CHOICE AND CHANCEAn Introduction to Inductive Logic

Third Edition

BRIAN SKYRMS

University of California at Irvine

Wadsworü Publishing ComPanY

Belnront, Calüomia' A Division <.¡f Wadsworth, lnc,

l¡¡úrrTUTo DltNvrsrrGActÉlN[5

F¡L O60rf Cl¡CLrSI F. BCf I

-: </ C.

4 (¿ .-/

ri^ ?

!3fit

l'lclitor: Kcrr¡¡ctlr Kirrg

Proch¡ctiott (',x¡rdi¡lation {r Dcsign: Wcndy Cal¡¡¡cnso¡l r

C)ovcr Dcsign: Kathcrirlc Tillotson

I)rint Rtrycr: Rtrth Cole

Copy Eclitor: Carol Do¡rdrca

o lf)g6 by warlsworth, Inc. All rights reservcd. No part of this book rnay be reproduced.

stc,rccl iria rctricval systcm, or iíanscr¡bcd, in any fonn or lly any means' elcctronic'

r¡rechanical, plrotocopying, recording,or.otherwisc, without thc prior.writtcn ¡1911is-

liion uf th" pi,rrlishcri Waás*orth eublishing Company, Belmont, California 94002, a

rlivisio¡¡ of Wadsworth, I¡¡c.

l'r i¡¡tr:tl ir¡ tlrc U¡¡itctl St¿tcs r¡f At¡rcrit a

.t 5 6 7 tJ 9 I0 - vJ 9ti 97 96 95 94

lsBN 0-5lq-05110-1,

l,ibrary of Congrcrs Cataloguing in Publicatiotr Da(a

Preface to the Third Edition

ln preparing this eclition of Clnice antl Chance, I have tried to update,

simplify, and sipplement the material in the earlier editions where needed and

to klep my hands off of parts üat would only be muddied by furüer tinkering.

Thcse jtrdgments arc not casy to nrakc. I was greatly hclped by suggestions

from Profássors Frans van der Bogert, Appalachian state universityi Ellery

Eells, u¡úversity of wisconsin, Madison; sidney Luckenbach, califomia st¿te

University at NorÜuidge; David Siemens, Jr., Los Angeles Pierce College; and

FredericÉ Suppe, University of Ma¡yland. They shouldn't, of course, be

bla¡¡recl for üe final ¡nix. I did¡¡'t take all üre ¿dvice offered, although I took a

lot of it.Ilcfercnccs arc upclutccl throughout, ncw exercises have becn added, some of

thc excrcises for üe advanccd student of üe second edition üat were too

¡clvar¡c,ed have bcen deleted, ar¡d some have been exptarned (e.g., section 2.3).

[xccpting thcsc cliangcs, ¿r¡d so¡ne minor rewriting here and üere, the first

four chap-ters are essentially the sanre as in üe second edition. Chapter V has

two new scctions: V.6r "Curt¡bling und thc Probubility Culculus" und V.8r

"l'robability and Causality," whoso titlcs ¡re self-dcscriptivc. Chapter VI, now

entitlecl coierence, contains ¡nost of üe material in chapter VI of üe second

edition togeücr with two new sections: VI.9: "Utility," which introduces von

Neu¡ilann-Morgenster¡) Utilitics, and VI.10: "Ramsey," which sketches Ra¡-¡r-

sey's rnethod for extracting sr,l rjectivr' probabilities and utilities from a coherent

p.tf"r"n"" ordering ovcr a ricl¡ sct r¡f fambles. Chapter VII, K'?1r'b of Proba-

»itity, itentirely new. Here, I try to give an even-handed survey of frequentist,

proferxity, and degree of belief pers¡rcctives on probability, togeüer with üe

inain problem areas for each view. \ r¡u will have to judge whether my own

philosophical bias is showing.

This -book

can be used in various ways. one is just to take üe chapters in

order, starting with the broad epistemological concerns o[ üe first thrcc

chaptcrs and working up to thc r¡¡ore formal ¡nethods of Chapters IV and V. O¡¡

üe oü"r hand, one of my fricnds has best results starting out wiü the dcvr:l-

opment of the probability calculus in Chapter V and then moving to Chaptcr I.

So¡nc irutnrctors omit üe trc¡tmcrrt of Mill's ¡ne tliods in Chapter IV entirtrly.

f AL,,n,CP(</a

'-5-Izl.

\{

8

Skynns, llrian,(lhcricc rt¡ttl cltr¡rcc.

llitrliograplry: P.Inclu<les indcx.l. Induction(l,ogic) 2. Probabilitics

IlCgl.Ss I!)l (i 16l

lsuN 0-134-05190-I

rr I I tl

1 ll1, Ll(li¡ll

l. Title8s-7 I 37

llI

t,l(.

,l l ,l l

, lr lr ll ,/i.,r 1lr { I 1{} t.[)

23291

lv I'I{EFACF] TO T}Iti 'TIiIIID EDITION

Others find that tlieir students arc interested in the applic¿tion <¡f Mill'snrcthods, and that Chaptcr IV introduces cnough truth-functional logic for the

strbset¡trcnt rrccds of Cha¡rtcr V. Sor¡¡cti¡ncs thc book is rrscti as a str¡rpletnetrt in

a philosophy of scicnce or un cpistcnrology coursc, in which casc thc instrtrctorcan pick and choosc.

Spccial thanks gocs to I)r. Ccorg l)orrr, wlro rvas ki¡ttl enotrglt to rcacl thc

cntirc rrutntscript vcry carcfrrlly, ancl who cattglrt iltntttneralrlc crrors of allki¡lds. 'fha¡lks also to Wcrrdy Caltncttst¡tt for secing tltc tttultttscril>t into 1>rint,

Brian Skyrrns

Univcrsity of C¿li[ornia at Irvi¡rc

rNVE:i7;;l;ll'i - , ir.-...: ¡ ¡ Fll-ilgoffLi.l:'í;lir;liU,

iii

I1

4

6l3r520

24

24

30

{i

47g

56)/BI

65

73

Contents

Prefsce

I. PROBABILITY AND INDUCTION

I. l. Introductionl. 2. ArgumentsL 3. tagicI. 4. Inductive Versus Deductive logicI. 5. The General and the SpecificI. 6. Epistemic ProbabilityI. 7. Probabiliry and tl¡e Problems of Inductive Logic

U. TI{E TRADMONAL PNOBLEM OF INDUCTION

fL l, IntroductionlI. 2. Hume's ArgumentII. 3. The lnductive Jtxtification of Induction

{, {, The Pragmatíc }ustification of InduetionU. 5. An Attempted Dissolution of the Traditional problem

of InductionII. B. Summary

Itr. TIIE GOODMAN PARADOX AND THE NEW RIDDLEOF INDUCTION

ffr. l. lntroductionIII. 2. Regularities and ProjectionIII. 3. The Goodm¿r¡ ParadoxItr. 4. The Goodman Pa¡adox, Regulariry, and üe Principle

of üe Uniformity of NatureIII. 5. Summary

TV. MN,I 'S ME-THODS OF EXPERIMENTAL INQUTRYAND TIIE NATI,IRE OF CAUSALTTY

fV. l. IntroductionIV. 2. The Structu¡e of Simple StatementsIV. 3. The Structure of Complex StatementstV. 4. Simple and Complex PropertiesIV. 5. Causality and Necessary and Sufficient Conditioru

757577

uu

vt

IV. 6. Mill's MethodsIV. 7. Thc Dircct Method o[ AgrcenrcntIY, 8. Tl¡e I¡¡vcrse lvlctho<i of AgreerncntIV. 9. 'Ite lvf ethod of DiflercnceIV.l0. Tlre Combined N{cthods

IV.ll. Thc Application of Míll's lvlethodsIV,l2. Sulficicnt Conditions and Futrctional Ilclationships

fV,l3. Lawlikc a¡rd Accidcntrtl Conditions

V. TIIE I'IIOBABILITY CALCUI,US

V. l. IntroductionV. 2, I'}robability, Argurlcnts, Statenrents, and PropertiesV. 3. Disjunction and Ncgation IlulesV. 4. Conjunction Rulcs anrl Co¡¡ditional ProbabilityV. 5. Expccted Value o[ a Car¡rblcV, 6. G.rnrl¡ling and the Prcrbability Calct¡lusV. 7. Bayes'TheorcrnV. 8. Probability and Causality

Appcntlix to Chaptcr V: SAMI'LINC AND STATISTICS

A. l. lntroductionA, 2. Dcscriptive StatisticsA, 3. Strnpliug altd |ro,cctivo Stutistics

VI. COHENENCE

VI. I. lntroductionVI. 2. Thc Probability C¿rlculus in ¿ Nt¡tshellVL 3, The Logical Consequetrce Principle Alone Is Not Enough

VI. 4. IletsVL 5. Fair Bets

VI. 6. The Dt¡tch IlookYI. 7. Conditionaliz.ationVI. 8. FallibilityVL 9. UtilityVI.l0. Ramsey

VII. KINDS OF PROBABILITY

VII. l. Introductio¡rVlI. 2. Rational Degrce of IlcliefVIL 3, Relative Frer¡rencyVII. 4. Chancc

h¡dex

IProbabilityand Induction

I.l. INTITODUCTION. lvlost people know rhat logíc h;u some-tlring to do with correct reasoning and with frarning compelling argu-nrcnts. Many also know tlrat there arc two l.¡asic brancl¡cs of logic: induc-tíve logic and deductive logic. Fewer people know that inductive Iogicis somchow tied up with the concept of probability. But just what logicis, what thc diflerence is betrvccn inductive and deductive logic, andwhat thc rclationship is betwecn probability and induction are c¡uestionso¡¡ wl¡icl¡ therc is not only widesprcad ignorance but llso widcly acceptednrisinformation. Answers in depth to these rluestions can only be attaincdby extensivc study of logic, But we can at this point give prel.iminaryanswers that will provide a pcrspcctive for the studcnt embarking on thestudy of inductive logic.

I.2. AIiCUMENTS. The word "argument" is uscd to rncan severaldillcrc¡rt tlrings in thc llrrgli.sh languagc. lVe spcak of two ¡rcoplc havingan i¡rgun)cnt, of onc ¡lcrson advurrcing au arguruont, ar¡d of thc valuc ofa ¡natl¡cm¿tical functir¡¡l as dcpcnding or¡ the valucs of its argunrcnts.O¡rc of tllcsc various scrrscs of "ar.guntcnt" is sclcctccl and rcfined by thelogician for the purposes at hand.

Whcn rvc speak of a pcrson advdncing ar¡ argurnent, we h¿ve in mindhis givirrg certair¡ reasons designcd to persuadc us that a ccrtain claimhe is making is correct. Let us c¿¡ll that claim which is being argued forthe conclusion of thc argurnent, and the ¡easo¡.ls advanced in support ofit thc preniscs. I[ wc norv abstract f¡orn thc cor)crotc situation in whichone person is trying to convince othcrs ar¡d consider thc bare boncs ofthis conccption of an argument, wc arrive at the following definition: Anargument is a list of sentences, onc of which is dcsignated as the con-clusion, and tlie rest of which are dcsignated as premises.

But if we consider the matter closely, we see üat this definition willnot do. Questions, cornrnands, and curses can all bo expressed by sen-tences, but thcy do not make factual claims nor can thcy .stand as reasonssupporting such claims. Suppose somcone said, "The Dirty Sox star

1

CONTEN'I'S

8B

89

95

l0l106

Iu115

t20

129

129

t32138

I48r52r53l,5(i

159

159

t62

167

t67L7lt75r78185

189I94198

202

205205208

211

2t1

I. r,rtc¡l¡¡ourY AND tNDUCTioNI. 2 ancul'r¡Nls

no t¡ouble idcntifying the prcntiscs rttld cottcltrsiott of a gtvc» argutrorrtif you ¡erncmbe¡ that:

Thc conclusi<.¡n statcs the point bcing argucd for and the prcnlises

state thc rcasons being advanced in supl)ort of the colrclusiorl.

Since irr logic rvc a¡e iutcrestcd in clarity rathc¡ thln in literary stylc,orrc simplc, clcar ¡¡ethod of stating alr argu¡nellt (and indicatirlg rvhiclr

statements are the premises and rvhich the conclusion) is prcferred tothc rich v:rriety of forms availablc in English. Wc will put an argument

into standard logical form sinrply by listing the prcntises, drawing a line

under thcm, and u,riting the conclusio¡r under thc line. For exanrple, the

argument "Diodo¡us was not an Eagle Scout, since Diodorus did not knowhow to tie a square knot and all Eagle Scouts know how to tie square

k¡¡ots" would be ¡tut into standard logical forrn as follows:

Diodorus did not know how to tie a srluarc knot.

AJI Eagle Scouts know how to tie squarc knots.

Iixcrciscs:

1. WIlich of the follorving setitcilccs are st]tcmcllts?l. i¡riends, lionrltns, countr)'¡nerl, lerrd mc yotll cxrs.

l¡. 'I'hc sum of tlie stluares of tlte sidcs oI a right triangle e<1uals the

s(iuuIc 0[ titu lrYPot0rluso.c. llrst tllou cc¡¡¡stdc¡cd nly scrvrlit Job, lr perfccI rrttcl att u1>rigltt rrtalr?

d. irly narne is l"attst; in all things tlry e<¡ual.

c. I) = t¡tc'.f. .lrlay he be boilecl iIr his orvlr pudding ¡rrd buried witir ¿ stick of holly

tlrrouglr hls ]leart.g. Ptolenry niai¡ttaÍned that thc strn revolved around the hl,r'th'h. Ouch!i. Did Sophocles rvrit¿: Elcctru?

¡. The sun lrevcr sets orl thc Ilritish Ern¡rire .

2. Which of the follorvi¡lg selections advance argurncnts? Ptrt ell argunrertts

in standald logical form.

a. All plofess<-rrs are absetlt-mindecl, and si¡lcc Dr. Wise is I profcssor he

must be abse¡rt-minded.b. Since three o'clock this afternoon I have felt ill, and norv I fcel worsc'

c. Canclidate X is certain to win the elcctio¡r bec¿usc hi's backe¡s have

more mo¡rey tha¡r do Ca¡ididate Y's, and furtLcr¡not'c C'rndidate X is

mole popular in the urban areas.

¡rrlclrt'r lrirs ¡r.rst llrokcn both liis arms allcl lcgs' thcir catchcr has gliru-

1u,,,,,, tl¡cir cntirc outfictld llrls cot¡.rc dorvn rvith l¡r¡b<¡nrc plaguc' and thcir

,i,,,rlt,rp has bccn cleportc<l. Thercforc they cannot possibly -win

the

;,; ;,,;;,ti." IIc u'r-rtrld cl.iarly bc aclva,cirrg ittt-argr'tmcnt' to thc cffcct that

tlrt: l)irty St¡x cnntiot wi,' ih" pcnn:rIlt' But if- sol¡reorlc said' "Ilow's your

.l,,t"rl Sir,r,1 trp o. tltc taSlc Nlay yo. pcrislr itr trttspcakaSlc slinrcl" he

r",r"f¿, whatcrir clsc ]re was doing, not bc adviincirlg an argumcnt'

'if,r, ir, lrt: woultl ,ot bc .civanci,g cJid"nce in support of a f actual c1airtr'

I-ettrscallltscnt0ncctlratrnakr:sadcfinitcfactualc}aiI¡raStatel¡tcllt."llarr¡ribal crossccl thc Alps," "socratcs was ¿ corrul)tor of youtll"'.'Evcry

ñ; ;;;;,; "r..y oth"' úody with a forcc dircctly proportio-nal to tl¡e

,.,,rr'of thcir rnasses ancl inversely Proportional to thc squarc of thcir dis-

tlrrcc," ¡¡lcl "Thc ¡¡¡oon is madc of avocado paste" are all statcments' sonrc

tn¡o, solne false. Wc,,,.y no'u formulate ihe logicialr''s dclilrition of an

lr rgrr ttrt'nt :

t)clirritio¡rl:Auargrtttrclltis¡]istt)[stutcntcttts'ol)cofwhichisclcsignated as thc conclusion anrl tl¡e rcst ol wliich are clcsig-

t¡,rtt'<l ¿ts l)r('l¡lis('s'

lrt or<littary s1>cccll rvc scldonr conre right out a¡ld say"'A' I1' C a-re rny

¡rrt'tttiscs a¡lcl l) is rtly cotlcltrsioll " llorucvcr' tltcrc flre scvcr¡i "irldicator

§'0rtls"tlrltt ltrt: cotlltttottly trst'tl irl lilrglislt ti> lloirrt otlt rvlliclr stttt:tltctlt

is tl¡t' c,¡tr:lttsit¡¡l itlttl tn'1,i,'l' ,'rc tlrt' ¡ticnliscs'. l'l¡1: rvQrd "thorcf,re" sig-

rrrls tlrltt rlrt: ¡lrt:ttrist's l"tt"ii"'t'" tu" tlrtot'g)'''.lrd tll.t tltc c.ncltrsion

il',,1,,,,,t to lrclrrc:st'rrtctl (as irr tlltr l)irtl'Sox exarrrpk:)"fllc rvords "tlrtrs"'

"const:t¡ttt:trtl1'," "ltt'ltcc," "so," lttttl tlrc pltresc "it foll0rr"s that" fur¡ction

irr cxrtctlY tllc salnc rvaY'

In ordinary <liscourse thc conclusiort is sotrlctintcs stttcd first' foilorvcd

by tlrc llrcrrtiscs iirlvatrctd irt strllport of it lrl thcsc c'ases' differe¡lt indi-

.,1,r, t"ur¿, a¡c uscd Considcr the follorving argumcnt: "So;raf,11 ,,is

lrasically sclfish, sincc aftcr all Socratcs is a trlan' a¡id all mcn arc bastcutty

sclfish." FIerc thc .u,]"iu'iu" is statcd first and the rvord "sittce" sig,als

tl¡ut re¿rsons in ,uppo.i of tllat conclusion follow' The words "because"

t¡¡cl "lor" ancl the phrase "by virtue of the fact that" are often uscd in

tlrt' s¿rnc rvay. There is a variation on this mode of presenting an argu-

,,,,,rt, ,ulr.r" tlle worcl "since" or "because" is followed by a list of prem-

isls rt¡ttl then tllc "nn"lu'iunt

for exanrple"'Sincc all mcn a¡e basically

slllislt :rntl St¡cnrtcs is l rrrlttt, Socrlttcs is lrltsicltlly st:lfislr'".l.lrt,sr. lrrt: tll0 rrrost

"o,u,r.,,r,., wirys of statíng lrguments in F'nglish, but

tlrr'¡.':rtt'rltlttlr $'ays, too n.,"'"'utt' to catalog llorvcver' you should llave

I. ¡,nor,r¡¡r¡-rrY AND INDUCTIoN

d, Iro¡r rvi]l not fk¡¿t rvlrcn pul. irl \vater because thc specific glavity of

water is less than that r-¡f irorl.c. In tlrc past, cvt)ry irtstlttlco oI sntr¡kc ll¿ts lrcc¡r accomllanicd l:1'firt:, so

thc ncxt insIancc of snrokt'u'i]l also bc ltccomltanietl by fiLe.

I.3, LOGIC. When we eDuluale an arguntcnt, rvc arc intcrcstcd intrvo things:

i. Arc thc prcnriscs truci)ii. Supposing that thc prctuiscs arc truc, rvltat sort of support do

they givc to tl¡e conclusio¡r?

TIle 6rst co¡lsideration i.s obviously of grcat intportancc. Thc argurncnt"All college students are higlrly intclligcnt, sincc all collcge studcnts are

insane, and all people who arc i¡rs¿tne arc lrighly intclligcnt" is not vcrycompelling, simply becausc it is a matter of common knowledge that the

premises arc false. Btrt irnportant as consideration (i) may bc, it is not tltcbusi¡:css of a logician to judgc whetlicr the prcntiscs of att argutnctlt arc

true or falsc.' Aftcr all, any statcnrents u'hatsocvcr can stancl as prcmisr:s

to some argulllcl)t, and tl¡e Iogiciarr has no spccial ¿rcccss to all hur¡'ran

kuowlcdge. If thc prcmiscs of an argur))cnt nrakc a clainl allout thc intcr-nal structurc of thc nuclcus of tllc carbo¡r atont, onc is likcly to gct nlolc

reliable judgmcnts as to their trutlr fro¡n a physicist than fron a logician.

If the prcrnises clai¡n that a ccrtain Irtccl¡auisrn is rcsponsiblc for the

chanrcloort's color cltattgc§, 0nc w0uld ask a biologi¡it, n0t a logieian,wl ether thcy are true.

Consideratio¡r (iÍ), horvcvcr, is the logician's stock in trade. Strpposing

that the premiscs arc truc, docs it follorv that tlrc concltrsion tnust bctrue? Do the premises provide strong but not conclusivc evidence forthe conclusion? Do thcy ¡trovidc any cviclcnce at all for it? Tl)esc arc

questions which it is the busi¡rcss of logic t<.¡ :¡nswer.

Definition 2: Lo¡¡íc is tlre stucly of tltc strcngth o[ the evidcrrtialIink betwcen tlre prcnrises and conclrrsions of argunlents.

In sor¡re argumcnts the Iink bctwccr.l tltc prcnriscs and thc conclusion is

the strongest possible in that thc truth of the premises guarantees tltetruth of tlic c<¡nclusion. Considcr tlrc following lrrgt¡r¡)cnt: "No Athcnia¡rever drank to excess, and Alcibiades was an Athcnian. Therefore, Alci-biades never drank to exccss." Now if wc supposc that thc prcnliscs "No

I. 3 loclc

Athenian cvcr clrar¡k to exccss" and "r\lcil¡iadcs \r,as an Átllcrriá1n" ar.ctruc, thcn wc ntust also suppose that thc cr¡uclusion "Alcibiadcs ncvcrdrank to cxccss" is also truc, for thcrc is no way i¡i rvhich thc co¡rclr¡sioucould be false while the ¡trenriscs rvcrc truc. Tllus for this argurrri.nt rve

. say that tho trutlr of üc premiscs gua¡'¿lntecs thc truth o[ tlrc conclusio»,and thc evidential link betwce¡t prcmiscs and conclusir-¡u is as strong aspossiblc.'I'hi.s is in llo way altcrcd lly tlrr: fact that tlrt: fir.st ¡rrcrrri.sc lrrrltltc couclusion arc falsc. What is irnpurtant for cvalrrating tlre.str.cngthof thc cvidc¡rtial link.is that, if tlrc prcr:riscs wclc truc, thc conclusio¡rwould also have to I¡e truc.

In other arguments the li¡tk betwcc¡¡ thc prclrri.ses and thc co¡rcltrsio¡ri.s not so strong, but tllc prenriscs ¡levc¡thclcss provide so¡nc cvidcntialsupport for the conclusion. Someti¡ncs the prcntiscs ¡>rovir)c .strorrg cv.i-dc¡lcc for the conclusion, sometinres rveak cvidc¡rce. In thc followirrgarguntc¡lt thc fruth of tire prenriscs docs not gua¡.ar)tcc thc truth of theconclusion, but the evidential link bctwecn the prentises a¡td the con-clusior¡ is still rluitc strong:

Black has confcssed to killing Whitc. Dr. Zcd has sigucd a .statc-nrcnt to thc effect that hc sa',v Illack shoot Whitc. A largc nunrberof witnesses hcard Whitc gasp with his dying brcath, "Black didit." Therefore Black kil)ed White.

But although thc premiscs arc good cvidcncc for thc co¡rclusion, wc knowthat thr.l truth of thc promises eloci not guarantel tho truth of thc con-c]usiol, for rcc ca¡"t inaginc circunstances in ultich tln ¡trcmises taouklbe truc and the conclusion false.

Suppose, for instance, that Black was insanc and tlrat ]rc confcsscdto every murder he ever hcard of, but that this fact rvas gr:ncrallyunknown bccause he had just rnovcd into thc nt'ighlrorhood. Tlris¡reculiarity was, howevcr, knowrr to Dr. Zed, rvho u,as \\¡lrite'spsychiatrist. For his own malevoicnt rcasons, l)r. Zcd dccidcrd toeliminate White and frame lllack. IIe convinced \\/hitc r¡ndcrhypnosis that Black was a homocidal ¡¡¡aniac bcnt on killing \Vlrite. 'Then one day Dr. Zed shot Whitc from bchind a pottcd plant andfled.

Lct it bc grantcd that thcsc ci¡'cu¡rrst¡nccs irrc lriglrll, irn¡rrolillrlc. IIthey were not, the premises could not provide strong t:vidential supportfor the conclusion. Nevcrthcless the circumsta¡rccs arc r)ot irn¡rossiblc arrdthus the trutil of the prcmises docs not guanrntce tho trrrth of theconclusion.

t Exccpt in ccrtai¡¡ vcry spccial cascs wlricl¡ nccti ¡tot conce¡¡r rts )¡crc

rl I. plroB,a.BrLrtY AND INDUC'rloN .

'l lrc loll,rrvrrrg is ert argumcnt irt rvhich the prcniises provide sottte

,.r¡rl(.n(('lor tlr()(()nclusiou,butirlu,lriclrtllcevidcrltiallirrkbet*'cctrtlrt'¡,r ('r¡r¡s(:s lr¡¡rl tltc ct¡llclusiotl is uruch wclrkc¡ th¿n irl thc forcgoirrgCIill¡¡l)lc;

Studc¡rt 777 arríved at the hcalth ccnter to obtain a medical cxcuse

fror¡¡ his final examination. He cornplaincd of nausea and a ircad-

aclrc. Tllc tlursc tt:portcd a tt'rrtpcrattrrc of I00 clcgrt'cs. l['lrcr<:lr¡rc

studcnt 777 was rcally ill.

Oivcu that thc prerr-riscs of this algur.ncnt al'c truc, it is not as ilrr¡rrobablctlrat the conclusio¡r is false as it was in tlie prcccding argurnent. Ile¡rctrthc argument is a wcakcr ortc, though not cntirely without mcrit.

Thtrs we sce tlrat arguments may havc variotts dagrces ot' strertgtlt'\Vhcll the prc¡nises preserrt absoltrtcly co¡rclttsivc evidencc for the con-

clusio¡l-that is, whcn thc truth of the prcrllises Strarantces thc truth o[

thc conclusion*then we h¿rvc thc strongcst ¡tossiblc type of argumcnt.There are cases ranging from this maximurn possiblc strerigth down to

argu¡nents where üe prernises contradict üe conclusiorl.

Exerciscs:

Arrange the following argumetlts iIr the ordcr of the stlength of the link

l¡etween premises a¡rd conclusion.

L No onc who is l)ot it mambcr r.¡f ths clrrb rvill l¡o atlmitterl to thc mscting'

I am not a mcrnber of the club.

2. The last thtee cars I have o-w¡red have all becn sports cars' They have all

¡rerfor.med beautifully and gíven ¡ne little troul¡le. Therefore I am sure that the

r)cxt sports car I own will also perform beautifully and give me little trouble.

3. My nose itclres; theleforc I am going to have a fight.

4. Brutus said that Caesar rvas ambitious, and Brutus was an l.ionorable man'

Tllerefore Caesat must l¡ave beerl ambitious.

5. The rveathetmt¡r has said that a low-pressute front is moving into tl¡e area.

The sky is gray and very ovet'cast. On the street I cltn see several people cart'y-

ing umllellas. Tlie weltherman is usually accurate. Therefore it will lain.

1.4 INDUCTIVE VERSUS DEDUCTM LOGIC, When an

elgume r)t is srrch that the truth of the prcrnises guarantces thc truth of the

tr¡r¡ch¡str.rn, u'c slr¿rll say that it is deductively valid. lVhcn an argument is

I. 4 rNuuc;r'lvu vI'r(sus l)ltl)tic'l'lvll Locl(

rur¡t dcductit,ol¡,valid but ncvcttlicless tlrc lllcririscs Providrt good cvltlctttt'[or tiltt cotrClusiou, t)tc argurrrt'nt is s¿icl to ltc itrcluctivt:11, strortg, I'lou'

strorrg iL is dcpr:rrcls otr lt,rtv tlluclt cl'idct¡tial strpporl tlrc prcrrtiscs giVt'

to thc concluSiotr. l¡l linc rvitll tllt: ctiscussion i» tlrc llst sectiort, \\'(i c¡l)dcfinc thcsc two cor)ccpts nrorc precisell' ¡s follow's:

Dcfinition 3: Atr argunrcnt is ¿lcrluctiucly oulitl if arrd only if it is

in¡tossiltlc that its coDclusiol¡ is f illsr: tvliilc its llrL'rltiscs lttc ttuc.

Defir¡itio¡'r.1: .Árr rtlgurtl(:ttt is i¡ultrctiot'ly sLtottg ii rrrril orrl¡' iI it rs

inrprobublc tltat its collclusic¡r¡ is f¡lsc rvhrlC its prt:uriscs llr(: trrr(',arrcl it is l)ot dcductivcly r,.rlid. 1']|.e dcgrct: oI iDdrrctivc strt'ngtlr

dc¡tcnrls oD lrOw,inrproba[.rlc it is tlrltt tl¡c concil¡sion is la]sc rvhilc

thc prcnriscs arc truc.r

'l'[c sr¡¡sc of "irnpossrblt:" irrtcnderl in l)t:lirritior¡ l] rt'rlLtittrs claliiicltiorr.

In a serrsc, it is irDltt.rssiblc Ic¡r lDe tt.r Il¡, ¡¡u,,,,¿ tirt: r'rrr;r¡r lry lla1ll>rrrg Irry

arnrs; tlris sensc <.¡f irnpossibility is cilll.'cl plty.:icaL iilt¡tossiltíl.ity.llLrt it is

not pliysical inr¡;ossibility th¡t u,c l¡ivc i¡r rrinrl in Dt'[initiorr 3. Cousirlt'r

the follow'ing argtutrcnt:

C,'ulgc is it tll:tlt.

Ccorge is 100 ycars olcl,

Ccorge has artlrritis.

Georgc \vill n0t run a tal¡r-lllint¡t'- rlri'lc tornor¡o\v'

Althougli it is physically,inrpossibic for thc ct¡nclrrsiotr ol tltc rtrgLrttrctrt to

be falsc (that is, tliat hc rvill indocd r'un ír fotrr-minutc nrilo) rvhilc thcr

prcmiscs rlrc t-ruc, thc arguntcnt, although a prctty gooci onc, is rrol dccluc-

tivcly valid.To uncover thc sensc of irnpossibilit)' i¡r thc dcfi¡riti<.rn of dcductive

validity, lct us look at an exanrple of a deductivcly vitlid rllgunrc¡)t:

No gourmcts cnjoy banana-tunll fislt sou[Ilés rvith clloc:o]atL' sittrc('.

Antoine cnioys banana-tuna fish souf{lés with chocol¡to slucc'

Antoirrc is not a gourlllct.

! AltiroLrgh tlrc "whilc" in j)efi¡¡ition 3 trrr¡' bc rc¡tl r¡s "i¡¡,t1" rviti¡ thc tleli¡ritio¡r

rc,rraining correct, tltc "whilc" iI Dcfinition 'l s]ror¡lrl bc rc¡cl ¡s "givcrr tlrrt"a¡tcl not "lr¡rl." Tlrc ¡c¡So¡rs for this cirn l¡e rDaric prccisc olly lfttr stlrttc ¡rrolr-

irltrlity tlrcory lr¡s bccn str¡djcrl. IIr¡rvcv(:r, tlr(i sc¡rs('ol l)c[irritio¡i 'l rv¡l] lrt'

crphilt,,l I¡rti't ill tlris scetit,l).

I. pno¡aBt¡-trY AND INDUCTIoN8

ln üis example it is impossiL:le in a strouger s€nse-wc shali say logically

¡Áiou¡aU-irr thc conclusion to be falsc rvhilc tltc pretnises are trttt:'

wlrat sort o[ irnpossibility is t]ris? For thc c-onc.h¡sio¡t to bc falst'Atrtoir¡c

wi¡ulcl havc to bc a go"á"i' For the seconcl prenrise to be trt¡e he would'

also havc to e.ioy b""u';-;;';;Á'l' 'ouffié' uiith chocolate sauce' But for

;;d; premisc io be true there must bc no such person Thus t<.r'suppose

the conclusion is false i' io fontto¿itt the factual claim ¡nade by the prcttr-

ises. To put the mattcr a" áiff"t""t *ty' thc f actual clairn nraclc by thc

cc¡nclusior.¡ is aircacly t'"pl'"ft tt ift" i'"n''i'"'' Tliis is a fcature o[ ¿ll

deductivclY valid arguments'

If an argurnent is dctluctivcly valici' its conclusion makcs no factual

clairntlratisnot,irtlclstinlplicitly,nla<lcbyitsprcrniscs.

'lhus it is logically ir»possiblc for thc cotrcltrsion of a deductivcly valid

argumer)t to bc false while its prcmiscs are true' because to sul)pose that

the conclusion is false tt'i"'""^ttt'átct so¡ne of the factual clai¡ls made

t'#Xff::l;r". rul,v tht: following .'rgttmunt is not dctluctively valicl:

GcolgeCcorgcCeorgc

is a ¡n¿rn.

is 100 ytrars old'

has arthritis.

C"*b" -tX ,ot ,un n fc¡ur-mirlutt: ltlilc tot¡rot-torv'

Thr: [actual clai¡¡r nracle by thc conclusiort is r¡oÚ.i¡rplicit in- thc prcrntscs'

for there is no prcnii'sc 'tníi"g that no l(X)-.ycar-old man rvith arthritis can

run a four-¡¡rirrutc milc' ói Lu""' we ali L"'lit''c this to bc a firct' but

thcre is nothing ln ttt" p'itit"''tlt"t "lnint' this to bc a factl if we atklacl

;;;;il;;i,i. "n"tt,l¡''" *t' 'uoul.

ha'c ¡t clctlt¡ctivclv vrlid argr¡¡nelrt'

The conclusion o[ an inducriuelg strong urgumenl, on the othcr ]larld,

ventures beyond t¡e fa"i'J "üinls"matl"

úy tirc prenrises' Thc conclusion

asscrts ¡nore than thc pt""'itü tttt" *" "átt dcsclibc situations in rvlrich

,fr" pl"",lt"t woukl bc truc and thc conclusio¡r false'

lf an argument is inductively strong'-its conclusion makes factual

claims that g" U"y"'J the fáctuai information given in the prenr-

ises. I

Thus an incluctively strong argument risks more than a cleductively valid

one; it risk the p,,'iúiiit if leading fron.r true prcrrrises to' a false

conclusion. But tliis rirt'ir-it" ptice thlt must be paid for the advantagc

Il, 4 ¡¡r'DucrlvE vEnsus DEDUCTIVL LoGIC

which inductively strong arguments have over deductively valid oncs: tl-re

possibility of discovery ani prediction o[ new facts on the basis of old

onf;"§r,i,ion 4 stated that an a¡:gument is inductively strong if and onlT if

it meets two conditions:

i. It is inrprob¿l¡l¿ lhat its colrclusion is falsc' rvlrilc its ple¡¡rrsos l¡rc

true.ii. It is not deductivelY valid'

Condition (ii) is required because aII deductively valid argun.rents mcet

condition (i). lt is impossiblz for üe ctnclusion of a deductively valid

;;;;;;, ü'b" f"lr" while its premiscs arc true' so thc probability that thc

coiclusion is false while the premises are true is zcro'

Condition (i), however, '"q'i'"s clarilicatio:r' The "while " in tlris co¡r-

clition should be read ut ;giu"" that," not as "and"' so that the condition

can be rephrased as:

i. It is inrprobablc that its conclusio¡¡ is false' gíoen tlwt its pretnises

are true.

But just rvhat do we mean by "givcn that"? A¡rd why is "It is irnprobable

tfrrt ltl "o.,.trsion

is false aiits"premises true" an incorrcct formulation

of condition (i)? What jt i¡" áiff"t"'ce, in this context' bctween "and"-';

':*lren ti¡at"? at this stage thcse qucstions aro best answercd by

examitring sevcral exampi;s;f Lg''''tnt' The following is an irrductively

strong argument:

Tirclc is intclligcnt lifc on Mcrcury'

There is intelligent life on Vcnus'

' Tl¡cre is rntclligent Iife on Earth

There is iutelligent life on Jupiter'There is intelligent lile on Saturn'

There is intelligent life on Uranus'

There is intelligent life on Neprunc'

There is intelligent life on Pluto'

ffiNotethattheconclusionisnotbyitselfprobable'Itis'infact'probablethat üe conclusion i, fuk" Strii'i' i"1p'obublt that the conclusion is false

giaen that th. pr"*ir., *'e true' Thai is' if the premises were true' then

onthebasisofthatinfo,mationitwou]dbeprobablethattheconclusio¡r

l0 I. i,nou¡¡¡¡-rrl AND INDUCTTON

would be hue (and thus improbable that it would bc false). This is not

afiected in the least by thc fact that some o[ the prernises themselves ¿rrt:

quite improbable. Thus although the corrclusion taken by itself is improb-able, and some of üe premises taken by themselves a¡c also improbable,üe conclusion is probable gíaen the premises. This cxample illustratesan important principle:

The type of probability that gradcs the inductive strerrgth of .rrgu-

ments-we shall call it inductiae probability-does not depend on

the premises alone or on the conclusion alone, but on the eoitlentialrekttion between the premises and the conclusion.

Hopefully we have now gained a certain intuitive understanding of the

phrase "given that." Let us now see why it is incorrect to replace it rvith"and" and thus incorrect to say that an argument is inductively strong ifand only if it is improbable üat its conclusion is false ond its premises irrc

true (and it is not deductively valid). Consider the following argument,which is not inductively strong:

There is a 2000-year-old nan in Cleveland.

There is a 2000-year-old man in Cleveland who has three heaCs.

Now it is quite probable that the conclusion is false giaen that the

premise is true. Given that there is a 2000-year-old man in Cleveland, itis quite likely that he has only one head. Thus the argument is nof induc'tively strong. But it is improbable that the conclusion is false and the

premise is true. For the conclusion to be false and the premise true, there

would have to be a non-three-headed 2000-year-old man in Clcveland,

and it.is quite improbable that there is ang 2000-year-old rnan in Cleve-

land. Thus it is improbable that the conclusion is false and the premise

is true, simply because it is improbable that the premise is true.'We now see that the inductive strength of arguments cannot depend

on the premises alone. Thus although it is improbable that the conclusionis false and the premises true, it is probable that the conclusion is false

! A conjunction, üat is, a compound sentence formed by the word "and," can

be no more probable tl¡an either of its coniuncts (the simple coostituent sen-

tences), es wül be shown in the section on probability' Thus if it is improbable

that üe premise is true, it must also be improbable that the conclusion is false

and the premise is true.

I. .1 i¡r»uc'nvE vt rrsLls IIDDUC'rrvu Locl(

gir.;ut tltat the prerniscs itrc truc ¿nd tlrc argur))cr)t is rrr.rt inductivclystrong.o

An:rr¡lrrrncnt nright bt: such that it is ilnprobublc that tltc ¡;rcrrtiscs are

truc and the conclusion falsc, sirrrply bccau.se it is irnprobablc that the

conclusion is false; that is, it is probablc that thc conclusiott is truc. lt.isiinportant to note that such conditions do not guararltec tliat thc argunretitis indrrctivcly strong. Considur the follorving t:xarnple of an argtrlnent thatlras a lrrol.,ablc cclnclusiort arrd yct is r¿ol iuductively strorrg:

Tlrcrc ¡s a rnrn in Cicvcland wlrr-¡ is 1999 ycars atrd 1I nlontl¡.s old¡nd in good l¡calth.

No rna¡r will live tc¡ [¡c 2000 years old.

Now the conclusion itself is highly probable. 'fhus it is improbable thatthc co¡rclusio¡r is false atrd cottscrlucntly irnproblble that thc conclusion

is false rir¡¿l tI¡e prcrrrise truc. But if thc prcmisc r erc trtre it would l-¡e

likcly that the concltrsior¡ would I;c fllse. By itsclf tl¡c ct.¡r¡clusiorr is prob-

ablc, but givcn the premise it is not.5

Tlie ¡nain points of this disci¡ssio¡r of incluctivc strength catl l¡c sulnmed

up as follows:

l. TIrc i¡lductivc probability oI an argttntcrtt is tlrc probability tlrat its

co¡rclusion is true givcrt tllat its pretnises arc true'

2. Thc inductivc probability of an argultrc¡it is detcr¡nined by tlrcevldcntill r'chtion br:twcutr its plutrtiscs lllld it§ collclusi0ll, u0t l,y tll0likelihood of the truth of its prerniscs alone or the likelihocrd of tlre truthof its conclu.sio¡ alone.

3. An argurncrtt is inductivcly strong if and only if :

a. Its irrductive probability is higlr.b. It is ¡rot deductivt,ly valid.

We dcfined logic as the study of the strcngth of the cvidcntial linkbctwecn thc plcnriscs and conclusions of arguntents. We havc scen that

there are two different standards against which to cvaluate the strength olthis link; dcductive validity and inductive strength. Corresponding to

¡'.fl¡us tire alralog of a principlc of dcdttctivc logic, Irutrrcly, tlrut any sta[c¡rre¡¡t

logically follows from a contradiction, does not hold in inductive logic,

5 Thus t-he analog to a principlc of dcductive logrc, namcly, that ¡ tautology

Iogically follows from any statemcnt whatsocvcr, tlocs not I¡old jn inductive

Iogic.

u

I. r,nou.uu.rry AND lNDUcrIoNt2

thesetwostanrlardsaretwobra¡rclresoflogic:decluctivtllogicarrdi"ár",*" logic. Deductiae logic is concerned with tests for dcductive

,"fiaiiy-,t,"i is, rules for clecitling whether or.not a given argument is

a"ár"íir"fy ,^lid-and rules for constructing deductively valid arguments'

l¡tductioe logic is concerned with tests for measuring the inductive'prob-

,il,ü, "r¿ tence the inductive strength' of arguments and with rules for

constructing inductively strong arguments'

Some books apPcar to t'gicti that tlrere are two difierent types of

argum€nts, dedtrctivc ot'á ¡'-¿"t?tiu', and that deductive logic is conccrtrcd

with deductive i¡rguments and inductivc logic with inductive argr¡ments'

;ñ;t, thcy sugfcst thc iollowing classification' together with thc as'

sumption that every argumclrt falls in one and only one category:

Dcductivc arguments Inductive argulncnts

Valid . StrongGood

Invalid WeakBad

Nothing, however, is further from the truth' for' as we have seen' all

inductively strong argrrments arc deductivcly invalid'u

It is more correct ao f'¡"i'i" arguments as being arranged on a scalc of

descending strength, as follows:

r\rguments

DeductivelY valid

Degrees oi inductivestrength

Worthless

IOn".lght suggest that wc dcfrnc a deductivc argumcnt as one that intends to

bc deductivcly valid an<l " i;;;;;; argurnent a' o'" thut -i"-Ts^lt,-tl" *:i:bo:ff

"'"Ji:;:'', Jl i,l, J -l,i l"i ñ,'io' u' ñ',""n * do not i nten d anv ü in g Peo pre

who advance arguments ¡nt"J''n"ny ihings' Sometimes they intend for the

argument to be deductivefy ""fiá' tá"t'rnls üey intend it to be inductively

strong; sometimes üey intend-ii-i" U" " clever sophistry; and sometimes they

don't know the difierence'

Ezekial's tail is the tail of a wolf'

I. 5 ruri ctrNEltAL AND TIM Sl'llctFtcl 13

Dcductive and inductivc logic arc not distingrrished by thc diflercnt types

ot arguments with which thcy deal but by the different standards against

which thc¡' evaluate argurnerlts'

Excrciscs;

Decic.le whether cach of the following lrrgurnonts is cleductívcly valid' in

tluctivcly stru¡rg, or ttcitllcr;

L Ceorge Washing:on §'as a contemporaly of Ad¡m Snlith

A<iarn Smith was a co¡ltcrnporary of Davitl Llunle'

Gcorge Washir)gton w¿s a contcmporary o[ Davicl I{unre'

z

Everyone rvho is slightly intelligent is slightly neurotic'

3. On all the birthdays I havc evcr had I have been less tha¡i 30 ycals old

O,, *y n"*t birthday I rvill be less than 30 years old

4. Bcing out in thc sun for long pcriods of timc rn;¡kes onc higlrly intolligent

Ivfa,,y Clalifornia¡rs are oLrt i¡t thc sun for Iong pcnods of timc'

IUany California¡rs are Irighly intclligent'

I.5. TflE GENERAL AND TIIE SPECIFIC' One of the most

widcspread misconccPtions.of logic is the bclicf that dcductive argumcnts

;;;;;;á t;* the gcncral ts thc s,ecrfic; and inductive argumcnts Pr,ceed

ir"- ii," specific á the gene,al' S,ch " view is nonsense' for' as we have

,""r, ^rgun',"nts

do not Iall ihto two categorics: deductive and inductive'

P"riupr",h.n all deductively oalid arguments proceed from the general

to the^spccilic and all inductively §'rong arguments procccd flom the spe-

"in" a if," general. Tl¡is view is not nonsense; it is simply rt¡i .¡rrect' There

are deducti"vely valid arguments that go from general to gencral:

All gorillas are apes.

, All apcs are manrrnals.

All gorillas are mammals.

from particular to Particular:

Ezekial is a wolf.Ezekial has a tail.

t4

and from particular to general:

Frederick is an aldenr-ran.

Frederick is a thief.

I. r,no¡,1,¡¡-lrrr A ND I¡-DUC'r IoN

Anyone who knows all aldermen knows some thief'

Similarly, inductively strong argumcnts do not fall into the narrow

category áí ".gr*"ut. having particular premises alrd a general conclu-

,ior7 *gu*oits by anatogy pioce"d {rom particular to particuiar' :rnd

althougñsuch a.gr-ents á.c-oft", misr.rsed, some are quite strong' The

following is an example of a strong argument by analogy:

l. Car A is a Hotmobile 66 and car B is a Hotmobile 66'

2. Car A has the súper'zazz cngine atrd car B has tlte sultcr-zazz

cngine.3. tar A's engine is in perfect condition and car B's engine is in

perfect condition'¿. SotL cars have the same type of transmission and the same final

d¡ive ratio.5, Car A's top spccd is ovcr 150 miles per hour

Car B's top speed is over 150 miles per hour'

Pre¡nises (1) through (4) are saicl to sct u¡t tlrc'analogy'-that is'.thcy

describe relevant si,r,ila¡lties bctween car A and car B whibh ¡nake it

likely that their top speed is thc §ame, and the¡efore licensc tlie move

frorn prenrise (5) to the conclusion.

It is not difficult to find inductively strong arguments that have general

premises and a general corrclusion; for cxarnple:

AII students in this class are higlrly intelligent'

All students in this class are strongly motivated to do well

No studcnt in this class has a heavy work l<¡ad'

No student in this class has psychological difficulties that would

interferc with his course work.

All students in this class will do well.

Inductively strong arguments with general premises and general conclu-

sions play an important role in advanced science' For example' Newtont

laws of motion were confirmed because they accounted for both caiileo's

lawsoffallingbodiesandKepler,slawsofplanetarymotion.Wecangive a rough approximation of the argumcnt supporting Newton's laws

of motion as t-ollows:

I. 6 oprs'r'urrlc PnorlAllrLIT)

All bodics frecly falling Irtr¡) tllc surfacc of thc carth obel' CaLleo's

I a rvs.

r\ll plrrrcts obcy KePlcr's lau's.@rWc can also giv<: atr cxrrttrple of an irlductivcly strong argumcnt with a

gencral lrrenti.se arrd a particular c<lllclusiotl;

All crner¡ld.s previously found llavc becn greerr.

People ofter-r u.se argumcDts of this type whcn they marsltal gerlerlliza-

tions about past c.rpcrience in ordcr to nlake a prcdictiorl about a par-

ticular irnpcndirrg cvcnt.l'hus thc diflcrence betwcen inductively strong and dcductively valid

argumcnts is not to be fou¡ld in thc gcnerality or ¡;artictrlarity of pretnises

aná c<¡nclusiorr but ¡athcr in thc defir¡Ítit¡ns of dcductivc validity aDd

inductivc strerrgth.

