindefiniteinextent,beinglimitedonly in its exclusion from some definite class or idea: as,Not to advance is torecede”(Logic,p.268).

Some writers go further and appear to deny that the so-called infinitejudgment has any meaning at all. This point is closely connected with aquestion thatwehavealreadydiscussed,namely,whether thenegative termnot-P has any meaning. If we recognise the negative term—and we haveendeavoured to 107 shew thatwe ought to do so—then the propositionS isnot-P isequivalent to thepropositionS isnotP,and theformerpropositionmust,therefore,havejustasmuchmeaningasthelatter.

Thequestionoftheutilityofsocalledinfinitepropositionshasbeenfurthermixed upwith the question as to the nature of significant denial. But it isbettertokeepthetwoquestionsdistinct.Whateverthetruecharacterofdenialmaybe,itisnotdependentontheuseofnegativeterms.

EXERCISES.


72.Determinethequalityofeachofthefollowingpropositions,andthedistributionofitsterms:(a)Afewdistinguishedmenhavehadundistinguishedsons;(b)Fewverydistinguishedmenhavehadverydistinguishedsons;(c)Notafewdistinguishedmenhavehaddistinguishedsons.[J.]

73.Examinethesignificanceoffew,afew,most,any,inthefollowingpropositions;Fewartistsareexemptfromvanity;Afew facts arebetter thanagreatdealof rhetoric;Mostmenareselfish; Ifanyphilosophershavebeenwise,SocratesandPlatomustbenumberedamongthem.[M.]

74.EverythingiseitherXorY ;XandYarecoextensive ;OnlyXisY ;TheclassXcomprisestheclassYandsomethingmore.Expresseachof thesestatementsbymeansofordinaryA,I,E,O categoricalpropositions.[C.]

75.Expresseachofthefollowingstatementsinoneormoreoftheformsrecognisedinthetraditionalschemeofcategoricalpropositions:(i)Noonecanberichandhappyunlesshe isalso temperateandprudent,andnotalwaysthen;(ii)Nochildeverfails tobetroublesomeif ill taughtandspoilt;(iii)Itwouldbeequallyfalsetoassertthattherichalonearehappy,orthattheyalonearenot.[V.]

76. Express, as nearly as you can, each of the following statements in the form of an ordinarycategoricalproposition,anddetermineitsqualityandthedistributionofitsterms: (a)Itcannotbemaintainedthatpleasureisthesolegood;108 (b)Thetradeofacountrydoesnotalwayssuffer,ifitsexportsarehamperedbyforeignduties; (c)Themanwhoshewsfearcannotbepresumedtobeguilty; (d)Oneorotherofthemembersofthecommitteemusthavedivulgedthesecret.[C.]

77.Findthecategoricalpropositions,expressedintermsofcasesofQornon-QandofRornon-R,whicharedirectlyorindirectlyimpliedbyeachofthefollowingstatements: (a)ThepresenceofQisanecessary,butnotasufficient,conditionforthepresenceofR ; (b)TheabsenceofQisanecessary,butnotasufficient,conditionforthepresenceofR ; (c)ThepresenceofQisanecessary,butnotasufficient,conditionfortheabsenceofR. In what respects, if any, does the categorical form fail to express the full significance of suchpropositionsastheabove?[J.]

78.“Honestyofpurposeisperfectlycompatiblewithblunderingignorance.” “Theaffairmighthaveturnedoutotherwisethanitdid.” “ItmaybethatHamletwasnotwrittenbytheactorknownbyhiscontemporariesasShakespeare.” Employtheabovepropositionstoillustrateyourviewsinregardtothemodalityofpropositions;andexamine the relationsbetweeneachof thepropositions andanyassertoricpropositionwhichmaybetakentobeitsgroundortobepartiallyequivalenttoit.[C.]


CHAPTERIII.

THEOPPOSITIONOFPROPOSITIONS.107

107 This chapter will bemainly concernedwith the opposition of categoricalpropositions;and,asregardscategoricals,complicationsarisinginconnexionwiththeirexistentialinterpretationwillforthepresentbepostponed.

79.TheSquareofOpposition.—Indealingwiththesubjectofthischapterit will be convenient to beginwith the ancient square of oppositionwhichrelates exclusively to the traditional schedule of propositions. It will,however,ultimatelybefounddesirabletogivemoregeneralaccountsofwhatis to be understood by the terms contradictory, contrary, &c., so that theymaybeadaptedtootherschedulesofpropositions.

Two propositions are technically said to be opposed to each other whentheyhavethesamesubjectandpredicaterespectively,butdifferinquantityorqualityorboth.108

108 Thisdefinition,accordingtowhichopposedpropositionsarenotnecessarilyincompatible with one another, is given byAldrich (p. 53 inMansel’s edition).Ueberweg (Logic, § 97) defines opposition in such a way as to include onlycontradictionandcontrariety;andManselremarksthat“subalternsareimproperlyclassed as opposed propositions” (Aldrich, p. 59). Modern logicians, however,usually adoptAldrich’s definition, and this seems on thewhole the best course.Some term iswanted to signify theabovegeneral relationbetweenpropositions;and though itmightbepossible to findamoreconvenient term,noconfusion islikelytoresultfromtheuseofthetermoppositionifthestudentiscarefultonoticethatitishereemployedinatechnicalsense.

TakingthepropositionsSaP,SiP,SeP,SoP,inpairs,wefindthattherearefourpossiblekindsofrelationbetweenthem.

(1)Thepairofpropositionsmaybesuchthattheycanneitherbothbetruenorbothfalse.This iscalledcontradictoryopposition,andsubsistsbetweenSaPandSoP,andbetweenSePandSiP.110

(2)Theymaybesuch thatwhilstbothcannotbe true,bothmaybefalse.


Thisiscalledcontraryopposition.SaPandSeP.

(3)Theymaybesuchthattheycannotbothbefalse,butmaybothbetrue.Subcontraryopposition.SiPandSoP.

(4)Froma givenuniversal proposition, the truth of the particular havingthesamequalityfollows,butnotviceversâ.109This issubalternopposition,theuniversalbeingcalledthesubalternant,andtheparticularthesubalternateorsubaltern.SaPandSiP.SePandSoP.

109 This result and some of our other resultsmay need to bemodifiedwhen,lateron,account is takenof theexistential interpretationofpropositions.But, asstatedinthenoteatthebeginningofthechapter,allcomplicationsresultingfromconsiderationsofthiskindareforthepresentputononeside.

Alltheaboverelationsareindicatedintheancientsquareofopposition.

Thedoctrine of oppositionmaybe regarded from twodifferent points ofview,namely,asarelationbetweentwogivenpropositions;and,secondly,asaprocessofinferencebywhichonepropositionbeinggiveneitherastrueor


asfalse,thetruthorfalsityofcertainotherpropositionsmaybedetermined.Takingthesecondofthesepointsofview,wehavethefollowingtable:—111 Abeinggiventrue,Eisfalse,Itrue,Ofalse ; Ebeinggiventrue,Aisfalse,Ifalse,Otrue ; Ibeinggiventrue,Aisunknown,Efalse,Ounknown; Obeinggiventrue,Aisfalse,Eunknown,Iunknown; Abeinggivenfalse,Eisunknown,Iunknown,Otrue ; Ebeinggivenfalse,Aisunknown,Itrue,Ounknown; Ibeinggivenfalse,Aisfalse,Etrue,Otrue ; Obeinggivenfalse,Aistrue,Efalse,Itrue.

80. Contradictory Opposition.—The doctrine of opposition in thepreceding section is primarily applicable only to the fourfold schedule ofpropositions ordinarily recognised.Wemust, however, look at the questionfrom a wider point of view. It is, in particular, important that we shouldunderstandclearlythenatureofcontradictoryoppositionwhatevermaybethescheduleofpropositionswithwhichwearedealing.

The nature of significant denial will be considered in some detail in theconcludingsectionof thischapter.At thispoint itwill suffice tosay that todeny the truth of a proposition is equivalent to affirming the truth of itscontradictory ;andviceversâ.Thecriterionofcontradictoryoppositionisthatof the twopropositions, onemustbe trueand theothermustbe false ; theycannotbe true together,buton theotherhandnomean ispossiblebetweenthem.Therelationbetweentwocontradictories ismutual; itdoesnotmatterwhich is given true or false, we know that the other is false or trueaccordingly.Everypropositionhas itscontradictory,whichmayhoweverbemoreorlesscomplicatedinform.

Itwillbefoundthatattentionisalmostinevitablycalledtoanyambiguityinapropositionwhenanattemptismadetodetermineitscontradictory.Ithasbeen truly said that we can never fully understand the meaning of apropositionuntilweknowpreciselywhatitdenies;andindeedtheproblemofthe import of propositions sometimes resolves itself at least partly into thequestionhowpropositionsofagivenformaretobecontradicted.

Thenatureofcontradictoryoppositionmaybeillustratedbyreferencetoa


discussionenteredintobyJevons(Studiesin112DeductiveLogic,p.116)asto the precisemeaning of the assertion that a proposition—say,Allgrassesareedible—is false.After raising thisquestion, Jevonsbeginsbygiving ananswer,whichmaybecalledtheorthodoxone,andwhich,inspiteofwhathegoesontosay,mustalsobeconsideredthecorrectone.WhenIassertthataproposition is false, I mean that its contradictory is true. The givenproposition is of the formA, and its contradictory is the correspondingOproposition—Some grasses are not edible.When, therefore, I say that it isfalse that all grasses are edible, I mean that some grasses are not edible.Jevons, however, continues, “But it does not seem to have occurred tologiciansingeneraltoenquirehowfarsimilarrelationscouldbedetectedinthe case of disjunctive and other more complicated kinds of propositions.Take, for instance, the assertion that ‘all endogens are all parallel-leavedplants.’ If thisbefalse,what is true?Apparently thatoneormoreendogensarenotparallel-leavedplants,orelsethatoneormoreparallel-leavedplantsarenotendogens.Butitmayalsohappenthatnoendogenisaparallel-leavedplantatall.Therearethreealternatives,andthesimplefalsityoftheoriginaldoesnotshewwhichofthepossiblecontradictoriesistrue.”

This statement is open to criticism in two respects. In the first place, insaying that one ormore endogens are not parallel-leaved plants,we do notmean toexclude thepossibility thatnoendogen isaparallel-leavedplantatall. Symbolically, Some S is not P does not excludeNo S is P. The threealternativesare,therefore,atanyratereducedtothetwofirstgiven.Butinthesecondplace,itisincorrecttospeakofeitherofthesealternativesasbeingbyitselfacontradictoryoftheoriginalproposition.Thetruecontradictoryistheaffirmationof the truthofoneor other of these alternatives. If the originalpropositionisfalse,wecertainlyknowthatthenewpropositionlimitingustosuchalternativesistrue,andviceversâ.

The point at issue may be made clearer by taking the proposition inquestion in a symbolic form. All S is all P is a condensed expression,resolvable into the form, All S is P and 113 all P is S. It has but onecontradictory,namely,EithersomeSisnotPorsomePisnotS.110Ifeitherofthese alternatives holds good, the original statementmust in its entirety be


false; and, on the other hand, if the latter is false, one at least of thesealternativesmustbetrue.SomeSisnotPisnotbyitselfacontradictoryofAllS isallP.These twopropositionsare indeed inconsistentwithoneanother;buttheymaybothbefalse.

110 ThecontradictoryofAllS isallPmay indeedbe expressed in a differentform,namely,SandParenotcoextensive,butthishaspreciselythesameforceasthecontradictorygivenin the text.Wegoontoshewthat twodifferentformsofthe contradictory of the same propositionmust necessarily be equivalent to oneanother.

ItfollowsthatwemustrejectJevons’sfurtherstatementthat“apropositionof moderate complexity has an almost unlimited number of contradictorypropositions,whicharemoreorlessinconflictwiththeoriginal.Thetruthofanyoneormoreofthesecontradictoriesestablishesthefalsityoftheoriginal,butthefalsityoftheoriginaldoesnotestablishthetruthofanyoneormoreofitscontradictories.”111Nodoubtapropositionwhich iscomplicated in formmayyieldanindefinitenumberofothernon-equivalentpropositionsthetruthofanyoneofwhichis inconsistentwith itsown. Itwillalsobe true that itscontradictorycanbeexpressed inmore thanoneform.But theseformswillnecessarily be equivalent to one another, since it is impossible for apropositiontohavetwoormorenon-equivalentcontradictories.Thispositionmaybeformallyestablishedasfollows.LetQandRbebothcontradictoriesofP.Theywillbeequivalentifitcan114beshewnthatifQthenR,andifRthenQ.SincePandQarecontradictories,wehaveIfQthennotP,andsinceP andR are contradictorieswehave If notP thenR. Combining these twopropositions we have the conclusion If Q then R. If R then Q followssimilarly.Hencewehaveestablishedthedesiredresult.

111 Itmustbeadmittedthat ithasnotbeenuncommonfor logicianstousethewordcontradictsomewhatloosely.Forexample,inthePortRoyalLogic,wefindthefollowing:“Exceptthewiseman(saidtheStoics)allmenaretrulyfools.Thismaybecontradicted(1)bymaintainingthatthewisemanoftheStoicswasafoolaswellasothermen;(2)bymaintainingthattherewereothers,besidestheirwiseman,whowerenotfools;(3)byaffirmingthat thewisemanof theStoicswasafool,and thatothermenwerenot” (p.140).Theaffirmationofanyoneof thesethree propositions certainly renders it necessary to deny the truth of the givenproposition, but no one of them is by itself the contradictory of the givenproposition.The truecontradictory is thealternativeproposition:Either thewisemanoftheStoicsisafoolorsomeothermenarenotfools.


Inconnexionwiththesamepoint,Jevonsraisesanotherquestion,inregardtowhichhisviewisalsoopentocriticism.Hesays,“Butthequestionariseswhetherthereisnotconfusionofideasintheusualtreatmentofthisancientdoctrineofopposition,andwhetheracontradictoryofapropositionisnotanyproposition which involves the falsity of the original, but is not the soleconditionofit.Iapprehendthatanyassertionisfalsewhichismadewithoutsufficient grounds. It is false to assert that the hidden side of the moon iscovered with mountains, not because we can prove the contradictory, butbecause we know that the assertor must have made the assertion withoutevidence.Ifapersonignorantofmathematicsweretoassertthat‘allinvolutesare transcendental curves,’ he would be making a false assertion, because,whethertheyaresoornot,hecannotknowit.”Weshould,however,involveourselvesinhopelessconfusionwerewetoconsiderthetruthorfalsityofapropositiontodependupontheknowledgeofthepersonaffirmingit,sothatthesamepropositionwouldbenowtrue,nowfalse.Itwillbeobservedfurtherthaton Jevons’sviewboth thepropositionsS isPandS isnotP would befalsetoapersonquiteignorantofthenatureofS.Thiswouldmeanthatwecould not pass from the falsity of a proposition to the truth of itscontradictory;andsucharesultasthiswouldrenderanyprogressinthoughtimpossible.

81.ContraryOpposition.—Seeking to generalise the relation betweenAandE, we might naturally be led to characterize the contrary of a givenpropositionbysaying that itgoesbeyondmeredenial,andsetsupa furtherassertionasfaraspossibleremovedfromtheoriginalassertion;sothat,whilstthecontradictoryofapropositiondenies itsentire truth, itscontrarymaybesaidtoassertitsentirefalsehood.Apairofcontrariesasthusdefinedmayberegardedasstandingattheopposite115endsofascaleonwhichthereareanumberofintermediatepositions.

On this definition, however, the notion of contrariety cannot verysatisfactorily be extendedmuch beyond the particular case contemplated intheordinarysquareofopposition.Forifwehaveapropositionwhichcannotitselfberegardedasstandingatoneendofascale,butonlyasoccupyinganintermediateposition,suchpropositioncannotberegardedasformingoneof


a pair of contraries. Plurative and numerically definite propositionsmaybetakenasillustrations.

Hence if it is desired todefine contrariety so that the conceptionmaybegenerallyapplicable,theideaoftwopropositionsstanding,asitwere,furthestapart from each othermust be given up, and any two propositionsmay bedescribedascontrariesiftheyareinconsistentwithoneanotherwithoutatthesame time exhausting all possibilities. Contraries must on this definitionalwaysadmitofamean,buttheymaynotalwaysbewhatweshouldspeakofasdiametricalopposites,andanygivenpropositionisnotlimitedtoasinglecontrary,butmayhaveanindefinitenumberofnon-equivalentcontraries.Atthesametime,itwillbeobservedthatthisdefinitionstillsufficestoidentifyAandEasapairofcontraries,andastheonlypairinthetraditionalschemeofopposition.

82. The Opposition of Singular Propositions.—Taking the propositionSocratesiswise,itscontradictoryisSocratesisnotwise ;112andsolongaswekeep to the same terms, we cannot go beyond this simple denial. Thepropositionhas,therefore,noformalcontrary.113Thisoppositionofsingularshasbeencalledsecondaryopposition(Mansel’sAldrich,p.56).

112 Thismust be regarded as the correct contradictory from the point of viewreachedin thepresentchapter.Thequestionbecomesa littlemoredifficultwhentheexistentialinterpretationofpropositionsistakenintoaccount.

113 We can obtain what may be called a material contrary of the givenproposition by making use of the contrary of the predicate instead of its merecontradictory;thus,Socrateshasnotagrainofsense.Thisisspokenofasmaterialcontrarietybecauseitnecessitatestheintroductionofafreshtermthatcouldnotbeformally obtained out of the given proposition. It should be added that thedistinction between formal andmaterial contrarietymight also be applied in thecaseofgeneralpropositions.

116If,however,thereissecondaryquantificationinapropositionhavingasingularsubject,thenwemayobtaintheordinarysquareofopposition.Thus,ifouroriginalpropositionisSocratesisalways(orinallrespects)wise,itiscontradictedbythestatementthatSocratesissometimes(orinsomerespects)notwise,while it has for its contrary,Socrates is never (or in no respects)wise,andforitssubaltern,Socratesissometimes(orinsomerespects)wise.Itmay be said that when we thus regard Socrates as having different


characteristicsatdifferent timesorunderdifferentconditions,oursubject isnotstrictlysingular,sinceitisnolongerawholeindivisible.Thisisinasensetrue, andwemight no doubt replace our proposition by one having for itssubject “the judgmentsor the acts ofSocrates.”But it doesnot appear thatthisresolutionofthepropositionisnecessaryforitslogicaltreatment.

The possibility of implicit secondary quantification, although no suchquantificationisexplicitlyindicated,isanotunfruitfulsourceoffallacyintheemployment of propositions having singular subjects. If we take suchpropositionsasBrowningisobscure,Epimenidesisaliar,Thisflowerisblue,andgiveastheircontradictoriesBrowningisnotobscure,Epimenidesisnotaliar,This flower isnotblue, shallwesay that theoriginalpropositionor itscontradictoryistrueincaseBrowningissometimes(butnotalways)obscure,orincaseEpimenidessometimes(butnotoften)speaksthetruth,orincasethe flower ispartly (butnotwholly)blue?There is certainlyaconsiderablerisk in such instances as these of confusing contradictory and contraryopposition,andthiswillbeavoidedifwemakethesecondaryquantificationofthepropositionsexplicitattheoutsetbywritingthemintheformBrowningis always (or sometimes) obscure, &c.114 The contradictory will then beparticularoruniversalaccordingly.


114 Or we might reduce them to the forms,—All (or some) of the poems ofBrowningareobscure,All(orsome)ofthestatementsofEpimenidesarefalse,All(orsome)ofthesurfaceofthisflowerisblue.

83. The Opposition of Modal Propositions.—So far in this chapter ourattentionhasbeenconfinedtoassertoricpropositions.Forthepresent,averybrief reference to theopposition117ofmodalswillsuffice.Themainpointsinvolvedwillcomeupforfurtherconsiderationlateron.

We have seen that the unconditionally universal proposition, whetherexpressed in theordinary categorical formAllS isP, or as a conditional Ifanything isS it isP, affirms anecessary connexion, bywhich ismeant notmerelythatalltheS’sareasamatteroffactP’s,butthatitisinherentintheirnature that they should be so. The statement that some S’s are not P’s isinconsistentwith thisproposition,but isnot itscontradictory,sinceboth thepropositionsmightbefalse:theS’smightallhappentobeP’s,andyettheremightbenolawofconnexionbetweenSandP.Thepropositioninquestionbeing apodeictic will have for its contradictory a modal of anotherdescription,namely,aproblematicproposition;andthismaybewrittenintheformSneednotbeP,orIfanythingisSstill itneednotbeP,accordingasouroriginalpropositionisexpressedasacategoricalorasaconditional

Similarly, thecontradictoryof thehypotheticalIfPis true thenQis true,thispropositionbeinginterpretedmodally, isIfP is truestillQneednotbetrue.

84. Extension of the Doctrine of Opposition.115—If we do not confineourselves to the ordinary square of opposition, but consider any pair ofpropositions (whatever may be the schedule to which they belong), itbecomes necessary to amplify the list of formal relations recognised in thesquare of opposition, and also to extend themeaning of certain terms.Wemaygivethefollowingclassification:

115 Theillustrationsgiveninthissectionpresupposeaknowledgeofimmediateinferences.Thesectionmayaccordinglyonafirstreadingbepostponeduntilpartofthefollowingchapterhasbeenread.

(1) Two propositionsmay be equivalent or equipollent, each propositionbeing formally inferable from the other. Hence if either one of the


propositions is true, theother is also true; and if either is false, theother isalsofalse.Forexample,aswillpresentlybeshewn,AllSisPandAllnot-Pisnot-Sstandtoeachotherinthisrelation.

(2)and(3)Oneofthetwopropositionsmaybeformallyinferablefromtheother,butnotviceversâ. Ifweare118consideringtwogivenpropositionsQandR, thisyields twocases: forQmaycarrywith it the truthofR,butnotconversely;orRmaycarrywithitthetruthofQ,butnotconversely.Ordinarysubalternpropositionswiththeirsubalternantsfall intothisclass;anditwillbeconvenienttoextendthemeaningofthetermsubaltern,soastoapplyittoany pair of propositions thus related, whether they belong to the ordinarysquareofoppositionornot. Itwill indeedbe found that anypair of simplepropositionsoftheformsA,E,I,O,thataresubalternintheextendedsense,are equivalent to some pair that are subaltern in themore limited sense.116ThusAllSisPandSomePisS,whicharesubaltern in theextendedsense,areequivalenttoAllSisPandSomeSisP.AllSisPandSomenot-SisnotPareanotherpairofsubalterns.Hereitisnotsoimmediatelyobviousinwhatdirectionwearetolookforapairofequivalentpropositionsbelongingtotheordinary squareof opposition.Nonot-P is S andSome not-P is not S will,however,befoundtosatisfytherequiredconditions.

116 This will of course not hold good when we apply the term subaltern tocompoundpropositions,e.g.,tothepairSomeSisnotPandsomePisnotS,SomeSisnotPorsomePisnotS.

(4)The propositionsmaybe such that they can both be true together, orbothfalse,oreitheronetrueandtheotherfalse.Forexample,AllSisPandAllP isS. Suchpropositionsmaybe called independent in their relation tooneanother.

(5)Thepropositionsmaybesuch thatoneorotherof themmustbe truewhilebothmaybe true.A pair of propositionswhich are thus related—forexample,SomeSisPandSomenot-SisP—may,byanextensionofmeaningasinthecaseofthetermsubaltern,besaidtobesubcontrary.ItcanbeshewnthatanypairofsubcontrariesoftheformsA,E,I,Oareequivalenttosomepairofsubcontrariesbelongingtotheordinarysquareofopposition;thus,theabovepairareequivalenttoSomePisSandSomePisnotS.


(6)Thetwopropositionsmaybecontrarytooneanother,inthesensethattheycannotbothbetrue,butcanbothbefalse.Itcanasbeforebeshewnthatanypairofcontrariesof119theformsA,E,I,Oareequivalenttosomepairofcontrariesinthemoreordinarysense.Forexample,thecontrariesAllSisPandAllnot-SisPareequivalenttoNonot-PisSandAllnot-PisS.

(7)Thetwopropositionsmaybecontradictorytooneanotheraccordingtothedefinitiongiven in section80, that is, they canneither both be true norboth false.All S is P and Some not-P is S afford an example outside theordinarysquareofopposition.ItwillbeobservedthatthesetwopropositionsareequivalenttothepairAllSisPandSomeSisnotP.

Two propositions, then, may, in respect of inferability, consistency, orinconsistency, be formally (1) equivalent, (2) and (3) subaltern, (4)independent, (5) subcontrary, (6) contrary, (7) contradictory, the termssubaltern, &c., being used in the most extended sense. What pairs ofcategorical propositions (into which only the same terms or theircontradictories enter) actually fall into these categories respectivelywill beshewninsections106and107.

Thesesevenpossiblerelationsbetweenpropositions(takeninpairs)willbefound to be precisely analogous to the seven possible relations betweenclasses(takeninpairs)asbroughtoutinasubsequentchapter(section130).

85. The Nature of Significant Denial.—It is desirable that, beforeconcluding this chapter, we should briefly discuss a more fundamentalquestionthananythathasyetbeenraised,namely,themeaningandnatureofnegationanddenial.

Weobserve, in the first place, that negation always finds expression in ajudgment,andthatitalwaysinvolvesthedenialofsomeotherjudgment.Thequestionthereforeariseswhethernegationalwayspresupposesanantecedentaffirmation.Thisquestionmustbeansweredinthenegativeifitisunderstoodto mean that in order to be able to deny a proposition we must begin byregarding it as true. The proposition which we deny may be asserted orsuggestedbysomeoneelse;or itmayoccur tousasoneofseveralpossiblealternatives;oritmaybeputintheformofaquestion.


It is, however, to be added that if a denial is to have any value as astatement of matter of fact, the corresponding 120 affirmation must beconsistentwiththemeaningofthetermsemployed.ThusifAconnotesm,n,p, and B connotes not-p, q, r, then the denial that A is B gives no realinformationrespectingA.FortheaffirmationthatAisBcannotbemadebyanyonewhoknowswhat ismeantbyAandB respectively.The samepointmaybeotherwiseexpressedbysayingthatjustastheaffirmationofaverbalproposition is insignificant regarded as a real affirmation concerning thesubject (and not merely as an affirmation concerning the meaning to beattached to the subject-term), so the denial of a contradiction in terms isinsignificantfromthesamepointofview.Suchadenialyieldsmerelywhatistautologousandpracticallyuseless.

For example, thedenial that the soul is a ship in full sail is insignificantregardedasastatementofmatteroffact;forsuchdenialgivesnoinformationtoanyonewhoisalreadyacquaintedwiththemeaningofthetermsinvolved.

Thenatureoflogicalnegationisofsofundamentalandultimateacharacterthat any attempt to explain it is apt to obscure rather than to illumine. Itcannot be expressed more simply and clearly than by the laws ofcontradictionandexcludedmiddle:a judgmentand itscontradictorycannotbothbetrue;norcantheybothbefalse.

Because every negative judgment involves the denial of some otherjudgment, ithasbeenargued thatanegative judgmentsuchasS isnotP isprimarilyajudgmentconcerningthepositivejudgmentSisP,notconcerningthesubjectS ; andhence that a negative judgment is not co-ordinatewith apositivejudgment,butdependentuponit.117

117 CompareSigwart,Logic,i.pp.121,2.

Passing by the point that a positive judgment also involves the denial ofsome other judgment, we may observe that a distinction must be drawnbetween“SisP”isnottrue(whichisajudgmentaboutSisP),andSisnotP(which is a judgment aboutS).Denial no doubt presents itself to themindmost simply in the first of these two forms. But in contradicting a givenjudgmentourmethodusually is toestablishanother judgment involving thesametermswhichstandsto thegivenjudgment in therelationexpressedby


thelawsofcontradiction121andexcludedmiddle;andwhenweoppose thejudgmentS isnotP to the judgmentS isPwehave reached the less directmodeofdenial inwhichwehaveagaina judgmentconcerningouroriginalsubject.

Theexampleheretakentendsperhapstoobscurethepointatissuebecausethedistinctionbetween“SisP”isnottrueandSisnotPmayappeartobesoslight as to be immaterial. That there is a real distinction will, however,appearclearifwetakesuchpairsofpropositionsas“AllSisP”isnot true,SomeSisnotP ;“AllSisallP”isnottrue,EithersomeSisnotPorsomePisnotS ;“IfanyPisQitisR”isnottrue,PmightbeQwithoutbeingR.

ItwillbeconvenientifingeneralweunderstandbythecontradictoryofapropositionP not its simple denial “P is not true,” but the propositionQinvolving the same terms, which is formally so related toP, thatP andQcannotbothbetrueorbothfalse.

Sigwartobservesthatthegroundofadenialmaybeeither(a)adeficiency,or (b) an opposition.118 Imay, for example, pronounce that a certain thingdoes not possess a given attribute either (a) because I fail to discover thepresence of the attribute, or (b) because I recognise the presence of someotherattributewhichIknowtobeincompatiblewiththeonesuggested.

118 Logic,i.p.127.

Thisdistinctionmaybeillustratedbyoneortwofurtherexamples.Thus,Imaydenythatamantravelledbyacertaintraineither(a)becauseIsearchedthe train through just before it started and found he was not there, or (b)becauseIknowhewaselsewherewhenthetrainstarted,—Imay,forinstance,have seenhim leave the station at the samemoment in another train in theopposite direction. Similarly, I may deny a universal proposition either (a)because I have discovered certain instances of its not holding good, or (b)because Iacceptanotheruniversalpropositionwhich is inconsistentwith it.Again,Imaydenythatagivenmetal,orthemetalcontainedinacertainsalt,iscopper(a)onthegroundofdeficiency,namely,thatitdoesnotanswertoacertaintest,or(b)ontheground122ofopposition,namely,thatIrecogniseittobeanothermetal,say,zinc.


Thegroundofdenialalwaysinvolvessomethingpositive,forexample,thesearch through the train,or thediscoveryof individualexceptions.But it isclearthatwhenweestablishanoppositionwegetaresultthatisitselfpositiveinawaythatisnotthecasewhenwemerelyestablishadeficiency.ThismayleaduptoabriefexaminationofadoctrineofthenatureofsignificantdenialthatislaiddownbyMrBosanquet.

MrBosanquetholds thatbare denialhas in itselfno significance, andheapparently denies that the contradictory of a judgment, apart from thegroundsonwhichitisbased,conveysanyinformation.119Forthemeaningofsignificantnegationwemust,hesays,looktothegroundsofthenegation;orelse for contradictory denial we must substitute contrary denial. As aconsequence, a judgment can, strictly and properly, “only be denied byanother judgment of the same nature; a singular by a singular judgment, agenericbyageneric,ahypotheticalbyahypothetical”;120and,presumably,aparticularbyaparticular,anapodeicticbyanapodeictic.

119 Logic,i.p.305.120 Ibid,p.383.

Itisofcoursetruethateverydenialmusthavesomekindofpositivebasis,but it is also necessary that a judgment should be distinguished from thegroundsonwhichitisbased.Wecannotsaythatajudgmentofgivencontentisdifferentfortwopeoplebecausetheyacceptitondifferentgrounds;andifit is said that this is to beg the question, since a difference in groundconstitutes in itself a difference in content, the reply is that such adoctrinemust render the content of every judgment so elusive and uncertain as tomakeitimpossibleofanalysis.

Theviewthatidentifiesthedenialofajudgmentwithitscontrarynotonlymixes up a judgment with its grounds, but also overlooks one of the twoprincipalgroundsofdenial.Whenthegroundofnegationisanopposition,wemaynodoubtbesaidtoreachdenialthroughthecontrary,thoughweshouldstillholdthatthedenialisinitselfsomethinglessthanthecontrary;butwhenthe ground of denial is a deficiency, even this cannot be allowed. If, forexample, I have arrived 123 at the conclusion that aman did not start by agiventrainbecauseIsearchedthetrain throughbefore itsdepartureanddid


notfindhimthere;orifIconcludethatagivenmetalisnotcopperbecauseitdoesnotsatisfyagiven test; Ihaveobtainednocontrary judgment,andyetmydenialisjustified.

Thesewouldbecasesofbaredenial.Ihavegainednopositiveknowledgeofthewhereaboutsofthemaninquestion,norcanIidentifythegivenmetal.Butsurelyitcannotbeseriouslymaintainedthatthedenialismeaninglessoruseless,say,toadetectiveinthefirstinstance,ortoananalyticalchemistinthesecond.

Ofcourseweseldomorneverrestcontentwithbaredenial.Thecontraryrather than thecontradictory representsourultimate aim.But it isoften thecasethat,temporarilyatanyrate,wecannotgetbeyondbaredenial;andweoughtnottoconsiderthatwehavealtogetherfailedtomakeprogresswhenallthat we have achieved is the exclusion of a possible alternative or theoverthrowofafalsetheory.Recentresearches,forexample,intotheoriginofcancer have led to no positive results; but it is claimed for them that bydestroyingpreconceived ideason the subject theyhave cleared theway forfutureadvance.Willanyoneaffirmthat thiswasnotworthdoingorthat thetimespentontheresearcheswaswasted?

Lookingat thequestionfromanotherpointofview, it issurelyabsurd tosaythatwecannotdenyauniversalunlessweareable tosubstituteanotheruniversal in its place.Various algebraical formulae have from time to timebeen suggested asnecessarilyyielding aprimenumber.Theyhave all beenoverthrown, and no valid formula has been established in their place. Butknowledgethattheseformulaearefalseisnotquiteappropriatelydescribedasignorance.

ElsewhereMrBosanquetsaysthatmereenumerativeexceptionsarefutileandcannottouchtheessenceoftheunconditionallyuniversaljudgmentstheyapparentlyoppose.121Heappearstohaveinviewcaseswherenothingmorethansomemodificationoftheoriginaljudgmentisshewntobe124necessary.But even so the enumerative exceptions have overthrown the originaljudgment. No doubt a scientific law which has had a great amount ofevidence in its favour is likely to containelementsof trutheven if it isnotaltogethertrue;andtheobjectofamanofsciencewhooverthrowsalawwill


betosetupsomeotherlawinitsplace.But,saysMrBosanquet,evenifthefirst generic judgmentwere a sheer blunder and confusion, as has been thecasefromtimeto timewith judgmentspropoundedinscience, it isscarcelypossibletorectifytheconfusionexceptbysubstitutingforitthetruepositiveconceptionsthatariseoutofthecaseswhichoverthrewit.”Hereitisadmittedthat the exceptions do overthrow the law, and the rest of the argument issurely an instanceof ignoratioelenchi. It ismoreover a pure, and inmanycasesanunjustifiable,assumptionthatthecaseswhichsufficetooverthrowafalselawwillalsosufficeasthebasisfortheestablishmentofatruelawinitsplace.

121 Logic,i.p.313.

EXERCISES.

86.Examinethenatureoftheoppositionbetweeneachpairofthefollowingpropositions:—NonebutLiberalsvotedagainstthemotion;AmongstthosewhovotedagainstthemotionweresomeLiberals;ItisuntruethatthosewhovotedagainstthemotionwereallLiberals.[K.]

87.Ifsomewereusedinitsordinarycolloquialsense,howwouldtheschemeofoppositionbetweenpropositionshavetobemodified?[J.]

88. Explain the technical terms “contradictory” and “contrary” applying them to the followingpropositions:FewSareP ;Hewasnottheonlyonewhocheated ;Two-thirdsofthearmyareabroad.[V.]

89.Givethecontradictoryofeachofthefollowingpropositions:—SomebutnotallSisP ;AllSisPandsomePisnotR ;EitherallSisPorsomePisnotR ;WhereverthepropertyAisfound,eitherthepropertyBorthepropertyCwillbefoundwithit,butnotbothofthemtogether.[K.]

12590.Givethecontradictory,andalsoacontrary,ofeachofthefollowingpropositions: Halfthecandidatesfailed; Wellingtonwasalwayssuccessfulbothinbeatingtheenemyandinutilisinghisvictory; Allmenareeithernotknavesornotfools; Allbuthehadfled; Fewofthemarehonest; Sometimesalloureffortsfail; Someofoureffortsalwaysfail.[L.]

91.Givethecontradictory,andalsoacontrary,ofeachofthefollowingpropositions: Iamcertainyouarewrong; SometimeswhenitrainsIfindmyselfwithoutanumbrella;


Whateveryousay,Ishallnotbelieveyou.[C.]

92.Definethetermssubaltern,subcontrary,contrary,contradictory,insuchawaythattheymaybeapplicabletopairsofpropositionsgenerally,andnotmerelytothoseincludedintheordinarysquareofopposition. Do the above exhaust the formal relations (in respect of inferability, consistency, orinconsistency)thatarepossiblebetweenpairsofpropositions? Illustrate your answer by considering the relation (in respect of inferability, consistency, orinconsistency) between each of the following propositions and each of the remainder: S and P arecoincident ;SomeSisP ;NotallSisP ;EithersomeSisnotPorsomePisnotS ;AnythingthatisnotPisS.[K.]

93. Given that the propositions X and Z are contradictory, Y and V contradictory, and X and Ycontrary,shew(withoutassumingthatX,Y,V,Zbelongtotheordinaryscheduleofpropositions)thattherelationsofVtoX,ZtoY,VtoZaretherebydeducible.[J.]

94. Prove formally that if two propositions are equivalent, their contradictories will also beequivalent.[K.]

95.Examine thedoctrine that a judgmentcanproperlybedeniedonlybyanother judgmentof thesame type. Illustrate by reference to (a)universal judgments, (b)particular judgments (c) disjunctivejudgments,(d)apodeicticjudgments.[K.]


CHAPTERIV.

IMMEDIATEINFERENCES.122

122 In thischapterweconcernourselvesmainlywith the traditionalschemeofpropositions, and exceptwhere an explicit statement ismade to the contraryweproceedon theassumption thateachclass representedbyasimple termexists intheuniverseofdiscourse,whileatthesametimeitdoesnotexhaustthatuniverse.Thisassumptionappearstohavebeenmadeimplicitlyinthetraditionaltreatmentoflogic.

96. The Conversion of Categorical Propositions.—By conversion, in abroadsense,ismeantachangeinthepositionofthetermsofaproposition.123Logic,however,isconcernedwithconversiononlyinsofarasthetruthofthenew proposition obtained by the process is a legitimate inference from thetruthoftheoriginalproposition.Forexample,thechangefromAllS isP toAllP isS isnota legitimate logical conversion, since the truthof the latterproposition does not follow from the truth of the former. In other words,logicalconversionisacaseofimmediateinference,whichmaybedefinedastheinferenceofapropositionfromasingleotherproposition.124

123 Ueberweg(Logic,§84)definesconversionthus.ComparealsoDeMorgan,FormalLogic,p.58.Ingeometry,allequiangulartrianglesareequilateralwouldbe regarded as the converse ofall equilateral triangles are equiangular. In thissenseofthetermconversion,whichisitsordinarynon-technicalsense,wemaysay—as we frequently do say—“Yes, such and such a proposition is true; but itsconverseisnottrue.”

124 Indiscussingimmediateinferenceswe“pursuethecontentofanenunciatedjudgment into its relations to judgments not yet uttered” (Lotze). Instead of“immediate inferences” Professor Bain prefers to speak of “equivalentpropositionalforms.”Itwillbefound,however,thatthenewpropositionsobtainedbyimmediateinferencearenotalwaysequivalenttotheoriginalproposition,e.g.,inconversionperaccidens.MissJonessuggests the termeduction asa synonymforimmediateinference(GeneralLogic,p.79);andshethendistinguishesbetweeneversionsandtransversions,aneversionbeinganeductionfromcategoricalformto categorical, or from hypothetical to hypothetical, &c., and transversion aneductionfromcategoricalformtoconditional,orfromconditional tocategorical,&c.Forthepresentweshallbeconcernedwitheversionsonly.


127Thesimplestformoflogicalconversion,andthatwhichisunderstoodin logicwhenwespeakofconversionwithout furtherqualification,maybedefinedasaprocessofimmediateinferenceinwhichfromagivenpropositionweinferanother,havingthepredicateoftheoriginalpropositionforsubject,anditssubjectforpredicate.Thus,givenapropositionhavingSforitssubjectandPforitspredicate,ourobjectintheprocessofconversionistoobtainbyimmediateinferenceanewpropositionhavingP for itssubjectandS for itspredicate. The original proposition may be called the convertend, and theinferredpropositiontheconverse.

Theprocesswillbevalidifthetwofollowingrulesareobserved: (1)The conversemust be the same in quality as the convertend (Rule ofQuality); (2)Notermmustbedistributedintheconverseunlessitwasdistributedintheconvertend(RuleofDistribution).

Applyingtheserulestothefourfundamentalformsofproposition,wehavethefollowingtable:—

Convertend. Converse.

AllSisP. A. SomePisS. I.SomeSisP. I. SomePisS. I.NoSisP. E. NoPisS. E.

SomeSisnotP. O. (None)

It is desirable at this stage briefly to call attention to a pointwhichwillreceive fuller consideration later on in connexion with the reading ofpropositions inextensionand intension,namely, that,generally speaking, inany judgmentwehavenaturallybefore themind theobjectsdenotedby the128 subject, but the qualities connoted by the predicate. In the process ofconverting a proposition, however, the extensive force of the predicate ismadeprominent,andanimportisgiventothepredicatesimilartothatofthesubject. At the same time the distribution of the predicate has to be madeexplicit in thought. It is inpassingfromthepredicative to theclass reading(e.g. from all men are mortal to all men are mortals), that the difficultysometimes found incorrectlyconvertingpropositionsprobablyconsists.We


shall at any rate dowell to recognise that conversion and other immediateinferencesusuallyinvolveadistinctmentalactoftheabovenature.

It follows from what has been said above that some propositions lendthemselvestotheprocessofconversionmuchmorereadilythanothers.Whenthepredicateofapropositionisasubstantivelittleornoeffortisrequiredinordertoconverttheproposition;moreeffortisnecessarywhenthepredicateis an adjective; and still more when in the original proposition the logicalpredicate is not expressed separately at all, as in propositions secundiadjacentis.Compareforpurposesofconversionthepropositions,Whalesaremammals,Lionsarecarnivorous,Astitchintimesavesnine. Insomecases,in consequence of the awkwardness of changing adjectives and verbalpredicates into substantives, theconversionofapropositionappears tobeaveryartificialproduction.125


97.SimpleConversionandConversionperaccidens.—Itwillbeobservedthat inthecaseofIandE, theconverse isof thesameformas theoriginalproposition;moreoverwedonotloseanypartoftheinformationgivenusbytheconvertend,andwecanpassbacktoitbyre-conversionoftheconverse.Theconvertendanditsconverseareaccordinglyequivalentpropositions.Theconversionundertheseconditionsissaidtobesimple.

In the case ofA, it is different; we cannot pass by immediate inferencefromAllSisPtoAllPisS,inasmuchasPisdistributedinthelatterofthesepropositionsbutundistributedintheformer.Hence,althoughwestartwithauniversal proposition, we obtain by conversion a particular 129 propositiononly,126andbynomeansofoperatingupon theconversecanweregain theoriginal proposition. The convertend and its converse are accordingly non-equivalentpropositions.Theconversioninthiscaseiscalledconversionperaccidens,127orconversionbylimitation.128

126 The failure to recognise or to remember that universal affirmativepropositionsarenotsimplyconvertibleisafertilesourceoffallacy.

127 TheconversionofAissaidbyManseltobecalledconversionperaccidens‘because it is not a conversion of the universal per se, but by reason of itscontaining the particular. For the proposition ‘Some B is A’ is primarily theconverseof‘SomeAisB,’secondarilyof‘AllAisB’”(Mansel’sAldrich,p.61).


ProfessorBaynesseemstodenythatthisisthecorrectexplanationoftheuseoftheterm (New Analytic of Logical Forms, p. 29); but however this may be, wecertainly need not regard the converse ofA as necessarily obtained through itssubaltern.ItispossibletoproceeddirectlyfromAllAisBtoSomeBisAwithouttheinterventionofSomeAisB.

128 Simpleconversionandconversionperaccidensarealsocalledrespectivelyconversiopuraandconversioimpura.CompareLotze,Logic,§79.

For concrete illustrations of the process of conversion we may take thepropositions,—A stitch in time saves nine; None but the brave deserve thefair.The firstof thesemaybewritten in the form,—Allstitches in timearethingsthatsaveninestitches.This,beinganAproposition,isonlyconvertibleper accidens, and we have for our converse,—Some things that save ninestitches are stitches in time. The second of the given propositions may bewritten,—Noonewhoisnotbraveisdeservingofthefair.This,beinganEproposition,maybeconvertedsimply,giving,Noonedeservingofthefairisnotbrave.Our resultsmaybeexpressed inamorenatural formas follows:Onewayofsavingninestitchesisbyastitchintime;Noonedeservingofthefaircanfailtobebrave.

No difficulty ought ever to be found in converting or performing otherimmediate inferences upon any given proposition when once it has beenbrought into the traditional logical form, its quantity and quality beingdetermined, its subject, copula, and predicate being definitely distinguishedfrom one another, and its predicate as well as its subject being read inextension. If, however, this rule is neglected, mistakes are pretty sure tofollow.

130 98. Inconvertibility of Particular Negative Propositions.—It followsimmediatelyfromtherulesofconversiongiveninsection96thatSomeS isnotPdoesnotadmitofordinaryconversion;forSwhichisundistributedintheconvertendwouldbecome thepredicateofanegativeproposition in theconverse, andwould therefore be distributed.129 Itwill be shewnpresently,however, thatalthoughweareunabletoinferanythingaboutP in thiscase,weareabletodrawaninferenceconcerningnot-P.

129 AsregardstheinconvertibilityofOseealsosections99and126.

Jevons considers that the fact that the particular negative proposition is


incapable of ordinary conversion “constitutes a blot in the ancient logic”(Studies in Deductive Logic, p. 37). There is, however, no sufficientjustification for this criticism.We shall find subsequently that just asmuchcanbeinferredfromtheparticularnegativeasfromtheparticularaffirmative(sincethelatterunliketheformerdoesnotadmitofcontraposition).Nologic,symbolicorother,canactuallyobtainmorefromthegiveninformationthantheancient logicdoes. Ithasbeensuggested thatwhatJevonsmeans is thatthe inconvertibility ofO results in a want of symmetry and that logiciansoughtspeciallytoaimatsymmetry.Withthislastcontentionwemayheartilyagree.Thewantofsymmetry,however,inthecasebeforeusisapparentonlyand results from takingan incompleteview. Itwill be found that symmetryreappearslateron.130

130 Seesections105,106.

99. Legitimacy of Conversion.—Aristotle proves the conversion of Eindirectly, as follows;131No S is P, therefore,No P is S ; for if not, SomeindividualP,sayQ,isS ;andhenceQisbothSandP ;butthisisinconsistentwiththeoriginalproposition.

131 “By the method called ἔκθεσις, i.e., by the exhibition of an individualinstance.”SeeMansel’sAldrich,pp.61,2.

HavingshewnthatthesimpleconversionofEislegitimate,wecanprovethattheconversionperaccidensofAisalsolegitimate.AllSisP,therefore,SomePisS ;for,ifnot,NoPisS,andtherefore(byconversion)NoSisP ;butthis131isinconsistentwiththeoriginalsupposition.ThelegitimacyofthesimpleconversionofIfollowssimilarly.

The above proof appears to involve nothing beyond the principles ofcontradiction and excluded middle. The proof itself, however, is notsatisfactory;foritpracticallyassumesthevalidityoftheveryprocessthatitseekstojustify,thatistosay,itassumestheequivalenceofthepropositionsSisQandQisS.

A better justification of the process of conversion may be obtained byconsideringtheclassrelationsinvolvedinthepropositionsconcerned.Thus,takinganEproposition,itisself-evidentthatifoneclassisentirelyexcludedfromanotherclass,thissecondclassisentirelyexcludedfromthefirst.132In


thecaseofanApropositionitisclearonreflectionthatthestatementAllSisP isconsistentwitheitheroftworelationsoftheclassesSandP,namely,SandPcoincident,orPcontainingSandmorebesides,andfurtherthattheseare the only two possible relations with which it is consistent. It is self-evident that ineachof thesecasesSomeP isS ; andhence the inferencebyconversionfromanApropositionisshewntobejustified.133InthecaseofanOproposition,ifweconsideralltherelationshipsofclassesinwhichitholdsgood,wefindthatnothingistrueofPintermsofSinallofthem.HenceOisinconvertible.134 The inconvertibility ofO can also be established 132 byshewingthatSomeSisnotP iscompatiblewitheveryoneof thefollowingpropositions—AllPisS,SomePisS,NoPisS,SomePisnotS.


132 ItisimpossibletoagreewithProfessorBain,whowouldestablishtherulesof conversion by a kind of inductive proof. He writes as follows:—“When weexaminecarefully thevariousprocesses inLogic,wefind themtobematerial totheverycore.TakeConversion.Howdoweknowthat,ifNoXisY,NoYisX?Byexaminingcases indetail,andfinding theequivalence tobe true.Obviousas theinferenceseemsonthemereformalground,wedonotcontentourselveswiththeformalaspect.Ifwedid,weshouldbeaslikelytosay,AllXisYgivesAllYisX ;we are prevented from this leap merely by the examination of cases” (Logic,Deduction,p.251).Butnoonewouldonreflectionmaintain it tobeself-evidentthat the simple conversionofA is legitimate; forwhen the case is put to uswerecogniseimmediatelythatthecontradictoryofAllPisSiscompatiblewithAllSisP.Ontheotherhand,noonecandenythatinthecaseofEthelegitimacyoftheprocessofconversionisself-evident.

133 Comparesection126,wherethisandothersimilarinferencesareillustratedbytheaidoftheEuleriandiagrams.

134 Again,comparesection126.

100. Table of Propositions connecting any two terms.—There are—connectinganytwotermsSandP—eightpropositionsof theformsA,E,I,O,namely,fourwithSassubject,andfourwithPassubject.Theresultsatwhichwehavearrivedconcerningtheconversionofpropositionsshewthatoftheseeight,thetwoEpropositionsareequivalenttooneanother,andthatthesameistrueofthetwoIpropositions,EandIbeingsimplyconvertible;alsothat these are the only equivalences obtainable. We have, therefore, thefollowingtableofpropositionsconnectinganytwotermsSandP:—

SaP,PaS,SeP=PeS,SiP=PiS,SoP,PoS.

ThepairofpropositionsSaPandPaSareindependent(seesection84);andthesameis trueof thepairsSoPandPoS,SaPandPoS,PaSandSoP.ThefirstpairtakentogetherindicatethattheclassesSandParecoextensive,andthey may be called complementary propositions. The second pair takentogether indicate that the classesS andP are neither coextensive nor eitherincluded within the other; they may be called sub-complementary


propositions.ThethirdpairtakentogetherindicatethattheclassSisincludedwithintheclassPbutthatitdoesnotexhaustthatclass;theymaybecalledcontra-complementary propositions. The fourth pair taken together indicatethattheclassPisincludedwithintheclassSbutthatitdoesnotexhaustthatclass;theyare,therefore,alsocontra-complementary.135

135 The new technical terms here introduced have been suggested by MrJohnson.

The above table will be supplemented in section 106 by a table ofpropositionsconnectinganytwotermsandtheir133contradictories,S,P,not-S,not-P. It will then be found that we have a symmetry that is at presentwanting.

101. The Obversion of Categorical Propositions.136—Obversion is aprocessofimmediateinferenceinwhichtheinferredproposition(orobverse),whilstretainingtheoriginalsubject,hasforitspredicatethecontradictoryofthe predicate of the original proposition (or obvertend). This process islegitimateforapropositionofanyformifatthesametimethequalityoftheproposition is changed. The inferred proposition is, moreover, in all casesequivalenttotheoriginalproposition,sothatwecanalwayspassbackfromtheobversetotheobvertend.

136 Theprocessofimmediateinferencediscussedinthissectionhasbeencalledbyagoodmanydifferentnames.Thetermobversion,whichisusedbyProfessorBain,isthemostconvenient.Othernameswhichhavebeenusedarepermutation(Fowler),aequipollence(Ueberweg),infinitation(Bowen),immediateinferencebyprivate conception (Jevons), contraversion (De Morgan), contraposition(Spalding). Professor Bain distinguishes between formal obversion andmaterialobversion.By formal obversion ismeant the kind of obversion discussed in theabove section, and this is the only kind of obversion that can properly berecognisedbytheformallogician.Materialobversion isdescribedas theprocessofmaking“obverse inferenceswhichare justifiedonlyonanexaminationof thematterof theproposition”(Logic,vol. i.,p.111);and the followingaregivenasexamples—“Warmth is agreeable; therefore, cold is disagreeable. War isproductive of evil; therefore, peace is productive of good. Knowledge is good;therefore, ignoranceisbad.”It isverydoubtful if thesearelegitimateinferences,formal or otherwise. The conclusions appear to require quite independentinvestigationstoestablishthem.Apartfromthis,however,itisamistaketoregardtheprocessasanalogoustoformalobversion.Inthelatter,theinferredpropositionhasthesamesubjectastheoriginalproposition,whilstitsqualityisdifferent;butneitheroftheseconditionsisfulfilledintheaboveexamples.Theprocessisreally


moreakintotheimmediateinferencepresentlytobediscussedunderthenameofinversion.

Wehavethefollowingtable:—

Obvertend. Obverse.

AllSisP. A. NoSisnot-P. ESomeSisP. I. SomeSisnotnot-P. O.NoSisP. E. AllSisnot-P. A.

SomeSisnotP. O. SomeSisnot-P. I.

134 Itwill beobserved that theobversionofAllS isP depends upon theprinciple of contradiction,which tells us that if anything isP then it is notnot-P;butthatwepassbackfromNoSisnot-PtoAllSisPbytheprincipleofexcludedmiddle,which tellsus that ifanything isnotnot-P then it isP.The remaining inferences by obversion also depend upon one or other ofthesetwoprinciples.

102.The Contraposition of Categorical Propositions.137—Contrapositionmaybedefined asa process of immediate inference inwhich from a givenproposition another proposition is inferred having for its subject thecontradictoryoftheoriginalpredicate.Thus,givenapropositionhavingSforitssubjectandPforitspredicate,weseektoobtainbyimmediateinferenceanewpropositionhavingnot-Pforitssubject.

137 Thisformofimmediateinferenceiscalledbysomelogiciansconversionbynegation ; Miss Jones suggests the name contraversion. More strictly we mightspeak of conversion by contraposition. The word contrapositive was used byBoethius for the opposite of a term (e.g.,not-A), the word contradictory beingconfined to propositional forms; and the passage fromAll S isP toAll not-P isnot-S was called Conversio per contrapositionem terminorum. In this usageBoethiuswasfollowedbythemedievallogicians.CompareMinto,Logic,pp.151,153.

Itwillbeobserved that in theabovedefinition it is leftanopenquestionwhether the contrapositive of a proposition has the original subject or thecontradictoryof theoriginal subject for its predicate; and everypropositionwhich admits of contraposition will accordingly have two contrapositives,eachofwhichistheobverseoftheother.Forexample,inthecaseofAllSisP there are the two formsNo not-P is S andAll not-P is not-S. For many


purposes the distinction may be practically neglected without risk ofconfusion. It will be observed, however, that when not-S is taken as thepredicate of the contrapositive, the quality of the original proposition ispreserved and there is greater symmetry.138 On the other hand, 135 if weregardcontrapositionascompoundedoutofobversionandconversioninthemannerindicatedinthefollowingparagraph,theformwithSaspredicateisthemore readilyobtained.Perhaps thebest solution (incases inwhich it isnecessary to mark the distinction) is to speak of the form with not-S aspredicate as the full contrapositive, and the formwithS as predicate as thepartialcontrapositive.139

138 The following is from Mansel’s Aldrich, p. 61,—“Conversion bycontraposition,whichisnotemployedbyAristotle,isgivenbyBoethiusinhisfirstbook,DeSyllogismoCategorico.HeisfollowedbyPetrusHispanus.Itshouldbeobserved,thattheoldlogicians,followingBoethius,maintainthatinconversionbycontraposition, as well as in the others, the quality should remain unchanged.Consequentlytheconverseof‘AllAisB’is‘Allnot-Bisnot-A,’andof‘SomeAisnotB,’‘Somenot-Bisnotnot-A.’Itissimpler,however,toconvertAintoE,andOintoI,(‘Nonot-BisA,’‘Somenot-BisA’),asisdonebyWallisandArchbishopWhately;andbeforeBoethiusbyApuleiusandCapella,whonoticetheconversion,but do not give it a name. The principle of this conversion may be found inAristotle,Top.II.8.1,thoughhedoesnotemployitforlogicalpurposes.”

139 In previous editions the form with S as predicate was called thecontrapositive, and the form with not-S as predicate was called the obvertedcontrapositive.

Thefollowingrulemaybeadoptedforobtainingthefullcontrapositiveofagiven proposition:—Obvert the original proposition, then convert thepropositionthusobtained,andthenoncemoreobvert.Forgivenapropositionwith S as subject and P as predicate, obversion will yield an equivalentpropositionwithS as subject andnot-P as predicate; the conversionof thiswill make not-P the subject and S the predicate; and a repetition of theprocessofobversionwillyieldapropositionwithnot-Passubjectandnot-Saspredicate.

Applyingthisrule,wehavethefollowingtable:—

OriginalProposition Obverse Partial

Contrapositive FullContrapositive

AllSisP. A. NoSisnot-P. E. Nonot-PisS. E. Allnot-Pisnot-S A.SomeSisnotnot-P.


SomeSisP. I. O. (None.) (None.)

NoSisP. E. AllSisnot-P. A. Somenot-PisS. I.

Somenot-Pisnotnot-S. O.

SomeSisnotP. O. SomeSisnot-P. I. Somenot-PisS.

I.Somenot-Pisnotnot-S.

O.

It will be observed that in the case of A andO, the contrapositive isequivalent to the original proposition, the quantity 136 being unchanged,whereasinthecaseofEwepassfromauniversaltoaparticular.140Inorderto emphasize this difference, and following the analogy of ordinaryconversion, the contraposition of A and O has been called simplecontraposition,andthatofEcontrapositionperaccidens.141

140 Inmosttext-books,nodefinitionofcontrapositionisgivenatall,anditmaybepointedoutthat,intheattempttogeneralisefromspecialexamples,Jevonsinhis Elementary Lessons in Logic involves himself in difficulties. For thecontrapositiveofAhegivesAllnot-P is not-S ;O he says has no contrapositive(butonlyaconversebynegation,Somenot-PisS);andforthecontrapositiveofEhegivesNoPisS.Itisimpossibletodiscoveranydefinitionofcontrapositionthatcan yield these results. Assuming that in contraposition the quality of theproposition is to remain unchanged as in Jevons’s contrapositive ofA, then thecontrapositiveofbothEandOisSomenot-Pisnotnot-S.

141 CompareUeberweg,Logic,§90.

That I has no contrapositive follows from the inconvertibility ofO. Forwhen Some S is P is obverted it becomes a particular negative, and theconversion of this proposition would be necessary in order to render thecontrapositionoftheoriginalpropositionpossible.

As regards theutilityof the investigationas to the inferences that canbedrawn from given propositions by the aid of contraposition, DeMorgan142pointsoutthattherecognitionthatEverynot-Pisnot-SfollowsfromEverySisP,whateverSandPmaystandfor,rendersunnecessarythespecialproofsthatEuclidgivesofcertainofhistheorems.143

142 SyllabusofLogic,p.32.143 Itwillbefoundthat,takingEuclid’sfirstbook,proposition6isobtainableby

contrapositionfromproposition18,and19from5and18combined;orthat5canbe obtained by contraposition from 19, and 18 from 6 and 19. Similar relationssubsistbetweenpropositions4,8,24,and25;and,again,betweenaxiom12andpropositions 16, 28, and 29.Other examplesmight be taken fromEuclid’s laterbooks.Insomeofthecasesthelogicalrelationsinwhichthepropositionsstandto


oneanotherareobvious;inothercasessomesupplementarystepsarenecessary.

InconsequenceofhisdislikeofnegativetermsSigwartregardsthepassagefromAllSisPtoNonot-PisSasanartificialperversion.ButherecognisesthevalueoftheinferencefromIfanythingisSitisPtoIfanythingisnotPitis not S.This distinction seems tobe littlemore thanverbal. It is to 137 beobservedthatwecanavoidtheuseofnegativetermswithouthavingrecourseto the conditional form of proposition: for example,Whatever is S is P,therefore,Whatever is not P is not S ; Anything that is S is P, therefore,AnythingthatisnotPisnotS.

103.TheInversionofCategoricalPropositions.—Indiscussingconversionandcontrapositionwehaveenquiredinwhatcasesitispossible,havinggivenapropositionwithSassubjectandPaspredicate, to infer(a)apropositionwith P as subject, (b) a proposition with not-P as subject. We may nowenquirefurtherinwhatcasesitispossibletoinfer(c)apropositionwithnot-Sassubject.

Ifsuchapropositioncanbeinferredatall,itwillbeobtainablebyacertaincombination of the more elementary processes of ordinary conversion andobversion.144 We will, therefore, take each of the fundamental forms ofpropositionandseewhatcanbe inferred (1)by firstconverting it,and thenperformingalternatelytheoperationsofobversionandconversion;(2)byfirstobvertingit,andthenperformingalternatelytheoperationsofconversionandobversion.Itwillbefoundthatineachcasetheprocesscanbecontinueduntilaparticularnegativepropositionisreachedwhoseturnitistobeconverted.

144 Itmightalsobeobtaineddirectly;bytheaid,forexample,ofEuler’scircles.Seethefollowingchapter.

(1)The results of performing alternately theprocesses of conversion andobversion,commencingwiththeformer,areasfollows:— (i) AllSisP, therefore(byconversion),SomePisS, therefore(byobversion),SomePisnotnot-S. Here comes the turn for conversion; but as we have to deal with anOproposition,wecanproceednofurther.

(ii) SomeSisP,


therefore(byconversion),SomePisS, therefore(byobversion),SomePisnotnot-S ;andagainwecangonofurther.138

(iii) NoSisP, therefore(byconversion),NoPisS, therefore(byobversion),AllPisnot-S, therefore(byconversion),Somenot-SisP, therefore(byobversion),Somenot-Sisnotnot-P. Inthiscaseeitherofthepropositionsinitalicsis theimmediateinferencethatwassought.

(iv) SomeSisnotP. Inthiscasewearenotableeventocommenceourseriesofoperations.

(2)The results of performing alternately theprocesses of conversion andobversion,commencingwiththelatter,areasfollows:— (i) AllSisP, therefore(byobversion),NoSisnot-P, therefore(byconversion),Nonot-PisS, therefore(byobversion),Allnot-Pisnot-S, therefore(byconversion),Somenot-Sisnot-P, therefore(byobversion),Somenot-SisnotP. Hereagainwehaveobtainedthedesiredform.

(ii) SomeSisP, therefore(byobversion),SomeSisnotnot-P.

(iii) NoSisP, therefore(byobversion),AllSisnot-P, therefore(byconversion),Somenot-PisS, therefore(byobversion),Somenot-Pisnotnot-S.

(iv) SomeSisnotP, therefore(byobversion),SomeSisnot-P, therefore(byconversion),Somenot-PisS, therefore(byobversion),Somenot-Pisnotnot-S.

Wecannowanswerthequestionwithwhichwecommencedthisenquiry.


The required proposition can be obtained only if the given proposition isuniversal;wethenhave,accordingasitisaffirmativeornegative,— AllSisP,therefore,Somenot-SisnotP(=Somenot-Sisnot-P);139 NoSisP,therefore,Somenot-SisP(=Somenot-Sisnotnot-P).

This form of immediate inference has been more or less casuallyrecognised by various logicians, without receiving any distinctive name.Sometimesithasbeenvaguelyclassedundercontraposition(compareJevons,ElementaryLessonsinLogic,pp.185,6),butitisreallyasfarremovedfromthe process to which that designation has been given as the latter is fromordinaryconversion.Theterminversionwassuggestedinanearliereditionofthiswork,andhassincebeenadoptedbysomeotherwriters.Inversionmaybe defined as a process of immediate inference in which from a givenproposition another proposition is inferred having for its subject thecontradictory of the original subject. Thus, given a proposition with S assubject andP as predicate, we obtain by inversion a new propositionwithnot-Sassubject.Theoriginalpropositionmaybecalledtheinvertend,andtheinferredpropositiontheinverse.

Intheabovedefinitionitisnotspecifiedwhethertheinverseistohaveforits predicateP or not-P. Hence two forms (each being the obverse of theother) have been obtained as in the case of contraposition. So far as it isnecessarytomarkthedistinction,wemayspeakoftheforminwhichPisthepredicateasthepartialinverse,andofthatinwhichnot-Pisthepredicateasthefullinverse.

104. The Validity of Inversion.—It will be remembered that we are atpresentworking on the assumption that each class represented by a simpletermexists in the universe of discourse,while at the same time it does notexhaust thatuniverse; inotherwords,weassumethatS,not-S,P,not-P,allrepresentexistingclasses.Thisassumption isperhapsspecially important inthe case of inversion, and it is connectedwith certain difficulties thatmayhavealreadyoccurredtothereader.InpassingfromAllSisP toitsinverseSomenot-SisnotP there isanapparent illicitprocess,which it isnotquiteeasy either to account for or explain away. For the term P, which isundistributed in the premiss, is distributed in the conclusion, and yet if the


universalvalidityofobversionand140conversionisgranted,itisimpossibletodetectanyflawintheargumentbywhichtheconclusionisreached.Itisintheassumptionoftheexistenceofthecontradictoryoftheoriginalpredicatethatanexplanationoftheapparentanomalymaybefound.Thatassumptionmaybeexpressed in the formSome thingsarenotP.TheconclusionSomenot-S is not P may accordingly be regarded as based on this premisscombinedwiththeexplicitpremissAllSisP ;anditwillbeobservedthat,intheadditionalpremiss,Pisdistributed.145

145 The question of the validity of inversion under other assumptionswill beconsideredinchapter8.

105.SummaryofResults.—Theresultsobtained in theprecedingsectionsaresummedupinthefollowingtable:—

A. E. I. O.i Originalproposition SaP SiP SeP SoPii Obverse SePʹ SoPʹ SaPʹ SiPʹiii Converse PiS PiS PeSiv ObvertedConverse PoSʹ PoSʹ PaSʹv PartialContrapositive146 PʹeS PʹiS PʹiSvi FullContrapositive146 PʹaSʹ PʹoSʹ PʹoSʹvii PartialInverse146 SʹoP SʹiPviii FullInverse146 SʹiPʹ SʹoPʹ

146 Inpreviouseditionswhatareherecalled thepartial contrapositiveand thefull contrapositive respectively were called the contrapositive and the obvertedcontrapositive;andwhatareherecalledthepartialinverseandthefullinversewerecalledtheinverseandtheobvertedinverse.

It may be pointed out that the following rules apply to all the aboveimmediateinferences:—141 Rule of Quality.—The total number of negatives admitted or omitted insubject,predicate,orcopulamustbeeven. Rules of Quantity.—If the new subject is S, the quantity may remainunchanged;ifSʹ,thequantitymustbedepressed;147ifP,thequantitymustbedepressedinAandO;ifPʹ,thequantitymustbedepressedinEandI.

147 Inspeakingofthequantityasdepressed,itismeantthatauniversalyieldsaparticular,andaparticularyieldsnothing.

106. Table of Propositions connecting any two terms and their


contradictories.—Takinganytwotermsandtheircontradictories,S,P,not-S,not-P,andcombiningtheminpairs,weobtainthirty-twopropositionsoftheformsA,E,I,O.Thefollowingtable,however,shewsthatonlyeightofthesethirty-twopropositionsarenon-equivalent.

(i) (ii) (iii) (iv)Universals

A SaP = SePʹ = PʹeS = PʹaSʹAʹ SʹaPʹ = SʹeP = PeSʹ = PaSE SaPʹ = SeP = PeS = PaSʹEʹ SʹaP = SʹePʹ = PʹeSʹ = PʹaS

ParticularsO SoP = SiPʹ = PʹiS = PʹoSʹOʹ SʹoPʹ = SʹiP = PiSʹ = PoSI SoPʹ = SiP = PiS = PoSʹIʹ SʹoP = SʹiPʹ = PʹiSʹ = PʹoS

Inthistable,columns(i)and(ii)containthepropositionsinwhichSorSʹissubject,andcolumns(iii)and(iv)thepropositionsinwhichPorPʹissubject.In columns (i) and (iv) we have the forms which admit of simplecontraposition(i.e.,AandO),andincolumns(ii)and(iii)thosewhichadmitof simpleconversion (i.e.,E and I). Contradictories are shewn by identicalplaces in the universal and particular rows. We pass from column (i) tocolumn (ii) by obversion; from column (ii) to column (iii) by simpleconversion;andfromcolumn(iii)tocolumn(iv)byobversion.

The forms in black type shew that we may take for our 142 eight non-equivalent propositions the four propositions connecting S and P, and asimilarsetconnectingnot-Sandnot-P.148Toestablishtheirnon-equivalencewe may proceed as follows: SaP and SeP are already known to be non-equivalent, and the same is true of SʹaPʹ and SʹePʹ ; but no universalproposition can yield a universal inverse; therefore, no one of these fourpropositionsisequivalenttoanyother.Again,SiPandSoParealreadyknowntobenon-equivalent,andthesameistrueofSʹiPʹandSʹoPʹ ;butnoparticularproposition has any inverse; therefore, no one of these propositions isequivalenttoanyother.Finally,nouniversalpropositioncanbeequivalentto


aparticularproposition.149148 TheformersetbeingdenotedbyA,E,I,O,thelattersetmaybedenotedby

Aʹ,Eʹ,Iʹ,Oʹ.149 MrsLaddFranklin,inanarticleonThePropositioninBaldwin’sDictionary

ofPhilosophyandPsychology,reachestheresultarrivedat inthissectionfromadifferentpointofview.MrsFranklinshewsthat,ifweexpresseverythingthatcanbe said in the form of existential propositions (that is, propositions affirming ordenying existence), it is at once evident that the actual number of differentstatementspossibleintermsofXandYandtheircontradictoriesxandy iseight.ForthecombinationsofXandYandtheircontradictoriesareXY,Xy,xY, xy, andwecanaffirmeachofthesecombinationstoexistortobenon-existent.Henceitisclearthateightdifferentstatementsoffactarepossible,andthattheseeightmustremaindifferent,nomatterwhattheforminwhichtheymaybeexpressed.

Itmaybeworthaddingthattheconditionalanddisjunctiveformsaswellasthecategoricalmayherebeincludedontheunderstandingthatallthepropositionsareinterpretedassertorically.Thus, thefour followingpropositionsare,on theaboveunderstanding,equivalenttooneanother:AllXisY(categorical);IfanythingisX,itisY(conditional);NothingisXy(existential);EverythingisxorY(disjunctive).

107.MutualRelations of the non-equivalentPropositions connecting anytwotermsandtheircontradictories.150—Wemaynowinvestigatethemutualrelationsofoureightnon-equivalentpropositions.SaP,SeP,SiP,SoPformanordinarysquareofopposition;andsodoSʹaPʹ,SʹePʹ,SʹiPʹ,SʹoPʹ.Referencetocolumns(iii)and(iv) in the tablewillshewfurther thatSaP,SʹePʹ,SʹiPʹ,SoPareequivalenttoanothersquareofopposition;andthatthesameistrueofSʹaPʹ,SeP,SiP,SʹoPʹ. This leaves only the following pairs unaccountedfor:143SaP,SʹaPʹ ;SeP,SʹePʹ ;SoP,SʹoPʹ ;SiP,SʹiPʹ ;SaP,SʹoPʹ ;SʹaPʹ,SoP ;SeP,SʹiPʹ ;SʹePʹ,SiP ;anditwillbefoundthatineachofthesecaseswehaveanindependentpair.

150 Thissectionmaybeomittedonafirstreading.

SaPandSʹaPʹ(whichareequivalenttoSaP,PaS,andalsotoPʹaSʹ,SʹaPʹ)takentogetherservetoidentifytheclassesSandP,andalsotheclassesSʹandPʹ. They are therefore complementary propositions, in accordance with thedefinitiongiveninsection100.Similarly,SePandSʹePʹ(whichareequivalentto SaPʹ, PʹaS, and also to PaSʹ, SʹaP) are complementary; they serve toidentifytheclassesSandPʹ,andalsotheclassesSʹandP.Itwillbeobservedthat the complementary of any universal proposition may be obtained byreplacingthesubjectandpredicaterespectivelybytheircontradictories.Anot


uncommon fallacy is the tacit substitution of the complementary of apropositionforthepropositionitself.

The complementary relation holds only between universals. Particularsbetweenwhichthereisananalogousrelation(thesubjectandpredicateoftheonebeingrespectivelythecontradictoriesofthesubjectandpredicateoftheother) will be found to be sub-complementary in accordance with thedefinition in section 100; this relation holds between SoP and SʹoPʹ, andbetweenSiPandSʹiPʹ.SoPandSʹoPʹ(whichareequivalenttoSoP,PoS,andalsotoPʹoSʹ,SʹoPʹ) indicatethat theclassesSandPareneithercoextensivenoreitherincludedwithintheother,andalsothatthesameistrueofSʹandPʹ ;SiP and SʹiPʹ (which are equivalent to SoPʹ,PʹoS, and also to PoSʹ, SʹoP)indicatethesamethingasregardsSandPʹ,SʹandP.

The four remaining pairs are contra-complementary, each pair servingconjointly to subordinate a certain class to a certain other class; or, rather,since each such subordination implies a supplementary subordination, wemay say that eachpair subordinates twoclasses to twoother classes.Thus,SaP andSʹoPʹ (which are equivalent toSaP,PoS, and also toPʹaSʹ,SʹoPʹ)takentogethershewthattheclassS iscontainedinbutdoesnotexhaust theclassP,andalsothattheclassPʹiscontainedinbutdoesnotexhausttheclassSʹ ;SʹaPʹandSoP(whichareequivalenttoSʹaPʹ,PʹoSʹ,andalsotoPaS,SoP)yieldthesameresultsasregardstheclassesSʹandPʹ,andtheclassesPandS ;SePandSʹiPʹ(whichareequivalent144toSaPʹ,PʹoS,andalsotoPaSʹ,SʹoP)asregardsSandPʹ,andPandSʹ ;andSʹePʹandSiP(whichareequivalenttoSʹaP,PoSʹ,andalsotoPʹaS,SoPʹ)asregardsSʹandP,PʹandS.

Denoting the complementaries ofA andE byAʹ andEʹ, and the sub-complementariesofIandObyIʹandOʹ, thevariousrelationsbetween thenon-equivalent propositions connecting any two terms and theircontradictoriesmaybeexhibitedinthefollowingoctagonofopposition:


Each of thedotted lines in the above takes the place of four connectinglines which are not filled in; for example, the dotted line marked asconnectingcontraries indicates the relationbetweenAandE,A andEʹ,AʹandE,AʹandEʹ.151


151 For the octagon of opposition in the form inwhich it is here given I amindebtedtoMrJohnson.

108. The Elimination of Negative Terms.152—The process of obversionenables us by the aid of negative terms to reduce all propositions to theaffirmative form; and the question may be 145 raised whether the variousprocesses of immediate inference and theuse,wherenecessary, of negativepropositionswillnotequallyenableustoeliminatenegativeterms.


Itisofcourseclearthatbymeansofobversionwecangetridofanegativetermoccurringasthepredicateofaproposition.Theproblemismoredifficultwhen the negative term occurs as subject, but in this case eliminationmaystillbepossible;forexample,SʹiP=PoS.Wemayevenbeabletogetridoftwonegative terms; forexample,SʹaPʹ=PaS. So long, however, aswe arelimitedtocategoricalpropositionsoftheordinarytypewecannoteliminateanegative term (without introducing another in its place)where such a termoccursassubjecteither(a)inauniversalaffirmativeoraparticularnegativewithapositivetermaspredicate,or(b)inauniversalnegativeoraparticularaffirmativewithanegativetermaspredicate.

Thevalidityoftheaboveresultsisatonceshewnbyreferencetothetableofequivalencesgiveninsection106.Atleastonepropositioninwhichthereis no negative term will be found in each line of equivalences except thefourthandtheeighth,whichareasfollows:

SʹaP = SʹePʹ = PʹeSʹ = PʹaS ;SʹoP = SʹiPʹ = PʹiSʹ = PʹoS.

InthesecaseswemayindeedgetridofSʹ(as,forexample,fromSʹaP),butit is only by introducingPʹ (thus, SʹaP =PʹaS); there is no getting rid ofnegative termsaltogether.Wemayhere referback to theresultsobtained insections100 and106;with two terms six non-equivalent propositionswereobtained, with two terms and their contradictories eight non-equivalentpropositions.Thegroundofthisdifferenceisnowmadeclear.

If, however, we are allowed to enlarge our scheme of propositions byrecognisingcertain additional types, and ifweworkon the assumption that


universalpropositionsareexistentiallynegativewhileparticularpropositionsare existentially affirmative,153 then negative terms may always beeliminated.146Thus,Nonot-Sisnot-PisequivalenttothestatementNothingisbothnot-Sandnot-P,andthisbecomesbyobversionEverythingiseitherSorP.Again,Somenot-Sisnot-PisequivalenttothestatementSomething isbothnot-Sandnot-P,andthisbecomesbyobversionSomethingisnoteitherSorP,or,asthispropositionmayalsobewritten,ThereissomethingbesidesSandP.Theeliminationofnegativetermshasnowbeenaccomplishedinallcases. It will be observed further that we now have eight non-equivalentpropositionscontainingonlySandP—namely,AllSisP,NoSisP,SomeSisP,SomeSisnotP,AllPisS,SomeP isnotS,Everything iseitherSorP,ThereissomethingbesidesSandP.

153 Itisnecessaryheretoanticipatetheresultsofadiscussionthatwillcomeatalaterstage.Seechapter8.

Following out this line of treatment, the table of equivalences given insection106mayberewrittenasfollows[columns(ii)and(iii)beingomitted,andcolumns(v)and(vi)takingtheirplaces]:

(i) (iv) (v) (vi)SaP = PʹaSʹ = NothingisSPʹ = EverythingisSʹorP.SʹaPʹ = PaS = NothingisSʹP = EverythingisSorPʹ.SaPʹ = PaSʹ = NothingisSP = EverythingisSʹorPʹ.SʹaP = PʹaS = NothingisSʹPʹ = EverythingisSorP.

SoP = PʹoSʹ = SomethingisSPʹ = Thereissomethingbesides

SʹandP.

SʹoPʹ = PoS = SomethingisSʹP = ThereissomethingbesidesS

andPʹ.

SoPʹ = PoSʹ = SomethingisSP = Thereissomethingbesides

SʹandPʹ.

SʹoP = PʹoS = SomethingisSʹPʹ = ThereissomethingbesidesS

andP.

Taking the propositions in two divisions of four sets each, the twodiagonalsfromlefttorightgivepropositionscontainingSandPonly.154

154 The first four propositions in column (v) may be expressed symbolicallySPʹ=0,&c.;thesecondfourSPʹ>0,&c.;thefirstfourincolumn(vi)Sʹ+P=1,


&c.;andthesecondfourSʹ+P<1,&c.;where1=theuniverseofdiscourse,and0=nonentity,i.e.,thecontradictoryoftheuniverseofdiscourse.Comparesection138.

147Theschemeofpropositionsgiven in thissectionmaybebrought intointeresting relation with the three fundamental laws of thought.155 Thescheme is based upon the recognition of the following propositional formsandtheircontradictories:

EverySisP ;

Everynot-Pisnot-S ;

NothingisbothSandnot-P ;

EverythingiseitherPornot-S ;andthesefourpropositionshavebeenshewntobeequivalenttooneanother.

155 CompareMrsLaddFranklininMind,January,1890,p.87.

IfintheabovepropositionswenowwriteSforP,wehavethefollowing:

EverySisS ;

Everynot-Sisnot-S ;

NothingisbothSandnot-S ;

EverythingiseitherSornot-S.

But the first two of these propositions express the law of identity, withpositiveandnegativetermsrespectively,thethirdisanexpressionofthelawofcontradiction,andthefourthofthelawofexcludedmiddle.Theschemeofpropositionswithwhichwehavebeendealingmay, therefore,besaid tobebased upon the recognition of just those propositional forms which arerequiredinordertoexpressthefundamentallawsofthought.

Sincethepropositionalformsinquestionhavebeenshewntobemutuallyequivalenttooneanother,thefurtherargumentmaysuggestitselfthatifthevalidityoftheimmediateinferencesinvolvedbegranted,thenitfollowsthatthe fundamental laws of thought have been shewn to bemutually inferablefromoneanother.Butitmay,ontheotherhand,beheldthatthisargumentisopentothechargeofinvolvingacirculusinprobandoonthegroundthatthevalidity of the immediate inferences themselves requires that the laws of


thoughtbefirstpostulatedasanantecedentcondition.

109. Other Immediate Inferences.—Some other commonly recognisedformsofimmediateinferencemaybebrieflytouchedupon.148

(1) Immediate inferences based on the square of opposition have beendiscussedintheprecedingchapter.

(2)Immediate inferenceby changeof relation is theprocesswherebywepassfromacategoricalpropositiontoaconditionaloradisjunctive,orfromaconditional to a disjunctive or a categorical, or from a disjunctive to acategoricaloraconditional.156Forexample,AllSisP,therefore,IfanythingisSitisP ;EverySisPorQ,therefore,AnySthatisnotPisQ.Referenceshavebeenalreadymadetoinferencessuchasthese,andtheywillbefurtherdiscussedlateron.

156 MissJonesspeaksofaninferenceofthiskindasatransversion.Seenote3onpage126.

(3)Immediate inferencebyaddeddeterminants isaprocessof immediateinferencewhichconsistsinlimitingboththesubjectandthepredicateoftheoriginalpropositionbymeansofthesamedeterminant.Forexample,—AllPis Q, therefore, All AP is AQ ; A negro is a fellow creature, therefore, Asuffering negro is a suffering fellow creature. The formal validity of thereasoning may be shewn as follows: AP is a subdivision of the class P,namely, that part of it which also belongs to the class A ; and, therefore,whateveristrueofthewholeofPmustbetrueofAP ;hence,giventhatAllPisQ,wecaninferthatAllAPisQ ;moreover,bythelawofidentity,AllAPisA ;therefore,AllAPisAQ.157

157 Itmustbeobserved,however,thatthevalidityofthisargumentrequiresanassumptioninregardto theexistential importofpropositions,whichdiffersfromthatwhichwehaveforthemostpartadopteduptothispoint.Ithastobeassumedthat universals do not imply the existence of their subjects. Otherwise thisinferencewouldnotbevalidinthecaseofnoPbeingA.Pmightexist,andallPmight beQ, but we could not pass to AP is AQ, since this would imply theexistenceofAP,whichwouldbeincorrect.Itisnecessarybrieflytocallattentiontotheaboveatthispoint,butouraimthroughalltheseearlierchaptershasbeentoavoidasfaraspossiblethevariouscomplicationsthatariseinconnexionwiththedifficultproblemofexistentialimport.

Theformalvalidityofimmediateinferencebyaddeddeterminantshasbeen


deniedonthegroundoftheobviousfallacyofarguingfromsuchapremissasanelephantisananimaltotheconclusionasmallelephantisasmallanimal,orfromsuchapremissascricketersarementotheconclusionpoorcricketersare poor men. In these cases, however, the fallacy really results from theambiguity of language, the added determinant 149 receiving a differentinterpretation when it qualifies the subject from that which it has when itqualifiesthepredicate.Atermofcomparisonlikesmallcanindeedhardlybesaid tohavean independent interpretation, its forcealwaysbeingrelative tosomeother termwithwhich it is conjoined.While then the inference in itssymbolicform(PisQ,therefore,APisAQ)isperfectlyvalid,itisspeciallynecessary to guard against fallacy in its use when significant terms areemployed.Allthatwehavetoinsistuponisthattheaddeddeterminantshallreceive the same interpretation in both subject and predicate. There is, forexample,nofallacyinthefollowing:Anelephantisananimal, therefore,Asmall elephant is an animal which is small compared with elephantsgenerally;Cricketersaremen,therefore,Poorcricketersaremenwhointheircapacityascricketersarepoor.

(4)Immediate inferencebycomplexconception isaprocessof immediateinferencewhich consists in employing the subject and the predicate of theoriginalpropositionaspartsofamorecomplexconception.Symbolicallywecanonlyexpressitsomewhatasfollows:PisQ,therefore,WhateverstandsinacertainrelationtoPstandsinthesamerelationtoQ.Thefollowingisaconcreteexample:Anelephantisananimal,therefore,theearofanelephantis the ear of an animal. A systematic treatment of this kind of inferencebelongstothespecialbranchofformallogicknownasthelogicofrelatives,anydetailedconsiderationofwhichisbeyondthescopeofthepresentwork.Attentionmay,however,becalledtothedangerofourcommittingafallacy,ifweperformtheprocesscarelessly.Forexample,ProtestantsareChristians,therefore,AmajorityofProtestantsareamajorityofChristians;Anegroisaman, therefore, thebest of negroes is thebest ofmen.The formerof thesefallaciesisakintothefallacyofcomposition(seesection11),sincewepassfromthedistributivetothecollectiveuseofaterm.

(5) Immediate inference by converse relation is a process of immediate


inference analogous to ordinary conversion but belonging to the logic ofrelatives. It consists in passing from a statement of the relation inwhichPstandstoQtoa150statementoftherelationinwhichQconsequentlystandstoP.The two terms are transposed and thewordbywhich their relation isexpressed is replaced by its correlative. For example,A is greater than B,therefore,BislessthanA ;AlexanderwasthesonofPhilip,therefore,PhilipwasthefatherofAlexander;Freedomissynonymouswithliberty,therefore,Libertyissynonymouswithfreedom.

Mansel gives the first two of the above as examples of materialconsequence as distinguished from formal consequence. “A MaterialConsequenceisdefinedbyAldrichtobeoneinwhichtheconclusionfollowsfromthepremissessolelybytheforceoftheterms.Thisinfactmeansfromsome understood Proposition or Propositions, connecting the terms, by theadditionofwhich themind isenabled to reduce theConsequence to logicalform……The failureof aMaterialConsequence takesplacewhenno suchconnexion exists between the terms as will warrant us in supplying thepremissesrequired;i.e.,whenoneormoreofthepremissessosuppliedwouldbefalse.ButtodeterminethispointisobviouslybeyondtheprovinceoftheLogician. For this reason, Material Consequence is rightly excluded fromLogic……Among thesematerial,and thereforeextralogical,Consequences,aretobeclassedthosewhichReidadducesascasesforwhichLogicdoesnotprovide; e.g., ‘Alexander was the son of Philip, therefore, Philip was thefatherofAlexander’;‘AisgreaterthanB,therefore,BislessthanA.’Inboththese it is ourmaterial knowledgeof the relations ‘father and son,’ ‘greaterandless,’thatenablesustomaketheinference”(Aldrich,p.199).

Thedistinctionbetweenwhatisformalandwhatismaterialisnotinrealityso simple or so absolute as is here implied.158 It is usual to recognise asformalonlythoserelationswhichcanbeexpressedbytheordinarycopulaisorisnot ;andthereisverygoodreasonforproceedinguponthisbasisinthegreater part of our logical discussions. No other relation is of the samefundamental importance or admits of an equally developed logicalsuperstructure.But it is important to recognise that there areother relationswhichmayremainthe151 samewhile the thingsrelatedvary;andwherever


this is the casewemay regard the relation as constituting the formand thethingsrelatedthematter.Accordinglywitheachsuchrelationwemayhaveadifferentformalsystem.Thelogicofrelativesdealswithsuchsystemsasareoutsidetheoneordinarilyrecognised.Eachimmediateinferencebyconverserelationwill, therefore,beformal in itsownparticularsystem.Thispoint isadmirably put by Miss Jones: “A proposition containing a relative termfurnishes—besides the ordinary immediate inferences—other immediateinferences to any one acquaintedwith the system towhich it refers. Theseinferencescannotbeeducedexceptbyapersonknowingthe‘system’;ontheother hand, no knowledge is needed of the objects referred to, except aknowledgeoftheirplaceinthesystem,andthisknowledgeisinmanycasescoextensive with ordinary intelligence; consider, e.g., the relations ofmagnitude of objects in space, of the successive parts of time, of familyconnexions,ofnumber”(GeneralLogic,p.34).

158 Comparesection2.

(6) Immediate inference by modal consequence or, as it is also called,inference by change of modality, is somewhat analogous to subalterninference.Itconsists innothingmorethanweakeningastatement inrespectofitsmodality;andhenceitisneverpossibletopassbackfromtheinferredtothe original proposition.Thus, from thevalidity of the apodeictic judgmentwecanpasstothevalidityof theassertoric,andfromthat tothevalidityoftheproblematic;butnotviceversâ.Ontheotherhand,fromtheinvalidityoftheproblematicjudgmentwecanpasstotheinvalidityoftheassertoric,andfromthattotheinvalidityoftheapodeictic;butagainnotviceversâ.159

159 CompareUeberweg,Logic,§98.

110.Reductionofimmediateinferencestothemediateform160—Immediateinference has been defined as the inference of a proposition from a singleotherproposition;mediateinference,ontheotherhand,istheinferenceofapropositionfromatleasttwootherpropositions.

160 StudentswhohavenotalreadyatechnicalknowledgeofthesyllogismmayomitthissectionuntiltheyhavereadtheearlierchaptersofPartIII.

We may briefly consider various ways of establishing the validity ofimmediateinferencesbymeansofmediateinferences.


152 (1) One of the old Greek logicians, Alexander of Aphrodisias,establishestheconversionofEbymeansofasyllogisminFerio.

NoSisP,therefore, NoPisS ;

for, ifnot, thenby the lawof contradiction,SomeP isS ; andwehave thissyllogism,—

NoSisP,SomePisS,

therefore, SomePisnotP,

areductioadabsurdum.161161 CompareMansel’sAldrich,p.62.TheconversionofAandtheconversionof

Imaybeestablishedsimilarly.

(2) It may be plausibly maintained that in Aristotle’s proof of theconversionofE(giveninsection99),thereisanimplicitsyllogism:namely,—QisP,QisS,therefore,SomeSisP.

(3)ThecontrapositionofAmaybeestablishedbymeansofasyllogisminCamestresasfollows:—

Given AllSisP, wehavealso Nonot-PisP, bythelawofcontradiction, therefore, Nonot-PisS.162

162 Similarly,grantingthevalidityofobversion,thecontrapositionofOmaybeestablishedbyasyllogisminDatisiasfollows:—

GivenSomeSisnotP,thenwehave

AllSisS, bythelawofidentity,and SomeSisnot-P, byobversionofthegivenproposition,

therefore, Somenot-PisS.

It will be found that, adopting the same method, the contraposition of E isyieldedbyasyllogisminDarapti.

(4)WemightalsoobtainthecontrapositiveofAllSisPasfollows:—

Bythelawofexcludedmiddle,Allnot-PisSornot-S,and,byhypothesis,AllSisP,


therefore, Allnot-PisPornot-S ;but,bythelawofcontradiction, Nonot-PisP,

therefore, Allnot-Pisnot-S.163163 Compare Jevons, Principles of Science, chapter 6, § 2; and Studies in

DeductiveLogic,p.44.

153(5)ThecontrapositionofAmayalsobeestablishedindirectlybymeansofasyllogisminDarii:—

AllSisP,therefore, Nonot-PisS ;

for,ifnot,Somenot-PisS ;andwehavethefollowingsyllogism,—

AllSisP,Somenot-PisS,

therefore, Somenot-PisP,

whichisabsurd.164164 Compare De Morgan, Formal Logic, p. 25. Granting the validity of

obversion,thecontrapositionofEandthecontrapositionofOmaybeestablishedsimilarly.

All the above are interesting, as illustrating the processes of immediateinference; but regarded as proofs they labour under the disadvantage ofdeducingthelesscomplexbymeansofthemorecomplex.

EXERCISES.

111. Give all the logical opposites of the proposition,—Some richmen are virtuous; and also theconverseofthecontraryofitscontradictory.Howisthelatterdirectlyrelatedtothegivenproposition? Doesitfollowthatapropositionadmitsofsimpleconversionbecauseitspredicateisdistributed?[K.]

112.Pointoutanyambiguitiesinthefollowingpropositions,andgivethecontradictoryand(wherepossible)theconverseofeachofthem:—(i)Someofthecandidateshavebeensuccessful;(ii)Allarenothappythatseemso;(iii)Allthefishweighedfivepounds.[K.]

113.Stateinlogicalformandconvert thefollowingpropositions:—(a)Hejestsatscarswhoneverfeltawound;(b)Axiomsareself-evident;(c)NativesalonecanstandtheclimateofAfrica;(d)NotoneoftheGreeksatThermopylaeescaped;(e)Allthatglittersisnotgold.[O.]


114. “The angles at the base of an isosceles triangle are equal.”What can be inferred from thispropositionbyobversion,conversion,andcontrapositionrespectively?[L.]

154115.Givetheobverse,thecontrapositive,andtheinverseofeachofthefollowingpropositions:—Thevirtuousalonearetrulynoble;NoAtheniansareHelots.[M.]

116.Give thecontrapositiveand(wherepossible) the inverseof thefollowingpropositions:—(i)Astitchintimesavesnine;(ii)Nonebutthebravedeservethefair;(iii)Blessedarethepeacemakers;(iv)Thingsequaltothesamethingareequaltooneanother;(v)Noteverytalewehearistobebelieved.[K.]

117.Ifitisfalsethat“Notonlythevirtuousarehappy,”whatcanweinfer(a)withregardtothenon-virtuous,(b)withregardtothenon-happy?[J.]

118.Writedownthecontradictory,andalso—wherepossible—theconverse,thecontrapositive,andtheinverseofeachofthefollowingpropositions: Abirdinthehandisworthtwointhebush; Nounjustactsareexpedient; Allarenotsaintsthatgotochurch.[K.]

119.Givethecontrapositiveandtheinverseofeachofthefollowingpropositions,—Theyneverfailwhodieinagreatcause;WhomtheGodslovedieyoung. IfAiseitherBorelsebothCandD,whatdoweknowaboutthatwhichisnotD?[K.]

120. Take the following propositions in pairs, and in regard to each pair state whether the twopropositions are consistent or inconsistentwith each other; in the former case, state furtherwhethereitherpropositioncanbeinferredfromtheother,and,ifitcanbe,pointoutthenatureoftheinference;inthelattercase,statewhetheritispossibleforboththepropositionstobefalse:—(a)AllSisP ;(b)Allnot-SisP ;(c)NoPisS ;(d)Somenot-PisS.[K.]

121.Transformthefollowingpropositionsinsuchawaythat,withoutlosinganyoftheirforce,theymayallhave the samesubjectand the samepredicate:—Nonot-P isS ;AllP is not-S ;SomeP is S ;Somenot-Pisnotnot-S.[K.]

122.Describethelogicalrelations,ifany,betweeneachofthefollowingpropositions,andeachoftheothers:— (i)Therearenoinorganicsubstanceswhichdonotcontaincarbon;155 (ii)Allorganicsubstancescontaincarbon; (iii)Somesubstancesnotcontainingcarbonareorganic; (iv)Someinorganicsubstancesdonotcontaincarbon.[C.]

123.“Allthatlovevirtueloveangling.” Arrangethefollowingpropositionsinthethreefollowinggroups:—(α)thosewhichcanbeinferredfromtheaboveproposition;(β)thosewhichareconsistentwithit,butwhichcannotbeinferredfromit;(γ)thosewhichareinconsistentwithit. (i)Nonethatlovenotvirtueloveangling. (ii)Allthatloveanglinglovevirtue. (iii)Allthatlovenotanglinglovevirtue. (iv)Nonethatlovenotanglinglovevirtue. (v)Somethatlovenotvirtueloveangling. (vi)Somethatlovenotvirtuelovenotangling (vii)Somethatlovenotanglinglovevirtue.


(viii)Somethatlovenotanglinglovenotvirtue.[K.]

124.Determinethelogicalrelationbetweeneachpairofthefollowingpropositions:— (1)Allcrystalsaresolids. (2)Somesolidsarenotcrystals. (3)Somenotcrystalsarenotsolids. (4)Nocrystalsarenotsolids. (5)Somesolidsarecrystals. (6)Somenotsolidsarenotcrystals. (7)Allsolidsarecrystals.[L.]


CHAPTERV.

THEDIAGRAMMATICREPRESENTATIONOFPROPOSITIONS.

125. The use of Diagrams in Logic.—In representing propositions bygeometrical diagrams, our object is not that we may have a new set ofsymbols, but that the relation between the subject and predicate of aproposition may be exhibited by means of a sensible representation, thesignification ofwhich is clear at a glance.Hence the first requirement thatoughttobesatisfiedbyanydiagrammaticschemeisthattheinterpretationofthe diagrams should be intuitively obvious, as soon as the principle uponwhichtheyarebasedhasbeenexplained.165

165 Hamilton’s “geometric scheme,” which he himself describes as “easy,simple, compendious, all-sufficient, consistent, manifest, precise, complete”(Logic, II. p. 475), fails to satisfy this condition. He represents an affirmativecopulabyahorizontaltaperingline( ),thebroadendofwhichistowardsthesubject.Negationismarkedbyaperpendicularlinecrossingthehorizontalone

( ). A colon (:) placed at either end of the copula indicates that thecorrespondingtermisdistributed;acomma(,)thatitisundistributed.Thus,forAllSisPwehave,—

S: ,P;

andsimilarlyfortheotherpropositions.

DrVenn rightly observes that this scheme is purely symbolical, and does notdeservetorankasadiagrammaticschemeatall.Thereisclearlynothinginthetwoendsof awedge to suggest subjects andpredicates,or in a colonandcomma tosuggest distribution and non-distribution” (Symbolic Logic, p. 432). Hamilton’sschememaycertainlyberejectedasvalueless.TheschemesofEulerandLambertbelongtoanaltogetherdifferentcategory.

A second essential requirement is that the diagrams should be adequate;thatistosay,theyshouldgiveacomplete,and157notapartial,representationoftherelationswhichtheyareintendedtoindicate.Hamilton’suseofEuler’sdiagrams, as described in the following section, will serve to illustrate thefailuretosatisfythisrequirement.


In the thirdplace, thediagrams shouldbe capable of representing all thepropositionalformsrecognisedinthescheduleofpropositionswhicharetobeillustrated,e.g.,particularsaswellasuniversal.Oneschemeofdiagramsmay,however, be better suited for one purpose, and another scheme for anotherpurpose.ItwillbefoundthatDrVenn’sdiagrams,tobedescribedpresently,are not quite so well adapted to the representation of particulars as ofuniversals.

Lastly, it is advantageous that a diagrammatic scheme should be as littlecumbrousaspossiblewhenitisdesiredtorepresenttwoormorepropositionsincombinationwithoneanother.ThisistheweakpointofEuler’smethod.Ascheme of diagramsmay, however, serve a very useful function inmakingclearthefullforceofindividualpropositions,evenwhenitisnotwelladaptedfortherepresentationofcombinedpropositions.

Afurtherrequirementissometimesadded,namely,thateachpropositionalform should be represented by a single diagram, not by a set of alternativediagrams.Thisis,however,bynomeansessential.Forifweadoptascheduleofpropositionssomeofwhichyieldonlyanindeterminaterelationinrespectof extensionbetween the terms involved, it is important that this shouldbeclearlybroughtout,andasetofalternativediagramsmaybespeciallyhelpfulfor the purpose. This point will be illustrated, with reference to Euler’sdiagrams,inthefollowingsection.166

166 Itmustbeborneinmindthatinalltheschemesdescribedinthischaptertheterms of the propositions which are represented diagrammatically are taken inextension,notinintension.

126.Euler’sDiagrams.—Wemay beginwith thewell-known scheme ofdiagrams, which was first expounded by the Swiss mathematician andlogician,LeonhardEuler,andwhichisusuallycalledafterhisname.167

167 Euler lived from 1707 to 1783. His diagrammatic scheme is given in hisLettresàunePrincessed’Allemagne(Letters102to105).

158Representing the individuals included inanyclass,ordenotedbyanyname,byacircle,itwillbeobviousthatthefivefollowingdiagramsrepresentallpossiblerelationsbetweenanytwoclasses:—


Theforceofthedifferentpropositionalformsistoexcludeoneormoreofthesepossibilities. AllSisPlimitsustooneofthetwoα,β ; SomeSisPtooneofthefourα,β,γ,δ ; NoSisPtoε ; SomeSisnotPtooneofthethreeγ,δ,ε.

Itwillbeobservedthat there isgreatwantofsymmetryin thenumberofcirclescorrespondingtothedifferentpropositionalforms;alsothatthereisanapparent inequality in theamountof informationgivenbyAandbyE, andagain by I and byO. We shall find that these anomalies disappear whenaccountistakenofnegativeterms.

ItismostmisleadingtoattempttorepresentAllSisPbyasinglepairofcircles,thus

orSomeSisPbyasinglepair,thus

159 for in each case the proposition really leaves uswith other alternatives.


This method of employing the diagrams has, however, been adopted by agoodmany logicianswho have used them, includingSirWilliamHamilton(Logic,I.p.255),andProfessorJevons(ElementaryLessonsinLogic,pp.72to 75); and the attempt at such simplification has brought their use intoundeserved disrepute. Thus, Dr Venn remarks, “The common practice,adopted in so many manuals, of appealing to these diagrams—Euleriandiagrams as they areoften called—seems tomeveryquestionable.TheoldfourpropositionsA,E,I,O,donotexactlycorrespondtothefivediagrams,andconsequentlynoneofthemoodsinthesyllogismcaninstrictproprietyberepresentedbythesediagrams”(SymbolicLogic,pp.15,16;comparealsopp.424,425).Thiscriticism,whileperfectlysoundasregardstheuseofEuler’scircles byHamilton and Jevons, losesmost of its force if the diagrams areemployedwithdueprecautions.Itistruethatthediagramsbecomesomewhatcumbrous in relation to the syllogism; but the logical force of propositionsand the logical relationsbetweenpropositionscan inmany respectsbewellillustratedbytheiraid.Thus,theymaybeemployed:—

(1)Toillustratethedistributionofthepredicateinaproposition.Inthecaseof each of the four fundamental propositionswemay shade the part of thepredicateconcerningwhichinformationisgivenus.

Wethenhave,—

160


Wesee thatwithAandI,onlypartofP is in someof thecases shaded;whereaswithEandO,thewholeofPisineverycaseshaded;anditisthusmade clear that negative propositions distribute, while affirmativepropositionsdonotdistribute,theirpredicates.

(2) To illustrate the opposition of propositions. Comparing twocontradictorypropositions,e.g.,A andO,we see that theyhaveno case incommon, but that between them they exhaust all possible cases.Hence thetruth,thattwocontradictorypropositionscannotbetruetogetherbutthatoneof them must be true, is brought home to us under a new aspect. Again,comparingtwosubalternpropositions,e.g.,AandI,wenoticethattheformergivesusall theinformationgivenbythelatterandsomethingmore,sinceitstillfurtherlimitsthepossibilities.Theotherrelationsinvolvedinthedoctrineofoppositionmaybeillustratedsimilarly.

(3)Toillustratetheconversionofpropositions.ThusitismadeclearbythediagramshowitisthatAadmitsonlyofconversionperaccidens.AllSisPlimitsustooneorotherofthefollowing,—

What thendoweknowofP? In the first casewehaveAllP isS, in thesecondSomePisS ;andsinceweareignorantastowhichofthetwocasesholdsgood,wecanonlystatewhatiscommontothemboth,namely,SomePisS.


Again, it ismadeclearhow it is thatO is inconvertible.SomeS isnotPlimitsustooneorotherofthefollowing,—

161 What then do we know concerning P? The three cases give usrespectively,—(i)AllPisS ;(ii)SomePisSandSomePisnotS ;(iii)NoPisS.But (i) and (iii) are inconsistentwith one another.Hencenothing canbeaffirmedofPthatistrueinallthreecasesindifferently.

(4) To illustrate the more complicated forms of immediate inference.Taking,forexample,thepropositionAllSisP,wemayask,Whatdoesthisenableustoassertaboutnot-Pandnot-Srespectively?Wehaveoneorotherofthesecases,—

Asregardsnot-P,theseyieldrespectively(i)Nonot-PisS ;(ii)Nonot-PisS.Andthusweobtainthecontrapositiveofthegivenproposition.

Asregardsnot-S,wehave(i)Nonot-SisP,(ii)Somenot-SisPandsomenot-SisnotP.168HenceineithercasewemayinferSomenot-SisnotP.


168 It isassumed in theuseofEuler’sdiagrams thatSandP bothexist in theuniverse of discourse, while neither of them exhausts that universe. Thisassumptionisthesameasthatuponwhichourtreatmentofimmediateinferencesintheprecedingchapterhasbeenbased.

E,I,Omaybedealtwithsimilarly.

(5) To illustrate the joint force of a pair of complementary or contra-complementary or sub-complementary propositions (compare section 100).Thus, thepairofcomplementarypropositions,SaPandPaS, taken together,limitusto

Similarly the pair of contra-complementary propositions, SaP and PoS,limit us to the relation marked β on page 158; and the pair of contra-complementary propositions,SoP and 162PaS, to γ ; while the pair of sub-complementarypropositions,SoPandPoS,giveusachoicebetweenδandε.

Theapplicationofthediagramstosyllogisticreasoningswillbeconsideredinasubsequentchapter.

Withregardtoalltheabove,itmaybesaidthattheuseofthecirclesgivesusnothingthatcouldnoteasilyhavebeenobtainedindependently.Thisisofcourse true; but no one, who has had experience of the difficulty that issometimes found by students in properly understanding the elementaryprinciples of formal logic, and especially in dealing with immediateinferences,will despise anymeans of illustrating afresh the old truths, andpresentingthemunderanewaspect.

The fact that we have not a single pair of circles corresponding to eachfundamental form of proposition is fatal if we wish to illustrate anycomplicatedtrainofreasoninginthisway;butinindicatingtherealnatureoftheinformationgivenbythepropositionsthemselves,itisratheranadvantagethan otherwise, inasmuch as it shews how limited in some cases thisinformationactuallyis.169

169 Dr Venn writes in criticism of Euler’s scheme, “A fourfold scheme of


propositionswillnotveryconvenientlyfitinwithafivefoldschemeofdiagrams…What the five diagrams are competent to illustrate is the actual relation of theclasses,notourpossiblyimperfectknowledgeofthatrelation”(EmpiricalLogic,p.229). The reply to this criticism is that inasmuch as the fourfold scheme ofpropositionsgivesbutanimperfectknowledgeoftheactualrelationoftheclassesdenotedbytheterms,theEuleriandiagramsarespeciallyvaluableinmakingthisclearandunmistakeable.Bytheaidofdottedlinesitisindeedpossibletorepresenteach proposition by a single Eulerian figure; but the diagrams then become somuchmoredifficulttointerpretthatthelossisconsiderablygreaterthanthegain.The first and second of the following diagrams are borrowed from Ueberweg(Logic,§71). In thecaseofO,Ueberweg’sdiagramisrathercomplicated;andIhavesubstitutedasimplerone.


InthelastofthesediagramswegetthethreecasesyieldedbyanOpropositionby(1)fillinginthedottedlinetotheleftandstrikingout theother,(2)fillinginbothdottedlines,(3)fillinginthedottedlinetotherightandstrikingouttheother.Thesethreecasesarerespectivelythosemarkedγ,δ,ε,onpage158.

163127.Lambert’sDiagrams.—A scheme of diagramswas employed byLambert170inwhichhorizontalstraightlinestaketheplaceofEuler’scircles.Theextensionofatermisrepresentedbyahorizontalstraightline,andsofaras two such lines overlap it is indicated that the corresponding classes arecoincident,whilesofarastheydonotoverlaptheseclassesareshewntobemutuallyexclusive.Boththeabsoluteandtherelativelengthofthelinesisofcoursearbitraryandimmaterial.

170 JohannHeinrichLambertwasaGermanphilosopherandmathematicianwholived from1728 to1777.HisNeuesOrganonwas published atLeipzig in 1768.Lambert’sowndiagrammaticschemedifferssomewhatfrombothofthosegiveninthe text; but it very closely resembles theone inwhichportionsof the lines aredotted. The modifications in the text have been introduced in order to obviatecertaindifficultiesinvolvedinLambert’sowndiagrams.Seenote2onpage165.

WemayfirstshewhowLambert’slinesmaybeusedinsuchamannerastobe precisely analogous to Euler’s circles. 164 Thus, the four fundamental


propositionsmayberepresentedasfollows:—

ThesediagramsoccupylessspacethanEuler’scircles.Buttheyseemalsotobelessintuitivelyclearandlesssuggestive.Thedifferentcasestooarelessmarkedly distinct from one another. It is probable that one would inconsequencebemoreliabletoerrorinemployingthem.

Thedifferentcasesmay,however,becombinedbytheuseofdottedlinesso as to yield but a single diagram for each proposition much moresatisfactorilythaninEuler’sscheme.Thus,AllSisPmayberepresentedbythediagram

wherethedottedlineindicatesthatweareuncertainastowhetherthereisoris not anyP which is not S.We obviously get two cases according as westrike out the dots or fill them in, and these are the two cases previouslyshewntobecompatiblewithanAproposition.

Again,SomeSisPmayberepresentedbythediagram


andherewegetthefourcasespreviouslygivenforanI165propositionby(a)fillinginthedotstotheleftandstrikingoutthosetotheright,(b)fillinginallthedots,(c)strikingthemallout,(d)fillinginthosetotherightandstrikingoutthosetotheleft.

Twocompleteschemesofdiagramsmaybeconstructedonthisplan,inoneof which no part of any S line, and in the other no part of anyP line, isdotted.171 These two schemes are given below to the left and rightrespectivelyofthepropositionalformsthemselves.

171 Itisimportanttogiveboththeseschemesasitwillbefoundthatneitheroneof them will by itself suffice when this method is used for illustrating thesyllogism.ForobviousreasonstheEdiagramisthesameinbothschemes.

ItmustbeunderstoodthatthetwodiagramsgivenaboveinthecasesofA,I,andOarealternative in thesense thatwemayselectwhichweplease to


representourproposition;buteitherrepresentsitcompletely.

We shall find later on that for the purpose of illustrating the syllogisticmoods,Lambert’smethod isagooddeal lesscumbrous thanEuler’s.172AnadaptationofLambert’sdiagramsinwhichthecontradictoriesofSandPareintroduced aswell 166 asS andP themselveswill be given in section 131.Thismoreelaboratedschemewillbefoundusefulforillustratingthevariousprocessesofimmediateinference.

172 DrVenn(SymbolicLogic,p.432)remarks,“AsawholeLambert’sschemeseemstomedistinctlyinferiortotheschemeofEuler,andhasinconsequencebeenvery little employedbyother logicians.”The criticismoffered in support of thisstatementisdirectedchieflyagainstLambert’sownrepresentationoftheparticularaffirmativeproposition,namely,

This diagram certainly seems as appropriate to O as it does to I; but themodification introduced in the text,and indeedsuggestedbyDrVennhimself, isnotopentoasimilarobjection.

