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volume20,no.31 november2020

Stoic Logic and

Multiple Generality

Susanne Bobzien & Simon ShogryOxford University

© 2020 SusanneBobzien&SimonShogryThis work is licensed under a Creative Commons

Attribution-NonCommercial-NoDerivatives 3.0 License. <www.philosophersimprint.org/020031/>

Introduction

Multiplegeneralityistheexistenceinasentenceorpredicateofonequantifierinthescopeofanother.Anexampleofasentencewithmul-tiplegeneralityis

Everyonelovessomeone.

WithFxyforthepredicate‘xlovesy’,onestandardsymbolizationis

∀x∃y Fxy

Thenumberofpossiblenestingsofquantifiersinthescopeofquanti-fiers is unlimited.Certainbasic natural language inferences requiremultiplegenerality.Forexample,withaforDioandbforPlato:

Everyoneloveseveryone. ∀x∀y Fxy

HencePlatolovesDio. Fab

Everyoneloveseveryone. ∀x∀y Fxy

Henceeveryonelovessomeone. ∀x∃y Fxy

ItisgenerallyagreedthatoneofFrege’scoreachievementswasthede-velopmentofalogicthatcanaccountformultiplegeneralityandthatforthispurposeheinstitutedrulesthatgovernthestackingofquan-tifiers.1Wedon’tquibblewiththis.Thereisnoexplicitsurvivingevi-dencethattheStoicshadafullyworked-outtheoryofmultiplegener-ality.Instead,wearguethattheStoicshadalltheelementsrequiredtointroducemultiplegenerality.Moreprecisely,thatamongthesparselysurvivingevidenceonStoiclogicthereissufficientmaterialtoestab-lishthattheStoicshadall thoseelementsforexistentialanduniver-salquantificationwithmorethanonequantifier,ifnotexactlyinthewayFregeintroducedthem.Rather, theirsystemofquantificationisvariable-free,notunlikethatintroducedbyQuinein1960.2Tothisend,

1. Frege1879,§11.SeeDummett1973,9;Rumfitt1994,599–607;Zalta2018.

2. Quine1960,cf.1971,1981.SeealsotheworkbyPaulineJacobson,e.g.Jacob-son1999.

susannebobzien&simonshogry Stoic Logic and Multiple Generality

philosophers’imprint –2– vol.20,no.31(november2020)

universals.WehopethatthefootnoteswillsatisfytheexpectationsofthosespecializinginStoiclogicthatnohistoricalormethodologicalcornershavebeencut.

Thepaperisstructuredasfollows:§1RelevantgeneralremarksonStoiclogic;§2Stoickatêgorêmataasmonadicpredicates;§3Monadicpredicatesasfunctions;§4Polyadicpredicates;§5Variable-freequan-tification I:monadic indefinitepropositions;§6Variable-freequanti-fication II:polyadic indefinitepropositions;§7Multiplegenerality I:scopeambiguity; §8Multiplegenerality II: anaphoric ambiguity; §9PolyadicpredicatesandStoicdeduction;§10Concludingremarks.

1. Some general remarks on Stoic logic

We remind readers of some very basic elements of Stoic logic. TheStoicssharplydistinguishlinguisticitemsor‘speech’(logos)fromwhatspeech signifies. Speech is a species of sound, namely, sound thatsignifiesmeaningor is significant (sêmantikê) (DL7.55–56,63). ‘Say-ables’(lekta),bycontrast,aretheincorporealitemsthataresignifiedbyspeech(DL7.57).Forexample,inuttering“Platowalks”,aspeaker‘says’ the sayable «Platowalks».Aswhat is signifiedby speech, say-ablesarecontentsofspeech.TheythusplayaroleanalogoustoFrege-ansenses.4Weusedoublequotationmarkstoindicatelinguisticitems(speech),guillemets («, ») to indicatecontent (sayables).Stoicgram-marstudiesthepropertiesandpartsofspeech,whileStoiclogicstudiesthepropertiesandpartsofsayables (DL7.43–44,63).5Althoughmanylaterancientsourcesconflatethisdistinction,orthodoxStoicsarecare-fultokeepapartthesubject-matterofgrammarandlogic.

Stoiccontentsarestructured,andtheirstructurecorresponds—tosomedegree—tothestructureoflanguage.Inclassifyingthevariouskindsofcontent,theStoicsrelyongrammaticalpropertiesofthelin-guistic itemsthatexpressthem.For instance,themonadicpredicate(katêgorêma)«…lovesPlato»issignifiedbyaverbandadeclinednoun,

4. Gaskin1997,94–95;Barnesetal.1999,95–96.

5. For an excellent, detailed introduction to Stoic grammar, see Atherton &Blank2003.AnexcellentintroductiontoStoiclogicisIerodiakonou2006.

andbuildingonexistingliterature,3weconsiderandreinterpretsomepassagesthathavenotyetbeengivenmuchattentioninthecontextofmultiplegeneralityandspecifyhowmultiplegeneralityplayedanactiveroleinStoiclogic.(WenotethatthesurvivingevidenceofStoiclogicofpredicatescoversonlyatinyfractionofwhatweknowtheSto-icswroteonthetopic.)

The relevance of our undertaking is threefold. First, it establish-es that Stoic logic, thoughmostly forgotten at thebeginningof theMiddleAges,ifnotalreadyinthefourthcenturyCE,sportsconsider-ableadvantagesoverAristotle’slogicinexpressinganddealingwithmultiple generality. Second, it indicates that fromAristotle to Frege,instead of one big step—a logical discovery to which all interven-ingphilosopherswere ‘simplyblind’ (Dummett 1973,9)—there isasomewhatmoregradualdevelopment.(Weremarkontherelationbe-tweenStoiclogicandmedievalattemptsatfiguringoutmultiplegen-eralitybrieflyinourconclusion.)Third,itisareminderthatthefocusonmultiplegeneralityexclusivelythroughthelensofFrege-inspiredvariable-bindingquantifier theory—as contrastedwith variable-freepredicatelogic—maypreventourappreciationofthedevelopmentofpre-Fregeantheoriesofmultiplegenerality.Additionally,weoffernewexplanationsofacoupleofpuzzlingelementsinStoiclogic.

Onegoalof thepaper is to introducea largeraudiencetothein-tricaciesofStoiclogicandtoitsmoregeneralpotential.Itisforthisreason thatweveryoccasionallyadda remarkabouthowtheStoictheorycouldbeextendedinanobviousmannertocovermoregeneralcases forwhich there is no evidence eitherway. Two examples arepolyadicpredicateswithhigherargumentnumbersandunrestricted

3. Some excellent work has been done on Stoic predicates (including someonmonadicquantification).WementioninparticularAtherton1993,44–48,259–264;Atherton&Blank, 2003, 314–316, 320–323; Barnes 1986, Barnes1999;Barnesetal.1999,111–114,197–206;Brunschwig1986,287–310=1994,63–67;Crivelli1994a,189–199;Frede1974,51–73;Gaskin1997,91–104;Lloyd1978. There are alsomost useful observations inDurand 2018; Egli 2000;Long&Sedley1987,v.1,199–200;Hülser1987–88;Hülseretal.2009.StoicpolyadicquantificationhasbeenconsideredbyUrsEgliinhis1993and2000,andweagreewithsomeofhisresults.



To removeambiguity innatural language, theStoics introduceasystemoflinguisticconventionsthatensurethattheformofspeechreveals thecontentsbeingexpressed.9Manyof themconcernwordorder.Most languages have either casemarking or rigid order (Mi-yagawa2012,ch.10).WhereasEnglish,forexample,hasbasicallynocasemarkingbutfairlyrigidorder,ancientGreekisfoundtowardtheotherendofthespectrum,withverylittlerigidorderbutfairlyarticu-latedcasemarking.ThisworkstotheadvantageoftheStoicsintheirattempt tostructurallydisambiguate languagebymeansof regimen-tation. It is far easier to introduce somedistinctive requirementsofrigidorderintoanatural languagewithextensivecasemarkingthanto introducecasemarkings intoanatural languagewith rigidorder.Forinstance,tosignifythenegationof«Platoiswalking»,theformula-tion“Platoisnotwalking”isdiscouragedbytheStoics.Itisreservedfortheaffirmation«Platoisnotwalking»,which,sinceitisassumedtoentailtheexistenceofPlato,isnotcontradictoryto«Platoiswalking».10 Instead,toexpressnegationstheStoicsrecommendprefixingthene-gationparticletothesentencethatsignifiesthepropositionitisnegat-ingthus“Not:Platoiswalking”(whichisgrammaticalinGreek).ThisaccuratelyreflectsthescopeoftheStoicnegationoperator.Generally,thereisplentifulevidencethattheStoicsusedthefollowingprinciple,whichwecallthe

Scope Principle: The expression that signifies the con-tentelementoroperatorwiththelargestscopeinaprop-ositionisthefirstexpressioninthesentence,orasclosetothebeginningofthesentenceasgrammarpermits.Ifanoperator consists ofmore thanonepart, the expres-sionthatsignifiesitsfirstpartisthefirstexpressioninthe

9. On Stoic natural language regimentation, see Bronowski 2019, 212–214;Atherton&Blank2003,314–316;Barnesetal.1999,96–97;andFrede1974.

10. Seee.g.Apul. Herm.191.6–11,Alex.An. Pr.402.3–19andBarnesetal.1999,102,andalso§7below.

andStoicpropositions(axiômata)tendtobesignifiedbydeclarativesentences.PropositionsarethefundamentalitemswithinStoiclogicand, as the sole non-derivative bearers of truth-value, can be com-pared to Fregean thoughts.6 So the relation between sayables andpropositionsisanalogoustothatbetweenFregeansensesandFregeanthoughts.ThemostbasicdistinctionofStoicpropositionsis thatbe-tweensimpleandnon-simpleones(DL7.68–69;SE M8.93,95,108).Thesimpleonesincludenegationsofsimplepropositions.Non-simplepropositionsarethosethatareputtogetherfrommorethanoneprop-ositionoronepropositiontakentwiceandthataregovernedbyoneormoreconnectivepartsoranegationoperator.Forexample,adisjunc-tionisgovernedbytheconnectiveparts‘either’(forthefirstdisjunct)and ‘or’ (for thesecond).Theprincipalnon-simplepropositionsareconjunction, conditional and exclusive disjunction.7 Negations andnon-simplepropositionsaredefinediteratively:thelanguageofStoicpropositionallogicissyntacticallyclosedundernegation,conjunction,disjunctionandconditional.

The Stoics do not posit a perfect one-to-one correspondence be-tweencontentandspeech.Thegrammaticalpropertiesofspeechareadefeasible,andpotentiallymisleading,guidetothecontentitsignifies.(“pandqorr” isanexample.)OntheStoicview,oneandthesameexpressionofnaturallanguage,ifitisambiguous,expressesmultiplecontents.8Moreover, the same content canbe signifiedbydifferentpiecesofspeech(Barnesetal.1999,96–97).

6. SeeBarneset al. 1999,93–96, for the limitationsof this comparison.NotealsothatthereisdisagreementamongscholarsonStoiclogicandlinguisticswithregardtothequestionwhethersayablesaremind-independent.Mind-dependencyisdefendedmostrecentlyindeHarven2018,228–230andbe-forethate.g.inLong1971,96–98,whilee.g.Barnes1999,211;Shogry2019,37fn.12;andBronowski2019,165–169,defendmind-independency.Thispaperisindependentofhowoneleansonthisquestion.

7. DL7.68–74,SE M8.93–94,108.Fordetailedtreatmentofthesimpleandnon-simplepropositions,seeBarnesetal.1999,96–111.

8. Fordetaileddiscussion,seeallofAtherton1993,butesp.131–133.



WorkonHellenisticphilosophyismethodologicallycomplex,andworkonStoic logic is so inparticular.Evidence isvery fragmentary.Ofhundredsofbooks(i.e.papyrusrolls)onStoiclogic,onlyonehassurvived(Chrysippus’Logical Investigations),13andinasorrystate.Ev-erything else is one ormore steps removed from the original texts.Sourcesaredependabletodifferentextentsforvariousreasons,andwewilloccasionallyremarkonthereliabilityofasource.Fordetails,the reader is referred to specialist secondary literature.14 Some guid-anceisgivenbyextensivelistsofbooktitlesonStoiclogicandade-tailed summary of Stoic logic inDiogenes Laertius.Many passagesofgreatinteresthavesurvivedquotedorparaphrasedbyoftenmuchlater authors,manyof themhostile to Stoicphilosophy.Wedoourbesttounscramblethescrapsofegg.Translationsareourown,unlessotherwisenoted.

2. Stoic katêgorêmata as monadic predicates: definitions

Stoicpredicates(theirtermiskatêgorêma,pluralkatêgorêmata)arecon-tents15andassuchbelong toStoic logic, rather thanStoicgrammar.Oursourcesindicatethat,probablystartingwithChrysippus,theSto-ics had an elaborate logical theory of katêgorêmata thatwas devel-opedover several generations.16Chrysippus’, andperhaps generally

13. PHerc. 307, col. xiii.19–22, ed. L.Marrone, ‘LeQuestioni Logici diCrisippo(PHerc307)’inCronache Ercolanesi27,1997,83–100.Alacunosebutlongpa-pyrusfragmentandtheonlyoneofChrysippus’booksonlogicofwhichwehavedirectevidence.Forexcellentgeneraldiscussionofthistext,seeBarnes1986;Barnesetal.1999,69–71;andMarrone1997.Earliereditionsofthepa-pyrusappearinFDSas698andinSVF2asfrag.298a.

14. E.g. Mansfeld 1999, 3–30; Hülser 1987–88, XXXII–LXVIII; Bobzien 1998,5–12;Barnes1999,69–76.

15. ThefirstStoic forwhomthere isevidence for thisview isChrysippus’pre-decessorCleanthes,whostatesthatkatêgorêmataarelekta(Clement,Strom.8.9.26.4).Post-CleantheanevidenceforthisclaimisadducedthroughoutthisSection.

16. Chrysippuswrotefourteenbooksonkatêgorêmata,oneonactive(ortha)andpassive (huptia) katêgorêmata, one on event-predicates (sumbamata, emen-dation,vonArnim)(DL7.191).Weexpecttreatmentofkatêgorêmatainhisbooks on indefinite and temporal propositions (DL 7.190). Katêgorêmata

sentence,orasclosetothebeginningofthesentenceasgrammarpermits.11

IntheStoicview,language(speech),suitablyregimented,isanappro-priatetooltorepresentthestructureofsayables,i.e.ofcontent.

Stoicpropositionsareof central importance for theStoic systemof deduction (Stoic syllogistic), which, unlike Aristotle’s syllogistic,isapropositionalsequentlogic.Thisnotwithstanding,theStoics, inparticularChrysippus, thirdheadof theStoaandbyfar thegreatestStoiclogician,displayedakeeninterestinthelogicalsignificanceofsub-propositionalelements.Theseinclude(i)thelogicalrelationsbe-tween‘saysthatx’, ‘x’and‘xistrue’,possiblyinconnectionwiththeLiar paradox; (ii) a sophisticated theory of demonstratives; (iii) thelogicofpluralexpressions;12and(iv),mostrelevanttoourpurposes,abasicsystemofvariable-freequantifyingoperatorsandadetailedclas-sificationofpredicatecontents.

11. Cf.,forexample,theStoicdefinitionsandexamplesofnegativeproposition(ἀποφατικόν, startingwithοὐχί:DL 7.69; SE M 8.89, 8.103; cf. Apul.Herm.191.6–11cui negativa particula praeponitur),eliminatingproposition(ἀρνητικόν, startingwithοὐδείς:DL7.70),privativeproposition(στερητικόν,startingwithpredicateexpressionwithalpha privativum:DL7.70),affirmativeproposition(κατηγορικόν,startingwithnoun/name:DL7.70),middleproposition(μέσον, startingwith noun/name: SE M 8.97), deiktic proposition (καταγορευτικόν, starting with demonstrative pronoun: DL 7.70), definite proposition(ὡρισμένον,startingwithdemonstrativepronoun:SE M8.96–97,Alex.An. Pr. 177–178),conditionalproposition(συνημμένον, startingwithεἰ:DL7.71,SE PH2.157–158, SE M 8.109–110), paraconditional (παρασυνημμένον, startingwithἐπεί:DL7.71), conjunctiveproposition (συμπεπλεγμένον, startingwithκαί:DL 7.72,Apoll.Dysc.On Conjunctions 218.15–19), negated conjunctiveproposition (ἀποφατικὴ συμπλοκής, startingwithοὐχὶ καὶ: SE M 8.226), dis-junctiveproposition (διεζευγμένον, startingwithἤτοι orἢ:DL 7.72; SE PH 1.69,2.158;SE M8.434;cf.Gellius16.8.13),causalproposition(αἰτιῶδες,start-ingwithδιότι:DL7.72),co-assumption(πρόσληψις,hasparticleδέassecondword:DL7.76),conclusion(ἐπιφορά,hasἄραassecondword:DL7.76,SE M 8.302),question(ἐρώτημα,startingwithἄρα:DL7.66,Ammon. Int. 199.20–23),inquiry(πύσμα,startingwithaninterrogativepronoun,e.g.ποῦ:SE8.71–72,Ammon.Int. 200.5–10),quasi-proposition(ὅμοιον ἀξιώματι,startingwithὡς:DL7.67,Ammon.Int.2.32–34).SeealsoBarnesetal.1999,101;Atherton1993,78–79;Frede1974,189–201.

12. See(i)Crivelli1994b;(ii)Frede1974,53–61;Lloyd1978;Barnesetal.1999,93–101;Durand2018,103–131;(iii)Cavini1993;Barnesetal.1999,152–155.



katêgorêma and an event predicate… as «is walking»yields forexample«Socrates iswalking».19 (Ammon. Int. 44.23–25)

(C) Thatwhich is predicated, then, is predicated of anupright noun or [upright] case-content … If [what ispredicatedofanuprightnounor[upright]case-content]producesacompletesentence,theycallitkatêgorêmaoreventpredicate.20(Stephanus,Int. 11.9–12)

(ThesetwotextsmixPeripateticwithStoicterminology;cf.fns.54,56.A canonically Stoic formulationwould not have ‘upright noun’ andwould have ‘proposition’ or ‘complete content’ for ‘complete sen-tence’.) Thedefinition of ‘katêgorêma’ asmonadic predicate is thuswell-attested.

AsecondStoicdefinitionof‘katêgorêma’allowsfortworeadings.First,areadingasanaccountofmonadicpredicates.Second,areadingasanaccountthatincludesmonadicandpolyadicpredicates:

(D)Akatêgorêma is…anobject21thatcanbeconnectedwith some thing or some things, asApollodorus22 says.(DL7.64)23

19. ἂν μὲν οὖν ὀνόματός τι κατηγορηθὲν ἀπόφανσιν ποιῇ, κατηγόρημα καὶ σύμβαμα παρ’ αὐτοῖς ὀνομάζεται…ὡς τὸ περιπατεῖ, οἷον Σωκράτης περιπατεῖ(Ammon.Int. 44.23–25).For theStoiceventpredicates (σύμβαμα), seebelow§3. (Squarebracketsinatranslationindicateaphrasesuppliedbycontext.Angledbrack-etsintextandtranslationareusedtoindicateatextualemendation.)