I.6. Ijl'IS'fEllIC PltOtsAI]lLITY. Wc lravc scen that thc conccpt

ol i¡rcltrctivc plobability applics to algr¡rncrrts. 'l'hc irrductivc prol;ability

of an argunrent is tlrc probebility that its conclltsio¡l is trtrc givcn that

its ¡rrcntiscs ¡rc truc. 'l'hus thc irrductivc probability of atr argunlent is a

,,,""rur" ol tltc stretrgtll of t|e cvidcr¡cc tlrat tl¡c llrt:ntist:s providc for the

cOnch¡sion, It is corrcct to speak of thc Ínilrrctivc Probability of an argu'

ment, bl¡t incorrcct to spcak of the inductive probability of statemcnts.

SillCe tl¡e prerniSes and c<¡lrclusiOn of any argultlerlt are statcrnentS, it is

incorrect tr¡ snl.ik <.¡[ tlrc iDdr-rctivC probabilrtl,of lr ¡rrcnrisc or of a

co¡ i cl trsi t¡l ¡.

There is, liorvcver, sonte sensc of probability .in rvhich it is intuitivclyacccptablc to s¡rt,ak of thc Probal;ility of rt ¡lrctttisc or concltrsion. Whc¡l

we sai<I that it is irnprobablc that there is a 2000-ycar-old ¡¡an in clcve-

land, u,e werc relying on sonlc such intuitivc sensc of probability'. 'l'lrere

¡llust the¡i bc a type of probability, other than inductivc ¡rrobabilit¡" tliat

applics to statctlrcnts rather than argulncnts.-Let us call this type of probability "epistemíc probability" bccause the

Greek stenr "e¡tistetne" means knowleclge, and the epistenric probability

of a state¡nent ciepencis on just what our stock of rclcvant knowledgc is,

Thus th¿ epistenúc probability ol a stotement car DaÍU lrom person to

person and lro¡n time to timn, sincc differcnt peoplc liavc diflercnt stocks

Lf knowledge at the sarnc time and thc sa¡nc person has diflcrcnt stocks

111,)

I6 I. pno¡.¡run-¡rv AND INDUCTIoN

I All thc supposed facts used in this illust¡aüon arc fictitious except for the fact

that Allentown, Pa', is in the United States.

of knorvlcdgc at diflcrcnt tirncs. For mc, th(j ePistenric probability that

üere is a 2000-year-old man now living in cleveland is quite low, since I

have certain batkgrouncl knowledge about the current normal iife span of

human being.s. I ]eel safe in using this statement as an exantple o[ a

statement whose epistemic probability is low because I feel safe in assutlr-

ing that your stock o[ background knowlcdgc is sirnilar in the retcvant

reipects ancl t¡us that |or Vou its epistenric probabil;ty is ¿¡lso low.

It is casy to imagine a situation in which the background knowlcdge

of two people would diflcr i¡r such a way as to generate a diffcrence in

the epistemic probability of a givcn statcment. For example, the epistemic

prob"billty that Pegasus will show in the third race may be different for

u f"r, in tú. g.andsiand than for Pegasus'jockey, owing to the difference

in thei¡ knowledge of the relevant factors involved'

It is also easy to see how the epistemic probability of a given, state-

nrent can change over time for a particular person' The fund of knowl-

edge that each-of us posscsscs is constantly in a state o[ flux. we are all

coistantly learning new thi.gs directly through experience and indirectly

through info'natián which is com¡nunicated to us. We are also, unfortu-

natelyl continually forgctting things that we once knew. This holds true

for sácieties an<I cultures as well as for iDdividuals, and hu¡nan knowl-

edge is continually in a dynamic process of simultaneous growth and

decay.It is important to §ec how upon the addition of new knowledge to a

previou§ boely of knowletlge the cpistomic probubility of a givcn stato'

ment could either increase or decrcase. Suppose we are interested in the

epistemic probability of the statement that Mr' X is an Armenian and

the only rilerant inior¡¡ation wc liave is that Mr. X is an Oriental rug

dealer in Allentown, Pa., that 90 per cent of the Oriental rug dealers in

the united states are Armenian, urr.I thnt Allentown, Pa., is in the united

states.' on the basis of this stock of relevanf knowledge, the epistemic

probability of the statement is equal to the inductive probability of the

following argttment:

Mr. X is an Oriental rug dealer in Allentown, Pa'

Allcntown, Pa., is in the United Statcs'

Ninety per cent of the oriental rug dealers in the united states are

Armenian.

L 6 sr,¡sreNrrc FnoBArlrLITY

The inductive probability of this argurnent is quitc high. If we arc rrow

given the new i¡rformation that although 90 pcr cent of the Oriental rugdealers i¡r the United States are Arnrenian, only 2 per cent of thc Oricntalrug dealcrs in Allentown, Pa., are Armcnian, rvhilc 98 per ccnt are Syrian,thc epistcnric probability that Mr. X is Armcnian rlecrcastrs clrastically, forit is now cqual to the inductivc probability of the following argument:

lvlr. X is an Oriental rug dcalcr in ¡\llcntowlr, i)a.

Alle¡itown, Pa.. is in thc United States.

Nincty per cent of the Oriental ntg dealr:rs in the Unitcd Statcs

arr: Armenian-Ninety-eight per cent of the Oriental rug dcalcrs in Allentown, Pa.,

are Syrian.Two per cent of the Orícr¡tal rüg dealers in Allentorvn, Pa., arcArmenian.

Mr. X is an Armenian,'

The inductive probability of üis argumcnt is rluitc low. Note that the

decrease in the epistemic probability of the statement ''Mr. X is a¡r

Armenian" results not from a change in the inductive probability of a

given argument but from the fact that, u¡ron the addition of ¡¡ew in(or'mation, a dif erent argument with more prernises becomes relevant inassessing its epistemic probability.

Suppose now wc are given still more information, to the effect that

Mr. X is 0 rncmber of the Arnlenian Club of Allcntown and that 0e par

cent of the me¡nbers of the Armenian Club are actually Armenians. Upon

addition of this information the epistemic probability that Mr. X is an

Arme¡rian again becomes quite high, for it is now equal to thc inductivcprobability of the following argument:

Mr. X is an Oriental rug dealer in Allentown, Pa.

Allentown, Pa., is in the United States.

Ninety per cent of the Oriental rug dealers in the United States arc

Armenian.Ninety-eight per cent of the Oriental rug dealers in Allentown, Pa.,

are Syrian.Trvo per cent of the Oriental rug dealers in Allentown, Pa., are

Armenian.lvf r. X is a member of the A¡menian CIub of Allentown, Pa.

Ninety-nine per cent of the members of the Armenian Cltrb are

Ar¡nenian.

L7

t¿J I. ¡,lroB,\¡]u-rrY ^ND

TNDUCTToN

Noticc orrcc r¡rorc that thc cpistenr.ic probability of thc statcrncnt changcslrt'clrrsr:, with the addition o[ ncw knowledge, it becamc equal to thci¡rrlrrc'tivt: prribability of a new argumcni with additional prerniscs.

li¡ristcrnic probabilities are im¡tortant to trs. They arc the probabilitiestu¡ron wltich we base our decisions. l'ro¡n a stock of krrowlcdge wc willarrivc at the associated epistemic probability of a staternent by the qp¡rli-cation of inductive logic. Exactly how inductive logic gets us epistenricprobabilities frorn a stock of knowlcdge depcnds on how wc characterizca stock of knowlcdgc. Just what knowlcdge is; how rve gct it; ,'vhat it is likconcc we have it; these are difficult epistenrological questions to rvhichrvc l¡avc no dcfinitive answcrs. At this stage we ¡nust r.r,ork rvithin sirn-plificd modcls of knowing ¡athcr than attcmpting a full analysis of man'scpistcnrologic¿l situatiorr.

The Ccrtainty Modcl: Suppose that our knowledge origirratcs inobservation; that observation lnakcs particular scntences (obscrvationreports) certain and that the probability of other scntences is att¡ibutal;leto thc ccrtainty of these. In such a situatior'¡ wc car) idcntify our stockof knowledge with a lisú ol sentences, those observation reports tl¡at havcbecn reudered certain by observational expcricncc. It is thcn natural toevaluate the probability of a statement by looking at urr argur»cnt withall our stock of knowlcdge as prcrniscs and thc statcrncnt in question lsthe conclusion. The inductive strength of th¡t argumer)t will detcrnrinethe probability of thc statc¡nent in <¡uestion, In thc eertainty rnoclcl, the

relation belween episternic probability ancl inductivc probábility is quitesimplc:

.

Definition 5: In the certainty model tbe epistenic probability o[a statement is the inductioe probability of that argument rvhichhas the statement in r¡uestion as its conclusion and whosc premisesconsist of all of the observation reports which comprise our stock

of knowledgc.

The ccrtainty model livcs up to its narne in assigrring c¡>istcrnic proba-bility of one to each observation report in our stc¡ck of knowlcdge. If an

observation report is part of our .stock of knowledgc, the argument rele-vant to its epistemic probability contains it both as conclusion. andpremise and is thus trivially deductivcly valid.

The simplicity of the certainty modcl is attractive, and somc philoso-phers have argucd that it really mirrors thc acttral huma¡r situation. The

I. 6 El,rsr'¡:rvttc r,¡loB^llrLr-fy

weight of cur¡crrt ,1>i,iorr, lrowcver, is that certai¡rf is not so easy tocome by and that whjlc we likc to pretcnd that we have it, we rarely(if ever) do. Such a view leads to a different kind of mo«lel.

Fallibility Models: Sup¡rosc that our knowledge originates in obser-vation and that obscrvation nrakcs the appropriate observation reportIikely, but not certain. For instance, I sce a bird that Iooks Iike a blackswan at the zoo. I am fairly confident in my ability to identify blackswans, but not so co¡lfident as to tlrink I could not make a mistake,My act of observation might thus confer Probabílity .gg on the obser-vation report: "The bi¡d in cage 3I of the zoo is a black swan," leavinga I per cent chance that I goofcd.

In a fallibility model, ou¡ stock of knowledge can no longer be repre-scntcd as sintply a list o[ observation reports. Now wc nced observationreports paired with the probabilitics that obsc¡vation has conferred uponthcm:

, pr (0,) : .99Pr (0,) = .97Pr10,) =.95

And calculating the cpistemic probability of a sentence, S, on such astock of knowledge will not be such a simple proposition. Certainly theinductive probability of

0,

0,0,

s

is relevant. But we stand a 13 per cent chance of hav.ing been mjstakenabout 0,1 We shall have to take into account the inductive probability of:

0,noi-0,

0,

S

and similarly for all other mistakes and conrbinations of mistakes wemight have made. Then we will havc to put together the inductiveprobabilities of thcse arguments in exactly the right way to get theepistemic probability of S.

19

20 L p¡ro¡anu_¡ry AND TNDUCTT()N

Fallibirity rnocrers arc too conrpricatccr for us to cliscus.s at trris ¡roint.Now that you know tlrcy c.xist, wc c¿¡n forget al>out thcm until yotrknow a littlc more ubo.,i p.obai,iliti"s. uniil't1.,";, I;; ,,rllrr"rr",fr,,,uabout cpistcmic probabilitic,s as i¡r tl¡c cc.tai,rti ,,,oclcl.

Exerciscs:

l. Construct several new cxamples in which the epistemic probability of astatement is increascd o¡ decrcase. by the acrdition of nerv information to aprevious stock of knowlcdge.

For the advanced stuclent;

2' An incorrigir¡re statcment is one which c¡n ,.r bc correctec.r. Does thecertainty modcl rerrder lr)y st¡tc¡¡te¡¡ts irrcorrigrblei

Suggested rcadings

wilriam K¡reare, probabiliry antl. r¡tducrk¡^ (Londorr: oxford Univer.sityPress, lg52), '.-l.hc Ilcl;itir.¡ri of l,rol_rabilitf to Evidc¡¡cc,,, pp. g_,l3.For ü¡e advanced stuclent:RicharC Jeffrey, TIrc

?trr_iI D(ci$it)n (2nd cd.) (Chicago; University ofChicago press, l9g3), chaj:. I l.

I,7, PITOBABILITY AND THE PROI],LEI\,ÍS OT, INDUCTIVEl9ClC: Deductivc logtc, ut least in its basic brauches, is *¿ J*J.p.¿.Tl¡e dcfinitions of its trorm u ra t ed,

", d,h ;;I:, n:[t,I!"X'ii"' ",,*' #T"ii ffi-":::Ji]Such.is not thc casc, howevcr,,,vith inductivc l<.rgic. Thcrc are no uni-versally accepted rules for con:tructing inciuctivelf r,rong rigu,nJil;;;generai agreement on a way of mcarurfng the inductive st¡ensth of rrs,,"nte¡lts; no prccrsc, unco¡ltrovcrsial dcfir¡itit¡¡l of il¡cluctivc p."frr"ü,i,ü iñr,i,ductive logic cannot be rea¡ned in the se¡xe i¡r which one rcarns argebraor the basic branches of deductive logic. This is not to say that inductive

1",.t:::.T,::".*atloline, in a sea of totil ig,ror"*"; many things are knowna.Do,ur lnductrve.logic, but_many problems still ¡emain to be solvcd. weshall try to get an idea of just what tL" p;;;l;_, are, as well as whatprogrcss has been ¡nadc toward thr:ir solutiori.

.. !or" of t'e main problcms of inr.'¡ctive l;;;; .", be franrecl in terms of(¡¡c concept ot inductive probability. I said that there is no prccisc,unco¡rtrovc¡sial definítion of inductive probabirity. I dici give , ¿"6nitio,.,of inductivc probability. Was it conrro*^rli i ,itrk not, bur, if you will

L 7 p¡ro¡¿nr¡.rl.\ ^ND.l.rIE

rnoBLEMs oI,. INDUCT.IvE Locr(j ZI

rcmenrber, it was ir,prccise. I said trrat tlle i,ductivc probability of a,argurncnt is the probabirity that its cor¡clusio¡l is t¡uc, given irrat itsprcrniscs are t¡uc, But at that point I cou.rcr ¡rot givc an cxact ircfiniti,nof "thc prrbability that a, argr.,rcnt,s conclu.sion is truc, givc, tl,at itspremiscs arc true." I was, instcad, rcducecl to givi.g e*rÁ¡rI", so tllatyou could get an intuitive.fecling for $e meaning óf ti.,i, ptrorc. Thelogicia,, howcver, is rict satisfiecl ,iit¡ n,i irtritir" fccring for the mcani.gof key words and phrases- He.wis-hes to anaryze thc concrpts inuor""J-ur¿arrivc at prccise, Li¡ambiguorls de,nitions, Thu, on. of ihc probrems ofinductive logic which remains outstanding is, rvhat, cxactly, i.' fr¿r",,".probability?This problem is intimately co¡rncctccr with two othcr probrems: How istlic inductivc probability of an :rrgrr'c¡rt ¡nt:asured? An., what arc thc

rules. for-constructing inductiveJy it.ong ,rgr,r,ents? Obviously we can_

'ot devclop an exact measurc of in<Iuctive p-robabirity if,ve do not knowprcciscly u,llat it is, And before wc car¡,lcrir" ¡ulcs for constructi,rg in".ductivcly strong arguments, wc ¡nust have rvays of tclling wlrich ¿lrgu-¡¡rc¡lts ¡¡¡casurc up to thc rcquircci clegrcc of iniuctive strcngth. 'rhus thcsolution to the problem of ¡:rovi<iing a pr"cisc dcfinition of inductivepr.bability dcter¡nines rvhat solutions arJ availabrc for thc probrcms ofdeternrining the inductive probabilitics of argur,cnts ancr constructirgsystc¡natic rules for generating inductively strong arg.uments.

- Let us call a precise defi¡ition of induciive prÁrUitity, together withtho ass,ciutcd method o.f dcterminirg thCI inrructivc

'pr,ñorrill.y ,]arguments and ruies for constructing inductively strong ;rgr;.r,E ;intluctiuc logic. Thus dillerent de'¡¡itiors of iniuctive p-u"rririiy ii""rise to ciffcrent inductive rogics. Now we are not intcrested in Ériiugjust any systen: of inductive logic. lve want a system that accr¡rds wellrvitli c<¡rnmon sense and scientific pr.actice. We lvant a system that givcs

the ¡cstrlt that most of the cascs tllat ,rc wourcl i¡ltuitively classify as i,-d-uctively strong arguments d_c inclced have a high i.,d,r"iire proúability.We want a system that accords with scicntiGc p.r'. ti"" ur,,l

"o.,.,nro, ..,nr",

but that is mor¿ prec,se, more clearly forrnu]aLd, rnd more rigorous thanthey are; a system that codifies, explains, ond ,.1 nes our intiitive ¡udg_¡nents' we shall call such a systcrn of in<ructiv r logic a scientiflc iiductiuc

.logic. Tlte prob.lem that we havc bccn rliscussing ca, n,iru l_r.,rcformulated as tlrc problem of corctructing a scrcntifc iid.ttctioe lo¿4ic.

The second major problem of inductive logic, an<1 the onc that has bcc¡rinore rvidely discussed in the lristory of philosoplry , is tlte proltLctn ol

i

i

I

,,) I. pr<t¡,r ¡tl-tt'y r\ND l¡*Dttc'f IoN

rutiotutLLy jrtsttlyitrg tltc ust: ol Lt stlslutr of t'cit't;tílic i¡ilttcti'sc /ogic trLtlrtr

tlra¡r sornt, otlrcr systcrlr r¡l rndrrctivt logic. Aftcr lll, tLcrc ¿lrc nlllr))' (lif-

fcrc¡t possiblc irrdrrctivc krgics. Sorrrr: rrriglrt ¡;ivc tirc rcstrlt tltrit ittgrrrrtcttts

that rvc thillk arc inductivcly stroDg arc, in lact, incluctivcly §'cak, rtttd

argurrlcnts that wc thi¡lk indlrctivcly u,ci1k itrc, itt [lct, irrdtrctivciy stro¡)g.

That is, tl¡crc aLc possible i¡rductivc logics rvhrclr rrc cliarrletr'ÍciIlly

opposed to scicntilic illductivc logíc, tvlriclt arc iD total disirgr'(iclncrlt

with scicntific 1>rircticc altd cornr¡ton st,nsc. Why sllolrltl u'c nOt ('l)lPloy

onr: of thcsc systcl¡ls r¿rthcr than scientific indtrc'tior¡?

Any a<lctlrratc A¡)s\vct to this rlrrcstiorr rutrsI t¿ikc itttt.¡ :tccottlrt tllt: tistrs tt¡

wliicll rvc ¡rut irrdtrctive logic (or, at prcsctlt, thc Vlgrrc ilrtuitio¡ts \\'c r.rse

in ¡tlacc of a prccisc systcrtr of inductivc logic), Onc t¡f tlrc nl«tst irnl>ot'titrtt

uscs of illductivc logic i¡^ to frarnc our e'xpcct¿Itiorts c¡f thc futurc r¡rl thc Lritsis

of our knowledge of the past and prcserlt. \Vc ¡lltrst usc our knorvledge of

thc past and preseDt as a guide to our cxl.tcctations of the future; it is tl¡e

or)ly guidc wc lravc. Ilut it is impossiblc t0 h¿rvcr a tlctluctiucly udlid argrr-

¡rrcl)t wllosc prcutiscs cO¡ttttit.t ¡,6¡,¡al irtlorlrlatit-rll solcly abotrt tltc ¡last

and prcscnt and wl)os0 coDclusion ¡-rakcs factrral clainls airout thc frrturc.

For thc conclusion of ¿ dcductively vali<1 urgu¡ncnt tn¿rkt:s t¡o factrr:rl

cl¡irn tllat is not llrcarly rnadc by thc plcrDiscs.'l'lrus tllc gall sc¡lalaLirrg

the pirst and prcserlt frorn thc futurc cilnrtot bc bridgcd in tltis rvay b1'

clcclrrctivcly valicl arguntt'ttts, al)(l if tllrr arglrnlt'llls rvt'trst'to briclgtr tllat

gilp ar0 to lr¡vc ilny stI(,ltgth \yllttts0cvcl tlrt'y lrrust irtr irrtlrrctivt'l¡'strollg,^

Lct us look ¿ littlc rtrorc closcl¡', tht'n, at tltc s'e¡'irr s'hich i¡rclLrctivc

logic tvould bc uscd to frarnc our cxpcctatiolts of tlrc futttrt:. Sttllllosc otrr

¡>lans dcpcnd critically on wl)ctlrcr it'"vill r.riIl torrlorro\\'. Tl¡crl thc lc:lso¡I-

ablc thing to clo, before wc (lccido r'¡lrat couLsc of action to takc, is to

asccrtain thc cpistenric probab.ility of tltc st¿tcrrrr:nt "It rvill r¡itl tc¡rt¡or-

row." This we do by puttillg all thc rclcvant irrfonriatiorl §'c ¡lorv lrrtv<'

into thc prcmiscs of an argumcnt wltosc co¡lclusior¡ is "It u'ill rain torrror-

,o*" ,,r,J asccftailling thc inductive probability of that argunrcnt' lf tllc

probability is high, we will have a strong .expectation of. rain arld rvill

8 TI¡c infc¡cnce from tllc p¡st ¿¡¡¡d l)rcscI¡t to tlte futurtr is ¡¡ot, of cottrsc, tltc ortll'

type of inference thi¡t cannot bc accc,mplislrcd by a deductivcly valid argulrrcrrt'

Wlrencver the conclLrsio¡r nlust ¡Drrkc t factttll (luir¡r that is trot nlade by tlrc

¡rrerrtis(s, ir tlcrlrrttivr'ly vrlitl argrrrrrL:ttt is ol¡t r¡[ tilc (llr('stlo¡l l"or irtstrtlcc, if rvc

rvisl¡cd to co¡rciudc sontctlring trl¡orrt tltc (lt5tJ¡¡t ¡).tst frortt ¡rrctttiscs tltut etrilrotil'

current geological dtta, ¡ deductively valid trgrrntt'nt cotrltl I¡ot do tlic iolr'

I'IIOI]AI]II,I'TY AND TIIIi PNOBLI'MS OI" INDUCTIvE LOCIC

rnake our plans on that basis. lf üc probability is near zero, wc will be

reasonably surc that it will not rain and act accordingly.'gNow although it is doubt{ul tltat anyonc carries out thc fo¡mal proccss

outli¡red aL¡ove rvhcn lre plans for thc [uturc, it is hard to dcny that, if wcwcre to r'¡.¡ake our rcasoning cxplicit, it would fall into this pattcrn. Thusthe nraking of rational dccisions is dependent, via the concept of epis-ternic probability, on our inductive logic. Thc sccond niain problcnr ofinductive logic, thcn, Icads us to thc follorving <luestion: How can wcrationally justify the usc of scientific inductivc logic, rathcr than some

other inductive logic, as an instrunlent for shaping our expectations oftlrc futurc?

TIre tu,o nrain problems of inductive logic are:

l. Thc const¡uction of a system of scientific inductive logic.

2. Thc rational justification of the use of that system rather than

so¡rrc other systcm o[ inductive logic.

It wouid sccm that the first problem must be solved before the sccond,

sincc §,c crn hardly justify the usc of a systcm of inductivc logic before

we knorv what it is. Nevertheless I shall discuss the second problem first.

It rrrakcs scnsc to clo this because we can sce why thc second problcm is

suclr a problcm without having to know all thc details of scientilic induc-

tive logic. Furthermore, philosophers historically came to appreciate tire

tliflicrrlty of thc second problem rnuch earlier than thcy ¡calizcd the full[r¡r't:t, ol tlrc fil'st pr'oblcut, 'l'lris sccortd prul-rlenl, thc truditiorlal problcrn

o[ inclrrctio¡r, is discussed in the next chaptcr.

Suggcsted rcadings

I(rrclolf Carrrap, Logical I¡ou¡dations ol Probabílíty (2rr.i ctl J, (Chicago:

Urrivcr.sity of Chictg<.r l']ress, 1962), pp. 1-15). (Advt'rt'c'.I students may

also rvish to see pi). 252-64.)

Irrvirt D. Bross, Design for Dccision (Nerv York: The Macmillan

OoInpany, 1953), chaPs. I and 2.

lloward Raifia, D¿cision Arulysís (Reading, M¿us: Addison-Wesley, 1968)'

e'I'lre account of dccision,making under unccrtainty skctchcd above is, ol neccssity,

drutically oveninrplified. Wc will see so¡ne morc detail in V.5' V' 6, and Chaptcr lV'

23

IIThe Traditional Problemof Induction

II.l. INTRODUCTION' In Chupter I we s¿w that inductivc logicis used to shapc our cxpcctations o[ that which is as yct unklrorvn otr tlic

basis of those f¿cts that arc alrcady knorvn; for itrstancc, to sltapc our

expectations of the future on the basis of our kr.¡owlcdge of tiie past and

prcsc¡rt, Our problcm is thc rational justification of thc usc of a systcm

of scic¡lti6c i¡rductivc logic, rather tl¡¿¡n sonte other systcrr of i¡rductive

logic, for this task.Thc Scottisl¡ ¡rhilosopher David Hume first raised this problem, rvhicli

we shall caü thc traditional problznt' of intluctian, in full force.' Hurtre

gave thc problem a cutting cdge, for he advanccd arguments dcsigned to

show that ¡ro such rational iustificabion of i¡rductive logic is possible, no

matter what the details of a systern of scientific inductive logic turn out

to bc. The history of philosophical discussion of i¡rductive Iogic sincc

Humc has bee¡r in largc measure occupied with attel:lpts to circutnvent

the difficultics he raiscd. This chapter examines thcse difficulties a¡rd thc

various attempts to overcome tllem.

1I.2. HUME'S ARGUI\fENT, Defo¡e wc can meaningfully discuss

argumcnts which purport to show that it is inrpossible to rationally justifyscientífic induction, we must be cloar on what would be recluired to

rationally justify a system of inductive logic. Presumably we could

rationally justify such a system if we could show that it is well suited forthe uses to which it is put. One of the most irlportant uses of i¡rductive

logic is in sctting up our predictions of the futurc.? Inductivc logic figurcs

in these predictions by way of epistemic probabilities. If a claim about

the future has high epistemic probability, we predict that it will prove

r I havc taken sonre libcrties with l{unrc and ltavc given tlre traditional ¡rroblcnrof induction a ncw twist for reasons that will become apparent, For Hume's own

statemcnt of the problem, see David Hume, Art lnquiry Concerning lluman

fJnderstauling, section lV, reprinted in A Motlen Inttoduction to Philosoplry

(rev. ed.), Paul Iidwartls and Arthur Pap, eds. (Clencoe, IIl.; Thc Free Prcss,

1965), pp. 12$-32.2 Iis ot)rcr uscs do not diffcr in ways essential to thc argrrmcnt.

24

lI. 2 riulrr's .'^t(cult{I'NT

true. Ánd, nore gencrally, we cxpect sonrcthitlg ¡llorc or lcss strongly as

its epistcrnic probability is higher oi lower. The epistemic probability of

a statcmcnt is just the in{qctivc probability of thc argumc¡rt which em-

boclies all uurilobl" infoinration itr its prenrises.' Tltus the cpistcn'ric

probability of a staternent dcpcnds on two things: (i) tirc stock of knowl-

edge and (ii) the inductive logic used to grade thc strcngth of tl.re argu-

ment from that stock oI knorvledgc to thc cortclusion.

Now obviously what we want is for ou¡ prcdictions to be co¡rect. lfwe could get by rvith dcductively valid arguments wc could be assured of

tnrc prcdictions all the timc. DcducLivcly valid arSurncnts lcad frcrn

trur: prcnriscs ahvays to tn¡e conchrsions and thr: statcmcnts comprising

or, ,to"k of knowledge are known to be true. But deductively valid

arg-u¡ncnts arc too conscrvaLivc to lcap front thc past a¡d prcscnt to tlicfuiurc. For this sort of daring bchavior wc will have to rcly on inductivcly

sfrong arguments-and we wilj havc to give u¡r the conlfortable assurallcc

that we will be right all thc tinre.FIr.¡w about nrost of üe timc? Let us call t-he sort of argurrtcnt uscd to

set up arr cpistcmic probability an c-argumcnt. That is, an e-urgurtrcrtt

is an arglrment which has, as its prcmiscs, some .stock o[ knowledge' Wc

might hope, tlrcn, that ir-rductivcly stro¡lg c-argumcnts rvill give us frue

coriclusio¡ts ntost of thc time.llemember that there are degrce s of induc-

tivc strcngtli and that, on thc basis o[ our prcscnt knowledgc, rvc do not

always simply predict or not-predict that an cvcnt will occur, but antici-

pata it witlt various degreec of confrlerwo' Wc migh¡ hope furthcr that

in<iuctivcly stfonger c-aiguments havc true co¡iclusion.s morc oltut tb',.rt

incluctively ueaker oncs. Finally, since we think that it is uscful to gather

evidcnce io enlarge our stock of knorvledge, we rnigirt hopc tlrat indr"rc-

tivcly strong e-argu¡ncnts givc us t¡uc conclu.sio¡ls nlorc often wlicn tlrc

stock of knowledgo embodicd in the premises is grcat than rvhcn it is

small.The last consideratio¡r rcally has to do with justifying epistcnric proba-

bilities as tools for prediction. Thc epistenric probability is thc inductive

probabilty of an argument embodying all r-)ur stock of knowlcdge in its

irc,rriscs. The requiicnjent that it crnbody all our knorvlcdgc, ancl ¡rot

iust sonre part of it, is knorvn as thc Total Eviclcncc C,,dition.'II rve

I I¡r tltc certi¡intY Inodcl.{ Somctimcs thc Total Eviclencc Condition is statctl as thc rcqtlircntcnt that a¡)

e-rrgumcnt cmbody only ¡ll <tttt rclctant knowlcrlgc This co¡nes to tlrc samc

thin;, however, since by definition, thc remainder of our stock of knowledge is

ir."llua.t just in casc its addition or delction fro¡r the prctniscs makcs no djf-

fe¡encc to the probabilitY.

25

9(, ll'rrlu'rnADllIONAL ltolll-liN{ o¡ Ih-DUC't'IoN

t ould slrorv that basing oul prcdictiorls orr r)to¡ c krrorvk'clgc givc.s rrs

ircttcr success ratios, wc would havc justificd tlrc totll cvitlr:¡lcr: condition.'fhc other considerations hiivc to do w,ith justilying tlrc otlrcr rlcter-

¡rlitrartt of cltistcinic probrrbility-tl¡r'incluctivt'logic rvlriclr irssigrrs irrrlrrc-tivc probabilitics to argunrcnts.

We are nr>w reacly to suggt:st rvhat js reqtrirccl to ratiorrrlly jrrstify ls¡'stcrrr oI irrdrrctivc logit':

Rational JustificationSuggastion 1: A systcrn of inductive logic is ratiorurlly justificd ifand only if it is slrorvn that tlrc arguutcnts to rvl¡ich it ussigtrs highindr¡ctivc probability yicld truc conclusions from truc prcntiscsI¡rost of tlic tirnc, and the c-iirgunrcnts to rvllich it assigns Iriglrcrinductivc probability yicld truc conclusic¡ns frorn trrre prcrniscs¡norc often tl¡an tlrc argument.s to which it assigns lowcr incluctiveprobability.

It is this sense of rational just.ificatiorr, or sornctlring cluite closc to rt, tl¡atI{u¡nc has i¡r ¡¡ind wllc¡i hc advanccs his argtrrlc»ts to plovc thlt a¡ational justification of scic¡itific iuductiorr is irnpt.r.ssiblc.

If scientific induction is to bc rationally justificd in thc sc¡rsc of Srrg-gcstion I, wc must cstablish that tllc argr¡rncr)rs to rvlrich it assigns highinductivc probability yicld truc conclrrsions fro¡n trtre prt:rrriscs rliost ofthe time. 13y what sort o[ ¡casoning, a.sks IIrrnre, could wc establisir suclra conclusi0nP lf thc argulncnt tlrrit we lnust use is tó lravc urry forc,cwllatsoev(:r, it rnust l-¡c citllcr dcductive ly valid or induc ti',,cly stror¡g.IIunre proceccls to sl¡orv tlr.rt ¡rt'itlrcr sort ,,i argr.rnlelrt coül(l do tlrc job.

Suppose rve try to rationally justify scientific inductivc logic by nreansof a deductively valid argumcnt. Tlre only prcmises we arc entitlcd touse in tl)is argurrtent are those that statc things wc know. Since wc dt¡not know what thc futtrre will bc likc (if we did, rvc would lritve no¡reed o[ atr inductive logic on whiclr to basc <¡ur predictions), the prcnriscscan contain knorvledge of only thc past and prescnt. But if the argurlrentis deductively valid, thcn the conclusion can ntake no factual claims thatare not already rnadc by thc premises. Thus tl¡c conclusior] o[ tlrc urgr:-mcnt can only refer to tlrc past and present, not to the frrturcr, fr¡¡ thcprcnri.ses macle no flrctt¡al clai¡¡rs abotrt tlrc ftrturc. Sucli a conclusio¡rcannot, lrowevcr, bc adt:qurte to rationally jrrstify scientific induction.

To rationaliy iustify scicntiEc induction wc must shorv that e-argumentsto which it assigns high inductive probability yield true conclusior:s frorntmc prenrise.; rnost of the timc. And "r¡rost <¡f the tirnc" does nr¡t ¡r¡ean

IL 2 uuv¡'s AuLUMENT

n¡ost r-¡f thc ti¡ne in orrly the pa.st and prcscrrt; it rncarrs most of tlrc tinrc,past, prcscnt, an(l futurc,lt i.s co¡rccivablc tllat a ccrtain typc of argun)cntnright lravc given us true conclusic.rrrs frorn truc prcrnises in tlrc past andnright ceasc to do so i¡l the futrrre. Sincc our conclusir¡lt c¡rnr¡ot tcll us horvsucccssful argurncnts will be in tlre futurc, it can¡rot establish tlrat thce-arguments to which scientific induction assigns higlr probability willgive us true conclusions from truc premises ntost of tlrc thnc. 'l'hus wecannot use a deductively valid argunrent to rationally justify i¡¡ductio¡r.

Suppose we tr),to rationally justify scicntiffc induction by nrcans of arr

inductivcly strong argurncnt. We construct our argr¡nrc¡rt, whatcvcr itrttay bc, and presertt it as- an inductively stror)g argun)ent. "Why do youthink that this i.s an inductively strong argurrrc¡rt?" IIur¡rc nright ask."Bccausc it h¿s a high inductivc probability," wc w<.¡ultl rcply. "And whatsystem of inductivc logic assigns it a Irigh probability?" "Scic¡rtific induc-tion, o[ course." What Hume lras pointcd out is that if rvc attcrnpt torationaliy justify scicntiffc induction by rrsc of a¡r irrductivt:ly strongargument, we are in thc position of having to assuntc that scicntifici¡rductio¡l is rcliablc in ordc¡ to provc tlrat scicrrtiflc incluction is rcliublclwc rre rcdt¡ccd to begging thc qr.rcstion. Thrrs wc car)r¡ot usc an induc-tively strong argurnent to rationally jtrsti[y scienti{ic induction.

A co¡nmon argument is that scicntific induction is justified bccausc ithas been cluitc successful in the past, On rcflcction, howevcr, we scc

that tlris argurncnt is really au attcnU)t to justify.inductio¡l by rnearrs of0lr induetivcly strong ¿rrgumcnt, alld thus lrcgs the qucstion, N{orc cr"plicitly, the argurnent reads sonrething likc this:

Argunrents that arc judged by scicntific i¡¡ductivc logic to havchigh inductive probability havc givcn us truc conclusions fro¡¡rtruc prernises rnost of the tinre in thc past.

Such argunrents will givc us truc co¡rclusir»s frotll truc prctttiscsmost of thc time, past, present, and future.

It should bc obvious that this argument is not dcductively valid. At best

it is assigrred high inductive probability by a systertr t.¡[ scicrrtific i¡rductivelogic. tsut the point at issue is whetller we should put our faith i¡r sr.rch

a systern.We can view thc trtditional proble¡n of inductior) Irorn ¿r diflcrent

pe¡spective by discussing it in tcrms of the princi¡ie ot' the unilornityof nature. Alüough we do not have the details of a system of scientificinductir.¡n in ha¡ld, we do know that it must accord rvell with cont¡non

28 II. rrtg rRÁDITIoNAL PnoBLEM oF INDUCT¡oN

sense and scienti6c practice, and wc are reasonably fami)iar with both.

A few examples will illustratc a general principle which appears to unde¡-

lie both scientific a¡tcl co¡nmon-sensc judgments of inductive strerrgth.

lf you wcrc' to ort.lcr lilct nrignon iIr a restaura¡rt, and a fricnd wore

to objcct tltat filet mignon would corrode your vitals and lead to quick

¡¡¡d violcnt cleath, it would see¡n quite sufficient to resporld that you had

often eaten filet rnignon without any of the dire consequclrces he pre-

dicted. Tl¡at is, yorr would intuitively judge the following argrrment to

be ürductively strong:

I have e¿ten filet mignon nrany tirnes and it has never corroded

my vitals.

suppose a scientist is asked whet)rer a rockct would work in rcachcs

,f ,pa..e bcyoncl the range of our telescopcs. [le replics that it wotrld,

and to b¡rcli r,p his aDsrvcr appcals to certain principles of theoretical

physics. Wl¡cn askcd rvh¡t evidcnce he has for these principlcs, hc c¡n

..ef"r to a grcat ¡¡r:rss <¡[ obscrvt,cl phcnontcna that corrol¡i¡ratc the¡n Thc

scientist is then ju<Jging tlre following argumcnt to l¡c inductivcly strong:

Irrir.rci¡.)lcs A, lJ, a¡rcl c corrcctly clcscribc thc l¡chavior of nratcrial

bodics i¡l all of tlrc rnany situatiolis wc havc obscrvcd'

Prilrciplcs A, B, tnd c corrcetly tlcscril¡c tltc llchavior of rnaterial

bodiei ¡n those reac¡es of space that wc have not as yet observed.

There appears to be a comrnon asstlnption undcrlying the judgments

that these arguments are inductively strong. As a steak eater you assume

that the futuie will be like the past, that typcs of food that proved health-

ful in the past will continue to prove so in the future. The scientist

assumcs that the distant reaches of space arc like the nearer ones, that

¡natcrial bo<Iics obcy thc sarne gcrrcral laws in all arcas of spacc. Thus

it scems that undcrlying our jtrdgnrents o[ inductive strcngth i¡r both

common sense and science .is the presupposition that nature is uniform

or, as it is somctimcs put, that likc causes produce like effects throtrghout

alí regíons of spacc ancl tinre. Tht¡s we can say that a system of scicntific

induciion will l¡asc its judgmcnts of inductive strcngth on the presup-

position that tnture is unifonn (and in particular that thc ftrturc will

resemble the past).

we ouglit to realize at this point that we have only a vaguc, intuitive

,¡rrrlcrsta¡r<ji¡rg of thc principlc of tht: uniforrnity of nattrrc, glcancd fronl

II. 2 uuu¡'s ARcuMENT

examples rather than spccified by precise definitions. This rough under-

standing is sufEcient for the purposcs at hand' But we should bear in

rnincl that the task of giving an exact deñnition of the principle, a deff-

nitio¡r of tlrc sort tfiat rvould bc prcsupposed by a system of scicntific

inductivc logic, is as difffcult as the construction of such a system itself.

Or¡c of thc problcrns is tl¡¿rt naturc is sirlply rot uDiform in all rcspects,

tl¡e futtrre docs nr¡t rcscmblc the past in all respects. Bcrtrand Russell

once speculated that the chicken on slaughter-day might rea§on that

whencver the hu¡nans came.it had been fcd, so when the hunlans would

conrc today it would also l¡e fed. The chicken thought that the futtrre

rvould resemble the past, but it was dcad wrong'The future may iesemblc the past, bt¡t ít does not do so in all respects'

A¡rcl rve do not know bcforchand what thosc rcspects arc nor to what

dcgree the future resembles the past. Our ignorance of what these respects

arc is a dcep reason behind thc total cvidence condition. Looking at more

a¡.¡d ¡¡rorc cvi<Icnce lrclPs us rcjcct spurious patterrls which we mightothcrrvisc project into the futrrre. Trying to say cxactly u.r/raú about¡¡aturcwc believe is tr¡rifon¡¡ tlius turns out to be a surprisingly delicate task."

IJut strppr.rsc tLat a subtlc aDd soplristicatcci vcrsion of thc principle of

thc uniformity of nature can be for¡nulated rvhich adequately explains

thc judgnrents of inductive strength rendcrcd by scicntific inductivc Iogic.

Thcn ifnature is indeed uniform in the required se¡rse (past, present, and

future), c-arguments judged strong by scientific induction will indecd

givc us truc conclusions most of thc tinle. Thclcforc the problern of

iation¡lly justifying scientilic induction could be reduced to the problem

of cstablishing that nature is uniform.But by what reasoning could we establish such a conclusio¡r? If an

argument is to have any force whatsoevcr it nrust be either deductivcly

,alid or incluctively strong. A deductively valid argument couid not be

adequate, for if the information in the premises consists solely of our

knowledge of the past and prcsent, then thc conclusion cannot tell us that

¡raturc wiil be uniform i¡r thc future. The conclusion o[ a dcductivcly

valid argument can ¡nake no facbual claims that are not alrcady made by

the premises, and factuai clairns about thc future are not factual clainrs

about thc past and present. But if we claim to havc established tl¡c

principle oi the unifonnity of nature by an argumcnt that is ratcd indrrc-

iively strong by scicntific inductivc logic, wc are open to a challengc as to

rvhy we shóuld place our faith in such argumcnts, But rvc cannot rcPly

oo

5 About which we will havc morc to say in Cltaptcr III

30 I] -'r'uu'l'liADlfr()N^r. prt()rlr-Erlr ()1.- INI)tJC'r'r()N

"Because nature is unifonrr," for that is prcciscly what wc arc tr¡,ing toestablish.

Lct us surnmarize the traditional problcni of induction. It appcars tlrttto rationally justify a sy.stcnr of scicnti6c inductivc logic rvc woulcl h¡veto establish that the e-arguntcnts it judges to bc inductivr:ly strr.¡ng giveus tmc conclusio¡rs nlost of thc ti¡nc. If rve try to provc tlrat tllis is tlrccase by means of a dedrrctively valid argun-rc¡rt whosc prcnriscs statctlrings wc alrcudy kr¡orv, thc¡r thc conclusion r¡llrst flll sllort <.¡f tlrc dc'sirt'rlgoal. But to try to rationally justify scicntific inductio¡¡ by r»cans of unÍrrqrurlent that scicntific induction irrdges to bc inductivcly strong is to bc'g(ltr'ilucrtt,ttt. l'lt.'s.ttrtc,lrtli.'rrltics .rris,'ii rvt'.tttctltltt to jrr:tiIr siir'rltificinductivc logic l-ry cstablislling tltiit r¡rttr.¡rc is utrifor¡u.

Exercise:

What problcnts l¡r'e there in try,irrg to jrrstily tlrc totll evrtlc¡tcc cort<iitiori?

Suggcstcd rcatlings

I).ryrrl Ilr¡¡¡,., .\r¡ lrt,¡rtrtt¡ (:t)ltt lt!ntE lltultnt (1¡¡r1r'1,11¿l¡¿,i¡lrA, ¡,t'. l\.tr'¡rtttrtIrl ¡r¡ .\ .\l,t/r'rr¡ lttltt'lttt ltt,tt to I'lrtlt'xt¡lty (rcr' ctl), t'ti' I''trrl

[rlrr.¡rtlr.rt¡,i.\¡tlr,rt l'.i¡,1()ltrrtrrt, lll,,'ll¡e l]¡cc l'tcrs, l116'r), ¡r¡r'

ll:]-:Il, .,,,,1 ,¡¡ /'rrr/rtrl,ltlt, r, I'tt'l'1, ttt¡ ,ttul l'tttilLlt)\(s, trtl' Sitlrrt'y

Ltrr'Lc¡rL,rcl¡ (llclrrtorrt, (l.rlrf.. l)rckt'rrs,rrr l'Lrblrri¡i¡¡g Co, I972), pp'

14-2t.

For the advanccd student:

l.J.Cood,"OnthcPrinciple<¡f'l'ot¿ll'lvide¡rce,"tsritishlournulfttrtlwPlú'losophy of Science l7 (1967)' 319-321.

' B. Skyrnrs, "Causal Decision Theory," '[hc ]t»trntil.ol Philosoplry 79 (1982)'

[,95-71 L

tI.3 TI{E INDUCTM JUSTIFICATIoN oF INDUCTION.Hu¡ne has prescnted us rvith a dilenrnra. lI rve üy to ¡ustify scicntifit:

incluctive logic by mcar)s of a dcductivcly vahd argumcnt with premiscs

knc¡rv¡l to bc truc, our coilclusion rvill bc too rvcak. l[ rVC try to usc atr

in<luctrvcly stro¡)g argumcnt, we arc reduccd to bcgging tlre qr.restion.

TIrc proponcnt of thc i¡rductive justification of iuduction tacklcs thc

sccond horn of thc clilcntnra. l{e niaintains that rvc carl ,ustify sciclrtific

iDduction lr1, arr inductivt'11, strong arilrirncnt *'ithout beggiDg tller <1rrcs-

tiorr. Altlrorrglr ltis ltttt:rrrpt is rrot lrltogc'tl¡ct stlcccssIr.rl, thcrc is e great

de¡l tc¡ L¡e Icuructl lrurtr it.

IL 3 rtr¡ rNDUC-rrvE Jus'l'tncA'r'roN or¡ rNDUC'r'loN 31

The answer to the question "Why should rve believe that scientificinduction is a reli¡blc guidc for our cxpcctations?" that irnmediatelyoccurs to everyone is "Because it has workcd wcll so far." Hume's objec-tio¡r to this answer was that it begs the qucstiou, tliat it assu¡ncs scicntifici¡rduction is leliablc in order to prove that scientific inclr¡ction is rcliablt,.The proponcnts of the inductivc justificatiorr of inductioll, howcver, clainrthat the answer only a¡tpears to beg the questiol), bccausc of a mistakenconccption of scicntific i¡rductiou. Thcy clliru tlr¡t if wc propcrly distirr-guish leoels of scicntific induction, ¡athc¡ than lumping all argumcntsthat sciend6c induction;udgcs to lrc strorry in ctnr; catt:gttry, wa will sct,that the inductivc justification of indt¡ction doc.s not bcg the question.

Just what thcn arc thcsc lcvcls of scicntific i»duction? And what is theirreleva»ce to the inductivc justification of i¡rcluction? We can distinguishdifferent lcacls ot' argunlent, in term.s of thc things they talk about,Arguntcuts on lcvcl I rvill trrlk about i¡rclividr.rrrl tlrirrgs or evcnts; folinstarrce:

i\lan,v jub-jLrb bircls l¡rrr.'c bt:c¡r obstrvcd, und tlrcy h¡vc all bccripurplc.

'l'lrc ¡rcxt jub-jrrb bird to bc observcd w,ill l¡e purl;le.

l.cvcl I oI scicntific irrtluctivc logic rvoulrl corrsist of ¡.ulcs [or.a.ssigningi¡rductivc plobabilitics to argurner)ts of lcvcl 1. Prcsurnably thc rulcs ofIevel I of scicntiflc irrduction would assign lrigli irrrluctive probal)ility tothc preccding argument. Argurncnts ou lcvcl 2 u,ill talk about argurnentson lcvcl 1; for instance:

Scinle deductively valid argurncnts r¡n lcvcl t have true prcrniscs.\ll deductively valid arguments on levcl 1 rvhich have tn¡c

prcrnises har,'t' truc conclusio¡ts.

Sornc dcductivc.ly r.rlid argurlcrrts on lcvci I llavc truc co¡rclusions.