128.DrVenn’sDiagrams.—In thediagrammaticschemeemployedbyDrVenn(SymbolicLogic,chapter5)thediagram

does not itself represent any proposition, but the framework into whichpropositionsmaybefitted.Denotingnot-SbySʹandwhatisbothSandPbySP,&c., it isclear thateverythingmustbecontained inoneorotherof thefour classes SP, SPʹ, SʹP, SʹPʹ ; and the above diagram shews fourcompartments (one being that which lies outside both the circles)corresponding to these four classes. Every universal proposition denies theexistence of one or more of such classes, and it may therefore bediagrammaticallyrepresentedbyshadingoutthecorrespondingcompartmentor compartments. Thus, All S is P, which denies the existence of SPʹ, isrepresentedby


NoSisPby

167Withthreetermswehavethreecirclesandeightcompartments,thus,—

AllSisPorQisrepresentedby

AllSisPandQby

It is in cases involving three or more terms that the advantage of thisscheme over the Eulerian scheme is most manifest. The diagrams are not,


however,quitesowelladaptedtothecaseofparticularpropositions.DrVenn(inMind,1883,pp.599,600)suggests thatwemightdrawabaracross thecompartmentdeclaredtobesavedbyaparticularproposition;173thus,SomeSisPwouldberepresentedbydrawingabaracrosstheSPcompartment.Thisplan can beworked out satisfactorily; but in representing a combination ofpropositions in this way special care is needed in the interpretation of thediagrams.Forexample, ifwehave thediagramfor three termsS,P,Q,andaregivenSomeSisP,168wedonotknowthatboththecompartmentsSPQ,SPQʹ, are to be saved, and in a case like this a bar drawn across the SPcompartmentisinsomedangerofmisinterpretation.

173 DrVenn’sschemediffersfromtheschemesofEulerandLambert,inthatitisnotbasedupontheassumptionthatourtermsandtheircontradictoriesallrepresentexistingclasses.Itinvolves,however,thedoctrinethatparticularsareexistentiallyaffirmative,whileuniversalsareexistentiallynegative.

129.ExpressionofthepossiblerelationsbetweenanytwoclassesbymeansofthepropositionalformsA,E,I,O.—AnyinformationgivenwithrespecttotwoclasseslimitsthepossiblerelationsbetweenthemtosomethinglessthanthefiveàprioripossibilitiesindicateddiagrammaticallybyEuler’scirclesasgivenat thebeginningofsection126. Itwillbeuseful toenquirehowsuchinformation may in all cases be expressed by means of the propositionalformsA,E,I,O.

Thefiverelationsmay,asbefore,bedesignatedrespectivelyα,β,γ,δ,ε.174Informationisgivenwhenthepossibilityofoneormoreoftheseisdenied;inother words, when we are limited to one, two, three, or four of them. Letlimitationtoα,orβ,theexclusionofγ,δ,εbedenotedbyα,β ;limitationtoα,β,orγ(i.e.,theexclusionofδandε)byα,β,γ ;andsoon.

174 Thus, the classes being S and P, α denotes that S and P are whollycoincident;β thatP containsS andmore besides;β thatS containsP andmorebesides;δ thatSandP overlap each other, but that each includes something notincludedbytheother;εthatSandPhavenothingwhateverincommon.

In seeking to express our information by means of the four ordinarypropositionalforms,wefindthatsometimesasinglepropositionwillsufficeforourpurpose;thusα,βisexpressedbyAllSisP.Sometimeswerequireacombination of propositions; thus α is expressed by the pair of


complementary propositions All S is P and all P is S, (since all S is Pexcludesγ,δ,ε,andallPisSfurtherexcludesβ).Someothercasesaremorecomplicated; thus thefact thatweare limited toαorδcannotbeexpressedmoresimplythanbysaying,EitherAllSisPandallPisS,orelseSomeSisP,someSisnotP,andsomePisnotS.

LetA=AllSisP,A1=AllPisS,andsimilarlyfortheotherpropositions.AlsoletAA1=AllSisPandallPisS,&c.Thenthefollowingisaschemeforallpossiblecases:—169

Limitationto denotedby Limitationto denotedbyα AA1 α,β,γ AorA1

β AO1 α,β,δ AorIO1

γ A1O α,β,ε AorE

δ IOO1 α,γ,δ A1orIO

ε E α,γ,ε A1orE

α,β A α,δ,ε AA1orOO1

α,γ A1 β,γ,δ IOorIO1

α,δ AA1orIOO1 β,γ,ε AO1orA1OorE

α,ε AA1orE β,δ,ε O1

β,γ AO1orA1O γ,δ,ε O

β,δ IO1 α,β,γ,δ I

β,ε AO1orE α,β,γ,ε AorA1orE

γ,δ IO α,β,δ,ε AorO1

γ,ε A1OorE α,γ,δ,ε A1orO

δ,ε OO1 β,γ,δ,ε OorO1

It will be found that any combinations of propositions other than thosegiven above either involve contradictions or redundancies, or else give noinformationbecauseallthefiverelationsthatareàprioripossiblestillremainpossible.

Forexample,AIisclearlyredundant;AO isself-contradictory;AorA1Oisredundant(sincethesameinformationisgivenbyAorA1);AorOgives


no information (since it excludes no possible case). The student isrecommended to test other combinations similarly. It must be rememberedthatI1=I,andE1=E.

170Itshouldbenoticedthatifwereadthefirstcolumndownwardsandthesecondcolumnupwardswegetpairsofcontradictories.

130.Euler’sdiagramsandtheclassrelationsbetweenS,not-S,P,not-P.—InEuler s diagrams, as ordinarily given, there is no explicit recognition ofnot-S and not-P; but it is of course understood that whatever part of theuniverseliesoutsideSisnot-S,andsimilarlyforP,anditmaybethoughtthatno further account of negative terms need be taken. Further consideration,however,will shew that this isnot thecase;and,assuming thatS,not-S,P,not-P all represent existing classes, we shall find that seven, not five,determinateclassrelationsbetweenthemarepossible.

Taking the diagrams given in section 126, the above assumption clearlyrequires that in the cases of α, β, and γ, there should be some part of theuniverselyingoutsideboththecircles,sinceotherwiseeithernot-Sornot-Porbothofthemwouldnolongerbecontainedintheuniverse.Butinthecasesofδandε it isdifferent.S,not-S,P,not-P are nowall of them representedwithinthecircles;andineachofthesecases,therefore,thepairofcirclesmayormaynotbetweenthemexhausttheuniverse.

Ourresultsmayalsobeexpressedbysayingthatinthecasesofα,β,andγ,theremustbesomethingwhichisbothnot-Sandnot-P;whereasinthecasesofδandε,theremayormaynotbesomethingwhichisbothnot-Sandnot-P.Euler’s circles, as ordinarily used, are no doubt a little apt to lead us tooverlookthelatterofthesealternatives.If,indeed,therewerealwayspartoftheuniverseoutsidethecircles,everyproposition,whetheritsformwereA,E,I,orO,wouldhaveaninverseandthesameinverse,namely,Somenot-Sisnot-P ; also, every proposition, including I, would have a contrapositive.TheseareerroneousresultsagainstwhichwehavetobeonourguardintheuseofEuler’sfivefoldscheme.

We find then that the explicit recognition of not-S andnot-P practicallyleavesα,β,andγunaffected,butcausesδandε each to subdivide into two


casesaccordingasthereisorisnotanythingthatisbothnot-Sandnot-P;andthe171Eulerianfivefolddivisionhasaccordinglytogiveplacetoasevenfolddivision.

In the diagrammatic representation of these seven relations, the entireuniverse of discourse may be indicated by a larger circle in which theordinary Eulerian diagrams (with some slight necessary modifications) areincluded.Weshallthenhavethefollowingscheme:—


172ItmaybeusefultorepeatthesediagramswithanexplicitindicationinregardtoeachsubdivisionoftheuniverseastowhetheritisSornot-S,Pornot-P.175Theschemewillthenappearasfollows:—


175 We might also represent the universe of discourse by a long rectangledividedintocompartments,shewingwhichofthefourpossiblecombinationsSP,SPʹ,SʹP,SʹPʹ are to be found.This planwill give the followingwhich preciselycorrespond,asnumbered,withthoseinthetext:—

(i) SP SʹPʹ


(ii) SP SʹP SʹPʹ

(iii) SP SPʹ SʹPʹ

(iv) SP SPʹ SʹP SʹPʹ

(v) SP SPʹ SʹP

(vi) SPʹ SʹP SʹPʹ

(vii) SPʹ SʹP

173ComparingtheabovewiththefiveordinaryEuleriandiagrams(whichmay be designated α, β &c. as in section 126), it will be seen that (i)correspondstoα;(ii)toβ;(iii)toγ;(iv)and(v)representthetwocasesnowyieldedbyδ;(vi)and(vii)thetwoyieldedbyε.

Oursevendiagramsmightalsobearrivedatasfollows:—Everypartoftheuniversemust be either S orSʹ, and alsoP orPʹ ; and hence the mutuallyexclusive combinations SP, SPʹ, SʹP, SʹPʹ must between them exhaust theuniverse. The case in which these combinations are all to be found isrepresentedbydiagram(iv);ifonebutoneonlyisabsentweobviouslyhavefourcaseswhicharerepresentedrespectivelyby(ii),(iii),(v),and(vi);ifonlytwoaretobefounditwillbeseenthatwearelimitedtothecasesrepresentedby(i)and(vii)orweshouldnotfulfiltheconditionthatneitherSnorSʹ,PnorPʹ, is tobealtogethernon-existent; for the same reason theuniversecannotcontainlessthantwoofthefourcombinations.Wethushavethesevencasesrepresentedbythediagrams,andtheseareshewntoexhaustthepossibilities.

174 The four traditional propositions are related to the new scheme asfollows:— Alimitsusto(i)or(ii); Ito(i),(ii),(iii),(iv),or(v); Eto(vi)or(vii); Oto(iii),(iv),(v),(vi),or(vii).


Workingoutthefurtherquestionhoweachdiagramtakenbyitselfistobeexpressedpropositionallywegetthefollowingresults: (i)SaPandSʹaPʹ ; (ii)SaPandSʹoPʹ ; (iii)SʹaPʹandSoP ; (iv)SoP,SoPʹ,SʹoP,andSʹoPʹ ; (v)SʹaPandSoPʹ ; (vi)SaPʹandSʹoP ; (vii)SaPʹandSʹaP.

ItwillbeobservedthatthenewschemeisinitselfmoresymmetricalthanEuler’s,andalso that it succeedsbetter inbringingout thesymmetryof thefourfold schedule of propositions.176A andE give two alternatives each, Iand O give five each; whereas with Euler’s scheme E gives only onealternative,A two,O three, I four, and it might, therefore, seem as if Eaffordedmoredefinite andunambiguous information thanA, andO than I,whichisnotreallythecase.Further,theproblemofexpressingeachdiagrampropositionally yields a more symmetrical result than the correspondingprobleminthecaseofEuler’sdiagrams.

176 Wehaveseenthat,similarly,inthecaseofimmediateinferencessymmetrycanbegainedonlybytherecognitionofnegativeterms.

This sevenfold scheme of class relations should be compared with thesevenfoldschemeofrelationsbetweenpropositionsgiveninsection84.

131.Lambert’sdiagramandtheclass-relationsbetweenS,not-S,P,not-P.—The following is a compact diagrammatic representation of the sevenpossible class-relations between S, not-S, P, not-P, based upon Lambert’sscheme.175


Inthisschemeeachlinerepresentstheentireuniverseofdiscourse,andthefirstlinemustbetakeninconnexionwitheachoftheothersinturn.Furtherexplanationwillbeunnecessary for thestudentwhoclearlyunderstands theLambertianmethod.

Onthesameprincipleandwiththeaidofdottedlinesthefourfundamentalpropositionalformsmayberepresentedasfollows:


176 In each case the full extent of a line represents the entireuniverseofdiscourse;anyportionofalinethatisdottedmaybeeitherSorSʹ(orPorPʹ,asthecasemaybe).

Thislastschemeofdiagramsisperhapsmoreusefulthananyoftheothersin shewingat aglancewhat immediate inferencesareobtainable fromeachpropositionbyconversion, contraposition, and inversion (on theassumptionthatS,Sʹ,P,andPʹallrepresentexistingclasses).Thus,fromthefirstdiagramwecanreadoffataglanceSaP,PiS,PʹaSʹ,SʹiPʹ ;fromthesecondSeP,PeS,PʹoSʹ,SʹoPʹ ;fromthethirdSiPandPiS ;andfromthefourthSoPandPʹoSʹ.ThelasttwodiagramsarealsoseenataglancetobeindeterminateinrespecttoPʹandSʹ,PandSʹ,respectively(thatistosay,Ihasnocontrapositiveandnoinverse,Ohasnoconverseandnoinverse).


EXERCISES.

132.IllustratebymeansoftheEuleriandiagrams(1)therelationbetweenAandE, (2) therelationbetweenIandO,(3)theconversionofI,(4)thecontrapositionofO,(5)theinversionofE.[K.]

133.AdeniesthatnonebutXareY ;BdeniesthatnonebutYareX.WhichofthefiveclassrelationsbetweenXandY(1)musttheyagreeinrejecting,(2)maytheyagreeinaccepting?[C.]

134.Takealltheordinarypropositionsconnectinganytwoterms,combinetheminpairssofarasispossiblewithoutcontradiction,andrepresenteachcombinationdiagrammatically.[J.]


CHAPTERVI.

PROPOSITIONSINEXTENSIONANDININTENSION.

135.FourfoldImplicationofPropositionsinConnotationandDenotation.—Indealingwiththequestionwhetherpropositionsassertarelationbetweenobjectsorbetweenattributesorbetweenobjectsandattributes,logicianshavebeen apt to commit the fallacy of exclusiveness, selecting some one of thegivenalternatives,andtreatingtheothersasnecessarilyexcludedthereby.Itfollows, however, from the double aspect of names—in extension andintension—thatthedifferentrelationsreallyinvolveoneanother,sothatallofthemareimpliedinanycategoricalpropositionwhosesubjectandpredicateare both general names.177 If any one of the relations is selected asconstitutingthemeaningoftheproposition,theotherrelationsareatanyrateinvolvedasimplications.

177 Inthediscussionthatfollowswelimitourselvestothetraditionalschemeofpropositions.

The problem will be made more definite if we confine ourselves to aconsideration of connotation and denotation in the strict sense, asdistinguished from comprehension and exemplification, our terms beingsupposed to be defined intensively.178 Both subject and predicate will thenhave a denotation determined by their connotation, and hence our 178propositionmaybeconsideredfromfourdifferentpointsofview,whicharenot indeed really independent of one another, but which serve to bringdifferentaspectsofthepropositionintoprominence.(1)Thesubjectmayberead in denotation and the predicate in connotation; (2) both termsmay bereadindenotation;(3)bothtermsmaybereadinconnotation;(4)thesubjectmaybereadinconnotationandthepredicateindenotation.

178 Withextensivedefinitionswemightsimilarlyworkouttherelationsbetweenthetermsofapropositioninexemplificationandcomprehension;andwitheitherintensive or extensive definitions, we might consider them in denotation andcomprehension.Thediscussioninthetextwill,however,belimitedtoconnotation


anddenotation,exceptthataseparatesectionwillbedevotedtothecaseinwhichbothsubjectandpredicatearereadincomprehension.

As an example, we may take the proposition, All men are mortal.179According to our point of view, this propositionmaybe read in anyof thefollowingways: (1)Theobjectsdenotedbymanpossesstheattributesconnotedbymortal ; (2) The objects denoted byman are includedwithin the class of objectsdenotedbymortal ; (3) The attributes connoted by man are accompanied by the attributesconnotedbymortal ; (4) The attributes connoted by man indicate the presence of an objectbelongingtotheclassdenotedbymortal.

179 A distinctionmay perhaps be drawn between the four following types ofpropositions;(a)Allmenaremortal ;(b)Allmenaremortals ;(c)Manismortal ;(d)Man is a mortal. Of these, (a) naturally suggests the reading of subject indenotationandpredicateinconnotationasmeaning,thethreeotherreadingsbeingimplications ;(b)issimilarlyrelatedtothereadingnumbered(2)above;(c)to(3);and(d)to(4).

Itshouldbespeciallynoticedthatadifferentrelationbetweensubjectandpredicate is brought out in each of these four modes of analysing theproposition,therelationsbeingrespectively(i)possession,(ii)inclusion,(iii)concomitance,(iv)indication.

Itmayveryreasonablybearguedthatacertainoneof theabovewaysofregarding the proposition is (a) psychologically the most prominent in themind in predication; or (b) the most fundamental in the sense of makingexplicitthatrelationwhichultimatelydeterminestheotherrelations;or(c)themost convenient for agivenpurpose,e.g., fordealingwith theproblemsofformallogic.Weneednot,however,selectthesamemodeofinterpretationineachcase.Therewould,forexample,benothinginconsistentinholdingthattoreadthe179subjectindenotationandthepredicateinconnotationismostcorrectfromthepsychologicalstandpoint; toreadbothtermsinconnotationthe most ultimate, inasmuch as connotation determines denotation and notvice versâ, and to read both terms in denotation the most serviceable forpurposesof logicalmanipulation.Tosay,however, thatacertainoneof thefour readings alone can be regarded as constituting the import of the


propositiontotheexclusionoftheotherscannotbutbeerroneous.Theyareintruthsomuchimplicated inoneanother, that thedifficultymayratherbe tojustifyatreatmentwhichdistinguishesbetweenthem.180

180 ThetruedoctrineisexcellentlystatedbyMrsLaddFranklininanarticleinMind,October,1890,pp.561,2.

(1)Subjectindenotation,predicateinconnotation. Ifwe read the subjectof aproposition indenotation and thepredicate inconnotation, we have what is sometimes called the predicative mode ofinterpreting theproposition.Thiswayof regardingpropositionsmostnearlycorresponds in the great majority of cases with the course of ordinarythought;181 that is to say,wenaturallycontemplate the subjectasaclassofobjects of which a certain attribute or complex of attributes is predicated.Such propositions as All men are mortal, Some violets are white, Alldiamondsarecombustible,maybetakenasexamples.DrVennputsthepointveryclearlywithreferencetothelastofthesethreepropositions:“IfIsaythat‘alldiamondsarecombustible,’Iamjoiningtogethertwoconnotativeterms,eachofwhich,therefore,impliesanattributeanddenotesaclass;butistherenotabroaddistinctioninrespectoftheprominencewithwhichthenotionofaclassispresentedtothemindinthetwocases?Asregardsthediamond,wethink at once, or think very speedily, of a class of things, the distinctiveattributes of the subject being mainly used to carry the mind on to thecontemplation of the objects referred to by them. But as regards thecombustibility, the attribute itself is the prominent thing … Combustiblethings, other than the diamond itself, come scarcely, if at all, under 180contemplation. The assertion in itself does not cause us to raise a thoughtwhethertherebeothercombustiblethingsthantheseinexistence”(EmpiricalLogic,p.219).


181 Thoughperhapswhatisactuallypresenttothemindisusuallyrathermorecomplexthanwhatisrepresentedbyanyoneofthefourreadingstakenbyitself.

Twopointsmaybenoticedas serving to confirm theview thatgenerallyspeakingthepredicativemodeofinterpretingpropositionsispsychologicallythemostprominent: (a)Themost striking difference between a substantive and an attributive(i.e.,anadjectiveoraparticiple)fromthelogicalpointofviewisthatintheformer thedenotation isusuallymoreprominent than theconnotation, eventhoughitmaybeultimatelydeterminedbytheconnotation,whilstinthelatterthe connotation is the more prominent, even though the name must beregardedasthenameofaclassofobjectsifitisentitledtobecalledanameinthestrictlogicalsenseatall.Correspondingtothiswefindthatthesubjectofapropositionisalmostalwaysasubstantive,whereasthepredicateismoreoftenanattributive. (b) It is always the denotation of a term that we quantify, never theconnotation.Whether we talk of allmen or of somemen, the complex ofattributesconnotedbyman istakeninitstotality;thedistinctionofquantityrelatesentirelyto thedenotationof theterm.Correspondingto this,wefindthat we naturally regard the quantity of a proposition as pertaining to itssubject,andnottoitspredicate.Itwillbeshewninthefollowingchapterthatthe doctrine of the quantification of the predicate has at any rate nopsychologicaljustification.

Thereare,however,numerousexceptionstothestatementthatthesubjectof a proposition is naturally read in denotation and the predicate inconnotation; for example, in the classificatory sciences. The followingpropositionsmaybe takenas instances:Allpalmsareendogens,Alldaisiesare compositae,None but solid bodies are crystals, Hindoos are Aryans,TartarsareTuranians.Insuchcasesasthesemostofuswouldnaturallythinkof a certain class of objects as included in or excluded from another classratherthanaspossessingornotpossessingcertaindefiniteattributes;inotherwords, as Dr Venn puts it, “the class-reference of the predicate is no lessdefinitethanthatofthesubject”(EmpiricalLogic,p.220).181InthecaseofsuchapropositionasNoplantswithoppositeleavesareorchids,thepositionisevenreversed,thatistosay,itisthesubjectratherthanthepredicatethat


weshouldmorenaturallyreadinconnotation.Wemaypassonthentootherwaysofregardingthecategoricalproposition.

(2)Subjectindenotation,predicateindenotation. Ifwereadboththesubjectandthepredicateofapropositionindenotation,we have a relation between two classes, and hence this is called the classmodeofinterpretingtheproposition.Itmustbeparticularlyobservedthattherelationbetween thesubjectand thepredicate isnowoneof inclusion inorexclusion from, not one ofpossession. Itmay at once be admitted that theclass mode of interpreting the categorical proposition is neither the mostultimate,nor—generallyspeaking—thatwhichwenaturallyorspontaneouslyadopt. It is, however, extremely convenient formanipulative purposes, andhence is the mode of interpretation usually selected, either explicitly orimplicitly, by the formal logician. Attentionmay be specially called to thefollowingpoints: (a) When subject and predicate are both read in denotation, they arehomogeneous. (b) In the diagrammatic illustration of propositions both subject andpredicatearenecessarilyreadindenotation,sinceitisthedenotation—nottheconnotation—ofatermthatwerepresentbymeansofadiagram. (c)Thepredicate of a propositionmust be read in denotation in order togiveameaningtothequestionwhetheritisorisnotdistributed. (d)Thepredicateaswellasthesubjectmustbereadindenotationbeforesuchaprocessasconversionispossible. (e)Inthetreatmentofthesyllogismbothsubjectandpredicatemustbereadindenotation(orelsebothinconnotation),sinceeitherthemiddleterm(firstandfourthfigures)orthemajorterm(secondandfourthfigures)ortheminorterm(thirdandfourthfigures)issubjectinoneofthepropositionsinwhichitoccursandpredicateintheother.

The class mode of interpreting categorical propositions is neverthelesstreatedbysomewritersasbeingpositively182erroneous.Buttheargumentsusedinsupportofthisviewwillnotbearexamination.

(i) It is said that to read both subject and predicate in denotation ispsychologically false. It has indeed been pointed out already that the class


modeof interpretation is not thatwhich as a rule first presents itself toourmindwhen a proposition is given us; butwe have also seen that there areexceptions to this, as, for example, in the propositions All daisies arecompositae.AllHindoosareAryan,AllTartarsareTuranians.Itis,therefore,clearly wrong to describe the reading in question as in all casespsychologicallyfalse.Onthesameshewing,anyotherreadingwouldequallybe psychologically false, for what is immediately present to the mind injudgment varies verymuch in different cases.Undoubtedly there aremanyjudgments in regard to which we do not spontaneously adopt the classreading.Still,analysisshewsthatinthesejudgments,asinothers,inclusioninor exclusion from a class is really implicated along with other things,althoughthisrelationmaybeneitherthatwhichfirstimpressesitselfuponusnorthatwhichismostimportantorcharacteristic.

(ii)Itisaskedwhatwemeanbyaclass,bytheclassofbirds,forexample,whenwesayAllowlsarebirds.“Itisnothingexistinginspace;thebirdsoftheworldarenowherecollectedtogethersothatwecangoandpickouttheowls from amongst them. The classification is a mental abstraction of ourown, foundedupon thepossessionofcertaindefiniteattributes.Theclass isnotdefiniteandfixed,andwedonotfindoutwhetheranyindividualbelongstoitbygoingoveralistofitsmembers,butbyenquiringwhetheritpossessesthenecessaryattributes.”182Insofarasthisargumentappliesagainstreadingthe predicate in denotation, it applies equally against reading the subject indenotation.ItisineffecttheargumentusedbyMill(Logic,i.5,§3)inorderto lead up to his position that the ultimate interpretation of the categoricalproposition requires us to read both subject and predicate in connotation,sincedenotationisdeterminedbyconnotation.Butifthisbegranted,itdoesnot prove the class reading of the 183 proposition erroneous; it only provesthatintheclassreading,theanalysisoftheimportofthepropositionhasnotbeencarriedasfarasitadmitsofbeingcarried.

182 Welton,Logic,p.218.

(iii) It is argued that when we regard a proposition as expressing theinclusion of one class within another, even then the predicate is onlyapparently read in denotation. “On this view,we do not really assertPbut


‘inclusion inP,’and this is therefore the truepredicate.Forexample, in theproposition‘Allowlsarebirds,’therealpredicateis,onthisview,not‘birds’but ‘included in the class birds.’ But this inclusion is an attribute of thesubject,andtherealpredicate,therefore,assertsanattribute.Itismeaninglesstosay‘Everyowlis theclassbirds,’anditisfalsetosay‘Theclassowlsistheclassbirds.’”183Thisargumentsimplybegsthequestioninfavourofthepredicativemodeofinterpretation.Itoverlooksthefactthattheprecisekindofrelationbroughtoutintheanalysisofapropositionwillvaryaccordingtothe way in which we read the subject and the predicate. An analogousargumentmightalsobeusedagainst thepredicative reading itself.Take theproposition,“Allmenaremortal.”Itisabsurdtosaythat“Everymanistheattributemortality,”orthat“Theclassmenistheattributemortality.”

183 Welton,Logic,p.218.

(iv)ItissaidthataclassinterpretationofbothSandPwouldleadproperlytoafivefold,notafourfold,schemeofpropositions,sincetherearejustfiverelations possible between any two classes, as is shewn by the Euleriandiagrams.Thiscontentionhasforce,however,onlyupontheassumptionthatwemust have quite determinate knowledge of the class relation betweenSand P before being able to make any statement on the subject; and thisassumption is neither justifiable in itself nor necessarily involved in theinterpretationinquestion.Itmaybeaddedthatifasimilarviewweretakenonthe adoption of the predicative mode of interpretation, we should have athreefold,notafourfoldscheme.For thenthequantityofoursubjectatanyratewouldhavetobeperfectlydeterminate,andwithSandPforsubjectandpredicate,thethreepossiblestatementswouldbe—AllSisP,SomeSisPand184someisnot,NoSisP.Thepointhereraisedwillpresentlybeconsideredfurtherinconnexionwiththequantificationofthepredicate.

(3)Subjectinconnotation,predicateinconnotation. If we read both the subject and the predicate of a proposition inconnotation,wehavewhatmaybecalledtheconnotativemodeofinterpretingtheproposition.InthepropositionAllSisP, therelationexpressedbetweentheattributesconnotedbySandthoseconnotedbyPisoneofconcomitance—“the attributes which constitute the connotation of S are always found


accompaniedbythosewhichconstitutetheconnotationofP.”184Similarly,inthecaseofSomeSisP,—“theattributes185whichconstitutetheconnotationof S are sometimes found accompanied by those which constitute theconnotation of P”; No S is P,—“the attributes which constitute theconnotation of S are never found along with those which constitute theconnotation of P”; Some S is not P,—“the attributes which constitute theconnotation of S are sometimes found unaccompanied by those whichconstitutetheconnotationofP.”

184 This is theonlypossiblereadinginconnotation,sofarasrealpropositionsareconcerned, if the termconnotation isused in thestrict senseasdistinguishedbothfromsubjectiveintensionandfromcomprehension.Unfortunatelyconfusionis apt to be introduced into discussions concerning the intensive rendering ofpropositions simply because no clear distinction is drawn between the differentpoints of viewwhichmay be takenwhen terms are regarded from the intensiveside. Hamilton distinguished between judgments in extension and judgments inintension,therelationbetweenthesubjectandthepredicateintheonecasebeingjust the reverse of the relation between them in the other. Thus, taking theproposition All S is P, we have in extension S is contained under P, and inintensionS comprehends P. On this view the intensive reading of All men aremortal is “mortality is part of humanity” (the extensive readingbeing “the classman is part of the class mortal”). This reading may be accepted if the termintensionisusedintheobjectivesensewhichwehavegiventocomprehension,sothat byhumanity is meant the totality of attributes common to all men, and bymortalitythetotalityofattributescommontoallmortals.Tothispointofviewweshallreturninthenextsection.Leavingitforthepresentononeside,itisclearthatif by humanity we mean only what may be called the distinctive or essentialattributesofman, then inorder that theabove readingmaybecorrect, thegivenpropositionmust be regarded as analytical. In otherwords, ifhumanity signifiesonly those attributes which are included in the connotation of man, then, ifmortalityisincludedinhumanity,weshallmerelyhavetoanalysetheconnotationof thenameman, inorder toobtainourproposition.Henceon thisview itmusteither bemaintained that all universal affirmative propositions are analytical, orelsethatsomeuniversalaffirmativescannotbereadinintension.Butobviouslythefirstof thesealternativesmustbe rejected,and thesecondpracticallymeans thatthereadinginquestionbreaksdownsofarasuniversalaffirmativesareconcerned.

Hamilton’sreadingbreaksdownevenmorecompletelyinthecaseofparticularsandnegatives.TheattributesconstitutingtheintensionsofSandPpartlycoincideisclearlynotequivalenttoSomeSisP ;forexample,theintension(inanysense)ofEnglishmanhassomethingincommonwiththeintensionofFrenchman,but itdoesnot followthatSomeEnglishmenareFrenchmen.Again, from the fact thattheintensionofShasnothingincommonwiththeintensionofP,wecannotinferthatNo S is P ; suppose, for example, that S stands for “ruminant,” and P for


“cloven-hoofed.”CompareVenn,SymbolicLogic,pp.391–5.

Itwillbenoticedthatintheconnotativereadingwehavealwaystotaketheattributeswhichconstitutetheconnotationcollectively.Inotherwords,bytheattributes constituting the connotation of a term we mean those attributesregardedasawhole.Thus,NoSisPdoesnotimplythatnoneoftheattributesconnotedbySareeveraccompaniedbyanyofthoseconnotedbyP.Thisisapparent ifwetakesuchapropositionasNooxygen ishydrogen. It followsthatwhen the subject is read in connotation the quantity of the propositionmustappearasaseparateelement,beingexpressedbytheword“always”or“sometimes,”andmustnotbeinterpretedasmeaning“all”or“some”oftheattributesincludedintheconnotationofthesubject.

It isarguedbythosewhodenythepossibilityoftheconnotativemodeofinterpreting propositions, that this is not really reading the subject inconnotationatall;alwaysandsometimesaresaidtoreduceustodenotationatonce. In reply to this, it must of course be allowed that real propositionsaffirm no relation between attributes independently of the objects towhichtheybelong.Theconnotativereadingimpliesthedenotative,andwemustnotexaggerate the nature of the distinction between them. Still the connotativereadingpresentstheimportofthepropositioninanewaspect,andthereisatany rate a prima facie difference between regarding one class as includedwithin another, and regarding one attribute as always accompanied byanother,even thougha little186 considerationmayshew that the two thingsmutuallyinvolveoneanother.185

185 Mill attaches great importance to the connotative mode of interpretingpropositions as compared with the class mode or the predicative mode, on thegroundthatitcarriestheanalysisastagefurther;andthismustbegranted,atanyratesofarasweconsidertheapplicationofthetermsinvolvedtobedeterminedbyconnotationandnotbyexemplification.Mill is,however, sometimesopen to thechargeofexaggeratingthedifferencebetweenthevariousmodesofinterpretation.Thisisapparent,forexample,inhisrejectionoftheDictumdeomnietnulloastheaxiomofthesyllogism,andhisacceptanceoftheNotanotaeestnotareiipsiusinitsplace.

(4)Subjectinconnotation,predicateindenotation. Taking the propositionAll S isP, and reading the subject in connotationandthepredicateindenotation,wehave,“TheattributesconnotedbySarean


indication of the presence of an individual belonging to the classP.” Thismodeof interpretation is always apossibleone,but itmustbegranted thatonly rarely does the import of a proposition naturally present itself to ourminds in this form. There are, however, exceptional cases in which thisreadingisnotunnatural.ThepropositionNoplantswithopposite leavesareorchidshasalreadybeengivenasanexample.Anotherexample isaffordedby the proposition All that glitters is not gold. Taking the subject inconnotationandthepredicate indenotationwehave,Theattributeofglitterdoesnotalwaysindicatethepresenceofagoldobject ; and itwillbe foundthat this readingof theproverbserves tobringout itsmeaningreallybetterthananyofthethreeotherreadingswhichwehavebeendiscussing.

Itisworthwhilenoticingherebywayofanticipationthatonanyviewofthe existential interpretation of propositions, as discussed in chapter 8, weshall still have a fourfold readingof categoricalpropositions in connotationanddenotation.Theuniversalnegativeandtheparticularaffirmativemaybetaken as examples, on the supposition that the former is interpreted asexistentiallynegativeandthelatterasexistentiallyaffirmative.Theuniversalnegativeyieldsthefollowing:(1)ThereisnoindividualbelongingtotheclassS and possessing the attributes connoted by P ; (2) There is no individualcommontothetwoclassesSandP ;(3)Theattributes187connotedbySandP respectively are never found conjoined; (4) There is no individualpossessingtheattributesconnotedbySandbelongingtotheclassP.Similarlythe particular affirmative yields: (1) There are individuals belonging to theclassSandpossessingtheattributesconnotedbyP ;(2)ThereareindividualscommontothetwoclassesSandP ; (3)TheattributesconnotedbySandPrespectively are sometimes found conjoined; (4) There are individualspossessingtheattributesconnotedbySandbelongingtotheclassP.Wesee,therefore, that the question discussed in this section is independent of thatwhichwillberaisedinchapter8;andthatforthisreason,iffornoother,nosolution of the general problem raised in the present chapter can afford acompletesolutionoftheproblemoftheimportofcategoricalpropositions.

136. The Reading of Propositions in Comprehension.—If, in taking theintensional standpoint, we consider comprehension instead of connotation,


our problem is to determine what relation is implied in any propositionbetween the comprehension of the subject and the comprehension of thepredicate. This question being asked with reference to the universalaffirmative proposition All S is P, the solution clearly is that thecomprehension of S includes the comprehension of P. The interpretation incomprehension is thus precisely the reverse of that in denotation (thedenotationofS is included in thedenotationofP); andwemightbe led tothinkthat,takingthedifferentpropositionalforms,weshouldhaveaschemeincomprehension,analogousthroughouttothatindenotation.Butthisisnotthe case, for the simple reason that in our ordinary statements we do notdistributivelyquantifycomprehensioninthewayinwhichwedodenotation;inotherwords,comprehensionisalwaystakeninitstotality.Thus,readinganIpropositionindenotationwehave—theclassesSandPpartlycoincide ;andcorrespondingtothisweshouldhave—thecomprehensionsofSandPpartlycoincide. But this is clearly not what we express by Some S is P ; for thepartialcoincidenceofthecomprehensionsofSandPisquitecompatiblewithNoSisP,thatistosay,theclassesSandPmaybemutuallyexclusive,andyet some attributesmay be common to the whole of S and 188 also to thewhole of P ; for example, No Pembroke undergraduates are also Trinityundergraduates.Again,givenanEproposition,wehave indenotation—theclassesSandPhavenopartincommon ;butforthereasonjustgiven,itdoesnot follow that the comprehension of S and the comprehension of P havenothingincommon.

ItisindeednecessarytoobvertIandEinordertoobtainacorrectreadingincomprehension.Wethenhavethefollowingscheme,inwhichtherelationof contradiction betweenA andO and between E and I is made clearlymanifest: AllSisP,ThecomprehensionofSincludesthecomprehensionofP ; NoSisP,ThecomprehensionofSincludesthecomprehensionofnot-P; SomeSisP,ThecomprehensionofSdoesnotincludethecomprehensionofnot-P; Some S is not P, The comprehension of S does not include thecomprehensionofP.



CHAPTERVII.

LOGICALEQUATIONSANDTHEQUANTIFICATIONOFTHEPREDICATE.

137.TheemploymentofthesymbolofEqualityinLogic.—Thesymbolofequality(=)isfrequentlyusedinlogictoexpresstheidentityoftwoclasses.Forexample, Equilateraltriangles=equiangulartriangles ; Exogens=dicotyledons ; Men=mortalmen.

Itis,however,importanttorecognisethatinthusborrowingasymbolfrommathematics we do not retain its meaning unaltered, and that a so-calledlogicalequation is, therefore,somethingverydifferent fromamathematicalequation. Inmathematics the symbolofequalitygenerallymeansnumericalor quantitative equivalence. But clearly we do not mean to express merenumerical equality when we write equilateral triangles = equiangulartriangles.Whateverthisso-calledequationsignifies,itiscertainlysomethingmorethanthattherearepreciselyasmanytriangleswiththreeequalsidesasthereare triangleswith threeequalangles. It is furtherclear thatwedonotintendtoexpressmeresimilarity.Ourmeaningisthatthedenotationsofthetermswhichareequatedareabsolutelyidentical;inotherwords,thattheclassofobjectsdenotedbythetermequilateraltriangleisabsolutelyidenticalwiththe class of objects denoted by the term equiangular triangle.186 It may,however, be objected that, if this 190 is what our equation comes to, theninasmuchasastatementofmereidentityisemptyandmeaningless,itstrictlyspeaking leaves us with nothing at all; it contains no assertion and canrepresent no judgment. The answer to this objection is thatwhilstwe haveidentityinacertainrespect,itiserroneoustosaythatwehavemere identity.Wehaveidentityofdenotationcombinedwithdiversityofconnotation,and,therefore,withdiversity of determination (meaning thereby diversity in theways inwhich theapplicationof the two terms identified isdetermined).187


The meaning of this will be made clearer by the aid of one or twoillustrations. Taking, then, as examples the logical equations already given,wemay analyse their meaning as follows. If out of all triangles we selectthosewhichpossessthepropertyofhavingthreeequalsides,andifagainoutof all triangles we select thosewhich possess the property of having threeequal angles,we shall find that ineither caseweare leftwithprecisely thesame set of triangles.Thus, each sideofour equationdenotesprecisely thesameclassofobjects,buttheclassisdeterminedorarrivedatintwodifferentways.Similarly,ifweselectallplantsthatareexogenousandagainallplantsthataredicotyledonous,ourresultsarepreciselythesamealthoughourmodeofarrivingatthemhasbeendifferent.Oncemore,ifwesimplytaketheclassofobjectswhichpossesstheattributeofhumanity,andagaintheclasswhichpossess both this attribute and also the attribute of mortality, the objectsselectedwill be the same; nonewill be excluded by our secondmethod ofselectionalthoughanadditionalattributeistakenintoaccount.

186 Itfollowsthatthecomprehensions(butofcoursenottheconnotations)oftheterms will also be identical; this cannot, however, be regarded as the primarysignificationoftheequation.

187 IhavepracticallyborrowedtheabovemodeofexpressionfromMissJones,who describes an affirmative categorical proposition as “a proposition whichassertsidentityofapplicationindiversityofsignification”(GeneralLogic,p.20).MissJones’smeaningmay,however,beslightlydifferentfromthatintendedinthetext, and I am unable to agree with her general treatment of the import ofcategoricalpropositions,asshedoesnotappeartoallowthatbeforewecanregarda proposition as asserting identity of application we must implicitly, if notexplicitly,havequantifiedthepredicate.

Sincetheidentityprimarilysignifiedbyalogicalequationisanidentityinrespectofdenotation, anyequationalmodeof readingpropositionsmustberegardedasamodificationofthe191“class”mode.Whathasbeensaidabove,however,willmake it clear that here as elsewhere denotation is considerednottotheexclusionofconnotationbutasdependentuponit;andweagainseehowdenotativeandconnotative readingsofpropositionsare really involvedin one another, although one side or the other may be made the moreprominentaccordingtothepointofviewwhichistaken.

Another point to which attention may be called before we pass on toconsiderdifferenttypesoflogicalequationsisthatinsofarasapropositionis


regardedasexpressingan identitybetweenits termsthedistinctionbetweensubjectandpredicatepracticallydisappears.Wehaveseenthatwhenwehavethe ordinary logical copula is, propositions cannot always be simplyconverted,thereasonbeingthattherelationofthesubjecttothepredicateisnot the same as the relation of the predicate to the subject. But when twoterms are connected by the sign of equality, they are similarly, and notdiversely, related to eachother; in otherwords, the relation is symmetrical.Such an equation, for example, as S = P can be read either forwards orbackwardswithout any alteration ofmeaning.There can accordingly be nodistinctionbetweensubjectandpredicateexceptthemereorderofstatement,andthatmayberegardedasformostpracticalpurposesimmaterial.

138.TypesofLogicalEquations188—Jevons(PrinciplesofScience,chapter3) recognises three types of logical equations, which he calls respectivelysimpleidentities,partialidentities,andlimitedidentities.


Simple identities are of the form S = P ; for example,Exogens = dicotyledons. Whilst this is the simplest case equationally, theinformationgivenby the equation requires twopropositions inorder that itmaybeexpressedinordinarypredicativeform.Thus,AllSisPandAllPisS ;AllexogensaredicotyledonsandAlldicotyledonsareexogens. If, however,we are allowed to quantify the predicate as well as the subject, a singlepropositionwillsuffice.Thus,AllSisallP,Allexogensarealldicotyledons.Weshallreturnpresentlytoaconsiderationofthistypeofproposition.

192 Partial identities are of the form S = SP, and are the expressionequationally of ordinary universal affirmative propositions. If we take thepropositionAllSisP,itisclearthatwecannotwriteitS=P,sincetheclassP, insteadofbeingcoextensivewith theclassS,may include it andagooddealmorebesides.Since,however,bythelawofidentityAllSisS,itfollowsfromAllSisPthatAllSisSP.Wecanalsopassbackfromthelatterofthesepropositionstotheformer.Hencethetwopropositionsareequivalent.ButAllSisSPmayatoncebereducedtotheequationalformS=SP.ForthisbreaksupintothetwopropositionsAllSisSPandAllSPisS,andsincethesecondof these is a mere formal proposition based on the law of identity, the


equationmustnecessarilyholdgoodifAllSisSPisgiven.Totakeaconcreteexample, the proposition All men are mortal becomes equationallyMen=mortalmen.SimilarlytheuniversalnegativepropositionSePmaybeexpressedintheequationalformS=Sp(wherep=not-P).

Limited identities are of the form VS = VP, which may be interpreted“WithinthesphereoftheclassV,allSisPandallPisS,”or“TheS’sandP’s,whichareV’s,areidentical.”SofarasV representsadeterminateclass,thereislittledifferencebetweentheselimitedidentitiesandsimpleidentities.This is shewn by the fact that Jevons himself gives Equilateraltriangles=equiangulartrianglesasaninstanceofasimpleidentity,whereasitsproperplaceinhisclassificationwouldappear tobeamongst the limitedidentities,foritsinterpretationisthat“withinthesphereoftriangles—alltheequilateralsarealltheequiangulars.”

The equationVS =VP is, however, used by Boole—and also by Jevonssubsequently—as the expression equationally of the particular proposition,andifitcanreallysufficeforthis,itsrecognitionasadistincttypeisjustified.IfwetakethepropositionSomeSisP,wefindthat theclassesSandPareaffirmed tohave somepart in common,butno indication is givenwherebythis part can be identified. Boole accordingly indicates it by the arbitrarysymbolV.ItisthenclearthatAllVSisVPandalsothatAllVPisVS,andwehavetheaboveequation.

193Itisnopartofourpresentpurposetodiscusssystemsofsymboliclogic;butitmaybebrieflypointedoutthattheaboverepresentationoftheparticularpropositionisfarfromsatisfactory.Inordertojustifyit,limitationshavetobeplacedupontheinterpretationofVwhichaltogetherdifferentiateitfromotherclass-symbols.Thus,theequationVS=VPisconsistentwithNoSisP(and,therefore,cannotbeequivalenttoSomeSisP)providedthatnoViseitherSorP,forinthiscasewehaveVS=0andVP=0.Vmust,therefore,belimitedbytheantecedentconditionthatitrepresentsanexistingclassandaclassthatcontains either S orP, and it is in this condition quite as much as in theequationitselfthattherealforceoftheparticularpropositionisexpressed.189

189 CompareVenn,SymbolicLogic,pp.161,2.

Ifparticularpropositionsaretruecontradictoriesofuniversalpropositions,


then it would seem to follow that in a system in which universals areexpressedasequalities,particulars shouldbeexpressedas inequalities.Thiswould mean the introduction of the symbols > and <, related to thecorresponding mathematical symbols in just the same way as the logicalsymbolofequalityisrelatedtothemathematicalsymbolofequality;thatistosay,S > SP would imply logicallymore thanmere numerical inequality, itwould imply that the class S includes thewhole of the class SP and morebesides. Thus interpreted, S > SP expresses the particular negativeproposition, Some S is not P.190 If we further introduce the symbol 0 asexpressingnonentity,NoSisPmaybewrittenSP=0,anditscontradictory,i.e.,SomeS isP,maybewrittenSP>0.We shall thenhave the followingscheme(wherep=not-P):

AllSisP expressedbyS=SPorbySp=0;SomeSisnotP ″ ″ S>SP ″ Sp>0;NoSisP ″ ″ SP=0 ″ S=Sp ;SomeSisP ″ ″ SP>0 ″ S>Sp.

190 SimilarlyX>Yexpressesthetwostatements“AllYisX,butSomeXisnotY,”justasX=Yexpressesthetwostatements“AllYisXandAllXisY.”

194 This scheme, it will be observed, is based on the assumption thatparticulars are existentially affirmative while universals are existentiallynegative.This introducesaquestionwhichwillbediscussed indetail in thefollowingchapter.Theobjectofthepresentsectionismerelytoillustratetheexpression of propositions equationally, and the symbolism involved has,therefore, been treated as briefly as has seemed compatible with a clearexplanation of its purport. Any more detailed treatment would involve adiscussionofproblemsbelongingtosymboliclogic.

139.TheexpressionofPropositionsasEquations.—Therearerarecasesinwhichpropositionsfallnaturallyintowhatispracticallyanequationalform;for example, Civilization and Christianity are co-extensive. But, speakinggenerally, the equational relation, as implicated in ordinary propositions, isnotonethatisspontaneously,oreveneasily,graspedbythemind.Henceasapsychological account of the process of judgment the equational renderingmayberejected.Itis,moreover,notdesirablethatequationsshouldsupersede


thegenerallyrecognisedpropositionalformsinordinarylogicaldoctrine,forsuchdoctrine shouldnotdepartmore thancanbehelped from the formsofordinary speech. But, on the other hand, the equational treatment ofpropositionsmustnotbesimplyputononesideaserroneousorunworkable.It has been shewn in the preceding section that it is at any rate possible toreduceallcategoricalpropositionstoaforminwhichtheyexpressequalitiesorinequalities;andsuchreductionisofthegreatestimportanceinsystemsofsymbolic logic. Even for purposes of ordinary logical doctrine, the enquiryhowfarpropositionsmaybeexpressedequationallyserves toaffordamorecomplete insight into their full import, or at any rate their full implication.Hence while ordinary formal logic should not be entirely based upon anequationalreadingofpropositions, itcannotaffordaltogether toneglect thiswayofregardingthem.

Wemaypassontoconsiderinsomewhatmoredetailaspecialequationalor semi-equational system—open also to special criticisms—by whichHamiltonandotherssoughttorevolutioniseordinarylogicaldoctrine.

195 140. The eight propositional forms resulting from the explicitQuantificationofthePredicate.—Wehaveseenthatintheordinaryfourfoldschedule of propositions, the quantity of the predicate is determined by thequality of the proposition, negatives distributing their predicates, whileaffirmatives do not. It seems a plausible view, however, that by explicitquantification thequantityof thepredicatemaybemadeindependentof thequalityoftheproposition,andSirWilliamHamiltonwasthusledtorecogniseeightdistinctpropositionalformsinsteadofthecustomaryfour:—

AllSisallP, U.AllSissomeP, A.SomeSisallP, Y.SomeSissomeP, I.NoSisanyP, E.NoSissomeP, η.SomeSisnotanyP, O.SomeSisnotsomeP, ω.

The symbols attached to the different propositions in the above schedule


are those employed by Archbishop Thomson,191 and they are those nowcommonlyadoptedsofarasthequantificationofthepredicateisrecognisedinmoderntext-books.

191 Thomsonhimself,however,ultimatelyrejectstheformsηandω.

The symbols usedbyHamiltonwereAfa,Afi, Ifa,Ifi,Ana,Ani, Ina, Ini.Here f indicates an affirmative proposition, n a negative; a means that thecorrespondingtermisdistributed,ithatitisundistributed.

ForthenewformswemightalsousethesymbolsSuP,SyP,SηP,SωP,ontheprincipleexplainedinsection62.

141. Sir William Hamilton’s fundamental Postulate of Logic.—Thefundamental postulate of logic, according to SirWilliamHamilton, is “thatwebeallowedtostateexplicitlyinlanguageallthatisimplicitlycontainedinthought”; and we may briefly consider the meaning to be attached to thispostulatebeforegoingon todiscuss theuse that ismadeof it inconnexionwiththedoctrineofthequantificationofthepredicate.

196Givingthenatural interpretation to thephrase“implicitlycontainedinthought,”thepostulatemightatfirstsightappeartobeabroadstatementofthegeneralprincipleunderlyingthelogician’streatmentofformalinferences.Inallsuchinferencestheconclusionisimplicitlycontainedinthepremisses;and since logic has to determine what inferences follow legitimately fromgivenpremisses,itmayinthissensebesaidtobepartofthefunctionoflogictomakeexplicitinlanguagewhatisimplicitlycontainedinthought.

It seemsclear,however, from theusemadeof thepostulatebyHamiltonand his school that he is not thinking of this, and indeed that he is notintendinganyreferencetodiscursivethoughtatall.Hismeaningratheristhatweshouldmakeexplicitinlanguagenotwhatisimplicitinthought,butwhatisexplicit in thought, or, as itmay be otherwise expressed, thatwe shouldmake explicit in language all that is really present in thought in the act ofjudgment.

Adopting this interpretation, we may come to the conclusion that thepostulate isobscurelyexpressed,butwecanhavenohesitationinadmittingits validity. It is obviously of importance to the logician to clear up


ambiguities and ellipses of language. For this reason it is, amongst otherthings, desirable that we should avoid condensed and elliptical modes ofexpression. But whether Hamilton’s postulate, as thus interpreted, supportsthe doctrine of the quantification of the predicate is another question. Thispointwillbeconsideredinthenexttwosections.

142. Advantages claimed for the Quantification of the Predicate.—Hamiltonmaintainsthat“inthoughtthepredicateisalwaysquantified,”andhence he makes it follow immediately from the postulate discussed in thepreceding section that “in logic the quantity of the predicate must beexpressed,ondemand,inlanguage.”“Thequantityofthepredicate,”saysDrBaynes in the authorised expositionofHamilton’sdoctrine contained inhisNew Analytic of Logical Forms, “is not expressed in common languagebecause common language is elliptical.Whatever is not really necessary totheclearcomprehensionofwhatiscontainedinthought,isusuallyelidedin197expression.Butwemustdistinguishbetweentheendswhicharesoughtbycommonlanguageandlogicrespectively.Whilst theformerseeks toexhibitwithclearnessthematterofthought,thelatterseekstoexhibitwithexactnessthe form of thought. Therefore in logic the predicate must always bequantified.”Itisfurthermaintainedthatthequantificationofthepredicateisnecessary for intelligible predication. “Predication is nothing more or lessthantheexpressionoftherelationofquantityinwhichanotionstandstoanindividual,ortwonotionstoeachother.Ifthisrelationwereindeterminate—ifwewereuncertainwhetheritwasofpart,orwhole,ornone—therecouldbenopredication.”

Amongst the practical advantages said to result from quantifying thepredicatearethereductionofallspeciesoftheconversionofpropositionstoone, namely, simple conversion; and the simplification of the laws ofsyllogism.Asregardsthefirstofthesepoints,itmaybeobservedthatifthedoctrine of the quantification of the predicate is adopted, the distinctionbetween subject and predicate resolves itself into a difference in order ofstatement alone. Each propositional form can without any alteration inmeaning be read either forwards or backwards, and every propositionmay,therefore,rightlybesaidtobesimplyconvertible.


It is further argued that the new propositional forms resulting from thequantificationof thepredicateare required inorder toexpress relations thatcannototherwisebesosimplyexpressed.Thus,Ualoneservestoexpressthefact that two classes are co-extensive; and evenω is said to be needed inlogical divisions, since if we divide (say) Europeans into Englishmen,Frenchmen,&c., thisrequiresustothinkthatsomeEuropeansarenotsomeEuropeans(e.g.,EnglishmenarenotFrenchmen).

143.ObjectionsurgedagainsttheQuantificationofthePredicate.—ThosewhorejectHamilton’sdoctrineofthequantificationofthepredicatedenyattheoutset the fundamentalpremissuponwhich it isbased,namely, that thepredicateofapropositionisalwaysthoughtofasadeterminatequantity.Theygofurtheranddenythatitisasarulethoughtofasa198quantity,thatis,asanaggregateofobjects,atall.Wehavealreadyinsection135indicatedgroundsfor the view that, while in the great majority of instances the subject of aproposition is in ordinary thought naturally interpreted in denotation, thepredicateisnaturallyinterpretedinconnotation.Thispsychologicalargumentis valid against Hamilton, inasmuch as he really bases his doctrine upon apsychologicalconsideration;anditseemsunanswerable.

Mill(inhisExaminationofHamilton,pp.495-7)putsthepointasfollows:“I repeat the appeal which I have already made to every reader’sconsciousness:Doeshe,whenhejudgesthatalloxenruminate,adverteveninthe minutest degree to the question, whether there is anything else whichruminates?Isthisconsiderationatallinhisthoughts,anymorethananyotherconsideration foreign to the immediate subject?One personmayknow thatthereareother ruminatinganimals,anothermay think that therearenone,athirdmaybewithoutanyopiniononthesubject:butiftheyallknowwhatismeantbyruminating,theyall,whentheyjudgethateveryoxruminates,meanexactlythesamething.Thementalprocesstheygothrough,sofarasthatonejudgmentisconcerned,ispreciselyidentical;thoughsomeofthemmaygoonfurther,andaddotherjudgmentstoit.Thefact,thattheproposition‘EveryAisB’ onlymeans ‘EveryA is someB,’ so far from being always present inthought, isnotat firstseizedwithoutsomedifficultyby the tyro in logic. Itrequiresacertaineffortofthoughttoperceivethatwhenwesay,‘AllA’sare


B’s,’weonlyidentifyAwithaportionoftheclassB.Whenthelearnerisfirsttoldthat theproposition‘AllA’sareB’s’canonlybeconverted in theform‘SomeB’sareA’s,’Iapprehendthat thisstrikeshimasanewidea;andthatthe truth of the statement is not quite obvious to him, until verified by aparticularexampleinwhichhealreadyknowsthatthesimpleconversewouldbefalse,suchas,‘Allmenareanimals,therefore,allanimalsaremen.’Sofaris it from being true that the proposition ‘AllA’s areB’s’ is spontaneouslyquantifiedinthoughtas‘AllAissomeB.’”

Awordmaybeaddedinreplytotheargumentthatifthe199quantityofthepredicate were indeterminate—if we were uncertain whether the referencewas to the whole or part or none—there could be no predication. This isperfectlytruesolongasweareleftwithallthreeofthesealternatives;butwemayhavepredicationwhichinvolvestheeliminationofonlyoneofthem,sothat there is still indeterminateness as regards the other two. To argue thatunlesswearedefinitelylimitedtooneofthethreeweareleftwithallofthemispracticallytoconfusecontradictorywithcontraryopposition.

A further objection raised to the doctrine of the quantification of thepredicate is that some of the quantified forms are composite not simplepredications.ThusAll S is allP is a condensedmodeof expression,whichmaybeanalysedintothetwopropositionsAllSisPandAllPisS.Similarly,ifwe interpretsomeasexclusiveofall, apoint towhichweshallpresentlyreturn,AllSissomeP isanexponiblepropositionresolvableintoAllS isPandSomePisnotS.Asarule,however,theuseofexponibleformstendstomakethedetectionoffallacythemoredifficult,andthisgeneralconsiderationapplieswithundoubtedforcetotheparticularcaseofthequantificationofthepredicate.Thebearingofthequantificationdoctrineuponthesyllogismwillbebrieflytoucheduponsubsequently,anditwillbefoundthattheproblemofdiscriminating between valid and invalidmoods is renderedmore complexanddifficult.Itmayindeedbedoubtedwhetheranylogicalproblem,withtheone exception of conversion, is really simplified by the introduction ofquantifiedpredicates.

Even apart from the above objections, the Hamiltonian doctrine ofquantification is sufficiently condemnedby itswant of internal consistency.


Its unphilosophical character in this respectwill be shewn in the followingsections.

144. The meaning to be attached to the word “some” in the eightpropositionalformsrecognisedbySirWilliamHamilton.—ProfessorBaynes,inhisauthorisedexpositionofSirWilliamHamilton’sdoctrine,wouldattheoutsetleadustosupposethatwehavenolongertodowiththeindeterminatesome of theAristotelianLogic, but that thisword is now to be used in themoredefinitesenseofsome,butnotall.Heargues,aswe200haveseen,thatintelligiblepredication requiresanabsolutelydeterminate relation in respectof quantity between subject and predicate, and that this ought to be clearlyexpressed in language.Thus,“if theobjectscomprisedunder the subjectbesome part, but not the whole, of those comprised under the predicate, wewriteAllXissomeP,andsimilarlywithotherforms.”

But if it is true thatweknowdefinitely the relativeextentof subjectandpredicate, and if some is used strictly in the sense of some but not all, weshouldhavebutfivepropositionalformsinsteadofeight,namely,—AllSisallP,AllSissomeP,SomeSisallP,SomeSissomeP,192NoSisanyP.


192 Usingsomeinthesensehereindicated,theinterpretationofthepropositionSomeS is someP is not altogether free fromambiguity.The interpretation I amadopting is to regard it as equivalent to the two following propositions withunquantifiedpredicates,namely,SomebutnotallSisPandSomebutnotallPisS.ItthennecessarilyimpliestheHamiltonianpropositionsSomeSisnotanyPandNoSissomeP.

We have already seen (in section 126) that the only possible relationsbetween two terms in respect of their extension are given by the followingfivediagrams,—

These correspond respectively to the five propositional forms givenabove;193 and it is clear that on the view indicated byDrBaynes the eightformsareredundant.194

193 NamelyU,A,Y, I,E.O and η cannot be interpreted as giving preciselydeterminateinformation;OallowsanalternativebetweenYandI,andηbetweenAandI.Fortheinterpretationofωseenote2onpage206.

194 CompareVenn,SymbolicLogic,chapterI.

It is altogether doubtful whetherwriterswho have adopted the eightfoldschemehavethemselvesrecognisedthepitfalls201surroundingtheuseofthewordsome.Manypassagesmightbequotedinwhichtheydistinctlyadoptthemeaning—somebutnotall.Thus,Thomson(LawsofThought,p.150)makesUandAinconsistent.Bowen(Logic,pp.169,170)wouldpassfromItoObyimmediate inference.195HamiltonhimselfagreeswithThomsonandBowenon these points; but he is curiously indecisive on the general question hereraised.Heremarks(Logic,II.p.282)thatsome“isheldtobeadefinitesomewhentheothertermisdefinite,”i.e.,inAandY,ηandO:but“ontheotherhand,when both terms are indefinite or particular, the some of each is left


whollyindefinite,”i.e.,inIandω.196Thisisveryconfusing,anditwouldbemostdifficulttoapplythedistinctionconsistently.Hamiltonhimselfcertainlydoesnotsoapplyit.Forexample,onhisviewitshouldnolongerbethecasethat twoaffirmativepremissesnecessitate an affirmative conclusion; or thattwonegativepremissesinvalidateasyllogism.197Thus,thefollowingshouldberegardedasvalid:

AllPissomeM,AllMissomeS,

therefore, SomeSisnotanyP.NoMisanyP,SomeSisnotanyM,

therefore, SomeorallSisnotanyP.195 “Thissortofinference,”heremarks,“Hamiltonwouldcallintegration,asits

effectis,afterdeterminingonepart,toreconstitutethewholebybringingintoviewtheremainingpart.”

196 CompareVeitch,InstitutesofLogic,pp.307to310,and367,8.“Hamiltonwould introduce some only into the theory of propositions, without, however,discarding the meaning of some at least. It is not correct to say that Hamiltondiscarded the ordinary logical meaning of some. He simply supplemented it byintroducing into thepropositional forms thatofsomeonly.” “Some, according toHamilton,isalwaysthoughtassemi-definite(someonly)wheretheothertermofthe judgment is universal.” Mr Lindsay, however, in expounding Hamilton’sdoctrine(Appendix toUeberweg’sSystemofLogic,p.580)saysmoredecisively,—“Since the subject must be equal to the predicate, vagueness in thepredesignationsmustbeasfaraspossibleremoved.Someistakenasequivalenttosomebutnotall.”Spalding(Logic,p.184)definitelychoosestheotheralternative.He remarks that in his own treatise “the received interpretation some at least issteadilyadheredto.”

197 Theanticipationofsyllogisticdoctrinewhichfollowsisnecessaryinordertoillustratethepointwhichwearejustnowdiscussing.

202Suchsyllogismsas these,however,arenotadmittedbyHamiltonandThomson; and, on the other hand, Thomson admits as valid certaincombinations which on the above interpretation are not valid. Hamilton’ssupremecanonof thecategorical syllogism is:—“Whatworse relation198 ofsubject and predicate subsists between either of two terms and a commonthirdterm,withwhichone,atleast,ispositivelyrelated;thatrelationsubsistsbetweenthetwotermsthemselves”(Logic, II.p.357).Thisclearlyprovides


that one premiss at least shall be affirmative, and that an affirmativeconclusion shall follow from two affirmative premisses.Thomson (Laws ofThought,p.165)explicitly laysdown thesame rules;andhis tableofvalidmoods (given on p. 188) is (with the exception of one obvious misprint)correctandcorrectonlyifsomemeans“some,itmaybeall.”

198 Thenegative relation is here considered “worse” than the affirmative, andtheparticularthantheuniversal.

145.Theuseof“some”inthesenseof“someonly.”—Jevons,inreplytothequestion, “What resultswould follow ifwewere to interpret ‘SomeA’sare B’s’ as implying that ‘Some other A’s are not B’s’?” writes, “Theproposition‘SomeA’sareB’s’isintheformI,andaccordingtothetableofoppositionIistrueifAistrue;butAisthecontradictoryofO,whichwouldbe the form of ‘Some otherA’s are notB’s.’ Under such circumstancesAcouldneverbetrueatall,becauseitstruthwouldinvolvethetruthofitsowncontradictory,which is absurd” (Studies inDeductiveLogic, 151). It is not,however, thecasethatwenecessarilyinvolveourselvesinself-contradictionifweusesomeinthesenseofsomeonly.Whatshouldbepointedoutisthat,ifweusethewordinthissense,thetruthofInolongerfollowsfromthetruthofA; and that, so far from this being the case, these two propositions areinconsistentwitheachother.

Takingthefivepropositionalforms,AllSisallP,AllSissomeP,SomeSisallP,SomeSissomeP,NoS isP,and interpretingsome in thesenseofsomeonly,itistobeobservedthateachoneofthemisinconsistentwitheachof theothers,whilstat thesame timenoone is thecontradictoryofany203one of the others. If, for example, on this scheme we wish to express thecontradictoryofU,wecandosoonlybyaffirminganalternativebetweenY,A, I, and E. Nothing of all this appears to have been noticed by theHamiltonian writers. Thus, Thomson (Laws of Thought, p. 149) gives aschemeofoppositioninwhichEandIappearascontradictories,butAandOascontraries.

Oneofthestrongestargumentsagainsttheuseofsomeinthesenseofsomeonly is verywell putbyProfessorVeitch,himself adiscipleofSirWilliamHamilton.Someonly,heremarks,isnotsofundamentalassomeatleast.The


formerimpliesthelatter;butIcanspeakofsomeatleastwithoutadvancingto themoredefinite stageof someonly. “Before I can speak of some only,mustInothaveformedtwojudgments—theonethatsomeare,theotherthatothersof thesameclassarenot?……Thesomeonlywould thusappearasthecompositeoftwopropositionsalreadyformed……Itseemstomethatwemust, first of all, work out logical principles on the indefinite meaning ofsome at least…… Some only is a secondary and derivative judgment.”(InstitutesofLogic,p.308).

Ifsomeisusedinthesenseofsomeonly, thefurtherdifficultyariseshowwearetoexpressanyknowledgethatwemayhappentopossessaboutapartofaclasswhenweareinignoranceinregardtotheremainder.Supposingforexample,thatalltheS’sofwhichIhappentohavehadexperienceareP’s, Iamnot justifiedinsayingeither thatallS’sareP’sor thatsomeS’sareP’s.TheonlysolutionofthedifficultyistosaythatallorsomeS’sareP’s.Thecomplexitythatthiswouldintroduceisobvious.

146. The interpretation of the eight Hamiltonian forms of proposition,“some”beingusedinitsordinarylogicalsense.199—Takingthefivepossiblerelations between two terms, as illustrated by the Eulerian diagrams, anddenoting themrespectivelybyα,β,γ,δ,ε, as in section126,wemaywriteagainst each of the propositional forms the relations which are compatiblewith204it,onthesuppositionthatsomeisusedinitsordinarylogicalsense,thatis,asexclusiveofnonebutnotofall:—200

U αA α,βY α,γI α,β,γ,δE εη β,δ,εO γ,δ,εω α,β,γ,δ,ε

199 The corresponding interpretationwhen some is used in the sense of someonlyisgiveninnotes1and2onpage200,andinnote2onpage206.

200 IftheHamiltonianwritershadattemptedtoillustratetheirdoctrinebymeans


oftheEuleriandiagrams,theywouldIthinkeitherhavefoundittobeunworkable,ortheywouldhaveworkeditouttoamoredistinctandconsistentissue.

Wehavethenthefollowingpairsofcontradictories—A,O;Y,η;I,E.ThecontradictoryofUisobtainedbyaffirminganalternativebetweenηandO.

Without the use of quantified predicates, the same information may beexpressedasfollows:—

U=SaP,PaS ;A=SaP ;Y=PaS ;I=SiP ;E=SeP ;η=PoS ;O=SoP.

Whatinformation,ifany,isgivenbyωwillbediscussedinsection149.

147. The propositions U and Y.—It must be admitted that thesepropositionsaremetwithinordinarydiscourse.205Wemaynot indeedfindpropositionswhichareactuallywrittenintheformAllSisallP ;butwehavetoallintentsandpurposesU,wheneverthereisanunmistakeableaffirmationthatthesubjectandthepredicateofapropositionareco-extensive.Thus,alldefinitionsarepracticallyUpropositions; soareall affirmativepropositionsofwhichboth the subjectand thepredicateare singular terms.201Takealsosuch propositions as the following: Christianity and civilization are co-extensive; Europe, Asia, Africa, America, and Australia are all thecontinents;202 The three whom I have mentioned are all who have everascended the mountain by that route; Common salt is the same thing assodiumchloride.203

201 Take theproposition,“MrGladstone is thepresentPrimeMinister.” IfanyonedeniesthatthisisU,thenhemustdenythattheproposition“MrGladstoneisanEnglishman”isA.Wehaveatanearlierstagediscussedthequestionhowfarsingular propositions may rightly be regarded as constituting a sub-class ofuniversals.

202 In this and the example that follows the predicate is clearly quantifieduniversally; so that if thesearenotU propositions, theymustbeYpropositions.Butitisequallyclearthatthesubjectdenotesthewholeofacertainclass,however


limitedthatclassmaybe.203 These are all examples of what Jevons would call simple identities as

distinguishedfrompartialidentities.Comparesection138.

Such propositions as the following, sometimes known as exclusivepropositions,maybegivenasexamplesofY:OnlyS isP ;Graduates aloneare eligible for the appointment; Some passengers are the only survivors.Thesepropositionsmaybe interpretedasbeingequivalent to the following:SomeSisallP ;Somegraduatesareallwhoareeligiblefortheappointment;Some passengers are all the survivors.204 This is, indeed, the only way oftreatingthepropositionswhichwillenableustoretaintheoriginalsubjectsassubjectsandtheoriginalpredicatesaspredicates.

204 In thesepropositions,some is tobe interpreted in the indefinite sense, andnotasexclusiveofall.

WecannotthenagreewithProfessorFowlerthattheadditionalforms“arenotmerely unusual, but are such aswenever douse” (Deductive Logic, p.31).Still in treating thesyllogism&c.on the traditional lines, it isbetter toretainthetraditionalscheduleofpropositions.Theadditionoftheforms206UandY does not tend towards simplification, but the reverse; and their fullforcecanbeexpressedinotherways.Onthisview,whenwemeetwithaUproposition,AllSisallP,wemayresolveitintothetwoApropositions,AllSisPandAllP isS,which taken together are equivalent to it; andwhenwemeetwithaYproposition,SomeSisallPorSaloneisP,wemayreplaceitbytheApropositionAllPisS,whichityieldsbyconversion.

148.Thepropositionη.—ThispropositionintheformNoSissomePisnotIthinkeverfoundinordinaryuse.Wemay,however,recogniseitspossibility;anditmustbepointedoutthataformofpropositionwhichwedomeetwith,namely.NotonlySisPorNotSaloneisP,ispracticallyη,providedthatwedonotregardthispropositionasimplyingthatanySiscertainlyP.

ArchbishopThomsonremarksthatη“hasthesemblanceonly,andnotthepowerofadenial.Truethoughit is, itdoesnotpreventourmakinganotherjudgmentoftheaffirmativekind,fromthesameterms”(LawsofThought,§79).This iserroneous; foralthoughA andηmaybe true together,U andηcannot, and Y and η are strictly contradictories.205 The relation of


contradiction inwhichY and η stand to each other is perhaps brought outmoreclearlyiftheyarewrittenintheformsOnlySisP,NotonlySisP,orSalone is P,Not S alone is P. It will be observed, moreover, that η is theconverseofO, and vice versâ. If, therefore, η has no power of denial, thesamewillbetrueofOalso.ButitcertainlyisnottrueofO.

205 Weareagaininterpretingsomeasindefinite.Ifitmeanssomeatmost,thenthepowerofdenialpossessedbyηisincreased.

149.Thepropositionω.—Thepropositionω,SomeSisnotsomeP,isnotinconsistentwithanyoftheotherpropositionalforms,notevenwithU,AllSis all P. For example, granting that “all equilateral triangles are allequiangular triangles,” still “this equilateral triangle is not that equiangulartriangle,”whichisall thatωasserts.SomeSisnotsomeP isindeedalwaystrue except when both the subject and the predicate are the name of anindividual and the same individual.206 De 207 Morgan207 (Syllabus, p. 24)observesthatitscontradictoryis—“SandParesingularandidentical;thereisbutoneS,thereisbutoneP,andSisP.”208Itmaybesaidwithouthesitationthatthepropositionωisofabsolutelynologicalimportance.

206 Somebeingagaininterpretedinitsordinarylogicalsense.MrJohnsonpointsoutthatifsomemeanssomebutnotall,weareledtotheparadoxicalconclusionthatωisequivalenttoU.Wemayregardastatementinvolvingareferencetosomebutnotallasastatementrelatingtosomeatleast,combinedwithadenialofthecorrespondingstatementinwhichallissubstitutedforsome.Onthisinterpretation,SomeSisnotsomePaffirmsthat“SandParenotidenticallyone,”butalsodeniesthat“someSisnotanyP”andthat“somePisnotanyS”; that is, itaffirmsSaPandPaS.

207 De Morgan in several passages criticizes with great acuteness theHamiltonianschemeofpropositions.

208 ProfessorVeitch remarks that inω “we assert parts, and that these canbedivided,orthattherearepartsandparts.Ifwedenythisstatement,weassertthatthethingspokenofisindivisibleoraunity……Wemaysaythattherearemenandmen.We say, aswedo every day, there are politicians and politicians, there areecclesiasticsandecclesiastics,therearesermonsandsermons.Thesearebutcovertformsofthesomearenotsome……‘Somevivisectionisnotsomevivisection’istrueand important; for theonemaybewithananaesthetic, theotherwithout it”(InstitutesofLogic,pp.320,1).ItwillbeobservedthatthepropositionTherearepoliticians and politicians is here given as a typical example of ω. Theappropriateness of this is denied byMrMonck. “Again, can it be said that thepropositionTherearepatriotsandpatriotsisadequatelyrenderedbySomepatriotsare not some patriots? The latter proposition simply asserts non-identity: the


former is intended to imply also a certain degree of dissimilarity [i.e., in thecharacteristicsorconsequencesofthepatriotismofdifferentindividuals].Buttwonon-identicalobjectsmaybeperfectlyalike”(IntroductiontoLogic,p.xiv).

150.SixfoldScheduleofPropositionsobtainedbyrecognisingYandη,inadditiontoA,E,I,O.209—ThescheduleofpropositionsobtainedbyaddingY and η to the ordinary schedule presents some interesting features, and isworthy of incidental recognition and discussion.210 It has been shewn insection100thatintheordinaryschemetherearesixandonlysixindependentpropositionsconnectingany two terms,namely, 208SaP,PaS,SeP (=PeS),SiP(=PiS),PoS,SoP.Ifwewritethesecondandthelastbutoneoftheseinforms inwhichSandP are respectively subject andpredicate,wehave theschedulewhichwearenowconsidering,namely,

SaP = AllSisP ;SyP = OnlySisP ;SeP = NoSisP ;SiP = SomeSisP ;SηP = NotonlySisP ;Sop = SomeSisnotP.

209 Inthisschedulesomeisinterpretedthroughoutinitsordinarylogicalsense.Uisomittedonaccountofitscompositecharacter;itsinclusionwouldalsodestroythesymmetryofthescheme.

210 It is not intended that this sixfold schedule should supersede the fourfoldschedule in the main body of logical doctrine. It is, however, important torememberthattheselectionofanyonescheduleismoreorlessarbitrary,andthatnoscheduleshouldbesetupasauthoritativetotheexclusionofallothers.

It will be observed that the pair of propositions, SyP and SηP, arecontradictories;sothatwenowhavethreepairsofcontradictories.Thereareofcourseotheradditionstothetraditionaltableofopposition,andsomenewrelations will need to be recognised, e.g., between SaP and SyP.With thehelp,however,ofthediscussioncontainedinsection107,thereaderwillhavenodifficultyinworkingouttherequiredhexagonofoppositionforhimself.

As regards immediate inferences, we cannot in this scheme obtain anysatisfactory obverse of either Y or η, the reason being that they havequantified predicates, and that, therefore, the negation cannot in thesepropositions be simply attached to the predicate. We have, however, the


followinginterestingtableofotherimmediateinferences:—211

Converse. Contrapositive. Inverse.SaP = PyS = PʹaSʹ = SʹyPʹSyP = PaS = PʹySʹ = SʹaPʹSeP = PeS = PʹyS = SʹyPSiP = PiS = PʹηS = SʹηPSηP = PoS = PʹηSʹ = SʹoPʹSoP = PηS = PʹoSʹ = SʹηPʹ

211 ItwillbeobservedthattheimpracticabilityofobvertingYandηleadstoacertainwantofsymmetryinthethirdandfourthcolumns.

Themainpointstonoticehereare(1)thateachpropositionnowadmitsofconversion,contraposition,andinversion;and(2)thattheinferredpropositionis ineverycaseequivalent totheoriginalproposition,sothat thereisnot inanyofthe209inferencesanylossoflogicalforce.Inotherwords,weobtainineachcaseasimpleconverse,asimplecontrapositive,andasimpleinverse.

EXERCISES.

151. Explain precisely how it is that O admits of ordinary conversion if the principle of thequantificationofthepredicateisadopted,althoughnototherwise.[K.]

152. Draw out a table, corresponding to the ordinary Aristotelian table of opposition, for the sixpropositions,A,Y,E,I,η,O(somebeinginterpretedinthesenseofsomeatleast).[K.]


CHAPTERVIII.

THEEXISTENTIALIMPORTOFCATEGORICALPROPOSITIONS.212

212 Itwillbeadvisableforstudents,onafirstreading,toomitthischapter.

153.ExistenceandtheUniverseofDiscourse.—Ithasbeenshewinsection49 that every judgment involves an objective reference, or—as it mayotherwisebeexpressed—areference tosomesystemofrealitydistinct fromtheactofjudgmentitself.Thereferencemaybetothetotalsystemofrealitywithout limitation,or itmaybe to someparticular aspectorportionof thatsystem. Whatever it may be, we may speak of it as the universe ofdiscourse.213Theuniverseofdiscoursemaybe limited invariousways; forexample,tophysicalobjects,ortopsychicalevents,oragainwithreferencetotimeorspace.Butinallcasesitisauniverseofrealityinthesenseinwhichthat term has been used in section 49. The nature of the reference inpropositions relating to fictitiousobjects, forexample, to thecharactersandoccurrences inaplayoranovel,maybe speciallyconsidered.Wemaysaythat in a case of this kind the universe of discourse consists of a series ofstatementsaboutpersonsandeventsmadebyacertainauthor;anditisclearthat such statementshaveobjective reality, although thepersons andeventsthemselvesarefictitious.Itfollowsthat,asregards211thereferencetoreality,suchapropositionas“HamletkilledPolonius”mustbeconsideredelliptical.Forthereferenceisnottorealpersonsortotheactualcourseofeventsinthepasthistoryoftheworld,asitiswhenwesay“MaryStuartwasbeheaded,”buttoaseriesofdescriptionsgivenbyShakespeareinaparticularplay.Thesedescriptionshave,however,arealityoftheirown,and(thedifferentnatureofthereferencebeingclearlyunderstood)IamnomorefreetosaythatHamletdid not kill Polonius (that is, that Shakespeare did not describe Hamlet askillingPolonius)thanIamtosaythatMaryStuartwasnotbeheaded.

213 “The universe of discourse is sometimes limited to a small portion of theactual universe of things, and is sometimes co-extensive with that universe”(Boole, Laws of Thought, p. 166). On the conception of a limited universe of


discourse,comparealsoDeMorgan,SyllabusofaProposedSystemofLogic, §§122,3, andFormalLogic, p.55;Venn,Symbolic Logic, pp. 127, 8; and Jevons,PrinciplesofScience,chapter3,§4.

Thesubstanceoftheabovehasbeenexpressedbysayingthatrealityistheultimatesubjectofeveryproposition.Everypropositionmakesanaffirmationaboutacertainuniverseofdiscourse,andtheuniverseofdiscourse(whateveritmaybe)hassomerealcontent.Inthissensetheneverypropositionhasanexistent subject.214 A further question may, however, be raised, namely,whether—usingtheword“subject”initsordinarylogicalsignification—allorany propositions should be interpreted as implying the existence (oroccurrence) of their subjectswithin the universe of discourse (or particularportionofreality)towhichreferenceismade.Itismainlywiththisproblem,andthewaysinwhichordinarylogicaldoctrinesareaffectedbyitssolution,thatweshallbeconcernedinthepresentchapter.

214 CompareBradley,PrinciplesofLogic,p.41.

In our discussion of existential import it will not be necessary that weshould make any attempt to determine the ultimate nature of reality. Thequestions at issue are, however, not exactly easy of solution, and varioussourcesofmisunderstandingareapttoarise.

There isone sense inwhich the existenceof somethingcorresponding tothe terms employed must be postulated in all predication. For in order tomakeuseofanyterminanintelligiblesensewemustmentallyattachsomemeaningto it.Hence theremustbesomething in themindcorrespondingtoevery term we use. Even in cases where there cannot be said to be anycorresponding mental product, there must at any rate 212 be somecorresponding mental process. This applies even to such terms as roundsquare ornon-humanman or root ofminus one.We are not indeed able toformanimageofaroundsquareoranideaofanon-humanman,norcanweevaluatetherootofminusone.Butweattachameaningtotheseterms,andthey must therefore have a mental equivalent of some sort. In the case of“round square” or “non-human man” this is not the actual combination inimagination or idea of “round”with “square” or “non-human”with “man,”for suchcombinationsare impossible.But it is the ideaof thecombination,regarded as a problem presented for solution, and perhaps involving an


unsuccessful effort to effect the combination in thought. It is apparently ofexistenceofthiskindthatsomewritersarethinkingwhentheymaintainthatof necessity every proposition implies logically the existence of its subject.Butourmeaningissomethingquitedifferentwhenwespeakofexistenceinthe universe of discourse. The nature of the distinctionmay bemademoreclearbythefollowingconsiderations.

Itwillbeadmittedthatwhateverelseisincludedinthefullimplicationofauniversal proposition, it at least denies the existence of a certain class ofobjects.NoSisPdeniestheexistenceofobjectsthatarebothSandP ;AllSisPdeniestheexistenceofobjectsthatareSwithoutalsobeingP. In thesepropositions,however,wedonotintendtodenytheexistenceofSP(orSPʹ)asobjectsofthought.Forexample,inthepropositionNorosesareblueitisnotour intention todenythatwecanforman ideaofblueroses ;nor in thepropositionAllruminantanimalsarecloven-hoofedisitourintentiontodenythatruminant animalswithout clovenhoofs can exist as objects of thought.Theseillustrationsmayhelpustounderstandmoreclearlywhatismeantbyexistenceintheuniverseofdiscourse.TheuniverseofdiscourseinthecaseofthepropositionNoS isP is theuniverse (whatever itmaybe) inwhich theexistenceofSPisdenied.Theuniverseofdiscourseinthecaseofauniversalaffirmative proposition may be defined similarly. As regards particulars itmay be best to seek an interpretation through the universals by which theparticulars213arecontradicted.Thus,theuniverseofdiscourseinthecaseofthepropositionSomeSisPmaybedefinedastheuniverse(whateveritmaybe) inwhich the existence ofSP would be understood to be denied in thecorresponding universal negative. The propositionSome S is notPmay bedealtwithsimilarly.

The question whether a categorical proposition is to be interpreted asformallyimplyingthatitstermsarethenamesofexistingthingsmaythenbeinterpreted as follows: Given a categorical proposition with S and P assubject andpredicate, is the existenceof SorofP formally implied in thatsphere(whateveritmaybe)inwhichtheexistenceofSP(orSPʹ)isdeniedbytheproposition(orbyitscontradictory)?

The questionmay be somewhat differently expressed as follows. Such a


proposition as No S is P denies the existence of a certain complex ofattributes, namely,SP. But with rare exceptions, S itself signifies a certaincomplex of attributes; and so does P. Does the proposition affirm theexistenceoftheselattercomplexesinthesamesenseasthatinwhichitdeniestheexistenceoftheformercomplex?

Nogeneralcriterioncanbelaiddownfordeterminingwhatisactuallytheuniverse of discourse in any particular case. It may, however, be said thatknowledgeastowhatistheuniversereferredtoisinvolvedinunderstandingthemeaningof anygivenproposition; andcases inwhich there canbe anypractical doubt are exceptional.215 Thus, in the propositions No roses areblue, All men are mortal, All ruminant animals are cloven-hoofed, thereference clearly is to the actual physical universe; in The wrath of theOlympiangodsisveryterribletotheuniverseoftheGreekmythology;216inFairiesareable toassumedifferent forms to theuniverseof folk-lore;217 inTwostraightlinescannotencloseaspacetotheuniverseofspatialintuitions.

215 Itmustatthesametimebeadmittedthatcontroversiessometimesturnuponanunrecognisedwantofagreementbetweenthecontroversialistsastotheuniverseofdiscoursetowhichreferenceismade.

216 The universe of the Greek mythology does not consist of gods, heroes,centaurs,&c.,butofaccountsofsuchbeingscurrentlyacceptedinancientGreece,andhandeddowntousbyHomerandotherauthors.Asregards thereference toreality, therefore, such a proposition asThewrath of theOlympian gods is veryterribleisellipticalinasensealreadyexplained.


217 Hereagainthereisanellipsis.Theuniverseoffolk-loredoesnotconsistoffairies, elves, &c., but of descriptions of them, based on popular beliefs, andconventionally acceptedwhen such beings are referred to.Of course for anyonewhoreallybelievedintheexistenceoffairies therewouldbenoellipsis,andtheuniverseofdiscoursewouldbedifferent.

214 With respect to the existential import of propositions the followingquestionsofferthemselvesforconsideration: (1) Is the problem one with which logic, and more particularly formallogic,isproperlyconcerned? (2)How should the propositions belonging to the traditional schedule beinterpretedasregardstheirexistentialimplications? (3)Canwe formulatea scheduleofpropositionswhichdirectlyaffirmordeny existence, and how will such a schedule be related to the traditionalschedule? (4)Howareordinarylogicaldoctrinesaffectedbytheanswergiventothesecondofthesequestions?

Itisclearthatthefirstandfourthofthesequestionsareconnected,sinceifthefourthadmitsofanypositiveansweratall,thefirstistherebyansweredintheaffirmative.Since,however, thefirstquestionblocksthewayandseemstodemandananswerbeforewecarrythediscussionfurther,itwillbewelltodealwithitbrieflyattheoutset.

Thesecondandthirdquestionsarealsocloselyconnectedtogether.

Betweenthesecondandfourthquestionsanimportantdistinctionmustbedrawn.Thesecondquestionisoneofinterpretation,andwithincertainlimitsthe answer to it is amatter of convention. Hence a given solutionmay bepreferredongroundsthatwouldnotjustifytherejectionofothersolutionsasaltogether erroneous, although they may be considered inconvenient orunsuitable.Buttheanswertothefourthquestionisnotsimilarlyamatterofconvention.On thebasisofanygiven interpretationofpropositional forms,the manner in which logical doctrines are affected can admit of only onecorrectsolution.

It is to be observed further that the fourth question can be dealt withhypothetically, that is to say, we can work out the consequences of


interpretationswhichwehavenointentionof215adopting;anditisdesirablethat we should work out such consequences before deciding upon theadoption of any given interpretation. Hence we propose to deal with thefourth question before discussing the second. The third question mayconvenientlybetakenafterthefirst.

154.FormalLogicand theExistential ImportofPropositions.—We havethen,inthefirstplace,brieflytoconsiderthequestionwhethertheproblemofexistentialimportisonewithwhichlogichasanyproperconcern.Itmaybeurgedthatformallogic,atanyrate,cannotfromitsverynaturebeconcernedwithquestionsrelatingtoexistenceinanyotherspherethanthatofthought.Thefunctionoftheformallogician,itmaybesaid,istodistinguishbetweenthat which is self-consistent and that which is self-contradictory; it is hisbusinesstodistinguishbetweenwhatcanandwhatcannotexistintheworldof thought. But beyond this he cannot go. Any considerations relating toobjectiveexistencearebeyondthescopeofformallogic.

Wemaymeettheaboveargumentbyclearlydefiningourposition.Itisofcourse no function of logic to determine whether or not certain classesactually exist in any given universe of discourse, any more than it is thefunction of logic to determinewhether given propositions are true or false.Butitdoesnotfollowthatlogichas,therefore,noconcernwithanyquestionsrelating toobjectiveexistence.For, just as, certainpropositionsbeinggiventrue,logicdetermineswhatotherpropositionswillasaconsequencealsobetrue, so given an assertion or a set of assertions to the effect that certaincombinations do or do not exist in a given universe of discourse, it candetermine what other assertions about existence in the same universe ofdiscourse follow therefrom.218 As a matter of fact, the premisses in anyargumentnecessarilycontaincertainimplicationsinregardtoexistenceintheparticularuniverseof216discoursetowhichreferenceismade,andthesameis trueof the conclusion; it is accordingly essential that the logician shouldmakesurethatthelatterimplicationsareclearlywarrantedbytheformer.

218 Thelatterpartofthisstatementisindeednothingmorethanarepetitionofthe former part from a rather different point of view. The doctrine that theconclusions reachedby the aidof formal logic cannever domore than relate towhat ismerelyconceivable isaverymischievouserror.Thematerial truthof the


conclusion of a formal reasoning is only limited by the material truth of thepremisses.

Withoutatpresentgoingintoanydetailwemayverybrieflyindicateoneor two existential questions that cannot be altogether excluded fromconsiderationinformallogic.Universalpropositions,aswehaveseen,assertnon-existence in some sphere of reality; and it is not possible to bring outtheirfull importwithoutcallingattentiontothisfact.Again, thepropositionAllSisPatleastinvolvesthatifthereareanyS’sintheuniverseofdiscourse,theremustalsobesomeP’s,whileitdoesnotseemnecessarilytoinvolvethatifthereareanyP’stheremustbesomeS’s.Butnowconverttheproposition.TheresultisSomePisS,andthisdoesinvolvethatifthereareanyP’stheremustbesomeS’s.219Howthen217cantheprocessofconversionbeshewntobe valid without some assumption which will serve to justify this latterimplication?Similarly, inpassing fromAllS isP toSomenot-S is not-P, itmust at leastbeassumed that ifS doesnot constitute theentireuniverseofdiscourse, neither doesP do so. It is indeed quite impossible to justify theprocessofinversioninanycasewithouthavingsomeregardtotheexistentialinterpretationofthepropositionsconcerned.220

219 Dr Wolf denies this. His argument is, however, based mainly on themisinterpretationofasingleconcreteexample.“Letus,”hesays,“takeaconcreteexample.Some things that children fear are ghosts.Does this proposition implythat if there is anything that children fear then there are alsoghosts?Surelyonemaylegitimatelymakesuchanassertionwhilebelievingthattherearethingsthatchildrenfear,andyetabsolutelydisbelievingintheexistenceofghosts.Infacttheabovepropositionmightverywellbeusedinconjunctionwithanexpressdenialoftheexistenceofghostsinordertoprovethat,whilesomethingsthatchildrenfeararereal,theyarealsoafraidofthingsthatdonotexist,butaremerelyimaginary”(StudiesinLogic,p.144).Anyspeciousnessthatthisargumentmaypossessarisesfromtheambiguityofthewords“thing”and“real.”Itisclearthatinordertomakethe proposition in question intelligible theword “things”must be interpreted tomean“things, realor imaginary.”Moreover “imaginary things”havea realityoftheirown,thoughitisnotaphysical,materialreality.Ghosts,therefore,doexistintheuniverseofdiscoursetowhichreferenceismade.Theobjectsdenotedbythepredicateofthepropositionhaveinfactjustthesamekindofexistenceascertainof the objects denoted by the subject. Looking at the matter from a slightlydifferentpointofview,itisclearthatifby“things”inthesubjectwemeanthingshaving material existence, then unless ghosts have a similar existence thepropositionisnottrue.

Bearing in mind the constant ambiguity of language, and the ways in which


verbal formsmayfail to representadequately the judgments theyare intended toexpress,itwouldinanycasebeunsatisfactorytoallowaquestionofthekindweareherediscussingtobedecidedbyasingleconcreteexample.DrWolf’sviewisthatSomeSisPdoesnot implythat if thereareanyS’s therearealsosomeP’s.SupposethenthattherearesomeS’sandthattherearenoP’s.Itfollowsthatthereare S’s but not a single one of them is P. What in these circumstances thepropositionSomeSisPcanmeanitisdifficulttounderstand.

SofarasDrWolf’sargumentisindependentoftheaboveconcreteexample,itappears todependuponan identificationof thepropositionSomeSisPwith thepropositionSmaybeP.Thelatterisamodalform,andisundoubtedlyconsistentwiththeexistenceofSandthenon-existenceofP.ButIventuretothinkthattheidentificationofthetwoformsrunsentirelycountertothecurrentuseoflanguage.I amquiteprepared to admit that ifAllS isP is interpreted as an unconditionaluniversal,meaningSassuchisP,itstruecontradictoryisSmaybeP,notSomeSisP.ButthisisjustbecauseIdonotthinkthatSomeSisPwouldbeunderstoodtoexpress merely the abstract compatibility of S andP. Certainly DrWolf’s ownconcrete example, referred to above, cannot bear this interpretation. For somefurtherobservationsonmodals inconnexionwithexistential import, seesections160and163.

220 Jevonsremarksthathedoesnotseehowtherecanbeindeductivelogicanyquestionaboutexistence,andobserves,withreferencetotheoppositeviewtakenbyDeMorgan,that“thisisoneofthefewpointsinwhichitispossibletosuspecthim of unsoundness “ (Studies in Deductive Logic, p. 141). It is, however,impossible to attach any meaning to Jevons’s own “Criterion of Consistency,”unlessithassomereferenceto“existence.”“Itisassumedasanecessarylawthatevery term must have its negative. Thence arises what I propose to call theCriterion of Consistency, stated as follows:—Any two or more propositions arecontradictorywhen,andonlywhen,afterallpossiblesubstitutionsaremade,theyoccasion the total disappearance of any term, positive or negative, from theLogicalAlphabet”(p.181).Whatcanthismeanbutthatalthoughwemaydenytheexistence of the combination AB, we cannot without contradiction deny theexistence of A itself, or not-A, or B, or not-B? This assumption regarding theexistential implication of propositions runs through the whole of Jevons’sequational logic.Thefollowingpassage, forexample, is takenalmostat random:“Thereremainfourcombinations,ABC,aBC,abC,abc.Butthesedonotstandonthesame logical footing,because ifwewere to removeABC, therewouldbenosuchthingasAleft;andifweweretoremoveabctherewouldbenosuchthingascleft.Nowitisthecriterionorconditionoflogicalconsistencythateveryseparatetermanditsnegativeshallremain.HencetheremustexistsomethingswhicharedescribedbyABC,andotherthingsdescribedbyabc”(p.216).

218155.TheExistentialFormulationofPropositions.—Wemaydefineanexistential proposition as one that directly affirms or denies existence (oroccurrence) in the universe of discourse (or portion of reality) to which


referenceismade.Suchpropositionsareofcoursemetwithinordinaryformsofspeech:forexample,Godexists,Itrains,Therearewhitehares,Itdoesnotrain,Unicornsarenon-existent.Thereisnorosewithoutathorn.Sometimestheaffirmationordenialofexistencetakesalesssimpleform,butisnonethelessdirect: forexample,TheassassinationofCaesar is anhistorical event,D’Artagnanisnotanimaginaryperson,Thecentaurisafictionofthepoets,Thelargecopperbutterflyisextinct.

Intheformalexpressionofexistentialpropositionsitwillbeconvenienttomake use of certain symbols described in the preceding chapter. Thus, theaffirmationof theexistenceofSmaybewritten in the formS > 0, and thedenial of the existence of S in the form S = 0. We shall then have anexistential schedule of propositions if we reduce our statements to one orotheroftheseformsortoaconjunctiveordisjunctivecombinationofthem.The relationbetween the traditional schedule and an existential schedule ofthiskindwillbediscussedinthenextsectionbutone.

It may here be pointed out that since the universe of discourse is itselfassumed to be real and hence cannot be entirely emptied of content, anydenialofexistenceinvolvesalsoanaffirmationofexistence.ForifwedenytheexistenceofS,wetherebyimplicitlyaffirmtheexistenceofnot-S, sincebythelawofexcludedmiddleeverythingintheuniverseofdiscoursemustbeeither S or not-S. It follows that every proposition contains directly orindirectlyanaffirmationofexistence.221

221 In an article in Baldwin’sDictionary of Philosophy and Psychology,MrsLadd Franklin points out that the proposition All S is P is equivalent to thepropositionEverythingisPornot-S,andhencenecessarilyimpliestheexistenceofeitherP ornot-S.Writex fornot-S andy forP, so that the original propositionbecomesAll but x is y ; it then implies, as its minimum existential import, theexistenceofeitherxory.

156.VariousSuppositionsconcerningtheExistentialImportofCategoricalPropositions.—Several different views may be 219 taken as to whatimplication with regard to existence, if any, is involved in categoricalpropositions of the traditional type. The following may be formulated forspecialdiscussion:—222

222 Thesuppositionsthatfollowarenotintendedtobeexhaustive.Wemight,for


instance, regardpropositionsas implying theexistencebothof their subjectsandtheir predicates, but not of the contradictories of these; or we might regarduniversalsasalwaysimplyingtheexistenceoftheirsubjects,butparticularsasnotnecessarilyimplyingtheexistenceoftheirs(seenote3onp.241);oraffirmativesasalwaysimplyingtheexistenceoftheirsubjects,butnegativesasnotnecessarilyimplying the existence of theirs. This last supposition represents the view ofUeberweg.StillanotherviewistakenbyLewisCarroll,whoregardsallcategoricalpropositions, except universal negatives, as implying the existence of theirsubjects.“Ineverypropositionbeginningwithsomeorall,theactualexistenceofthe subject is asserted. If, for instance, I say ‘allmisers are selfish,’ Imean thatmisersactuallyexist.IfIwishedtoavoidmakingthisassertion,andmerelytostatethelaw thatmiserlinessnecessarily involvesselfishness, I should say ‘nomisersareunselfish,’whichdoesnotassertthatanymisersexistatall,butmerelythat,ifanydid exist, theywould be selfish” (Gameof Logic, p. 19). Itwould take toomuchspace,however,togiveaseparatediscussiontosuppositionsotherthanthosementionedinthetext.

(1)Itmaybeheldthateverycategoricalpropositionshouldbeinterpretedas implying the existence both of objects denoted by the terms directlyinvolved and also of objects denoted by their contradictories; that, forexample,AllSisPshouldberegardedasimplyingtheexistenceofS,not-S,P,not-P.ThisviewisimpliedinJevons’sCriterionofConsistencymentionedinthenoteonpage217.ItisalsopracticallyadoptedbyDeMorgan.223

223 “By the universe (of a proposition) is meant the collection of all objectswhicharecontemplatedasobjectsaboutwhichassertionordenialmaytakeplace.Leteverynamewhichbelongstothewholeuniversebeexcludedasneedless:thismustbeparticularlyremembered.Leteveryobjectwhichhasnot thenameX(ofwhichtherearealwayssome)beconceivedas thereforemarkedwith thenamexmeaningnot-X”(Syllabus,pp.12,13).Compare,also,DeMorgan’sFormalLogic,p.55.

(2)Itmaybeheldthateverypropositionshouldbeinterpretedasimplyingsimply the existence of its subject. This is Mill’s view (as regards realpropositions); for he holds that we cannot give information about a non-existentsubject.224This is nodoubt theview that, at any rateon a first 220considerationofthesubject,appearstobeatoncethemostreasonableandthemostsimple.

224 “Anaccidentalornon-essentialaffirmationdoesimplytherealexistenceofthe subject,because in thecaseofanon-existent subject there isnothing for thepropositiontoassert”(Logic,I.6,§2).

(3) Itmay be held thatwe should not regard propositions as necessarily


implyingtheexistenceeitherof theirsubjectsorof theirpredicates.Onthisview, the full implication ofAll S is Pmay be expressed by saying that itdenies the existence of anything that is at the same time S and not-P.SimilarlyNoS isP implies the existenceneither ofS nor ofP, butmerelydeniestheexistenceofanythingthatisbothSandP.SomeSisP(orisnotP)may be readSome S, if there is any S, is P (or is not P). Herewe neitheraffirmnordenytheexistenceofanyclassabsolutely;225thesumtotalofwhatweaffirmisthatifanySexists,thensomethingwhichisbothSandP(orSandnot-P)alsoexists.Onthisinterpretation,therefore,particularpropositionshaveahypotheticalandnotapurelycategoricalcharacter.

225 Jevonslaysdownthedictum that“wecannotmakeanystatementexceptatruismwithout implying thatcertaincombinationsof termsarecontradictoryandexcludedfromthought”(PrinciplesofScience,2ndedition,p.32).Thisistrueofuniversals(thoughsomewhatlooselyexpressed),butitdoesnotseemtobetrueofparticularpropositions,whateverviewmaybetakenofthem.

(4)Itmaybeheldthatuniversalpropositionsshouldnotbeinterpretedasimplying the existence of their subjects, but that particular propositionsshouldbeinterpretedasdoingso.226OnthisviewAllSisPmerelydeniestheexistenceofanythingthatisbothSandnot-P;NoSisPdeniestheexistenceof anything that is both S and P ; Some S is P affirms the existence ofsomething that is both S andP ;Some S is not P affirms the existence ofsomethingthatisbothSandnot-P.Thus,universalsareinterpretedashavingexistentially a negative force, while particulars have an affirmative force.Thishypothesiswillbefoundtoleadtocertainparadoxicalresults,butitwillalso be shewn to lead to amore satisfactory and symmetrical treatment oflogicalproblemsthanisotherwisepossible.227

226 DrVennadvocates thisdoctrinewithspecial reference to theoperationsofsymbolic logic;but there isno reasonwhy it shouldnotbeextended toordinaryformallogic.

227 The hypothesis in question has been already provisionally adopted in thescheme of logical equivalences given in section 108, and also in the symbolicschemeofpropositionsgivenonpage193.

221157.Reduction of the traditional forms of proposition to the form ofExistential Propositions.—Without at present attempting to decide betweenthe different possible suppositions as to the existential import of the


traditional forms of proposition, we may enquire how on the differentsuppositions they may be reduced to existential form. It will be assumedthroughout that both the traditional forms and the existential forms areinterpreted assertorically. In the caseof eachof the traditional forms itwillsufficetodealwiththetwofundamentalsuppositions,namely,thatitdoesandthatitdoesnotimplytheexistenceofitssubject.

Theuniversalaffirmative. (1) IfSaP is interpretedasnotcarryingwith itanyexistential implication in regard to its separate terms, it isequivalent totheexistentialpropositionSPʹ=0.DrWolfdeniesthisonthegroundthatSaPcontains further the implication“If thereareanyS’s, theymustallbeP’s”;andhence that,whileon thesupposition inquestionSPʹ=0 isan inferencefromSaP,itisnotequivalenttoit.Itisofcourseaveryelementarytruththatinferencesarenotalwaystheexactequivalentsoftheirpremisses.Butintheabove argument DrWolf has apparently overlooked the fact that SPʹ = 0,equallywithSaP, contains the implication“If there are anyS’s theyare allP’s.”228Bythelawofexcludedmiddle,everyS(ifthereareanyS’s)mustbePornotP, and sinceSPʹ = 0, the above inference clearly follows.SPʹ = 0carrieswithitinfactthetwoimplicationsIfS>0thenP>0, IfP>0 thenSʹ>0.ThesemayalsobewrittenintheformsEitherS=0orP>0,EitherPʹ=0orSʹ>0.

228 DrWolf perhaps draws a distinctionbetween the proposition “If there areanyS’stheymustallbeP’s”andtheproposition“IfthereareanyS’stheyareallP’s,”givingtotheformeranapodeictic,andtothelatteramerelyassertoric,force.But if so, then the former is implied by All S is P, only if this proposition isapodeictic,notifitismerelyassertoric.TheargumentisinthiscaseirrelevantsofarasthepositionwhichItakeisconcerned,sinceitisonlytheassertoricSaPthatIregardasequivalenttoSPʹ=0.DrWolfcanhardlymaintainthatallpropositionsoftheformAllSisPareapodeictic.Hiswholetreatmentofthesubjectwithwhichwe are now dealing appears, however, to be valid only if it relates to a modalschedule of propositions. At the same time he nowhere clearly indicates alimitationofthiskind,andmanyofthedoctrineswhichhecriticisesareintendedbythosewhoadoptthemtoapplyonlytoanassertoricschedule.

222(2)IfSaPisinterpretedasimplyingtheexistenceofS,thenitmaybeexpressedexistentiallyS>0andSPʹ=0.TheseexistentialformscarrywiththemtheimplicationsP>0,EitherPʹ=0orSʹ>0.

The universal negative. Taking the same two suppositions the


correspondingexistentialswillbe:— (1)SP=0(carryingwithittheimplicationsEitherS=0orPʹ>0,EitherP=0orSʹ>0); (2)S>0andSP=0(withtheimplicationsPʹ>0,EitherP=0orSʹ>0).

Theseresultsneednoseparatediscussion.

Theparticularaffirmative. (1)On thesupposition thatSiPdoesnotcarrywith it any implication as to the separate existence of its terms, it can beexpressedexistentiallyEitherS=0orSP>0.ItmightalsobewrittenintheformIfS>0 thenSP>0.Complicationsresultingfromthe introductionofconsiderations of modality will, however, be more easily avoided if thehypotheticalformisnotmadeuseof.

(2)OnthesuppositionthattheexistenceofSisimplied,SiPisreducibletotheformSP>0.

Theparticularnegative.Herethecorrespondingresultsare(1)EitherS=0orSPʹ>0;(2)SPʹ>0.

Wemay sumup our resultswith reference to the third and fourth of thesuppositionsformulatedintheprecedingsection.

Letnopropositionbeinterpretedasimplyingtheexistenceofitsseparateterms.Thencorrespondingtothetraditionalschedulewehavethefollowingexistentialschedule:—

A,—SPʹ=0;E,—SP=0;I,—EitherS=0orSP>0;O,—EitherS=0orSPʹ>0.

Thisrepresentswhatmayberegardedastheminimumexistentialimportofeachofthetraditionalpropositions(interpretedassertorically).

ItmustberememberedthatSPʹ=0carrieswithit theimplicationsEitherS=0orP>0,EitherPʹ=0orSʹ>0.

Let particulars be interpreted as implying, while universals are notinterpretedasimplying,theexistenceoftheirsubjects.223Wethenhave:—


A,—SPʹ=0;E,—SP=0;I,—SP>0;O,—SPʹ>0.

158. Immediate Inferencesand theExistential ImportofPropositions.—Ithasbeenalreadysuggestedthatbeforecomingtoanydecisioninregardtotheexistential import of propositions, it will be well to enquire how certainlogical doctrines are affected by the different existential assumptions uponwhichwemayproceed.Thisdiscussionwillasfaraspossiblebekeptdistinctfrom the enquiry as to which of the assumptions ought normally to beadopted.Thelatterquestionisofahighlycontroversialnature,butthelogicalconsequences of the various suppositions ought to be capable ofdemonstration,soastoleavenoroomfordifferencesofopinion.

We shall in the present section enquire how far different hypothesesregarding the existential import of propositions affect the validity ofobversion and conversion and the other immediate inferences based uponthese. In the next section we shall consider inferences connected with thesquareofopposition.

Wemaytakeinorderthesuppositionsformulatedinsection156.

(1)Leteverypropositionheunderstood to imply theexistenceofboth itssubjectanditspredicateandalsooftheircontradictories. It is clear that on this hypothesis the validity of conversion, obversion,contraposition, and inversion will not be affected by existentialconsiderations. The terms of the original proposition together with theircontradictories being in each case identical with the terms of the inferredproposition together with their contradictories, the latter cannot possiblycontainanyexistentialimplicationthatisnotalreadycontainedintheoriginalproposition.229

229 Thereadermayberemindedthatinourfirstworkingoutoftheseimmediateinferencesweprovisionallyassumed,apartfromanyimplicationcontainedinthepropositions themselves, that the terms involved and also their contradictoriesrepresentedexistingclasses.

224(2)Leteverypropositionheunderstoodtoimplysimplytheexistenceof


itssubject. (a)Thevalidityofobversionisnotaffected. (b)TheconversionofAisvalid,andalsothatofI.IfAllSisPandSomeSisP implydirectly theexistenceofS, then theyclearly imply indirectly theexistenceofP ; and this is all that is required inorder that their conversionmaybe legitimate.The conversionofE is not valid; forNoS isP impliesneither directly nor indirectly the existence of P, whilst its converse doesimplythis. (c)ThecontrapositionofEisvalid,andalsothatofO.NoSisPandSomeSisnotPbothimplyonourpresentsuppositiontheexistenceofS,andsincebythelawofexcludedmiddleeverySiseitherPornot-P,itfollowsthattheyimplyindirectlytheexistenceofnot-P.ThecontrapositionofAisnotvalid;for it involves the conversion ofE, whichwe have already seen not to bevalid.230 (d)Theprocessofinversionisnotvalid;foritinvolvesinthecaseofbothAandEtheconversionofanEproposition.231 IfalongwithanEpropositionwearespeciallygiventheinformationthatPexists, or if this is implied in some other proposition given us at the sametime, then theE propositionmay of course be converted. In correspondingcircumstances the contraposition and inversionofA and the inversionofEmaybevalid.232Oragain,givensimplyNoSisP,wemayinferEitherPisnon-existentornoPisS ;andsimilarlyinothercases.

230 OrwemightarguedirectlythatthecontrapositionofAisnotvalid,sinceAllSisPdoesnotimplytheexistenceofnot-P,whilst itscontrapositivedoes implythis.

231 OragainwemightarguedirectlyfromthefactthatneitherAllSisPnorNoSisPimpliestheexistenceofnot-S.

232 Forexample,given(α)NoSisP,(β)AllRisP,wemayunderourpresentsuppositionconvert(α),since(β)impliesindirectlytheexistenceofP ;andwemaycontraposit(β), since(α) implies indirectly theexistenceofnot-P. Itwillalsohefoundthat,giventhesetwopropositionstogether,theybothadmitofinversion.

(3)Let no proposition he understood to imply the existence either of itssubjectorofitspredicate. 225Havingnowgotridoftheimplicationoftheexistenceeitherofsubjectorpredicateinthecaseofallpropositions,wemightnaturallysupposethatin


nocaseinwhichwemakeanimmediateinferenceneedwetroubleourselveswith any question of existence at all. As already indicated, however, thisconclusionwouldbeerroneous. (а)Theprocessofobversionisstillvalid.Take,forexample,theobversionofNoSisP.TheobverseAllSisnot-PimpliesthatifthereisanySthereisalsosomenot-P.ButthisisnecessarilyimpliedinthepropositionNoSisPitself.IfthereisanySitisbythelawofexcludedmiddleeitherPornot-P;therefore,given thatNoS isP, it follows immediately that if there isanySthereissomenot-P. (b)TheconversionofE isvalid.SinceNoS isPdenies theexistenceofanything that isbothSandP, it implies that if there is anyS there is somenot-P and that if there is anyP there is somenot-S ; and these are theonlyimplicationswithregardtoexistenceinvolvedinitsconverse.TheconversionofA,however, isnotvalid;nor is thatofI.ForSomeP isS implies that ifthereisanyPthereisalsosomeS ;butthisisnotimpliedeitherinAllSisPorinSomeSisP. (c) That the contraposition ofA is valid follows from the fact that theobversion of A and the conversion of E are both valid.233 That thecontrapositionofE and that ofO are invalid follows from the fact that theconversionofAandthatofIarebothinvalid. (d)Thatinversionisinvalidfollowssimilarly. Onourpresentsuppositionthenthefollowingarevalid:theobversionandcontrapositionofA, theobversionofI, the obversion and conversionofE,theobversionofO;thefollowingareinvalid:theconversionandinversionofA, the conversion of I, the contraposition and inversion of E, thecontrapositionofO.234

233 Orwemightarguedirectlyasfollows;sincethepropositionAllSisPdeniestheexistenceofanythingthatisbothSandnot-P,itimpliesthatifthereisanySthereissomePandthatifthereisanynot-Pthereissomenot-S ;andthesearetheonlyimplicationswithregardtoexistenceinvolvedinitscontrapositive.