20. τὸ κατηγορούμενόν τινος ἢ ὀνόματος κατηγορεῖται ἤγουν εὐθείας ἢ πτώσεως.….καὶ εἰ μὲν αὐτοτελῆ τὸν λόγον ἀπεργάζεται, καλοῦσιν αὐτὸν κατηγόρημα ἢ σύμβαμα (Stephanus,Int. 11.9–12).

21. Intheirlogic,theStoicsappeartouselekton(‘sayable’)andpragma (whichwetranslateas‘object’) synonymously,e.g.DL7.57,63.SeealsoBronowski2019,118–119;Atherton1993,250;Barnesetal.1999,197–198.

22. Reference is to thesecond-centuryBCEStoicApollodorusofSeleucia, stu-dentofDiogenesofBabylon.

23. Ἔστι δὲ τὸ κατηγόρημα … πρᾶγμα συντακτὸν περί τινος ἢ τινῶν, ὡς οἱ περὶ Ἀπολλόδωρόν φασιν (DL7.64).TheancientGreekphraseὡς οἱ περὶ X φασιν

theearlyStoic,notionofkatêgorêmatawasoneofmonadicpredicates.AmatchingStoicdefinitionis

(A)Akatêgorêma is…anincompletesayablethatcanbeconnectedwithanupright case-content (orthê ptôsis) toyieldaproposition.17 (DL7.64)

Thedefinition (A) classifies katêgorêmata as a kindof content thatcanbeconnectedwithotherthings.Itspecifiesthesethingsasuprightcase-contents,andtheresultingcontentasaproposition.AStoiccase-content (ptôsis)canbethoughtofasthecontentsignifiedbyanoun.18 An upright case-content is the case-content signified by a nominative noun.Thustext(A) suggeststhatakatêgorêmaisakintoamonadicpredicate,exceptthatcontemporarytheoriesdonotrequirethattheanaloguetoStoiccase-contentbeinthenominative(seealsobelow§3).Twolaterancienttextscontainvariantsofthisdefinition.

(B) Now,ifsomethingispredicatedofanounandyieldsanassertiblecontent,itiscalledbythem[i.e.theStoics]

alsofeatureprominentlyinChrysippus’Log. Inv.Apollodorusofferedadefi-nitionofkatêgorêma(below).CleanthesandSphaerusofBorysthenes(pu-pilofZenoandCleanthes)authoredoneworkeachonkatêgorêmata (DL 7.175,178),thoughthesemayconcerntheircausalaspect.Fortheseaspectsofkatêgorêmata,seeBobzien1998,18–21,andHankinson1999,483–486.Wesetthemasidehere.

17. Ἔστι δὲ τὸ κατηγόρημα…λεκτὸν ἐλλιπὲς συντακτὸν ὀρθῇ πτώσει πρὸς ἀξιώματος γένεσιν.We here translate suntaktos as ‘can be connected’, rather than thealternative ‘connected’, since a katêgorêma is not necessarily always con-nectedwith something (DL7.63and§3). (A) is the third ina listof threeindependentdefinitions,mostprobablyoriginatingindifferentworksbydif-ferentStoics,asiscommonincompendiaandepitomesofStoicdoctrine.Thefirstdefinitionis‘akatêgorêmaisthatwhichissaidofsomething’(Ἔστι δὲ τὸ κατηγόρημα τὸ κατά τινος ἀγορευόμενον). It is lessspecificthan(A),butindi-catesthatkatêgorêmataaremonadicpredicates.Theseconddefinitionisourtext(D).

18. ThesubjectofStoiccases or case-contents (πτῶσις)isdifficult.Weassume,withGaskin1997,94–101,andDurand2018,73–78,thatptôseisarecontents.Foralternativeviews,seeBronowski2019,352–359;Long&Sedley1987,200v.1;Frede1994,13–17;seealsothediscussioninAtherton&Blank2003,324–326.Nothinghangsonthisquestionhere.



So (E) confirms thefirst reading. Further corroborationmaybe thefactthatChrysippushadaninterestinsingularandpluralexpressions(DL7.192)andpossiblyconsideredsingularandpluralkatêgorêmatainLog. Inv. cols. I.5–7, I.15–20, II.21–26.28Singularandpluralexpres-sions as possible arguments for Stoic monadic predicates are alsomentionedinalatertext.29

On the second reading of (D), the definition would refer to whatin contemporary logicwould be the arguments the predicate takes(‘this onewalks’,‘DioseesPlatoconversewithSocrates’).Thedefinitionwouldthencoverbothmonadicandpolyadicpredicates.30Thisread-ingissometimesthoughttobesupportedbythesecondoftwoStoicdefinitionsof‘verb’(rhêma),whichdisplayssalientparallelsto(D).31

(F)Averb is… ,or as somesay, a caselesselementofspeech that signifies something that can be connectedwithsome thingorsome things, suchaswrite, speak.32 (DL7.58)

This definitiondoes notmention katêgorêmata. Theparallel to (D) suggeststhatthesignified‘something’isakatêgorêmainthesenseofthattext.Now,averbmayleaveroomformorethanonecase-content,andtheStoicswereawareofthis.So,thisdefinitionof‘verb’mayfa-vourthereadingof(D) asadefinitionofkatêgorêmaasonethatcov-ersmonadicandpolyadiccases.

28.AllpointedoutbyBarnes1999,204n.167.Cf.alsoFrede1974,53.

29. ‘Itruesthis one,itrues these’inAmmonius’reportoftheStoicparasumbamata (Ammon.Int.44.32,forwhichseebelow§§3and4.1).

30.ThisreadingisadoptedbyGaskin1997,93.

31. NotedbyBarnes1999,203,fn.65;Atherton1993,45–46.

32. ῥῆμα δέ ἐστι…ἤ, ὥς τινες, στοιχεῖον λόγου ἄπτωτον, σημαῖνόν τι συντακτὸν περί τινος ἢ τινῶν, οἷον Γράφω, Λέγω· (DL7.58).ἄπτωτονisoftentranslatedas‘unde-clinable’orthelike.AsStoiccase-contents(πτώσεις)arenotlinguisticitems,themeaningismorelikelythatverbsdonotsignifycase-contents.

Onthe first reading,thephrase‘somethingorsomethings’referstoasingularorpluralcontentofwhichthepredicateispredicated(‘this one walks’,‘thesewalk’).24Thisreadingiscompatiblewiththedefinitionintexts(A), (B)and(C),whichsaysnothingaboutthenumber(singularorplural)ofthecase-content.25Itmakesthedefinitionof‘katêgorêma’asoneofmonadicpredicate,exceptthatittakespluralitiesinadditiontosingularitiesinsubjectplace(e.g.‘these’or‘some’).Strongindepen-dentevidencesupportsthisreading.Thustext

(E)Moreover, they alsomake this distinction: desire isforthosethingswhicharesaidofsomepersonorseveralpersons,whichthe[Stoic]logicianscallκατηγορήματα,forexample«tohaveriches»or«toreceivehonours».26(Cic.Tusc. Disp.4.21)

containsimplicitlyaverysimilarStoicdefinition,namely:

(1)Akatêgorêmaisthatwhichissaidofoneormorepeo-ple(orthings).27

Theexamplesin(E)areanaloguestomonadicpredicatesthatcanbesaidofoneormorepersons:‘someonehasriches’,‘somehaveriches’.

is almost always just another way of saying ‘as X says’, and we translateaccordingly.

24. Long&Sedley1987,v.2,199,andBarnes1999,204,notethepossibilityofthisinterpretationwithout endorsing it.Hicks 1925;Hülser 1987–88, 809, 933;andMensch2018adoptthisreading.

25. Pace Barnes1999,204,whoassumesthatthedefinitionin(A)presupposestheadditionofasingularnominativecase.

26. Distinguunt illud etiam, ut libido sit earum rerum, quae dicuntur, de quodam aut qui-busdam, quae κατηγορήματα dialectici appellant, ut habere divitias, capere honores.Thede quodam aut quibusdam couldalsobetranslatedasneuter(Graver2002,46): ‘desire is for thosethingswhicharesaidofsome thingorsome things’.Thisproducesamoregeneraldefinitionofmonadicpredicates.WenotethatCiceroisgenerallyareliablesourceforearlyStoicdoctrine.

27.WecanimaginetheGreek:κατηγόρημα τὸ κατά τινος ἢ τινῶν ἀγορευόμενον.Ci-cero’squodam aut quibusdam clearlyreferstotheuprightcase-content,theup-rightπτῶσις.(1)appearstobeafusionoftheseconddefinitionofkatêgorêmaatDL7.64withthefirst(giveninfn.17above).



3. Monadic predicates (katêgorêmata) as functions

It has been suggested that one can think of Stoic katêgorêmata asfunctionsroughlyinthesenseinwhichFregeconsideredpredicatesasfunctions.35Weagreewiththis.Infact,wearguethatthesimilaritiesgofurtherthanhasbeenpointedout.ThereisnodirectevidencethattheStoicshadatermfor‘function’.Rather,oneneedstoshowthattherolekatêgorêmata (andrelatednotions)play inStoic logicprovidessufficientevidenceforthissuggestion.GoodinitialevidencecanbefoundintheStoicdefinitionsofthreekindsofsimpleaffirmativeprop-ositions.Ofthesetheresurvivetwosets.Theyappeartomatchincon-tent,andpartiallyinterminology.36A‘definite’propositionconsistsofademonstrativeuprightcase-content(ptôsis)andakatêgorêma.A‘mid-dle’propositionconsistsofanuprightcase-contentandakatêgorêma.An‘indefinite’propositionconsistsofoneormoreindefinitepartsandakatêgorêma.37Examplesfordefinitesare«thisoneiswalking»,«thisone is sitting»; formiddle ones «Dio iswalking», «Socrates iswalk-ing»,«ahumanbeingissitting»;andforindefinites«someoneiswalk-ing»,«someone issitting».38Thesetofdefinitionsandthematching

somefurtherimplausibleconjectures,suchasthattheStoicsbelievedthatanuprightcase-contentwould,intheparakatêgorêmataorparasumbamata,beex-pressedbyanobliquelinguisticexpression(95,98).Gaskin’sviewalsocannotaccountforthemorecomplexmonadicpredicates(katêgorêmata)attestedforChrysippus(PHerc307;seeour§3).Forourviewontherelationsbetweensumbamata, parasumbamataandkatêgorêmata,seebelowfn.54.

35. Frege1891,1904. SeeGaskin1997,94–101;Hülseretal.2009,378.Cf.alsoEgli1986.Athertonquestionstheanalogy(1993,45–46,n.10).

36.Sextushas‘indefinite’(aoriston),‘middle’(meson)and‘definite’(hôrismenon) (M8.97–100);Diogenes‘indefinite’(aoriston),‘categorical’(katêgorikon)and‘categoreutical’ (katêgoreutikon). (DL7.70).We followSextus’ usage, inpartbecauseoftheobscurityofDiogenes’terms.Ourfocuswillbetheindefinitepropositions,whoseterminologyisagreedonbybothsources.Forgeneraldiscussionofthisclassification,seeFrede1974,53–67;Durand2019,§§2–6;Barnesetal.1999,97–100;Brunschwig1994;Crivelli1994a;andEbert1993.

37. ThereisatextualprobleminDiogenes’definitionoftheindefinite,forwhichseebelow§5.

38. DL7.70,SE M8.96–98.Anthrôposcouldexpressthegeneric,‘humanbeing’,orwhatisexpressedinEnglishbyanindefinitearticleandanounphrase,‘a

Weseetwopossibilities.33EitherallStoicsusedtheterm‘katêgorê-ma’throughoutformonadicpredicatesonly.Ortheterm‘katêgorêma’wasoriginally,andbyChrysippusamongothers(see§3),usedformo-nadicpredicatesonly,butacoupleofgenerationslater,theStoicApol-lodorusaddedanalternativenotionofkatêgorêmaas thatwhich issignifiedbyaverbandthathenceincludesbothwhatwecallmonadicandwhatwecallpolyadicpredicates.Thisalternativenotionwouldhavebeenmotivatedbyconsiderationsofgrammarand,fromalogical perspective,wouldbeaseriousstepbackcomparedtotheearlierno-tion,sinceitnolongerprovidesforone-placepredicatesofcomplexformslikeFxa∧p(see§3).Eitherway,itiscertainthatChrysippusandotherearlyStoicshadalogicalnotionofmonadicpredicatesandused‘katêgorêma’toexpressthatnotion,andmoreoverthatthisnotionwastheprevalentStoicone ((A), (B), (C), (E) above,and§3).Hereafter‘monadicpredicate’translatestheStoic‘katêgorêma’asandwhenitisusedforthatnotion.34

33. Ultimately,theevidenceisnotconclusiveforeither,asBarnes1999,204,alsoconcludes.

34.Gaskin1997arguesthat ‘katêgorêma’wasusedinabroadandinanarrowway: in thenarrowsense,onlyaverb that lacksnothingbutanominativecase-content forcompletion isakatêgorêma; in thebroadsense,anyverbsignifiesakatêgorêma(93,103).WebelievethatGaskin’sviewrestsontwoerrorsandsomeunprovenconjectures.HismainerrorisinhisargumentthatDL7.64(ourtext(A))impliesthattheStoicsinsistedthateverypropositionrequiresanominativecase,i.e.thatitisanecessaryconditionforsomethingtobeapropositionthatitcontainanominativecase(91–92andthenrepeat-edpassim).However,DL.7.64 isadefinitionnotofpropositions(axiômata) butofkatêgorêmata. Itonlycommitsoneto theviewthat theStoicsmain-tained that it isanecessaryandsufficientcondition forsomething tobeakatêgorêma,thatcompletionwithanominativecasegeneratesaproposition.Gaskin’smistakenassumptionguideshisentireargument,andwithoutittheargumentcollapses.WebelievethatGaskinisalsoinerrorwhenassumingthatwhatisgenerallyacceptedtobealacunainDL7.64containedafourfold distinction(forwhichseeour§4)andthatthisisusuallyassumed(92withfn.3).ThefourfolddistinctionisnotreportedforChrysippus;thetwofolddis-tinctionbetweensumbamataandparasumbamatais;anditisparasumbamata alonethatareusuallyassumedtohavegonemissinginthelacuna.SoGaskincannotusehisconjectureofa fourfolddistinction in the lacuna tobackuphis –mistaken– assumption that a nominative case is necessary for some-thingtobeaStoicproposition.Gaskin’sincorrectassumptionleadshimto



justasaFregeanpredicatefunctioncanbecompletedbyitsargument(including,inthatcase,avariable-bindingquantifier).42

As to thevalue of a completedpredicate function, in the caseofpropositions,itseemstobeatruth-evaluablecompletecontent,or,inotherwords,thepropositionitself:itisonly(andprecisely)assertiblecomplete contents thathavea truth-value,eitherbeing trueorbeingfalse.Bycompletingakatêgorêmawithoneof theoptions forargu-ments fromabove,aproposition,and that is a contentwitha truth-value,isgenerated(cf.text(A)).AndonedefinitionofStoicproposi-tionsisthattheyarecompletecontentsthatareeithertrueorfalse(SE M8.73;DL7.65).

ThustherearegoodreasonstoassumethattheStoicsthemselvesthought of a katêgorêma as something similar to a function: some-thingthat,whencompletedbyanargument,producesatruth-evalu-ablecompletecontent.43This‘modern’understandingofkatêgorêmaasfunctionisfurtherconfirmedbyexamplesofkatêgorêmatawhoselogicalstructureisofgreatercomplexity.In§2wearguedthattheStoicterm‘katêgorêma’isbesttranslatedasmonadic predicate.Thesimplestexampleswere expressedbyfinite verbal phrases, such as “iswalk-ing”,“iswriting”,“isthinking”(DL7.63,64).AsecondStoicdefinitionof‘verb’as‘apartofspeechthatsignifiesanuncompounded(asunth-eton)monadicpredicate’(DL7.58)44suggeststhattheStoicshavean

42. Since,inStoicsources,theargumentsthemselves(«Dio»,«ahumanbeing»,etc.)arenevercalled,oradducedasexamplesof,incompletecontents,weas-sumethattheyarenotincomplete.

43. Insomecases inwhichakatêgorêma iscompletedbyanargument, there-sultingvalueisacompletecontentthatisnottruth-evaluable,oratleastnotinanystraightforwardmanner.Examplesareapromiseoracommand.SeeChrysippus,Log. Inv. col.XIII;DL7.66–68; andSE M 8.70–74.Wedonotdiscusssuchnon-propositionalcompletecontentshere.SeeBarnes1986fordiscussionofthelogicofcommands.

44. ῥῆμα δέ ἐστι μέρος λόγου σημαῖνον ἀσύνθετον κατηγόρημα, ὡς ὁ Διογένης (DL 7.58).ThedefinitionisattributedtoDiogenesofBabylon,pupilofChrysippus,logicteacherofCarneades.

examplesofthethreekindsofaffirmativesimplepropositionssuggestthatthekatêgorêma(monadicpredicate)islikeafunctionthatcanbecompletedbyslottingintoitsargumentplaceanargumentofoneofthreekinds.39Theseshowsomesimilarity,inorder,toademonstrative,anindividualconstantandsomethinglikeavariableboundbyanex-istentialquantifier.Thekindofargumentthatfillstheargumentplaceofthekatêgorêmadeterminesthekindofpropositiononegets.

FurtherevidenceisprovidedbytheStoicnotionsofcompleteandincompletecontents.Wearetoldthat

(G) Incomplete isa content thathas theexpressionun-finished, forexample«…iswriting»40, forweask ‘who?’.Complete is a content that has the expressionfinished,forexample«Socratesiswriting».Katêgorêmataarethenamongtheincompletecontents,andpropositionsamongthecompletecontents.41 (DL7.63)

The incomplete content «iswriting» can be completed variously by«Dio»,«ahumanbeing»,«thisone»,«someone»or«theteacherKallias»,

humanbeing’.Thecontextmakesclearthathereitisthelatter.SeeBarnesetal.1999,98.Theco-classificationofpropernamesandgenericnounsasoneclassofexpressionsmayseemodd. It ismotivatedbyStoicmetaphys-ics.SeeLong&Sedley1987,182,v.1;Bailey2014,285–290.Thereasonswhy«a human being is sitting» ismiddle and not indefinite, and is lumped to-getherwith«Dioiswalking»,areagainmetaphysicalratherthanlogical,andwedisregardthem,sinceweareaftermultiplegenerality.Fordetails,seee.g.Bronowski2019,304–340;Durand2018,66n.6,169;Bailey2014,295–298;andCaston1999,187–192.

39.Anotherexample(SE M8.308):‘Ifsome godtellsyouthatthisonewillberich,thisonewillberich;butthis godhere(IpointatZeus,byhypothesis)tellsyouthatthisonewillberich.’SeefurtherdiscussioninDurand2019,§§46–47.