This is a deductively valid argument on levcl 2 which talks aboutdeductively valid arguments on level 1. Thc following is also an argumenton lcvel 2 which talks about argunrents on levcl I:

Some argunre¡rts on levcl I wllicl¡ thc rulcs of levcl I of scicntificinductive logic say are inductivcly strong havc truc premises.'I'he denial of a true statcment is a false statentent.

Some arguments o¡'r level I which the rulcs of lcvel I o[ scientificinductive logic say arc inductivcly strong havc prenriscs whoscdcnial is falsc.

32 U. r-¡¡c rnAr)r'rIoNAL prtoBLuM oF' tNDUCTIoN

This is a deductivcly valid argurnent o, lcvcl 2 whicrr tarks about argu-rntnt.s on levcl l, wlricl¡ tlrc rulcs of lcvel I of scicntific irrductivc logicclassify as inductively strong.

There are, of coursc, arguments o, level 2 which arc not deductivelyvalid, and thcre is a corrcsponding second lcvel o[ scicntific indr¡ctjvelogic which consists of rules that assign degrecs of indt¡ctir c st¡.cngth tothesc arguments. Tl¡crc are argur¡rcnts on lcvcl 3 wlricl¡ talk aboutlrgu-n¡cnts on Icvcl 2, argu¡¡rcnts on lcvcl 4 wlriclr talk ll-¡r,¡ut argtrrnr:nls orrlevel 3, etc. For each level of argrrmcnt, scientiGc inductivc logic has acorresponding level of rules.

This charactcrization of the levels of argument, and thc corrcspondinglcvcls of scicntific induction, is sunr¡narized in Table 1. As thc tar¡le shows,

l ablc I

Levcls of argun)cnt

l': Algurnt:nts about ar-gtrrnt'rrt: r¡¡¡ IcvclÁ.- 1.

:

Levels of scicntificinductivc Iogic

I:(ulr:s for assigning inductivcprobabilitics to arguntcnts otllcvel l'.

Arguntcnts about argrr¡¡lcnts r¡ll lcvcl l.

Argumclrts about indi-viduals.

Rulcs for assigning inductiveprobabilitics to ilrgu¡r)cnts o¡rlcvcl 2.

Rules for assigrring ir¡ductiveprobabilitics to argume¡)ts onlevel 1..

2:

l:

scie¡rtific inductivc logic is sccn not as a simple, homogencous system butrather as a cor¡1:lcx structt¡rc cornposed of a¡¡ infinite r¡uml¡cr of stratao[ distinct scts of rulcs. The scts c¡[ rules o¡r dillcrcnt levcls are rrot, how-evcr, totally unrelatcd. Thc rules on each lcvel prcsupposc, in sorr.rc sc'se,that naturc is uniforrn and that thc futurc will resemble thc past. If thiswere nr¡t thc casc, rve rvould have no reaso¡r for calling the wholc systemof lcvels rr .sy,stcrrr <,[ ,sctetútftc irrtitrctivc logic.

\\/c.¡rt: ¡ro\v l¡l;r ¡losrtrorr to st,t,l¡r.nv tlrc systern r¡f lt,vels r,I scrc¡rtr6cI¡ttltlt tt,r¡¡ tr lr¡ lrt' t ttt¡rl,,r, tl ¡rr tlrt rlilr¡rllrt. lrrstrlir,rtr,rrr ol r¡rclLlctto¡t.It¡ ¿¡¡rut r lt¡ llrr r¡r¡r rr¡,¡r "\\ l,r rlr,,rrltl rl't' ¡rl.r.t, r,r¡r f¿¡tlr ¡r¡ tltt.r¡le: sl

II. 3 rrrs rNDUC'rrvE JUS'rlr.r(;^'r'loN or,- rNDUc.r.roN

Icvcl I of scic¡rtific i¡rductive logici'" thc pro¡roncnt of tl¡c inductive justi-fic¿¡tio¡l of i¡rdr¡ctio¡¡ will ¿rdvancc ¿¡n rrgr¡rncnt o¡l Icvcl 2:

Anrong rrgurrrr:nts usecl to rnake prcdictions i¡t the past, e-argu-¡¡¡c¡its on lcvcl 1 (rvhiclt according to lcvol 1 of scicl¡tific inductivclogic are ingluctivc)y strorrg) hiive givc,n tn¡c collclusiorrs l¡li¡stof thc tinrc.

\\'itlr rcgurd t, tl¡c r¡cxt ¡trcrlicti,rr, .¡r .-lr.g.r¡)c¡rt jrrclgcd irrduc-tivcly strong by thc rules of scicntific indrrctive logíc will yield a

trt¡e conclrrsion.

Tlrc ¡rroponcnt rvill maintain that the premi.sc of this argumer.,t is true,lncl if wc ask why he thinks that thi.s is an indrrctivcly strong argument,lrc rvill lcply that tltc ntlc¡; ol l.cacl 2 of scíutlifir: i¡xlt¿ctioe logjc assigrr ita lrigh irrductivc probability. If wc now ask wh¡, ,"vc shor.rld put our iaithin tlrcse rr-rlcs, hc will adv¿r¡rce a si¡nilar argurncr¡t on levcl 3, justify that¡rgr¡n¡c¡lt l>y rppcal to tlrc n¡lcs of scicntific iDductive logic on level 3,jtrsti[¡, tltosc nrlcs l;y an argrrlncrrt o¡l lcvcl 4, e tc.

Tlrr: inrlrrctivc justificatiorr oI i¡rductio¡l is.sur-r¡rnl.izcd il¡ T¡blc 2. TIle;¡rrows i¡r tlrc table slrow thc c¡rdcr of justification, Tlrus the rulcs <¡f .levcl Iarc justificd by an argrrrucnt on lcvel 2, which is ju.stificd by tlre rulcs onIcvcl 2, rvlrich arc justified lty an argurncnt on lcvcl 3, ctc.

Lct ,s ¡¡ow scc how rt is tllat the proponent of the i¡iductive justificationof i¡lrluction cnn plcad not guilty to flu¡uc's clrlrgc of llcggirrg thc qucs-tion, tllirt is, of prcsupposing cxactly what onc is trying t<., prove. In¡'ustifyirrg thc rulcs of Icvel l, tlre proponcnt of thc irductivc juit.iÉcationof induction does not presupposc thattlrcse ruk:.s w.ill work thc next time;in firct, hc ¡dv¿inces an argunlclt (on Ievcl 2) to shorv tlrat they will workncxt ti¡nt'. No*'it is trrre that t]lc use o[ this aigurne¡]t prcsupposes thattllc rulcs of lcvcl 2 will work rrext tirnc. But tl¡crc is anothcr ¿rrgumc¡rtrvaitirrg o¡r levcl 3 to show that the rules of lcvci 2 will wor.k. The use oftlrat argument does not prcsuppose what it is trying to cstablish; it prc-.sr¡PPoscs that thc rules on lcvcl 3 will work. Tlrus uoDc o[ tlic argurnrrnts.scd in thc inductive justification of induction presupposc rvhat ihey aretrying to prove, and the inductive ju.stification of indt¡ction does nottechnically beg the question.

I).rl'r¡;s Irorv thcsc lc'cls rvork c¡. bc ¡nadc cl.arr:r by rookirrg at rL

sirn¡rle cxrnr¡rle. suppose our orrly ob.scrvations c¡f tl¡e world have bee..f 10í) jtrb-jrrb l;ircls a¡rd tlr.y h^vc all bce¡r pur¡rlc. After observirg99 lul;-¡ub birds, rvc advanccd arguntcnt jj-gg:

u I I. -r.ltu -rrrADr.rroNAL prtoBI-¡rN{ oF rNDUC.t.loN

\\/c h:rvc secn gg jub-jub birds and they werc all ¡rur¡rle,'l'hc nerxt juL-jub bird we see will be purple.

T,blc 2

Lcvcls of argurnentLcvels of scic¡rtific

inductive logic

They will work wellncxt ti¡lc. \2: llules r.¡f levcl l of sci-----+2:entific inductive logicllrivc rvorkt.d rvt:ll intltc ¡tast.

o

'l'lrt'y rvrll rvork ri.cll,rrc.rt tr¡¡)r.. \t:

:'l-

liult.s <,¡[ Ievr.l 2 o¡ r"¡--] 3c¡rt,fic inductivc logichavc workcd'well inthc ¡tast.

:Ilules for assigning inductiveprobabilities to argumcrrts onlevel 3.

Iiules for assigning inductiveprobabilitics to argurnents onIt:r,el 2.

l: ilulcs for assrgrnrrg irlrluctiveprobilbilitios to argumcnts onlcvel I.

o 'I'lrc s(atcrrrcr,r ''rulcs <¡f lcvcl I .f scic¡rtilic in<luctivc logic lravc w,orkc<l w,cllirr tlrc past," is to be tuken as sho¡tLrr¡d lor "argurncnts on lcvcl I, whicltaccording to thc rulcs of Ievcl I of scicntific inductive kr¡iic rrc incluctivelystro¡¡g dnd wl¡icl¡ havc b.r'n Lrscd to ¡r.rLe ¡;rcclictio.s i. tlrc ¡rast, havc givcn

' .s tru'i conclusions, whcrr th. ¡rrcrniscs scre true, most of the time.,,Thr¡s thergrrrrrcrrt t¡¡r lcvcl 2,sctl t, jrr rrfy thc ¡ules of lcvcl I is eractiy thc s¡nlc onc

rs ¡rut forth in tl¡c sccond ¡rarugraplr on ¡ragc 33.

This argrrr»ent was given high in<,luctivc probability by rules of level Iof scientrfic inductive logic. We knew its premises to be true, and wetook its conclusion as a prediction. The l00th jub.jub bi¡d can thus becorrectly dcscribed as purple-or ¿rs the color that ¡nakes the conclusio¡rof argunrent j;-99 true-or as the color that results ln a successful pre-dr.tion by tlre ¡ules of level I of sciertiÉc i¡rducti'e logic. Let u.s alsosupptl5e tlrat sirnilar argur¡lcnts hacl beerr advanccd in the ¡tast: ij gS, jj-97,clt. Euch clf tlr.se urgrrrrr.rrts \vlls ar) .-argurrrcrrt to rvhich tlri' rLrles ofIrvcl I.ssrgrcd lrrglr irrdrrctrrc probability. l'hus tlre r-¡l¡scrr,.itrons of

IL 3 rrrr TNDUCTTVE Jusrr¡'rcA'r'loN or. TNDUCTIoN

jrrb-jrr[r bircls 98 lr¡d 99, ctc., arc ulso r-rbscrvation.s of succcssfr¡l outcon]csto prcdictions bascd on as.signments of ¡rrobabilitics to c-argurncnts l)yrrrles c,f level l. This gives rise to an argumcnt on level 2:

e-argumc¡tts on lcvel l, whicl¡ arc assigned high inductive probt-bility by rules of level 1, have had thcir conclusion.s predicted 98tinrcs and all those predictions wcrc successful.

It",l.t",g tt- ""*is assigncd high inductivc probability will also lead to success.

'l'his argurncnt is assigned high inductive probrbility by rulcs of level 2.

If thc next jub-jub l¡ird to bc observcd is pur.plc, it rnakes this level 2

argurncnt st¡cccssful in addition to making the appropriatc level I argu-n.tcnt succcssful. A string of sucll successcs givc.s rise to a similar argumer)ton lcvcl 3 and so on, up thc ladder, as indicated in Table 2.u

If sornco¡rc rvcrt: tc¡ objr.ct that r,,,ltat is wantcd is a justification of sci-e¡rtific induction as a *,lrr.¡le ¿uld that this lras ¡rot been given, the pro-poncnt of thc inductivc justification of inducti«-rn would reply that, f<.rr

t'vcr¡, lcv.'l oI rules r.¡f scicr¡tific inductive logic, lre has a justification( orr a lrighcr levcl), arrd tlrat certainly if cvcry levcl of rules is justi6ed,tltun tlrc rv)tolc systcrn is ¡ustificcl. Ilc woulcl rnaintain tl¡at it rnakes no.scr¡5c to irsk lor a jrrstificatiorr Ior tllc systcl¡t oucr u¡ul ul¡ooc a justificatio»fr-¡r raclr o[ its part-s. 1'lris position, it ¡nust bc adrnittcd, has a good detrlof plausibiiity; a final cvalr¡ation o[ its ntcrits, )rowcver, must await somefrrrtlrcr dcveloprnents.

1'lrc ¡rositiorr lreld by thc pro¡roncnt of tlrc, inductivc justification ofi¡rduction contrasts with thc position held by Ilurne in that it scts difie¡e¡rtrcrluircrnents for the ¡ational justification of a systcrn of inductive logic.1'lrc lollow,ing is inrplicit in tlrc irrrlrrctive justification of i¡rduction:

llatio¡¡al Just ification

Sugi¡c.stion /I; A systcrn of irrltrctivc logic is rationally justificd if,for every lcvel ( k) of rules ol that syst€ur, there is an e-argumenton tirc next higlrcst lcvel (l."l 1) rvlriclr:

i. Is ldjtrdgcd inductivcly strong by its own systcm's rules (the.se

will be rtrlcs of level A -i' 1).

ii. Ilas as its concltrsion the state¡nent that thc syste¡n's rules on tlreoriginul level (k) u,ill rvork rvcll ncxt trmc.

,,Irr'fable 2, "works rvell" is used as slrorihand for "assig¡rs lrigli irrdrrctivc ¡rrob-abiIty to ar¡ c-argu¡ncr)t whose conclusion turns out to be t¡ue."

36 lI. r¡le 'r'¡rADrl'roNAL pHo¡ILEM ot' TNDUCTToN

It is irnportant to see that rclrcth¿r o. stJstc¡n ol induction ntccls tlrcsec<¡¡rlilio¡ts depcruls not orly on tlrc :;ystetn of inductiorr itscLf but aLl;o o¡tlhe facts, on tlrc wa7 that the world is. We can imagine a situation inwhich scie¡rtific induction would indccd not ¡ncet thesc condítions.Imaginc a world which has bcen so chaotic that scientific irrduction o¡rlcvcl I lrls ¡tot rvorkcd wcll; that is, suppose tli¿t tho c-arg-umcnts onlevel l, which according to thc rules of level I of scic¡rtific inductivc logicare inductively strong and which have been used to make predictions inthc past, have givcn us lalsc conclusions from t¡uc premiscs most of thctime. In such a .situatio¡r thc inductivc justification of induction could notl;c carricd tlrrorrglr. For irltlrorrglr thc argurncrrt on lt:vcl 2 u.sc<l to jtrstifythc rules of lcvcl I of scientific induction, that is:

Rulcs of lcvcl I of scienti6c inductivc logic havc rvorkcd wcll inthc past.

They will work well next timc. ,

would still bc adjrrdgcd irrductivcly strong by tlic rules of lcvcl 2 ofscic¡:tific i¡rductivt: logic, its ¡rrt:rrrisr: wor¡ld not lrc truc. I¡¡dcecl i¡t tlrcsitu¡tic¡n unrlcr con.sidcration tht: following arguntcnt oll lcvcl 2 taouldItave a prelnisc tllat was knowu to bc trut: alrcl rvould also bc ad¡udgcdu¡dL¡c'tir clr' -,¡¡¡¡g l¡r, thc n¡lcs of lc",cl 2 of .scic¡rtjfic inductivc logic:

Iir¡lr'¡ irl ir'r't'l I t¡f sc,it.¡it¡lic lr¡tlr¡clivt.loit¡c h.ire not \!orkcd \r..,Ji

in the pa.st.

l-llr'r'l ¡ll ¡rot lork lr.ell ¡te'rt ti¡¡lc'.

l'i¡,.s r't'cir, (()¡)cci'c t.ri sitr¡¡tro¡¡s,l r'hrclr.ie'el 2 of scienti6c induc-tio¡¡, instcad of justifyi,g Ievel I of scienti6c induction, would tcll usthat levcl I is unrcliable.

we are not, in fact, in such a situation. Levcl I of scientific inductio¡rhas served trs rltritc wcll,' and it is upon this fact that thc inductive j.sti-fication of induction capitalizes. This is indced an important fact, but itremains to bc scer¡ whethcr it is sufEcient to rationally justify a systenlof scicnti6c inductivc logic.

TIre propo'c¡¡t of the i¡rductivc jtrstification of scicntific inductivcI.gic ha"- d<¡r<' ..s ¿¡ scrvicc in disti.grrislrirrg tlrt: varior¡s lcvt:l-- of i.clrrc-tion. llc Ila.s also rradr: ari irn¡;ortant co¡ltribution by pointing out thatthcrc arc po-ssible sjtu¿tio¡rs in which thc higher lcvcls oi scjcnñlic i¡rrlrrc,-

r Cr)¡¡s¡(l( rllt( tltc lrrrlrrr; ,,I st rr'¡¡tr', ll¡r5 sl.rlcr¡¡(.nt rs urr ovcrsrrr¡rlific¿rtro¡¡ rr]¡r¡stsl.rll-s rlr s(rv(i sl,n{)il5 (r¡r¡rtrict.Ilttr¡r

II. 3 'r¡lc rNDUCTrvr:JUSlrFrcA'rlo¡.v o!- TNDUCT-roN

tion do not always suppbrt the Iower levels and that we are, in fact, not insuch a situation. But as a justification of the systenr of scie¡rtific ináuctionhí.s reasoning is not totally satisfactory. while he has not technicallybegged the question, he has come very close to it. Although he has anargumcnt to justify every Ievel of scientific induction, and although noneof his argumcnts presuppose cxactly what tl:ey arc trying to prove,lhc ¡us-tification of each level presup¡roses the corrcct¡rcss of the level abore it.Lower levcls are jrrstified by higher levels, but arways higher Ievels ofscientific induction. No nr^tt.r how far we go in the justifying proccss, weare a)ways within the system of scientific induction. Now, isni ihis loadingtl.rc clicc? couldn't sonrcono r.vith a com¡;lctely difiercnt systcm of induc-tive logic cxccr¡tc tlre sarne mancuver? couldn't hc justify cach level ofñu logic by ap¡rcal to highor Ievels of lrrs logic? Indecd he could. civentlre sarnc factrral situation in which the inductive justification of scientificindrrction is carricd out, an cntirely diffcrcnt system of inriuctive logiccor¡ld also meet thc condition.s laid down undcr Rational Justificatiin,s.ggcstion IL Lct us takc a clo.scr look at such a contrasting system ofindrrcttvc logic.

we said that scientific inclr¡ction assumes that, in some scr)se, natureis r¡¡rifor¡n a¡rd thc [uturc rvill bc like tlrc past. some such assumption isto be found iracking the rulcs on cacli lcvcl of scientific inductivc Iogic.Tlre as.sunrptions a¡e not exactly the samc on cacli level; they must bedilfcrcnt bccauso \vc ean imlgino a situation in whiclr sciontific inductionorr lcvcl 2 rvould tel.l us that scicrtific induction o. lcvcl I will not workwell. Thus different principles of the uniformity of nature are presupposedon diffcrcnt levels of scicnti6c inductivc logic. But althotrgh they are notexactly thc sarne, they are similar; they are all principles of the uniformityof natu¡c. Thus each level of scientific inductivc logic presupposes that,ir'¡ so¡ne sense, nature is uniforr¡r and the futurc will be like the past.A s1'stenr of Ínductive logic that would be diamztrícallg o!:posed. to scien-trfic inductivc logic would bc .r'rc wlrich presupposed on alr revels thatthe future will not be like the past. we shall call this system a system ofcou nter inductioe Io gic.

Let us see how counterinductive logic would r.vork on level l. Scien-tific i¡rdrctivc logic, which as.sumes that the future will be like thc past,rvould as.sign the following argu¡ne¡rt a high inductive probability:

Ivfany jub-jub birds havc bcen observed and they have all beenpurple.

37

Thc next jub-jub bird to be observed will be purple.

r ]¡t II. r¡rB TnADI'I'IoN^L pnoBLEM oF INDUCTToN

(-:r)ur¡terir)(luctive logic, rvhich assunres that the future will r¡of be like thepa.st, would assign it a low i¡lductivc ¡trobal;ility and woukl insteacl assigna high inductive prt-rbability to the followirrg argurnent:

Many jub-jub birds have bccn obse¡ved and thcy havc all beenpurplc.

'I-he ncxt jub-jub bird to be observed will not be purple.

I¡r gt.,!rniral, countc¡iuductivc logic assigns low inductive probabi.lities toargurnents that are assigned high inductive probabilities by scientifrcinductive logic, and high inductive probabilities to arguments that areassigrred low inductive probabilities by scientific inductive logic.

Now su¡rpose that a countcrinductivist decided to give an inductivejustification of counterinductive logic. The scientific inductivist wouldjustify his rules of level I by the following level 2 argument:

Rulcs of level I of scientific induction have worked well in thepast.

'l-lrcy rvill work rvt:ll ncxt tinrc.

Thc countcri¡rductivist, on thc othcr hand, would justify his rules of levclI b1, ¿¡16¡¡,"r ki¡rcl of lcvcl 2 irrgurnt:nt:

liulcs of lcvcl I of cor¡l¡t('ri¡)([uctivc logic llavc nr¡l work(.d wcll intlrc past.

They will work wcll ncxt tin.¡c.

By thc cou¡rtcrinductivi.st's ¡ules, tlris is a¡r rrrductively strong argument,for,on level 2 he also assumes that tlrc future will be unlike the past. Thusthe countcrinductivist is not at all bothe¡ed by thc fact that his level Irules have been failurcs; indeed hc takcs this as evidence that they willbe successful in thc future. Cranted his argument appears absurd to us,for we are all at heart scicntific inductivists. But if the scientific inductivistis allowed to use his own ¡ules on levcl 2 to justify his rules on level l,how can we deny the samc right to thc counterinductivist? If asked tojustify his rules on level 2, the counterinductivist will advance a similarargument on level 3, etc. If an inductive justification of scientific induc-tive logic can be carried through, then a parallel inductive justi-ficationo[ counterinductive logic ca¡r be c¿rrried through. Table 3 summarizeslr<,¡w tlris w,c,uld bc donc.

II. 3 rue INDUCTIVE JUSTIFIcATIoN oti INDUc:l'IoN

Tablc 3

39

Lcvcl of Justifying argurncnts o[ the

argr¡ment scie¡rtiffc inductivistJustifying argumcnts of the

cou nterconducti vist

¿.

llules of lcvel 2 of scientificinductive logic havc worke dwcll in thc past.

llulcs <.¡f level 2 of counter-.inductive logic have trot

rvorkcd well in the past,

Thcy will work well nexttir¡c.Rules of level I of scientificinductive logic workcd wellin the past.

They will wo¡k well nexttime.Rules of lcvel 1 of counter-inductive logic have notworkcd well in thc past.

They will work well next

time.They will work well next

time.

'l'hc countcrinductivist is, of course a fictitious character. No one goes

tlrrorrgh lifc consistcntly adhcring to thc canons o[ countcrinductive logic,

although some of us do occasionally slip into counterinductive reasoning.

Tlrc poor poker player who thinks that his luck is due to change because

lre has bccn losing.so hcavily is a prirnc cxan)plc. But asidc from a de-

scription oI garrrblcrs'r'utior¡alizrttiorts, coutrtt:rittdtrctivc logic has littlc

¡rracticnl signiff cancc.

It rlucs, Iiowcvcr, have grcat thco¡ctical sigrrificancc. For what we hlvt:shorvn is that if scicntific irrductive logic mects the conditions laid down

under Rational Justification, Suggcstion II so does counterinductive logic.'I'llis is sufficicnt to .show that Suggcstiorr Il is inadec¡uate as a dcfinitionfo¡ rational justification. A rltional justification of a system of inductrvelogic must provide reasons for using that svstem rather than any other.

Tlius if two inconsistent systcms, scicntific inductiorl and countcrinduction,can nreet the conditions of Suggcstion II, thcn Suggestion II cannot be

an adequate definition of rational iustification' The arguments examined

in this section do show that scicntific inductivc logic meets the conditions

of Suggcstiorr II, but tlicse argurncnts do I)ot rationally justify scientific

induction,This is not to say that what has been pointed out is not both important

and intcresting. Let us say that any system of inductive logic that meets

the conrlitions of Suggestion Il is inductiocly colrcrcnt with the facts.

It rnay bc true that for a systcttr o[ inductivc logic to be rationally justiñed

40 II. r¡r¡ rrrADlrroNAL t)noüLEllr o!_ rNDUC.r.roN

it Inust be inductivciy cohcrcnt with thc facts; that is, that i¡rcluctivecohercncc with tllr, fac.ts. lnay lrc ¡i ,cces.sary. conditio. fr. ,r,io,,J-¡,,r,i_6cation. But thc examplc oi the counte¡inir"tivist shr_,w.s conclusivclythat inductive cohcrc¡rcc with tht: facts is not by itsclf suflicic.t to ration-ally justify ¿¡ systc¡n of i,cluctivc l.,glc. Co,i."qucntly thc inductivc justi-6cation of scjentific inductive togic"faits, '

wc may summarizc our d.iscrission of the inciuctive justification ofinduction as follows:I' Thc p¡'oponcnt of thc_inductivc justification of scie¡¡tific i¡rauctionpoints out that scientific indrrctive logic is incructivcly

""¡"."",-*¡i., ,rr"facts.2' Hc'clairns that this is sufficient to rationa,y justify scientific inductiveIogic.3' But it is not sufficient since cou¡rterinductive Iogic is arso incructivcrycolrcrcnt with the facts.4' Nevertheless it i's important and .infor¡nativc sincc wc can ir.rgirrecircumstanccs in which scie.tific incructrve.'rogic wourd not bc inrirrctiíely

t olrcrclrt witlr tlrc l;¡r l:.. 5.'l'l¡c ¡rr.p,rrt'rrt r¡f lr¡t.i¡¡tr*c.ti'c justificatio¡¡ oI sclc,tific ¡r¡drrctir.l¡:rs l¡ls,r sr¡rt,,,.rl,.tl ltr (.r¡¡lr¡i{ 1,, ,,r,r ,,tt,,rrtr,r,, tl,r, f,,.t t}¡¡t tht,r.t,arc: v¿¡ri_out lt,vt'ls r¡l rrrtlr¡r lrr¡¡r

[xcrci¡cs for ü¡c udva¡rccd studc¡¡tr

l 'fte disct¡ssicrn in tr¡is scctir¡n is ail carricd on ir¡ tcrms of in.sluttc. ¡tr«rictiitrr:lác ncxr jub-jub L¡ird will bc purplc.'r'rx: ¡tcrt e-argurnent ixsigncrl high i.ducti\ c ¡rrobability by r,lcs of rever2 wilrl¡ave a concluion tlat turns out to l¡e truc.

raüer than gencrolily prediclion:

All jub-jub birü are purplc,,lr'ost ju[;-jub lrirds are purple.Mosl c-arg.rrrc't'§ r:isigncd high intructive pro'alririties by rules o[ lcvcl 2 willhave true conclusions.

co.sidcr ¿ rittre lnrxrcr worr<l.consrsting of tcrr flips of a frrnny-lo.ki.g c.in'y anodd-shaped flipping nlachinc. srr¡rposc ihot t-h" "ui,,

hr,, corne up heads on thc firstt-hreo tosscs. what sort uf gcncrility pre.iction could an intltrctivist rnake on thisevidcncc? A cor¡nterindt¡ctivist?

.2' \vl¡¿trrrrtrr[gclltr.rlrly¡rrcrlrclrorrti¡rrl¡ltl¡c¡or¡¡terrrrclr¡ctivist¡r¿kt:,r¡Icvcl2¿l¡otrt the sr¡t.ccss uf crlr¡¡rtert¡¡tlr¡r tt0r¡ ,¡¡ lcvel I i,

II. 4 rn¡: pnAGMATrc JUSnrrcATroN oF TNDUCTIoN 4I

3. suppose üat the counterinductivist predicts th¡t all the remaining trials will behcatls íf the observed number of iails exceeds the observed number of hcÁ, that all thercmaining trials will .be tails if thc observed number of heads exceeds the observednumber of tails, ancl makes no prediction in case of ties; likewise for predictions ofsuccess of counterinduction. Then he predicts, on üe eüdence of three heads, that thenext sevcn tosses will l¡e tails, and on the evidence of two failures of counterin<luctionüat üe remaining predictions of counterinduction will be successes, Notice üat thesetwo predictions cannot both be t¡ue. ll tl¡e next seven tosses all come up taits, then ontoss s€ven t}¡e observed numl¡er of tails will exceed the obserued number of heads andcounterinduction will predict heads on tosses eight to ten. These predictiors cannot besuccesses if, as the counterinductivist originally predicted, the results are tails.Therefore, the particular form of counteri¡iductive skepticism considered in thisexample is inconsis tent!

Suggcstcd rcadings

John Stuart Mill, "Thc Crou¡rd of Induction,,' reprinted ia A lvloclernIntroduction to Philosophy (rev. ed), paul Edwards and Arthur pap, eds.(Glcncoe, Ill.: Thc Frec Prcss, 1965), pp. I33-41.L', L. Will, "\Vill tlie I.'r¡tr¡re Be Like the past?", reprintcd in A MotlernIrtrt¡drctirtt to Plilo,soplry (rev. ed.), p¡rtrl Edwards a,cl Arthur p^p,cds. (Clerrcoc, lll.: 'I'he Frec prcss, Ig65), pp. l4g-Sg.Nl¿x Illack, "lnrluctivc Support r.¡f Inductivc Rules,', problems of Analy_sis (Ithaca, N.Y.: Cornell University press, lg54), pp. I9l-20g.All of thcse arthors are argui.g for some type o[ inductive iustifica-t¡r¡lt of i¡ldr¡clirrrr, althorrgh nonc of thcm hokls the exact position out-lined in this scctior, which is a synthesis of several viewpoints.'rhestude¡rt cannot only l_¡roade¡l his knowlccl(c of attcmpts to justify scien-tific induction inrluctively, but also tcsr l¡is knowledge of the majorpoints of this section, by cr-itically examining the positions taken by

, lhese authors.

For the arlvanced student:

Arthur Burks, Cuusc, Cltance, and Reason (Chicago: University of ChicagoPress, I979).

II.4. THE PRACMATIC JUSTIFICATION OF INDUCTION.Remember that the traditional problem of induction can be formulated as aclilcmnra: If thc rcasonir¡g wc usc to rationally justify scientific inductivelogic is to havc any strength at all it rnust bc either deductively valicl orinductivcly strong. But if rvc try to justify scicntific induciive Iogic byn]e¿rr¡s of a dcductivcly valicl argumcnt with prcmises that are known to bt:true, orr conclusio¡r will be too rveak. And if we try to use an inductivclystrong argument, we are reduced to begging the question. Whereas the

,ll II. r¡t¡ I-nADrf.roNAL pnotsLEM o¡- INDUCTIoN

l)roponcnt ol lht'. irulucllue justification of scicntific induction atter)rPts to¡{o ovcr thc sccor¡cl hol. of the dilcrnrna, t}rc propo,crt of thc 7-tra[rnuticjusti6catio' of inductio¡i attacks thc fi¡st hor,;ire itte,npts to ¡ustlff scicn-tific.ilductivc logic by rnc¡rns of a dcductively valicl argurnánt.

The Pragrnatic justificatio, .f i,ciuction r.vas proposr:cl by Ilcrbcrt l.cigla'd elaboratcd by IJans Hcichcnbach, bc,th founác* of ir," logical ern-Piricist ¡¡r<¡vc¡r¡cut." llcjclrt.rrl-¡ach's pragrrrltic justificatiorr of incl"uctio¡¡ istluitc'conrPIic¡tcrl, f<.¡r it clt.pt'rrcl.: orr \vlllt Ir,.bcligvt,s are tl¡e clctuils (atlcast thc b.sic dctails) of sciertific i,ductivc L-rgic, 1'hus ¡lo onc can fullyu¡ttltrst¿llcl llt'ichclrbacl¡'s lr¡jun)(fr)ts until lrc has stu<-[ictI ilciclrg¡l¡¿rcrh'sdc6¡ritii.¡¡l of probal;ility a.d tl¡c rr¡etrrocl hc prescribcs for criscovcrirg¡rrobabilitics. we shall rctu, to thcse qucsti.ns latcr; at this point wc willdiscuss a sinrplificd vcrsion of thc pragnratic justification of incluction. TIrisversio¡r is c.¡rrcct as far as it gocs. only bcar ir¡ rni¡id that thcre is nro¡c to l-¡cIcal'ncd.

Ilcichcnbach rvishcs to justify scientific i»ductivc logic b¡, a dcductivr:lyvalid argunre¡lt. Yet he agrccs with Hurnc that n<¡ dár"iiu. valid argu-¡rcnt rvith prcrrriscs tlrat arc knor'll to bc t¡uc can givc us the conclusiontlr..i 5c¡c'¡¡¡j[i¡ inductir-r¡l rvill givc us tr.uc co¡rclusions nrost of thc tinlc, Hcru(r ecs ruitl¡ IIur¡rc tlr.lt t]rc, cont-litions ol Ilatio¡i¡l Justification, suggcstionI, ca.¡iot bt: r.ct, sirct'lrt,Iull¡, irtt'rds to ratiorrally justify scicriüfic ir-tluclivt'lr-rgir', tlrt'orrl¡ p:rtlr ol)L.r to lri¡¡r is to arguc tlrut tlre conrliti¡¡s ofllatiorr.l Justificirtio., srrggcsti,rr I. nt:r'cl ,ot bc,rct i. orcler to justify as.vstt'rrr o[ inductivc' lugic. IIc l)r'or, r rls Iu lidvu¡rce lris own suggestiori asto what is rurlLrirccl for rati«rral justification and to attcrnpt Io justifyscicntific inrluctivc logic in thc.se terr¡s.

II IIt¡t¡rt:'s lrrgrrrrrcrrt.s;rrt,Cr>rn.ct, tl¡t.rt,is t)o !va)/ of slrowing tl¡at scit,n-tific i¡lduction u,ill giv<'us trr¡r,crllrcltrsi<¡¡.¡s fronl true prenriscs nrost of tlretir,c. IJut si¡¡cc'IIurrr"s argrrrrrcrrts rrlrp.ly t.<¡ual.ly rvclr to:rry systc, ofincluttivr: logic thcrt'is.o *,ay.f slrorvi¡rrr that any comp(.trr¡g systcn.r ofinductive logic will givc u.s tnrc cr¡nclusiors fror¡l tnrc ¡rrcnris<-,s rnost of thctinlc citllcr.'I-lrus scicntiflc incluctivc logic hts thc sanrc status as all othcrsystems of i.ducti'c logic i, tl¡is rr-¡attcr. No other systcm of inductivc logiccan lrc rlr:¡¡.rr¡n.stratcd to lx: su¡rt.rior.to scicntific i¡rcluctivc logic irr thc.sc¡xt:of showi.g tlrat it ¡1ii,.s true co¡rclusi,¡rs fro,l truc prcnriscs nlo¡e oftc¡rtlran sc'it'rrti6c. inrluetiut, logit.

n llr'*,'r,¡, tlrr rr'1,'llrrlr.¡l,r¡rr,rr¡r ,l'ri,ri r)r.r(r¡¡lrr(,1rr\trfirrtrrrrr, , lr. tr.rrt,rl1,.r,It,rtl,.r¡l,rS.,,,r,,1,¡'l',¡¡,, rl,,lour¡,1,rolA¡¡¡t.¡rr..¡¡¡l)r.r¡.1¡]¡.rl¡r¡,,¡¡rrl,rrtltl, ¡ r,r,,¡¡, I ¡,,.,,,, tl,, . r, ,t, .t t,¡ i,. ,¡,i,, r tl,.,t Ar,¡, rr,., L.,r ¡,r,,.1,,,,,1

lI. 4 r¡is puACMATrc JUSTr!-tcATroN o¡- rNDUC.l.roN 43

llcichenbach clainrs that although it is impossible to show that anyi.ductive nrcthod wiil l¡c successful, it ca¡r bc show¡r that scientific induc-tion rvill bc succcssful, if any method of induction will bc successful. I¡rothcr words, it is possiblc that no inductive logic will guide us to e-argu_nrr:¡lts that givc us truc conclusions nlost of tllc tirne, but if any method *illthcn scientific i,ductivc log;c will also. If this ca¡r bc shown, ihen it woulclscc¡¡r fair to say that scicr¡tific induction has bcen r.ationally justificd. Aftcrall wc rnr¡st ¡¡rrLc so¡nc sort <¡f ju<lgrncrt.s, cr-¡¡iscious or unconscior.rs, asto thc inductivc stre,gth o[ argumcnts if rvc a¡e to live at all. wc musrbasc <lur dccisior¡s on our.xl)cctatio¡rs of thc futurc, ar.¡d rvc l¡¡sc our.cxpcctations of thc futurc on our knowledgc of the past an<I present. Weare all ganrblers, with the stakcs bcing the success or iailurc of our plans ofaction. Lifc is an cxploration of the u,known, and every huma¡l actionprcsun)cs a wagcr with naturc.

But if our decisions are a garnblc and if no nlcthocl is guarantced to bcsuccessful, tlrcn it would scc¡¡ rrtio¡ral to bct o¡r tirat nrcthod which will bcsucccssful, if any rnetlrod will. supposc that you wcrc forcibly taken intoa lockcd roonr and told that rvhethcr or not you will be allowed to livedcpcnds o, rvlrcthcr you win or losc a wager. Thc object of the wager is ubox with red, blue, yellorv, arrd or.ange lights o, it. you know nothing aboutthe construction of thc box but are told that cithcr all of thc lights, some oftlicnr, or ¡ronc of thc¡lr will co¡ne on. You arc to bct on onc of the colors, Ifthc colored light you choose comes on, you livc; if not, you dic. But beforcyou tnakc your choicc you flr., also told that ncithe r thc blue, nor thc yellow,nor thc orange light can co,,c on rvithout thc rt:d light also coming on. lfthis is the only infornratiorr you have, thcn you u,ill srrrcly bct on red. Foralthough you lravc no guara¡ltcc tlrat your bct r-¡u rcd will t¡c successful(aftcr all, lll tlr. light.s nright r'<'rnain dark) you krrorv thlt, if nny bct will b<,

succc:;sful, ¿ bcI ou rcd will b, succcssful. Rcic]rcnl¡acli claims that scier¡-tific i¡-¡ductivc logic is in thc sa¡r¡c privilcged position vis-ir-vis other sys-tcnrs r¡f inductivc logic as is thc rcd Iight vis-i-r,is thc othcr liglrts.

This lcads us t. a .cw proposal ¿rs to what is rcrluircd to rationally justifya s)'st(,m of inductivc logic:

Rational Justification

Suggestiort /1/r A systcnt of inductivc logic is rationally justified ifwc can show that thc e-arguments that it adjudgcs inductivclystrong will givc us true co¡rclusions rnost of tl¡e ti¡nc, if c-argumentsrdjudged inductivcly strong by any rnctlrod will.

44 II. rr¡B TRADTTToNAL pnoBLEM oF INDUCTIoN

Reichenbach attempts to show that scientific inductive logic rnects tlicco'ditions of Rational Justification, Suggestion III, by a deductively validargument. The arguntent goes roughly like this:

Eithcr ¡rature is u¡rifo¡¡n or it is not.l[ ¡raturc is uniforrn, scic¡ltific induction will L¡c.succcssful.If rr.rture is not unifor¡n, then no method will bc succcssfull

If any rncthod of induction will be succcssful, thcn scicntific i¡lcluc-tion will bc succcssful.

"l'h,.:rc is no (lr(,\tion that this argunrc,t is dcductivcly v¿lid, and thc firstand sccond prcrniscs arc surely know¡r to bc t¡uc. Br¡t how do wr: knorv thattl.¡e third premise is true? Couldn't there be some strange inductive methodthat would l¡c sucsessf ul cven if nature were not u¡riform? How do wc knorvthat for anv nrcthod to bc successful nature mUst bc uniform?

Iiciclrcrl¡ach has a rcs¡ro'sc rcady for this ch,llc,gc. su¡r¡rosc that i, a

cornplctcly ch¿¡otic urriverse, sorne r¡rethod, call it rnethod X, rvere succcss-lul.'I'hcn thcrc is.still at lcast o¡rc outsta¡lding uniformity in naturc: thcur¡ilor¡nity of ¡rrctlrc¡d X's succt,ss. Al¡d scic¡rtific i¡rdt¡ctio¡r tvotrlrl discovcll/¡at u¡rifornt¡ty.1'lr;rt ¡s, if rlctlrod X is succcssful on thc rvholc, if it givcs ustrtrr'¡rrt'tlictror)s r¡lost c.¡f thc tu¡rt', tt¡r,¡l sooncr or Iatcr thc st¡tc¡rrr,¡it "Nletl¡-.,1 \ lr.r.s bcr,¡¡ rt'li,rblc rrr tlre ¡>ast" rvill bc truc, ¡n<l the follow,irrg:lr!1r¡¡)rcr)tu',rultl lrc adjuclgt'd irrrluctivcly strorrg [:y scicntific inductivc logic:

Itfctlrod X lr:rs I¡ct,rr rt:liirblc irr tlrc ¡rast.

lvlcthod X will l¡c rclir¡blc,,in thc future.'fhus if ntcthod X is succcssfü1, scicntific i¡lduction rvill also be succcssfuli¡r that it rvill discovcr ¡ncthorl X's rclial¡ilitr,, anrl, so to spc:rr1.,liccnsc ¡ncth-r¡cl X ls rr srrbsirliary,¡lcthotl .r[ ¡rlcdictrrrr. TIris corn¡rlctcs thc 1;rooI tlratscicntil: inductio¡r r,,ill bc succcssl,rl if any nretllod u,ill.

Tlrc job may appear to l-¡c dolic, but in fact therc is a great dcal morc tol-¡c said. In ordcl to analyzc ju.st wlurt has i¡cen provcd a¡¡d rvllat lils not,*,c shall nrr,tli'c irlca oI lcvi'ls of inductivc logic, which rvas ricvclo¡tccl intht'last st:c'tiotr. WIror rvc talk about a nrt:thocl, wr: arc rcally talking rborrta systc¡n o[ inductivc logic, rvhilc glossing ovcr t]re fact that a systcrn ofinductivc logic is composccl of cli.stinct lr:vcls of rules. Lct us now pay atten-tiorr to this [act. Sincc a syster)) of i¡rductive logic is contposccl o[ distinct

" Note that *r' are l¡t,rc usrrrg gcrrcrality, ratlrcrcrcrrrsl l, ¡r .10

tlt.tn ¡¡rsl¡¡¡e, l,r(,dlcttr¡r¡. Scc

IL 4 rt¡r rRAGMATIC fusrrFrcATroN or INDUCTIoN

Ievcls of ¡ules, in order to justify that system we rvoulcl have to justify eachlevel of its rules. Thus to justify scientific inductive logic we would have tojustify level I rules of scientific inductive logic, level 2 rules of scientificinductivc logíc, levcl 3 rulcs of scicntific inducti'c logic, ctc. If each of theselcvcls of rulcs is to bc justificd in accordancc with thc princi¡rlc "It i.s

ratio¡lal to rely on a method that is successful if any method is successful,"then the pragmatic justification of induction must establish the following:

I: Lcvel 1¡ulcs of scicntific induction will be successful if Icvel 1

rulcs of any system of inductivc logic rvill bc succcssful.2: Lcvcl 2 rulcs of scicntific induction will be succcssful if lcvel 2

rulcs of any systenr of inductivc logic rvill bc succcssful.

li, Level /< rules of scienti6c induction will be succcssful if level krules of any systcrn of inducdvc logic will l¡c succcssful.

Ilut if wc look closely at thc pragrnatic justification of induction, wc seethat it docs not cstablish this but rather somcthing quite different.

su¡r¡rose that systern X of inductive logic is strccessful on level r. Thatis, thc argunrcnts that it adjudges to be inductivcly strong give us trueconclusions f¡om true premises most of the time. Tlien sooner or later anargurncnt on level 2 which is adjudged inductively strong by scienti6cinductivc logic, that is:

Ilulcs of levcl I of systcnr X havc bccn reliable in thc ¡rast.

Ilules of lcvel I of system X rvill be ¡eliable in the future.

rvill corne to havc a prcmise that is known to be true. If the rules onievel I .f systcm X give truc prcdicti,ns most ol the ti¡,e, then sooner orlatcr it rvlll bc true tliat thcy havc grvcn us truc prcdictio¡rs ¡nost of thetirrre irr the past. And once we hrrvc this premisc, scicntific induction on.levcl 2 leads us to the conclusion that they will be reliable in the future.

Thus what has been shown is that if any systcm of inductive logic hassucccssful rules on levcl I, thcn scicnti6c induction provides a justifyingargu¡r'¡ent for thcse rulcs on lcvel 2. Indecd we can gcneralize this princi-ple and say that if a system of inductive logic has successfuj rules on agiven level, then scienti6c induction proüdes a justifying argument onthe next highest level. More precisely, the pragmatist has demonstratedthe following: If system X of. inductive logic has rules on levcl /c whichpick out, as induótively strong arguments of level k, those which givctrue predicüons nrgst of the time, then there is an argument on levei

11AJ

46 II' run TIIADITIoNAL Plt()uLENr ol¡ INDUC'I'IoN

k'll,whichisadjudgedinductivelystrongbythert¡lesoflevclk-lluf scientifi" induciive-logic, whic¡ has as its conclusiotr thc statcr.e,t

that the rules of systcm X on lcvel I¡ are reliable, ancl rvhich has a prcrr-rise

that will sooner or later be kr¡own to be true'

Now this is quite difiercnt fronr showing that if ahy rnethod rvorks orr

any levcl then icicntific irductio. will alst.r rvork o¡i l/¡¿l lt:vel, or cvcir

frin sho*ing that if a.y l¡rctho<-l 'v,rks t¡rr lcvel I tlrerr scic¡rtific i¡rcltrctiotr

will rvr¡rk o,i Icrel 1. Instea«l wlrat has 5ce¡r s¡,rvrr is tllat if arl)' ot¡cr

rncthod is gcncrally succcssful oh lt:vel 1 then scit:nti[ic indrrction rvill

Ilavc at l"al on" notal¡lc st¡cccss o¡r level 2; it rvill t'r'cnttrally ¡rrctlitt tirc

continucd success of that other Incthod on lcvcl 1'

Althouglr tllis is ar¡ irrtcrcstirrg rrrrd irrtPortltrtt cottclttsiott, it is ¡rt¡t suf[i-

cient for i|c task at hancl. Srppoi" rvc wish to choosc a set of rulcs for I.vt:l

I. I¡¡ c¡rclcr to [-¡c ill a positi,.,,, r,,,rlogous trr tlrt' wltgt t lrlrtrtrt tlrtr Iror rvitlr

thc colorcd liglrts, rvc woulcl ilavc t0 knorv that sciclrtific illcluctir¡rl rVouid

bc succcssf ul án lcvct I if rrny mctllr¡rl werc succt'ssfr.rl r.¡rr lcvcl l. llut rvt' <ltr

not knorv this. For all rvc know, scir,¡rtific irltluction rrriglrt Iail c¡lr lt'r't'l I

aDcl anotht.r ¡nt'tl¡r¡d rnight bt' rlltitt'stttlc t'ssful. lI t]¡is r'r'r'rt' tlit' CiIs(', s('i('¡l-

tilic inductio¡t on lcvCl 2 rvould cvt'rrtually tt'll us so, l-¡trt tlris is tltritc a

il i lIt'rcrrt Initttcr.Irr su,rrnilry, tlrc attcutl;t ut;t llt.agrttatit jtrstilicatrorr ol'itttlttctiolt lt¡s ltt:ttlt'

rrs r.:alize tl¡ai a «letluctiu.]l), u,,li.l jrrstilicatiorr ol'st it'r¡tilic i¡ttltrt tiotr rvottltl lrtr

"i.,.,.,,i.f,f" if it could "stablirl, tl¡ilt: i[ ilrty systcrtt r¡f i¡rtlr¡ctivt: logrc hrrs

s,,..csrlirl rulcs on ü givcn levcl, tlrcrr scicrrtific irrrltrctlvc logic rvill lr[r't:

srrcccssft¡l rr¡lt:s on thai Icvcl. llt¡t thc lIl'¡¡trlnellts ¿¡tlr'¡trcctl itt tltt PrltgrttirticjÑiti"",in,.I fsil to csta[:¡lislr this ct¡nclt¡stotr. I¡tstc¿ttl tltcv shorv tltut il'arry

l;.;;; oiin,lu.ti"" logic has successlul rulcstr¡t r gi!crr levcl' thcn sc'icr¡tiflc

inciuctiie logic *.iil 1,..,,rr" ;r ju-stifr ing ¡rgLrnlcrtt lor tho.se rtries o¡t tllt- rrt'rl

higl¡er level.