234 Dr Wolf holds in opposition to the view here expressed that on thesupposition in question all the ordinary immediate inferences remain valid. ThisconclusionisbasedonthedoctrinethatSomeSisPdoesnotimplythatifthereisanySthereisalsosomeP.“AllSisPandSomeSisP,itistrue,donotimplythat‘ifthereisanyPthereisalsosomeS.’ButthenSomePisSdoesnotnecessarilyimply that either. There can, therefore, be no objection, on that score, against


inferring, by conversion,SomeP is S fromAll S isP orSome S isP.With thevindication of conversion all the remaining supposed illegitimate inferencesconnectedwith it are alsovindicated.Wemay, therefore, conclude that to letnopropositionalformassuchnecessarilyimplytheexistenceofeitheritssubjectoritspredicate innowayaffects thevalidityofanyof the traditional inferencesoflogic”(StudiesinLogic,p.147).IhavedealtwithDrWolf’spositioninthenoteonpage 216; and it is unnecessary to repeat the argument here. If importance isattached to concrete examples, Imay suggest, as anexample for conversion,Allbluerosesareblue (aformalpropositionwhichmustberegardedasvalidontheexistential supposition under discussion); and, as an example for inversion, AllhumanactionsareforeseenbytheDeity.Thereare,moreover,certaindifficultiesconnectedwithsyllogisticandmorecomplexreasoningsthatneedabriefseparatediscussion,evenwhenthecaseofconversionhasbeendisposedof.

226 (4) Let particulars be understood to imply, while universals are notunderstoodtoimply,theexistenceoftheirsubjects. (a)Thevalidityofobversionisagainobviouslyunaffected.235 (b)TheconversionofEisvalid,andalsothatofI,butnotthatofA.236 (c)ThecontrapositionofAisvalid,andalsothatofO,butnotthatofE. (d)Theprocessofinversionisnotvalid. These results areobvious; and the final outcome is—asmight havebeenanticipated—thatwemay infer a universal froma universal, or a particularfromaparticular,butnotaparticularfromauniversal.237 227Animportantpointtonoticeisthatintheimmediateinferenceswhichremainvalidonthissupposition(namely,obversion,simpleconversion,andsimplecontraposition) there isno lossof logical force;whileat thebest thereverse would be the case in those that are no longer valid (namely,conversionperaccidens,contrapositionperaccidens,andinversion).


235 Obversion thus remains valid on all the suppositions which have beenspeciallydiscussedabove.If,however,affirmativesareinterpretedasimplyingtheexistenceoftheirsubjectswhilenegativesarenotsointerpreted,thenofcoursewecannotpassbyobversionfromEtoA,orfromOtoI.

236 Butfromthetwopropositions,AllSisP,SomeRisS,wecaninferSomePisS ;andsimilarlyinothercases.

237 On the assumption, however, that the universe of discourse can never beentirelyemptiedofcontent,SomethingisPmaybeinferredfromEverythingisP,andSomethingisnotPmaybeinferredfromNothingisP.Again,asisshewnbyDrVenn(SymbolicLogic,pp.142–9),thethreeuniversalsAllSisP,Nonot-SisP,All not-S is P, together establish the particular Some S is P. Any universe ofdiscoursecontainsàpriorifourclasses—(1)SP,(2)Snot-P,(3)not-SP,(4)not-Snot-P.AllSisPnegatives(2);Nonot-SisPnegatives(3);Allnot-SisPnegatives(4).Giventhesethreepropositions,therefore,weareabletoinferthatthereissomeSP,forthisisallthatwehaveleftintheuniverseofdiscourse.Asalreadypointedout,theassumptionthattheuniverseofdiscoursecanneverbeentirelyemptiedofcontentisanecessaryassumption,sinceitisanessentialconditionofasignificantjudgmentthatitrelatetoreality.Iftheuniverseofdiscourseisentirelyemptiedofcontentwemusteitherfailtosatisfythiscondition,orelseunconsciouslytranscendtheassumeduniverseofdiscourseandrefertosomeotherandwideroneinwhichtheformerisaffirmednottoexist.

159.TheDoctrineofOppositionandtheExistentialImportofPropositions.—The ordinary doctrine of opposition, in its application to the traditionalscheduleofpropositions,isasfollows:(a)ThetruthofSomeSisP followsfromthatofAllSisP,andthetruthofSomeSisnotPfromthatofNoSisP(doctrineofsubalternation);(b)AllSisPandSomeSisnotPcannotbothbetrueand theycannotbothbe false, similarly forSomeS isPandNoS isP(doctrineofcontradiction);(c)AllSisPandNoSisPcannotbothbetruebuttheymaybothbefalse(doctrineofcontrariety);(d)SomeSisPandSomeSisnot P may both be true but they cannot both be false (doctrine of sub-contrariety).Wewillnowexaminehowfartheseseveraldoctrinesholdgoodunder various suppositions respecting the existential import ofpropositions.238

238 Ofcoursethedoctrineofcontradictionalwaysholdsgoodinthesensethatapairofrealcontradictoriescannotbothbetrueorbothfalse;andsimilarlywiththeotherdoctrines.Thedoctrinesthatwehavetoconsiderarenotthese,butwhetherSaPandSoParereallycontradictoriesirrespectiveoftheexistentialinterpretationofthepropositions,whetherSaPandSeParereallycontraries,andsoon.

It should be added that, throughout the discussion, the propositions are


supposedtobeinterpretedassertorically,ashasalwaysbeenthecustomwiththetraditionalschedule.Thenecessityforthisprovisowillfromtimetotimebepointedout.

(1)Leteverypropositionbeinterpretedasimplyingthe228existencebothofitssubjectandofitspredicateandalsooftheircontradictories.239

239 ItwouldbequiteadifferentproblemifweweretoassumetheexistenceofSand P independently of the affirmation of the given proposition. A failure todistinguishbetweentheseproblemsisprobablyresponsibleforagooddealoftheconfusion and misunderstanding that has arisen in connexion with the presentdiscussion.But it is clearlyone thing to say (a)“AllS isPandS is assumed toexist,”andanothertosay(b)“allSisP,”meaningthereby“SexistsandisalwaysP.”Incase(a)itisfutiletogoontomakethesuppositionthatSisnon-existent;incase(b),ontheotherhand,thereisnothingtopreventourmakingthesupposition,andwefindthat,ifitholdsgood,thegivenpropositionisfalse.

Onthissupposition,ifeitherthesubjectorthepredicateofapropositionisthe name of a classwhich is unrepresented in the universe of discourse orwhichexhauststhatuniverse,thenthatpropositionisfalse;foritimplieswhatis inconsistentwith fact. It follows that a pair of contradictories as usuallystated,andalsoapairofsub-contraries,maybothbefalse.Forexample,AllSis P andSome S is not P both imply the existence ofS in the universe ofdiscourse. In thecase then inwhichS doesnot exist in thatuniverse, thesepropositionswouldbothbefalse.

Ifaconcreteillustrationisdesired,wemaytakethepropositions,Noneofthe answers to the question shewed originality,Some of the answers to thequestion shewed originality, and assume that each of these propositionsincludesaspartof its implication theactualoccurrenceof its subject in theuniverseofdiscourse.Thenourpositionisthatiftherewerenoanswerstothequestionatall,thetruthofboththepropositionsmustbedenied.Thefactoftherehavingbeennoanswersdoesnotrender thepropositionsmeaningless;butitrendersthemfalse,theirfullimportbeingassumedtobe,respectively,There were answers to the question but none of them shewed originality,Therewereanswerstothequestionandsomeofthemshewedoriginality.

We must not of course say that under our present supposition truecontradictories cannot be found; for this is always possible. The truecontradictoryofAllSisPisEithersomeSisnotP,orelseeitherSornot-S


orPornot-Pisnon-existent.Similarlyinothercases.Theordinarydoctrinesofsubalternationandcontrarietyremainunaffected.

229(2)Leteverypropositionbeinterpretedasimplyingtheexistenceofitssubject. For reasons similar to those stated above, the ordinary doctrines ofcontradiction and sub-contrariety again fail to hold good. The truecontradictoryofAllSisPnowbecomesEithersomeSisnotP,orSisnon-existent. The ordinary doctrines of subalternation and contrariety againremainunaffected.

(3)Letnopropositionbeinterpretedasimplyingtheexistenceeitherofitssubjectorofitspredicate. (a)Theordinarydoctrineofsubalternationholdsgood. (b)Theordinarydoctrineofcontradictiondoesnotholdgood.AllS isP,forexample,merelydeniestheexistenceofanyS’sthatarenotP’s;SomeSisnotPmerelyassertsthatifthereareanyS’ssomeofthemarenotP’s.InthecaseinwhichSdoesnotexistintheuniverseofdiscoursewecannotaffirmthefalsityofeitherofthesepropositions.240 230(c)Theordinarydoctrineofcontrarietydoesnotholdgood.Forifthereisnoimplicationoftheexistenceofthesubjectinuniversalpropositionsweare not actually precluded from asserting together twopropositions that areordinarilygivenascontraries.AllS isPmerely denies that there are anySnot-P’s, No S is P that there are any SP’s. We may, therefore, withoutinconsistencyaffirmbothAllSisPandNoSisP ;butthisisvirtuallytodenytheexistenceofS.241 (d)Theordinarydoctrineofsub-contrarietyremainsunaffected.

240 DrWolf(StudiesinLogic,p.132)deniesthevalidityof thisreasoning.Headmits apparently that the existential propositions SPʹ = 0 andEither S = 0 orSPʹ > 0 are not contradictories; but he denies that on the supposition underdiscussionSaPandSPʹ=0areequivalent.HismaingroundfortakingthisviewisthatSaPcarrieswithittheimplicationIf thereareanyS’s theyareallP’s,whileSPʹ=0doesnotcarrywithitanysuchimplication.Thispositionhasbeenalreadycriticizedinsection157.DrWolfreliespartlyuponconcreteexamples,but insodoing he complicates the discussion by introducingmodal forms of expression.Thusfortheproposition“Somesuccessfulcandidatesdonotreceivescholarships,”we find substituted in the course of his argument “If there are any successfulcandidatesthensomeofthemdonot(orneednot) receivescholarships,”and the


insertionofthewordsinbracketsyieldsapropositionwhich,althoughaninferencefrom the original proposition, is not really equivalent to it, unless the originalproposition is itself interpreted modally. Later on Dr Wolf explicitly alters thewholeproblembyassumingthatwhatisunderconsiderationisamodalscheduleof propositions. Thus he goes on to say, “What SaP and SeP really expressseverallyisthenecessityandtheimpossibilityofSbeingP”;andforthepurposeofcontradictingSaPandSeP,“SiPandSoPneedmeannomorethanSmaybePandSneednotbeP.”ThequestionhowfarSaPandSePshouldbeinterpretedmodallyisdiscussedelsewhere.All Iwouldpointouthere is that it isadistinctquestionfromthatraisedinthetext,whichisaquestionrelatingtothetraditionalscheduleofpropositionsinterpretedassertorically.Thewholequestionofexistentialimportis indeed one that cannot be discussed to any purpose until the character of thescheduleofpropositionsunderconsiderationhasbeendefined.Fromthemixingupofschedulesandinterpretationsnothingbutconfusioncanresult.Inthefollowingsectiontheoppositionofmodalswillbebrieflyconsideredinconnexionwiththeirexistentialimport.

241 Ofcourseontheviewunderconsiderationweoughtnottocontinuetospeakofthesetwopropositionsascontraries.

(4) Let particulars be interpreted as implying, while universals are notinterpretedasimplying,theexistenceoftheirsubjects. (a)Theordinarydoctrineofsubalternationdoesnotholdgood.SomeS isP,forexample,impliestheexistenceofS,whilethisisnotimpliedbyAllSisP. (b)Theordinarydoctrineofcontradictionholdsgood.AllSisPdeniesthatthereisanySthatisnot-P;SomeSisnotPaffirmsthatthereissomeSthatisnot-P. It is clear that thesepropositions cannotbothbe true; it is also clearthattheycannotbothbefalse.SimilarlyforNoSisPandSomeSisP. (c)Theordinarydoctrineofcontrarietydoesnotholdgood.AllSisPandNoSisParenotinconsistentwithoneanother,buttheforceofassertingbothofthemistodenythatthereareanyS’s.242Thisfollowsjustasinthecaseofourthirdsupposition.243 231 (d) The ordinary doctrine of sub-contrariety does not hold good.244SomeSisPandSomeSisnotParebothfalseinthecaseinwhichSdoesnotexistintheuniverseofdiscourse.

242 If,however,wearegivenNoSisPandalsoSomeSisP,thenweareabletoinferthatAllSisPisfalse.ThesecondofthesepropositionsaffirmstheexistenceofS,andthereforedestroysthehypothesisonwhichalonethefirstandthirdcanbetreatedascompatible.

243 The above doctrine has been criticized on the ground that it practically


amountstosayingthatneitherofthegivenpropositionshasanymeaningwhatever,butthateachisamereshamandpretenceofpredication;andarequestismadeforconcrete examples. The following examplemay perhaps suffice to illustrate theparticularpointnowatissue:“Anhonestmillerhasagoldenthumb”;“Well,Iamsure that no miller, honest or otherwise, has a golden thumb.” These twopropositions are in the form of what would ordinarily be called contraries; buttaken together they may quite naturally be interpreted as meaning that no suchperson can be found as an honest miller. The former proposition would indeedprobably be intended to be supplemented by the latter or by some propositioninvolvingthe latter,andso tocarry inferentially thedenialof theexistenceof itssubject.

Another example is contained in the following quotation from Mrs LaddFranklin: “All x is y,No x is y, assert together thatx is neithery nornot-y, andhence that there is no x. It is common among logicians to say that two suchpropositions are incompatible; but that is not true, they are simply togetherincompatible with the existence of x. When the schoolboy has proved that themeetingpointoftwolinesisnotontherightofacertaintransversalandthatitisnotontheleftofit,wedonottellhimthathispropositionsareincompatibleandthat one or other of themmust be false, but we allow him to draw the naturalconclusion that there is no meeting point, or that the lines are parallel” (Mind,1890,p.77n.).

DrWolf (Studies in Logic, p. 140), criticizing Mrs Ladd Franklin’s concreteexample,maintainsthatthetwopropositionsgivenbyheraresub-contraries(IandO),notcontraries(AandE).Amoment’sconsiderationwill,however,shewthatthisisnotthecasesinceneitherofthepropositionsisparticular.Atthesametimeitis true that a littlemanipulation is required tobring them to the formsAandE.There is also the assumption that “on the right” and “on the left” exhaust thepossibilities and are therefore contradictory terms.Granting this assumption, thetwopropositionsmaybeexpressedsymbolicallyintheformsNoSisP,NoSisnotP,anditthenneedsonlytheobversionofoneofthemtobringthemtotheformsAandE.

244 Itmaybeworthobservingthat,given(b),(d)mightbededucedfrom(c)orviceversâ.

Therelationbetweencontradictories isby far themost important relationwithwhichweareconcernedindealingwiththeoppositionofpropositions,anditwillbeobservedthatthelastoftheabovesuppositionsistheonlyoneunderwhichtheordinarydoctrineofcontradictionholdsgood.

160.TheOppositionofModalPropositionsconsidered inconnexionwiththeir Existential Import.—The propositions discussed in the precedingsections have been the propositions belonging to the traditional scheduleinterpretedassertorically.Turningnowtothecorrespondingmodalschedule,


wemaybrieflyconsiderhowthedoctrineofoppositionisaffected, ifatall,on the supposition that the propositions included in the schedule are notinterpretedasimplyingtheexistenceoftheir232subjects.WefindthatonthissuppositionSassuchisPandSneednotbeParetruecontradictories.

SassuchisP (interpretedasnotnecessarilyimplyingtheexistenceofS)does more than deny the actual occurrence of the conjunction S not-P, itdeniesthepossibilityofsuchaconjunction;andallthatisnecessaryinordertocontradict this is toaffirmthepossibilityof theconjunction.Thisisdoneby the proposition S need not be P (also interpreted as not necessarilyimplying theexistenceofS).On the samesupposition,Sas such isP,SassuchisotherthanP,aretruecontraries.

Here, however, another problem suggests itself. Leaving on one side thequestion as to any implication of actuality, are modal propositions to beinterpretedascontainingany implication in regard to thepossibilityof theirantecedents? And, further, how does our answer to this question affect theopposition of modals? The consideration of this problem may be deferreduntil we come to deal with the opposition of conditional propositions (seesection176).

161. Jevons’s Criterion of Consistency.—In passing to the explicitdiscussion of the existential import of categorical propositions, we mayconsider first the Criterion of Consistency, which is laid down by Jevons(following De Morgan):—Any two or more propositions are contradictorywhen,andonlywhen,afterallpossiblesubstitutionsaremade,theyoccasionthe total disappearance of any term, positive or negative, from the LogicalAlphabet. The criterion amounts to this, that every proposition must beunderstood to imply the existence of things denoted by every simple termcontained in it, and also of things denoted by the contradictories of suchterms. If, for example,wehave thepropositionAllS isP, this implies thatamongthemembersoftheuniverseofdiscoursearetobefoundS’sandP’s,not-S’s,andnot-P’s.IndefenceofthisdoctrineJevonsappearstorelymainlyupon thepsychological lawof relativity,namely, thatwecannot thinkatallwithoutseparatingwhatwe thinkabout fromother things.Hence ifeitheraterm or its contradictory represents nonentity, that term cannot be either


subject or predicate in a significant 233 proposition.245 It is clear, however,that thispsychologicalargument fallsawayas soonas it isallowed thatwemaybeconfiningourselvestoalimiteduniverseofdiscourse,orindeedifweconfine ourselves to any universe less extensive than thatwhich covers thewhole realmof theconceivable.Of course themore limited theuniverse towhichourpropositionissupposedtorelatethemoreeasilymaySorPeitherexhaustitorbeabsentfromit;butwithverycomplexsubjectsandpredicatesthe contradictory of one or both of our terms may easily exhaust even anextended universe. Take, for example, the proposition, No satisfactorysolutionoftheproblemofsquaringthecirclehaseverbeenpublishedbyMrA.Here the subject is non-existent; and itmay happen also thatMrA. hasneverpublishedanythingatall.246Further,ifIamnotallowedtonegativeX,whyshouldIbeallowedtonegativeAB?ThereisnothingtopreventXfromrepresentingaclassformedbytakingthepartcommontotwootherclasses.IncertaincombinationsindeeditmaybeconvenienttosubstituteXforAB,orvice versâ. It would appear then that what is contradictorywhenwe use acertainsetofsymbolsmaynotbecontradictorywhenweuseanothersetofsymbols. This argument has a special bearing on the complex propositionswhichareusuallyrelegatedtosymboliclogic,buttowhichJevons’scriterionisintendedparticularlytoapply.

245 ThispointisputsomewhattentativelyinapassageinJevons’sPrinciplesofScience(chapter6,§5)whereheremarks:“IfAwereidenticalwith‘Bornot-B,’itsnegativenot-Awouldbenon-existent.Thisresultwouldgenerallybeanabsurdone,andIseemuchreasontothinkthatinastrictlylogicalpointofviewitwouldalwaysbeabsurd.Inallprobabilityweoughttoassumeasafundamentallogicalaxiomthateverytermhasitsnegativeinthought.Wecannot thinkatallwithoutseparating what we think about from other things, and these things necessarilyformthenegativenotion.Ifso,itfollowsthatanytermoftheform‘Bornot-B’isjustasself-contradictoryasoneoftheform‘Bandnot-B’.”

246 Otherexampleswillbegiveninthefollowingsection.

NodoubtJevons’scriterionissometimesaconvenientassumptiontomake;provisionally, for example, in working out the doctrine of immediateinferencesonthetraditionallines.Butitisanassumptionthatshouldalwaysbeexplicitlyreferredtowhenmade;anditoughtnottoberegardedashavingan 234 axiomatic and binding force, so as tomake it necessary to base thewholeoflogicuponit.


162.TheExistentialImportofthePropositionsincludedintheTraditionalSchedule.—Wemaynowturntotheconsiderationofthequestionwhetherthepropositions SaP, SeP, SiP, SoP should or should not be interpreted asimplyingtheexistenceoftheirsubjectsintheuniverseofdiscoursetowhichreferenceismade.Inthissectionitwillbeassumedthattheimportofallthepropositionsunderdiscussionisassertoric,notmodal.

A brief reference may be made to two sources of misunderstanding towhichattentionhasalreadybeencalled. (а)Allpropositionscontainaffirmationsrelatingtosomesystemofreality;andbyanalysiseverypropositionmaybemadetoyieldan“ultimatesubject”whichisreal,namely, thesystemofreality towhichthepropositionrelates.Thissystemofrealityiswhatwemeanbytheuniverseofdiscourse;and,aswe have seen, the universe of discourse can never be entirely emptied ofcontent.Itmustthenbeunderstoodthatifwedecidethatcertainpropositionalforms are not to be interpreted as containing as part of their import theaffirmation of the existence of their subjects, it is far from being therebyintended that propositions falling into these forms contain no affirmationrelatingtoreality.247 (b)We must put on one side a very summary solution of our problem,which,ifitwerecorrect,wouldrenderanyfurtherdiscussionneedless.How,itmaybeasked,canwepossiblyspeakaboutanythingandatthesametimeexclude it from the universe of discourse? This question suggests a certainambiguitywhichmay attach to the phraseuniverseof discourse, butwhichcan hardly remain an ambiguity after the explanations already given. Theansweristhatwecancertainlythinkandspeakaboutathingwithreferencetoa given universe of discourse without implying, or even believing in, itsexistenceinthatuniverse.Suppose,forexample,thatIsaytherearenosuchthingsasunicorns. If thisstatement is tobeaccepted, itmustbe interpretedliterally(notelliptically);anditisclearthattheuniverseofdiscoursereferredto is thematerial235universe.248 I speak thenofunicornswith reference tothematerialuniverse,butdenythatsuchcreaturesaretobefound(orexist)init.

247 CompareSigwart,Logic,i.p.97n.248 Itishardlynecessarytopointoutthatideasofunicornsexistinimagination,


andthatstatementsaboutunicornsaretobemetwithinfairytales.

The question we have to discuss is one of the interpretation ofpropositional forms,249 and the solution will therefore be to some extent amatter of convention. We shall be guided in our solution partly by theordinary usage of language, and partly by considerations of logicalconvenienceandsuitability.

249 Seesection48.

As regards theordinaryusageof language therecanbenodoubt thatweseldomdoasamatteroffactmakepredicationsaboutnon-existentsubjects.For such predicationswould in general have little utility or interest for us.“Thepracticalexigenciesoflife,”asDrVennremarks,“confinemostofourdiscussions to what does exist, rather than to what might exist” (SymbolicLogic,p.131).Wemust,however,considerwhethertherearenotexceptionalcases;andifwecanfindanyinwhichit isclearthatthespeakerwouldnotnecessarily intend to imply the existence of the subject, we may draw theconclusion that the propositional form of which he makes use is not inpopularusageuniformlyintendedtoconveysuchanimplication.

UniversalAffirmatives.Ifauniversalaffirmativepropositionisobtainedbyaprocessofexhaustiveenumeration(e.g.,AlltheApostleswereJews,Allthebookson that shelfarebound inmorocco),or if it isobtainedbyempiricalgeneralisation based on the examination of individual instances (e.g., Allruminantanimalsarecloven-hoofed),thenitisclearthattheexistenceofthesubjectisapresuppositionoftheaffirmation.Wemay,however,notecertainotherclassesofcasesinwhichsuchapresuppositionisnotnecessary.

(a) We may affirm an abstract connexion of attributes, based onconsiderationsofadeductivecharacteroratany ratenotobtainedbydirectgeneralisationfromobservedinstancesofthesubject,andtheexistenceofthesubjectisthennotessential.Forexample,Theimpactoftwoperfectlyelastic236bodiesleadstonodiminutionofkineticenergy ;Everybody,notcompelledby impressed forces to change its state, continues in a state of rest or ofuniformmotioninastraightline.

Itmayperhapsbesaidthatallpropositionsfallingwithinthiscategorywillbe really apodeictic, and that our present discussion has been limited to


assertoricpropositions.Thereissomeforceinthiscriticism.Itis,however,tobe remembered that the assertoricSaP can be inferred from the apodeicticSaP, so that if we can have the latter without any implication as to theexistence of S we may have the former also, unless indeed we decide todifferentiate between them in regard to their existential implication. Theexamples thatwe have given aremoreover expressed in ordinary assertoricform,andnotinanydistinctiveapodeicticform,suchasSassuchisP,It isinherentinthenatureofStobeP.

(b)ThepropositionSaPmay express a rule laid down, and remaining inforce, without any actual instance of its application having arisen. Forexample,Allcandidatesarrivingfiveminuteslatearefinedoneshilling,Allcandidates who stammer are excused reading aloud, All trespassers areprosecuted.

If it is argued that, in such cases as these,250 the propositions oughtproperlytobewrittenintheconditionalandnotinthecategoricalform(e.g.,Ifanycandidatearrivesfiveminuteslate,thatcandidateisfinedoneshilling),thereplyisthatthisistomisunderstandthepointjustnowatissue,whichiswhether we meet with propositions in ordinary discourse which arecategorical in form and yet are hypothetical so far as the existence of theirsubjects is concerned. It is of course open to us to decide that for logicalpurposeswewillsointerpretcategoricalpropositionsthatinsuchcasesastheabovethecategoricalformcannolongerbeused.Butforthepresentwearemerelydiscussingpopularusage.

250 Thisargumentmightbeusedwith,referencetocasescomingunder(a)or(c)aswellaswithreferencetothosecomingunder(b).

(c)AssertionsinregardtopossiblefutureeventsaresometimesthrownintotheformSaP.Forexample,Whostealsmypursestealstrash,Thosewhopassthisexaminationan237luckymen.Thefirstof thesepropositionswouldnotbe invalidated supposingmy purse never to be stolen, and the latter, asDrVennremarks,251would be tacitly supplemented by the clause “if any suchtherebe.”

251 SymbolicLogic,p.132.

(d)Thereare cases inwhich the intended implicationof apropositionof


theformAllSisP is todenythat thereareanyS’s;forexample,Anhonestmillerhasagoldenthumb,AllthecartsthatcometoCrowlandareshodwithsilver.252

252 Both these propositions are naturally to be interpreted as containing anindirect denial of the existence of their subjects. “Crowland is situated in suchmoorishrottengroundintheFens,thatscarceahorse,muchlessacart,cancometoit”(Bohn’sHandbookofProverbs,p.211).Itwouldappear,however,thatthisproverbhasnowlost its force, inasmuchas“since thedraining, insummer time,cartsmaygothither.”

UniversalNegatives.Itisstilleasiertofindinstancesfromcommonspeechinwhichuniversalnegativepropositions,thatis,propositionsoftheformNoSisP, arenot tobe regardedasnecessarily implying theexistenceof theirsubjects.

(a)Thereareagaincasesinwhichthepropositionisreachedbyaprocessof abstract reasoning about a subject the actual existence or occurrence ofwhichisnotpresupposed;forexample,Aplanetmovinginahyperbolicorbitcanneverreturntoanypositionitonceoccupied.253

253 ThisexampleistakenfromDixon,EssayonReasoning,p.62.

(b) The import of the proposition may be distinctly to imply, if notdefinitelytoaffirm,thenon-existenceofthesubject;forexample,Noghostshavetroubledme,Nounicornshaveeverbeenseen.254

254 The universe of discoursemust here be taken to be thematerial universe.Withreferencetothisexample,however,acriticwrites,“Butsurelytheuniverseofimaginationistheonlyoneapplicable;forunicornshavelongbeenknownnottobelong to the actual material universe.” The universe of imagination may berequiredinordertosustainthepositionthatthesubjectofthepropositionexistsintheuniverseofdiscourse;butanypersonmakingthestatementwouldcertainlynotbereferringtotheworldofimaginationortheuniverseofheraldry,forthesimplereasonthatineitherofthesecasestheproposition(whichmustthenbeinterpretedelliptically) would obviously not be true. On the other hand, we can quite wellsuppose the statement made with reference to the material universe: “Whetherunicornsexistornot,atanyratetheyhaveneverbeenseen.”Again,totakeanotherexampleofasimilarkindwherethereferenceisalsotothephenomenaluniverse,wecanquitewellsupposethestatementmade:“Whetherthereareghostsornot,atany rate none have ever troubledme.” In order to avoid misapprehension, it isimportanttodistinguishtheaboveexamplesfromsuch(elliptical)propositionsasthefollowing:“ThewrathoftheHomericgodsisveryterrible,”“Fairiesareabletoassume different forms.” In each of these cases, the subject of the proposition


(properlyinterpreted)existsintheparticularuniversetowhichreferenceismade.Seenotes2and3onpage213.

238(c)AdenialoftheconjunctionABCmaybeexpressedintheformNoABisCwithout any intentionof thereby affirming the conjunctionAB ; forexample,No satisfactory solution of the problemof squaring the circle hasbeen published,No woman candidate for the Theological Tripos has beeneducatedatNewnhamCollege,NoAdvancedStudentinLawisontheboardsofTrinityCollege.255

255 “Asaninstanceofapossiblynon-existentsubjectofanegativeproposition,take the following: ‘No person condemned forwitchcraft in the reign ofQueenAnnewasexecuted.’”(Venn,SymbolicLogic,p.132.)

Particulars.Inthecaseofparticularpropositions,itisfarlesseasytogiveexamples,suchasmightbemetwithinordinarydiscourse,inwhichthereisnoimplicationoftheexistenceofthesubjectsofthepropositions.Theremaybe exceptions, but at any rate the cases are exceedingly rare in which inordinaryspeechwepredicateanythingofanon-existentsubjectwithoutdoingsouniversally.Themain reason for this is, asDrVennpoints out, that “anassertion confined to ‘some’ of a class generally rests upon observation ortestimony rather than on reasoning or imagination, and therefore almostnecessarilypostulatesexistentdata,thoughthenatureofthisobservationandconsequent existence is, as already remarked, a perfectly open question.‘Sometwiningplantsturnfromlefttoright,’‘Somegriffinshavelongclaws,’bothimplythatwehavelookedintherightquarterstoassureourselvesofthefact.InonecaseImayhaveobservedinmyowngarden,andintheotheroncrestsor in theworksof thepoets,butaccording to theappropriate testsofverification, we are in each case talking of what is.”256 If we look at thequestion239 fromtheotherside,wefind thatwhenourprimaryobject is toaffirmtheexistenceofaclassofobjects,ourassertionverynaturallytakestheform of a particular proposition. If, for example, we desire to affirm theexistence of black swans, we say Some swans are black. The existentialimplication of a proposition of this kind in ordinary discourse is one of itsmostfundamentalcharacteristics.

256 SymbolicLogic,p.131.Again,insuchapropositionas“Somesea-serpentsarenothalfamilelong”(meaningyourso-calledsea-serpents),thesubjectofthepropositionexistsintheuniversetowhichreferenceismade,namely,theuniverse


whichmaybedescribedastheuniverseoftravellers’tales.Wearehereregardingthepropositionasellipticalinasensethathasbeenalreadyexplained.

Onthewholeitcannotbesaidthattheusagesofordinaryspeechaffordadecisive solution of the problem under discussion. It has, however, beenshewn (1) that we seldom or never make statements about non-existentsubjects in the form Some S is P or the form Some S is not P ; (2) that,althoughitisalsotruethatwedonotasaruledosointheformAllSisPortheformNoSisP,stillthereareseveralclassesofcasesinwhichtheuseofthese latter forms isnot tobeunderstoodasnecessarilycarryingwith it theimplicationthatS isexistent.Henceweshouldbedepartingvery littlefromordinaryusage ifwewere to decide to interpret particulars as implying theexistenceoftheirsubjects,butuniversalsasnotdoingso(thatis,asnotdoingsobytheirbareform).

Idonot,however,regardthissolutionasnecessitatedbypopularusage.Itis,forinstance,stillopentoanyonetoadopttheconventionthat,forlogicalpurposes,thecategoricalformshallonlybeusedwhentheimplicationoftheexistenceofthesubjectisintended.Onthisinterpretation,theconditionalorhypothetical formmustbeadoptedwhenever theexistenceof the subject isleftanopenquestion.Thus,ifwearedoubtfulabouttheexistenceofS(or,atany rate, do notwish to affirm its existence),wemust be careful to say, IfthereisanyS,thenallSisP,insteadofsimplyAllSisP ;inotherwords,thehypotheticalcharacterofthepropositionsofarastheexistenceofitssubjectisconcernedmustbemadeexplicit.

The problem then not being decided by considerations of popular usagealone, we must go on to enquire how the question is affected byconsiderationsof logicalconvenienceandsuitability.Hereagain there isnoonesolution that is inevitable.Reasonscan,however,beurged in favourofinterpreting particulars as implying, but universals as not implying, theexistenceoftheir240subjects;257andthis,aswehaveseen,isasolutionthatderivessomesanctionfrompopularusage.

257 Onthisviewwhenever it isdesiredspecially toaffirmtheexistence in theuniverseofdiscourseofthesubjectofauniversalproposition,aseparatestatementtothiseffectmustbemade.Forexample,ThereareS’s,andallofthemareP’s.If,on the other hand, it is ever desired to affirm a particular proposition withoutimplyingtheexistenceofthesubject,thenrecoursemustbehadtothehypothetical


orconditionalformofstatement.Thus,ifwedonotintendtoimplytheexistenceofS,insteadofwritingSomeS’sareP’s,wemustwrite,IfthereareanyS’s,theninsomesuchcasestheyarealsoP’s.

(1) A consideration of the manner in which the validity of immediateinferencesisaffectedbytheexistentialimportofpropositionsaffordsreasonsfor the adoption of this interpretation.258 The most important immediateinferences are simple conversion (i.e., the conversion of E and of I) andsimple contraposition (i.e., the contraposition ofA and ofO). If, however,universals are regarded as implying the existence of their subjects, then, asshewninsection158,neithertheconversionofEnorthecontrapositionofAis valid, irrespective of some farther assumption;whereas, if universals arenot regarded as implying the existence of their subjects, then both theseoperations are legitimate without qualification. On the other hand, theconversionof I and the contrapositionofO are valid only if particularsdoimplytheexistenceoftheirsubjects.259


258 It has been objected that to base our view of the existential import ofpropositionsuponthevalidityorinvalidityofimmediateinferencesistoargueinacircle.“Whether,”itissaid,“theimmediateinferencesarevalidornotmustbeaconsequence of the view taken of the existential import of the proposition andshould not, therefore, be made a portion of the ground on which that view isbased.”Thisobjectioninvolvesaconfusionbetweendifferentpointsofviewfromwhich the problemof the relation between the existential import of propositionsandthevalidityof logicaloperationsmayberegarded.Insection158 the logicalconsequencesofvariousassumptionswereworkedoutwithoutanyattemptbeingmadetodecidebetweentheseassumptions.Ourpointofviewisnowdifferent;weareinvestigatingthegroundsonwhichoneoftheassumptionsmaybepreferredtotheothers,andthereisnoreasonwhytheconsequencespreviouslydeducedshouldnotformpartofourdatafordecidingthisquestion.Theargumentcontainsnothingthatisofthenatureofacirculusinprobando.

259 Thus, the table of equivalences given in section 106 is valid on theinterpretationwithwhichwearenowdealing.Thedependenceofthetablegiveninsection108upon the same supposition is stillmoreobvious. It hasbeen alreadypointed out that the remaining immediate inferences based on conversion andobversionareofmuchlessimportance;seepage227.

241 Turning to immediate inferences of another kind, it is clear that ifuniversal propositions formally imply the existence of their subjects, wecannot legitimatelypass fromAllX isY toAllAXisY.260For it ispossiblethattheremaybeX’sandyetnoAX’s,andinthiscasetheformerpropositionmaybetrue,whilethelatterwillcertainlybefalse.Again,giventhatAisX,BisY,CisZ,wecannotinferthatABCisXYZ.Suchrestrictionsasthesewouldconstitute an almost insurmountablebar toprogress in inference as soon aswehavetodowithcomplexpropositions.261

260 Itwillbeobserved further thatupon the sameassumptionwecannotevenaffirm the formal validity of the proposition All X is X. For X might be non-existent,andthepropositionwouldthenbefalse.

261 HenceMrsLaddFranklinisledtotheconclusionthat“noconsistentlogicofuniversalpropositionsispossibleexceptwiththeconventionthattheydonotimplytheexistenceoftheirterms”(Mind,1890,p.88).

(2) We may next consider the existential import of propositions withreferencetothedoctrineofopposition.Ithasbeenshewninsection159thatifparticularsareinterpretedasimplyingtheexistenceoftheirsubjects,whileuniversals are not so interpreted, then A and O, E and I, are truecontradictories; but that this is not the case under any of the othersuppositions discussed in the same section.262 There can, however, be no


doubtthatoneofthemostimportantfunctionsofparticularpropositionsistocontradicttheuniversalpropositionsofoppositequality;andhencewehaveastrongargumentinfavourofaviewoftheexistentialimportofpropositionswhichwillleavetheordinarydoctrineofcontradictionunaffected.

262 A and O, E and I, will also be true contradictories if universals areinterpretedasimplyingtheexistenceoftheirsubjects,whileparticularsarenotsointerpreted. Itwouldbe interesting, if spacepermitted, toworkout the resultsofthissuppositionindetail.Ifthestudentdoesthisforhimself,hewillfindthatthisistheonlysupposition,underwhichtheordinarydoctrineofoppositionholdsgoodthroughout. All other considerations, however, are opposed to its adoption. Italtogether conflicts with popular usage; it renders the processes of simpleconversion and simple contraposition illegitimate; and whilst making universalsdoublejudgments,itdestroysthecategoricalcharacterofparticularsaltogether.Inregardtothislastpointseepage220.

Asregards thedoctrinesofsubalternation,contrariety,andsubcontrariety,ourresults(namely,thatIdoesnotfollowfromA,orOfromE,thatAandEmay both be true, and that I 242 andO may both be false) are no doubtparadoxical.But thisobjection is farmore thancounterbalancedby the factthatthedoctrineofcontradictionissaved.Forascomparedwiththerelationbetweencontradictories,theseotherrelationsareoflittleimportance.Wemayspecially consider the relation betweenA and I. Some S is P cannot nowwithout qualification be inferred fromAll S isP, since the former of thesepropositions implies the existence of S, while the latter does not. But as amatteroffactthisisaninferencewhichweneverhaveoccasiontomake.Iftheirexistentialimportisthesamewhyshouldweeverlaydownaparticularpropositionwhenthecorrespondinguniversalisatourservice?Ontheotherhand,theviewthatweareadvocatinggivesSomeSisPastatusrelativelytoAll S is P as well as relatively toNo S is P which it could not otherwisepossess;andsimilarlyforSomeSisnotP.Ourresultasregardstherelationbetween SaP and SiP has been described as equivalent to saying “that astatementofpartialknowledgecarriesmorerealinformationthanastatementof fullknowledge; since ifweonlypossess limited information,andsocanonlyassertSiP,wetherebyaffirmtheexistenceofS ;butifwehavesufficientknowledgetospeakofallS(Sremainingthesame)thestatementofthatfullknowledge immediately casts a doubt upon that existence.” This way ofputtingitis,however,misleadingifnotpositivelyerroneous.Ontheviewin


questionitisincorrecttosaysimplythatSiPandSaPgive“partial”and“full”knowledgerespectively,forSiPwhilegivinglessknowledgethanSaPinonedirectiongivesmoreinanother.Inotherwords,theknowledgewhichis“full”relativelytoSiP isnotexpressedbySaPbyitself,butbySaP togetherwiththestatementthattherearesuchthingsasS.263

263 ThepositiontakenaboveinregardtosubalternationisverywellexpressedbyMrsLaddFranklin. “Nothingof course isnow illogical thatwasever logicalbefore. It ismerely a question ofwhat convention in regard to the existence oftermsweadoptbeforeweadmit thewarm-bloodedsentencesofreal life intotheironmoulds of logicalmanipulation.With the old convention (whichwas neverexplicitly stated) subalternation ran thus:No x’s are y’s (andwe herebymean toimplythattherearex’s,whateverxmaybe),therefore,Somex’sarenon-y’s.Withthenewconventiontherequirementissimplythatifitisknownthattherearex’s(asitisknown,ofcourse,inbyfarthegreaternumberofsentencesthatitinterestsustoform)thatfactmustbeexpresslystated.Theargumentthenis:Nox’sarey’s,Therearex’s,therefore,Therearex’swhicharenon-y’s.”

243(3)Thereisonefurtherpointofimportancetobenoted,andthatis,thatthe interpretationofA,E,I,O propositionsunder consideration is theonlyinterpretationaccording towhicheachoneof thesepropositions is resolvedintoasinglecategoricalstatement.ForifAandEimplytheexistenceoftheirsubjects they express double, not single, judgments, being equivalentrespectivelytothestatements:ThereareS’s,buttherearenoSPʹ’s ;ThereareS’s,but therearenoSP’s ;whereason the interpretationhereproposed theysimplyexpress respectively thesingle judgments:TherearenoSPʹ’s ;TherearenoSP’s.Ontheotherhand,ifIandOdonotimplytheexistenceoftheirsubjects,insteadofexpressingcategoricaljudgments,theyexpresssomewhatcomplexhypotheticalones,beingequivalentrespectivelytothestatement:IfthereareanyS’sthentherearesomeSP’s ;IfthereareanyS’sthentherearesome SPʹ’s ; whereas on our interpretation they express respectively thecategoricaljudgments:ThereareSP’s ;ThereareSPʹ’s.264

264 Comparesections156,157.

On the whole, there is a strong cumulative argument in favour ofinterpretingparticulars,butnotuniversals,asimplyingformallytheexistenceoftheirsubjects.265Thissolution244istoberegardedaspartlyofthenatureofaconvention.Wearrive,however,attheconclusionthatnoothersolutioncanequallywellsufficeasthebasisofascientifictreatmentofthetraditional


scheduleofpropositions,solong,atanyrate,asthepropositionsincludedinthescheduleareregardedasassertoricandnotmodal.

265 Wemaybrieflydiscuss inanoteoneor twoobjections to thisviewwhichhavenotyetbeenexplicitlyconsidered.

(а) Mill argues that a synthetical proposition necessarily implies “the realexistence of the subject, because in the case of a non-existent subject there isnothing for the proposition to assert” (Logic, i. 6, § 2). In answer to this it issufficient to point out that a non-existent thing will be described as possessingattributeswhichareseparatelyattributesofexistingthings,althoughthatparticularcombinationofthemmaynotanywherebefound,andifweknow(aswemaydo)thatcertainoftheseattributesarealwaysaccompaniedbyotherattributeswemaypredicate the latterof thenon-existent thing, therebyobtaininga realpropositionwhich does not involve the actual existence of its subject. As an argument adhominem itmayfurtherbepointedoutthatMill inclinestodenytheexistenceofperfectstraightlinesorperfectcircles.Wouldhethereforeaffirmthatwecanmakenorealassertionsaboutsuchthings?

(b)MrWelton repeats several times that apropositionwhich relates toanon-existentsubjectmustbeamerejumbleofwords,apredicationinappearanceonly.“Thatthemeaningofauniversalpropositioncanbeexpressedasadenialistrue,but this is not its primary import.And thisdenial itselfmust rest uponwhat theproposition affirms. Unless SaP implies the existence of S, and asserts that itpossessesP,we have no data for denying the existence ofSPʹ. For ifS is non-existent the denial that it is non-P can have no intelligible meaning” (Logic, p.241).Theexampleswhichwehavealreadygivenaresufficienttodisposeofthisobjection;butitmaybeworthwhiletoaddafurtherargument.AccordingtoMrWelton, an E proposition implies the existence of its subject but not of itspredicate.WecannottheninferPeSfromSePbecausewehavenoassuranceoftheexistenceofP.ButinaccordancewiththepositiontakenbyMrWelton,weoughtto go further and say thatPeS must be a mere jumble of words unless we areassuredof theexistenceofP. It is impossible,however, to regardPeSasamereunmeaning jumble of words, a predication in appearance only, when SeP is asignificantandtrueproposition.PeSmaybefalse,oritmaybeanunnaturalformofstatement,butitcannotbemeaninglessifSePhasameaning.Take,forexample,thepropositions—NowomanisnowhangedfortheftinEngland,NopersonnowhangedfortheftinEnglandisawoman.Thesecondofthesepropositionsisfalseifit is taken to imply that thereareat thepresent timepersonswhoarehangedfortheft in England, but how it can possibly be regarded as meaningless I cannotunderstand.

(c)Miss Jones argues that if some carrieswith it an implication of existence,whenusedwithasubject-term,itmustdosoequallywhenusedwithapredicate-term; but the predicate of an A proposition being undistributed is practicallyqualifiedbysome ;hence,ifSomeSisPimpliestheexistenceofSandthereforeofP,All S isPmust imply the existence ofP and therefore ofS. In reply to thisargumentitmaybepointedout,first,thatadistinctionmayfairlybedrawnwithout


anyriskofconfusionbetweenatermexplicitlyquantifiedbythewordsomeandatermwhichwecanshewtobeundistributedbutwhichisnotexplicitlyquantifiedat all; and, secondly, that the position which we have taken is based upon aconsiderationoftheimportofpropositionsasawhole,notupontheforceofsignsof quantity considered in the abstract. The irrelevancy of the argument will beapparent if it is taken in connexion with the reasons which we have urged forholding that particulars should be interpreted as implying the existence of theirsubjects.

163. The Existential Import of Modal Propositions.—Of apodeicticpropositions it may be said still more emphatically than of assertoricuniversals that theydonotnecessarily imply theexistenceof their subjects.Fortheyassertanecessaryrelationbetweenattributes,thegroundofwhichisfrequently 245 to be sought in abstract reasoning rather than in concreteexperiences.Andthesameistrueofthedenialofapodeicticpropositions.Wemayonabstractgroundsassert thepossibilityofacertainconcomitance (ornon-concomitance)ofattributeswithouthavinghadactualexperienceofthatconcomitance (or non-concomitance), and without intending to imply itsactuality.HenceweshouldnotinterpretthepropositionSmaybeP,anymorethanthepropositionSmustbeP,asbyitsbareformaffirmingtheexistenceofS.

IthasbeenshewnthatinorderthatthepropositionsAllSisPandSomeSisnotPmaybetruecontradictories,oneorotherofthemmustbeinterpretedasimplyingtheexistenceofS.Itfollows,however,fromwhathasbeensaidabovethatthesameconditionneednotbefulfilledinorderthatSmustbePandSneednotbePmaybetruecontradictories.266

266 It isbecauseDrWolfidentifiestheordinaryparticularpropositionwiththeproblematicpropositionthatheisledtotheconclusionthatSaPandSoParetruecontradictoriesalthoughneitherofthemisinterpretedasimplyingtheexistenceofS.

Buttothisithastobeaddedthat,inorderthatthesetwopropositionsmaybetruecontradictories,oneorotherofthemmustbeinterpretedasimplyingthepossibleexistenceofS.Thislineofthoughthasbeensuggestedinsection160,anditwillbepursuedfartherinsections176and179.


EXERCISES.

164. The particular judgment has, from different stand-points, beenidentified (a) with the existential judgment, (b) with the problematicjudgment,(c)withthenarrativejudgment.Commentoneachoftheseviews.[C.]Thestudentmayfindthattowriteadetailedanswertothisquestionwillhelptoclearuphisviews

respectingtheparticularproposition.Nodetailedanswerwillherebegiven;butattentionmaybecalledtooneortwopoints. (a)Twokindsofexistentialjudgmentsmaybedistinguished. (i)Thosewhichaffirmexistenceindefinitely,thatis,246somewhereintheuniverseofdiscourse;forexample,Therearewhitehares,Thereisadevil. (ii)Thosewhichaffirmexistencewithreferencetosomedefinitetimeandplace;forexample,Itrains,Iamhungry. Theparticularmayperhapsbeidentifiedwith(i),hardlywith(ii). (b)We may be justified in affirming the problematical S may be P, when we cannot affirm theparticularSomeSisP.Therearereasonsforinterpretingthelatterjudgmentexistentiallyasregardsitssubject,whichdonotapplytotheformerjudgment. (c)Thenarrativejudgmentneednothavetheindefinitecharacteroftheparticular.Wemay,however,hold that the twokindsof judgmenthavethis incommonthat therearegroundsfor interpretingbothexistentiallyasregardstheirsubjects.

165.DiscusstherelationbetweenthepropositionsAllSisPandAllnot-SisP.Thisisaninterestingcasetonoticeinconnexionwiththediscussionraisedinsections158and159.

Wehave

SaP=SePʹ=PʹeS ;SʹaP=SʹePʹ=PʹeSʹ=PʹaS.

Thegivenpropositionscomeout,therefore,ascontraries. Ontheviewthatweoughtnottoenterintoanydiscussionconcerningexistenceinconnexionwithimmediateinference,wemust,Isuppose,restcontentwiththisstatementofthecase.Itseems,however,sufficientlycurious todemand further investigationandexplanation.Wemayasbefore takedifferentsuppositionswithregardtotheexistentialimportofpropositions. (1)Ifeverypropositionimpliestheexistenceofbothsubjectandpredicateandtheircontradictories,then it isatonceclear that the twopropositionscannotbothbe true together; forbetween them theydenytheexistenceofnot-P. (2)Ontheviewthatpropositionsimplysimplytheexistenceoftheirsubjects,ithasbeenshewninsection158thatwearenotjustifiedinpassingfromAllnot-SisPtoAllnot-PisSunlesswearegivenindependently the existence of not-P. But it will be observed that in the case before us the givenpropositionsmakethisimpossible.SinceallSisPandallnot-SisP,andeverythingiseitherSornot-Sbythe lawofexcludedmiddle, it followsthat247nothingisnot-P. Inorder, therefore, to reduce the


givenpropositions to sucha form that theyappear as contraries (andconsequently267as inconsistentwitheachother)wehavetoassumetheverythingthattakentogethertheyreallydeny. (3)and (4).On theview thatatany rateuniversalpropositionsdonot imply theexistenceof theirsubjects, we have found in section 159 that the propositionsNonot-P is S,All not-P is S, are notnecessarilyinconsistent,fortheymayexpressthefactthatPconstitutestheentireuniverseofdiscourse.Butthisfactisjustwhatisgivenusbythepropositionsintheiroriginalform. Undereachhypothesis,then,theresultobtainedissatisfactorilyaccountedforandexplained.

267 It will be remembered that under suppositions (1) and (2) the ordinarydoctrineofcontrarietyholdsgood.

166.“Theboyisinthegarden.” “Thecentaurisacreationofthepoets.” “Asquarecircleisacontradiction.” Discuss theabovepropositionsas illustratingdifferent functionsof theverb“tobe”;orasbearinguponthelogicalconceptionofdifferentuniversesofdiscourseorofdifferentkindsofexistence.[C.]

167.Discusstheexistentialimportofsingularpropositions. “TheKingofUtopiadidnotdieonTuesdaylast.”Examinecarefullythemeaningtobeattachedtothedenialofthisproposition.[K.]

168. Some logicians hold that from All S is P we may infer Some not-S is not-P. Take as anillustration,AllhumanactionsareforeseenbytheDeity.[C.]

169. Discuss the validity of the following inference:—All trespassers will be prosecuted, Notrespassershavebeenprosecuted,therefore,Therehavebeennotrespassers.[C.]

170. On the assumption that particulars are interpreted as implying while universals are notinterpretedasimplyingtheexistenceoftheirsubjectsintheuniverseofdiscourse,examine(statingyourreasons)thevalidityofthefollowinginferences;AllSisPandSomeRisnotStherefore,Somenot-SisnotP ;AllSisPandSomeRisnotP,therefore,Somenot-Sis248notP ;AllSisPandSomeRisS,itis,therefore,falsethatNoPisS ;AllSisPandSomeRisP,itis,therefore,falsethatNoPisS.[K.]

171.Discusstheformalvalidityofthefollowingarguments,(i)onthesuppositionthatallcategoricalpropositionsaretobeinterpretedasimplyingtheexistenceoftheirsubjectsintheuniverseofdiscourse,(ii)onthesuppositionthatnocategoricalpropositionsaretobesointerpreted: (a)AllPisQ,therefore,AllAPisAQ ; (b)AllAPisAQ,therefore,SomePisQ.[K.]

172.WorkoutthedoctrineofOppositionandthedoctrineofImmediateInferencesonthehypothesisthatuniversalsaretobeinterpretedasimplying,whileparticularsarenottobeinterpretedasimplying,theexistenceoftheirsubjectsintheuniverseofdiscourse.[K.]


CHAPTERIX.

CONDITIONALANDHYPOTHETICALPROPOSITIONS.

173.The distinction between Conditional Propositions and HypotheticalPropositions268—PropositionscommonlywrittenintheformIfAisB,CisDbelongtotwoverydifferent types.Fortheymaybetheexpressioneitherofsimplejudgmentsorofcompoundjudgments(asdistinguishedinsection55).

268 ForthedistinctionindicatedinthepresentsectionIwasinthefirstinstanceindebted to an essay,written in 1884, byMrW.E. Johnson.This essay has notbeen published in its original form; but the substance of it has been included insomepapersonTheLogicalCalculusbyMrJohnsonwhichappearedinMindin1892.

In the first place,A being B andC being D may be two events or twocombinationsofproperties,concerningwhichitisaffirmedthatwheneverorwhereverthefirstoccursthesecondwilloccuralso.Forexample,Ifanimportdutyisasourceofrevenue,itdoesnotaffordprotection ;Ifachildisspoilt,his parents suffer ; If a straight line falling upon two other straight linesmakes the alternate angles equal to one another, the two straight lines areparalleltooneanother ;Ifalightedmatchisappliedtogunpowder,therewillbe an explosion ;Where the carcase is, there shall the eagles be gatheredtogether.Whatisaffirmedinallsuchcasesastheseisaconnexionbetweenphenomena;itmaybeeitheraco-inherenceofattributesinacommonsubject,or a relation in time or space between certain occurrences. Propositionsbelongingtothistypemaybecalleddistinctivelyconditional.

But again,A is B andC is D may be two propositions of independentimport, the relation betweenwhich cannot be 250 directly resolved into anytimeorspacerelationorintoanaffirmationoftheco-inherenceofattributesinacommonsubject.Inotherwords,arelationmaybeaffirmedbetweenthetruthoftwojudgmentsasholdinggoodonceandforallwithoutdistinctionofplaceortimeorcircumstance.Forexample,Ifitbeasintocovethonour,Iam


themostoffendingsoulalive ;Ifpatienceisavirtue,therearepainfulvirtues ;IfthereisarighteousGod,thewickedwillnotescapetheirjustpunishment ;Ifvirtueisinvoluntary,soisvice ;If theearthisimmoveable, thesunmovesround the earth. Propositions belonging to this type may be calledhypotheticalasdistinguishedfromconditional,ortheymaybespokenofstillmoredistinctivelyastruehypotheticalsorpurehypotheticals.269

269 The abovedistinctionhasbeen adopted in some recent treatisesonLogic,but it must be borne inmind thatmost logicians use the terms conditional andhypotheticalassynonymousorelsedrawadistinctionbetweenthemdifferentfromtheabove.

Thepartsoftheconditionalandalsoofthetruehypotheticalarecalledtheantecedentandtheconsequent.Thus,inthepropositionIfAisB,CisD, theantecedentisAisB,theconsequentisCisD.

It is impossible formally to distinguish between conditionals andhypotheticalssolongaswekeeptotheexpressionIfAisB,CisD,sincethismay be either the one or the other. The following forms, however, areunmistakeablyconditional:WheneverAisB,CisD ;InallcasesinwhichAisB,CisD ;IfanyPisQthenthatPisR.270TheformIfAistruethenCistrue is,on theotherhand,distinctivelyhypothetical.AandCherestandforpropositionsorjudgments,not terms,andthewords“is true”areintroducedinordertomakethisexplicit.Itisquitesufficient,however,towritethetruehypotheticalintheformIfAthenC.

270 Conditionalscangenerallybereducedtothelastofthesethreeformswithoutmuch difficulty, and such reduction is sometimes useful. A consideration of theconcrete examples already given will, however, shew that a certain amount ofmanipulationmaybe required inorder toeffect the reduction.The followingareexamples:Ifanychildisspoilt,thenthatchildwillhavesufferingparents ;If anytwostraightlinesaresuchthatanotherstraightlinefallinguponthemmakesthealternateanglesequaltooneanother,thenthosetwostraightlinesareparalleltooneanother.

251 Since a conditional proposition usually contains a reference to someconcurrence in time or space, the if of the antecedent may as a rule bereplacedeitherbywhenorbywhere,asthecasemaybe,withoutanychangeinthesignificanceoftheproposition;butthesamecannotbesaidinthecaseof the truehypothetical.Thisconsiderationwilloftensuffice to resolveany


doubtthatmayariseinconcretecasesastotheparticulartypetowhichanygivenpropositionbelongs.Another andmore fundamental criterionmaybefound in the answer to the question whether or not the antecedent andconsequentarepropositionsof independent import,whosemeaningwillnotbeimpairediftheyareconsideredapartfromoneanother.Iftheanswerisintheaffirmative,thenthepropositionishypothetical.Thus,takingexamplesofhypotheticalsalreadygiven,wefindthattheantecedents,It isasintocovethonour,Patienceisavirtue,Virtueisinvoluntary,andtheconsequents,Iamthemostoffendingsoulalive,Therearepainful virtues,Vice is involuntary,all retain their full meaning though separated from one another. If, on theother hand, the consequent necessarily refers us back to the antecedent inorder that it may be fully intelligible, then the proposition is conditional.Thus, taking by itself the consequent in the first conditional given on page249,namely,itdoesnotaffordprotection,weareatonce led toaskwhat isheremeantbyit.Theansweris—thatimportduty.Butwhatimportduty?Anadequate answer can be given only by introducing into the consequent thewholeoftheantecedent,—animportdutywhichisasourceofrevenuedoesnotaffordprotection.Wenowhavethefullforceofouroriginalconditionalproposition in the formofasinglecategorical. Itwillbe found that ifotherconditionals are treated in the sameway, they resolve themselves similarlyintocategoricalsoftheformAllPQisR.271252Theproblemofthereductionof conditionals and hypotheticals to categorical formwill be considered inmore detail later on in this chapter, and it will be shewn that whilst suchreductionisalwayspossible,andgenerallysimpleandnatural,inthecaseofconditionals,itisnotpossibleatall(withtermscorrespondingtotheoriginalantecedentandconsequent)inthecaseofhypotheticals.272

271 Asanotherexample,wemaytaketheconditionalproposition,Iftheweatheris dry, theBritish root-crops are light.Here itmay at first sight appear that theconsequent is a proposition of independent import. The proposition,The Britishroot-cropsarelight,is,however,ajudgmentincompletelystated.Foritcontainsatime-referencethatneedstobemadeexplicit.Theconditionalreallymeans,If inanyyeartheweatherisdry,theBritishroot-cropsinthatyeararelight ;andthisisequivalent to the categorical,Any year in which the weather is dry is a year inwhichtheBritishroot-cropsarelight.Bylookingattheconditionalinthisway,wesee thenecessityof referringback to theantecedent inorder that theconsequentmaybefullyexpressed.


272 Thequestionmayberaisedwhetherapropositionoftheform,IfthisPisQ,it isR, isproperly tobedescribedasa singularconditionalorasahypothetical.Theansweristhatapropositionofthisformaffordsakindofjunctionbetweentheconditionalandthehypothetical;itisderivablefromtheconditional,IfanyPisQ,it is R ; but it is itself hypothetical. The antecedent and the consequent arepropositionsofindependentimport;andthepropositionasawholeisnotdirectlyreducible(asistheconditional,IfanyPisQ,itisP)tocategoricalform.Thus,theproposition,IfanyPisQ,itisR,mayprimafaciebereducedtotheformAnyPthat is Q is R ; but the proposition, If this P is Q, it is R, certainly cannot beidentifiedwiththesingularcategorical,ThisPwhichisQisR.

174.TheImportofConditionalPropositions.—ItissometimesheldthattherealdifferentiaofallpropositionsoftheformIfAisB,CisDis“toexpresshuman doubt.” Clearly, however, there is no intention to express doubt asregardstherelationbetweentheantecedentandtheconsequent;andthedoubtmust, therefore, be supposed to relate to the actual occurrence of theantecedent. But so far at any rate as conditionals are concerned, the doubtwhich they may thus imply must be considered incidental rather than thefundamentalordifferentiatingcharacteristicbelongingtothem.Theifof theconditional may, as we have seen, usually be replaced by when withoutaltering the significance of the proposition, and in this case the element ofdoubt is no more prominent than in the categorical proposition. From thematerial standpoint, conditionals may or may not involve the actualoccurrence of their antecedents. Whenever the connexion between theantecedent and the consequent can be inferred from the nature of theantecedent independently of specific experience (and thismay be themoreusualcase), then theactualhappeningof the253 antecedent isnot involved;butifourknowledgeoftheconnexiondoesdependonspecificexperience(asit sometimes may), then such actual happening is materially involved. Forexample, the statement, “If we descend into the earth, the temperatureincreasesat anearlyuniform rateof1°Fahr. for every fifty feetofdescentdowntoalmostamile,”isbaseduponknowledgegainedbyactualdescentsintotheearthhavingbeenmade,andapartfromsuchexperiencethetruthofthestatementwouldnothavebeenknown.

The question of main importance in regard to the import of conditionalpropositionsiswhethersuchpropositionsaretobeinterpretedasmodalorasmerely assertoric. Confining ourselves for the present to the universal


affirmative,thatis,totheformIfanyPisQthenitisR,areweaffirminganecessary relation between P beingQ and its being R, or are we merelyaffirmingthatitsohappensthateveryPthatisQisalsoR?Thisisreallyinanotherformthedistinctionalreadydrawnbetweenunconditionallyuniversalpropositions and empirically universal propositions, and our answer mustagainbe that thesameformofwordsmayexpress theone judgmentor theother.Therecanbenodoubtthattheproposition,Iftheanglesatthebaseofatriangleareequaltooneanother,thattriangleisisosceles,isintendedtobeinterpreted modally as expressing a necessary connexion, while theproposition,Ifanybookistakendownfromthatshelf,itwillbefoundtobeanovel,wouldbeintendedtobeinterpretedmerelyassertorically.

In ordinary discourse conditionals are as a rule modal; but this is notuniversally the case. Unless, therefore, we are prepared to depart fromordinaryusage (and there is a gooddeal to be said for suchdeparture),wemustrecognisebothassertoricconditionalsandmodalconditionals,andthisdistinction must be borne in mind in all that follows. We shall find thatpractically the same problem arises in regard to true hypotheticals, andweshallhavetoconsideritfurtherinthatconnexion.

175.ConditionalPropositionsandCategoricalPropositions.—Wemaygoon to consider what is the essential nature of the distinction betweenconditional propositions and 254 categorical propositions, and in particularwhetherthedistinctionisoneofverbalformonlyoronethatcorrespondstoarealdistinctionbetweenjudgments.

Ifavitaldistinction is tobedrawnbetween the twoforms, itmustbeonone or other of the two following grounds, namely, either (i) that thecategorical is to be interpreted assertorically while the conditional is to beinterpretedmodally,or(ii)thatthecategoricalistobeinterpretedasimplyingthe existence of its subjectwhile the conditional is not to be interpreted asimplyingtheoccurrenceofitsantecedent.

(i) There is much to be said for adopting a convention by which thecategoricalformwouldbeinterpretedassertoricallyandtheconditionalformmodally.Theadoptionof thisconventionwould,however,necessitate somemodificationof the formsofordinary speech, for, aswehavealready seen,


the propositionAllS isP is in current use sometimes apodeictic,while thepropositionIfanySisPthenitisQmay(thoughperhapsrarely)bemerelyassertoric.Whether theone formor theother isused reallydependsagooddealonlinguisticconsiderations.Consider,forinstance,thepropositions,Allisoscelestriangleshavetheanglesat theirbaseequaltooneanother,If theangles at the base of a triangle are equal to one another, that triangle isisosceles. These propositions fall naturally into the categorical andconditionalformsrespectively,simplybecausetherehappenstobenosingleadjective (like “isosceles”)which connotes “having two equal angles.” It isclear, however, that the use of the one form rather than the other is notintendedtoimplyanyfundamentaldifferenceinthecharacteroftherelationasserted.Ifeitherofthepropositionsinitsordinaryuseisapodeictic,soistheother;ifeitherismerelyassertoric,soistheother.

It is tobeaddedthat ifweadopttheconventionunderconsiderationthentheuniversal categorical is inferable from theuniversal conditional, butnotvice versâ ; while, on the other hand, the problematic conditional (whichcorrespondstotheparticular)isinferablefromtheparticularcategorical,butnotviceversâ.Thus,AllPQisRissubalterntoIfanyPisQit255isR,whileIfanyPisQitmaybeRissubalterntoSomePQisR.

(ii)Wemaypassontoconsiderwhethercategoricalsandconditionalsaretobedifferentiatedinrespectoftheirexistentialimport.

We have seen in section 163 that if categoricals are interpretedmodallythey are not to be regarded as necessarily implying the existence of theirsubjects; and certainly conditionals, interpreted modally, are not to beregardedasnecessarilyimplyingtheoccurrenceoftheirantecedents.Henceifbothpropositional formsare interpretedmodally,wehavenodifferentiationasregardstheirexistentialimport.

Itfurtherseemsclearthat,sofarasuniversalareconcerned,aconditionalproposition—even though interpreted as merely assertoric—is not to beregarded as necessarily implying the actual occurrence of its antecedent.Hence,whether,ontheassertoricinterpretationofboth,thetwoformsaretobe existentially differentiated depends upon our existential interpretation ofthecategorical.


(a) If a universal categorical is interpreted as necessarily implying theactualexistenceofitssubject,thenwehaveamarkeddistinctionbetweenthetwoforms.273IfanyPisQthenitisalsoRcannotberesolvedintoAllPQisR,sincethelatterimpliestheexistenceofPQ,whiletheformerdoesnot.


273 ThisisUeberweg’sview,“Thecategoricaljudgment,indistinctionfromthehypothetical,always includes thepre-suppositionof theexistenceof thesubject”(Logic,§122).

(b) If, on the other hand, universal categoricals are not interpreted asnecessarily implying the existence of their subjects, then universalconditionalsanduniversalcategoricals(bothbeinginterpretedassertorically)maybe resolved intooneanother.Wemaysay indifferentlyAllS isPorIfanythingisSit isP ;IfeverAisBthenonallsuchoccasionsCisBorAlloccasionsofAbeingBareoccasionsofCbeingD.

Particular conditionals, so far as they are merely assertoric, are almostwithout exception based upon specific experience. Hence they may notunreasonablybe interpretedas implyingtheoccurrenceof theirantecedents,as,forexample,inthe256proposition,“SometimeswhenParliamentmeets,itis opened by the Sovereign in person.” The existential interpretation ofcategoricals forwhich a preferencewas expressed in the preceding chaptermay therefore be adopted for conditionals also, so far as they are merelyassertoric;andthetwoformsbecomemutuallyinterchangeable.

Onthewhole,exceptinsofarasweadopttheconventionindicatedunder(i) above, there seems no reason for drawing a vital distinction betweenjudgments according as they are expressed in the conditional or thecategorical form.274 Many of the conditionals of ordinary discourse areindeed so obviously equivalent to categoricals that they hardly seem torequireaseparateconsideration.275Atthesametime,aswehaveseen,somestatementsfallmorenaturallyintotheoneformandsomeintotheother.Themorecomplexthesubject-term,thegreateristheprobabilitythatthenaturalformofthepropositionwillbeconditional.

274 Ithasbeenarguedthat,startingfromthecategoricalform,wecannotpasstotheconditional,ifthesubjectofthepropositionisasimpleterm.Thebasisofthisargumentisthattheantecedentofaconditionalrequirestwoterms,andthatinthecase supposed these are not provided by the categorical. Thus, Miss Jones(ElementsofLogic,p.112)takestheexample,“Alllionsarequadrupeds.”Itwillnotdo, she says, to reduce this to the form, “If any creatures are lions, they arequadrupeds,”sincethisinvolvestheintroductionofanewterm,andpassingbackagain to the categorical form,we shouldhave“All creatureswhichare lions arequadrupeds,”apropositionnotequivalenttoouroriginalproposition.If,however,“creature” is regardedaspartof theconnotationof“lion,” there isno reason for


refusingtoallowthatthetwopropositionsareequivalenttooneanother.Similarly,inanyconcreteinstance,bytakingsomepartoftheconnotationofthesubjectofour categorical proposition, we can obtain the additional term required for itsreduction to the conditional form. Where we are dealing with purely symbolicexpressions,andthisparticularsolutionofthedifficultyisnotopentous,wemayhaverecoursetotheall-embracingterm“anything,”suchapropositionasAllSisPbeingreducedtotheformIfanythingisSitisP.

275 Theexamplesgivenatthecommencementofsection173arereducibletothefollowingcategoricals:Importdutieswhichare sourcesof revenuedonotaffordprotection ;All spoilt children have suffering parents ;All pairs of straight lineswhich are such that another straight line falling upon themmakes the alternateangles equal to one another are parallel ;All occasions of the application of alightedmatchtogunpowderareoccasionsofanexplosion ;Anyplacewherethereisacarcaseisaplacewheretheeagleswillgathertogether.

176.TheOppositionofConditionalPropositions.—Thisquestionneedsaseparatediscussionaccordingasconditionalsareinterpreted(a)assertorically,or(b)modally.

257 (a) If conditionals are interpreted assertorically, then the ordinarydistinctionsbothofqualityandofquantitycanbeappliedtotheminjustthesamewayas to categoricals.Wemay regard thequalityof a conditional asdeterminedbythequalityofitsconsequent;thus,thepropositionIfanyPisQthen that P is not R may be treated as negative.276 As regards quantity,conditionals are to be regarded as universal or particular, according as theconsequentisaffirmedtoaccompanytheantecedentinallormerelyinsomecases.

276 ThenegativeforceofthispropositionwouldbemoreclearlybroughtoutifitwerewrittenintheformIfanyPisQthenitisnotthecasethatitisalsoR.ThecategoricalequivalentisNoPQisR.

Wehavethenthefourtypesincludedintheordinaryfour-foldschedule:—

IfanyPisQ,itisalsoR ; AIfanyPisQ,itisnotalsoR ; E

SometimesifaPisQ,itisalsoR ; ISometimesifaPisQ,itisnotalsoR. O

These propositions constitute the ordinary square of opposition, and ifconditionalsareassimilatedtocategoricalssofarastheirexistentialimportisconcerned,thentheoppositionofconditionalsontheassertoricinterpretation


seems to requireno separatediscussion.277 Itmay,however,bepointedoutthatthereismoredangerofcontradictoriesbeingconfusedwithcontrariesinthecaseofconditionalsthaninthecaseofcategoricals.IfAisBthenCisnotDisveryliabletobegivenasthecontradictoryofIfAisBthenCisD.Butitis clear on consideration that both these propositions may be false. Forexample, the two statements—If theTimes saysone thing, theWestminsterGazette says another; If theTimes says one thing, theWestminsterGazettesays thesame, i.e., doesnot sayanother—mightbe, andasamatterof factare, both false; the two papers are sometimes in agreement and sometimesnot.

277 Thefourpropositionsarepreciselyequivalenttothefourcategoricals,—AllPQisR,NoPQisR,SomePQisR,SomePQisnotR.

(b)Onthemodalinterpretation,thedistinctionbetween258apodeicticandproblematictakestheplaceofthatbetweenuniversalandparticular;andifwemaintain thedistinctionbetweenaffirmativeandnegative,wehave the fourfollowingpropositionscorrespondingtotheordinarysquareofopposition:

IfanyPisQ,thatPmustbeR ; Am

IfanyPisQ,thatPcannotbeR ; Em

IfanyPisQ,thatPmaybeR ; ImIfanyPisQ,thatPneednotbeR. Om

It will be convenient to have distinctive symbols to denote modalpropositions,and those thatwehavehere introducedwillserve tobringouttheanalogiesbetweenmodalsandtheordinaryassertoricforms.

Intheaboveschedule,subjecttoacertainconditionmentionedbelow,Am

andOm, and alsoEm and Im, are contradictories according to the definitiongiveninsection84;AmandEmarecontraries;AmandIm,andalsoEmandOm,aresubalterns;andImandOmaresubcontraries.

Theconditionreferredtorelatestotheinterpretationofthepropositionsasregardstheimplicationofthepossibilityoftheirantecedents.Thus, inorderthatAmandOm(orEmandIm)maybetruecontradictoriesitisnecessarythatapodeicticandproblematicpropositionsshallbeinterpreteddifferentlyinthisrespect.If,forexample,Amisinterpretedasnotimplyingthepossibilityofits


antecedentthenitsfullimportistodenythepossibilityofthecombinationPandQwithoutR. Its contradictorymust affirm thispossibility.Omwill not,however, do this unless it is interpreted as implying the possibility of thecombinationP,Q.

Itisnecessarytocallattentiontothiscomplication,buthardlynecessarytowork out in detail the results which follow from the various principles ofinterpretationthatmightbeadopted.Ifthestudentwilldothisforhimself,hewill find that the results correspond broadlywith those obtained in section159.278

278 In connexionwith theproblemofoppositionwemay touchbrieflyon therelationbetweentheapodeicticpropositionIfanyPisQthatPmustbeRandtheassertoricpropositionSomePQisnotR.Thesepropositionsarenotcontradictories,fortheymaybothbefalse.Theycannot,however,bothbetrue;andthelatter,ifitcan be established, affords a valid ground for the denial of the former. MrBosanquetappearsnot toadmit this,but tomaintain, inopposition to it, that theenumerative particular is of no value as overthrowing the abstract universal.“WhenwehavesaidthatIf(i.e.,insofaras)amanisgood,heiswise,itisidletoreplythatSomegoodmenarenotwise.Thisistoattachanabstractprinciplewithunanalysed examples.What wemust say in order to deny the above-mentionedabstractjudgmentissomethingofthiskind:IforThoughamanisgood,yetitdoesnotfollowthatheiswise,thatis,Thoughamanisgood,yetheneednotbewise”(Logic,i.p.316).Butsurelyifwefindthatsomegoodmenarenotwise,wearejustifiedinsayingthatthoughamanisgoodyetheneednotbewise.Ofcoursetheconverse does not hold. We might be able to shew that wisdom does notnecessarily accompany goodness by some other method than that of producinginstances. But if we can produce undoubted instances, that amply suffices toconfutetheapodeicticconditional.

259 177. Immediate Inferences from Conditional Propositions.—In aconditional proposition the antecedent and the consequent correspondrespectively to the subject and thepredicateof a categoricalproposition. Inconversion,therefore,theoldconsequentmustbethenewantecedent,andincontrapositionthenegationoftheoldconsequentmustbethenewantecedent.

(a) On the assertoric interpretation, the analogy with categoricals is soclosethatitisunnecessarytotreatimmediateinferencesfromconditionalsinanydetail.Oneortwoexamplesmaysuffice.TakingtheAproposition,IfanyPisQthenitisR,wehaveforitsconverseSometimesifaPisRitisalsoQ,and for its contrapositive If any P is not R then it is not Q. Taking theE


propositionIfanyPisQthenitisnotR,wehaveforitsconverseIfanyPisRthenit isnotQ,andfor itscontrapositiveSometimes ifaP isnotR it isQ.The validity of these inferences is of course affected by the existentialinterpretationofthepropositionsjustasinthecaseofthecategoricals.Itwillbenoticedthatinsomeimmediateinferences(forexample,thecontrapositionofA) the conditional form has an advantage over the ordinary categoricalform inasmuch as it avoids the use of negative terms, the employment ofwhichissostronglyobjectedtobySigwartandsomeotherlogicians.279

279 Comparepage136.

(b) Ifconditionalsare interpretedmodally, then theapodeictic form takesthe place of the universal, and the 260 problematic takes the place of theparticular.Onthisbasis,theconverseofIfanyPisQthatPmustbeRwouldbeIfanyPisRthatPmaybeQ,andthecontrapositivewouldbeIfanyPisnotRthatPcannotbeQ.

Are these inferences legitimate? On the interpretation that a modalproposition implies nothing as to the possibility of its antecedent, then ouranswermustbe in the affirmative, as regards the contrapositionofAm. Thefullimportbothoftheoriginalpropositionandofthecontrapositiveistodenythe possibility of the combination P and Q without R. On the sameinterpretation,however, the conversionofAm is not valid.For the converseimpliesthatifPRispossiblethenPQispossible,whilethepossibilityofPRcombinedwith the impossibility ofPQ is compatible with the truth of theoriginalproposition. It canbeshewnsimilarly that,while theconversionofEmisvalid,itscontrapositionisinvalid.

Ifweweretovarytheinterpretation,theresultswouldbedifferent.

The correspondence between the results shewn above and our resultsrespecting the conversion and contraposition of the assertoric A and Epropositions,ontheinterpretationthatnopropositionimpliestheexistenceofits subject (seepage225), is obvious.The truth is that the interpretationofmodalsinrespecttothepossibilityoftheirantecedentsgivesrisetoproblemsprecisely analogous to those arising out of the interpretation of assertoricpropositionsinrespecttotheactualityoftheirsubjects.Itisunnecessarythatweshouldworkoutthedifferentcasesindetail.


Amongst immediate inferences from a conditional proposition, itsreduction to categorical form, so far as this is valid, is generally included.Thisisacaseofwhathasbeencalledchangeofrelation,meaningtherebyanimmediate inference inwhichwe pass from a given proposition to anotherwhichbelongstoadifferentcategoryinthedivisionofpropositionsaccordingtorelation(seesection54).ThemoreconvenienttermtransversionisusedbyMissJonesforthisprocess.

How far conditionals can be inferred from categoricals and vice versâdepends on their interpretation. If both types of 261 propositions areinterpretedassertoricallyorbothmodally,andiftheyareinterpretedsimilarlyasregardstheimplicationoftheexistence(orpossibility)oftheirsubjects(orantecedents),thenthevalidityofpassingfromeithertypetotheothercannotbecalled inquestion.Somedoubtmay,however,beraisedas towhether inthis case we have an inference at all or merely a verbal change. This is adistinctiontowhichattentionwillbecalledlateron.

Ifconditionalsareinterpretedmodallyandcategoricalsassertoricallythen(apartfromanycomplicationsthatmayarisefromexistentialimplications)AcanbeinferredfromAmorEfromEm,butnotviceversâ.Ontheotherhand,ImcanbeinferredfromI,orOmfromO,butnotviceversâ.

We have another case of transversionwhenwe pass from conditional todisjunctive,orfromdisjunctivetoconditional.Theconsiderationofthiscasemustbedeferreduntilwehavediscusseddisjunctives.

178.TheImportofHypotheticalPropositions.—ThepurehypotheticalmaybewrittensymbolicallyintheformIfAistruethenCistrue,ormorebriefly,IfAthenC,whereAandCstandforpropositionsofindependentimport.Itisclear that this proposition affirms nothing as regards the truth or falsity ofeitherAorCtakenseparately.Wemayindeedframetheproposition,knowingthatC isfalse,withtheexpressobjectofshowingthatA isfalsealso.WhatwehaveisofcourseajudgmentnotabouteitherAorCtakenseparately,butaboutAandCinrelationtooneanother.

The main question at issue in regard to the import of the hypotheticalpropositioniswhetheritismerelyassertoricorismodal.Thecontrastmaybe


simply put by asking whether, when we say If A then C, our intention ismerelytodenytheactualityoftheconjunctionofAtruewithCfalseoristodeclarethisconjunctiontobeanimpossibility.

The contrast between these two interpretations can be brought out mostclearlybyaskinghowthepropositionIfAthenCistobecontradicted.IfourintentionismerelytodenytheactualityoftheconjunctionofA truewithCfalse,thenthecontradictorymustasserttheactualityofthisconjunction;if262our intention is to deny the possibility of the conjunction, then thecontradictory will merely assert its possibility. In other words, on theassertoric interpretation thecontradictorywillbeAis truebutC is false ;280onthemodalinterpretationitwillbeIfAistrueCmaybefalse.281

280 Wemaylookatitinthisway.LetACdenotethetruthofbothAandC,ACʹthe truth of A and the falsity of C, and so on. Then there are four à prioripossibilities,namely,AC,ACʹ,AʹC,AʹCʹ, oneor other ofwhichmust holdgood,butanypairofwhicharemutuallyinconsistent.ThepropositionIfAthenCmerelyexcludesACʹ,andstillleavesAC,AʹC,AʹCʹ,aspossiblealternatives.Indenyingit,therefore,wemustdefinitelyaffirmACʹ,andexcludethethreeotheralternatives.Hencethecontradictoryasabovestated.

281 Acertain assumption isnecessary, inorder that this resultmaybecorrect.The opposition of hypotheticals on themodal interpretationwill be discussed inmoredetailinsection179.

Hypotheticalsintendedtobeinterpretedassertoricallyaretobemetwithinordinary discourse, but they are unusual.There appear to be two cases: (a)Whenweknowthatoneorotheroftwopropositionsistruebutdonotknow(ordonotremember)which,wemayexpressourknowledgeintheformofahypothetical, If X is not true then Y is true, and such hypothetical will bemerelyassertoric.Forexample,If the flowers Iplanted in thisbedwerenotpansiestheywereviolets.Heretheintentionismerelytodenytheactualityoftheflowersbeingneitherpansiesnorviolets.(b)Wemaydenyapropositionemphatically by a hypothetical in which the proposition in question iscombined as antecedent with a manifestly false consequent; and suchhypotheticalwillagainbemerelyassertoric.Forexample,Ifwhatyousayistrue,I’maDutchman ;If thatboycomesback,I’lleatmyhead(videOliverTwist);I’mhangedifIknowwhatyoumean.Intheseexamplestheintentionis todeny theactuality (not thepossibility)of theconjunctions,—Whatyou


sayistrueandIamnotaDutchman ;ThatboywillcomebackandIshallnoteatmy head ; I am not hanged and I knowwhat you mean; and since theelementsoftheconjunctionsprintedinitalicsareadmittedlytrue,theforceofthepropositions is todeny the truthof theother elements, that is to say, toaffirm,—Whatyousayisnottrue,Thatboywillnotcomeback,Idonotknowwhat you mean. Similarly 263 we may sometimes employ the hypotheticalform of expression as an emphatic way of declaring the truth of theconsequent (an antecedent being chosen which is admittedly true); forexample, If he cannot act, at any rate he can sing. Here once more thehypotheticalismerelyassertoric.

It cannot, however, be maintained that any of the above are typicalhypotheticals;andtheclaimthatournaturalinterpretationofhypotheticalsisordinarily modal may be justified on the ground that we do not usuallyconsiderittobenecessarytoaffirmtheantecedentinordertobeabletodenyahypothetical.Wehaveseenthat,inordertodenytheassertorichypotheticalIfAthenC,wemustaffirmAanddenyC ;butweshouldusuallyregarditassufficient for denial if we can shew that there is no necessary connexionbetweenthetruthofAandthatofC,whetherAisactuallytrueornot.

We shall then in the main be in agreement with ordinary usage if weinterpret hypotheticals modally, and the adoption of such an interpretationwillalsogivehypotheticalsamoredistinctivecharacter.Inwhatfollowsthehypotheticalformwillaccordinglyberegardedasmodal,exceptinsofarasanexplicitstatementismadetothecontrary.282

282 Either C is true or A is not true is usually regarded as the disjunctiveequivalentofthehypotheticalIfAistruethenCistrue.Therelationbetweenthesetwopropositionswillbediscussedfurtherlateron.Itis,however,desirabletopointoutatoncethat,iftheequivalenceistoholdgood,boththepropositionsmustbeinterpreted assertorically or both modally. There is a good deal to be said fordifferentiating the two forms by regarding the hypothetical as modal and thedisjunctiveasmerelyassertoric.ThismethodoftreatmentisexplicitlyadoptedbyMrMcColl.Hewrites(usingthesymbolism,a:bforIfathenb,a+bforaorb,aʹ for thedenialofa)—“Theexpressiona :bmaybereada impliesb or If a istrue,bmustbetrue.Thestatementa:bimpliesaʹ+b.Butitmaybeaskedarenotthetwostatementsreallyequivalent;oughtwenotthereforetowritea:b=aʹ+b?Now if the two statements are really equivalent their denials will also beequivalent.Letusseeifthiswillbethecase,takingasconcreteexamples:‘Ifhepersists in his extravagance he will be ruined’; ‘He will either discontinue his


extravaganceorhewillberuined.’Thedenialofa:bis(a:b)ʹandthisdenialmaybe read—‘Hemay persist in his extravagancewithout necessarily being ruined.’Thedenialofaʹ+bisabʹwhichmayberead—‘Hewillpersistinhisextravaganceandhewillnotberuined.’Nowitisquiteevidentthattheseconddenialisamuchstronger and more positive statement than the first. The first only asserts thepossibility of the combination abʹ ; the second asserts the certainty of the samecombination. The denials of the statements a : b and aʹ + b having thus beenproved to be not equivalent, it follows that the statements a :b and aʹ + b arethemselvesnotequivalent,and that, thoughaʹ+b isanecessaryconsequenceofa:b,yeta:bisnotanecessaryconsequenceofaʹ+b”(seeMind,1880,pp.50to54;oneortwoslightverbalchangeshavebeenmadeinthisquotation).

264Somewriterswhoadoptthemodalinterpretationofhypotheticalsspeakof the consequent as being an inference from the antecedent. There are nodoubtsomehypotheticals towhichthisdescriptionaccuratelyapplies.Thus,wemayhavehypotheticalswhichareformalinthesenseinwhichthattermhasbeenusedinsection31,theconsequentbeing,forinstance,animmediateinference from the antecedent, or being the conclusion of a syllogism ofwhich the premisses constitute the antecedent.The following are examples,—Ifallisoscelestriangleshavetheanglesatthebaseequaltooneanother,thennotriangletheanglesatwhosebaseareunequalcanbeisosceles ;IfallmenaremortalandthePopeisaman,thenthePopemustbemortal.

Butmore usually the consequent of a hypothetical proposition cannot beinferred from the antecedent alone. The aid is required of suppressedpremisseswhicharetakenforgranted,thepremisswhichaloneisexpressedbeingperhapstheonlyoneastothetruthofwhichanydoubtisregardedasadmissible.Itwould,therefore,bebettertospeakoftheconsequentasbeingthenecessaryconsequenceoftheantecedent,thanasbeinganinferencefromit.WhenwespeakofCasbeingan inferencefromA, there is a suggestionthatAaffordsthecompletejustificationofC,whereaswhenwespeakofitasanecessaryconsequence,thissuggestionisatanyratelessprominent.283

283 MissJones(GeneralLogic,p.45)divideshypotheticalsintoformalorself-contained hypotheticals and referential hypotheticals. In the former, “theconsequent is an inference from the antecedent alone”; in the latter, “theconsequentisinferrednotfromtheantecedentalone,butfromtheantecedenttakeninconjunctionwithsomeotherunexpressedpropositionorpropositions.”

179. The Opposition of Hypothetical Propositions.—Regardinghypotheticalsasalwaysaffirminganecessaryconsequence,itmayreasonably


beheldthattheydonotadmitofdistinctionsofquality.Sigwartaccordinglylays it down that all hypotheticals are affirmative. “Passing to hypotheticaljudgments265containingnegations,wefind that theform‘IfA is,B isnot’represents thenegationofapropositionas thenecessaryconsequenceofanaffirmation,thusaffirmingthatthehypothesesAandBareincompatible.”284The force of this argument must be admitted. There is, however, someconvenience in distinguishing between hypotheticals according as they leadup, in the consequent, to an affirmation or a denial; and in the formaltreatmentofhypotheticals,weshallbebetterabletopreserveananalogywithcategoricalsandconditionalsifwedenotethepropositionIfXistruethenYistruebythesymbolAm,andthepropositionIfXistruethenYisnottruebythesymbolEm.

284 Logic,i.p.226.

Whether or notwedecide thus to recognise distinctionsof quality in thecaseofhypotheticals,wecertainlycannotrecognisedistinctionsofquantity.Theantecedentofahypotheticalisnotaneventwhichmayrecuranindefinitenumber of times, but a propositionwhich is simply true or false.We havealready seen that the same proposition cannot be sometimes true andsometimes false, since propositions referring to different times are differentpropositions.285

285 This,asMrJohnsonhaspointedout,mustbe taken inconnexionwith therecognition of propositions as involvingmultiple quantification. “Thus we mayindicateaseriesofpropositionsinvolvingsingle,double, triple…quantification,whichmay reach any order of multiplicity: (1) ‘All luxuries are taxed’; (2) ‘Insome countries all luxuries are taxed’; (3) ‘At some periods it is true that in allcountriesallluxuriesaretaxed’.…withrespecttoeachofthetypesofproposition(1), (2), (3).… I contend that,whenmade explicitwith respect to timeor place,etc.,itisabsurdtospeakofthemassometimestrueandsometimesfalse”(Mind,1892,p.30n.).

Donotdistinctionsofmodality,however, taketheplaceofdistinctionsofquantity?Up to thispoint,wehavepracticallyconfinedourattention to theapodeictic hypothetical, If A then C. This proposition is denied by thepropositionIfAistruestillCneednotbetrue(thatistosay,ThetruthofCisnotanecessaryconsequenceofthetruthofA).Canthislatterpropositionbedescribedasaproblematichypothetical?Clearlyitisnotahypotheticalatall


if we begin by defining a hypothetical as the affirmation of a necessaryconsequence. There seems, however, no need for this limitation. We maydefinea266hypotheticalasapropositionwhichstartingfromthehypothesisofthetruth(orfalsity)ofagivenpropositionaffirms(ordenies)thatthetruth(or falsity) of another proposition is a necessary consequence thereof. But,whether or not we adopt this definition, there can be no doubt that theproposition If A then possibly C appropriately finds a place in the sameschedule of propositions as If A then necessarilyC. In such a schedulewehavethefourforms,—

IfAistruethenCistrue ; Am

IfAistruethenCisnottrue ; Em

IfAistruestillCmaybetrue ; Im

IfAistruestillCneednotbetrue. Om

Thesefourpropositionscorrespondtothoseincludedintheordinarysquareofopposition;and,ifwestartwiththeassumptionthatAispossiblytrue,286theordinaryrelationsofoppositionholdgoodbetweenthem.AmandOm,EmandImarepairsofcontradictories;AmandEmarecontraries;AmandIm,Em andOm,arepairsofsubalterns;ImandOmaresubcontraries.

286 By this ismeant thatwe startwith the assumption thatA is possibly trueindependently of the affirmation of any one of the propositions in question. Thereader must particularly notice that this assumption is quite different from theassumptionthateachofthepropositionalformsimpliesaspartofitsimportthatAispossiblytrue;otherwisetheresultsreachedinthisparagraphmayappeartobeinconsistentwiththosereachedinthefollowingparagraph.

If,however, it isnotassumed thatA ispossibly true, then theproblem ismore complicated, since the character of the relations is affected by themannerinwhichthepropositionsareinterpretedinrespecttothepossibilityof theirantecedents.The resultsare substantially thesameas in thecaseofmodal conditionals (section 176), and correspond with those obtained insection 159, where the analogous problem in regard to categoricals(assertorically interpreted) is discussed.Thus, in order thatAm, andOm,Em

andIm,maybecontradictories,apodeicticandproblematicpropositionsmustbeinterpreteddifferentlyasregardstheimplicationornon-implicationofthe


possibletruthoftheirantecedents;while,ontheotherhand,inorderthatAm

andIm,EmandOm,maybesubalterns,267problematicpropositionsmustnotbe interpreted as implying the possible truth of their antecedents unlessapodeicticpropositionsareinterpretedsimilarlyinthisrespect.Ifweinterpretneither apodeictic nor problematic hypotheticals as implying the possibletruth of their antecedents, then the contradictory of If A, then C may beexpressedintheformPossiblyA,butnotC(or,asitmayalsobeformulated,Aispossiblytrue,andifitistrue,stillCneednotbetrue).

Itwouldoccupytoomuchspace todiscuss indetailall theproblemsthatmight be raised in this connexion. The principles involved have beensufficiently indicated; and the readerwill find no difficulty inworking outother cases for himself. We may, however, touch briefly on the relationbetweenthepropositionsIfAthenCandIfAthennotC,shewinginparticularthatonnosuppositionaretheytruecontradictories.

If these two propositions are interpreted assertorically, then so far frombeingcontradictories,theyaresubcontraries.For,supposingAhappensnottobetrue, thenitcannotbesaidthateitherofthemisfalse: thestatementIfAthenCmerelyexcludesACʹ,andIfAthenCʹmerelyexcludesAC ;hencetwopossibilitiesareleft,AʹCorAʹCʹ,neitherofwhichisinconsistentwitheitherof the propositions.287 On the other hand, the propositions cannot both befalse,sincethiswouldmeanthetruthofbothACʹandAC.

287 The validity of the above result will perhaps be more clearly seen bysubstitutingforthehypotheticalstheir(assertoric)disjunctiveequivalents,namely,EitherAisnottrueorCistrue,EitherAisnottrueorCisnottrue.Asaconcreteexample, we may take the propositions, “If this pen is not cross-nibbed, it iscorrodedbytheink,”“Ifthispenisnotcross-nibbed,itisnotcorrodedbytheink.”Supposing that the pen happens to be cross-nibbed, we cannot regard either ofthesepropositionsasfalse.Itwillbeobservedthattheirdisjunctiveequivalentsare,“Thispeniseithercross-nibbedorcorrodedbytheink,”“Thispeniseithercross-nibbedornotcorrodedbytheink.”Takeagainthepropositions,“Ifthesunmovesroundtheearth,someastronomersarefallible.”“Ifthesunmovesroundtheearth,allastronomersareinfallible.”Thetruthofthefirstofthesepropositionswillnotbe denied, and on the interpretation of hypotheticals with which we are hereconcerned thesecondcannotbesaid tobe false. Itmaybe takenasanemphaticwayofdenyingthatthesundoesmoveroundtheearth.

Returningtothemodalinterpretationofthepropositions,thenifinterpreted


as implying the possible truth of their 268 antecedents, they are contraries.Theycannotbothbe true,butmaybothbe false. Itmaybe thatneither thetruthnorthefalsityofCisanecessaryconsequenceofthetruthofA.288

288 IthasbeenarguedthatIfAthenCmusthaveforitscontradictoryIfAthennotC,sincetheconsequentmusteitherfollowornotfollowfromtheantecedent.ButtosaythatCdoesnotfollowfromAisobviouslynotthesamethingastosaythatnot-CfollowsfromA.

Oncemore,ifinterpretedmodallybutnotasimplyingthepossibletruthoftheirantecedents,thepropositionsmaybothbetrueaswellasbothfalse.Thiscase is realised when we establish the impossibility of the truth of apropositionbyshewingthat,ifitweretrue,inconsistentresultswouldfollow.

180. Immediate Inferences from Hypothetical Propositions.—The mostimportantimmediateinferencefromthepropositionIfAthenCisIfCʹthenAʹ.This inference isanalogous tocontraposition in thecaseofcategoricals,andmaywithoutanyriskofconfusionbecalledbythesamename.Wemayaccordingly define the term contraposition as applied to hypotheticals asaprocessofimmediateinferencebywhichweobtainanewhypotheticalhavingfor its antecedent the contradictory of the old consequent, and for itsconsequent the contradictory of the old antecedent. If we recognisedistinctions of quality in hypotheticals, then (as regards apodeictichypotheticals)thisprocessisvalidinthecaseofaffirmativesonly.Itwillbeobserved that from the contrapositive we can pass back to the originalproposition; and from this it follows that the original proposition and itscontrapositiveareequivalents.289Thefollowingareexamples:“Ifpatienceisavirtue,therearepainfulvirtues”=“Iftherearenopainfulvirtues,patienceisnotavirtue”;“IfthereisarighteousGod,thewickedwillnotescapetheirjust punishment”= “If thewicked escape their just punishment, there is norighteousGod.”

289 Thisholdsgoodwhetherweadopttheassertoricorthemodalinterpretation.Ontheformerinterpretation,theimportofboththepropositionsIfAthenCandIfCʹthenAʹ istonegativeACʹ ;on the latter interpretation, the importofboth is todenythepossibilityoftheconjunctionACʹ.

FromthenegativehypotheticalIfAistruethenCisnottruewecaninferIfC is true thenA is not true. This is analogous toconversion in the caseof


categoricals.

269FromtheaffirmativeIfAthenC,wemayobtainbyconversionIfCthenpossiblyA ;butthisisonlyontheinterpretationthatbothpropositionsimplythepossibilityofthetruthoftheirantecedents.290Thereaderwillnoticethatto pass from If A then C to If C then A would be to commit a fallacyanalogous to simply converting a categorical A proposition; and this isperhaps the most dangerous fallacy to be guarded against in the use ofhypotheticals.291


290 Compare section 158.The various results obtained in section 158may beappliedmutatis mutandis to modal hypotheticals. The reader may consider forhimselfthecontrapositionofEm.

291 OntheassertoricinterpretationIfAthenCmerelynegativesACʹ,whileIfCthenAmerelynegativesAʹC,andhenceitisclearthatneitherofthesepropositionsinvolvestheother;onthemodalinterpretationtheresultisthesame,forthetruthofCmaybeanecessaryconsequenceofthetruthofA,whiletheconversedoesnotholdgood.

Aconsiderationofimmediateinferencesenablesustoshewfromanotherpointof view that If A then C and If A then Cʹ are not true contradictories. For thecontrapositives If A then Cʹ, If C then Aʹ, are equivalent to one another; andwhenever two propositions are equivalent, their contradictories must also beequivalent.ButIfAthenCisnotequivalenttoIfCthenA.

If distinctions of quality are admitted, then the process of obversion isapplicabletohypotheticals.Forexample,IfAistruethenCisnottrue=IfAistruethenCʹistrue.Itisnearlyalwaysmorenaturalandmoreconvenienttotakehypotheticalsintheiraffirmativeratherthanintheirnegativeform;andhenceinthecaseofhypotheticalsmoreimportanceattachestotheprocessofcontrapositionthantothatofconversion.

IfthefalsityofCisassumedtobepossible,thenwemaypassbyinversionfromIfAthenCtoItispossibleforbothAandCnottobetrue ;or,puttingthesamethinginadifferentway,wemaybyinversionpassfromIfAthenCtoIfthefalsityofCispossiblethenthefalsityofbothAandCispossible.292ItisofcourseafallacytoarguefromIfAthenCtoIfAʹthenCʹ.

292 The inversionofEmmaybeworkedout similarly.Here, as elsewhere, theprocessofinversion,althoughoflittleornopracticalimportance,raisesproblemsthatareofconsiderabletheoreticalinterest.

Turningtoproblematichypotheticals,wefindthatfromthepropositionIfAistrueCmaybetrue,weobtainbyconversionIfCistrueAmaybetrue ;andfrom the proposition If A is 270 true C need not be true we obtain bycontraposition If C is true A need not be true. Here the analogy withcategoricalsisagainveryclose.

181. Hypothetical Propositions and Categorical Propositions.—A truehypothetical proposition has been defined as a proposition expressing arelationbetween twootherpropositionsof independent import,notbetweentwo terms; and it follows that a true hypothetical is not, like a conditional,


easily reducible to categorical form. So far aswe can obtain an equivalentcategorical,itssubjectandpredicatewillnotcorrespondwiththeantecedentand consequent of the hypothetical.Thus, the proposition IfA thenCmay,according to our interpretation of it, be expressed in one or other of thefollowingforms;Aisapropositionthetruthofwhichisincompatiblewiththefalsity of C ; A is a proposition from the truth of which the truth of Cnecessarily follows. It will be observed that, apart from the fact that thesepropositionsarenotoftheordinarycategoricaltype,293thepredicateisnotineitherofthemequivalenttotheconsequentofthehypothetical.294Nodoubtahypothetical proposition may be based on a categorical proposition of theordinary type. But that is quite a different thing from saying that the twopropositionsareequivalenttooneanother.

293 Since they are compound, not simple, propositions. The expression ofcompound propositions in categorical form is not convenient, and it is better toreserve the hypothetical and disjunctive forms for such propositions, thecategoricalandconditionalformsbeingusedforsimplepropositions.

294 Amongst other differences the contrapositives of both these propositionsdiffer from thecontrapositiveof thehypothetical.For,oneither interpretationofthehypothetical,itscontrapositiveisIfCisnottruethenAisnottrue,whilstthecontrapositives of the above propositions are respectively,—ApropositionwhosetruthiscompatiblewiththefalsityofthepropositionCisnotthepropositionA,ApropositionfromwhichthepropositionCisnotanecessaryconsequenceisnotthepropositionA.

The relation between hypothetical and disjunctive propositions will bediscussedinthefollowingchapter.

182. Alleged Reciprocal Character of Conditional and HypotheticalJudgments.—MrBosanquetarguesthatthehypotheticaljudgment(andunderthis designation hewould include the conditional aswell aswhatwe havecalledthetrue271hypothetical)“whenideallycompletemustbeareciprocaljudgment.IfAisB,itisCmustjustifytheinferenceIfAisC,itisB.Weareofcourseinthehabitofdealingwithhypotheticaljudgmentswhichwillnotadmitofanysuchconversion,andtherulesoflogicacceptthislimitation…Ifinactualfact…ABisfoundtoinvolveACwhileACdoesnotinvolveAB,it isplainthatwhatwasrelevanttoACwasnotreallyABbutsomeelementαβwithinit…Apartfromtimeontheonehandandirrelevantelementsontheother, I cannot seehow the relationof conditioningdiffers from thatof


beingconditioned…Inotherwords,ifthereisnothinginAbeyondwhatisnecessary toB, thenB involvesA just as much asA involvesB. But ifAcontainsirrelevantelements,thenofcoursetherelationbecomesone-sided…The relationofGround is thus essentially reciprocal, and it is onlybecausethe‘grounds’allegedinevery-daylifeareburdenedwithirrelevantmatterorconfusedwithcausationintime,thatweconsidertheHypotheticalJudgmenttobeinitsnaturenotreversible”(Logic,I.pp.261–3).

Thequestionhereraisedisanalogoustothatofthepossibilityofpluralityofcauseswhichisdiscussedininductivelogic.Itmayperhapsbedescribedasawideraspectofthesamequestion.Solongasagivenconsequencehasapluralityofgrounds,itisclearthatthehypotheticalpropositionaffirmingittobeaconsequenceofaparticularoneofthesegroundscannotadmitofsimpleconversion,fortheconvertedpropositionwouldholdgoodonlyifthegroundinquestionwerethesoleground.

MrBosanqueturgesthattherelationbetweengroundandconsequencewillbecome reciprocal by the elimination from the antecedent of all irrelevantelements. It should be added that we can also secure reciprocity by theexpansionoftheconsequentsothatwhatfollowsfromtheantecedentisfullyexpressed. Thus, if we have the hypothetical If A then γ, which is notreciprocal, it ispossiblethatAmaybecapableofanalysis intoαβ,andγofexpansionintoγδ,sothateitherofthehypotheticalsIfαthenγ,Ifαβthenγδ,is reciprocal. In the former case we have a more exact statement of theground, all extraneous 272 elements being eliminated; in the latter case wehaveamorecomplete statementof theconsequence.Sometimes,moreover,thelatterofthesealternativesmaybepracticablewhiletheformerisnot.

Thismaybetestedbyreferencetoaformalhypothetical.ThepropositionIfallSisMandallMisP,thenallSisPisnotreciprocal.WemaymakeitsobyexpandingtheconsequentsothatthepropositionbecomesIfallSisMandallMisP,thenwhateveriseitherSorMisPandisalsoMornotS.Buthowin this case it would be possible to eliminate the irrelevant from theantecedent it is difficult to see. Our object is to eliminate M from theconsequent,andifinadvanceweweretoeliminateitfromtheantecedentthewhole forceof thepropositionwouldbe lost.And the same is trueofnon-


formal hypotheticals, at any rate in many cases. Instances of reciprocalconditionalsmaybegivenwithoutdifficulty, forexample, Ifany triangle isequilateral, it is equiangular. Such propositions are practically Upropositions. We may also find instances of pure hypotheticals that arereciprocal; but, on the whole, while agreeing with a good deal that MrBosanquet sayson the subject, I amdisposed todemur tohisview that thereciprocal hypothetical represents an ideal atwhichwe should always aim.We have seen that there are two possible ways of securing reciprocity,whether or not they are always practicable; but the expansion of theconsequentwouldgenerallyspeakingbeextremelycumbrousandworsethanuseless, while the elimination from the antecedent of everything notabsolutely essential for the realisation of the consequent would sometimesempty the judgment of all practical content for a given purpose. WithreferencetothecasewhereABinvolvesAC,whileACdoesnotinvolveAB,Mr Bosanquet himself notes the objection,—“But may not the irrelevantelementbejusttheelementwhichmadeABintoABasdistinctfromAC, sothatbyabstractingfromitABisreducedtoAC,andthejudgmentismadeatautology, that is, destroyed?” (p. 261). This argument, although somewhatoverstated,deservesconsideration.ThepointuponwhichIshouldbeinclinedtolaystressisthatincriticisingajudgmentweoughttohaveregard273tothespecialobjectwithwhich ithasbeenframed.Ourobjectmaybe toconnectAC with AB, including whatever may be irrelevant in AB. Consider theargument,—If anything is P it is Q, If anything is Q it is R, therefore, Ifanything isP it isR. It is clear that ifwe compare the conclusionwith thesecondpremiss,theantecedentoftheconclusioncontainsirrelevanciesfromwhichtheantecedentofthepremissisfree.Yettheconclusionmaybeofthegreatestvaluetouswhilethepremissisbyitselfofnovalue.Ifouraimwerealwaystogetdowntofirstprinciples,therewouldbeagooddealtobesaidforMrBosanquet’sview, though itmightstillpresentsomedifficulties;butthereisnoreasonwhyweshouldidentifytheconditionalorthehypotheticalpropositionwiththeexpressionoffirstprinciples.

It is to be added that, ifMrBosanquet’s view is sound,weought to sayequallythattheAcategoricalpropositionisimperfect,andthatincategoricalsthe U proposition is the ideal at which we should aim. In categoricals,


however, we clearly distinguish betweenA andU; and so far as we giveprominencetothereciprocalmodal,whetherconditionalorhypothetical,weoughttorecogniseitsdistinctivecharacter.WemayatthesametimeassigntoitthedistinctivesymbolUm.

EXERCISES.

183. Give the contrapositive of the following proposition: If either noP isR or noQ isR, thennothingthatisbothPandQisR.[K.]

184.Therearethreemeninahouse,Allen,Brown,andCarr,whomaygoinandout,providedthat(1)theynevergooutallatonce,andthat(2)AllennevergoesoutwithoutBrown. CanCarrevergoout?[LEWISCARROLL.]

185.Therearetwopropositions,AandB.Letitbegrantedthat IfAistrue,Bistrue.(i)LettherebeanotherpropositionC,suchthat IfCistrue,thenifAistrueBisnottrue.(ii)274(ii)amountstothis,— IfCistrue,then(i)isnottrue.But,exhypothesi,(i)istrue. Therefore,Ccannotbetrue;fortheassumptionofCinvolvesanabsurdity. Examinethisargument.[LEWISCARROLL.]

[If the problem in section 184 is regarded as a problem in conditionals, this is the correspondingprobleminhypotheticals.]

186. Assuming that rain never falls in Upper Egypt, are the following genuine pairs ofcontradictories? (a)TheoccurrenceofraininUpperEgyptisalwayssucceededbyanearthquake;theoccurrenceofraininUpperEgyptissometimesnotsucceededbyanearthquake. (b) If it is true that it rained inUpper Egypt on the 1st of July, it is also true that an earthquakefollowedonthesameday;ifitistruethatitrainedinUpperEgyptonthe1stofJuly,itisnotalsotruethatanearthquakefollowedonthesameday. Iftheabovearenottruecontradictories,suggestwhatshouldbesubstituted.[B.]

187.Givethecontrapositiveandthecontradictoryofeachofthefollowingpropositions: (1) If any nation prospers under aProtectiveSystem, its citizens reject all arguments in favour offree-trade; (2)IfanynationprospersunderaProtectiveSystem,weought torejectallarguments infavouroffree-trade.[J.]

188.Examinethelogicalrelationbetweenthetwofollowingpropositions;andenquirewhetheritislogicallypossibletohold(a)thatbotharetrue,(b)thatbotharefalse:(i)Ifvolitionsareundetermined,


thenpunishmentscannotrightlybeinflicted;(ii)Ifpunishmentscanrightlybeinflicted,thenvolitionsareundetermined.[J.]


CHAPTERX.

DISJUNCTIVE(ORALTERNATIVE)PROPOSITIONS.

189.The termsDisjunctive andAlternative as applied toPropositions.—PropositionsoftheformEitherXorYistrueareordinarilycalleddisjunctive.It has been pointed out, however, that twopropositions are reallydisjoinedwhen it isdenied that theyareboth true rather thanwhen it isasserted thatoneorotherof themis true;andthe termalternative,assuggestedbyMissJones (Elements of Logic, p. 115), is obviously appropriate to express thelatter assertion. We should then use the terms conjunctive, disjunctive,alternative,remotive,forthefourfollowingcombinationsrespectively:XandYarebothtrue,XandYarenotbothtrue,EitherXorYistrue,295NeitherXnorYistrue.

295 Some writers indeed regard the proposition Either X or Y is true asexpressingarelationbetweenXandYwhich isdisjunctive in theabovesenseaswell as alternative; but the disjunctive character of this proposition as regardsXand Y is at any rate open to dispute, whilst its alternative character isunquestionable(seesection191).

Whilst, however, thenamealternative is preferable todisjunctive for thepropositionEither X or Y is true, the latter name has such an establishedposition in logical nomenclature that it seems inadvisable altogether todiscontinue its use in the old sense. It may be pointed out further that analternative contains a veiled disjunction (namely, betweennot-X andnot-Y)even in the stricter sense; for the statement that Either X or Y is true isequivalent to the statement thatNot-X and not-Y are not both true. Hence,althoughgenerallyusingthetermalternative,Ishallnotentirelydiscardthetermdisjunctiveassynonymouswithit.

276190.TwotypesofAlternativePropositions.—Inthecaseofpropositionswhich are ordinarily described as simply disjunctive a distinction must bedrawnsimilartothatdrawnintheprecedingchapterbetweenconditionalsand


true hypotheticals. For the alternatives may be events or combinations ofproperties one or other of which it is affirmed will (always or sometimes)occur,e.g.,Everybloodvesseliseitheraveinoranartery,Everyprosperousnationhaseitherabundantnaturalresourcesoragoodgovernment ;ortheymaybepropositionsof independent importwhose truthor falsitycannotbeaffected by varying conditions of time, space, or circumstance, and whichmust therefore be simply true or false, e.g.,Either there is a future life ormanycrueltiesgounpunished,EitheritisnosintocovethonourorIamthemostoffendingsoulalive.

Anypropositionbelongingto thefirstof theabovetypesmaybebroughtunderthesymbolicformAll(orsome)SiseitherPorQ,andmay,therefore,beregardedasanordinarycategoricalpropositionwithanalternativetermaspredicate.Itisusualandforsomereasonsconvenienttodeferthediscussionof the import of alternative terms until propositions of this type are beingdealtwith.Suchpropositionsmightotherwisebedismissedafteraverybriefconsideration.296

296 ItshouldbeparticularlyobservedthatalthoughthepropositionEverySisPorQmaybesaidtostateanalternative,itcannotberesolvedintoatruealternativecombination of propositions. Such a resolution is, however, possible if theproposition(whileremainingaffirmativeandstillhavinganalternativepredicate)issingularorparticular:forexample,ThisSisPorQ=ThisSisPorthisSisQ ;SomeSisPorQ=SomeSisPorsomeSisQ.

Corresponding to this,wemaynote thatanaffirmativecategoricalpropositionwithaconjunctivepredicate isequivalent toaconjunctionofpropositions if it issingularoruniversal,butnotifitisparticular.Thus,ThisSisPandQ=ThisSisPandthisSisQ ;AllSisPandQ=AllSisPandallSisQ.FromthepropositionSomeS isPandQwemay indeed inferSome S isP and some S isQ ; butwecannotpassback from thisconclusion to thepremiss,andhence the twoarenotequivalenttooneanother.