40.As Gaskin 1997, 102–103, points out, the Greek γράφει is ambiguous andleavesunclearwhetheroneshouldbetranslated“iswriting”or“heiswriting”.Inanyevent,itisclearthatthecontentexpressedisanincompletecontent,i.e.thekatêgorêma,«…iswriting».Cf.Barnesetal.1999,203.

41. ἐλλιπῆ μὲν οὖν ἐστι τὰ ἀναπάρτιστον ἔχοντα τὴν ἐκφοράν, οἷον Γράφει· ἐπιζητοῦμεν γάρ, Τίς; αὐτοτελῆ δ’ ἐστὶ τὰ ἀπηρτισμένην ἔχοντα τὴν ἐκφοράν, οἷον Γράφει Σωκράτης.ἐν μὲν οὖν τοῖς ἐλλιπέσι λεκτοῖς τέτακται τὰ κατηγορήματα, ἐν δὲ τοῖς αὐτοτελέσι τὰ ἀξιώματα.



juxtapositionofwhatisexplicitlyreferredtoasaproposition(axiôma) andwhatisexplicitlyreferredtoasamonadicpredicate(katêgorêma) (Col.XIII,17–22):

(3)Theproposition«Dioiswalking,but ifnot,he issit-ting.»(Col.XIIIlines17–19)

(4)Thekatêgorêma «towalk,but ifnot, tosit» (Col.XIII lines19–22)

In the vicinity, there are similar constructions, whichmust also bekatêgorêmata(Col.XI).Ofparticularinterestis

(5)«towalk,sinceitisday»(Col.XIlines24–25) 47

Itisthepeculiartechnicalformulationwithinfinitiveswhichindicatesthatwhatisatissueisapredicate(quaincompletesayable,notquapartofa—complete—proposition).48(4)and(5)areofcriticalimpor-tance,sincetheyshowthatforChrysippusamonadicpredicateisnot thatwhichissignifiedbyaverb(rhêma).(Thosearetheuncompounded monadicpredicates.)Rather,amonadicpredicatecanbesignifiedbyarathercomplexexpressionandcontainbothapluralityofkatêgorê-mataandtwo-placeconnectivesandanegator(4)andevenproposi-tions(5).(Reader,letthissinkin,please,becauseitisextraordinary!)Incontemporarysymbolism,theanalogousmonadicpredicateto(4)wouldbeperhaps

(6)Fx ∧ (¬Fx → Gx)

andto(5)

47. The Greek is (3) περιπατεῖ Δίων, εἰ δὲ μὴ, κάθηται; (4) περιπατεῖν, εἰ δὲ μὴ, καθῆσθαι;and(5)περιπατεῖν, ἐπεὶδ’ἡμέρα ἐ[στι]ν.Thecases inCol.XI arereferredtowithoutaspecificnoun.Weassumethatτὸ ὅλον τοῦτο(Col.XI.23)isshortforτὸ ὅλον τοῦτο κατηγόρημα.

48.Cf. theexamplesinfn.46above.SeealsoBarnes1999,203n.164,whoas-semblesyetmoreexamples(SE PH3.14,M9.211;SenecaEpist.117.3,12;and,forZeno,StobaeusEcl.1.13.1c),andInwood1985,65:‘Stealingandnotsteal-ingarepredicatesandareindicatedintheGreekbytheinfinitiveformoftheverb,whichisoftenusedtostandforpredicates.’

expressionforsuchbasicmonadicpredicates:uncompoundedmonadicpredicates.45Theirgeneralformwouldbe

(2)…F or,incontemporaryterms, Fx

There are also a considerable number of examples of monadicpredicates, explicitly calledkatêgorêmata, that consist ofmore thanwhat is signified by a verb. First, there are severalwhich have twocomponents: “to sail through rocks”, “tohave riches” and “toobtainhonours”,and“todrinkabsinthe”,allexpressedwiththeverbintheinfinitive.46Justasinthecaseofuncompoundedmonadicpredicates,theirformcanbesymbolizedas

(2)…F or,incontemporaryterms, Fx

thoughF itselfissignifiedbyacomplexexpressionthatcombinesaverbwith a generic singularorpluralnoun, eitherdeclinedor as aprepositionalphrase.Thisisinaccordancewiththefactthat‘katêgorê-ma’isalogicalexpressionascomparedwithagrammaticalone.Itisitsincompleteness,notitscorrespondingtowhatisexpressedbyaverb,thatdeterminesitscharacter.Butwecandobetter.

Chrysippus is our best source for monadic-predicate analoguesto contemporary logic.One crucial bit of evidence is fromhisLogi-cal Investigations. Here we find, in consecutive lines, the following

45. Asuntheta katêgorêmata:Forthisexpression,seee.g.Mich.Sync.79:‘Ingeneral,everyverbissaidtobeakatêgorêma(i.e.bytheStoics);specifically,thereare compounded katêgorêmata and uncompounded katêgorêmata.A com-pounded[katêgorêma]isonewhichiscombinedwiththecaseofanameorapronoun,whereasanuncompounded[katêgorêma]istheverbitselfsaidonitsown.TheStoicscallthecombiningcompounding.’(πᾶν ῥῆμα γενικῶς λέγεται κατηγόρημα· εἰδικῶς δὲ σύνθετον κατηγόρημα καὶ ἀσύνθετον κατηγόρημα· σύνθετον μέν ἐστι τὸ συντεταγμένον πτώσει ὀνόματος ἢ ἀντωνυμίας· ἀσύνθετον δὲ αὐτὸ τὸ ῥῆμα τὸ καθ’ ἑαυτὸ λεγόμενον· σύνθεσιν δὲ οἱ Στωϊκοὶ τὴν σύνταξιν λέγουσιν.)Notethatthislatesource,asisquitecommon(see§1),confusesthelevelsoflanguageandcontentanddoesnotdistinguishbetweenverbandkatêgorêma.ThiskindofStoicuseofasunthetosinDL7.58isalsoconfirmedatSE M8.136,‘nopropositionisuncompounded’(οὐδὲν ἀξίωμα ἀσύνθετον). SeealsoGaskin1997,93,fortheterm‘uncompounded’usedofkatêgorêmata.

46. DL7.64:διὰ πέτρας πλεῖν;cf.SE M8.297.Tusc. Disp. 4.21:habere divitias, capere honores;SE PH2.230,232:τὸ ἀψίνθιον πιεῖν.



monadic predicates ((B) and (C)) and secondary-event predicateswithmonadicsecondarypredicates(parakatêgorêmata).Theydefinethesecondary-eventpredicateasthatwhich‘yieldsanassertiblecon-tentwhenpredicatedofan[oblique]case-content(ptôsis)’andstatethat‘itislikeasecondarypredicate,asinthecaseof«to…isregret»,forexample«toSocratesisregret»’(Ammon.Int. 44.25–27).53Now,theStoicsfindthemselvesinalongphilosophicaltraditionthattakesitasgiven that the subject-predicate distinction provides the basic com-ponents of sentences, with the subject expression generally in thenominative.Byintroducingthesecondary-event predicatesasakindofmonadicpredicateandlabellingit ‘secondary’, theStoics increase the scope of what is logically treatable. Inlogic, itdoesnotmatterwhetherasentencethatexpressesapropositionhasanoun(ornounclause)inthenominative.Whatmattersisthatthesentencereflectsthestruc-tureof thepredicate function,here«to…isregret»,andhowitcanbecompletedintoaproposition,here«toSocratesisregret».Thattheargumentofthefunctionisanobliquecase-contentandisexpressedwiththedativebecomeslessimportantforitsbeingapredicate.54(Be-

53. ἂν δὲ[πλαγίου]πτώσεως[τι κατηγορηθὲν ἀπόφανσιν ποιῇ],παρασύμβαμα[παρ’ αὐτοῖς ὀνομάζεται]…καὶ ὂν οἷον παρακατηγόρημα, ὡς ἔχει τὸ«μεταμέλει»,οἷον «Σωκράτει μεταμέλει» (Ammon. Int. 44.25–27). Barnes’ translation ‘it ruesSocrates’ is more elegant. Our clumsy rendering is designed to make ex-plicittherelationwiththeobliquecase-content,hereexpressedbyadative.Bronowski 2019, 425–426, maintains that the Stoics did not consider thecombination of parakatêgorêmata (and implied less-than-parakatêgorêmata) withobliquecase-contents aspropositions(axiômata).HerreasonsseemtobethatDiogenesLaertiusdoesnotmentionparakatêgorêmata andthattheydonotoccur inStoicarguments inoursurvivingevidence.Theseseemtoustobeinsufficientreasons.NosourcedirectlysupportsBronowski’sclaimthatpropositionscannotbeconstitutedfromparakatêgorêmata andobliquecase-contents.Onthecontrary,Ammoniusreferstothecombinationofanyspeciesofpredicate(κατηγόρημα, παρακατηγόρημα, ἔλαττον ἢ κατηγόρημαandἔλαττον ἢ παρασύμβαμα)withoneormorecasesas‘thatwhichweassert’(apo-phansis)(Ammon.Int.44.19–45.6);cf.theStoicdefinitionoftheproposition(axiôma)asthatsayingwhichweassert(apophainesthai) (DL7.66).

54. DL 7.64 seems to report that some katêgorêmata are sumbamata and someareparasumbamata(ifwefollowthemostcommonemendationofthetext).Latersourcesassociatesumbamatawithkatêgorêmata, andforparasumbamata we find the new term ‘parakatêgorêma’ (see §4.1 for some of those texts).

(7)p ∧ (p → Fx)49

Itdoesnotmatterwhatcontemporarysymbolismexactlywouldcap-turethesetwopredicates.50Itmattersthat,eithertime,wehaveasmo-nadicpredicate a logical item that, until Frege,wouldgenerallynothavebeencalledapredicate.51Howdoweknowthattherearetobethree(ortwo)x’sratherthananxanday(incontemporaryspeak),inwhich lattercasewewouldhaveadyadicpredicate?First,weknowthisbecausewearegiven(3)asthecorrespondingproposition(com-pletecontent)for(4),withtheanalogouscontemporaryform

(8)Fa ∧ (¬Fa → Ga).

Second,akatêgorêma is,byitsdefinition,amonadicpredicate(§2).SothereisundoubtableevidencethattheStoics(andinparticularChrys-ippus)havemonadicpredicatesthatmaycontainascomponentscon-nectives,morethanonepredicateandwholepropositions.

TheStoicsmadeprogressover theirpredecessorswith regard totheunderstandingofpredicatesas ‘objectsof logic’alsoinafurtherrespect.Thisistheirdistinctionbetweeneventpredicates(sumbamata) (DL7.64) and secondary-eventpredicates (parasumbamata).Thedis-tinction isChrysippean,52 andwe know little about its origin. Laterancient texts unquestionably understand both Stoic notions as ana-logues tomonadic predicates. They associate event predicateswith

49. The corresponding complete content is a quasi-conditional (parasunêmme-non).DL7.71providesits(Fregean-sounding)truth-conditions.SeeBarnesetal.1999,108–109.

50.Alternatively,onecoulduseChurch’slambdacalculus.Alambdaabstractorwould thenbind thevariables in (4) and (5), andquantificationwouldbeappliedtotheresultingexpression.Inthisway,quantificationitselfdoesnotbindvariables,asituationthatarguablycomesclosertoStoicquantificationthanFrege-stylequantificationdoes.

51. SuchcomplexmonadicpredicatesalsoseemtofeatureintheStoictheoryofaction,onwhichactions(hormaiinStoicterms)aredirectedatkatêgorêmata(AriusinStobaeusEcl.2.9b=88Wachsmuth;cf.Cic.Tusc. Disp. 4.21,quotedabove,anddiscussioninInwood1985,118–126).

52. Cf.Lucian,Vit. Auct. 21.Chrysippusalsowroteaworkonsumbamata—ifvonArnim’semendationiscorrect(DL7.191).SeealsoBarnes1996.



katêgorêmataandcertainincompletecontentsthattheyreferredtoasless-than-katêgorêmata.55

(H)Andagain, ifwhat ispredicatedofanounrequirestheadditionofacaseofanountoproduceanassertion,itiscalled(or‘saidtobe’)lessthanakatêgorêma,asinthecase of «likes» and «favours», for example «Plato likes».Foronlyifwhomisaddedtothis,forexample«Dio»,doesitproduceasadefiniteassertion«PlatolikesDio».56Am-mon.Int.44.32–45.3.57

Thecontrast isbetweenkatêgorêmata ((B) above)and less-than-katêgorêmata. A katêgorêma plus a noun (or rather a case-content

55. Less-than-katêgorêmata arealsomentionedinStephanus,Int. 11.18–21;Apoll.Dysc.,On Syntax 3.402–403; StobaeusEcl. 2.76; andScholia to Lucian, Vit. Auct. (27) 21.56–60.Thesetextsareofdifferentlevelsofreliability(seefn.57).OtherthanStobaeus,allcontainlaterancientPeripateticorPlatonisttermi-nologyandconflatethelevelsofsignifiersandsignified.

56. καὶ πάλιν ἂν μὲν τὸ τοῦ ὀνόματος κατηγορούμενον δέηται προσθήκης πτώσεως ὀνόματός τινος πρὸς τὸ ποιῆσαι ἀπόφανσιν, ἔλαττον ἢ κατηγόρημα λέγεται, ὡς ἔχει τὸ «φιλεῖ» καὶ τὸ «εὐνοεῖ», οἷον «Πλάτων φιλεῖ» τούτῳ γὰρ προστεθὲν τὸ τινά, οἷον «Δίωνα», ποιεῖ ὡρισμένην ἀπόφανσιν τὴν «Πλάτων Δίωνα φιλεῖ».Weheretrans-late ‘philei’with ‘likes’ rather than ‘loves’ tobringoutmore clearly that inancientGreektherewouldbeanexpectationoftheobjectofbeinglikedorlovedtobeprovided.

57. TheAmmoniuspassage(fromwhich(H)and(B)aboveareexcerpted)ex-plicitlysaysthatitreportsStoictheory,andindicatesthebeginningandendofthatreport. ItusesStoicterminologyandcanonicalStoicexamples. It isnotafragmenttakenfromanearlyStoictext,sinceitinterspersescompari-sonwithPeripatetictheoryandrepeatedlyconflatestheStoicdistinctionbe-tweencontentand linguisticexpression.On thepassageand its reliability,seealsoBarnes1999,205;Gaskin1997,95;andLong&Sedley1987,v.2203–204.Therearesixsurvivingparallelstothispassage:Stephanus,Int. 11.2–21;Anon. Int. 3.6–17;Apoll.Dysc.,On Syntax 3.402–403;Apoll.Dysc.,On Syntax 3.429;Apoll.Dysc.,On Pronouns115.9–13;andScholiatoLucian, Vit. Auct. (27) 21.56–60.Cf.alsoPriscian,Inst. Gram. 18.4;andSudas.v.sumbama.Mostareless reliable than Ammonius. The distinction between katêgorêmata andless-than-katêgorêmata isrecordedmoreorlessaccuratelyinallofthese:seeesp.Stephanus,Int. 11.2–21,Apoll.Dysc.,On Syntax 3.155,andtheScholiatoLucian,Vit. Auct.SeealsoGaskin1997,106.Bronowski2019,429,assumeswithoutargumentthatless-than-katêgorêmataarekatêgorêmata,somethingwedonotfindplausible.

low in§7and§8wewill see that thedistinctionbetween sumbama and parasumbamaisrelevanttothestructureofStoicpropositions:initsabsence,propositionswouldbeambiguous,whichisincompatiblewithStoictheory.)

4. Stoic polyadic predicates

4.1 Wehavearguedthat theStoickatêgorêmataarebestunderstoodastheStoicanalogues tomonadicpredicates,andwehavenotedhowadvancedthisStoicnotionisasalogicalnotion,fromtheperspectiveofcontemporaryfirst-orderlogic.ItisgenerallyassumedthatthelackofasolutiontotheproblemofmultiplegeneralityintraditionalAristo-telianlogicisinpartduetotheabsenceofpolyadicpredicates.So,theobviousnextquestioniswhethertheStoicshaddyadicandgenerallypolyadic predicates. In fact, for the nontrivialmanifestation ofmul-tiplegenerality,onlydyadicpredicatesarenecessary.Andinfact,theyhad. It is amply documented that the Stoics distinguished between

Parasumbamata in those later sources are completed with an oblique case-content. The third definition of katêgorêma (text (D), DL.7.64) entails thatkatêgorêmatarequireanuprightcase-contenttogenerateaproposition.Soareoraren’tbothsumbamataandparasumbamata katêgorêmata?Weoffertwoal-ternativeanswers.(1)ForChrysippusandsomeofhissuccessors,bothcountaskatêgorêmata.Atsomelaterpoint,theassociationsjustmentionedarein-troduced,andparasumbamataarenolongerkatêgorêmata,butareparakatêgorê-mata.Thedefinitions inDL7.64areassumedtobe inchronologicalorder(Atherton1993,253fn.31).Thefirsttwodefinitionsallowforparasumbamata asasubclassofkatêgorêmata.Thethirdonereflectsthelaterassociations,inwhich thenotionofkatêgorêmahasnarrowed. (2)TheStoicsdistinguishedbetweenagenericandaspecificsenseofmonadicpredicate(katêgorêma):inthespecificsense,itdoesnotincludeparasumbamataorparakatêgorêmata;inthegenericsense,itdoes.ThisdistinctiondifferssubstantiallyfromGaskin’s,abovefn.34.(Itisquitepossiblethatthecontentoftheexpressions‘sumbama’ and‘parasumbama’changedsomewhatovertime,andstartedoutrelatedtothecausativeaspectofkatêgorêmata,butwedisregardthispointhere,sinceevidenceistooscarceforreasonableconjecture.Generally,thestatusofthepairofexpressionssumbama/parasumbamaandtheirrelationshiptothepairkatêgorêma/parakatêgorêma isnotatall clear-cut,andseveralof the sourcesseemconfused.However, the specifichistorical and textualdifficultiesarenotrelevanttoourtopic,i.e.multiplegenerality.)



when explaining Zeno’s and Cleanthes’ views of the end or telos.59 Here‘toliveinagreement(with)’issaidtobesomethingless<thana>katêgorêma,which(sothetextimplies)becomesakatêgorêmaif‘withnature’(têi phusei)isaddedassecondargument,namely‘tolivein agreementwith nature’. This application of ‘less than a katêgorê-ma’ in conjunction with its definition and the descriptions as ‘lessthana katêgorêma’and‘notacompletekatêgorêma’suggestthattheearly Stoics conceived of katêgorêmata and less-than-katêgorêmataas ‘nested’ functions. First the (non-subject) argument place in less-than-katêgorêmataisfilled(fromFxytoFxa).Thisyieldsa(complete)katêgorêma(Fxa).Thenthesubject-argument-placeisfilled(fromFxa toFba).Thisyieldsacompletecontent.60 InourStoicnotation,thisisfrom… ⁄---Fto … /aF tob /aF,filledfrominsideout.