BotIüeattemPtataPragmatiCjustificationanc]theattenlPtatanillductir,e

;*tifr.rtlon have iailed io p,o"d"''n al¡v)u.te.lu5tjfication ¡¡[ scienlihc irtduc-

ton. Neverüeles, both of ihem have brought forth,seful facts. For instance,

üepragmaticjustificationofinductionshowsoneclearadvantageofscientificincuction over countcrinduction. The counterinductivist cannc-rt prove tliat.if

".y rr",f,"a i, successful on level 1, counte ri¡ir'luction on lcvel 2 will cvcntually

práü", its continued success. A¡rd the exerciscs at the er¡d of the previous

lection suggest that some care is rer¡uired to even givc a logícally consistent

for¡nulation o[ countcrinduction as a gcrrcral policy'

Itsct:¡lts,tlrcrr,tll¿ttlrcrt:isstillrtlrtlllforctllrstrlrctivctlrtrtrglrtt-rrrt}rclrrtllllcrrr, iu¡d t}r¡t wc calt lcarlt ¡ttttcl¡ [rt¡ttt llr,cvit-ltts attcrlrllts tt¡ scllvc it.

41II.5 rnao¡rtoNAL PnoBLErlr ol' INDUCTIoN

Suggested readings

Ha¡rs Reichenbach, "O¡¡ üe Justification of lnducti<¡n," in Readings in

Phitosophícal Anolysis, ed. Herbert Feigl and Wilfrid Sellars (Ncw

York: Appleton-Century-Crofts, Inc., 1949), ¡tp- j24-29.

F'or the advanced student¡

B. Skyrms, Prugnutics and Enqtiricinr (New Ilaven and Lo¡r<'lon: Yale Unr-

versity Press, 1984), chap.3, "Learning from Ex¡rerience."

iI.5. AN ATTEMPTED DISSOLUTION OF THE TRADI.

TIONAL PROBLEM OF INDUCTION. Thc incluctivc iustification of

irlciuctiou attackcd ot¡c h«¡rt¡ oI flunrc's dilc¡¡rrna by atternpting to rationallyjustify induction by mcatrs of inductivcly strong :Irgumc¡¡ts' Thc pragmaticjustification of inductio¡r attackcd tlrc othcr Irorn by attcrnpting to rationally

]ustify inciuction by nrcans of a ticductivcly valid argument. Thosc who

"tt"n.,pt to ciissolvc the problcrn of induction takc the t¡ird alternative.

Tlrey att"mpt to go bctwecn thc horns of the dilenrma by clainring that

no argurttcttt whatsocvcr is trcccssary to iustify induction'Thc¡c arc nrany variations on tlre philoso¡rhical tlrc¡nc of tlissolving the

traditional problem o[ inductio¡r. we shall not try to survey cvcry nuancc of

cvary possibl0 vüri[tiou, but ruthur g,(ul¡tiue lltosc cutrsiderutioll§ wltich aL¡r

at present nrost frcquently advanccd to show ihat no argum{jnt whatsOcvcr

is necessary to justify scientific inductivt iogic.

Those rvho tuirh to dissolve the t¡adiiional problenr of i¡¡duction §ay thrt

the question t¡at Senerates üis prob,lem, üat is, "\\/hr- is it rational to

acceit scientiÉc ináuctive logic?" is a silly qucstron, a question born of

confr]Jio¡). There a¡e thrce n¡ain contentiot)s ¡s to ryh,r'it ís .r silly'cluestion:

(1) It asks us to turn induction into deduct.ion. (2) Someone w'ho doubts tlrc

rationaliry, of accepting screnti6c inductive logic simpl¡' does not u¡de¡'

stand the words he is using. (3) It ask for a justification beyond the limits

where justiEcation rnakes scnse. Let us survey these co¡rtcntions and thc

considerations adduced in their favor.

I. It is a silly question because it asks us to turn induction into deduc-

tionr The problem of justifying scientific inductive logic arises bccause it,is

logically possible for án argu*ent adiudgcd inductively strong by scicntific

ináuctivc logic to lead us frorn true prcmises to ;r f¿lsc conclusio¡r. Now ifone is accus-tomed to üinking of dcductive validity as the only standard

48 IL ruB TItADtrloNAL PTIoBLEM or INDUCTIoN

against which the strcngth of argutttctrts can be- mcasurcd' lie will be un-

,itirfi",l with this sitLraiion a¡rd rvish to providc somc logical guarantcc

that inductively strong arguments will give true conclusions fronr true

fr.rnir., g" *llt clem*and-certainty where certainty cannot be. had..And

ih",, lr. ¡51a5 why it is ratio¡ral to acccpt scic¡rtific inductivc logic hc is

demandillgaproofthatargurncntsacljuclgcdstrorigbyscicr¡tificinductivelogic rvill g,u" u, true concl-usions fronr true prernises all the time. o[ course,

itIr" i. nJ.,r"h prooí bccausc certainty that the conclusion rvill be true if

thc prcrniscs arc tr.ue is tl¡e hallmark of dcductivc validity, not incluctivc

,,r.,rg,t,. Thus he is dcnrancling that thc argutnents adiudgcd.strong by

,.i",,iifi" i¡¡t^¡ctivc logic shoulrl i.,ally l-'c dccluctivcly valid' which is inrpos'

siblc. Once ,"c realii" that cleductive validity is not the only. stand.ard

rugainst rvhicl¡ thc streugth o[ argurn-cnts ca¡r bc nrcasurcd and that induc-

ti"vd strcngth is a legitiÁte sta¡ráarcl in its owD right; o¡rcc wc rcalizc that

urgu,r,.n,r that givi high probability rather.than certair.rty arc still good

arlun'tents, rvc s-ec tlrat thii deniand is ridiculous' The rcquest for a iustifi'

caiion of scit.utific i¡lcluctive logic iirises fron-r thc nristakc¡l opinion that

clccluctivc validity is tlrc only stanclarcl irgainst rvhich thc stlcngtlt of

argurnCt)ts can bc t¡cirsurCcl, ancl Iron] atr adolcscc¡rt clcsire fol cCrtaitr§'irt

¡ rvorld oI cha¡lcc,2. lt rs a silly <¡ucstio¡r l¡ccause sonrconc rvllo doubts tlrat it is ratio¡ral to

acccli)t scicntific ,ndrrctiuc logic sirn¡tl¡, rlocs rtot u¡¡dcrstand thc wolds hc is

usirrg: Srrp¡>osc Jo¡lcontr ,rk.,l yo' *úy h" should lrclieve that thc fathcr of

ó.,,g1,* iü,,tr wrls rlrul.. You wuultl pro¡rr¡ly be ut n loss to know \vlltlt wits

trorüli,,g hinl. Therc is no tlucstion o[ g;'ilrcring cvidcnce for thc asscrtion

it',,,t t¡,,?otn,,r of Gcnglris K'hrn,un, ,r.1". Th.r" is no r¡ucstio, of advanc-

i,rl-,,rgr,,.lt.',rts in its favor' I]cirrg nrltlc is sirnply part of rvhat wc tnclrt by

being a father. sorneor.¡e who giants that thcr.r, rvas such a person as the

fathlr of Gc,¡ihis Kha¡r a¡¡¿ ycipr.tcsts thtt lrc is trot surc tli.t that pcrson

r¡,as malc sinrply clocs not unclcrstaud thc rvords hc is usi»g'

Wchavcaparallclcascrviththctrac]itionalproblemofinduction,forparto[whatwcmcalibybt:ingrrtúíorrolisacceptingscicntificinductiveiogic. Supposc that a p",.on do"' not form his expcctations of thc future

,o-ughly i,iaccorclanc" with scicntific inductive logic' but ¡ather i¡r accord-

nn"I *i,t, countcrinductive logic. Would we ¡1ot, on this basis, iudge hinr

to be irrational? Supposc that another pcrson based all his maior decisions

o¡-r visions of the future hc has whcn aslcep' F-urthermorc' he has ahvays

br:cn wrong. When wc point out this fact to him' he rcplies thirt hc docs

Irot carc I-,."",,. lre lras just l¡ir<,I a visiorr tlrut assurcd lrirll tllat alI lris

II. 5 ru¡¡rr¡oNAl PRoBLEM oF INDUCTIoN 49

future visio¡rs will be accurate. would we not, on tlús Lrasis, judge hinl to

l-¡e irrational? Exampies could be multiplicd to show that the u§e of scien-

tific i¡rcluctive logic is a standard of lationality, that part of what we mean

by being rationai is accepting scientific inductive logic. Thus the question"Why is it rational to acccPt scie¡rtific i¡rductivc logic?" is as silly as the

.lu"riior, "Why was Genghis KL¡an's father malc?" Someonc who doubts

ü¿t it is rati,onal to accept scientific inductive logic sinrply does not

understand the words he ís using.

3, It is a silly <lucstion becausc it asks for a iustilication beyorrd the limits

whcrc justi6cation makcs scnsc: Supposc somc unrcdccrrlablc skcptic wcrc

to ask why it is ratio¡ral to acccpt any argumcnt at all. o¡re could not

aclv¡ncc any ¿fgunrent to convince hirn witliout begging the question, for'

lrc lras callcd in[o doubt thc acccptability of all argttmcuts. Clcarly thcrc is

rro possibili§ of rational cliscussion rvith such an individual, for l¡e has

c¡llcd into cloubt all the machincry of rational discussion. He has asked for

justification beyon<] thc limits where justification r¡akcs scuse. For jtrstifica-

iir¡rr to ¡n¡kc sc¡tsc tl¡crc ¡rust bc.so¡nc machincry lr:ft for that justification,

a¡rtl scic,ntilic incluctivt¡ logic is ¡rl cssclttial part of thc lllachi¡rery for ra-

tionrl cliscr¡ssio¡r. To call it into rlucstion is also to ask for a iustificationb.:voncl thc Ii¡»its rvhcrc rati<l¡ral justification nrrkes sense For if induc-

tivcly stroDg argu¡ncnts are callcd into clucstiorl, then thc orlly nlachincry

lcft for r¡tional rliscussion consists in dccluctivcly valid argunrcnts, ¡nd, as

IIu¡¡rc Iras show¡r, thcsc arc palticulally unsuitable for tlic task of rationally

justifying iucluction. 'l'hc rcqrrt,st fol ratiotrll justiflcrttiotr tlocs not niakc

s"¡,r" rvl,, rr it is dircctccl at tlrc: nrachirlcry of rational justi6cation it.sclf.

'fhr¡s thc r¡rrcstion "Why is it rirtional to acc('pt scit:¡rtific intluctivc logic?" is

:rs silly lsihc <1r.rcsti<.ln "Wliy is it lrrtiolill to xcccl)t illly lt¡'gutlrclrt at ¿¡ll?"

Thc foregoing contcntions arg I sant¡llc ol ti¡c considcrations advanced

by thosc nuñ-o *;rh tb dissolvc tlrc tr.rditional problcnr of i¡rductiou. If they

,,r" "orr"ct

in statirig. that no argunrcnt whatsot:vcr is necessary to justify

scic¡¡tific inductivc iogic, then Hu¡nc's dilenrma is a paper tiger and thc

prol)onents of the inductive ancl pragniatic iustifications of induction have

l."n tittirg at windruills. Thc considcratio¡rs so far irdvanccd in favor of thc

dissolution of the traditional probleln of inductio¡r do have a certairr

tmou¡rt of rvcight. But it is doubtful that they are sufncient to pull the teeth

out of Hurlc's problcnr. Lct us cxaminc thc thrcc main obicctions to thc

qucstiotr "Why is it ration¿rl to acccpt scientific inductive logic?"

L It is a silly question because it u;ks us to turn indr¡ctio¡r into dcdrrctio¡r:

This is the nrosf popular and the nrost unsophisticated obiection to thrr

50 II. rrtE 1'ltADI'I'roNAL prtout-trM orr INDUCTtoN

Ir'aditional llroblcrn of inductiorr; it sirrr¡rl¡'rttisr-cplcst'rtts tlrc problurr.Suggcstion I for rational justification did rrot clcntand that e-argumcrttsadjudged inductively strong by scientific inductivc logic shoulcl aluays givcus true conclusions. What it rctluilccl f<.¡r ratio¡ral justi[ication is that it bc

sl¡orvn that thcsc argunrcnts givc true conclusiol)s ¡nost of thc til¡rc. Tlrtrs

tlrr¡sc rvho ask for a rittionirl justification irt tllc settsc of Suggcstion I a¡c ltotasking us to turn induction into deduction and they arc not asking that thclink bctwceli prcmiscs arld cotlclusion irl irlductivcly strong argutnctlts btr

t¡rrc of ct:rtai¡rty ratlrcr tlrau high probll;ility.1'lrt:y lcilizc that ir¡tluctivclysuollg argumcnts cal), ott occasiotr, lcad us fl'orrl truc prenrisc's to l¡ fllst'conclusion, a¡rd that tliis is sinr¡rly tltc ¡lalure of thc L¡east. What tlrcy rvantis son.re sort of guarantr:c that rvith rcspcct to our c-argurncl'rts, tllis u'illprovc thc exccptior.t rathcr than thc rulc.'I'hus tlrc problcnr ol jtrstificatiorrariscs not sinrpl¡,bccausc it is logically possiblc for lrt incluctively strongargunicut to lcad us fl'o¡Il truc ¡)rcntiscs to li falsc conclusioll sottlc of tlrctirrrc', lrut l.¡ecausc it is logicalll'¡rt-rssiblc [ol irrclLrctjvt'11'strorrg ('-arl]trrrr( r1ts

to lcacl rrs to falsc conclusions ntr¡st ol tlrc tirtrc..'\ltlroutlr tlrir ulrjlctiorr is r'.rsilv tli:¡losctl r,[, rt tlot's lrir\'('il i)oil)[. Sor]l('

¡rl,rl,rso¡rlrcrs l¡.rt,'1,,',¡, \o.r\\'((l lrv ti¡t'tr,rtiitrorial prolrlt'ttt t¡[ i¡rt]ttttlc¡tltlr.rt tlrt'r' l¡.¡r r' .,,r¡r, t() tlr( r()r( lrirru¡t tl¡.tt rl tll,'r't' is :trt1' strL rtgtlr lt ell ttlll¡rr:t' .r¡llunr( r¡t\ rrt' tl¡t¡r[r .,t, t¡,tltt, ltrlll st.rotl(, tllt'tl tlio:t' irl{t¡¡¡]( l)t\rrrrirt rt'.rllr lrt tlttlttr'(rr,'lr r'.tlltl .illltlr)l( trt5 itr tlisgttt:t.r" 'l'lrt' olrjt'ttioll*'r'l¡itve lrct'¡t cotrsttlertttg il; a tt'1tl¡'trr tlrosc plrilosopllt'rs, altlrorrglr rt

sllt¡ul<l ¡rot lrt'rlir<'r'tctl lt tl¡r: tri¡clitiorrt) pt,rblt'trt of inrlr¡ctiorr itst'l[.Orrc cr.rn still frcl tlrc forcc of tlic trarlitioriul ¡rroblcrti of irltluctiorl cvt:rr iI

onc ¡caliz.c.s tlrlt clt.ductivt,valirlity is ¡¡ot tlrt: otrly,statttlltttl:rgainst rvlrich

thc strcngth of argurnerrts c¡¡l bc nrt'rrsrrrt'cl, unrl thut iniluctivcly strong

argurncnts grvc us high probability ratlrcr thun ct:rtrrinty. Iior thc traditit¡rr¡lprol,llcrn c¡f i¡lrluction is gr.rrcrlrtr:d lry rt rt'r1rrt'st Iol Lrsttrltttct'tli:rt i¡rtltlctivt'probability is ¿r uscful tool for its ultinratc plrrposG: prcdictiorr 0f tl¡c futurc.The dcsirc is ¡tot to charrge iDductivc probability irrto dccluctivc ccrtitil§;it is to jLrstify thc usc of inductive probability as a guidc to ratiorurl dt'cision

nraking.2. lt is a silly <lucstiot¡ I¡cclusc so¡ttcol)(' rvlro clc¡u[¡ts that it is ration¡l to

accept scientific indr¡ctive logic sirnply dr¡es r¡ot undcrstand tlic rvords he is

using: 'I'hosc rvho advancc this objcction ag:riDst thc tradition¡l problenr of

induction rvoulcl contcntl tlrat part o[ rvllat \\/c lllcAn b1'¡¡t'o¡1'l decision

rrraking is the usc o[ scit,rrtific inducti',,r'logic rincl thirt tlrrt is thc cnd of tlrt'

I.'l'l¡.rt ls, tiltlrrt trvt l¡' rrlirl ltt ¡r,rrr, r,tr rr ¡llr .t rrrrrsrrtI prt o¡¡sr

II.5 rnlo¡rloNAl pRoBLEM o!- TNDUCTIoN

(lucstion. l¡r a scnse they are right, rlthough this is uot thc cnd of tht: ques-

tion. They are correct in claiming that if a person consistently based his

expectations of the future on methods that conflict with scientiEc inductivelogic, then he would be judged, on that l¡asis, as acting irrationally. So itsecms that accept¿rncc of scienti6c induction is part of what we mean byra fionality.

But, to pwsue this exarnplc a iittle further, suppose we came upon a

wholc cultu¡e that based its cxpcctations of the future on methods that con-flicted with scicntific inductio¡r. Lct us call thc nrcmbcrs of this culture"Omegas." Whcncvc¡ the Omcgas have a particularly important decision tomake, they base their predictions of relevant future happenings on thepronouncenre¡¡ts of a witch doctor rather than on scientifrc induction. Un-fortunatcly for the Ornegas the witch doctors are usually wrorlg, althoughtlrcy havc occasionally nrade succcssful prcdictions; trevel'theless the

Orncgas continue to place faith in their witch di.¡ctors. Now we wouldinclccd ;udgc tlrc Orncgas' reliancc o¡¡ thc witch cloctor fo be irrational.

Suppose, having lcarncd thcir languagc and wishing to speed the "pro-

grcss oI civilizltion," rve tlccicle to cotrvcrt thc Orncgas to scicnti6c induc-

tir¡n. \!'r'poi¡¡t out th¿rt thc witch doctr¡rs havc not bcen vcry successful illtlrc p.rst and tlrcy rcply that they have indcecl had a very long period ofl.¡¡d luck but that thcy rrc surc tltat witch doctors will l-re successful irltlrc fulurc. Su¡rposc tht'rr wc rsk thc¡n to justify this faith. Thcy reply thatthis is a silly question, that relyin¡; on the \vitch doctor is part of what thcy

rncan by lrcing rational; if wc doubt that it is rational, wc must simplync¡t unrlerstand tlre words wc arc using.

Wh¡t arc wc now to say to thcnr? Pcrhaps we rlil nol understand the

words we used, since the discussion took place in their language, not ours.

Perhap, their conccption of ratio¡rality is diflcrt'ut from ours, and theiru,ord wl¡ich we thotrght was a¡r cxact translatior¡ of thc English word"rutional" is not. Suppose their word is "brational."'fhcn part of the mean-

ing of "brational" is to rely on thc witch doctor, and part of the meaningof "¡atio¡ral" is to accept scientific inductive logic. They proudly call them-sclves brational and we proudly call ourselves rational. Oncc this is under-stood, the Ornegas will even agrec that thcy arc irrational, but will maintai¡¡

that what is inrportant is that they are brational and we are unbrational. Inthis situation, what sort of co¡rsidcrations could rve advance to convinccthem that they should accept scientific induction? In other words, hoúcould we conoince them that rationalitA is superior to bratiorwlity?

We could convince them if we could show them that scientific inductionis better suited to the task of predicting the future than reliance on the witch

5I

52 II. r¡rr TnADITIoNAL pnog¡.Elvf oF INDUC'rloN

doctor, for tlr«:y too ¿ll'c ir¡tt:¡'t,stcd in corrcctly prc<licting thc futurc. Wccor¡ld ccrtainly convirrcc tlicn¡ iI rvc could sl¡orv that scic¡rtific i¡rductio¡lrvould br,r'ight nrorc oftclt tha¡l the rvitch doctor. Tllus wc arc blck to thctraditior¡al irroblcrl o[ i¡¡cluction.

Thc point is that if wc sirn¡;ly .sly that thc acccptancc of scicntific induc-tivc logit'is pllt ol Ilrr,rrrt':rrrirrg oI r'ltiorurlity unr'l rt.It¡st'to 1;rrr.strt'itsjustification utty f ultlrcr, Ilrcrr rvc arc lroldiug scit:ntific i¡rduction as a rlogrnlbuilt into our larrguirgc,.jtrst as thc Or»cgas arc lrolcling ri'liancc on thcrvitch rloctor as u rLrgrna built irrto tht'irs. Thr¡s t'vcn i[ rvc irgrc(' that plrt oftltc rtrcarting o[ ratiorrality'is acccptirrg scicntific inductivc logic, thc tladi-ticnal problt'nl of ilrrluctio¡) rcapp('ar.s to hautrt us as thc r¡rrcstiorr "Wlry is

r;rtiorrality.str¡¡t,rior Io lrrltiorral ity?"3. It is a sill¡,<1rrc'stion lrcr:¿rrrsc it asks ft¡r a justificatiorr bcyorr«1 tlrr:

Ii¡nits wlrcrt' jtr.stificatiorr ¡rrlkcs sc¡.¡se : TIrosc rvho rvisll to rais«r tlris trlrit'c-tir.rn agairrst tlrc traclitionul problcnr of inductio¡¡ lnvc un ansrvcr rt'a<lyconct'rnirrg our diflicrrltics rvith tlrc Orncgas. Thc Onr«rgas, tlicy will con-tt'rrtl, rtrt'irr ir ¡lositiorr ¡rrrrirllcl tt> tlrr: Iry¡lr>tlrctitrrl ¡rt:rsorr rvl¡r¡ w'oulrl ¡rot

ncc('l)t itt)\'itrgrrtrrt'rrt rvluttsot'vt't. -flrc¡'r'is rt,lrlll,lrt pt>ssilrrlity'r¡[ [rrritftr]rlr , rrssirl¡r ivillr ()rrrt'qrrs r>r¡ tl¡is ¡rr.rttlr si¡¡r'r' tl¡r'r, rclu.sr, to lt'r,t'¡tt tlrt,rrr,rclrirrt'r'¡'¡)('(('.\s.lrv lor Irr¡itfr¡l rliscr¡ssio¡¡.'l-lrt'rc is r)o w¡),tr.¡ co¡tvi¡rcc//r¿ lr tl¡;rt r;rtio¡rirlilv is srr¡rcrior lo lrrrtiorrrrlit¡'. br¡t tllt, rrltir¡l:rti' :rns\v(rIri; t¿.r ¡rrt¡st sirrr¡rlv lrt: tlil¡t ¡t is rutio¡r:rl to l.¡t'rlrtio¡¡irl lrrcl irrutior,,rl to l¡r:

lrrirtiorrirl, -l'lrir

rLrr';, tlot in('il¡r tlrrrt r¡rrr flr'c('ptílnc(' ol igicntific i¡rrlrrctiyr:iogic is t dogrrttr, lor rvl¡at w(, ur('a¡r Iry dogrna is lrn irntti<;¡rally ll<:ltl

I¡rlit'f. ff r¡¡¡c'rt'Ir¡st,s tr.¡ tt:st ]ris l;clir,fs lry scic¡ltific i¡¡rlt¡ctivc logic, tltcsc

lrclit'ls ln,srrirl to l¡t: rlognrirticirlly lrt,ld. Il «r¡rr: ¡s rvillirrg tr¡ tcst lris l.lclit'[.s

l;y scit:rrtific i¡r<luctivc lo¡iic, tlrt'n Irc is ¡rot dogrnatic.'l'hus tho rrccr:¡rt:rnccr

<¡[ scicnti0c i¡rdt¡ctivc logic is rr<lt u clogrrra; it is onc oI tl¡t'ulti¡rratc ¡rlirr-cip'les of rcu.son.

Noticc tllirt so¡r'¡cor)c rvlto argrrcs in this way is rcrrlly srrggcstirig anoth()rtype oI ratio¡ral jrr.stificltiorr; l¡c is cllirning tl)¿t lris typc of ration¿rljtrstificatiorr is tlrt: onl¡'t¡'pt: possil>lc an<l tlrr: only ty1>«.that ¡nakcs scnsc.

This typc <;I ratio¡¡¿rl ju.stificatiorr is r'¡rbr¡diccl in thc following srrggcstion:

Ilatiunal Justilication

.Srrg;1r'.rlion /\'. ,{ sl'str'¡¡r of in«lr¡r'tivc logic is rltiorrtlly jtrsti[it:d ilrl t.rtr lrt :lrt¡rr¡r 1,, l¡t .rrr , r¡,l,,rrlr¡rrt rrt r¡l tl¡ost' i¡¡tlrr, tivc rr¡lt's of\( r't¡(i .lrtl,,,lr,t¡r,,¡, sr'¡rsl llr.rl rrt'l.rll lo l¡t':t stlu¡tl.rrtl of r;ttio¡l-.,i¡lr

II. 5 rna»rr¡oNAl pnoBLEM oF TNDUCTToN

Since what we mean by scientific inductive logic is a system that accordswell with common sense and scientiGc practice, scientific inductive Iogicis automatically justiGed in the sense of Suggestion IV.

But let us remember that it has nowhere been demonstrated that rationaljustification in the sense of Suggestion IV is the best that we can do.Humc rnay havc show¡r th¿rt rational justification in thc sense of Sugges-tion I is inrpossible, but there may be a sense of rational justification thatis weaker than Suggestion I but stronger than Suggestion IV in whichrational justification is possible. After all, wl:en we attempt to justifyscicntiEc inductive logic without bcgging the question, we are not dtl-prived of all machinery for rational discussion (as we would be if weattempted to answer the question "Why should one accept any argumentat all?"). Thc discussion of the atternpts at inductivc and prag¡natic justi-ficatio¡rs of induction showed that we could still do quite a bit withor¡tbcgging the question.

Suppose that we could somehow show that scie¡rtific inductive togic isbcttcr suited for accorn¡;lislring thc purposcs <-¡f inductive logic than ar:yother system. Wc would then havc a stronger justification than Sugges-tion IV, and rathcr than rnercly condernning the Omegas as irrational wecould give them rcasons why our way of thinking is superior to theirs.lVc would not have to be satis6cd sinrply with commitment to scientificinductive logic as an integral part of the nrachinery of reason; we couldoxplilin wlry it is iind ought to bC an integral part 0f that machinery, Thusa ratio¡ral justification that is stronger than that proposed by Suggestion IVwill nrake sense if wé can discovcr one.

Wc can sumntarizc the main conclusions of this sectio¡r as follows:l. Those who rvish to dissolve the tradition¿l problem of induction

claim that scic¡rtific inductivc logic is rafionallv justificd sirnply becauseacceptarce of scientific inductivc logic is part oi what we mean by beingrational, and that it does not make sense to look for a different kind ofrational justifi cation.

2. There is a sense of rational justification in which they are correct ir.i

saying that scientífic inductivc logic is justified bccause it is an essentialpart of the macl¡inery of rational thought.

3. But they are incorrect in saying that it does not make sense to lookfor a stronger type of rational justification because:

a. A stronger type of justification would be valuable if we could findone,

b. No one has demonstrated that a stronger type of justification rs

impossible.

53

Í¡.1 II.'l'ltu'l rr^t,I'l roN^L I'rtoulu¡f oF tNDUCTIoN

llClrr¡t'rvt'lcavc tl¡c attcrtt¡rtcd dissr¡luti<¡n of thc traditional problcrn of

rrrrlrrttiorr, onc final point slroultl bc ¡lotcd. Evcn if wc arc satisficd rvith

Srrggr:stiorr lV, a proltlcrli rc¡naill.s. OIrcc rvc havc a systet¡l ,ri scicntific

i¡rrlrrctivc logic wc ca¡r say thtt, in tcrrns of Suggcstion IV, it is auto'

rrraticalll, jrrsti[icrl. l]ut corrstn¡ctirrg u systt:nr r¡[ scicutific incluctiv<: Iogic

i.s lrn ir¡u¡¡(.¡)st,ly cli{ficult tlrsk ir¡ itscl[. \Vc shul] scc in tllc nt,rt cllrlttcrthat the job of nraking cxplicit atrd systernizing "thosc inductive nrlc:s of

scicncc and cotnt¡rotr sc¡tsc tl¡¡t wc takc to be a sta¡lcllrd of rationality" is

a corrrltlcx a¡¡d dclicatc r:ntcr¡rrisc. 1ir jrrstily a systcrrr oI irrductivr: lt.rgic

i¡i tcr¡ns of Srrl'r1t:sti«-rn IV rvc rntr.st cotnplcttl tlris crrtcrprist:, [or wc ttlttst

sllow tlrat it is a sy.stcn¡ r.¡f sciclrtific irrductivc logic'. Tlrtrs cvctt iI tltc

t¡aditi<¡nal ¡rroblcnr of induction is di.ssolvcd, the new riddlc of induction,rvl¡icll rvc rvill c¡rcountcr in tlrr: trcxt chitptt:r, rvill..rcrnain.

Suggcstcd rcadings .

l't'tcr S(¡rtrt'sott, I¡ttrttlttclitttt lo l-ogitul 7'lllory (Nt'rr Yorl: Jolrll \Vilt'r'

{ir So¡rs, Irrc., l1)5J), ¡r¡r. !'15-(ii),

l'.rtrl l',rirt',rltl:, "llcrt¡.rtrrl llr¡:sl il t l)or¡lrlr ltirot¡t lt¡tlt¡ttio¡r," ¡l) /.()A¡a'

,ttrl I.uttyrttgr',.'\trlltorrr l"lt'ti, til 1(i.rrilt'ri (,ltr, N.\'; i\t¡t'llor lJooks,

111(i,i), 1r1r 5i.t-5i

1I.6. SUNII\{AltY. \\/t: I¡rtvc tlt'r'cltipcd the traditiorral problc¡I) of

i¡r<ltrctio¡r,rnil rliscr,.st.<l sr.vclrrl ults\\/('rs to it. \\/c fotrtrd tllat cltclr positir>ll

rvc cli.sr.L¡sst'cl l¡:rtl rr clilicrcr)t s('t of stur¡tlurds lor mtiorlitl justilication of

.t systt ttt ol rrttlrrttivr' logi,.

l. [r,osiliott:1'lrr,or.igin:rl lllt nt;rliorr rrl t,.'t¡:rrlitior¡ul ltroblern oI irrcluc-

t ,¡tt. Slttntlurtl. lor lktlittrt,tl luslifitrttiL.rrt: A systt:ttl oI ilttltrctivt:Iogir: is rationlrlly ju.sti[ir'<l if lrrd orrly if it is slrorvn tlrat tlrt:

c-ilrgur)rcr)ts tlrat it lrtljtrtlgr:s intlttc'tivt:lY strorrg yiclcl trut: col¡clu-

siorrs ¡¡rr.¡st ol tlri' tirrlc.

ll. l' ositiott :'i'lrt: ir¡cltrctivc iustificrrtior¡ oI irrductiorr.

Strt¡ttktrd t'or llutittruil lrrstilicul;orr: A systcrtr of irlcluctivc logic isrlrtiorrally ju.stilicd il lor cVc'ry lt'vcl (A) ol rulc-s of tllat systcttt tlttrei.s an c-rtrguttr(:lrt o¡l tllc Ircrt lriglrtst lcvcl (A l-1) rvlrirh:

i. is l<ljutlgt'rl irrtlrrctrvt'ly strorrg lry its r>rv¡t syst('r])'s rtrlcs.

ii, IIlr.s us it.s ct¡tttlt¡.siort tl¡c stritt'¡¡tclit tllirt tlre systelrt's rulc: t.ttt

tlrt'origirr.tl Itrel (A) uill w'irrk lvi:ll I¡cxt tj¡rrtr.

IL 6 suv¡raliv

IIL P¡¡silio¡¡; Thc pragrnatic justification of indtrctio¡r.Standard for l\utionul Justificatiort: A systcn-r of inductive Iogic is

rrtionally justificd if it is sliown that the e-argumcnts tirat it adjudge-s

inductively strong yitld true conclusions ¡nost of thc tinle, iie-nrgunlcnts adjuclgcd irrductivcly strong by any rncthod will.

IV. P¿¡,r'ilir.¡rr: 'l-l¡r: dissolutior) of tlle traditior¡al prol)lcru of ir¡duction'Stu'u!anl f or Rutionul Justiftcatiort: A s¡'stern of inductive logic is

ratiorrally jrrstificcl iI it is show¡r t<¡ l¡c rrrr ctnbodiment of tho.sc

ir¡ductivc rulcs o[ scic¡'¡cc ilt)(l col])¡))on scrlse which wc take to trc

a sta¡rdard of ratiorrality.

'I'lre attcrrrpt itt aIr i¡rcltrctivc jtrstilicati<-r¡r c¡[ scicntific irlductivc lr-rgic

taught us to recognizc different levcls of argurrlents and colrespondinglcvcls of inductive rulc.s. It also shorved that scietrtific inductivc logic

rnects thc standards for llatir-¡rral Justification, Strggcstiorr II. Howcver,

\vc sa\\/ tlrat Suggcstion II is rcally not a scr)sc of rational justification at

¡ll, [or l¡r¡th scit,¡rtific ill<lr¡ctivc logic trrrd cottrttcrillclttctívt: logic clln !ncet

its c«¡rrclitiorrs. 'l'hus it clrrrrot ltrstify thc cltoicc o[ «rlle ovcr thc other.'l'lrt'Jttcrrrpt itt it ¡rritgtttatrc jtrstificütior¡ of sciclrtific inductivc logic

slro*'t'd trs tlltt Strggcstion IlI, propcrly interprcted itl tertns of levels of

Ír¡t.lLrctit-rn, rvotrld lrc ar) acccl)tal)lc setrsc oI I'ational justification, althougliit *or¡ltl l-,r'u u'.'akcr st't¡se thitll tltat proposcd irr Suggcstion I. I:lowever,

tlrt, prrrgruatic justificatiort fuils to tlc¡nonstratc that scientific indt¡ction

nrc0t) tlre co¡rditioils of Srrggcstion lIl,Tlre attcrrrpt at a dissolutiorr of thc traditional probleni of i¡iductiorr

sllorvs that Suggcstion IV is a scnsc ()[ r¡tir.¡¡i¿ii jtrstification, although rr

vcry wcak oDc. Furtlterrtt¡rc, it.sholvs that scicrltlfic indr¡ctive logic nreets

tlrc c<¡¡rclitit¡rrs of Strg.gcstio¡r l\'. llc¡rvcvor, it fails to slrorv that it is sensc-

rcss tt-¡ l,rok lor a stt'otrgt'r jtrstilicatiorr tlrarr Strggcstio¡r lV.'fl¡c rr:uraining problcrtr is tt¡ ct.r¡¡:ttuct a dctailcd systetn of i¡rductive

logic rvhiclr ¡nccts thc conditio¡r, oI Suggcstion IV, that is, which is a

systcrn of .scicntific i¡lductive logic, llrrd to fi¡rd ir stroDgcr justification of

tlris systcrn if one is possible .

55

ilIThe Goodman Paradoxand The New Riddleof Induction

III.l. INTRODUCTION. In Clraptcr II wc prcscnted somc gcn-eral s¡;ccifications for a systern of scientific inductive logic. We said itshoulcl bc a systcrn of rulcs for assigning inductivc probabilitics to argu-n)cnt)-, \vitlr cliffcrc¡¡t lcvcls of nrlcs corrcsponrling to thc cliffcrr:¡tt lcvcls of;trgurtrcrtts. TIris systr:rn ¡»ust accorcl fairly rvcll rvith conrmon sr.¡rsc au<l

scic¡itific practicc. lt ¡¡lust o¡r trach lcvcl plcsupposc, i¡r sonrc scrrsc, that¡rltt¡r'r'is t¡¡¡i[orr¡r arrrl tlrat tlrc It¡turc rvill rcscrnl¡lt: tlrt: past.'l'hr:sc gcrrt:mlspcrcificatious wcrc sufficicr¡t to give us a fourrdation for survcyinq tht'trl-clitiorr¡l ¡.lroblcnr of inductio¡r a¡rrl thc nrljor attcnrpts to solve or clis.solvc it.

Ilorvcvcr, to br: ¡lrlc to appll'scitntific inductive¡ logic, as a rigorousclisci¡rlinc, \r,e n)r¡st krrorv 1>rcciscly rvhat its rt¡les are. Unfortrr¡ratcly noonc llus yct llrodrrct'rl lrrr arlt'rlrnrtc [r.¡¡r¡¡ulutio¡i oI t]¡c rules of scit:¡¡tifici¡¡rlr¡ctivt'logic. Irr [;rtt, irrclr¡ctivc ltlgic is irl rnrrch thc sa¡ne statc as

tl,'rlLrctivc logic rvas [¡clore Aristotlc.'i-his unhirppy statc of aflairs is notdrre to a.scurcity of brrrirrporvcr in thc fluld of ilductivn logic, Somc oftlre grcat r¡rirrds r¡f Iristory lravc attackcd its problcnrs. The distance byrvlrich thcy havc fallc¡r slrort of tlrcir goals is a nrcasure of thc difficultyof tlrc strbjcct. l¡or¡¡¡trlating tlrc rrrlc.s of intluctivc logic, in fact, appcursto bc a ¡¡lorc difficr¡lt crrtcri;rise than doing thc same for dcdrrctive logic.Dcductivc logic is a "ycs or no" aflair; an argument is eithcr deductivelyvalid or it is not. But inductlvc strength is a mattcr of degrce. Thus whiledcductivc logic must classify arguments as valid or not, inductive logicmust meas¿rc thc inclr¡ctive strcngth o[ argurncnts.

Setting up such rules of measurcrnent is not an easy task. It is in factbeset'with so rnar)y problerns that so¡nc plrilosophcrs have bccn convi¡rccdit is impossible. Thcy nraintain that a systcm of scientific induction cannotbe constrt¡ctcd; that prediction of the,future is an art, not a scicnce; andtl¡at rve rnust rcly o¡¡ the intuitiorrs o[ cxperts, rather than on scicnti6cirlductivc logic, to prcdict thc [uturc, We c¿¡n only hope that this gloonryd<¡ctrinc is l¡s nlistakcr¡ irs tllc vicu,of thosc carly Crccks rvlro bclicvedde<lr¡ctivc Lrgic cotrltl ¡rcvcr l¡c rctlrrccd tr.r a prccisc systcrn of nrlcs ¿¡¡¡d

ff

III. 2 rrEcuLAnlrrEs AND plioIECTroN

must forever remain the domain of professional experts on reasoning.If constructing a system of scientiEc inductive logic were totally impos-

sible, we would be left with an intellectual vacuum, which could not befilled by appeal to "experts." For, to decide whether someone is an expertpredictor or a charlatan, we must assess the evidence that his predictionswill be correct. And to assess this evidence, we must appeal to the secondIevcl of scientific inductive logic.

Fortunately there are grounds for hope. Those who have tried to con-struct a system of scientific inductivc logic have made some solid advanccs.Although thc intellcctual jigsaw puzzle has not bccn put together, we atleast know what some of the lticces look like. Later we shall examinesome of these "building blocks" of inductive logic, but first we shall tryto ¡rut tht: problem of con.stn¡cti¡)g a systcm of scientific induction i¡¡per.spcctive by examining o¡re of thc main obstaclcs to this goal.

III.2. REGULARITIES AND PIIOJECTION. At this point younray be puzzled as to why the co¡rstruction of a system of scienti§c in-ductivc logic is so. difficult. Aftcr all, we know that scientific inductionassu¡ncs tli¿t nature is uniform ahd that tlre future will be like the past,so if, for example,, all observed emeralds have been green, the premiseenrbodying this infórhation confcrs high probability on the conclusio¡rthat the ncxt e¡ncrald to bc observcd will bc grecn. We say that scientificinductive logic proiects an glzss¡ps¡J regularity i¡rto the future becauseit assigns high inductive probability to the argument:

AII observed cme¡alds have been green.

The next emerald to be observed will be green,

In contrast, counterinduction would assume that the observed regularconnection between being an cmcrald and being green would not hold inthe future, and thus would assign high inductive probability to theargument:

Ali observed emeralds have been green.

The next emerald to be observed will not be green.

So it seems that scientific induction, in a quite straightforward manner,takes observed patterns or regularities in nature and assumes that theywill hold in the future. Along these same lines, the premise that 99 percent of the observed emeralds have been green would confer a slightlylowcr probability on the conclusion that the next emerald to be observed

57

58 III. rtrr GooDMAN PAnADox

would be green. Why c.In we not simply say, then, that argr.rments of the

forrn

All obsc¡ved X's have been Y's.

The ncxt ob.servcd X will be a Y.

have an inductive probability of I, and that all arguments of thc form

Nincty-nine pcr ccnt of the observed X's have l¡cen I's.

The next observed X will be a Y.

have an inductive probability of 99/100?That is, why can wc uot sinrply construct a systcni of scicntific induc-

tion by giving the following rulc on cacli lcvcl?

Rule S: An argurnctit of tlle forrn

N pr.r cent of thc observed X's havc bce¡t Y's

'l'l¡c tlcxt obscrvctl X will L¡c a )'.

is t<-¡ l¡c assrgrrcd tlrc irrductivc probal;ility N/1fi).

Ilulc S docs prolcct obscrvt:d rcgularitics intr¡ the future. But thcre arc

several rcasons wlty it cattnot coltstitute ll syster¡i <-¡f sCie¡rtific irrductivc

logrc.

Thc nrost obvious inadetluacy of llul, S is that it only applics to argu-

ments of a spccific [orm, and we are intcrested in asscssing tl¡e inductive

strcngth of argunrents of diffcrent forms. Consider argumcnts which, in

addition to .a premise stating the percentagc of obseryed X's that have

been Y's, have another prcmisc stating how Irir,lty X's have bec¡r observed'

Herb the rule does not apply, for the argumcnts are not of the requiredform. For example, Rule S does not tell us how to assign inductive prob-

abilities to the following arguments:

.ITen eme¡alds have been observed.

Ninety per cent of thc observcdemeralds have been green.

IT

One million emeralds have beenobserved.

Nine§ per cent of the observedemeralds have been green.

Tlre next emeraldrvill lrt: grecn.

to be obscrved The ncxt emcrald to be observed

will be green.

III. 2 ¡recr,r-ARITIES AND pItoJECTroN

Obviously scientific inductive logic sltotrld tell us how to a.ssig» inductiveprobabilitics to these argunrents, and i¡r assig¡rirg tlresc probabilities itshould take into account that the premises of Argumerit II bring a much

greater amount of cvidcnce to bcar than the ¡trcmises of Argurnent I.Anothcr type of argumcnt that Rule S dt¡cs Irot tcll us liow to evaluate

is one that includes a prernise stating in what varrcty of circumstances the

rcgularity has been found to hold. That is, Ilulc S does ¡rot tell trs how to

assign inductive probabilitics to the following arguments:

59

UIEvery person who has takcn drugX has exhil¡ited no advcrsc sidtr

¡eactions.

Drug X has only bccn ldnrinis-tcred to pcrsons bctwccn 20 a¡ld25 y'cars of agc rvh«-r arc in goocl

hcalth.

IVEvcry pclson who Itas takcn drugX has cxhibitcd ¡ro advcrse sidt:

rc¿tctio¡ts.

Drug X l¡as Lrccn irdnlirlistcrcd to

pcrsol)s of all agcs ancl varyingclegrt:cs of Ircaltlr.,

'l-he rrcxt pcrson to take drug X l'lrc ncxt pc¡'sor) to takc drug Xrvill havc ¡ro ¡dvcrse sidc reactions. will h¿rvc ¡lo advcrsc sidc lcactions

Again scicntific inductivc logic should tcll us how tr.r assign irrductivt:

probubilitics to tl¡esc urgunrcllt§, and irr rloing sr¡ it slroultl ti¡kc irrto i¡ccoul)t

the fact tlrut the prcmises of Argunrcnt IV tell trs th¿¡t thc rcgularity has

l¡ecn found to hold in a great v:rricty of circu¡r.¡stur¡ccs, whcrcas tlre prern-

iscs of Argurnent IIi infornr us that tl¡c rcgularitv has becn found to hold

in only a limitcd area.

Therc are rnany other types «-rf arguntelrt that ltLrlc S docs not tcll us llou'to evaluate, including most of the arguments lclvatrced as cxamples iIrChapter I. We can now appreciate rvhy an adcriuate system of rules for

scicntific inductive logic nrust be a fairly cottlplt:x structurc. But thcre is

another shortcoming of Rulc S which has to do with argutrcrrts to which itdoes apply, that is, arguments of the form:

N pcr cent of thc observcd X's have bccr¡ )"s.

The next observed X will be a Y.

The following two argunlents are of that form, so wc can apply Rulc S to

evaluate thenr:

60

vOne hundred pcr cent of thc ol¡-servcd samplcs of pu¡.c watcr havchad a frceziug point of *S2 <le-grees Fahrenhcit.

IIL r¡le GooDMAN rArtlDox

VIO¡rc hundrcd per ccnt of thc rc-corded ccono¡nic dcprcssions hirve«¡ccurrcd at tllc s¡¡llc tirnc as lirr.gt,sunspots.

Thc rrext observcd samplc of purewater will have a freezing point off 39 dcgrces Fahrenhcit.

The ncxt cconomic dcprcssion willoccur at thc sanrc tinte as a lar.gesunspot.

If we apply Rulc s wc fincr that it assigns an incluctivc probabiliry of I toeach of these argunrents. But surcry Arlument v has

" *r"h hrghc'r cicgree

of inductivc strcngth tha, Argurncnt v-il wc focr pcrfcctry ¡u.stificcl in ixo-jccting i¡rto thc futu«r tlrc ol¡.s<,rvc<l rr:gul¡r cr¡n¡rcction bc¡ws1.¡ a ccrtai¡¡typc of chcmical compourrd and its frcczíng ¡>oirrt. Ilut wc fccr that tlrt,obscrvcd rcgular connccti«¡n bctwccn

""o,,J.]" cycrcs and sunspots is acoincidcncc, an accidcntal rcgularity or.spurious corrcration, rvrrici¡ shourclnot bc proj.ctcd i¡rto tl¡c fr¡tt¡r.. wr: sr¡ali s^y th.t tr¡c t¡bscrvc<l rr.grrlaritl,rcportcd irr tlic ¡rrt'rrrist.«rf Argu,r:.t v i.s prol.clil¡l¿, w,l¡ilc tl,,, r"[ul,,r,tyrr,¡>ortc'cl irr tlrt.¡irr.rrrist,of Argurrrt,rrt Vl is ¡¡,¡i. \\,r,¡¡rust rro*,s,rplristicrtr.our (ol¡((.1)t¡o¡¡ of st¡r.rrlrfit,r¡¡rlr¡ctir.t. lo(i¡ 5¡¡ll f r¡rtl¡c.r. Sr,it,lltificll¡ductivcIogrt d0r's ¡lr01r'tt ol»crrlrl r.r{r¡rlir¡t¡(,r i¡¡tr¡ tl¡t,ftrturr,, lrut o,,r1,pr¡¡,r..tibl.rcgullritics. ltdocs l.s.\u¡r¡('tlr.¡t naturt.is u¡rifon¡r a¡ld that thc iuturc rvillrcscn¡[¡lc tirc part, Lrut orrly i¡r ccrt¡jrr rc¡l)c(,(¡, lt docs as¡t¡rnc tllat ol¡st:rvcclpattcr¡ls in naturc will bc repcatcd, but only ccrt¿lin typcs of patterns. ThusR-ule.s is not adequate for scicntific incluctive logic becauseit is incapableof taking i,to .ccou¡rt dillcrt'¡rct's i' projcctibirity of rcgularitics.