It may be added that a negative categorical proposition with an alternativepredicatecannotbesaidtostateanalternativeatall,sincetodenyanalternationisthesamethingastoaffirmaconjunction.ThusthepropositionNoSiseitherPorQcanonlyberesolvedintoaconjunctivesynthesisofpropositions,namely,NoSisPandnoSisQ.

277Alternativepropositionsofthesecondtypearecompound(asdefinedinsection 55). They contain an alternative combination of propositions ofindependentimport:andtheyhavefortheirtypicalsymbolicformEitherXis


trueorYistrue,ormorebriefly,EitherXorY,whereXandYaresymbolsrepresentingpropositions(notterms).Sofarasitisnecessarytogivethemadistinctivename,theyhaveaclaimtobecalledtruealternativepropositions,sincetheyinvolveatruealternativesynthesisofpropositions,andnotmerelyanalternativesynthesisofterms.

ItwillbeconvenienttospeakofPandQasthealternantsofthealternativetermPorQ,andofXandYas thealternantsof thealternativepropositionEitherXorY.

191.TheImportofDisjunctive(Alternative)Propositions.—The twomainquestionsthatariseinregardtotheimportofalternativepropositionsare(1)whetherthealternantsofsuchpropositionsarenecessarilytoberegardedasmutually exclusive, (2) whether the propositions are to be interpreted asassertoricormodal.

(1)Weaskthen,inthefirstplace,whetherinanalternativepropositionthealternantsaretobeinterpretedasformallyexclusiveofoneanother;inotherwords,whether in thepropositionAllS iseitherAorB it isnecessarily (orformally)impliedthatnoSisbothAandB,297andwhetherinthepropositionXistrueorYistrue it isnecessarily(orformally)impliedthatXandYarenotbothtrue.Itisdesirabletonoticeattheoutsetthatthequestionisoneoftheinterpretationofapropositionalform,andonethatdoesnotariseexceptin connexion with the expression of judgments in language. Hence thesolutionwillbe,atanyratepartly,amatterofconvention.

297 Thisisanalternativepropositionofthefirsttype,andthesamequestionisraised by asking whether the term A or B includes AB under its denotation orexcludesit;inotherwords,whetherthedenotationofAorBisrepresentedbytheshadedportionofthefirstorofthesecondofthefollowingdiagrams:

278Thefollowingconsiderationsmayhelptomakethispointclearer.LetXandYrepresenttwojudgments.ThenthefollowingaretwopossiblestatesofmindinwhichwemaybewithregardtoXandY:


(a)wemayknow thatoneorotherof them is true,and that theyarenotbothtrue; (b)wemayknowthatoneorotherofthemistrue,butmaybeignorantastowhethertheyareorarenotbothtrue.

Now whichever interpretation (exclusive or non-exclusive) of thepropositionalformXorYisadopted,therewillbenodifficultyinexpressingalternativelyeitherstateofmind.Ontheexclusiveinterpretation,(a)willbeexpressedintheformXorY,(b)intheformXYorXYʹorXʹY(XʹrepresentingthefalsityofX,andYʹ thefalsityofY).Onthenon-exclusiveinterpretation,(a)willbeexpressedintheformXʹYorXYʹ,(b)intheformXorY.Therecan,therefore,benointrinsicgroundbasedonthenatureofjudgmentitselfwhyXorYmustbeinterpretedinoneofthetwowaystotheexclusionoftheother.

As thenwe are dealingwith a question of the interpretation of a certainform of expression, we must look for our solution partly in the usages ofordinary language.Weask, therefore,whether inordinaryspeechwe intendthat the alternants in an alternative proposition should necessarily beunderstoodasexcludingoneanother?298Averyfewinstanceswillenableustodecidein thenegative.Take,forexample, theproposition,“Hehaseitherused bad text-books or he has been badly taught.”No onewould naturallyunderstand this to exclude the possibility of a combination of bad teachingandtheuseofbadtext-books.Orsupposeit laiddownasa279conditionofeligibilityforsomeappointmentthateverycandidatemustbeamembereitherof the University of Oxford, or of the University of Cambridge, or of theUniversityofLondon.Wouldanyoneregardthisasimplyingtheineligibilityof persons who happened to be members of more than one of theseUniversities?Jevons(PureLogic,p.68)instancesthefollowingproposition:“Apeeriseitheraduke,oramarquis,oranearl,oraviscount,orabaron.”Wedonotconsiderthisstatementincorrectbecausemanypeersasamatteroffact possess two or more titles. Take, again, the proposition, “Either thewitnessisperjuredortheprisonerisguilty.”Theimportofthisproposition,asitwouldnaturallybeinterpreted,isthattheevidencegivenbythewitnessissufficient,supposingitistrue,toestablishtheguiltoftheprisoner;butclearlythere is no implication that the falsity of this particular piece of evidence


wouldsufficetoestablishtheprisoner’sinnocence.298 Therearenodoubtmanycasesinwhichasamatteroffactweunderstand

alternantstobemutuallyexclusive.Butthisisnotconclusiveasshewingthatevenin these cases the mutual exclusiveness is intended to be expressed by thealternativeproposition.Foritwillgenerallyspeakingbefoundthat insuchcasesthefactthatthealternantsexcludeoneanotherisamatterofcommonknowledgequite independently of the alternative proposition; as, for example, in theproposition,Hewasfirstorsecondintherace.ThispointisfurthertoucheduponinPartIII,Chapter6.

Butitmaybeurgedthat thisdoesnotdefinitelysettle thequestionof thebest way of interpreting alternative propositions. Granted that in commonspeech the alternants may or may not be mutually exclusive, it maynevertheless be argued that in the use of language for logical purposesweshouldbemoreprecise,andthatanalternativestatementshouldaccordinglynot be admitted as a recognised logical proposition except on the conditionthatthealternantsmutuallyexcludeoneanother.

Wemay admit that the argument from the ordinary use of speech is notfinal. But at any rate the burden of proof lies with those who advocate adivergencefromtheusageofeverydaylanguage;foritwillnotbedeniedthat,otherthingsbeingequal,thelesslogicalformsdivergefromthoseofordinaryspeech thebetter.Moreover,condensedformsofexpressiondonotconduceto clearness, or even ultimately to conciseness.299 280 For where ourinformation ismeagre, a condensed form is likely to expressmore thanweintend, and in order to keep within the mark we must indicate additionalalternatives.On thisground,quiteapart fromconsiderationsof theordinaryuse of language, I should support the non-exclusive interpretation ofalternatives. The adoption of the exclusive interpretation would certainlyrenderthemanipulationofcomplexpropositionsmuchmorecomplicated.

299 Obviouslyadisjunctivepropositionisamorecondensedformofexpressionon the exclusive than on the non-exclusive interpretation. Compare Mansel’sAldrich,p.242,andProlegomenaLogica,p.288.“Letusgrantforamomenttheoppositeview,andallow that thepropositionAllC iseitherAorB implies as acondition of its truthNoC can be both. Thus viewed, it is in reality a complexproposition,containingtwodistinctassertions,eachofwhichmaybethegroundoftwo distinct processes of reasoning, governed by two opposite laws. Surely it isessentialtoallclearthinkingthatthetwoshouldbeseparatedfromeachother,andnotconfoundedunderoneformbyassumingtheLawofExcludedMiddle tobe,


whatitisnot,acomplexofthoseofIdentityandContradiction”(Aldrich,p.242).It may be added that one paradoxical result of the exclusive interpretation ofalternativesisthatnoteitherPorQisnotequivalenttoneitherPnorQ.

AfurtherparadoxicalresultispointedoutbyMrG.R.T.RossinanarticleontheDisjunctive Judgment inMind (1903, p. 492), namely, that on the exclusiveinterpretationthedisjunctivesAiseitherBorCandAiseithernotBornotCareidenticalintheirimport;forineachcasetherealalternantsareBbutnotCandCbutnotB.Thus,totakeanillustrationborrowedfromMrRoss,thetwofollowingpropositions are (on the interpretation in question) identical in their import,—“Anyonewhoaffirms thathehas seenhisownghost is eithernot saneornottellingwhathebelievestobethetruth,”“Anyonewhoaffirmsthathehasseenhisownghostiseithersaneortruthful.”

MrBosanquetandotherwriterswhoadvocatetheexclusiveinterpretationof disjunctives appear to have chiefly in view the expression in disjunctiveformofalogicaldivisionorscientificclassification.Ishouldofcourseagreethatsuchadivisionorclassificationis imperfect if themembersofwhichitconsists are notmutually exclusive aswell as collectively exhaustive. Thiscondition must also be satisfied when we make use of the disjunctivejudgment in connexionwith thedoctrineofprobability.300 Itwill, however,hardly be proposed to confine the disjunctive judgment to these uses. Wefrequentlyhaveoccasiontostatealternativesindependentlyofanyscientificclassification or any calculation of probability; andwemust not regard thebareformofthedisjunctivejudgmentasexpressinganythingthatwearenotpreparedtorecogniseasuniversallyinvolvedinitsuse.

300 Inthisconnexionthefurtherconditionofthe“equality”inacertainsenseofthealternantshasinadditiontobesatisfied.

It is of course always possible to express an alternative 281 statement insuchaway that thealternantsare formally incompatibleorexclusive.Thus,notwishingtoexcludethecaseofAbeingbothBandCwemaywriteAisBorbC ;301or,wishing toexclude thatcase,A isBcorbC.But inneitheroftheseinstancescanwesaythattheincompatibilityofthealternantsisreallygivenbythealternativeproposition.ItisamerelyformalpropositionthatNoAisbothBandbCorthatNoAisbothBcandbC.ThepropositionEveryAisBcorbCdoes,however,tellusthatnoAisbothBandC ;andwhenfromourknowledge of the subject-matter it is obvious that we are dealing withalternants that aremutually exclusive (and no doubt this is a very frequent


case),we have in the above form ameans of correctly and unambiguouslyexpressingthefact.Whereitisinconvenienttousethisform,itisopentoustomakeaseparatestatementtotheeffectthatNoAisbothBandC.AllthatisherecontendedforisthatthebaresymbolicformAiseitherBorCshouldnotbeinterpretedasbeingequivalenttoAiseitherBcorbC.

301 Whereb=not-B,andc=not-C.What iscontained in thisparagraph is tosomeextentarepetitionofwhatisgivenonpage278.

(2)Wemay pass on to consider the secondmain question that arises inconnexionwith the import of disjunctive (alternative) propositions, namely,whether such propositions are to be interpreted as modal or as merelyassertoric.

In chapter 9 it was urged that the modal interpretation of the typicalhypotheticalpropositionIfAthenCmustberegardedasthemorenaturalone,on theground thatwe shouldnotordinarily think itnecessary toaffirm thetruthofA inorder tocontradict theproposition,aswouldbenecessary if itwereinterpretedassertorically.302SimilarlytheenquiryastohowweshouldnaturallycontradictthetypicalalternativepropositionsEverySiseitherPorQ,EitherXorY is true,mayhelpus indecidingupon the interpretationofthesepropositions.

302 Seepage263.

On the assertoric interpretation, the contradictories of the propositions inquestionareSomeSisneitherPnorQ,NeitherXnorYistrue ;onthemodalinterpretation,theyareAnSneednotbeeitherPorQ,PossiblyneitherXnorYistrue.282Therecanbenodoubtthat thislastpairofpropositionswouldnotasaruleberegardedasadequatetocontradictthepairofalternatives;andon thisgroundwemayregard theassertoric interpretationofalternativesasmost in accordance with ordinary usage. There is also some advantage indifferentiating between hypotheticals and alternatives by interpreting theformer modally and the latter assertorically, except in so far as a clearindication is given to the contrary. It is not of course meant that modalalternativesareneverasamatterof fact tobemetwithor that theycannotreceive formal recognition; they can always be expressed in the distinctiveformsEverySmustbeeitherPorQ,EitherXorYisnecessarilytrue.


192. Scheme of Assertoric and Modal Propositions.—By differentiatingbetweenformsofpropositionsinthemannerindicatedinprecedingsectionswehaveaschemebywhichdistinctiveexpressioncanbegiventoassertoricandmodalpropositionsrespectively,whethertheyaresimpleorcompound.

Thus the categorical form of proposition might be restricted to theexpression of simple assertoric judgments; the conditional form to that ofsimple modal judgments; the disjunctive (alternative)303 form to that ofcompound assertoric judgments; and the hypothetical form to that ofcompoundmodaljudgments.

303 Weareofcoursereferringheretodisjunctive(alternative)propositionsofthesecond type only, alternative propositions of the first type being treated ascategoricalswithalternativepredicates.Seesection190.

I have not in the present treatise attempted to adopt this scheme to theexclusion of other interpretations of the different propositional forms; but Ihavehaditinviewthroughout,andIputitforwardasaschemetheadoptionof which might afford an escape from some ambiguities andmisunderstandings.

193.TheRelationofDisjunctive(Alternative)PropositionstoConditionalsandHypotheticals.—Itmaybeconvenientifwebrieflyconsiderthisquestionindependently of the distinctions indicated in the preceding section, theassumption being made that these different types of propositions areinterpreted either all assertorically or all modally. On this assumption,alternative propositions are reducible to the conditional or the truehypothetical formaccording to the type towhich theybelong.Thus,283 theproposition, “Every blood vessel is either a vein or an artery,” yields theconditional, “If anybloodvessel isnot avein then it is anartery”; the truecompound alternative proposition, “Either there is a future life or manycruelties gounpunished,”yields the truehypothetical, “If there is no futurelifethenmanycrueltiesgounpunished.”

It may be asked whether an alternative proposition does not require aconjunctionoftwoconditionalsorhypotheticals inorder fully toexpress itsimport.Thisisnotthecase,however,ontheviewthatthealternantsarenottobe interpreted as necessarily exclusive. It is true that even on this view an


alternativeproposition, suchasEitherXorY, is primarily reducible to twohypotheticals, namely, If not X then Y and If not Y then X. But these arecontrapositivestheoneoftheother,andthereforemutuallyinferable.Hencethefullmeaningofthealternativepropositionisexpressedbymeansofeitherofthem.

On the exclusive interpretation, the alternative propositionEither X or Yyieldsprimarilyfourhypotheticals,namely,IfXthennotYandIfYthennotXinadditiontothetwogivenabove.Buttheseagainarecontrapositivestheoneoftheother.Hencethefullimportofthealternativepropositionwillnowbeexpressedbyaconjunctionofthetwohypotheticals,IfXthennotYandIfnotXthenY.

ThisisdeniedbyMrBosanquet,whoholdsthatthedisjunctivepropositionyieldsapositiveassertionnotcontainedineitherofthehypotheticals.“‘Thissignallightshewseitherredorgreen.’Herewehavethecategoricalelement,‘This signal light shews some colour,’ and on the top of this the twohypotheticaljudgments,‘Ifitshewsreditdoesnotshewgreen,’‘Ifitdoesnotshew red it does shew green.’ You cannot make it up out of the twohypothetical judgments alone; they do not give you the assertion that ‘itshews some colour.’”304 But surely the second of the two hypotheticalscontains this implication quite as clearly and definitely as the disjunctivedoes.305

304 EssentialsofLogic,p.124.305 MrBosanquet’sopinionthat“thedisjunctionseemstocompletethesystem

ofjudgments,”andthatinsomewayitrisessuperiortootherformsofjudgment,isapparentlybasedontheviewthatitisbytheaidofthedisjunctivejudgmentthatwe set forth the exposition of a system with its various subdivisions. Apart,however, from the fact that a disjunctive judgment does not necessarily containsuchanexposition,MrBosanquet’sdoctrineappears to regardaclassificationofsomekindasrepresentingtheidealofknowledge;andthiscanhardlybeallowed.Wecannot,forexample,regardtheclassificationsofsuchascienceasbotanyasofequal importance with the expressions of laws of nature, such as the law ofuniversalgravitation.Andtheultimatelawsonwhichallthesciencesarebasedarenotexpressedintheformofdisjunctivepropositions.

284 Returning to the distinctions indicated in the preceding section, it ishardlynecessarytoaddthatifthehypotheticalIfnotXthenYisinterpreted


modally,whilethealternativeEitherXorYisinterpretedassertorically,thenthealternativecanbeinferredfromthehypothetical,butnotviceversâ.

EXERCISES.

194. Shew how an alternative proposition in which the alternants are not known to be mutuallyexclusive(e,g.,EitherXorYorZistrue)maybereducedtoaforminwhichtheynecessarilyareso.Writethenewpropositioninassimpleaformaspossible.[K.]

195.Shewwhythefollowingpropositionsarenotcontradictories:WhereverAispresent,BispresentandeitherCorBisalsopresent ;InsomecaseswhereAispresent,eitherBorCorBisabsent.Howmusteachofthesepropositionsinturnbeamendedinorderthatitmaybecomethetruecontradictoryoftheother?[K.]

196.NoPisbothQandR.Reducethisproposition(a)totheformofaconditionalproposition,(b)tothe form of an alternative proposition. Give the contradictory of the original proposition, of itsconditionalequivalent,andofitsalternativeequivalent;andtestyourresultsbyenquiringwhetherthethreecontradictoriesthusobtainedareequivalenttooneanother.[K.]


PARTIII.

SYLLOGISMS.


CHAPTERI.

THERULESOFTHESYLLOGISM.

197. The Terms of the Syllogism.—A reasoning which consists of threepropositionsofthetraditionalcategoricalform,andwhichcontainsthreeandonlythreeterms,iscalledacategoricalsyllogism.

Ofthethreetermscontainedinacategoricalsyllogism,twoappearintheconclusionandalsoinoneorotherofthepremisses,whilethethirdappearsinthepremissesonly.Thatwhichappearsasthepredicateoftheconclusion,andinoneof thepremisses, iscalledthemajorterm ; thatwhichappearsasthesubjectoftheconclusion,andinoneofthepremisses,iscalledtheminorterm ;306 and that which appears in both the premisses, but not in theconclusion(beingthattermbytheirrelationstowhichthemutualrelationofthetwoothertermsisdetermined),iscalledthemiddleterm.


306 Themajor andminor terms are also sometimes called theextremes of thesyllogism.

Thus,inthesyllogism,—

AllMisP,AllSisM,

therefore, AllSisP ;

Sistheminorterm,Mthemiddleterm,andPthemajorterm.

286Theserespectivedesignationsofthetermsofasyllogismresultedfromsuch a syllogism as that just given being regarded as typical. With theexceptionofthesomewhatrarecaseinwhichthetermsofapropositionarecoextensive, the above syllogism may be represented by the followingdiagram.Here

clearly themajor term is the largest in extent, and theminor the smallest,whilethemiddleoccupiesanintermediateposition.

Butwe have no guarantee that the same relation between the terms of asyllogismwillhold,whenoneofthepremissesisnegativeorparticular.Thus,thesyllogism—NoMisP,AllSisM,therefore,NoSisP—yieldsasonecase


where the major term may be the smallest in extent, and the middle thelargest.Again,thesyllogism—NoMisP,SomeSisM,therefore,SomeS isnotP—yieldsasonecase

wherethemajortermmaybethesmallestinextentandtheminorthelargest.

Whilst,however,themiddletermisnotalwaysamiddleterminextent,itisalwaysamiddleterminthesensethatbyitsmeansthetwoothertermsareconnected,andtheirmutualrelationdetermined.

287198.The Propositions of the Syllogism.—Every categorical syllogismconsistsofthreepropositions.Oftheseoneistheconclusion.Thepremissesarecalledthemajorpremissandtheminorpremissaccordingastheycontainthemajortermortheminortermrespectively.

Thus, AllMisP (majorpremiss),AllSisM (minorpremiss),

therefore, AllSisP (conclusion).

It isusual (as in theabovesyllogism) tostate themajorpremissfirstandthe conclusion last. This is, however, nothingmore than a convention.Theorderofthepremissesinnowayaffectsthevalidityofasyllogism,andhasindeed no logical significance, though in certain cases it may be of somerhetoricalimportance.Jevons(PrinciplesofScience,6,§14)arguesthatthecogencyofasyllogismismoreclearlyrecognisablewhentheminorpremississtatedfirst.Butitisdoubtfulwhetheranygeneralruleofthiskindcanbelaiddown. In favour of the traditional order, it is to be said that inwhat isusuallyregardedasthetypicalsyllogism(AllMisP,AllSisM,therefore,AllSisP)thereisaphilosophicalgroundforstatingthemajorpremissfirst,sincethatpremissgivesthegeneralrule,ofwhichtheminorpremissenablesusto


makeaparticularapplication.

199.TheRulesoftheSyllogism.—Therulesofthecategoricalsyllogismasusuallystatedareasfollows:— (1)Everysyllogismcontainsthreeandonlythreeterms. (2)Everysyllogismconsistsofthreeandonlythreepropositions.

Thesetwoso-calledrulesarenotproperlyspeakingrulesforthevalidityofanargument.Theysimplyservetodefinethesyllogismasaparticularformofargument.Areasoningwhichdoesnotfulfiltheseconditionsmaybeformallyvalid,butwedonotcall itasyllogism.307Thefourrulesthatfollow288arereallyrulesinthesensethatif,whenwehavegotthereasoningintotheformofasyllogism,theyarenotfulfilled,thenthereasoningisinvalid.308

307 Forexample,BisgreaterthanC,AisgreaterthanB,therefore,AisgreaterthanC.

Here is a valid reasoningwhich consists of threepropositions.But it containsmorethanthreeterms;forthepredicateofthesecondpremissis“greaterthanB,”while the subject of the first premiss is “B.” It is, therefore, as it stands, not asyllogism.Whether reasoningsof thiskind admitofbeing reduced to syllogisticformisaproblemwhichwillbediscussedsubsequently.

308 Apparentexceptionstotheseruleswillbeshewninsections205and206toresult from the attempt to apply them to reasonings which have not first beenreducedtosyllogisticform.

(3)Nooneofthethreetermsofasyllogismmaybeusedambiguously;andthemiddletermmustbedistributedonceatleastinthepremisses.

This rule is frequently given in the form: “The middle term must bedistributedonceatleast,andmustnotbeambiguous.”Butit isobviousthatwehavetoguardagainstambiguousmajorandambiguousminoraswellasagainstambiguousmiddle.Thefallacyresultingfromtheambiguityofoneofthetermsofasyllogismisacaseofquaternioterminorum,thatis,afallacyoffourterms.

ThenecessityofdistributingthemiddletermmaybeillustratedbytheaidoftheEuleriandiagrams.Given,forinstance.AllPisMandAllSisM,wemayhaveanyoneofthefivefollowingcases:—


HereallthefiverelationsthatareàprioripossiblebetweenSandParestillpossible.Wehave,therefore,noconclusion.

If inasyllogismthemiddle termisdistributed inneitherpremiss,wearesaidtohaveafallacyofundistributedmiddle.

289 (4) No term may be distributed in the conclusion which was notdistributedinoneofthepremisses.

Thebreachofthisruleiscalledillicitprocessofthemajor,orillicitprocessoftheminor,asthecasemaybe;or,morebriefly,illicitmajororillicitminor.

(5)Fromtwonegativepremissesnothingcanbeinferred.

Thisrulemay,likerule3,beverywellillustratedbymeansoftheEuleriandiagrams.

(6) If one premiss is negative, the conclusion must be negative; and toproveanegativeconclusion,oneofthepremissesmustbenegative.309

309 This rule and the second corollary given in the following section aresometimescombinedintotheonerule,Conclusiosequiturpartemdeteriorem ;i.e.,theconclusionfollowstheworseorweakerpremissbothinqualityandinquantity,a negative being considered weaker than an affirmative and a particular than auniversal.

200.CorollariesfromtheRulesoftheSyllogism.—Fromtherulesgivenin


theprecedingsection,threecorollariesmaybededuced:—310

310 The formulation of these corollaries may in some cases help towards themoreimmediatedetectionofunsoundsyllogisms.

(i)Fromtwoparticularpremissesnothingcanbeinferred. Twoparticularpremissesmustbeeither (α)bothnegative,or (β)bothaffirmative,or (γ)onenegativeandoneaffirmative. Butincase(α),noconclusionfollowsbyrule5. Incase(β), sinceno termcanbedistributed in twoparticular affirmativepropositions,themiddletermcannotbedistributed,andthereforebyrule3noconclusionfollows. Incase(γ),ifanyvalidconclusionispossible,itmustbenegative(rule6).Themajorterm,therefore,willbedistributedintheconclusion;andhencewemusthavetwotermsdistributedinthepremisses,namely,themiddleandthemajor (rules 3, 4). But a particular negative proposition and a particularaffirmativepropositionbetweenthemdistributeonlyoneterm.Therefore,noconclusioncanbeobtained.

(ii)Ifonepremissisparticular,theconclusionmustbeparticular. 290Wemusthaveeither (α)twonegativepremisses,butthiscaseisrejectedbyrule5;or (β)twoaffirmativepremisses;or (γ)oneaffirmativeandonenegative. Incase(β)thepremisses,beingbothaffirmativeandoneofthemparticular,candistributebutone termbetween them.Thismustbe themiddle termbyrule3.Theminor term is, therefore,undistributed in thepremisses, and theconclusionmustbeparticularbyrule4. In case (γ) the premisseswill between them distribute two and only twoterms.Thesemustbethemiddlebyrule3,andthemajorbyrule4(sincewehave a negative premiss, necessitating by rule 6 a negative conclusion, andtherefore the distribution of the major term in the conclusion). Again,therefore,theminorcannotbedistributedinthepremisses,andtheconclusionmustbeparticularbyrule4.


De Morgan (Formal Logic, p. 14) gives the following proof of thiscorollary:—“IftwopropositionsPandQtogetherproveathirdR, it isplainthatPandthedenialofRprovethedenialofQ.ForPandQcannotbetruetogetherwithoutR.Now,ifpossible, letP (aparticular)andQ (auniversal)proveR (a universal). Then P (particular) and the denial ofR (particular)provethedenialofQ.Buttwoparticularscanprovenothing.”311

311 Furtherattentionwillbecalledinalaterchaptertothegeneralprincipleuponwhichthisproofisbased.Seesection264.

(iii)Fromaparticularmajorandanegativeminornothingcanbeinferred. Since theminorpremiss isnegative, themajorpremissmustbyrule5beaffirmative.But it is also particular, and it therefore follows that themajortermcannotbedistributedinit.Hence,byrule4,itmustbeundistributedintheconclusion, i.e., theconclusionmustbeaffirmative.But also, by rule6,since we have a negative premiss, it must be negative. This contradictionestablishesthecorollarythatfromthegivenpremissesnoconclusioncanbedrawn.

The following mnemonic lines, attributed to Petrus Hispanus, 291

afterwardsPope JohnXXI., sumup the rules of the syllogismand the firsttwocorollaries:

Distribuasmedium:necquartusterminusadsit:Utraquenecpraemissanegans,necparticularis:Secteturpartemconclusiodeteriorem;Etnondistribuat,nisicumpraemissa,negetve.

201. Restatement of the Rules of the Syllogism.—It has been alreadypointed out that the first two of the rules given in section 199 are to beregardedasadescriptionofthesyllogismratherthanasrulesforitsvalidity.Again,thepartofrule3relatingtoambiguitymayberegardedascontainedintheprovisothatthereshallbeonlythreeterms;for, ifoneofthetermsisambiguous,therearereallyfourterms,andhencenosyllogismaccordingtoourdefinitionofsyllogism.Therulesmay,therefore,bereducedtofour;andtheymayberestatedasfollows:—

A.Tworulesofdistribution: (1)Themiddletermmustbedistributedonceatleastinthepremisses;


(2) No term may be distributed in the conclusion which was notdistributedinoneofthepremisses; B.Tworulesofquality: (3)Fromtwonegativepremissesnoconclusionfollows; (4) Ifonepremiss isnegative, theconclusionmustbenegative;and toproveanegativeconclusion,oneofthepremissesmustbenegative.312

312 Therulesofqualitymightalsobestatedasfollows;Toproveanaffirmativeconclusion, both premissesmust be affirmative;Toprove a negative conclusion,onepremissmustbeaffirmativeandtheothernegative.

202.Dependence of the Rules of the Syllogism upon one another.—Thefourrulesjustgivenarenotultimatelyindependentofoneanother.Itmaybeshewnthatabreachof thesecond,orof the third,orof thefirstpartof thefourthinvolvesindirectlyabreachofthefirst;or,again,thatabreachofthefirst, or of the third, or of the first part of the fourth involves indirectly abreachofthesecond.

292 (i)The rule that two negative premisses yield no conclusion may bededucedfromtherulethatthemiddletermmustbedistributedonceatleastinthepremisses.

ThisisshewnbyDeMorgan(FormalLogic,p.13).HetakestwouniversalnegativepremissesE,E.Inwhateverfiguretheymaybe,theycanbereducedbyconversionto

NoPisM,NoSisM.

Thenbyobversiontheybecome(withoutlosinganyoftheirforce),—

AllPisnot-M,AllSisnot-M ;

andwe have undistributedmiddle.Hence rule 3 is exhibited as a corollaryfromrule1.ForifanyconnexionbetweenSandPcanbeinferredfromthefirstpairofpremisses,itmustalsobeinferablefromthesecondpair.

The case inwhich one of the premisses is particular is dealtwith byDeMorgan as follows;—“Again, No Y is X, Some Ys are not Zs, may beconvertedinto


EveryXis(athingwhichisnotY),Some(thingswhicharenotZs)areYs,

inwhichthereisnomiddleterm.”

This is not satisfactory, since wemay often exhibit a valid syllogism insucha form that thereappear tobe four terms;e.g.,AllMisP,All S isM,maybereducedtoAllMisP,NoSisnot-M,andthereisnownomiddleterm.

The case in questionmay, however, be disposed of by saying that ifwecannot infer anything from two negative premisses both of which areuniversal,àfortioriwecannotfromtwonegativepremissesoneofwhichisparticular.313

313 Thisargumentholdsgoodinthespecialcaseunderconsiderationevenifweinterpretparticulars,butnotuniversals,asimplyingtheexistenceoftheirsubjects.Forthevalidityoftheaboveproofthattwouniversalnegativesyieldnoconclusionremains unaffected even if we allow to universals the maximum of existentialimport.

(ii)Therulesthatfromtwonegativepremissesnothingcanbeinferredandthat ifonepremiss isnegativetheconclusionmustbenegativearemutuallydeduciblefromoneanother.

The following proof that the second of these rules is deducible from thefirst is suggested byDeMorgan’s deduction of 293 the second corollary asgiveninsection200.IftwopropositionsPandQtogetherproveathirdR,itisplainthatPandthedenialofRprovethedenialofQ.ForPandQcannotbe true together withoutR. Now, if possible, letP (a negative) andQ (anaffirmative)proveR(anaffirmative).ThenP(anegative)andthedenialofR(a negative) prove the denial ofQ. But by hypothesis two negatives provenothing.

Itmaybe shewnsimilarly that ifwe startbyassuming the secondof therulesthenthefirstisdeduciblefromit.

(iii)Any syllogism involving directly an illicit process ofmajor orminorinvolvesindirectlyafallacyofundistributedmiddle,andviceversâ.314

314 ForthistheoremanditsproofIamindebtedtoMrJohnson.

LetPandQbethepremissesandRtheconclusionofasyllogisminvolving


illicitmajororminor,atermXwhichisundistributedinPbeingdistributedinR.ThenthecontradictoryofRcombinedwithPmustprovethecontradictoryof Q. But any term distributed in a proposition is undistributed in itscontradictory.X is therefore undistributed in the contradictory ofR, and byhypothesis it is undistributed in P. But X is the middle term of the newsyllogism,whichisthereforeguiltyofthefallacyofundistributedmiddle.Itisthusshewnthatanysyllogisminvolvingdirectlyafallacyofillicitmajororminorinvolvesindirectlyafallacyofundistributedmiddle.

Adoptingasimilarlineofargument,wemightalsoproceedintheoppositedirection,andexhibittherulerelatingtothedistributionofthemiddletermasacorollary from the rule relating to thedistributionof themajorandminorterms.

203.Statementof the independentRulesof theSyllogism.—The theoremsestablishedintheprecedingsectionshewthatthefirstpartofrule4(asgivenin section 201) is a corollary from rule 3, and that rule 3 is in its turn acorollaryfromrule1;alsothatrules1and2mutuallyinvolveoneanother,sothateitheroneofthemmayberegardedasacorollaryfromtheother.Weare,therefore,leftwitheitherrule1orrule2andalsowiththesecondpartofrule4; and the independent rules of the syllogismmay accordingly be stated asfollows:294 (α)Rule of Distribution:—The middle term must be distributed once atleastinthepremisses[or,asalternativewiththis,Notermmaybedistributedintheconclusionwhichwasnotdistributedinoneofthepremisses]; (β)RuleofQuality:—Toproveanegativeconclusiononeofthepremissesmustbenegative.315

315 Onexaminationitwillbefoundthattheonlysyllogismrejectedbythisruleandnotalsorejecteddirectlyorindirectlybytheprecedingruleis thefollowing:—AllP isM,AllM is S, therefore,Some S is not P. In the technical languageexplainedinthefollowingchapter,thisisAAOinfigure4.Sofar,therefore,asthefirst three figuresareconcerned,weare leftwitha single rule,namely,a ruleofdistribution,whichmaybestatedineitherofthealternativeformsgivenabove.

It should be clearly understood that it is not meant that every invalidsyllogismwilloffenddirectlyagainstoneofthesetworules.Asadirecttestfor thedetectionof invalid syllogismswemust still fallbackupon the four


rulesgiveninsection201.316All thatwehavesucceededinshewingis thatultimatelythesefourrulesarenotindependentofoneanother.

316 If,forexample,forourruleofdistributionweselecttherulerelatingtothedistributionofthemiddleterm,thentheinvalidsyllogism,

AllMisP,NoSisM,

therefore, NoSisP,

doesnotdirectlyinvolveabreachofeitherofourtwoindependentrules.Butifthissyllogismisvalid,thenmustalsothefollowingsyllogismbevalid:

AllMisP(originalmajor),SomeSisP(contradictoryoforiginalconclusion),

therefore SomeSisM(contradictoryoforiginalminor);

andherewehaveundistributedmiddle.Hencetherulerelatingtothedistributionofthemiddletermestablishesindirectlytheinvalidityofthesyllogisminquestion.Theprinciple involved is thesameas thatonwhichweshall find theprocessofindirectreductiontobebased.

Take, again, the syllogism:PaM,SeM,∴ SaP. This does not directly offendagainsttherulesgivenabove;butthereaderwillfindthatitsvalidityinvolvesthevalidityofanothersyllogisminwhichadirecttransgressionoftheserulesoccurs.

204.ProofoftheRuleofQuality.—Forthefollowingveryinterestingandingeniousproofof theRuleofQuality (asstated in theprecedingsection) IamindebtedtoMrR.A.P.Rogers,ofTrinityCollege,Dublin.Inthisproofthesymbolfn( )isusedtodenotetheformofaproposition,thetermswhichthe 295 proposition contains in any given case being inserted within thebrackets.Thus,iffx(P,M)symbolisesAllMisP,thenfx(B,A)willsymboliseAllAisB:or,again,iffy(S,M)symbolisesSomeSisnotM,thenfy(B,A)willsymboliseSomeB is notA. Itwill be observed that the order inwhich thetermsaregivendoesnotnecessarilycorrespondwiththeorderofsubjectandpredicate.

Let f1( ), f2( ), f3( )bepropositionsbelonging to the traditional schedule.Then“f1(P,M), f2(S,M),∴ f3(S,P)”will be the expression of a syllogism;and, since the syllogism is a process of formal reasoning, if the abovesyllogismisvalidinanycase,itwillholdgoodifothertermsaresubstitutedforS,M,P(oranyofthem).Thus,substitutingSforM,andSforP,if“f1(P,M),f2(S,M),∴f3(S,P)”isavalidsyllogism,then“f1(S,S),f2(S,S),∴f3(S,S)”


willbeavalidsyllogism.

It follows, by contraposition, that if “f1(S,S), f2(S, S),∴ f3(S, S)” is aninvalid syllogism, then “f1(P,M), f2(S,M),∴ f3(S, P)” will be an invalidsyllogism.

Ifpossible, let f1( )and f2( ) be affirmative,while f3( ) isnegative.Thenf1(S, S) and f2(S, S) will be formally true propositions, while f3(S, S) isformally false.Hence f3(S,S) cannot be a valid inference from f1(S,S) andf2(S, S); in other words, “f1(S, S), f2(S, S),∴ f3(S, S)” must be an invalidsyllogism. Consequently, “f1(P.M), f2(S,M),∴ f3(S,P)” cannot be a validsyllogism;thatis,wecannothaveavalidsyllogisminwhichbothpremissesareaffirmativeandtheconclusionnegative.

205. Two negative premisses may yield a valid conclusion; but notsyllogistically.—Jevons remarks: “The old rules of logic informed us thatfrom twonegativepremissesno conclusion couldbedrawn,but it is a factthat the rule in this bare formdoes not hold universally true; and I amnotaware that any precise explanation has been given of the conditions underwhichitisorisnotimperative.Considerthefollowingexample,—Whateverisnotmetallic isnotcapableofpowerfulmagnetic influence,Carbon isnotmetallic, therefore,Carbon is not capable of powerful magnetic influence.Herewehavetwodistinctlynegativepremisses,andyettheyyieldaperfectly296 valid negative conclusion.The syllogistic rule is actually falsified in itsbareandgeneralstatement”(PrinciplesofScience,4,§10).317

317 Lotze (Logic, § 89;Outlines of Logic, §§ 40-42) holds that two negativepremisses invalidate a syllogism in figure 1 or figure 2, but not necessarily infigure 3. The example upon which he relies is this,—No M is P,No M is S,therefore,Somenot-S is notP. The argument in the textmay be applied to thisexampleaswellastotheonegivenbyJevons.

This apparent exception is, however, no real exception. The reasoning(whichmaybeexpressedsymbolicallyintheform,Nonot-MisP,NoSisM,therefore,No S is P) is certainly valid; but if we regard the premisses asnegativeithasfourtermsS,P,M,andnot-M,andisthereforenosyllogism.Reducingittosyllogisticform,theminorbecomesbyobversionAllSisnot-M, an affirmative proposition.318 It is not the case, therefore, thatwe have


succeededinfindingavalidsyllogismwithtwonegativepremisses.Inotherwords, while we must not say that from two negative premisses nothingfollows, it remains true that if a syllogism regularly expressed has twonegative premisses it is invalid.319 It must not be considered that this is ameretechnicality,andthatJevons’sexampleshewsthattheruleisatanyrateofnopracticalvalue.Itisnotpossibletoformulatespecificrulesatallexceptwith reference to some defined form of reasoning; and no given rule isvitiated either 297 theoreticallyor for practical purposesbecause it doesnotapplyoutsidetheformtowhichaloneitprofessestoapply.320

318 Itmaybeadded that it is in this form that thecogencyof theargument ismost easily to be recognised.Of course every affirmation involves a denial andviceversâ ;butitmayfairlybesaidthatinJevons’sexampletheprimaryforceoftheminor premiss, considered in connexionwith themajor premiss, is to affirmthatcarbonbelongstotheclassofnon-metallicsubstances,ratherthantodenythatitbelongstotheclassofmetallicsubstances.

319 Byasyllogismregularlyexpressedwemeanareasoningconsistingofthreepropositions,whichnotonlycontainbetweenthemthreeandonlythreeterms,butwhich are also expressed in the traditional categorical forms.Attentionmust becalled to thisbecause, ifwe introduceadditionalpropositional formsof thekindindicated on page 146, we may have a valid reasoning with two negativepremisses, which satisfies the condition of containing only three terms; forexample,

NoMisP,SomeMisnotS,

therefore, ThereissomethingbesidesSandP.

Itwill be found that this reasoning is easily reducible to a valid syllogism inFerison.

320 A case similar to that adduced by Jevons is dealt with in thePort RoyalLogic (Professor Baynes’s translation, p. 211) as follows:—“There are manyreasonings, of which all the propositions appear negative, and which are,nevertheless, very good, because there is in themonewhich is negative only inappearance,andinrealityaffirmative,aswehavealreadyshewn,andaswemaystill further see by this example:That which has no parts cannot perish by thedissolutionofitsparts;Thesoulhasnoparts;therefore,Thesoulcannotperishbythedissolutionofitsparts.Thereareseveralwhoadvancesuchsyllogismstoshewthatwehavenorighttomaintainunconditionallythisaxiomoflogic,Nothingcanbe inferred from pure negatives ; but they have not observed that, in sense, theminorofthisandsuchothersyllogismsisaffirmative,sincethemiddle,whichisthesubjectofthemajor,isinittheattribute.Nowthesubjectofthemajorisnotthatwhichhasparts,butthatwhichhasnotparts,andthusthesenseoftheminoris, The soul is a thing without parts, which is a proposition affirmative of a


negative attribute.”Ueberwegalso,whohimself gives a clear explanationof thecase,shewsthatitwasnotoverlookedbytheolderlogicians;andhethinksitnotimprobable that thedoctrineofqualitativeaequipollencebetween two judgments(i.e., obversion) resulted from the consideration of this very question (SystemofLogic,§106).Compare,further,Whately’streatmentofthesyllogism,“Nomanishappywhoisnotsecure;notyrantissecure;therefore,notyrantishappy”(Logic,II.4,§7).

The truth is that by the aid of the process of obversion the premisses ofevery valid syllogismmay be expressed as negatives, though the reasoningwillthennolongerbetechnicallyintheformofasyllogism;forexample,thepropositionswhichconstitutethepremissesofasyllogisminBarbara—AllMisP,AllSisM,therefore,AllSisP—maybewritteninanegativeform,thus,NoMisnot-P,NoSisnot-M,andtheconclusionAllSisPstillfollows.

206.OtherapparentexceptionstotheRulesoftheSyllogism.—Itiscuriousthatthelogicianswhohavelaidsomuchstressonthecaseconsideredintheprecedingsectiondonotappear tohaveobserved that,assoonasweadmitmore than three terms, other apparent breaches of the syllogistic rulesmayoccurinwhatareperfectlyvalidreasonings.Thus, thepremissesAllPisMandAllSisM,inwhichMisnotdistributed,yieldtheconclusionSomenot-Sisnot-P;321and298hencewemightarguethatundistributedmiddledoesnotinvalidateanargument.Again,fromthepremissesAllMisP,Allnot-MisS,wemayinferSomeSisnotP,322althoughthereisapparentlyanillicitprocessof the major. It is unnecessary after what has been said in the precedingsection to give examples of valid reasonings in which we have a negativepremisswithanaffirmativeconclusion,or twoaffirmativepremisseswithanegativeconclusion,or aparticularmajorwithanegativeminor.Anyvalidsyllogismwhich isaffirmative throughoutwillyield thefirstand, if ithasaparticularmajor,alsothelastofthesebytheobversionoftheminorpremiss,andthesecondbytheobversionoftheconclusion.Theonlysyllogisticrules,indeed,whichstillholdgoodwhenmorethanthreetermsareadmittedaretheruleprovidingagainstillicitminorandthefirsttwocorollaries.

321 Bythecontrapositionofbothpremissesthisreasoningisreducedtothevalidsyllogistic form,All not-M is not-P,All not-M is not-S, therefore,Somenot-S isnot-P.

322 By the inversionof thefirstpremiss, this reasoning is reduced to thevalidsyllogisticform,Somenot-MisnotP,Allnot-MisS,therefore,SomeS isnotP.


Comparesection104.

Butofcoursenoneof theaboveexamplesreally invalidate thesyllogisticrules;fortheseruleshavebeenformulatedsolelywithreferencetoreasoningsofacertainform,namely,thosewhichcontainthreeandonlythreeterms.Ineverycasethereasoninginevitablyconformsto therulewhichitappears toviolate, as soon as, by the aid of immediate inferences, the superfluousnumberoftermshasbeeneliminated.

207.Syllogismswith twosingularpremisses.—Bain(Logic,Deduction,p.159)arguesthatanapparentsyllogismwithtwosingularpremissescannotberegardedasagenuinesyllogisticordeductiveinference;andheillustrateshisviewbyreferencetothefollowingsyllogism:

SocratesfoughtatDelium,SocrateswasthemasterofPlato,

therefore, ThemasterofPlatofoughtatDelium.

Theargumentisthat“theproposition‘SocrateswasthemasterofPlatoandfought at Delium,’ compounded out of the two premisses, is nothingmorethanagrammaticalabbreviation,”whilstthestephencetotheconclusionisamereomissionofsomething thathadpreviouslybeensaid.“Now,wenever299considerthatwehavemadearealinference,astepinadvance,whenwerepeatlessthanweareentitledtosay,ordropfromacomplexstatementsomeportionnotdesiredatthemoment.SuchanoperationkeepsstrictlywithinthedomainofEquivalenceor Immediate Inference. Innoway, therefore, canasyllogismwith twosingularpremissesbeviewedasagenuinesyllogisticordeductiveinference.”

Thisargumentleadsuptosomeinterestingconsiderations,butitprovestoomuch. In the following syllogisms the premisses may be similarlycompoundedtogether:

Allmenaremortal, ⎱Allmenaremortalandrational ;

Allmenarerational, ⎰

therefore,Somerationalbeingsaremortal.Allmenaremortal, ⎱

Allmenincludingkingsaremortal ;Allkingsaremen, ⎰


therefore,Allkingsaremortal.323323 Compare with the above the following syllogismwhich has two singular

premisses:—TheLordChancellorreceivesahighersalarythanthePrimeMinister;LordHerschellistheLordChancellor;therefore,LordHerschellreceivesahighersalarythanthePrimeMinister.ThesepremisseswouldpresumablybecompoundedbyBainintothesingleproposition,“TheLordChancellor,LordHerschell,receivesahighersalarythanthePrimeMinister.”

Do not Bain’s criticisms apply to these syllogisms as much as to thesyllogismwith twosingularpremisses?Themethodof treatmentadopted isindeed particularly applicable to syllogisms in which the middle term issubjectinbothpremisses.Butwemayalwayscombinethetwopremissesofasyllogisminasinglestatement,anditisalwaystruethattheconclusionofasyllogismcontainsapartof,andonlyapartof,theinformationcontainedinthetwopremissestakentogether;hencewemayalwaysgetBain’sresult.324Inotherwords,intheconclusionofeverysyllogism“werepeatlessthanweareentitledtosay,”or,ifwecaretoputitso,“dropfromacomplexstatementsomeportionnotdesiredatthemoment.”


324 ItmaybepointedoutthatthegeneralmethodadoptedbyBooleinhisLawsofThoughtistosumupallhisgivenpropositionsinasingleproposition,andtheneliminatethetermsthatarenotrequired.ComparealsothemethodsemployedinAppendixCofthepresentwork.

300208.Chargeofincompletenessbroughtagainsttheordinarysyllogisticconclusion.—This charge (a consideration of which will appropriatelysupplement thediscussioncontained in theprecedingsection) isbroughtbyJevons (Principles of Science, 4, § 8) against the ordinary syllogisticconclusion.ThepremissesPotassium floats onwater,Potassium is ametalyield,accordingtohim,theconclusionPotassiummetalispotassiumfloatingonwater.But“Aristotlewouldhaveinferredthatsomemetalsfloatonwater.Hence Aristotle’s conclusion simply leaves out some of the informationaffordedinthepremisses ;itevenleavesusopentointerpretthesomemetalsinawidersensethanwearewarrantedindoing.”

In reply to this itmaybe remarked: first, that theAristotelianconclusiondoes not profess to sum up the whole of the information contained in thepremisses of the syllogism; secondly, that somemust here be interpreted tomean merely “not none,” “one at least.” The conclusion of the abovesyllogismmight perhaps better bewritten “somemetal floats onwater,” or“somemetalormetals&c.”LotzeremarksincriticismofJevons:“Hiswholeprocedure is simply a repetition or at the outside an addition of his twopremisses; thus itmerelyadheres to thegiven facts,andsuchaprocesshasneverbeentakenforaSyllogism,whichalwaysmeansamovementofthoughtthat uses what is given for the purpose of advancing beyond it…… Themeaningof theSyllogism,asAristotle framed it,would in thiscasebe thattheoccurrenceofafloatingmetalPotassiumprovesthatthepropertyofbeingsolightisnotincompatiblewiththecharacterofmetalingeneral”(Logic,II.3, note). This criticism is perhaps pushed a little too far. It is hardly a fairdescription of Jevons’s conclusion to say that it is the mere sum of thepremisses; for it brings out a relation between two terms which was notimmediatelyapparentinthepremissesastheyoriginallystood.Stilltherecanbe no doubt that the elimination of the middle term is the very gist ofsyllogisticreasoningasordinarilyunderstood.

Itmaybe added, as anargumentumadhominem against Jevons, that his


own conclusion also leaves out some of the information afforded in thepremisses.Forwecannotpass301backfromthepropositionPotassiummetalispotassiumfloatingonwatertoeitheroftheoriginalpremisses.

209.TheconnexionbetweentheDictumdeomnietnulloandtheordinaryRulesoftheSyllogism.—ThedictumdeomnietnullowasgivenbyAristotleas the axiomonwhich all syllogistic inference is based. It applies directly,however,tothosesyllogismsonlyinwhichthemajortermispredicateinthemajorpremiss,andtheminortermsubjectintheminorpremiss(i.e.,towhatare called syllogisms in figure 1). The rules of the syllogism, on the otherhand, apply independently of the position of the terms in the premisses.Nevertheless,itisinterestingtotracetheconnexionbetweenthem.Itwillbefoundthatalltherulesareinvolvedinthedictum,butsomeoftheminalessgeneralform,inconsequenceofthedistinctionjustpointedout.

Thedictummaybe stated as follows:—“Whatever ispredicated,whetheraffirmatively or negatively, of a term distributedmay be predicated in likemannerofeverythingcontainedunderit.”

(1)Thedictumprovidesforthreeandonlythreeterms;namely,(i)acertaintermwhichmust be distributed, (ii) something predicated of this term, (iii)somethingcontainedunderit.Thesetermsarerespectivelythemiddle,major,andminor.Wemayconsidertherulerelatingtotheambiguityoftermstobealsocontainedhere,sinceifanytermisambiguouswehavepracticallymorethanthreeterms.

(2)Thedictumprovidesforthreeandonlythreepropositions;namely,(i)aproposition predicating something of a term distributed, (ii) a propositiondeclaring something to be contained under this term, (iii) a propositionmaking the original predication of the contained term. These propositionsconstitute respectively the major premiss, the minor premiss, and theconclusion,ofthesyllogism.

(3) The dictum prescribes not merely that the middle term shall bedistributedonceatleastinthepremisses,butmoredefinitelythatitshallbedistributed in the major premiss,—“Whatever is predicated of a termdistributed.”325


325 Thisisanotherformofwhatwillbefoundtobeaspecialruleoffigure1,namely,thatthemajorpremissmustbeuniversal.Comparesection244.

302 (4) Illicit process of the major is provided against indirectly. Thisfallacycanbecommittedonlywhentheconclusionisnegative;butthewords“in like manner” declare that if there is a negative conclusion, the majorpremissmustalsobenegative;andsinceinanysyllogismtowhichthedictumdirectly applies, the major term is predicate of this premiss, it will bedistributed in its premiss aswell as in the conclusion. Illicit process of theminorisprovidedagainstinasmuchasthedictumwarrantsusinmakingourpredication in the conclusion only of what has been shewn in the minorpremisstobecontainedunderthemiddleterm.

(5)The proposition declaring that something is contained under the termdistributed must necessarily be an affirmative proposition. The dictumprovides,therefore,thatthepremissesshallnotbothbenegative.326

326 Itreallyprovidesthattheminorpremissshallbeaffirmative,whichagainisoneofthespecialrulesoffigure1.

(6)Thewords“inlikemanner”clearlyprovideagainstabreachoftherulethat if one premiss is negative, the conclusion must be negative, and viceversâ.

EXERCISES.327

327 Thefollowingexercisesmaybesolvedwithoutanyknowledgebeyondwhatiscontainedintheprecedingchapter,theassumptionhoweverbeingmadethatifnoruleofthesyllogismasgiveninsection199orsection201isbroken,thenthesyllogismisvalid.

210.IfPisamarkofthepresenceofQ,andRofthatofS,andifPandRareneverfoundtogether,amIrightininferringthatQandSsometimesexistseparately?[V.]Thepremissesmaybestatedasfollows:

AllPisQ,AllRisS,NoPisR ;


andinordertoestablishthedesiredconclusionwemustbeabletoinferatleastoneofthefollowing,—SomeQisnotS,SomeSisnotQ. Butneitherofthesepropositionscanbeinferred;fortheydistributerespectivelySandQ,andneitherof these terms isdistributed in thegivenpremisses.Thequestion is, therefore, tobeanswered in thenegative.

303211. If itbeknownconcerninga syllogism in theAristotelian systemthatthemiddletermisdistributedinbothpremisses,whatcanweinferastotheconclusion?[C.]Ifbothpremissesareaffirmative,theycanbetweenthemdistributeonlytwoterms,andbyhypothesis

themiddletermisdistributedtwiceinthepremisses;hencetheminortermcannotbedistributedinthepremisses,anditfollowsthattheconclusionmustbeparticular. Ifoneofthepremissesisnegative,theremaybethreedistributedtermsinthepremisses;thesemust,however,bethemiddletermtwice(byhypothesis)andthemajorterm(sincetheconclusionmustnowbenegativeandwillthereforedistributethemajorterm);hencetheminortermcannotbedistributedinthepremisses,anditagainfollowsthattheconclusionmustbeparticular. Buteitherbothpremisseswillbeaffirmative,oroneaffirmativeandtheothernegative;inanycase,therefore,wecaninferthattheconclusionwillbeparticular.

212. Shew directly in how many ways it is possible to prove theconclusions SaP, SeP ; point out those that conform immediately to theDictumdeomnietnullo ;andexhibit theequivalencebetweentheseandtheremainder.[W.](1)ToproveAllSisP.

Bothpremissesmustbeaffirmative,andbothmustbeuniversal. Sbeingdistributedintheconclusionmustbedistributedintheminorpremiss,whichmustthereforebeAllSisM. Mnotbeingdistributedintheminormustbedistributedinthemajor,whichmustthereforebeAllMisP. SaPcanthereforebeprovedinonlyoneway,namely,

AllMisP,AllSisM,

therefore, AllSisP ;

andthissyllogismconformsimmediatelytotheDictum. (2)ToproveNoSisP. Bothpremissesmustbeuniversal,andonemustbenegativewhiletheotherisaffirmative;i.e.,onepremissmustbeEandtheotherA. First, let the major beE, i.e., eitherNoM is P orNo P is M. In each case the minor must beaffirmativeandmustdistributeS ;therefore,itwillbeAllSisM. 304Secondly,lettheminorbeE,i.e.,eitherNoSisMorNoMisS.IneachcasethemajormustbeaffirmativeandmustdistributeP ;therefore,itwillbeAllPisM. WecanthenproveSePinfourways,thus,—


(i) MeP, (ii) PeM, (iii) PaM, (iv) PaM, SaM, SaM, SeM, MeS,

⎯⎯ ⎯⎯ ⎯⎯ ⎯⎯

SeP. SeP. SeP. SeP.

Ofthese,(i)onlyconformsimmediatelytothedictum,andwehavetoshewtheequivalencebetweenitandtheothers. Theonlydifferencebetween(i)and(ii)isthatthemajorpremissoftheoneisthesimpleconverseofthemajorpremissof theother; they are, therefore, equivalent.Similarly theonlydifferencebetween(iii) and (iv) is that theminorpremissof theone is the simpleconverseof theminorpremissof theother;theyare,therefore,equivalent. Finally,wemayshewthat (iv) isequivalent to (i)by transposing thepremissesandconverting theconclusion.

213. Given that the major term is distributed in the premisses andundistributedintheconclusionofavalidsyllogism,determinethesyllogism.[C.]Since the major term is undistributed in the conclusion, the conclusion—and, therefore, both

premisses—mustbeaffirmative.Hence,inordertodistributeP,themajorpremissmustbePaM ;andinordertodistributeM(whichisnotdistributedinthemajorpremiss),theminorpremissmustbeMaS.Itfollowsthatthesyllogismmustbe

AllPisM,AllMisS,

therefore, SomeSisP.

214.Provethatif threepropositionsinvolvingthreeterms(eachofwhichoccurs in two of the propositions) are together incompatible, then (a) eachtermisdistributedatleastonce,and(b)oneandonlyoneofthepropositionsisnegative. Shewthattheserulesareequivalenttotherulesofthesyllogism.[J.]Notwoofthepropositionscanbeformallyincompatiblewithoneanother,sincetheydonotcontain

the same terms.But eachpairmustbe incompatiblewith the third, i.e., the contradictoryof anyonemustbededuciblefromtheothertwo.Itfollowsthat305weshallhavethreevalidsyllogisms,inwhichthegivenpropositionstakeninpairsarethepremisses,whilstthecontradictoryofthethirdpropositionisineachcasetheconclusion.328

Then (a) each term must be distributed once at least. For if any one of the terms failed to bedistributedatleastonce,weshouldobviouslyhaveundistributedmiddleinoneofoursyllogisms;and(sinceatermundistributedinapropositionisdistributedinitscontradictory)illicitmajororminorinthe two others. If, however, the above condition is fulfilled, it is clear that we cannot have eitherundistributed middle, or illicit major or minor. Hence rule (a) is equivalent to the syllogistic rulesrelatingtothedistributionofterms. Again,(b)oneofthepropositionsmustbenegative,butnotmorethanoneofthemcanbenegative.Forifallthreewereaffirmative,then(sincethecontradictoryofanaffirmativeisnegative)weshouldin


eachofoursyllogismsinferanegativefromtwoaffirmatives;andiftwowerenegative,weshouldhavetwo negative premisses in one of our syllogisms, and (since the contradictory of a negative isaffirmative) anaffirmativeconclusionwithanegativepremiss in eachof theothers. If, however, theabove condition is fulfilled, it is clear that we cannot have either two negative premisses, or twoaffirmativepremisseswithanegativeconclusion,oranegativepremisswithanaffirmativeconclusion.Hencerule(b)isequivalenttothesyllogisticrulesrelatingtoquality.

328 Every syllogism involves two others, in each ofwhich one of the originalpremisses combined with the contradictory of the conclusion proves thecontradictoryoftheotheroriginalpremiss.Hencethethreesyllogismsreferredtointhetextmutuallyinvolveoneanother.Comparesections264,265.

215. Explainwhat ismeant by a syllogism ; and put the following argument into syllogistic form:—“Wehavenorighttotreatheatasasubstance,foritmaybetransformedintosomethingwhichisnotheat,andiscertainlynotasubstanceatall,namely,mechanicalwork.”[N.]

216.Putthefollowingargumentintosyllogisticform:—Howcananyonemaintainthatpainisalwaysanevil,whoadmitsthatremorseinvolvespain,andyetmaysometimesbearealgood?[V.]

306217.IthasbeenpointedoutbyOhmthatreasoningtothefollowingeffectoccursinsomeworkson mathematics:—“A magnitude required for the solution of a problem must satisfy a particularequation,andasthemagnitudexsatisfiesthisequation,itisthereforethemagnituderequired.”Examinethelogicalvalidityofthisargument.[C.]

218.Obtainaconclusionfromthetwonegativepremisses,—NoPisM,NoSisM.[K.]

219. If it is false that the attributeB is ever found coexisting withA, and not less false that theattributeCissometimesfoundabsentfromA,canyouassertanythingaboutBintermsofC?[C.]

220.Giveexamples(insymbols—takingS,M,P,asminor,middle,andmajorterms,respectively)inwhich, attempting to infer a universal conclusion where we have a particular premiss, we commitrespectivelyonebutoneonlyofthefollowingfallacies,—(a)undistributedmiddle,(b)illicitmajor,(c)illicitminor.Givealsoanexampleinwhich,makingthesameattempt,wecommitnoneoftheabovefallacies.[K.]

221.Cananapparentsyllogismbreakdirectlyalltherulesofthesyllogismatonce?[K.]

222. Can you give an instance of an invalid syllogism in which the major premiss is universalnegative,theminorpremissaffirmative,andtheconclusionparticularnegative?Ifnot,whynot?[K.]

223.Shewthat (i)Ifbothpremissesofasyllogismareaffirmative,andonebutonlyoneofthemuniversal,theywillbetweenthemdistributeonlyoneterm; (ii)Ifbothpremissesareaffirmativeandbothuniversal,theywillbetweenthemdistributetwoterms; (iii) If one but only one premiss is negative, and one but only one premiss universal, they willbetweenthemdistributetwoterms; (iv)Ifonebutonlyonepremissisnegative,andbothpremissesareuniversal,theywillbetweenthemdistributethreeterms.[K.]

224.Ascertainhowmanydistributedtermstheremaybeinthepremissesofasyllogismmorethanintheconclusion.[L.]

225. If theminor premiss of a syllogism is negative,what doyouknowabout the positionof the


termsinthemajor?[O’S.]

307226.If themajortermofasyllogismisthepredicateofthemajorpremiss,whatdoyouknowabouttheminorpremiss?[L.]

227.Howmuchcanyou tell aboutavalid syllogism ifyouknow(1) thatonly themiddle term isdistributed; (2)thatonlythemiddleandminortermsaredistributed; (3)thatallthreetermsaredistributed?[W.]

228.Whatcanbedeterminedrespectingavalidsyllogismundereachofthefollowingconditions:(1)thatonlyonetermisdistributed,andthatonlyonce;(2)thatonlyonetermisdistributed,andthattwice;(3) that two termsonly are distributed, eachonlyonce; (4) that two termsonly are distributed, eachtwice?[L.]

229.Twopropositionsaregivenhavingatermincommon.IftheyareIandA,shewthateithernoconclusionortwocanbededuced;butifIandE,alwaysandonlyone.[T.]

230.Findout, from the rulesof the syllogism,whatare thevalid formsof syllogism inwhich themajorpremissisparticularaffirmative.[J.]

231.Given(a)thatthemajorpremiss,(b) that theminorpremiss,ofavalidsyllogismisparticularnegative,determineineachcasethesyllogism.[K.]

232. Given that themajor premiss of a valid syllogism is affirmative, and that themajor term isdistributedbothinpremissesandconclusion,whiletheminortermisundistributedinboth,determinethesyllogism.[N.]

233.ShewdirectlyinhowmanywaysitispossibletoprovetheconclusionsSiP,SoP.[W.]

234.Shewthatiftherulethatanegativeconclusionrequiresanegativepremissbeomittedfromthegeneralrulesofthesyllogism,theonlyinvalidsyllogismtherebyadmittedissuchthat,ifitsconclusionbefalsewhilstitspremissesaretrue,thethreetermsofthesyllogismmustbeabsolutelycoextensive.[O’S.]

235.Find,bydirectapplicationof the fundamental rulesof syllogism,whatare thevalid formsofsyllogism inwhichneitherof thepremisses isauniversalpropositionhaving thesamequalityas theconclusion.[J.]

308236. Inwhat caseswill contradictorymajor premisses both yield conclusionswhen combinedwiththesameminor? Howaretheconclusionsrelated? Shewthatinnocasewillcontradictoryminorpremissesbothyieldconclusionswhencombinedwiththesamemajor.[O’S.]

237.(a)Alljustactionsarepraiseworthy;(b)Nounjustactionsareexpedient;(c)Someinexpedientactionsarenotpraiseworthy;(d)Notallpraiseworthyactionsareinexpedient. Do(c)and(d)followfrom(a)and(b)?[K.]

238.Reducethefollowingargumentstoordinarysyllogisticform: (i)NoMisS,WhateverisnotMisP,therefore,AllSisP ; (ii)Itcannotbethatnonot-SisP,forsomeMisPandnoMisS ; (iii)Itisimpossibleforthethreepropositions,AllMisP,AnythingthatisnotMisnotS,SomethingsthatarenotPareS,alltobetruetogether;


(iv)EverythingisMorP,NothingisbothSandM,therefore,AllSisP.[K.]

239.Shewthatthefollowingsyllogismsbreakdirectlyorindirectlyalltherulesofthesyllogism: (1)AllPisM,AllSisM,therefore,SomeSisnotP ; (2)AllMisP,AllMisS,therefore,NoSisP.[K.]

[The so-called rules that every syllogism contains three and only three terms, and that everysyllogism consists of three and only three propositions, are not here included under the rules of thesyllogism.]

240.Inacircularargumentinvolvingtwovalidsyllogisms,QandUareusedaspremissestoproveR,whileRandVareusedaspremissestoproveQ ;shewthatUandVmustbeapairofcomplementarypropositions,i.e.,oftheformsAllMisNandAllNisMrespectively.[J.]

241. Shew that if two valid syllogisms have a common premiss while the other premisses arecontradictories,boththeconclusionsmustbeparticular.[K.]

242. Given the premisses of a valid syllogism, examine in what cases it is (a) possible, (b)impossible,todeterminewhichistheminortermandwhichthemajorterm.[J.]


CHAPTERII.

THEFIGURESANDMOODSOFTHESYLLOGISM.

243.FigureandMood.—Bythefigureofasyllogismismeantthepositionofthetermsinthepremisses.Denotingthemajor,middle,andminortermsbythelettersP,M,S respectively,andstating themajorpremiss first,wehavefourfiguresofthesyllogismasshewninthefollowingtable:—

Fig.1. Fig.2. Fig.3. Fig.4.

M–P P–M M–P P–MS–M S–M M–S M–S⎯⎯ ⎯⎯ ⎯⎯ ⎯⎯

S–P S–P S–P S–P

By the mood of a syllogism is meant the quantity and quality of thepremisses and conclusion. For example,AAA is amood inwhich both thepremissesandalsotheconclusionareuniversalaffirmatives;EIOisamoodinwhich themajor is a universal negative, theminor a particular affirmative,andtheconclusionaparticularnegative.Itisclearthatiffigureandmoodarebothgiven,thesyllogismisgiven.

244. The Special Rules of the Figures; and the Determination of theLegitimateMoodsineachFigure.329—Itmayfirstofallbeshewnthatcertaincombinationsofpremissesareincapableofyieldingavalidconclusioninanyfigure. A priori, there are possible the following sixteen differentcombinationsofpremisses,themajorpremissbeingalwaysstatedfirst:—AA,AI,AE,AO,IA,II,IE,IO,EA,EI,EE,EO,OA,OI,OE,OO.Referringback,however, to the syllogistic rules and corollaries (as given in sections 199,200), we find that EE, 310 EO,OE,OO (being combinations of negativepremisses)yieldnoconclusionbyrule5;thatII,IO,OI(beingcombinationsofparticularpremisses)areexcludedbycorollaryi.;andthatIE isexcludedbycorollary iii.,which tellsus thatnothing follows fromaparticularmajor


andanegativeminor.329 The method of determination here adopted is only one amongst several

possiblemethods.Anotherissuggested,forexample,insections212,233.

Weareleftthenwiththefollowingeightpossiblecombinations:—AA,AI,AE,AO,IA,EA,EI,OA ;andwemaygoontoenquireinwhichfiguresthesewill yield conclusions. Inpursuing this enquiry, special rulesof thevariousfiguresmaybe determined,which, taken togetherwith the three corollariesestablished in section 200, replace the general rules of distribution. Thesespecial rules, supplemented by the general rules of quality and thecorollaries,330willenablethevalidityofthedifferentmoodstobetestedbyamereinspectionoftheformofthepropositionsofwhichtheyconsist.

330 The general rules of quality and the corollaries can be directly appliedwithoutreferencetothepositionofthetermsinthepremissesofasyllogism.Thisisnotthecasewiththegeneralrulesofdistribution.Theobjectofthespecialrulesis, in the case of each particular figure, to substitute for the general rules ofdistributionspecialrulesofquantityandquality.

Thespecialrules331andthelegitimatemoodsofFigure1.331 Asindicatedinsection209,thespecialrulesoffigure1followimmediately

fromthedictumdeomnietnullo.

Thepositionofthetermsinfigure1isshewnthus,—

M–PS–M

⎯⎯

S–P

and it can be deduced from the general rules of the syllogism that in thisfigure:— (1) The minor premiss must be affirmative. For if it were negative, themajor premiss would have to be affirmative by rule 5, and the conclusionnegative by rule 6. The major term would therefore be distributed in theconclusion, and undistributed in its premiss; and the syllogism would beinvalidbyrule4. (2) The major premiss must be universal. For the middle term, beingundistributed in the affirmative minor premiss, must be distributed in the


majorpremiss.

311Rule (1)shews thatAEandAOandrule (2) thatIAandOA,yieldnoconclusions in this figure. We are accordingly left with only fourcombinations,namely,AA,AI,EA,EIFromtherulesthataparticularpremisscannot yield a universal conclusion or a negative premiss an affirmativeconclusion, while conversely a negative conclusion requires a negativepremiss,itfollowsfurtherthatAAwilljustifyeitheroftheconclusionsAorI,EAeitherEorO,AIonlyI,EIonlyO.Therearethensixmoodsinfigure1which do not offend against any of the rules of the syllogism,332 namely,AAA,AAI,AII,EAE,EAO,EIO.

332 Rule (2) provides against undistributedmiddle, and rule (1) against illicitmajor.Wecannothaveillicitminor,unlesswehaveauniversalconclusionwithaparticularpremiss,andthisalsohasbeenprovidedagainst.

MrJohnsonpointsout that the followingsymmetrical rulesmaybe laiddownfor the correct distribution of terms in the different figures; and that these rules(three in each figure) taken together with the rules of quality are sufficient toensurethatnosyllogisticruleisbroken.

(i) To avoid undistributedmiddle: in figure 1, If theminor is affirmative, themajormustbeuniversal;infigure4,Ifthemajorisaffirmative,theminormustbeuniversal;infigure2,Onepremissmustbenegative;infigure3,Onepremissmustbe universal. (The last of these rules is of course superfluous if the corollariescontainedinsection200aresupposedgiven.)

(ii)Toavoid illicitmajor: infigures1and3, If theconclusion isnegative, themajormustbenegativeand,therefore,theminoraffirmative;infigures2and4,Iftheconclusionisnegative,themajormustbeuniversal.

(iii) To avoid illicit minor: in figures 1 and 2, If the minor is particular, theconclusionmustbeparticular; in figures3and4, If theminor isaffirmative, theconclusionmustbeparticular.(Thefirstofthesetworulesisagainsuperfluousasaspecialruleifthecorollariesaresupposedgiven.)

Theaboverulesaresubstantiallyidenticalwiththosegiveninthetext.

Theactualvalidityofthesemoodsmaybeestablishedbyshewingthattheaxiomof thesyllogism, thedictumdeomnietnullo, applies to them;orbytaking them severally and shewing that in each case the cogency of thereasoningisself-evident.

ThespecialrulesandthelegitimatemoodsofFigure2.

Thepositionofthetermsinfigure2isshewnthus,—


P–MS–M

⎯⎯

S–P ;

312anditsspecialrules(whichthereaderisrecommendedtodeducefromthegeneralrulesofthesyllogismforhimself)are,— (1)Onepremissmustbenegative ; (2)Themajorpremissmustbeuniversal.

Theapplicationoftheserulesagainleavessixmoods,namely,AEE,AEO,AOO,EAE,EAO,EIO.

Recourse cannot now he had directly to the dictum de omni et nullo inordertoshewpositivelythatthesemoodsarelegitimate.Itmay,however,beshewn in each case that the cogency of the reasoning is self-evident. Theolder logiciansdidnotadopt thiscourse; theirmethodwas toshew that,bytheaidofimmediateinferences,eachmoodcouldbereducedtosuchaformthat thedictum did apply directly to it. The doctrine of reduction resultingfromtheadoptionofthismethodwillbediscussedinthefollowingchapter.


Thepositionofthetermsinthisfigureisshewnthus,—

M–PM–S

⎯⎯

S–P ;

anditsspecialrulesare,— (1)Theminormustbeaffirmative ; (2)Theconclusionmustbeparticular.

Proceedingasbefore,weare leftwithsixvalidmoods,namely,AAI,AII,EAO,EIO,IAI,OAO.



Thepositionofthetermsinthisfigureisshewnthus,—

P–MM–S

⎯⎯

S–P ;

andthefollowingmaybegivenasitsspecialrules,— (1)Ifthemajorisaffirmative,theminormustbeuniversal ; (2)Ifeitherpremissisnegative,themajormustbeuniversal ;313 (3)Iftheminorisaffirmative,theconclusionmustbeparticular.333

333 Thespecialrulesofthefourthfigurearevariouslystated.TheyaregivenintheaboveforminthePortRoyalLogic,pp.202,203.See,also,section255.

The result of the application of these rules is again six valid moods,namely,AAI,AEE,AEO,EAO,EIO,IAI.

Ourfinalconclusion then is that thereare24validmoods,namely,six ineachfigure.

InFigure1,AAA,AAI,EAE,EAO,AII,EIO. InFigure2,EAE,EAO,AEE,AEO,EIO,AOO. InFigure3,AAI,IAI,AII,EAO,OAO,EIO. InFigure4,AAI,AEE,AEO,EAO,IAI,EIO.

245.WeakenedConclusionsandSubalternMoods.—Whenfrompremissesthat would have justified a universal conclusionwe content ourselveswithinferringaparticular(as,forexample,inthesyllogismAllMisP,AllSisM,therefore,SomeSisP),wearesaidtohaveaweakenedconclusion,and thesyllogism is said to be aweakenedsyllogism or to be in a subalternmood(becausetheconclusionmightbeobtainedbysubalterninference334fromtheconclusionofthecorrespondingunweakenedmood).

334 IntreatingthesyllogismonthetraditionallinesitisassumedthatS,M,Pallrepresentexistingclasses.Subalterninferenceis,therefore,avalidprocess.

Intheprecedingsectionithasbeenshewnthatineachfiguretherearesixmoodswhichdonotoffendagainstanyofthesyllogisticrules:sothatinallthere are 24 distinct valid moods. Five of these, however, have weakened


conclusions; and, since we are not likely to be satisfied with a particularconclusionwhenthecorrespondinguniversalcanbeobtainedfromthesamepremisses,thesemoodsareofnopracticalimportance.Accordinglywhenthemoodsofthevariousfiguresareenumerated(asinthemnemonicverses)theyareusuallyomitted.Still,theirrecognitiongivesacompletenesstothetheoryofthesyllogism,whichitcannototherwisepossess.Thereisalsoasymmetryintheresultof314 theirrecognitionasyieldingexactlysixlegitimatemoodsineachfigure.335

335 It has been remarked that 19 being a prime number at once suggestsincompletenessorartificialityinthecommonenumeration.

Thesubalternmoodsare,— InFigure1,AAI,EAO ; InFigure2,EAO,AEO ; InFigure4,AEO.

ItisobviousthattherecanbenoweakenedconclusioninFigure3,sinceinnocaseisitpossibletoinfermorethanaparticularconclusioninthisfigure.

AAIinFigure4issometimesspokenofasasubalternmood.Butthisisamistake.WiththepremissesAllPisM,AllMisS,theconclusionSomeSisPiscertainly inonesenseweaker than thepremisseswouldwarrantsince theuniversalconclusionAllPisSmighthavebeeninferred.ButAllPisSisnotthe universal corresponding to Some S is P. The subjects of these twopropositionsaredifferent;andweinferallthatwepossiblycanaboutSwhenwesaythatsomeSisP.Inotherwords,regardedasamoodoffigure4,thismood is not a subaltern.AAI in figure 4 is thus differentiated fromAAI infigure1,anditsinclusioninthemnemonicversesjustified.

246.StrengthenedSyllogisms.—If inasyllogismthesameconclusioncanstillbeobtainedalthoughforoneofthepremisseswesubstituteitssubaltern,thesyllogismissaidtobeastrengthenedsyllogism.Astrengthenedsyllogismisthusasyllogismwithanunnecessarilystrengthenedpremiss.336


336 Compare De Morgan, Formal Logic, pp. 91, 130. De Morgan calls asyllogismfundamental,whenneitherofitspremissesisstrongerthanisnecessarytoproducetheconclusion(FormalLogic,p.77).

Forexample,theconclusionofthesyllogism—

AllMisP,AllMisS,

therefore, SomeSisP,

could equally be obtained from the premissesAllM isP,SomeM is S ; orfromthepremissesSomeMisP,AllMisS.

By trial we may find that every syllogism in which there 315 are twouniversalpremisseswithaparticularconclusionisastrengthenedsyllogism,withthesingleexceptionofAEOinthefourthfigure.337

337 Ageneralproofofthispropositionwillbegiveninsection351.

Inafullenumerationtherearetwostrengthenedsyllogismsineachfigure:—

InFigure1,AAI,EAO ;InFigure2,EAO,AEO ;InFigure3,AAI,EAO ;InFigure4,AAI,EAO.

It will be observed that in figures 1 and 2, a syllogism having astrengthenedpremissmayalsoberegardedasasyllogismhavingaweakenedconclusion,andviceversâ ;butthatinfigures3and4, thecontraryholdsinbothcases.Theonlysyllogismwithaweakenedconclusionineitherofthesefigures isAEO in figure 4; and in thismoodno conclusion is obtainable ifeitherofthepremissesisreplacedbyitssubaltern.

If syllogisms containing either a strengthened premiss or a weakenedconclusionareomitted,weareleftwith15validmoods,namely,4ineachofthefirstthreefiguresand3infigure4.

247. The peculiarities and uses of each of the four figures of thesyllogism.338—Figure1. In this figure it ispossible toproveconclusionsofall the forms A,E, I, O; and it is the only figure in which a universal


affirmative conclusion can be proved. This alonemakes it by far themostuseful and important of the syllogistic figures. All deductive science, theobjectofwhichistoestablishuniversalaffirmatives,tendstoworkinAAAinthisfigure.

338 On the distinctive characteristics of the different figures, see also sections269to271.

Anotherpointtonoticeisthatonlyinthisfigureisitthecasethatboththesubjectoftheconclusionissubjectinthepremisses,andthepredicateoftheconclusion predicate in the premisses; in figure 2 the predicate of theconclusion is subject in the major premiss; in figure 3 the subject of theconclusionispredicateintheminorpremiss;andinfigure4thereisadoubleinversion.339Thisnodoubtpartly 316 accounts for the fact that a reasoningexpressed in figure 1 so often seemsmore natural than the same reasoningexpressedinanyotherfigure.340

339 Thedoubleinversioninfigure4isoneofthereasonsgivenbyThomsonforrejectingthatfigurealtogether.Comparesection262.

340 CompareSolly,SyllabusofLogic,pp.130to132.

Figure 2. In this figure,onlynegativescanbeproved; and therefore it ischieflyusedforpurposesofdisproof.Forexample,Everyrealnaturalpoemisnaïve ;thosepoemsofOssianwhichMacphersonpretendedtodiscoverarenot naïve (but sentimental); hence they are not real natural poems(Ueberweg,SystemofLogic,§113). Ithasbeencalled theexclusive figure;becausebymeansofitwemaygoonexcludingvarioussuppositionsastothenature of something under investigation, whose real character we wish toascertain(aprocesscalledabscissioinfiniti).Forexample,Suchandsuchanorder has such and such properties, This plant has not those properties ;therefore,Itdoesnotbelong to thatorder.Asyllogismof thiskindmayberepeated with a number of different orders till the enquiry is so narroweddown that theplaceof theplant iseasilydetermined.Whately (ElementsofLogic,p.92)givesanexamplefromthediagnosisofadisease.

Figure 3. In this figure, only particulars can be proved. It is frequentlyusefulwhenwewishtotakeobjectiontoauniversalpropositionlaiddownbyanopponentbyestablishinganinstanceinwhichsuchuniversalproposition


doesnotholdgood.

Itisthenaturalfigurewhenthemiddletermisasingularterm,especiallyiftheothertermsaregeneral.Ithasbeenalreadyshewnthatifoneandonlyonetermofanaffirmativepropositionissingular,thattermisalmostnecessarilythesubject.Forexample,suchareasoningasSocratesiswise,Socrates isaphilosopher, therefore, Some philosophers are wise, can only with greatawkwardnessbeexpressedinanyfigureotherthanfigure3.

Figure 4.This figure is seldomused, and some logicianshavealtogetherrefusedtorecogniseit.Weshallreturntoadiscussionofitsubsequently.Seesection262.

Lambert in his Neues Organon expresses the uses of the differentsyllogistic figures as follows: “The first figure is suited to the discovery orproofofthepropertiesofathing;317thesecondtothediscoveryorproofofthe distinctions between things; the third to the discovery or proof ofinstances and exceptions; the fourth to the discovery or exclusion of thedifferentspeciesofagenus.”

EXERCISES.

248.WhyisIEan inadmissible,whileEI isanadmissible,mood ineveryfigureof thesyllogism?[L.]

249.Whatmoodsaregoodinthefirstfigureandfaultyinthesecond,andviceversâ?Whyaretheyexcludedinonefigureandnotintheother?[O.]

250.(i)ShewthatOcannotstandaspremissinfigure1,asmajorinfigure2,asminorinfigure3,aspremissinfigure4. (ii)ShewthatitisimpossibletohavetheconclusioninAinanyfigurebutthefirst.Whatfallacieswouldbecommittedifthereweresuchaconclusiontoareasoninginanyotherfigure?[C.]

251.Twovalidsyllogismsinthesamefigurehavethesamemajor,middle,andminorterms,andtheirmajor premisses are subcontraries; determine—without reference to themnemonic verses—what thesyllogismsmustbe.[K.]

252.Prove,bygeneralreasoning,thatanymoodvalidbothinfigure2andinfigure3isvalidalsoinfigure1andinfigure4.[C.]

253.Shew,withoutindividualreferencetothedifferentfigures,thatEAOisastrengthenedsyllogism


ineveryfigure,andthatAAIisastrengthenedsyllogismwheneveritisvalid.[K.]

254. Shew, by general reasoning, that every valid syllogism in which the middle term is twicedistributed contains a strengthened premiss. Does it follow that it must have also a weakenedconclusion?[K.]

255.Shewthatthefollowingtworuleswouldsufficeasthespecialrulesforthefourthfigure:(i)Theconclusionandmajorcannothavethesameformunlessitbeparticularaffirmative;(ii)Theconclusionandminorcannothavethesameformunlessitbeuniversalnegative.[J.]


CHAPTERIII.

THEREDUCTIONOFSYLLOGISMS.

256.TheProblemofReduction.—Byreductionismeantaprocesswherebythereasoningcontainedinagivensyllogismisexpressedinsomeothermoodor figure.Unless an explicit statement ismade to the contrary, reduction issupposedtobetofigure1.

Thefollowingsyllogisminfigure3maybetakenasanexample:

AllMisP,SomeMisS,


It will be seen that by simply converting the minor premiss, we havepreciselythesamereasoninginfigure1.

Thisisanexampleofdirectorostensivereduction.

257. IndirectReduction.—Aproposition is established indirectlywhen itscontradictory is proved false; and this is effected if it can be shewn that aconsequenceofthetruthofitscontradictorywouldbeself-contradiction.

ThemethodofindirectproofisinseveralcasesadoptedbyEuclid;anditmaybeemployed in thereductionofsyllogismsfromonemood toanother.Thus,AOOinfigure2isusuallyreducedinthismanner.Theargumentmaybestatedasfollows:— Fromthepremisses,—

AllPisM,SomeSisnotM,

itfollowsthat SomeSisnotP ;

forifthisconclusionisnottrue,then,bythelawofexcluded319middle,itscontradictory(namely,AllSisP)mustbeso;and,thepremissesbeinggiven


true,thethreefollowingpropositionsmustallbetrue,namely,

AllPisM,SomeSisnotM,AllSisP.

Butcombiningthefirstandthethirdofthesewehaveasyllogisminfigure1,namely,

AllPisM,AllSisP,

yieldingtheconclusion AllSisM.

SomeSisnotMandAllSisMare,therefore,truetogether;but,bythelawofcontradiction,thisisabsurd,sincetheyarecontradictories. HenceithasbeenshewnthattheconsequenceofsupposingSomeSisnotPfalseisaself-contradiction;andwemayaccordinglyinferthatitistrue.

It will be observed that the only syllogism made use of in the aboveargument is in figure 1; and the process may, therefore, be regarded as areductionofthereasoningtofigure1.

ThismethodofreductioniscalledReductioadimpossibile,orReductioperimpossibile,341orDeductioadimpossibile,orDeductioadabsurdum.ItistheonlywayofreducingAOOinfigure2orOAOinfigure3tofigure1,unlessnegative terms are used (as in obversion and contraposition); and it wasadoptedbytheoldwritersinconsequenceoftheirobjectiontonegativeterms.

341 CompareMansel’sAldrich,pp.88,89.

Itwillbeshewnlateron in thischapter thatbyemploying themethodofindirect reduction systematically we can bring out with great clearness therelationbetweenthedifferentmoodsandfiguresofthesyllogism.

258. The mnemonic lines Barbara, Celarent, &c.—The mnemonichexameterverses(whicharespokenofbyDeMorganas“themagicwordsbywhich the different moods have been denoted for many centuries, wordswhichItaketobemorefullofmeaningthananythateverweremade”)areusuallygivenasfollows:320

Barbără,Cēlārent,Dărĭi,Fĕrĭōqueprioris:


Cēsărĕ,Cāmēstres,Festīnŏ,Bărōcŏ,secundae:Tertia,Dāraptī,Dĭsămis,Dātīsĭ,Fĕlapton,Bōcardō,Fērīsŏn,habet:QuartainsuperadditBrāmantip,Cămĕnes,Dĭmăris,Fēsāpŏ,Frĕsīson.

Each validmood in every figure, unless it be a subalternmood, is hererepresentedbyaseparateword;andinthecaseofamoodinanyof theso-called imperfect figures (i.e., figures 2, 3, 4), the mnemonic gives fullinformationforitsreductiontofigure1,theso-calledperfectfigure.

Theonlymeaninglesslettersareb(notinitial),d(notinitial),l,n,r,t ; thesignificationoftheremainderisasfollows:—

Thevowelsgive thequalityandquantityof thepropositionsofwhich thesyllogismiscomposed;and,therefore,reallygivethesyllogismitself, if thefigureisalsoknown.Thus,Camenesinfigure4representsthesyllogism—

AllPisM,NoMisS,

therefore, NoSisP.

Theinitiallettersinthecaseoffigures2,3,4shewtowhichofthemoodsof figure 1 the givenmood is to be reduced, namely, to thatwhich has thesameinitialletter.ThelettersB,C,D,Fwerechosenforthemoodsoffigure1asbeingthefirstfourconsonantsinthealphabet.

Thus,CamestresisreducedtoCelarent,—

AllPisM, ⟍ ⟋ NoMisS,NoSisM, ⟋ ⟍ AllPisM,

therefore, NoSisP. therefore, NoPisS,therefore, NoSisP.342

342 Theorderofinferenceinthisandinotherreductionsmightbemadeclearbytheuseofarrows,representinginference,asfollows:

AllPisM, ⟍ ↗ NoMisS,NoSisM, ⟋ ↘ AllPisM,

↓NoSisP. ← NoPisS,

s (in themiddle of aword) indicates that in the process of reduction the


preceding proposition is to be simply converted. 321 Thus, in reducingCamestres to Celarent, as shewn above, the minor premiss is simplyconverted.

s(attheendofaword)shewsthattheconclusionofthenewsyllogismhastobe simply converted inorder that thegivenconclusionmaybeobtained.This again is illustrated in the reductionofCamestres.The finals does notaffect the conclusion ofCamestres itself, but the conclusion ofCelarent towhichitisreduced.343

343 Thispeculiarityinthesignificationofsandpwhentheyarefinal letters issometimes overlooked. The point to be noted is that the conclusion of thesyllogismoriginallygivenisnot,liketheoriginalpremisses,adatumfromwhichwesetout,butaresultthatwehavetoreach.Itfollowsthattheconclusiontobemanipulated,ifany,mustbetheconclusionofthesyllogismobtainedbyreduction,not theconclusionof theoriginal syllogism.This isclearlyshewn in thecaseofCamestres by the method adopted in the last preceding note to illustrate thereduction of Camestres to Celarent. The reduction of Disamis, Bramantip,Camenes,Dimaristofigure1mightbeillustratedsimilarly.

p(inthemiddleofaword)signifiesthattheprecedingpropositionistobeconvertedperaccidens ;as,forexample,inthereductionofDaraptitoDarii,—

AllMisP, AllMisP,AllMisS, SomeSisM,

therefore, SomeSisP. therefore, SomeSisP.

p (at the end of a word344) implies that the conclusion obtained byreductionistobeconvertedperaccidens.Thus,inBramantip,thepdoesnotrelate to the I conclusion of the mood itself;345 it really relates to theAconclusionofthesyllogisminBarbarawhichisgivenbyreduction.Thus,—

AllPisM, ⟍ ⟋ AllMisS,AllMisS, ⟋ ⟍ AllPisM,

therefore, SomeSisP. therefore, AllPisS,therefore, SomeSisP.

344 Seethelastprecedingnote.345 Compare,however,Hamilton,Logic,I.p.264,andSpalding,Logic,pp.230,

1.


m indicates that in reduction the premisses have to be transposed(metathesispraemissarum);asjustshewninthecaseofBramantip,andalsointhecaseofCamestres.

csignifiesthatthemoodistobereducedindirectly(i.e.,by322reductioperimpossibileinthemannershewnintheprecedingsection);andthepositionoftheletterindicatesthatinthisprocessofindirectreductionthefirststepistoomit thepremissprecedingit, i.e., theotherpremiss is tobecombinedwiththe contradictory of the conclusion (conversio syllogismi, or ductio percontradictoriampropositionemsiveperimpossibile),Theletterc isbysomewriters replaced byk,Baroko andBokardo being given as themnemonics,insteadofBarocoandBocardo.

The following lines are sometimes added to the verses given above, inordertomeetthecaseofthesubalternmoods:—

QuinqueSubalterni,totidemGeneralibusorti,Nomenhabentnullum,nec,sibenecolligis,usum.346

346 Themnemonicshavebeenwritteninvariousforms.ThosegivenabovearefromAldrich,andtheyaretheonesthatareingeneraluseinEngland.WallisinhisInstitutio Logicae (1687) gives for the fourth figure, Balani, Cadere, Digami,Fegano,Fedibo. P. vanMusschenbroek inhis InstitutionesLogicae (1748) givesBarbari, Calentes, Dibatis, Fespamo, Fresisom. This variety of forms for themoodsoffigure4isnodoubtduetothefactthattherecognitionofthisfigureatallwasquiteexceptionaluntilcomparativelyrecently.Comparesections262,263.

AccordingtoUeberweg(Logic,§118)themnemonicsrun,—

Barbara,Celarentprimae,DariiFerioque.Cesare,Camestres,Festino,Barocosecundae.TertiagrandesonansrecitatDarapti,Felapton,Disamis,Datisi,Bocardo,Ferison.QuartaeSuntBamalip,Calemes,Dimatis,Fesapo,Fresison.

UeberweggivesCamestrosandCalemosfortheweakenedmoodsofCamestresandCalemes. This is not, however, quite accurate. The mnemonics should beCamestropandCalemop.

ProfessorCarvethRead(Logic,pp.126,7)suggestsaningeniousmodificationof the verses, so as to make eachmnemonic immediately suggest the figure towhich the corresponding mood belongs, at the same time abolishing all theunmeaningletters.Hetakeslasthesignofthefirstfigure,nofthesecond,rofthethird,andtofthefourth.Thelines(tobescanned,saysProfessorRead,discreetly)thenrun


Ballala,Celallel,Dalii,Felioqueprioris.Cesane,Camesnes,Fesinon,Banocosecundae.TertiaDarapri,Drisamis,Darisi,Ferapro,Bocaro,Ferisorhabet.QuantainsuperadditBamatip,Cametes,Dimatis,Fesapto,Fesistot.

ProfessorMackenziesuggeststhat,ifthisplanisadopted,itwouldbebettertotakerforthefirstfigure(figurarecta,thestraightforwardfigure),nforthesecondfigure(figuranegativa),t for the thirdfigure(figuratertiaorparticularis), and lforthefourthfigure(figuralaeva,theleft-handedfigure).ComparealsoMrsLaddFranklin,StudiesinLogic,JohnsHopkinsUniversity,p.40.

323259.ThedirectreductionofBarocoandBocardo.—Thesemoodsmaybe reduced directly to the first figure by the aid of obversion andcontrapositionasfollows.347

Baroco:—


therefore, SomeSisnotP,

is reducible to Ferio by the contraposition of the major premiss and theobversionoftheminor,thus,—

Nonot-MisP,SomeSisnot-M,

therefore, SomeSisnotP.347 AnothermethodistoreduceBarocoandBocardobytheprocessofἔκθεσις

to other moods of figures 2 and 3, and thence to figure 1. Ueberweg writes,“Baroco may also be referred toCamestres when those (some) S of which theminorpremissistrueareplacedunderaspecialnotionanddenotedbySʹ.Thentheconclusionmust holdgooduniversallyofSʹ, and consequently particularly ofS.Aristotle calls such a procedure ἔκθεσις” (Logic, § 113). As regards Bocardo,“Aristotle remarks that this moodmay be proved without apagogical procedure(reductioadimpossibile)bytheἐκθέσθαιorλαμβάνεινof thatpartof themiddlenotionwhichistrueofthemajorpremiss.IfwedenotethispartbyN,thenwegetthepremisses;NeP ;NaS:fromwhichfollows(inFelapton)SoP ;whichwastobeproved”(§115).Theprocedureis,however,rathermorecomplicatedthanappearsin the above statements. In the case ofBaroco (PaM,SoM,∴ SoP), let the S’swhicharenotM(ofwhichbyhypothesistherearesome)bedenotedbyX ;thenwehavePaM,XeM,∴XeP (Camestres); butXaS, and hencewe have furtherXeP,XaS,∴SoP(Felapton). In thecaseofBocardo(MoP,MaS,∴SoP), let theM’swhicharenotP(ofwhichbyhypothesistherearesome)bedenotedbyN ;thenwe


haveMaS,NaM,∴NaS (Barbara);andhenceNeP,NaS,∴SoP (Felapton). Theargumentinbothcasessuggestsquestionsconnectedwiththeexistentialimportofpropositions; but the consideration of such questions must for the present bedeferred.

Faksokohasbeensuggestedasamnemonicforthismethodofreduction,kdenoting obversion, so that ks demotes obversion followed by conversion(i.e.,contraposition).

Whately’smnemonicFakoro (ElementsofLogic,p.97)doesnot indicatetheobversionoftheminorpremiss(rbeingwithhimanunmeaningletter).

324Bocardo:—

SomeMisnotP,AllMisS,

therefore, SomeSisnotP,

is reducible to Darii by the contraposition of the major premiss and thetranspositionofthepremisses,thus,—

AllMisS,Somenot-PisM,

therefore, Somenot-PisS.

Somenot-PisSisnotindeedouroriginalconclusion,butthelattercanbeobtained from it by conversion followed by obversion. This method ofreduction may be indicated by Doksamosk (which again is obviouslypreferabletoDokamo, suggestedbyWhately,since the latterwouldmake itappearasifweimmediatelyobtainedtheoriginalconclusioninDarii.)

260.Extension of theDoctrine of Reduction.—The doctrine of reductionmaybeextended,anditcanbeshewnnotmerelythatanysyllogismmaybereducedtofigure1,butalso that itmaybereducedtoanygivenmood(notbeing a subaltern mood) of that figure.348 This position will obviously beestablished if we can shew that Barbara, Celarent, Darii, and Ferio aremutuallyreducibletooneanother.

348 Compare,further,sections284,285.

BarbaramaybereducedtoCelarentbytheobversionofthemajorpremissandalsoofthenewconclusiontherebyobtained.Thus,usingarrows,asinthe


noteonpage320,

AllMisP, → NoMisnot-P,AllSisM, → AllSisM,

↓AllSisP. ← NoSisnot-P.

Conversely,Celarent is reducible toBarbara ; and in a similar manner, byobversionofmajorpremissandconclusion,DariiandFerioarereducibletooneanother.

It will now suffice if we can shew thatBarbara andDarii aremutuallyreducibletooneanother.Clearlytheonlymethodpossiblehereistheindirectmethod.

TakeBarbara,

MaP,SaM,⎯⎯

∴ SaP ;

325for,ifnot,thenwehaveSoP ;andMaP,SaM,SoPmustbetruetogether.FromSoPbyfirstobvertingandthenconverting(anddenotingnot-PbyPʹ)wegetPʹiS,andcombiningthiswithSaMwehavethefollowingsyllogisminDarii,—

SaM,PʹiS,⎯⎯

∴ PʹiM.

PʹiM by conversion and obversion becomesMoP ; and thereforeMaP andMoPare true together;but this is impossible, since theyarecontradictories.Therefore,SoPcannotbetrue,i.e.,thetruthofSaPisestablished.

Similarly,DariimaybeindirectlyreducedtoBarbara.349

MaP, (i)SiM, (ii)


⎯⎯

∴ SiP. (iii)

Thecontradictoryof(iii)isSeP,fromwhichweobtainPaSʹ.Combiningwith(i),wehave—

PaSʹ,MaP,⎯⎯

∴ MaSʹ inBarbara.

But fromthisconclusionwemayobtainSeM,which is thecontradictoryof(ii).

349 It has beenmaintained, that this reduction is unnecessary, and that, to allintentsandpurposes,DariiisBarbara,sincethe“someS”intheminoris,andisknowntobe,thesamesomeasintheconclusion.Comparesection269.

261. Is Reduction an essential part of the Doctrine of the Syllogism?—Accordingtotheoriginaltheoryofreduction,theobjectoftheprocessistobesurethattheconclusionisavalidinferencefromthepremisses.Thevalidityofasyllogisminfigure1maybedirectlytestedbyreferencetothedictumdeomnietnullo:but thisdictumhasnodirect application to syllogisms in theremainingthreefigures.Thus,Whatelysays,“Asitisonthedictumdeomnietnullothatallreasoningultimatelydepends,soallargumentsmaybeinonewayorotherbroughtintosomeoneofthefourmoodsinthefirstfigure:andasyllogism is, in that case, said to be reduced” (Elements of Logic, p. 93).ProfessorFowlerputs thesamepositionsomewhatmoreguardedly,“Aswehaveadoptednocanonforthe2nd,3rd,and4thfigures,wehaveasyet326nopositiveproofthatthesixmoodsremainingineachofthosefiguresarevalid:wemerelyknowthat theydonotoffendagainstanyof thesyllogisticrules.Butifwecanreducethem,i.e.,bringthembacktothefirstfigure,byshewingthat they are only different statements of itsmoods, or in otherwords, thatpreciselythesameconclusionscanbeobtainedfromequivalentpremissesinthe first figure, their validity will be proved beyond question” (DeductiveLogic,p.97).

Reduction is, on the other hand, regarded by some logicians as bothunnecessaryandunnatural.Itis,inthefirstplace,saidtobeunnecessary,on


thegroundthatthedictumdeomnietnullohasnoclaimtoberegardedastheparamount law for all valid inference.350 In sections 270 to 272 it will beshewn that dicta can be formulated for the other figures, which may beregardedasmakingthemindependentofthefirst,andputtingthemonalevelwithit.Itmayalsobemaintainedthatinanymoodthevalidityofaparticularsyllogismisasself-evidentasthatofthedictumdeomnietnullo itself;andthat,therefore,althoughaxiomsofsyllogismareusefulasgeneralisationsofthesyllogisticprocess, theyareneedless inorder toestablish thevalidityofanygivensyllogism.ThisviewisindicatedbyUeberweg.

350 CompareThomson,LawsofThought,p.172.

Reductionis,inthesecondplace,saidtobeunnatural,inasmuchasitofteninvolvesthesubstitutionofanunnaturalandindirectforanaturalanddirectpredication.Figures2 and3 at any rate have their special uses, and certainreasoningsfallnaturallyintothesefiguresratherthanintothefirstfigure.351

351 CompareaquotationfromLambert(NeuesOrganon,§§230,231)givenbySirW.Hamilton(Logic,II.p.438).

The following example is givenbyThomson (LawsofThought, p. 174):“Thus,whenitwasdesirabletoshewbyanexamplethatzealandactivitydidnotalwaysproceed fromselfishmotives, thenatural coursewouldbe somesuchsyllogismasthefollowing.TheApostlessoughtnoearthlyreward,theApostleswerezealousintheirwork;therefore,327somezealouspersonsseeknotearthlyreward.”Inreducingthissyllogismtofigure1,wehavetoconvertourminorinto“SomezealouspersonswereApostles,”whichisawkwardandunnatural.

Takeagainthissyllogism,“EveryreasonablemanwishestheReformBilltopass,Idon’t,therefore,Iamnotareasonableman.”ReducedintheregularwaytoCelarent,themajorpremissbecomes,“NopersonwishingtheReformBilltopassisI,”yieldingtheconclusion,“NoreasonablemanisI.”

Further illustrations of this pointwill be found ifwe reduce to figure 1,syllogismswithsuchpremissesasthefollowing:—Allorchidshaveoppositeleaves,Thisplanthasnotoppositeleaves;Socratesispoor,Socratesiswise.

Theaboveargumentsjustifythepositionthatreductionisnotanecessary


partofthedoctrineofthesyllogism,sofarastheestablishmentofthevalidityofthedifferentmoodsisconcerned.352

352 Hamilton(Logic,I.p.433)takesacuriouspositioninregardtothedoctrineof reduction. “The last three figures,” he says, “are virtually identical with thefirst.”Thishasbeenrecognisedbylogicians,andhence“thetediousanddisgustingrules of their reduction.” But he himself goes further, and extinguishes thesefiguresaltogether,asbeingmerely“accidentalmodificationsofthefirst,”and“themutilatedexpressionsofacomplexmentalprocess.”AsomewhatsimilarpositionistakenbyKantinhisessayOntheMistakenSubtiltyoftheFourFigures.Kant’sargument isvirtuallybasedon the twofollowingpropositions: (1)Reasonings infigures2,3,4requiretobeimplicitly,ifnotexplicitly,reducedtofigure1,inorderthattheirvaliditymaybeapparent;forexample,inCesarewemusthavecovertlyperformed the conversion of the major premiss in thought, since otherwise ourpremisseswouldnotbeconclusive;(2)Noreasoningseverfallnaturallyintoanyof themoodsoffigures2,3,4,whichare, therefore,amereuseless inventionoflogicians.Ongroundsalreadyindicated,boththesepropositionsmustberegardedaserroneous.Afurthererrorseemstobeinvolvedin thefollowingpassagefromthe same essay of Kant’s: “It cannot be denied that we can draw conclusionslegitimately in all these figures. But it is incontestable that all except the firstdetermine the conclusion only by a roundabout way, and by interpolatedinferences,andthattheverysameconclusionwouldfollowfromthesamemiddleterm in the first figure by pure and unmixed reasoning.” The latter part of thisstatementcannotbejustifiedinsuchacaseasthatofBaroco.

At the same time, no treatment of the syllogism can be 328 regarded asscientific or complete until the equivalence between the moods in thedifferentfigureshasbeenshewn;andforthispurpose,aswellasforitsutilityasa logicalexercise,a full treatmentof theproblemof reductionshouldberetained.353


353 See,further,sections266,268.

262. The Fourth Figure.—Figure 4 was not as such recognised byAristotle;anditsintroductionhavingbeenattributedbyAverroëstoGalen,itisfrequentlyspokenofastheGalenianFigure.ItdoesnotusuallyappearinworksonLogicbeforethebeginningoftheeighteenthcentury,andevenbymodern logicians its use is sometimes condemned. Thus Bowen (Logic, p.192) holds that “what is called the fourth figure is only the first with aconvertedconclusion;thatiswedonotactuallyreasoninthefourth,butonlyinthefirst,andthenifoccasionrequires,converttheconclusionofthefirst.”Thisaccountoffigure4cannot,however,beaccepted,sinceitwillnotapplytoFesapoorFresison.Forexample,fromthepremissesofFesapo(NoPisMandAllMisS)noconclusionwhateverisobtainableinfigure1.354

354 ForthemostpartthecriticsofthefourthfigureseemtoidentifyitaltogetherwithBramantip. The following extract fromFatherClarke’sLogic (p. 337)willservetoillustratethecontumelytowhichthispoorfigureissometimessubjected:“Oughtwetoretainit?Ifwedo,itshouldbeasasortofsyllogisticHelot,toshewhow low the syllogism can fall when it neglects the laws on which all truereasoning is founded, and to exhibit it in themost degraded formwhich it canassume without being positively vicious. Is it capable of reformation? Not ofreformation, but of extinction……Where the same premisses in the first figurewouldproveauniversal affirmative, this feeble caricatureof it is contentwith aparticular;wherethefirstfiguredrawsitsconclusionnaturallyandinaccordancewiththeformsintowhichhumanthoughtinstinctivelyshapesitself,thispervertedabortionforcesthemindtoanawkwardandclumsyprocesswhichrightlydeservestobecalled‘inordinateandviolent.’”FatherClarke’sownviolenceappearstobeattributable mainly to the fact that figure 4 was not, as such, recognised byAristotle.

Thomson’s ground of rejection is that in the fourth figure the order ofthought is wholly inverted, the subject of the conclusion having been apredicateinthepremisses,andthepredicateasubject.“Againstthisthemindrebels; andwecanascertain that the conclusion isonly the converseof therealone,byproposingtoourselvessimilarsetsofpremisses,to329whichweshall always find ourselves supplying a conclusion so arranged that thesyllogism is in the first figure, with the second premiss first” (Laws ofThought,p.178).Asregardsthefirstpartofthisargument,Thomsonhimselfpointsout that the sameobjectionappliespartially to figures2and3. Itnodoubthelpstoexplainwhyasamatteroffactreasoningsinfigure4arenot


oftenmetwith;355butitaffordsnosufficientgroundforaltogetherrefusingtorecognisethisfigure.ThesecondpartofThomson’sargumentis,forareasonalreadystated,unsound.Theconclusion,forexample,ofFresisoncannotbe“theconverseoftherealconclusion,”since(beinganOproposition)itisnottheconverseofanyotherpropositionwhatsoever.

355 The reasons why figure 4, “with its premisses looking one way, and itsconclusionanother,”isseldomused,areelaboratedbyKarslake,AidstotheStudyofLogic,I.pp.74,5.

It isindeedimpossibletotreatthesyllogismscientificallyandcompletelywithoutadmittinginsomeformorotherthemoodsoffigure4.Inanàprioriseparationoffiguresaccordingtothepositionofthemajorandminortermsinthepremisses,thisfigurenecessarilyappears,andityieldsconclusionswhicharenotdirectlyobtainablefromthesamepremissesinanyotherfigure.Itisnot actually in frequent use, but reasoningsmay sometimes not unnaturallyfallintoit;forexample,NoneoftheApostleswereGreeks,SomeGreeksareworthyofallhonour,therefore,SomeworthyofallhonourarenotApostles.

263. IndirectMoods.—The earliest form in which the mnemonic versesappearedwasasfollows:—

Barbara,Celarent,Darii,Ferio,Baralipton,Celantes,Dabitis,Fapesmo,Frisesomorum,Cesare,Camestres,Festino,Baroco,Darapti,Felapton,Disamis,Datisi,Bocardo,Ferison.356

356 First published in theSummulaeLogicales of PetrusHispanus, afterwardsPopeJohnXXI.,whodiedin1277.ThemnemonicsoccurinanearlierunpublishedworkofWilliamShyreswood,whodiedasChancellorofLincolnin1249.

Aristotlerecognisedonlythreefigures:thefirstfigure,whichheconsideredthe type of all syllogisms and which he 330 called the perfect figure, thedictumdeomnietnullobeingdirectlyapplicabletoitalone;andthesecondandthirdfigures,whichhecalledimperfectfigures,sinceitwasnecessarytoreducethemtothefirstfigure,inordertoobtainatestoftheirvalidity.

Before the fourth figure,however,wascommonly recognisedas such, itsmoodswererecognisedinanotherform,namely,asindirectmoodsofthefirstfigure; and theabovemnemonics—Baralipton,Celantes,Dabitis,Fapesmo,


Frisesomorum—representthesemoodssoregarded.357357 Fromthe14thtothe17thcenturythemnemonicsfoundinworksonLogic

usually give the moods of the fourth figure in this form, or else omit themaltogether. Wallis (1687) recognises them in both forms, giving two sets ofmnemonics.

Theconceptionofindirectmoodsmaybebestexplainedbystartingfromadefinition of figure,which contains no reference to the distinction betweenmajor and minor terms, and which accordingly yields only three figuresinsteadoffour,namely:Figure1,inwhichthemiddletermissubjectinoneofthepremissesandpredicateintheother;Figure2,inwhichthemiddletermispredicateinbothpremisses;Figure3,inwhichthemiddletermissubjectinbothpremisses.Themoodsoffigure1maythenbedistinguishedasdirectorindirectaccordingasthepositionofthetermsintheconclusionisthesameastheirpositioninthepremissesorthereverse.358Thus,with331thepremissesMaP,SaM,wehaveadirectconclusionSaP,andanindirectconclusionPiS.These are respectively Barbara and Baralipton. Similarly, CelantescorrespondstoCelarent,andDabitistoDarii.WiththepremissesMeP,SiM,weobtainthedirectconclusionSoP,butnothingcanbeinferredofPintermsofS. There is, therefore, no indirectmood corresponding toFerio. On theother hand, Fapesmo and Frisesomorum (the Fesapo and Fresison of thefourthfigure)havenocorrespondingdirectmoods.

358 It follows that ifwecompare the conclusionof an indirectmoodwith theconclusionofthecorrespondingdirectmood(wheresuchcorrespondenceexists),weshallfindthatthetermshavechangedplaces.Mansel’sdefinitionofanindirectmoodas“oneinwhichwedonotinfertheimmediateconclusion,butitsconverse”(Aldrich,p.78)must,however,berejectedforthereasonthatitcannotbeappliedtoFapesmoandFrisesomorum,whichareindirectmoodshavingnocorrespondingvaliddirectmoodsatall.Inthesewecannotbesaidtoinfer“theconverseoftheimmediate conclusion,” for there is no immediate conclusion.Manseldealswiththese two moods very awkwardly. “Fapesmo and Frisesomorum,” he remarks,“havenegativeminorpremisses,andthusoffendagainstaspecialruleofthefirstfigure; but this is checked by a counterbalancing transgression. For by simplyconvertingO, we alter the distribution of the terms, so as to avoid an illicitprocess.” But the notion that we can counterbalance one violation of law bycommitting a second cannot be allowed. The truth of course is that, in the firstplace, the special rulesof the first figure asordinarilygivendonot apply to theindirect moods; and in the second place, the conclusionO is not obtained byconversionatall.


Clearly it is nomore thana formaldifferencewhether the fivemoods inquestion are recognised in the manner just indicated, or as constituting adistinct figure; but, on thewhole, the latter alternative seems less likely togiverisetoconfusion.

Thedistinctionbetweendirectandindirectmoodsasaboveexpressedisforobviousreasonsconfinedtothefirstfigure.Itwillbeobserved,however,thatin the traditional names of the indirectmoods of the first figure theminorpremiss precedes themajor, and if we seek to apply a distinction betweendirectand indirectmoods in thecaseof the secondand third figures, it canonlybewithreferencetotheconventionalorderofthepremisses.Thus,inthesecond figure, taking the premissesPeM,SaM,wemay infer eitherSeP orPeS, and if we call a syllogism direct or indirect according as the majorpremiss precedes the minor, or vice versâ, thenPeM,SaM, SeP will be adirect mood, and PeM, SaM, PeS an indirect mood. The former of thesesyllogisms is Cesare, and the latter is Camestres with the premissestransposed.359 Hence the latter will immediately become a direct mood bymerely changing the order of the premisses; and the artificiality of thedistinctionisatonceapparent.Theresultwillbefoundtobesimilarinothercases,andthedistinctionmay,therefore,berejectedsofarasfigures2and3areconcerned.