Consistentwith their distinction between event predicates (sum-bamata) and secondary-event predicates (parasumbamata) (§3), theStoics also defined less-than-secondary-event-predicates. The latterstandtosecondary-eventpredicatesasless-than-katêgorêmatastand

59.AriusinStobaeusEcl.II.7.6,tr.Long&Sedleymodified:‘Zenorenderedtheendas:“livinginagreement”…Hissuccessors,furtherarticulatingthis,ex-presseditthus:“livinginagreementwithnature”,sincetheytookwhatZenosaidtobeless<thana>predicate.Cleanthes…added“withnature”,andren-dereditthus:“theendislivinginagreementwithnature”.’(Τὸ δὲ τέλος ὁ μὲν Ζήνων οὕτως ἀπέδωκε, τὸ ὁμολογουμένως ζῆν…Οἱ δὲ μετὰ τοῦτον, διαρθροῦντες, οὕτως ἐξέφερον, ὁμολογουμένως τῇ φύσει ζῆν· ὑπολαβόντες ἔλαττον εἶναι <ἢ>κατηγόρημα τὸ ὑπὸ τοῦ Ζήνωνος ῥηθέν.Κλεάνθης γὰρ…προσέθηκε τῇ φύσει, καὶ οὕτως ἀπέδωκε, τέλος ἐστὶ τὸ ὁμολογουμένως τῇ φύσει ζῆν.)Stobaeus’excerptsfromAriusareaveryreliablesourceforearlyStoictheory.

60.Could it not be theotherway about, fromFxy toFay toFab?Possiblebutunlikely.TheStoicshadatermfortheFxaanalogue(katêgorêma).TheydonotofferoneforanFay analogue.«…likesDio»isa(complete)katêgorêma,morepreciselyanupright(oractive)one.ThereisnoanalogueterminStoictheoryfor«Platolikes…».Thepurposeofpassage(H) istomakeitclearwhyitwouldbewrongtothinkof«…likes---»asakatêgorêma:itwouldbewrongsince«Platolikes…»isnotaproposition;i.e.applicationofthedefinitionofkatêgorêma (as defined in text (A)) fails. This reasoning in (H) says noth-ingabouttheorderof logicalconstruction.Thefactthat«…likes---» is ‘notacompletekatêgorêma’,(fn.58)plustheStobaeuspassage(fn.59),stronglysuggeststhatyougofromsomethingthatisnotacompletekatêgorêmatosomethingthatisacompletekatêgorêma(ofthekindwehavein§§2and3),andfromtheretoaproposition.

expressedbyanoun,DL7.64,70,§2)yieldsacompletecontent.Anin-completecontentthat,in addition torequiringanuprightcase-content,requiresanobliquecase-contenttoyieldacompletecontent,isnotakatêgorêma.Itislessthanakatêgorêmainthesensethatakatêgorê-maneedsonenoun-content forproducingaproposition,whereasaless-than-katêgorêma needs(atleast)two.

Asothersnoticed,theseless-than-katêgorêmataareperfectcandi-datesforStoicdyadicpredicates(Barnes1999,205–206).(H)couldbeapassageinacontemporarylogictextbookthatexplainsdyadicpredi-cates:Thestudentmayexpect«Platolikes»tobeapropositionorcom-pletecontent,butitisnot.Justas«…iswalking»requires«Socrates»forcompletion,so«Platolikes…»stillrequiressomething,forinstance«Dio»,forcompletion.«likes»and«favours»havetwoargumentplac-es.Our textsarevery clear that this ishow theStoics conceivedofthem.‘Theycallitless than a katêgorêma,becauseitisnotacompletekatêgorêma.’58TheStoicstandardformisaplaceforanuprightcase-content(…),followedbyaplaceforanobliquecase-content(/---),fol-lowedbyless-than-a-katêgorêma;withFfor‘loves’:

(9)…/---F

WithindividualvariablesandFfor‘loves’,thiscorrespondsroughlytothecontemporary

(10)Fxy

In our most reliable evidence on less-than-katêgorêmata, some(probablyearly)Stoicsappliedthenotionof less-than-katêgorêmata

58.Cf.e.g.‘…but«Socrateslikes»,sincethe‘whom’ismissingeventhoughthesubjectistakenintheupright,[thatis]sincethepropositionisnotcomplete,theycallitless than a katêgorêma,becauseitisnotacompletekatêgorêma.’(τὴν δὲ Σωκράτης φιλεῖ, ἐπειδὴ λείπει τὸ τίνα, κἂν κατ’ εὐθεῖαν ἐλήφθη ὁ ὑποκείμενος, ἀλλ’ οὖν, ἐπεὶ μὴ αὐτοτελὴς ἡ πρότασις, ἔλαττον ἢ κατηγόρημά φασιν αὐτήν, ὅτι μὴ τέλειόν ἐστι κατηγόρημα.ScholiatoLucian,Vit. Auct.(27)21.54–58.)Thepas-sageanditscontextstronglysuggestthattheauthorconfoundspropositionsandpredicates;alternativelysomethingelsehasgonewrongwiththetext.



constantsineitherargumentplace.Withthedefinitionsofthethreekindsofsimpleaffirmativepropositionsinmind(§3),weexpectthis.63 Theillustrationsofless-than-katêgorêmatacontainonlypropernames.ThereareseveralStoicexamplesofwhatarelikelytobeStoicpolyadicpredicatesthathaveindefinitepartsinsubjectplace.64PerhapsmostvaluableisthisStoicexampleofaplausible proposition,whichwillen-gageusrepeatedly:65

(I)Apropositionisplausibleifitprovokesassent,forex-ample«Ifsomeonegavebirthtosomething,thensheisitsmother».66 (DL7.75)

(Thesentencethatexpresses theproposition in(I) iswhat linguistscallanif-clausedonkeysentence.67)Inits‘antecedent’,thepropositionin(I)givesanillustrationofapropositionthatcontainsa(potential)two-placepredicatewithtwoindefiniteparticlesasarguments:‘gavebirthto’with‘someone’and‘something’.Inform,thispredicateisanal-ogousto

63.Theextantdefinitionsof‘beinglessthanakatêgorêma’(textsinfns.56,58)usenon-Stoic terminology for thearguments:onoma, ptôseôs onomatos,etc.,whereasDL7.70has theStoicorthê ptôsis, deiktikê orthê ptôsis andaoriston morion.

64. «Ifsome godtellsyouthatthis onewillberich,this onewillberich»(SE M8.308,Greek in fn. 69) combines indefinitepart, case-content anddemonstrativecase-content.TheChrysippean-based «if someonewasborn in the signofSirius,thatonewillnotdieatsea»(Cic.Fat.12–14)arguablycontains«…borninthesignof---»and«…willdie(at)---».«WhensomeoneisinMegara,heisnotinAthens»mighthavebeenthoughttocontain«…isin---».

65.Aplausibleproposition(pithanon axiôma)isaStoicpropositionthatinclinesustowardsassent,eveniffalse(DL7.75).Cf.theChrysippeanbooktitle:On plausible conditionals (DL7.190),whichsuggestsexamplesliketheoneintext(I)mayoriginatewithChrysippus.SeealsoBarnes1985.

66. DL7.75:εἴ τίς τι ἔτεκεν, ἐκείνη ἐκείνου μήτηρ ἐστί.

67. Donkey sentencescontainapronounwhoseanaphoricreferenceisintuitivelyclearbutwhosesyntactic functionescapes straightforward linguisticanaly-sis.Astandardexampleis‘Everymanwhoownsadonkeybeatsit’;seee.g.Geach1962.Anif-clause donkey sentenceisadonkeysentencethatstartswithanif-clauseinsteadofe.g.auniversalquantifyingexpression.

tokatêgorêmata.AStoicexample is ‘cares for’ (‘to…for --- there iscare’), completedas ‘Socratescares forAlcibiades’,withG for ‘caresfor’:61

(11)/…/---G

Evidently, these are also dyadic predicates. The contemporary ana-logue,withGfor‘caresfor’,is

(12)Gxy

Wehavebeenunable tofindexamples inwhichboth argumentplacesarefilledwiththesameargument,e.g.DiolovesDio(Δίων Δίωνα φιλεῖ),butwekeeplooking.TheStoicdefinitionsoftheirtwokindsofdyadicpredicatesarecompatiblewiththesameargumentfillingbothargumentplaces.Asanalogy,weofferthatinStoicdefinitionsofnon-simplepropositionsitisstatedthatthesamepropositioncanbeusedtwice,e.g.p → p (SE M8.93), andintheirtheoryofindemonstrables,argumentsoftheformp → p, p ⱶ pcountasfirstindemonstrablesjustasthoseofformp → q, p ⱶ q(Barnesetal1999,136).Sincenoneofthesepointsiscompelling,weleavethequestionopen.62

4.2 Nextweargue that, in theStoicview, less-than-katêgorêmatacouldtakeStoicquantifyingexpressions(tis, tis-ekeinos)insteadofindividual61. ThusAmmon.Int. 45.3–5,tr.Barnesmodified:‘Ifwhatispredicatedofacase-

content(ptôsis)needstobeputtogetherwithanotherobliquecase-contenttomakeanassertion,itissaidtobeless than a secondary-event predicate:thus“thereiscare”e.g.“toSocratesforAlcibiadesthereiscare”’(i.e.‘SocratescaresforAlcibiades’). (ἂν δὲ τὸ τῆς πτώσεως κατηγορούμενον ᾖ τὸ δεόμενον ἑτέρᾳ συνταχθῆναι πλαγίᾳ πτώσει πρὸς τὸ ποιῆσαι ἀπόφανσιν, ἔλαττον ἢ παρασύμβαμα λέγεται, ὡς ἔχει τὸ μέλει, οἷον ‘Σωκράτει Ἀλκιβιάδου μέλει’.)See furtherdiscus-sioninBarnes1999,205.

62.Arguably, someone’s killing oneself, cutting one’s own hair, etc. are differ-entkindsofrelationsfromsomeone’skillingsomeoneelse,cuttingsomeoneelse’shair(tryit!),etc.Thegeneralassumptionincontemporaryfirst-orderlogicthatonecanfillallargumentplaceswiththesameargumentseemsnat-ural if, likeFrege,oneusesmathematicalexampleswhenintroducingfunc-tionswithtwoarguments(Frege1891,27–28).Itisnotessentialtothenotionofapolyadicpredicate.



Doesitnotcontainapredicate«… tells--- that*** willberich»,rep-resentableas

(14)…/---***H

andanalogousinformtothecontemporary

(15)Hxyz ?

We doubt it: one would expect the Stoics to have treated indirectspeechdifferently(cf.thefunctionof‘said’intheStoicunmethodical ar-guments,e.g.Alex.An. Pr.22.17–26).ClearevidenceforStoicthree-or-more-placepredicatesisthusabsent.Ofcourse,foralogictocontainmultiplegeneralityand to reflecton itsproblems,dyadicpredicatessufficeentirely.

5. Variable-free predicate logic: monadic predicate logic

TheStoicshavenoindividualvariables.(Thequestionwhethertheyhadpropositionalvariablesisofnoconcernhere.)A fortiori,theyhavenovariable-bindingquantifiers.Wearelookingatavariable-free predi-cate logic.Suchathingisnotunknownincontemporarylogic.Quine’sshort and splendid paper ‘Variables Explained Away’ provides amethodthatallowsonetoreplaceFrege-styleindividualvariablesandquantifierswithasetofoperators.

TheStoicsappear todistinguishbetweentwokindsof indefinitepropositions. The first is an analogue to contemporary existentialpropositions.

(K) According to them (i.e. the Stoics), indefinite arethose<propositions>inwhichanindefinitepart(morion) governs,suchas«someoneissitting».70 (SE M8.97)

70. SE M8.97:ἀόριστα δέ ἐστι κατ’ αὐτοὺς ἐν οἷς ἀόριστόν τι κυριεύει μόριον, οἷον τὶς κάθηται.

(13)Fxy

InStoicterms,«…to---gavebirth»wouldbelessthanakatêgorêma.Itappearsthenthat, intheStoicview, less-than-katêgorêmatacouldtakeStoicquantifyingexpressionsalongwithindividualconstantsordemonstrativesineitherargumentplace.SotheStoicshadthekindofnotionofdyadicpredicatethatisrequiredformultiplegenerality.

4.3. Logiciansareboundtoaskatthispoint:Whatabout(analoguesto)predicateswithmorethantwoargumentplaces?Alas,noStoicsourceexplicitlymentionsordiscussessuchcases.68TheStoicscouldhaveeasilydefinedthree-or-more-placepredicatesrecursivelyonthebasisof theirdefinitionofdyadicpredicates.For the termofa three-placepredicate, the phrase ‘less-than-a’ could be prefixed to ‘less-than-a-katêgorêma’. Generally, if a (less-than-a)n-katêgorêma is a polyadicpredicate, then a (less-than-a)n+1-katêgorêma is a polyadic predicate.Wecouldalsosaythata(less-than-a)n-katêgorêmaistheanaloguetoan(n+1)-placepredicate,withn≥0.Thisisnotthatfarfetched:theSto-icsuseiterativedefinitionsinmanypartsoftheirlogic.Conjunctivepropositionsaresaidtoconsistoftwopropositionsandaconjunctiveconnective(DL7.72),andnegationsaredefinedaspropositionsthatstartwith theprefix ‘not:’ (Apul.Herm. 191.6–11;SE M 8.103; cf. SE M8.89;DL7.73).Thisaccountsforconjunctiveandnegativepropo-sitionsofanycomplexity(Barnesetal.1999,105–106).Still,thereisnodirectproofthattheStoicsdefinedthree-or-more-placepredicates.WhataboutthefollowingStoicexample?

(J)Ifsome godtellsyouthatthis onewillberich,thisonewillberich.69 (SE M8.308,cf.PH2.141,italicsours)

68.Somemay think that the secondStoicdefinitionof katêgorêma (text (D)) attestsagenericuseofkatêgorêmaandreferstopossibleargumentplaces(above §2). Accordingly they may suggest that such polyadic predicateswouldbecovered.Wehaveargued(above§2) that, ifunderstoodasearlyStoic,(D)shouldnotbereadinthisway.

69. SE M8.308:εἴ τίς σοι θεῶν εἶπεν ὅτι πλουτήσει οὗτος, πλουτήσειοὗτος.



thing, since they can be—truthfully—expressed in the presence ofany particularwalking, sitting, etc. personor thing. It is immaterialwhichpersonorthing.74

Non-referenceisconfirmedbytheStoictruth-conditionsforsimple(affirmative)existential indefinite propositions.Theindefiniteproposition«Someoneiswalking»issaidtobetruepreciselyifthecorrespondingdefinite proposition («this one iswalking» accompaniedby an indi-catingofaspecificperson) is true(SE M8.98).75 (Byacorresponding definite or middle propositionwemeanhenceforwardapropositionthatdiffers from the indefinite proposition at issue only in that it has acase-contentordemonstrativecase-content inplaceofan indefinitepart.)Wecanconfidentlyconcludethattheindefinitepropositionsofform(16)areStoicexistentialpropositionswithmonadicpredicatesandonequantifierexpression.Wecallthemmonadicindefiniteexis-tentialpropositions.

Thesecondkindof indefiniteproposition issomewhatharder topindown.BasedonwhatwesaidabouttheStoicnotionofmonadicpredicate,wearguethatthefullsentencebehind(L)alsoprovidesanexampleofamonadicindefiniteuniversalproposition.Thisrequiresacloselookatthatfullsentence.

(N)Anindefinite[simpleproposition]isonethatiscom-posedfromanindefinitepartor indefinitepartsand<a

74.Asnotedabove,Stoicreferringexpressionsappear tohavea ‘case-content’(ptôsis)atthecontentlevel.Propernames,demonstrativesandnouns(incer-tain functions) eachhave a correspondingptôsis. Fordemonstratives, theyarecalled‘demonstrativecase-contents’.Theparticlestisandtis-ekeinoshavecorresponding‘indefiniteparts’(aorista moria)atthecontentlevel.Noindefi-niteptôsisorcase-contentisevermentioned.Neither‘tis’northeanaphoric‘ekeinos’(forwhichseebelow)northecombinationofthetwoareevercalledptôsisinStoictexts.SeealsoCrivelli1994a,189.

75. Adefinitepropositionistrueifthatwhichispointedatfallsunderthepredi-cate(SE M8.100).Theaccountofthetruth-conditionsofindefiniteproposi-tions leads towell-knownproblems for certainStoicpropositions, suchas«someoneisdead»and«thisoneisdead»,whichwecanhereignore,seeforavarietyof interpretationBarnesetal.1999,98–101;Bailey2014,281–284;Durand2019,§§29–34;Bronowski2019,415–418.

(L) Anindefinite[simpleproposition]isonethatiscom-posed from an indefinite part … and <a katêgorêma>,suchas«someoneiswalking»….71 (DL7.70)

Thisfirstkindofindefinitepropositioniscomposedofoneindefinitepart72andamonadicpredicate.Theindefinitepartgoverns—thatis,ithaswidestscope—inthesepropositions.GiventheStoicScopePrin-ciple(§1), ‘someone’or‘something’(tis, ti)isthusthecharacterizingexpressioninsentencesthatsignifytheseindefinitepropositions.Theexamplesin(K)and(L)areexamplesofsuchindefinitesimplepropo-sitions.With τ (fromtis, tinos, etc.)fortheindefinitepart,wecanrepre-senttheir(Stoic)formas

(16)τF

Thecontemporaryanalogueis

(17)∃x Fx

This isconfirmedby theexplanationofwhy indefinitepropositionsareindefinite

(M)«Someoneiswalking»isindefinite,sinceitdoesnotdemarcateanyoneoftheparticularwalkingindividuals.Foritcangenerallybeexpressedinthepresenceofanyoftheparticularwalkingindividuals.73 (SE M8.97)

Weunderstand(M)asconveying that indefinitepropositionsdo not refer.Thereasonisthattheydonotpickoutanyparticularpersonor

71. DL7.70:ἀόριστον δέ ἐστι τὸ συνεστὸς ἐξ ἀορίστου μορίου…καὶ κατηγορήματος, οἷον τὶς περιπατεῖ….WeacceptvonArnim’sadditionof<καὶ κατηγορήματος>,anemendationwhichseemsuncontested.

72. Texts(K) and(L)implythattheindefinitemorionispartofthepropositionandhenceatthelevelofcontent.Wetranslatemorionas‘part’anduse‘par-ticle’forthecorrespondinglinguisticexpression.ThereisnoStoicequivalenttothisdistinction;itismerelyforclarity.

73. SE M8.97:τὸ μὲν οὖν «τὶς περιπατεῖ» ἀόριστόν ἐστιν, ἐπεὶ οὐκ ἀφώρικέ τινα τῶν ἐπὶ μέρους περιπατούντων· κοινῶς γὰρ ἐφ’ ἑκάστου αὐτῶν ἐκφέρεσθαι δύναται· (‘inthepresenceof’:LSJs.v.ἐπίA.1.2.e).



call an indefinite conditional.Of theseagoodnumberof examplessurvive.79Thevery sameexampleisalsoattestedinAugustineDial.3,which is basedon Stoic logic.80 (It speaks in favour of the emenda-tionthatitresultsinaStoiccanonicalformulationandaknownStoicexample.)