Exercises:

I' construct five inductivery str.rrg *rqumer)tr to which Iiurc S croes notapp¡y.

- 2' cive two r¡ew exirm¡>les .f projcctibre rcgur:.'itrc.s

^¡¡<] two .cw exumprcs

of unprojectible rcgrr Iarities.' 3. For each <.¡f tlrc f,ll<-rwing x'grrrnc¡it.s, st.tc whether Rulc s is appric,brc.

If it is applicable, what i,ductive prorrabitity does it assign to the ar.gumcnt?

a. O¡re lrur¡drcd l)cr ccr¡t of tlrc crows c¡bscrvcd h¡vc l¡ecn l;l¡ck.'I lre r)ext crow to be ol_¡served will be black.

b. Onc l¡undred per cer)t of the crows observccl havc been black.

A ¡¡ cruws ¡re l¡llck.

III.3 r¡¡c cooDMAN pAnADox

Every tirne I have lc¡okcd lt a calenci¡rr, the ciate lrirs been befor.eJanuary l, 1984.

The ¡¡ext time I look at a cirle¡¡dar tt," ¿@I 984.

Every time fire has l¡een observed, rt has coutinued to l¡urn accord-ing to the laws of riature u¡ttil extinguished.

All unobserved fires continue to tr.nffiuntil extinguished.

Eighty-five per cent of the time when I have droppecl a piece ofsilverware, company has subseqrrently arrived.

Thc next time I drop a ¡;iccc "f .,;mqucntly arrive.

6l

C-

d.

III.3. THE COODMAN PAIIADOX. If onc trics to construct vari-,,'s cx;r,rI,lcs of .¡rrojcctibrc a*cl urrprojcctibrc rcguraritics, I¡c will soonc()r¡)('to tlrc t.,clr¡si.rr trrat ¡.,rrjcc:tirririty rs rrot si'i¡rry.,,y.,, o, nu,,,ifl"i,l¡ut .itl¡c¡ :r ¡¡r.ttcr,I rregrr:c, soi,c rcgularitics lirc higl,ly ¡xo¡""tibre, soulcIr¡r'e.a r,iddl,g clcgrcc of prolcctibiriiy, a,d so,re irre quite unprojectibrc.Just Jrrrv. urr¡rro¡cctibrc a rcgurarity .o,i [r. h* bcc¡l crc.lor¡s,roaJ rrf

'ñ.r-so¡ r Ct¡ocl ¡¡r;r r r i ¡r I ¡ is f lrnlcr u.s,.gr.uc_ltlccn', paradox,

Coodnta. invit.s us to cor¡sidcr n n"* *lo, rvord, ,.grue.,, It is to have thegcrrcral logical fcrrturcs o[ ou¡..old cslol rvt¡rds suclr ui.gr0c¡1,,,,,blua,,iand"rcd." TI¡at is, wr: can speak of things being ¿i cc¡tain color at a certain time-for examplc, "John,s face is rcd ,**,,_r-na wc can speak of things eithcrt'cmai,ing thc sanre color or cha.ging colors. Thc ,.,"w

"olo, *ord Yg.u";; i,dcfi¡lcd ü¡ tcrms of thc fa¡,iria.

"olur-*o.crr "grcer" ancr..brue,,as folro,,vs:

Definition 6: A certain thing X is said to bc gruc at a certain time ú ifancl only if :

X is grccn at t arul f ís bcfore tlie ycar 2000

X is bluc at t arul f is during or af ter the ycar 2000.

Let us see how this definition works. If you see a g¡.een grasshopper today,you ca¡r corrcctly ¡¡raintai¡r that you l¡ave scc¡r l gruc grasshopicr toda¡,.Today"is before the ycar 2000, ancl before the y""i200dro,r"tru"g i, frl"just when it is green. But if you or o¡re of you, descendrntr-,il"u';;;;,.,grasshoppe'during or after the year 2000, it would then b" in*r.J"iiu¡naintain that a gruc grasshopper had been seen. During and after th" y"r,

62 III. rs¡ GooDMAN pAnArrox

2000, so¡¡¡ethirrg is gruc just rvlrcn it is I¡luc. f'hus aftcr tlrc yi,ar 2üX), a l.¡lucsky rvould also be a grue sky.

supposc ¡row ü¡t a cliamelco, werc kcpt on a grcen cloth untir tlrebegin,ing of the year 2000 and thcn t¡ansfer¡ed to a blue cloth. In tcrms ofgreen and blue we rvould say that thc chameleon changed color fronr greento bluc. But i, terrns of thc ¡rew color *,orcl "g..,." ,u. woul<l say tÉat itrcmair¡ed the same color': "grue." The othcr side of the coin is that rvhe,:,omething renrains thc same color in terms of the old color words, it rvillch:rnge c'olor irr tcrrx of the new one. suppose rve h.vc a piece of glass thatis green ,ow and that rvill remain green during and after the ycar 2000.'rhen rve rvould have to say that it was a grue before the year 2000 but wasr)ot grue during and after the year 2000. At the beginning of the year 2000it changed color f¡onr grue to some other color. To na¡ne the color that itchanged to rvc introduce the new color word "bleen.','lBleen" is defined intcrrls of "grcc¡r" and "blue" as follorvs:

Dcfi¡rition 7: A ct'rt:rin tlrirrg X is slitl t<-¡ 1,.. 1rl¿1.,r¡ rrt ¡ ccrtain tin¡t,f iI arrcl orrly il:

X is [.¡luc at ú arxl f is l¡cforc thc ycar 2(X)r0

or,\ is gr't,t,rr irt I ¿¿rx/ / rs tlurirrrl or lrItcr tllc t,r:ur 2000.

'llrrr.rlrt:lir¡ctltttycitt 2(XX)sor¡rctl¡i¡rgisgruc.lrrst wlrerrit isgrct'rr¿¡r,l i¡lt.¡¡.irrst rvlrcrr it is l)lu('. l¡, or .¡ltr:r'llrt. \ r,.rr 2(x)0 sr¡r¡rt,tlri¡rg is grrrt,jrrst w,lrcrr it isbltrc alrrl blccrr jtrst wircrr it is grccrr. ln tcnr¡s of'tlrc t¡ld ct¡Lrr rvorcls tlrc ¡ric¡3of gfass rt'rnuins tllt's:r¡¡lc coltrr (gn'r,rr), lrut i¡r ter¡ns r¡[thr: ¡rc,*,color w«r¡.rlstlrc ¡rit:cc ol'girrss clrangcs cokrr (fi.olrr gruc to l>lccrr).

Ir:raginc a tribe of people speaking a language that had "grue,, and"blccn" as basic color w<¡rds rathcr tha¡r thc morc familiar oncs that we usc.Suppose we clcscribc a situation in our larrguage-for cxample. the piece ofglass bcing grcen beforc thc ycar 200O and renraining grecn afterward-in which wc would say that there is

'o change in colo¡. But if they correctrydcscribe thc same situation in their language, then, in tlrcir terms, there is achangc. This lead.s to the irnport¿rnt and rather startling conclusio. that;vhether a certai¡r situation involves change or not may depend on thedt'scriptivc rnaclrincry of thc languagc used to discuss that situatio¡r.

Onc nright object that "grue" and "blecn" are not acceptable color wordsl¡ccausc they iravc rcf.rc¡rcc to a spccific date in their de6nitions. It is quitetrrrt'tl¡rrt irrour Ltngut¿ic, ¡¡r rvl¡icll lrluc u¡ld grc(,n ilrc thc basic color words,r'.r r¡r .urrl lrl, r'rr rrrrrrt l,r'rl, lirrrl rrot orl¡, ir¡ rt'r¡rrs oI l-,luc atrd grt., rr br¡t alsorrr t.r¡,r r,{ tl¡r,rl.rtr,"l(X[) r r¡ " l]ut .r r¡rr.r].r.r uf tht,gruc.blccn languagc

III. 3 rlrn GooDMAN pAnADox ¡

could rrlai¡itairl tl¡at dcfinitit¡tls of «¡u¡ colc¡r'rvorcls ir lris larrgtrugc rnust alsohavc

'cfercnce to a specific datc. In thc gruc-l_rlcc,, languag", .,gruc',

anrl"blecn" arc l-¡asic, and "l¡luc" und "grccn" a¡c dc.[incd,,s; fbllu,ur,

-

Definition 8: A ccrtain tlrirrg x is said to bc grccn at a ccrtain ti¡rc ú

if and only if ;

X isrgruc at t ancl ú is bcforc tlrc ycar 2000

X is l¡lccn ¡t t u¡uI I is dur.ing or lftt,r tlre yt,.rr.0000.

Dofinition 9: A ccrtairr tlring X is said to bc bluc at li ccrtain tir¡rc rif and orrly if :

X is l¡lec¡r ¡t t ail f is bcfor.c tlrc ycrrr 2000or

X is gruc ¡t t and I is cluring or n[tcr tlrc ycaL 2000.

Dc[irri,g tlre old colol rvortl.s irr t.L¡rs of tl¡c rrc,r,r'r'rlrrir.cs ¡cIcrc,cc to as¡lt'cifit tllttt'tts ttlt¡cll lts dtlirrirrg Ilrc rrt'rv rvo¡'tls ir¡ [t.rrrrs r-¡l tl¡t,9lrl. S9 tl¡t,forrrral stltrcturc of tlrcir clcfi¡ritiuns givcs uo Lt,usorr [r¡ bclicvc thut "gr.ue"ru¡ttl "blc'c:rr" lir('¡lot lcgitiruatc, rrltlrorrglr .¡rlurlilirrr, cr¡kl¡ r'or.ds.

l.ct tts st't'rt'lrlt crtr¡ bc l<':tr¡lccl rrborrl lt'(ullrritit,s lr¡¡cl 1lr'ojt'ctibility frorntlrt'sc rrt'*'colul rvords. wc lravc allclrdy slrt¡r'n tl¡at rvlrctllcr tlrcrc isclrliltgc rrr;t givctt situltiort nriy clt'pt'n<l o¡t n,lrut lirrgtrr.stic tr¡rclrirr¡r.y isusc(l to dcsclibc tlrat situutio¡r, Wt, slr,rll llo\y iillo\v tl¡lt lr,lrlt r,:¡¡ulariticsrvc find in ii givcn srtuatic¡n also nray dupcnd on otrl tlcscriptivc r»aclrincry.Su¡rposc tlrat at o¡rt: nlir¡utt: to nridrright on Dccr.rubcr Sl, l9gg, a gcmcxpcrt is lskcd to predict rvhat thc color of u ccltai¡r cmcr¡ld rvill be aftcrnridnight. fIe k.orvs thrrt rll obscr"'ecl c,icr¡lcls I¡^r,. L¡.c¡-r grcen. Hcprojccts this rcgulirritl, into tllc frrturc and prcdic,ts thnt rlrc crncrald willrc¡lrai¡.r grccrr. Noticr: that this is,¡r irccor(lnncc rvrtir llulc s, rvhich assignsan inductivc probability of I to thc argu¡ncu[:

O¡rc hundled pcr ccnt of thc ti¡ucs that cnrcr.rlds lravc bccn obscrvcdtlrcy hirve bccn grccn.

The next time that an entcnild is obsclvcd it lvill bc green.

Ilut if the gc¡l cxpcrt were ¿ speaker of thc gruc-bleen languagc, lrcwould find a diffcrcnt'cgrrlarity i¡r thc color of oL¡scrvcd cr¡rcr.irlds. Hcu'ould uoticc that cvcly tirrc arr onclald Irad bccn obscrvcd it ha«l bcc¡¡gluc. (llcnrcrnbcr that l-¡t'[olc tlrc ycar'2000 cvcrytlrirrg tlrlt is glccn is als<-r

gruc.) Norv if hc follorved Rule S Irc rvould projcct t/ris rcgularity into the

63

M III. rgs cooDMAN PArtADox

futu¡e, for Rule S also assigns an inductive probability of I to the arguntent:

One hundreC per cent of the times emeralds have becn observedthey have becn "gruc."

The next time an emerald is observed it will be "gruc."

And if hc projected thc regularity that all observed e¡neralds h'ave beengrue into the future hc would predict that the emerald will remain gruc.But during thc year 2000 a thing is "grue" only if it is l¡luc. So by projcctingthis regularity hc is in effect predicting that the emerald will change fromgreen to bluc,

Now, we will all agrec that this is a ridiculous prediction to nrakc on tlrebasis of thc evidence. A¡rd no one is really clainring that it should bc made.But it can¡rot [¡c dcnicd that this prediction rcsults fro¡n tlre projcction intothe futurt'of an obscrvcd rcgularity in accordancc witll Iiulc S. The pointis that thc rcgulari§ of cvcry observed cmcrald having been grue is a

totally unprojcctible regularity. A¡rd the prediction of our hy¡xrthcticalgruc-blccn-spcaking gcrn cxpcrt is an extrcrnc casc of tllc troublc wc gctinto whe¡l wc try to ¡rrojcct, r,i¡ so¡¡rc rukr such as Rulc S, rcgularitics tlratarc i¡r fact urr¡rrojt'r'trl;1r..

'l-l¡c trr¡ul¡lc rvc ¡4t't irrto rs ¡lrtl,:t'tl,-lt,t'p, for tlrc prcrliction so ¡rrivcd atrr'¡ll corrllrct rv¡tlt tl¡c ¡trcdiction;rrrivt,d at by pro¡ectirrg a llro¡cctibleregularity. If we projcct thc projcctible rcgularity that every timc ancnrerald l¡as bccrr oL¡st'rvcd it l¡as bce¡¡ grccrr, thcn wc arrivc at the prc-diction that the emerald will rcrnain grecn. If we projcct the unprojectibleregularity tlrat cvcry tinle an crncrald has l-¡ecn obscrved, it has bccn grue,then we arrive at thc prcdÍction that the cmerald will change front green toblue. Thcse two prcdictions clcarly arc in conflict.,

Thus thc ¡nistakc o[ projcctirrg an urrprojcctiblc rcgularit¡ rriay not onlylead to a riCiculous prcdictir,rr. lt nray, [urthcrmorc, Ic¡d to a predictionthat corrllicts with a legitimatc prcdiction rvhich results frorn projecting a

projectible regulari§ discovered in tlte same set of dala. An acceptablesystem of scie¡rti6c inductive logic must provide some rneans to escape thisconflict. It must incorporatc rules that tell us which rcgularities arc projcc-tible. From the discussion of accidental regularities and the sr¡nspot theoryof economic cycles, rvc alrcady know tl¡at scicnrific inductivc logic musthave rulcs for detcrmining projcctibility. But the Coodman paradox gives

¡ Acttrally tlrcy nrr: inco¡rsistcr¡t only rurdcr tlrc assrrmptiorr tllat tl¡e c¡nerald rvillrrot bc dcstroycd br:[r¡rc 2000 r.r¡., brrt prcsumal>ly wc will Irave indepcnclentrnductrvc cvrtlcncc lor tlris asstrmptr,rn.

[II.4 n¿r;ul.rnITy AND uNr!'onMr-r'y orr NAME

this point r)ew urgcncy by dcnro»strating how unprojcctiblc a rcgularitycan br: and how serious arc tlic conscquenccs of projccting a totally unpro-jectiblc regularity.

Lct us summarizc what is to be lcarncd from thc discussion of "grue"

and "blecn":1, Whcther wc find clrangc or not in u ccrtain situation nray dcpcncl on

thc linguistic nraclrincry \ve usc to describc that situation.2. Wlrat rcgulrritics w,c lind in u scrlucncc of occ urrcnccs r»uy dcpcrrcl on

thc linguistic rlachincry uscd tc¡ dcscribc that scrlucncc.3. Wc may find two rcgularitics in a scqucr:cc of occurrcnces, one

proj<:ctiblc ¿ncl onc unprojcctiblc, suclr that tlrc pr cclictions th¿¡t arisc fronrprojccting thcnr both arc in conflict.

Iixerciscsl

.1. 'l-rir¡rslrtc the folk;rvrrrg tlcscliptions i¡r Icrrns oi "l;ltrc" arrd "grccrr" intocr¡rrivllclrt dcscliptiuns i¡l tc¡¡¡ls uf "grrrc" arrd "l¡lcc¡¡":

r. 'l-c¡r ¡njllio¡¡ ¡,eurs frorn norv tl¡c grass rvill be glcerr ar¡d the sca bluc.Ir. f rr 50 y'c.rrs frorn ¡rorv tl¡c follorving sorrgs rvi)l bc vcry popular: "Crccn

[:iycr" rrrcl "NIy l]luc llc¡vcr¡."t. In tlrc 1950's thcsc so¡)g.s wcre vcly po¡tular'; "SI¡e W<.¡rc Illue Velvet "

"'flrc Wearing r-if thc Cree¡¡," ar¡d "Birth of thc Blues."d. There will be a miracle at the beginning of tlre year 2000: the colt¡r of

the sky will clrangc from bluc to grccn,

2. Dcfine "gnre" in tcrms of "blue," "green," irnd "bleerr" without mentioningtlre ycar 2000.

IIl.4. THE GOODIV{AN PAIiADOX, IUIGULÁRITY, AND TtlEPIUNCIPLE Ol'TIIE UNIFOIINII'I'Y OF NATURE. Wc s¿w, in thc lastscction, that projccting obscrvcd rcgularitics into the future is not as simpleas it first appcars. The rcgularitics found ín a ccrtain sequcnce of eventsmay depend on thc language uscd to dcscribc tlrat sequcnce of events. TheGood¡¡ran paradox shorvcd tlrat if wc try to plojcct all rcgulnrities that canbc fou¡rd by using any languagc, our predictions ¡nay conflict with oncanotl¡er. 'Ihis is a startling rcsult, ¡¡rd it dranratizcs the nccd for rulcs fordctcrnrini»g projcctibility i¡.¡ scicntific i¡rduction. (This rnight I¡c accour-plishcd through thc spccification of tlic rnost fruitful language for scicntificdcscription of cvcnts.)

This ¡rccd is furthcr dlanratizcd by the followrng, cvcn urorc startlirrgrcsult: Iior any prcdiction lvhatsocvcr, we can find a rcgularity rvhose pro-

65

III. rsr cooDMAN PARADox

jection liccnses that predictiolr. Of coursc, rnost of thcse rcgularitics will be

unprojcctible. The point is that we nced rulcs to eliminate those predictionsl¡ased on unprojectible rcgularitic.s. I sha]l illust¡ate this principlc in threcways: (1) in an example that closcly resembles Coodrnan's "gruc-bleen"

paradox, (2) with refercncc to tire extrapolation of curvcs on graphs, (3)

with reference to thc problcrn, often encou¡rtc¡cd on intelligence tests, ofcontinuing a sequence of numbers. The knowlcdgc gaincd f¡om this discus-

sion rvill then be applicd to a reéxamination of the principlc of theuniforrnity of naturc.

Example ISupposc you arc presentcd with four boxes, cach labelcci "Excelsior!" In

the first box you discovcr a green insect; in thc second, a yellow ball of wax;

irr the third, a purple feathcr. You arc now told that thc fourth box contains

a mask and are asked to preclict its color. You must look for a regularity in

tlris scrlucncc o[ discovcric.s, rvhosc projcction rvill licc¡rsc a prcdiction as to

thc color.oI tllc r¡rask. Altlrough on thc [acc of it, this sccrrrs irnpossiblt:, rvitlr

rr littlc irrgcnuity t rt:gtrlurity crrrr lrc fou¡ld. Wlrat is nlorc, for any ¡.)rt'<lictiortyou rvislr to rrrlkc, tlrr.rc is a rcgulrrrity whosc plojcctit¡¡l rvill liccrlsc tl¡at

predrctiorr. Sup¡rosc you \\'rr,IIt to prcdict tlrat tlrc mask w'ill I.¡c rcd. TIre

rcgullrity is found in thc foll<-¡rvi¡rg nralr¡)(:r.

Lt , Lls dcfin.r a llc\v word, "sl)aI[," A stlarf is so¡rrt:tllillg pr('§c¡ltcd to you

in a box lal¡clcd "Excclsior!" urrd is cithcl an inscct, a ball of wax, a

[r'atllcr, or ¿r rllsk. Norv you l¡avc t¡l.¡scrvcd tlrrctr snat[s atld arc ltbout tcr

observc a {ou¡th. This is a stcp tow¡rd rcgularity, I.¡ut thcre is still thc

problenr tlr¿rt thc thrcc obscrvetl snarfs l¡¿rve bceD dillcrent colors. O¡rc

more definition is requircd in orde¡ to find regularity irl apparent chaos.

A tlririg X is said to bc "nrurklt:" ;ust u'hcn:

X is an inscct and X is grcr:n

orX is n ball of rvar arul X is yclloiv

ofX is a f cathcr antl X is lturple

oÍX is st.¡¡rrc othcr typc of thrrrg r¡rrtl X is rcd

Nr¡rv *r'l¡,rvt ft¡tt¡ttl tlr,'rt'gul:rrity: :rll ttbst'rvt'tl s¡tttrfs ltltlt'lrc'trlt tnt¡rklt:.

llrrr.¡rrLi¡r'rltlrr:rt.(ul.r¡rtvlritotllclr¡turt,ilsstlt¡tit)gtllnttlrctlcxtstr.trf ttr

III. 4 nrcur,Anrry AND UNIFoRMTTY or NATURE

bc, obscrvcd will bc nrurklc, wc obtai¡r thc rcquircd prcdiction.'? Thc next

snarf to be observed will bc a mask, and for a mask to be murkle it must be

¡cd. Necdless to say, this rcgularity is quitc unprojcctiblc. But it is inrpot'-

tant to sce that we could discovcl an unprojectiblc rcgularity that, if itwerc projcctcd, would lcad to thc prcdictiorr that thc nrask is rccl. And it is

easy to sec that, if we wantcd to discovcr a legularity that would lead to a

prcdiotion that thc mask rvill Lrc a diffclt:nt color, a fcw altcrations to tltcdc6nition of "nrurklc" would ¿rccomplish thi.s ainr. Tltis.sort of thing carr

always bc r-lonr: and, ls wc shull scc, itr sotnc arcits w(: ncccl ¡rot cvc¡l r<:sot't

to such cxotic rvords as "snarf," "¡nurklo," "gruc," lncl "blccn."

Examplc 2

Whcn Lrasing predictions on statistical data, wc oftcu make usc o[ graphs,

which hclp summarizc thc eviclcncc and guidc us in making our prcdic-tions. To illustrate, supposc a cc¡tai¡'r srnall country t¡kcs a ccnsus cvery l0

Population innrillions

Tinrc in ycars, with 0the year oI the

first census

l0 20 30

Figure I

2 This p¡olectton is in acco¡dance wiü Rulc S, u'hich assigns an inductiveprobabllity of I to the argurncnt:

AU ol¡serucd s¡ra¡fs havc bcc¡r ¡¡rrrrkle.

Thc ¡lext snarf to bo observed will bo ¡nurkle.

67

68 III. rus GooDrvlAN pAnADox

ycars, arcl lras takc, trrrcc so far. r'hc poPulation rvas I rui.lrio, at tlrctime of thc first census, 12 ¡nilrion at tl¡e second census, and 13 nriilion atthc third. This infor¡natio, is represcntcd on a graprr i, Figure r. Eachdot .c¡:rcscnts thc information as to populatio, size gainecr from oncccnsus' Fo. c.r,mplc, tho ¡niddle dot reprcsc.ts tr¡e scco]lri cc,sus, takcni, tlrc ycar I0, ¿¡rd slrorvirrg a popuratio. of l2 ¡nir.ri,¡r. f-lrus it i. pracedat the i,tcrscction of trrc verticai une drarv¡r fronr .he ycar I0 o,1,I tl,"l¡oriz.o¡rtrrl Iir¡<: drr¡rvrr Ir¡¡rr ll¡<r ¡xrpulltiorr of l2 nri]lio¡1.

str¡r¡>osc ¡row you arc u.skctr topicdict tlrc p«rp.r,ti,n ,f tr¡is crurrtry atthc timc of tl¡c fourth ccr¡sus, that is, in thc ycnr 30, you woulcr rravc tolook for a rcgula'ity that courd bc p'ojectccr into trre futurc. Li thc ab-sc¡¡ce of a.y f,rthcr infor¡natio¡r, you wourd pr.ol;abry i;roccccl ", rorio*.,FirsL you would noticc that thc poi,ts rci,'cscnting thc first tru.ee ecnsus:rll fall <¡¡r tlrc st.aight IiLrc labckcl rr i, Irigu*r 2, rlrd w,oultl thcn pr.o¡cctthis

'cgularity irrto thc futurc. Tlris is in ¿Icco¡.dancc rvith Rule s, rvl¡icl¡assigrrs an incl'.rctivc probability of I to tlrc follorving irrgunrcnt:

All poi.ts rcprcsc.trrrg c(,nsus so far takc, havc fallc¡i o¡l linc A'l'lrt'¡roirrt rt'¡rru's.rtirrg tllr,¡rc,rt r.cr¡su.s tt¡ lrt: tirkt,rr *,ill f;rll on li¡1c..1.

Population inmillions

l514

l3121ll0

I

I

I

I

I

I

I

I

20

Irigrrrc 2

IIL 4 nrcu¡,Anrry AND uNrFonMtry oT NATURE 69

This projection would lead you to thc prcdicüon that the population atthe time of the fourth census will be 14 million, as shown bv the dottedIines in Figure 2. The process by which you would ar¡ive at your predic-tion is called extrapobtion. If you had used similar reasoning to estimatethe population during thc year LS at L2l5 million, the process woul<i becallc<l intcrpolation. lnterpolation is cstimating the positíon of a pointthat lies bctueen the points reprcsenting the data. Extrapolation is esti-mating the position of a point that lies outside the points representing tledata. So your prcdictio¡r would lrc obtaincd by cxtrapolation, and yourextrapolation would be a projection of the rcgularity that all the pointsplottcd so far fell on line A.

But it is obvious that thcre arc quite a few other regularities to befound i¡r the data rvhich you did not choose to project. As shown inFigure 3 there is the regularity that all the points plotted so far fall oncurvc B, and the regularity tl¡at all rhr: ¡:oints plotted so far fall on curyeC. The projection of one of these regularities will lead to a differentprediction.

If you cxtrapolate along curvc B, you can predict that the populationin thc year 30 will be back to 11. million. If you extrapolate along curveC, you can prcdict that tlic population rvill leap to 17 million. There are

Population inmillions

t7l61514

t3121l10

20

Figure 3

Time in years Time in years

7ll III. r¡¡r cooDMAN pAnADox

,rrk't'tl ¿u¡ r¡llinite nur¡¡lrcr o[ curvcs that pass through all tlre po!nts andtl¡rrs,¡¡ rr¡li¡ritc number of regularitics ir-r the data. whatever predictionyotr r,vislr to r¡r:rkc, a rcgrrlarity cau l¡r: [ourrd wlrosc Projcctioll will Iicc¡¡sctlrat prcdiction.

Exarnplc 3

often intclligcnce a.d aptitudc tests co»tain prob.lenrs whcrc orc is givena -sc(lucncc o[ ¡lrrrlr[rcr.s nnd asked to cor¡tí¡ruc thc scrlrrcnce; for cxa,D¡rlc,

j.1,2,3,4,5,...;it.2,4.6,8, 10,...;

iii. 1,3,5,7,9,. ..

Thc natulal "vay

in rvllicll to cor.lti¡ruc scquc.cc (i) is to adcl 6 to thc cnci,for se<1uc.cc (ii) to add 12, ard {or sc<1ue.cc (iii) to add 11. l'hescproblcrns arc rcally problcrns of inductive logic on the intuitive levcl; oneis asked to di.scovcr a rcgularity in thc scgrnent of the scrics give. an<i to¡rrojcct that rcgrrlarity in ordcr to find thc ¡.rcxt ¡.ru¡»bcr of thl se¡ics,

L.t us r¡lrk. llris rr':rs,rrirg t:x¡rlicit ft¡r tlrt'tllrcc sr:rics grvcrr. In t,x-irrrr¡rlr'(i) tl¡r'[rrst urt'¡rrl¡,,r,rf tl¡t s,.rrt.s is l, t]lr,sct.r¡¡rd ¡¡¡cnlbcr is g, tlrcllrr¡rl r¡r.r¡rl^'¡ rr i. .rr¡rl rri ¡,,, rrr.r.rl, 1,,¡ .rll ll¡,. ¡rrt..llrllr.rs grvcrr, tlrc Atlr¡r,( r¡¡l)( r rr ( II r,(' l)¡i,l{ ( r tlur r, grrl.rrrt;,. to Iirrti tl¡t. r¡t,xt lr)c¡ll)r.r ol tl¡c5r'r'l(:, \\'( rv¡ll rc.rst.¡¡¡ tlr.rt tllc:rxtl¡ r¡lc¡rrbcl rs 6, rvlrrcll is tl'¡c ¡rlsu,crrrrtLrrtivt l¡ lrrrrvtrl .rt lrtl<.lrr', lrr t'xunrplr. (ir) tlrc first rnc:r¡rlrcr is tivicc l,tllc.scc.¡¡tl is t*'icr'2, arrrl. irr gcrr,-,r'.1, [<¡r rill tht, r¡rcrnLers givc., thc lthr¡tcr¡rl¡t'r is trvicc A. If rvc ¡rroicct this rr.gtrlrrrity, u,c rvj]l rt,ason that tfrt:sixth ¡»crn[.¡cr is trvitc 6, or I2, rvliiclr is tlrc ansrvcr intrritivcly arriveci atbclore. In exaruplc (iii) thc first r'¡rcnrbcr is twice I lcss r, thc scco,dn¡cr¡¡l¡cr i.s tw,icc 2Ic.ss l, ii¡¡rl tllc third n-rtlnl;c,.i.s trvicc 3lcs.s 1. In gcrr-cral, for all thc nlcrlrbers give., tllc kth rr)c¡ltbcl is trvicc Ir lr,ss L It'rvc¡;rojcct tliis regularity, rvc will rcaso¡r thr,t tlre sixtir mcrnbcr.,rf tho sericsis t,'vice 6 less l, or lt, rvhich is the rcsult i,tuitively arrived at. \{/e saythat l- is a gurcrutirt¡i lunctktrt fr.¡r the first scries, 2A a gencrating functionfor t]lc scco.rl sclic's, rtrrd 2k - 1 lr gcrr:ratirrg frrrrction for thc third scries.Althotrgh "ge.crati.g f.rrction" ,ray so.rd lrkc a vcry technical ternr, itsnrearrirrg is quitc sirrrplc. It is a forrr¡ula r'itlr k i¡r it, such tllat if I is sub-stitutcd for A- it givcs thc first r¡renrlrcr r¡f tllc scrics, if 2 is substrtutccl forA it givcs tlic .sccuncl rncrnber, ctc.

'llit¡s tlrt'rt'grrllrrity'*'t'f,rurrrl irr t',rr'li r¡1 tlrr'.sr,si,rits is tllat lr ct,rtuirigt'ttt'rlttirrg lLu¡ctron trlldr,rl l¡ll tlrr.gr\,(,¡r ¡n(r¡rlrcrs oI tlrc scrit,s.'l'l¡jsrtgrrirrrt¡'rra: Irol,ttctl lr¡'.nsrrrrrirrg tlr.it tlrt,sinno gcrr(,¡irtrlrg ILrrrcti()rr

III. 4 nrcur,A-Rrry AND uNIFoRMtry or NATunE 7l

would yield the next member of the series, and so we were able to fill inthe ends of the se¡ies. For example, the prediction that the sixth membe¡of serics (iii) is l1 irnplicitly rcsts on thc following argumcnt:

For every given member of series (iii) the kth member of that

series was 2k - 1.

For the next member of series (iii) the kth nrember will be 2k-1.

But, as you may cxpect, therc i.s a fly in the oir¡tmcnt. If we look moreclosely at thesc examples, wc c.rn find otlrcr rtgularities in the given

members of the various series. And the projection of these other regu-larities conflicts with the projcction of the regularities we have alreadynotcd. The generating function (k-l) (k-2) (fr-3) (k-4) (k-5)*k also

yields the five given members of series (i). (This can be checked bysubstituting I for k, which gives L; 2 f.or k, which gives 2; and so on, up

through 5.) But if we proiect this rcgularity, the rcsult is that the sixthmember of thc series is 126l

Indced whatever number u,e wish to prcdict for the sixth member ofthc scrics, thcrc is a gcncrating function that will fit tlrc given menrbers

of tlrc serics arrd that will yicld the prcdiction u'c want. It is a mathe-r¡¡atical fact th¡t irr gcncral tliis is truc. For any finite string of numberswhrch [rcgins r scries, thcrc arc gcncrating functions tlrat 6t that string ofgiverr numbcrs and yield whatever ¡rext membc¡ is desired. Whateve¡p(cdiq!,ion wc \Yish to make, we can fincl a rt:grrlarity whose projectionwill licensc that prediction.

Thus if thc intelligcnce tcsts wcre sim¡tly looking for the proiection of

a rcgularity, any nurnbcr at thc end of thc serics would bc correct. Whattl.rey are looking for is not simply the projection of a regularity but the

projcction,of an intuitivcly projcctiblc rcgularity.If we have perhaps belal¡ored the point in Er.rnrples (l), (2), and (3)

rvr have done so because tlri: principle they illustrate is so hard to accept.

Any prediction whatsoever can be obtained by projecting regularities. As

Goodman puts it, "To say üat valid predictions are those based on past

regularities, without being able to say rohich rcgularities, is thus quitepointless. Regularities are where you find them, and you can find them

anywhere."' An acceptable scientiÉc inductive logic must have rules fordetermining the projecribility of regularities,

¡ Nclsn Cmdm zn, h'oct, l'tction ond tbrecost (Carnbridgc, Miss.: llaruard

Univesity t'res, 1955), p. 82;2nd cd. (lndiana¡rclis ud New York: Robbs Mcrrill, 1965);

3rd cd. (lnümapolis: Ilackett, 1979).

t¿ III. rup cooDNr^N pA¡t^Dox

It rcnlai,s to bc sllor". I¡orv tliis di.sc'ussio¡r of rcgur.ritics arrd ¡>r,jccti-l'ility.bear.s orr tlrr: ¡rrincipr«r of thc rrniforrnity of ,"atr¡rc. J,,rt ar'ru,r'.".r,tl¡:rt tllr' ¡¡:r¡vt' cl*r¿rctt riz.tir¡¡¡ tif scic¡ltific indrrctivc ],16"

", " ,frt.,,,tlrlrt ilrojccts r¡llsc¡vccl rcgtrlaritit:.s i¡lto tllc ft¡turc w:rs,rirrtlt,ss r¡¡¡lt,ss rvc.ra¡r say wlrich regularitics it projects, so wc .sllall sc:g ¡h¡¡ tl¡e statcntcntthat scie¡'rti6c inductivc rogic ¡rrcsu,poscs trrc u,ifor,rity .r ,r"i.r" i,equally lrointlcss unlcs.. we arc abrc to say ín wrrut respccts r,*rrlo-u-pr"-srr¡r¡rosctl to bc.,ifo',. For it i.s scrf-contradictor.y t,,ry,t,ri,rri,,.'" r,r¡¡rifor¡¡r i, all rcs¡rccts, and,triviar to say it is u¡liior¡¡r i,, ,o¡,r" ,:,*jr..tr.I, thc original statc¡nc¡rt of thc Goodrnau ¡rar.adox, tho gc.r cx¡rt,rt] ,,urrospoke our ordinary Iangtragc, assu¡,ed nature to bc unifo"r¡ll rriti, ,.,sp""tto tllc bl.c¡le.ss or grcer)ncss of cr¡eralds. since observcd c,.¡eralds llad.lrva1's becrr grcen, ir¡rd si¡rcc rrc w¿rs assurnirrg that naturc is r¡nift>r¡¡l andtll¿rt tl¡«r fr¡turc rvorrld rcsc¡nl¡r. trrc'pa.st in this rcs¡rt,ct, lrc pr.cdictccl tlrattllc e¡¡lcrald rvot¡ld rcrrair¡ grccr). lfut thc hy¡rotl.,"ti",ir g"u.,

"*¡r"rt *r,ospokc tlrc gruc-bloc, lnrrg'irgc assr¡¡rccr nat.rc to bc t¡nifo"r.nr uit'h rc,¡»cctlo tltc i:tu('t¡(:.st; or l¡ltctt¡tr,,t.:; ol crncuir/,r. Sincc olrst,rvt,rl t,¡nt.r..l<ls IrrrlItlw'11's lrcclt gtttt: i¡¡¡tl si¡tcc Itt, rr'¿ts rrssr¡¡)ri¡rg tltitt lr¡tr¡rc i.s rr¡ri[r¡.¡rr a¡¡cltllr¡t tl,t'lr¡trrrt'rvor¡l<l ¡(,s(,¡r¡l)l(, tlrt'¡rLst i,r tliis r.,,sp,,t.t, Irt,¡rri,rli.tt,ri tlr^tll,t,t'¡t¡,:¡l¡lcl rV0r¡ltl ¡.t,rn:¡ilr grrre. lJtrt \\,c:,lt\\,tlr¿¡t tilcsc trr,,,, ¡rra,l,"tur,.,,"rr'¡rr i¡r.r¡¡¡flict. Tlrt,frtr¡¡t,cil¡)¡)ot rt,st'¡r¡bl. tlic ¡r.st ir¡ b.tlr tlrcsr,r"a1,s,.\s r't lr;i't's.t',, strcll co¡rllicts crrr lrt: rrrrrlti¡rlir«l ¿¡ri irrfittitrtnt. -l-lrt.f,,rrrrctiuulot rcsc¡¡rl¡lr.tlrc ¡rrsl irr all rcs¡>t:q.15. It is.sclf-corrtraclictory to.su¡,that:lrtrlrc is ur¡jfolnl irr lrll rcs¡rccts.

"vc rniglrt try to rct't'rrt to trl. crair¡r tr¡ut scic¡rtific irrductio¡r prcsrp-

r)oso.§ tll¿rt rratr¡;c is,r¡ilorn¡ in.so¡¡rc rcsIects. Bt¡t this clai¡¡r is ro *"akrs to be l¡o clai¡¡r at all.'r-o say trrat natt¡rc i.s u¡.¡ifor¡, i. so¡,e r"rp".,,i.s to say tlrirt it cx)lil¡its sol¡)(: lruttcrns, tl¡¿rt thcrc lrc so¡r.r() rcgrrlaritics innatr¡rc.takc¡l as a wholc (irr botlr the ol;scrvr:cl ancl t¡nol>scnl.,cl ¡.,arts ofrratrrrc). Ilut as wc havc -scc¡l i¡r this scctio., in any scqr¡c]¡¡c(, rir r>rrscrva-tjo¡ls, ¡¡o mattcr lrow chaotic thc dat¿r ¡nily sce¡lr, ih"r" nr" arrvays regu-Iarities. Tl¡is hold.s,ot ,rrly [,rr r",¡u,,,,ccs of obscrvations btrt ,lro ?o,,)aiurc as a wll,lc. N. ¡¡lattcr rlow cllaotic rratur.c nrigrrt be, it wouldrll.*oyr cxhibit sonrc l)¿rtter¡rs; it wotrrd arways bc unifor¡n in ,o,.,,," .".p"",r.I'l¡csc r¡¡rilc¡r¡,ities r,iglrt s.crrr lrighly artifici¿1, st¡cl¡ as " ,,,,i[ur,rrity i,rtcrr¡ls of gruc,rrd blcc¡r or s.urf ,rrd ¡»urkrc. Trrcy r,igrrt l;c ficrrdishryc.r,plcx. I-Jut ¡lr.r ¡n¿ttcr Irrw,atu'c rrriglrt bchave, tlrcic rv.rrl<l ulrv;ryslrC s0r¡,r: rrrri[0rrrrity, '¡r.rtrrr.rl",r '',r¡tific.irrl," sirrrlllc or.corrrlllr:r. Ir i, t1,,,,,,-l,¡r' tr¡r l'rl t() s,r\ tlr.rr r¡.rlr¡r' ¡s rr¡tr,r*l r¡¡ \or¡¡(. tt s¡r.t ts. 'r'rr rr tl¡trt.¡l.r¡rr ¡rt rl¡.rt rt rr r¡rrl¡r r¡¡,lr¡r {r,'a ¡ri,\r¡l)l)()5r,s rli.rt rr.rtrr¡r. ls lr¡r¡1.¡¡a rs

III.5 sur"ru,rnv

to convey any information at all, it must specify in what respects scienti8cinduction presupposes üat nature is uni-form.

The points about regularities and proiectibility and thc uniformity ofnaturc arc rcally two sides of the same coin. There are so many regu-larities in any sequence of observations and so many ways for nature tobe uniform üat the statements "scientiGc induction projects observedregularities into the future" and "Scientiffc induction presupposes theuniformity of nature" lose all meaning. They can, however, be reinvestedwith meaning if we can formulate rules of proiectibility for scientific induc-tive logic. Then we could say that scientific inductive logic projectsregularities that meet these standards. And that would be saying some-thing informative. We could reformulate the principle of the uniformityof nafure to mean: Nature is such that projecting regularities that meetthese standards will lead to correct predictions most of the time. Thusthe whole concept of scientific inductive logic rests on the idea of pro-jectibility. The problem of formulating precise rules for determiningprojcctibility is thc ncw riddlc of induction.

Excrcisc:

In the example of the four boxes labeled "Excelsiorl" find a regularity in theobservations whose projection would lead to the prediction that the mask rvillbe blue.

ULs. SUMMARY. This chapter described the scope of the prob-Iem of constructing a system of scientific i¡¡ductive logic. We began withthe supposition that scienti§c inductive logic could be simply character-ized as the projection of observed regularities into the future in accor-dancc wiü some rule such as Rrrle S. We saw thrt this characterizationof scientific inductive logic is inadequate for several reasons, the mostimportant being that too many regularities are to be found in any givenset of data. In one set of data we can find regularities whose projectionleads to conflicting predictions. In fact, for any prediction we choose,there will be a regularity whose projection licenses that prediction,

Scienti6c inductive logic must select from the multitude of regularitiespresent in any sequence of observations, for indiscriminate projectionleads to paradox. Thus in order to characterize scientific inductive logicwe must specify the rules uscd to determine which regularities it con-siders to be projectible. Thc problem of formulating these rules is calledthc new riddle of índuction.

/.)

7.1 III. -r'ur cooDr\rAN P^nAD()x

lisscntially thc sl¡nre problcur rcappears if we try to cl¡ar¿ctcrizc scicn-tific inductive logic as a system that presupposes that nature is uniform.To say that nature is uniform in some rcspects is trivial. To say that naturcis uniform in cll respects is not only false but self-contradictory. Thus ifwe are to characterize scientific i¡rductive logic in te¡ms of some principleof the uniformiry of nature which it presupposes, we must say in whatrespects nature is preslrpposed to be uniform, which in turn determincswlrat regularities sc.ientific inductive logic takes to be projectible. So theproblem about the uniformity of nature is just a difierent facet of thenew riddle of induction.

The problem of constructing a system of scientific inductive logic willnot be solved until the ncw riddle of induction and othcr problerns havcbeen solved. Although these solutions have not yet been found, therehave been developments in the history of inductive logic which constituteprogress towards a system.

In the next chapter we shall pursue an analysis of causality which castssome light on well-known fcaturcs of the experimcntal method. Tl¡err wcwill discuss thc major achievcmcnt of thc ficld, thc probability calculus.

Suggcstcd rcadiugs

Nclson Coodnran, F¿¿t, Fiction and Forccasl (Cambridge, Mass.: FIar-vard University Press, 1955), clrap.3, "Tl¡e Nerv Riddle of I¡rductron."Bcrtrand Russell, "On the Notion of Cause," in Mysticism and Logic(New York: Anchor Books, 1957), pp. 174-201.

For üe advanced studenh

IUchard JelIrey, Tlw bgic of Decision (2nd ed.) (Chicago: University of Chi-cago Press, 1983), Sec. 12.3.

IV

Mill's Methods ofExperimental Inquiryand the Nature of Causality

IV.l. INTRODUC'TION. One of the purposes of scientific induc-tive logic is, prcsumably, to asscss the cvidcntial warrant for statements ofcause and efiect. But what exactly do statements claiming causal connec-tion mean, and what is their relation to statements describing de factoregularities? These are old and dcep questions and at best we can giveonly partial answers to them.

Yet evcn partial answers can be illuminating. In his System ol Logic,published in 1843, John Stuart Mill discussed 6ve "methods of experimen-tal inqtriry" airncd at discovcring cruscs. Although not original with Mill,these nrcthods have bccomc famous as Mill's metlwds.It so happens thata relativcly sirnple distinction behvecn different typcs of causal relation-ships rvill allow a complete analysis of thc logic of lr4ill's methods.

But before this elementary an'alysis can be made to pay its dividends,and before more profound problems can be raised, it is necessary to intro-duce the basic logical machinery of simple and cornplex statements andpropcrties.

IV.2. THE STRUCTURE oI¡ SIMPLE STA'IEMENTS. A sú¿ü¿-

nent is a sentence üat makes a deli¡rite claim. A straíghtforward way otmaking a claim is to (l) identify what you are talking about and (2) make aclaim al¡out it. Thus in the simple statement "Socrates is bald," the propername "Socrates" identifies who we are talking about and üe predicate "isbald" makes our claim about him. In general, expressions üat identify what we

are talking about a¡e called refeñng elpressioru and üe exPressions tsed tomake factual claims about the things we are talking about are calledcharacterizing erpressiotu. Ttrus üe name "Socrates" refers to a certainindiüdual, and the predicate "is bald" characterizes üat individual.

75

76 IV. ¡¡l¡,¡.'s METHoDS AND cAUSALtry

Although proper na¡ncs arc au important typc of refcrring expression,thele are others. P¡onouns such as "I," "you," "he," and "it" are rcferringcxprcssions oftcn uscd in ordinary specch, rvhere contcxt is rclicrl upon tomake clear r.,,lrat is being talkcd about. Sometinres whole plirascs are usedas referring expressior.rs. In the statement "The first President of the UnitedStatcs had rvoodcn falsc teeth," the phrase "The first President of thc UnitcdStatcs" is uscd to refer to George Washington. He is then characterized as

having woodcn falsc tccth (as in fact he did).Although .stfltcnrL'¡¡ts lrc oftcn constructed out of onc rcfcrring ('xprcs-

sion, ils in tlrc exarrrplcs abovc, so¡nctinles they arc constructcd out of moretl¡¿¡r our: rcfcrring cxprcssion, plus an cxprcssion that ch¿¡ractcrizcs the rcla-tionslrip betrveen the things refe¡red to. For instance, the statenrent"i\'lc¡g¡¡y is hotter than Pluto" contains trvo referring exprcssions-"Mercu-ry" and "Pluto"-and one characterizing expression-"is hotter than."Cli¿rr¡ctcrizing cxprcssions that charactcrize an individual thing are calledTsro¡tctiy crpressions ot one-pl4ce predicates, "Is bald," "is red," "conductscicctricity" arc cxamples of pro¡terly expressions. Characterizing exprcs-sions that characterize two or more individual things in relation to one anotheri¡¡c callcd rcüttiotul c4rn-ssrc»u or twny-pktcepredicotes. "Is hotter than," "is¿¡ lrrother of," "is to the north of," "is between" are examples of relationalex[x( \lOnS.

Tl¡c l¡asic rvly to co¡tstruct a siruplc statcntcnt is to comL¡inc rcfcrringand chsrocterizing cxprossions to mükc thc appropriote fuctual clni¡n, Inthe ncxt scctíon it rvill bc secn how thcse sinrple statemer)ts can bc conr-bined with logical corrnectives to fornt complex statements.

. Excrciscs;

Pick or¡i thc rcferring and chalacte,jzing ex¡tressio¡rs in the following state-¡nents. State whether each characterizing expression is a propc.ty expressionor a rel¡tior¡al cxpression.

l. It ís th¡ East and Julict is the sun.

2. lvtozirrt is far sultcrrol to Chopin.