359 Take,again,thepremissesMaP,MoS.Herethereisnodirectconclusion,butonly an indirect conclusion PoS. This, however, is merely Bocardo with thepremissestransposed.

264. Further discussion of the process of Indirect Reduction.—Thediscussionoftheproblemofreductionintheprecedingpageshasinthemainfollowedthetraditionallines.It332is,however,desirabletotreattheprocessofindirectreductioninarathermoreindependentandsystematicmanner.Bydoingso,weshallfindthattheprocessenablesustoexhibitveryclearlyandsymmetrically the relations between the first three figures, and also thedistinctivefunctionsofthesefigures.

Theargumentonwhichindirectreductionisbasedisoneofwhichwehaveseveral timesmade use (e.g., in the proof of the second corollary adoptedfromDeMorgan in section 200, and in certain of the proofs contained insection202),namely,thatifXandYtogetherproveZ,thenXandthedenial


ofZmustprovethedenialofY,andviceversâ.

The process may conveniently be exhibited as the contraposition of ahypothetical.Thus, fromthepropositionXbeinggiven, ifY thenZwemayinferbycontrapositionXbeinggiven,ifnotZthennotY ;andwecanequallypassbackfromthecontrapositivetotheoriginalproposition.

Sincethecontradictoryoftheconclusionofasyllogismmaybecombinedwith either of the original premisses, it follows that every valid syllogismcarrieswithitthevalidityoftwoothersyllogisms.Henceallvalidsyllogismsmust be capable of being arranged in sets of three which are mutuallyequivalent.

The three equivalent syllogisms may be symmetrically expressed asfollows(wherePandPʹ,QandQʹ,RandRʹarerespectivelycontradictories):

(i)premisses,PandQ ;conclusionRʹ ;(ii)premisses,QandR ;conclusionPʹ ;(iii)premisses,RandP ;conclusionQʹ.

Itmustbeunderstoodthattheorderofthepremissesinthesesyllogismsisnotintendedtoindicatewhichismajorandwhichminor.

265.TheAntilogism.—Eachof the three equivalent syllogisms just giveninvolvesfurther theformal incompatibilityof the threepropositionsP,Q,R(compare section 214). Three propositions, containing three and only threeterms,whicharethusformallyincompatiblewithoneanother,constitutewhathas been called by Mrs Ladd Franklin an antilogism.360 Thus, 333 thesyllogism, “MaP, SaM, therefore, SaP,” has for its equivalent antilogism,“MaP,SaM,SoP are threepropositions that are formally incompatiblewithoneanother.”

360 SeeBaldwin’sDictionaryofPhilosophy,art.SymbolicLogic.Itisshewninthis article that the whole of syllogistic reasoning may be summed up in thefollowingantilogism,thesymbolismofsection138beingmadeuseof,—

[(AB=0)(bC=0)(AC>0)]=0.

The fifteen moods containing neither a strengthened premiss nor a weakenedconclusionmay,by theaidofconversionsandobversions,beobtained from thisantilogismaccordingasthecontradictoryofoneorotherofthethreeincompatiblesistakenastheconclusion.


266. Equivalence of the Moods of the first three Figures shewn by theMethodofIndirectReduction.—Ifoneofourthreeequivalentsyllogismsisinoneofthefirstthreefigures,thenitcanbeshewnthatthetwootherswillbeintheremainingtwoofthesefigures.

Thus,letP,Q,∴Rʹbeinfigure1,theminorpremissbeingstatedfirst.Itmaythenbewritten

S ⎯ M,M ⎯ P,∴(S ⎯ P)ʹ. (1)

Thesecondsyllogismbecomes

M ⎯ P,S ⎯ P,∴(S ⎯ M)ʹ; (2)

andthethirdis

S ⎯ P,S ⎯ M,∴(M ⎯ P)ʹ. (3)

Itwillbeseenthat(2)isinfigure2,and(3)infigure3.

Next,letP,Q,∴Rʹbeinfigure2,themajorpremissbeingstatedfirst.Wethenhaveforourthreesyllogisms,—

P ⎯ M,S ⎯ M,∴(S ⎯ P)ʹ; (1)S ⎯ M,S ⎯ P,∴(P ⎯ M)ʹ; (2)S ⎯ P,P ⎯ M,∴(S ⎯ M)ʹ. (3)

Here(2)isinfigure3,(3)infigure1.

Finally,letP,Q,∴Rʹbeinfigure3,themajorpremissbeingstatedfirst.Wehave

M ⎯ P,M ⎯ S,∴(S ⎯ P)ʹ; (1)M ⎯ S,S ⎯ P,∴(M ⎯ P)ʹ; (2)S ⎯ P,M ⎯ P,∴(M ⎯ S)ʹ. (3)

Here(2)isinfigure1,(3)infigure2.

Hencewe see that, startingwith a syllogism in anyoneof the first threefigures(theminorpremissprecedingthemajorinfigure1,butfollowingitinfigures2and3),andtakingthe334propositionsintheabovecyclicorder,thenthefigureswillalwaysrecurinthecyclicorder1,2,3.361

361 If wewere to start with a syllogism in figure 1, themajor premiss being


statedfirst,thenthecyclicorderoffigureswouldbe1,3,2,andinfigures2and3theminorpremisswouldprecedethemajor.

Itfollowsthat(aswealreadyknowtobethecase)theremustbeanequalnumberofvalidsyllogismsineachofthefirstthreefigures,andthattheymaybearrangedinsetsofequivalenttrios.Theseequivalenttrioswillbefoundtobe as follows (sets containing strengthened premisses or weakenedconclusionsbeingenclosedinsquarebrackets);

Barbara,Baroco,Bocardo;

[AAI,AEO,Felapton;]

Celarent,Festino,Disamis;

[EAO,EAO,Darapti;]

Darii,Camestres,Ferison;

Ferio,Cesare,Datisi.

The corresponding antilogisms are AAO, [AAE,] EAI, [EAA,] AIE,EIA.362

362 Thepositionofthetermsintheseantilogismscorrespondstothatoffigure1,themajorpremissbeingstatedfirst.

267.TheMoods of Figure 4 in their relation to one another.—We haveseen that in the equivalent trios of syllogisms yielded by the process ofindirectreductionweneverhaveinanyonetriomorethanonesyllogisminfigure1,orinfigure2,orinfigure3.Figure4is,however,self-containedinthe sense that if we start with a syllogism in this figure, both the othersyllogismswill be in the same figure. Proceeding as in the last section,wemayshewthisasfollows,themajorpremissbeingstatedfirst:363

P ⎯ M,M ⎯ S,∴(S ⎯ P)ʹ; (1)M ⎯ S,S ⎯ P,∴(P ⎯ M)ʹ; (2)S ⎯ P,P ⎯ M,∴(M ⎯ S)ʹ. (3)

363 Itwillbefoundthatitcomestojustthesamethingiftheminorpremississtatedfirst.

It follows that in figure 4 the number of valid syllogismsmust be somemultipleofthree.Thenumberis,asweknow,six.Thereare, therefore, two


equivalenttrios;andtheywillbefoundtobeasfollows:335

[Bramantip,AEO,Fesapo;]Camenes,Fresison,Dimaris.

The equivalent antilogisms are [AAE,]AEI. Comparing this result withthatobtainedintheprecedingsection,weseethattheonlyvalidantilogisticcombinationsareAAOandAEI,withtheadditionofAAE(inwhichoneofthethreepropositionsisunnecessarilystrengthened).364

364 Thisresultmightbeinferredfromtherulesgiveninsection214.

268.EquivalenceoftheSpecialRulesoftheFirstThreeFigures.—Letthefollowingbeavalidsyllogisminfigure1,—

(minor) S ⎯ M, (1)(major) M ⎯ P, (2)

(conclusion) ∴ (S ⎯ P)ʹ. (3)

Thenthecorrespondingvalidsyllogisminfigure2willbe

(major) M ⎯ P, (2)(minor) S ⎯ P, contradictoryof(3)

(conclusion) ∴ (S ⎯ M)ʹ; contradictoryof(1)

andthecorrespondingvalidsyllogisminfigure3willbe

(major) S ⎯ P, contradictoryof(3)(minor) S ⎯ M, (1)

(conclusion) ∴ (M ⎯ P)ʹ. contradictoryof(2)

Thespecialrulesoffigure1are

minor affirmative,major universal,

thatis,(1)mustbeaffirmative,(2)mustbeuniversal.

Infigure2,(2)isthemajor,andthecontradictoryof(1)istheconclusion.Therefore,infigure2wemusthavetherules,—

major universal,conclusion negative[andhenceonepremissnegative].


Infigure3,(1)istheminor,andthecontradictoryof(2)istheconclusion.Therefore,infigure3wemusthavetherules,—

minor affirmative,conclusion particular.

Thusthespecialrulesoffigures2and3areshewntobededuciblefromthespecialrulesoffigure1.Wemightequally336wellstartfromthespecialrulesoffigure2oroffigure3anddeducetherulesofthetwootherfigures.365

365 Thecompleterulesfortheantilogismsofthefirstthreefigures,asgivenattheendofsection266,are (a) firstpropositionuniversal, (b) second propositionaffirmative,(c) third propositionopposite in quality to the first, and (unless it isstrengthened) opposite in quantity to the second. These rules replace all generalrules.

269.Schemeof theValidMoodsofFigure l.—Sofaras thenatureof thereasoning involved is concerned, there is practically no distinction betweenBarbaraandDarii,orbetweenCelarentandFerio.For ineachcase, ifS istheminorterm,theS’sreferredtointheconclusionarepreciselythesameS’sasthosereferredtointheminorpremiss.

Again, the only difference between Barbara and Celarent, or betweenDariiandFerio,isthattheuniversalrulewhichtheminorpremissenablesustoapplytoaparticularcaseisinBarbaraandDariiauniversalaffirmation,whileinCelarentandFerioitisauniversaldenial.

Wemay,therefore,sumupallfourmoodsinthefollowingscheme:366

AllBisC(orisnotC), (Rule)All(orsome)AisB, (Case)

therefore, All(orsome)AisC(orisnotC). (Result)366 CompareC. S. Peirce in the JohnsHopkins Studies in Logic, p. 148, and

Sigwart,Logic,i.p.354.Sigwartgivesthefollowingformula:

IfanythingisMitisP(orisnotP),CertainsubjectsSareM,

therefore, TheyareP(orarenotP).

Thiswayofsettingoutthevalidmoodsoffigure1shewsclearlyhowtheyareallincludedunderthedictumdeomnietnullo.

270.Schemeof theValidMoodsofFigure 2.—Applying the principle of


indirectreduction,wemayimmediatelyobtainfromtheschemegiveninthelastprecedingsectionthefollowingscheme,summingupthevalidmoodsoffigure2:367337

AllBisC(orisnotC), (Rule)Some(orall)AisnotC(orisC), (DenialofResult)

therefore, Some(orall)AisnotB. (DenialofCase)367 Sigwart’swayofputting it (Logic, i. p.354) is that in figure2, insteadof

inferringfromgroundtoconsequence,weinferfrominvalidityofconsequencetoinvalidityofground;andhegivesthefollowingscheme:

IfanythingisPitisM(orisnotM),CertainsubjectsSarenotM(orareM),

therefore, TheyarenotP.

This scheme may be expressed in the following dictum,—“If a certainattributecanbepredicated,affirmativelyornegatively,ofeverymemberofaclass,anysubjectofwhichitcannotbesopredicateddoesnotbelongtotheclass.”368 This dictum may, like the dictum de omni et nullo, claim to beaxiomatic, and it is related to the valid syllogisms of figure 2 just as thedictumdeomnietnulloisrelatedtothevalidsyllogismsoffigure1.369

368 Thedictumforfigure2,sometimescalledthedictumdediverso,isexpressedintheaboveformbyMansel(Aldrich,p.86).ItwasgivenbyLambertintheform,“If one term is contained in, and another excluded from, a third term, they aremutually excluded.” This is at least expressed loosely, since it would appear towarrant a universal conclusion, if any conclusion at all, inFestino andBaroco.Bailey(TheoryofReasoning,p.71)givesthefollowingpairofmaximsforfigure2,—“When the whole of a class possess a certain attribute, whatever does notpossess the attribute does not belong to the class.When thewhole of a class isexcludedfromthepossessionofanattribute,whateverpossessestheattributedoesnotbelongtotheclass.”

369 Lambertisusuallyregardedastheoriginatoroftheideaofframingdictathatshall be directly applicable to figures other than the first. Thomson, however,pointsout that it isanerror tosuppose thatLambertwas the first to inventsuchdicta.“Morethanacenturyearlier,Keckermannsawthateachfigurehaditsownlawanditsownpeculiaruse,andstatedthemasaccurately,iflessconcisely,thanLambert”(LawsofThought,p.173,note).Distinctprinciples for thesecondandthirdfiguresare laiddownalsoin thePortRoyalLogic,whichwaspublished in1662.

271.SchemeoftheValidMoodsofFigure3.—Dealingwithfigure3inthesame way as we have done with figure 2, we get the following scheme,


summingupthevalidmoodsofthatfigure:

Some(orall)AisnotC(orisC), (DenialofResult)All(orsome)AisB, (Case)

therefore, SomeBisnotC(orisC). (DenialofRule)

It is not easy to express this scheme in a single self-evident maxim.370Separatedictaofanaxiomaticcharactermay,338however,beformulatedfortheaffirmativeandnegativemoodsrespectivelyoffigure3,namely,“Iftwoattributescanbothbeaffirmedofaclass,andoneatleastofthemuniversallyso, then these two attributes sometimes accompany each other,” “If oneattribute can be affirmed while another is denied of a class, either theaffirmation or the denial being universal, then the former attribute is notalwaysaccompaniedbythelatter.”371

370 Lambert gave the followingdictumdeexemplo for figure3:—“Two termswhichcontainacommonpartpartlyagree,orifonecontainsapartwhichtheotherdoesnot,theypartlydiffer.”Thismaximisopentoexception.Theproposition“Ifone term contains a part which another does not, they partly differ” applied toMeP,MaS,wouldappear to justifyPoS justasmuchasSoP, or else toyield analternative between these two.Mr Johnson gives a single formula for figure 3,namely,“Astatementmaybeappliedtopartofaclass,ifitapplieswholly[oratleastpartly]toasetofobjectsthatareatleastpartly[orwholly]includedinthatclass.”Thisiscorrect,butperhapsnotveryeasytograsp.

371 These dicta (or dicta corresponding to them) are sometimes calledrespectivelythedictumdeexemploandthedictumdeexcepto.

272.Dictum for Figure 4.—The following dictum, called the dictum dereciproco,wasformulatedbyLambertforfigure4:—“IfnoM isB,noB isthisorthatM ;ifCis(orisnot)thisorthatB,thereareB’swhichare(orarenot)C.”ThefirstpartofthisdictumisintendedtoapplytoCamenes,andthesecondpart to theremainingmoodsof thefourthfigure;but theapplicationcan hardly in either case be regarded as self-evident. Several other axiomshavebeenconstructedforfigure4;buttheyare,asarule,littlemorethanabareenumerationof thevalidmoodsof that figure,whilst at the same timethey are less self-evident than these moods considered individually. Thefollowing axiom, however, suggested byMr Johnson, is not open to thesecriticisms: “Three classes cannot be so related, that the first is whollyincluded in the second, the secondwholly excluded from the third, and the


thirdpartlyorwhollyincludedinthefirst.”Thisdictumaffirmsthevalidityoftwoantilogisms;inotherwords,itdeclaresthemutualincompatibilityofeachof thefollowingtriosofpropositions:XaY,YeZ,ZiX ;XaY,YeZ,ZaX ; and itwillbefoundthattheseincompatiblesyieldthesixvalidmoodsofthefourthfigure.372


339

EXERCISES.

273.ReduceBarbaratoBocardo,BocardotoBaroco,BarocotoBarbara.[K.]

274.ReduceFeriotofigure2,Festinotofigure3,Felaptontofigure4.[K.]

275.ReduceCamestrestoDatisi.WhycannotCamestresbereducedeitherdirectlyor indirectly toFelapton?CanFelaptonbereducedtoCamestres?[K.]

276.Assumingthatinthefirstfigurethemajormustbeuniversalandtheminoraffirmative,shewbyreductio ad absurdum that the conclusion in the second figure must be negative and in the thirdparticular.[J.]

277.Statethefollowingargumentinasyllogismofthethirdfigure,andreduceit,bothdirectlyandindirectly, tothefirst:—Somethingsworthyofbeingknownarenotdirectlyuseful,foreverytruthisworthyofbeingknown,whilenoteverytruthisdirectlyuseful.[M.]

278.Statethefigureandmoodofthefollowingsyllogism;reduceittothefirstfigure;andexaminewhetherthereisanythingunnaturalintheargumentasitstands:— Nonewhodishonourthekingcanbetruepatriots;foratruepatriotmustrespectthelaw,andnonewhorespectthelawwoulddishonourtheking.[J.]

279.“Rejecting thefourthfigureandthesubalternmoods,wemaysaywithAristotle:A isprovedonlyinonefigureandonemood,Eintwofiguresandthreemoods,Iintwofiguresandfourmoods,Oin three figures and six moods. For this reason,A is declared by Aristotle to be the most difficultpropositiontoestablish,andtheeasiesttooverthrow;O,thereverse.”Discussthefitnessofthesedatatoestablishtheconclusion.[K.]

280.Prove,fromthegeneralrulesofthesyllogism,thatthenumberofpossiblemoods,irrespectiveofdifferenceoffigure,is11. Inthe19moodsofthemnemonicverses,only10outofthepossible11moodsarerepresented.Findthemissingmood,andaccountforitsabsencefromtheverses.[L.]

281.Given (1)theconclusionofasyllogisminthefirstfigure, (2)theminorpremissofasyllogisminthesecondfigure, (3)themajorpremissofasyllogisminthethirdfigure,340 examine in each case how far thequality andquantity of the two remainingpropositionsof the


syllogismcanbedetermined(itbeinggiventhatthesyllogismdoesnotcontainastrengthenedpremissoraweakenedconclusion). Expresstheresult,asfaraspossible,ingeneraltermsineachfigure.[J.]

282. Find out in which of the valid syllogistic moods the combination of one premiss with thesubcontraryoftheconclusionwouldestablishthesubcontraryoftheotherpremiss.[L.]

283.Constructasyllogisminaccordancewitheachofthefollowingtwodicta:— (1)Anyobject that is foundto lackapropertyknowntobelong toallmembersofaclassmustbeexcludedfromthatclass; (2) Ifanyobjects thathavebeen included inaclassare found to lackacertainproperty, then thatpropertycannotbepredicatedofallmembersoftheclass. Assignthemoodandfigureofeachargument,andshewtherelationsbetweentheabovedictaandthedictumdeomnietnullo.[L.]

284.Shew that anygivenmoodmaybedirectly reduced to anyothermood,provided (1) that thelattercontainsneitherastrengthenedpremissnoraweakenedconclusion,and(2)thatiftheconclusionoftheformerisuniversal,theconclusionofthelatterisalsouniversal[K.]

285.Shewthatanygivenmoodmaybedirectlyor indirectlyreducedtoanyothermood,providedthatthelatterhasnoteitherastrengthenedpremissoraweakenedconclusion,unlessthesameistrueoftheformeralso.[K.]

286.ExaminethefollowingstatementofDeMorgan’s:—“Therearebutsixdistinctsyllogisms.Allothers aremade from them by strengthening one of the premisses, or converting one or both of thepremisses, where such conversion is allowable; or else by first making the conversion, and thenstrengtheningoneofthepremisses.”[K.]

287.Shew,bytheaidoftheprocessofindirectreduction,thatthespecialrulesforFigure4giveninsection244aremutuallydeduciblefromoneanother.[RR.]


CHAPTERIV.

THEDIAGRAMMATICREPRESENTATIONOFSYLLOGISMS.

288.TheapplicationoftheEuleriandiagramstosyllogisticreasonings.—InshewingtheapplicationoftheEuleriandiagramstosyllogisticreasoningswemaybeginwithasyllogisminBarbara:

AllMisP,AllSisM,

therefore, AllSisP.

The premisses must first be represented separately by means of thediagrams.Eachyieldstwocases;thus,—

Toobtain the conclusion, eachof thecasesyieldedby themajorpremissmustnowbecombinedwitheachof thoseyieldedby theminor.Thisgivesfourcombinations,373andwhateveristrueofSintermsofPinallofthemistheconclusionrequired.


373 These combinations afford a complete solution of the problem as towhatclass-relations between S, M, and P are compatible with the premisses; andsimilarlyinothercases.ThesyllogisticconclusionisobtainedbytheeliminationofM.

342

IneachcaseSeithercoincideswithPorisincludedwithinP ;henceallSisPmaybeinferredfromthegivenpremisses.

Next,takeasyllogisminBocardo.Theapplicationofthediagramsisnowmorecomplicated.Thepremissesare

SomeMisnotP,AllMisS.

Themajorpremissyieldsthreecases,namely,


andtheminorpremisstwocases,namely,

343 Taking them together we have six combinations, some of whichthemselvesyieldmorethanonecase:—


344SofarasSandPareconcerned(Mbeingleftoutofaccount)theseninecasesarereducibletothefollowingthree:


Theconclusion,therefore,isSomeSisnotP.

Itmustbeadmittedthatthisisverycomplex,andthatitwouldbeaseriousmatterifinthefirstinstancewehadtoworkthroughallthedifferentmoodsinthismanner.374Still,forpurposesofillustration,thisverycomplexityhasacertain advantage. It shews how many relations between three terms inrespectofextensionarelefttous,evenwithtwopremissesgiven.

374 Ueberweg,however,takesthetroubletoestablishinthiswaythevalidityofthevalidmoodsinthevariousfigures.Thomson(LawsofThought,pp.189,190)introduces comparative simplicity by the use of dotted lines. His diagrams are,however,incorrect.

289. The application of Lambert’s diagrammatic scheme to syllogisticreasonings.—As applied to syllogisms, Lambert’s lines are much lesscumbrousthanEuler’scircles.Themainpointtonoticeisthatitisingeneralnecessarythatthelinestandingforthemiddletermshouldnotbedottedoverany part of its extent.375 This condition can be satisfied by selecting theappropriatealternativeforminthecaseofA,I,andOpropositions,asgiveninsection127.AsexampleswemayrepresentBarbara,Baroco,Datisi,andFresisonbyLambert’smethod.

375 ThefollowingrepresentationofBarbara,

illustrates the kind of error that is likely to result if the above precaution isneglected. If this representationwere correct we should be justified in inferringSomePisnotSaswellasAllSisP.

345


290. The application of Dr Venn’s diagrammatic scheme to syllogisticreasonings.—SyllogismsinBarbara,Camestres,Datisi,andBocardomaybetaken in order to shew how Dr Venn’s diagrams can be used to illustratesyllogisticreasonings.

ThepremissesofBarbara,

AllMisP,AllSisM,

excludecertaincompartmentsasshewninthefollowingdiagram:


ThisyieldsatoncetheconclusionAllSisP.

346SimilarlyforCamestreswehavethefollowing:

ForDatisiwehave

Bocardoyields


It will be remembered that this scheme is based upon a particularinterpretationofpropositionsasregards theirexistential import.Thestudentwill find it useful to attempt to represent by Dr Venn’s diagrams a moodcontainingastrengthenedpremiss,forexample,Darapti.

347

EXERCISES.

291.RepresentCelarentbytheaidofEuler’sdiagrams.Willthesamesetofdiagramsserveforanyotherofthesyllogisticmoods?[K.]

292.RepresentbymeansoftheEuleriandiagramsthemoodsFestino,Datisi,andBramantip.[K.]

293.Determine(i)bytheaidofEuler’sdiagrams,(ii)byordinarysyllogisticmethods,whatisallthatcanbeinferredaboutSandPintermsofoneanotherfromthefollowingpremisses,SomeMisP,SomeMisnotP,SomePisnotM,SomeSisnotM,AllMisS.[K.]

294.RepresentinLambert’sschemethemoodsDarii,Cesare,Darapti,Bocardo,Fesapo.[K.]

295.RepresentinDrVenn’sdiagrammaticschemethemoodsFerio,Cesare,Baroco,Dimaris.[K.]

296.Shew(i)bymeansofEuler’sdiagrams,(ii)bymeansofDrVenn’sdiagrams,thatIEyieldsnoconclusioninanyfigure.[K.]

297. Shew diagrammatically that no conclusion can be obtained from IA in figure 1, fromAA infigure2,fromAEinfigure3,fromAOinfigure4.[K.]

298.Determine,bytheaidofEuler’sdiagrammaticscheme,alltherelationsthatareàprioripossiblebetweenthreeclassesS,M,P.[K.]

299.Testthefollowingargument(i)byDrVenn’sdiagrammaticscheme,(ii)byordinarysyllogisticmethods: “Allbravepersonsarewell-disciplined;nopatriotsaremercenary;butsomemercenarypersonshavebeen found to be brave, and not all patriots can be consideredwell-disciplined; it follows that somebraveandwell-disciplinedpersonshavebeenbothmercenaryandunpatriotic,whileothers thathave


beenpatrioticandunmercenarywerebutill-disciplinedcowards.”[C.]

300.GivenAllXisYorZ,AllYisZorX,AllZisXorY,AllYZisX,AllZXisY,AllXYisZ,prove(a) by the aid ofDrVenn’s diagrammatic scheme, (b)without the aid of diagrams, thatX,Y,Z arecoextensive.[RR.]


CHAPTERV.

CONDITIONALANDHYPOTHETICALSYLLOGISMS.

301. The Conditional Syllogism, the Hypothetical Syllogism, and theHypothetico-Categorical Syllogism.—The forms of reasoning in whichconditionalorhypotheticalconclusionsare inferred from twoconditionalortwohypotheticalpremisses are apparentlyoverlookedby some logicians; atany rate, they frequently receive no distinct recognition, the term“hypotheticalsyllogism”beinglimitedtothecaseinwhichonepremissonlyishypothetical.

Thefollowingdefinitionsmaybegiven: (1) A conditional syllogism is a reasoning consisting of two conditionalpremissesandaconditionalconclusion;376

e.g., IfanyAisC,itisD,IfanyAisB,itisC,

therefore, IfanyAisB,itisD.

(2) A hypothetical syllogism (or, more distinctively, a pure hypotheticalsyllogism) is a reasoning consisting of two hypothetical premisses and ahypotheticalconclusion;376

e.g.,— IfQistrue,Ristrue,IfPistrue,Qistrue,

therefore, IfPistrue,Ristrue.

(3) A hypothetico-categorical syllogism (or, as it may also be called, amixedhypotheticalsyllogism) isareasoningconsistingof threepropositionsin which one of the premisses is 349 hypothetical in form, while the otherpremissandtheconclusionarecategorical;377

e.g.,— IfPistrue,Qistrue,Pistrue,


therefore, Qistrue.376 To be quite exact, the condition should be added that the premisses and

conclusioncontainbetweenthemthreeandonlythreeelements(correspondingtothetermsofthecategoricalsyllogism).

377 It seemsunnecessary todiscuss separately thecase inwhicha conditionalpremiss and a categorical premiss are combined: e.g., All selfish people areunhappy;Ifachildisspoilt,heissuretobeselfish;therefore,Ifachildisspoilthewillbeunhappy.Suchasyllogismasthisisresolvableintoanordinarycategoricalsyllogismbyreducingtheconditionalpremisstothecategoricalform,“Allspoiltchildren are selfish”; or it may be resolved into a conditional syllogism bytransformingthecategoricalpremissintothecorrespondingconditional,“Ifanyoneisselfish,heissuretobeunhappy.”Thefollowingisanotherexample:Ifwaterissaltitwillnotboilat212°;Seawaterissalt;therefore,Seawaterwillnotboilat212°.CompareMrF.B.TarbellinMind,1883,p.578.Thehypothetico-categoricalsyllogismasabovedefinedcannotbesosummarilydisposedof.

This nomenclature, so far as concerns the distinction between thehypothetical and the hypothetico-categorical syllogism, is adopted bySpalding and Ueberweg. Sigwart uses the terms “pure hypotheticalsyllogism,” and “mixed hypothetical syllogism.” Some logicians (e.g.,Fowler) give the name “hypothetical syllogism” to all the above forms ofreasoning without distinction. Others (e.g., Jevons) define the hypotheticalsyllogismsoastoincludethelastformonly,theothersnotbeingrecognisedasdistinctformsofreasoningatall.Thisviewmaybetosomeextentjustifiedby the very close analogy that exists between the syllogism with twoconditionalor twohypotheticalpremissesandthecategoricalsyllogism:butthedifferenceinformisworthatleastabriefdiscussion.

302. Distinctions of Mood and Figure in the case of Conditional andHypothetical Syllogisms.—In the conditional, and in the hypothetical,syllogism,theantecedentoftheconclusionisequivalenttotheminortermofthecategoricalsyllogism,theconsequentoftheconclusiontothemajorterm,andtheelementwhichdoesnotappearintheconclusionatalltothemiddleterm. Distinctions of mood and figure may be recognised in precisely thesameway as in the caseof the categorical syllogism.Thus, the conditionalsyllogism given in the preceding section is inBarbara. The following areexamplesofothermoods:350

Festino,— NeverwhenEisF,isitthecasethatCisD,


SometimeswhenAisB,CisD,therefore, SometimeswhenAisB,itisnotthecasethatEisF.Darapti,— WheneverCisB,EisF,

WheneverCisD,AisB,therefore, SometimeswhenAisB,EisF.Camenes,— WheneverEisF,CisD,

NeverwhenCisD,isitthecasethatAisB,therefore, NeverwhenAisB,isitthecasethatEisF.

In these threeexamples the form inwhich thepropositionsareexpressedsuggests an assertoric interpretation. On the modal interpretation, either ofconditionalsorofhypotheticals,theproblematicpropositionmayberegardedas taking the place of the particular, and we shall then again have all theordinary distinctions of mood and figure. We may illustrate fromhypotheticals:

Darii,— IfQistrue,Ristrue,IfPistrue,Qmaybetrue,

therefore, IfPistrue,Rmaybetrue.Baroco,— IfRistrue,Qistrue,

IfPistrue,Qmaybefalse,therefore, IfPistrue,Rmaybefalse.Disamis,— IfQistrue,Rmaybetrue,

IfQistrue,Pistrue,therefore, IfPistrue,Rmaybetrue.378Camenes,— IfRistrue,Qistrue,

IfQistrue,Pisnottrue,therefore, IfPistrue,Risnottrue.

378 The reader may possibly hesitate to admit the validity of this reasoning,althoughhefeelsnodifficulty inregardto thevalidityofanordinarycategoricalsyllogism inDisamis. This apparent anomaly is connected with the problem ofexistential import. It will be shewn in section 342 that the validity ofDisamisdepends on our interpretation of propositions as regards their existential import,andwemayperhapsnotregardcategoricalsandhypotheticalsasanalogousinthisrespect.

303.Fallacies inHypothetical Syllogisms.—On themistaken suppositionthatapurehypotheticalpropositionisequivalenttoacategoricalproposition


in which both the subject 351 and the predicate are singular terms, andtherefore ipso facto distributed, it has been argued that the syllogistic rulesrelating to the distribution of terms have no application to hypotheticalsyllogisms;andthattheonlyruleswhichneedbeconsideredintestingsuchsyllogisms are those relating to quality, namely, the rule forbidding twonegative premisses, and the rule insisting that a negative premiss and anegativeconclusionmustalwaysbefoundtogether.Butitisclearlyanerrorto regard the consequent of a hypothetical proposition as equivalent to asingular term occurring as the predicate of a categorical proposition. Anaffirmative hypothetical is not simply convertible, and in respect ofdistribution, its consequent practically corresponds to the undistributedpredicateofanaffirmativecategoricalinwhichthetermsaregeneral.Ontheotherhand,anegativehypotheticalissimplyconvertible;anditsconsequentcorresponds to the distributed predicate of a negative categorical.We mayaccordingly have fallacies in hypothetical syllogisms corresponding to (1)undistributed middle, (2) illicit major, (3) illicit minor. The following areexamplesofthesefallaciesrespectively:— (1) IfRthenQ,IfPthenQ,therefore,IfPthenR ; (2) IfQthenR,IfPthennotQ,therefore,IfPthennotR ; (3) IfQthenR,IfQthenP,therefore,IfPthenR.

304. The Reduction of Conditional and Hypothetical Syllogisms.—Conditionalandhypotheticalsyllogismsinfigures2,3,and4maybereducedtofigure1justasinthecaseofcategoricalsyllogisms.Thustheconditionalsyllogism inCamenes given in section 302 may be reduced as follows toCelarent:

NeverwhenCisD,isitthecasethatAisB,WheneverEisF,CisD,

therefore, NeverwhenEisF,isitthecasethatAisB,therefore, NeverwhenAisB,isitthecasethatEisF.

Accordingtotheordinaryruleasindicatedinthemnemonic,thepremisseshave here been transposed, and the conclusion of the new syllogism isconvertedinordertoobtaintheoriginalconclusion.

352 Similarly the hypothetical syllogism in Baroco given in section 302


maybereducedasfollowstoFerio:

IfQisfalse,Risfalse,IfPistrue,Qmaybefalse,

therefore, IfPistrue,Rmaybefalse.

305. The Moods of the Mixed Hypothetical Syllogism.—It is usual todistinguishtwomoodsofthemixedhypotheticalsyllogism,themodusponensandthemodustollens.379

379 Ueberweg remarks that it would bemore accurate to speak of themodusponens as themodus ponendo ponens, and of themodus tollens as the modustollendotollens(Logic,p.452).

(1) In the modus ponens (also called the constructive hypotheticalsyllogism)thecategoricalpremissaffirmstheantecedentof thehypotheticalpremiss,therebyjustifyingasaconclusiontheaffirmationofitsconsequent.Forexample,

IfPistruethenQistrue,Pistrue,

therefore, Qistrue.

(2)Inthemodustollens(alsocalledthedestructivehypotheticalsyllogism)the categorical premiss denies the consequent of the hypothetical premiss,therebyjustifyingasaconclusionthedenialofitsantecedent.Forexample,

IfPistruethenQistrue,Qisnottrue,

therefore, Pisnottrue.

Thesemoodsfallintolinerespectivelywiththefirstandsecondfiguresofthe categorical syllogism. For we have seen that in figure 1 we pass fromgroundtoconsequence,andinfigure2fromdenialofconsequencetodenialof ground.380 It has, however, been shewn in section 266 that to everysyllogisminfigure1therecorrespondsnotonlyasyllogisminfigure2,butalsoasyllogisminfigure3;andthequestionmaythereforebeaskedwhatthemixedhypotheticalsyllogism353yields thatwill fall into linewithfigure3.Theanswer is that, taking theplaceof figure3,wehavea reasoningwhichconsists in disproving a connexion of ground and consequence by shewing


thatthesupposedgroundholdstruebutnotthesupposedconsequence.Thismaybeillustratedbywritingdownthetwootherreasoningscorrespondingtotheordinarymodusponens.Wehave

(1) IfP,Q ; (a)butP ; (b)∴Q. (c)

(2) IfP,Q ; (a)butnotQ ; contradictoryof(c)∴notP. contradictoryof(b)

(3) P ; (b)butnotQ ; contradictoryof(c)

∴QisnotanecessaryconsequenceofP. contradictoryof(a)380 Themixed hypothetical syllogismmay be reduced to the form of a pure

hypotheticalsyllogismbywritingthecategoricalPistrueintheformIfanythingistrue,Pis true. If this isdone, itwillbeseenfromanotherpointofviewthat themodusponensmayberegardedasbelongingtofigure1andthemodustollens tofigure2.

If(1)isconsideredtobeinfigure1,then(2)isinfigure2,and(3)infigure3. It is true that (3) departs toomuch from the ordinary type of themixedhypotheticalsyllogismtojustifyusincallingitbythatname.Butitisaformofreasoningthatmaywellreceivedefiniterecognition.

306.FallaciesinMixedHypotheticalSyllogisms.—Therearetwoprincipalfallacies that may be committed in arguing from a hypothetical majorpremiss: (1)It isafallacy toregard theaffirmationof theconsequentas justifyingtheaffirmationoftheantecedent.Forexample,

IfPistruethenQistrue,Qistrue,

therefore, Pistrue.

(2) It is a fallacy to regard the denial of the antecedent as justifying thedenialoftheconsequent.Forexample,

IfPistruethenQistrue,Pisnottrue,


therefore, Qisnottrue.

These fallacies may be regarded as corresponding respectively toundistributedmiddleandillicitmajorinthecaseofcategoricalsyllogisms.381

381 Given“IfPandonlyifPthenQ,”thenwemayofcoursearguefromQtoPor fromnot-P tonot-Q; and no doubt in the case of ordinary hypotheticals it isoften tacitlyunderstood that theconsequent is trueonly if theantecedent is true.This must, however, be expressly stated if the argument based upon it is to beformallyvalid.

354Theresults reached in thisand theprecedingsectionmaybesummedup in the following canon for the mixed hypothetical syllogism: Given ahypothetical premiss expressed affirmatively, then the affirmation of theantecedent justifies the affirmation of the consequent; and the denial of theconsequentjustifiesthedenialoftheantecedent;butnotconverselyineithercase.

307.The Reduction ofMixedHypothetical Syllogisms.—Any case of themodustollensmaybereducedtothemodusponens,andviceversâ.

Thus,

IfPistruethenQistrue,Qisnottrue,

therefore, Pisnottrue,

becomes,bycontrapositionofthehypotheticalpremiss,

IfQisnottruethenPisnottrue,Qisnottrue,

therefore, Pisnottrue ;

andthisisthemodusponens.382382 AcategoricalsyllogisminCamestresmaysimilarlybereducedtoCelarent

withouttransposingthepremisses.Thus,AllPisM,NoSisM,therefore,NoSisP,becomes,bycontrapositionofthemajorandobversionoftheminorpremiss,Nonot-MisP,AllSisnot-M,therefore,NoSisP.

308. Is the reasoning contained in the mixed hypothetical syllogismmediate or immediate?383—Kant,Hamilton,384 Bain, and others argue thatinferencesof thekind thatwehave justbeenconsideringareproperly tobe


regardednotasmediate,butasimmediate,inferences.383 Similarargumentsonbothsidesmaybeusedinthecasewhereaconditional

premissandacategoricalpremissarecombined.384 Logic, ii.p.383.Onpage378,however,Hamiltonseems to take theother

view.

Now,takingthesyllogism—

IfPistruethenQistrue,butPistrue,

therefore, Qistrue,

355theconclusionisatanyrateapparentlyobtainedbyacombinationoftwopremisses, and the process is moreover one of elimination, namely, of thepropositionPistrue.Hencetheburdenofproofcertainlylieswiththosewhodenytheclaimsofsuchaninferenceasthistobecalledmediate.

Bain (Logic, Deduction, p. 117) seems to argue that the so-calledhypothetical syllogism is not reallymediate inference, because it is “apureinstanceofthelawofconsistency”;inotherwords,because“theconclusionisimpliedinwhathasalreadybeenstated.”Butisnotthisthecaseinallformalmediate inference? It cannotbemaintained that the categorical syllogism ismorethanapureinstanceofthelawofconsistency;orthattheconclusioninsuchasyllogismisnotimpliedinwhathasbeenalreadystated.ButpossiblyBain may mean that the conclusion is implied in the hypothetical premissalone.Indeedhegoesontosay,“’Iftheweathercontinuesfine,weshallgointo the country’ is transformable into the equivalent form ‘The weathercontinuesfine,andsoweshallgointothecountry.’Anypersonaffirmingtheone,doesnot, inaffirmingtheother,declareanewfact,but thesamefact.”Surelythisisnotintendedtobeunderstoodliterally.Takethefollowing:—Ifwarisdeclared,Imustreturnhome;Ifthesunmovesroundtheearth,modernastronomyisadelusion.Aretheserespectivelyequivalenttothestatements,Warhasbeendeclared,andsoImustreturnhome;Thesunmovesroundtheearth,andsomodernastronomyisadelusion?Besides,ifthepropositionIfPis true thenQ is true implies the truth ofP,what becomes of the possiblereasoning,“ButQisnottrue,therefore,Pisnottrue”?

Furtherargumentsthathavebeenadducedonthesamesideareasfollows:


— (1) “There is nomiddle term in the so-called hypothetical syllogism”.385The answer is that there is an element 356 in the premisseswhich does notappearintheconclusion,andthatthiscorrespondstothemiddletermofthecategoricalsyllogism.

385 This isKant’sargument.Amoreplausibleargumentwouldbethat there isnominor term. It will be found, however, that, in the reduction of the mixedhypothetical syllogism to the form of a pure hypothetical syllogism, somethingcorrespondingtoaminortermhastobeintroduced.Comparenote2onpage352.

(2)“Intheso-calledhypotheticalsyllogism,theminorandtheconclusionindifferentlychangeplaces”.386Thisstatementiserroneous.Takingthevalidsyllogismgivenatthecommencementofthissectionandtransposingtheso-calledminorandtheconclusion,wehaveafallacy.

386 This argument isHamilton’s.He remarks that, in hypothetical syllogisms,“thesamepropositionisreciprocallymediumorconclusion”(Logic,ii.p.379).DrRay(Deductive Logic,NoteC) holds thatHamilton is herewrongly interpreted;andthathemeantnomorethanthatwithahypotheticalpremissIfAisB,CisD,arelationbetweenAandBmaybeeithertheotherpremiss(asinthemodusponens)or the conclusion (as in themodus tollens). Dr Ray is possibly right. But if so,Hamiltondoesnotexpresshimselfclearly.ForAisB (thepremissof themodusponens)iscertainlynotthesamepropositionasAisnotB (theconclusionof themodustollens).Itmaybeaddedthattheargumentinitsnewformisirrelevant.Inthe categorical syllogism we have something precisely analogous. For given amajorpremissAllMisP,arelationbetweenMandSmaybetheminorpremiss(inwhichcaseMwillbethemiddleterm),oritmaybetheconclusion(inwhichcaseM will be the major term). Compare the syllogisms: All M is P, All S is M,therefore,AllSisP ;AllMisP,NoSisP,therefore,NoSisM.

(3) “The major in a so-called hypothetical syllogism consists of twopropositions,thecategoricalmajoroftwoterms.”Thismerelytellsusthatahypotheticalsyllogismisnotthesameinformasacategoricalsyllogism,butseemstohavenobearingonthequestionwhethertheso-calledhypotheticalsyllogismisacaseofmediateorofimmediateinference.

Turningnowtotheothersideofthequestionnosatisfactoryanswersseempossible to the following arguments in favour of regarding the mixedhypotheticalsyllogismasacaseofmediateinference.Inanysuchsyllogism,thetwopremissesarequitedistinct,neithercanbeinferredfromtheother,butbotharenecessaryinorderthattheconclusionmaybeobtained.Againifwe


compare with it the inferences which are on all sides admitted to beimmediate inferences from the hypothetical proposition, the differencebetweenthetwocasesisapparent.FromIfPistruethenQistrue,IcaninferimmediatelyIfQisnottruethenPisnottrue ;butIrequirealsotoknowthatQisnottrueinordertobeabletoinferthatPisnottrue.

357 And whether the mixed hypothetical syllogism can or can not beactually reduced to pure categorical form, it can at least be shewn to beanalogous to the ordinary categorical syllogism, which is admitted to be acase of mediate reasoning. Moreover there are distinct forms—themodusponensandthemodustollens—whichareanalogous todistinct formsof thecategorical syllogism; and fallacies in the mixed hypothetical syllogismcorrespondtocertainfallaciesinthecategoricalsyllogism.

Theargumentinfavourofregardingthemodustollens—IfPistruethenQistrue,butQisnottrue,therefore,Pisnottrue—asmediateinferenceisstillmoreforcible;butofcoursethemodusponensandthemodustollensstandorfalltogether.387

387 Insection316itwillbeshewnfurtherthatthehypotheticalsyllogismandthedisjunctivesyllogismalsostandorfalltogether.

Professor Croom Robertson (Mind, 1877, p. 264) has suggested anexplanationastothemannerinwhichthiscontroversymayhavearisen.Hedistinguishes thehypothetical “if” from the inferential “if,” the latter beingequivalent to since, seeing that,because. No doubt by the aid of a certainaccentuationtheword“if”maybemadetocarrywithitthisforce.ProfessorRobertsonquotesapassagefromClarissaHarlowe inwhichtheremark,“Ifyouhavethevalueformycousinthatyousayyouhave,youmustneedsthinkherworthytobeyourwife,”isexplainedbythespeakertomean,“Sinceyouhave &c.” Using the word in this sense, the conclusion C is D certainlyfollowsimmediatelyfromthebarestatementIfAisB,CisisD;orratherthisstatementitselfaffirmstheconclusion.When,however,theword“if”carrieswithitthisinferentialimplication,wecannotregardthepropositioninwhichit occurs as merely hypothetical. We have rather a condensed mode ofexpression including twostatements inone; itmay indeedbeargued that inthe single statement thus interpreted we have a hypothetical syllogism


expressedelliptically.388388 CompareMansel’sAldrich,p.103.

358

EXERCISES.

309.Shewhowthemodusponensmaybereducedtothemodustollens.[K.]

310.Testthefollowing:“Ifallmenwerecapableofperfection,somewouldhaveattainedit;butnonehavingdoneso,nonearecapableofit.”[V.]

311.Examinetechnicallythefollowingargument:— Ifyouneededfood,Iwouldgiveyoumoney;butasyoudonotcaretowork,youcannotneedfood;therefore,Iwillgiveyounomoney.[J.]

312.Shewwhatconclusioncanbeinferredfromthepremisses:Healwaysstaysinwhenitrains,butheoftengoesoutwhenitiscold.[J.]

313.ConstructconditionalandhypotheticalsyllogismsinCesare,Bocardo,Dimarisandreducethemtothefirstfigure.[K.]

314.Name themoodand figureof the following, and shew that eitheronemaybe reduced to theotherform:

(1) IfRistrue,Qistrue,IfPistrue,Qisnottrue,

therefore, IfPistrue,Risnottrue ;(2) IfYistrue,Zisnottrue,

IfYistrue,Xmaybetrue,therefore, IfXistrue,Zneednotbetrue.

[K.]

315.LetX,Y,Z,P,Q,Rbesixpropositions. Given(1) IfXistrue,Pistrue ; (2) IfYistrue,Qistrue ; (3) IfZistrue,Ristrue ; (4) OfX,Y,Zoneatleastistrue ; (5) OfP,Q,Rnotmorethanoneistrue ;provesyllogistically (i) IfPistrue,Xistrue ; (ii) IfQistrue,Yistrue ; (iii) IfRistrue,Zistrue ; (iv) OfP,Q,R,oneatleastistrue ; (v) OfX,Y,Z,notmorethanoneistrue.[K.]


CHAPTERVI.

DISJUNCTIVESYLLOGISMS.

316.TheDisjunctive Syllogism.—Adisjunctive (oralternative) syllogismmay be defined as a formal reasoning in which a categorical premiss iscombinedwithadisjunctive(alternative)premisssoastoyieldaconclusionwhich is either categorical or else disjunctive (alternative) with feweralternantsthanarecontainedinthedisjunctivepremiss.389


389 Archbishop Thomson’s definition of the disjunctive syllogism—“Anargument inwhich there isadisjunctive judgment” (LawsofThought,p.197)—must be regarded as toowide if, as is usually the case, an affirmative judgmentwith a disjunctive predicate is considered disjunctive. It would include such asyllogismasthefollowing,—BiseitherCorD,AisB,thereforeAiseitherCorD.Theargumenthereinnowayturnsuponthealternationcontainedinthemajorpremiss,andthereasoningmayberegardedasanordinarycategoricalsyllogisminBarbara,themajortermbeingcomplex.

Logicians have not, as a rule, given any distinctive recognition to argumentsconsisting of two disjunctive premisses and a disjunctive conclusion; and MrWelton goes so far as to remark that “both premisses of a syllogism cannot bedisjunctive since from two assertions as indefinite as disjunctive propositionsnecessarilyare,nothingcanbeinferred”(Logic,p.327).Itis,however,clearthatthis is erroneous, if an argument consisting of two hypothetical premisses and ahypothetical conclusion is possible, and if a hypothetical can be reduced to thedisjunctiveform.Asanexamplewemayexpress indisjunctivesthehypotheticalsyllogismgivenonpage348:EitherQisnottrueorRistrue,EitherPisnottrueorQistrue,therefore,EitherPisnottrueorRistrue.Herequestionsofmodalityare left on one side.Theywould not, however, in any casematerially affect theargument.

Forexample,

AiseitherBorC,AisnotB,

therefore, AisC ;EitherPorQorRistrue,Pisnottrue,

therefore, EitherQorRistrue.

360Thecategoricalpremissineachoftheabovesyllogismsdeniesoneofthe alternants of the alternative premiss, and the conclusion affirms theremaining alternant or alternants. Reasonings of this type are accordinglydescribedasexamplesofthemodustollendoponens.

Itfollowsfromtheresolutionofdisjunctivepropositionsintoconditionalsorhypotheticalsgiveninsection193that(questionsofmodalitybeingleftononeside)theforceofadisjunctiveasapremissinanargumentisequivalenteithertothatofaconditionalortothatofahypotheticalproposition.

Thus,


EitherAisBorCisD,AisnotB,

therefore, CisD ;

mayberesolvedintotheform

IfAisnotB,CisD,AisnotB,

therefore, CisD ;

orintotheform

IfCisnotD,AisB,AisnotB,

therefore, CisD.

Acorollaryfromtheaboveisthatthosewhodenythecharacterofmediatereasoning to the mixed hypothetical syllogism must also deny it to thedisjunctive syllogism, or elsemust refuse to recognise the resolutionof thedisjunctivepropositionintooneormorehypotheticals.

In the above example it is not quite clear from the form of the majorpremiss whether we have a true hypothetical or a conditional. But in thefollowingexamples,whichareaddedtoillustratethedistinction,itisevidentthatthealternativepropositionsareequivalenttoatruehypotheticalandtoaconditionalrespectively:

EitherallA’sareB’sorallA’sareC’s,ThisAisnotB,

therefore, AllA’sareC’s ;AllA’sareeitherBorC,ThisAisnotB,

therefore, ThisAisC.390390 Whenthealternativemajorpremissisequivalentnot toatruehypothetical

buttoaconditional(asinthesecondoftheaboveexamples),thesyllogismmaybereduced topurecategorical form(unless thecategoricalandconditional formsofpropositionareinsomewaydifferentiatedfromoneanother).Thus,

EveryAwhichisnotBisC,ThisAisanAwhichisnotB,

therefore, ThisAisC.


361317.Themodus ponendo tollens.—In addition to themodus tollendoponens,somelogiciansrecogniseasvalidamodusponendotollensinwhichthe categorical premiss affirms one of the alternants of the disjunctivepremiss,andtheconclusiondeniestheotheralternantoralternants.Thus,

AiseitherBorC,AisB,

therefore, AisnotC.

The argument here proceeds on the assumption that the alternants aremutuallyexclusive;but this,on the interpretationofalternativepropositionsadoptedinsection191, isnotnecessarilythecase.Hencetherecognitionordenial of the validity of the modus ponendo tollens in its ordinary formdependsuponourinterpretationofthealternativeformofproposition.391

391 It will be observed that, interpreting the alternants as not necessarilyexclusive of one another, the modus ponendo tollens in the above form isequivalenttooneofthefallaciesinthemixedhypotheticalsyllogismmentionedinsection306.

Nodoubtexclusivenessisoftenintendedtobeimpliedandisunderstoodtobeimplied.Forexample,“Hewaseitherfirstorsecondintherace,Hewassecond, therefore, He was not first.” This reasoning would ordinarily beacceptedasvalid.Butitsvalidityreallydependsnotontheexpressedmajorpremiss,butontheunderstoodpremiss,“Noonecanbebothfirstandsecondinarace.”Thefollowingreasoningisinfactequallyvalidwiththeonestatedabove, “He was second in the race, therefore, He was not first.” Thealternativepremissis, therefore,quiteimmaterialtothereasoning;wecoulddo just aswellwithout it, for the reallyvitalpremiss, “Noonecanbebothfirst and second in a race,” is true, and would be accepted as such, quiteirrespectiveofthetruthofthealternativeproposition,“Hewaseitherfirstorsecond.”Inother362casesthemutualexclusivenessofthealternantsmaybetacitlyunderstood,althoughnotobviousàprioriasintheaboveexample.Butinnocasecanaspecial implicationof thiskindberecognisedwhenwearedealing with purely symbolic forms. If we hold that the modus ponendotollensasabovestatedisformallyvalid,wemustbepreparedtointerpretthealternantsasineverycasemutuallyexclusive.


If, however, we take a major premiss which is disjunctive, not in theordinary sense (inwhichdisjunctive is equivalent to alternative), but in themoreaccuratesenseexplainedinsection189, thenwemayhavea formallyvalid reasoningwhich has every right to be described as amodusponendotollens.Thus,

PandQarenotbothtrue ;butPistrue ;

therefore, Qisnottrue.392392 This is in the stricter sense a disjunctive syllogism, the modus tollendo

ponens being an alternative syllogism. The reader must, however, be careful torememberthatthelatteriswhatisordinarilymeantbythedisjunctivesyllogisminlogicaltext-books.

Thefollowingtableoftheponendoponens,&c.,intheirvalidandinvalidformsmaybeuseful:

Valid Invalid

PonendoPonensIfPthenQ,

butP,∴Q.

IfPthenQ,butQ,∴P.

TollendoTollensIfQthenP,butnotP,∴notQ.

IfQthenP,butnotQ,∴notP.

TollendoPonensEitherPorQ,butnotP,

∴Q.

NotbothPandQ,butnotQ,

∴P.

PonendoTollensNotbothPandQ,

butP,∴notQ.

EitherPorQ,butQ,

∴notP.

Theabovevalidformsaremutuallyreducibletooneanotherandthesameistrueoftheinvalidforms.

363 318. The Dilemma.—The proper place of the dilemma amongsthypotheticalanddisjunctiveargumentsisdifficulttodetermine,inasmuchasconflicting definitions are given by different logicians. The followingdefinitionmay be taken as perhaps on thewhole themost satisfactory:—Adilemma is a formal argument containing a premiss in which two ormorehypotheticalsareconjunctivelyaffirmed,andasecondpremiss inwhich the


antecedents of these hypotheticals are alternatively affirmed or theirconsequents alternatively denied.393 These premisses are usually called themajorandtheminorrespectively.394

393 In the strict use of the term, adilemma implies only two alternants in thealternativepremiss;iftherearemorethantwoalternantswehaveatrilemma,oratetralemma,orapolylemma,asthecasemaybe.

394 This applicationof the termsmajor andminor is somewhat arbitrary.Thedilemmatic force of the argument is indeed made more apparent by stating thealternativepremiss(i.e.,theso-calledminorpremiss)first.

Dilemmas are called constructive or destructive according as the minorpremiss alternatively affirms the antecedents, or denies the consequents, ofthemajor.395

395 A further formof argumentmaybe distinguished inwhich the alternationcontained in the so-calledminor premiss is affirmed only hypothetically, and inwhich,therefore,theconclusionalsoishypothetical.Forexample,

IfAisB,EisF ;andifCisD,EisF ;IfXisY,eitherAisBorCisD ;

therefore, IfXisY,EisF.

This might be called the hypothetical dilemma. It admits of varietiescorrespondingtothevarietiesoftheordinarydilemma;butnodetailedtreatmentofitseemscalledfor.

Since it is a distinguishing characteristic of the dilemma that the minorshouldbe alternative, it follows that thehypotheticals intowhich themajorpremissofaconstructivedilemmamayberesolvedmustcontainatleasttwodistinct antecedents. Theymay, however, have a common consequent. Theconclusionofthedilemmawillthencategoricallyaffirmthisconsequent,andwill correspondwith it in form.396Thedilemma itself is in this case calledsimple. If, on the other hand, the major premiss contains more than oneconsequent,theconclusionwillnecessarilybealternative,andthedilemmaiscalledcomplex.

396 Itwillusuallybeasimplecategorical;butseethefollowingnote.

364 Similarly, in a destructive dilemma the hypotheticals into which themajorcanberesolvedmusthavemorethanoneconsequent,buttheymayormay not have a common antecedent; and the dilemma will be simple orcomplexaccordingly.


Wehavethenfourformsofdilemmaasfollows: (i)Thesimpleconstructivedilemma.

IfAisB,EisF ;andifCisD,EisF ;butEitherAisBorCisD ;

therefore,EisF.

(ii)Thecomplexconstructivedilemma.

IfAisB,EisF ;andifCisD,GisH ;butEitherAisBorCisD ;

therefore,EitherEisForGisH.397

(iii)Thesimpledestructivedilemma.

IfAisB,CisD ;andifAisB,EisF ;butEitherCisnotDorEisnotF ;

therefore,AisnotB.

(iv)Thecomplexdestructivedilemma.

IfAisB,EisF ;andifCisD,GisH ;butEitherEisnotForGisnotH ;

therefore,EitherAisnotBorCisnotD.398397 Thefollowingisasimple,notacomplex,constructivedilemma:

IfAisB,EisForGisH ;andifCisD,EisForGisH ;butEitherAisBorCisD ;

therefore,EitherEisForGisH.

The hypotheticals which here constitute the major premiss have a commonconsequent; but since this is itself alternative, the conclusion appears in thealternativeform.Thiscaseisanalogoustothefollowing,—AllMisPorQ,AllSisM,therefore,AllSisPorQ,—wheretheconclusionofanintrinsicallycategoricalsyllogismalsoappearsinthealternativeform.Comparethenoteonpage359.

398 Thefollowingisasimple,notacomplex,destructivedilemma:

IfbothPandQaretruethenXistrue,andunderthesamehypothesisYistrue ;butEitherXorYisnottrue ;

therefore,EitherPorQisnottrue.

In thecaseofdilemmas,as in thecaseofmixedhypothetical syllogisms,theconstructiveformmaybereducedtothedestructiveform,andviceversâ.Allthathastobedoneistocontrapositthehypotheticalswhichconstitutethe


major365premiss.Oneexamplewillsuffice.Takingthesimpleconstructivedilemmagivenabove,andcontrapositingthemajor,wehave,—

IfEisnotF,AisnotB ;andifEisnotF,CisnotD ;butEitherAisBorCisD ;

therefore,EisF ;

andthisisadilemmainthesimpledestructiveform.

ThedefinitionofthedilemmagivenaboveispracticallyidenticalwiththatgivenbyFowler(DeductiveLogic,p.116).Mansel(Aldrich,p.108)definesthe dilemma as “a syllogism having a conditional (hypothetical) majorpremisswithmorethanoneantecedent,andadisjunctiveminor.”EquivalentdefinitionsaregivenbyWhatelyand Jevons.According to thisview,whilethe constructive dilemmamay be either simple or complex, the destructivedilemmamustalwaysbecomplex,sinceinthecorrespondingsimpleform(asintheexamplegivenonpage364)thereisonlyoneantecedentinthemajor.Thisexclusionseemsarbitraryandisagroundforrejectingthedefinitioninquestion.Whately,indeed,regardsthenamedilemmaasnecessarilyimplyingtwo antecedents ; but it should rather be regarded as implying twoalternatives, either of which being selected a conclusion follows that isunacceptable.Whatelygoeson to assert that the excluded form ismerely adestructivehypotheticalsyllogism,similartothefollowing,

IfAisB,CisD ;CisnotD ;

therefore, AisnotB.

But thetworeallydifferpreciselyas thesimpleconstructivedilemmagivenonpage364differsfromtheconstructivehypotheticalsyllogism,—

IfAisB,EisF ;AisB ;

therefore, EisF.

Besides,itisclearthattheformunderdiscussionisnotmerelyadestructivehypotheticalsyllogismsuchashasbeenalreadydiscussed,sincethepremisswhich is combined with the hypothetical premiss is not categorical butalternative.


Thefollowingdefinitionissometimesgiven:—“Thedilemma(ortrilemmaorpolylemma)isanargumentinwhichachoiceisallowedbetweentwo(orthree ormore) alternatives, but it is 366 shewn thatwhichever alternative istaken the same conclusion follows.” This definition, which no doubt givespoint to the expression “the horns of a dilemma,” includes the simpleconstructive dilemma and the simple destructive dilemma; but it does notallowthateitherofthecomplexdilemmasisproperlyso-called,sinceineachcaseweareleftwiththesamenumberofalternantsintheconclusionasarecontainedinthealternativepremiss.Ontheotherhand,itembracesformsthatare excluded by both the preceding definitions; for example, the followingreasoning—which should rather be classed simply as a destructivehypothetico-categoricalsyllogism—

IfAis,eitherBorCis ;butNeitherBnorCis ;

therefore, Aisnot.399399 CompareUeberweg,Logic,§123.

Jevons(ElementsofLogic,p.168)remarksthat“dilemmaticargumentsaremoreoftenfallaciousthannot,becauseitisseldompossibletofindinstanceswhere two alternatives exhaust all the possible cases, unless indeed one ofthembethesimplenegativeoftheother.”Inotherwords,manydilemmaticarguments will be found to contain a premiss involving a fallacy ofincomplete alternation. It should, however, beobserved that in strictness anargument is not itself to be called fallacious because it contains a falsepremiss.

EXERCISES.

319.Whatcanbeinferredfromthepremisses,EitherAisBorCisD,EitherCisnotDorEisF?Exhibitthereasoning(a)intheformofahypotheticalsyllogism,(b)intheformofadilemma.[K.]

320.Reducethefollowingargument,consistingof threedisjunctivepropositions, to theformofanordinary categorical syllogism: Everything is either M or P, Everything is either not S or not M,therefore,EverythingiseitherPornotS.[K.]


321.Discussthelogicalconclusivenessoffatalisticreasoninglikethis:—IfIamfatedtobedrownednow,thereisnouseinmystruggling;ifnot,thereisnoneedofit.ButeitherIamfatedtobedrownednoworIamnot;sothatitiseitheruselessorneedlessformetostruggleagainstit.[B.]


CHAPTERVII.

IRREGULARANDCOMPOUNDSYLLOGISMS.

322.TheEnthymeme.—Bytheenthymeme,Aristotlemeantwhathasbeencalledthe“rhetoricalsyllogism”asopposedtotheapodeictic,demonstrative,theoretical syllogism.The following is fromMansel’s notes toAldrich (pp.209 to 211): “The enthymeme is defined by Aristotle, συλλογισμὸς ἐξεἰκότωνἤσημείων.Theεἰκὸςandσημεῖων themselvesarepropositions; theformerstatingageneralprobability,thelatterafact,whichisknowntobeanindication, more or less certain, of the truth of some further statement,whether of a single fact or of a general belief. The former is a propositionnearly,thoughnotquite,universal ;as‘Mostmenwhoenvyhate’:thelatterisa singular proposition, which however is not regarded as a sign, exceptrelatively to some other proposition, which it is supposedmay be inferredfrom it. The εἰκός, when employed in an enthymeme,will form themajorpremissofasyllogismsuchasthefollowing:

Mostmenwhoenvyhate,Thismanenvies,

therefore, Thisman(probably)hates.

“The reasoning is logically faulty; for, the major premiss not beingabsolutelyuniversal,themiddletermisnotdistributed.

“Theσημεῖωνwillformonepremissofasyllogismwhichmaybeinanyofthethreefigures,asinthefollowingexamples:

Figure1. Allambitiousmenareliberal,Pittacusisambitious,

therefore, Pittacusisliberal.368Figure2. Allambitionsmenareliberal,

Pittacusisliberal,therefore, Pittacusisambitious.


Figure3. Pittacusisliberal,Pittacusisambitious,

therefore, Allambitiousmenareliberal.

“The syllogism in the first figure alone is logically valid. In the second,there is an undistributedmiddle term; in the third, an illicit process of theminor.”400

400 Onthissubjectthestudentmaybereferredtotheremainderofthenotefromwhich the above extract is taken, and toHamilton,Discussions, pp. 152 to 156.ComparealsoKarslake,AidstotheStudyofLogic,BookII.

Anenthymeme isnowusuallydefinedasasyllogismincompletelystated,oneofthepremissesortheconclusionbeingunderstoodbutnotexpressed.401The arguments of everyday life are to a large extent enthymematic in thissense;and thesamemaybesaidof fallaciousarguments,whichareseldomcompletely stated, or their want of cogency would be more quicklyrecognised.

401 Thisaccountoftheenthymemeappearstohavebeenoriginallybasedontheerroneousideathatthenamesignifiedtheretentionofonepremissinthemind,ἐνθυμῷ.Thus,inthePortRoyalLogic,anenthymemeisdescribedas“asyllogismperfect in the mind, but imperfect in the expression, since some one of thepropositions is suppressed as too clear and toowell known, and as being easilysuppliedby themindof those towhomwe speak” (p. 229).As regards the trueoriginofthenameenthymeme,seeMansel’sAldrich,p.218.

Anenthymeme is said tobeof the firstorderwhen themajorpremiss issuppressed;ofthesecondorderwhentheminorpremississuppressed;andofthethirdorderwhentheconclusionissuppressed.

Thus, “Balbus is avaricious, and therefore, he is unhappy,” is anenthymeme of the first order; “All avaricious persons are unhappy, andtherefore, Balbus is unhappy,” is an enthymeme of the second order; “Allavariciouspersonsareunhappy,andBalbusisavaricious,”isanenthymemeofthethirdorder.

323.ThePolysyllogismandtheEpicheirema.—Achainofsyllogisms,thatis, a series of syllogisms so linked together that the conclusion of onebecomesapremissofanother, iscalledapolysyllogism. Inapolysyllogism,anyindividualsyllogism369theconclusionofwhichbecomesthepremissof


asucceedingoneiscalledaprosyllogism,anyindividualsyllogismoneofthepremisses ofwhich is the conclusion of a preceding syllogism is called anepisyllogism.Thus,—

AllCisB, ⎫AllBisC, ⎬ prosyllogism,

therefore, AllBisD, ⎭⎫but AllAisB, ⎬episyllogism.

therefore, AllAisD, ⎭

The same syllogism may of course be both an episyllogism and aprosyllogism,aswouldbe thecasewith theaboveepisyllogismif thechainwerecontinuedfurther.

A chain of reasoning402 is said to be progressive (or synthetic orepisyllogistic)whentheprogressisfromprosyllogismtoepisyllogism.Herethepremissesarefirstgiven,andwepassonbysuccessivestepsofinferenceto theultimateconclusionwhich theyyield.Achainof reasoning is,on theother hand, said to be regressive (or analytic or prosyllogistic) when theprogressisfromepisyllogismtoprosyllogism.Heretheultimateconclusionisfirstgivenandwepassbackbysuccessivestepsofprooftothepremissesonwhichitmaybebased.403

402 The distinctionwhich follows is ordinarily applied to chains of reasoningonly; but the readerwill observe that it admits of application to the case of thesimplesyllogismalso.

403 On the distinction between progressive and regressive arguments, seeUeberweg,Logic,§124.

Anepicheiremaisapolysyllogismwithoneormoreprosyllogismsbrieflyindicated only. That is, one or more of the syllogisms of which thepolysyllogismiscomposedareenthymematic.Thefollowingisanexample:

AllBisD,becauseitisC,AllAisB,

therefore, AllAisD.404404 A distinction has been drawn between single and double epicheiremas

accordingasreasonsareenthymematicallygiveninsupportofoneorbothof thepremisses of the ultimate syllogism. The example given in the text is a singleepicheirema;thefollowingisanexampleofadoubleepicheirema:


AllPisY,becauseitisX ;AllSisP,becauseallMisP ;

therefore, AllSisY.

Theepicheiremaissometimesdefinedasifitwereessentiallyaregressivechainofreasoning.Butthisishardlycorrect,if,asisusuallythecase,examplessuchasthe above are given; for it is clear that in these examples the argument is onlypartlyregressive.

370 324. The Sorites.—A sorites is a polysyllogism in which all theconclusions are omitted except the final one, the premisses being given insuchanorder thatany twosuccessivepropositionscontainacommonterm.Two forms of sorites are usually recognised, namely, the so-calledAristotelian sorites and the Goclenian sorites. In the former, the premissstatedfirstcontainsthesubjectoftheconclusion,whilethetermcommontoanytwosuccessivepremissesoccursfirstaspredicateandthenassubject;inthe latter, the premiss stated first contains the predicate of the conclusion,while the term common to any two successive premisses occurs first assubjectandthenaspredicate.Thefollowingareexamples:

AristotelianSorites,— AllAisB,AllBisC,AllCisD,AllDisE,

therefore, AllAisE.GoclenianSorites,— AllDisE,

AllCisD,AllBisC,AllAisB,

therefore, AllAisE.

Itwillbefoundthat,inthecaseoftheAristoteliansorites,iftheargumentis drawn out in full, the first premiss and the suppressed conclusions allappear asminor premisses in successive syllogisms. Thus, the Aristoteliansoritesgivenabovemaybeanalysedintothethreefollowingsyllogisms,—

(1) AllBisC,AllAisB,

therefore, AllAisC ;


(2) AllCisD,AllAisC,

therefore, AllAisD ;371(3) AllDisE,

AllAisD,therefore, AllAisE.

Here the premiss originally stated first is the minor premiss of (1), theconclusionof(1)istheminorpremissof(2),thatof(2)theminorpremissof(3); and so it would go on if the number of propositions constituting thesoriteswereincreased.

In theGoclenian sorites, the premisses are the same, but their order isreversed,andtheresultofthisisthatthepremissoriginallystatedfirstandthesuppressed conclusions becomemajor premisses in successive syllogisms.Thus, the Goclenian sorites given above may be analysed into the threefollowingsyllogisms,—

(1) AllDisE,AllCisD,

therefore, AllCisE ;(2) AllCisE,

AllBisC,therefore, AllBisE ;

(3) AllBisE,AllAisB,

therefore, AllAisE.

Here the premiss originally stated first is the major premiss of (1), theconclusionof(1)isthemajorpremissof(2);andsoon.

The so-called Aristotelian sorites405 is that to which the 372 greaterprominence is usually given; but it will be observed that the order ofpremissesin theGoclenianformis thatwhichcorrespondstothecustomaryorderofpremissesinasimplesyllogism.406

405 This form of sorites ought not properly to be calledAristotelian; but it isgenerallysodescribedinlogicaltext-books.ThenamesoritesisnottobefoundinanylogicaltreatiseofAristotle,thoughinoneplaceherefersvaguelytotheform


ofreasoningwhichthenameisnowemployedtoexpress.ThedistinctexpositionofthisformofreasoningisattributedtotheStoics,anditisdesignatedsoritesbyCicero;butitwasnottillmuchlaterthatthenamecameintogeneraluseamongstlogiciansinthissense.TheformofsoritescalledtheGoclenianwasfirstgivenbyProfessorRudolfGocleniusofMarburg(1547to1628)inhisIsagogeinOrganumAristotelis, 1598. Compare Hamilton,Logic, I. p. 375; and Ueberweg, Logic, §125.Itmaybeaddedthatthetermsorites(whichisderivedfromσωρὸς,aheap)wasusedbyancientwritersinadifferentsense,namely,todesignateaparticularsophism,basedon thedifficultywhich is sometimes found inassigninganexactlimittoanotion.“Itwasasked,—wasamanbaldwhohadsomanythousandhairs;youanswer,No:theantagonistgoesondiminishinganddiminishingthenumber,till either you admit that he who was not bald with a certain number of hairs,becomesbaldwhenthatcomplementisdiminishedbyasinglehair;oryougoondenyinghimtobebald,untilhisheadbehypotheticallydenuded.”Asimilarpuzzleisinvolvedinthequestion,—Onwhatdaydoesalambbecomeasheep?Soritesinthissenseisalsocalledsophismapolyzeteseosorfallacyofcontinuousquestioning.SeeHamilton,Logic,i.p.464.

406 Themistake is sometimesmade of speaking of theGoclenian sorites as aregressiveformofargument.Itisclear,however,thatinbothformsofsoriteswepass continuously from premisses to conclusions, not from conclusions topremisses.

Asoritesmayofcourseconsistofconditionalorhypotheticalpropositions;anditisnotatallunusualtofindpropositionsofthesekindscombinedinthismanner.Theoreticallyasoritesmightalsoconsistofalternativepropositions;butitisnotlikelythatthiscombinationwouldeveroccurnaturally.

325.TheSpecialRulesoftheSorites.—ThefollowingspecialrulesmaybegivenfortheordinaryAristoteliansorites,asdefinedintheprecedingsection:— (1)Onlyonepremisscanbenegative;andifoneisnegative,itmustbethelast. (2)Onlyonepremisscanbeparticular;andifoneisparticular,itmustbethefirst.

AnyAristoteliansoritesmayberepresentedinskeletonform,thequantityandqualityofthepremissesbeingleftundetermined,asfollows:—

S M1

M1, M2

M2, M3


……… ……………… ………Mn−2, Mn−1

Mn−1, Mn

Mn, P⎯⎯⎯ ⎯⎯⎯

S P

373(1)Therecannotbemorethanonenegativepremiss,foriftherewere—sinceanegativepremissinanysyllogismnecessitatesanegativeconclusion—we should in analysing the sorites somewhere come upon a syllogismcontainingtwonegativepremisses.

Again, if one premiss is negative, the final conclusionmust be negative.HenceP must be distributed in the final conclusion. Therefore, it must bedistributed in its premiss, i.e., the last premiss, whichmust accordingly benegative.Ifanypremissthenisnegative,thisistheone.

(2)Sinceithasbeenshewnthatallthepremisses,exceptthelast,mustbeaffirmative,itisclearthatifany,exceptthefirst,wereparticular,weshouldsomewherecommitthefallacyofundistributedmiddle.

The special rules of the Goclenian sorites, as defined in the precedingsection,maybeobtainedbytransposing“first”and“last”intheabove.

326.ThepossibilityofaSorites inaFigureother than theFirst.—Itwillhave been noticed that in our analysis both of the Aristotelian and of theGocleniansoritesalltheresultingsyllogismsareinfigure1.Suchsoritesmayaccordinglybesaidtobethemselvesinfigure1.Thequestionariseswhetherasoritesispossibleinanyotherfigure.

Theusualanswertothisquestionisthatthefirstorthelastsyllogismofasoritesmaybeinfigure2or3(e.g.,infigure2wemayhaveAisB,BisC,CisD,DisE,FisnotE,therefore,AisnotF)butthatitisimpossiblethatallthestepsshouldbeineitherofthesefigures.407“Everyone,”saysMill,“who374understandsthelawsofthesecondandthirdfigures(oreventhegenerallawsofthesyllogism)canseethatnomorethanonestepineitherofthemisadmissible in a sorites, and that it must either be the first or the last”


(ExaminationofHamilton,pp.514,5).


407 SirWilliamHamiltonindeedprofessestogivesoritesinthesecondandthirdfigures,whichhave,hesays,beenoverlookedbyotherlogicians(Logic,II.p.403).It appears, however, that by a sorites in the second figure he means such areasoningasthefollowing,—NoBisA,NoCisA,NoDisA,NoEisA,AllFisA,therefore,NoB,orC,orD,orE,isF ;andbyasoritesinthethirdfiguresuchasthefollowing,—AisB,AisC,AisD,AisE,AisF,therefore,SomeB,andC,andD,andE,areF.Hedoesnothimselfgivetheseexamples;butthattheyareofthekind which he intends may be deduced from his not very lucid statement, “Insecond and third figures, there being no subordination of terms, the only soritescompetent is thatbyrepetitionof thesamemiddle. Infirst figure, there isanewmiddle termforeverynewprogressof the sorites; in secondand third,onlyonemiddletermforanynumberofextremes.Infirstfigure,asyllogismonlybetweenevery second term of the sorites, the intermediate term constituting the middleterm. In the others, every two propositions of the commonmiddle term form asyllogism.”Butitisclearthatintheacceptedsenseofthetermthesearenotsoritesat all. In each case the conclusion is amere summation of the conclusions of anumberofsyllogismshavingacommonpremiss;inneithercaseisthereanychainargument.Hamilton’sowndefinitionof the sorites, involvedas it is,mighthavesaved him from this error. He gives for his definition, “When, on the commonprincipleofall reasoning,—that thepartofapart isapartof thewhole,—wedonotstopatthesecondgradation,oratthepartofthehighestpart,andconcludethatpartofthewhole,butproceedtosomeindefinitelyremoterpart,asD,E,F,G,H,&c.,which,onthegeneralprinciple,weconnectintheconclusionwithitsremotestwhole,—thiscomplexreasoningiscalledaChain-SyllogismorSorites”(Logic,I.p.366).InconnexionwithHamilton’streatmentofthisquestion,Millveryjustlyremarks,“IfSirW.Hamiltonhadfoundinanyotherwritersuchamisuseoflogicallanguage as he is here guilty of, he would have roundly accused him of totalignoranceoflogicalwriters”(ExaminationofHamilton,p.515).

This treatment of the question seems, however, open to refutationby thesimple method of constructing examples. Take, for instance, the followingsorites:—

(i) SomeSisnotM1,AllM2isM1,AllM3isM2,AllM4isM3,AllPisM4,

therefore, SomeSisnotP.(ii) SomeM4isnotP,


AllM4isM3,AllM3isM2,AllM2isM1,AllM1isS,

therefore, SomeSisnotP.

Analysingthefirstof theabove,andinsertingthesuppressedconclusionsinsquarebrackets,wehave—375

SomeSisnotM1,AllM2isM1,

[therefore, SomeSisnotM2,]AllM3isM2,

[therefore, SomeSisnotM3,]AllM4isM3,

[therefore, SomeSisnotM4,]AllPisM4,


This is the only resolution of the sorites possible unless the order of thepremisses is transposed, and itwill be seen that all the resulting syllogismsareinfigure2andinthemoodBaroco.Thesoritesmayaccordinglybesaidtobeinthesamemoodandfigure.ItisanalogoustotheAristoteliansorites,the subject of the conclusion appearing in the premiss stated first, and thesuppressedpremissesbeingallminorsintheirrespectivesyllogisms.

Thecorrespondinganalysisof(ii)yieldsthefollowing:—

SomeM4isnotP,AllM4isM3,

[therefore, SomeM3isnotP,]AllM3isM2,

[therefore, SomeM2isnotP,]AllM2isM1,

[therefore, SomeM1isnotP,]


AllM1isS,therefore, SomeSisnotP.

These syllogisms are all in figure 3 and in the moodBocardo ; and thesoritesitselfmaybesaidtobeinthesamemoodandfigure.ItisanalogoustotheGocleniansorites,thepredicateoftheconclusionappearinginthepremissstated first, and the suppressed premisses beingmajors in their respectivesyllogisms.

Itwill beobserved that the rulesgiven in thepreceding sectionhavenotbeensatisfiedineitheroftheabovesorites,thereasonbeingthattherulesinquestioncorrespondtothespecialrulesoffigure1,anddonotapplyunlessthesoritesis376inthatfigure.Forsuchsoritesasarepossibleinfigures2,3,and4,otherrulesmightbeframedcorrespondingtothespecialrulesofthesefiguresinthecaseofthesimplesyllogism.

Itisnotmaintainedthatsoritesinotherfiguresthanthefirstarelikelytobemet with in common use, but their construction is of some theoreticalinterest.408

408 Theexamplesgiveninthetexthavebeenpurposelychosensoastoadmitofonlyoneanalysis,whichwasnotthecasewiththeexamplesgiveninthefirsttwoeditionsof thiswork.Theoriginal exampleswere, however, perfectlyvalid, andfurther light may be thrown on the general question by a brief reply to certaincriticismspasseduponthoseexamples.Thefollowingwasgivenforfigure2(thesuppressed conclusions being inserted in square brackets), and itwas said to beanalogoustotheAristoteliansorites:—

AllAisB,NoCisB,

[therefore, NoAisC],AllDisC,

[therefore, NoAisD],AllEisD,

therefore, NoAisE.

It has, to begin with, been objected that the above is Goclenian, and notAristotelian,inform,“thesubjectofeachpremissafterthefirstbeingthepredicateofthesucceedingone.”ThisoverlooksthemorefundamentalcharacteristicoftheAristotelian sorites, that the first premiss and the suppressed conclusions are allminors in their respective syllogisms. It has further been objected that thefollowing analysis might serve in lieu of the one given above:—AaB,CeB, [∴CeA,]DaC,[∴DeA],EaD,∴AeE.Nodoubt thisanalysis isapossibleone,but


the objection to it is its heterogeneous character. The first premiss and the firstsuppressedconclusionaremajors,whilethelastsuppressedconclusionisaminor.Again, the first syllogism is in figure 2, the second in figure 1, and the third infigure 4. It must be granted that what has been above called a heterogeneousanalysisisinsomecasestheonlyoneavailable,butitisbettertoadoptsomethingmore homogeneouswhere possible. If the first premiss of a sorites contains thesubject, and the last the predicate, of the conclusion, then the last premiss isnecessarilythemajorofthefinalsyllogism;andhencetherulemaybelaiddownthat we can work out such a sorites homogeneously only by treating the firstpremiss and all the suppressed conclusions as minors, and all the remainingpremissesasmajors, in their respectivesyllogisms.Acorrespondingrulemaybelaiddownifthefirstpremisscontainsthepredicate,andthelastthesubject,oftheconclusion.

Itwillbefoundthatasoritesinfigure4cannothavemorethanalimitednumberofpremisses.Thispointisraisedinsection335.

327.Ultra-totalDistributionoftheMiddleTerm.—Theordinarysyllogisticrule relating to thedistributionof the377middle termdoesnot contemplatetherecognitionofanysignsofquantityotherthanallandsome ;andifothersignsarerecognised,therulemustbemodified.Forexample,theadmissionof the signmost yields the following valid reasoning, although the middletermisnotdistributedineitherofthepremisses:—

MostMisP,MostMisS,


Interpretingmost in the senseofmore thanhalf, it clearly follows from theabovepremisses that theremustbesomeMwhich isbothSandP.ButwecannotsaythatineitherpremissthetermMisdistributed.

Inorder tomeetcasesof thiskind,Hamilton (Logic, II.p.362)gives thefollowingmodification of the rule relating to the distribution of themiddleterm:“Thequantificationsofthemiddleterm,whetherassubjectorpredicate,taken together, must exceed the quantity of that term taken in its wholeextent”;inotherwords,wemusthaveanultra-totaldistributionofthemiddleterminthetwopremissestakentogether.

DeMorgan (Formal Logic, p. 127) writes as follows: “It is said that ineverysyllogismthemiddletermmustbeuniversalinoneofthepremisses,inorderthatwemaybesurethattheaffirmationordenialintheotherpremiss


maybemadeofsomeorallof the thingsaboutwhichaffirmationordenialhasbeenmadeinthefirst.Thislaw,asweshallsee,isonlyaparticularcaseof the truth: it is enough that the two premisses together affirm or deny ofmore thanall the instancesof themiddle term.If therebeahundredboxes,intowhichahundredandonearticlesoftwodifferentkindsaretobeput,notmorethanoneofeachkindintoanyonebox,someonebox,ifnotmore,willhavetwoarticles,oneofeachkind,putintoit.Thecommondoctrinehasit,that an article of one particular kindmust be put into every box, and thensomeoneormore of another kind into oneormore of the boxes, before itmaybeaffirmedthatoneormoreofdifferentkindsarefoundtogether.”DeMorgan himself works the question out in detail in his treatment of thenumerically definite syllogism 378 (Formal Logic, pp. 141 to 170). Thefollowingmaybetakenasanexampleofnumericallydefinitereasoning:—If70percent.ofMareP,and60percent.areS,thenatleast30percent.arebothSandP.409Theargumentmaybeputasfollows:Ontheaverage,of100M’s70arePand60areS ;supposethatthe30M’swhicharenotPareS,still30S’saretobefoundintheremaining70M’swhichareP’s;andthisisthedesired conclusion. Problems of this kind constitute a borderland betweenformal logicandalgebra.Somefurtherexampleswillbegiven inchapter8(section345).

409 Usingotherletters,thisistheexamplegivenbyMill,Logic,ii.2,§1,note,and quoted by Herbert Spencer, Principles of Psychology, II. p. 88. The moregeneralproblemofwhichtheaboveisaspecialinstanceisasfollows:GiventhattherearenM’sinexistence,andthataM’sareSwhilebM’sareP,todeterminewhat is the least number of S’s that are also P’s. It is clear that we have noconclusionatallunlessa+b>n,i.e.,unlessthereisultra-totaldistributionofthemiddleterm.Ifthisconditionissatisfied,thensupposingthe(n−b)M’swhicharenot-P are all of them foundamongst theMS’s, therewill stillbe someMS’s leftwhichareP’s,namely,a−(n−b).HencetheleastnumberofS’sthatarealsoP’smustbea+b−n.

328.The Quantification of the Predicate and the Syllogism.—It will beconvenienttoconsiderbrieflyinthischaptertheapplicationofthedoctrineofthequantificationofthepredicatetothesyllogism;theresultisthereverseofsimplification.410Themostimportantpointsthatarisemaybebroughtoutbyconsideringthevalidityofthefollowingsyllogisms:infigure1,UUU,IUη,AYI; in figure2,ηUO,AUA; in figure3,YAI. In thenext sectionwewill


proceedmoresystematically,Uandωbeingleftoutofaccount.410 In connexion with his doctrine of the quantification of the predicate,

Hamiltondistinguishesbetweenthefiguredsyllogismandtheunfiguredsyllogism.Inthefiguredsyllogism, thedistinctionbetweensubjectandpredicateisretained,as in the text.Bya rigidquantificationof thepredicate,however, thedistinctionbetween subject and predicate may be dispensed with; and such being the casethereisnogroundleftfordistinctionoffigure(whichdependsuponthepositionofthemiddletermassubjectorpredicateinthepremisses).ThisgiveswhatHamiltoncalls the unfigured syllogism. For example:—Any bashfulness and anypraiseworthy are not equivalent, All modesty and some praiseworthy areequivalent, therefore, Any bashfulness and any modesty are not equivalent; Allwhales and some mammals are equal, All whales and some water animals areequal, therefore, Some mammals and some water animals are equal. A distinctcanonfortheunfiguredsyllogismisgivenbyHamiltonasfollows:—“Inasfarastwonotionseitherbothagree,oroneagreeingtheotherdoesnot,withacommonthirdnotion;insofarthesenotionsdoordonotagreewitheachother.”

(1)UUUinfigure1isvalid:—

AllMisallP,AllSisallM,

therefore, AllSisallP.

ItwillbeobservedthatwheneveroneofthepremissesisU,theconclusionmaybeobtainedbysubstitutingSorP(asthecasemaybe)forMintheotherpremiss.379

Without the use of quantified predicates, the above reasoning may beexpressedbymeansofthetwofollowingsyllogisms:

AllMisP, AllMisS,AllSisM, AllPisM,

therefore, AllSisP ; therefore, AllPisS.

(2)IUηinfigure1isinvalid,ifsomeisusedinitsordinarylogicalsense.ThepremissesareSomeMissomePandAllSisallM.Wemay, therefore,obtainthelegitimateconclusionbysubstitutingSforMinthemajorpremiss.ThisyieldsSomeSissomeP.

If,however,some ishereusedin thesenseofsomeonly,NoS issomePfollowsfromSomeSissomeP,andtheoriginalsyllogismisvalid,althoughanegativeconclusionisobtainedfromtwoaffirmativepremisses.


ThissyllogismisgivenasvalidbyThomson(LawsofThought,§103);butapparentlyonlythroughamisprintforIEη.Inhisschemeofvalidsyllogisms(thirty-sixineachfigure),Thomsonseemsconsistentlytointerpretsomeinitsordinarylogicalsense.Usingthewordinthesenseofsomeonly,severalothersyllogismswouldbevalidthathedoesnotgiveassuch.411


(3) AYI in figure 1, some being used in its ordinary logical sense, isequivalenttoAAIinfigure3intheordinarysyllogisticscheme,andisvalid.Butitisinvalidifsomeisusedinthesenseofsomeonly,fortheconclusionnow implies thatS andP are partially excluded fromeachother aswell aspartially coincident,whereas this is not implied by the premisses.With 380thisuseofsome, thecorrectconclusioncanbeexpressedonlybystatinganalternativebetweenSuP,SaP,SyP,andSiP.Thiscasemayservetoillustratethecomplexitiesinwhichweshouldbeinvolvedifweweretoattempttousesomeconsistentlyinthesenseofsomeonly.412

412 CompareMonck,Logic,p.154.

(4)ηUOinfigure2isvalid:—

NoPissomeM,AllSisallM,

therefore, SomeSisnotanyP.

Without the use of quantified predicates, we can obtain the sameconclusioninBocardo,thus,—

SomeMisnotP,AllMisS,


Itwillbeobservedthatboth(3)and(4)arestrengthenedsyllogisms.

(5)AUAinfigure2runsasfollows,—

AllPissomeM,AllSisallM,

therefore, AllSissomeP.

Herewehaveneitherundistributedmiddlenor illicit processofmajoror


minor,nor isanyruleofqualitybroken,andyet thesyllogismis invalid.413Applyingtherulegivenabovethat“wheneveroneofthepremissesisU,theconclusionmaybeobtainedbysubstitutingSorP(asthecasemaybe)forMin the other premiss,”we find that the valid conclusion isSome S is all P.Moregenerally,itfollowsfromthisruleofsubstitutionthatifonepremissisUwhile in theotherpremiss themiddle termisundistributed, then the termcombinedwiththemiddletermintheUpremissmustbeundistributedintheconclusion.Thisappears tobe theoneadditional syllogistic rule required ifwerecogniseUpropositionsinsyllogisticreasonings.

413 WeshouldhaveacorrespondingcaseifweweretoinferNoSisPfromthepremissesgivenintheprecedingexample.

AlldangeroffallacyisavoidedbybreakinguptheUpropositionintotwoApropositions.Inthecasebeforeuswe381have,—AllPisM,AllMisS ;AllP is M,All S is M. From the first of these pairs of premisses we get theconclusionAllPisS ;inthesecondpairthemiddletermisundistributed,andthereforenoconclusionisyieldedatall.

(6)YAIinfigure3isvalid:—

SomeMisallP,AllMissomeS,

therefore, SomeSissomeP.

Theconclusion ishoweverweakened, since fromthegivenpremisseswemightinferSomeSisallP.414Itwillbeobservedthatwhenwequantifythepredicate, the conclusion of a syllogismmay beweakened in respect of itspredicateaswell as in respectof its subject. In theordinarydoctrineof thesyllogismthisisforobviousreasonsnotpossible.

414 Or,retainingtheoriginalconclusion,wemightreplacethemajorpremissbySome M is some P ; hence, from another point of view, the syllogism may beregardedasstrengthened.

Without quantification of the predicate the above reasoning may beexpressedinBramantip,thus,

AllPisM,AllMisS,



Wecouldgetthefullconclusion,AllPisS,inBarbara.

329.Table of valid moods resulting from the recognition ofY and η inaddition toA,E, I,O.—If we adopt the sixfold schedule of propositionsobtainedbyaddingOnlyS isP (Y)andNotonlyS isP (η) to theordinaryfourfoldschedule,asinsection150,everypropositionissimplyconvertible,and,therefore,avalidmoodinanyfigureisreducibletoanyotherfigurebythe simple conversion of one or both of the premisses. Hence if the validmoodsofanyonefigurearedetermined,thoseoftheremainingfiguresmaybeimmediatelydeducedtherefrom.

Itwillbefoundthatineachfiguretherearetwelvevalidmoods,whichareneitherstrengthenednorweakened.This resultmaybeestablishedbyeitherofthetwoalternativemethodswhichfollow.382

I. We may enquire what various combinations of premisses will yieldconclusionsoftheformsA,Y,E,I,O,η,respectively.

Itwill suffice, aswehavealreadyseen, toconsider someone figure.Wemay,therefore,takefigure1,sothatthepositionofthetermswillbe—

M PS M⎯⎯⎯⎯

S P

(i)ToproveSaP,bothpremissesmustbeaffirmative;and,inordertoavoidillicitminor,theminorpremissmustbeSaM. It followsthat themajormustbeMaPortherewouldbeundistributedmiddle.HenceAAAistheonlyvalidmoodyieldinganAconclusion.

(ii) To prove SyP, both premisses must be affirmative; and, in order toavoidillicitmajor,themajorpremissmustbeMyP.ItfollowsthattheminormustbeSyM,inordertoavoidundistributedmiddle.HenceYYYistheonlyvalidmoodyieldingaYconclusion.

(iii)ToproveSeP, themajormustbe(1)MePor (2)MyPor (3)MoP inordertoavoidillicitmajor.If(1),theminormustbeSaMortherewouldbeeithertwonegativepremissesorillicitminor;if(2),itmustbeSeMorthere


wouldbeundistributedmiddleor illicitminor; if (3), itmustbeaffirmativeanddistributebothSandM,whichisimpossible.HenceEAEandYEEaretheonlyvalidmoodsyieldinganEconclusion.

(iv) To prove SiP, both premisses must be affirmative, and since SaMwouldnecessarilybeastrengthenedpremiss,theminormustbe(1)SiMor(2)SyM.If(1),themajormustbeMaPortherewouldbeundistributedmiddle;and if (2), itmustbeMiPor therewouldbeastrengthenedpremiss.HenceAII and IYI are the only valid (unstrengthened and unweakened) moodsyieldinganIconclusion.

(v)ToproveSoP, themajormustbe (1)MeP or (2)MyP or (3)MoP ortherewouldbeillicitmajor.If(1),theminormustbeSiMortherewouldbeastrengthened premiss; if (2), it must be SoM or there would be either twoaffirmativepremisseswithanegativeconclusionorundistributedmiddleora383 strengthenedpremiss; and if (3), itmust beSyM or therewould be twonegativepremissesorundistributedmiddle.HenceEIO,YOO,OYOaretheonly valid (unstrengthened and unweakened) moods yielding an Oconclusion.

(vi)ToproveSηP, theminormustbe (1)SeM or (2)SaM or (3)SηM ortherewouldbeillicitminor.If(1),themajormustbeMiPortherewouldbeastrengthened premiss; if (2), the major must be MηP or there would beundistributedmiddleortwoaffirmativepremisseswithanegativeconclusionorastrengthenedpremiss;andif(3),themajormustbeMaPortherewouldbeundistributedmiddleortwonegativepremisses.HenceIEη,ηAη,Aηηarethe only valid (unstrengthened and unweakened) moods yielding an ηconclusion.

Byconvertingoneorbothof thepremisseswemayatoncededucefromthe above a table of valid (unstrengthened and unweakened)moods for allfourfiguresasfollows:—

Fig.1. Fig.2. Fig.3. Fig.4.

AAA YAA AYA YYAYYY AYY YAY AAYEAE EAE EYE EYEYEE AEE YEE AEE


AII YII AII YIIIYI IYI IAI IAIEIO EIO EIO EIOYOO AOO YηO AηOOYO ηYO OAO ηAOIEη IEη IEη IEηηAη OAη ηYη OYηAηη Yηη AOη YOη

II.Theabovetablemayalsobeobtainedby(1)takingallthecombinationsofpremisses that areàpriori possible, (2) establishing special rules for theparticularfigureselected,which(takentogetherwiththerulesofquality)willenableustoexcludethecombinationsofpremisseswhichareeitherinvalidorstrengthened whatever the conclusion may be, (3) assigning the validunweakenedconclusionintheremainingcases.

384 The following are all possible combinations of premisses, valid andinvalid:

AA(b) YA IA EA(b) OA ηA(b)(c)AY YY(a) IY(a) EY OY(a) ηYAI YI(a) II(a) EI OI(a) ηI(c)AE(b) YE IE [EE](b) [OE] [ηE](b)AO YO(a) IO(a) [EO] [OO](a) [ηO]Aη(b)(c) Yη Iη(c) [Eη](b) [Oη] [ηη](b)(c)

Thecombinationsinsquarebracketsareexcludedbytherulethatfromtwonegativepremissesnothingfollows.

Takingthethirdfigure,inwhichthemiddletermissubjectineachpremiss,andrememberingthatthesubjectisdistributedinA,E,ηandintheseonly,whilethepredicateisdistributedinY,E,Oandintheseonly,thefollowingspecialrulesareobtainable:

(a) One premiss must beA,E, or η, or the middle term would not bedistributedineitherpremiss;

(b) One premiss must be Y, I, or O, or the middle term would bedistributed in both premisses, and there would hence be a strengthened


premiss;

(c)Ifeitherpremissisnegative,oneofthepremissesmustbeY,E,orO,forotherwise (since theconclusionmustbenegative,distributingoneof itsterms)therewouldbeillicitprocesseitherofmajororminor.

Theserulesexcludethecombinationsofpremissesmarkedrespectively(a),(b),(c)above.

Assigning the valid unweakened conclusion in the case of each of thetwelvecombinationswhichremain,wehavethefollowing;AYA,AII,AOη,YAY,YEE,YηO,IAI,IEη,EYE,EIO,OAO,ηYη.Fromthis, the tableofvalid (unstrengthened and unweakened) moods for all four figures may beexpandedasbefore.

330. Formal Inferences not reducible to ordinary Syllogisms.415—Thefollowingisanexampleofwhatisusuallycalledtheargumentàfortiori:385

BisgreaterthanC,AisgreaterthanB,

therefore, AisgreaterthanC.

As this stands, it is clearly not in the ordinary syllogistic form since itcontainsfour terms;anattempt is,however,sometimesmadetoreduceit toordinarysyllogisticformasfollows:

BisgreaterthanC,therefore, WhateverisgreaterthanBisgreaterthanC,

but AisgreaterthanB,therefore, AisgreaterthanC.

415 Attempts to reduce immediate inferences to syllogistic form have beenalreadyconsidered in section110. In thepresent section,non-syllogisticmediateinferenceswillbeconsidered.

With De Morgan, we may treat this as a mere evasion, or as a petitioprincipii.TheprincipleoftheargumentàfortioriisreallyassumedinpassingfromBisgreaterthanCtoWhateverisgreaterthanBisgreaterthanC. Itmayindeedbeadmittedthatbytheabovereductiontheargumentàfortioriisresolved into a syllogism together with an immediate inference. But thisimmediateinferenceisnotonethatcanbejustifiedsolongaswerecognise


only such relationsbetween termsor classes as are impliedby theordinarycopula;andifanyonedeclinedtoadmitthevalidityoftheargumentàfortiorihe would decline to admit the validity of the step represented by theimmediateinference.

Thefollowingattemptedresolution416mustbedisposedofsimilarly:

WhateverisgreaterthanagreaterthanCisgreaterthanC,AisgreaterthanagreaterthanC,therefore,AisgreaterthanC.

416 CompareMansel’sAldrich,p.200.

Atanyrate,itisclearthatthiscannotbethewholeofthereasoning,sinceBnolongerappearsinthepremissesatall.

The point at issuemay perhaps bemost clearly indicated by saying thatwhilsttheordinarysyllogismmaybebaseduponthedictumdeomnietnullo,the argumentà fortiori cannot bemade to rest entirely upon this axiom.Anew principle is required and onewhichmust be placed on a parwith thedictumdeomnietnullo,notinsubordinationtoit.Thisnewprinciplemaybeexpressedintheform,Whatever is386greater thanasecond thingwhich isgreaterthanathirdthingisitselfgreaterthanthatthirdthing.

Mansel(Aldrich,pp.199,200)treatstheargumentàfortioriasanexampleof a material consequence on the ground that it depends upon “someunderstoodpropositionorpropositions,connectingtheterms,bytheadditionofwhichthemindisenabledtoreducetheconsequencetologicalform.”Hewould effect the reduction in one of the ways already referred to. This,however,begsthequestionthatthesyllogisticistheonlylogicalform.Asamatter of fact the cogency of the argument à fortiori is just as intuitivelyevident as that of a syllogism inBarbara itself.Why shouldno relationberegarded as formal unless it canbe expressedby theword is?Touchingonthiscase,DeMorganremarksthattheformallogicianhasarighttoconfinehimselftoanypartofhissubjectthathepleases;“buthehasnorightexcepttherightoffallacytocallthatpartthewhole”(Syllabus,p.42).

There are an indefinite number of other arguments which for similarreasonscannotbereducedtosyllogisticform.Forexample,—AequalsB,B


equalsC, therefore,A equalsC ;417X is a contemporary ofY, and Y of Z,therefore,XisacontemporaryofZ ;AisabrotherofB,B isabrotherofC,therefore,A is abrotherofC ;A is to the rightofB,B is to the right ofC,therefore,AistotherightofC ;AisintunewithB,andBwithC,therefore,AisintunewithC.Alltheseargumentsdependuponprincipleswhichmaybe387placedonaparwith thedictumdeomnietnullo,andwhichareequallyaxiomaticintheparticularsystemstowhichtheybelong.

417 In regard to this argumentDeMorganwrites, “This is not an instance ofcommonsyllogism:thepremissesare‘AisanequalofB ;BisanequalofC.’Sofar as common syllogism is concerned, that ‘an equal of B’ is as good for theargument as ‘B’ is amaterial accident of themeaning of ‘equal.’ The logiciansaccordingly,toreducethistoacommonsyllogism,statetheeffectofcompositionofrelationinamajorpremiss,anddeclarethatthecasebeforethemisanexampleof thatcomposition inaminorpremiss.As in,A isanequalofanequal (ofC);Everyequalofanequalisanequal ;therefore,AisanequalofC.ThisItreatasamere evasion.Amongvarious sufficient answers thisone is enough:mendo notthinkasabove.WhenA=B,B=C,ismadetogiveA=C, thewordequals isacopula in thought, andnotanotionattached toapredicate. There are processeswhicharenot thoseofcommonsyllogismin the logician’smajorpremissabove:butwaivingthis, logic isananalysisof theformof thought,possibleandactual,and the logician has no right to declare that other than the actual is actual”(Syllabus,pp.31,2).

The claims that have been put forward on behalf of the syllogism as theexclusiveformofalldeductivereasoningmustaccordinglyberejected.

Suchclaimshavebeenmade,forexample,byWhately.Syllogism,hesays,is “the form to which all correct reasoning may be ultimately reduced”(Logic,p.12).Again,heremarks,“AnargumentthusstatedregularlyandatfulllengthiscalledaSyllogism;which,therefore,isevidentlynotapeculiarkind of argument, but only a peculiar form of expression, in which everyargumentmaybestated”(Logic,p.26).418

418 ComparealsoWhately,Logic,pp.24,5,and34.

Spalding seems to have the same thing in view when he says,—“Aninference,whoseantecedentisconstitutedbymorepropositionsthanone,isamediate inference. The simplest case, that in which the antecedentpropositions are two, is the syllogism. The syllogism is the norm of allinferenceswhoseantecedent ismorecomplex; andall such inferencesmay,


by thosewho think itworthwhile, be resolved into a series of syllogisms”(Logic,p.158).

J.S.Millendorsestheseclaims.“Allvalidratiocination,”heobserves,“allreasoning by which from general propositions previously admitted, otherpropositionsequallyorlessgeneralareinferred,maybeexhibitedinsomeoftheaboveforms,”i.e.,thesyllogisticmoods(Logic,II.2,§1).

What is required inorder to fill the logicalgapcreatedby the admissionthatthesyllogismisnotthenormofallvalidformalinferencehasbeencalledthe logic of relatives.419 The function of the logic of relatives is to takeaccountofrelationsgenerally,andnot“thosemerelywhichareindicatedbythe ordinary logical copula is” (Venn,Symbolic Logic, p. 400).420 The linewhich this branch of logic may take, if it is ever fully 388 worked out, isindicatedby the followingpassage fromDeMorgan (Syllabus, pp.30,31):—“Aconvertiblecopula isone inwhich thecopular relationexistsbetweentwonamesbothways:thus‘isfastenedto,’‘isjoinedbyaroadwith,’‘isequalto,’&c.areconvertiblecopulae.If‘XisequaltoY’then‘YisequaltoX,’&c.A transitive copula is one in which the copular relation joins X with Zwhenever it joinsX with Y and Y with Z. Thus ‘is fastened to’ is usuallyunderstoodasatransitivecopula:‘XisfastenedtoY’and‘YisfastenedtoZ’give ‘X is fastened to Z.’” The student may further be referred to Venn,Symbolic Logic, pp. 399 to 404; and also to Mr Johnson’s articles on theLogicalCalculusinMind,1892,especiallypp.26to28and244to250.


419 Comparepages149to151.420 Ordinaryformallogicisincludedunderthelogicofrelativesinterpretedin

the widest sense, but only in a more generalised form than that in which it iscustomarilytreated.

EXERCISES.

331.Shewthatifeitheroftwogivenpropositionswillsufficetoexpandagivenenthymemeofthefirstorsecondorderintoavalidsyllogism,thenthetwopropositionswillbeequivalenttoeachother,providedthatneitherofthemconstitutesastrengthenedpremiss.[J.]

332.Given one premiss and the conclusion of a valid syllogismwithinwhat limitsmay the otherpremissbedetermined?Shewthattheproblemisequallydeterminatewiththatinwhichwearegivenboththepremissesandhavetofindtheconclusion.Inwhatcasesisitabsolutelydeterminate?[K.]

333.ConstructavalidsoritesconsistingoffivepropositionsandhavingSomeAisnotBas itsfirstpremiss.Pointoutthemoodandfigureofeachofthedistinctsyllogismsintowhichthesoritesmayberesolved.[K.]

334.Discuss thecharacterof the following sorites, in eachcase indicatinghow farmore thanoneanalysisispossible:(i)SomeDisE,AllDisC,AllCisB,AllBisA,therefore,SomeAisE ;(ii)SomeAisB,NoCisB,AllDisC,AllEisD,therefore,SomeAisnotE ;(iii)AllEisD,AllDisC,AllCisB,AllBisA,therefore,SomeAisE ;(iv)NoDisE,SomeDisC,AllCisB,AllBisA,therefore,SomeAisnotE.[K.]

389335.Discussthepossibilityofasoriteswhichiscapableofbeinganalysedsoastoyieldvalidsyllogismsallofwhichareinfigure4.Determinethemaximumnumberofpropositionsofwhichsuchasoritescanconsist.[K.]

336.Examinethevalidityofthefollowingmoods: infigure1,UAU,YOO,EYO; infigure2,AAA,AYY,UOω; infigure3,YEE,OYO,AωO.[C.]

337. Enquire in what figures, if any, the following moods are valid, noting cases in which theconclusionisweakened:—AUI;YAY;UOη;IUη;UEO.[L.]

338.Ifsomeisusedinthesenseof“some,butnotall,”whatcanbeinferredfromthepropositionsAllMissomeP,AllMissomeS?[K.]

339. Giving to some its ordinary logical meaning, shew that, in any syllogism expressed withquantifiedpredicates,apremissoftheformUmayalwaysberegardedasastrengthenedpremissunlesstheconclusionisalsooftheformU.[K.]

340.Isitpossiblethatthereshouldbethreepropositionssuchthateachinturnisdeduciblefromtheothertwo?[V.]

341.Determinespecial rules for figures1,2,and4,corresponding to thespecial rules for figure3


giveninsection329.[K.]


CHAPTERVIII.

PROBLEMSONTHESYLLOGISM.

342. Bearing of the existential interpretation of propositions upon thevalidity of syllogistic reasonings.—We may as before take differentsuppositionswithregardtotheexistentialimportofpropositions,andproceedtoconsiderhowfarthevalidityofthevarioussyllogisticmoodsisaffectedbyeachinturn.

(1)Leteverypropositionbeinterpretedasimplyingtheexistencebothofitssubjectandofitspredicate.421Inthiscase,theexistenceofthemajor,middle,andminortermsis ineverycaseguaranteedbythepremisses,andthereforeno further assumptionwith regard to existence is required in order that theconclusion may be legitimately obtained.422 We may regard the abovesupposition as that which is tacitly made in the ordinary doctrine of thesyllogism.

421 Itwillbeobservedthatthisisnotquitethesameassupposition(1)insection156.

422 If,however,wearetobeallowedtoproceedasinsection206(wherefromallPisM,allSisM,weinferredsomenot-Sisnot-P)wemustposittheexistencenotmerelyofthetermsdirectlyinvolved,butalsooftheircontradictories.

(2)Let every proposition be interpreted as implying the existence of itssubject. Under this supposition an affirmative proposition ensures theexistence of its predicate also; but not so a negative proposition. It followsthat any mood will be valid unless the minor term is in its premiss thepredicateofanegativeproposition.Thiscannothappeneitherinfigure1orinfigure2,sinceinthesefigurestheminorisalwayssubjectinitspremiss;norinfigure3,sinceinthisfiguretheminor391premissisalwaysaffirmative.Infigure4,theonlymoodswithanegativeminorareCamenesanditsweakenedform AEO. Our conclusion then is that on the given supposition everyordinarilyrecognisedmoodisvalidexceptthesetwo.423


423 Reduction to figure 1 appears to be affected by this supposition, since itmakes the contraposition ofA and the conversion ofE in general invalid. ThecontrapositionofA is involved in thedirect reductionofBaroco(Faksoko). Theprocessis,however,inthisparticularcasevalid,astheexistenceofnot-Misgivenbytheminorpremiss.TheconversionofEisinvolvedinthereductionofCesare,Camestres,andFestinofromfigure2;andofCamenes,Fesapo,andFresisonfromfigure 4. Since, however, one premiss must be affirmative the existence of themiddle term is therebyguaranteed, andhence the simple conversionofE in thesecondfigure,andinthemajorofthefourthbecomesvalid.Alsotheconversionofthe conclusion resulting from the reduction ofCamestres is legitimate, since theoriginalminor term is subject in its premiss.HenceCamenes (and itsweakenedform) are the only moods whose reduction is rendered illegitimate by thesuppositionunderconsideration.Thisresultagreeswiththatreachedinthetext.

(3)Letnopropositionbeinterpretedasimplyingtheexistenceeitherofitssubjectorof itspredicate.TakingS,M,P, as theminor,middle, andmajorterms respectively, the conclusionwill imply that if there is any S there issome P or not-P (according as it is affirmative or negative). Will thepremissesalsoimplythis?Ifso,thenthesyllogismisvalid;butnototherwise.

Ithasbeenshewninsection212thatauniversalaffirmativeconclusion,AllSisP,canbeprovedonlybymeansofthepremisses,AllMisP,AllSisM ;anditisclearthatthesepremissesthemselvesimplythatifthereisanySthereis some P. On our present supposition, then, a syllogism is valid if itsconclusionisuniversalaffirmative.

Again,asshewninsection212,auniversalnegativeconclusion,NoSisP,canbeprovedonlyinthefollowingways:—

(i) NoMisP(orNoPisM),AllSisM,⎯⎯⎯⎯

therefore, NoSisP ; (ii) AllPisM,

NoSisM(orNoMisS),⎯⎯⎯⎯

therefore, NoSisP.

392 In(i) theminorpremissimpliesthat ifSexists thenMexists,and themajor premiss that ifM exists then not-P exists. In (ii) the minor premissimpliesthatifSexiststhennot-Mexists,andthemajorpremissthatifnot-M


existsthennot-Pexists(asshewninsection158).Henceasyllogismisvalidifitsconclusionisuniversalnegative.

Next, let the conclusion be particular. In figure 1, the implication of theconclusionwithregardtoexistenceiscontainedinthepremissesthemselves,since theminor termis thesubjectofanaffirmativeminorpremiss,and themiddle term the subjectof themajorpremiss. In figure2,wemayconsidertheweakenedmoodsdisposedofinwhathasbeenalreadysaidwithregardtouniversal conclusions; for under our present supposition subalternation is avalidprocess.TheremainingmoodswithparticularconclusionsinthisfigureareFestino andBaroco. In the former, theminor premiss implies that ifSexists thenMexists,and themajor that ifMexists thennot-Pexists; in thelatter, theminor premiss implies that ifS exists thennot-M exists, and themajorthatifnot-Mexiststhennot-Pexists.