It isoftenassumed that the indefinite conditionalsareStoicnon-simple propositions,moreprecisely, conditionals (sunêmmena).81 (Re-callthatStoicpropositionsareeithersimpleornon-simple,§1.)Thisishoweverbothimplausibleandproblematic.Norisitentailedbythename: ‘indefinite’mayfunctionasanalienansadjective. It is implau-sible,sincetheStoics,intheirlanguageregimentation,requirethatinaproperconditionalthesubjectexpressionoftheantecedentsentence

impliesthatekeinoshasademonstrativeuse(Gal.Plac. Hipp. Plat.2.2.7–12)asdetrimentaltotheclaimthatinlogictheStoicsregimentedtheuseofthepairtis/ekeinosinsuchawaythatitcountedasapairofindefinitepart(icle)sinwhichthesecondpartdoesanaphoricduty:Thepairofparticlesworkstogetherasaunit,justliketheStoicpairsofconnectives kai…kai---andêtoi … ê---.SeealsoCrivelli1994a,196;Caston1999,196–197;Durand2019,§§43–45,51–54.WenotethatekeinosseemsnevertooccurindemonstrativefunctioninthemanydemonstrativeexamplesinStoiclogic.

79.For someof them, seeCic. Fat. 15;DL7.75; SE M 11.8–13, threeexamples,for helpful discussion ofwhich, seeCrivelli 1994b, 498.Cf. alsoCic.Acad. 2.20–21;SE M1.86;Plutarch,Com. Not. 1080c.Epict.,Diss. 2.20.2–3seemstoprovideevidenceastothereformulationofanegativeuniversal,‘NoFisG’.Cf.Crivelli1994a,n.36.

80.Cf.theconditionalsmentionedatDL7.78,«IfDioiswalking,Dioismoving»(withanuncontestedemendationbyvonArmin);andGellius16.8.9«IfPlatoiswalking,Platoismoving».ConsistentwiththestandardpracticeinStoiclogic,theseconditionalsareofferedasstockexamplesandareassumedtrue.Weargue that (N) provides (18) as another stockexampleof a true condi-tional,butnowonethatis indefinite.Simpleprobabilitycalculationsshowthatthisismorelikelythanthatwehave«someoneiswalking»and«thatoneismoving»astwoisolatedexamples,whichjusthappentoformafamiliarstockexamplewhenputtogether.Cf.SE M8.100,where«someoneiswalk-ing»and«someoneissitting»aregivenastwoisolatedexamplesofindefinitepropositions.

81. SoFrede1974,59fn.11;Goulet1978,205;Crivelli1994a,198–199;Bobzien1998,156–159.ExceptionsareDurand2018,167–169;Frede1974,64–67(notentirelyclear);andpossiblyLong&Sedley1987,v.1,207(notentirelycleareither).

katêgorêma>, such as «someone iswalking», «<if some-oneiswalking>,heismoving».76 (DL7.70,continuationof(L)withgapin(L)filled)

Weassumethatthesecondexampleisofthesecondkind,thatis,anindefinite simplepropositionwithmore than one indefinite part andamonadicpredicate.Weagreewithandadopt(in(N))asuggestedemendationthatassumesatextuallacunaby—veryplausible—hap-lography.77Thenwehaveassecondexample

(18)«Ifsomeoneiswalking,heismoving.»

Itcontainstheindefiniteparts‘someone’,whichisnon-anaphoric,and‘thatone’,whichisanaphoricwithcross-referencetothefirstindefinitepart.78Theresultis,inawell-attestedcanonicalform,whattheStoics

76. DL7.70:ἀόριστον δέ ἐστι τὸ συνεστὸς ἐξ ἀορίστου μορίου ἢ ἀορίστων μορίων <καὶ κατηγορήματος>, οἷον τὶς περιπατεῖ,<εἰ τὶς περιπατεῖ,> ἐκεῖνος κινεῖται.Morelit-erally,ἐκεῖνος κινεῖταιtranslatesas‘that oneismoving’.Sinceweargue(below)thattheregimentedsentencesthatexpressuniversalpropositionsuseformsofἐκεῖνοςasanaphoric pronouns, thenatural translation inEnglish iswithapersonalpronoun.(HerewefollowCrivelli1994a.)Thishastheadditionaladvantage thatelementsofcaseandgender in theGreekcanberendereddirectly.Insomecasesinwhichitincreasesclarity,werender‘thatone’.

77. The textwithout emendation ishighlypuzzling.Cf. thedetailed fn. Frede1974,59–60,andCrivelli 1994a, 189: ‘Onewonderswhybothexamplesof-fered byDiogenes are simple indefinite propositions consisting of one in-definiteparticleandonepredicate,whilstnoexample isgivenofasimpleindefinitepropositionconsistingofmorethanoneindefiniteparticleandonepredicate.’TheemendationwassuggestedbyEgli1981,inconjunctionwithanalternative tovonArnim’semendation(fn.71above),whichwedonotadopt,andwhichdoesnotchangethesenseofthedefinitionasgivenhere.Egli’semendationisalsoadoptedbyHülser1987–88,1142(=FDS914).Criv-ellitriestomakesenseofthetextbystipulatingaStoicdistinctionbetweenanaphoric and non-anaphoric simple indefinite propositions (1994a, 189).However,notextsattestsuchadistinction,andweconsidertheconjectureofanaphoricsimpleindefinitepropositionsphilosophicallyawkward.

78.Kneale&Kneale1962,146,suggestthatekeinosmustbeanaphoric,cross-re-ferringtoanantecedentinaconditional.WedonotclaimthattheStoicscon-sideredoccurrencesofekeinoswithoutaprecedingformoftisinaprecedingclauseasanaphoric.(Thatwouldbesilly.)Equallydowenotconsiderthefactthat,inaworkonpsychologyinastrangeetymologicalexplanationof‘egô’uti-lizedtocorroboratetheStoicplacementofthemindintheheart,Chrysippus



moregenerallybetweenconditionalsanduniversalpropositions.(Forexample, intermsof traditionalFregean-Russellian logic, in(19)theconditionalconnectivehaswidescope;in(18)theuniversallyquanti-fyingindefinitepartsdo.)Thattheywerenotunawareofthisisclearfromthefactthat,intheStoicview,

(20)Ifsomethingishuman,itisamortal,rationalanimal.

expressesauniversalproposition(katholikon [axiôma]),inthiscaseadefinition,andsomeonewhoutters(20)saysthesamepropositionassomeonewhoutters

(21)(Every)humanisamortalrationalanimal.85

In(21),andaccordingtotheStoicScopePrinciple,‘every’wouldhavelargest scope.86Given the stated synonymy of (20) and (21), ‘some-thing… it…’must have largest scope in (20). Accordingly, in ourStoic examples, the two-particle quantifying expression always haseachpart as far to the frontof its clause as is possible. For theSto-ics,(21)expressesasimpleproposition,andwith(20)and(21),onesaysthesameproposition.Butthesamepropositioncannotbeboth

85. SE M11.8:‘fortheonesaying“Manisamortalrationalanimal”saysthesamethinginmeaning,thoughdifferentinexpression,astheonesaying“ifsome-thingisaman,itisamortalrationalanimal”’(ὁ γὰρ εἰπὼν“ἄνθρωπός ἐστι ζῷον λογικὸν θνητόν”τῷ εἰπόντι“εἴ τί ἐστιν ἄνθρωπος, ἐκεῖνο ζῷόν ἐστι λογικὸν θνητόν”τῇ μὲν δυνάμει τὸ αὐτὸ λέγει, τῇ δὲ φωνῇ διάφορον).Whatonesays(legei)istheproposition(e.g.DL7.66).Hencethosetwospeakerssaythesameproposi-tion.Thefollowingsentence(SE M11.9)leavesnodoubtthatthesingularnounwithoutarticle‘man’isunderstooduniversally,ascoveringeveryman(i.e.humanbeing).Forkatholikonusedforanindefiniteconditional,cf.SE M 1.86,Epict.Diss.2.20.2–3,Plut.Com. Not. 1080c;seealsoCrivelli1994a,fn.36,andCaston1999,195–199,esp.197.

86.NotetheexampleSE M11.10–11:‘Forthepersonwhocarriesoutadivisioninthismanner,“amonghumanbeings,someareGreeksandsomearebar-barians[i.e.non-Greekspeakers]”sayssomethingequal to“ifsomethingsare human, they either are Greeks or are barbarians”’ (ὁ γὰρ τρόπῳ τῷδε διαιρούμενος“τῶν ἀνθρώπων οἱ μέν εἰσιν Ἕλληνες, οἱ δὲ βάρβαροι”ἴσον τι λέγει τῷ“εἰ τινές εἰσιν ἄνθρωποι, ἐκεῖνοι ἢ Ἕλληνές εἰσιν ἢ βάρβαροι”).Herethelead-ingexpression‘amonghumanbeings’(τῶν ἀνθρώπων)haswidestscope.ThisalsoshowsthattheStoicsconsidersuchpropositions,withagenericnounwithoutarticleassubjectexpression,asauniversalpropositionofsorts.

berepeatedintheconsequentsentence,ifthesubjectisthesame;e.g.“IfPlatolives,thenPlatobreathes”.Thepractise(wecallitanaphora re-moval)appliestoallnon-simplepropositionsandisoften,thoughnotuniformly,followedinthesources.82ItensuresthatitisdiscernibleatthesentencelevelthatwehavetwoindependentsimplepropositionsthatarecombinedinaccordancewiththeStoicdefinitionofacondi-tionalintoanon-simpleone.Itbecomesthusdiscerniblethatthecon-sequentisdetachablee.g.withaStoicfirstindemonstrable,whichhastheformofmodus ponens.Therelevantnon-simpleproperconditional,composedoftwosimpleindefinitepropositions,isthisconditional:

(19)Ifsomeoneiswalking,someoneismoving.83

(19)satisfies theStoicdefinitionofconditionalsasnon-simpleprop-ositions that are composedwith the conditional connective ‘if’ (DL 7.71).Bycontrast,Stoicindefiniteconditionalsresistanaphoraremov-al,andneitherdotheysatisfythedefinitionoftheconditional,norisitpossibletodetachtheir‘consequent’withmodus ponens.Itwouldbeabsurd toassumethat theStoicsdidnotsee thedifferencebetween(18)and(19).Andthereisevidencethattheydidseeit.84

Theassumptionthattheindefiniteconditionalsarenon-simpleisproblematic,sincetheStoicswouldbeshowntobeunawarenotjustofthedifferencebetweenthesentencesexpressing(18)and(19),but

82.Cf.e.g.DL7.77,78,80;SE M8.246,252,254,305,308,423;SE PH2.105,106,141;Gellius16.8.9;Gal.Inst. Log.iv.1;Simp.Phys.1300;Alex.An. Pr.345,Cic.Fat12.NosurvivingancientsourceexplicitlydiscussestheStoicconventionof anaphora removal.Webelieve that the frequencywithwhichanaphoraremovaloccurs, togetherwiththefactthat itsoundsasunidiomaticandisasrareinancientGreekasinEnglish,issufficientevidence.Itislikelythatbecause it isnot idiomatic insomecases inwhichthisregimentationisnotfollowed,thesearescribalchanges;andthatforthesamereasoninother,es-peciallylater,ancientcases,someauthorsareunawareoftheconventionanddisregardit.SeeBarnesetal.1999,104–105,fordiscussionofStoicanaphoraremoval.

83.Cf.Barnesetal.1999,112;alsoEgli2000,20–21.

84.Cic.Fat.15;seeBobzien1998,156–159.Egli2000,20–21,arguesplausiblythattheStoicsolutiontotheNobodyParadoxshowsthattheStoicsdistinguishedbetweenapairofsentencessimilarto(18)and(19).



theproposition,plusamonadicpredicateasdefinedbytheStoics,forwhichwehaveevidenceelsewhere.87

There are thenStoic indefiniteuniversal propositions to comple-menttheStoicindefiniteexistentialpropositions;bothkindsaresim-plepropositions.88There is evidence for their semantics, too (SE M 11.8–13).89Itisbasedonwhatarecalledsubordinateinstancesofuni-versalpropositions.90Thesearethemselvespropositions:Theycanbefalse(ibid.),andaccordingly,true.Theinstancesarepropercondition-alswithdefinite(ormiddle,seebelow§6)componentpropositions,inwhichthecase-contentisthesameinbothcomponentpropositions.Iftherecanbeafalsesubordinateinstance,theindefiniteuniversalisfalse.Weinferthatiftherecannot beanyfalsesubordinateinstances,theindefiniteuniversalistrue.

Why ‘cannotbeany’ andnot ‘areno’?Weassume that the ‘if’ intheuniversals,orindefiniteconditionals,hasthesamelogicalstrengthas in theStoicproper conditionals (cf.Cic.Fat 11–17;Bobzien 1998,156–159).Thisisalsosuggestedbytheirname.So‘canbe’isrequiredbythe—Chrysippean—truth-conditionsforStoicconditionals,whichcontainamodal element: a conditional is truewhen its antecedentandthenegationofitsconsequentareincompatible(DL7.73).How-ever,Chrysippuscouldanddidalsoaccountforakindofnon-simplepropositionthatcorrespondstomaterialconditionals,i.e.tothePhilo-nian conditional.He(andsomeotherStoics,itseems)usednegationsof conjunctions with the antecedent of a Philonian conditional asfirstconjunctandthecontradictoryoftheconsequentassecondcon-junctasthecorrectformofthatPhilonianconditional.(ThePhilonian

87.ForadditionalevidencethattheStoicswereawarethatindefinitecondition-alswerenotnon-simplepropositions(andassuchnotthekindofcondition-alsasarepartofpropositionallogic),see§9.

88.Wedonotherediscussthequestionwhetherthetwoaredualsandinterde-finable.Fortheirnegations,seeBarnesetal.1999,113–114;Egli2000.

89.SeealsoCrivelli1994a,193–194,199–202;Barnesetal.1999,113;confirmedbyDL7.75,seebelow§6.

90. ὑποτασσομένου (SE M11.11);ὑποταχθέντος (SE M11.9).

simpleandnon-simple.Hence(20),andbygeneralization(18),andStoic indefinite conditionals generally, express simple propositions.As such, they shouldconsistof indefinitepartsandapredicate (DL 7.70,above).Thisisexactlywhattheyconsistof.Theindefiniteparts‘something’and‘it’worktogether as a unit(justliketheparts‘either…or…’,‘and…and…’innon-simplepropositions(§1)).Theremainderofthecontent

(22)«If…iswalking,…ismoving»

isamonadicpredicatebythedefinitionofmonadicpredicate(katêgorê-ma).WehaveevidencethattheStoics,Chrysippusinparticular,hadthatsortofpredicate (above(4), (5)).The logical formof thepropo-sitions(andtheregimentedsentencesexpressingthem)canthenberepresentedas

(23)Conditionalconjunction+indefiniteparticle(τi)(+cases) + finite verb + corresponding anaphoric particle(εi)(+cases)+finiteverb.

We always assume that any Stoic regimentation is (inGreek) gram-maticallycorrect.Incontemporarysymbolismwithvariables,(22)hastheformofamonadicpredicate

(24)Fx → Gx

andtheindefiniteproposition(18)hastheform

(25)∀x (Fx → Gx)

SowecanmakesenseofaperplexingStoicdefinitionwithexamples(in(N))if(i)weacceptaveryplausibleemendationthatestablishesaStoicstockexampleinStoiccanonicalform,and(ii)wethentaketherestoredtext to mean exactly what it says.ThissolvestheconundrumofStoic indefiniteconditionals:Theyarenotnon-simplebutsimplepropositions,justasthetextimplies.(iii)Theseareconstructedwithtwoindefinitepartsthatformalogicalunitandhavewidestscopein



‘ ’ indicateswhatevermodalforceChrysippeanindefiniteuniversalconditionalshave,andthesecondto∀x (Fx ⊃ Gx)or∀x¬(Fx ∧ ¬Gx), with‘⊃’formaterialimplication.

Just as the indefinite part «someone» in Stoic existential propo-sitions is non-referring (above), so are the indefinite parts «some-one—that one». The relation between the existential and its corre-lateddefinitepropositionhasaparallel in the relationbetween theuniversalanditssubordinatedproperconditionalswiththesameref-erentforthesubjectexpressioninbothcomponentpropositions.Intheexistentialcase,fortruth,thepresenceofatruecorrelateddefiniteisrequired.Intheuniversalcase,fortruth,theabsenceof(thepossi-bilityof)afalsesubordinatedconditionalisrequired.

WenowreturntoQuine’s‘VariablesExplainedAway’.Quinenotesthat insimplemonadicexistentialpredicates (corresponding tooneoccurrence of the variable), the variable can be dropped, and that‘some(G)isF’isabetterformulationthan‘some(G) x issuchthatxisF’(343).OntheStoicside,forsuchcases,theScopePrincipledoesallthatisnecessary.Positionismarkedbytheindefinitepart‘someone/something’,andthedesiredmonadicpredicateisabstractedasbeingeverythingotherthanthispart,asperthedefinitionofindefiniteprop-ositions.SotheStoic ‘someone(orsomething)F’(forshortτF,with‘τ’ for tis, tinos, etc.) combines theexistenceprefixwith thevariableitbinds—inmodernterms‘(somethingx issuchthatx) F’.ForStoicmonadicuniversalpredicates,invariable-bindingquantifierformula-tions,thesamevariableoccurstwice.HeretheScopePrinciplecom-binedwithanaphoricreferencedoesall that isnecessary. ‘(If)some-one—thatone’combinesmarkingthepositionsof(whatwouldbe)the twooccurrencesof thevariable(oneposition isgivenby ‘some-one’,theotherby‘thatone’)withtheanaloguetotheuniversalitypre-fix‘everythingxissuchthat’.WegivetheformofsentencesexpressingsuchpropositionsasτiF → εiG,wherethepredicatecomesinthethreeparts→, F and G,with‘ε’fortheanaphoricexpressions‘he’(ekeinos),

(below§§6and9).(ii)Webelievethetextssuggestthatuniversalquantifica-tionhaslargestscope(below§7).

conditionals are thus accurately rendered as truth-functional, sincetheyarereallynegatedconjunctionsandStoicnegationandconjunc-tionarebothtruth-functional.)Moreover,Chrysippus’logiccontainedcorrespondingindefinitenegationsofconjunctions,too.Weassume,byanalogytoChrysippeanindefiniteconditionals,thattheChrysippeannegated conjunctions that replace Philonian conditionals are truewhenthereare nofalsesubordinateinstances.(Cf.Cic.Fat11–17andBobzien1998,156–159,onthispoint.)Fromacontemporaryperspec-tive,theStoicshavetwokindsofuniversalpropositions:onecontain-ingamodalelement,theotherbeingnon-modalandtruth-functional.Fordifferentiation,wecallthemtheChrysippean universalandthe‘Phi-lonian’ universal.(Nosuchnamesareknownfromantiquity.)Wegen-erallyconsiderChrysippeanconditionalsandChrysippeanindefiniteconditionalsasparadigm.Thecasefornegatedconjunctionsisanalo-gous,andwementionthemonlyoccasionally.