3. Thc Pyrcnces are located between Spain and Fra¡lce.

-1. ,\ll tl¡c point! alorrg the cilcu¡nfercnce ofa circle are equitlistarrt from thc(('tlt,

l ¡¡¡ L¡¿ck.

3 r1,5

IV.3 srnucruRE oF coMpLEx sTATEMENTS

IV.3. THE STRUCTURE OF COMPLEX STATEMENTS. Con-sider the two simple statements "socrates is bald" and "socrates is wise.',Each of these statements is composed o[ one referring expression and onecharacterizing expression. From these statements, together with the words"not," "and," and "or," we can construct a variety of complex statements:

Socrates is noú bald.Socrates is bald ¿nd Socrates is wise.Socratcs is bald or Soc¡ates is wise.Socrates is nof bald or Socrates is wise.Socrates is bald and Socrates is wise or Socrates is rwt balcl and,Socrates is noú wise.

The words "not," "and," and "or" are neither referring nor characterizingexpressions. They are called logical coniectioes and are used together withreferring and characterizing expressions to make complex factual claims,

We can see how the logical connectives are used in the making of com-plex factual claims by investigating how the truth or falsity of a complexstatement depends on the truth or falsity of its sinrple constituent state-ments. A sinrple statement is true just when its characterizing expressioncorrectly characterizes the thing or things it refers to. For instance, thestatemc¡rt "soc¡ates is bald" is true if and only if socrates is in fact bald;otherwise it is false. whether a complex statement is true or not dependson the truth or falsis of is simple constituent statcments and the way thatthoy aro put togother with the logicol connecfivcs, Let us see how thisprocess works for each of the connectives.

Noú. We derry or negate a simple statement by placing the word "not" atthe appropriate place within it. For instance, the denial or negation of thesimplc statcment "Socrates is bald" is thc comltlex statement "socrates isnot bald." Often wc abl¡rcviatr: a stat('rncnt by using a singlc letter; forexample, we may let thc .le tter "s" stand for "socrates is bald." And we maydeny a statement by placing a sign lor negation, ",.," in front of the letterthat abbreviates that statement, Thus "-s" stands fo¡ "socrates is not bald."Now it is obvious that when a statement is true its dcnial is false, and whena statcment is false its denial is true. Using the shorthand int¡oduced above,we can symbolize this information in the following truth table, where Tstands for true and F for false:

P-PTF

TI

FT

I V. ur¡-l's lvrrrl'HoDs AND cAUsALr-i y

Wllat this tal¡le tells us is that if thc statcmcnt "p" is true, thcn its dcnial,'-p," is false. If the statement "p" is false, then its denial, "-p," is true.The truth table is a sunmary of the way in which the truth or fals.ity of thecomplex statement depcnds on the truth or falsity of its constituentstatements,

And. We form the conjurrction of two statenrents by putting the word"and" between them. Each of the original statenrents is then called a

coniurv-t, A conjuncbion is true just rvhen botl¡ of thc conjurrcts are truc.Using the symbol'&" to abbreviate the word "and" we can represent thisin üe following truth table:

p&q

TTFF

TFFF

He¡e we have four possible combinations of truth and falsity that theconstifuent statements "p" and "g" might have, and correspondirrg t«.¡ each

combination we have an entry telling us whether the complex statertrc¡tt"ñq" is true or false for that combination. Thus in the case where "p" is

t¡ue and "g" is true, "ñq" is also true. Wherc "p" is true and"q" is falsc,

"p&q" is false. Where "p" is false end "q" is true, "p&g" ls again false, Andwhere both "p" and"q" are false,"p&q" remains false.

Or.Tbe word "or" has trvo distinct uses in English, Sontetimes "p ot q"nrc¿rns "eithe¡'p or q,l¡ut ¡wt both," as in "I rvill go to the nrovies or I rvillstay honre and study." This is calied tbe erclusi¡¡e sense of "or." So¡neti¡ncs"p or,q" means "p ot q or both," as in "Club members or their wives mayattend." This is called tl¡c itrcluitse use of "or." We are cspecially intercstcdin the inclusive sense of "or," which we shall represent by the symbol "v.""pvq" is called a disiunction (or alternation), with "p" and' "q" being thed.biuncls. The truth table for disjunction is:

q pvq

TTFF

TTTF

7tJ

p

TFTF

TFTF

IV, 3 srnucrt¡nr oF coMpLEx sTATEMENTS

By rcfcrcncc to thc truth tablcs for "-," '&" and "v" wc can construct a

truth tablc for any complcx statcment. Conside¡ the complcx statemcnt"Socratcs is not bald o¡ Socratcs is wisc." This conrplex statement containstwo simple constituent statoments: "Socratcs is bald" and "socrates is wise."Wc may abbreviate the fi¡st statemcnt as "s" and the second as "rr." Wecan then syrnbolize the complex statement as "-svlo." We may use thefo.llowing procedure to construct a truth table fo¡ this comple.'r statentent:

Step l: List all the possible conrbinatio¡rs of t¡uth and falsity for thcsimple constituent statements, "s," "to."Step 2: For each of thcse combinations, find whethc¡ "-s" is true orfa,lse from the truth tal¡lc for negation.Step 3: For each of the combinations, 6nd whethe¡ "-sVru" is true orfalse from step 2 and the truth tablc for disjunction.

Thc result is the following truth tab.le for "-svtD":

Stcp I Stcp 2 Stcp 3.*§ ,-,ó'vt0

Casc 1;

Case 2:Casc 3:Case 4:

TFTF

This truth table tells us exactly rvhat factua.l clai¡n the complex statementmakes, fo¡ it shorvs us in rvhich cases that stateme¡rt is t¡ue and in which it isfalse.

Since a truth table tells us rvhat factual clai¡l is nrade by a complexstatenrent, it can tcll us when hvo statements make the same factual claim.Let us exa¡¡rinc thc trutl¡ tal-¡le for "(s&u,,)v(,--s&tr,)":

+s s&r¿.r .--s&u' (s&u.:)v(-s&u)

Case 1:

Case 2:Case 3:

Case 4:

TFTF

FFTF

TFTF

Note that in reading across thc truth table we start with the sinrple con-stituent stiltcr¡)ents, procced to tl¡e ncxt largest cornplcx staterncnts, until

IJLMtxrCO

É'

79

ttr,tsI,t<()F':lFr:,c:))-r

á.-:ll^tt1c.¡t.:s|-rt-,

TFTT

FFTT

TTFF

C)(_)|t' Ir

TTrF

FFTT

TFFF

:t)

lo

Fo

\'. olnsl i li,to CL liry,l:. i,,;¿cr0;r¿s l jiosúfic¿r

"DR Fnl,A t¡.,^ / r ..-.-:i¿r

80 IV. ltu,'s METHoDS AND cAUSALITY

wc finally arrivc at thc conrplex staternent that is the goal' Thc truth tablc

shows that the 6nal cornplex statement is true in cases i and 3 and false in

cases 2 and 4, But nptice that üe simple statement "t¿" is also true in cases Iand 3 and false in cases 2 and 4. This shows üat the simple state¡nent "ur"

and the complcx statement "(s&tr.r) v (-s&o)" make the same factual claim'To claim that socrates is either bald and wise or not bald and wise is itrst a

complicated way of claiming that Socrates is wise, Whcn two statemcnts

make the same factL'al claim, they are logically equioalent,Truth tables may also be used to show that two complex statcments make

conflicting factual claims. For example, the claim made by the statemcnt"^ s&-¡r" obviously conflicts with the claim made by thc statement"s&ur." Soc¡ates cannot boü be bald al¡d wise a¡ld be not bald a¡¡d not

wise. This conflict is rcflccted in a huth table for both statemcnts:

s&tu ,*5§.-u)

Case 1:

Case 2:

Case 3:Casc 4:

FTFT

Thc state¡lent "s&u.r" is true orrly in case l, while the statement "-s&-uis true only in case 4. Therc is no case in which botl¡ statcments are true.

Thus the two statements make conflicting factual claims. When two statc-

mt¡rts make conflicting factual clair¡s. they are lnconsistcnt with each

othcr, c¡r mutually exclusioe.

Thcre are some peculiar complex statements that ntake no factual claim

whatsoever. If we say "Either Socrates is bald or Socrates is not bald" we

have really not said anything at all about S,¡crates. Let us see how this

situation is reflected in the truth tablc for "sv-§":

-s §v,s

Case l:Case 2:

TTFF

TFFF

FFTT

TFTF

FFFT

TF

FT

TT

Thc reason why the statement "§v-s" makes no factual claim is that it is

true no mattcr what the facts are. This is illustrated in thc truth table by the

statement being true in all cases. When a complex staternent is truc no

matter what the truth values of its constituent statements are, that state-

rncnt is called'a tautolagY.

IV. 3 srnucruRE oF coMpLEx STATEMENTS

At the oppositc cnd o[ thc scalc fro¡n a tautology is tlrc §pc of st¡tcmcntthat nrakes an impossiblc claim. For instancc, thc statcmerrt "Socrates is

bald and Socrates is not bald" must be false no ¡nattcr what thc state of

Socrates'head, This is reflectcd in thc truth tablc by thc statemcnt bcingfalse in all cases:

."§ s&-.s

Case I:Case 2:

TF

Such a statemcnt is callcd a self-contradiclion. Sclf-contradictions arc falscno rnatter what the facts are, in co¡rtrast to tautologies, which at'c true no

matter rvhat the facts arc. Statcrncnts that arc ncithcr tautologics nor self-

contrirdictions arc callcd contingent statentents bccausc whcther tllcy altr

t¡uc or not is contingeut o¡r rvhat thc facts are' A contingent statemcnt is

truc i¡r some cases and false in othcrs.The purposc of this scctio¡r has bccn to convey an understanding of the

basic idcas behind truth tables and the logical connectivcs. We shall apply

these idc¿rs in our discussion of Mill's n.rethods and tltc thcory of probabili§.Thc nlain points of this scction arc:1. Complex statemer¡ts are constructed from sinrplc statcnrc¡rts a¡rd thc

logical connectives "-," "&," "v."

2. The truth tables for "-," "&," "v" show how thc truth or falsity of

cornplcx state¡¡cnts depends on the truth or falsity o[ thcir sinrple con'

stitucnt statcments.3. With the aid of the truth tables for "-,"'&," "v," 4 truth table may

be constructed for any complex statcment'4. The truth table for a conrplex statemcnt will havc a case for each

possible co¡nbination of truth or falsi§ of its simplc constituent statcments.

it *ill .ho* in each case whether the complex statement is true or false'

5. The factual clai¡n made by a complcx statcmcnt ca¡r be discovercd by

examining the cases in which it is true and those in which it is falsc.

6. If two statements arc truc in exactly the same cases, they make the

same factual claim and are said to bc logically equivalent.

7. If two statements are such that there is no case in which they are both

truc, they ¡nake conflicting factual claíms and arc said to bc inconsistcnt

with cach other, or mutually exclusive.

8. If a statement is true in all cases, it is a tautology; if it is false in all

cases, it is a sclf-contradiction; other"vise it is a contingcnt statement'

BI

FT

FF

ri:l

l,lx t.rr.i.scs:

l. (lsurli t¡r¡tl¡ tal¡lcs, find wh.ich ofl,r¡ir,.rlly r.r¡rrtvaL:ttt, wluch are mutually

i\. l), - -P.b. -pv-q, p&.q.

c. ¡r&-(/, -(p&q).d. -pvq, p&.-q.e, (pvp)&q,p6,(Svq).f , -pvq, p&-q.

IV. ¡rrr¡-¡-'s ME'rIIoDS AND cAUsALny

2. Using truth tables, find which of the followir)g statements are tautologies,

which are self-contradictions, and which are contingent statements:

a. -*pv-p.b, pvqvr.c. (t¡vp)&-(pvp).d. (pv-q)v-(pr-q).e. p&q&r.t.--(pu-p).g.-pvpvq.

IV.4. SIIUPLE A¡,iD COtltPL[X PROI')ERTIES.' Just as conrplcxstatcrnc¡¡ts rnay be co¡lstructed out of srnrple ones with tlrc logical connec-tives, so complex propcrtics may bc constructed out of sinrple oncs. Fronlüe simple propertics "is red," "is black," "is fat," "is short," and the logicalconnectives, we can construct the complex properties "is not red," "is ¡cd orblack," "is fat and short," "is not fat and is short," ctc. We shall use capiialletters to abb¡eviate properties; for example, "R" stands for "is red."

We can use a method to examine complex properties which is quitesiniilqr to the method of truth tables used tc examine complex statements,

Whether a complex property is present o: ¿l¡st',rt in a given 'l,ing or eventdepends on whether its constituent simple propcrties are present or absent,just as the truth o¡ falsi§ of a complcx statement depends on the trutl¡ orfalsi§ of its simple qonstituent statements. When the logical conncctivesare used to construct complex properties, we can refer to the following

I In this and the following section the wo¡d "property" is usetl ambiguously to

mean "property" or "propcrty expression," but the¡e should be no diflicultyresolving any instance of the ambiguity. Things erhibít properties; üat is,

properties are present in things. Propertg erptessions, on the oüer hand, describe

things. A property expression conectly describes a thing or event just whcn thecorresponding propcrly ¡s in fact prcsent in that dring or ev('¡rt,

the following pairs of st¡terner¡ts ar0

exclusive, a¡¡d whrch are ¡reither:

]V. 4 srup¡-r AND coMpLEx pRopEnrlEs

presence tables, where "F" and "C" stand for simplc properties and where"P" stands for "present" and "A" for "absent";

83

Table IIIFGFvG

P

AAP

PAAA

P

P

P

A

Note that these tables are exactly the same as the truth tables for the logicalconnectives except that "present" is substituted for "true" and "absent" issubstituted fo¡ "false." With the aid of these presence tables fo¡ the logicalconncctives, we can construct a presence table for any complex property incxactly tlie same way as we co¡rstructed truth tables for complex statements.Thc presence tal¡le for a complex propcrty will have a case for each possiblecorrrbinatio¡l of prcscncc or abscncc of its simple constituent properties. Forcach casc, it will tcll whcthcr thc conrplex property is prescnt or absent. As

a¡r illustrati<¡n, we may construct a presence table for "-FvC":

c -F -FvGCasc I:Case 2:

Casc 3:

Case 4:

There are other parallels between the t¡eatment of complex statementsand the treatment of compler prope*ies. Two complex properties arc bgi-cally equioalent if they are present in exactly üe same cases; two propertiesare mutually erclusioe if there is no case in which they are boü present.

When a property is present in all cases (such as "Fv-F") it is called a ¿ni-oersol property. A universal property is analogous to a tautology. When a

property is absent in all cases, it is called a nulJ proper§. A null property is

analogous to a self-contradiction. The properties in which we are most

interested in inductive logic are üose which are neither universal nor null.Tliese are called confingent properties. Mill's methods of experimental in-quiry are directed toward discovering causal ¡elations between contingentproperties. These causal relations are discussed further in the next section.

Table IF-F

Table IIFCF&G

P

AP

A

P

P

AA

P

AP

A

P

P

AA

r:lt

F

P

P

AA

P

AP

A

AAP

P

P

A

P

P

84 IV. ur¡-¡-'s METHoDS AND cAUSALrry

Excrciscs:

l. Using presence tal,¡les, find which of the following pairs o[ properties arelogically equivalent, which are mutually exclusive, and which are neither:

a. -FvC, --Cv-F.b, -Fv-G, -(F&C).c. -FvG, F&-C.d. Fv-(F&C), -(17&G)&F.e. -F&-C, -(FvC),f. -(FvGvH) ,FvCvH.g. F&-G, -(r&C).

2. Using presence tables, find out which of the following properties arerrniversal, which are null, and which are contingent:

a. -FvCvF.b. (FvF)&-(FvF).c. -(Fv-F).d. ([v-C)&(Cv-r').

'e. FvCvH,f. -(F&-G)v-(Gv-F).

IV.5, CAUSALI'I-Y AND NECIj§SAITY AND SUITFICIENTCONDITIONS. tvlany of the inquir.ies of both scientific rescarch and prac-tic, ' allairs rnay be cha¡acterized as the scarch for the causes of 'rertainellects, Thc practical application of knowledge of causes consists either inproducing thc causc in ordcr to prouucc the effect or ln removing the causein order to prcvent tl¡e ellcct. Knowledgc of causes is the key to control ofcllccts. Thus physicia¡rs search for the cause of certain diseases so that theyma)' rcr¡rovc the cause and prevent thc effect. On the other hand, adve¡tis-ing rrrcn cngage in motivational rescarch into the causcs of consumer<lema¡rd so that they can produce the cause and thus produce the cffect ofco¡lsulne r dc¡¡rand for the ir products.

However, tlre word "causc" is used in English to mean several differenttlrirrgs. For tlris reason, it is more useful to talk about necessary contlitior»utd stt{ficienl conditio¡» rather t}ran about causes.

Definition l0: A properiy F ís a suffcient conilition for a property Cif arxl only if t¡ltctcocr F is Ttrcsent,C is present.Definitio¡'¡ ll: A propcrty II is a neccss¿ry condition for a property Iif and only if oltencaer I is prescnt, H is ¡tresent.

llt'rrrg r r¡r¡ ()\'( r [)) .r st( .r¡¡¡r()llt'r ¡s ¡ sullieit ¡¡t cor¡ditior¡ [or dc.rtlr, but rt rs

tr()t.r t¡r(r\\.¡r\ (r)t¡(lrt¡()¡¡ \\'l¡r'¡¡tr(r §()¡¡t('o¡Ic l¡iu bcc¡r rtttt tlvcr by a

IV. 5 caus¿,¡,rry AND suFFrcrEñT coNDITIoNs

stcamroller, hc is dcad. But it is not tile case that anyonc who is dcad has

L¡een ru¡r over by a steamroller. On thc othcr hand, the prcsence of oxygenis a necessary condition, but not a sufficient condition for combustion.Wl¡enever conrbustion takes place, oxygen is present. But happily it is nottruc that whencvcr oxygen is present, combustion takcs place. When we saythat A causes B we sometimes mean that A is a sufficient condition for B,sometimes that A is a necessary condition for B, sometimes that A is bothncccssary and sufficient for B, and so¡nctimcs nonc of thcse things.

If we are looking for causes in order to produce an effect, it is reasonableto look for sufficient conditions for that efiect. If we can manipulate circum-stances so that the sufficient condition is prcsent, the eficct will also bepresent. If we are looking for causes in order to prevent an eEect, it is

rcasonable to look for necessary conditio¡rs for that effect. If we prevent a

necessary condition from materializing, we can prevent thc effect. Thcoradicatio¡r of ycllorv fcver is a striking illustration of this stratcgy, Doctorsüscove¡ed that being bittcn by a certain type of mosquito was a necessarycondition for contracting yellow fever. It was not a sufficient condition, forsornc pcoplc who wcre bittcn by thcse mosquitos did not contract ycllowfcvcr. Conscquently a campaign was institutcd to destroy these rnosquitosthrough thc widesprcad usc of i¡rsecticidc and tl¡us to prcvcnt yellow fever.

F¡om the dc6nitions of necessary and sufficient conditions, we can provesevcral irnportant principles. It follows immediately from the definitionsthat r

l. If A is a sufficicnt condition fo¡ B, then B is a necessary conditionfor A.2. ln C is a necessary condition for D, thcn D is a sufficient conditionfor C.

To say that A is ¿ sufiicient corr.lition for B is, by definition, to say thatwhe¡lever A is present, B i.s prescnt. But to say that B is a necessary condi-tion for A is, by dcfinition, to say the same thing.

Let us look at some illustrations of thcse principles. Since the presence ofoxygcrr is a necessary conditior) for combustion, tlicn by principle 2 com-bustion is suffcient to ensure the presence of oxygen. Thus suppose some-one lowers a burning candle into a deep mine shaft he proposes to explore.If the candlc continues to bur¡r, he will know that the shaft contains suffi-cient oxygcn to breathe. To illustrate princi¡rle L, let us suppose that a

profcssor has constructed a tcst such that a high grade on the tcst is saff-cient to guarantee that the student has studied the material. Then studyingtlrc niaterial is a necessan¡ condition for doing well on the test.

85

8{I IV, un¡-'s METHoDs AND cAUsALrrY

'l u'o;rtltliIit¡¡¡al ¡trinciplcs rcquirc a litt]c ¡no¡c thougltt:

3. lf A is a sufficient condition for B, then .*B is a suflicicnt ct¡ndi-tiorr for -4.4. lt C is a ne ccssary co¡rdition fo¡ D, tlien -I) is a necessary condi-tir,n for -C. r

Using thc defi¡¡ition of sufficient condition, principlc 3 beco¡ncs:

3'. If whenever A is present B is present, thcn whenever -B ispresent -A is present.

Nc¡w ¡c¡¡lct¡r[.¡t:¡ {tor¡r tlre ¡tlcscrrctr taL¡lc [or rrcgatit-ur tll¡t -lJ is prcseutju.st whcn B is abse¡rt aud -A is prcscrrt just whcn A is lil¡scnt. So principle3 can be rewrittcn again as:

3". lf wl¡c¡rcvcr A is prcscnt B is plcscnt, tlrcn wlicncver IJ is abscntA is abse¡rt.

Wc can now scc why this prirrciplc is corrcct. Supposc that lvhcncvcr A is

prescnt, B is prcscnt. Sup¡tose furthcr tllat lJ is al¡si:nt ut ¿i ccrtein case.

Tlrt'¡¡ A ¡nust also bc absertt, for if r\ w('rc prcscr)t, lJ rvoulcl btr prcscrrt, andit is ¡¡ot. Lct us scc lrorv tl¡is work-s ür a corrcrete casc. Supp<.rsr: tliat a ccrt¡inir,ír'ctit¡¡r is a sufficic¡rt co¡rditio¡¡ [or a ]righ fever; tlrat is, cvcryonc rvho suf-fers fro¡rt tlris i¡rfcctiorr rurrs a higli fcvcr.'l'hcn tlrt,al-¡scncc of a high fcvcris sufficicnt tcr ¡luarantcc thlt a ircrsun is not suflcri¡lg from thir infcction,

That principle 4 is corrcct can bc clcnronstratcd in the sanrc way. Usingthe definitio¡r of nt'ccssary condition, \!'c carr rcrvritc principkr 4 as:

4'. If whencver D is prcsc¡)t C is present, thcn whenever .-C is

prcscnt -D is prescnt.

Arid using ihe prcscnr:c tablc for ncgatiort, wc calt rcrvritt' 't agliin as:

4". If whencvcr D is prcscr)t C is prescnt, then rvl¡cttcvcr C is absc¡rt

I) is abscnt.

And this is simpiy a restatcrncnt of principlc 3" using differe¡rt letters.Wc can usc thc sarnc cxarnplc to illustrate principlc 4. Since suflcring

fronr tlrc infection is a sulficic¡rt co¡rditro¡r fol running a high fcvcr, runningthc fcvcr is a ¡rcccsstry condition for heving tl¡c infection (principle 1). By

¡rr irrciltlc 4, sincc rurrning tlrc fcvcr is a lrcccssary cr¡ntlitio¡r for havrng tlrcir,li'ction, not having thc infection is a ncccssary condition for not runni¡19

¡ [t'vcr. (lt is not a srrf[icicnt co¡lditio¡l sutcc t¡tlrcl discascs nright rcsult in a

fcver.)

IV. 5 causelrry AND suFFrcrENT coNDrrIoNS

Two niorc principles will complcte this survey of the basic principlesgovcrning neccssary and sufficient conditio:rs:

5. If A is a sufficient condition for B, then -A is a necessary condi-tion for -IJ.6, If C is a neccssary condition for D, thcn -C is a sufficient condi-tion for -D.

Using the rl¡finitions and the presence table for negation we can rewriteprinciple 5 as:

5'. If rvhencver A is prcsent B is present, the¡r whenever B is absentA is absc¡tt.

But this is cxactly what we established in 3". In the same manner, we canrcwritc 6 as:

6'. If whcnevcr I) is prcsent C is present, tlte¡r whencver C is absent,

D is absent.

But this is cxactly what wc established in principle 4". A concrete illustra-tion o[ principlc 5 is that, sincc bcing run over by a steamroller is a sufficientcondition for death, not being run over by a steaniroller is a necessary con-dition for staying alive. And principle 6 can be illustrated by the observa-

tion that, if studying is a ncccssary condition for passing a test, not studyingÍs a sufficicut conditiorr fol fuiling it,

Whcn rve spcak of tile cause of an effect in ordinary language, we some-ti¡ncs ¡nean a sufficient condition, as whetr we say that the infectio¡t was thecause of the fever or that being run over by a steamroller was the cause ofdcatli. Sorneti¡nes we mcan a nccessary condition, as when we say thatyellow lcver was caused by thc bite c,f the mosquito or a high score on thetcst was due to diligent study. OIr tltc other hand, necessary and sufficientconditions are sometimes not causcs at all but rather sympÚonrs or sigru.The continued burning of the candle \{'rs a sign of the presence of oxygen.

The high fever was a slJmptom of the infection. When we analyze Mill'smethods, it will be seen that the precise language of necessary and suffi-

cicnt co¡rditions is much more useful than the vague language of cause

a¡rd cffcct, sign and synrptorn.

Exercises:

Show that the following principles are correct and give a cot¡crete illustlationof each

'

87

;i'i

88 IV. Ir.r¡¡-l's ME'rHoDs AND cAUSALTTY

1. If -B is a sullicicr¡t co¡¡ditio¡¡ for -A, tltc¡r A is ¿t sulHcie¡¡t con<litior¡

for iJ,

2. lf -D is rr ncccssary condition for -C, tl¡cn C is l¡ ¡lcccssary cr¡trclitit¡¡t

f,or D.

3. If -A is u neccssary co¡rditit¡¡t for -0, tlrc¡r A is a sufficiclrt cor¡ditit¡¡t

for B.

4. lf -C ís r sufficie¡rt cor¡dition for -D, t}¡cn C is a neccssary conditiontot D. I

5. If A is a Irecessary co¡rdition for E and B is l trecessary colrdition for E,

then A&B is a necessary condition for E.

6. If A&B is a necessary condition for E, then A is a necessary condition for

E and B ís a necessary condition for E.

7. lf A is a strlficient conditio¡r for E and B is a sulficient cor¡clition for Ij,tl¡cn ávB is a sufficicnt condition for E.

8. lf AvB is a sulficient condition for E, then A is a sr¡lñcient conditíor¡ for

E and B is a sufficie¡rt condition for E.

9. If ^

is ir rrcccssrry cr.¡ntlitior¡ fc¡r'/i, tl¡e¡r wl¡atevcr thc pro¡lclty I;, Avl" is

a ¡lcccssary co¡lditiol¡ for E.

I0. If A is ¡ sulllcic¡rt con<litio¡t f,¡r E, then w[¡¡tever thc propelty I;, A&1"

is a sulficicnt co¡¡d¡tion for E.

Srrggestctl rciidirrg

.lsr¿cl Scl¡elllcr,'l'hc Attulonty ol lrtr¡uiry (Nerv York: AlIrotl A Krlopl,

1963), pp. l9-25.

[¡or the udvanced studcntr

Emest Sosa (e<J.) Cauotion ond Conditio¡tttls (London: Oxford University

Prcss, l1)75).

, IV.6. MILL'S METIIODS. N{ill presented fivc mcthods designecl to

guidc thc experimcnter in his search for causes. They are the method olugreernenl,l\c ntetlrctl ol diffetence, the ioint Twthod, the nwtlod of con-

cornitant Darir;tion, and the metlod. ol residues. However, Mill did not actu-

ally originate thesc methods, nor did he fully understand them.

The theoretical basis of Mill's methods has only recently bcen fully ex-

plored by the philosopher C. tI. von Wright''? Following von Wright, we

will prcscnt lr4ill's ¡ncthods a littlc diflcrently than Mill did. We will be ablc

2 C"org Ilenrik von Wright, A Trcotise ott

( Pattcrson, N.J.; Littlclicld, Atlams & Co., 1960 ).

l¡d,ucüor urtd Probobility

lY. / DIRDCT N{ETHoD o! AcxEtsMENr

to uncovcr tl¡c thcorctical basis t¡f Mill's ¡ncthods in a discussion of thc

method of agrccmcnt, the method of difterence, and the ioint method. since

tl¡ere is nothing essent¡ally nerv i¡t the ¡nethod of cc¡ncomita¡¡t variation and

thc method of rcsidues, we shall not discuss thenr. flowever, we shall still be

lcft with fivc rnctlrods, for therc arc two variations of thc method of agree-

¡nent and two variations of the joint method'

Thcse nlcthods are to be viewcd as methods of 6nding the necessary or

sufEcicnt conditio¡ls of a givcn property. The property whosc nccessary

or sufficient conditions are being sought is callcd the conditioned property.

A conditionecl properfy may have more than one sufficient condition. If the

conditioned property is death, being run over by a steam¡oller is one suffi-

cient condition for it, but there are many othcrs' A conditioned propertynray also havc rnorc than o¡rc nccessary condition' If the conditioned prop-

crty is thc occur¡ence of combustio¡r, the prcsencc of oxygen is a necessary

condition for it, but so is the presence of an oxidizable substance. Those

properties suspected of being necessary or sumcient conditions for a

givcn conditioncd propcrty arc callt:d possible con¿itioning proqerties.Thegencral problcm is, "How is the information gaincd from obscrving various

occurre¡¡ccs uscd to pick out thc ncccssary a¡rd sufficient co¡rditions from

thc possiblc conditioning propt:rtics?" Thc following nrcthods alc attcmpt§

to ¿rnswer this question.

IV.7. TfIE DIRECT METHOD OF AGREENIENT. Suppose that

r-»rc of thc possiblc conditioning propertics A, B, C, o¡ D is suspectcd of

bcing a r)ecessary condition for the conditioned properry E, but which one

is not k¡lown, Supposc furtlter that, aithcr by cxpcrirnentll rnanipulotion or

simply by studious observation, a rvide variety of occttrrences a¡e observed

in $,lrich E is present, and that the only possible conditioning property that

is present on all these occasions is C. The set of obscrvatio¡rs show¡r in

Exanrplc I

Possible conditioningproperties

ABC

Conditionedproperty

DEOccurrence 1¡

Occu¡rence 2:

Occurrence 3:

PP

A

PAP

Exarnple I corresponds to this desc¡iption. Occurrencc I shows that Dcannot be a nccessary condition for E. The dcfinition of nccessary condi-

PP

P

AP

A

P

P

P

90 IV. ül¡-¡.'s MnrHoDs ^ND

cAUSALITy

tion tells us that a nccessary condition for I must be prcsent rvhcncver ris prcsent. But in l, r is prescnt while D is abscnt. Thus occurrcncc I elimi-nates D frorn thc list of possiblc ncccssary conditions. ln the sa¡nc ¡na¡lncr,occurrencc 2 shows that B cannot be a neccssary condition for ri, since E isprescnt whilc ü is absent. occurrence B climinatcs A and eliminatcs D oncemo¡e, Thc orriy candidate lcft for thc olEce of necessary co¡rclition for E is c.The observations shou'that if one of the possibrc conáitioning propcrties isin fact a necessary condition for E, then c must bc that n"""rrrry cóndition,

In Exarnplc L tl¡rcc occurrer)ccs wcrc rccluircd l¡cfore A, B, ancr D coulclbe eliminated as possible necessary conditions for E. Actually we mighthave done without occurrence I since occurrence s also elíminated-D.Horvevcr, in the occurrencc shown in Examplc 2 alr courcl be eriminatedat one stroke. The principlc of elimination is the sanre: Any property thatis abscnt when E is present cannot bc a nccessary condition foi ¿.

Suppose somcone werc to objcct that thc absencc of D might be neces-sary for tlrc prcscncc of E, tl¡at is, tl¡at --D night bc a ncccss,rry contJitiorrfor E, a¡¡d that the data in Exarnplr: g havc ¡rot eliminatcd that possibility.This is corrcct, but it shows ¡ro dcfcct i' thc argunrcnt. only thc sirnprclrropcrtit's A, B,c, rr¡¡tl I) r'r'rt'i¡¡cludr,cl in tlrc possiblt: conditioning ¡:rop-

lixarrr¡rlc 2

I'«¡ssi[¡lc corrdi tiorringpropt'rtics

ABCD

Co¡¡clit ioncdproperty

E

Occurrcnce l,

Example 3

Poss iblc condit ion i ng ¡rropcrticsConditioned

Simple Conr¡>lex proper§A B C D -A-B*C-D E

Occurrencc I: AAPA

crties; thc cornplex property *D rvus not, And all that was clairned wasthat if one of the possible co¡iditio¡ring pro¡>erties is a neccssary conditionf.or 8., thett C is tl¡at r)cccssary condition. But if rvc \\,crc to adi thc nr:ga-tio¡rs of A, ts, C, and D to our list of possiblc conditioning properties, then

IV. 7 omrcr METHoDoF AGREEMENT

occurrcnce I of Exarnple 2 would not sufficc to eliminate all the alternativesbut C. This is readily shown in Examplc 3.

We can tcll whethcr a complex propcrty is prcscnt or absc¡rt in a givcnoccurrence from the information as to whether its constituent simple prop-erties are prese¡rt or absent. This information will be found in the presencetabie fo¡ that complex propcrty, Here it need only be remembered, fromthe presence table for negation, that the negation of a property is absentwhen that property is present, and present when that property is absent.Now in Example 3, occwrence L shows üat A, B, D, and ,-C cannot be nec-essary conditions for E. This leaves C, -,{, -8, and *D as likely candi-dates. If the field is to be narrowed down, some nrore occurrences nrust beobserved. These might give the results shown in Example 4. Again occur-rence 1 eliminates A, B, D, and -C. Occurrence 2 further eliminates -4,occ\urence 3 eliminates -8, and occrurence 4 elinlinates *D. Thus theonly possible conditioning property lcft is C. If any one of the possible con-ditioning properties is a neccssary co¡rdition for E, thcn C is that necessarycondition.

Example 4

Possible conditioning properties

ConditionedSimple Complex property

A B C D-A -B-C *D E

91

Occurrence 1:

Occurrence 2:Occl-rrrence 3:Occrrrrence 4:

AAPAAPAA

PAPAPAPP

AAAA

P

AP

P

P

P

AP

P

PP

P

P

P

P

A

Example 5

Possible conditioning propcrties

ConditionedSimple Complex property

A B C D-A -B.C -D E

Occurrence l:Occurrence 2:

APPP P A A A PPAPA A P A P P

92 IV. u¡¡-¡-'s ME'rlroDs ANIJ cAUS^LI-I y

I, Iixarnple 4 it took four occu¡¡enccs to elimi¡rate all thc possible concli-tioni,rg properties but onc. However, the two occurrences observcd inExample 5 would have donc thc job. Occurrence 1 shows that A, -8, -C ,

and -D cannot be necessary co¡rditions fo¡ .0 since thcy arc abscnt when Eis present. Occur¡ence 2 f urther eli¡ninates B, D, -A, and -C, leaving onlyc. Thus in this exarnple if onc of the possiblc conditioning pr.pcrties is anecessary condition fo¡ ^0, thcn C is that necessary condition. It is truc, ingeneral, that if we admit both simple properties and their negation aspossible conditioning properties, then the minimum number of occurrencesthat can eliminate all but o¡re of the possible conditioning propcrties is 2. Aswe saw before, when only simple properties are admitted as possible con-rlitioning propclties, the ¡ninimum number of occurrcnces that carr elirni-nate all but o¡rc of thcrn is l. Ilut the basic principlc of climination remai¡lsthcsame ín both cascs: A property th¿rt is absent when D is prcse't cannotbc a ncccssary condition for E.

Wc were able to cxtcnd Mill's nrcthod of agrccrncnt to cover ncgativepossiblc conditioning properties, a¡rd this makcs scnsc, for ncgativc proper-tic.s arc r¡uitc ofte'r ¡)ccessary conditions. Not I;cing ru. ovcr l;y a stca,t-rollt' is a ¡)cccssary condition for rernairrirrg alivc ard ,ot lcttirrg orrc'sgratlc a'era{c f¿rll bclorv ¡ cr:rtair¡ lroirrt rrray bc a nect:ssary condition forrcrrrlrirring irr collcgc. Wc a¡c irrtcrestecl irr rrt:gativc r¡cccssüry colrrlitiorrsL.,er..,,sc thr:¡,tcll us rvh¿t we rrlust avoid in order to atttin our goals. Ilutncgations o[sirrrplc ¡rmpcrtrcs ilrc not t]rr: only corn¡rlex propcrties that nraybe inrportant neccssary conditions.

Lct us co¡lsider disjunctions of sirnplc propcrties as neccssary cor¡ditions.Either having high graclcs in high school or scoring wcll c¡n the entranceexan¡ination rnight be a ncccssary condition for gctting into collcge. Itmight not be a sufficient condition since someone who mcets this qualifica-tion mightstill be rejccted on the grounds that he is crirnirrally insa¡re. Totakc another cxarnplt', in football cithcr rnaking a touchdowri or a ficlcl goal

Example 6

Possible conditioning propcrtiesConditioned

Cornplex propcrt)'DBvCE

()t crrrrr'¡tt c I( )( ( tlr r|n( r' :

SinrplcABC

P

P

APAA,{ ,.\ l, ,{

IV. 7 ourscr METlroD oF AGnTiEMENT

Example 7

or a convc¡'sio, or a safety is a nccessary condition for scoring. In this caseüe nccessary condition is also a sufficie,t condition. we are interested indisjunctive ncccssary conclitions bccausc they lay out a ficld of alternatives,one of which ¡nust be realized if we a¡e to achieve ccrtain ends.

The question of what happens when disjunctions (alternations) of simplepropertics arc allowed into a sct of possiblc conditioning propcitics is üoinvolved to be t¡cated fully here. But the principle of

"ll*inriion remainsthe same. we can see how this principre operates in trvo simplified exampresthat allow oniy simple properties and one crisjunction ,, porribr" condition-ing properties, In Example 6 the complex properry BvC is the only propertythat is always present when E is present. occu¡rences l and 2 eriminaie alltlrc sirnl:le propcrtics as ncccssary conditions. Thus if onc of thc possibrecorrditioriing propcrties is a ncccssary co'dition for Ii, Bvc is thai neces-sary condition.

I, Exar,plc 6 thc disju.ctio, was thc propi:rty rcft aftcr..il ttrc othcrs ha<lbccn clirninated. Let us now look ¿t Examplc 7, whcr-c trrc disjunction itselfis elirninated. occu¡rc¡rce l clinlinates A and c as nccessary conditions, and

93

Possib.lc con<litioning propertics

SirnplcABCD

ConditioncdCornplcx propcrfy

BvC EOccu¡rence l:Occurrence 2:

PP

P

AAA

AP

P

AP

P

occu.iler)cc 2 sl¡ows tlrat ncither IJ,or c ¡ror Bvc carr be a neccssary condi-tio¡r for ^0. This leaves only D, so if o¡ri: of the possiblc co.rlitioning proper-tics is a nccessary condition for .0, then D is that ncccssary conditión. weshall not cxplorc further th.c treatmcnt of complcx possiblc conclitioningproperties by the direct method of agreement.3 But you cannot go wrong if

r The direct mcthod of agrccn)ent can bc expanded to include simple propcrties,ncgations of simplc properties, and disjunctions of sinrple prop"rli"r-rr,l tl,"i.negations as possible conditioning propertics. Thcrc is no need to worry aboutconjunctions since a conjunction, that is, F&C, is a necessary condition for E ifurtl onlg if F is a necessary condition for E an<i c is a neccssary condition for E.Thus il wc can discover all the individuar nccessar¡,con<1itions, we arrtomaticaryhave all thc conjunctive necessary,conditions.

94 IV. l.r¡lr-'s ME.rlroDS ÁNrJ cAUSALrl.y

yorr rcrrr.rrrbcr tl¡at thc pri.ciplc of clirnination i¡r tlrc dircct nrctlrocl ofagrccnrent is: A ¡rroirerty that is abse¡rt rvhc¡r.E is prestrnt cannot bc ancccssary co¡lditio¡r fo¡ .0.

usc of thc dircct mcthod of agrccmcnt rcquircs looking for. occurrencesof tllc cr¡,ditiorrcd property ir¡ circurnstanccs as variccl as possiblc. If thesccircunr.sr¿r.ccs arc so varied tl¡at ouly onc of thc ¡rossiblc uo,ditioningprop.rtics is prcscrrt r'hcrlr:vr:r thc conditionccl propcrty is ¡rrcscnt, it rnaybc sus¡rcctcd tlrat tl¡at ¡;ropcrty is a noccssury co¡rclition for. thc conclitio¡reclpropcrty. lt lras b.t:, shown tl¡at thc logic bchincl this ¡nr.thod is thc samc asthc Iogic bt'l¡ir¡tl tl¡t'r¡¡ctlrocl of thc nrastcr dctcctívc rvlro cli¡ninatcs suspcctsorrc by onc i¡l ordcr to find thc rnurdcrcr. If only o¡ic o[ thc possibret orrclitioníng ¡rropr.r.tics i.s prcscnt wlrc¡lcvcr thc co¡¡rlitioncrl ¡rro1;crty isprcscnt, thr:¡l all the othcr possiblc co,ditioning propcrties arc climinirtecl ¿sll( (('ssilr) co¡¡tlítio¡ls silrct'tlrr'y:rrt't'lrcl¡:rbst,r¡t ¡n:¡t lt.,rst ()¡)r'r)r.(.r¡¡.r(.n(.(,ir¡rvl riclr tl¡r' cr¡n<litio¡r<'<l ¡;rop«,rty is prcsr.nt.

IJr¡t the ¡nctlro<l <>[ agrr:r'¡ncrrt rr,.sc¡l¡l¡lcls thr: rrrctlro<] <.¡[ tlrt: rr¡astcr clt,tcc-tivt'in trvo fr¡rtllcr rvu1,s. \!/lrt,rr st.u.tirrg o¡r a ¡¡rurrlcr clis¡, thc clt,tt,ctiv¡t'r¡tt¡lot lx, sr¡rr. tlrrrt lrt, u ril lrc ;rblt, to cli¡nilt;rtc rll tlrt, suspcr.ts br¡t or¡r:.Altcr:¡ll, tl¡t'nrurtlcr rrriglrt lrlrvt,lrt.r,¡l rlr¡¡rc Iry trvo pt,o¡rlr, ,",,,]ki,,,r tgg¡t¡cr,'l l¡t'si¡r¡rt'is t¡ut t¡l tl¡t' ¡¡rt,tlrr¡rl ol irg¡cc¡¡)('¡rt, for u r:orrrlitro¡,,,.1 pru¡r"rt¡r.",,.,¡\'c r¡)ol('tl¡.r¡¡ r¡¡¡t.¡)('c(,ss¡r') ct¡nditio¡¡. N{orcovt,r tlrc rnastCr clctective¡nny nat havc thc nru¡dc¡q¡ 0r niurd(.rcrs in his jn¡tjal li;t of suspects andrnay c,cl up olirrri,atirrg all tlr. po.ssibilitics. In this casc Irc rvill have to goI¡ack ¿r¡ltl Iook for ¡rlol'c su.sl)ccts tr¡ i¡lclurlc ir¡ l niorc corn¡lr.chc¡sivc list. Ina si¡niliir nla¡Incr rlrc scic¡ltist n)ay not havc incluclcd thc uccessary condi-tio, or corrditio.s for a co¡rditioncd proPerty in his initial list of possiblcconditioning Propcrtics. Thus his o[>scrvrtions rnight ,,lirninatc all Iri.s

¡xrssi blc crinclitio¡ling ¡rropcr.tics.

Excrcises;

l. Irr Exarn¡rle I rvlrich o[ tl¡c fr¡llorvirrg conr¡rlex ¡rropcrtics i¡re eli¡ni¡¡atcdas r¡eccsslrry co¡¡<litir¡¡rs for li by occt¡rrc¡¡ces 1,2, ar¡d 3?

¡. -,4. d, -D.I¡. - iJ. c. AvD.c.-c. f.uvc.

2 lr¡ tlrc [olklrvirrg cxrttrt¡rlc, [o¡ cucll ()ccurrcr)co firrcl ivlictlrgr the complcx¡tlo¡lcttit:s i¡rc l)r(lscl¡t ol ltlisc¡¡t ar¡d w'lrrch oI tlrc ¡tossi[rle corrriitiorrirrg ¡rro¡r-crlir.s .rrc t.l¡nl¡¡¡rrtcrl irs n('c(,ssirry cr¡¡rtlitio¡rs [rlr L':

IV.8 ¡Nvsnsn METHoD oF AGuEEMT,NT 9s

SimpleABC-A

Possible conditioning properticsConditioncrl

Complex p¡.operty

-B -C AvC -Rl,C EOccurrence l:Occurrence 2:O<'currence 3:Occurrence 4:Occurrence 5:Occu¡'rence 6:

PPPP PAP APPPAAAPAAP

P

P

PP

PP

3' In Exercise 2 o¡re of the possiblc conditioning properties was not elinr-inated. Describe an occurrence which would eli¡ninatc it.

1V.8, THE INVERSE METTIOD OF AGREETIÍENT. Thc inver.scnrcth<.¡d of agrccnrcnt is a nrcthod for fi,ding suflicicnt con«litio¡rs. To fi¡¡d ¡sufficie.t conditio. for a givc^ propcrty, E, wc look.for a pr.operty that isabsent whcncvcr.L- is absc¡rt.'i-his is illustratc<.I i. Exarnplc g,. ó ls itre onlypossiblc cc-rnditioning propcrty tlrat is absc¡lt rvllcncver thc conclitionecl

Exanrple 8

Possible conditioning Co¡rditio¡redpropcrtics prope üy

ABCDEOccu¡rence l; POccurrence 2: AOccurrence 3: P

APA

AAP

AAA

AAA

property is absent. Thus by thc i.r,crse method of agrccment, if one of thcpossible conditioning propertics is a sufficicnt condition for E, then D isthat suffi cient condition.

The inversc *"ihod of agrecmcnt .perates in thc folrowing manner: wcknow from the dcfinition of sufficicnt co¡rdition üat a sufficicnt conditionfor E cannot be prese,t when E is absent. To say that a ccrtain property isa sufficicnt condition for E means that whenever that property is-prcs"ni, tiis also present. Thus in Example 8 occurrence I shows that A cannot be asulficient condition for E si¡rcc it is prcscnt whcn E is abscnt. occurrcnce 2shows that B cannot bc a sufficient co¡rdition for E for the same reason, ancloccurrence 3 does the same for c a¡rd A once again. D is therefore thc onlyproperty lcft that can be a sufficient condition for E. In this way thc i¡rverst:

!)0 IV. ulu¡-'s NIulltoDs AND cAUsALll Y

rrrr.tl¡otl o[:rg«:t'rrrr:ttt, likc the direct method, works by eliminating possible

llnrlirlatt:s ottc by one,

lrxanrplc 9

Possiblesuflicient

conditiorrsfor Il

AI]C DD

I'ossible r¡cccssllry

corrditions for -L-A

.B _C -D

-*¡l

Occurrence I: AOccurrence 2: AOccurrence 3: P

APPPPAAPAAPAAAPAPA

AAPPAP

P

P

P

The inverse mcthod of agrcement nray bc viewcd as an applicatiorl of the

direct rncthod to negativc prolrcrties. This is possiblc in light of tlrc prin'

ciple: if -A is a neccssary colrdition for .-.8, thcn A is a sufncicnt condi-

tion E,' Example 9 illustratcs this nrctlrod in action. The only possible

necessary conclition for .-E that is ¡rrcst,Dt rvhcnevcr ,"-E is prcselt is -D.Noticc that this conrcs to tllt,sa¡rlt'thrrrg lrs sri¡'ing that tlrc onll'onc of tlit'

possiblc sufficie¡rt conditions for ij tlrat is abscnt rvheneVcr I is abscnt is D

Tlrus by tltc clirt:ct ¡n(,tllo(l of irgrccl)l('l)t, il t¡¡tt'oI tlrc possiblt'r]('ccssilry

co¡i<litions for -D is actually a ¡)ccessirr'),r'or¡clitit¡¡l [or -- L, tllcrl --D is tlr¿t

nceersary conditi0n, But by thc principlc colln(rctillg ncgiltivc nCCCS§aly

corr<litit¡r¡s for -/j antl po.sitivc sufllcit'nt corrtlitit¡trs [o¡ ,Ii, tlris is tl¡c srtlrlt'

as saying if onc of ther possiblc sufficir,nt co¡rditio¡ls for.[l is actullly a

sufficicnt co¡trlition fc¡r' Ii, tlit'n D is that sufficicnt condition. Tltus lve ¿rrrivtr

at thc inversc rncthod t.rf agt'ecrnclrt.At this poilit it rnay bc ust,ful to cotnparc tl¡c clilect ¿rncl ittvcrsc ¡rletlrods

o[ aglccrrrcrrt. The cíirect r¡ictlrr.¡cl is a n]ctlro<] of íirrding nccessury coruIí'

fior¡s. To find a necessary condition for E, rvc look for a propcr§ that is

prescnt whe¡revcr E is prcscnt. Thc direct nlcthod clcl>cnds orr the follorving

principle of eliminatiou: A propcrty thtt is abseDt whe¡r D is present callnot

L" n n"""rro.y conclitioll for E. ]'hc i¡¡vcrsc nlct|od is a ¡¡rctlrod for firldilrg

sufficient conditions. To find a sumcicnt condition for E, we look for a

. This principle can be easily provecl with the aid of the preccding principlcs

guu"..i,.g .J.".rnry ¿rrcl sttfficit trt corrtliti.trs. lf *A is a nccessary condition for

-8, then E is a ncccssary condition for A. And i[ l] ls ¿ ncccss:rry condition for

A, then A is a st¡ñ'icie¡lt contlition for E. If we find l>y the direct method of

i¡gr(.(.t¡¡(,t¡l tl¡¡t -zl is rr rrt,r.t,ssirry <r,r,rliti,,¡¡ lrtr -ll, w< tti.ty ttltr.ltltlc tl¡at A iS

a srrflicicnt contlition for ll.