Alltheordinarilyrecognisedmoods,then,offigures1and2arevalid.Butitisotherwisewithmoodsyieldingaparticularconclusioninfigures3and4,with thesingleexceptionof theweakenedformofCamenes(whichisitselfthe onlymoodwith a universal conclusion in these figures). Subalternationbeingavalidprocess,thelegitimacyofthelatterfollowsfromthelegitimacyofCamenesitself.Butinallothercasesinfigures3and4,theminortermisthepredicateofanaffirmativeminorpremiss.Itsexistence,therefore,carriesnofurtherimplicationofexistencewithitinthepremisses.Itdoessointheconclusion. Hence all themoods of figures 3 and 4, with the exception ofAEE and AEO in the latter figure, are invalid. Take, as an example, asyllogisminDarapti,—

AllMisP,AllMisS,⎯⎯⎯⎯⎯


TheconclusionimpliesthatifSexistsPexists;but393consistentlywiththepremisses, S may be existent while M and P are both non-existent. Animplicationis,therefore,containedintheconclusionwhichisnotjustifiedbythepremisses.


Henceonthesuppositionthatnopropositionimpliestheexistenceeitherofitssubjectorofitspredicatealltheordinarilyrecognisedmoodsoffigures1and2arevalid,butnoneofthoseoffigures3and4exceptingCamenesandtheweakenedformofCamenes.424

424 Anexpressstatementconcerningexistencemay,however,rendertherejectedmoods legitimate. If, for instance, the existence of themiddle term is expresslygiven,thenDaraptibecomesvalid.

(4) Let particulars be interpreted as implying, but universals as notimplying, the existence of their subjects. The legitimacy of moods withuniversal conclusions may be established as in the preceding case. Takingmoodswithparticularconclusions,itisobviousthattheywillbevalidiftheminor premiss is particular, having theminor term as its subject; or if theminorpremissisparticularaffirmative,whethertheminortermisitssubjectorpredicate.Disamis,Bocardo,andDimaris are alsovalid, since themajorpremissineachcaseguaranteestheexistenceofM,andtheminorimpliesthatifMexiststhenSexists.Theabovewillbefoundtocoverallthevalidmoodsinwhich one premiss is particular. There remain only themoods inwhichfromtwouniversalsweinferaparticular.Itisclearthatallthesemoodsmustbe invalid, for theirconclusionswill imply theexistenceof theminor term,andthiscannotbeguaranteedbythepremisses.425

425 Hypotheticalconclusions(oftheformIfSexiststhen&c.)willofcoursestillbelegitimate.

On the supposition then that particulars imply, while universals do notimply, theexistenceof theirsubjects, themoodsrendered invalidareall theweakened moods, together with Darapti, Felapton, Bramantip, andFesapo,426eachofwhichcontainsastrengthenedpremiss.Morebriefly,anyordinarilyrecognised394moodisonthissuppositionvalid,unlessitcontainseitherastrengthenedpremissoraweakenedconclusion.427

426 Itwillbeobservedthattheletterpoccursinthemnemonicforeachofthesemoods,indicatingthattheirreductiontofigure1involvesconversionperaccidens.Onthesuppositionunderdiscussionthisprocessisinvalid,andwemayfindhereaconfirmationoftheaboveresult.

427 Thisresultmayberegardedasaffordinganadditionalargumentinfavouroftheadoptionofsupposition(4).

343.Connexionbetweenthetruthandfalsityofpremissesandconclusion


inavalidsyllogism.—Bysaying that a syllogism isvalidwemean that thetruth of its conclusion follows from the truth of its premisses; and it is animmediateinferencefromthisthatiftheconclusionisfalseoneorbothofthepremissesmustbefalse.Theconversedoesnot,however,holdgoodineithercase. The truth of the premisses does not follow from the truth of theconclusion;nordoes the falsityof theconclusion follow from the falsityofeitherorbothofthepremisses.

Theabovestatementswouldprobablybeacceptedasself-evident;stillitismoresatisfactorytogiveaformalproofofthem,andsuchaproofisaffordedbymeansofthethreefollowingtheorems.428

428 Itisassumedthroughoutthissectionthatourscheduleofpropositionsdoesnot include U. The theorems hold good, however, for the sixfold schedule,includingYandη,aswellasfortheordinaryfourfoldschedule.

(1)Givenavalidsyllogism,theninnocasewillthecombinationofeitherpremisswiththeconclusionestablishtheotherpremiss.

Wehavetoshewthatifonepremissandtheconclusionofavalidsyllogismare taken as a new pair of premisses they do not in any case suffice toestablishtheotherpremiss. Wereitpossibleforthemtodoso,thenthepremissgiventruewouldhaveto be affirmative, for if it were negative, the original conclusionwould benegative,andcombiningtheseweshouldhavetwonegativepremisseswhichcouldyieldnoconclusion. Also, themiddle termwould have to be distributed in the premiss giventrue.This is clear if it is not distributed in theother premiss; and since theotherpremissistheconclusionofthenewsyllogism,ifitisdistributedthere,itmust also be distributed in the premiss given true orwe should have anillicitprocessinthenewsyllogism.395 Therefore, the premiss given true, being affirmative and distributing themiddle term, cannot distribute the other term which it contains.429 Neitherthereforecanthistermbedistributedintheoriginalconclusion.Butthisistheterm which will be the middle term of the new syllogism, and we shallconsequentlyhaveundistributedmiddle. Hencethetruthofonepremissandtheconclusionofavalidsyllogismdoesnot establish the truth of the other premiss; and à fortiori the truth of the


conclusioncannotbyitselfestablishthetruthofboththepremisses.430429 Thisstatement,thoughnotholdinggoodforU,holdsgoodforYaswellas

A.430 Othermethods of solutionmore or less distinct from the abovemight be

given.Asomewhat similarproblem isdiscussedbySolly,SyllabusofLogic, pp.123to126,132to136.Wehaveshewnthatonepremissandtheconclusionofavalidsyllogismwillneversufficetoprovetheotherpremiss,butitofcoursedoesnot followthat theywillneveryieldanyconclusionatall; foraconsiderationofthisquestion,seethefollowingsection.

(2)Thecontradictoriesofthepremissesofavalidsyllogismwillnotinanycasesufficetoestablishthecontradictoryoftheoriginalconclusion.

Thepremissesoftheoriginalsyllogismmustbeeither(α)bothaffirmative,or(β)oneaffirmativeandonenegative. In case (α), the contradictories of the original premisses will both benegative;andfromtwonegativesnothingfollows. In case (β), the contradictories of the original premisses will be onenegativeandoneaffirmative;andifthiscombinationyieldsanyconclusion,itwill be negative. But the original conclusion must also be negative, andthereforeitscontradictorywillbeaffirmative. In neither case then can we establish the contradictory of the originalconclusion.431

431 Itispossible,however,thatsomeconclusionmaybeobtainable.Seesection359.

(3)One premiss and the contradictory of the other premiss of a validsyllogism will not in any case suffice to establish the contradictory of theoriginalconclusion.432

432 Itdoesnotfollowthatonepremissandthecontradictoryoftheotherpremissof a valid syllogism will never yield any conclusion at all. See the followingsection.

396This followsatonce from the firstof the theoremsestablished in thissection.LetthepremissesofavalidsyllogismbePandQ,andtheconclusionR,PandthecontradictoryofQwillnotprovethecontradictoryofR ;forifthey did, it would follow thatP andR would proveQ ; but this has beenshewnnottobethecase.


We have now established by strictly formal reasoningAristotle’s dictumthatalthough it isnotpossible syllogistically toget a falseconclusion fromtrue premisses, it is quite possible to get a true conclusion from falsepremisses.433Inotherwords,thefalsityofoneorbothofthepremissesdoesnotestablish the falsityof theconclusionofa syllogism.Thesecondof theabovetheoremsdealswiththecaseinwhichboththepremissesarefalse;thethirdwiththatinwhichoneonlyofthepremissesisfalse.

433 Hamilton(Logic,I.p.450)considersthedoctrine“thatiftheconclusionofasyllogism be true, the premisses may be either true or false, but that if theconclusionbefalse,oneorbothofthepremissesmustbefalse”tobeextralogical,ifitisnotabsolutelyerroneous.Heisclearlywrong,sincethedoctrineinquestionadmitsofapurelyformalproof.

344.Arguments from the truthofonepremissand the falsityof theotherpremissinavalidsyllogism,orfromthefalsityofonepremisstothetruthofthe conclusion, or from the truth of one premiss to the falsity of theconclusion.—In this section we shall consider three problems, mutuallyinvolved in one another, which are in a manner related to the theoremscontainedintheprecedingsection.Ithas,forexample,beenshewnthatonepremissandthecontradictoryoftheotherpremisswillnotinanycasesufficetoestablishthecontradictoryoftheoriginalconclusion;theobjectofthefirstofthefollowingproblemsistoenquireinwhatcasestheycanestablishanyconclusionatall.

(i)Tofindapairofvalidsyllogismshavingacommonpremiss,suchthatthe remaining premiss of the one contradicts the remaining premiss of theother.434

434 ThisproblemwassuggestedbythefollowingquestionofMrO’Sullivan’s,whichputs thesameprobleminanother form:Given thatonepremissofavalidsyllogism is false and the other true, determine generally in what cases aconclusioncanbedrawnfromthesedata.

397WehavetofindcasesinwhichPandQ,PandQʹ(thecontradictoryofQ)arethepremissesoftwovalidsyllogisms.Inworkingoutthisproblemandtheproblemsthatfollow,itmustberememberedthatiftwopropositionsarecontradictories,theywilldifferinquality,andalsointhedistributionoftheirterms, so that any term distributed in either of them is undistributed in theotherandviceversâ.Wemay,therefore,assumethatQisaffirmativeandQʹ


negative.LetPcontainthetermsXandY,whileQandQʹcontainthetermsYandZ,sothatY is themiddle term,andXandZ theextreme terms,ofeachsyllogism. Since Qʹ is negative, P must be affirmative; and since Y must beundistributedeitherinQorinQʹ,itmustbedistributedinP. HenceP=YaX. Qʹmust distributeZ: for the conclusion (being negative) must distributeoneterm,andXisundistributedinP.ItfollowsthatZisundistributedinQ. HenceQ=YaZorYiZorZiY ; Qʹ=YoZorYeZorZeY. Ifthedifferentpossiblecombinationsareworkedout,itwillbefoundthatthe followingare the syllogismssatisfying thecondition that ifonepremiss(thatinblacktype)isretained,whiletheotherisreplacedbyitscontradictory,aconclusionisstillobtainable:— Infigure1:AII; Infigure3:AAI,AAI,IAI,AII,EAO,OAO; Infigure4:IAI,EAO.

(ii)To findapair of valid syllogismshavinga commonconclusion, suchthatapremissintheonecontradictsapremissintheother.

LetQ andQʹ (whichwemay assume to be respectively affirmative andnegative)bethepremissesinquestion,andPʹ theconclusion;alsoletQandQʹcontainthetermsYandZ,whilePʹcontainsthetermsXandZ,sothatZisthemiddleterm,andXandYtheextremeterms,ofeachsyllogism. It follows immediately that Pʹ is negative; also that Y 398 must beundistributedinPʹ,sinceitisnecessarilyundistributedeitherinQorinQʹ. HencePʹ=YoX. SinceXisdistributedinPʹitmustalsobedistributedinthepremisswhichiscombinedwithQʹ ;andas thispremissmustbeaffirmative, itcannotalsodistributeZ,whichmust thereforebedistributed inQʹ (andundistributed inQ). HenceQ=YaZorYiZorZiY ; Qʹ=YoZorYeZorZeY. Ifthedifferentpossiblecombinationsareworkedout,itwillbefoundthat


the following are the syllogisms satisfying the condition that the sameconclusion is obtainable from another pair of premisses, of which onecontradictsoneoftheoriginalpremisses(namely,thatinblacktype):— Infigure1:EAO,EIO; Infigure2:EAO,AEO,EIO,AOO; Infigure3:EIO; Infigure4:AEO,EIO.

(iii)Tofindapairofvalidsyllogismshavingacommonpremiss,suchthattheconclusionofonecontradictstheconclusionoftheother.435

435 This problem was suggested by the following question of Mr Panton’s,whichputsthesameprobleminanotherform:Iftheconclusionbesubstitutedforapremissinavalidmood,investigatetheconditionswhichmustbefulfilledinorderthatthenewpremissesshouldbelegitimate.

Let P be the common premiss, Q and Qʹ (respectively affirmative andnegative)thecontradictoryconclusions;alsoletPcontainthetermsXandY,whileQandQʹcontainthetermsYandZ,sothatXisthemiddleterm,andYandZtheextremeterms,ofeachsyllogism. SinceQisaffirmative,Pmustbeaffirmative;andsinceeitherQorQʹwilldistributeY,PmustdistributeY. HenceP=YaX. Thepremisswhich, combinedwithP, provesQmust be affirmative andmustdistributeX ;itcannotthereforedistributeZ,andZmustaccordinglybeundistributedinQ(anddistributedinQʹ).399 HenceQ=YaZorYiZorZiY ; Qʹ=YoZorYeZorZeY. Ifthedifferentpossiblecombinationsareworkedout,itwillbefoundthatthefollowingarethesyllogismssatisfyingtheconditionthatthecontradictoryof theconclusion isobtainable,althoughoneof thepremisses(that inblacktype)isretained:— Infigure1:AAA,AAI,EAE,EAO; Infigure2:EAE,EAO,AEE; Infigure4:AAI,AEE.436

436 It will be observed that each of the above problems yields nine cases.Between them they cover all the 24 valid moods; but there are three moods(namely,EAOinfigures1and2andAAIinfigure3)whichoccurtwiceover.The


15unstrengthenedandunweakenedmoodsareequallydistributed,namely,thefouryielding I conclusions (togetherwithOAO) falling under (i); the six yieldingOconclusions(exceptOAO) under (ii); the fiveyieldingA orE conclusions under(iii).Allthemoodsoffigure1(exceptthosewithanIpremiss)fallunder(iii);allthemoodsoffigure2(exceptthosewithanEconclusion)under(ii);allthemoodsoffigure3(excepttheonenothavinganApremiss)under(i).

Thethreesetsofmoodsworkedoutabovearemutuallyderivablefromoneanother.Thus,

(i) (ii) (iii)PandQ∴R = QandRʹ∴Pʹ = RʹandP∴QʹPandQʹ∴Tʹ = QʹandT∴Pʹ = TandP∴Q

In this table (i) represents thepossiblecases inwhich,onepremissbeingretained,theotherpremissmaybereplacedbyitscontradictory.Wecanthendeduce (ii) the cases in which, the conclusion being retained, one premissmaybereplacedbyitscontradictory;and(iii)thecasesinwhich,onepremissbeingretained,theconclusionmaybereplacedbyitscontradictory.Wemightofcourseequallywellstartfrom(ii)orfrom(iii),andthencededucethetwoothers.

Comparingthefirstsyllogismof(i)withthesecondsyllogismof(iii)andviceversâ,weseefurtherthat(i)givesthecasesinwhich,onepremissbeingretained, theconclusionmaybereplacedby theotherpremiss;and that (iii)givesthecasesinwhich,onepremissbeingretained, theotherpremissmaybereplacedbytheconclusion.

400 The following is another method of stating and solving all threeproblems:Todetermineinwhatcasesitispossibletoobtaintwoincompatibletriosofpropositions,eachtriocontainingthreeandonlythreetermsandeachincludingapropositionwhichisidenticalwithapropositionintheotherandalsoapropositionwhichisthecontradictoryofapropositionintheother.

LetthepropositionsbeP,Q,RʹandP,Qʹ,T ;andletPcontainthetermsXandY ;QandQʹthetermsYandZ ;RandT,thetermsZandX.SupposeQtobeaffirmative,andQʹnegative. Thensinceoneofeachtrioofpropositionsmustbenegative,andnotmorethanonecanbeso (asshewn insection214),PandTmustbe affirmative,


andRʹnegative. Again,sinceeachofthetermsX,Y,Zmustbedistributedonceatleastineach trio of propositions (as shewn in section 214), and since Y must beundistributedeitherinQorinQʹ,YmustbedistributedinP. HenceP=YaX. X,beingundistributedinP,mustbedistributedinRʹandT. HenceT=XaZ. Z, being undistributed in T, must be distributed in Qʹ, and thereforeundistributedinQ,anddistributedinRʹ. HenceQ=YaZorYiZorZiY ; Qʹ=YoZorYeZorZeY ; Rʹ=XeZorZeX. Wehavethenthefollowingsolutionofourproblem:—

YaZ,YaZorYiZorZiY,XeZorZeX ;YaZ,YoZorYeZorZeY,XaZ.

345.NumericalMoodsof theSyllogism.437—The following are examplesofnumericalmoodsinthedifferentfiguresofthesyllogism:—401

Figure1. (i) AllM’sareP’s,AtleastnS’sareM’s,

therefore, AtleastnS’sareP’s ;(ii) LessthannM’sareP’s,

AllS’sareM’s,therefore, LessthannS’sareP’s ;

(iii) LessthannM’sareP’s,AtleastnS’sareM’s,

therefore, SomeS’sarenotP’s ; Figure2. (iv) AllP’sareM’s,

LessthannS’sareM’s,therefore, LessthannS’sareP’s ;

(v) LessthannP’sareM’s,AllS’sareM’s,

therefore, LessthannS’sareP’s ;


(vi) LessthannP’sareM’s,AtleastnS’sareM’s,

therefore, SomeS’sarenotP’s ; Figure3. (vii) LessthannM’sareP’s,

AtleastnM’sareS’s,therefore, SomeS’sarenotP’s ;

(viii) AllM’sareP’s,AtleastnM’sareS’s,

therefore, AtleastnS’sareP’s ;(ix) AtleastnM’sareP’s,

AllM’sareS’s,therefore, AtleastnS’sareP’s ; Figure4. (x) AtleastnP’sareM’s,

AllM’sareS’s,therefore, AtleastnS’sareP’s ;

(xi) AllP’sareM’s,LessthannM’sareS’s,

therefore, LessthannS’sareP’s ;402(xii) LessthannP’sareM’s,

AtleastnM’sareS’s,therefore, SomeS’sarenotP’s.

437 This section was suggested by the following question of Mr Johnson’s:—“Shewthevalidityofthefollowingsyllogisms:(i)AllM’sareP’s,AtleastnS’sareM’s,therefore,AtleastnS’sareP’s;(ii)AllP’sareM’s,Less thannS’s areM’s,therefore,LessthannS’sareP’s;(iii)LessthannM’sareP’s,AtleastnM’sareS’s,therefore,SomeS’sarenotP’s.Deducefromtheabovetheordinarynon-numericalmoodsofthefirstthreefigures.”

Theabovemoodsmaybeestablishedasfollows:— (i)FromAllM’sareP’s,itfollowsthatEverySwhichisMisalsoP,andsinceAtleastnS’sareM’s,itfollowsfurtherthatAtleastnS’sareP’s. Denotingthemajorpremissof(i)byA,theminorbyB,andtheconclusionbyC,weobtainimmediatelythefollowingsyllogisms:—

A, Cʹ,Cʹ, B,⎯ ⎯


∴ Bʹ ; ∴ Aʹ ;

andthesearerespectivelyequivalentto(iv)and(vii). (v)isobtainablefrom(iv)bytransposingthepremissesandconvertingtheconclusion; (ii)from(v)byconvertingthemajorpremiss; (iii)from(vii)byconvertingtheminorpremiss; (vi)from(iii)byconvertingthemajorpremiss; (viii)from(i)byconvertingtheminorpremiss; (ix)from(viii)bytransposingthepremissesandconvertingtheconclusion; (x)from(i)bytransposingthepremissesandconvertingtheconclusion; (xi)from(iv)byconvertingtheminorpremiss; (xii)from(vii)byconvertingthemajorpremiss.

Theordinarynon-numericalmoodsofthedifferentfiguresmaybededucedfromtheaboveresultsasfollows:— Figure1.(i)Puttingn=totalnumberofS’s,wehaveMaP,SaM,∴SaP,thatis,Barbara ;andputtingn=1,wehaveMaP,SiM,∴SiP,thatis,Darii. (ii) Puttingn=1,MeP,SaM,∴SeP(Celarent). (iii)Puttingn=1,MeP,SiM,∴SoP(Ferio). AAIandEAOfollowàfortiori.

Figure2(iv)Puttingn=totalnumberofS’s,PaM,SoM,∴SoP(Baroco);puttingn=1,PaM,SeM,∴SeP(Camestres). 403(v) Puttingn=1,PeM,SaM,∴SeP(Cesare). (vi) Puttingn=1,PeM,SiM,∴SoP(Festino). AEOandEAOfollowàfortiori.

Figure 3. (vii) Putting n = total number of M’s, MoP, MaS, ∴ SoP(Bocardo);puttingn=1,MeP,MiS,∴SoP(Ferison). (viii)Puttingn=1,MaP,MiS,∴SiP(Datisi). (ix) Puttingn=1,MiP,MaS,∴SiP(Disamis). DaraptiandFelaptonfollowàfortiori.

Figure4.(x)Puttingn=1,PiM,MaS,∴SiP(Dimaris). (xi) Puttingn=1,PaM,MeS,∴SeP(Camenes). (xii)Puttingn=1,PeM,MiS,∴SoP(Fresison).


Bramantip,AEO,andFesapofollowàfortiori.

EXERCISES.

346.“WhateverPandQmaystandfor,wemayshewàpriorithatsomePisQ. ForAll PQ is Q by the law of identity, and similarly All PQ is P ;therefore,byasyllogisminDarapti,SomePisQ.”Howwouldyoudealwiththisparadox?[K.]


Asolution isaffordedby thediscussioncontained insection342;and thisexample seems to shewthattheenquiry—howfarassumptionswithregardtoexistenceareinvolvedinsyllogisticprocesses—isnotirrelevantorunnecessary.

347.Whatconclusioncanbedrawnfromthefollowingpropositions?Themembers of the boardwere all either bondholders or shareholders, but notboth;andthebondholders,asithappened,wereallontheboard.[V.]Wemaytakeasourpremisses: Nomemberoftheboardisbothabondholderandashareholder, Allbondholdersaremembersoftheboard;andthesepremissesyieldaconclusion(inCelarent), Nobondholderisbothabondholderandashareholder,thatis,Nobondholderisashareholder.

348.Thefollowingrulesweredrawnupforaclub:—(i) The financial committee shall be chosen from amongst the 404 generalcommittee; (ii) No one shall be a member both of the general and librarycommittees,unlesshebealsoonthefinancialcommittee;(iii)Nomemberofthelibrarycommitteeshallbeonthefinancialcommittee. Is there anything self-contradictory or superfluous in these rules? [VENN,SymbolicLogic,p.331.]LetF=memberofthefinancialcommittee,

G=memberofthegeneralcommittee, L=memberofthelibrarycommittee. Theaboverulesmaythenbeexpressedsymbolicallyasfollows:— (i) AllFisG ; (ii) IfanyLisG,thatLisF ; (iii) NoLisF. From(ii)and(iii)weobtain(iv)NoLisG. Therulesmaythereforebewrittenintheform, (1) AllFisG, (2) NoLisG, (3) NoLisF. Butinthisform(3)isdeduciblefrom(1)and(2). Henceallthatiscontainedintherulesasoriginallystatedmaybeexpressedby(1)and(2);thatis,therulesasoriginallystatedwerepartlysuperfluous,andtheymaybereducedto (1)Thefinancialcommitteeshallbechosenfromamongstthegeneralcommittee; (2)Nooneshallbeamemberbothofthegeneralandlibrarycommittees. If (ii) is interpreted as implying that there are some individualswho are on both the general andlibrarycommittees,thenitfollowsthat(ii)and(iii)areinconsistentwitheachother.

349.Given that themiddle termisdistributed twice in thepremissesofa


syllogism, determine directly (i.e., without any reference to the mnemonicverses or the special rules of the figures) inwhat differentmoods itmightpossiblybe.[K.]

Thepremissesmustbeeitherbothaffirmative,oroneaffirmativeandonenegative. Inthefirstcase,bothpremissesbeingaffirmativecandistributetheirsubjectsonly.Themiddletermmust,therefore,bethesubjectineach,andbothmustbeuniversal.Thislimitsustotheonesyllogism,—405

AllMisP,AllMisS,


Inthesecondcase,onepremissbeingnegative,theconclusionmustbenegativeandwill,therefore,distribute the major term. Hence, the major premiss must distribute the major term, and also (byhypothesis) the middle term. This condition can be fulfilled only by its being one or other of thefollowing,—NoMisPorNoPisM.Themajorbeingnegative,theminormustbeaffirmative,andinordertodistributethemiddletermmustbeAllMisS. Inthiscasewegettwosyllogisms,namely,—

NoMisP,AllMisS,

therefore, SomeSisnotP ;NoPisM,AllMisS,


Thegivenconditionlimitsus,therefore,tothreesyllogisms(oneaffirmativeandtwonegative);andbyreferencetothemnemonicverseswemayidentifythesewithDaraptiandFelaptoninfigure3,andFesapoinfigure4.

350.Ifthemajorpremissandtheconclusionofavalidsyllogismagreeinquantity,butdifferinquality,findthemoodandfigure.[T.]Sincewecannothaveanegativepremisswithanaffirmativeconclusion,themajorpremissmustbe

affirmativeandtheconclusionnegative.Itfollowsimmediatelythat,inordertoavoidillicitmajor,themajorpremissmustbeAllPisM(whereMisthemiddletermandPthemajorterm).Theconclusion,therefore, must be No S is P (S being the minor term); and this requires that, in order to avoidundistributedmiddleandillicitminor,theminorpremissshouldbeNoSisMorNoMisS.HencethesyllogismisinCamestresorinCamenes.

351.Givenavalidsyllogismwithtwouniversalpremissesandaparticularconclusion,suchthatthesameconclusioncannotbeinferred,ifforeitherofthe premisses is substituted its subaltern, determine themood and figure ofthesyllogism.[K.]LetS,M,Pberespectivelytheminor,middle,andmajortermsofthegivensyllogism.Then,since


theconclusionisparticular,itmustbeeitherSomeSisPorSomeSisnotP.406 First,ifpossible,letitbeSomeSisP. The only termwhich need be distributed in the premisses isM. But sincewe have two universalpremisses, two termsmust be distributed in them as subjects.438 One of these distributionsmust besuperfluous;anditfollowsthatforoneofthepremisseswemaysubstituteitssubaltern,andstillgetthesameconclusion. TheconclusioncannotthenbeSomeSisP. Secondly,ifpossible,lettheconclusionbeSomeSisnotP. IfthesubjectoftheminorpremissisS,wemayclearlysubstituteitssubalternwithoutaffectingtheconclusion.ThesubjectoftheminorpremissmustthereforebeM,whichwillthusbedistributedinthispremiss.M cannot also be distributed in the major, or else it is clear that its subaltern might besubstituted for the minor and nevertheless the same conclusion inferred. The major premiss must,therefore,beaffirmativewithMforitspredicate.Thislimitsustothesyllogism—

AllPisM,NoMisS,

therefore, SomeSisnotP ;

andthissyllogism,whichisAEOinfigure4,doesfulfilthegivenconditions,foritbecomesinvalidifeitherofthepremissesismadeparticular. Theaboveamountstoageneralproofofthepropositionlaiddowninsection246:—EverysyllogisminwhichtherearetwouniversalpremisseswithaparticularconclusionisastrengthenedsyllogismwiththesingleexceptionofAEOinfigure4.

438 Wehereincludethecaseinwhichthemiddletermisitselftwicedistributed.

352. Given two valid syllogisms in the same figure in which themajor,middle,andminortermsarerespectivelythesame,shew,withoutreferencetothe mnemonic verses, that if the minor premisses are subcontraries, theconclusionswillbeidentical.[K.]TheminorpremissofoneofthesyllogismsmustbeO,andthemajorpremissofthissyllogismmust,

therefore,beAandtheconclusionO.Themiddleandthemajortermshavingthentobedistributedinthepremisses,thissyllogismisdetermined,namely,—



407Sincetheothersyllogismistobeinthesamefigure,itsminorpremissmustbeSomeSisM ;themajormustthereforebeuniversal,andinordertodistributethemiddletermitmustbenegative.Thissyllogismthereforeisalsodetermined,namely,—

NoPisM,SomeSisM,


Theconclusionsofthetwosyllogismsarethusshewntobeidentical.


353. Find out in which of the valid syllogisms the combination of onepremiss with the subcontrary of the conclusion would establish thesubcontraryoftheotherpremiss.[J.]In the original syllogism (α) let X (universal) and Y (particular) prove Z (particular), the minor,

middle,andmajortermsbeingSM,andP,respectively.Thenwearetohaveanothersyllogism(β)inwhichXandZ1(thesub-contraryofZ)proveY1(thesub-contraryofY).Inβ,SorPwillbethemiddleterm. Itisclearthatonlyonetermcanbedistributedinαiftheconclusionisaffirmative,andonlytwoiftheconclusionisnegative.HenceScannotbedistributedinα,anditfollowsthatitcannotbedistributedinthepremissesofβ.ThemiddletermofβmustthereforebeP,andasXmustconsequentlycontainPitmustbethemajorpremissofαandYtheminorpremiss. ZmustbeeitherSiPorSoP.First,letZ=SiP.ThenitisclearthatX=MaP,Z1=SoP,Y1=SoM,Y=SiM.Secondly,letZ=SoP.ThenZ1=SiP,X=PaMorMePorPeM(sinceitmustdistributeP),Y1=SiM(ifXisaffirmative)orSoM(ifXisnegative),Y=SoMorSiMaccordingly. Hencewehavefoursyllogismssatisfyingtherequiredconditionsasfollows:—

MaP MeP PeM PaMSiM SiM SiM SoM⎯⎯ ⎯⎯ ⎯⎯ ⎯⎯

SiP SoP SoP SoP

Itwillbeobservedthattheseareallthemoodsofthefirstandsecondfigures,inwhichonepremissisparticular.

354.Isitpossiblethatthereshouldbeavalidsyllogismsuchthat,eachofthe premisses being converted, a new syllogism is obtainable giving aconclusion in which the old major and minor terms have changed places?Provethecorrectnessofyouranswerbygeneralreasoning,andifitisinthe408 affirmative, determine the syllogism or syllogisms fulfilling the givenconditions.[K.]If sucha syllogismbepossible, it cannothave twoaffirmativepremisses,or (sinceA canonlybe

convertedperaccidens)weshouldhavetwoparticularpremissesinthenewsyllogism. Therefore, the original syllogism must have one negative premiss. This cannot beO, sinceO isinconvertible. Therefore,onepremissoftheoriginalsyllogismmustbeE. First, let this be themajor premiss.Then theminor premissmust be affirmative, and its converse(beingaparticularaffirmative),willnotdistributeeitherofitsterms.Butthisconversewillbethemajorpremissofthenewsyllogism,whichalsomusthaveanegativeconclusion.Weshouldthenhaveillicitmajorinthenewsyllogism;andhencetheabovesuppositionwillnotgiveusthedesiredresult. Secondly, let the minor premiss of the original syllogism be E. The major premiss in order todistribute theoldmajor termmust beA,with themajor term as subject.We get then the following,satisfyingthegivenconditions:—

AllPisM,


NoMisS,orNoSisM,therefore,NoSisP,orSomeSisnotP ;

thatis,wereallyhavefoursyllogisms,suchthatbothpremissesbeingconverted,thus,

NoSisM,orNoMisS,SomeMisP,

wehaveanewsyllogismyieldingaconclusioninwhichtheoldmajorandminortermshavechangedplaces,namely,

SomePisnotS.

Symbolically,—

PaM, SeM, ⎱

MeS, ⎱ or MeS, ⎰

or SeM, ⎰ MiP,⎯⎯ ⎯⎯

∴or

SePSoP

⎱⎰

∴ PoS.

Ifitberequiredtoretainthequantityoftheoriginalconclusion,thatconclusionmustbeSoP,inthiscasethenwehaveonlytwosyllogismsfulfillingthegivenconditions.

355.ShewthatiftheproportionofB’soutoftheclassAisgreaterthanthatoutoftheclassnot-A,thentheproportion409ofA’soutoftheclassBwillbegreaterthanthatoutoftheclassnot-B.439[J.]

439 Thisand thefollowingproblemcannotproperlybecalledproblemson thesyllogism.Theyaregivenasexamplesinnumericallogic.

LetthenumberofA’sbedenotedbyN(A),thenumberofAB’sbyN(AB),&c. Then,sinceEveryAisABorAb(bythelawofexcludedmiddle)andNoAisbothABandAb(bythelawofcontradiction),itfollowsthat

N(A)=N(AB)+N(Ab).

Wehavetoshewthat

N(AB) N(Ab)⎯⎯ > ⎯⎯

N(B) N(b)

followsfrom

N(AB) N(aB)⎯⎯ > ⎯⎯ .N(A) N(a)

Thiscanbedonebysubstituting

N(AB)+N(Ab)forN(A),&c.

Thus,


N(AB) N(aB)⎯⎯ > ⎯⎯,N(A) N(a)N(a) N(A)

∴ ⎯⎯ > ⎯⎯,N(aB) N(AB)N(aB)+N(ab) N(AB)+N(Ab)

∴ ⎯⎯ > ⎯⎯,N(aB) N(AB)N(ab) N(Ab)

∴ ⎯⎯ > ⎯⎯,N(aB) N(AB)N(ab) N(aB)

∴ ⎯⎯ > ⎯⎯,N(Ab) N(AB)N(Ab)+N(ab) N(AB)+N(aB)

∴ ⎯⎯ > ⎯⎯,N(Ab) N(AB)N(b) N(B)

∴ ⎯⎯ > ⎯⎯,N(Ab) N(AB)N(AB) N(Ab)

∴ ⎯⎯ > ⎯⎯.N(B) N(b)

356.Giventhenumber(U)ofobjects intheUniverse,andthenumberofobjects ineachof theclassesx1,x2,x3,…xn, shewthat the leastnumberofobjectsintheclass(x1x2x3…xn)

=U−N(x1)−N(x2)−N(x3)…−N(xn).410

whereN(x1)meansthenumberofthingswhicharenotx1;N (x2)means thenumberofthingswhicharenotx2;&c.[J.]

GivenN(x1),N(x2),&c., thenumberofobjects in theclass (x1orx2…orxn) isgreatestwhennoobjectbelongstoanypairoftheclassesx1,x2,…;andinthiscaseit=N(x1)+N(x2)…+N(xn). Hencetheleastnumberinthecontradictoryclass,x1x2x3…xn,

=U−N(x1)−N(x2)…−N(xn).

357.Provethatwiththreegivenpropositions(oftheformsA,E,I,O)itisneverpossibletoconstructmorethanonevalidsyllogism.[K.]

358. On the supposition that no proposition is interpreted as implying the existence either of itssubjectorofitspredicate,findinwhatcasesthereductionofsyllogismstofigure1isinvalid.[K.]

359. Given a valid syllogism, determine the conditions under which the contradictories of the


premisseswillfurnishpremissesforanothervalidsyllogismcontainingthesameterms.Howwill theconclusionsofthetwosyllogismsberelatedtooneanother?[K.]

360.Shewthatthenumberofpauperswhoareblindmalesisequaltotheexcess,ifany,ofthesumofthewholenumberofblindpersons,addedtothewholenumberofmalepersons,addedtothenumberofthosewhobeingpaupersareneitherblindnormales,abovethesumofthewholenumberofpaupers,added to thenumberof thosewhonot beingpaupers areblind, and to thenumberof thosewhonotbeingpaupersaremale.[Jevons,PrinciplesofScience.]

361.Shewthat,ifXandYareanytwopropositionscontainingacommonterm,then(a)oneofthefourcombinationsXY,XYʹ,XʹY,XʹYʹwillalwaysformunstrengthenedpremissesforavalidsyllogism;(b)eitheronlyoneofthefourcombinationswilldoso;or,iftwo,thesyllogismssoformedwillbeofthesamemood.[RR.]

362. Two arguments whose premisses are mutually consistent but which contain sub-contraryconclusionsareformedinthesamefigurewiththesamemiddleterm. Findoutdirectlyfromthegeneralrulesofsyllogismwhatcanbeknownwithregard to themoodsandfigureofthetwogivenarguments.[J.]

411363.SomeMisnotP,AllS isallM.What conclusion follows from thecombinationof thesepremisses? CanyouinferanythingeitheraboutSintermsofPoraboutPinterms,ofSfromtheknowledgethatboththeabovepropositionsarefalse?[K.]

364.(i)EitherallMisallPorSomeMisnotP ;(ii)SomeSisnotM.Whatisallthatcanbeinferred(a)aboutSintermsofP,(b)aboutPintermsofS,fromtheknowledgethatboththeabovestatementsarefalse?[K.]

365.(a)“Agoodtemper isproofofagoodconscience,and thecombinationof these isproofofagooddigestion,whichagainalwaysproducesoneortheother.”Shewthatthisispreciselyequivalenttothefollowing:“Agoodtemperisproofofagooddigestion,andagooddigestionofagoodconscience.” (b)Examine(bydiagramsorotherwise)thefollowingargument:—“Patriotismandhumanitarianismmust be either incompatible or inseparable; and though family-affection and humanitarianism arecompatible, yet either may exist without the other; hence, family affection may exist withoutpatriotism.”Reducetheargument,ifyoucan,toordinarysyllogisticform;anddeterminewhetherthepremissesstateanythingmorethanisnecessarytoprovetheconclusion.[J.]

366. “All scientific persons are willing to learn; all unscientific persons are credulous; therefore,somewhoarecredulousarenotwillingtolearn,andsomewhoarewillingtolearnarenotcredulous.” Shew that the ordinary rules of immediate andmediate inference justify this reasoning; but that acertainassumptionisinvolvedinthusavoidingtheapparentillicitprocess.Shewalsothat,acceptingthevalidityofobversionandsimpleconversion,wehaveananalogouscaseinanyinferenceofaparticularfromauniversal,[J.]

367.Aninvalidsyllogismofthesecondfigurewithaparticularpremissisfoundtobreakthegeneralrulesofthesyllogisminthisrespectonly,thatthemiddletermisundistributed.Iftheparticularpremissisfalseandtheothertrue,whatdoweknowaboutthetruthorfalsityoftheconclusion?[K.]

368. A syllogism is found to offend against none of the syllogistic rules except that with twoaffirmativepremissesithasanegativeconclusion.Determinethemoodandfigureofthesyllogism.[K.]

412369.Giventwovalidsyllogismsinthesamefigureinwhichthemajor,middle,andminorterms


arerespectivelythesame,shew,withoutreferencetothemnemonicverses,thatiftheminorpremissesarecontradictories,theconclusionswillnotbecontradictories.[K.]

370. Find two syllogisms, having neither strengthened premisses nor weakened conclusions, andhavingM andN respectively as theirmiddle terms,which satisfy the following conditions: (a) theirconclusions are to be subcontraries; (b) their premisses are to prove that Some M is N, and to beconsistentwiththefactthatSomeMisnotN.[J.]

371. Is it possible that there should be two syllogisms having a common premiss such that theirconclusions,beingcombinedaspremissesinanewsyllogism,maygiveauniversalconclusion?Ifso,determinewhatthetwosyllogismsmustbe.[N.]

372. Three given propositions form the premisses and conclusion of a valid syllogism which isneitherstrengthenednorweakened.Shewthat if twoof thepropositionsarereplacedby theircontra-complementaries,theargumentwillstillbevalid,providedthatthepropositionremainingunalterediseitherauniversalpremissoraparticularconclusion.[J.]

373.AcertainpropositionstandsasminorpremissofasyllogisminthesecondfigurewhosemajortermisX.Thesamepropositionstandsalsoasmajorpremissofasyllogisminthethird figurewhoseminor termisY. If thegivensyllogismsareboth formallyandmateriallycorrect, shewhowineverycasewemayconcludesyllogisticallythat“someYisnotX”[J.]

374.Findoutthevalidsyllogismsthatmaybeconstructedwithoutusingauniversalpremissofthesamequalityastheconclusion. Shewhowthesesyllogismsmaybedirectlyreducedtooneanother;andrepresentdiagrammaticallythecombinedinformationthattheyyield,onthesuppositionthattheyhavethesameminor,middle,andmajortermsrespectively.[J.]

375.Expresstheexactinformationcontainedinthetwopropositions,AllSisM,AllMisP,bymeansof(1)twopropositionshavingSandnot-Srespectivelyassubjects;(2)twopropositionshavingMandnot-Mrespectivelyassubjects;(3)twopropositionshavingPandnot-Prespectivelyassubjects.[K.]


CHAPTERIX.

THECHARACTERISTICSOFINFERENCE.

376.TheNatureofLogical Inference.—Thequestionas to thenatureandcharacteristicsofinference,sofarasitssolutiondependsonthemoreorlessarbitrarymeaningthatwechoosetoattachtotheterm“inference,”isamerelyverbal question. The controversies to which the question has given rise donot,however,dependmainlyonverbalconsiderations;andthefactthattheypartlydosohasincreasedrather thandiminishedthedifficultieswithwhichtheproblemisbeset.

It will be generally agreed that inference involves a passage of thoughtfromagivenjudgmentorcombinationofjudgmentstosomenewjudgment.This alone, however, is not sufficient to constitute inference in the logicalsense. The formation of new judgments by the unconscious association ofideas is a psychological process which might be brought under the abovedescription;butitisnotwhatwemeanbylogicalinference.

(1) It is, in the first place, an essential characteristic of logical inferencethat the passage of thought should be realised as such. The connexionbetween the judgment or judgments from which we set out and the newjudgment atwhichwe arrivemust be one ofwhichwe are, at any rate onreflection,explicitlyconscious.

(2) But this again is not in itself sufficient. There must further be anapprehension that thepassageof thought isone that isvalid ; theremust, inother words, be a recognition that the acceptance of the judgment orjudgments originally given 414 constitutes a sufficientground or reason foracceptingthenewjudgment.

Inlogicalinference,then,IdonotmerelypassfromPtoQ ;IrealisethatIamdoingso.And Iapprehend further that the truthofPbeinggranted, thetruthofQ necessarily follows. For logical inference, in short, it is required


that thereshouldbea logical relationbetweenapremissorpremissesandaconclusion, not merely a psychological relation between antecedents andconsequentsinatrainofthought.

Thisdistinctionbetween the logicaland thepsychologicalmaybebrieflyillustrated by reference to what are known as acquired perceptions.Psychologistsare,forexample,agreedthatourperceptionofdistancethroughthesenseofsightorthesenseofsoundisnotimmediate,butacquiredinthecourse of experience. Here then we have a case in which one perceptiongenerates another; but there is no conscious passing from premisses to aconclusion,andnothingthatcanproperlybecalledinference.Hencewemustreject Mill’s dictum that “a great part of what seems observation is reallyinference” (Logic, iv. 1, § 2), so far as the dictum is based—as to a largeextentitis—onthepositionthatagreatpartofourperceptionsareacquired,notimmediate.Here,aswellasinconnexionwithsomeofhisotherandmoreimportantlogicaldoctrines,Millisopentothechargeoffailingtodistinguishbetweenthecauseofabeliefanditsgroundorreason.

377.TheParadoxofInference.—Thedescriptionoflogicalinferencegivenin the preceding section leads up immediately to the fundamental difficultywhichanydiscussionofthesubjectmustinevitablybringtotheforefront.Weareinfactface tofacewithwhathasaptlybeendesignatedthe“paradoxofinference.” On the one hand, we are to advance to something new; theconclusionof an inferencemustbedifferent from thepremisses, andhencemustgobeyondthepremisses.Ontheotherhand,thetruthoftheconclusionnecessarilyfollowsfromthetruthofthepremisses,andtheconclusionmustthereforeinsomesensebecontainedinthepremisses.

Theremayappeartobeacontradictionhere;andthisview415tendstobeconfirmedwhenitisfoundthatthetwocharacteristicsofinferencereferredtoarebydifferentschoolsof logiciansusedinsuchawayasbetweenthemtodeprivethecategoryofinferenceofanycontentwhatsoever.

Ontheonehand,bylayingstressonthecharacteristicofnovelty,wemaybeledtodoubtwhetherformalinferenceofanydescriptioncanproperlybesocalled.For inall such inference theconclusion is implicitlycontained inthe premisses, and in uttering the premisses we have virtually committed


ourselvestotheconclusion.Howthencanwebesaidtomakeanyadvancetowhatisreallynew?

On theother hand, by laying stress on the characteristic of necessity,wemay be led to doubt whether any inductive inference can properly be socalled.Forinsuchinferencethefalsityoftheconclusionisnotdemonstrablyinconsistentwiththetruthofthepremisses.Wemayholdthatifthepremissesare true the conclusionwill be true. But can we hold that itmust be true,unlesswealsoholdthatinaffirmingthepremisseswehavevirtuallyaffirmedtheconclusiontoo?Andthenwearebackontheotherhornofthedilemma.

This is not the place atwhich to discuss the difficulty from the point ofviewofinductivelogic.Wemust,however,attemptasolutionfromthepointofviewofformallogic.

378.Thenatureofthedifferencethattheremustbebetweenpremissesandconclusioninaninference.—Inordertofindasolutionofthedifficulty,sofaras formal inference is concerned, wemust pursue our analysis further.Wehavesaidthattheconclusionmustbedifferentfromthepremissorpremisses.But we have not yet asked what must be the nature of the difference orwherein it must consist; and it is on the answer to this question thateverythingturns.

Ifweconsider twosentencesweshall find that theymaydiffer fromoneanother from three distinct standpoints, representing three degrees ofdifference.

(1)Inthefirstplace,twosentencesmaydifferfromoneanotherfromtheverbalstandpointonly;thatistosay,thoughdifferentinthewordsofwhichthey are made up, they may have the same meaning, and what the one isintended to convey 416 to the mind may be precisely what the other isintended to convey. In this case, regarded as propositions and not asmeresentences, they cannot be said to be really different at all; for they do notrepresentdifferentjudgments.

This (to take an example from Jevons) applies to two such sentences asVictoriaistheQueenofEngland,VictoriaisEngland’sQueen.Itappliesalsoto a statement expressed in a given language and the same statement


translated into a second language, assuming that an absolutely literaltranslationispossible.

It has indeed been maintained by some writers that a difference ofexpression necessarily involves some difference of thought.But this at anyrate appears not to be the case where one single word is substituted foranother completely coincidentwith it both indenotation and in connotation(as thought by the speaker). Where one complex term is substituted foranother(forexample,England’sQueenforQueenofEngland) theremaynodoubt be involved some change in the order of thought; but this does notnecessitate any change of meaning in the thought considered as a whole.Againweoughtperhapsnottosaythatthesamepropositionexpressedintwodifferent languages has absolutely the same mental equivalent, since aconsciousness of the actual words of which a proposition consists mayconstitute part of its mental equivalent. But, as before, this makes nodifferenceinthemeaningthatthepropositionisintendedtoconvey.

It should be added that when we have a judgment expressed in twodifferentlanguagesorintwodifferentformsinthesamelanguage,thereis(ormaybe)involvedthefurtherjudgmentthatthetwomodesofexpressionareequivalent.Adistinctissueis,however,hereraised.440

440 This issue is, I think, involved in an argument used byMiss Jones (in anarticle inMind, April, 1898) in support of the doctrine that we have inferencewheneverwe pass from a given proposition to another that is verbally differentfromit; forexample,fromVictoria isQueenofEngland toVictoria isEngland’sQueen.Thepassagefromoneofthesepropositionstotheotheris,inMissJones’sview, not indeed a formal immediate inference, but a syllogism in which anunderstood premiss has to be supplied: thus,Victoria isQueen of England,TheQueenofEngland isEngland’sQueen, therefore,Victoria isEngland’sQueen. Itmay, Miss Jones adds, seem futile or even puerile to set out at length whateverybodyornearlyeverybodyknowswithouttelling;buttheremaybecases(e.g.,the case of a child or of a foreigner learning the English language) in which areasoningofthiskindhastobegonethrough.

Itappearstomethatthereishereafailuretodistinguishbetweentwodifferentpoints of view. We may no doubt draw an inference as to the equivalence ofmeaningoftwotermsortwoexpressions,wherethewholeargumentisconcernedwith themeaning of terms or the force of expressions. Thus, to take (or, rather,adapt) another of Miss Jones’s examples, we may readily admit that there isinference if a German argues that because the word Valour is equivalent inmeaning to the word Tapferkeit, and the word Bravery is also equivalent in


meaning to the word Tapferkeit, therefore, the words Valour and Bravery areequivalent inmeaning.Again, a child or a foreignermay arrive by a process ofinference at the equivalence of such forms asQueen of England andEngland’sQueen.But in thesyllogismgivenabovethefirstpremissandtheconclusionarestatements of fact, while the second premiss is a statement as to modes ofexpression,itsimportbeing“TheexpressionQueenofEnglandisequivalenttotheexpressionEngland’sQueen.”Hencetherearemorethanthreetermsandwehavenotproperlyanysyllogismatall.Sofarasthereisinferenceinthecasesupposed,itwillbesomethinglikethefollowing,—“TheformofwordsQueenofEnglandisequivalent in meaning to the form of wordsEngland’sQueen,” therefore, “ThejudgmentwhichisexpressedintheformVictoriaisQueenofEnglandmayalsobeexpressedintheformVictoriaisEngland’sQueen.”This is the inference, ifany,that a foreigner studying the languagewouldmake;and it isverydifferent fromprofessingtopassfromthejudgmentVictoriaisQueenofEnglandtothejudgmentVictoriaisEngland’sQueen.

417 (2)Inthesecondplace,wemayhaveadifferencewhichgoesbeyondmere difference of expression, and constitutes a difference in subjectivemeaning, though there may still be no difference from the objectivestandpoint. In this case we have two distinct propositions, not merely twodifferentsentences,andthesepropositionsaretheexpressionsoftwodifferentjudgments.

This relation holds in my view between a proposition and itscontrapositive; for example, between Euclid’s twelfth axiom, “If a straightlinemeettwostraightlinessoastomakethetwointerioranglesonthesamesideofittakentogetherlessthantworightangles,thesestraightlines,beingcontinuallyproduced,shallatlengthmeetonthatsideonwhicharetheanglesthatare less than tworightangles,”and thesecondpartof the twenty-ninthproposition of his first book, “If a straight line fall on two parallel straightlines,itshallmakethetwo418interioranglesonthesamesidetogetherequaltotworightangles.”Itcannotbesaidthatinsuchacaseasthiswehaveanyobjectivedifference,anydifference in thematterof factasserted;butat thesame timewehold that the two judgments towhichexpression isgivenarenottoberegardedasidenticalquâjudgments.

Tothisdistinctionweshallreturnshortlyfromamorecontroversialpointofview.

(3)Inthethirdplace,oursentencesmaydiffernotmerelyfromtheverbalandsubjectivestandpoints,butalsofromtheobjectivestandpoint; theymay


affirmdistinctmattersoffact.As,forexample,ifoneofthemstatesthatallpotassium with which we have experimented takes fire when thrown onwater, and the other that a piece of potassiumwithwhichwe have not yetexperimentedwilldothesame.

Now in all three of these caseswe have novelty, and the question to bedecided iswhich of the three kinds of novelty is requisite in order thatwemay have inference. I hold the right answer to be that, for inference,subjectivenoveltyisnecessaryandsufficient.

There is practically universal agreement that something more than meredifferenceofverbalexpressionisrequisiteforinference.441

441 MissJonesholdsthatverbaldifferencesuffices;butthisisonlybecauseshealsoholds, aswehave seen, thatwecannothavemereverbaldifference, that is,differenceofexpressionwithoutdifferenceofthought.

Objectivenoveltyiscertainlysufficient,butisitrequisite?ItisaffirmedtobesobywritersoftheschoolofMill.Thismayofcoursebeamerequestionofdefinition;thatistosay,inferencemaybedefinedabinitioinsuchawayastorequirethattheconclusionreachedshallexpresssomeobjectivefactnotcontainedinthedataonwhichitisbased.Thematterbeingthusdecidedbydefinition, it followswithoutcontroversy thatcontraposition,syllogism,andother formal inferences (so called) are not properly to be spoken of asinferences at all. But there a good deal more than a mere question ofdefinitioninvolved.Thosewhodemandobjectivenoveltyappeartoholdthatwithoutitwecannothavemorethanmere419verbalnovelty.Theyoverlook,oratanyratepracticallydeny,thepossibilityoftakinganintermediatecoursewherebywemay have somethingmore than verbal novelty, but somethinglessthanobjectivenovelty.

Herethenwehaveoneforminwhichthepointmainlyatissueinregardtothenatureof inferencepresents itself. Is itpossiblefor twojudgments tobedifferentquâjudgments,althoughfromtheobjectivestandpointoneofthemstates nothing that is not also stated by the other? Or, to put the questiondifferently,cantwojudgments(orsetsofjudgments)bedistinctasjudgmentsalthough they are not logically independent, that is, although self-evidentrelationsexistbetweenthemsuch that the truthofoneof theminvolves the


truthoftheother?

Iamready toadmit that it isnoeasymatter todrawahardand fast linedeterminingwheremereverbal novelty ends and subjectivenoveltybegins.Before attempting to dealwith this difficulty, however, Iwill endeavour toshewthatthereundoubtedlyarecasesinwhichwehaveprogressinthoughtwithoutreachinganythingthatisobjectivelynew.

Mill,aftergivingexamplesofso-calledimmediateinferences,says,“Inallthesecasesthereisnotreallyanyinference;thereisintheconclusionnonewtruth,nothingbutwhatwasalreadyassertedinthepremisses,andobvioustowhoeverapprehendsthem”(Logic,ii.1,§2).Nowitiscertainlythecasethatin any formal inference the conclusion is implicitly contained in thepremisses,andaffirmsnoabsolutelynewfact.Butitisonethingtosaythataconclusionisvirtuallycontainedincertainpremisses,andquiteanothertosaythatitisobvioustowhoeverapprehendsthepremisses.Theidentificationofthese twopositions isoneof theunfortunateconsequencesof takingsimpleconversionas the typeofall immediate inference,andasinglesyllogisminBarbara as the typeof allmediate formal inference. Itmaybedifficult foranyonetoapprehendthatnoSisPwithoutatonceapprehendingthatnoPisS,or toapprehend thepremissesofa syllogism inBarbarawithout atonceapprehendingtheconclusionalso.Thesecaseswillneeddiscussion;butjustnow we are more concerned to point out 420 that there are other formalinferences against which any similar charge of obviousness cannot bebrought.

Allthetheoremsofgeometryarevirtuallycontainedincertainaxiomsandpostulates, and if we can exhaustively enumerate the axioms there is in asensenonewgeometricalfactleftforustoassert.Yetnoonewouldsaythatthe whole of geometry is at once obvious to anyone who has clearlyapprehended the axioms.We shall, however, dealwith syllogistic inferencemoreindetail inalatersection.Forthepresentwewillinthemainconfineourselvestoimmediateinferences.

In order to shew that the conclusion of an immediate inference is notalways immediately obvious to anyone who clearly apprehends the givenpremiss,itmaybepointedoutthatitisEuclid’spracticetogiveindependent


proofsofcontrapositives.442Forexample, thesecondpartofEuclid I.29 isthecontrapositiveofaxiom12.ButitisimpossibletosupposethatifEuclidhad regarded I. 29 as not really distinct from axiom 12, but merely as arepetition of that axiom in other words, he would have given an elaborateproofofit.Thefollowingaretwootherfairlysimpleexamplesofimmediateinferences:WhereBisabsent,eitherAandCarebothpresentorAandDarebothabsent,therefore,WhereCisabsent,eitherBispresentorDisabsent ;WhereAispresent,eitherBandCarebothpresent,orCispresentwithoutD,orCispresentwithoutF,orHispresent,therefore,WhereCisabsent,weneverfindHabsent,Abeingpresent.


442 Seenote4onpage136.

In such cases as these, and they are comparatively simple ones of theirkind,itcannotbemaintainedthattheconclusionisatonceobviouswhenthepremissisgiven.Asamatteroffact,mistakesarenotunfrequentlymadeinimmediateinferencesofastillsimplerandmoreelementarycharacter.

379.The Direct Import and the Implications of a Proposition.—At thispoint a question may fairly be raised as to how we determine what is theexplicit forceof agivenproposition, assuming theproposition tobeclearlyunderstoodandfullygraspedbythemind.Thisquestionisbynomeanseasy421 toanswer,and thedifficultywhich itpresents is thesourceof thedoubtwhichsometimesariseswhenweattempttodrawthelinebetweenimmediateinferencesandmereverbaltransformations.

If immediate inferences are possible, we must be able to discriminatebetweenthedirectlogicalimport(ormeaning)ofapropositionanditslogicalimplications;anditmustbepossibletograspfullythemeaningwithoutatthesame time necessarily realising all the implications.443 We may begin bydistinguishingbetween(1)thecontentofthejudgmentactuallypresenttoourmind when we utter or accept a proposition in ordinary discourse or inordinaryreading;(2)thecontentofthejudgmentwhichonreflectionweareabletoregardasconstitutingthefulllogicalmeaningoftheproposition;(3)the content of this judgment together with the content of other judgmentswhichitlogicallyimplies.


(1) is a psychological product which may be, and usually is, logicallyimperfect;thatistosay,itneedstobeamplifiedifwearefullytorealisethemeaning of the proposition. Such amplification cannot be regarded asconstituting inference.For, inmakingany inference,ourstartingpointmustbethepropositionconsideredinitslogicalcharacter.Theinferencecomesinwhenwepassfrom(2)to(3).Thequestion,however,arisesastohowfartheamplificationistoextendifourobjectistostopshortat(2).Inotherwords,wheredoesmeaningendandimplicationbegin?

It has been pointed out at an earlier stage that in assigning to given


combinations of words their logical import there is a certain element ofarbitrariness.There isoftenasimilarelementofarbitrariness informulatingthefundamentalaxiomsofascience,aswellasinframingdefinitions.Thus,in geometry we cannot do without some special axiom relating to parallelstraightlines,butwehavesomechoiceastowhattheaxiomshallbe.Hencewhatisanaxiominonesystemmaybeatheoreminanother,andviceversâ.Similarly,whetherQ is toberegardedaspartof themeaningofP,orasaninference from P, may be relative to the interpretation 422 adopted of thescheduleofpropositionstowhichPbelongs.Someillustrationsofthispointwillbegivenshortly.

Wehavecitedcases inwhich itappearsclear thatwehave inferenceandnotmereverbaltransformation.Butinmostofthesecasesintermediatestepsmaybeinserted;andifthisisdonetothefullestpossibleextent,theprogressat each stepmay be so slight that itmay not be at all easy to saywhereinpreciselytheinferenceistobefound.

Wemustthenproceedtoconsiderthelimitingcasesinwhichtheremaybelegitimatedoubtastowhetherwehaveinferenceornot.Oneofthesecasesisthatofconversion.Thequestionwhetherthereisinferenceinconversionmaybeinitself,asMrBosanquetputsit,“apointoflittleinterest”(EssentialsofLogic,p.141).Nevertheless,asalimitingcase, it isnotlightlytobeputonone sidewhenwe are attempting to decidewhat fundamentally constitutesinference.

Itappearstomethatconversionisaprocessofinferenceifwearedealingwithascheduleofpropositionsinwhichthepredicativereadingisadopted.Insuch a schedule the primary import of the various propositions involves adifferentiation between subject and predicate, and to predicateP ofS or todenythatPcanbepredicatedofSisadifferentthingfrompredicatingSofPor denying thatS can be predicated ofP.Moreoverwemay grasp the onerelationwithoutnecessarilyrealisingwhetheritdoesordoesnotinvolvetheother.Butinanequationalsystemitisdifferent.Iftwoclassesareaffirmedtobe identical it ismerely averbalquestionwhich ismentioned first, andwecannotconsiderthatwehavemadeanyprogressinthoughtwhenwemerelyaltertheorderinwhichtheyarenamed.Itfollowsthatwemustconsiderthat


wehave inferencewhenwe reduceapropositionexpressedpredicatively totheequationalform.

Ineitherschedule,contraposition(oraprocessanalogoustocontraposition)presentsitselfasaninference.Intheonecase,wehaveAllSisP,therefore,AnythingthatisnotPisnotS ;intheothercase,S=SP,therefore,Pʹ=PʹSʹ.

Supposeagainthatwehaveanexistentialschedule,andthatwestartfromthepropositionSPʹ=0[Thereisnothingthatis423SandatthesametimenotP].Herewhatcorresponds toconversion is thepassage toEitherPS>0orS=0[ThereissomethingthatisbothPandSorelseSisnon-existent];and,whatcorrespondstocontrapositionisthepassagetoPʹS=0[ThereisnothingthatisnotPandatthesametimeS].Conversion,butnotcontraposition,nowappearsasaprocessof inference.Itfollowsthat thereis inferencewhenwepasstothisschedulefromeitheroftheothers,orviceversâ.

Afurtherconsequencetobedrawnfromtheaboveconsiderationsisthatifpropositionsaregivenatrandom,inferencemayat theoutsetberequiredinordertoadaptthemtoagivenlogicalschedule,thoughasarulethiswillnotbenecessary.Thispointhasalreadybeentoucheduponinsection48.

380. Syllogisms and Immediate Inferences.—In the above argument wehaveconfinedourselvesmainlytotheconsiderationofimmediateinferences.The same question in relation to the syllogism usually presents itself in aslightly different form, namely, whether every, syllogism involves a petitioprincipii ;andweshalldiscussitinthisforminthefollowingsection.Inthemeantime, we may observe that if there is no such thing as immediateinference properly so called, then the claims of the syllogism to containinference become very hard to maintain. For by the aid of immediateinferences the premisses of a syllogism can be combined into a singleproposition, and the conclusion can then be obtained as an immediateinferencefromthecombination.444


Asanexample,wemaytakeasyllogisminBarbara:445

AllMisP, (1)AllSisM, (2)


therefore, AllSisP.

From(1),

EverythingismorP,therefore, EverySismorP.

Combiningthiswith(2)wehave

EverySisM,andalsomorP ;(3)

therefore,EverySisMP(sincenothingcanbeMm);

therefore,EverySisP.445 Intheargumentthatfollowsm=not-M,s=not-S.

424All the above steps are immediate inferences, except the combinationwhichyields(3).Hence,ifweholdthatsyllogismisinferencewhileso-calledimmediate inference is not, we must regard the whole of the inference asconcentrated in the mere combination of two propositions into a singleproposition;andthisishardlyapositionthatcanbeaccepted.

Thegivensyllogismmightalsobereducedasfollows: From(1)itfollowsthatEverythingismorP ;(4) andfrom(2)wegetEverythingissorM.(5) Combining(4)and(5),Everythingis(sorM)and(morP); therefore,EverythingissmorsPorMP ; therefore,EverySisP.

Wemaynote inpassing that ifelimination is regardedasconstituting theessenceofinference,thenineachoftheaboveresolutionsofthesyllogismallthe inference is concentrated in the last step, and this again seemsparadoxical.

381. The charge of Petitio Principii brought against SyllogisticReasoning.446—The objection to syllogistic reasoning that it necessarilyinvolvespetitioprincipiiisofconsiderableantiquity.ThusSextusEmpiricus(circa200A.D.),oneoftheLaterSkeptics,seekingtodisprovethepossibilityofdemonstration,urged,asoneofhisarguments,thateverysyllogismmovesinacircle,sincethemajorpremiss,uponwhichtheproofof theconclusion


depends, requires in order that it may be itself established a completeenumeration of instances, amongst which the conclusion must itself beincluded.447 The same objection to the syllogism is raised bymany recentlogicians,includingMillandhisfollowers.“Itmust,”saysMill,“begrantedthat ineverysyllogism,consideredasanargument toprove theconclusion,thereisapetitioprincipii”(Logic,ii.3,§2).

446 ThereisaverygooddiscussionofthisquestioninVenn’sEmpiricalLogic,chapter15.ThereadermayalsobereferredtoMansel’seditionofAldrich,noteE,andtoLotze’sLogic,§§98–100.

447 CompareUeberweg,HistoryofPhilosophy(Englishtranslation,i.p.216).

ItmaybesaidattheoutsetthattheplausibilityoftheargumentbywhichMill seeks to justify this position depends a 425 good deal upon a certainambiguity that attaches to the phrase petitio principii.When the charge ofpetitioprincipiiisbroughtagainstareasoning,isitmerelymeant(1)thatthepremisses would not be true unless the conclusion also were true, or is itmeant (2) that the conclusion is necessary for the proof of one of thepremisses? It is clearly one thing to say that the premisses of a certainreasoning cannot be true unless the conclusion is true, and quite another tosaythatwecannotknowthepremissestobetrueunlesswepreviouslyknowtheconclusiontobeso,ortosaythattheproofofthepremissesnecessitatesthattheconclusionshallhavebeenalreadyestablished.Onlyinthesecondoftheabovesensescanpetitioprincipiibe regardedasa fallacy ; andanyonewho,seeking toprove thateverysyllogismisguiltyof the fallacyofpetitioprincipiimerelyshewsthatsyllogisticreasoninginvolvespetitioprincipii intheothersense,himselfcommitsthefallacyofignoratioelenchi.

In his systematic treatment of fallacies, Mill classifies petitio principiiamongst fallacies of confusion, and quotes with approval Whately’sdefinition:itisthefallacy“inwhichoneofthepremisseseitherismanifestlythesameinsensewiththeconclusion,orisactuallyprovedfromit,orissuchasthepersonsyouareaddressingarenotlikelytoknow,ortoadmit,exceptas an inference from the conclusion” (Logic, v. 7, § 2n.). This fallacy hasbeendescribedasbeinga fallacyofproof rather thana fallacyof inference ;that is tosay, itariseswhenweaskhowagiven thesis is tobeestablished,rather thanwhenweaskwhat followsfromagivenhypothesis.Wehave to


enquirewhethereverysyllogismisopento thechargeofpetitioprincipii inthissense.

It isobvious that theanswer to thequestion in thecaseof anyparticularsyllogismdependsuponthegroundsonwhich thepremissesare themselvesaffirmed;andwemaybeginbycallingattentiontocertaincasesinwhichthejusticeofthechargemustbeadmitted,theconclusionofthesyllogismbeingregardedasathesistobeproved.

Onecaseiswhenthemajorpremissisananalyticproposition.448ForifMbydefinitionincludesPamongstits426properties,Iamnotjustifiedinsayingof S that it isM unless I have already satisfied myself that it is P. Thefollowingisanexample:All triangleshave threesides; thefigureABCisatriangle;therefore,ithasthreesides.

448 ThiscaseisnoticedbyLotze,Logic,§99.

A second obvious case of circulus in probando is where we seek toestablishoneofthepremissesofasyllogismbymeansofanothersyllogismin which the ultimate conclusion itself appears as a premiss. For example,—AllMisP(forallSisP,andallMisS);andallSisM ;therefore,allSisP.

A third case, which for our immediate purpose is more important thaneitherof theabove, iswherethemajorpremiss isanenumerativeuniversal,summing up a number of individual instances each one ofwhich has beenseparatelyconsidered.Forexample,AlltheapostleswereJews;Peterwasanapostle; therefore, Peter was a Jew. A universal proposition relating to alimitedclass, suchas the apostles, isusually establishedbyconsidering themembers individually; and if the truth of a universal proposition could beestablished in this manner only, then the charge that syllogistic reasoningnecessarily involves petitio principii would not admit of refutation. ThisappearstobeassumedintheargumentofSextusEmpiricusquotedabove.Itisalsoassumedinthefollowingdilemma,whichhasbeengivenassummingupMill’sdoctrine:“Ifallthefactsofthemajorpremissofanysyllogismhavebeenexamined,thesyllogismisneedless;andifsomeofthemhavenotbeenexamined,itisapetitioprincipii.Buteitherallhavebeenexaminedorsomehavenot.Therefore,thesyllogismiseitheruselessorfallacious,”Mill’sown


argumentmayalsobequoted:“Wecannotbeassuredofthemortalityofallmen,unlesswearealreadycertainofthemortalityofeveryindividualman”(Logic,ii.3,§2).449

449 Bain (Logic,Deduction, p. 208) taking as an example the syllogism, “Allmen are mortal, All kings are men, therefore, All kings are mortal,” asks“Supposing therewere any doubt as to the conclusion that kings aremortal, bywhatrightdoweproclaim,inthemajor,thatallmenaremortal,kingsincluded?”Hethencontinues,“Inordertosay,‘Allmenaremortal,’wemusthavefoundinsomeotherwaythatallkingsandallpeoplearemortal.Sothattheconclusionfirstcontributesitsquotatothemajorpremiss,andthentakesitbackagain.”ThereplytoBain’schallenge is that ifweare indoubtas towhetherkingsaremortal,wemay resolve our doubt by shewing that kings belong to a class themortality ofwhich isadmitted.Thequestion thenresolves itself intowhether it ispossible toestablishthemortalityofmankindingeneralwithoutanyexplicitconsiderationoftheparticularcaseofkings.

427Itcannot,however,foramomentbeallowedthatuniversalpropositionsadmitofproofonlybyenumeration.Propositionsthatdoadmitofsuchproofareindeedgenerallyspeakingoflittleimportance.Thesyllogismischieflyofvalue inferentially where the major premiss is universal in the fullest andmostunlimitedsense,thatis,unconditionallyuniversal,expressingagenerallawdependentonqualitativerelations.Thetruecharacterandvalueofsuchapremiss, though ordinarily written in the formAll S is P, would be betterbroughtoutbytheuseofoneoftheformsAnySisP,WhateverisSisalsoP,ItisthenatureofStobeP,IfanythingisSitisP.450

450 Sigwart holds that, in order properly to understand the value of thesyllogism,we should takeasour type theconditional (or, asheexpresses it, thehypothetical),ratherthanthecategorical,syllogism.Weneed,hesays,butglanceatanymathematicalorphysicaltext-booktoassureourselvesthatbyfarthegreaternumber of propositions which are used as major premisses are hypothetical innature,ifnotinexpression.“Propositionssuchas‘twocircleswhichintersecthavenocommoncentre’arehypotheticalinnature;thepropositionstatestheconditionuponwhichthepredicateisdenied.…Itisthesamewiththeformulaeofanalyticalmechanics; these and others of the same description are hypothetical judgments,andinferencesaremadeinaccordancewiththembysubstitutingdefinitevaluesforthegeneralsymbols”(Logic,§55).Sigwartperhapsattachesundueimportancetothemerequestionofform.Ifourmajorpremissisunconditionallyuniversal,andisunderstood tobeso, itdoesnotaffect thecharacterof the reasoningwhetherweadopt thecategoricalmodeofexpressionor theconditional.Sigwart’s reasonfordwellingonthehypotheticalforceofthemajorpremissistobefoundlargelyinthetrivial nature of the examples that it has been customary to give of the purely


categoricalsyllogism.

The following may be noted as typical cases in which the grounds foraccepting the truthof thepremissesofa syllogismarequite independentofanyexplicitknowledgeofthetruthoftheconclusion.

(1)Themajor premissmay itself be accepted as axiomatic, or itmaybededucible (without the assistance of the conclusion) from more ultimateprinciplesthatareacceptedasaxiomatic.Ithasindeedbeenarguedthataself-evident maxim cannot be used, or is at any rate superfluous, as a proof,becauseanyconclusionthatitmightbeemployedtoestablishwouldbeitselfequally self-evident.451 A consideration of ordinary 428 geometrical proofswill,however,atonceshewthat this isnotnecessarilythecase,andthatbythe aid of self-evident premisses conclusions may be reached that arecertainlynotthemselvesself-evident.

451 CompareBailey,TheoryofReasoning,p.74.

(2)Themajorpremissmaybebasedonauthority,ormaybeacceptedontestimony;oritmaybetheexpressionofacivillaw,orofacommand,orofarule of conduct;452 and in none of these cases can it be in any degreegroundedupontheconclusion.

452 “Wefind,”saysSigwart,“awidefieldforourinferencesintheapplicationofgenerallawswhichhavetheirorigininourwillandaremeanttoregulatethatwill.Inlayingdownageneralruleofconduct,ourwilldeterminesthatthereshallbeauniversally valid connexion between certain conditions and certain modes ofaction.Ifwewillthegenerallaw,itislogicallynecessarythatweshouldwilltheparticularactionsprescribedbythelaw,ifourwillistobeconstantandconsistent,andvalid foreveryonewhoagrees inwilling thegeneral law.Allpenalcodes inimposingapenaltyofimprisonmentfortheft,ofcapitalpunishmentformurder,laydown a series of hypothetical judgments which establish a universal connexionbetween committing the crime and incurring the penalty. These judgments,moreover,mayalsoberegardedastheoreticalpropositionsinsofarastheyexpressthegeneralobligationof the judge togive sentence in accordancewith the law”(Logic,i.p.337).

(3)Themajorpremissmaybean imperfect induction,basedonevidencethat does not include the conclusion. As an example, we may take thereasoning involved in testing the nature of a given substance in practicalchemistry.Inareasoningofthiskindourimmediatestartingpointisgeneralknowledgeofthepropertiesofchemicalsubstances.Thisknowledgehasbeen


inductivelyobtained,butitisimpossiblethatitshouldintheslightestdegreedependonanyantecedent acquaintancewith thepropertiesof theparticularsubstancewhich is now to be investigated for the first time.Or, again,wemay takeastronomical inferencesbasedon the lawofuniversalgravitation.That law is an induction based on particular observations, but it implies aninfinite number of facts that form no part of the evidence on which it isacceptedastrue;andmanyofthesefactsareinthefirst instancebroughttoour notice as inferences from the law, not as data leading up to it. If it isaffirmedthat,incasessuchasthese,themajorpremissescannotlegitimatelybeestablished independentlyof the conclusions syllogisticallyderived fromthem, then429 thevalidityof imperfect inductionasaprocessofarrivingatknowledgemustbedenied.

Ifasked tomeet theargumentcontained in theprecedingparagraph,Millwoulddoubtlessrefertohisdoctrineofthefunctionofthemajorpremissinasyllogism.Therealproofoftheconclusionofasyllogism,hewouldsay,istobe found, not in themajor premiss itself, but in the evidence onwhich themajor premiss is based: the major premiss is nothing more than amemorandum of evidence from which the conclusion might be directlyobtained: the interventionof themajorpremiss isoftenconvenient,but it isnotanessentiallinkinthepassagefromtheultimatedatatotheconclusion.Inreply,itmaybesaidthatthereisatanyrateashiftingofthegroundhere,andthatMill’s doctrine, even if accepted, fails to justify the charge that everysyllogisminvolvespetitioprincipii ;foritisadmittedthattheconclusiondoesnot itself constitute any part of the data from which the major premiss isobtained.Wemust,however,gofurtherandrejectthedoctrineonthegroundthatthereareatanyratesomecasesinwhichthegenerallawexpressedbythemajorpremissisanabsolutelynecessarylinkintheargument.Thus, totakebut one illustration, there are many consequences of the law of universalgravitation which it would be quite impossible to infer directly from theevidencelyingbehindthatlawwithouttheinterventionofthelawitself.

Having regard then to instances such as those adduced above, we mustrejecttheviewthatsyllogisticreasoningessentiallyinvolvespetitioprincipii,inthesenseofcirculusinprobando.Anyplausibility that theopposedview


maypossessdependsuponsomeconfusionbetweenthestatementthateverysyllogism is guilty ofpetitioprincipii in the above sense and the statementthatineverysyllogismthepremissespresupposetheconclusioninthesensethattheycouldnotbetrueunlesstheconclusionweretrue.

The latter statement is applicable not only to syllogistic, but to alldemonstrative,inference.Thequestionmayindeedberaisedwhetheritisnotapplicable to all valid inference whatsoever. It is in fact one horn of thedilemmareferredtoinsection377.

430 At any rate it is a misuse of language to speak of a reasoning asinvolving petitio principii on this ground. By petitio principii is alwaysunderstood a certain form of fallacy. But inmaking explicit what to beginwithismerelyimplicitthereisnothingthatcanbyanystretchoflanguagebetermed fallacious. To say that all deductive science is nothing but a hugepetitioprincipiiisclearlyanabsurdity.Themostthatcanbesaidisthatinalldemonstrative reasoning (so-called) there is really no inference frompremisses but only the interpretation of premisses. So far as this is amerequestion of language, itmay suffice to note the paradoxical conclusions towhichitleads;forexample,thatinthewholeofEuclidthereisnosuchthingasinferenceorproof.Sofarasitisnotamerequestionoflanguage,itturnsonpointsthatwehavealreadydiscussed,forinstance,thepossibilityoftherebeing an advance in knowledge subjectively considered although from theobjective standpoint the conclusions reached contain nothing new. It isunnecessarytorepeatthediscussionwithspecialreferencetothesyllogism.


CHAPTERX.

EXAMPLESOFARGUMENTSANDFALLACIES.

382.Inhowmanydifferentmoodsmaytheargumentimpliedinthefollowingpropositionbestated? “No one canmaintain that all persecution is justifiablewho admits that persecution is sometimesineffective.” Howwould the formalcorrectnessof the reasoningbeaffectedby reading“deny” for“maintain”?[V.]

383.Noonecanmaintain that all republics securegoodgovernmentwhobears inmind thatgoodgovernmentisinconsistentwithalicentiouspress. What premisses must be supplied to express the above reasoning in Ferio, Festino and Ferisonrespectively?[V.]

384.Writethefollowingargumentsinsyllogisticform,andreducethemtothefirstfigure:— (α)Falklandwasaroyalistandapatriot;therefore,someroyalistswerepatriots. (β)Allwhoarepunishedshouldberesponsiblefortheiractions;therefore,ifsomelunaticsarenotresponsiblefortheiractions,theyshouldnotbepunished. (γ)AllwhohavepassedtheLittle-GohaveaknowledgeofGreek;henceA.B.cannothavepassedtheLittle-Go,forhehasnoknowledgeofGreek.[K.]

385. “It is impossible to maintain that the virtuous alone are happy, and at the same time thatselfishnessiscompatiblewithhappinessbutincompatiblewithvirtue.” Statetheaboveargumentsyllogisticallyinasmanydifferentmoodsaspossible.[J.]

432386.Givethetechnicalnameofthefollowingargument:—Paymentbyresultssoundsextremelypromising;butpaymentbyresultsnecessarilymeanspaymentforaminimumofknowledge;paymentforaminimumofknowledgemeansteachinginviewofaminimumofknowledge;teachinginviewofaminimumofknowledgemeansbadteaching.[K.]

387.FromPfollowsQ ;andfromRfollowsS ;butQandScannotbothbetrue;shewthatPandRcannotbothbetrue.[DeMorgan.]

388. If (1) it is false thatwheneverX isfoundY is foundwith it, and (2)not lessuntrue thatXissometimesfoundwithouttheaccompanimentofZ,areyoujustifiedindenyingthat(3)wheneverZisfound there also you may be sure of finding Y? And, however this may be, can you in the samecircumstancesjudgeanythingaboutYintermsofZ?[R.]

389.Canthefollowingargumentsbereducedtosyllogisticform? (1)Thesunisathinginsensible; ThePersiansworshipthesun; Therefore,thePersiansworshipathinginsensible. (2)TheDivinelawcommandsustohonourkings;


LouisXIV.isaking; Therefore,theDivinelawcommandsustohonourLouisXIV.[PortRoyalLogic.]

390.Examine the followingarguments;where theyarevalid, reduce themifyoucan tosyllogisticform;andwheretheyareinvalid,explainthenatureofthefallacy:— (1)WeoughttobelievetheScripture; TraditionisnotScripture; Therefore,weoughtnottobelievetradition. (2)Everygoodpastorisreadytogivehislifeforhissheep; Now,therearefewpastorsinthepresentdaywhoarereadytogivetheirlivesfortheirsheep; Therefore,thereareinthepresentdayfewgoodpastors. (3)ThoseonlywhoarefriendsofGodarehappy; Now,therearerichmenwhoarenotfriendsofGod; Therefore,therearerichmenwhoarenothappy.433 (4)ThedutyofaChristianisnottopraisethosewhocommitcriminalactions; Now,thosewhoengageinaduelcommitacriminalaction; Therefore,itisthedutyofaChristiannottopraisethosewhoengageinduels. (5)ThegospelpromisessalvationtoChristians; SomewickedmenareChristians; Therefore,thegospelpromisessalvationtowickedmen. (6)Hewhosaysthatyouareananimalspeakstruly; Hewhosaysthatyouareagoosesaysthatyouareananimal; Therefore,hewhosaysthatyouareagoosespeakstruly. (7)YouarenotwhatIam; Iamaman; Therefore,youarenotaman. (8)Wecanonlybehappy in thisworldbyabandoningourselves toourpassions,orbycombatingthem; Ifweabandonourselvestothem,thisisanunhappystate,sinceitisdisgraceful,andwecouldneverbecontentwithit; Ifwecombatthem,thisisalsoanunhappystate,sincethereisnothingmorepainfulthanthatinwardwarwhichwearecontinuallyobligedtocarryonwithourselves; Therefore,wecannothaveinthislifetruehappiness. (9)Eitheroursoulperisheswiththebody,andthus,havingnofeelings,weshallbeincapableofanyevil;orifthesoulsurvivesthebody,itwillbemorehappythanitwasinthebody; Therefore,deathisnottobefeared.[PortRoyalLogic.]

391.Examinethefollowingarguments:— (1)“HethatisofGodhearethmywords:yethereforehearthemnot,becauseyearenotofGod.” (2)Allthefishthatthenetinclosedwereanindiscriminatemixtureofvariouskinds:thosethatweresetasideandsavedasvaluable,werefishthatthenetinclosed:therefore,thosethatweresetasideandsavedasvaluable,wereanindiscriminatemixtureofvariouskinds. (3)Testimonyisakindofevidencewhichisverylikelytobefalse:theevidenceonwhichmostmenbelieve that there are pyramids in Egypt is testimony: therefore, the evidence on which most menbelievethattherearepyramidsinEgyptisverylikelytobefalse.434 (4)IfPaley’ssystemistobereceived,onewhohasnoknowledgeofafuturestatehasnomeansofdistinguishingvirtueandvice:nowonewhohasnomeansofdistinguishingvirtueandvicecancommit


nosin: therefore, ifPaley’ssystemistobereceived,onewhohasnoknowledgeofafuturestatecancommitnosin. (5)IfAbrahamwerejustified,itmusthavebeeneitherbyfaithorbyworks:nowhewasnotjustifiedbyfaith(accordingtoJames),norbyworks(accordingtoPaul):therefore,Abrahamwasnotjustified. (6)Forthosewhoarebentoncultivatingtheirmindsbydiligentstudy,theincitementofacademicalhonours is unnecessary; and it is ineffectual, for the idle, and such as are indifferent to mentalimprovement:therefore,theincitementofacademicalhonoursiseitherunnecessaryorineffectual. (7)Hewhoismosthungryeatsmost;hewhoeatsleastismosthungry:therefore,hewhoeatsleasteatsmost. (8)Amonopolyofthesugar-refiningbusinessisbeneficialtosugar-refiners:andofthecorn-tradetocorn-growers: and of the silk-manufacture to silk-weavers,&c.,&c.; and thus each class ofmen arebenefitedbysomerestrictions.Nowalltheseclassesofmenmakeupthewholecommunity:therefore,asystemofrestrictionsisbeneficialtothecommunity.[Whately,Logic.]

392.The following are a fewexamples inwhich the reader can tryhis skill indetecting fallacies,determiningthepeculiarformofsyllogisms,andsupplyingthesuppressedpremissesofenthymemes: (1)Nonebut thosewhoarecontentedwith their lot in lifecan justlybeconsideredhappy.But thetrulywisemanwillalwaysmakehimselfcontentedwithhislotinlife,and,therefore,hemayjustlybeconsideredhappy. (2)All intelligiblepropositionsmustbeeither trueor false.The twopropositions“Caesar is livingstill,”and“Caesarisdead,”arebothintelligiblepropositions;therefore,theyarebothtrue,orbothfalse. (3)Manythingsaremoredifficultthantodonothing.Nothingismoredifficulttodothantowalkonone’shead.Therefore,manythingsaremoredifficultthantowalkonone’shead. (4)NonebutWhigsvoteforMrB.AllwhovoteforMrB.areten-poundhouseholders.Therefore,nonebutWhigsareten-poundhouseholders.435 (5)IftheMosaicaccountofthecosmogonyisstrictlycorrect,thesunwasnotcreatedtillthefourthday.Andifthesunwasnotcreatedtillthefourthday,itcouldnothavebeenthecauseofthealternationofdayandnight for thefirst threedays.Buteither theword“day” isused inScripture inadifferentsense to that inwhich it iscommonlyacceptednow,orelse thesunmusthavebeen thecauseof thealternationofdayandnightforthefirstthreedays.HenceitfollowsthateithertheMosaicaccountofthecosmogonyisnotstrictlycorrect,orelsetheword“day”isusedinScriptureinadifferentsensetothatinwhichitiscommonlyacceptednow. (6)Sufferingisatitletoanexcellentinheritance;forGodchastenseverysonwhomhereceives. (7)Itwillcertainlyrain,fortheskylooksveryblack.[Solly,SyllabusofLogic.]

393.Examinethefollowingarguments;sofarastheyarevalid,reducethemtosyllogisticform;andwheretheyareinvalid,explainthenatureofthefallacyinvolved:— (1)Ifyouargueonasubjectwhichyoudonotunderstand,youwillproveyourselfafool;forthisisamistakethatfoolsalwaysmake. (2) It is not the case that anymetals are compounds, and it is incorrect to say that everymetal isheavy; it may, therefore, be inferred that some elements are not heavy, and also that some heavysubstancesarenotelements. (3)Noyoungman iswise; foronlyexperiencecangivewisdom,andexperiencecomesonlywithage.[K.]

394.Examinetechnicallythefollowingargument:— Everyoneiseitherwellinformedofthefactsoralreadyconvincedonthesubject;noonecanbeatthesametimebothalreadyconvincedonthesubjectandamenabletoargument:henceitfollowsthatonly


thosewhoarewellinformedofthefactsareamenabletoargument.[J.]

395. Dr Johnson remarked that “amanwho sold a penknifewas not necessarily an ironmonger.”Againstwhatlogicalfallacywasthisremarkdirected?[C.]

396.Examinethefollowingarguments,pointingoutanyfallaciesthattheycontain:436 (a)Themorecorrectthelogic,themorecertainlywilltheconclusionbewrongifthepremissesarefalse.Therefore,wherethepremissesarewhollyuncertainthebestlogicianistheleastsafeguide. (b)Thespreadofeducationamongthelowerorderswillmakethemunfitfortheirwork:forithasalwayshadthateffectonthoseamongthemwhohappentohaveacquireditinprevioustimes. (c)Thispamphletcontainsseditiousdoctrines.ThespreadofseditiousdoctrinesmaybedangeroustotheState.Therefore,thispamphletmustbesuppressed.[C.]

397.Examinethefollowingarguments:— (1) A telescope with the eye-piece at one side of the tube is probably a reflector; Lord Rosse’stelescopeisareflector;therefore,LordRosse’stelescopeprobablyhastheeye-pieceatonesideofthetube. (2)Goodworkmendonotcomplainoftheirtools;mypupilsdonotcomplainoftheirtools;therefore,mypupilsareprobablygoodworkmen. (3)If,ontheonehand,theheathen,throughwantofbetterknowledge,cannothelpbreakingtheTenCommandments,thentheydonotstandcondemned;if,ontheotherhand,theyarecondemned,itisfordoing thatwhich theywell knewwaswicked, andwhich theywerewell able to refrain fromdoing;therefore,whateverhappenstothem,justiceissatisfied.[K.]

398. Discuss the nature of the reasoning contained, or apparently intended, in the followingsentences:— It is impossible toprove thatpersecution is justifiable ifyoucannotprove that somenon-effectivemeasuresarejustifiable;fornopersecutionhaseverbeeneffective. Thisdeedmaybegenuinethoughitisnotstamped,forsomeunstampeddeedsaregenuine.[C.]

399.Statethefollowingargumentsinlogicalform,andexaminetheirvalidity:— (1) Poetrymust be either true or false: if the latter, it ismisleading; if the former, it is disguisedhistory,andsavoursofimpostureastryingtopassitselfoffformorethanitis.Somephilosophershavethereforewiselyexcludedpoetryfromtheidealcommonwealth.437 (2)Ifweneverfindskinsexceptasthetegumentsofanimals,wemaysafelyconcludethatanimalscannotexistwithoutskins. Ifcolourcannotexistby itself, it follows thatneithercananything that iscolouredexistwithoutcolour.Soiflanguagewithoutthoughtisunreal,thoughtwithoutlanguagemustalsobeso. (3)Had an armistice been beneficial to France andGermany, itwould have been agreed upon bythosepowers;butsuchhasnotbeenthecase;itisplainthereforethatanarmisticewouldnothavebeenadvantageoustoeitherofthebelligerents. (4)Ifwearemarkedtodie,weareenow Todoourcountryloss:and,iftolive. Thefewermen,thegreatershareofhonour.[O.]

400.Examinelogicallythefollowingarguments:— (a)If truthfulnessisneverfoundsavewithscrupulousness,andif truthfulnessisincompatiblewithstupidity,itfollowsthatstupidityandscrupulousnesscanneverbeassociated. (b)Yousaythatthereisnorulewithoutanexception.Ianswerthat,inthatcase,whatyouhavejustsaidmusthaveanexception,andsoprovethatyouhavecontradictedyourself.


(c)Knowledgegivespower;consequently,sincepowerisdesirable,knowledgeisdesirable.[L.]

401.Examinethefollowingarguments,statingtheminsyllogisticform,andpointingoutfallacies,ifany:— (a) Some who are truly wise are not learned; but the virtuous alone are truly wise; the learned,therefore,arenotalwaysvirtuous. (b)Ifalltheaccusedwereinnocent,someatleastwouldhavebeenacquitted;wemayinfer,then,thatnonewereinnocent,sincenonehavebeenacquitted. (c)Everystatementoffactdeservesbelief;manystatements,notunworthyofbelief,areassertedinamanner which is anything but strong; we may infer, therefore, that some statements not stronglyassertedarestatementsoffact. (d) That many persons who commit errors are blameworthy is proved by numerous instances inwhichthecommissionoferrorsarisesfromgrosscarelessness.[M.]

402.Examinetechnicallythefollowingarguments:— (1)Thosewhohold that the insaneshouldnotbepunishedought inconsistency toadmitalso thatthey should not be threatened; 438 for it is clearly unjust to punish any one without previouslythreateninghim. (2)Ifhepleadsthathedidnotstealthegoods,why,Iask,didhehidethem,asnothiefeverfailstodo? (3)Knaveryandfollyalwaysgotogether;so,knowinghimtobeafool,Idistrustedhim. (4)Howcanyoudenythattheinflictionofpainisjustifiableifpunishmentissometimesjustifiableandyetalwaysinvolvespain? (5)IfIdenythatpovertyandvirtueareinconsistent,andyoudenythattheyareinseparable,wecanatleastagreethatsomepoorarevirtuous.[V.]

403.Detectthefallacyinthefollowingargument:— “Avacuumisimpossible,forifthereisnothingbetweentwobodiestheymusttouch.”[N.]

404.Considerthefollowingargument:— GrantedthatAisB,toprovethatBisA.B(likeeverythingelse)iseitherAornotA.IfBisnotAthenbyourfirstpremisswehavethesyllogism—AisB,BisnotA,therefore,AisnotA,whichisabsurd.HenceitfollowsthatBisA.[ProfessorJastrow,intheJournalofEducationFebruary,1897.]

405.Examinethefollowingargument:— Itis impossible toprovethatsocietycancontinuetoexistwithoutcompetitionunlessyoucanalsoprovethat theabsenceofcompetitionwouldnot leadto thedeteriorationof individuals;forasocietywhosemembersdeterioratecannotlongcontinuetoexist.[M.]

406. Express the following propositions in their simplest logical form; examine their mutualconsistencyorinconsistency,andthevalidityofthefinalconclusion:— SomeofMrN’spublishedviewsarenew,andsometrue;infact,theyarealloneortheother;and,thoughitcannotbemaintainedingeneralthataviewthatisnotnewisonthataccountnecessarilynottrue,yetitcanbeconfidentlyassertedthateverypossiblefalseviewonthissubjectwaspropoundedbysomeoneorotherbeforeMrN.wrote:fromwhichitwouldappearthatwhileitmayormaynotbethatMrN.’sviewsareallnew,itiscertainthattheyarealltrue.[J.]

439407.Examinetechnicallythefollowingarguments:—

(a)

“’Tisonlythepresentthatpains,


Andthepresentwillpass.”

(b) All legislative restraint is either unjust or unnecessary; since, for the sake of a single man’sinterests, to restrain all the rest of the community is unjust, and to restrain the man himself isunnecessary. (c)OnlyConservatives—andnotallofthem—areProtectionists;onlyLiberals—andnotallofthem—areHomeRulers;butbothpartiescontainsupportersofwomen’sfranchise.HenceonlyUnionists—and not all of them—are Protectionists, while the supporters of women’s franchise contain bothUnionistsandFree-traders. (d)Noschool-boycanbeexpected tounderstandConstitutionalHistory,andnonebutschool-boyscan be expected to remember dates; so that no one can be expected both to remember dates and tounderstandConstitutionalHistory. (e)Tobewealthyisnottobehealthy;nottobehealthyistobemiserable;therefore,tobewealthyistobemiserable. (f) Whatever any man desires is desirable; every man desires his own happiness; therefore, thehappinessofeverymanisdesirable.[J.]

408.Examinethevalidityofthefollowingarguments:— (1) I knew he was a Bohemian, for he was a good musician, and Bohemians are always goodmusicians. (2)Bulliesarealwayscowards,butnotalwaysliars;liars,therefore,arenotalwayscowards. (3)IfallthesoldiershadbeenEnglish,theywouldnotallhaverunaway;butsomedidrunaway;andwemay,therefore,inferthatsomeofthematleastwerenotEnglish. (4)Nonebutthegoodarereallytobeenvied;alltrulywisemenaregood;therefore,alltrulywisemenaretobeenvied. (5)Youcannotaffirmthatallhisactswerevirtuous,foryoudenythattheywereallpraiseworthy,andyouallowthatnothingthatisnotpraiseworthyisvirtuous. (6)Sincetheendofpoetryispleasure,thatcannotbeunpoeticalwithwhichallarepleased. (7)MostMisP,MostSisM,therefore,SomeSisP.440 (8)OldParr,healthyasthewildanimals,attainedtotheageof152years;allmenmightbeashealthyasthewildanimals;therefore,allmenmightattaintotheageof152years. (9)Itisquiteabsurdtosay“Iwouldrathernotexistthanbeunhappy,”forhewhosays“Iwillthis,rather than that,”choosessomething.Non-existence,however, isnosomething,butnothing,and it isimpossibletochooserationallywhentheobjecttobechosenisnothing. (10)Becausethequalityofhavingwarmredbloodbelongstoallknownbirds,itmustbepartoftheirspecificnature;butunknownbirdshavethesamespecificnatureasknownbirds;therefore,thequalityofhavingwarmredbloodmustbelongtotheunknownaswellastheknownbirds,i.e.,beauniversalandessentialpropertyofthespecies.[K.]


APPENDIXA.

THEDOCTRINEOFDIVISION.

409.Logical Division.—The term division, as technically used in logic,maybedefinedasthesettingforthofthesmallergroupswhicharecontainedundertheextensionofagiventerm.It isalsodefinedastheseparationofagenus into its constituent species. These two definitions are practicallyequivalent to one another.Division is to be distinguished from the settingforth of the individual objects belonging to a species, which is technicallydescribedasenumeration.

In logical division, the larger class which is divided is called the totumdivisum, the smaller classes into which it is divided being the membradividentia (dividing members). By the ground or principle of division(fundamentum sive principium divisionis) is meant that attribute orcharacteristicofthetotumdivisumuponwhosemodifications thedivision isbased.Agivenclassmayofcoursebedividedindifferentwaysaccordingtothe particular attribute or attributes whose variations are selected asdifferentiating its various species. Thus, having regard to the equality orinequality of the sides, trianglesmay be divided into equilateral, isosceles,and scalene; or, having regard to the size of the largest angle, into obtuse-angled, right-angled, and acute-angled. Again, propositions are divisibleaccording to their truth or falsity, or according to their quantity, or theirquality,andsoon.

It is sometimes said that the principle of division must be presentthroughout the dividing members, though constantly varied. On the otherhand, it is said that in divisionwe invariably try to think of some attributewhich ispredicableofcertainmembersof thegroup,butnotofothers.The


former of these statements does not 442 very well apply when we simplydivide a class according to the presence or absence of some attribute (forexample,candidatesfortheCivilServiceintosuccessfulandunsuccessful)orwhen the attribute in question may be entirely wanting in some instanceswhilstpresentinvaryingdegreesinotherinstances.Inotherwords,giventheattributewhosevariationsconstituteourprincipleofdivision,wemayhavetorecognise a limiting case inwhich it is altogether absent; thus, in dividingundergraduatesaccordingtotheircolleges,wemayhavetorecogniseaclassof non-collegiate students. The second statement is always true when wesimplycontrastanygivenspecieswithalltheremainingspecies,anditmaybeconsidered adequatewherewehavedivisionbycontradictories. Inothercases,however,it isinadequate;as,forinstance,whenwedividecandidateswhoaresuccessfulintheIndianCivilServiceExaminationaccordingtotheprovincetowhichtheyareassigned.

410. Physical Division, Metaphysical Division, and Verbal Division.—Followingtheolder logicians,wemaydistinguishdivisionasdefinedinthepreceding paragraph, that is, logical division in the strict sense, from othersensesinwhichthetermisused.

Thedivisionofanindividualthingintoitsseparatepartsiscalledphysicaldivision orphysicaldefinition (Whately,Logic, p. 143) orpartition ; as, forexample, ifwedivideawatch intocase,hands, face, andworks;or abookintoleavesandbinding.Wehave,ontheotherhand,alogicaldivisionifwedividewatchesintogold,silver,&c.,orintoEnglish,Swiss,American,&c.;or ifwedividebooks intofolios,quartos,&c.Bain(Logic, II.p.197)givestheanalysisofachemicalcompoundasaninstanceof logicaldivision.It isratheraninstanceofphysicaldivision.Inlogicaldivisionthetotumdivisumisalways predicable of all the individuals belonging to each of themembradividentia ;forexample,Allmenareanimals,Allsquaresarerectangles.Butthisisnotthecaseinchemicalanalysis.Wecannotsaythatoxygeniswater,orthatsulphurisvitriol,orthatsodiumissalt.

Distinctbothfromlogicaldivisionandfromphysicaldivisionisthementaldivision of a thing into its separate qualities. This is called metaphysicaldivision.Wehaveanexamplewhenweenumeratetheseparatequalitiesofa


watch,itssize,accuracy,thematerialofwhichitscaseiscomposed,&c.;orwhenwespecify thesizeofabook, its thickness,colour, thematerialof itsbinding,thequalityofthepaperofwhichitsleavesarecomposed,andsoon.443Aphysical division canbe actuallymade; awatch, for example, canbetakentopieces.Ametaphysicaldivision,ontheotherhand,isonlypossiblementally. It should be added that the metaphysical division of individualobjectsmaybemadethebasisofalogicaldivisionoftheclasstowhichtheybelong.

One further kind of divisionmay be noticed, namely, the division of anambiguous or equivocal term into its several significations. This is calledverbal division (Clarke, Logic, p. 331) or distinction (Mansel’sAldrich, p.37).Forexample,wehave todistinguishbetweenawatch in thesenseofavigil,inthesenseofaguard,andinthesenseofatime-piece.

411.RulesofLogicalDivision.—Thefundamentalrulesoflogicaldivisionare(1)thatthemembersofthedivisionshallbemutuallyexclusive;and(2)that collectively they shall be exactly coextensive with the class that isdivided. Thus if the class X is correctly divided into XA, XB, XC, thefollowingpropositionsmustholdgood,namely,NoXAisBorC,NoXBisCorA,NoXCisAorB,EveryXisAorBorC.

The two following rules are generally added: (3) Each distinct act ofdivisionshouldproceedthroughoutupononeandthesamebasisorprinciple;(4) If thedivision involvesmore thanone step, it shouldproceedgraduallyandcontinuouslyfromthehighestgenustothelowestspecies,thatistosay,itshouldnotpasssuddenlyfromahighgenustoalowspecies.

It may be objected that (1) and (2) ought not in a strict sense to bedescribedasrules,butratherasconstitutingbetweenthemaprecisestatementofwhatis impliedwhenwespeakofa logicaldivision.Theybecomerules,however, in thesense thataprofessed logicaldivisionwhich fails tosatisfyeitherof themimplies relationsbetween themembersof thedivisionwhichdo not as a matter of fact hold good. Rules (3) and (4) are of a differentcharacter. They are rules in the sense that theymust be compliedwith if adivisionistohavepracticalutility.


Rule(3)isnotintendedtocondemntheprocessesofsub-divisionandco-division. Having made a division upon one principle, we may proceed tosubdividetheclassesthusarrivedatinaccordancewithanotherprinciple,andsoonindefinitely.Ascientificclassificationwillalwaysconsistofahierarchyofclassesthusobtained.Thereisagainnoreasonwhythesameclassshouldnotfordifferentpurposesbedividedinaccordancewithtwoormoredifferent444 principles, so long as these are kept distinct from one another, and themembersofthedifferentresultingdivisionsnotconfusedtogether.

Ithasbeensaid thatabreachof rule (1)necessarily involvesabreachofrule(3),sincetherecannotbeanyoverlappingofclassessolongasadivisionproceeds correctly upon a single principle. This does not, however, alwayshold good unless we interpret the word “correctly” as implying thatprecautions are taken to avoid any overlapping, which of course begs thequestion.Thus,ifwedividetrianglesintothosewhichhave(a)arightangle,(b)anobtuseangle,(c)anacuteangle,wemaybesaidtoproceedupononeprinciple, and yet the resulting classes are not mutually exclusive. It may,again, be argued that the classes equilateral triangle, isosceles triangle,scalenetriangle(whichresultfromadivisionbaseduponasingleprinciple)arenotmutuallyexclusive,sinceallequilateraltrianglesareisosceles.

Thisargumentcanonlybemetbysayingthat,inthefirstcase,wearenotproceeding upon any clear principle unless we make our division intotriangleswhose largest angle is an obtuse angle, a right angle, or an acuteangle, respectively; nor unless, in the second case, our principle is themaximumnumberofsidesthatareequaltooneanother,sothatanisoscelestriangle isdefinedasa triangle thathas twoandonly twosidesequal.Anyoverlappingofclassesisthenineachcaseprovidedagainst;butonly,itmaybeargued,becausespecialprecautionshavebeentakentoattainthisend.Bythe adoption of similar precautions, a division which proceeds “correctly”uponasingleprinciplewillalsobeexhaustive.

Looking at the question from the other sidewemay note that a divisionwhichsatisfiesbothrule(1)andrule(2)mayneverthelessbeacross-division;for itmay happen that two different principles of division yield coincidentresults.Forexample,anisoscelestrianglebeingdefinedasatrianglethathas


twoandonlytwosidesequal,thereisacross-division,butnooverlappingofclasses,oromissionofanyclasscontainedinthetotumdivisum,ifwedividetriangles intoscalene, isosceles,andequiangular;or ifwedivideplants intoacotyledons,monocotyledons,andexogens.

Asregardsrule(4),itistobeobservedthatadivisionwhichproceedspersaltumwillusuallybemuchlesseffectivethanoneinwhichtheintermediatestepsarefilledin.Theworstviolationofthisruleoccurswhenthedivisionisdisparate, that is, when “one of the classes into which we divide is animmediate and proximate 445 class, while others are mediate and remote”(Clarke, Logic, p. 242); as, for example, if we divide animals intoinvertebrates,fishes,amphibians,reptiles,birds,elephants,horses,dogs,&c.

Another rule of division is sometimes added, namely, that “none of thedividing members must be equal in extent to the divided whole” (Clarke,Logic,p.236).Whenthisruleisbroken, thedivisionissaidtobecomenulland void, because one of the sub-divisions contains nomembers. From theformal point of view, however, the observance of this rule can hardly beinsisted upon. We need not regard a division as necessarily implying theactualoccurrenceofallitsmembersintheuniverseofdiscourse;andtherulein questionwould deprive the logician of the right to employ the powerfulmethodofdivisionbycontradictories.Itmaybeadifferentmatterwhenweareconsideringscientificclassificationfromthematerialstandpoint.

412.DivisionbyDichotomy.—Divisionbydichotomyor,asitissometimescalledmoredistinctively,dichotomybycontradictionisthedivisionofaclasssimplywithreferencetothepresenceorabsenceofagivenattributeorsetofattributes; as, for example, when X is divided into XA and Xa (wherea = not-A). An illustration is afforded by the Tree of Porphyry or RameanTree, in which Substances are first divided into Corporeal Substances(Bodies)andIncorporealSubstances,BodiesbeingthendividedintoAnimateBodies (Living Beings) and Inanimate Bodies, Living Beings being nextdivided into Sensitive Living Beings (Animals) and Insensitive LivingBeings,andAnimalsbeingintheirturndividedintoRationalAnimals(Men)and Irrational Animals. At each step in this scheme we proceed by takingcontradictories.ItwasinpraiseofdichotomaldivisionthatJeremyBentham,


whoisherequotedwithapprovalbyJevons(PrinciplesofScience,30,§12),spoke of “thematchless beauty of theRameanTree.”When thismethod isemployed we ensure formally that the members of our division shall bemutually exclusive and collectively exhaustive. For, by the law ofcontradiction,No X is both A and a ; and, by the law of excludedmiddle,EveryXiseitherAora.

It ispointedoutbySpalding(Logic,p.146)andbyJevons(PrinciplesofScience,30,§9) thatall logicallyperfectdivision isultimately reducible todichotomy, usuallywith the implication that someof the sub-classeswhichareàprioripossiblearenotasamatteroffacttobefoundintheuniverseofdiscourse.Thus,446 ifwe take theclassXanddivide it intoXAandXBweimplythatintheclassX,AandBareneverfoundeitherbothpresentorbothabsent.Hencethedivisionisequivalenttothefollowingdichotomaldivision:—

Anyotherdivision,howevercomplicatedinitscharacter,maybereducedtodichotomy in a similarway.This is interesting and important andbringsout the value of dichotomy as a method of testing divisions. It must beunderstood,however, that inspeakingofalldivisionasultimatelyreducibletodichotomy,itisnotintendedtoimplythatdichotomyusuallyrepresentsouractual procedure in making divisions. Each sub-class is usually arrived atimmediatelybyreferencetosomepositivemodificationofthefundamentumdivisionis ; and the different sub-classes are co-ordinate with one another.Consider, for example, the division of conic sections into parabolas,hyperbolas,ellipses,circles,andpairsofstraightlines.Itmustbeaddedthatfromthematerialstandpoint,puredivisionbydichotomyisoflittlescientificvalue, because of the indefinite character of the sub-classes which are


determinednegatively.

413. The place of the Doctrine of Division in Logic.—The doctrine ofdivision,astreatedbytheolderlogicians,receiveslittlerecognitionbysomemodernwritersontwoverydifferentgrounds:(1)byMill,takingthematerialstandpoint, it is regarded as too purely formal, and hence ismerged in thedoctrine of scientific classification; (2) by some writers belonging to theconceptualist school, e.g., Mansel, it is rejected as not being sufficientlyformal.

(1) It is true that the rulesof logicaldivision leadusavery littleway inpracticalscience.Theygivecertainconditionswhichmustbecompliedwith;buttheyneitherhelpustowardsmakinggooddivisions,norprovideuswithatestwhich is capableof being formally applied.Leavingdichotomyononeside, we cannot, without the aid of material knowledge, even determinewhether the members of a given division are mutually exclusive andcollectively447exhaustive.When,however,weavowedlypassbeyondpurelyformalconsiderationsandtakeupamaterialstandpoint,thenthedoctrineofdivisionshouldrightlygiveplacetoadoctrineofclassification,whichisnotcontentwith such rules as those laid down above, but seeks to indicate theprinciples that should serve as a guide in the classification of objectsscientifically.

In regard to the use of the terms division and classification, Miss Jonesdraws a distinction which is of value and to which it might be wellsystematically to adhere. “Division and classification are the same thinglooked at from different points of view; any table presenting a divisionpresentsalsoaclassification.Adivisionstartswithunityanddifferentiatesit;aclassificationstartswithmultiplicity,and reduces it tounity,orat least tosystem”(ElementsofLogic,p.123).

(2)Itremainstobeconsideredhowfaranytreatmentofdivisionwhatevercanproperlyfallunder theconsiderationofformal logic.Fromthispointofviewdivisionisusuallycontrastedwithdefinition.Thelatterofthese—usingthephraseologyof the conceptualist logicians—expounds the intensionof aconcept;theformerexpoundsitsextension.Buttheintensionofaconceptissaid to be far more intrinsic to it than its extension. Given a concept its


intension is necessarily given; but knowledgeof its extension, such asmayserve to determine its division, will require a fresh appeal to the subject-matter.“Division,”saysMansel,“isnot,likedefinition,amentalanalysisofgivenmaterials:thespecificdifferencemustbeadded tothegivenattributesofthegenus;andtogainthisadditionalmaterial,itisnecessarytogooutoftheactof thought, toseekfornewempiricaldata”(ProlegomenaLogica,p.192).Forexample, thedivisionofmembersof theUniversityofCambridgeintothoseinstatupupillariandmembersoftheSenatecouldnotbeobtainedwithoutsomethingmorebeinggiventhanthemereconceptionofamemberoftheUniversity.Moreover,unlessweproceedbycontradictories,wecannot,whenwehavegotourdivision,formallydeterminewhetheritcomplieswithourrulesornot.

Theabovepositionmaybeaccepted, ifanexceptionismadefordivisionbydichotomy.Mansel,however,andsomeotherlogicianswillnotevenallowthatdivisionbydichotomyisaformalprocess;andheretheylaythemselvesopentocriticism.Thegroundsonwhichtheirviewisbasedaretwofold:—(i)Itisnotsufficientthat448 thegenustobedividedbegiven; theprincipleofdivisionmustbegivenalso.“Eveninthecaseofdichotomybycontradictionthe principle of division must be given, as an addition to the attributescomprehended in the concept, before the logician can take a single step”(Prolegomena Logica, p. 207). “The division ofA intoB and not-B is notstrictlyformal;forthedividingattribute,notbeingpartofthecomprehensionofA, has to be sought for out of themere act of thought, afterA has beengiven”(Mansel’sAldrich,p.38).(ii)Wecannottellàpriorithatboththesub-classesobtainedbydichotomyreallyexist.How,forexample,canwedivideAintoBandnot-BwhenforanythingweknowtothecontraryallAmaybeB?“Logically,thedivisionofanimalintomortalandimmortalisasgoodasthat into rational and irrational” (Mansel’s Aldrich, p. 38). Both theseargumentsaresummedupinthefollowingquotationfromMrMonck:“ItisallegedindeedthatLogicenablesustodivideall theB’s into theB’swhichareC’sandtheB’swhicharenotC’s……ButLogicdoesnotsupplyuswiththetermCandafterwehaveobtainedthistermtherearetwocasesinwhichtheproposeddivisionfails,namely,wherealltheB’sareC’sandwherenoneof them are so. In either of these events the classB remains aswhole and


undivided as before; and whether they have occurred or not cannot beascertainedbyLogic.ThisDivisionbyDichotomy,asitiscalled,isasmuchoutsidetheprovinceofLogicasanyotherkindofdivision”(Logic,p.174).