Virtually all extant examplesof Stoicuniversals are restricted uni-versalsinthesensethatuniversalquantifiershaveonlyconditionalsornegatedconjunctions in their immediate scope:Not ‘everyone ismoving’,but‘ifsomeoneiswalking,thatoneismoving’or‘not:bothsomeoneiswalkingandnot:thatoneismoving’.Theclosesttonon-restricted universals is ‘if some things exist, those things either aregoodorarebadorareindifferent’.91(FortheStoics,‘existence’(einai) isreservedtobodies,andsoisnarrowerthantheontologicalcategory‘something’(ti),whichincludesbothbodiesandnon-bodies.)Still,theexample suffices to illustrateby analogyhowunrestricted universalscouldbeexpressedwithintheStoicframework:‘ifsomethingissome-thing,thenthatthing…’(ei ti ti estin, ekeinon…).If so,allChrysippeanuniversals could be expressed canonically in conditional form.Cor-respondingly,all‘Philonian’universalscouldbeexpressedcanonicallyintheformofnegatedconjunctions(Cic.Fat15–16).Veryroughly,thefirstwouldcorrespondtothecontemporary∀x (Fx ⊃ Gx),92where

91. SE M 11.11:εἴ τινά ἐστιν ὄντα, ἐκεῖνα ἤτοι ἀγαθά ἐστιν ἢ κακά ἐστιν ἢ ἀδιάφορα.

92.Whynot ∀x (Fx ⊃ Gx)?(i)Thesurvivingargumentformdoesnotrequireit



‘Someone is walking’ is true precisely if the corresponding definitepropositionistrue(SE M8.98;§5above;Crivelli1994a,190–193).Bysimplegeneralization,weobtainconvincingStoictruth-conditionsfordyadic(double)existentialindefinitepropositions,bywhichwemeanthosethatcontainadyadicpredicateandtwoindefiniteexpressions.Themultiplygeneral «Someone loves someone» is true if the corre-spondingdoublydefinite«Thisone(indicating a specific person)lovesthisone(indicating a specific person)»istrue.Heretherearetwoactsofindicatingdirectedattwodifferentpeople(cf.PHerc307Col.IV),orpossiblyatthesameperson.Weused, e,…asindividualconstantsindemonstrativeatomicpropositions that are accompaniedby the rel-evantadmissibleactsofindicating.Then,incontemporaryterms,∃x∃y FxyistrueiffsomeFdeistrue.

WhatabouttheStoicdyadicuniversalsor∀∀cases?Itisrealisticto assume that in (26) «…of --- is themother» (usingGreekword-orderagain)isalsoadyadicpredicate(less-than-a-katêgorêma).94Asawhole,(26)thenillustratestheStoicversionofadyadicpredicatewithtworestricteduniversalquantifications.95Thepredicate«If…to---gavebirth,…of---isthemother»isinformanalogousto

(29)Fxy → Gxy…/---F→…/---G

Theproposition (26)includestwopairsofquantifyingparts,‘someone-she’ and ‘something-it’, that take the two pairs of argument places.This ispreciselywhatweexpect fromwhatweknowabout themo-nadiccases.96Theformof(26)isthenanalogousto

94.Notethatthemainclause‘thatoneismothertoanotherone’containstwoanaphoricexpressionswithouttheirreferent-expressions,andassuchdoesnotexpressaStoicproposition.Itisnotacompletecontent.

95. Therearenoexamplesofunrestricteduniversalizationwithtwo-placepredi-cates(analoguesto∀x∀y Fxy).However,recalltheuniversalpropositionatSE M11.11,discussedabovein§5andbelowin§7:«ifsomethingexists,thenitiseithergoodorbadorindifferent».

96.ItalsogivesadditionalsupporttotheDL7.70readingof‘someone’and‘he’(ekeinos)abovein§2.

she(ekeinê),etc.,andtheanaphoricrelationindicatedbysubscripts‘i’, ‘k ’,….Generally,inthetwoStoicmonadiccases,acombinationoftheScopePrincipleandanaphoricreferencesuffices.Problemsofmulti-plegeneralitycanoccuronlyinindefinitepropositionswithmorethanonequantification.

6. Variable-free predicate logic: Polyadic predicate logic

ToseewhetherStoiclogiccouldhandletheproblemofmultiplegen-erality,weneedtoconsiderpropositionswithmorethanonequantify-ingexpression.Inkeepingwiththeevidence,weconfineourselvestodyadicpredicatesandpropositionswithtwoquantifyingexpressions.Therearefoursuchcases,intermsofmodernquantifierexpressions:∃∃, ∀∀, ∃∀, ∀∃.Allissuesofambiguitiesarepostponedto§7and§8.

For thefirst twocases, consideragain theStoic if-clausedonkeysentencefromtext(I).Forourpurposeitispreferable,hereandbelow,toretaintheGreekwordorder,eventhoughinEnglishitisunidiom-aticandpotentiallycringe-inducing:

(26) «If someone (male or female) to something gavebirth,thensheofitisthemother».93

The‘antecedent’providesuswithaStoic∃∃propositionthatcontainsadyadicpredicate(less-than-a-katêgorêma):

(27)«Someonetosomethinggavebirth.»

Thisappearstobethecanonicalorregimentedpositioningofthein-definiteparts,paralleltothatoftheuprightandobliquecase-contentsinmiddlepropositionswithdyadicpredicates(§4).WithF for ‘gavebirthto’,theformof(27)isanalogousto

(28)∃x∃y Fxy τ1 /τ2 F

We have an explicit account of the Stoic truth-conditions for in-definitesonlyfor(affirmative)monadic existential indefinite propositions:

93. DL7.75:εἴ τίς τι ἔτεκεν, ἐκείνη ἐκείνου μήτηρ ἐστί.



(32)and(33a)canbeexpressedwith‘every’inGreekasinEnglish,e.g.‘everyhumanloveseveryhuman’.100

We saw that there are no explicit accounts of the truth-condi-tions for Stoicmonadic universal propositions, analogous in form to∀x(Fx→Gx),butthatthereissufficientevidencetoreconstructthese:Amonadicuniversalindefinitepropositionistruepreciselywhentherecan be no false subordinate instances (§5). By generalization to dy-adicuniversal indefinitepropositions,aStoicpropositionanalogousinformto∀x∀y(Fxy→Gxy)shouldbetruepreciselywhentherecanbenofalsesubordinateinstancesofthese:NoFab→Gabcanbefalse.Inaddition,wehaveanexplicitsemanticpresentationofacounterex-ampleinvokedtoshowthefalsehoodoftheplausibleproposition(26).

(O)Butthisisfalse.Foritisnotthecasethatthehenofaneggisthemother.101

Here we have, implied, a sufficient condition for the falsehoodof (26): Given the truth-conditions for indefinite conditionals (seeBarnesetal.1999,112–113;SE M11.9–11),(26)isfalsebecauseoneofits subordinatednon-indefiniteconditionals is false,presumably: «ifahentoanegggivesbirth,thenahentoaneggisamother»,follow-ingtheexamplein(O).AnecessaryandsufficientconditionforthisChrysippeangenuineconditional tobe true is that it cannothaveatrueantecedentandafalseconsequent.102Thissuggeststhatthetruth-conditionsfor(26)arethatitcannothaveacounterexample;andthat(O) gives a counterexample in the form of a false subordinated in-

100. πᾶς ἄνθρωπος πάντα ἄνθρωπον φιλεῖ. Sentences expressing propositionslike(32)and(33a)areknownasbishop sentences,afterKamp(seeHeim1990and,forabriefexposition,Elbourne2010,65–68).SincenoStoicexamplesofthiskindsurvive,wementionthisonlyinpassinganddonotspeculateabouttheStoictakeontheirsemantics.

101. DL7.75,continuationof(I):ψεῦδος δὲ τοῦτο· οὐ γὰρ ἡ ὄρνις ᾠοῦ ἐστι μήτηρ.

102. Barnes et al. 1999, 106–108.We saw that for the truth of an indefinitepropositionwemayneedthetruthofacorrespondingdefiniteproposition(orpartiallydefiniteproposition).Asexpected,forthedemonstrationbycoun-terexampleofthefalsehoodofaproposition,acorrespondingmiddleproposi-tionseemstosuffice.

(30)∀x∀y (Fxy → Gxy)97

Itisnotclearifandhowsuchpropositionscouldbeexpressedwith‘every’(as‘ifsomethingishuman,itismortal’wassaidtobesynony-mous to ‘(every)human ismortal’ in themonadic case in§5)—notevenifoneresortstoacrobaticdistortionsofnaturallanguage.98Thusherewehavea logically motivatedreasonwhytheStoicsintroducedtheconditional form for expressinguniversalpropositions: it is required whentherearetwouniversalizationsinaproposition.Ofcourse(26)isnotwhatonewouldexpecttherestrictedparallelto

(31)∀x∀y Fxy

tobe.Stoicdyadicpredicatesprovidethematerialforarestrictedver-sionof(31),too.Anexamplewouldbe

(32) If someoneτ1 is human and someoneτ2 is human,thentheyε1lovethemε2.

99

(33)∀x∀y ((Fx ∧ Fy) → Gxy)

Or,withtheexistencepredicateE,

(33a) If someoneτ1 exists and someoneτ2 exists, thentheyε1lovethemε2.

(34)∀x∀y ((Ex ∧ Ey) → Gxy)

97. InourStoicizedsymbolism:τi /τkF → εi /εkG,withtheregimentedwordorderandτiforthefirsthalfoftheindefinite(upright)part, εi forthecorrespond-inganaphoricsecondhalf,/τkforthefirsthalfoftheindefiniteobliquepartand/εk forthecorrespondinganaphoricobliquesecondhalf.Itisofcoursenotnecessarythatanuprightfirsthalfhasanuprightanaphoricpart,oranobliquefirsthalfanobliqueanaphoricpart(see§8).

98.Cf. thewell-knowncriticismsof thesuppositio theories developedbyMedi-evallogicians: Dummett1973,8,andonancientAristotelianlogic,seeBarnes2007,159–165.

99.Perhaps as a regimented form for ‘some are human’ (τινες ἄνθρωποί εἰσιν), givenChrysippus’interestinplurals,seefn.to(48).



whichstartwiththeGreekfor‘every’(pas).((35)restrictsonlytheuni-versal,nottheexistentialpart.)AssumesuchasentenceisintendedtoexpressaStoicpropositioninwhichtheuniversalparthaswidescope.Then, in principle, the formulation with ‘every’ (pas) as first wordwouldfindStoicapprovalbecauseof theirScopePrinciple, that thefirstexpressionofasentence indicateswidest scope(see§1).104Pos-siblybecauseoftheperceivedambiguityin(35)betweenChrysippeanand ‘Philonian’universal (§6), theStoicswouldadvocateregimenta-tion.Forbrevity,wehereconsideronlytheChrysippeanversions.So,theStoicfullyregimentedformofsuchsentencesasChrysippean uni-versalswouldalmostcertainlybe:

(36)Ifsomeoneisaman,helikessomething.105

Ifsomeoneτ1isaman,heε1likessomethingτ2.

Thisgiveswidescopeto‘someone-he’(tis-ekeinos),before‘something’(ti).Thesimilaritytothecontemporaryformalanalogueisobvious:

(37)∀y (Fy → ∃x Gyx)

104. Objection:TherearenoattestedexamplesofStoicsentenceswithpas asthesubjectexpression.ThereforethesegenerallydonotseemtobecasesthattheStoicswereinterestedinorhaddealtwith.Reply:Thisobjectionisun-convincing.WeassumethatitwastheStoicviewthatuniversalpropositionsshouldbeexpressedintheformofoneoftwospecifictypesofindefinitesen-tences,sincetheStoicstookuniversalsentencestobeambiguous,andthattheyprobablyregimentedsuchsentencestodisambiguatethem.Dependingontheintendedmeaning,theyweretobeformulatedasindefiniteconjunc-tionsor indefinite conditionals. (Asmainevidencewe takeCic.Fat. 11–15,alsoDL7.82(Sorites),Augustine,Civ. 5.1;forfurtherevidence,seeBobzien1998,156–167,andthediscussioninFrede1974,101–106).SoitisnotthecasethattheStoicswerenotinterestedinsentenceswithpas.Rather,theyfoundsuchsentenceslacking,sinceambiguous,andintroducedadisambiguatingregimentation,presumablyforpurposesofscientifictheoryanddialectic.Itwouldhencebesurprising,ifwefoundsuchsentencesasexamplesinStoiclogic,andtheirabsencecannotbetakenasevidencethattheStoicswerenotinterestedinthecontentofwhattheythoughtwasintendedtobeexpressedinnaturalGreeklanguageinsentenceswithpas.

105. εἴ τίς ἐστιν ἄνθρωπος, ἐκεῖνος τι φιλεῖ.Cf.SE M11.8–9.

stance.Weassumethatthesamewouldbethecaseforpropositionslike(32)and(33a).

7. Scope ambiguity

The kindof ambiguity that is standardly offered as evidenceof theingenuityofFrege’squantifierlogicisthatofscope ambiguityinquan-tifyingexpressions(e.g.Dummett1973,9–15).Sincetheorderoftwoormoreofthesamequantifyingexpressionsisconsideredimmaterialineverylogicalrespect(thoughsee§8),suchambiguityisdiscernedwhenquantifying expressions aremixed.103 Themost basic contem-porary forms are∀y∃x Fyx and ∃x∀y Fyx. The problem ofmultiplegenerality isoftenexplainedwithexamplesof just these forms. ‘Ev-eryonelovessomeone’istheparadigmintheEnglishlanguage.Here‘everyone’may havewide or narrow scope. There are no survivingexamplesofStoicpropositionswithmixedquantification, that is,ofpolyadicpredicatesthataregovernedbyamixtureofexistentialanduniversalindefiniteparts.Atthesametime,thereisno reasontothinkthatStoicdyadicpredicatesexcludecompletionsthatgeneratepropo-sitionswithmixedindefinitequantification.Theonlyprovisoisthat,asinthecaseofmonadicpredicates,universal‘quantifying’isusuallyrestricted,andStoiclanguageregimentationwouldaskforexpressioninconditionalform.UsingwhatinformationthereisonStoicmonadicexistential and restricted universal quantifying, it is straightforwardto(re)constructstepbysteptheStoicversionofthemixedcases.TheGreekfor‘Everyhumanbeinglovessomeone’hasascopeambiguitysimilar to the English. In this sectionwedisregard anaphoric ambi-guity.Ourexamplesaredeliberatelychosensoastoavoidthatissue.ThusancientGreekcontainssentencesequivalentinformto

(35)Everymanlikessomething.

103. In Stoic notation of dyadic universal propositions, it is not immaterialwhichsecondhalfofatwo-partquantifyingexpressiongoeswithwhichfirsthalf.See§§6and8.



twoclauses.Wesuggestthatinourcaseofmixedquantifiedsentenc-eswithwide-scopeexistential, Stoic logiciansmayhaveusedapar-allelGreekformulationwithanindefiniteparticleinsteadofanounordemonstrativepronounasastep towardsregimentingthemixedsentences:

(39)Thereexistssomethingthateverymanlikes(esti ti ho pas anthrôpos philei).

Thissortofformulation‘thereexistssomethingthat…’comesasclosetoanexistentiallyquantifyingexpressioninnaturallanguageasonemaywishfor.TheScopePrinciplegivesitwidescopein(39).Itisalsoveryhard,ifnotimpossible,toreadsuchsentencesashaving‘every’withwide scope.TheStoics recommend (38) asparsingof the sim-pleproposition‘Kalliasisnotwalking’inthecontextofevading scope ambiguity,i.e.ofnegation.107Itisthusentirelyplausiblethattheyalsorecommend formulations like (39) as parsingof a simple indefinitepropositiontoevadescopeambiguity(cf.Egli2000).

SomeoftheStoicdiscussionofexistentialimportinsimpleaffirma-tivepropositionssuggestsaparsingof(38)asa‘conjunction’withananaphoricexpressionlikethisone

(40)Socratesexistsandhe(thatone)iswalking.108

ThisshowsthattheStoicshadalltheelementsavailabletoparsesen-tences like(39)as indefiniteconjunctions109 thathaveanexistentialfirstconjunctandanindefiniteconditional,orChrysippeanuniversal,assecondconjunct:

(41)Thereexistssomething(esti ti),andifsomeoneisaman,helikesit.

107. Seepreviousnote.

108. ἔτι ὁ λέγων ‘Σωκράτης περιπατεῖ’ ἴσον λέγει τῷ ‘ἔστι τις Σωκράτης, κἀκεῖνος περιπατεῖ’(Alex.An. Pr.404.27–28).

109. SeeCic.Fat. 15anddiscussion inBobzien 1998, 156–159;Crivelli 1994a,201–202.

Nextassumethat(theGreekequivalentto)(35)isintendedtoex-pressaStoicpropositioninwhichtheexistentialparthaswidescope.TheScopePrinciplesuggeststhat,inthiscase,theformulationof(35)wouldnoteveninprinciplefindStoicapproval—oratleastnotforlog-icalpurposes.TheimmenseflexibilityofwordorderinGreekseemstostopshortofallowingalternativesthatstartthesentencewithanexistentialparticlefollowedbyauniversalparticle(ti pas,say).WeareluckytohavesomeevidencethatguidesusinreconstructinghowtheStoicswouldhave regimentedsuchsentenceswithwide-scopeexis-tential.ItisbasedonaparallelcaseinwhichtheStoicsintroduceanexistentialpropositionaspartoftheirparsingasimplemiddleordefi-niteproposition:‘Kalliasiswalking’isparsedas‘ThereexistsacertainKalliaswhoiswalking’and‘Kalliasisnotwalking’as

(38)‘ThereexistsacertainKalliaswhoisnotwalking’.106

(TheStoicpointofthisrephrasingistoremovethescopeambiguityof negationbymaking it explicit that Stoic affirmative definite andmiddlepropositions,whichinclude«Kalliasisnotwalking»,haveex-istentialimport.TheStoicnegation«not:Kalliasiswalking»doesnot.)Herethestructureofasimplemiddlepropositionismadeapparentbyparsingthestructureinasentencethatconsistsofacombinationof

106. SeeAlex. An. Pr. 402.15–18,reportingaStoicview:‘fortheysayifKalliasdoesnotexist,«Kalliasisnotwalking»isnolessfalsethan«Kalliasiswalking»,since inbothwhat issignifiedis thatacertainKalliasexistsandthatwalk-ing(ornot-walking)holdsofhim’(μὴ γὰρ ὄντος Καλλίου οὐδὲν ἧττόν φασι τῆς «Καλλίας περιπατεῖ»ψευδῆ εἶναι τὴν«Καλλίας οὐ περιπατεῖ»·ἐν ἀμφοτέραις γὰρ αὐταῖς εἶναι τὸ σημαινόμενον ἔστι τις Καλλίας, τούτῳ δὲ ὑπάρχει ἢ τὸ περιπατεῖν ἢ τὸ μὴ περιπατεῖν).Alexander’sphrasing‘thatwalking(ornot-walking)holdsofhim’isPeripatetic.WethinkthattheformulationoftwoStoicexampleswith relative clauseslaterinthepassageisthecanonicallyStoicone(Alex.An. Pr. 402.29–33):‘sothustheonesaying“thisoneisn’twalking”sayswhatisequiv-alentto“there exists this one indicated here, who isn’t walking”…theonesaying“theteacherKalliasisn’twalking”sayswhatisequivalentto“there exists a cer-tain teacher Kallias who isn’t walking”’(οὕτως γὰρ τὸν λέγοντα‘οὗτος οὐ περιπατεῖ’ ἴσον λέγειν τῷ‘ἔστιν ὁ δεικνύμενος οὗτος, ὃς οὐ περιπατεῖ’.…τὸν λέγοντα‘Καλλίας ὁ γραμματικὸς οὐ περιπατεῖ’ ἴσον λέγειν τῷ ‘ἔστι τις Καλλίας γραμματικός, ὃς οὐ περιπατεῖ’).SeealsoAlex.An. Pr. 404.27–29.OnAlex.An. Pr . pp.402–404,seealsoLloyd1978.