IV. 8 wvnnsE METHoD oF AGITEEMENT

property that is absent whenever E is absent. The inverse method depends

on the following principle of elimination: A property that is present whenE is abscnt cannot be a sufficicnt bondition for E.

In Example 8 it required three occurrences to narrow down the 6eld to D'However, the occurrence shown in Example 10 would alone eliminate A, B,

and C. In the inverse nrethod of agreement, as in the direct method, if weonly adrnit siniplc propcrtics as possiblc c<¡nditioning propcrtics, then the

.least number of occurrences that can elirninatc all l¡ut one of the possible

conditioning propcrtics is l.

Example I0

Possibleconditioning Conditionedpropcrties property

ABCDEOccurrence 1: PP

Suppose, horvever, we wish to admit negative properties as possible

conditioning propcrties. This is a rcasonable steP to take, for negativesu{Iicierrt conditio¡rs ca¡l bc <1uitc irnportant, Not staying aw¡ke rvhiledrivirrg may bc a sufficient condition for having an accidcnt. Not being able

to sc(: r)ray bc a sufficicnt con(lition for not bcing callcd for rnilitary scrvice.

(By thc prirrciplc that if .-F is sufficie¡rt for -C, thcn F is ¡lecessary for C,

thi.s rvould nrcan that being ablc to scc would bc r rtcccssary condf tion forl-icing callcd for nrilitary servicc.) Wc will intrr-¡tlucc nc8¿rtive possible

conditioning ¡rroperties as in the scction on the dircct rncthod of agreement.

But this tir¡re we will rely on thc principle of elinrination of the invcrserncthod of agrccnrcnt: A propcrty tlrat is prcsent whcn E is abscnt cannot be

a sufficicnt condition for E. In Example ll the only possible conditioning

Exarnple ll

Possible conditioning propertiesConditioned

Simple Conrple x propertyA B C D-A *B -C _D E

Occurrencel: A P A P P

Occurrence2: A P A A P

Occur¡ence3: P P A A AOccurrence4: A P P'A P

97

AA

APAAAPPAAPPAAAPA

9rl IV. lr¡ll's METTIoDS AND cAUSALITY

property that is not eliminatcd is a negative one. *B is the only possible

"r,,iitiorirg propcrty that is absent in cvcry occurrencc in which E is

abscnt, so iI oDc 0f tlrc possiblc conditioniDg propcrtics is a suflicicnt con-

dition for E, then -B is that sufficicnt co¡rdition,It ncccl not takc as many occurrcr)ccs as in Iixarnplc ll to elirninate all thc

possil;lc conclitioning propcrtic.s but onc. Two occurrcrlccs o[ th<: right kind

co.rl<,I d,, tlrt: jolr,,,s slrr¡wn in Exarnplc 12. In tltis (xatnplc, if onc of the

Exarnplc 12

Possiblc conditioning propcrtiesCorrd itionccl

Sirnplc ComPlcx PropcrtyABCD-A-B-C-DE

Occrrrrcncc l:Oc ct¡rrcncc 2:

A

P

APPPA A A

APAP P A

P

A

possiLrlc concliticrrrirrg propt:rtics is lt suf[icicilt co¡rdition for Ii, tlrcrt --B is

tlrat suflicicr¡t co¡rrliti<¡u. Irr thc i¡lvcrsc ntt:thocl of agrccmcnt, as in thc

dircct ¡uctllotl, if only sirrrplc propcrties and tlrcir trcgatior¡s are adrtlittcd

as possiblc conditiorring pro¡lcrtics, the¡r the le¡st nunrbcr of occurrences

tlr¡t can clirninatc all but onc of the possible conditioning propcrtics is 2.

Wc, runy furthcr cxtcnrl thc invcrsc Iltcthod of agrccmcrtt to ullorv cott'

jurrctiorrs of sinrplc propertics as pU,siblc conditiorling propertics. For

c.xa,rrpl", supposo wc arotolcl tl¡at cating good food arul getting plcnry of

rcst alxl gctiiirg ¡ nrodcratc ¿¡mou¡rt of cxcrcisc is ¡ sulficicnt condition for

good hcJth. Thc invcrsc ¡Dcthod of agrce nrt'rtt would supporf this contcn-

Iion if *c found that whcncvcr goocl hc'alth rvas abscnt, this cornplex con-

clition was also absc¡¡t (that is, if cvcryotie who was in poor licalth had not

c¡tcn ¡¡oocl food or lr¡<l not gott(.n crloullh rcst or had rlot cxcrciscd). The

i,rrcrs" ntctl¡ocl of agrcclncnt would riisproo¿ tltis contc¡ltion if an occur-

rcncc wi.rs [ound whcrc goocl hcalth rva.s absent and thc complcx condition

was Prcscl¡t (that is, if sonreorrc was founcl in poor hcalth who had eaten

guoi fon,l, and gottcn plenry of rcst, and gotten a modcrate amount of

cxercis<:).Let us look iit two cxamplcs of the inversc mcthod of agleenrent where a

conjurrction is aclrnittccl as a lx.rssiblc conditionirrg property. Irt Exanrplc 13

all ihc possible conditioDing propcrtics exccpt thc coniunction-are climi-

¡ratccl. 'l-hr.' only possiblc conditioning propcrty that is abscnt whenevcr E

is ¡bscnt is the córnplcx propcrty C&D lf one of tlre possible conditioning

IV. 8 NvsRsE METrroD oT aGREEMENT

Example 13

99

Possible conditioning properties

ASimpleBCD

ConditionedComplcx propertyC&D E

P

P

PA

PA

propcrties is a sufHcient condition for E, C&D is that sufficicnt condition.In Exanrplc 14 the coniunction itself is eliminatcd. If one of the possible

conditioning propcrtics is a sufficicnt condition for D, then D is that suffi-

cit:nt co¡rdition. We shall not explore furthc¡ the treatment of complexpossible conditionir:g propcrties by the inverse nlethod of agreement.s But

you cannot go wrong if you remember that the principle of elimination i¡r

Example l4

Possible conditioning propertiesConditioned

Complex propertyDB&CE

Occurrcnco I l

Occurrc¡rce 2:

P

P

the inverse method of agreement is: A property that is present when E

is absent cannot be a sufficient cor¡dition for E.The parallels drawn between the method of üc master detective and the

direct rnethod of agreement hold also for the inversc method of agreement.

It should not be thought that the ficld of possible sufficient conditions can

always be narrowed down to onc, for a propcrty can have seve¡al sufficientconditÍons. We should also be prepared for thc eventuality that the ob-

served occurrenccs will eliminate all possible conditioning properties in the

5 The inverse meüod of agreement can be expandcd to include simple pro¡>erties,

negations of simple properties, and conjunctions of simple properties and theirnegations as possible conditioning properües. Thcre is no need to worry ¿bolrt

disjunctions, since a disjunction, FvC, is a suficient condition for E íf and onl:¡ ilF is a sufficient condition for E and G is a sufficient condition for E. For this

reason if one can discover all t}¡e sulñcient conditions that are not disirrDctiotrs,

he will automatically have all the sufficient conditions that are disjunctions

Occurrcncc l:Occurrc¡rce 2;

AP

AA

AA

ASimplcBC

A

P

AA

AP

P

AAA

l(x) IV. rvul¡-'s METIIoDS AND cAUSALITY

lrrt All,'¡ ;rll, ,r srrllit'ir'¡rt condition may Irot have been included iu thc list of

¡r,r,,rrlrlc lorrrlittottirtg ltropcrties. ln such a casc wc would havc to col)struct

.r nro¡r' r',rrrrprclrt'nsivc list of possible conditioning propertics. In somc

cirscs tlris r¡rorc comprehensivc list might bc constructed by considcring

,'orrr¡rllx prolx'rties that werc not included in thc original list.Irr l'ixrrrrr¡ Ic 15 thc fivc occurrcnces show th¿t none of thc possible con-

,lititrrrirrg ¡rro¡rct'tics catr bc a sufllcicrlt conditir¡n for [. llut thcy suggcst

tlr:rt tlrt: c«rrrrplcx prol)crfy B&C rnight be addcd to tl¡t'list of ¡rossiblt't:orr-rlrtiorrirrg propcrtics. Tlris propcr§ is illrvays absc¡rt rvlrc¡t I is absc»t.

'l'lrc situatio¡r, horvever, might be morc problcnlatic. Tlrc obscrved

Example 15

Possible conditioningpropcrtics

ABCD

Conditioncdi)ropcrtY

E

Occurrcttct: IOccur¡t'¡tcr'2Occttr¡t rti, ll( )t t ru ¡ r't¡r'r' 'l()ccr¡¡ r t ¡rt r' 5

AI'AI'AI)[,.\A

l,l'Á

A

AP

P

I)

l)A

()(('r.¡il( ¡¡L'L-5 ilugl)t ¡rol ()t)ly ('l¡t¡linirtc itl! tl¡rr ¡il¡rflf Pr0!crti('5 irl ths li¡t but

:rlso ¡li tlrt: (contingt'nt) corn¡rlt'x ¡rropcrtics that can l-re constrt¡ctcd out of

tl¡t,rrr. Suclt is tlre case in Exarrrplc i6, It is irnl;ossiblc to discovcr in this list

Exanrplc l6

Possiblc corrtlitiotrtrrqproPr ' ti( s

AlJC

(louditione d

l)r()pcrfyti

Occurrc¡rcc I r

Occurrcncc !:Occrtll etrcr: 3:

Occulrcncc 4:()ccurrcr.¡cc 5:

Or'cu¡ rcrrct: 6:()r'currctrct' 7:()r'< rrrrt:rtcr'8:

PPPPPAPAAPAP

P

A

P

AP

AP

AAAAAAAA

any (contingent) complex property c$nstructed out of A' B' arrtl C wl¡icl¡

i, not "llrninated

as u ,r?fi"ii¡t condition for E by thcse eight occ'rrr.

rences. ln such a case some new simple properties would have to be added

to the list of possible conditioning properties'

Exercises:

L In Example I which of the following prop€rties are eliminated as suficient

conditions Íor Eby @currencei t' 2, and 3?

e. A&8.f. B&c.g. AeD.

2. In Erample l0 which of üe following properties are eliminated by occurrence l?

a. A. e' A&C'

b. B. f' B&C'

c. C. g' A&C'

IV.9 ¡lErHoP oF DIFFEIIENCE

a. A.b. B.

c. C.d.D

d. D.

3. In ExamPle lI which of

mcurre¡rces?a. A&8.b. B&c.c, B&D.

4. ln ExamPle 13 are there

othcr thrn C&D which ure not

l0r

h. A&D,

üe following proP€rties are eliminated by the four

d. A&D.e. A&C.f. C&D,

any coniunctions of the simple properties listed

eliinÍnutud l-ry oceurrences I nnd 9?

IV.g. THE METI{OD OF DIFFERENCE' The direct method of

agrccnrc¡Jt was a rneüod for finding the necessary conCitions of a given

prol)crty. The inverse method of agieenrent was a rnethod for 6nding the

iuffi^"i"nt conditions of a given property' Suppose' however' that our in-

qri.y f,"a a niore restricüd goul' Suppot" that wc wanted to find out

which of the propertie, pr"r",,I in a ceiiain oc,currence of the conditioned

Or"n"ra, are sufficient conditions for it' To illustrate, let us suppose we

ñna " á"ua man with no marks of violence on his body' In trying

-toá"t".-irr" the cause of death, we are looking for a suficient condition for

death, But we are not looking for ang sufficient condition for death. Being

;; ;;", by a steamroller is "a

sufficient condition for death, but that fact

is i¡relevant to our inquiry since this particular man was not run over by

a steam¡oller. The conditiáning p'operty "being run over by a steanrroller"

i, uUr"rr, in this particrrl", o"Jt'i'""ce' What we are looking for is a suf -

,&

Lt¿2 IV. u¡¡-¡.'s M[Tl¡oDS AND cAUsALrry

ficicnt condition for dcath_amor)g thc propt:rtics that arc prcsont in trrisparticular occurrc'cc i. rvhicl¡ dcath ii prcsc,rt. It is tl¡.is rort of inquiryfor wl¡ich the nrctlrod of clifft:rcncc is clcsignerl.

It i's i¡nportaut to notc rvhy no analogáus question can bc raised for.cccss¿rry conclitions. It follow.s from thc rlt:fi¡riiion of rr.ccssary conclitio¡lth¡t all thc,cccss.ry co.ditio¡rs for a givcn propcrty mr¡st bc prescrrtrvhcncver-that ¡rro¡,crty is prcscnt. If losi of c.¡nsciousncss is a ,rcc.,s.r,rrycondition for dcatl¡, it rvirr rrc prcscnt in cvery cas. of clenth. Thc c¡,.sti..s"wlrat pronertics urc ¡¡cccs.§ary co¡¡clition.s for E?,'ar¡d ,.wI¡ich of thr,propertics tl¡at arc prcscnt in this particr¡lar occurrencc of I arc ¡rccessaryco¡lrlitions for D?" Iravc cxactly thc sarnc a¡rswcr. I. co,tr.ast, rvlrcn a givcrrPro¡rcrty is prcscnt, sr-r¡nc of its st¡lllcic,t conclitio,s l,ay bc absc,t. üa.ypcople who die have not_bcc¡r run ovc,.by a stcar,roiler nor bccn crcc.pí-t¿ltcd o¡1 thc grrilloti,c. 'fhc qrr.stion "úh¡"h of thc prop.rtics that irr.r.prcsc.t in this particulirr occurr.¡lcc of L- arc st¡fficicnt co¡rrlitions for ^0?',rvill, in gencral, lravc

^ shortcr list of propcrtics as its answer than tlrt:question "Which propcrtics arc sufficient conclitions for E?,,L lix,,r¡ric l7 occu¡rt'¡rcc o dot,s ,ot cri*iratc arry of thc pos.siblc con-<iiti.rrirg ¡rro¡>r'rti.s ¡s srrffici<'¡rt cr¡¡¡<ritir¡¡¡s for D. Iit¡t if thc qtrcstion ofirrtt ¡r'st is "wlricl¡ of trrr' ¡rr',¡r.r'ti.s tr¡;rt a'r, prr,st:,t in occurrcnct, o ar(,

lixarn¡rlc l7

I'o¡sil rl, r'orrrlitiorrirrgpt oltr,t'tit,s

ABCD

Co¡rrlitirlrrctlpro¡rcrty

EOccurrcr)cc o: I) A Í) P

.sulllcicnt co,ditio¡rs for D?" thcn thc candidatcs arc lirnit.cl to A, c, a¡rrlD' Lct us now look for otlrcr occurrc.ccs that wiil narro* ,ror.u, th,, §crd.Thc ¡rrincipl. of clinrination is thc s¿rmc as that cmployccr in thc inversc¡ltcthocl of agrccrncnt: z\ propr:rty tlrat is prcscnt whcn L'is al¡scl¡t cilnr¡otbc a sufllci.¡¡t co¡iclitir,¡r for c. Thcrcforc lct us rook for aclclitio¡lar occur-rcnccs wl¡cn E is absr:.t. supposc that the results of our investiga,ion ,."¡¡s shc¡rv¡r irr Iixarrple 18, I, this cxanr¡rrc occurrcnccs I and 2 criminateA and D as st¡fHcicnt co,critio¡rs for [. of thc possiblc .n,,,litrourng frop-crtit's tllr¡t w('rc l)r'(',i(lrrI i¡r,ccil.t.¡rct, ", o.ly ó i, l"ft. -fhu.s if o,,""of tü.,¡rossilrL'corrrlrti.rrrrrg ¡rrr¡rt.rtir.s tl'rt rvl¡s I)r(.s(,r)t i¡l n.cr¡rr.rrcc o is a

'rrlli<¡r'¡¡l rorlri¡r¡r'¡r lu¡ 1. rlr,'¡r c ls tr¡.rt surficit'r¡t co¡¡crrtit¡¡¡. Nott'tl¡at Blrrt¡-lrl r1.,,1,, .r srrllr, rr rrl ¡r,¡rrlrlr,,r, f,,r /.. l¡r¡l rt rs ¡¡ot ()r(, r't. rv,rrltl lrr.

IV. 9 vrrrroD or DrrFE¡TENCE t03

Example l8

Possible conditiouingproperties

ABCDOccurre¡lce o:

Occurre¡rce l:Occurrcncc 2:

D

I)A

AAA

P

AA

P

AP

i¡rterested in, since we are looking for a sufficie¡lt condition which w¿spresent in occurrenoe o. Occurrcnces I and 2 cli¡ninate candidates incxactly the same way as in the i¡ivcrse mcthod of agrcenrcnt. In theinverse nrethod of agreentcnt, however, we started with all the possiblecondtioni'g properties as candidates, while in thc method of diffcrence,wc start rvith the possiblc conditioning propcrtics that are ¡trcsent in aparticular occurrence in which tl¡e conditioned property is prcsent. (Weshall always call the.occurrence that defines the candidates "occurrence o"

and rvill nur¡ber as before thc occurrc¡rces t-liat elimitwtc some of thccandidates.)

If only sirnple propertics a¡e adnrittcd as possible couditioning prop-crtics, then occurrencc o nright orly lcavc onc c¿rndidutc as show¡r i¡¡Exarnple I9. In this t:xarnplc the only possil-:le conditioning property

Iixornplc l0

Co¡rditio¡redpropcrty

E

P

AA

Possible conditioning¡rropcrties

ADCI)Occur¡e ¡rce o: I'

present is D, Thus without looking for eliminating occurre¡rces it maybc concluded that if one of the possible conditioning properties that isprcserlt in occurrcncc o is a sufEcicnt condition for E, then D is thatsufficient condition. In fact, if only simple propcrties are admitted aspossible conditioning properties, then occurrence o nright leave no ca¡r-didates whatsoever.

But there is no reason why negations of the simple properties cannot bead¡nitted as possible conditioning properties, as was done in the treatrnclto[ the direct and inverse ¡nethods of agreement. If both sirnplc propertit.sarrd thcir ncgations are allowed as possiblc conditioning Propcrties, thc¡),

Conditionedpropcrty

E

PAA

l().1 lV. rul¡l-¡-'s ¡vrE'IIIoDS AND cAUSALrry

ur ,ur ()( r'unlnt't' o, r'x:rt't[y lurlf of tlre possiblc conditioning propcrtíes will1,,'lt.ll ;r: r'.¡¡¡,lirl¿rtts,.sirrce cxactly half of them must bc prcsent in anyr¡( ('ur rr'¡¡( ('. l'rr¡ tlllr (/ccurrcl¡ccs l¡iust bc sougllt thcr) in <.¡rdcr to cli¡nirtatcr,,u rl oI t lrcsr' ¡rropcrtics, as in the inverse method of agreernent.

l,r.t rrs rlctcrr¡¡inc which of thc properties present i¡l occur¡cnce o a¡esr¡llir:it.rrt ,'o¡¡ditio¡rs for E. In Examplc 20 D, *A, -8, and -C are the

Iixarnple 20

Possible conditioning propcrties

SimpleABCD

ConditionedComplex property

-A-B-C-D E

Occurrence o: A A A P

Occurrcnce1: A P A P

Occurrencc2; P P P A

P

AP

possible conrlitioning propcrtics preser)t in occurrence o and are the candi-dates. Ocrurrc¡rccs I arrt.l 2 eli¡¡li¡rate D, -4, artd -C. Tlre only one oftlrr candid:rtcs u,lrich rc'¡nai¡rs ¡s -lJ, leadir¡g to tlrc corlclusior) that if one

of the possiblc conditio¡ri¡rg propcrtrcs that is prcsent in occurrence o isa sL,ilicicnt conilition for [, tht'n -B is tllat suflicicnt condition.

As has l-¡cc¡l shorvrr, if sirrr¡rlc ¡rro¡rcrtics a¡¡d tl¡cir ncgatiorts arc allowcd

as possll-rlc conditioding propcrtics, thcn occu¡rence 0 will leave cxactlyhalf of the possiblc conditioning properties as candidates. The least nurn-

ber of additional occurrences needcd to climinatc all thesc candidates butonc is one, if that onc occurrcncc is o{ the right kind, as shorvtt in Exallr¡rle21. Tl¡c possible conditioning propcrtics tliat arc plesent ir, occurrence "constitute ihe ca¡rdidttcs, atttl tltcy arc B, C, -4, and -D. Occu¡'rence Iclimi¡r¿t, s B, -A, and -D sincc thcy are present when E is absent. Thus if

Exarnple 2l

Possible conditior:ing propertiesConditioned

propertyE

Occurrcncc o

Occt¡rrcrrcc l

P

P

AA

P

AP

AAP

P

AA

Sinrple CornplexABCD-A-B-C-D

APPA P A AAPAAPAP

IV. 9 urrroD oF DIFFERENcE r05

onc of the possiblc conditioning Propcrtics that .is prcscnt i¡r c¡ccurrc¡lce o

is a sufficient condition for Ij, then C is tlrlt sufficient co¡rdition'

lf you look closely at Exarnplc 2J., you will nr-rticc that the rcason all

car-rrliclatcs but c were eliminated is that c is thc only possible condition-

ing property that was both prescnt in occur¡ence o (where E was present)

ancl abscnt in occurrencc 1 (whcre .E was abscnt). All other possible con-

ditioning propcrties prcscnt in occurlcncc o wcr'c also present in occur-

rence l,-rvlreic E was abscnt, and thus wcrc elirni¡rated. lt follows that

all the possil-rlc conditionirrg propertics that wcrc absent in occurrence "

were also absent in occturence I (cxcept ior -C). In other words, there

rvas o¡ly o¡c clrangc in the prcsc¡cc o¡' abscttcc of t|c possiblc,co¡ldi-tioning propcr.ties fiom occurre.ce o to occurrc¡tce l: the c¡ange from C

lr"ing pi'"s"rt and -C bei.g apsent itt occurrc,ce o to C bcing absent

an.l .-b bci.g prescnt in occurrcncc 1.,, Whc¡r both sinrple propcrties and

their ncgatioris arc allo*er] to be possiblc conditioning properties in the

method o[ differcncc, this is thc only way in which oDc clinrinatirlg occur'-

rencc can elirninate all but o¡ie of the possible conditioning properties.

This rather special case of thc nrcthod of difference is what N{ill describes

as "thc nrctliod o[ c]ifferencc." Howcvcr, N{ill's view of the mcthod of

di{Icrcnce \vas too narrow, for, as has bcen sltolvtl, thc rncthod has appli-

c¿rtion when seve¡al eliminating occurrences, rather than iust one, narrow

down üe field.Thc rncthod of dil[crcnce Dray be cxpanded, in cxactly the sarne way as

in thc invcrsc mcthod of ugrccnrr:nt, by ullowitrg coniullctions of lirnplcpropertics as possible conditioning properties. Sinrply remember that we

staü with a particular occurrrellcc, occurrcncc ", in rvhich E is present'

Thc candiclatcs will thcn bc all thc possiblc coDditioning ¡lroperties that

arc prcscnt in occurrc¡lcc.. wc tlrcrr look for occurrcnccs where E is

al-¡sent, so tl-¡at some of thc c.,ndi.lrrtt:s caD I-¡c eliminated. A candidate is

climinatcd if it is prcscnt iD an occurrcncc rvltcrc L' is absent, since a

sufficient conditio¡r for E can¡ri.¡t be prcscnt rvhcn E is absent. If all can-

clidatcs but one are clintinatcd, wc clil concludc th¡t, if one of the possible

conditioning properties prcsc¡rt in occurrence o is a sufficie¡lt condition

for E, then the rernaining candidate is üat sufficient conditiorr' But, as

in the dircct and inverse nrcthods of agreernent, it is not always possible

to narrow down the field to one candidate, lrfore than one sufficient

¿ This one change iD the possible con<.litioning propcrties corresponds to the

chunge in tlre conditi,¡,¡.r.1 propcrty: E is prcscnt i¡l occurrc¡¡cc o u¡¡«l absc¡¡t in

occurrence 1

r06 IV. rr,ull's ME'rrroDS AND cAUSALtry

condition for E may l:c prescnt i¡l occurrence ". When a man is simul-taneously behcadcd, shot through tlrc hcart, and exposcd to a lcthaldose of nerve gas, several sufficient condilions for death are prcsent.On the other hand, the climinatirlg occurrences might eliminate all tl¡ecandidates. This would show üat the list of possible conditioning prop-erties did nót include a propcrty that was both prcsent in occurrcncc "and a sufficicnt condition for [. In such a case other factors that werepresent in occurrence o must be sought and included in a new, expandedIist of possible conditioning propcrties.

Exercises:

Consider the follorving example:

Poss jble conditioning propertiesConditioned

propertyE

Simple ComplcxAÜCD -A -B -C -D

Occr¡rrencc": PAAP

l. Wlrat arc tlre c¡¡¡<iidatcs?

2. Descril.¡e an elimirr:rtirrg occurrence that would elimi¡rate all thc candidatesbut one.

3. I)eseribe aa eliminílting oecurre ncc thnt rvould olirninels all the e andidate¡,4. Describe three eliminating occurrences, each of which would eliminate

exactly one of the candidates.

5. What wor:ld yorr corrclr¡de if you obselved the occurrence that you de-scribed in Exercise 2?

6. What would you co¡rclude if you observed the occurrence you described irr

Exercise 3?

7. What would you conclude if you observed the three occurrences youdescribed in Exercise 4?

8. What would you conclude i[ you observed all the occurrences that youdescribed in Exercises 2 and 4, There are several correct answers to Exercises 2

and 4, and the answer to this qucstion will depend on which ones you chose.

IV.I0. TIIE COI\IIIINED IvIETIIODS. Sometimes a proporty is

both a ncccssary ancl sufllcicnt c«.¡r¡dition for another property. It has

alrcady bcen pointcd out thlt in football tlte cornplex propurty "rnaking

a touchdow¡l or nr¡king u field goal or rnaking a convcrsio¡r or rlaking

AP

IV. l0 r¡rB co¡vftllNED NfEl'ltol)s

Exanrple 22

t07

a safcty" is both a necessary ancl sufficicnt conditiorr for scoring. tvlcdical

auttroritics thought until rcccDtly thxt stoppagc o[ thc lrcart for more

than a few minutes was both a nccessary and sullicicnt conditon for death.

In elerncDtary physics. bcing actcd on by a nct forcc is both a nccessary

and sufficie¡it condition for a chhnge in a body's vclocity. Since there

is a mcüocl for fincling,ncccssary conditions-tlrc tli¡'cct ¡nctl¡od of agree-

mcnt-and fwo methods for finding sufficicnt conditions-thc invcrse

mcthod of agreemcnt and the method of differcucc-they may be com-

bincd in ordcr to find conditi<-¡¡rs that ¡rc both ncccssnry and sulficicnt.

In Exarnplc 22 tlre dircct nr¡d i¡tversc m<:thotls of tgrccmcnt are com-

Possiblc conditionirtg propcrtics

Sir»plr:t\BCD

CJonrplcx

-A *B -C

Conditio¡rcdpropcrty

-DEOccurrc¡rcc l: l'Occttrrt'rlcc 2: AOccurrc¡rcc 3; A

Occurre¡icc ,1: P

APPPPAAA

AAl,PPPAA

PAAP

A

P

P

PP

AA

PA

AP

bineei i¡lto what is ealled tl¡c cloublc nwtlñcl oÍ agrcement. Occurrence Ielimi¡rntcs ts, D, *A, and .*C and occurre¡¡ce 9 clinrilrates A, --8, -C,ancl -D as necessary conditio¡ts for E iD accorclirnco with thc direct

method of agreement, for thcy are abscnt when .0 is prcsent We can

conclude then from occurrences 1 and 2, by thc direct mcthocl of agrce-

ment, that if one of the possible conditioning propcrtics is a nccessary

conclition for [, tlro¡r C is tl¡at nccossary condition. In accordancc with

tltc inoerse ¡nctlrorl of agrcetllcut, occurt'c¡lcc 3 climinatcs B, D, -4,and -C at¡d occurrcnce 4 climirrates A, -8, -C, and -D as sufficient

conditio¡rs for I, since thcy are prcscnt when /j is al¡sc¡rt. Tlris again

leaves only C. We can concludc fronr occurrcnccs 3 and 4, by thc i¡rvcrse

ntcthod of agrccnrcnt tliat i[ onc of thc possiblc conr]itioning propcrties

is a suf[icir:nt condition for E, thcrr C is thrt sufficicnt condition. Putting

thcsc results togethcr lcacls to the conclusion that if onc of tlic possiblt:

conciitionir-rg propertics is both a ncccssxry a¡rd a suflicit:trt cor¡ditir¡n

for.L, thcn c is that ltroPcrty. IIowcvcr, a strongcr conclusic¡D Inay bc

drawn: If one of thc possíble conditioning properties is a ncccssary

condition for E, ancl one of thc possible conditioning Propcrtitrs i.s ¿t

I08 lV. v¡r-¡.'s METIIoDS AND cAUSALITY

',rrllit;rr:¡rt cr.¡rtditiotr for E, tlicn one and the same possiblc conditioning

¡rropurty is botli a necessary and sufficient conditiorl for E' and that

¡rro¡rcrty is C.' ,l:lr. ioint metlrcd of agreement and. difference, which is illustrated in

I-*r*pi" 23, combines thá direct method of agrecment and t¡e mcthod of

diflcre,rce. The 6¡st stcp is to apply the method of difiercnce to Exam¡rlc

23. Occurrence o sets up as candidates for thc sufncicnt condition for E

those properties tlrat are present in occurrence o, namely, A, C, -B' and

-D. But occurrence I shows that neither A nor .-B nor -D can be a

sufficient condition for E, since they are all present whcn E is absent'

Example 23

Possible conditioning properties

Simple ComPIex

A B C D -A -B*C_D

C«rrrditio¡reclpropertY

E

Occurreucco: P A P AOccurre¡rce l: P A A AOccurrence 2: A P P P

AAP

P

P

A

AP

A

P

AP

PP

A

lhis leaves only c, Thus wc can conelude from oceurrcncc o and sccur-

rcncc 1 that i-f onc of the possiblc conditioning propcrties prescnt in occtu-

re¡)c,, o is a su{ñcient "oÁdition

for l:, then C is that sufncient condition.

Now let us apply üe direct meüod of agreement to Example. 23'

O""u.r"r". ' *"y 'U"

uscd again since, in accordance with the di¡ect

method of agreement, it eliminatcs B, D, -A, and .-C as nccessary con-

ditions for E. A.rd occurrence 2 further eliminates A, -B' and ,-D as

necessary conditions for E since üey also are absent in an occurrence

*¡".".e'it present. This leaves only C' So from occurrence o and occur-

rence 2, byih" dir""t meüod of agreement, we can conclude that if one

of tl" poríiUte conditioning properties is a neces§ary condition for- E'-th.en

c is üat necessary "orditlo-r,.

iutüng the results of the method of dif-

ference and the direct method of 'agieement together leads to the con-

clusion that: If one of the possible conditioning properties present. in

occurrence o is a sufficient condition {or E and if o¡re of the possible

condirioning properties is a necessary condition for E' then one and

,1," ,^rn" pJtriUf" conditioning propcrty that is prcsent in r¡ccurrcucc. o is

both r nc'lcssary and sufiicieit-condition for [, and tlrat property is C'

IV, l0 rHr coMBINED METHoDS 109

In con'rparing the exanrpic of ürc joint ¡¡lctlrod of agrecrtrcnt a¡¡d drt-

ferencc rvith thc previous cxamplc of thc doublc nlcthod of agrccnlcnt,notc th:rt occurrenccs o, l, and 2 of Exarnplc 23 arc tlic samc, rcspcctively,as occurrcnces l, 4, and 2 of Example 22. Notice also tl.¡at Example 22,

usirrg the dr:ul¡lc nrctliod of agrcetnctrt, takcs four occurrc¡lccs to narrow

clown tlie ficld to C, while Example 23, using the ioint method of agree-

rncnt and dific¡encc, takcs only thrcc occurrcnccs. Docs tllis rncan tllltthe joint rnethod of agrccnrent and difference is, in sonre wily, a morc

cf[icic¡rt r»ct]rod tha¡r tllo doublc ¡rictl¡od of agrccrncnt? Not at all. Lcss

occurrer)ccs arc ncedcd iu Examplc 23 than irt Exarnple 22 bccausc tlleconclusion draw¡r from Example 23 is weaker than that drawn from

Exrimple 22. Fronl Exanrplc 23 wc Inay concludc that if onc of the

possible conditioning properties whiclt is present in occurrenc¿ o is a

sufficicnt condition for E and one of thc possible conditioning propcrties

is a ncccssary condition for [, thcn C is both tlrc nccessaly arld thc strf-

ficicnt condition. If wc rvallt to rentove tlie rcstrictiou "which is preseüt

in occurrcncc o" then thc cxtra occurrellcc that appears in Exanrplc 22

is needcd, and the doul¡lc nrcthod o[ agrecnrcnt Inust be used. Conse-

r¡ucrrtly front Exanipl c 22 tlte strollger conclusion ntay be drrrwn that ifonc of the possible conditioning properties is a sulEcient condition for Eancl one of the possible conditioning propcrtics is a ncccssary condition

for E, then C is boü the necessary and thc sufHcicnt condition.Wht:ther ¡he joint ¡)lathod of agrscrtic¡¡t ¡¡ltl diffcrc¡lcs or tllP dsuble

nlcthod of agrccnrent is choscn rlcpcutls on whrrt prcvious knowlcdgc t*'c

havc. S,Lrpl:osc wc l¡uvc obst:rvcd an occurrc¡lcc lrld havc good rcason to

bclicvc that or¡e of the possiblc coDditio»ing plopcrtics which is pr'cscnt in

that occur¡.c¡rce is a sulñcicnt condition for E. lVc would thcrl dcsignatc tliltoccurrencc as Occurrence o aDd procccd witl¡ thc joint nrctlrod of agrcc-

¡nent a¡rd difference. If, howevcr, we had good tcasor'¡ to believe oilly that

one or another of the possible conditioning propertics is a sufficient co¡r-

dition for E, we would have to rely on the double method of agreernent.

Thc cornbined methods are cqually efficient, but thcy arc appropriatc in

diff erent circumstauces.The conrbined methods may be expandcd to include other conrplex proP-

crties (disjunctions and conjunctions of simple propertics a¡rd the negations

of sinrple properties), but a discussion of thcse more involved forms of

Mill's rncthods belongs itt more advanced tcxts. llcmcmber, howevcr, that

cvcr.ything that has beer.r said about Mill's methods, and everything that can

Lr. ,ri,l about thcir rnort: involvcd foilDs, rcsts on two sinrplc principlcs

o[ clinri¡lation: ,

..{

ll0 IV. u¡¡-t's t!{ETHoDS AND cAUSALITY

Exercises;

Suppose you have observed the follorving occurrellccs:

Possiblc conclitioning propcrticsConditioned

Complex Property

-A-B-C-DEOccurrence l:Occurrence 2:

Occrrrrc¡rce 3:

A

A

P

AP

A

P

P

A

P

P

A

AI]CD -A-B-C-D E

AAPP

i, A nccessary condition for n cannot bc absent whcn '0 is prcscnt'

ii, A sufficic¡lt condition for E cannot be prcscnt rvltcll I is absent.

These hvo principles are more important to remember than Miil's methods

ttr",nr.lu.r, und ih"y should alrvays be borne i¡r mind rvhen a mass of

data is being analYzed.

SimpleABCD

P P AAPAAAAPPP

P

AP

L Supposc yott know tl¡¡t one of the ¡rossible conditioning propcrtics ts a

,,"."rr"ry'.u,,,Jrtio¡r f,,r [. \Vhich o¡¡o ls iti \Vl¡at occurrc¡lces did you usc and

ri l¡rtlr r¡l lrlrll's nrctl¡txjs dttl ¡rtu a¡r¡lly?

3. Su¡r¡xrsc lt,tt kttttu'tl¡¿t o¡¡e t-rl thc liossr[;lc coutlitionirrg prol)crtics wliiclr is

¡rr"r",.t i,, u""urr"n"" I ts a sufGc¡c¡¡t co¡ltlitron tor Ii Wl¡ich one is jt? What

,r."urr",,.", drtl you usc ir¡¡d wltic-h t¡r¡c of lvl¡ll's nrctl¡ods did you apply?

3. Sup¡rosc you know that óne of the possible conditioning properties is a

n""ers.ry'cn,.,.lltion fu, E and that one of the ¡rossible conditioni'g properties

which is present iI) occurrence I is a sufñcient condition for E' Do you know

whether áne possible conditioning property is both a necessary and sufficient

condition for '¿l lf so, which on" ir it an<I which orre o[ Mill's methods did you

use?

4. Suppose you know that one of the pt' siblc r:onJrtioning l'¡''l:crties is a

,.I"*rr*y'"onaitiun fo, E. you also knorv that o¡¡e of tl¡e possible conditioning

frop"rtlá, is a suf§cient condition for E, but you do Irot know whether it is a

prop"r,y that is present in occttrrence l' Furtl¡ern¡ore' you have obse¡ved an

additional occurrence ;

AAOccrrrrence 4

Do you know whether one possiblc conditioning property is both a necessary. and

a su{Ecient condition for ¿i lf so, which one is ii and which one o[ Mill's methods

did you use?

IV. Il app¡-lcATtoN or NrrLL.s lvrE'rIIoDs It1

5. Suppo.se you had only obscrvcd occurrences I and 2 bt¡t you krrcw that one

,rf tlrc lir».stble corrclitiorrirrg ¡rlopcltics s'irs btttlt ll rlt'ccssilt)'l¡lrcl ¡ stlf]icic¡¡t

conditi<.)I¡ for C. Using thc trvo principlcs <-¡f cljrninatrr-¡n, ci¡n vou tcll rvhich one it

is?

IV.ll. TI{E APPLICATION OIr ivlILL'S l\fETHODS' Tltc con-

clusions rvc drcrv fronr valious applications of i\lill's rlethods alrvays began

rvith phrirscs suclt rrs "lf orrc of the possiblc conditio¡lirlg propt'rties is a

nccessliry co¡idition for E . . . ," or "lf o¡]c of tlre possible coltditionillgpropcrtics §,lticll is prt'scnt iu occurrcncc o.is rr.st¡liicic¡¡t couditiot¡ fo¡

E...," ctc. It would scem that otrr co¡lfide¡rcc that lvlill's methods have

found a ncccssary condition, or a sufficient conditioll, or a ucccssary aud

sufficicnt condition dcpcnds on our confidct'rcc thirt thc list of possible

couditioning l:ropcrtics colltaills the rcquisitc kirtd of condition. But how

can wc bc,surc tl¡at this list docs coDtirin thc type of condition bcing sought?

Onc suggcstion nright bc to i¡rcludc o/l propclties as ¡;ossible condition-ing propt:rtics rrucl to rcly orr so¡nc ltlirrci¡tlc of tlrc uniformity of natule toguarantee üat each conütioncd Prol)erty has so¡ne nccessary and some

sufllcicnt conditions, Thcre arc nl¡ny thi¡rgs \4,rong with this suggestion,

l>ut thc r)rost l)rrrcticirl objcction is tltat thcrc alc sinrply too rnatry Propeltiest() t¡kr,ir¡tr¡ ecct-)t¡¡lt. Ir,cn if \\,(].u'('i¡ttclcstcd only in tlre propcltics tlrat

arc pr('sc¡lt irr u givcn ocCLu'¡'(,!tcC, as irl tl¡c nit:thod Of difTcrcncc, IrOt all

proltcrtics that arc pt'csctlt cii¡r bc co¡lsiclcrcd, ln atry occut'x:¡lcc tllerc arc

eouirtless propgtics pr¡§ent. W|ctr you slr.uz¡, t|eru ttrc hund¡eds of

chc¡ltical rcactiotis going on u'ithin your body: various trlcctric¿rl curlellts

arc cir.culating itr your rrcrvc fibcrs; you i¡l'c bciDg l,:onrbirlclecl by vtrrious

ty¡re-s of "1""t,n,.,.,rg,]"tic

radiation; divcrsc happenings, grc¿rt and-small,

.uiruu,,,l 1.orr, It rváulcl b. an intpossiblc tusk to mcirst¡r('and catalog all

thcsc thirrgs ¡'¡rcl r:lÍr¡-ri¡l¡tc thcur, onc by onc, by IVlill':; nlcthods in order to

find r suflicic¡lt conclitirtr, lor slr,'r'zitrg.For ñlill's rlrctl¡ods to bc of rlll)/ !.tsc, thcrc rl.¡t¡st l:c so¡nc rv¡y of ascer-

tairring whiit f¿rctors are likt,ly tO bc rclt:r,¡rit to thc conditio¡tccl property

i¡r which rvc are ir¡tercstcd; thCrc mt¡st bc sontc w¿ry of scttirlg up a list of

rc¡sonal¡lc lcngtlr of possiblc cortclitiotrirtg proptrrtics rvhich prolrably con-

tains tltc ¡lcccssu¡'y or sr¡fficicnt conclitions ltt'irlg sought. 1'hc only way to

rlo this is to irpply incluctivc logic to a pIr",,iously ¡crluilcd bocly of cvitlcllcc.

Mill's ¡nctlioás o,.t of no Lrsr: r¡¡tless rvc lrllcady havc sor¡.tc inductive knowl-

edgc to guide us in sctting up thc trist'of possiblc conditio¡lirlg propcrtics.

Ñtill's nrcthods ar.c uscful iD scicncc, but thcir uscfulncss dcpcrlds on

inductivcly based judgnrents as to what factors arc likcly to bc relcvar.rt to

I l:.1 IV. rurll-¡-'s MIITHoDS AND CAUSALITY

., ¡iivr.rr cr¡r¡tlitioned property. Of course, inductively based judgments are

¡rot rr¡l¿lliblc. We may be mistaken in beüeving üat the list of possiblc

t.orrtlitioning properties contains a necessary or a suftcient condition. The

occurrcnccs observed may eliminate ali of üe possible conditioning pro¡t-

crtics. If this happens, all the evidence at our disposal niust be reéxamined,

ancl perhaps new evidence must be sought, in o¡der to ffnd ncr'r' properties

that are probably reler'ant to the conditioned property rtnder investigatioD.()ncc thc inductivc judgrncnt has bccn made as to which additional prop-

erties trust be considered, Mill's methods may then be rcapplied. In tlresc¿rrch for ¡x:ccssary and sufncicnt conditions, Mill's nrethods arc part of the

picture, but they are not the whole picture. The most basic, and lcast

i,nderstood, part of the process is the setting up of lists of possible condi-

tioning properties.Imagine the following scientific experimental situation in which Mill's

nrcthods rnight be applied. Suppose wc have threc new drugs that hold

pronrise for the cure of a hithcrto incurablc discase: drug A, drug B, and

drug C, We administer various combinations of these drugs and notewhcther the patient is cured or not. The results are tabulatcd in Exarnple 2'1,

u'ht':c "A" nrcans drug A llas bectr adrni¡tistcred, "B" tnca¡ls drug B has

bcc¡r adr»rnistcrod, and "C" ¡ncatts drug C has beel'r adrtli¡iistercd. "["

Exam¡rlc 2l

Possible condrtioningproperties

ABC

Conditionedproperty

t-

Occur¡enie l:Occur¡ence 2:

' Occu¡rence 3:Occurrence 4:

Occurrcncc 5:

means the patient has been cured. Occu¡rence 5 represents the cases of all

the previous patients who had not taken any of these drugs and who had

not been cured. The cases of patients to whom various combinations of the

ncrv drugs have been administered are tabulated under occurrences 1

through 4. Example 24 constitutcs a case of thc double mcthod of agree-

¡nt,nt a¡¡d it warra¡lts th¡ co¡lclusir.r¡t that il onc oI thc ¡rossible cnrrditioning

¡lropt rtrt: rs r t¡('(('s\.rry t'ot¡dtt¡r.¡¡t lor [, al¡d t¡¡¡c <¡f thc lxtssiblc t'o¡rdition-

P

P

AAA

P

AP

AA

AP

AP

A

P

P

AAA

IV. 1l app¡-¡cATroN oF MrLL's MrirrroDs

Example 25

ing propertics is a sufficient condition for E, thcn A is both thc necessaryand the sufEcient condition fo¡ E. Thus these results leacl to the conclusionthat, to the best of our knowledge, the administration of drug A is both anecessary and a sufficient condition fo¡ a cure of tlc discasc.

But suppose that someone tries other combinations of the drugs and getsthe results shown in Example 25. occurrencc 7 shows that A is not a suffi-cie¡rt condition for E. Therefore if we wished to 6nd a sufficient conditionfor E, wc would have to cxpand our list of pt-rssiblc conditioning ¡rrop-ertics. suppose now that biochemical thcory suggcsts that there may be a

I13

Possible conditioningproperties

ABC

Conditior\cdproperty

E

Occur¡ence 6:Occurrcnce 7:Occur¡ence 8:

P

PAP

P

APP

PAA

chcmical intcraction if the thrce drugs arc aclnrinistcrcd simultaneously,and that such a chemical interaction might cancel out thcir ellectivenessagai.st the disease. The sum of our observations wourd then suggest thatwhat may be happening in occurrence z is that drugs B and c ará i¡rteract.ing and preventing drug A from curing the diseaie. (We could imaginedifre¡ent occurrences that would rugg"ri that two drugs a." effcctive ínlyin combination.) Drug A seems to be effective whc. iaken alone (occur-rence 6), or when taken rviü B but rvithout C (occurrencc 1), oi whentaken u'ith c but without B (occurrencc 2). This sugscsts tlrat trre comprexpropcrry, A&-(B&t,), that is, tak:,rg drug Á but not in conjunction *¡thboth drug B and drug c, is rcally thc sufficicnt conclitio¡r for E. And if wcwerc to add this complex properry to our list of possiblc conclitioningproperties, and use all the eight occurrences, we would find that it is thenthe only possible conditioning property that is present whenever E ispresent and absent whenever E is absent. In this way we can reapply thedouble method of agreement to an enlarged set of possible .or,áiiioningproperties, in the face of additional occurrences, in order to revise ourconclusion and make it more sophisticated.