As regards the first of the above arguments, there is no reason why theprinciple of division (A) should not be assumed given aswell as the totumdivisum(X).ThequestioniswhetherwecanthenformallydivideXintoXAandXa. The fact thatA must be given as well as X does not prevent thepossibilityof formaldivisionbydichotomy,anymore than the fact that theconclusionofasyllogismisnotcontainedinonepremissalonepreventsthesyllogismfrombeingaformalprocess.

Theforceofthesecondargumentdependsupontheimplicationthatallthesub-classes obtained as the result of a division necessarily exist in theuniverse of discourse. If this implication is granted, then dichotomy iscertainlynotaformalprocess;butwhyneedweassumetheexistenceofallthe sub-classes obtained by dichotomy? Without such an assumption, ourdivision may not have much practical utility, but its formal validity willremainunaffected.449Wehaveonlytomakeitclearthatwearedividingtheextension of a term,not itsdenotation, in the sense inwhich extension anddenotation have been already distinguished.453 This is in keeping with thegeneral standpoint of formal logic, which can deal with classes withoutregarding theirexistenceasnecessarilyguaranteed inanyassigneduniverseofdiscourse.IfwearenotallowedtoapplytheprincipleofexcludedmiddleinformallogicandsayEveryXisAora,untilweknowthat thereactuallyexistbothXA’sandXa’s,we shallbeexceedinglyhampered, andcanmakebut little progress, especially in the treatment of complex inferences. Someschemesof symbolic logic (e.g., Jevons’s)dependessentially andexplicitlyuponanantecedentschemeofdichotomaldivision.


453 Seesection21.

Wemaythenregarddivisionbydichotomyasaformalprocess,butonlyonthe understanding (1) that the principle of division is given as well as thegenus to be divided; (2) that the division is not assumed to be more thanhypothetical so far as concerns the existence of the resulting sub-classes inanyassigneduniverseofdiscourse.


APPENDIXB.

THETHREEFUNDAMENTALLAWSOFTHOUGHT.

414. The three Laws of Thought.—The so-called fundamental laws ofthought(thelawofidentity,thelawofcontradiction,andthelawofexcludedmiddle)aretoberegardedasthefoundationofallreasoninginthesensethatconsecutive thought and coherent argument are impossible unless they aretakenforgranted.Thefunctionwhichtheythusperformis,however,negativerather than positive. Whilst constituting necessary postulates, apart fromwhichourthoughtwouldbecomechaotic,theydonotbythemselvesadvanceusonourway.Ontheonehand,wecannotwithout theirsupportproceedastep in reasoning; on the other hand, ifwewere to rely on their aid alone,thoughtwouldimmediatelycometoastandstill.

This isatany rate theview takenof the three laws in theexposition thatfollows. It is true thatmany logicians have ascribed to them functions of amore positive character, and—starting from the position that they are thefundamentalassumptionsoflogic—havegoneontoregardthemasthebasisuponwhichaloneall logicaldoctrine,atany rate in itsmore formalaspect,canbeestablished.Theattempt to justify thisviewhasnecessitatedreadingintothelawsmuchmoremeaningthantheycanproperlybemadetocontain,andtheirinterpretationhasinconsequencebecomehighlycomplexandevenconfused.

At the outset the question arises whether the laws are to be regarded asreferringtoterms(orconcepts)ortojudgments.Myownviewisthat,inallthreecases,thelatterreferenceisthemorefundamental,butthatareferenceoftheformerkindisinvolvedsecondarily.ThisIshallendeavourtobringoutin dealing with 451 the laws individually. The distinction is one to which


considerableimportanceisrightlyattachedbySigwart.

Thequestionofthemutualrelationsbetweenthethreelawsmaybebrieflytoucheduponbeforeweproceedtoconsiderthelawsseparatelyandindetail;it isnot,however,aquestionthatcanbedisposedofuntilalaterstage.Themainpointtowhichattentionmayconvenientlybecalledatonceisthatitisonly in relation to theother laws that the full forceof eachof themcanbebrought out. The laws of identity and contradiction may be regarded aspositive and negative statements of the same principle, namely, theunambiguity of the act of judgment; and the laws of contradiction andexcludedmiddlearesupplementarytooneanotherinsofarasbetweenthemtheyexpress thenatureofnegation.At thesametime,anendeavourwillbemadetoestablishtheindependenceofthelawsinthesensethattheycannotbededucedonefromanother.

415.TheLawofIdentity.—FollowingSigwart,Ithinkitmostconvenienttointerpretthislawasexpressingtheunambiguityoftheactofjudgment.Truthis something fixed and invariable. In thewords ofMrBradley, “Once truealwaystrue,oncefalsealwaysfalse.Truthisnotonlyindependentofme,butitdoesnotdependuponchangeandchance.Noalterationinspaceortime,nopossibledifferenceofanyeventorcontext,canmaketruthfalsehood.IfthatwhichIsayisreallytrue, thenitstandsforever”(Logic,p.133).454Hence,sinceajudgmentistheexpressionoftruth,thecontentofajudgmentisfixedand invariable; and only when our judgments are so regarded can ourthoughtsandreasoningsbevalid.Itisinthissensethatthelawofidentityisafundamental principle of logic (which is the science of valid thought andreasoning);foritisclearthatifforagivenjudgmentwewereallowed—whenitsuitedus—tosubstituteanother,orifthecontentofagivenjudgmentcouldberegardedasnowthisandnowsomethingelse,all thoughtwouldbecomechaoticandreasoningwouldbeasham.Of thevalidityofnosinglestepofreasoning452couldwebesure,sinceaswe took thestep thecontentof theoriginal judgment might change, and on this ground it would be open toanyone to admit the original judgment and at the same time deny theinferenceattemptedtobedrawnfromit.

454 Comparewhathasbeenalreadysaidinsection50abouttheuniversalityofjudgments. In particular, the bearing ofMr Bosanquet’s distinction between the


timeofpredicationandthe time inpredicationmustbeborne inmind.Whenwesay that the truthaffirmed inany judgment is independentof time,wemean thetimeofpredication,andweassumethat thejudgmentisfullyexpressed: inorderthat it may be fully expressed, the time in predication, if any, must be madeexplicit.

Itmaybesaidthat,asthusinterpreted,thelawofidentitymerelystatesthatwe cannot both affirm a judgment and deny it, and that this is what isexpressedinthelawofcontradiction.Thereisforceinthis,totheextentthatthe laws of identity and contradiction may be regarded as expressing thepositiveandnegativeaspectsofthesameprinciple.Itis,asSigwarthassaid,only through the rejectionof simultaneousaffirmationandnegation thatwebecome conscious of the unambiguity of the act of judgment.At the sametime, the positive formulation of the principle in the form of the law ofidentity—apart from its negative formulation in the form of the law ofcontradiction—isjustifiableandhelpful.

The unambiguity of the act of judgment may be expressed somewhatdifferently(anditspositiveaspect,asdistinctfromwhatisexpressedbythelaw of contradiction, may thereby be mademore clear) by saying that therepetitionofajudgmentneitheraddstonoraltersitsforce.Onthisbasiswemayperhaps justify thepassageof thoughtwhichconsists in the repetition,notofacompletejudgment,butofpartofitscontentonly.Inotherwords,wemaythusjustifyformalreasoning,sofarasitinvolvesmereelimination;andinthemajorityofformalreasoningseliminationisinvolved,thoughitmaybequestionedwhethermereeliminationfromasingleproposition(asinpassingfromAllSisMPtoAllSisP)isbyitselfentitledtothenameofreasoningatall.

Mill gives an enunciation of the law of identity which must bedistinguishedfromtheabove:“Whateveristrueinoneformofwordsistrueineveryotherformofwordswhichconveysthesamemeaning”(ExaminationofSirWilliamHamilton’sPhilosophy,p.466).Thisisapostulatewhichitisnecessarytomakeinconnexionwiththeuseoflanguageasaninstrumentofthought. So long as the judgment expressed is the same, the form ofexpressionwhichwegivetoitisimmaterial;and,sinceinlogicaldoctrinewecannot explicitly recognise more than a limited number of distinct


propositionalforms,wehavetoclaimtobeallowedtosubstituteforanynon-recognisedformthatrecognisedformwhich453expressesthesamejudgment.Mill’s postulate, however, goes beyond the law of identity regarded asexpressingtheunambiguityoftheactofjudgment,anditcannotberegardedas equally fundamental. It is sometimes given as the justification ofimmediateinferences:tothispointweshallreturn.

Wemay now turn to the law of identity in the form inwhich it ismoreordinarilystated,namely,AisA,Everythingiswhatitis.Thisformisopentocriticismifregardedasprofessingtogiveinformationwithregardtoobjects.In another sense, however, it may be taken to express an unambiguity ofterms or concepts which is involved in the unambiguity of the act ofjudgment. For it is clear that unless in any given process of thought orreasoningourtermsorconceptshaveafixedsignificationandreference,theunambiguity of the act of judgment cannot be realised. We have here thesecondaryreferencetotermsorconceptswhichiscontainedinallthelawsofthoughtinadditiontotheirprimaryreferencetojudgments.

As the repetitionofa judgmentneitheradds tonoralters its force, sowemay say the same of terms (or concepts),meaning thereby that to refer toanythingasbothAandAisthesamethingastorefertoitsimplyasA.ThisyieldsBoole’sfundamentalequationx2=x(whichitselfadmitsofatwofoldinterpretationaccordingasxstandsforatermoraproposition).

ThereasonswhyweshouldnotinterprettheformulaAisAasexpressingajudgment respecting the object A have to be considered. The fundamentaldifficulty is that this so-called judgment is, if interpreted literally, notthinkable at all. For all actual thought implies difference of some kind.Wheneverwethinkofanything,itisasdistinguishedfromsomethingelse,oras having properties in common with other things, or at any rate as itselfexistingatdifferenttimes.Henceinnocasecanwethinkpureidentity.

Therearetwowaysofavoidingthisdifficulty.

(a)Wemaysay thatwhat is intendedby identity isnotpure identity,butexact likeness in some assigned respect or respects, the likeness sometimesamounting, so far at any rate as our apprehension is concerned, to


indistinguishablenessexceptinthepropertyofoccupyingdifferentportionsofspace(as,forexample,inthecaseofanumberofpinsorbulletsofthesamemakeandsize).Onthisinterpretation,thelawofidentitymayberegarded454asequivalenttoJevons’sprincipleoftheSubstitutionofSimilars—“Whateveris true of a thing is true of its like”—or to the axiom that “Things that areequal to the same thing are equal toone another.”Mansel indeedexplicitlygivesthisaxiomasequivalenttothelawofidentity.

Itseemsclear,however,onreflectionthatitisamisnomertospeakoftheseprinciplesaslawsof identity,andthatatanyratetheycannotbeadequatelyexpressedbythebareformulaAisA.Norcananyanalogousinterpretationbegiventothelawsofcontradictionandexcludedmiddle.Wemust, therefore,reject this interpretation of the law of identity regarded as one of the threetraditionallawsofthought.

(b)Wemayattempt toevade thedifficultybyexplaining thatby identitywemeancontinuousidentity,aswhenIsay“Thispenisthesameastheonewithwhich Iwaswriting yesterday.”Here there is no longer pure identity,sincethereisadifferenceoftime.

If,adoptingthisinterpretation,wemeanbythelawofidentitythatwhatistrueofanythingatagiventimeis trueof itatother timesalso,wehavenoself-evidentlaw,butafallacy.Forthepropertiesofobjectsarenotconstant.Inotherwords,thepossessionbyanobjectofanygivenpropertyisnot,likethetruthofajudgment(fullyexpressed),independentoftime.

Wemustthenbythelawofidentity,asthusinterpreted,meantoassertnotanyidentityofproperties,buttheidentityofthesubjectofpropertiesamidstallthechangesthatmaytakeplaceinthepropertiesthemselves.Thismayberegardedasatheoryastothenatureofindividualityandcontinuousidentityinthemidstofchange,andisofgreatimportanceinitsproperplace.Butitcannot properly stand as one of the traditional laws of thought whichconstitutethefoundationoflogicaldoctrine.

416. The Law of Contradiction.—The principle of contradiction is bestregarded as expressing one aspect of the relation between contradictoryjudgments,namely,thattheycannotbothbetrue.Theessentialcharacteristic


ofajudgmentisthatitclaimstobetrue.Butwecannotdeclareanythingtobetruewithout implicitly declaring something else to be false.All affirmationimpliesdenial;andwecannotclearlygrasptheimportofanygivenjudgmentunlessweunderstandpreciselywhatitdenies.

Therelationbetweenajudgmentanditsdenialismadeexplicitbythelawof contradiction and the law of excluded middle, the 455 first of whichdeclaresthattwocontradictoryjudgmentscannotbothbetrue,andthesecondthattheycannotbothbefalse.

It isclearthatthelawofcontradiction,asthusinterpreted,doesnotcarryusveryfar,andthatitcannotfulfilthefunction,whichHamiltonassignedtoit, of serving as the principle of all logical negation. It serves, however, toexpressthesignificanceofnegation,andatthesametimetosetforth(fromadifferentpointofviewfromthattakenbythelawofidentity)afundamentalpostulatewhichmustbegrantedifourprocessesofthoughtandreasoningaretobevalid.Forvalidityofthoughtandreasoningdemandthatfalsejudgmentsshallberefuted;andonlybythehelpofthelawofcontradictionisanysuchrefutation possible. The refutation requires that another judgmentcontradictoryofthefirstshallbeestablished;butthiswouldgofornothing,iftwocontradictoriescouldbetruetogether.

The law of contradiction thus takes its place by the side of the law ofidentityasafirstprincipleofdialecticandreasoning:notindeedadvancingusonourway,but servingas apostulate,withoutwhich itwouldnot evenbepossibleforustomakeastart.

WemaypasstoaconsiderationoftheformulaAisnotnot-A,bywhichthelawofcontradiction ismoreusuallyexpressed.Here,asSigwartpointsout,wehavenolongeranexpressionofarelationbetweentwojudgments,butanaffirmation that in a given judgment the predicate must not contradict thesubject;andinasmuchasdenialandcontradictionhaveprimarilynomeaningexcept in relation to judgments, this interpretation of the principle ofcontradictioncanatanyratenotberegardedasequallyfundamentalwiththatwhichwehavepreviouslygiven.At the same time, it is clear that if anyAwerenot-A,then,understandingnot-AtodenotewhateverdoesnotbelongtotheclassA,we shouldhave twocontradictory judgments, forwe shouldbe


abletoassertofsomethingboththatitbelongedtotheclassAandthatitdidnotbelongtotheclassA.

TheformulaAisnotnot-Aneednot,therefore,berejected,ifitssecondarycharacterisrecognised.

Mill’s attitude towards the law of contradiction involves an apparentinconsistency.Hebeginsbyregardingitasamodeofdefiningnegation.455Itis, he says, a mere identical proposition that if the negative be true, theaffirmativemustbefalse;forthe456negativeassertsnothingbutthefalsityofthe affirmative, and has no other sense or meaning whatever. He goes on,however,bothintheLogicandintheExaminationofSirWilliamHamilton’sPhilosophy,tospeakofthelawasageneralisationfromexperience.Hefindsits original foundation in the fact that belief and disbelief are two differentmentalstates,excludingoneanother,thisbeingafactwhichweobtainbythesimplestobservationofourownminds.Weobserve,moreover,thatlightanddarkness, sound and silence, equality and inequality, in short any positivephenomenon whatever and its negative, are distinct phenomena, pointedlycontrasted, and the one always absent when the other is present. From allthesefactsthelawofcontradictionis,inMill’sopinion,ageneralisation.

455 Logic,ii.7§5.

Twodistinctpointsappeartobeinvolvedinthisargument.Asregardsthereferencetobeliefanddisbelief,wemustagreethatthefoundationofthelawof contradiction is to be found in the nature of judgment. The essentialcharacteristicofajudgmentisthatitclaimstobetrue,andtheaffirmationofatruthimpliesbyitsverynatureadenial.Itis,however,difficulttoseewhereanygeneralisationcomesinhere.

TheotherpointthatMillraises,namely,thefactthatallourknowledgeisofcontrastsisageneralisationwhichisordinarilyknownasthepsychologicallawofrelativity.Thefact,however,thatwecannotapprehendlightexceptasdistinguishedfromdarkness,soundexceptasdistinguishedfromsilence,etc.,cannotbe regardedasequivalent to the lawofcontradiction.What that lawasserts is,asMillhimselfputs it, that“thesamepropositioncannotbebothfalseandtrue.”


Boole maintains that “the axiom of metaphysicians which is termed theprincipleofcontradiction,andwhichaffirmsthatitisimpossibleforanythingtopossessaqualityandatthesametimenottopossessit,isaconsequenceofthe fundamental law of thought, whose expression is x2 = x.” The law ofcontradictionisexpressedinBoole’ssystemintheformx(1−x)=0,wherexmaystandeitherforthetruthofajudgmentorforaterm;anditisofcourseclearthatx(1−x)=0followsfromx2=x.Itwill,however,beobservedthattheconversealsoholdsgood,sothatthequestionastowhichofthetwolawsis really themore fundamental remainsopen todiscussion.Apart fromthis,any attempt to deduce the law of contradiction from any other principlewhatsoever is open to the fundamental objection that unless the law ofcontradiction is 457 accepted as a postulate no single step in reasoning ispossible:forassoonasitisopentoustoaffirmajudgmentandatthesametimetodenyit,itisàfortioriopentoustoaffirmajudgmentandtodenyanyinferencethatmaybedrawnfromit.Tothequestionoftheinterdependenceofthelawsofthoughtweshallreturn.

It has been denied that the law of contradiction is a necessary law ofthought,onthegroundthatnotonlydoweoftenmeetwithself-contradiction,but that sometimes people have even boasted of holding contradictoryopinions.456If,however,thelawofcontradictionistoberejected,itmustbeshewnnotmerelythatwesometimescontradictourselves,butthatwedosowith perfect clearness of thought, and that we do not thereby stultifyourselves.

456 CompareBain,Logic,Deduction,p.223.

Themere fact of our holding contradictory opinions goes for nothing solong as the self-contradiction is not realisedbyus. In such cases itmaybeassumedthatoneorotherof thecontradictorydoctrineswillbegivenupassoonasthecontradictionbetweenthemismademanifest.Ifthetruthofbothisstillmaintained,itwillprobablybefoundthatthereissomereservation—as, for example, bymeansof a distinctionbetweendifferent kindsof truth,onedoctrinebeingheldtobetrueliterallyandtheotherinsomepoeticalorallegorical sense—whereby consistency is restored at the expense ofambiguityandwantofclearness.Apart fromsomeexplanationof thiskind,


the problemof accounting for theway inwhich someof us appear to holdinconsistentbeliefs isone for thepsychologist rather than the logician.Theultimate explanation must be sought in confusion of thought, or lack ofintellectualsincerity,orinthesetwocausescombined.Fromalogicalpointofview to rest in an unresolved contradiction is to stultify ourselves and toconfessfailure.

417.TheSophismof“TheLiar.”—ThesophismknownasΨευδόμενοςorThe Liar has been thought by some writers to present an exception to theuniversalapplicabilityofthelawofcontradiction.457

457 CompareUeberweg,Logic,p.245.

“Epimenides, theCretan, says that all Cretans are liars.He is, therefore,himselfa liar.Hencewhathesays isnot true,and theCretansarenot liars.But if so,his statementmaybeaccepted, and theyare liars.And soon,adinfinitum.”

The solution is simple ifwe interpret the statement ofEpimenides 458 tomeanmerely thatCretansusuallyspeakfalsehood.Lethisassertion thenbeunderstood in a stricter sense than this, and as meaning that Cretans arealways inall things liars, thatnoassertionmadebyaCretan iseverbyanychancetrue.

Again the solution is simple if we merely suppose the assertion false.Epimenides here speaks falsely, butCretans frequently or sometimes speakthe truth.Weareobviously confusing the contradictorywith the contrary ifwe pass from the position that it is not true that Cretans are always in allthingsliarstothepositionthatwhataCretansaysmustthereforebetrue.

Thesophismisalittlemorepuzzlingifwebeginbyassumingittobetruethat Cretans never speak the truth. Such an assumption contains no self-contradiction, and there is therefore nothing to prevent our taking it as ourstarting-point.Thisbeingso,letEpimenidesmakehisassertion.Becauseitistrue,here isaCretanwhohasspoken the truth,and therefore it is false. Itsown truth proves its own falsity.But, again, because it is true, Epimenidescannotbespeakingthetruth,andthereforeitisfalse.Oncemoreitsowntruthprovesitsownfalsity.


Theargumentmayalsobeputasfollows.AssumeittobetruethatCretansarealwaysinallthingsliars,andthenletEpimenides,theCretan,makethisassertion.Eitherhespeakstrulyorhespeaksfalsely.Butifhespeakstruly,ittherebyfollowsthathespeaksfalsely;whilst,ontheotherhand,ifhespeaksfalsely,hemerelyaffordsadditionalevidenceofthetruthofwhathesays.

The problem offering itself for solution is how an apparently validargument can thus yield as its result nothing but a bare contradiction. Theexplanation is that we have commenced with premisses that are implicitlycontradictory, and that our subsequent reasoning has fulfilled its properfunction in making the contradiction explicit. There is nothing self-contradictory in assuming that Cretans never speak the truth; but havingcommencedwith this assumption,we cannotwithout implicit contradictionsupposeaCretantomaketheassertion.Inotherwords,thetwopremisses—Cretansarealwaysinall thingsliars;andEpimenides,theCretan,saidso—cannotbetruetogether.

418. The Law of Excluded Middle.—The law of excluded middlesupplements thelawofcontradictioninexplainingthenatureof therelationbetweentwocontradictoryjudgments.Thelawofcontradictiontellsusthat,of two contradictory judgments one or other 459must be false, the truth ofeither implying the falsity of the other; the lawof excludedmiddle tells usthatof twocontradictoryjudgmentsoneorothermustbetrue, thefalsityofeither implying the truth of the other. It is only by the aid of the two lawscombinedthatthemeaningofnegationcanbefullyexpressed.

Sigwart regards the law of excluded middle as a derivative principledependentupontheprincipleofcontradictionandanotherprinciplewhichhedesignatestheprincipleoftwofold(ordouble)negation.Heobserves that tointerpret the nature of negation completelywemust add to the principle ofcontradiction the further principle that the negation of the negation isaffirmative, that to deny a negation is equivalent to affirming the samepredicateof thesamesubject.Tothisfurtherprinciplehegives thenameofdoublenegation;anditis,hesays,onlybecausethedenialofthenegationisthe affirmation itself that there is no medium between affirmation andnegation.


The deduction is as follows. LetX =A is B, and X = A is not B. Theprincipleofcontradiction tellsus thatof the twojudgmentsXandX,one isnecessarilyfalse.Itfollowsthatoneisnecessarilytrue.ForifIdenyX thenbysodoingImaintainX,while ifIdenyX then(bytheprincipleofdoublenegation) I maintain X. Therefore, the denial of both is equivalent to theaffirmation of both, that is, it involves a contradiction. Hence there is nomiddlestatementbetweenaffirmationandnegation.

In criticism of the above it may be questioned whether the bare law ofcontradiction justifies us in passing explicitly from the denial of X to theaffirmationofX.Sigwart’sownstatementoftheprincipleofcontradictionisthat X and X cannot be true together. This enables us to pass from theaffirmationofXtothedenialofX,orfromtheaffirmationofXtothedenialofX ;butnothingmore.Thereappears,moreover,tobeawantofsymmetryinSigwart’streatmentofthematter.Hemakesthelawofcontradictionyield(1)affirmationofXisdenialofX,(2)affirmationofXisdenialofX,(3)denialofX isaffirmationofX ;while theprincipleofdoublenegationyieldsonly(4)denialofXisaffirmationofX.

All four of these relations are required in order that the nature ofcontradiction may be fully expressed; but unless we sum up all four in asingle statement, it seems better to express (1) and (2) by means of theprinciple of contradiction, and (3) and (4) bymeans of a second principle,whetherwecallthelatterbythenameofthe460principleofexcludedmiddleor by any other name. It will be observed thatwe can express (1) and (2)togetherintheformNotbothXandX,and(3)and(4) together in theformEitherXorX.

Sigwart’sprincipleofdoublenegationthusappearstoexpressone-halfofwhatisordinarilyexpressedbymeansofthelawofexcludedmiddle;anditsseparate recognitionmaybe regarded as unnecessary. I agreewithSigwart,however,inholdingthatthelawofexcludedmiddledoesnomorethanhelptounfoldthemeaningofnegation.

ItisnotnecessarytooccupyspaceindiscussingtherelationoftheformulaEveryAisBornot-Btotheprincipleofexcludedmiddleasabovedescribed.Thisformulaexpressesasecondaryrelationbetweenso-calledcontradictory


termswhichfollowsfromthecorresponding,butmorefundamental,relationbetweencontradictoryjudgments.

For what is ordinarily known as the law of excluded middle, Jevonsproposesthenamelawofduality.458Thishedoesonthegroundthatthelawinquestionasserts thatateverystep thereare twopossiblealternatives,andhence gives to all the formulae of reasoning a dual character. The law ofduality occupies an important position in Jevons’s system of formal logic,which is based on the repeated application of the principle of dichotomaldivision.Itmay,however,bequestionedwhether,asthusemployed,thelawofdualityoughtnottoincludethelawofcontradictionaswellasthelawofexcluded middle. It is as important at each stage that the alternatives areexclusiveasthattheyareexhaustive.

458 PrinciplesofScience,1,§3.

419.Groundsonwhichtheabsoluteuniversalityandnecessityofthelawofexcludedmiddlehavebeendenied.—Theuniversalapplicabilityofthelawofexcludedmiddle has beenmore frequently denied than that of either of thetwolawspreviouslydiscussed.Thedenialusuallydependsuponaconfusionbetween contradictory opposition and contrary opposition. It is said, forexample,thatthereisameanbetweengreaterandless.This is true;but thelawofexcludedmiddledoesnotexcludethepossibilityofsuchamean.Thatlawdoesnot tellus thatagivenquantitymustbeeithergreateror less thananother given quantity; it only tells us that itmust be either greater or notgreater.

Closelyconnectedwiththisisthecasewhereourinability461(throughlackof the requisiteknowledgeorpowerofdiscernment) todecide in favourofeitheroftwocontradictoryalternativesissupposedtoyieldathirdalternative;as, for example, where to the two alternatives “guilty” and “not guilty” isaddedthethirdalternative“notproven.”“Guilty”and“notguilty,”consideredpurely in relation to the supposed culprit, are true contradictories, and theyadmitofnomean.But“provedtobeguilty”and“provedtobenotguilty”arecontraries, not contradictories; and it is here that the third alternative “notproven”comesin.

Somedifficultymayalsoarisefromambiguityoruncertaintyintheuseof


language.Thusitmayperhapsbesaidthataprisonermaybeneither“guilty”nor “not guilty,” but “partially guilty.” By “guilty,” however, we mustunderstandeither“entirelyguilty”or“guiltyinanydegree”;andwhicheverofthesemeaningsweadoptthedifficultyisresolved.

Wemay deal similarly with the questionwhether an action occupying afiniteintervaloftimeforitscompletionhasorhasnottakenplacewhenitisactuallyproceeding;forexample,whetherabattlehasorhasnotbeenfoughtwhenitishalfthrough,orwhetherthesunhasorhasnotrisenwhenhalfitscircumferenceisabovethehorizon.

The difficulties which arise in such cases as these are really verbaldifficulties.

Other difficulties arising from uncertainty as to the precise range ofapplicationoftermsarepartlyverbalandpartlydependentuponourimperfectpowersofdiscrimination.Wemayperhapshesitate tosayofagivencolourwhether it is “blue” or “green,” and therefore whether it is “blue” or “notblue.” If, however, by means of the spectrum or otherwise we are able todeterminequitepreciselywhatwemeanby“blue,”thedifficultyisobviated.

Mill remarks, on a different ground from any of the above, that theprincipleofexcludedmiddleisnottrueunlesswithalargequalification.“Apropositionmust be either true or false,provided that the predicate be onewhichcaninanyintelligiblesensebeattributedtothesubject.‘Abracadabraisasecondintention’isneithertruenorfalse.Betweenthetrueandthefalsethereisathirdpossibility,theunmeaning”(Logic,ii.7,§5).

The reply to this is that the law of excluded middle applies only topropositionsproperlyso-called,thatis,topropositionsregardedastheverbalexpressionsof judgments,aconditionwhichclearly is462notsatisfiedbyasentence (falsely called a proposition) which is unmeaning. If we define aproposition as the verbal expression of a judgment, then an “unmeaningproposition”—amerefortuitousjumbleofwordsthatconveysnothingtothemind—isinrealityacontradictioninterms.

Byan“unmeaningproposition”intheaboveargumentwehaveunderstoodaso-calledpropositionwhichhasnomeaningforthepersonwhouttersitor


foranyoneelse.Toagivenindividualastatementmadebysomeoneelsemaybe unmeaning because he does not understand the force of the termsemployed;butthisinnowayaffectstheprinciplethatthestatementwillasamatteroffactbeeithertrueorfalse.

Whilst, however, every judgmentmust be either true or false, it is quitepossible thatunsuitablequestionsmaybeput, the correct answers towhichwillbenegative,butwillbefelttobebarrenandinsignificantbecauseanyonewhounderstands themeaningof the termsemployedwill recogniseatoncethat the predicate cannot in any intelligible sense be attributed to thesubject.459


Isvirtuecircular?Thisquestionisfelttobeabsurd;butitisnotunmeaning.Bysayingthatanythingiscircularwemeanthatithassomefigureandthatitsfigure is circular. If, therefore, the question of circularity is raised inconnexionwith something that is immaterial, and thereforehasno figureatall,theanswermustbeinthenegative.460

460 CompareBradley,PrinciplesofLogic,p.145.MrBradleyputsthequestion,“When a predicate is really knownnot to be ‘onewhich can in any intelligiblesensebeattributedtothesubject,’isnotthatitselfgroundenoughfordenial?”

This pointmay perhaps hardly seemworth raising. It helps, however, toexplainhowMillisledtohisdenialoftheuniversalapplicabilityofthelawof excludedmiddle. Inhis criticismofHamilton’sdoctrineofnoumena thequestionisraisedwhethermatterinitselfhasaminimumofdivisibilityorisinfinitely divisible. Mill’s answer is that although we appear here to havecontradictoryalternatives,bothmayhavetoberejected,sincedivisibilitymaynotbepredicableatallofmatterinitself.Inotherwords,thepropositionthatmatter in itself has a minimum of divisibility is neither true nor false, butunmeaning.

It is tobeobserved,however, that“havingaminimumof463divisibility”and “being infinitely divisible” are not contradictories except within thesphereofthedivisible.Ifawiderpointofviewbetaken,thecontradictoryof“havingaminimumofdivisibility”mustbeexpressedsimplyintheform“nothaving aminimum of divisibility,” the latter including the case of “infinite


divisibility,”andalso thatof “theabsolute inapplicabilityof theattributeofdivisibility.”

420.AretheLawsofThoughtalsoLawsofThings?—Ontheviewtakenofthe lawsof thought in theprecedingpages, thequestionwhether these lawsarealso lawsof thingsmustbe regardedas somewhatmisleading.Wehavedescribed the laws aspostulateswhich are fundamental in all valid thoughtand reasoning, and we have regarded them as concerned essentially withjudgments.Ourresultsmaybeverybrieflysummarisedasfollows.

Thetruthaffirmedinanyjudgment,whenfullyexpressed, is independentoftimeandcontext.Itisaccordinglynotopentoustoacceptajudgmentatonestageofanargumentorcourseofreasoningandrejectitatanother.Thisunambiguity of the fact of judgment is declaredby the lawof identity, andagain by the lawof contradiction, the one looking at the question from thepositive,and,theotherfromthenegative,pointofview.Again,alljudgmentinvolves both affirmation and denial; and the force of any judgment is notfullygraspedbyusuntilwerealiseclearlywhat itdeniesaswellaswhat itaffirms. The law of contradiction, in conjunctionwith the law of excludedmiddle, has the function of making explicit what wemean by denial. Thethree lawsmaybeexpressedby these formulae: Iaffirmwhat Iaffirm,anddenywhatIdeny ;IfImakeanyaffirmation,Itherebydenyitscontradictory ;IfImakeanydenial,Itherebyaffirmitscontradictory.

Itfollowsthatwecannotmakeanyprogressinmaterialknowledgeexceptin subordination to these laws. But at the same time they do not directlyadvance our knowledge of things. They are distinctly laws relating tojudgments,andnotdirectlytothethingsaboutwhichwejudge.

Nodoubtwhenitissaidthatthelawsofthoughtarealsolawsofthings,thelawsarecontemplatedinwhatwehaveregardedastheirsecondaryforms:AisA ;Aisnotnot-A ;Everything isAornot-A.But even so it is difficult togive themanymeaningregardedas realpropositions.By“A”wemean“A”neither more nor less; and by “not-A” we mean “that which is not A butincludes everything else.” The laws do not profess to give any 464materialknowledge, and their validity is in no way dependent upon materialconditions.


Thequestionraisedinthissectionhasinsubstancebeenalreadydealtwithinrathermoredetailinspecialconnexionwiththelawofidentity.

421.MutualRelationsofthethreeLawsofThought.—Ifthevalidityoftheordinaryprocessesofimmediateinferenceisgranted,itcanbeshewnthatthethreelawsofthoughtmutuallyinvolveoneanother.

Startingfromthehypotheticalproposition,

IfAistruethenCistrue (i),

weobtainasits(true)disjunctiveequivalent,

ItcannotbethatAistrueandCisnottrue (ii),

andasitsalternativeequivalent,

EitherCistrueorAisnottrue (iii).

IfnowforC,wewriteAwehavethefollowingsetofequivalentpropositions:

IfAistrue,itistrue ;ItcannotbethatAisbothtrueandnottrue ;

Aiseithertrueornottrue ;

andtheseareexpressionsofthelawofidentity,thelawofcontradiction,andthelawofexcludedmiddlerespectively.

Ithasbeenalreadyshewninsection108thatasimilarresultisobtainableifwewriteSforPinthefollowingtrioofequivalentpropositions:

EverySisP ;NothingisbothSandnotP ;EverythingisPornotS.

Theseresultsindicatethecloserelationsthatexistbetweenthethreelaws.But it is a mistake to suppose that we can regard one only of them asfundamentaland the twoothersasdeducible from thisone.For the lawsofthought standat the foundationofallproof, and theymustbepostulated inorder that the equivalences above assumedmay themselvesbe shewn tobevalid.

422.TheLawsofThoughtinrelationtoImmediateInferences.—Granting


that the laws of thought stand at the foundation of all proof, it is a furtherquestionwhatinferences,ifany,canbeshewntobevalidbytheiraidalone.

Hamilton claims that the law of identity is the principle of all logicalaffirmation,thelawofcontradictionofalllogicalnegation,465andthelawofexcludedmiddleofalllogicaldisjunction.Bylogicalaffirmationwemayhereunderstand affirmationwhich canbe basedonpurely formal considerationswithout reference to the matter of thought, and we may interpret logicalnegation and logical disjunction similarly. The three laws of thought areaccordinglyheldbyHamiltontojustifywhatwehaveelsewherecalledformalpropositions, according as they are affirmative, negative, or disjunctiverespectively. The division into affirmative, negative, and disjunctive is,however,ofthenatureofacrossdivision;andthequestionariseswhereweare to place formal hypotheticals such as the following:—If it is true thatwhateverisSisP,thenitistruethatwhateverisnotPisnotS ;Ifit istruethatallSisMandthatallMisP,thenitistruethatallSisP.Apparently,since theyare affirmative, theyare tobebroughtunder the lawof identity;and inasmuch as the principle of any formal inference whatsoevermay beexpressed in a formal proposition similar in character to the abovepropositions,wefindthatHamiltonpracticallylaysdownthedoctrinethatinthe three laws of thought (if not in the law of identity alone) we have asufficientfoundationuponwhichtobasealllogicalinference.

This doctrine may, in the first place, be briefly considered with specialreferencetoimmediateinferences.

Itmaybegrantedthattheprocessofobversioncanbebasedexclusivelyonthelawsofcontradictionandexcludedmiddle.FromAllSisPwepasstoNoSisnot-Pbythelawofcontradiction;andfromNoSisPwepasstoAllSisnot-Pbythelawofexcludedmiddle.

But it is a different matter when we pass to the consideration of theprocessesofconversionandcontraposition;anditwillbefoundthatattemptsto base these processes exclusively on the three laws of thought usuallyresolvethemselveseitherintobareassertionsorelseintopracticaldenialsthatconversionandcontrapositionareprocessesofinferenceatall.


De Morgan observes, “When any writer attempts to shew how theperceptionofconvertibility‘AisBgivesBisA’followsfromtheprinciplesofidentity, difference, and excluded middle, I shall be able to judge of theprocess;asitis,Ifindthatothersdonotgobeyondthesimpleassertion,andthatImyselfcandetectthepetitioprincipiiineveryoneofmyownattempts”(SyllabusofLogic,p.47).

466ThetestthatIshouldbedisposedtoapplytoanyattemptedproofofthevalidityoftheprocessofconversionistoaskwhereintheprincipleinvolvedintheproofmakesmanifesttheinconvertibilityofanOproposition,andtheillegitimacyofthesimpleconversionofA.Itisclearthatwehavenorighttoassume that any self-evident principles that we may call to our aid461 areequivalenttothelawofidentity.


461 Forexample,—Ifoneclassiswhollyorpartiallycontainedinasecond,thenthesecondisatleastpartiallycontainedinthefirst;Ifoneclassiswhollyexcludedfromasecond,thenthesecondiswhollyexcludedfromthefirst.

ThefollowingattempttoestablishtheconversionofAandofIbymeansof the law of identity may be taken as an example: “Every affirmativepropositionmaybeconsideredasassertingthattherearecertainthingswhichpossess the attributes connoted both by the subject and the predicate—theclass SP. Hence the principle of identity justifies the conversion of anaffirmativeproposition.ForifthereareS’swhichpossesstheattributeP,theprincipleof identitynecessitates thatsomeof theobjectswhichpossess thatattributeareS’s.”Thelawofidentityisreferredtohere,butwemayfairlyaskinwhat form that law really comes in.Does the argument amount tomorethan that as thus analysed thevalidityof the conversion inquestion is self-evident?462Mightwenotforthewords“theprincipleofidentitynecessitates”substitutethewords“itisself-evident”?463

462 Insofarastheargumentisintendedtoamounttomorethanthis,itcontainsapetitioprincipii.

463 Compare, further, thediscussionof the legitimacyofconversion insection99.

Nodoubtifimmediateinferencesarenomorethanverbaltransformations,thentheycanallbebasedontheprincipleofidentityasinterpretedbyMill,namely,ontheprinciplethatwhateveristrueinoneformofwordsistrueinanyotherformofwordshavingthesamemeaning.Butifconversion(oranyotherformofimmediateinference)ismorethanmereverbaltransformation,theequivalenceof theconvertendand theconverse is justwhatwehave toshew; they are not merely two different forms of words having the samemeaning.

423.TheLawsofThoughtandMediateInferences.—Manselexpressestheviewthatsyllogisticreasoning—andindeedallformalreasoningwhatsoever—can be based exclusively on the laws of 467 identity, contradiction, andexcludedmiddle.Theprincipleofidentityis,hesays,immediatelyapplicableto affirmative moods in any figure, and the principle of contradiction tonegatives.464Hisproofofthispositionconsistsinquantifyingthepredicatesof the propositions constituting the syllogism, and then making use—for


affirmatives—oftheaxiomthat“whatisgivenasidenticalwiththewholeorapartofanyconcept,mustbeidenticalwiththewholeorapartofthatwhichis identicalwith the same concept,” and—for negatives—of the axiom that“someorallS,beinggivenas identicalwithallorsomeM, isdistinctfromeverypartofthatwhichisdistinctfromallM.”

464 ProlegomenaLogica,p.222.

These formulae, however, go distinctly beyond the laws of identity andcontradictionasordinarilystated.Theymayindeedberegardedasequivalentto thedictumdeomnietnullo, adaptedsoas tobeapplicable tosyllogismsmadeupofpropositionswithquantifiedpredicates;andifit isassumedthatthe dictum is only another form of stating the laws of identity andcontradictionthenthequestionneedsnofurtherdiscussion.Onlyinthiscasewemustnolongerexpressthelawofidentityeitherintheform“Whatistrueistrue,”orintheform“AisA”;northelawofcontradictioneitherintheform“Ifajudgmentistrue,itscontradictoryisnottrue,”orintheform“Aisnotnot-A.” The laws as thus formulated cannot be regarded as adequateexpressionsoftheaxiomuponwhichsyllogisticreasoningproceeds.Theydonot bring out the function of the middle term which is the characteristicfeatureofthesyllogism,norcouldtherulesofthesyllogismbededucedfromthem.

Of course syllogistic reasoning, like all other reasoning, presupposes thelawsofthought,andintheprocessofindirectreduction,whichoccupiesanotunimportant place in the doctrine of the syllogism, these laws come inexplicitly.

Itisnotnecessarytoconsiderindetailformalinferencesbelongingtothelogicofrelatives,e.g.,BisgreaterthanC,AisgreaterthanB,therefore,Aisgreater thanC.Herewe require the principle thatwhatever is greater thananythingthatisgreaterthanathirdthingisitselfgreaterthanthethirdthing;anditwouldbestillmoredifficultthaninthecaseofthedictumdeomnietnullotoevolvethisprincipleimmediatelyoutofthethreelawsofthought.


APPENDIXC.

AGENERALIZATIONOFLOGICALPROCESSESINTHEIRAPPLICATIONTOCOMPLEXPROPOSITIONS.465


CHAPTERI.

THECOMBINATIONOFTERMS.

465 Thefollowingpagesdealwithproblemsthathaveordinarilybeenrelegatedto symbolic logic.Theydonot, however, treat of symbolic logicdirectly, if thattermisunderstoodinitsordinarysense,namely,asdesignatingthatbranchofthescience inwhich symbols ofoperation are used.Of course in a broad sense allformallogicissymbolic.

424.ComplexTerms.—Asimpletermmaybedefinedasatermwhichdoesnot consist of a combination of other terms.Wedenote a simple termby asingleletter;forexample,A,P,X.Thecombinationofsimpletermsyieldsacomplexterm;andthecombinationmaybeeitherconjunctiveoralternative.

Acomplextermresultingfromtheconjunctivecombinationofothertermsmaybecalledaconjunctive term,and itwillbe foundconvenient todenotesucha termbythesimple juxtapositionof theother termsinvolved.466Thiskindofcombinationissometimescalleddetermination;andwemayspeakoftheelementscombinedinaconjunctivetermasthedeterminantsofthatterm.Thus,AandBarethedeterminantsoftheconjunctivetermAB.

466 The conjunctive combination of terms is in symbolic logic usuallyrepresentedbythesignofmultiplication.

Acomplextermresultingfromthealternativecombinationofother termsmay be called an alternative term; and we may speak of the elementscombinedinsuchatermasthealternantsofthatterm.Thus,AandBarethealternantsofthealternativetermAorB.467

467 Thealternativecombinationoftermsisinsymboliclogicusuallyrepresentedbythesignofaddition.

469Inthefollowingpages,inaccordancewiththeviewindicatedinsection191, the alternants in an alternative term are not regarded as necessarilyexclusive of one another (except of course where they are formalcontradictories). Thus, ifwe speak of anything as beingA or B we do not


intendtoexcludethepossibilityofitsbeingbothAandB.Inotherwords,AorBdoesnotexcludeAB.

It isnecessaryat thispoint toconsiderbriefly the logical significationofthewordsand,or.Inthepredicateofapropositiontheirsignificationisclear;they indicate conjunctive and alternative combination respectively; forexample,PisQandR,PisQorR.Butwhentheyoccurinthesubjectofaproposition there is in each case an ambiguity to which attention must becalled.

Thus,therewouldbeagaininbrevityifwecouldwriteapropositionwithanalternative termas subject in the formPorQ isR.This last expressionwould,however,morenaturallybeinterpretedtomeanPisRorQisR, theforce of the or being understood, not as yielding a single categoricalproposition with an alternative subject-term, but as a brief mode ofconnecting alternatively twopropositionswith a commonpredicate.Hence,whenweintendtheformer,themoredefinitemodeofstatement,WhateveriseitherPorQisR,orAnythingthatiseitherPorQisR,shouldbeadopted.

ThereisalsoambiguityintheformPandQisR.Thiswouldnaturallybeinterpreted,notasasinglecategoricalpropositionwithaconjunctivesubject-term (PQ is R), but as a brief mode of connecting conjunctively twopropositionswithacommonpredicate,namely,PisRandQisR. Inorder,therefore,toexpressunambiguouslyapropositionwithaconjunctivesubject-term, it will be well either to adopt the method of simple juxtapositionwithoutanyconnectingwordas,forexample,PQisR,orelsetoemployoneofthemorecumbrousforms,WhateverisbothPandQisR,orAnythingthatisbothPandQisR.468

468 Itwillbeobservedthatbothinthiscaseandinthecaseofor,wegetridofthe ambiguity by making the words occur in the predicate of a subordinatesentence.Mr Johnsonexpresses the substanceof the last threeparagraphs in thetext by pointing out that “common speech adopts the convention: Subjects areexternally synthesisedandpredicatesare internally synthesised”(Mind, 1892, p.239). In other words, and and or occurring in a predicate are understood asexpressingaconjunctiveoranalternativeterm ;butoccurringinasubjecttheyareunderstoodasexpressingaconjunctiveoranalternativeproposition.

425. Order of Combination in Complex Terms.—The order of 470


combination in a complex term is indifferent whether the combination beconjunctiveoralternative.469

469 Thisissometimesspokenofasthelawofcommutativeness.CompareBoole,LawsofThought,p.31,andJevons,PrinciplesofScience,2,§8.

Thus,ABandBAhave thesamesignification. Itcomes to thesame thingwhetheroutoftheclassAweselecttheB’soroutoftheclassBweselecttheA’s.

Again,A or B andB or A have the same signification. It is amatter ofindifferencewhetherweformaclassbyaddingtheB’stotheA’sorbyaddingtheA’stotheB’s.

426.TheOpposition of Complex Terms.—However complex a termmaybe, the criterion of contradictory opposition given in section 40 must stillapply: “Apairof contradictory termsare so related thatbetween them theyexhausttheentireuniversetowhichreferenceismade,whilstinthatuniversethere is no individual ofwhich both can be affirmed at the same time.” Inwhat follows itwillbe foundconvenient todenote thecontradictoryofanysimpletermbythecorrespondingsmallletter.Thusfornot-Awemaywritea,andfornot-Bwemaywriteb.

NowwhateverisnotABmustbeeitheraorb,whilstnothingthatisABcanbeeitheraorb.Hence

⎰AB, ⎱aorb,

constituteapairofcontradictories.Similarly,

⎰AorB,⎱ab,

areapairofcontradictories.AndthesamewillholdgoodifAandBstandfortermswhich are already themselves complex (although relatively simple ascomparedwithABorAorB).

If, then, two terms are conjunctively combined into a complex term (ofwhichtheywillconstitutethedeterminants),thecontradictoryofthiscomplexterm is found by alternatively combining the contradictories of the two


determinants.And,conversely,iftwotermsarealternativelycombinedintoacomplexterm(ofwhichtheywillconstitutethealternants),thecontradictoryofthiscomplextermisfoundconjunctivelycombiningthecontradictoriesofthetwoalternants.

In each case,we substitute for the relatively simple terms involved theircontradictories,and(asthecasemaybe)change471conjunctivecombinationinto alternative combination, or alternative combination into conjunctivecombination.

Butwhateverdegreeofcomplexitya termmay reach, itwillconsistofaseriesofconjunctiveandalternativecombinations;anditmaybesuccessivelyresolved into the combinationof pairs of relatively simple terms till it is atlast shewn to result from the combination of absolutely simple terms. Forexample,—ABC or DE or FG results from the alternative combination ofABCorDEwithFG ;ABCorDEresultsfromthealternativecombinationofABCwithDE ;FGresultsfromtheconjunctivecombinationofFwithG ;andABC,DEmayberesolvedsimilarly.

Hence the successive application of the above rule, for finding thecontradictoryofacomplex termwherewearedealingwitha singlepairofdeterminantsoralternants,will result inourultimately substituting foreachsimple term involved its contradictory, and reversing the nature of theircombination throughout.470Wemay, therefore, lay down the following ruleforobtainingthecontradictoryofanycomplexterm:Replaceeachconstituentsimple term by its contradictory and throughout substitute conjunctivecombination for alternative combination and vice versâ.471 This rule is ofsimple application, and it is of fundamental importance in the treatment ofcomplexpropositionsadoptedinthefollowingpages.

470 Thus,takingthetermABCorDEorFG,andinthefirstinstancedenotingthecontradictoryofacomplextermbyabardrawnacrossit,wehavesuccessively,—

ABCorDEorFG= ABC(DEorFG)= (ABorc)DE.FG= (aorborc)(dore)(forg).

471 CompareSchröder,DerOperationskreisdesLogikkalkuls,p.18.


Thus,thecontradictoryofAorBC

is aand(borc),i.e., aborac ;

andthecontradictoryofABCorABD

is (aorborc)and(aorbord),

which,bytheaidofrulespresentlytobegiven,isreducibletotheform

aorborcd.

It is possible for two complex terms to be formally inconsistent orrepugnant without being true contradictories. This will be the case if theycontain contradictory determinants without between them exhausting theuniverseofdiscourse.ThetermsABandbCaffordanexample:nothingcanbebothABandbC (for, if this472wereso,somethingwouldbebothBandnot-B),butwecannotsayàpriori thateverything iseitherABorbC (sincesomethingmaybeAbc,whichisneitherABnorbC).

427.Duality of Formal Equivalences in the case of Complex Terms.—Itwill be shewn in the following sections that certain complex terms areformallyequivalenttoothercomplextermsortosimpleterms(forexample,AoraB=AorB,AorAB=A);anditisimportanttonoticeattheoutsetthatsuchformalequivalencesalwaysgoinpairs.Foriftwotermsareequivalent,theircontradictoriesmustalsobeequivalent;andhence,applyingtheruleforobtaining contradictories given in the preceding section, we are enabled toformulatethesimplelawthattoeveryformalequivalencetherecorrespondsanother formal equivalence inwhichconjunctive combination is throughoutsubstituted for alternative combination and vice versâ.472 This lawmay bemorepreciselyestablishedasfollows:—Aformalequivalencethatholdsgoodforanygivensetoftermsmustequallyholdgoodforanyothersetofterms;and,therefore,whateverholdsgoodforthetermsA,B,&c.mustholdgoodfortheircontradictoriesa,b,&c.Hence,givenanyequivalence,wemayfirstreplaceeachsimpletermbyitscontradictory,andthentakethecontradictoryofeachsideoftheequivalence.Theresultofthisdoubletransformationwillbe that we shall obtain another equivalence in which every conjunctivecombinationhasbeenreplacedbyanalternativecombination,andconversely,


whiletheterm-symbolsinvolvedhaveremainedunchanged.Thisproveswhatwasrequired.

472 ThisispointedoutbySchröder,DerOperationskreisdesLogikkalkuls,p.3.The two equivalenceswhich are thusmutually deducible the one from the othermaybesaidtobereciprocal.

Theapplicationoftheabovelawwillbefullyillustratedinthesectionsthatimmediatelyfollow.

428. Laws of Distribution.—In order to combine a simple termconjunctivelywithanalternativeterm,wemustconjunctivelycombineitwitheveryalternantofthealternative.473Aand(BorC)474denoteswhateverisAandat thesametimeeitherBorC,andhence isequivalent toABorAC. Itfollowsthatinordertocombinetwoalternativetermsconjunctively,wemustconjunctivelycombineeveryalternantoftheonewitheveryalternantoftheother.Thus,473(AorB)(CorD)denoteswhateveriseitherAorBandatthesametimeeitherCorD,andisequivalenttoACorADorBCorBD.475

473 CompareJevons,PrinciplesofScience,5,§7.474 In such a case as this the use of brackets is necessary in order to avoid

ambiguity.Thus,AandBorCmightmeanABorC,orasaboveABorAC.475 Whether or notwe introduce algebraic symbols into logic, there is here a

verycloseanalogywithalgebraicmultiplicationwhichcannotbedisguised.

Wehavethen

A(BorC)=ABorAC,

andapplyingthelawofdualityofformalequivalencesgivenintheprecedingsection,wehaveatonceanotherequivalence,namely,

AorBC=(AorB)(AorC).476476 This equivalence might also be established independently by the aid of

certainoftheequivalencesgiveninthefollowingsections.

ThesetwoequivalencesarecalledbySchrödertheLawsofDistribution.477Theyareofthegreatestimportanceinthemanipulationandsimplificationofcomplexterms.

477 DerOperationskreisdesLogikkalkuls,pp.9,10.

429.LawsofTautology.—The following rulesmay be laid down for the


omissionofsuperfluoustermsfromacomplexterm: (a)Therepetitionofanygivendeterminantissuperfluous. OutoftheclassAtoselecttheA’sisaprocessthatleavesusjustwherewebegan.Inotherwords,whatisbothAandAisidenticalwithwhatisA.Thus,suchtermsasAA,ABB,aretautologous;theformermerelydenotestheclassA,andthelattertheclassAB.Hencetheaboverule,whichiscalledbyJevonstheLawofSimplicity.478 (b)Therepetitionofanygivenalternantissuperfluous. Tosay thatanything isAorA is equivalent to sayingsimply that it isA.HencesuchtermsasAorA,AorBCorBC,aretautologous;andwehavetheaboverule,whichiscalledbyJevonstheLawofUnity.479

478 SeePureLogic,§42;andPrinciplesofScience,2,§8.Thecorrespondingequationx2=xisinBoole’ssystemfundamental;seeLawsofThought,p.31.

479 SeePureLogic,§69;andPrinciplesofScience,5,§4.

Itwillbeseenbyreferencetotherulegiveninsection427thattheLawofSimplicity(AA=A)andtheLawofUnity(AorA=A)arereciprocal;thatis,theformerisdeduciblefromthelatterandviceversâ.Fortheonlydifferencebetween them is that conjunctive combination in the one is replaced byalternativecombinationintheother.480

480 Itmayassistthereaderinfollowingthereasoninginsection427ifweworkthroughthisparticularcaseindependently.IfAA=A,thenaa=a,forwhateverisformallyvalidinthecaseofAmustalsobeformallyvalidinthecaseofanyotherterm. But if two terms are equivalent their contradictories must be equivalent.Hencefromaa=a,itfollowsthatAorA=A.AnditisclearthatwemightpasssimilarlyfromAorA=AtoAA=A.

474 430. Laws of Development and Reduction.—Important formalequivalencesareyieldedbythelawsofcontradictionandexcludedmiddle.

By the lawof contradiction a termcontaining contradictory determinants(for example, Bb) cannot represent any existing class. Hence A or Bb isequivalent to A simply; in other words, the conjunctive combination ofcontradictoriesmaybeindifferentlyintroducedoromittedasanalternant.

Again, by the law of excluded middle a term containing contradictoryalternants (for example,Borb) represents the entire universe of discourse.HenceA (Borb) is equivalent toA simply; in otherwords, thealternative


combinationofcontradictoriesmaybeindifferentlyintroducedoromittedasadeterminant.

Itwillbeobservedthattheaboveequivalences,namely,

AorBb=A,A(Borb)=A,

arereciprocal.

ApplyingfurthertheLawsofDistributiongiveninsection428wehavethefollowing:

A=AorBb=(AorB)(Aorb),A=A(Borb)=ABorAb.

Thesemaybetakenasformulaeforthedevelopmentandthereductionofterms. Thus, the substitution of (A or B) (A or b) forA may be called thedevelopment of a term by means of the law of contradiction; and thesubstitutionofABorAbforAthedevelopmentofatermbymeansofthelawof excludedmiddle. In both the above cases the termA is developed withreferencetothetermB.SimilarlybydevelopingAwithreferencetoBandC,weshouldhave(AorBorC)(AorBorc)(AorborC)(AorBorc)ifwemakeuseof the lawofcontradiction,orABCorABcorAbCorAbc ifwemakeuseofthelawofexcludedmiddle.Developmentbymeansofthelawofexcludedmiddleisthemoreusefulofthetwoprocessesinthemanipulationof complex terms, and it may be understood that this is meant when thedevelopmentofatermisspokenofwithoutfurtherqualification.

Conversely,theprocessofpassingfrom(AorB)(Aorb)toA,orfromABorAbtoA,maybecalledthereductionofatermbymeans475ofthelawofcontradictionorthelawofexcludedmiddle,asthecasemaybe.

FollowingJevons,wemayspeakofanalternativetermofthetypeABorAbasadualterm,andofthesubstitutionofAforABorAbasthereductionofadualterm.481

481 PureLogic, § 103. The conjunctive term (A or B) (A or b) may also bespokenofasadualterm,anditsreductiontoAasthereductionofadualterm.

431.Laws of Absorption.—It may be shewn that any alternant which is


merelyasubdivisionofanotheralternantmaybeindifferentlyintroducedoromittedfromacomplexterm.Thus,ABbeingasubdivisionofA,thetermsAorABandAareequivalent.Thisrule(whichiscalledbySchrödertheLawofAbsorption482)maybeestablishedasfollows:BythedevelopmentofAwithreferencetoB,AorABbecomesABorAborAB ;but,by the lawofunity,thisisequivalenttoABorAb ;andbyreductionthisisequivalenttoA.

482 Der Operationskreis des Logikkalkuls, p. 12. This Law of Absorption isequivalenttooneofBoole’s“MethodsofAbbreviation”(LawsofThought,p.130).Compare,also,Jevons,PureLogic,§70.

Applying the rule given in section 427 we obtain a second law ofabsorption,namely,A(AorB)=A,whichisthereciprocalofthefirstlawofabsorption,AorAB=A.

432.LawsofExclusionandInclusion.—Thecontradictoryofanyalternantinacomplextermmaybeindifferentlyintroducedoromittedasadeterminantof any other alternant; that is to say, the terms A or aB and A or B areequivalent.Thismaybeestablishedasfollows:BythelawofabsorptionAoraBisequivalenttoAorABoraB,andbyreductionthisyieldsAorB.TheaboveequivalencemaybecalledtheLawofExclusiononthegroundthatbypassingfromAorBtoAoraBwemakethealternantsmutuallyexclusive.

ThereciprocalequivalenceA(aorB)=ABmaybeexpressedasfollows:Thecontradictoryofanydeterminantinacomplextermmaybeindifferentlyintroduced or omitted as an alternant of any other determinant. ThisequivalencemaybecalledtheLawofInclusiononthegroundthatbypassingfromABtoA(aorB)wemakethedeterminantscollectivelyinclusiveoftheentireuniverseofdiscourse.

433.SummaryofFormalEquivalencesofComplexTerms.—Thefollowingis a summary of the formal equivalences contained in the five precedingsections (those that are bracketed together being 476 in each case related tooneanotherreciprocallyinthemannerindicatedinsection427):—

(1) A(BorC)=ABorAC, ⎱LawsofDistribution ;

(2) AorBC=(AorB)(AorC), ⎰

(3) AA=A, ⎱LawsofTautology(LawofSimplicityandLawofUnity) ;(4) AorA=A, ⎰


(5) A=AorBb=(AorB)(Aorb), ⎱ LawsofDevelopmentandReduction ;(6) A=A(Borb)=ABorAb, ⎰

(7) AorAB=A, ⎱LawsofAbsorption ;

(8) A(AorB)=A, ⎰

(9) AorB=AoraB, ⎱ LawofExclusionandLawofInclusion.(10) AB=A(aorB), ⎰

434.TheConjunctiveCombinationofAlternativeTerms.—Thefirstlawofdistribution gives the general rule for the conjunctive combination ofalternatives.Butwith a view to such combination special attentionmay becalled(i)tothesecondlawofdistribution,namely,(AorB)(AorC)=AorBC ;and(ii)totheequivalence(AorB)(ACorD)=ACorADorBD,whichmaybeestablishedasfollows:Bythefirstlawofdistribution(AorB)(ACorD) isequivalenttoAACorABCorADorBD ;butbythelawofsimplicityAAC=AC,andbythelawofabsorptionACorABC=AC ;henceouroriginaltermisequivalenttoACorADorBD,whichwastobeproved.

From the above equivalencesweobtain the two followingpractical ruleswhich are of great assistance in simplifying the process of conjunctivelycombiningalternatives: (1) If two alternatives which are to be conjunctively combined have analternant in common, this alternant may be at once written down as onealternantoftheresult,andweneednotgothroughtheformofcombiningitwithanyoftheremainingalternantsofeitheralternative; (2)Iftwoalternativesaretobeconjunctivelycombinedandanalternantofoneisasubdivisionofanalternantoftheother,thentheformeralternantmaybe at oncewritten down as one alternant of the result, andweneednot gothrough the formofcombining itwith the remainingalternantsof theotheralternative.483

483 TheserulesareequivalenttoBoole’ssecondMethodofAbbreviation(LawsofThought,p.131).

477

EXERCISES.


435.Simplifythefollowingterms:(i)ADoracD ;(ii)AdorAeoraBoraCoraEorbCorbdorbEorbeorcdorce.[K.](i) By rule (1) in section 433,AD or acD is equivalent to (A or ac)D ; and this by rule (9) is

equivalentto(Aorc)D ;whichagainbyrule(1)isequivalenttoADorcD.484 (ii)ThedualtermbEorbemaybereducedtob,andhenceAdorAeoraBoraCoraEorbCorbdorbEorbeorcdorce=AdorAeoraBoraCoraEorborbCorbdorcdorce.Bysection433,rule(7),wemay nowomit all alternants inwhichb occurs as a determinant, and by rule (9),Bmay beomittedwhereveritoccursasadeterminant;accordinglyourtermisreducedtoAdorAeoraoraCoraEorborcdorce.Sinceaisnowanalternant,afurtherapplicationofthesamerulesleavesuswithaorborcdorceordore ;andthisisimmediatelyreducibletoaorbordore.

484 Wemightalsoproceedasfollows:ADoracD=ADorAcDoracD[byrule(7)]=ADorcD[byrule(5)].

436.ShewthatBCorbDorCDisequivalenttoBCorbD.[K.]

437.Give thecontradictoriesof the following terms in their simplest formsas seriesofalternants:—ABorBCorCD ;ABorbCorcD ;ABCoraBc ;ABcDorAbcdeoraBCDeorBCde.[K.]

438.Simplifythefollowingterms: (1)AboraCorBCdorBcorbDorCD ; (2)ACDorAcorAdoraBorbCD ; (3)aBCoraBeoraCDoraDeorAcDorabDorbcDoraDEorcDE ; (4)(Aorb)(Aorc)(aorB)(aorC)(borC).[K.]

439.Provethefollowingequivalences: (1)ABorACorBCoraBorabcorC=aorBorC ; (2)aBCoraBdoracdorABdorAcdorabdoraCdorBCdorbcd=aBCoradorBdorcd ; (3)PqrorpQsorpqorprsorqrsorpSorqR=porq.[K.]


CHAPTERII.

COMPLEXPROPOSITIONSANDCOMPOUNDPROPOSITIONS.

440.ComplexPropositions.—Acomplexpropositionmaybedefinedasapropositionwhichhasacomplex termeither for its subjector itspredicate.Theordinarydistinctionsofquantityandqualitymaybeappliedtocomplexpropositions; thus All AB is C or D is a universal affirmative complexproposition.SomeABisnotEFisaparticularnegativecomplexproposition.In the following pages propositions written in the indefinite form will beinterpretedasuniversal,sothatABisCDwillbeunderstoodtomeanthatallAB is CD. It is to be added that in dealing with complex propositions weinterpretparticularsasimplying,butuniversalsasnotimplying,theexistenceoftheirsubjectsintheuniverseofdiscourse.

441. The Opposition of Complex Propositions.—The opposition ofcomplex terms has been already dealtwith, and the opposition of complexpropositionsinitselfpresentsnospecialdifficulty.Itmust,however,bebornein mind that as we interpret particulars as implying the existence of theirsubjects, but universals as not doing so,wehave the followingdivergencesfromtheordinarydoctrineofopposition:(1)wecannotinferIfromA,orOfromE;(2)AandEarenotnecessarilyinconsistentwitheachother;(3)IandOmaybothbefalseatthesametime.Theordinarydoctrineofcontradictoryopposition remainsunaffected.The followingareexamplesofcontradictorypropositions:AllXisbothAandB,SomeXisnotbothAandB ;SomeXisYandatthesametimeeitherPorQorR,NoXisYandatthesametimeeitherPorQorR.

442. Compound Propositions.485—A compound proposition may bedefined as a proposition which consists in a combination of otherpropositions.Thecombinationmaybeeitherconjunctive(i.e.,when479 twoor more propositions are affirmed to be true together) or alternative (i.e.,whenanalternativeisgivenbetweentwoormorepropositions);forexample,


All AB is C and some P is not either Q or R is a compound conjunctiveproposition;EitherallABisCorsomePisnoteitherQorRisacompoundalternativeproposition.Propositionsconjunctivelycombinedmaybespokenof asdeterminants of the resulting compound proposition; and propositionsalternatively combined may be spoken of as alternants of the resultingcompound proposition. In what follows, both conjunctive and alternativepropositionsareinterpretedasbeingassertoric.



Only two types of compound propositions are here recognised, theconjunctiveandthealternative.Purehypotheticalpropositionsarecompound,but(exceptinsofarasweinterprethypotheticalsandalternativesdifferentlyin respect of modality) they are equivalent to alternative propositions, andmay be regarded as constituting one mode of expressing an alternativesynthesis.Thus (takingxandy as symbols representingpropositions, andxand y as their contradictories) the hypothetical proposition If x then yexpressesanalternativebetweenxandy and is, therefore, equivalent to thealternativepropositionxory.Combinationsof the truedisjunctive type(forexample,notbothxandy)mayalsoberegardedasamodeofexpressinganalternative synthesis; thus, the true disjunctive proposition just given isequivalenttothealternativepropositionxory.486

486 Theabovemayseemtoimplythatanalternativesynthesismaybeexpressedinagreaternumberofwaysthanaconjunctivesynthesis.This,however,isnotthecase. It has been shewn that an alternative synthesis may be expressed by ahypotheticalorby thedenialofaconjunctive(that is,bya truedisjunctive).Butcorrespondingtothis,aconjunctivesynthesismaybeexpressedbythedenialofahypotheticalorby thedenialofanalternative.Thus, representing thedenialofapropositionbyabardrawnacrossit,wehave

xy=x̅ory̅=Ifx,y ̅;xy=xory=Ifx,y.

Mr Johnson shews that any ordinary proposition with a general term assubject may be regarded as a compound proposition resulting from theconjunctive or alternative combination of singular (molecular) propositions,withacommonpredication,butdifferentsubjects.LetS1,S2,…S∞representa number of different individual subjects; and letS represent the aggregatecollectionofindividualsS1,S2,…S∞.Then

S1andS2,andS3…andS∞=EveryS ;

S1orS2,orS3……orS∞=SomeS.

480“Thuswearriveatthecommonlogicalforms,EverySisP,SomeS isP. The former is an abbreviation for a determinative, the latter for analternative,synthesisofmolecularpropositions.”487

487 Mind,1892,p.25.MrJohnsonofcourserecognisesthataquantifiedsubject-


term(allS)isnotusuallyamereenumerationofindividualsfirstapprehendedandnamed. But he points out that “however the aggregate of things, to which theuniversalnameapplies,ismentallyreached,thepropositionalforceforpurposesofinferenceorsynthesisingeneralisthesame”(p.28).

Inotherwords,EverySisP=S1isPandS2isPandS3isP…andS∞isP ;SomeSisP=S1isPorS2isPorS3isP…orS∞isP.

443.TheOpposition of Compound propositions.—The rule for obtainingthecontradictoryofacomplextermgiveninsection426maybeappliedalsotocompoundpropositions.Thus,thecontradictoryofacompoundpropositionisobtainedby replacing theconstituentpropositionsby theircontradictoriesand everywhere changing the manner of their combination, that is to say,substituting conjunctive combination for alternative and vice versâ.488 Thefollowingareexamples:AllAisBandsomePisQhasforitscontradictoryEithersomeAisnotBornoPisQ ;EithersomeAisbothBandC,orallBiseitherCorbothDandEhasforitscontradictoryNoAisbothBandC,andsomeBisnoteitherCorbothDandE.

488 Ithasbeenshewnintheprecedingsectionthatthewordsallandsome areabbreviationsofconjunctiveandalternativesynthesisrespectively.Hencetherulethat, in the ordinarily recognised propositional forms, contradictories differ inquantityaswellas inquality is itselfonlyaparticularapplicationof thegenerallawherelaiddown.

Itfollows,asinsection427,thatthereisadualityofformalequivalencesinthecaseofcompoundpropositions,eachequivalenceyieldingareciprocalequivalence inwhich conjunctive combination is throughout substituted foralternativecombinationandviceversâ.

444.FormalEquivalencesofCompoundPropositions.—Thelawsrelatingto the conjunctive or alternative synthesis of propositions are practicallyidenticalwiththoserelatingtotheconjunctiveoralternativecombinationofterms; and we have accordingly the following propositional equivalencescorrespondingtotheequivalencesoftermsgiveninsection433.Thesymbolsherestandforpropositions,not terms;andnegation is representedbyabaroverthepropositiondenied.481

(1) x(yorz)=xyorxz, ⎱


LawsofDistribution ;(2) xoryz=(xory)(xorz), ⎰

(3) xx=x, ⎱ LawsofTautology(LawofSimplicityandLawofUnity) ;(4) xorx=x, ⎰

(5) x=xoryy=(xory)(xory), ⎱ LawsofDevelopmentandReduction ;(6) x=x(yory)=xyorxy, ⎰

(7) xorxy=x, ⎱LawsofAbsorption ;

(8) x(xory)=x ⎰

(9) xory=xorxy, ⎱ LawofExclusionandLawofInclusion.(10) xy=x(xory),489 ⎰

489 Itisnotmaintainedthatalltheabovelawsareultimateorevenindependentof one another. The synthesis of propositions is admirably worked out by MrJohnson in his articles on the Logical Calculus (Mind, 1892). He gives fiveindependent laws which are necessary and sufficient for propositional synthesis.These laws are briefly enumerated below; for a more complete exposition thereadermustbereferredtoMrJohnson’sowntreatmentofthem. (i)TheCommutativeLaw:Theorderofpuresynthesisisindifferent(xy=yx). (ii)TheAssociativeLaw:Themodeofgroupinginpuresynthesisisindifferent(xy.z=x.yz). (iii)TheLawofTautology:Themererepetitionofapropositiondoesnotinanywayaddtooralteritsforce(xx=x). (iv) The Law of Reciprocity: The denial of the denial of a proposition isequivalent to its affirmation (x̅ =x). “In this principle are included the so-calledLawsofContradictionandExcludedMiddle,viz.,‘Ifx,thennotnot-x’,and‘Ifnotnot-x,thenx’.” (v)TheLawofDichotomy: The denial of any proposition is equivalent to thedenialofitsconjunctionwithanyotherpropositiontogetherwiththedenialofitsconjunctionwiththecontradictoryofthatotherproposition(x=xyxy)̅.“ThisisafurtherextensionoftheLawofExcludedMiddle,whenappliedtothecombinationofpropositionswithoneanother.Thedenial thatx isconjoinedwithycombinedwith the denial that x is conjoined with not-y is equivalent to the denial of xabsolutely.For,ifxweretrue,itmustbeconjoinedeitherwithyorwithnot-y.Thislaw,which (itmustbe admitted) looks at first a little complicated, is the specialinstrument of the logical calculus. By its means we may always resolve apropositioninto twodeterminants,orconverselywemaycompoundcertainpairsofdeterminantsintoasingleproposition.”

445.TheSimplificationofComplexPropositions.—Thetermsofacomplexproposition may often be simplified by means of the rules given in theprecedingchapter,andtheforceof theassertionwillremainunaffected.Forthefurthersimplificationofcomplexpropositionsthefollowingrulesmaybe


added: (1) In a universal negative or a particular affirmative proposition anydeterminant of the subject may be indifferently introduced or omitted as adeterminantofthepredicateandviceversâ.

482TosaythatNoABisACisthesameastosaythatNoABisC,orthatNoB isAC.For to say thatNoAB isAC is the same thing as to deny thatanything is ABAC ; but, as shewn in section 429, the repetition of thedeterminantA issuperfluous,andthestatementmaythereforebereducedtothedenial thatanythingisABC.AndthismayequallywellbeexpressedbysayingNoABisC,orNoBisAC.490

490 Seealso thesections in the followingchapter relating to theconversionofpropositions.

Again,SomeABisACmaybeshewntobeequivalenttoSomeABisC,ortoSomeBisAC ;foritsimplyaffirmsthatsomethingisABAC,andtheprooffollowsasabove.

(2)Inauniversalaffirmativeoraparticularpropositionanydeterminantofthesubjectmaybeindifferentlyintroducedoromittedasadeterminantofanyalternantofthepredicate.

AllAisABmayobviouslyberesolvedintothetwopropositionsAllAisA,AllA isB.491But the former of these is amerely identical proposition andgives no information. All A is AB is, therefore, equivalent to the simplepropositionAllAisB.Similarly,AllABisACorDEisequivalenttoAllABisCorDE.

491 The resolution of complex propositions into a combination of relativelysimpleoneswillbeconsideredfurtherinthefollowingsection.

Again,SomeAisnotABaffirmsthatSomeAisaorb ;492butbythelawofcontradictionNoAisa ;therefore,SomeAisnotB,andobviouslywecanalsopassbackfromthispropositiontotheonefromwhichwestarted.Similarly,SomeABisnoteitherACorDEisequivalenttoSomeABisnoteitherCorDE.

492 Theprocessofobversionwillbeconsideredindetailinchapter3.

(3) In a universal affirmative or a particular negative proposition any


alternant of the predicatemay be indifferently introduced or omitted as analternantofthesubject.

IfAllAisBorC,thenbythelawofidentityitfollowsthatWhateverisAor B is B or C ; it is also obvious that we can pass back from this to theoriginalproposition.

Again,ifSomeAorBisnoteitherBorC,thensincebythelawofidentityAllBisBitfollowsthatSomeAisnoteitherBorC ;anditisalsoobviousthatwecanpassbackfromthistotheoriginalproposition.

(4) In a universal affirmative or a particular mgative proposition thecontradictory of any determinant of the subject may be indifferentlyintroducedoromittedasanalternantofthepredicate,andviceversâ.

483 By this rule the three following propositions are affirmed to beequivalenttooneanother:AllABisaorC ;AllBisaorC ;AllABisC ;andalsothethreefollowing:SomeABisnoteitheraorC ;SomeBisnoteitheraorC ;SomeABisnotC.

Therulefollowsdirectlyfromrule(1)byaidof theprocessofobversion(seechapter3).

(5) In a universal negative or a particular affirmative proposition thecontradictory of any determinant of the subject may be indifferentlyintroducedoromittedasanalternantofthepredicate.

Bythisrulethetwofollowingpropositionsareaffirmedtobeequivalenttooneanother:NoABisaorC ;NoABisC ;andalsothetwofollowing:SomeABisaorC ;SomeABisC.

Therulefollowsdirectlyfromrule(2)byobversion.

(6) In a universal negative or a particular affirmative proposition thecontradictory of any determinant of the predicate may be indifferentlyintroducedoromittedasanalternantofthesubject.

Thisrulefollowsfromrule(3)byobversion.

446.The Resolution of Universal Complex Propositions into EquivalentCompoundPropositions.—Wemayenquirehowfarcomplexpropositionsare


immediately resolvable into a conjunctive or alternative combination ofrelatively simple propositions. Universal propositionswill be considered inthissection,andparticularsinthenext.

UniversalAffirmatives.Universalaffirmativecomplexpropositionsmaybeimmediately resolved into a conjunctionof relatively simpleones, so far asthereisalternativecombinationinthesubjectorconjunctivecombinationinthepredicate.Thus, (1)WhateverisPorQisR=AllPisRandallQisR ; (2)AllPisQR=AllPisQandallPisR.

Universal Negatives. Universal negative complex propositions may beimmediately resolved into a conjunctionof relatively simpleones, so far asthereisalternativecombinationeitherinthesubjectorinthepredicate.Thus, (3)NothingthatisPorQisR=NoPisRandnoQisR ; (4)NoPiseitherQorR=NoPisQandnoPisR.

So far as there is conjunctive combination in the subject or alternativecombination in the predicate of universal affirmative propositions, orconjunctivecombinationeitherinthesubjectorinthepredicateofuniversalnegativepropositions,theycannotbe484immediately493resolvedintoeitheraconjunctive or an alternative combination of simpler propositions. It may,however, be added that propositions falling into this latter category areimmediatelyimpliedbycertaincompoundalternatives.Thus, (i) AllPQisRisimpliedbyAllPisRorallQisR ; (ii) AllPisQorRisimpliedbyAllPisQorallPisR ; (iii)NoPQisRisimpliedbyNoPisRornoQisR ; (iv) NoPisQRisimpliedbyNoPisQornoPisR.

493 It will be shewn subsequently that even in these cases universal complexpropositionsmayberesolvedintoaconjunctionofrelativelysimpleronesbytheaidofcertainimmediateinferences.

447.The Resolution of Particular Complex Propositions into EquivalentCompound Propositions.—Particular complex propositions cannot beresolvedintocompoundconjunctives,buttheymayundercertainconditionsbeimmediatelyresolvedintoequivalentcompoundalternativepropositionsinwhich thealternantsare relativelysimple.This is thecaseso faras there is


alternative combination in the subject or conjunctive combination in thepredicate of a particular negative, or alternative combination either in thesubjectorinthepredicateofaparticularaffirmative.Thus, (1)SomePorQisnotR=SomePisnotRorsomeQisnotR ; (2)SomePisnotQR=SomePisnotQorsomePisnotR ; (3)SomePorQisR=SomePisRorsomeQisR ; (4)SomePisQorR=SomePisQorsomePisR.

Particular complex propositions cannot be immediately resolved intocompound propositions (either conjunctive or alternative) so far as there isconjunctive combination in the subject or alternative combination in thepredicate if the proposition is negative, or so far as there is conjunctivecombination either in the subject or in the predicate if the proposition isaffirmative. In these cases, however, the complex proposition implies acompound conjunctive proposition, though we cannot pass back from thelattertotheformer.Thus, (i)SomePQisnotRimpliesSomePisnotRandSomeQisnotR ; (ii)SomePisnoteitherQorRimpliesSomePisnotQandsomePisnotR ; (iii)SomePQisRimpliesSomePisRandsomeQisR ; (iv)SomePisQRimpliesSomePisQandsomePisR.

It must be particularly noticed that, although in these cases the 485compoundpropositioncanbeinferredfromthecomplexproposition,stillthetwoarenot equivalent.Forexample, fromSomeP isQandsomeP isR itdoesnotfollowthatSomePisQR,forwecannotbesurethatthesameP’sarereferredtointhetwocases.

Alltheresultsofthissectionfollowfromthoseoftheprecedingsectionbytheapplicationoftheruleofcontradictiontothepropositionsthemselvesandtheruleofcontrapositiontotherelationsofimplicationbetweenthem.

448.TheOmissionofTermsfromaComplexProposition.—Fromthe twopreceding sections we may obtain immediately the following rules forinferringfromagivenpropositionanotherpropositioninwhichcertaintermscontainedintheoriginalpropositionareomitted: (1)Anydeterminantmaybeomittedfromanundistributedterm ;494


(2)Anyalternantmaybeomittedfromadistributedterm.495494 Thesubjectofaparticularorthepredicateofanaffirmativeproposition.495 Thesubjectofauniversalorthepredicateofanegativeproposition.

Forexample,— WhateverisAorBisCD,therefore,AllAisC ; SomeABisCD,therefore,SomeAisC ; NothingthatisAorBisCorD,therefore,NoAisC ; SomeABisnoteitherCorD,therefore,SomeAisnotC.

Theabove rulesmayalsobe justified independently, aswillbe shewn inthe following section. The results which they yield must be distinguishedfromthoseobtainedinsection445.Inthecasesdiscussedinthatsection,thetermsomittedwere superfluous in the sense that theiromission leftuswithpropositions equivalent to our original propositions; but in the aboveinferenceswecannotpassbackfromconclusiontopremiss.FromSomeAisC,forexample,wecannotinferthatSomeABisC.

449. The Introduction of Terms into a Complex Proposition.—Corresponding to the rules laiddown in theprecedingsectionwehavealsothefollowing: (1)Anydeterminantmaybeintroducedintoadistributedterm ; (2)Anyalternantmaybeintroducedintoanundistributedterm.

These rules, and also the rules given in the preceding section, may beestablished by the aid of the following axioms: What is true of all(distributively)istrueofeverypart ;Whatistrueofpartofapartistrueofapartofthelargerwhole.

486 When we add a determinant to a term, or remove an alternant, weusuallydiminish,andatanyratedonot increase, theextensionof the term;when,on theotherhand,weaddanalternant,or removeadeterminant,weusually increase, and at any rate do not diminish, its extension. Hence itfollowsthatifatermisdistributedwemayaddadeterminantorremoveanalternant,whilstifatermisundistributedwemayaddanalternantorremoveadeterminant.Thus, AllAisCD,therefore,AllABisC ;


NoAisC,therefore,NoABisCD ; SomeABisC,therefore,SomeAisCorD ; SomeABisnoteitherCorD,therefore,SomeAisnotC.

Fromtheaboverulestakeninconnexionwiththerulesgiveninsection445wemayobtainthefollowingcorollaries: (3)Inuniversalaffirmatives,anydeterminantmaybe introduced into thepredicate, if it isalsointroducedinto thesubject;andanyalternantmaybeintroducedintothesubjectifitisalsointroducedintothepredicate. GivenAllA isC, thenAll AB is C by rule (1) above; and from thisweobtainAllABisBCbyrule(2)ofsection445. Again,givenAllAisC,thenAllAisBorC ;andtherefore,byrule(3)ofsection445,WhateverisAorBisBorC. (4)Inuniversalnegativesanyalternantmaybeintroducedintosubjectorpredicate, if its contradictory is introduced into the other term as adeterminant. GivenNoA isC, thenNoAB isC ; and, therefore, by rule (5)of section445,NoABisborC. Again, givenNoA isC, thenNoA isBC ; and, therefore, by rule (6) ofsection445,NoAorbisBC.

In none of the inferences considered in this section is it possible to passbackfromtheconclusiontotheoriginalproposition.

450. Interpretation of Anomalous Forms.—It will be found thatpropositionswhichapparentlyinvolveacontradictionintermsandarethusindirectcontraventionofthefundamentallawsofthought—forexample,NoABis B, All Ab is B—sometimes result from the manipulation of complexpropositions.Ininterpretingsuchpropositionsasthese,adistinctionmustbedrawn between universals and particulars, at any rate if particulars areinterpretedasimplying,whileuniversalsarenotinterpretedasimplying,theexistenceoftheirsubjects.

487ItcanbeshewnthatauniversalpropositionoftheformNoABisBorAllAbisBmustbeinterpretedasimplyingthenon-existenceintheuniverseofdiscourseofthesubjectoftheproposition.Forauniversalnegativedeniestheexistenceofanythingthatcomesunderbothitssubjectanditspredicate;


thus,NoABisBdeniestheexistenceofABB,thatis,itdeniestheexistenceofAB.Again,auniversalaffirmativedeniestheexistenceofanythingthatcomesunder its subjectwithout also coming under its predicate; thus,AllAb isBdeniestheexistenceofanythingthatisAbandatthesametimenot-B,thatis,b ;butAbisAbandalsob,andhencetheexistenceofAbisdenied.

Sincetheexistenceofitssubjectisheldtobepartoftheimplicationofaparticular proposition, the above interpretation is obviously inapplicable inthe case of particulars.Hence if a proposition of the formSomeAb isB isobtained, we are thrown back on the alternative that there is someinconsistency in the premisses; either some one individual premiss is self-contradictory,orthepremissesareinconsistentwithoneanother.

EXERCISES.

451.ShewthatifNoAisbcorCd,thenNoAisbd.[K.]

452.Give thecontradictoryofeachof the followingpropositions:—(1)Floweringplantsareeitherendogensorexogens,butnotboth;(2)Floweringplantsarevascular,andeitherendogensorexogens,butnotboth.[M.]

453.Simplifythefollowingpropositions:— (1)AllABisBCorbeorCDorcEorDE ; (2)NothingthatiseitherPQorPRisPqrorpQsorpqorprsorqrsorpSorqR.[K.]


CHAPTERIII.

IMMEDIATEINFERENCESFROMCOMPLEXPROPOSITIONS.

454.TheObversionofComplexPropositions—Thedoctrineofobversionisimmediatelyapplicable tocomplexpropositions;andnomodificationof thedefinition of obversion already given is necessary. From any givenpropositionwemayinferanewonebychanging itsqualityand takingasanewpredicatethecontradictoryoftheoriginalpredicate.Thepropositionthusobtainediscalledtheobverseoftheoriginalproposition.

Theonlydifficultyconnectedwith theobversionofcomplexpropositionsconsistsinfindingthecontradictoryofacomplexterm;butasimpleruleforperforming this process has been given in section 426:—Replace all thesimple terms invoked by their contradictories, and throughout substitutealternativecombinationforconjunctiveandviceversâ.

ApplyingthisruletoABorab,wehave(aorb)and(AorB),thatis,AaorAboraBorBb ;butsincethealternantsAaandBbinvolveself-contradiction,theymaybyrule(5)ofsection433beomitted.Theobverse,therefore,ofAllXisABorabisNoXisAboraB.

As additional examples we may find the obverse of the followingpropositions:(1)AllAisBCorDE ;(2)NoAisBcEorBCF ;(3)SomeA isnoteitherBorbcDEforbcdEF.

(1)AllAisBCorDEyieldsNoAis(borc)andatthesametime(dore),or,bythereductionofthepredicatetoaseriesofalternants,NoAisbdorbeorcdorce.

(2)NoAisBcEorBCF.Herethecontradictoryofthe489predicateis(borCore)and(borcorf),whichyieldsborCcorCforceoref.Ccmaybeomittedbyrule(5)ofsection433;alsoefbyrule(7),sinceefiseitherCeforcef.HencetherequiredobverseisAllAisborCforce.


(3)SomeAisnoteitherBorbcDEforbcdEF.TheobverseisSomeAisband(BorCordoreorF)and(BorCorDoreorf);andbytheapplicationoftherulessummarisedinsection433thiswillbefoundtobeequivalenttoSomeAisbCorbDForbdforbe.

455.TheConversionofComplexPropositions.—Generalising,wemaysaythatwehaveaprocessofconversionwheneverfromagivenpropositionweinfer a new one in which any term that appeared in the predicate of theoriginalpropositionnowappearsinthesubject,orviceversâ.

Thus the inference fromNo A is BC toNo B is AC is of the nature ofconversion.Theprocessmaybesimplyanalysedasfollows:—

NoAisbothBandC,therefore,NothingisatthesametimeA,B,andC,

therefore,NoBisbothAandC.

Thereasoningmayalsoberesolvedintoaseriesofordinaryconversions:—

NoAisBC,therefore(byconversion),NoBCisA,thatis,withinthesphereofC,noBisA,

therefore(byconversion),withinthesphereofC,noAisB,thatis,NoACisB,

therefore(byconversion),NoBisAC.

Or,itmaybetreatedthus,

NoAisBC,therefore,bysection445,rule(1),NoACisBC,

therefore,alsobysection445,rule(1),NoACisB,therefore(byconversion),NoBisAC.

SimilarlyitmaybeshewnthatfromSomeAisBCwemayinferSomeBisAC.

Henceweobtainthefollowingrule:Inauniversalnegativeoraparticularaffirmativepropositionanydeterminantof thesubjectmaybetransferredtothepredicateorviceversâwithoutaffectingtheforceoftheassertion.


Wehavejustshewnhowfrom

NoAisBC,

wemayobtainbyconversion

NoBisAC.

490Similarly,wemayinfer

NoCisAB,NoABisC,NoACisB,NoBCisA.

Thepropositionmayalsobewrittenintheform

ThereisnoABC,or,NothingisatthesametimeA,B,andC.

The last of these is a specially useful form to which to bring universalnegativesforthepurposeoflogicalmanipulation.

InthesamewayfromSomeAisBCorBDwemayinfer

SomeABisCorD,SomeACorADisB,SomeBisACorAD,SomeCorDisAB,SomeBCorBDisA,

SomethingisABCorABD.

Thereisnoinferencebyconversionfromauniversalaffirmativeorfromaparticularnegative.

456. The Contraposition of Complex Propositions.—According to ouroriginal definition of contraposition,we contraposit a propositionwhenweinferfromitanewpropositionhavingthecontradictoryoftheoldpredicateforitssubject.Adoptingthisdefinition,thecontrapositiveofAllAisBorCisAllbcisa.

The process can be applied to universal affirmatives and to particularnegatives.Byobversion,conversion,andthenagainobversion,itisclearthat


ineachofthesecaseswemayobtainalegitimatecontrapositivebytakingasanewsubjectthecontradictoryoftheoldpredicate,andasanewpredicatethe contradictory of the old subject, the proposition retaining its originalquality.Forexample:AllAisBC,therefore,Whateverisborcisa ;SomeAisnoteitherBorC,therefore,Somebcisnota.

Theabovemaybecalledthefullcontrapositiveofacomplexproposition.It should be observed that any proposition and its full contrapositive areequivalenttoeachother;wecanpassbackfromthefullcontrapositivetotheoriginalproposition.

Indealingwithcomplexpropositions,however,itisconvenienttogivetothe term contraposition an extendedmeaning.Wemay say that we have aprocessofcontrapositionwhenfromagivenpropositionweinferanewoneinwhichthecontradictoryofanytermthatappearedinthepredicateof theoriginalpropositionnowappears491 in the subject, or the contradictory ofanytermthatappearedinthesubjectoftheoriginalpropositionnowappearsinthepredicate.

Threeoperationsmaybedistinguishedallofwhichareincludedundertheabove definition, and all of which leave us with a full equivalent of theoriginalproposition,sothatthereisnolossoflogicalpower.

(1) The operation of obtaining the full contrapositive of a givenproposition,asabovedescribedanddefined.496

496 In somecaseswemaydesire todroppartof the informationgivenby thecompletecontrapositive.Thus,fromAllAisBCorEmayinferWhateverisbeorceisa ;butinagivenapplicationitmaybesufficientforustoknowthatAllbeisa.

(2) An operation which may be described as the generalisation of thesubject of a proposition by the addition of one or more alternants in thepredicate.Thus,fromAllABisCwemayinferAllAisborC ;fromSomeABisnoteitherCorDwemayinferSomeAisnoteitherborCorD.

For inferencesof this type the followinggeneral rulemaybegiven:Anydeterminantmaybedroppedfromthesubjectofauniversalaffirmativeoraparticularnegativeproposition,ifitscontradictoryisatthesametimeaddedasanalternantinthepredicate.


Thisrulemaybeestablishedasfollows:GivenAllABisC(orSomeABisnotC)—andthesemaybetaken,sofarastheruleinquestionisconcerned,astypesofuniversalaffirmativesandparticularnegativesrespectively—wehaveby obversionNo AB is c (or Some AB is c), and thence, by the rule forconversiongiven in section455,NoA isBc (orSome A is Bc); then againobvertingwehaveAllAiseitherborC(orSomeAisnoteitherborC),therequiredresult.

Itwill be observed that, as stated at the outset, these operations leaveuswith a proposition that is equivalent to our original proposition. There is,therefore,nolossoflogicalpower.

By the application of the above rule with regard to all the explicitdeterminants of the subject any universal affirmative proposition may bebroughttotheformEverythingisX1orX2…orXn ;anditwillbefoundthatby means of this transformation, complex inferences are in many casesmateriallysimplified.

(3)An operationwhichmay be described as the particularisation of thesubject of a proposition by the omission of one or more alternants in thepredicate.Thus,fromAllAisBorCwemayinferAllAbisC ;fromSomeAisnoteitherBorCwemayinferSomeAbisnotC.

492Forinferencesofthistypethefollowinggeneralrulemaybegiven:Anyalternantmaybedroppedfromthepredicateofauniversalaffirmativeoraparticular negative proposition, if its contradictory is at the same timeintroducedasadeterminantofthesubject.497

497 Theapplicationofthisruleagainleavesuswithapropositionequivalenttoouroriginalproposition.Thefollowingrule,whichmayberegardedasacorollaryfrom the above rule, or which may be arrived at independently, does notnecessarilyleaveuswithanequivalent:Ifanewdeterminantisintroducedintothesubjectofauniversalaffirmativeproposition (seesection449)everyalternant inthepredicatewhichcontainsthecontradictoryofthedeterminantmaybeomitted.Thus,fromWhateverisAorBisCorDXorEx,wemayinferWhateverisAXorBXisCorD.

Theapplicationofthisrulemaysometimesresultinthedisappearanceofallthealternantsfromthepredicate;andthemeaningofsucharesultisthatwenowhaveanon-existentsubject.

Thus,givenAllPisABCDorAbcdoraBCd,ifweparticularisethesubjectby


making it PbC, we find that all the alternants in the predicate disappear. TheinterpretationisthattheclassPbCisnon-existent,thatis,NoPisbC ;aconclusionwhichmightofcoursehavebeenobtaineddirectlyfromthegivenproposition.

This rule is the converse of that given under the preceding head; and itfollows from the fact that the application of that rule leaves us with anequivalentproposition.

Thefollowingmaybetakenastypicalexamplesofthedifferentoperationsincludedaboveunderthenamecontraposition:—

AllABisCDorde ;therefore, (1)AnythingthatiseithercDordEisaorb ;

(2)AllAisborCDorde ;(3)WhateverisABDorABEisCD.

CombinationsofthesecondandthirdoperationsgiveAnythingthatisAcorAdisborde ;AnythingthatisBDorBEisaorCD ;

&c.

Inalltheabovecasesoneormoretermsdisappearfromthesubjectorthepredicateoftheoriginalproposition,andarereplacedbytheircontradictoriesin the predicate or the subject accordingly. Only in the full contrapositive,however,iseverytermthustransposed.

The importance of contraposition as we are now dealing with it inconnexionwithcomplexpropositionsisthatbyitsmeans,givenauniversalaffirmative proposition of any complexity, we may obtain separateinformationwithregard toany termthatappears in the493subject, orwithregardtothecontradictoryofanytermthatappearsinthepredicate,orwithregardtoanycombinationofsuchterms.

Thus, given All AB is C or De, by the process described as thegeneralisationofthesubjectwehaveAllAisborCorDe,AllBisaorCorDe,EverythingisaorborCorDe ;theparticularisationofthesubjectyieldsAllABcisDe,WhateverisABdorABEisC,&c.;andbythecombinationoftheseprocesseswehaveAllAcisborDe,&c.

Again,thefullcontrapositiveoftheoriginalpropositionisWhateveriscd


orcEisaorb ;fromwhichwehaveAllcisaorborDe,WhateverisdorEisaorborC,&c.

457. Summary of the results obtainable by Obversion, Conversion, andContraposition.—Thefollowingisasummaryoftheresultsobtainablebytheaidoftheprocessesdiscussedinthethreeprecedingsections: (1)Byobversionanypropositionmaybechangedfromtheaffirmativetothenegativeform,orviceversâ. Forexample,AllABisCDorEF,therefore,NoABisceorcfordeordf ;SomePisnotQR,therefore,SomePiseitherqorr. (2) By the conversion of a universal negative proposition separateinformationmaybeobtainedwithregardtoanytermthatappearseitherinthesubjectorinthepredicate,orwithregardtoanycombinationoftheseterms. Forexample,fromNoABisCDorEFwemayinferNoAisBCDorBEF,NoCisABDorABEF,NoBDisACorAEF,etc. Alsobyconversionanyuniversalnegativepropositionmaybereducedtothefollowing:NothingiseitherX1orX2…orXn. Forexample,theabovepropositionisequivalenttothefollowing:NothingiseitherABCDorABEF. (3) By the conversion of a particular affirmative proposition separateinformationmaybeobtainedwithregardtoanydeterminantofthesubjectorofthepredicate,orwithregardtoanycombinationofsuchdeterminants. Forexample,fromSomeABorACisDEorDFwemayinferSomeA isBDEorBDForCDEorCDF,SomeDisABEorABForACEorACF,SomeADisBEorBForCEorCF,etc. AlsobyconversionanyparticularaffirmativepropositionmaybereducedtotheformSomethingiseitherX1orX2…orXn. 494 For example, the above proposition is equivalent to the following:SomethingiseitherABDEorABDForACDEorACDF. (4) By the contraposition of a universal affirmative proposition separateinformation may be obtained with regard to any term that appears in thesubject, orwith regard to the contradictory of any term that appears in thepredicate,orwithregardtoanycombinationoftheseterms. Forexample,fromAllABisCDorEFwemayinferAllAisborCDorEF,AllcisaorborEF,AllBeisaorCD,Allceisaorb,AllAdfisb,&c.


Also by contraposition any universal affirmative proposition may bereducedtotheformEverythingiseitherX1orX2…orXn. For example, the above proposition is equivalent to the following:EverythingisaorborCDorEF. (5) By the contraposition of a particular negative proposition separateinformationmaybeobtainedwithregardtoanydeterminantofthesubjectorwith regard to the contradictory of any alternant of the predicate or withregardtoanycombinationofthese. For example, fromSome AB or AC is not eitherD or EF wemay inferSomeAisnoteitherbcorDorEF,SomedisnoteitheraorbcorEF,SomeAeorAfisnoteitherbcorD,&c. AlsobycontrapositionanyparticularnegativepropositionmaybereducedtotheformSomethingisnoteitherX1orX2…orXn. For example, the above proposition is equivalent to the following:SomethingisnoteitheraorbcorDorEF.

EXERCISES.

458.Nocitizenisatonceavoter,ahouseholder,andalodger;noristhereanycitizenwhoisnoneofthethree. Everycitizeniseitheravoterbutnotahouseholder,orahouseholderandnotalodger,oralodgerwithoutavote. Arethesestatementspreciselyequivalent?[V.]


Inmaybeshewnthateachofthesestatementsisthelogicalobverseoftheother.Theyare,therefore,preciselyequivalent.

Let V=voter, v=notvoter;H=householder, h=nothouseholder;L=lodger, l=notlodger.

495The first of thegiven statements isNocitizen isVHLor vhl ; therefore (by obversion),Everycitizen is either vorhor l and isalsoeitherVorHorL ; therefore (combining these possibilities),EverycitizeniseitherHvorLvorVhorLhorVlorHl. But(bythelawofexcludedmiddle),HviseitherHLvorHlv ;therefore,HvisLvorHl.Similarly,LhisVhorLv ;andVlisHlorVh. Therefore,EverycitizenisVhorHlorLv,whichisthesecondofthegivenstatements. Again,starting fromthissecondstatement, it follows(byobversion) thatNocitizen isat thesametime v orH, h or L, l or V ; therefore,No citizen is vh or vL or HL, and at the same time l or V ;therefore,NocitizenisvhlorVHL,whichbringsusbacktothefirstofthegivenstatements.

459.Given“AllDthatiseitherBorCisA,”shewthat“Everythingthatisnot-Aiseithernot-Bandnot-Corelseitisnot-D.”[DeMorgan.]Thisexampleandthosegiveninsection466areadaptedfromDeMorgan,Syllabus,p.42.Theyare

also given by Jevons,Studies, p. 241, in connexion with his Equational Logic. They are all simpleexercisesincontraposition. WehaveWhatiseitherBDorCDisA ;therefore,Allais(bord)and(cord);therefore,Allaisbcord.

460.Inferallthatyoupossiblycanbywayofcontrapositionorotherwise,fromtheassertion,AllAthatisneitherBnorCisX.[R.]Thegivenpropositionmaybethrownintotheform

EverythingiseitheraorBorCorX ;

and it isseen tobesymmetricalwithregard to the termsa,B,C,X, and thereforewith regard to thetermsA,b,c,x.WearesurethenthatanythingthatistrueofAistruemutatismutandisofb,c,andx,that anything that is true ofAb is truemutatismutandis of any pair of the terms, and similarly forcombinationsthreeandthreetogether. Wehaveatoncethefoursymmetricalpropositions:

AllAisBorCorX ; (1)AllbisaorCorX ; (2)AllcisaorBorX ; (3)AllXisaorBorC. (4)

496Thenfrom(1)byparticularisationofthesubject:

AllAbisCorX ; (i)

withthefivecorrespondingpropositions;

AllAcisBorX ; (ii)


AllAxisBorC ; (iii)AllbcisaorX ; (iv)AllbxisaorC ; (v)AllcxisaorB. (vi)

Byarepetitionofthesameprocess,wehave

AllAbcisX(whichistheoriginalpropositionoveragain); (α)andcorrespondingtothis: AllAbxisC ; (β)

AllAcxisB ; (γ)Allbcxisa. (δ)

Itwillbeobservedthatthefollowingarepairsoffullcontrapositives;—(1)(δ),(2)(γ),(3)(β),(4)(α),(i)(vi),(ii)(v),(iii)(iv).

Afurtherseriesofpropositionsmaybeobtainedbyobvertingalltheabove;andastherehasbeennoloss of logical power in any of the processes employed we have in all thirty propositions that areequivalenttooneanother.

461.IfABiseitherCdorcDe,andalsoeithereForH,andifthesameistrueofBH,whatdoweknowofthatwhichisE?[K.]WhateverisABorBHis(CdorcDe)and(eForH);

therefore,WhateverisABorBHisCdeForcDeForCdHorcDeH ;therefore,WhateverisABEorBHEisCdH ;therefore,AllEisahorborCdH.

462.GivenAisBCorBDEorBDF, inferdescriptionsof the termsAce,Acf,ABcD.[Jevons,Studies,pp.237,238.]Inaccordancewithrulesalreadylaiddown,wehaveimmediately—

AceisBDF ;AcfisBDE ;

ABcDisEorF.

463.Findtheobverseofeachofthefollowingpropositions:— (1)NothingisA,B,orC ; (2)AllAisBcorbD ; (3)NoAbisCDEforCdorcDforcdE ; (4)NoAisBCDorBcd ; (5)SomeAisnoteitherbcdorCdorcD.[K.]

497464.Shewthatthetwofollowingpropositionsareequivalenttoeachother:—NoAisBorCDorCEorEF ;AllAisbCdeorbcEforbce.[K.]

465.Contraposittheproposition,AllAthatisneitherBnorCisbothXandY.[L.]

466.Findthefullcontrapositiveofeachofthefollowingpropositions: (1)WhateverisBorCDorCEisA ; (2)WhateveriseitherBorCandatthesametimeeitherDorEisA ;


(3)WhateverisAorBCandatthesametimeeitherDorEFisX ; (4)AllAiseitherBCorBD.[DeMorgan.]

467.Findthefullcontrapositiveofeachofthefollowingpropositions:— AllAisBCDeorbcDe ; SomeABisnoteitherCDorcDEorde ; WhateverisABorbCisaCdorAcd ; WhereAispresentalongwitheitherBorC,DispresentandCabsentorDandEarebothabsent ; SomeABCorabcisnoteitherDEFordef.[K.]

468.WhatinformationcanyouobtainaboutAf,Be,c,D,fromthepropositionAllABisCDorEF?[M.]

469.Establishthefollowing:WhereBisabsent,eitherAandCarebothpresentorAandDarebothabsent;therefore,whereCisabsent,eitherBispresentorDisabsent.[K.]

470.Establishthefollowing:WhereAispresent,eitherBandCarebothpresentorC ispresentDbeingabsentorCispresentFbeingabsentorHispresent;therefore,whereC isabsent,AcannotbepresentHbeingabsent.[K.]

471.GiventhatWhateverisPQorAPisbCDorabdEoraBCdEorAbcd,shewthat(1)AllabPisCDordEorq ;(2)AllDPisbCoraq ;(3)WhateverisBorCdorcDisaorp ;(4)AllBisCorporaq ;(5)AllABisp ;(6)Ifaeiscorditisporq ;(7)IfBPiscorDoreitisaq.[K.]

472.BringthefollowingpropositionstotheformEverythingiseitherX1orX2…orXn:— WhateverisAcoraboraCisbdfordeF ; NothingthatisAandatthesametimeeitherBorCisDordE.[K.]

473. Shew that the results in section 447 follow from those in section 446 by the rules ofcontradictionandcontraposition.[K.]


CHAPTERIV.

THECOMBINATIONOFCOMPLEXPROPOSITIONS.

474. The Problem of combining Complex Propositions.—Two or morecomplex propositions given in simple combination, either conjunctive oralternative,constituteacompoundproposition.Hencetheproblemofdealingwith a combination of complex propositions so as to obtain from them asingleequivalentcomplexproposition,whichistheproblemtobeconsideredin the present chapter, is identical with that of passing from a compoundproposition to an equivalent complex proposition; and it is, therefore, theconverseof theproblemwhichwaspartiallydiscussedinsections446,447.Thelatterproblem,namely,thatofpassingfromacomplextoanequivalentcompoundproposition,willbefurtherdiscussedinchapter6.

475. The Conjunctive Combination of Universal Affirmatives.—We mayheredistinguishtwocasesaccordingasthepropositionstobecombinedhaveorhavenotthesamesubject.

(1)Universalaffirmativeshavingthesamesubject.

AllXisP1orP2……orPm,AllXisQ1orQ2……orQn,

may for our present purpose be taken as types of universal affirmativepropositions having the same subject. By conjunctively combining theirpredicates,thus,

AllXis(P1orP2…orPm)andalso(Q1orQ2…orQn),thatis,AllXisP1Q1orP1Q2…orP1Qn

orP2Q1orP2Q2…orP2Qn

or…… ……

orPmQ1orPmQ2…orPmQn,499


we may obtain a new proposition which is equivalent to the conjunctivecombinationof the twooriginalpropositions; it sumsupall the informationwhichtheyjointlycontain,andwecanpassbackfromittothem.

In almost all cases of the conjunctive combination of terms there arenumerous opportunities of simplification; and, after a little practice, thestudent will find it unnecessary to write out all the alternants of the newpredicateinfull.Thefollowingareexamples:—

(i) AllXisABorbce,AllXisaBCorDE ;

therefore, AllXisABDE.

It will be found that all the other combinations in the predicate containcontradictories.

(ii) AllXisAorBcorD,AllXisaBorBcorCd ;

therefore, AllXisACdoraBDorBc.(iii) EverythingisAorbdorcE,

EverythingisACoraBeord ;therefore, EverythingisACorAdorbdorcdE.

(2)Universalaffirmativeshavingdifferentsubjects.

Given the conjunctive combination of two universal affirmativepropositions with different subjects, a new complex proposition may beobtainedbyconjunctivelycombiningboththeirsubjectsandtheirpredicates.Thus,ifAllXisP1orP2andAllYisQ1orQ2,itfollowsthatAllXYisP1Q1

orP1Q2orP2Q1orP2Q2.Butinthiscasethenewpropositionobtainedisnotequivalenttotheconjunctivecombinationoftheoriginalpropositions;andwecannotpassbackfromittothem.

Asinglecomplexpropositionwhichsumsupalltheinformationcontainedintheoriginalpropositionsmay,however,beobtainedbyfirstreducingeachofthemtotheformEverythingisX1orX2…orXn,and thenconjunctivelycombiningtheirpredicates.

476.The Conjunctive Combination of Universal Negatives.—Here again


wemaydistinguish twocasesaccordingas thepropositions tobecombinedhaveorhavenotthesamesubject.

(1)Universalnegativeshavingthesamesubject

NoXisP1orP2……orPm,NoXisQ1orQ2……orQn,

may for our present purpose be taken as types of universal negativepropositions having the same subject. Given these two propositions 500 inconjunctive combination, a new complex proposition may be obtained byalternativelycombiningtheirpredicates.Thus,

NoXisP1orP2……orPmorQ1orQ2……orQn.

This new proposition is equivalent to the two original propositions takentogether,sothatwecanpassbackfromittothem.Theprocessofcombiningthe predicates is again likely to give opportunities of simplification. Thefollowingareexamples:

(i) NoXiseitheraBoraCoraEorbCorbE,NoXiseitherAdorAeorbdorbeorcdorce ;therefore,NoXiseitheraorbordore.498

(ii) NothingisaBCoraBeoraCDoraDe,NothingisAcDorabDoraDEorbcDorcDE ;therefore,NothingisaBCoraBeoraDorcD.


(2)Universalnegativeshavingdifferentsubjects.

Giventheconjunctivecombinationoftwouniversalnegativepropositionswith different subjects a new complex proposition may be obtained byconjunctively combining their subjects and alternatively combining theirpredicates.Thus,ifNoXisP1orP2andNoYisQ1,orQ2,itfollowsthatNoXY is P1 or P2 or Q1 or Q2. In this case the inferred proposition is notequivalenttothepremisses;andwecannotpassbackfromittothem.

Asinglecomplexpropositionwhichsumsupalltheinformationcontainedintheoriginalpropositionsmay,however,beobtainedbyfirstreducingeachof them to the formNothing is X1,or X2 … or Xn, and then alternatively


combiningtheirpredicates.

477.The Conjunctive Combination of Universals with Particulars of thesameQuality.—Wemayhereconsider,first,affirmatives,andthen,negatives.

(1) Affirmatives. From the conjunctive combination of a universalaffirmative and a particular affirmative having the same subject, a newparticular affirmative proposition may be obtained by conjunctivelycombining their predicates. IfAllX isP1orP2 andSome X isQ1orQ2, itfollows thatSome X is P1Q1or P1Q2or P2Q1or P2Q2. Here the particularpremissaffirmstheexistenceofXandofeitherXQ1orXQ2;andtheuniversalpremissimpliesthatifXexiststheneitherXP1orXP2exists.

Wecanpassbackfromtheconclusiontotheparticularpremiss,butnottotheuniversalpremiss.Theconclusionis,therefore,notequivalenttothetwopremissestakentogether.

501 A new complex proposition cannot be directly obtained from theconjunctive combination of a universal affirmative and a particularaffirmative having different subjects. The propositions may, however, bereducedrespectivelytotheformsEverythingisP1orP2…orPm,Somethingis Q1 or Q2 … or Qn, and their predicates may then be conjunctivelycombinedinaccordancewiththeaboverule.

(2)Negatives. From the conjunctive combination of a universal negativeandaparticularnegativehavingthesamesubject,anewparticularnegativeproposition may be obtained by the alternative combination of theirpredicates. IfNoX is eitherP1orP2 andSome X is not eitherQ1orQ2 itfollows thatSomeX isnoteitherP1orP2orQ1orQ2.Thevalidity of thisprocessisobvioussincetheparticularpremissaffirmstheexistenceofX.Byobversionitcanalsobeexhibitedasacorollaryfromtherulegivenaboveinregard to affirmatives.We can again pass back from the conclusion to theparticularpremiss,butnottotheuniversalpremiss.

With regard to the conjunctive combination of universal negatives andparticular negatives having different subjects, the remarksmade concerningaffirmativesapplymutatismutandis.


478. The Conjunctive Combination of Affirmatives with Negatives.—Byfirst obverting one of the propositions, the conjunctive combination of anaffirmativewithanegativemaybemadetoyieldanewcomplexpropositioninaccordancewiththerulesgivenintheprecedingsections.Forexample,

(1) AllXisAorB, NoXisaC,therefore,AllXisAorBc ;

(2) EverythingisPorQ, NothingisPqorpR,therefore,NothingispRorq ;

(3) AllXisABorbce, SomeXisnoteitheraBCorDE,therefore,SomeXisABdorABeorbce.