(44) is aproper Stoic conjunctionwithaStoic indefinite conditionalassecondconjunct.(Itcanbe‘cut’—see§9.)For(44)tobetrue,bothitsconjunctsneedtobetrue.Thesemanticsforthefirstconjunctareclear(§5,fn.75).Soarethoseforthesecond,whichisamonadicuni-versalindefiniteproposition(above§6).Theindefiniteconditionalistruewhennocorrespondingdefiniteconditionalcanbefalse.So(41)as awhole is true,when thepropositionobtainedby replacing theindefiniteconditionalin(44)withadefiniteonethathastwooccur-rencesofadefinitepronounthatincludesindicatingaparticularthingtwice(‘thisδ2’)cannotbefalse:

(45) Both thisδ1 thing exists and if thisδ2 one is aman,thisδ2onelikesthisδ1thing.

SotheStoicsemanticsofthepropositionsofkind(41)isstraightforward.For the semantics of propositions with wide scope—Chrysippe-

an—universal(36)(‘Ifsomeoneisaman,helikessomething’)truth-conditionsmaynot fall into place as easily. Even so, it is clear thattheywould logicallydiffer fromthenarrow-scopeone.Theproposi-tionexpressedby(36)istrueifitcannot havefalsesubordinatecases.Subordinatecasesareofthekind

(46)Ifthisoneδ1isaman,thisoneδ1likessomething.

Theseareproperconditionalsandfalseifitcan bethattheirantecedentistrueandtheirconsequentfalse.Soeverysubordinatecaselike(46)must betrue.Proposition(46)istrueifitcannot bethecasethat«thisoneδ1isaman»istrueand«thisoneδ1likessomething»isfalse.Sowecannothavethat.Inotherwords,wecannothave«thisoneδ1isaman»trueand«not:thisoneδ1 likessomething»true.Hencewecannothaveanypropositionbetruethatisofthekind

(47) Both this oneδ1 is a man and not: this oneδ1 likessomething.

Theparallelwith(40)shows that the formulationwithan initialex-istential‘conjunct’isnotconjuredoutofthinair.Paralleltothecaseof (40), theparsingof (35)withwide-scopeexistentialas (41)givesusthestructureofthemixedexistential-wide-scopepropositionwithrestricteduniversalwithout making it a genuine Stoic conjunction or a con-ditional.Again,thesimilaritytocontemporaryanalogueswithrestric-tion is obvious.Dependingonwhether existencewas considered apredicate(withExfor‘xexists’),110theycorrespondtooneofthese:

(42)∃x∀y (Fy → Gyx)

(43)∃x(Ex ∧∀y (Fy → Gyx))

So,theStoicsareabletoindicatethefactthattheexistentialparthaswide scope in the proposition both with the non-‘conditional’ sen-tenceplusScopePrinciple(39)andwiththe‘conditional’expressionoftheuniversal(41).

Basedonthesemanticsformonadicindefinitepropositions,wecanofferareconstructionofthesemanticsofStoicdyadicindefinitemixedpropositions.Forthesemanticsof thosewithwide-scopeexistential,wecombineStoic truth-conditions for theexistentialwith those fortheuniversal,inorder.Theproposition(41)istrue,ifacorrespondingdefiniteproposition is true.Thecorrespondingdefinite conjunctionhastwooccurrencesofadefinitepronounthatincludesindicatingaparticularthingtwice(‘thisδ1’),inlieuof‘something’and‘thatthing’:

(44)Boththisδ1existsandifsomeoneisaman,helikesthisδ1.

110. Wedonotherediscuss thequestionwhether in these cases the Stoicsconsider ‘exists’ (εἶναι /ὄντα) as a predicate. It appears, though, that theydo.Chrysippusthinkstheclaimthat‘ofallthingsthatexist,somearegood,somebad,someinbetween’isequivalenttotheuniversal(indefinitecondi-tional)claim,‘ifsomethingsexist,theyareeithergoodorbadorindifferent’(τῶν ὄντων τὰ μέν ἐστιν ἀγαθά, τὰ δὲ κακά, τὰ δὲ τούτων μεταξύ δυνάμει κατὰ τὸν Χρύσιππον τοιοῦτόν ἐστι καθολικόν· εἴ τινά ἐστιν ὄντα, ἐκεῖνα ἤτοι ἀγαθά ἐστιν ἢ κακά ἐστιν ἢ ἀδιάφορα) (SE M11.11,mentionedin§5).Notealsoanotherregu-larexampleinStoiclogic: ‘Itisday’,whichtranslatesliterallyas ‘dayexists’(ἡμέρα ἐστίν, DL7.68et passim).



Insum,theStoicsthushadallthemeansavailabletodistinguishboth syntactically and semantically between the twobasic kinds ofmixedmultiplegenerality,andwithoutvariables.WithinStoiclogicofpredicatesandindefinitepropositions,theparadigmscopeambiguityofmixed-quantifierpropositionswithdyadicpredicates (andrestric-tiononuniversalpropositions)canberemoved.Thisisachieved(i)bylanguageregimentationwith theScopePrinciple forsentenceswith‘every’(pas)and‘someone’(tis)and(ii)atthelevelofparsingproposi-tions,bytheparsingofuniversalpropositionsintotheformofindefi-niteconditionalsorindefinitenegatedconjunctions.ThustheStoicsareabletoexpressunambiguously(andwithoutvariables)multiplegeneralityforalanguagewithrestricteduniversality—intheabsenceofanyanaphoricambiguity.

8. Anaphoric ambiguity in Stoic quantifying expressions

Fregeanvariable-bindingquantifierseliminatetwokindsofambiguity:scopeambiguityandanaphoricambiguity.112Inaproposition(ormoregenerallycompletecontent),quantificationwithvariablesunambigu-ouslybindseveryoccurrenceofeveryvariable,andanaphoricambi-guitydoesnotoccur.TheStoics,though,havenovariables.DoesorcanStoiclogicaccountforanaphoricambiguity?Considerthefollow-inggenericstripped-down(contemporary)formoftheStoicdonkeysentencethatexpressesproposition(26),withtwouniversalquantify-ingexpressionthatincludetwooccurrencesofanaphora

(51)IfsomeonesomeoneF,thenthatonethatoneG.

Suchsentencesarepotentiallyambiguousbetween

(52)Ifsomeone1someone2 F,thenthatone1thatone2 G.

(53)Ifsomeone1someone2 F,thenthatone2thatone1 G.

112.Wehereuse‘anaphoricambiguity’asshortfor‘contextsensitivityofanaphor-icexpressionsthatcross-refertonon-referringexpressionswithinthesamesentence,consideredinisolation,withthecontextbeingalinguisticone’.

(For the corresponding semantics of propositions with wide-scope‘Philonian’universal, simplyreplace ‘cannothave’with ‘hasno’, ‘canbe’with‘is’,‘mustbe’with‘is’,etc.,andmutatismutandisbelow.)

Asforthe logical relations between the two kinds of dyadic mixed indefi-nites,weexpectproposition(36)tobetruewheneverproposition(41)istrue,butnotviceversa.Thisisindeedthecase.Iftherecanbenocounterexampleto(36),therecanbenoneto(41).Moreover,(36)canbetruewhen(41)isnot.In(36)‘something’isnotanaphoric,whereasin (41) ‘it’ is anaphoric on ‘something’ (ti).And in the definite con-ditional(45)thatcorrespondsto(41),‘thisthing’hastwicethesamedemonstratedobject,whereasfor(36)thereisnosuchdouble-demon-strationrequirement.

Therelevantlogicalrelationbetweenthecases∀∀and∀∃(seein-troduction)canalsobeexpressed.Takefor∀∀proposition(32)from§6.(32)istruewhentherecanbenocounterexample(oftwohumanswhodon’tloveeachother,i.e.):

(48) Both (both this oneδ1 is human and this oneδ2 ishuman)111andnot:(thisoneδ1 lovesthisoneδ2).

Takefor∀∃

(49) If someoneτ1 is human, then that oneε1 lovessomeoneτ2andthatoneε2ishuman.

Proposition(49)istruewhentherecannotbeacounterexample(ofahumanwhodoesn’tloveahuman,i.e.):

(50)Both thisoneδ1 ishumanandnot: (both thisoneδ1 lovessomeoneτ2andthatoneε2ishuman).

Clearly,iftherecanbenocounterexample(48)to(32),thentherecanbenocounterexample(50)to(49).

111.Thiswouldcorroborate,withanadditionalcase,theStoicregimentationoftheplural(οὗτοι ἄνθρωποί εἰσιν).Cf.theexampleinSE M10.99andtheex-tendeddiscussioninCrivelli1994b.SeealsoPHerc307Col.IVforacaseofthespeakerexpressingadefinitepropositionbymeansofpointingat twopeople.



Stoiccorrespondingmiddleanddefinitepropositionsare«thisone δ1 isthisone δ2»and«DioisTheon».

114In(55)declensionisofnouse,andpositiontheonlyoptionfordisambiguation.Alternativelyonemightargue that,due to symmetry combined with indefiniteness, there isnoambiguity, sinceexchanging ‘someone1’and ‘someone2’doesnotaf-fectwhatissaid.(However,weherereservejudgementontheques-tionwhether«thisone δ1isthisone δ2»isthesamepropositionas«thisone δ2isthisone δ1»and«DioisTheon»thesameas«TheonisDio»,asopposedtosimplybeingequivalent.)

Theproposition(26)(from§§6and7)isanexamplewithtwo pairs of universally quantifying parts.(52)and(53)showthatandhowsuchsentencesarepotentiallyambiguous.Insentence(26),genderindica-torsavertambiguity.115Weusesubscriptindices,M, F, MFandNasindi-catorsofmale,female,maleorfemale,andneutergendertoexpressthisfor(26):

(56) If someoneMF to somethingNgivesbirth, then thatoneFtothatthingN ismother.

Genderthenworkssimilarlytoselectional restriction.Butgenderdoesnotalwayspreventambiguity.Takethefollowingcase:

(57) If someoneMF to someoneMF gives birth, then thatoneFtothatoneFismother.116

114. ἐστιν οὗτος οὗτος, εἶναι τοῦτον τοῦτον and Δίων Θέων ἐστὶν, Δίωνα Θέωνα εἶναι, cf.PHerc307ColsIVandV.ItisalsofoundinGalen,On Fallacies 4.7–9,forwhichseediscussioninEdlow2017,64–65,andAtherton1993,383.

115.SomeStoicsconsideredthegenderofademonstrativecase-contenttobese-mantically relevantand thuspartofaproposition: «this oneδM iswalking»and«thisoneδMisnotwalking»arebothfalse(falsebyparemphasis,Alex.An. Pr. 402.25–26),iftheobjectoftheactofindicating(deixis)isfemale.Thesetwopropositionsaresaidtobeequivalentto«thisoneδM,whoisbeingindi-cated,existsandis(not)walking».Fordiscussion,seeDurand2019,§§21–24.Cf.Alex.An. Pr.402,21–30,citedin§7above.SeealsoAlex.An. Pr.404.31–34.

116.Hereandinthefollowingsentences(58),(59),(60),(61),(58a)and(59a),thefemininegender in the ‘consequent’ (ekeinê,…)disambiguates thegeneric,feminineormasculine,genderinthe‘antecedent’(tis,…)asfeminine.Our

Again,‘1’and‘2’areusedtoindicatewhichanaphoricexpressionreferstowhichindefinitepart(icle).Withtwoindefinite‘someone’particlesfollowedbytwoanaphoric‘thatone’particles,itisambiguouswhichanaphora isbound towhichparticle.Similarambiguitiescanbeob-servedinmixedquantification.

Intheirlogic,theStoicshaveattheirdisposalfourelementsthatcan inprincipleplay the rolesof operatorsof thekindQuine intro-duces:(i)positionororderofStoicquantifyingexpressions,(ii)active-passive formulations, (iii)declensionand (iv)gender. IthasbaffledhistoriansoflogicthattheStoicsconsiderallfouratthelevelofStoicpropositions, as contrasted with linguistic items.113 Stoic theory of(variable-free)quantificationmayhelpexplainwhy.Weconsiderdif-ferentcases.

Fortwoexistentialquantifiers,ourexistingexamplesare‘someonegivesbirthtosomeone’and,implied,‘someoneissomeone’smother’.Inprinciple,therecanbeambiguity.Ifweaddindicestotheindefiniteexpressions,wehave

(54)Someone1someone2givesbirthto.

What determines, informally speaking, who gives birth to whom?Word position is extremely flexible inGreek.Nonethelesswe haveseen that the Stoics regiment position.Another disambiguating ele-mentisdeclension.In(54),declensionissufficientintheGreek,sincethenominativeanddativepronounswoulddiffer(e.g.‘someone’,‘tosomeone’).TheScopePrinciplemayhave required that theuprightcase (orthê ptôsis) beplacedfirst, therebydeterminingwhat kindofpropositionwehave.This isconsistentwithallourexamplesfor in-definitepropositions.Declensiondoes,however,notworkincasesofequalityandidentity,forexample

(55)Someone1someone2is.

113.Seee.g.Frede1994.Somehavesimplyassumedtheymustbe functionsoflanguage(ibid.13–14).



Stoic active predicates have by definition as content element anoblique case-content, i.e. that which would be expressed by a de-clinednounexpression.Theexamplesforthepassiveemploytwoofthethreeverbsthataregivenfortheactive.This,togetherwiththetalkofapassiveparticle,120suggeststhattheStoicsmayhavethoughtthatonecanconvertonetotheother.(Alatertextsaysthisexplicitly.121) Wemaythinkofthepassiveparticleassomethinglikeatransformation operatorthat,appliedtotheactive,providesthepassive(atthecontentlevel).122AcombinationofActive-to-passive ConversionwiththeScopePrinciplepositioningallowstheremovaloftheambiguityin(57):123

(60)IfsomeoneMF1tosomeoneMF2givesbirth,thenthatoneF1tothatoneF2ismother.

(61) If someoneMF1 by someoneMF2 is born, then thatoneF1tothatoneF2ismother.

APrinciple of Anaphoric Congruencethatanaphoricreferenceisinorderofoccurrence(thefirst indefiniteexpressionhas thefirstanaphoricexpression referring to it; the second indefinite expression has thesecond anaphoric expression referring to it) can then supplementtheScopePrincipleasasecondprincipleforposition.ThuspositiontogetherwithActive-to-passive conversiondetermines the ‘referent’oftheanaphoricexpression.(57)isdisambiguatedinto(60)and(61)by use of Anaphoric Congruence with Active-to-passive Conver-sion. Strictly speaking, in this example the Principle of Anaphoric

γένεσιν, οἷον Ἀκούει, Ὁρᾷ, Διαλέγεται· ὕπτια δ’ ἐστὶ τὰ συντασσόμενα τῷ παθητικῷ μορίῳ, οἷον Ἀκούομαι, Ὁρῶμαι.

120. Presumablyὑπό,cf.Gaskin1997,92.

121. Scholium in Dionysius Thrax, 401.1–20. The ‘philosophers’ mentionedmustbeStoicphilosophers,sincetheterminologyisStoic.

122. Cf.Alex.An. Pr. 403.14–24, reportingStoicviewson the transformation(enklisis)ofpresenttopasttense.SotheStoicswerefamiliarwithanotionoftransformationofthiskind.

123. NotethesimilarityofthisapproachtoPeterLudlow’streatmentofbishopsentencesinLudlow1994,171–172.

This isambiguousinthat itcoulddenoteeitherofthefollowingun-regimentedpropositions



Heregenderanddeclensiondonothelp;neitherdoespositionalone.We know that de facto listeners will, by considerations of charity,choosetheinterpretationthatmakesthesentencetrue(e.g.Davidson1974,Stalnaker1978,Grice1989).However,fromaStoicperspective,suchpragmaticconsiderationswouldnotbepartoflogic—norissuchinformationalwayssemanticallyorcontextuallyprovided.Thisleavesactive andpassive.Chrysippuswrote a bookon active andpassivepredicates, and discussed them elsewhere.117 Perhaps the followingpassagegivesushisview.Inanycase,itgivesaStoicview.

(P) Of(monadic)predicates,someareactive(ortha),somepassive(huptia),andsomeneither.Activearethosethat,being connectedwithoneof theoblique case-contents(ptôseis), yield amonadic predicate, for example ‘hears’,‘sees’,‘isconversingwith’.118Passivearethosethat,beingconnectedwithapassiveparticle[yieldamonadicpredi-cate],forexample‘amheard’,‘amseen’.119 (DL7.64)

subscriptnumeralsdeterminetheanaphorarelationbetweenthepronounsindependentoftheirspecificgenderindicators.

117. DL7.192,Chrysippus,Log. Inv.fr.3lines4–18;Col.I.23,II.17–21,allnotedbyBarnes1999,206andMarrone1984.

118.Wereadthisassayingthatanactivepredicatecomestobebytheconnec-tionwithanobliquecase-content,inthesensethattheactivepredicateisthemonadicpredicatethatistheresultofthiscombination.SoalsoBarnes1999;Hülser1987–88,809;Hicks1925;Mensch2018.

119. DL7.64:καὶ τὰ μέν ἐστι τῶν κατηγορημάτων ὀρθά, ἃ δ’ ὕπτια, ἃ δ’ οὐδέτερα.ὀρθὰ μὲν οὖν ἐστι τὰ συντασσόμενα μιᾷ τῶν πλαγίων πτώσεων πρὸς κατηγορήματος



282)—activeandpassivearepartofthecontent.Itisnotsomuchthate.g. ‘DiolovesPlato’and ‘Platois lovedbyDio’, takenontheirown,giveusdetectablydifferentcontent.Whatisassertedwithonedoesnotdiffer fromwhat is assertedwith theother.Rather, it is that, in certain more complex propositions,theactiveandpassivearestructuringelementsthatarepartofwhatstructurestheproposition,andthusareadetectablepartofthesimplepropositionin so far asitispartofthemorecomplexproposition.With(60a)‘someoneF1tosomeoneF2givesbirth’,thesamepropositionmaybeexpressed(andasserted)aswith(61a)‘someoneF1bysomeoneF2isborn’.However,withthesentences(60) and (61), which contain (60a) and (61a) respectively, differentcomplexpropositionsareexpressed(andasserted).IftheStoicsmadeuseoftheirdistinctionofactiveandpassivepredicatesinthisway,wecanobservea similarity toDummett’sdistinctionbetweenassertoric contentandingredient sense(ase.g.setoutinDummett1991,47–50).126

yieldstheproposition«PlatohearsDio».Thenanactive-to-passivetransfor-mationyieldstheproposition«DioisheardbyPlato»,formedfrom«Dio»andthepassivemonadicpredicate«---isheardbyPlato»,whichyieldsthedyadicpredicate (less-than-katêgorêma) «--- isheardby…»and theoblique case-content(ptôsis)«Plato».

126. Howwouldthisworkforpolyadiccaseswithanadicityhigherthantwo?FirstnotethatnaturallanguageslikeEnglishgenerallyhaveahardtimedeal-ingwithsuchcaseswhenuseisrestrictedtosimpleanaphoricexpressions.Taketheparadigm‘xisbetweenyandz’asexample:‘Ifsomethingislocatedbetweensomethingandsomething,itisneitherthesameasitnorthesameasit.’TheStoicuse,intheirlogicofordinalsforschematicrepresentationofpropositions in arguments and in their formulationof inference rules (the-mata),suggeststhattheywouldhaveresortedtonumeralsforsuchcases,asindeedEnglishspeakersmight, too.TheStoicsgenerallycounteachoftherelevant elements and count them in order of occurrence (cf.DL 7.80–81andBobzien2019). For the caseat issue, thiswouldyield ‘if something isbetweensomethingandsomething,thenitisneitherthesecondthingnorthethirdthing’,andthree-dimensionally,‘ifsomethingisbetweensomethingandsomethingandsomething, then it isneither thesecond thingnor thethirdthingnorthefourththing’,etc.Jointly,theScopePrincipleandthePrin-cipleofAnaphoricCongruencearethensufficientforsuchcases,althoughtheremaybeinstanceswheresomelanguageregulationisrequiredthatgoesagainst idiomatic formulations. Recall, though, that Greek is very flexiblewithwordorder.

Congruence suffices without Active-to-passive Conversion, if oneallowsforvariablecaseorder,sothatanominativecantakesecondplace.124However,positionalonedoesnot suffice in exampleswithaneuternominativeandanaccusativeexpressionintheconsequentclause,sincethesearegenerallyidenticalinform.Incontrast,Active-to-passive Conversionalwaysallowsreverseorderoftheantecedentex-pressionsandthuscanhandlesuchexamples:

(60a) If somethingN1 somethingN2 attacks, that thingN1 (ekeino)thatthingN2 (ekeino)kills.

(60b) If somethingN1 by somethingN2 is attacked, thatthingN2 (ekeino)thatthingN1 (ekeino)kills.

OfcoursewedonotknowwhethertheStoicssuggestedsuchdisam-biguation by use of Anaphoric Congruence with Active-to-passiveConversion.However, itdoesgenerallywork foractiveandpassivepredicates. Assume that «… hearsDio» is an activemonadic predi-cate(theresultofconnecting«…hears---»with«Dio»asanobliquecase-content(DL7.64)),andthatwith«Plato»itformstheproposition«Plato hears Dio». Then an active-to-passive transformation yieldstheproposition«Dio isheardbyPlato», formedfrom«Dio»andthepassivemonadicpredicate«---isheardbyPlato».WithStoicallyregi-mentedGreekwordorder,weget from ‘PlatoDiohears’ to ‘DiobyPlato isheard’andthus, if required,canobtain thedesiredorderofcontent-cases.125Thismayhelpexplainwhy—unlikeforFrege!(1897,

124. (58a)IfsomeoneMF1tosomeoneMF2 F,thenthatoneF1tothatoneF2 G.(59a)IfsomeoneMF1tosomeoneMF2 F,then to that oneF1thatoneF2 G.

Thisprovidesanalternativeforthosewhopreferadifferentreadingoftext

(P),ifatacost.FortheStoicsmayhavewishedtokeeptheplacementofcasesfordetachmentinderivations(forexamples,seethederivationsdiscussedin§9). Instantiationbecomessemi-automaticwithActive-to-passive Conversion.Without,detachmenteitherlosesthisfeatureorproducesnon-canonicalfor-mulations,e.g.‘toHebeHeraismother’ratherthan‘HeratoHebeismother’.

125. If theactivecontent isadyadicpredicate(less thanakatêgorêma), thisworksmuchthesame.«…hears---»connectedwiththeobliqueptôsis«Dio»yieldsthemonadicpredicate«…hearsDio»,whichwiththeupright«Plato»



ThereisalsoampleevidencethattheStoicsacknowledgetheva-lidityofarguments(andargumentformsormodes(tropoi))thathaveaStoicuniversalindefinitepropositionasfirstpremise,thathaveaninstantiationofthe‘antecedent’assecondpremiseandthatdeduceasconclusionaninstantiationofthe‘consequent’,wherethesamecase-content,demonstrativeornot, is instantiated in thesecondpremiseandtheconclusion:

(62) If someone is walking, he is moving; this man iswalking; thereforethismanismoving.(SoinAug. Dial. 3.84–86;cf.Cic. Fat.11–15)129

1999,116–121with86–88;Gaskin1995;Mignucci1978;Frede1974,107–117;Kneale&Kneale1962,123–128;Mates1961,36–41.Tense logic:Denyer1999;Gaskin1995;Crivelli1994b;Long&Sedley1987,v.1,51;Vuillemin1985.Sup-positional inferences: Bobzien 1997.Nobody Paradox:Mansfeld 1984; Caston1999,187–192.Paradoxes of Presupposition:Bobzien2012.Liar Paradox:Barnesetal.1999,163–170;Cavini1993.Unmethodically conclusive arguments:Barnesetal.1999,151–155.

129. Question:Cannot (62)be taken toshowthat itsfirstpremise is in factaStoicconditional, that is,anon-simpleproposition,since it seemsthat itcouldfeatureinasyllogismofthefirstindemonstrableform?Reply:No.TheStoicfirst indemonstrable isdescribedasbeingcomposedofaconditionalanditsantecedentaspremisesanditsconsequentasconclusion(DL7.80,SE M8.224).(62)isnotofthatform.Follow-upquestion:Canwenotassumethatthefirstpremisein(62)isinfactaStoicconditional,thatis,anon-simpleproposition,since inafirststep theStoics infer from itadefiniteormiddleconditionalandthengenerateasyllogismofthefirstindemonstrableformwiththatconditional?Reply:ThereisnoevidencethattheStoicshadavalidinferenceformthatpermitsaninferencefromanindefiniteconditionaltoacorrespondingdefiniteormiddleconditional.Onthecontrary,theevidencesuggeststhat(62)asitstandswasconsideredtoillustrateavalidinferenceform.BoththeAugustinepassageandCiceroFat11–15suggestthattheSto-ic inferencewentdirectly fromthe indefiniteconditionalandadefiniteormiddlecorrespondenttoitsantecedenttothematchingcorrespondenttoitsconsequent.InadditionwenotethatStoicvalidinferenceforms(Antipater’scontroversialone-premiseargumentsaside)(i)allcontainatleasttwoprem-ises;moreover(ii)itseemsthattheso-calledwhollyhypotheticalsyllogisms,which infer a conditional from twoconditionals,werenot acceptedas syl-logismsbytheStoics,cf.e.g.Bobzien2000,sothereisnoprecedentfortheinferenceofaconditionalfromoneormoreconditionalsofthesamesize(i.e.withthesamenumberoftwo-placeconnectives).Webelievethenthatweareonsafegroundinassumingthat(62)doesnotinanywayshowthat,for

Jointly,position,declension,genderandactive-passiveappear tosufficetoresolveallanaphoricambiguitiesthatmayoccurwithinthevariable-free frameworkofStoicparsingofuniversalandexistentialquantification.Forthree-or-more-placepredicates,therulesoropera-torsbasedonactive-passive,position,genderanddeclensionat thelevelofcontentwouldhavetobegeneralizedforsuchpredicates.Webelievethatthisistosomeextentpossible.127

9. Polyadic predicates in Stoic theory of inference

JonathanBarnescontendsthattheStoicdistinctionbetweenmonadicandpolyadicpredicates‘remainedinert’andthat‘noStoicexploitedthe distinction in his account of inference and syllogism’ (Barnes1999,206).Stoic syllogisticorsequentlogic(orprooftheoryortheoryof deduction) does indeed not include the distinction betweenmo-nadicandpolyadicpredicates.It isapropositional logicwhichdoesnotanalysethecontentofatomic(i.e.simpleaffirmative)propositions(Bobzien2019).

However, Stoic logic (and that includes inferences) was by nomeansrestrictedtoStoicsyllogistic.Wehaveevidenceabouttherudi-mentsofalogicof imperatival inferences,ofamodallogic,oftenselogic, of a theoryof suppositional inferences, of discussionsof vari-ouslogicalparadoxes,ofcertainarguments(probablydiscussedinthecontext of theLiar) that included intensional expressions like ‘says’,andmore.128Andasshouldbynowbeclear,wehavetherudimentsofavariable-freepredicatelogicformonadicanddyadicpredicates.

127. With this suggested reconstruction and elaboration on Stoic indefinitepropositions,wecanalsoanswerthequestionhowtodeterminewhether,inthecaseofsimplepropositionswithdyadicpredicatesandmixed arguments (definite, middle, indefinite), the propositions themselves are definite ormiddleor indefinite(Brunschwig1994,67).Firstasentence thatexpressessuchapropositionisputincanonicalform.Thisrequirestheexpressionthatsignifies thesubjectargument to takefirstplace,which, in thecaseof sec-ondaryeventpredicates(parasumbamata),wouldbelinguisticallyexpressedbythedative.Theexpressionwiththelargestscope,accordingtotheScopePrinciple,thendetermineswhatkindofpropositionwehave.

128. For example: Imperatives: Barnes 1986, 21–26.Modal logic: Barnes et al.



withthosewhoareagainstthecutoftheindefinites’(DL7.197).Here‘indefinites’ is likelytorefer to indefiniteconditionalsandindefinite(negationsof)conjunctions,and‘tocut’islikelytobeatechnicalterm.Therearetwoplausibleandrelatedmeanings:(i)Onecannotcutanindefiniteconditionalintotwocomponentpropositions.(ii)Onecan-notdetachaconclusion.Thatis,from‘ifsomeoneF,thatoneG’onecannotdetach,byusingassecondpremise‘someoneF’,‘thatoneG’asconclusion.Sincethetitlesareinthesectionoftitlesoninferences,thelatterismorelikely.Thereisnoreasontoassumethatwhateverisdis-cussedinalltheseChrysippeanbooksislimitedtoindefiniteproposi-tionswithmonadicpredicates.

LetusaddtothisindirectsupportforStoicdiscussionofargumentswithnon-idlepolyadicpredicatesthatitisimpliedby(26)thattheSto-icshaveinferenceschemataorargumentformsthatcontainnon-idlemultiplygeneralizedsentences.Herearetext(I)(whichcontains(26))andtext(O)insuccession:

(I)Apropositionisplausibleifitprovokesassent,forex-ample«Ifsomeonegavebirthtosomething,thensheisitsmother».(O)But this is false.For thehen isnot themotherofanegg.(DL7.75)

ThispointstowardtheStoicsacceptingasvalidargumentssuchas:

(63)Ifsomeonetosomethinggavebirth,thatoneofthatthingisthemother.

NowHeratoHebegavebirth.

HenceHeraofHebeisthemother.131

and:(64)Ifsomeonetosomethinggavebirth,thatoneofthatthingisthemother.

Nowthishentothisegggavebirth.

Hencethishenofthiseggisthemother.131.Thequestionsinfn.129couldberehashedhere.Theanswerscanaswell.

Moreover, there is evidence that the Stoics were specifically con-cerned with certain questions about inferences that involve indefi-nitepropositions.Firstinlineistheso-calledNobodyparadox(outis), aboutwhichChrysippuswroteeightbooks(DL7.198).Itisdescribedas ‘an argument composed of an indefinite and a definite [proposi-tion]thathasitssecondpremiseandconclusionconnected’(DL7.82).Asmostancientparadoxes,itcameinseveralvariants.Oneisappar-ently‘ifsomeoneishere,heisnotinRhodes;butamanishere;hencenot:amanisinRhodes’(DL7.82).130Itisnothardtoguessthatthenon-referringcharacterof ‘someone’and ‘nobody’,andtheresultingrestrictions on inference, played a central role in the discussion ofthisparadox.Second,inthesamesectionofChrysippeanbooktitlesoninferences,twobooks‘onargumentsfromindefiniteanddefinite[propositions]’aresandwichedbetweentitlesofbooksontheNobody(DL7.198).ThissuggeststheirformisrelatedtotheformoftheNo-bodyarguments.Weassume that either (62) is anexampleof such‘arguments[composed]fromanindefiniteandadefinite[proposition](i.e.aspremises)’,orthattheyareofthemorebasickind

(62a) ‘If someone iswalking, he ismoving; this one iswalking;thereforethisoneismoving.’

(ItisnotunusualforStoicillustrativeargumentstobetransmittedinslightvariations.)Third,Chrysippusalsowrotebooksentitled‘Proofsthatonemustnotcuttheindefinites’and‘Replytothosethatdisagree

theStoics,indefiniteconditionals(andindefinitenegatedconjunctions)wereconsiderednon-simplepropositions.

130. ReadingσυντακτικὸςwithLong:οὔτις δέ ἐστι λόγος συντακτικὸς ἐξ ἀορίστου καὶ ὡρισμένου, συνεστώς πρόσληψιν δὲ καὶ ἐπιφορὰν ἔχων, οἷον “εἴ τίς ἐστιν ἐνταῦθα, οὐκ ἔστιν ἐκεῖνος ἐν Ῥόδῳ. <ἀλλὰ μήν ἐστί ἄνθρωπος ἐνταῦθα· οὐκ ἄρα ἄνθρωπός ἐστιν ἐν Ῥόδῳ>” (DL 7.82).We choose this reading based on theparallelwithDL7.198(seemaintext),andchoose‘aman’(ἄνθρωπος)ratherthanDorandi’s‘someone’(τίς)inthesupplementedsecondpremiseandcon-clusion,sinceallotherextantversionsoftheNobodyhaveἄνθρωπος.ForthevariantsoftheNobody,seee.g.DL7.82and7.187;furtherevidenceinHülser1987–88astexts1205–1207,1209,1247–1251;cf.alsoMansfeld1984;Caston1999,187–192.



formulationsinordertoavoidtheproblemsofmultiplegenerality,orwhethertheyweresimplychosentoaccuratelyrepresentthestructureofpropositionswithmonadicanddyadicpredicates(includingthedif-ferentstructuresoftheirtwokindsofuniversals),andtheproblemsofmultiplegeneralityconsequentlyjustdidnotarise,wedonotknow.Forcasesofanaphoricambiguityinpropositionswithpolyadicpredi-cates, rigid position and conditional formulation of universals aresupplementedwithcasemarking,gendermarkingandactive-passivetransformation,allofwhichtheStoicsplaceatthelevelofcontent,andthuslogic,asopposedtolinguisticexpressionsandgrammar.

Atleastfordyadicgenerality,thecombinationofcasemarking,gen-dermarkingandactive-passivetransformationwithrigidpositionandconditionalformulationofuniversalsmakesitpossibletodevelopasystemthatcoversthegroundFregeanvariable-bindingquantifierssoniftilycontrol.TheStoicplacementofcases,genderandactive-passiveatthelevelofcontentisthusjustified.Thefivefactorsconvergeintoamethodthatensuresasystemofequalstrengthasvariable-bindingquantifiers for dyadic predicates that can be expressed in (ancientGreek)natural languagewithoutunduedeformations.The fact thatGreek iscase-basedandhasveryflexiblewordordermakes thead-ditionofrulesforrigidpositioningeasy.SomeevidenceimpliesthattheStoicsconsideredargumentformsthatcontainpropositionswithmultiplegenerality,andthatpolyadicpredicateswerenotinferentiallyidleinStoiclogic.

We have not offered a worked-out formal Stoic predicate logic.ThiswouldrequireanintegrationofStoicindefinitepropositionswiththeir propositional logic. Neither havewe ventured beyond dyadicpredicates,althoughitseemstousthat,uptoapoint,generalizationtopolyadicpredicatesispossible.NorhavewediscussedtherelationbetweenStoic logicandmedievalandearlyRenaissance treatmentsofmultiplegenerality.132Patently, there is space for further research.

132. WearenoexpertsinmedievalorearlyRenaissancephilosophy.Itseemstous,though,thattheattemptsatsolvingtheproblemsofmultiplegeneral-itybymedievalphilosophersaregenerallyimpededbytheircommitmentto

Moregenerally, itpoints toward theiraccepting thevalidityofargu-mentformslike:

(65)Ifsomething1something2 F,thatthing1thatthing2 G.

NowabF.

HenceabG.

Whyisthisimplied?Itisimpliedsincethehen-eggcaseisadducedasacounterexampletothetruthoftheindefiniteconditional(26).Butthehen-eggcaseproducessuchacounterexampleonly becausethereisanassumedvalidargumentformofthekind(64)thatadmitsinstantia-tion,withmiddleordefinitepropositionsfor‘antecedent’and‘conse-quent’.Weconcludethatitisexceedinglylikelythatmultiplegeneral-itywasnotinferentiallyidleinStoiclogic.

10. Conclusion

We have seen that the Stoics used regimented variable-free formu-lations for expressing existential and universal quantifying proposi-tionswithmonadicanddyadicpredicates.Thestructureofexistentialpropositions with simple monadic predicates is determined by thepositioningoftheStoicquantifyingexpression,whichcanbeunder-stoodretrospectivelyascombiningtheexistenceprefixwiththevari-able it binds.All universal propositions areparsedby theStoics ashavingindefiniteconditionalforminwhichtheindefinitepartinthe‘consequent’anaphoricallycross-referstothenon-referringindefinitepartinthe‘antecedent’.Thatis,theyhavetheformofif-clausedonkeysentences. (For ‘unrestricted’universals, the ‘antecedent’ couldhaveusedanexistencepredicate.)Thecombinationofrigidpositionwiththeparsingofuniversalsasindefiniteconditionalsisadequateforthemonadiccases.

The combination of rigid position and conditional formulationofuniversalswithoutbindingvariablesalso suffices for casesofdy-adic predicates, as long as there is no ambiguity introducedby theanaphoric expressions. Whether the Stoics introduced conditional



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Acknowledgements

We thankPaolo Fait,MattMandelkern,MarkoMalink, JamesAllen,AlexanderBown,andatleastoneanonymousrefereeforveryhelpfulwritten comments.Wearegrateful toVanessadeHarven forusefuloralfeedback,toStephenMennforofferingtotalkwithusaboutthepaperonSkype, and to theeditorsof this journal for their efficientpracticalassistance.

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