479. The Conjunctive Combination of Particulars with Particulars.—Particularscannottoanypurposebeconjunctivelycombinedwithparticularssoastoyieldanewcomplexproposition.ItistruethatfromSomeXisP1orP2andsomeXisQ1orQ2,wecanpasstoSomeXisP1orP2orQ1orQ2.Butthisisamereweakeningoftheinformationgivenbyeitherofthepremissessingly;andbytherulethatanalternantmayatanytimebeintroducedintoanundistributed term (section449), it could equallywell be inferred 502 fromeitherpremisstakenbyitself.AgainfromSomeXisnoteitherP1orP2andsomeX isnoteitherQ1orQ2wecanpass toSomeX isnot eitherP1Q1orP1Q2orP2Q1orP2Q2.Butsimilarremarksagainapply,sincewehavealreadyfound thata determinant may at any time be introduced into a distributedterm.

480. The Alternative Combination of Universal Propositions.—Given anumber of universal propositions as alternants in a compound alternativepropositionwecannotobtainasingleequivalentcomplexproposition.FromthecompoundpropositionEitherallAisP1orP2orallAisQ1orQ2wecanindeedinferAllAisP1orP2orQ1orQ2;butwecannotpassbackfromthistotheoriginalproposition.499



481.TheAlternativeCombinationofParticularPropositions.—It followsfrom the equivalences shewn in section 447 that a compound alternativeproposition in which all the alternants are particular can be reduced to theformof a single complexproposition. If all the alternantsof the compoundproposition have the same subject and are all affirmative, their predicatesmustbealternativelycombined in thecomplexproposition; if theyallhavethesamesubjectandareallnegative, theirpredicatesmustbeconjunctivelycombinedinthecomplexproposition.Ifthealternantshavedifferentsubjects,theymustallbereducedtotheformSomethingis…beforetheirpredicatesarecombined;iftheydifferinquality,recoursemustbehadtotheprocessofobversion.Itisunnecessarytodiscussthesedifferentcasesindetail,butthefollowingmaybetakenasexamples:

(i) SomeXisPorsomeXisQ=SomeXisPorQ ; (ii) SomeXisnotPorsomeXisnotQ=SomeXisnotPQ ;(iii) SomeXisPorsomeYisQ=SomethingisXPorYQ ;(iv) SomeXisPorsomeYisnotQ=SomethingisXPorYq.

482.TheAlternativeCombinationofParticularswithUniversals.—Fromacompound alternative proposition in which some of the alternants areparticularandsomeuniversal,wecaninferaparticularcomplexproposition;but in this case we cannot pass back from the complex proposition to thecompoundproposition.Thefollowingareexamples:

(1) AllAisPorsomeAisQ,therefore,SomethingisaorPorQ ;500(2) AllAisPorsomeBisnotQ,therefore,SomethingisaorBqorP.

500 We cannot inferSomeA is P orQ, since this implies the existence ofA,whereasthenon-existenceofAiscompatiblewiththepremiss.

503

EXERCISES.

483.Reduce thepropositionsAllP isQ,NoQ isR to such a form that the universe of discourseappears as the subject of each of them; and then combine the propositions into a single complexproposition.HowisyourresultrelatedtotheordinarysyllogisticconclusionNoPisR?[K.]

484.Combinethefollowingpropositionsintoasingleequivalentcomplexproposition:AllXiseither


Aorb ;NoXiseitherACoracDorCD ;AllaisBorx.[K.]

485.Everyvoterisbotharatepayerandanoccupier,ornotaratepayeratall;Ifanyvoterwhopaysratesisanoccupier,thenheisonthelist;Novoteronthelistisbotharatepayerandanoccupier. Examinetheresultsofcombiningthesethreestatements.[V.]

486.EveryAisBCexceptwhenitisD ;everythingwhichisnotAisD ;whatisbothCandDisB ;andeveryDisC.Whatcanbedeterminedfromthesepremissesas tothecontentsofouruniverseofdiscourse?[M.]


CHAPTERV.

INFERENCESFROMCOMBINATIONSOFCOMPLEXPROPOSITIONS.

487.Conditionsunderwhichauniversalpropositionaffordsinformationinregard toanygiven term.—Theproblem tobe solved inorder todeterminethese conditions may be formulated as follows: Given any universalproposition,andanytermX,todiscriminatebetweenthecasesinwhichthepropositiondoesandthoseinwhichitdoesnotaffordinformationwithregardtothisterm.

Inthefirstplace,itisclearthatifthepropositionistoaffordinformationinregardtoanytermwhateveritmustbenon-formal.Ifitisnegative,letitbyobversionbemadeaffirmative.Thenitmaybewrittenintheform

WhateverisA1A2…orB1B2…or&c.isP1P2…orQ1Q2…or&c.,

whereA1,B1,P1,Q1,&c.areallsimpleterms.501501 So that both subject and predicate consist of a series of alternants which

themselves containonly simpledeterminants; that is, there is no alternant of theform(AorB)(CorD).

As shewn in section 446, this may be resolved into the independentpropositions:—

AllA1A2…isP1P2…orQ1Q2…or&c. ;AllB1B2…isP1P2…orQ1Q2…or&c. ;

&c. &c. &c.;

innoneofwhichisthereanyalternationinthesubject.

These propositionsmay be dealtwith separately, and if any one of themaffordsinformationwithregardtoX,thentheoriginalpropositiondoesso.

Wehavethentoconsiderapropositionoftheform

AllA1A2…AnisP1P2…orQ1Q2…or&c. ;


andthispropositionmaybycontrapositionbereducedtotheform505

Everythingisa1ora2…oranorP1P2…orQ1Q2…or&c. ;

fromwhichmaybeinferred

AllXisa1ora2…oranorP1P2…orQ1Q2…or&c.

Any alternant in the predicate of this proposition which contains x mayclearlybeomitted.

Ifallthealternantscontainx,thentheinformationaffordedwithregardtoXisthatitisnon-existent.

If some alternants are left, then the proposition will afford informationconcerningX unless, when the predicate has been simplified to the fullestpossibleextent,502oneofthealternantsisitselfXuncombinedwithanyotherterm, in which case it is clear that we are left with a merely formalproposition.

502 All superfluous termsbeingomitted, but the predicate still consisting of aseriesofalternantswhichthemselvescontainonlysimpledeterminants.

Nowoneof thesealternantswillbeX in thefollowingcases,andonlyinthesecases:— First, If oneof the alternants in thepredicateof theoriginalproposition,whenreducedtotheaffirmativeform,isX. Secondly, If any set of alternants in the predicate of the originalproposition,whenreducedtotheaffirmativeform,constitutesadevelopmentofX,sinceanydevelopment(forexample,AXoraX,ABXorAbXoraBXorabX)isequivalenttoXsimply.503 Thirdly,Ifoneofthealternantsinthepredicateoftheoriginalproposition,whenreducedtotheaffirmativeform,containsXincombinationsolelywithsomedeterminantthatisalsoadeterminantofthesubjectorthecontradictoryof someother alternant of the predicate; since in either of these cases suchalternantisequivalenttoXsimply.504 Fourthly, Ifoneof thedeterminantsof thesubject isx ; since in thatcaseweshallaftercontrapositionhaveXasoneofthealternantsofthepredicate.

503 Seesection430.504 Bysection445,rule(2),AllABisAXorDisequivalenttoAllABisXorD ;


andbythelawofexclusion(section432)AoraXisequivalenttoAorX.

The abovemay be summed up in the following proposition:—Any non-formaluniversalpropositionwillaffordinformationwithregardtoanytermX, unless, after it has been brought to the affirmative form, (1) one of thealternantsof thepredicate isX, or (2) any set of alternants in the predicateconstitutesadevelopmentofX,or(3)anyalternantofthepredicatecontainsXincombination506solelywithsomedeterminantthatisalsoadeterminantofthesubjectorthecontradictoryofsomeotheralternantofthepredicate,or(4)xisadeterminantofthesubject.

If, after the proposition has been reduced to the affirmative form, allsuperfluoustermsareomittedinaccordancewiththerulesgiveninchapters1and2, then the criterion becomesmore simple:—Any non-formal universalpropositionwillaffordinformationwithregardtoanytermX,unless(afterithas been brought to the affirmative form and its predicate has been sosimplifiedthatitcontainsnosuperfluousterms)Xisitselfanalternantofthepredicateorxisadeterminantofthesubject.505

505 Itmaybeaddedthateveryuniversalproposition,unlessitbepurelyformal,willaffordinformationeitherwithregardtoXorwithregardtox.For ifbothXandxappearasalternantsof thepredicate,orasdeterminantsof thesubjectofauniversalaffirmativeproposition,thenthepropositionwillnecessarilybeformal.

IfinsteadofXwehaveacomplex termXYZ, thennodeterminantof thistermmustappearbyitselfasanalternantofthepredicate,andtheremustbeat leastonealternant in thesubjectwhichdoesnotcontainasadeterminantthecontradictoryofanydeterminantofthiscomplexterm;i.e.,noalternantinthe predicate must be X, Y, or Z, or any combination of these, and somealternantofthesubjectmustcontainneitherx,y,norz.

Theabovecriterionisofsimpleapplication.

488.Informationjointlyaffordedbyaseriesofuniversalpropositionswithregardtoanygiventerm.—Thegreatmajorityofdirectproblems506involvingcomplex propositions may be brought under the general form, Given anynumberofuniversalpropositionsinvolvinganynumberofterms,todeterminewhat isall the information that they jointlyaffordwith regard toanygiventermorcombinationofterms.IfthestudentturnstoBoole,Jevons,orVenn,


hewill find that this problem is treated by them as the central problem ofsymboliclogic.507

506 Inverseproblemswillbediscussedinthefollowingchapter.507 “Boole,”saysJevons,“firstputforth theproblemofLogicalSciencein its

completegenerality:—Givencertainlogicalpremissesorconditions,todeterminethedescriptionofanyclassofobjectsunderthoseconditions.Suchwasthegeneralproblem of which the ancient logic had solved but a few isolated cases—thenineteen moods of the syllogism, the sorites, the dilemma, the disjunctivesyllogism,andafewotherforms.Booleshewedincontestablythatitwaspossible,bytheaidofasystemofmathematicalsigns,todeducetheconclusionsofalltheseancientmodesof reasoning, and an indefinitenumberof other conclusions.Anyconclusion, in short, that itwaspossible todeduce fromany setofpremissesorconditions, however numerous and complicated, could be calculated by hismethod”(PhilosophicalTransactions,1870).ComparealsoPrinciplesofScience,6,§5.

507Ageneralmethodofsolutionisasfollows:— Let X be the term concerning which information is desired. Find whatinformationeachpropositiongivesseparatelywithregardtoX,thusobtaininganewsetofpropositionsoftheformAllXisP1orP2…orPn. This is always possible by the aid of the rules for obversion andcontrapositiongiveninchapter3.Bytheaidoftherulegivenintheprecedingsection those propositions which do not afford any information at all withregardtoXmayatoncebeleftoutofaccount. Next let the propositions thus obtained be combined in the mannerindicatedinsection475.Thiswillgivethedesiredsolution. Ifinformationisdesiredwithregardtoseveralterms,itwillbeconvenienttobringallthepropositionstotheform

EverythingisP1orP2…orPn ;

andtocombinethematonce,thussummingupinasinglepropositionalltheinformation given by the separate propositions taken together. From thispropositionall that isknownconcerningXmay immediatelybededucedbyomitting every alternant that contains x, all that is known concerningY byomittingeveryalternantthatcontainsy,andsoon.

Themethodmaybevariedbybringingthepropositionstotheform

NoXisQ1orQ2…orQn,


ortotheform

NothingisQ1orQ2…orQn,

then combining them as in section 476, and (if an affirmative solution isdesired)finallyobvertingtheresult.Itwilldependontheformoftheoriginalpropositionswhetherthisvariationisdesirable.508


508 This secondmethod isanalogous to thatwhich isusuallyemployedbyDrVenninhisSymbolicLogic.BothmethodsbearacertainresemblancetoJevons’sIndirectMethod;butneitherofthemisidenticalwiththatmethod.

In an equational systemof symbolic logic, a solutionwith regard to anyterm X generally involves a partial solution with regard to x also. In theemployment of the abovemethods, x must be found separately. It may beaddedthatthecompletesolutionsforXandxsumupbetweenthemthewholeof the information given 508 by the original data; in otherwords, they are,takentogether,equivalenttothegivenpremisses.509

509 Having determined that All X is P and that All x is q, we may bycontrapositionbringthelatterpropositiontotheformAllQisX,anditmaythenbefoundthatPandQhavesomealternantsincommon.Thesealternantsarethetermswhich(inBoole’ssystem)aretakenintheirwholeextentintheequationgivingX ;andthesolutionthusobtainediscloselyanalogoustothatgivenbyanyequationalsystemofsymboliclogic.

Thefollowingmaybetakenasasimpleexampleofthefirstoftheabovemethods.ItisadaptedfromBoole(LawsofThought,p.118).

“Given1st, thatwhereverthepropertiesAandBarecombined,eitherthepropertyC,orthepropertyD,ispresentalso,buttheyarenotjointlypresent;2nd,thatwhereverthepropertiesBandCarecombined,thepropertiesAandD are either both presentwith them, or both absent; 3rd, thatwherever thepropertiesA andB are both absent, thepropertiesC andD are both absentalso;andviceversâ,wherethepropertiesCandDarebothabsent,AandBarebothabsentalso.FindwhatcanbeinferredfromthepresenceofAwithregardtothepresenceorabsenceofB,C,andD.”

Thepremissesmaybewrittenasfollows:(1)AllABisCdorcD ; (2)AllBCisADorad ;(3)Allabiscd ;(4)Allcdisab.

Then, from(1), AllAisborCdorcD ;andfrom(2), AllAisborcorD ;

therefore (bycombiningthese), AllAisborcD ;(3)givesnoinformationregardingA(seetheprecedingsection);

butby(4), AllAisCorD ;therefore, AllAisbCorbDorcD ;

and,sincebDisbydevelopmenteitherbCDorbcDthisbecomes


AllAisbCorcD.

Thissolvestheproblemasset.Proceedingalsotodeterminea,wefindthat(1)givesnoinformationwithregardtothisterm;butby(2),Allaisborcord ;andby(3),AllaisBorcd ;therefore,AllaisBcorBdorcd.Againby(4),All a is b or C or D. Therefore, All a is BCd or BcD or bcd ; and bycontraposition,WhateverisBcdorbCorbDorCDisA.510

510 Taking intoaccount the resultarrivedatabovewith regard toA, itwillbeseenthatthismayberesolvedintoWhateverisbCorbDisAandNothingisBCDor Bcd. These two propositions taken together with the solution for A areequivalenttotheoriginalpremisses.

489.The Problem of Elimination.—By elimination in logic is meant theomissionofcertainelementsfromapropositionorsetof509propositionswiththeobjectofexpressingmoredirectlyandconcisely theconnexionbetweenthe elements which remain. An example of the process is afforded by theordinarycategoricalsyllogism,wheretheso-calledmiddletermiseliminated.Thus,giventhepremissesAllMisP,AllSisM,wemayinferAllSisMP ;butifwedesiretoknowtherelationbetweenSandPindependentlyofMwearecontentwith the lessprecisebutsufficientstatementAllS isP ; inotherwords,weeliminateM.

Eliminationhasbeenconsideredbysomewriterstobeabsolutelyessentialto logical reasoning. It is not, however, necessarily involved either in theprocessofcontrapositionorintheprocessdiscussedintheprecedingsection;andifformalinferencesarerecognisedatall,thenameofinferencecertainlycannot be denied to these processes. We must, therefore, refuse to regardelimination as of the essence of reasoning, although it may usually beinvolvedtherein.511

511 Comparesections207,208.

490.EliminationfromUniversalAffirmations.—Anyuniversalaffirmativeproposition(or,bycombination,anysetofuniversalaffirmativepropositions)involvingthetermXanditscontradictoryxmaybycontrapositionbereducedtotheformEverythingisPXorQxorR,whereP,Q,RarethemselvessimpleorcomplextermsnotinvolvingXorx ;andsincebytherulegiveninsection448adeterminantmayatanytimebeomittedfromanundistributedterm,we


mayeliminateX(andx)fromthispropositionbysimplyomittingthem,andreducingthepropositiontotheformEverythingisPorQorR.512

512 Wemightalsoproceedasfollows:SolveforXandforx,asinsection488,sothatwehaveAllXisA,AllxisB,whereAandBaresimpleorcomplextermsnotinvolvingeitherXorx.Then,sinceEverythingisXorx,weshallhaveEverythingisAorB,andthiswillbeapropositioncontainingneitherXnorx.

Wemust,however,hereadmitthepossibilityofP,Q,RbeingoftheformsA or a, Aa. These are equivalent respectively to the entire universe ofdiscourseandtonothing.Thus,ifPisoftheformAora,andQisoftheformAa, our proposition will before elimination more naturally be writtenEverythingisXorR ;ifQisoftheformAora,andRoftheformAa,itwillmorenaturallybewrittenEverythingisPXorx.ItfollowsthatifeitherPorQ is of the formAora (that is, if eitherP orQ is equivalent to the entireuniverseofdiscourse),thepropositionresultingfromelimination510willnotaffordanyrealinformation,sinceitisalwaystrueàpriorithatEverythingisAoraor&c.ThusweareunabletoeliminateXfromsuchapropositionasAllAisXorBC.

Thefollowingmaybegivenasanexampleofeliminationfromuniversalaffirmatives.

LetitberequiredtoeliminateX(togetherwithx)fromthepropositionsAllP is XQ or xR, Whatever is X or R is p or XQR. Combining thesepropositions, we have Everything is XQR or p ; therefore, by elimination,EverythingisQRorpthatis,AllPisQR.ItwillbeobservedthatP(togetherwithp)cannotbeeliminatedfromtheabovepropositions.

491. Elimination from Universal Negatives.—Any universal negativeproposition (or, by combination, any set of universal negative propositions)containingthetermXanditscontradictoryxmaybyconversionbereducedtotheformNothingisPXorQxorR,whereP,Q,RarethemselvessimpleorcomplextermsnotinvolvingeitherXorx.Herewemight,inaccordancewiththerulegiven insection448,simplyomit thealternantsPX,Qx, leavinguswith the proposition Nothing is R. This, however, is but part of theinformationobtainablebytheeliminationofX.WehavealsoNoXisP,andNoQisx,thatis,AllQisX ;whencebyasyllogisminCelarentwemayinfer


No Q is P. The full result of the elimination is, therefore, given by thepropositionNothingisPQorR.513

513 CompareMrsLadd Franklin’s Essay onTheAlgebra of Logic (Studies inLogicbyMembersoftheJohnsHopkinsUniversity).Thesameconclusionmaybededucedbyobversionfromtheresultobtainedintheprecedingsection.NothingisPXorQxorRbecomesbyobversionEverythingisprXorqrx.Therefore,bytheelimination of X, Everything is pr or qr ; and this proposition becomes byobversionNothingisPQorR.

Anothermethod bywhich the same resultmay be obtained is as follows:BydevelopingthefirstalternantwithreferencetoQandthesecondwithreferencetoP,NothingisPXorQxorRbecomesNothingisPQXorPqXorPQxorpQxorR.ButPQX or PQx is reducible to PQ, and on omitting PqX and pQx, we haveNothingisPQorR.

ItisinterestingtoobservethattheaboveruleforeliminationfromnegativesisequivalenttoBoole’sfamousruleforelimination.InordertoeliminateXfromtheequation F(X) = 0, he gives the formula F(1) F(0) = 0. Now any equationcontainingX can be brought to the formAX +Bx +C = 0, where A, B, C areindependent ofX.ApplyingBoole’s rulewe have (A+C)(B+C) = 0, that is,AB+C=0;andthisispreciselyequivalenttotherulegiveninthetext.

The following is an example: Let it be required to eliminateX from thepropositionsNoPisXqorxr,NoXorR isxPorPqorPr.511CombiningthesepropositionswehaveNothingisXPqorXProrxPorPqR ; therefore,byeliminationinaccordancewiththeaboverule,NothingisPqorPr,thatis,NoPisqorr.

492.EliminationfromParticularAffirmatives.—AnyparticularaffirmativepropositioninvolvingthetermXmaybyconversionbereducedto theformSomethingiseitherPXorQxorR,whereP,Q,RareindependentofXandx.We may here immediately apply the rule given in section 448 that adeterminantmayatanytimebeomittedfromanundistributedterm;andtheresultofeliminatingXisaccordinglySomethingiseitherPorQorR.514

514 Thus the rule for elimination from particular affirmatives is practicallyidenticalwiththeruleforeliminationfromuniversalaffirmatives.

493. Elimination from Particular Negatives.—Any particular negativepropositioninvolvingthetermmaybycontrapositionbereducedtotheformSomething is not either PX or Qx or R. By the development of the firstalternantwithreferencetoQandthatofthesecondalternantwithreferencetoP, thispropositionbecomesSomethingisnoteitherPQXorPqXorPQxor


pQxorR.ButPQXorPQx is reducible toPQand thealternantsPqX,pQxmaybytherulegiveninsection448beomitted.HencewegetthepropositionSomethingisnoteitherPQorR,fromwhichXhasbeeneliminated.515

515 Thustheruleforeliminationfromparticularnegativesispracticallyidenticalwith the rule for elimination from universal negatives. The same rule may bededucedbyobversionfromtheresultobtainedintheprecedingsection.Somethingis not either PX orQx or R ; therefore,Something is either prX or qrx or pqr ;therefore,Somethingiseitherprorqr ;therefore,SomethingisnoteitherPQorR.

494.Order of procedure in the process of elimination.—Schröder (DerOperationskreisdesLogikkalkuls,p.23)pointsoutthatfirsttoeliminateandthencombineisnotthesamethingasfirsttocombineandtheneliminate.For,as a rule, if a term X is eliminated from several isolated propositions thecombinedresultsgivelessinformationthanisaffordedbyfirstcombiningthegivenpropositionsandtheneffectingtherequiredelimination.

Thereareindeedmanycasesinwhichwecannoteliminateatallunlesswefirstcombinethegivenpropositions.Thisisofcourseobviousinsyllogisms;andwe have a similar case ifwe take the premissesEverything is A or X,EverythingisBorx.WecannoteliminateXfromeitherofthesepropositionstakenbyitself,sinceineachofthemX(orx)appearsasanisolatedalternant.But by 512 combination we haveEverything is Ax or BX ; and this by theeliminationofXbecomesEverythingisAorB.516

516 Working with negatives we get the same result. Taking the propositionsNothing is ax,Nothing is bX, separately, we cannot eliminate X from either ofthem.ButcombiningtheminthepropositionNothingisaxorbX,weareabletoinferNothingisab.

Thereareothercasesinwhicheliminationfromtheseparatepropositionsispossible,butwhere thisorderofprocedure leads toaweakenedconclusion.Take the propositionsEverything isAXorBx,Everything isCX orDx. ByfirsteliminatingXandthencombining,wehaveEverythingisACorADorBC or BD. But by first combining and then eliminating X our conclusionbecomes Everything is AC or BD, which gives more information than isaffordedbythepreviousconclusion.


EXERCISES.

495. Suppose that an analysis of the properties of a particular class ofsubstanceshasledtothefollowinggeneralconclusions,namely: 1st,ThatwhereverthepropertiesAandBarecombined,eitherthepropertyC,orthepropertyD,ispresentalso;buttheyarenotjointlypresent; 2nd,ThatwhereverthepropertiesBandCarecombined,thepropertiesAandDareeitherbothpresentwiththem,orbothabsent; 3rd,ThatwhereverthepropertiesAandBarebothabsent,thepropertiesCandDarebothabsentalso;andviceversâ,wherethepropertiesCandDarebothabsent,AandBarebothabsentalso. Shew thatwherever thepropertyA ispresent, thepropertiesB andCarenot both present; also that wherever B is absent while C is present, A ispresent.

[Boole,Laws of Thought, pp. 118 to 120; compare also Venn, SymbolicLogic,pp.276to278.]

Asolutionofthisproblemhasalreadybeengiveninsection488.Wemayalsoproceedasfollows.Thepremissesare:

AllABisCdorcD,(i)

AllBCisADorad,(ii)

Allabiscd,(iii)

Allcdisab.(iv)

513By(i),NoABisCD,therefore,NoAisBCD.(1)

By(ii),NoBCisAd,therefore,NoAisBCd.(2)

Combining(1)and(2),itfollowsimmediatelythatNoAisBC.

BoolealsoshewsthatAllbCisA.Thisisapartialcontrapositiveof(iii).Wehavesofarnotrequiredtomakeuseof(iv)atall.

496.Giventhesamepremissesasintheprecedingsection,provethat:— (1)WhereverthepropertyCisfound,eitherthepropertyAorthepropertyBwillbefoundwithit,butnotbothofthemtogether; (2)IfthepropertyBisabsent,eitherAandCwillbejointlypresent,orCwillbeabsent; (3) If A and C are jointly present, B will be absent.[Boole, Laws of


Thought,p.129.]First,By(i),AllCisaorbord ;by(ii),AllCisaorborD ;therefore,AllCisaorb.

Also,by(iii),AllCisAorB ;therefore,AllCisAboraB.(1)

Secondly,By(iii).AllbisAorc,therefore,bysection432,AllbisACorc.(2) Thirdly,from(1)itfollowsimmediatelythat

AllACisb.(3)

Thegivenpremissesmayallbesummedupintheproposition:EverythingisAbCorAbDoraBCdorabcdorBcD.Fromthis,theabovespecialresultsandothersfollowimmediately.

497.GiventhateverythingiseitherQorR,andthatallRisQ,unlessitisnotP,provethatallPisQ.[K.]Thepremissesmaybewrittenasfollows:(1)AllrisQ,(2)AllPRisQ.

By(1),AllPrisQ,andby(2),AllPRisQ ;butAllPisProrPR ;therefore,AllPisQ.

498.WhereAispresent,BandCareeitherbothpresentatonceorabsentatonce;andwhereCispresent,Aispresent.Describetheclassnot-Bundertheseconditions.[Jevons,Studies,p.204.]Thepremissesare(1)AllAisBCorbc,(2)AllCisA.

By(1)Allbisaorc,andby(2)AllbisAorc,therefore,Allbisc.

499.Itisknownofcertainthingsthat(1)wherethequalityAis,Bisnot;(2) whereB is, and only whereB is,C andD are. 514 Derive from theseconditionsadescriptionoftheclassofthingsinwhichAisnotpresent,butCis.[Jevons,Studies,p.200.]Thepremissesare:(1)AllAisb ;(2)AllBisCD ;(3)AllCDisB.

NoinformationregardingaCisgivenby(1).Butby(2),AllaCisborD ;andby(3),AllaCisBord. Therefore,AllaCisBDorbd.

500. Taking the same premisses as in the previous section, drawdescriptionsoftheclassesAc,ab,andcD.[Jevons,Studies,p.244.]By(1),Everythingisaorb,andby(2),EverythingisborCD.Therefore,EverythingisaCDorb ;

andby(3),EverythingisBorcord.Therefore,EverythingisaBCDorbcorbd. HenceweinferimmediatelyAllAcisb,Allabiscord,AllcDisb.

501.ThereisacertainclassofthingsfromwhichApicksoutthe‘XthatisE,andtheYthatisnotZ,’andBpicksoutfromtheremainder‘theZwhichisYandtheXthatisnotY.’Itisthenfoundthatnothingisleftbuttheclass‘Z


which is not X.’ The whole of this class is however left. What can bedeterminedabouttheclassoriginally?[Venn,SymbolicLogic,pp.267,8.]The chief difficulty in this problem consists in the accurate statement of the premisses. Call the

originalclassW.Wethenhave

AllWisXZorYzorYZorXyorxZ,thatis,AllWisXorYorZ ;(1)

AllxZisW ;(2)NoxZisWXZorWYzorWYZorWXy,

thatis,NoxZisWYZ.(3)

Wemaynowproceedasfollows:—By(1),AllWisXorYorZ ; andby(3),AllWisXoryorz.Therefore, All W is X or Yz or yZ. (2) affords no information regarding the classW, except thateverythingthatisZbutnotXiscontainedwithinit.

502.(1)Ifanationhasnaturalresources,andagoodgovernment,itwillbeprosperous. (2) If it has natural resourceswithout a good government, or agood government without natural resources, it will be contented, but notprosperous. (3) If it has neither natural resources nor a goodgovernment itwillbeneithercontentednorprosperous. ShewthatthesestatementsmaybereducedtotwopropositionsoftheformofHamilton’sU.[O’S]515LetanationwithnaturalresourcesbedenotedbyR,anationwithagoodgovernmentbyG,a

prosperousnationbyP,andacontentednationbyC.Thenthegivenstatementsmaybeexpressedasfollows:—(1)AllRGisP ;(2)AllRgorrGisCp ;(3)Allrgiscp. Bycontraposition,(2)mayberesolvedintothetwopropositions,AllcpisRGorrg,AllPisRGorrg.But by (1)No cp isRG ; and by (3)NoP is rg.Hence the two propositions intowhich (2)wasresolvedmaybereducedtotheform,Allcpisrg,AllPisRG. ThethreeoriginalstatementsareaccordinglyequivalenttothetwoUpropositionsAllRGisallP,Allrgisallcp.

503.Let theobservationofaclassofnaturalproductionsbe supposed tohaveledtothefollowinggeneralresults. 1st.That inwhichsoeverof theseproductions thepropertiesA andC aremissing,thepropertyEisfound,togetherwithoneofthepropertiesBandD,butnotwithboth. 2nd.Thatwherever thepropertiesAandD are foundwhileE ismissing,thepropertiesBandCwilleitherbothbefound,orbothbemissing. 3rd.ThatwhereverthepropertyAisfoundinconjunctionwitheitherBorE,orbothofthem,thereeitherthepropertyCorthepropertyDwillbefound,butnotbothofthem.Andconversely,whereverthepropertyCorDisfound


singly,therethepropertyAwillbefoundinconjunctionwitheitherBorEorbothofthem. Shew that it follows that Inwhateversubstances thepropertyA is found,therewillalsobefoundeitherthepropertyCorthepropertyD,butnotboth,orelsethepropertiesB,C,andEwillallbewanting.Andconversely,WhereeitherthepropertyCorthepropertyDisfoundsinglyorthepropertiesB.C,andDaretogethermissing,therethepropertyAwillbefound.ShewalsothatIfthepropertyAisabsentandCpresent,Dispresent.

[Boole,Laws of Thought, pp. 146–148. Venn, Symbolic Logic, pp. 280,281.JohnsHopkinsStudiesinLogic,pp.57,58,82,83.]

Thepremissesareasfollows:—

1st, AllacisBdEorbDE ; (i) 2nd, AllAdeisBCorbc ; (ii) 3rd, WhateverisABorAEisCdorcD ; (iii)

WhateverisCdorcDisABorAE. (iv)

516Wearerequiredtoprove:—

AllAisCdorcDorbcd ; (α)AllCdisA ; (β)AllcDisA ; (γ)AllbcdisA ; (δ)AllaCisD. (ε)

First,By(iii),AllAisCdorcDorbc.Butby(ii),AllAbeiscord ;andby(iv),AllAbeisCDorcd ;therefore,AllAbeiscd.Hence,AllAisCdorcDorbcd.(α)

Secondly,(β)and(γ)followimmediatelyfrom(iv). Thirdly,from(i),wehavedirectly,Noacisbd ;therefore(byconversion),Nobcdisa ;therefore,AllbcdisA.(δ)

Lastly,by(iv),AllCdisA ;therefore,bycontraposition,AllaCisD.(ε)

WemayobtainacompletesolutionsofarasAisconcernedasfollows:

By(ii),517AllAisBCorbcordorE ;by(iii),AllisbeorCdorcD ;

therefore,AllAisCdorcDEorbcDorbceorbde ;by(iv).AllAisBorEorCDorcd ;

therefore,AllAiscDEorbcdeorBCdorCdE.

ThisincludesthepartialsolutionwithregardtoA,—AllAisCdorcDorbcd.BoolecontentshimselfwiththisbecausehehasstartedwiththeintentionofeliminatingEfromhisconclusion. Wemaynowsolvefora.(ii)and(iii)givenoinformationwithregardtothisterm.Butby(i),AllaisBdEorbDEorC ;andby(iv),AllaisCDorcd.Therefore,AllaisBcdEorCD.Andthisyieldsbycontraposition,WhateverisbcorCdorcDorceisA.


517 NoinformationwhateverwithregardtoAisgivenby(i),sinceaappearsasadeterminantofthesubject.Seesection487.

504.Giventhesamepremissesasintheprecedingsection,shewthat,— 1st. If the property B be present in one of the productions, either thepropertiesA,C,andDareallabsent,orsomeonealoneof themisabsent.Andconversely,iftheyareallabsentitmaybeconcludedthatthepropertyBispresent. 2nd. IfA andCare both present or both absent,Dwill be absent, quiteindependentlyof thepresenceorabsenceofB. [Boole,LawsofThought,p.149.]Wemayproceedherebycombiningall thegivenpremissesin517themanner indicatedinsection

475.Fromthe result thusobtained theaboveconclusionsaswellas thosecontained in theprecedingsectionwillimmediatelyfollow.

By(iii),EverythingisaorbeorCdorcD ;andby(iv).EverythingisABorAEorCDorcd ;

therefore,EverythingisABCdorABcDorACdEorAcDEoraCDoracdorbCDeorbcde ;thereforeby(i),EverythingisABCdorABcDorAbcdeorACdEorAcDEoraBcdEoraCDorbCDe ;

thereforeby(ii),EverythingisABCdorAbcdeorACdEorAcDEoraBcdEoraCD.(v)

Hence,AllBisACdorAcDEoracdEoraCD ; AllacdisBE ; AllACisBdordE ; AllacisBdE. EliminatingEfromeachoftheabovewehavetheresultsarrivedatbyBoole. EliminatingbothAandEfrom(v)wehave

EverythingisBCdorbcdorCdorcDorBcdorCD ;

that isEverything isCorDorcd,which isan identity.This isequivalent toBoole’sconclusion that“thereisnoindependentrelationamongthepropertiesB,C,andD”(LawsofThought,p.148). Anyfurtherresultsthatmaybedesiredareobtainableimmediatelyfrom(v).

505.GivenXY=A,YZ=C,findXZintermsofAandC.

[Venn,SymbolicLogic, pp. 279, 310–312. JohnsHopkins Studies in Logic,pp.53,54.]Thepremissesmaybewrittenasfollows:

EverythingisAXYoraxoray ;(1) EverythingisCYZorcyorcx.(2)

By(1),AllXZisAYoray,andby(2),AllXZisCYorcy ;therefore,AllXZisACYoracy.Hence,eliminatingY,AllXZisACorac. Thissolvestheproblemasset.ButinordertogetacompletesolutionequivalenttothatwhichwouldbeobtainedbyBoole,thefollowingmaybeadded:Solvingasaboveforxorz,andeliminatingY,we


haveAllthatiseitherxorzisAcXzoraCxZorac.Whence,bycontraposition,WhateverisACorAxorAZorCXorCzisXZ.Inotherwords,WhateverisACorAZorCXisXZ ;andNothingisAxorCz.

518 506. Shew the equivalence between the three following systems ofpropositions:(1)AllAbiscd ;AllaBisCe ;AllDisE ;(2)AllAisBorcorD ;AllBEisA ;AllBeisAdorCd ;AllbDisaE ;(3)WhateverisAoreisBord ;AllaisbEorbdorBCe ;AllbCisa ;AllDisE.[K.]Byobversion,thefirstsetofpropositionsbecomeNoAbisCorD ;NoaBiscorE ;NoDise ;and

thesepropositionsarecombinedinthestatement,NothingiseitherAbCorAbDoraBcoraBEorDe.(1)

By obverting and combining the second set of propositions,we haveNothing isAbCdor aBEoraBceorBDeorAbDorbDe.(2)

ButAbCdorAbD isequivalent toAbCorAbD ;aBEoraBcetoaBEor aBc ;BDeor bDe toDe.Hence(1)and(2)areequivalent. Again,byobvertingandcombiningthethirdsetofpropositions,wehaveNothingisAbDorbDeoraBcoraBEorabDeoracDeorAbCorDe.(3)

ButsincebDe,abDe,acDeareallsubdivisionsofDe,(3)immediatelyresolvesitselfinto(1).

507.Fromthepremisses(1)NoAxiscdorcy,(2)NoBXiscdeorcey,(3)NoabiscdxorcEx,(4)NoAorBorCisxy,deduceapropositioncontainingneitherXnorY.[JohnsHopkinsStudies,p.53.]By(2),NoXisBcde,andby(1)and(3),NoxisAcdorabcdorabcE ;therefore,bysection491,No

AcdorabcdorabcEisBcde ;therefore,NoAcdisBe. ItwillbeobservedthatsinceYdoesnotappearinthepremisses,ycanbeeliminatedonlybyomittingallthetermscontainingit.

508. Themembers of a scientific society are divided into three sections,whicharedenotedbyA,B,C.Everymembermustjoinone,atleast,ofthesesections,subjecttothefollowingconditions:(1)anyonewhoisamemberofAbutnotofB,ofBbutnotofC,orofCbutnotofA,maydeliveralecturetothemembersifhehaspaidhissubscription,butotherwisenot;(2)onewhoisamemberofA but not ofC, ofC but not ofA, orofB but not ofA,mayexhibit an experiment to the members if he has paid his subscription, butotherwisenot;but(3)everymembermusteitherdeliveralectureorperformanexperimentannuallybeforetheothermembers.Findtheleastadditiontotheseruleswhichwillcompeleverymembertopayhissubscriptionorforfeithismembership.[JohnsHopkinsStudies,p.54.]LetA =member of sectionA,&c.;X = one who gives a lecture; 519Y = one who performs an

experiment;Z=onewhohaspaidhissubscription.


Thepremissesare (1)AllAboraCorBcisxorZ ; (2)AllAcoraBoraCisyorZ ; (3)EverymemberisXorY ; (4)EverymemberisAorBorC. TheproblemistofindwhatistheleastadditiontotheseruleswhichwillresultintheconclusionthatEverymemberisZ. By(1),Allziseitherxorelse(aorB)(Aorc)(borC);therefore,AllzisxorABCorabc. Similarly,by(2),AllzisyorACorabc ;therefore,AllzisxyorxACorABCorabc. By(3),AllzisXorY ;therefore,AllzisXABCorXabcorxYACorYABCorYabc. By(4),AllzisAorBorC ;therefore,AllzisXABCorxYACorYABC ;butAllYABCiseitherXYABCorxYABC ;therefore,AllzisXABCorxYAC. Hence,wegainthedesiredresultifweaddtothepremisses,Noz isXABCorxYAC.Therequiredruleisthereforeasfollows:Noonewhohasnotpaidhissubscriptionmayjoinall threesectionsanddeliveralecture,normayhejoinAandCandexhibitanexperimentwithoutdeliveringalecture.

509.WhatmaybeinferredindependentlyofXandY fromthepremisses:(1)EithersomeAthatisXisnotY,orallDisbothXandY ;(2)EithersomeYisbothBandX,orallX iseithernotYorCandnotB?[JohnsHopkinsStudies,p.85.]Thepremissesmaybewrittenasfollows:(1)EithersomethingisAXy,oreverythingisXYord ;(2)

EithersomethingisBXY,oreverythingisxoryorbC. Bycombining thesepremisses as in chapter4,Either something isAXy and something isBXY, orsomething isAXyandeverything isxoryorbC,orsomething isBXYandeverything isXYord,oreverythingisbCXYorbCdordxordy.518 Therefore,eliminatingXandY(seesections490and492),EithersomethingisAandsomethingisB,orsomething isA,or520something isB,oreverything isbCord ; andbycombining the first threealternantsasinsection481,thisbecomes

EithersomethingisAorBoreverythingisbCord.

Thisconclusionmayalsobeexpressedintheform

Ifeverythingisab,theneverycisd.518 Wecannot, ifweare tobe leftwithanequivalentproposition,express the

firstthreeofthesealternantsinanon-compoundform.Seesections477,479.

510.Sixchildren,A,B,C,D,E,Farerequiredtoobeythefollowingrules:(1) onMonday andTuesday no four can go out together; (2) onThursday,Friday, and Saturday no three can stay in together; (3) on Tuesday,


Wednesday,andSaturday,ifBandCaretogether,thenA,B,E,andFmustbetogether;(4)onMondayandSaturdayBcannotgooutunlesseitherD,orA,C,andEstayathome.AandBarefirst todecidewhat theywilldo,andCmakes his decision beforeD,E, andF. Find (α)whenC must go out, (β)whenhemuststayin,and(γ)whenhemaydoashepleases.[JohnsHopkinsStudies,p.58.]LetA=caseinwhichAgoesout,a=thatinwhichhestaysin,&c.

Thenthepremissesareasfollows: (1)OnMondayandTuesday,—threeatleastmuststayin ; (2)OnThursday,Friday,andSaturday,—nothreecanstayintogether ; (3)OnTuesday,Wednesday,andSaturday,—EverycaseisABEForabeforBcorbC ; (4)OnMondayandSaturday,—Everycaseisaceorbord. Inordertosolvetheproblem,wemustcombinethepossibilitiesforeachday,theneliminateD,E,andF,andfindinwhatwaysthemovementsofAandBdeterminethoseofC. (i)OnMonday,—wehaveEverycaseisaceorbord,combinedwiththeconditionthatthreeatleastmust stay in. One alternant therefore is def without further condition, and it follows that we candeterminenoindependentrelationbetweenA,B,andC. HenceonMondayCmaydoashepleases. (ii)OnTuesday,—wehaveEverycase isABEForabeforBcorbC, combinedwith theconditionthatthreeatleastmuststayin.Therefore,EverycaseisabeforBcorbC ;519andeliminatingD,E,andF,EverycaseisaborBcorbC.

519 The twoalternantsBcandbCmightherebemademoredeterminate, thus,aBcdoraBceoraBcforBcdeorBcdforBcefandabCdorabCeorabCforbCdeorbCdforbCef.Butsinceweknowthatwearegoingonimmediatelytoeliminated, e, and f, it is obvious, evenwithoutwriting themout in full, that thesemoredeterminateexpressionswillatoncebereducedagaintoBcandbCsimply.

521HenceitfollowsthatonTuesday(α)ifAgoesoutwhileBstaysin,Cmustgoout,and(β)ifBgoesout,Cmuststayin. (iii)OnWednesday,—EverycaseisABEForabeforBcorbC ;or,eliminatingD,E,andF,EverycaseisABoraborBcorbC.Therefore,AllAbisCandAllaBisc. HenceonWednesday(α)ifAgoesoutwhileBstaysin,Cmustgoout,and(β)ifAstaysinwhileBgoesout,Cmuststayin. (iv)OnThursdayandFriday,—theonlyconditionisthatnothreecanstayintogether. HenceonThursdayandFridayifAandBbothstayin,Cmustgoout. (v) On Saturday,—Every case is ABEF or abef or Bc or bC ; alsoEvery case is ace or b or d.Combining these premisses,Every case is ABdEF or abef or aBce or Bcd or bC. But we have thefurther condition that no three can stay in together. Therefore,Every case is ABdEF or ABcdEF orAbCDEorAbCDForAbCEForbCDEF.Therefore,eliminatingD,E,andF,EverycaseisABorbC. HenceonSaturdayifBstaysin,Cmustgoout.


511.Given(1)AllPisQR,(2)Allpisqr ;shewthat(3)AllQisPR,(4)AllRisPQ.[K.]

512.EliminateRfromthepropositionsAllRisPorpq,AllqisProrR,AllqRisP.[K.]

513.Shewtheequivalencebetweenthefollowingsetsofpropositions:—(1)aisBC ;bisAC ;CisAboraB ;(2)aisBC ;BisAcoraC ;cisAB ;(3)AisBcorbC ;bisAC ;cisAB.[K.]

514.Sayby inspection, statingyour reasons,whichof the followingpropositionsgive informationconcerningA,aB,b,bCd,respectively:AllAbisbCdorc ;AllbdisAorbCorabc ;WhateverisaorBiscorD ;WhateverisAborbcisbDorcDore ;EverythingisAoraborBcorCd.[K.]

515.Determinetheconditionsunderwhichaparticularpropositionaffordsinformationinregardtoanygiventerm.[K.]

516.ItisknownofcertainthingsthatthequalityAisalwaysaccompaniedbyCandD,butneverbyB ;andfurther,thatthequalitiesCandDneveroccurtogether,exceptinconjunctionwithA.WhatcanweinferaboutC?[M.]

522517.Giventhateverythingthat isQbutnotS iseitherbothPandorneitherPnorRand thatneitherRnorSisbothPandQ,shewthatnoPisQ.[K.]

518.WhereCispresent,A,B,andDareallpresent;whereD ispresent,A,B,andCareeitherallthreepresentorallthreeabsent.ShewthatwheneitherAorBispresent,CandDareeitherbothpresentorbothabsent.Howmuchof thegiven information issuperfluoussofaras thedesiredconclusion isconcerned?[K.]

519.Given(i)AllPqrisST ;(ii)QandRarealwayspresentorabsenttogether ;(iii)AllQRSisPTorpt ;(iv)AllQRsisPt ;(v)AllpqrSisT ;thenitfollowsthat(1)AllPqisrST ;(2)AllPsisQRt ;(3)AllpQisRSt ;(4)AllpTisqr ;(5)AllQsisPRt ;(6)AllQTisPRS ;(7)AllqSisrT ;(8)Allqsispr ;(9)Allqtisprs ;(10)AllsTispqr.[K.]

520.WhatcanbedeterminedaboutPintermsofQandRfromthepremissesAllPisQorX,SomePisnotRX?[K.]

521.Giventhatallhonestmenarehappy,andthatalldishonestmenareunwise;andassumingthathonestanddishonest,happyandunhappy,wiseandunwise,arepairsoflogicalcontradictories;whatisallthatcanbeinferredaboutmenwhoarehappy,unhappy,wise,unwise,respectively?[K.]

522.Ifthriftlessnessandpovertyareinseparable,andvirtueandmiseryareincompatible,andifthriftbeavirtue,cananyrelationbeprovedtoexistbetweenmiseryandpoverty?Ifmoreoverallthriftlesspeopleareeithervirtuousornotmiserable,whatfollows?[V.]

523.Atacertainexamination,all thecandidateswhowereentered forLatinwerealsoentered foreitherGreek,French,orGerman,butnotformorethanoneoftheselanguages;allthecandidateswhowerenotenteredforGermanwereenteredfortwoatleastoftheotherlanguages;nocandidatewhowasentered forbothGreekandFrenchwasentered forGerman,but all candidateswhowereentered forneitherGreeknorFrenchwereenteredforLatin.Shewthatallthecandidateswereenteredfortwoofthefourlanguages,butnoneformorethantwo.[K.]

524. (1)Wherever there is smoke there isalso fireor light; (2)Wherever there is lightandsmokethereisalsofire;(3)Thereisnofirewithouteithersmokeorlight.

523 Given the truth of the above propositions, what is all that you can infer with regard to (i)circumstanceswhere there is smoke; (ii) circumstanceswhere there isnot smoke; (iii) circumstances


wherethereisnotlight?[W.]

525. Inacertainwarehouse,when thearticlesofferedareantique, theyarecostly,andat thesametimeeitherbeautiful orgrotesque, butnotboth.When they arebothmodern andgrotesque, they areneither beautiful nor costly. Everythingwhich is not beautiful is offered at a low price, and nothingcheapisbeautiful.Whatcanweassert(1)abouttheantique,and(2)aboutthegrotesquearticles?[M.]

526.Shewthatthefollowingsetsofpropositionsareequivalenttooneanother:— (1)Allaisborc ;AllbisaCd ;AllcisaB ;AllDisc. (2)AllAisBC ;AllbisaC ;AllCisABdorabd. (3)AllAisB ;AllBisAorc ;AllcisaB ;AllDisc. (4)AllbisaC ;AllAisC ;AllCisd ;AllaCisb. (5)AllcisaB ;AllDisaB ;AllAisB ;AllaBisc. (6)AllAisBC ;AllBCisA ;AllDisBc ;AllbisC.[K.]

527.Shewthatacertainsetoffourpropertiesmustbefoundsomewhere together, if thefollowingfactsareknown:“Everythingthathasthefirstpropertyoriswithoutthelasthasthetwoothers;andifeverythingthathasboththefirstandlasthasoneorotherbutnotbothofthetwoothers,thensomethingthathasthefirsttwomustbewithoutthelasttwo.”[J.]

528.Giventhepropositions:(i)allmaterialgoodsareexternal;(ii)nointernal(=non-external)goodsaredispropriable;(iii)alldispropriablegoodsareappropriable;(iv)nocollectivegoodsareappropriableor immaterial (= non-material); what is all that we can infer about (a) appropriable goods, (b)immaterialgoods?[J.]

529.EliminateXandYfromthefollowingpropositions:AllaXisBcYorbcy ;NoAXisBY ;AllABisY ;NoABCDisxY.ShewalsothatitfollowsfromthesepropositionsthatAllXYisAboraBc.[K.]

530.Given(1)AllAisBcorbC,(2)AllBisDEorde,(3)AllCisDe ;shewthat(i)AllAisBcDEorBcdeorbCDe,(ii)AllBcDisE,(iii)Allabdisc,(iv)AllcdisaborBe,(v)AllbCDise.[Jevons,PureLogic,§160.]

524531.Given(1)AllaBiscorD,(2)AllBEisDForcdF,(3)AllCisaBorBEorD,(4)AllbDiseorF,(5)AllbfisaorCorDE,(6)AllbcdEisAforaF,(7)AllAisBorCDEforcDforcdE ;shewthat(i)AllAisB,(ii)AllCisD,(iii)AllEisF.[K.]

532.Shewtheequivalencebetweenthetwofollowingsetsofpropositions:

(1) AllAisBCorBEorCEorD ; AllBisACDEorACdeorcdE ; AllCisABorAEoraD ; AllDisABCEorAceoraC ; AllEisACoraCBorBc.

(2) AllaisBcdEorbcdeorbD ; AllbisaorceordE ; AllcisAbDeorabdeorBdE ; AlldisabceorBcEorBeorbE ; Alleisaborbcord.

[K.]

533.Given


(1) AllbcisDEorDforhk,(2) AllCisaBorDEFGorBFH,(3) AllBcdiseLorhk,(4) AllAcfisd,(5) AllkisBCorCdorCforH,(6) AllABCDEFGisHorK,(7) AllDEFGHisB,(8) AllABlisforh,(9) AllADFKlisH,(10) AllADEFHisBorCorGorL ;

shewthatAllAisL.[K.]


CHAPTERVI.

THEINVERSEPROBLEM.

534.NatureoftheInverseProblem.—Bytheinverseproblemisheremeanta certain problem so-called by Jevons. Its nature will be indicated by thefollowingextracts,whicharefromthePrinciplesofScienceandtheStudiesinDeductiveLogicrespectively.

“In the Indirect process of Inference we found that from certainpropositionswecouldinfalliblydeterminethecombinationsoftermsagreeingwiththosepremisses.Theinductiveproblemisjusttheinverse.Havinggivencertain combinations of terms, we need to ascertain the propositions withwhichtheyareconsistent,andfromwhichtheymayhaveproceeded.Nowifthereadercontemplatesthefollowingcombinations,—

ABC abCaBC abc,

hewillprobablyrememberatoncethattheybelongtothepremissesA=AB,B =BC. If not, hewill require a few trials before hemeetswith the rightanswer, and every trialwill consist in assuming certain laws and observingwhether the deduced results agree with the data. To test the facility withwhichhecansolvethisinductiveproblem,lethimcasuallystrikeoutanyofthe possible combinations involving three terms, and say what laws theremaining combinations obey. Let him say, for instance, what laws areembodiedinthecombinations,—

ABC aBCAbc abC,

“Thedifficultybecomesmuchgreaterwhenmore termsenter526 into thecombinations.Itwouldbenoeasymattertopointoutthecompleteconditionsfulfilledinthecombinations,—


ACeaBCeaBcdEabCeabcE.

AftersometroublethereadermaydiscoverthattheprincipallawsareC=e,andA=Ae ;buthewouldhardlydiscovertheremaininglaw,namelythatBD=BDe”(PrinciplesofScience,1sted.,vol.I.,p.144;2nded.,p.125).

“The inverseproblem is always tentative, and consists in inventing laws,and trying whether their results agree with those before us” (Studies inDeductiveLogic,p.252).

Theproblemmaypreferablybestatedasfollows:— Givenacomplexpropositionoftheform

EverythingisP1P2…orQ1Q2…or…,

to find a set of propositions not involving any alternative combination ofterms,whichshalltogetherbeequivalenttoit.520

520 Theproblemmayalsobestatedasfollows:—Givenauniversalaffirmativecomplexpropositioncontainingalternativetermstofindanequivalentcompoundconjunctivepropositionallthedeterminantsofwhichareaffirmativeandfreefromalternativeterms.

ItmaybeobservedthatJevonsdoesnotdefinitelyexcludealternativetermsinhis solutions of inverse problems, thoughhe generally seeks to avoid them.Theproblem cannot, however, be defined with accuracy unless such terms areexplicitlyexcluded.

Theinverseproblemisinasenseindeterminate,forwemayfindanumberof sets of propositions, not involving any alternative combination of terms,which are precisely equivalent in logical force, and hence any inverseproblemmayadmitofanumberofsolutions.Butitisnotnecessarytohaverecoursetoaseriesofguessesinordertosolveanyinverseproblem,norneedthemethodof solutionbedescribed aswholly tentative.Several systematicmethodsofsolutionapplicabletoanyinverseproblemareformulatedin thefollowing sections. Since, however, more solutions than one are possible,someofwhicharesimplerthanothers,theprocessmayberegardedasmore


orlesstentativeinsofarasweseektoobtainthemostsatisfactorysolution.

Thefollowingmaybetakenasourcriterionofsimplicity.Comparingtwoequivalentsetsofpropositions,notinvolvingany527alternativecombinationofterms,thatsetmayberegardedasthesimplerwhichcontainsthesmallernumberofpropositions.Ifeachsetcontainsthesamenumberofpropositions,then we may count the number of terms involved in their subjects andpredicates takentogether,andregard thatoneas thesimplerwhich involvesthefewerterms.

535.AGeneral Solution of the Inverse Problem.—Let us suppose, then,thatwe are given a complex proposition involving alternative combination,and that we are to find a set of propositions, not involving alternativecombination,whichshalltogetherbeequivalenttoit.

Thedatamaybewrittenintheform

EverythingisPorQorSorTor&c.,

whereP,Q,&c., are themselves complex terms involving conjunctive, butnotalternative,combination.521

521 The proposition in its original form may admit of simplification inaccordance with the rules laid down in chapter 1. It will generally speaking befound advantageous to have recourse to such simplification before proceedingfurtherwiththesolution.

Bycontrapositiononeormoreofthesecomplextermsmaybebroughtoverfromthepredicateintothesubject,sothatwehave

WhateverisnoteitherPorSor&c.isQorTor&c.

Theselectionofcertaintermsfortranspositioninthiswayisarbitrary(anditisherethattheindeterminatenessoftheproblembecomesapparent);butitwillgenerallybefoundbesttotaketwoorthreewhichhaveasmanycommondeterminantsaspossible.

WhatisnoteitherPorSor&c.isQorTor&c.

will, when the subject is written in the affirmative form, be immediatelyresolvable into a series of propositions, which taken together give all theinformationoriginallygiven.522Anyofthesepropositionswhichstillinvolve


alternative combination may be dealt with in the same way, until noalternativecombinationremains.

522 Seesection446.

We shall now be left with a set of propositions which will satisfy therequiredconditions.Thepossibilityofvarious simplificationshas,however,to be considered. Thus, it will be necessary to make sure that each of thepropositionsisitselfexpressedinitssimplestform;523andtoobservewhetheranytwoormoreofthepropositions528admitofasimplerecombination.524Itmay alsobe found that someof thepropositions canbe altogether omitted,inasmuch as they add nothing to the information jointly afforded by theremainder;orthat,consideredintheirrelationtotheremainingpropositions,theymay, at any rate, be simplified by the omission of one ormore of thetermswhichtheycontain.525Whenthesesimplificationshavebeencarriedasfarasispossibleweshallhaveourfinalsolution.526

523 Forexample,AllABisBCmaybereducedtoAllABisC.524 Forexample,AllacisdandAllBcisdmaybecombinedintoAllcDisAb.525 Thus,forthepropositionsAllABisCDandAllAbisCwemaysubstitute

thepropositionsAllABisDandAllAisC.526 It may be observed that it is no part of our object to obtain a set of

propositionswhicharemutuallyindependent.Asamatteroffact,itwillgenerallybefoundthatthemaximumsimplificationinvolvestherepetitionofsomeitemsofinformation.Thus,intheexamplegivenintheprecedingnotethepropositionsAllABisCDandAllAbisCarequiteindependentofoneanother;butthepropositionAllAisCrenderssuperfluouspartoftheinformationgivenbythepropositionAllABisD.

The solution may, if we wish, be verified by recombining into a singlecomplexpropositionthepropositionsthathavebeenobtained,anoperationbywhich we shall arrive again at a series of alternants substantially identicalwiththoseoriginallygivenus.Suchverificationis,however,notessentialtothevalidityofourprocess,which,ifithasbeencorrectlyperformed,containsnopossiblesourceoferror.

Thefollowingexampleswillservetoillustratetheabovemethod.

I.ForourfirstexamplewemaytakeoneofthosechosenbyJevonsintheextractquotedintheprecedingsection.


Giventheproposition,EverythingiseitherABCorAbcoraBCorabC,wearetofindasetofpropositionsnotinvolvingalternativecombinationwhichshallbeequivalenttoit. BythereductionofaBCorabCtoaC,followedbycontraposition,wehaveWhatisneitherABCnorAbcisaC ;therefore,WhatisaorBcorbCisaC ;andthismayberesolvedintothethreepropositions:—

⎧ AllaisC,⎨ Bcisnon-existent,⎩ AllbCisa.

Bcisnon-existentisreducibletoAllBisC ;andthispropositionandAllaisCmaybecombinedintoAllcisAb.

529Hencewehaveforoursolutionthetwopropositions:—

⎰ AllcisAb,⎱ AllbCisa.

Itwillbefoundthatbytherecombinationofthesepropositionsweregaintheoriginalproposition.

II.Wemay next take themore complex example contained in the sameextractfromJevons.

The given alternants are ACe, aBCe, aBcdE, abCe, abcE ; and by thereductionof dual terms, theybecomeaBcdE,abcE,Ce. Therefore,What isnotaBcdEorabcEisCe ;andthispropositionmayberesolvedintothefourpropositions:—

⎧⎨⎩

AllAisCe ; (1)AllBDisCe ; (2)AllCise ; (3)AlleisC. (4)

But since by (3)All C is e, (1) may be reduced toAll A is C ; and thispropositionmaybecombinedwith(4)yieldingAllc isaE.Alsoby(3), (2)maybereducedtoAllBDisC.

Henceoursolutionbecomes


⎧ AllBDisC,⎨ AllCise,⎩ AllcisaE.

This solution may be shewn to be equivalent to the solution given byJevonshimself.

III.ThefollowingproblemisfromJevons,PrinciplesofScience,2nded.,p.127(Problemv).

ThegivenalternantsareABCD,ABCd,ABcd,AbCD,AbcD,aBCD,aBcD,aBcd,abCd.

Bythereductionofdualsthesealternantsmaybewrittenasfollows:ABCorABcdorAbDoraBCDoraBcorabCd.

Therefore,bycontraposition,WhateverisnotABCorAbDoraBcisABcdoraBCDorabCd.

ButWhateverisnotABCorAbDoraBcisequivalenttoWhateverisABcor aBC or ab or bd. Hence we have for our solution the following set ofpropositions:

(1) AllABcisd, (2) AllaBCisD,(3) AllabisCd, (4) Allbdisa.527

ThisisequivalenttothesolutiongivenbyJevons,Studies,p.256.527 We first obtainAll bd is aC ; but since by (3)All abd is C, this may be

reducedtoAllbdisa.

530 IV.Thefollowingexample isalsofromJevons,PrinciplesofScience,2nd edition, p. 127 (Problem viii). In hisStudies, p. 256, he speaks of thesolutionasunknown.Afairlysimplesolutionmay,however,beobtainedbytheapplicationofthegeneralruleformulatedinthissection.

ThegivenalternantsareABCDE,ABCDe,ABCde,ABcde,AbCDE,AbcdE,Abcde,aBCDe,aBCde,aBcDe,abCDe,abCdE,abcDe,abcdE.

Bythereductionofdualsthesealternantsmaybewritten:ABCeorABcdeorAbcdorACDEoraBCdeorabdEoraDe.

Therefore, by contraposition,Whatever is not either ABCe or ABcde or


AbcdorabdEoraDeisACDEoraBCde.

But it will be found that, by the application of the ordinary rule forobtainingthecontradictoryofagiventerm,WhateverisnoteitherABCeorABcdeorAbcdorabdEoraDe isequivalenttoWhateverisAbCoradeorBEorAcDorDE.

Henceourpropositionisresolvableintothefollowing:

(i) AllAbCisDE ;(ii) AlladeisBC ;(iii) AllBEisACD ;(iv) AcDisnon-existent ; (v) AllDEisAC.

Butby(v)AllBEisACord ;therefore,(iii)maybereducedtoAllBEisD.Againby(iv),AllDEisaorC ;therefore,(v)maybereducedtoAllDEisA.

Hencewehavethefollowingasourfinalsolution:—

(1) AllAbCisDE ;(2) AlladeisBC ;(3) AllBEisD ;(4) AllcDisa ;(5) AllDEisA.

536.AnotherMethodofSolutionoftheInverseProblem.—Anothermethodofsolvingtheinverseproblem,suggestedtomebyDrVenn,istowritedowntheoriginalcomplexpropositioninthenegativeform,i.e.,toobvertit,beforeresolving it. It has been already shewn that a negative propositionwith analternative predicate may be immediately broken up into a set of simplerpropositions.

Insomecases,especiallywherethenumberofdestroyedcombinationsascomparedwiththosethataresavedissmallthisplanisofeasierapplicationthanthatgivenintheprecedingsection.

531 To illustrate this method we may take two or three of the examplesalreadydiscussed.


I.EverythingisABCorAbcoraBCorabC ;therefore,byobversion,NothingisAbCoracorBc ;andthispropositionisatonceresolvableinto

⎰ AllAbisc,⎱ AllcisAb.528

528 The equivalence between this and our former solution is immediatelyobvious.EquationallyitwouldbewrittenAb=c.

II.Everything isACeoraBCeoraBcdEorabCeorabcE ; therefore, byobversion,NothingisAcorBcDorCEorce.

Thispropositionmaybesuccessivelyresolvedasfollows:

⎧ NocisAore,⎨ NoEisC,⎩ NoBDisc.⎧ AllcisaE,⎨ AllEisc,⎩ AllBDisC.

III.EverythingisABCDorABCdorABcdorAbCDorAbcDoraBCDoraBcDoraBcdorabCd ;therefore,byobversion,NothingisABcDorAbdoraBCdorabcorabD ; and thispropositionmaybe successively resolvedasfollows:

⎧⎨⎩

NoABcisD ;NobdisA ;NoaBCisd ;NoabiscorD.

⎧⎨⎩

AllABcisd ;Allbdisa ;AllaBCisD ;AllabisCd.

Itisratherinterestingtofindthatnotwithstandingtheindeterminatenessoftheproblemweobtainbyindependentmethodsthesameresultineachoftheabovecases.


537.AThirdMethodofSolutionoftheInverseProblem.—Thefollowingisa third independentmethod of solution of the inverse problem, and it is insomecaseseasierofapplicationthaneitherofthetwoprecedingmethods.

532Anypropositionoftheform

Everythingis……

mayberesolvedintothetwopropositions:

⎰ AllAis……⎱ Allais……

which taken together are equivalent to it; similarly All A is …… may beresolved into the twoAll AB is……,All Ab is…… and it is clear that bytakingpairsofcontradictoriesinthiswaywemayresolveanygivencomplexproposition into a set of propositions containing no alternative terms.Redundanciesmustofcourseasbeforebeasfaraspossibleavoided.

Toillustratethismethodwemayagaintakethefirstthreeexamplesgiveninsection535.

I.EverythingisABCorAbcoraBCorabCmayberesolvedsuccessivelyasfollows:

⎰ AllCisABoraBorab ;⎱ AllcisAb.⎰ AllbCisa ;529⎱ AllcisAb.

529 TakingBCasoursubjectwehaveAllBCisAora,andsincethisisamerelyformalproposition,itmaybeomitted.

II.EverythingisACeoraBCeoraBcdEorabCeorabcEmayberesolvedsuccessivelyasfollows:

⎰ AllCisAeoraBeorabe ;⎱ AllcisaBdEorabE.⎧ AllCise ;⎨ AllcisaE ;⎩ AllcisBdorb.⎧ AllCise ;


⎨ AllcisaE ;⎩ AllBcisd.

III.EverythingisABCDorABCdorABcdorAbCDorAbcDoraBCDoraBcDoraBcdorabCdmayberesolvedsuccessivelyasfollows:

⎰ AllBisACDorACdorAcdoraCDoracDoracd ;⎱ AllbisACDorAcDoraCd.

⎰ AllBisACoraDorcd ;⎱ AllbisADoraCd. 533

⎧⎨⎩

AllBCisAoraD ;AllBcisaDord ;AllAbisD ;AllabisCd.

⎧⎨⎩

AllBCdisA ;AllABcisd ;AllAbisD ;AllabisCd.

Theabovesolutionsarepractically thesameas thoseobtained in the twoprecedingsections.

538.MrJohnson’sNotationfortheSolutionofLogicalProblems.—InhisarticlesontheLogicalCalculusMrJohnsonproposesanotationbytheaidofwhich the solution of inverse problems may be facilitated. It consists inrepresenting conjunctive combination by horizontal juxtaposition, andalternativecombinationbyverticaljuxtaposition.Abar—drawnhorizontally

orvertically—servesthepurposeofabracketwherenecessary.Thus,

representsABorCD; represents(AorC)and(BorD).These two

forms are of course not equivalent to each other.But if contradictories areplacedinapairofdiagonallyoppositecorners, then thecombination is the

sameinwhicheverwaywereadit.Thus, representsABoraC ;

represents(AorC)and(aorB).Buttheseareequivalenttoeachother;for(Aor C) and (a or B) is equivalent to AB or aC or BC, and—since BC by


development isABC or aBC—this is equivalent toAB or aC. Mr Johnsoncontinues as follows:—“By adopting the plan of placing successive letter-symbols in opposite corners we may solve the inverse problem withsurprisingease.Themethodofsolutioncloselyresembles the thirdof thoseadoptedbyDrKeynes,anditwasthisthatsuggestedmine.Iwill,therefore,illustratebytakingDrKeynes’sthreeexampleswhicharethefollowing:—

534Herethecolumnsordeterminantsmaybereadoff:—

(CorAb)and(Boraorc)=(Ifc,thenAb)and(IfAC,thenB).

Thisisread:(Ifc,thenaE)and(IfBD,thenC)and(IfC,thene).

That is: (Ifab, thenCd)and (Ifbd, thena)and (IfABD, thenC)and (IfBCd,thenA). In this lastproblem,wefirstplaceBandbopposite; thenfortheBalternants,weplaceCandcopposite,andforthebalternantsAanda.Togetthesimplestresult,weshouldaimatdividingthecolumnsintoasequaldivisionsaspossible.

Thenotationthusexplainedenablesus tosolveanyproblemsinasimple


manner.Theexpressioninitsfinalformmaybereadequallywellincolumnsorinrows,i.e.,asadeterminativeorasanalternativesynthesis.Ofcourse,aprecisely similar process may be used, if we started with determinativelygivenormixeddata”(Mind,1892,p.351).

539.TheInverseProblemandSchröder’sLawofReciprocalEquivalences.—Theinverseproblemmayalsobesolved,thoughsomewhatlaboriously,bythe aid of the reciprocal relation between the laws of distribution given insection 428, this reciprocal relation depending upon the law that to everyequivalence there corresponds another equivalence in which conjunctivecombination is throughout substituted for alternative combination and viceversâ.Thus,bythefirst lawofdistribution,(AorB)and (CorD)=ACorADorBCorBD,andhencefollowsthecorrespondingequivalenceABorCD=(AorC)and(AorD)and(BorC)and(BorD). In thiswayany inverseproblemmay be practically resolved into themore 535 familiar problem ofconjunctivelycombiningaseriesofalternativeterms.530

530 Itwillbeobservedthattheinverseprobleminvolvesthetransformationofalogicalexpressionconsistingofaseriesofalternantsintoanequivalentexpressionconsistingof a seriesofdeterminants.Schröder’sLawofReciprocity shews thattheprocessrequiredforthistransformationispracticallythesameasthatbywhichan expression consisting of a series of determinants is transformed into anequivalentexpressionconsistingofaseriesofalternants.

Taking as an example the first problem given in section 535, we mayproceedasfollows:(AorBorC)and(Aorborc)and(aorBorC)and(aorborC)=(AorBcorbC)and(aorC)=ACoraBcorbC.Therefore,wehavethecorrespondingequivalenceABCorAbcoraBCorabC= (AorC)and(aorBorc)and(borC).HencethepropositionEverythingisABCorAbcoraBCorabCmayberesolvedintothethreepropositions,EverythingisAorC,EverythingisaorBorc,EverythingisborC ;andwehaveforoursolution of the inverse problem: All c is A, All bC is a, All c is b ; or,combiningthefirstandlastofthesepropositions,AllcisAb,AllbCisa.

Similarly,thesecondprobleminsection535maybesolvedasfollows:—(AorCore)(aorBorCore)(aorBorcordorE)(aorborCore)(aorborcorE)=aCorbCdorCEorce.HencethecorrespondingequivalenceACeoraBCeoraBcdEorabCeorabcE=(aorC)(borCord)(CorE)(c


ore);andwehaveforoursolutionoftheinverseproblem,AllAisC,AllBDis C, All c is E, All C is e ; or, combining the first and third of thesepropositions,AllcisaE,AllBDisC,AllCise.

EXERCISES.

540.Findpropositionsthatleaveonlythefollowingcombinations,ABCD,ABcD,AbCd,aBCd,abcd.[Jevons,Studies,p.254.]


Jevonsgivesthisasthemostdifficultofhisseriesofinverseproblemsinvolvingfourterms.Itmaybesolvedasfollows:— Everything is ABCD or ABcD or AbCd or aBCd or abcd ; therefore, by contraposition and thereductionofdualterms,WhateverisnoteitherAbCdoraBCdisABDorabcd. 536Therefore,WhateverisABoraborcorDisABDorabcd ;andthisisresolvableintothefourfollowingpropositions:

⎧⎨⎩

AllABisD, (1)Allabiscd, (2)AllcisABDorabd, (3)AllDisAB. (4)

Sinceby(4)AllDisAB,andby(2)Allabisd,(3)maybereducedtoAllcisDorab,andthereforetoAllcdisab.Also,by(4)Allabisd,andhence(2)maybereducedtoAllabisc.

Oursetofpropositionsmaythereforebeexpressedasfollows:—

⎧⎨⎩

AllABisD,Allabisc,Allcdisab,AllDisAB.531

531 Restoring the second of these propositions to the formAll ab is cd, andwriting the propositions equationally, the solution may be expressed in a stillsimplerform,namely,AB=D,ab=cd.

541. Resolve the proposition Everything is ABCDeF or ABcDEf orAbCDEF or AbCDeF or AbcDeF or aBCDEf or aBcDEf or abCDeF orabCdeF or abcDef or abcdef into a conjunction of relatively simplepropositions.

[Jevons,PrinciplesofScience,2nded.,p.127(Problemx.)]Thefollowingisasolution:—

(1)AllAisD ;(2)AllABCise ;(3)AllaFisbCe ;(4)AllBfisDE ;(5)Allbfisace ;(6)AllcFisbe.

ThisissomewhatlesscomplexthanthesolutionbyDrJohnHopkinsongiveninJevons,StudiesinDeductiveLogic,p.256,namely:—

(i) Alldisab ;(ii) AllbisAForae ;(iii) AllAfisBcDE ;(iv) AllEisBforAbCDF ;


(v) AllBeisACDF ;(vi) Allabcisef ;(vii) Allabefisc.

537542.Howmanyandwhatnon-disjunctivepropositionsareequivalenttothe statement that “What is eitherAb orbC isCd or cD, and vice versâ”?[Jevons,Studies,p.246.]Thegivenstatementisatonceresolvableintothefourfollowingpropositions:

⎧⎨⎩

AllAbisCdorcD, (i)AllbCisCdorcD, (ii)AllCdisAborbC, (iii)AllcDisAborbC. (iv)

(i)mayberesolvedinto⎰ AllAbcisD, (v)⎱ AllAbDisc. (vi)

But (vi) is inferable from (ii); and observing some other obvious simplifications we obtainimmediatelythefollowingsolution:

(1)AllAbcisD ;(2)AllbCisd ;(3)AllCdisb ;(4)AllcDisAb.

543.Shewtheequivalencebetweenthetwosetsofpropositionsgiveninsection541.[K.]

544. Find which of the following propositions may be omitted without affecting the informationgivenbythepropositionsasawhole:AllAbiscDE ;AllAcisbDE ;AllAdisBCe ;AllAeisBCd ;NoaEisBorC ;NoBisc ;AllBdisACe ;NobDisCore ;NobEisAdorC ;AllCisB ;AllCdisABe ;AllcDisbE ;AllcEisAbDorab ;AlldeisABCorabc.[K.]

545. Resolve each of the following complex propositions into a conjunction of propositions notcontaininganyalternativecombinationofterms:

(1)EverythingisABCDorAbCdoraBcDorabcd ; (2)EverythingisAbCDorAbCdorAbcdoraBcdorabCDorabCdorabcd ; (3)EverythingisAbcDEoraBCdoraBCEoraBcdoraBdeorabCeorabceorabDeorabdeorBcdEorbCDe ; (4)Everything isABCEorABcdorABcEorABdeorAbcdorabCEorabcEorabdEorabdeorBCde ; (5)EverythingisABCDEorABCdEorABcDEorABcDeorABcdeorAbCdEorAbcdeoraBCDEoraBCdeorabCDEorabcDe ; 538(6)EverythingisABDeorABDForAcDeorAceforaBDeoraBDForabCDorabCdorabcDorabcdoraCDEoraCDeoraCdEoraCdeoracDeoraDEForaDEforaDeForaDeforBcDForbceForbcef ; (7)EverythingisAbdEorAbeforAbForAcdeforaBDForabCForaCdEoradeorbCDeorbCdforbDEF ; (8)EverythingisABCEforAbeoraBCdforaBcdEoraBcdeForabeforbceF.[K.]


546.Expressthefollowingpropositioninassmallanumberasyoucanofpropositionsinwhichnoalternative combination of terms occurs: Everything is ABCDe or ABCdE or ABcDe or AbCdE orAbCdeoraBCdEoraBcDEoraBcdeoraBcdEorabCdeorabCdE.[J.]

547.Solvethefourthproblemgiveninsection535,(α)bythemethoddescribedinsection536,(β)bythatdescribedinsection537.[K.]

548.Solvetheproblemgiveninsection540andalsothefourthproblemgiveninsection535byaidofthenotationdescribedinsection538.[K.]

549.Solvethethirdandfourthproblemsgiveninsection535bythemethoddescribedinsection539.[K.]

550.Shewthatanyuniversalcomplexpropositionmayberesolvedintoasetofpropositionsinwhichnoconjunctivecombinationoftermsoccurs.[K.]


INDEX.


Abscissioinfiniti,316.

AbsoluteName,63.

Absorption,Lawsof,475.

AbstractNames,16–19;canthedistinctionbetweengeneralsandsingularsbeappliedtothem,19–21.

AccidentalProposition,49.

AcquiredPerceptions,414.

AddedDeterminants,ImmediateInferenceby,148,9.

Addition,signof,insymboliclogic,468n.

Aequipollence,133n.

AffirmativeProposition,92.

Aldrich,109n. ;322n.

All,asasignofquantity,97–100.

Alternant,277;468;479.

AlternativeCombinationofTerms,468,9;ofPropositions,479.

Alternative Judgments and Propositions, 84, 275; two types, 276, 7; their import, 277–82; theirreductiontotheformofconditionalsorhypotheticals,282–4.

AlternativeSyllogisms,359–62.

AlternativeTerms,276;468.

AmbiguousMiddle,288.

AmbiguousTerm,Fallacyof,288.

AmpliativeProposition,49.

AnalyticPropositions,50–2;natureoftheanalysisinvolvedinthem,53–6.

And,itslogicalsignification,469.

Antecedent,250.

Antilogism,332;334;335;336n.

ApodeicticJudgmentsandPropositions,86–91;98–100.SeealsoModalPropositions.

Argumentàfortiori,384–6;467.

AristoteliandoctrineofModals,85,6.

AristotelianSorites,370–3.

Aristotle,130;329;367;396.

AssertoricJudgmentsandPropositions,86–91;schemeofassertoricandmodalpropositions,282.

AttributiveTerm,180.


Bailey,S.,337n. ;427n.

Bain,A.,ongeneralandsingularnames,12,14n. ;onconnotation,26n. ;onverbalpropositions,50n. ;ondefinition,55;126n. ;onconversion,131n.,onobversion,133n. ;onsyllogismswithtwosingularpremisses,298,9;onthemixedhypotheticalsyllogism,354,5;426n. ;442;457n.

Barbara,Celarent,&c.,319–22.

Baynes,T.S.,96;129n. ;onthequantificationofthepredicate,196,199.

Benecke,E.C.,25;44.

Bentham,Jeremy,445.

Boethius,134n.,134n.2.

Boole,LawsofThought,192;210n. ;299n. ;453;456;470n. ;473n. ;475n. ;476n. ;506;508;510n. ;512,13;515,16.

Bosanquet,B.,onthepartsoflogic,8;onlogicalmeaningandpsychicalidea,28;onlanguage,29;onpartsinintension,36n. ;ontheconnotationofpropernames,45n. ;46n. ;onthereferenceto time in judgments,77; his classification of judgments, 80; on the particular proposition,101; on the nature of significant denial, 122–4; 259 n. ; on the reciprocal character ofconditionalsandhypotheticals,270–3;ontheimportofdisjunctives,280,283;onconversion,422;451n.

Bowen,F.,133n. ;201;328.

Bradley,F.H.,53,4;211n. ;451;462n.

CategoricalPropositions,82;seealsoPropositions.

CategoricalSyllogism,seeSyllogism.

ChangeofRelation,Inferenceby,148;260,1.

Clarke,R.F.,102n. ;106n. ;443;445.

Classmodeofinterpretingpropositions,181–4.540

Classification,447.

Co-division,443.

CollectiveNames,14,15.

Collectiveuseofnames,15,16;ofthewordall,97,8.

CombinationofComplexPropositions,498–501

Commutativeness,Lawof,470n.

ComplementaryNames,62.

ComplementaryPropositions,132;143,4;161.

ComplexConception,ImmediateInferenceby,149.

ComplexConstructiveDilemma,364.


ComplexDestructiveDilemma,364.

Complex Propositions, 478; their opposition, 478; their simplification, 481–3; resolution intoequivalentcompoundpropositions,483–5;omissionofterms,485;introductionofterms,485,6; interpretationofanomalousforms,486,7; theirobversion,488,9; their conversion,489,90; their contraposition, 490–3; their combination, 498–502; inferences from theircombination,504–8;eliminationfromcomplexpropositions,508–12.

Complex Terms, 468–477; order of their combination, 469, 70; their opposition, 470–2; theirsimplification,472–6;summaryofformalequivalences,476.

Composition,Fallacyof,16n.

CompoundJudgmentsandPropositions,82–4;theirmodality,90,1;478–80; theiropposition,480;theirformalequivalences,480,1.

Comprehension,26,7;30;31–3;lawofvariationwithexemplification,37;relationtodenotation,38,9;readingofpropositionsincomprehension,187,8.

Concept,notthelogicalunit,9.

Concepts,empirical,metaphysical,andlogical,27,8.

Conceptsandnames,10.

ConceptualisttreatmentofLogic,4,5;10,11;66–8.

ConcreteNames,16–19.

ConditionalPropositions,distinguishedfromhypotheticalpropositions,249–52;theirimport,252–6;theirrelationtocategoricals,253–6;theiropposition,256–8;immediateinferencesfromthem,259–61;theirallegedreciprocalcharacter,270–3.

ConditionalSyllogisms,348–51.

Conjunctivecombinationofterms,468;ofpropositions,478,9.

ConjunctiveJudgmentsandPropositions,83.

ConjunctiveTerms,468.

Connotation,24–7;distinguishedfrometymology,28;howfarvariable,28,9;31–3;lawofvariationwithdenotation,37.

Connotativemodeofinterpretingpropositions,184–6.

ConnotativeNames,40–7.

Consequent,250.

ConstructiveDilemma,363,4.

ConstructiveHypotheticalSyllogism,352.

ContingentJudgments,85.

ContinuousQuestioning,Fallacyof,372n.

Contra-complementaryPropositions,132;143,4;161.

Contradiction,Lawof,147;454–8;474.


Contradictioninterms,53n.

Contradictory Opposition, 109; 111–14; 119; 121; how affected by the existential import ofpropositions,227–32.

ContradictoryPropositions,seeContradictoryOpposition.

ContradictoryTerms,61,2;470,1.

ContrapositionofPropositions,134–7;attemptstoreducecontrapositiontosyllogisticform,151–3;illustratedbyEuler’sdiagrams,161;howaffectedby theexistential importofpropositions,223–7; of conditionals, 259, 60; of hypotheticals, 268–70; is contraposition a process ofinference,422,3;ofcomplexpropositions,490–3.

Contrapositionperaccidens,136.

Contrapositive,seeContraposition.

ContraryOpposition,119;114,5;118;howaffectedbytheexistential importofpropositions,227–32.

ContraryPropositions,seeContraryOpposition.

ContraryTerms,62,3.

Contraversion,133n. ;134n.

ConventionalIntension,23;26,7.

Converse,127.

ConverseRelation,ImmediateInferenceby,149–51.

ConversionbyContraposition,seeContraposition.

ConversionbyLimitation,129.

ConversionbyNegation,134n.

Conversion of Propositions, 126–130; legitimacy of the process, 130–2; attempts to reduceconversiontosyllogisticform,152;illustratedbyEuler’sdiagrams,160,1;how541affectedby the existential import of propositions, 223–7; of conditionals, 259, 60; of hypotheticals,268,9;isconversionaprocessofinference,422,3;nottobebasedexclusivelyonthethreelawsofthought,465,6;ofcomplexpropositions,489,90.

Conversionperaccidens,128,9.

Conversiopuraetimpura,129n.

ConversioSyllogismi,322.

Convertend,127.

ConvertibleCopula,388.

Copula,93.

CorrelativeName,63.

CriterionofConsistency,Jevons’s,217n. ;219;232,3.


Deductioadimpossibile,oradabsurdum,319.

Definitionbytype,34.

De Morgan, A., use of the terms contrary and contradictory, 62 n. ; 101 n. ; 104; 104 n. ; onconversion,126n. ;133n. ;oncontraposition,136;153n. ;onthepropositionω,206,7;210n. ;217n. ;on theexistential importofpropositions,219,232;on thesyllogistic rules,290,292;314;on themnemonicverses,319;on thenumericallydefinite syllogism,377; on theargumentàfortiori,385,6;onthelogicofrelatives,387,8;onimmediateinferencesandthelawsofthought,466;495.

Denial,Natureof,119–24.

Denotation,29–31;31–3;lawofvariationwithconnotation,37;relationtocomprehension,38,9.

DestructiveDilemma,363,4.

DestructiveHypotheticalSyllogism,352.

Determinant,468;479.

Determination,468.

DevelopmentofTerms,474.

Diagrams, their use in Logic, 156, 7; Euler’s, 157–62; Lambert’s, 163–6; Venn’s, 166–8;development of Euler’s diagrams, 170–4; of Lambert’s diagrams, 174–6; application ofdiagramstosyllogisticreasonings,341–6.

Dichotomy,seeDivisionbyDichotomy.

Dictaforthesecond,third,andfourthfigures,337,8.

Dictumdediverso,337n.

Dictumdeexcepto,338n.

Dictumdeexemplo,337n.,338n.

Dictumdeomnietnulloandtheordinaryrulesofthesyllogism,301,2.

Dictumdereciproco,338.

Dilemma,363–6.

Directreduction,318;ofBarocoandBocardo,323,4.

DisjunctiveJudgmentsandPropositions,83,4;275–84.

DisjunctiveSyllogisms,359–62.

DisjunctiveTerms,seeAlternativeTerms.

Distinction,443.

Distribution,Lawsof,472,3.

Distributionoftermsinaproposition,95,6;illustratedbyEuler’sdiagrams,159,60.

Distributiveuseofnames,15,16;ofthewordall,97,8.

Division,seeLogicalDivision,MetaphysicalDivision,DivisionbyDichotomy,&c.

Division by Dichotomy, 445; all valid division reducible to dichotomy, 445, 6; is division by


dichotomyaformalprocess,447–9.

Division,Fallacyof,16n.

Dixon,E.T.,237n.

DoubleNegation,Principleof,459.

Duality,Lawof,460.

DualityofFormalEquivalences,472.

DualTerms,475.

Eduction,127n.

ἔκθεσις,130n. ;323n.

Elimination,involvedinsyllogisticreasoning,300;theproblemofeliminationinlogic,508,9;rulesforelimination,509–12.

EmpiricalConcepts,27,8.

EmpiricallyUniversalPropositions,99.

Enthymeme,367,8.

Enumeration,441.

EnumerativeUniversalPropositions,98.

Epicheirema,369.

Episyllogism,369.

Equality,Symbolof,189–91.

EquationsinLogic,189–91;theirtypes,191–4;.expressionofpropositionsasequations,194.

EquipollentPropositions,117.

EquivalentPropositions,117;tablesofequivalentpropositions,141;146;208;481.

EquivalentTerms,Tableof,476.

EquivocalTerm,65.

EssentialProposition,50.

EtymologyandConnotation,28.

Euclid,136;420;430.

Euler’sdiagrams,five-foldscheme,157–62;seven-foldscheme,170–4;theirapplicationtothe542quantificationofthepredicate,200–4;tosyllogisticreasonings,288,341–4.

Eversion,127n.

ExcludedMiddle,Lawof,61n. ;147;458–63;474.

Exclusion,Lawof,475.

ExclusiveFigure,316.


ExclusiveProposition,205.

Exemplification,31–5;lawofvariationwithcomprehension,37.

ExemplicativeName,41.

ExistenceandtheUniverseofDiscourse,210–13.

Existential Import of Propositions, nature of the questions involved, 214; how far formal logicconcerned with them, 215–17; various suppositions, 218–20; bearing on immediateinferences, 223–7; on the doctrine of opposition, 227–32; existential import of thepropositions included in the traditional schedule,234–44;ofmodalpropositions,244, 5; ofconditionalpropositions,255,6;probleminconnexionwithhypotheticals,256,7;bearingoftheexistentialimportofpropositionsuponthevalidityofsyllogisticreasonings,390–4.


ExistentialPropositions,218;theirrelationtothetraditionalformsofproposition,221–3.

ExplicativeProposition,50.

ExponibleProposition,104n.

Extension of Names and Concepts, 22; distinguished from denotation, 29, 30; how related tointension,31–40;propositionsinextensionandintension,177–88.

ExtensiveDefinition,31–5.

ExtensivelyVerbalProposition,51n.

Few,asasignofquantity,103,4.

Figures of the Syllogism, 300; their special rules, 309–13; their peculiarities and uses, 315–17;equivalence of the special rules of the first three figures, 335; schemes of validmoods infigures 1, 2, and 3, 336–8; dicta for figures 2, 3, and 4, 337, 8; figures of the conditionalsyllogism, 349, 50; of the hypothetical syllogism, 349, 50; of the hypothetico-categoricalsyllogism,352,3.

Folk-lore,Universeof,213n.

FormofaProposition,3;92;150,1.

FormandMatter,2,3.

FormalContradictories,62n.

FormalLogic,1–3.

FormalObversion,133n.

FormalPropositions,52,3.

FourthFigure,328,9;itsmoodsregardedasindirectmoodsofthefirstfigure,329–31;moodsofthefourthfigure,334,5;dictum,338.

Fowler,T.,133n. ;205;325,6;349;365.

FundamentalSyllogism,314n.

Fundamentumdivisionis,441.

Fundamentumrelationis,64.

GalenianFigure,328.

GeneralNames,11–13.

GeneralPropositions,103.

GoclenianSorites,370–3.

GrammaticalAnalysisofaProposition,92n.

GreekMythology,Universeof,213n.

Green,T.H.,42n. ;54n.


Groundorreasonofabelief,distinguishedfromcauseofabelief,414.

Hamilton,SirW.,onsingularpropositions,102,3;104;105;hisschemeofdiagrams,156n. ;hisuseofEuler’sdiagrams,159;onjudgmentsinextensionandintension,184n. ;hisdoctrineofthequantification of the predicate, 195 ff.; his fundamental postulate of logic, 195, 6; on theinterpretationofsome,200,1;321n. ; 326n. ; on the doctrine of reduction, 327n. ; on themixedhypotheticalsyllogism,354–6;368n. ;371n. ; on figure of sorites, 373n. ; on ultra-totaldistributionofthemiddleterm,377;ontheunfiguredsyllogism,378n. ;396n. ;onthelawofcontradiction,455;462;basesformalinferencesonthethreelawsofthought,464,5.

Hamiltonianschemeofpropositions,79;195ff.

Hobhouse,L.T.,69n.

HypotheticalDilemma,363n.

Hypothetical Judgments andPropositions,83,4;distinguished fromconditionalpropositions,249–52; their import,261–4; theiropposition,264–8; immediate inferences from them, 268–70;theirrelationtocategoricals,270;theirallegedreciprocalcharacter,270–3.

HypotheticalSyllogisms,348–57.

Hypothetico-CategoricalSyllogism,348,9;352–7.

Identity,Lawof,147;451–4.

Illicit major and illicit minor, 289; involve indirectly undistributed 543 middle, 298; apparentexceptionstotheruleagainstillicitmajor,298.

ImmediateInferences,126–53;howaffectedby theexistential importofpropositions,223–7;fromconditionalpropositions,259–61;fromhypotheticalpropositions,268–70;cantheybebasedexclusivelyonthethreelawsofthought,464–6;fromcomplexpropositions,488–494.

ImperfectFigures,330.

ImplicationandMeaning,71,2;177;178n. ;421–3.

ImportofPropositions,natureoftheenquiry,70–4.

Inclusion,Lawof,475.

IndefiniteName,59–61.

IndefiniteProposition,105.

IndependentPropositions,118.

IndesignateProposition,105.

IndirectMoods,329–31.

IndirectReduction,318,9;331–7.

IndividualName,11.

IndividualProposition,102.

Inequality,Symbolsof,193.


Inference,natureof,413,4;paradoxof,414,5;conclusioninwhatsensedifferentfrompremisses,415–20;limitingcases,422,3;isconversionaprocessofinference,422,3;iscontrapositionaprocessofinference,422,3;issyllogismaprocessofinference,423–30.

Infinitation,133n.

InfiniteName,59–61.

InfiniteProposition,106,7.

Integration,201n.

IntensionofNamesandConcepts,22;conventional,subjective,andobjective,23–7;howrelatedtoextension,31–40;propositionsinintensionandextension,177–88.

IntensiveDefinition,32–5.

IntensivelyVerbalProposition,51n.

Inverse,139.

InverseProblem,525–535.

Inversion of Propositions, 137–9; validity of the process, 139, 40; illustrated byEuler’s diagrams,161;howaffectedbytheexistentialimportofpropositions,223–7;ofhypotheticals,269.

Invertend,139.

Jevons,W.S.,12n. ;19n. ;20n. ;hisuseofthetermconnotation,26;37n. ;regardspropernamesasconnotative, 41–3; on relative names, 63, 4; on contradictory opposition, 111–14; onconversion,130;133n. ;oncontraposition,136n. ;139;152n. ;hisuseofEuler’sdiagrams,159;ontypesoflogicalequations,191,2;ontheinterpretationofsome,202;205n. ;210n. ;onquestionsaboutexistenceinlogic,217n. ;hiscriterionofconsistency,217n.,219,232,3;220n. ;ontheimportofdisjunctives,279;ontheorderofpremissesinasyllogism,287;onnegativepremisses,295;ontheordinarysyllogisticconclusion,300;349;365;366;416; ondivisionbydichotomy,445,6;449;hisprincipleofthesubstitutionofsimilars,453,4;onthelawofduality,460;470n. ;472n. ;473;475;495;onBoole’sSystemofLogic,506n. ;507n. ;ontheinverseproblem,525,6,529,30.

Johnson,W.E.,31n. ;ontheimportofpropositions,70n. ;ontheformulationofpropositions,72;onmultiplequantification,106n. ;132n. ;144n. ;onthepropositionω,206n. ;onthedistinctionbetweenconditionalandhypotheticalpropositions,249n. ;265n. ;293n. ;onthespecialrulesofthesyllogisticfigures,311n. ;ondictaforthethirdandfourthfigures,338;388;469n. ;onthe analysis of ordinary categorical propositions, 479, 80; on the synthesis of propositions,481n. ;hisnotationforthesolutionofinverseproblems,533,4.

Jones,MissE.E.C.,126n. ;134n. ;148n. ;151;190n. ;on theexistential importofpropositions,244n. ;onconditionalpropositions,256n. ;260;onhypotheticalpropositions,264n. ;ontheuseofthetermalternative,275;onthenatureof inference,416n. ;418n. ; on division andclassification,447.

Judgment,thelogicalunit,8,9.

Judgments, as related to propositions, 66–8; their essential characteristics, 70; their objectivereference,74–6;theiruniversality,76,7;theirreferencetotime,76,7;theirnecessity,77,8;


theirclassification,79–81;theirdivisionaccordingtorelation,82;intosimpleandcompound,82–4;theirmodality,84–91;theirquantityandquality,91,2.SeealsoPropositions.

Judgmentsofactuality,88.

Judgmentsofnecessity,88.

Judgmentsofpossibility,88.

Kant,hisclassificationofjudgments,81;hisdoctrineofmodality,86;54491,2;104n. ;106;onthefiguresofthesyllogism,327n. ;onthemixedhypotheticalsyllogism,354,5.

Karslake,329n. ;368n.

LaddFranklin,Mrs,onnegativeterms,60n. ;142n. ;147n. ;ontheimportofpropositions,179n. ;on the existential import of propositions, 218 n. ; 231 n., 241 n., 242 n. ; 323 n. ; on theantilogism,332;510n.

Lambert, J. H., his diagrammatic scheme, 163–6, 174–6; on the uses of the different syllogisticfigures, 316, 7; 326 n. ; on dicta for the different figures, 337 n., 338; application of hisdiagrammaticschemetosyllogisticreasonings,344,5.

Languageastheinstrumentofthought,3–5.

LawsofThought,147;450,1;lawofidentity,451–4;lawofcontradiction,454–8;lawofexcludedmiddle,458–63; are the lawsof thought also lawsof things,463, 4; theirmutual relations,464;howfartheyestablishimmediateinferences,464–6;mediateinferences,466,7.

LewisCarroll,GameofLogic,219n.

Liar,Sophismofthe,457,8.

LimitativeProposition,106.

LimitedIdentities,192.

Lindsay,T.M.,201n.

Logic, definition of, 1; formal andmaterial, 1–3; its connexionwith language, 3–5; its relation topsychology,5,6;itsutility,6,7;itsabstractcharacter,68–70.

LogicalDivision,441,2;itsrules,443–5;allvaliddivisionreducibletodichotomy,445,6;placeofthedoctrineofdivisioninlogic,446–9;divisionandclassification,447.

LogicalConcepts,27,8.

LogicalDoctrine,itsthreeparts,8,9.

Lotze,H.,onnegativeterms,59n.,61n. ;ongeneralanduniversaljudgments,99n. ;126n. ;129n. ;onnegativepremisses,296n. ;criticismofJevons,300;424n. ;425n.

McColl,H.,263n.

Mackenzie,J.S.,322n.

MajorPremiss,287;Mill’sviewofitsfunction,429.


MajorTerm,285,6.

Mansel,H.L., 51n. ; on opposition, 109n. ; 115; on conversionper accidens, 129n. ; 130n. ; oncontraposition,134n. ;onmaterialconsequence,150;152n. ; on the importofdisjunctives,279n. ;319n. ;onindirectmoods,330n. ;337n. ;357n. ;onthedilemma,365;367;368n. ;ontheargumentàfortiori,385n.,386;424n. ;443;ontheplaceofdivisioninlogic,446–8;onthelawofidentity,454;basessyllogisticinferencesonthelawsofthought,466,7.

MaterialConsequence,150;386.

MaterialContradictories,62n.

MaterialContrariety,115n.

MaterialObversion,133n.

MatterofaProposition,3;92;150,1.

MeaningandImplication,71,2;177;178n. ;421–3.

MediateInference,151;andthelawsofthought,466,7.

Membradividentia,441.

MetaphysicalConcepts,27,8.

MetaphysicalDivision,412,3.

MetaphysicalUniversality,105n.

Metathesispraemissarum,321.

MethodsofAbbreviation,Boole’s,475n. ;476n.

MiddleTerm,285,6;itsultra-totaldistribution,376–8.

Mill,J.S.,onnames,9n. ;20n. ; onconnotation,24,5;onconnotativenames,40; regards propernamesasnon-connotative,41,2;hisdistinctionbetweenrealandverbalpropositions,54n. ;on negative names, 61 n. ; his classification of propositions, 80, 1; on the import ofpropositions,182;186n. ;onthequantificationofthepredicate,198;ontheexistentialimportof propostions, 219; 243n. ; on figure of sorites, 373, 4; 378n. ; 387; 414; on immediateinferences,419; his doctrine that in every syllogism there is apetitioprincipii, 424–30; ondivisionandclassification,446;onthelawofidentity,452;466;onthelawofcontradiction,455,6;onthelawofexcludedmiddle,461–3.

MinorPremiss,287.

MinorTerm,285,6.

Minto,W.,134n.

MixedHypotheticalSyllogism,348,9;352–7.

Mnemonicsforthevalidmoodsofthesyllogismandtheirredactiontothefirstfigure,319–22; forthedirectreductionofBarocoandBocardo,323,4;fortheindirectmoodsofthefirstfigure,329,30.545

ModalConsequence,ImmediateInferenceby,151.

ModalPropositions,90n. ;theiropposition,116,7;231,2;theirexistentialimport,244,5;258;266,7;distinctivesymbolsforthem,258;schemeofassertoricandmodalpropositions,282.See


alsoConditionalPropositionsandHypotheticalPropositions.

ModalityofJudgments,84–91.

Modusponendoponens,352n. ;362.

Modusponendotollens,361,2.

Modusponens,352.

Modustollendoponens,360;362.

Modustollendotollens,352n. ;362.

Modustollens,352;itsreductiontothemodusponens,354.

Monck,W.H.S.,30n. ;56n. ;207n. ;380n. ;448.


Moodsof theSyllogism,309;whatmoodsare legitimate ineachfigure,309–13; subalternmoods,313,14;strengthenedmoods,314,15;equivalenceofthemoodsofthefirstthreefigures,333,4;moodsoffigure4,334,5;schemeofvalidmoodsoffigure1,336;offigure2,336,7;offigure3,337,8;moodsoftheconditionalsyllogism,349,50;ofthehypotheticalsyllogism,349,50;ofthehypothetico-categoricalsyllogism,352,3;of thedisjunctivesyllogism,359–62.

MoralUniversality,105n.

Most,asasignofquantity,103,4;effectofitsrecognitionasasignofquantityontherulesofthesyllogism,376,7.

MultipleQuantification,105,6;265n.

Multiplication,signof,insymboliclogic,468n.

Musschenbroek,P.van,InstitutionesLogicae,322.

NamesandConcepts,10,11.

NecessaryJudgments,85–91.

NecessityofJudgments,77,8.

NegativePremisses,289;292,3;295–7.

NegativePropositions,92.

NegativeTerms,57–61;theireliminationfrompropositions,144–6.

NominalisttreatmentofLogic,4,5;10,11;66–8.

NumericallydefinitePropositions,104.

NumericallydefiniteSyllogism,377,8.

NumeericalMoodsoftheSyllogism,400–3.

ObjectivedistinctionsofModality,87–90.

ObjectiveExtension,30.

ObjectiveIntension,24;26,7.

ObjectivereferenceinJudgments,74–6.

Obverse,133.

ObversionofPropositions,133,4;howaffectedbytheexistentialimportofpropositions,223–7;ofhypotheticalpropositions,269;ofcomplexpropositions,488,9.

Obvertend,133.

OctagonofOpposition,144.

OppositionofComplexTerms,470–2.

Opposition of Propositions, 109–19; illustrated by Euler’s diagrams, 160; how affected by theexistential import of propositions, 227–31; of modal propositions, 231, 2; of conditional


propositions, 256–8; of hypothetical propositions, 264–8; of complex propositions, 478; ofcompoundpropositions,480.

Or,itslogicalsignification,469.

OstensiveReduction,318.

PartialIdentities,192.

ParticularPropositions,100–2;theirexistentialimport,238,9;245,6.

Partition,442.

Peirce,C.S.,336n.

PerfectFigure,329,30.

Permutation,133n.

PetitioPrincipiiandtheSyllogism,424–30.

PetrusHispanus,290,1;329n.

PhysicalDefinition,442.

PhysicalDivision,442,3.

PlurativePropositions,103.

Polylemma,363n.

Polysyllogism,368,9.

PopeJohnXXI,291;329n.

Porphyry,Treeof,35n. ;445.

PortRoyalLogic,105n. ;113n. ;297n. ;313n. ;337n. ;368n. ;432,3.

PositiveName,57.

PostulateofLogic,Hamilton’s,195,6.

PredicateofaProposition,92;howtobedistinguishedfromthesubject,96,7.

PredicativeInterpretationofPropositions,179-81.

Principiumdivisionis,441.

PrivativeConception,ImmediateInferenceby,133n.

ProblematicJudgments,86–91.SeealsoModalPropositions.

ProgressiveArgument,369.

ProperNames,13,14;15n. ;haveno546correspondingabstracts,17n. ;arenon-connotative,41–7;have subjective intension and comprehension, 42; may become connotative when used todesignateacertaintypeofperson,45.

Propositiosecundiadjacentis,93;tertiiadjacentis,93.

Propositionalforms,53;theirinterpretation,70–2.

Propositions,asrelatedtoJudgments,66–8;their interpretation,68,70–2;problemof their import,


70–4;theirformulation,72,3;theirclassification,79–81;theirdivisionaccordingtorelation,82;theirdivisionintosimpleandcompound,82–4;theirdivisionaccordingtomodality,84–91; their division according to quantity, 91, 2; their division according to quality, 92; thetraditionalscheme,92–95; their opposition, 109–19; theirmutual relations, 117–19, 142–4;connecting two terms, 132; connecting two terms and their contradictories, 141, 146; theirdiagrammatic representation, 156–76; in extension and in intension, 177–88; predicativemode of interpretation, 179–81; class mode of interpretation, 181–4; connotative mode ofinterpretation,184–6;subjectinterpretedinconnotationandpredicateindenotation,186,7;incomprehension,187,8;propositionsexpressedasequalitiesandinequalities,193,4;sixfoldscheduleincludingYandη,207–9;existential importofpropositions,234–45;direct importand implications of a proposition, 420–3. See also Complex Propositions, ConditionalPropositions,Judgments,&c.

Prosyllogism,369.

Psychology,itsrelationtoLogic,5,6.

Quality of Propositions, 92; 106; of conditional propositions, 257, 8; of hypothetical propositions,264,5.

QuantificationofthePredicate,195–209;itsapplicationtothesyllogism,378–84.

QuantityofPropositions,91,2;howaffectedbytheirquality,95n. ;ofconditionalpropositions,257,8;ofhypotheticalpropositions,265.

Quaternioterminorum,288.

RameanTree,445.

Ray,P.K.,356n.

Read,C.,62n. ;322n.

RealPropositions,49.

ReciprocalEquivalences,Schröder’sLawof,472;bearingofthislawontheinverseproblem,534.

Reductioadimpossibileorperimpossibile,319.

ReductionofDualTerms,474,5.

Reduction of Syllogisms, nature of the process, 318; direct and indirect reduction, 318, 9; directreductionofBarocoandBocardo,323,4; extensionof thedoctrineof reduction,324, 5; isreductionanessentialpartofthedoctrineofthesyllogism,325–8;indirectreduction,331–7;reduction of conditional and hypothetical syllogisms, 351, 2; of mixed hypotheticalsyllogisms,354.

RegressiveArgument,369.

Relation,Divisionofpropositionsaccordingto,82.

RelativeNames,63–5.

Relatives,Logicof,149–51;387,8.


Relativity,Lawof,456.

RemotivePropositions,84.

RepugnantTerms,63;471.

Robertson,G.C.,357.

Rogers,R.A.P.,294.

Ross,G.R.T.,280.

Schröder,DerOperationskreisdesLogikkalkuls,471n. ;472n. ;473;475;511;534.

SecondaryOpposition,115.

SecondaryQuantification,105;116.

Self-contradiction,457.

SextusEmpiricus,424;426.

Shyreswood,W.,329n.

Sigwart,onempirical,metaphysical,andlogicalconcepts,27,8;onthenamesofultimateelements,34;onapparentlytautologouspropositions,52n. ;onnegativenames,57–60;onthereferenceto time in judgments, 77; on compound judgments, 82 n., 83 n. ; on modality, 86, 7; onuniversaljudgments,99n. ;onnegativejudgments,120n. ;onthegroundsofdenial,121;128n. ; on contraposition, 136; 234 n. ; on hypotheticals, 264, 5; on figures 2 and 3 of thesyllogism,366n. ;349;onthevalueofthesyllogism,427n.,428n. ;onthelawsofthought,451;onthelawofidentity,461,2;onthelawofcontradiction,455;onthelawofexcludedmiddleandthelawoftwofoldnegation,459,60.

SimpleConstructiveDilemma,364.

SimpleContraposition,136.

SimpleConversion,128.547

SimpleDestructiveDilemma,364.

SimpleIdentities,191.

SimpleJudgmentsandPropositions,82;theirmodality,86–90.

SimpleTerm,468.

Simplicity,Lawof,473.

SingularNames,11–13;maybeconnotative,41,2.

SingularPropositions,102,3;theiropposition,115,16;aspremissesinasyllogism,298,9.

Solly,SyllabusofLogic,316n. ;395n. ;434,5.

Some,asasignofquantity,100,1;inthedoctrineofthequantificationofthepredicate,199–204.

Sophismapolyzeteseos,372n.

Sorites,370–6.

Spalding,W.,133n. ;201n. ;321n. ;349;387;445.


Spencer,H.,378n.

SquareofOpposition,110.

StrengthenedSyllogism,314,15.

StudiesinLogicbyMembersoftheJohnsHopkinsUniversity,323n. ;510n. ;517–20.

SubalternMoods,313,14.

SubalternOpposition,110;117,18;howaffectedbytheexistentialimportofpropositions,227–31.

SubalternantandSubalternate,Propositions,110.

Sub-complementaryPropositions,132;143,4;161.

SubcontraryOpposition,110;118;howaffectedbytheexistentialimportofpropositions,227–31.

Sub-division,443.

SubjectofaProposition,92;howtobedistinguishedfromthepredicate,96,7.

SubjectivedistinctionsofModality,86,7;90n.

SubjectiveExtension,30.

SubjectiveIntension,23,4;26,7;29.

SubstantialTerms,12n. ;15n.

Syllogism,285; its terms and propositions, 285–7; its rules as ordinarily stated, 287–9; corollariesfrom the rules, 289–91; restatement of the rules, 291; their dependence upon one another,291–3;statementoftheindependentrules,293,4;proofoftheruleofquality,294,5;apparentexceptionstotherules,295–8;syllogismswithtwosingularpremisses,298,9;istheordinarysyllogisticconclusionopentothechargeofincompleteness,300;figuresandmoods,309–17;reduction of syllogisms, 318–38; diagrammatic representation of syllogisms, 341–6;syllogismswithquantifiedpredicates,378–84;areallformalinferencesreducibletoordinarysyllogisticform,384–8;validityofsyllogisticreasoningshowfaraffectedbytheexistentialimport of propositions, 390–4; true conclusion obtainable from false premisses, 394–6;numericalmoods,400–3;syllogismsandimmediateinferences,423,4;syllogisticreasoningand the charge ofpetitio principii, 424–30. See alsoConditional Syllogism,Figures of theSyllogism,&c.

SymbolicLogic,189–94;468n.

SymbolsforPropositions,93,4.

SynonymousProposition,50.

SyntheticChainofReasoning,369.

SyntheticProposition,49.

Tarbell,F.B.,349n.

Tautology,Lawsof,473.

Terms,Logicof,11.

Tetralemma,363n.


Thomson,W.,195;201,2;203;206;315n. ;326;328,9;337n. ;344n. ;359n. ;379.

Timeofpredicationandtimeinpredication,77;451n.

Totumdivisum,441.

TraditionalSchemeofPropositions,79;92–5;234–44.

TransitiveCopula,388.

Transversion,127n. ;148n. ;260.

Trilemma,363n.

TwofoldNegation,Principleof,459.

Ueberweg, P., on opposition, 109n. ; on conversion, 126 n. ; 133 n. ; 136 n. ; 151 n. ; on Euler’sdiagrams, 162 n. ; on the existential import of propositions, 219 n. ; 255 n. ; on negativepremisses,297n. ;316;forminwhichhegivesthemnemonicverses,322n. ;onthereductionofBarocoandBocardo,323n. ;326;344n. ;349;352n. ;366n. ;369n. ;371n. ;424n. ;457n.

Ultra-totaldistributionofthemiddleterm,376–8.

UnconditionallyUniversalPropositions,99.

UndistributedMiddle, Fallacy of, 288; involves indirectly illicit process of major or minor, 293;apparentexceptiontotheruleagainstundistributedmiddle,297,8.

UnfiguredSyllogism,378n.

Unity,Lawof,473.548

UniversalPropositions,97–100;theirexistentialimport,235–8.

UniversalityofJudgments,76,7.

UniverseofAttributes,31n.

UniverseofDiscourse,29,30;75,6;210–13;226n. ;234,5.

UnivocalName,65.

Veitch,J.,54n. ;201n. ;203;207n.

Venn, J., 15n. ; 30n. ; 44n. ; on verbal disputes, 50n. ; on contradictory terms, 62n. ; 96 n. ; onHamilton’s geometric scheme, 156 n. ; on Euler’s diagrams, 159, 162 n. ; on Lambert’sdiagrams, 165 n. ; his own scheme of diagrams, 166–8; on the predicative mode ofinterpretingpropositions,179,180;185n. ;193n. ;200n. ;210n. ;ontheexistentialimportofpropositions,220n. ;ontheinferenceofparticularsfromuniversals,226n. ;235; 237; 238;application of his diagrammatic scheme to syllogistic reasonings, 345, 6; on the logic ofrelatives,387,8;424n. ;506;507n. ;530.


VerbalDispute,50n.

VerbalDivision,443.

VerbalPropositions,49–52.

Wallis,InstitutioLogicae,322n. ;330.

WeakenedConclusion,313,14.

WeakenedSyllogism,313,14.

WeakerPremiss,289n.

Welton,J.,182;183;243n. ;359n.

Whately,R.,297n. ;323,4;on thedoctrineof reduction,325;on thedilemma,365;holds thatallvalid reasoning is reducible to syllogistic form,387;hisdefinitionofpetitioprincipii, 425;433,4.

Wolf,A.,216n. ;221;225n. ;229n. ;231n.


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