But we might not be finishcd even at this point. suppose that anotherrescarchcr were to point out that all our tcsts have been made on patients

Li4 IV. u¡t-¡-'s METHoDS AND CAUSALTTY

in whom the diseasc was at an carly stagc, and that many discascs arc morceasily curcd in their carly stagcs than in their advanccd stages. This rvouldsuggest that our complcx ¡rropcrty only appcars to bc a sufficicnt ,'onditio¡r

for E, because we have ¡rot tested our drugs on advanced cases of tlrc dis-casc. What wc would now lravc to do is to takc üris additional f¡ctor ir:toaccount i¡r r..¡rrr Iist of possible conditioning propcrtics. We could int¡oducea new property, D, which is said to be prescnt when tho discase is in itsadvanced stages and abscnt otherwise. In all tlte occu¡rences whcrc drugshave becn administered so far, D has been abscnt. Now we would have tofind various occurronccs rvhcre D was present, That is, wc should admi¡r-istcr va¡'ious co¡nl.¡inations of drugs to patients in advanced stagcs of the

disea.e and note the results. If üe treatment that effected a cure beforeu,ere to still effect a cure, then we would not have to rcvise our bclicf thatA&-(B&C) is a sufficient condition for E. But if our treatmcnt failcd in

advanced cascs, then we might havc to say that the sufficient condition fo¡bcing curcd is having the discase in an carly stagc and rccciving thc corrcct

c,r¡rL¡ination of drugs. That is, wc would havc to say tliat A&-(B&C) is

not a sulñcien¡ co¡rdition for L', l¡ut that -D&A&-(B&C) is ¡ sulficicntcr>nd¡tio¡r for Ij.

\Vt'could irrragirrr,:ur c¡rtllcss strt l¡¡t t¡f tlevcloprttt'ttts rvliiclr Irriglrt lolctr

u.s tr,.rdd r¡lorc a¡rcl rrrort'corrtplcx attd sirnplt: prolrcrtit:s to our list of

pos.srlrlc corrditioriirrg llropcrtics ¡rtd to co¡rti¡turlly rccvtlulltc otlr rt:.srtlts

Sonrco¡¡c nriglrt dcvclop a r)cw drtrg that cfl:cts a curc i¡¡ tltc aL¡sc¡rcc t¡f

drug A and thus shorv that A is not a lrcccssary condition for [. Additionalrcscarch rnight suggt'st othcr frtctors tltrrt rtriglrt bc rclcvr¡rt ¿¡ttd rvh<¡sc

relationship to E wc might wish to examinc. It is by suclt a process thatirf ill's methods, in conjunction with a co¡ltinual scarch for ncw occurrences,

and new relevant possible conditioning propertics, corrtributc trr the growthof scientific kncwledge .

Suggcsted readings

Georg Henrik von Wright, A Treatise o¡ lnduction and Probability

(Pattcrson, N.J.: Littleñeld, Adams & Co., 1960), chap.4 ancl chap. 6, pt.

4. (This book is qrritc difficult nnd reqtrircs a tlrorough grasp of dedrrctive

logic.)

Thc following books may l¡c co¡¡sultcrl for a r¡lore historicl] t¡cirtmtrltt of

Nlill's mctlrods:

Irvirrg lvl. Cctpi, !ntrtxlu(;tiott to Logic (2rr<l crl.) (Ncrv Yt¡rk: TIre

Itlacnrillan Corrrparry, I 96 I ).

IV.12 surr¡crENT coNDrrroNs AND FUNCTToNAL RELATToNSHTps IIs

Morris Cohen and Erncst Nagel, An lntroduction to Logic and, ScientificMcthod (New York: Harcourt, Brace & World, Inc., lg34).Njcholas Rescher, l¡t¡oductiott to Logic (Ncrv York: Saint Martin's,1964).

IV,I2. SUFIIICIENT CONDITIONS AND FUNCTIONAL RE.LATIONSHIPS. The preceding treatmcnt of Mill's methods in termsof necessary and sufEcient conditions proceeded entirely in qualitativeterms. One may wonder what relevance, if any, that discussion has forsciences which have moved from qualitative to quantitative language,Here, ascriptions of cause or statements of necessary and sufficient con-ditiorx have been replaced by functional relationships expressed bymathematical equations. The basic logic of the situation, however, is notas difierent as it may seem. An equation expressing a functional relation-ship between physical quantitics is tantamount to not one but an inffnitenumber of statements to thc eflect that onc physical property is a sufficientconditio¡r fc¡r anot]¡cr.

To u¡rdcrst¿irrd this, wc nrust look first at the relation between properticstncl plrysical quantities, Considcr a physical quantity, for exarnplc, tem-peraturc, as ¡neasured on a given scale (e.g., degrees Kelvin). We makea factual clairn about a state of a physical systcrn rvhen we say that itstcrnpcraturc (in degrecs Kclvln) has a ccrtain valuc, Temperature (§omeasurcd) is thus a relntion between states of physical systems and(non-negative rcal) nunibers. This is to say ¡ro ¡nore than:

For every non-negative real number, ¡, there is associated a uniquephysical propeüV, lwoing.tl'rc tentperature x in degrees Keli¡n.

A physical quantity can. thus be se¿n rs not one but rather an infinitefamily of physical p.opertier. The propcrtics in such a family are rnutuallyexclusioe (a physical systcm cannot h¿rve two difierent temperatures atthe same time) and iointl4 exhausti,se (a physical systcm in a given statemust have some temperatu¡e or other) over the states of the appropriatetype of physical system (the concept of temperature l¡as no meaning whenapplicd, for instance, to tl-¡e nucleus of an atom). An appropriate set of realnumbers serves as a fruitful fling system for the physical properties insuch a family, We can thus say that:

A physical q:nntity is a family of physical qualities, mutually exclu-sive and jointly exhaustivc ovcr the statcs of the intended class ofphysical systems, indexed by some set of real numbers.

ll(i IV. lu¡-r-'s METHoDS AND cAUSALtry

\Vr, s¡ri<l tlrat thc indcxing of the physical quantitics by tlrc indcx st:t

ol rr.r¡l r¡rrrr¡lrcrs for¡ns alruitful filing systcm. It is fruitful just in that it,to¡1t:tlrcr with the filing systems of othcr physical r¡uantitics, enablcs t¡s tt¡fr¡r¡nulate physical laws in terrns of n¡athcmatical equations. To sec howtlris works, let us considcr a fcw sirnplc cquations. First thc equatiottx : 2y. This equation givcs concise expression to an infinitc nu¡¡rl¡cr o[stírtements, of which a few are:

Ifyis0,xis0.Ifyis.l,xis2.Ifyis3i,xis7.

In gencral, for each valuc of y, thc er¡uutiott corrclutes ,r unirlue valuc of ¡.Wc givt: cxprcssion to tlris fact by saying that hcrc x is t lunction of y.

Tlris cquation also ¡nakcs y a function of r, sincc for evcry value of r itcorrelates a unique value for y (i.e., rr). It does not always follow, horv-

cver, that if ¡ is a function of y, y is a function of ¡. Considcr thc ccluatiorrx: g'. flerc, cach valuc of y dctcrrrrincs a uniclue valuc c¡f ¡, L¡ut tlreconvcrso is not truc. lf ¡ is *4, 17 may be eithcr *2 or' -2. 'fhus ¡ is a

f unctiorr of y, l-rut y is rrot a fut¡ctio¡i ol ¡.What clc¡es this lrrcan irr physical tcrr¡is whe ¡r thc variablcs of the c<1uuti<,»r

rcprust:nt plrysical quarrtitics? If thc variablcs reprcscnt plrysical qualrtiticsnlc¡rsr.¡¡'crl on fixcd scillcs (c.g., tcnll)cft¡tl¡r'c Kclvirr) tlrctt, as wt: hi¡vt'

.tccr), cilch ¡rurncricill valuc of a vilriilblc rcpr()sct¡ts ir plrysical <luality(c.g., having a tcnr¡rcrature of 10 degrees Kelvin). If a physical quantity,(),, is a lutrctktrt <.¡f anothcr, Q,, thcn for cvcry value o¡ of Q, tltert: is

a uniqucly dctc¡nrincrl value r-:¡ of Q, such that whe¡rever a physical sys-

tc¡n lras o, ol Q, it lrus o¡ of Q,. That is, having r.r; of Q, is a suffcientcoruLítio¡t for liaving o¡ of Qr. Thus, in terms of our view of physicalquantitics as fa¡nilics of physical qualitics, wc may say tbat:

If Q, is a function of Q,, then for every mcmber of ttrc fanrily Q,,tlrere is somc nie¡¡rbe¡ of the farnily Q, for olúch it is a suf-

ficient condition.

This generalizes itr a straiglrtforward way to functions of several vari-ables. For exarnplc, considcr the idcal gas law:

'rt'- LP

u,lrcrc V is volunrc, ',i' is tcnrperatLrrc and P is prcssurc. k is called a

systanr-tlcpcrxlctú r:o¡»tu¡tJ sincc it v:rries from systc¡n to systcm (e.g, dif-

IV. 12 sur¡rcmNT coNDITIoNS AND FUNCTToNAL RELATroNsrrrps l lz

fercnt l¡alloons 6lled with gas) but remains constant over difierent statesof the same system (e.g., heating a balloon, or submerging it to 200fathoms). The equaüon establishes V as a function of k, T, and P in thateach triple of values for k, T, and P uniquely determines a value for V.Thus in conjunction:

k=LandT:l00and,P:50is a sufiicicnt condition Íor V = 2.

In general, we can say that if a quantity, Qo, is a function of sev-eral others, Q, . . Q", then for every conjunctive physical propertywhich contains as conjuncts iust one member from each of the familiesQ, . . . Q,, üere is a membe¡ of the farnily Qo for which it is a sufficientcondition. Mathematical equations establishing functional relations be-tween physical quantities thus allow succi¡rct expression of extremelyrich clai¡ns about sufficient conditions

Sincc we are still, at basis, dealing with sufficie,t conditions, the funda-nrental prirrciples tl¡at we r»cd to analyze Mill's rnethods must still apply,alüough in a slightly more complicated way.llnstead of one cond.itionedproperty, rvc have the family of properties comprising a physical quan-tity (tlre depcndent variable). We rnust find a way for establishing, foreach member of this family, a list of possible conditioning properties.This is a two-stage process, Thc first stage is to construct a list of physicalquantities whose vaiues are likely to bc rcleva¡)t in dctcrmining thcvalue of the dependent va¡iable. We can call this c,ur list of independentaariables. The second stage is to construct a list of likely looking func-üioru which make the depcndent variable a function of our independentva¡iables. For each conditioned property (value of the dependent vari-able), each of these functions determines one or more complex propertieswhich are possible su.fficient conditions for it. Fo¡ example, consider thetwo functions:

TT2(i) v:t,and(ii)Y:Á p-

Function (i) would make the following, among others, sufficient condi-tionsforV:1:

k:Land.T=Land.P:Ik: I ond.T :2 o¡1d,P = 2

Function (ii) also makes

k:LandT:LarulP:I

118 IV. n¡¡-¡-'s METI{oDS AND CAUsALITY

a sufficicnt conclition fo¡ v = I, but disagrccs witlr function (i) in

maI'ing

k: l unilT - 2a¡xl P :2

a sufficie¡rt conditio» f.or V = 2 ratlrcr tlrr¡r for V - l' Si¡rce V = Ia¡rd V = 2 arc ¡»utually cxclusivc plrysical propt:rtics, itt ittly occurrencc

at lcast onc of thcnr r¡ust bc abscnt. wc h¡ver, then, only to look for

¿rn occu¡'rc¡tccwhcrt, k: la¡-ttlT =2antlP =2it't order to climinate

eitl,cr the hypothcsis that function (i) givcs tllc corrcct sulllcicnt condi-

tions or üe hypothcsis tl¡at condition (ii) docs. Thc ¡¡rctltocl opcrativtr

here is thus a straightforward application of the i¡rvcrsc mcthod of agree-

ment. The only new twist is tl¡at wc ltavc a farrlily of condiüo¡rerl

propcrties rvhich are rnutually cxclusiv<r, so that if onc is prcscnt, thc

rest nrust be abscnt.'Although thc ¡neclranis¡l] for the cli¡rri¡ration of the pro¡roscd func-

tior¡s is quit.,clcnr ht:rr:, thc ltroccss [or s('ttirrg rtP tlttr list of possibltr

fu¡rctio¡u is, ls bc[orc, rluitc tnurky. srrcll is to br: t'xpt'ct<:<1, s¡¡it:e tltt:

first ltroccss is rcirlly <lt:tlrrctivt: rvl¡,rrt'as tlrc lr¡ttcr is gt:Dtrinelf irrdt¡ctivtl.

Nevcrtlrclcss, thc urrlly,sis givcrr so fllr'c,,:r[¡lt,s t¡s tt.¡ shcd sonrc light orl

tlrc proccss of isolrrtirrg relcvl¡lt irlclc¡rcrtdcrrt r':rri¡blcs

Itc¡rrc¡ril-¡c¡. tl¡at if P is a sul[icicDt cor¡ditio¡r for C, thc¡r so is P anrl Q(rrnC, of coursc, P arul not-Q). Thcrc al'e, thcrr, so¡nc vcry cumbcrso¡rle

sufllcicrrt collLlitiolls aroutrd, l-rut ol-rviorrsly tltc ltttlst krrowletlgc is guincd

by firrdirrg tlrc shor.tcst oues. I¡l tcr¡ns of functions, tlris nreans that if

a quxntity, Q,,, is a fuDction of arrotlrcr, Q,, it is rtlso ¡ furrction of Q'o,.,á e, for any quantity Qr, Again, t¡c ¡.ost i.tercsti,g functions are

stateJ in tcuns of thc nrininlunr ¡ru¡nbcr 6f vxriablcs needcd to do the

job.. Supposc wc st¿rt witlr a list of likcly corrclitioniDg cltrantitics'Q,,

Q., . . , q,,, vltty tltcrn inclcpcndcntly rnd find tllat one of them' say

Q,, doesu't rnake any cliflcr.cncc in the colditíoncd quantity. That is,

for diffcrent fixed cornbinations of valucs fol Q. ' ' Q,,, thc valuc of

the conditio¡iccl qulntiry ¡c¡nains thc sanlc rvlrerl tltc value of Q r is

varicd. Then rve lrave gooci rcasc¡¡r to bclicvc that vllucs of Q, rvould

be cxccss fat in statcments of sufEcicnt conditio¡rs for tllc conditioned

quantity. I¡r othcr rvords, rvc havc good (inductivc) rc¿tson for belicving

? Notc llrrrt tlrs rrrtr¡r¡s tlrlrt tl¡,.c0rrr¡ri,r c0rrtliti,rrrtttl; ¡lro¡lcrtics inclt¡tlc cot¡-

lilnctrot)s oI srnrplc ¡rr0pertics ¡¡rd tllc tre¡.iutiorts of :irnplc pro¡;ertics-wlric]l is

¡rr:t rr,lr,rt w'c t¡cttl (stt lrxrtttott"'r, ¡1. 99 ).

IV. 12 surncrENT coNDrrloNs AND FUNCTToNAL RFr ATIoNsHTPs ll9

that Q¡ is not a releoant variable. This process of reducing ou¡ list ofconditioning quantities is called isolating the reboant oariables,

Thc second stagc in sctting up the conditioning properties was tosclect a likely list of functions which make the dependent variable a

function of the rcnraining independcnt variables. It is difficult to say

anything very informative about the selection process. Sometimes we

are guided by the sorts of functional relationships which have alreadybeen found to hold in similar physical situations, but it is difrcult tosay what "si¡niltr" means here. Somctimes we seem to be guided byconsiderations oÍ. timplicity of the expressions which designate the func-tion. But, except in certain spccial cases, simplicity is a highly elusiveconcept.

To the question, "Why do we have this two-stage meüod of for-nrulating statements of suflicicnt conditions?" we have already seen a

relativcly superficial (though corrcct) answer. That is, this method allowsthc succinct fornrul¿rtion of statemc¡rts of such power and scope thattlrcy rvould otherwisc excecd thc rcsourccs of our language. 'But thereis ¿urothr:r, nrorc profound, reason; that ül¡e two stages ore not induciioelyiulcpculcnt, To sec what I rncan, consider a new physical quantity, T',which is just like temperature Kelvin except that when temperature indcgrccs Kelvin cquals 10, To : 90 and when temperature in degrees Kel-vin cqual.s 90, To = 10. ?o conrprehe¡rds exactly the same physical quali-ties as ternperature, and uses the same set of numbers to index üem, butthe llling systcm is different. Imrgine now, fornluláting the fdeal gas lawin terms of 7'o rather than ternperature, Suddenly, the simple becomes

more cornplcxs (and by thc same token, the complex can become moresimple). Our basic physical magnitudes come to us, then, not simply as

artless vehicles for the expression of factual claims, but raüer as bearers

of inductive wisdom. It is they, rather than other families of physicalqualities, which have found expression in our ianguage precisely because

they have been found to enter into simply expressible functional rela-

tionships in a wide variety of physical contexts,o Language comes tous with inductive commitments-commitments so deeply ingrained thatit is easy to overlook them. But, as we learned from Goodman in thelast chapter, overlooking them leads to an excessively simple-mindedview of the nature of the inductive process.

s Thc student who knows some advanccd logic is invited to consider üe proposi-

tion that unde¡ zuch a change, the formerly simple may become ineffable.

r The most basic of sucli relationships consist of the corrclation of ürdcpcrtdclrt

methods of measuring a physical quantity.

t20 IV. ul¡-¡-'s METHoDs AND cAUSALI-I'Y

Suggested readings

On physical quantitics :

S. S. Stcvcns, "O¡r the Tlreory c,f Scales of Mcasurcmerrt," Scirrrrc¿;, 1946,

pp. 677-80.

B. lillis, Basic Conccpls ol Llcusurcnrctrl (London; Canrbridge Urriv.

Press, I966).

Kranz, Luce, Suppes, and Tversky, Foundutio¡ts of Measutatrcnt, Vol. I(New York: Acadcmic Press, l97l), chap. l.On simplicity:W. V. O. Quine, "On Simple Theories of a Complex World," Synthese

Vol, 15 (1963): 10&-6, reprinted in Probability, Confrmatiorr, and Sinr'plicítg, ed. Foster and Martin (New York: Odyssey Press, 1966).

Nelson Goodman, "Safety, Strength and Simplicity," Philosophy olSticnce, Vol. XXVIII (1961), pp. lSG-151.

Roger Rosenkrantz, Inference, Method and Decision (Dordrecht, Ilolland: D.

Reidel, 1977), chap.5.

l\'.t3. t,.t\l LIKL AND ACCII)IiN]'AI- CONDITIONS. Ir¡ Scc

tror¡ l\'.5 rvt'dtfrrrcd a sul[ic¡t'rrt ct¡¡ttl¡t¡t.¡t¡ ¿s follo*'s:

A plr.ll;crt¡', I;, is a sullicrelrt cc¡t¡dittt-¡tt lor.t ¡rropt'rty, (-J, rl .rrrtl

o¡rly rf ultutwucr li it pruscrtl, C i; prcscrü,

flence, the following are all legitin)ate statetnctlts of sufficie¡rt corlditic¡rrs:

L. Being o brothcr is a sulficicnt co¡rdition for being male .

2. Being ooer six feet tall ¡t " ,¡flicicnt co¡.¡dibion for being oaer

fae feet tall.3. Being purc @ateÍ at a pressl ,r' ot' one atnnsTtl ,.¿ aul o tem'

perature of lAO degtees cefttigrale is a sufficicnt co¡rditio¡r for

boíling.4. Haoing an inertial nlr.ss of one kilogram is a sulficient condition

for haoirtg a grü)itatioilal nwss ol one kilogran.t"

5. Eating tlinn¿r at tlw Paris Nootlle on lo¡Luaru 12, 1C72, is a suf-

ficient condition for being uruler six f eet tall.

ro l¡rerti¿l r»rss is ¡trc:¡surcd wtth rcs¡lcct to Ncwto¡r's second law, gravitatiorlrl

¡¡russ u'itlt rcsl)cct to Irts Iaw oI ul¡tvc¡s¿l grrvitution.

IV. 13 ¡-,Lw¡-IKE AND ACCIDENTA¡, coNDrrloNs tzt

6. Nelson Goodman had only dimes, quarters, and half-dollars inhis pocket on VE day," so being a coin in N¿lson Cood¡nan'spocket on YE day is a sufficient condition for being made olsiloet.

It is obvious fro¡n these examples that üere are strikingly di.fferent gradesof sufficiency. The sufñciency of the condition in Example I is due, in a

most transparent way, to the meanings of the tcrms involved. We maycall it an arul4tic sumcient condition. Example 2 also depends on thcconcepts involved, rather than the way we ffnd the world to be, so weshall also call it an analytic sumcient condition. However, Example 2

should remind us üat an account of anal¡icity is not always so easy togive as in the case of Example 1. It is not clear whether Example 3 isanalytic or not, Is having a certain boiling point part of what wemcan by being pure water? Is the boiling point of pure water involvedin the definition of the centigrade scale? If the answer is yes to either ofthese qucstions, thcn we may havc an analytic suflicicnt condition. Ifwo havc indopcndcnt definitit¡¡rs of purc watcr and tenrperature centi-gradc tlrcn Examplc 3 statcs ¡ sufficicnt condition which is inforrnativcal-¡out the rvay the world operates. Actual practice tends to shift fromor)c sct of mcanings to another depending on what is most convenientto tlrc occasior¡. Thus, actual practice does not providc an unambiguousans\\,cr as to whcthcr Example 3 is an analytic suflicient condition or not.

Such u scn)ulrticully rrruddlcd stute of alluirs ls cornnron in human Inn-guagc, ordinary and scicntific, and in such cascs an unambiguous answeris orily to be had by making a decision as to how the words are to bcused on a particular occasion. Examples 4, 5, and 6 are all synthetic. but4 is clcarly diflerent in kind from 5 and 6. Example 4 states a conditionwhich is süfiicient by oirtue of physical l.aw. ln this respect it reseml¡lesExanrple 3, when Example 3 is interpreted as a synthetic statement.Examples 5 and 6, however, state conditions which are sufficient simplyby lwpperutarrce. It simply lnpperwd that no one over six feet tall came

to the Paris N<¡odle to eat dinner that day. lt iust happen¿d that Nelson

Goodman had no pennies in his pocket on VE day. We say that these

truths are accidental rather than lntolíke.Although each of üe distinctions between grades of sufficiency raiscs

important and interesting questions, we shall focus here on the last one:the distinction betaeen accidental ard l¿tolike swfficient condirioru. This

"d,

I t Victory in Europe, World War II.

IV. ¡'tr¡,r-'s t\f l'lrllol)s ¡\ND cAUSALITYtoo

is arr inrportant clistinctir.rn for irrductivc Iogic.'l'hc cstirblislrtrtctrt r.¡[

citht.rsortofsullicicntcor-¡clitionnruybca¡rint]uctivoa[fair.,I.¡uttlrcro}cs;;;; oi"y aro so diflcrcnt tlrat onc rvoulcl susl)cct that i»dtrctivc logic

,i.I"'rlá ar.", thcm diffcrc¡rtly. lt is li¡rvlikcr st-]Hcicnt conditions rvl¡ich

,*t" u1, thc bocly of scicnccstatcnlcnts of accidcrrtal sufncicnt conditions

ilt" S ;"¿ 6 t."¡ likc any othcr factu¿il statetuctrt' sct up thc 'rpplication

of a scic¡rtific thcory, irutihoy ncvcr ft¡r¡n part of strclr a tllt:ory itsclf' Thc

,r-r-,,ri"t,t"t *c.,i",1 to illustrate Mill's-r,cthorls w'crc' accortlingly' all

;;;rr;i;; oi lo*liL" sulficie¡rt coDditi<.r¡rs (or nt'ccssil'y conditi<¡¡rs-r¡ll tl¡c

JJ,i,l'"alon, being made here obviously apply to nccessary co.ditic¡ns

also).Now, clocs this irnpcril our aturlysis of Mill's nrcthods? Not at all Our'

,,rriyri, clr:pcndcd u,ily o,, thc pririciplc that a 'suffcicnt conditio¡'t ca.ntnt

b," grr"r"nr'rulrcn tlw conditiaicd ¡tttryert't is ..ul¡senl' Tlris ¡rrinciple of

"li,.li,ratio,, f<.¡llows fronr the clcfi,ritio,i of su[Jicíent co¡u]itío¡t ¡nd thus

holds[c¡r¿llsullicic¡rtconclitions;iiccidc¡¡ta],lrrrvlikc,oranalytic(likc-*tr" f", ncccssary contlitio'rs)' Tlic story tltirt rvc lravc told about N'lill's

¡nctltods is nothing but thc trutl¡' It is' ltorvcvcr' far f¡o¡u thc lvhc¡lc trutlr'

Hi,rr,, urcthocls apply to ünaiytic sullicit.ut conditions, but cliurinuting

t-¡tilurcollditiorrirtglrropcrt,icsissurcll,lttlttllcl¡lostcfficicntway[o¡rrl\/eat /.rrrrrg u ltrotltcr r, ,-,'ln"ittt't ct¡itdition fot beitg male lt rvould bc

a lr,rlrclessly i,rc<.r,l,pct"'rt ¡natlrcr¡latician or logician lv]ro rcliccl oll tl¡c

"-p"ii,.",.riíf rnctlrorl fu' l'is tltco¡t:¡¡ls' A so^ientist' orl tltc othcrr I¡a¡rd'

rvoulc.l not cvcn war)t arr anulytic sullicicnt conditio¡r in his list o( condl-

ti",rirg propcrties, for knowiirg :rn analy.tic sufficient co¡rdition givcs us

,ro i,,firnr,,,t,r¡r about tlrc wly fh" *orltl L¡cltltvcs' Ncitl¡cr wot¡ld a scic¡l-

Ur,'-"t, a propcrty tik" bcÍng o coi¡t itt Nch¿''r¡ Coodt¡tu¡t's pockct on

VE, clag on his iist of iátti-f'fJt"fñc1cnt conditions for lteírtg composed

"r ,¡iru Hc knows that ti¡c overwhchrring likclihood is thlt if this turns

out to be a sulücie¡rt conclitio¡r' it rvill turn out to bc a¡r iLcciclcntal suf-

6cient co¡rdition."Ti¡is raiscs I rvo qut'stiotrs:

tAl Ho* do wc clistinguish luwlike fro¡n accidc¡ital sufflcient con-

ditions?(B) Why is it that iawlikc con<Iitions fincl a placc in thc body of

scic¡rcc rvltcrcas acciclcrrt¡l ottcs cl<l t¡ot?

Thc a¡lswcr to the seconcl <lucstion, if it is to )r¿vc any philosophical

i*forio,r"., must flow from considcrations of thc function of scicntific

i"*. ¡r¿ anything bettcr than an ad hoc r¡rswcr. to thc first question

must flow from a satisfactory .rllswcr to thc sccond'

Looking at Examples I through 6 it is casy to conjecture that thc dif-

fr:rence b-"t*""r, accident "n,l i"* is the dillcrence bctween part and

whole; that laws are truths about the wholc univcrse, throughout space

arr«l tirlrc, whereas truths which arc:¡l¡orrt rcstrictctl ¡;ilrtsolit (e.g-, thc l'aris

N<xrdlc and Nelson Goocl¡na¡¡'s ¡rockct ft>r tlrc: s¡lccilicd pcriorls of tinrc) rnay

be accidcrttal.'"

such a view has its attractions. surely the ¡nost striking cxamples o[

accidental conditions stem from generalizations of spatio-temporally lim-

ited scope. And the preoccupation of science witlt l,awlíke sufficient con-

ditions is neatly explained by the unioersality of scicnce' Science- is

concerned with patterns which recur throughout thc universc, rathcr

üan with gossip about a particular spatio-tcmporal rcgion' This concern

flows from the essential pursuits of science: explanotion and predictio.n'

Science always explains an event by showing it, in some way'1r to be

an instance of a general pattern, rather than just a freak occr¡üence'

As for prediction, áur geneializations about the Paris Noodle and Nelson

Coodman's pocket are obviously not very powerful predictive instru'

ments, becaise üey don't covei much tcrritory and typically ue don't

knsw'sbQut the wffiaicnt Gonditilns un\il ue haaa alreody eoasred. the

tter,ritory! Both the contcnts of Gooclman's pocket and the clientele of

the Paris Noodle at the times in question had to bc completely surveyed

bcfore confidcnce could bc placcá in our statcmcnts of sufficic¡rt condi-

tions. And givcn such a complete survcy, therc is nothing lcft for thenr

to preclict. Since no complete survcy of the univcrsc is possible' generali-

zation, about it nrust be }:nown, if rt all, while there is still predictive

Iife left in them.Suppose we accordingly try to define a law as a true generalization

which does not name .páin" iirn"r, pl'""', or individuals' Isn't it possible

that even the general descriptive machinery we have left may pick out

IV. 13 l¡rwuIKE AND AccIDENTAL coNDITIoNS L23

¡3 Truths about rcstricted parts of thc univcrsc that are logicol consct¡ucnccs ol

Iaus, e.g., the statement that cvery object in Nelson Coodman's pocket on VIi

day'obü the law of gravitation, w<¡uld of coursc be lawlike rathcr th¡¡¡

accidental.

rr In t1jh{.¡, tloy the particular event must bc related to tlre ¡¡eneral pattcr¡r is

a matter of extensive debate.

l:l,l IV. u¡¡-l-'s METTIoDS AND cAUsALll.\.

¡:'Sayrrrg oI t:ucli t]rirr¡¡ tlrat lt ¡5 cr(l¡cr ¡r<¡t ¿ cor¡r c¡rel,rscd ¡¡r ir st¡rit.lr¡rc oft¡'¡rc I or it rs silver ( or Loth ).

,r r¡r¡i¡ll lir¡itr'<:li¡ss of objects? For instance, isn't it possible that a dcscrip-Ir.¡¡ r¡l Nclso¡¡ coodrnan's pocket on vE day down to the 6nest cletall,tl.w¡¡ t, tlrc trajectorics of subatomic particles, could bc so specific with-r¡rt c«rrrtai¡ring names for times, places, or individuals, that tlle only thing,¡ tl¡<: wholc universe which would answcr to it woukl be Nclson iood-rrrarr's pocket on VE day. Tl-ren, according to our definition, being a coinetclosed in such a structure would have to be a lawlike .ufficiáit

"rr.,-dition for being cottrposed of siloer. But it is crcarly accidcntal. Lr [act,it is doubly accidental, for it would be somcthing of an accidcnt thatcoodman's pocket would be the only structure in the univc¡se answcringthe description iri cluestion.

What has gone wrong? One natural line of thought is to con.iccturcthat the trouble lies in defining spatio-tcmporal lirnitation of sco'pe uiathe terms in which the generalization is couched, rather tha, úy ti.,"objects to which it rcfers. why not say that a law is a true generalizatio¡rwhich does not refer to any spatio-temporaily limited (orlltcrnativery,to an¡' finite) class of objects? Then

All coins encloscd in a strucrurc of type I arc conrposecl of silvcr.*c¡u.ld fail to bc a l¿u,, cvc¡¡ rI truc, i[ Nclso, C<.¡odr¡.ian,s pockct on VEda1, «rnstituted tlre orrly structurc of ty¡ic I.

IJr,. waitl why d'wc assur¡lc that tl¡is gcricralizati,rt is onlg ultoutcoins cnclosed in structu¡es of type I? To be surc, if we know that acertain object is such a coin, wc know that it is c¡ucíal to the gcncraliza-tion, It is crucial in that, if it tu¡ns out not to be silver, it falsifics thcgeneralizalion. But if we know of anotl¡cr object that it is not silaer,then similarly we know it to be crucial to the generalization. It is cruciali¡r that if it turns out to bc a coin encloscd i, a structure of type I, itrvill falsify ihe generalization. In all fairrr,..ss, then, we,ught iá allowtlrat our rlcncralization is a/so about objccts not conlpost.l of silvel..r\rrotlrer way to put the sarne point is to note tl-tat being a coi¡t e¡closeclin-a structure of type I is a sufficie¡rt condition Íor being composed olsiluer just in case not lteing composed of siloer is a sulñcient conditionfor not being a coín etlcbsed in a structure ol type I. Thus our gencrali-zation refe¡s l-roth to coins enclosed in a st¡ucture of type I ancl to objectsnot made of silver. A littlc further discussion niight convince us that itrefers to cverything clse as well.'t But we l,ave alreacly gorc far e,ough

IV. 13 lawuxE AND ACCTDENTAL coND¡TroNs l2S

to see that we are on the wrong track, The class of objects rcferred to byour gencralization is no longcr spatio-temporaliy limiied or finite.

The attcmpt to locate üe clivicring line úctwcen acciclcntal ancr lawlikesullicient conditions in conside¡ations of spatio-tcmporal limitation ofscope seems to havc conre to a dcad end. AncI if the problcms so farraised for this approach are not enough, considc¡ thc following

"*urnpl"(due to Professor Carl Hempcl) |

7' It scc,s likcly that thcrc is no body of pur.e gord in the universewhosc mass cquals or cxceeds 100;000 Lilogárnr. tt ,o,-biiig oltorly compos.ed of pure golcl is a suflicicnt condition for lruaing amass of Lxs than 100,0N kibs.

Note that our belief in the forcgoing is quite conrpatibrc with the beriefi¡r an infinite universe st¡ewn with a¡r infinite nr*Ir". of bodies

"o*for"aof purc gold aul an infi.ite ¡ru,rbcr of bodics having ¿r rnass of ',nár"

than 100,000 kg.Yct, for all that, rve rvoulcr co.sidcr such a suflicient co,dition not a

mattcr of law but rathcr a¡¡ accident_a ,,global acciclc,t,,, if you please.

A rvorld ,riglrt obcy tlrc sanrc plrysical l"rrr .r, ours, arci yet có,tain hugenrasscs of gold just as a rvqrld with the samc larvs rnigút hor" portiJ",nro'i,g with diffcrcnt vcrocitics. what thcn is the diflcrcnce Letr"eensuch global accidc¡rts and t¡uc l¡ws?,0

-A major difference seems to be that laws arc crucial to the structureof our rvhole view of the worrd in a rvay that accicrental generariz;ii;nsare not. If .stro¡ionrers u,.ou.cccl thc discovt,ry ,rf a lar:gt, i,tcrstcl.lar.body of purc gold, we would- find it surprising, but ,ot áirtu.bing. ttwould arouse our curiosity and our desire for an explanation. The f"alsi-fic¡tion of a physical law, on the othcr hanr'I, rvouid crll for revisionthrou¡frout a whole systcrn of bcliefs rri,cr rvourcr destroy a *ht¡le tissueof cxplanahons. Tra,quillity is rcstorcc.l orrry whe,

" ,r.* row ,.oi¿..,the chaos.

, 9n: yo, of. view.ing this difference is to regard laws not merely asbeliefs about üe world but, ür addition, as contingent rules fo, "tungirgour b.eliefs under the pressure of new evidence. I iow believe _y ;;;ñ-

chord to be safcly at rest in coleta, california. If I would lea¡n th¿t ,,hugc net forcc werc being applicd to it, say by a hurricane, I wouldrcvise that bclief in accorrrancc witr¡ th. larvs ,f plrysi"r r,r,l f"., fu, it,

rx

ii

ili

r0'I'hc distinction is not mercly a phikrsophical nicety, but is ¡mportant for tl)eanalysis of scientific thcories; for instance, in th" fouuj"t¡ons of thcrmodynanria.

L26 IV. rrr¡¡-u's METIIoDs AND cAUsALITy

s:rfcty. lt is rrot srrr'¡rrisirrg tlrat our systc¡¡I ol l¡r'licls sl¡ot¡ltl sr¡l[r'r rr

greatcr disturbancc whctr rulcs norrna.lly uscd fo¡' changing bcliefs ¡rrust

thcmsclvcs bc rcviscd in conrparison to situatio¡ls i¡l rvhicl¡ tltcy rcrrtainintact.

It is <¡f coursc truc tllat ¡n lcciclcntr¡l gcncri.rlizatiotr, or irtclccd lr-rystatcnrcnt wc l¡elicvc to bc truc, plays a rolc i¡r dctcrrnining how wcchangc our beliefs undcr thc pressurc of ncrv evide¡:ce. But thc lolcappc¡rs to bc diffcrcnt frorrr that playcd by laws. Lt:t us cottrparc. If Iarn told on good autlrority that :r new hcavenly body of rrtass gt'cirtcl tltrtrr100,000 kg. has bccn discovcred, I rvill assl¡¡ne that this is ¡)ot a tnass ofpurc gold. But if latcr investigrtions convince ¡rie thrt it is, i¡¡ fact, puregold I will not (as i¡r the case of the harpsicliord) revise nty previous

belicf and conclude that it must rcally wcigh less than 100,000 kilos.Rathcr, I will give up my belief that ali bodics cor»posed of purc gold

have a n¡ass of less than 100,000 kilos.But consider thc coius in Gooclr»arl's pockct on \¡E clay. You rtrly ltitve

c.rtrcnrcll'strong g¡'ourxls [o¡'bcUcvi¡tg tlrrrt:rll thcsc coins lrrc silvt'r; slty

you wcrc 1:rcsent lt tllc tirnc, ol¡scrvctl Cood¡tr¿r¡l ttrlrting Itis lttrckctsinsiclc out yiclding just tlrrcc coins, that ypu tcstcd tltcrrt chcnrically andlourrd tllc¡n to bc silvel , r:tc. Norv if sortlconc co¡tvir¡ces you that IIC l)its

o¡rc of thc coins i¡l Coodrn¡r¡'s pockct on VE day you will assunlc thatit is silvcr. And if hc thc¡r fishcs out a coppcr cct¡t, cxclainting "Tlris is

itl" ¡'ou will rcvisc youl opinions both as to the origin of thc coi¡r and

thc vcracity of its posscssor. Thus, the fact tlrat a bclief is hcld rvithcxtrc¡¡rc tcnacity docs not gua¡.tlltcc tltat it is furlctionirrg its ii lilrv, cvctr

though laws arc typicirlly nto¡'c stable nrld ccntritl picces o[ r.¡ur i¡ttcllectu¿rl

c.r1ui1-rnrcrrt tlran ntcrrr l¡ctual judgnrcnts.I'crlraps tl¡c r¡rattc¡ can L¡c clarificcl iI rvc considcr the l'¡r-fctcltcd so¡t

of circumstanccs under which belng a coi¡t in Cootlmart's pocket on Vlirlay would bc co¡rsidcred a lawlike sufficient co¡rdition f.or lteirrg nndeol siloer, Supposc that our grounds for belicving that all tltese coi¡rs are

silver is that wc know Coodman's pockets had a ccrtain physical struc-turc; that this structure sets trp a force field which allows only silver

articles to enter (or, rrrore fancifully, onc whicl) ttatts¡llutes all othcr ele-

mcnts to silver). lf the suggcstion that laws have a special placc as lulcs

for changing belicfs has any ctrrrency then rve shoulcl be able to find

diflcrc¡rccs bctrvccn such application of this sufficie¡rt condition rrt thc

larvlikc a¡rd accident¡l c¿rscs,

II,¡¡¡ur¡r ol-.¡sctvrtitl¡¡ is frrliilrlt'. .rr¡tl tl¡r're is so¡r¡c likt'lilr<.¡otl, l¡os't'r't'tsr¡¡.rll, tllat rvc ¡¡tisscd ¡ coi¡l \vl)etl txalttitrÍttg Curdnran's pockt'ts. (I)cr'-

IV, 13 l¿w¡-rKE AND AccTDENT^L coNDrrroNs 127

ha¡:s it stuck in tlrat littlc cr¡¡¡lcr of tlrc pockct tlri¡t <lt¡csr¡'t lrrrrr irrsi<l«r otrt;perhaps this pair of 1)ants had ,u.ro watch pockcts; ctc.) Suppose thltthcrc is such a coin. If wc arc suddcnly infornrcd of its r:xistencc, rvhutare we tcr think of its eompositio¡r? /n tlw accklental case we have nocluc rvlrat to tl¡ink and if it tunrs out to bc coppt'r rvo will ¡rot fi¡lcl thi.sdisturbing ovcr and above our initial disturbance at htving misscd it,ln tlw bulike case, the infc¡e¡rce rule still applics and we will be quitcconfident thtt it is silver. I[ it turns <-¡ut to bc coppcr, u,e rvill l¡asten toreexamine tlre structu¡e of Goodntan's pocket und if wc find no fault inour previous beliefs about it, we will bc forced to seck for some revisedphysical ücory to account for thcse facts.

Laws then, do seem to have a special status as rulcs for revising ourbeliefs. This special status is perhaps most casily scen in our reasoningabout n¡h¿ú might haoe been. We will say, of a glass of purc, cold water(at a prcssure of onc atmosphcrc):

(A) If this watcr /¡¿rl 1.r¿cn Irc¡tcd to 100 dcgr.ct:s ccrrtigr.adc rtuoukl lrcoe boilcd.

because we believe that:

(B) All pure water at a pressurc o[ onc atnrospherc and a tcm-perature of 100 dcgrees centigrade boils.

is a. lau. (A) is said to be a coultcrfachnl conditional sincc, as thc watcr.has ¡¡oú been hcated, its if-clausc is contrary to fact. Thc law (lJ) is saidto support the counterfactual co¡rdition (A). If wc revicrv our cxamplcs,rve will find that laws support cou¡rtcrfactuals i¡r a rvay that accidentalgeueralizations do ¡rot. Su1:posc I havc a box rvith an incr.tial nrass of3/5 of. a kilogram. I say without trcpidation that if this box had had an iner.-tial mass of I kg., it taoukl lwoe had a gravitational rrass of I kg. Ilutif a ccrtpin ¡nan is 6r fcet tall I will certainly not say that if he liad eate¡rdinner in the P¿ris Noodle on January 12, 1972, he would have beenunder 6 feet tall. I will say that if a net force had been applied to myharpsichord, it would have moved. But I will not say that if this pennyhad L¡ecn in Coodman's pockct on VE day it rvould have bccn silvcr norwill I say that if Jupitcr were made of pure gold it would have ¿r massof less than 100,000 kilograms.

Sonrc rnetaphysicians have held that state¡¡rents of what rnight havc been areotrjective stateme¡rts alrout parallel worlds or branches of ti¡ne. Other thir¡kcrshold tl¡at correct counterfactuals are fables constructed according to otrr

I

I

l:'Jl I\/. rtr¡¡-r,'s N{ETIIoDs AND C^USALITY

( (r¡ltr¡llcr¡l tt¡lrr l,r¡ ,lr;rtr¡lrrr¡', lx:lit:ls llcasoning about what might have been

l*r url,," lr¡t ll¡r'¡¡¡ o¡¡lt' rrr ¡rra<'tic<r for reasoning about what might be'

It ¡l¡rrrrltl l,r't¡,rv r:lt.er tlrat lawlike and accidental conditions are

rl¡llr.rr'¡rl, irtrtl 1'ott l¡:t','t'sol¡lt: f{cneral indication of how they are different'

lrr¡t tl¡r r¡r.r'ific'ltrott <-¡f cliffciences has not been precise' How- cxactly

rlo luu':. ftrr¡ctir¡n a-s contingent rules of inference? What are the rules

frrr t'lurrt¡1irr¡1 our beliefs about laws? Just what is needed for a law to

luplx)rt a given counterfacfuai? Despite an enornous amount of work

tl¡r:rc Lr, as yct, no generally sarisfaátory solution to these and related

I)rol)lclrrs. Thcy rcmain a ma¡or area of concern for the philosophv of

rcicncc.

Suggested readingr

Ernest Adams, The l-ogic of ConditbruLs (Dordrecht, Holland: D' Reidel'

re75).

Roderick Chisholm, "The Contrary-to-Fact Conütioosl," in Reodings in

PhilosophUal Analysis, ed. Feigl & Sellan (New York: Appleton{entury'

Crofu, Lnc,, l9a9), PP. 482-97.

Nelson Coodm en, Fact, Fiction and Forecast (Cambridge' Mass'; Harvard

Universiry Press, 1955).

William Harper, et al. (eds.) I/s (Dordrecht, Holland: D' Reidel' l9Bl)'

Ernest Nagel, The Stntclu¡e of Sciance (New York: Harcourt' Brace & World'

I96l), chap.4, "The L,ogrcal Character of Scienti6c [¡ws.''

Brian Skyrms, Pragmatics and Empiriclsm (New Haven and l¡¡ndon: Yale

Úniu"rriry Press, té81), chap.5, "Á Bayesi"n Theory of Subjturctive Condi'

tionals."

jl'.a-,t-,,.^gI

The ProbabilityCalculus

V.l. INTIODUCIION. The theory of probability resulted fromt'e cooperation of two eminent r"""ri""ril,-"entuD, nlatiernaticians andif l,l?fl I[ :; :3I:,Í*I; :l L1iÍ, h ; r,;; " "..f

i,'"',, o, "

oto Braise po,".i *¡o in turn entere,,;;,;.;:*:i:,ffi::"Ilf

F,,"..?fT:Fermat, in order to air"u$ ih";:*.;;,;rvas born in the pascat-F";;;';r;;;;:"r[,,"al theory or probability

We have used the rvord .,probalriliJ:'rr,¡", freely in the ,iscussionso far, rvith only a roug_', lrriiil". s.ríp li ,,, _"oring. rn it,is ctraprcrrve rvill learn the mathcmatical .j", ilrri .a quantit), must satisfy inorde¡ to qualify as a.probabilrry. \41ü tilrt uo*l*¿ge in lran., wc rr.illproceed in chapter vi,to examií" ,h;;;;;;r for berieving that episrómic

:;1J,ro""n,e probabititie,,;;;rd";^

;;o"0r 0,,,,,", jn the mathemat ica r

Y,2, PNOBABILITY, ANGU]\ÍENTS, STATE]\{ENTS. ANDPROPERTIES. The *o.d..prolrriiftir;,^""^.0 for a number of distinctconcepts' Earrie¡ ] nointed out the á,r"r""". bchvcen inductive prorr_llllllt u'hich appties to arsuments, anct crapplÍes to strtcÁentr. r¡.."' ir"y;;";;;;":''stemic probabilitv, rvhich

lp3,lies to p.op".J.r. .when. ;"' ;;;";i,,,??""ff;?,}r t

l,|,;,J,lrÍla natura.l" in dice, or th.e prob.ability ,f li"ir' ^to

age 65, rve are ascribingp¡obabilities to properries, Wt.r l" lp.ri'?f ,f," probability tlot lohnV. Jones will live to a'e 65., o, tt," proíJüiiti tt,ot thc ncxt ti,.oru oi th"dice wil.l come up . nr't*rl, ,u. ...'rr.".]iingrp.olrrlrilities to statements.lhus üere are at lea.st t¡.". Jig;r"ii"iro.r"", probability rvhich appi.vto three diflerent t-* _or

tlinff 'ü;'á::,'rt.t"-.rtr, ancl properties.

^.1:",.,,r, there ií , .orron core to theseabili§: Each of these,¡.," tñ,",,,

",,, ffi F fl l* # fJ,"rr.* 3'' f:#iiJl,il;ijor probabilitv arc inrerrclar".i ,, "ii"r'írr;;'r:;" of *,hich werc broughtt¡ut in the discussion o[ ínductive ari "p,ri.r,,. probability. In chapterw'r wiil be shown how th.r. aig...ri:;;;:;; or probabilit.r. put fles'

129

v

V. rrg PRoBABILITY CALCILUS130

on tI¡e skeleton o[ the mathematical theory of probability' Here' horvevcr'

*,e shall rerrict ""rr.r*l',o á.r.toping-the mathematical thcory'

Tbe mathemrti.l tr'Jof i' "fit" "iltd th" probability. calculus' ln

order to facirirate tr-r. jr"rirü oi-.*"*pr"r we shall develop the prob-

abüW calculus,, " tnni'"t'io t"'"^i"ls' But we shall see later how

il;; "k" ;;"orntnod"it arguments and .propcrties'

Remember that the;d';;i;; ro' "-"t"|l" and "v" enablc us to find

out wbeüer " *'pi"''iüi";;;i is true or false if rve know rvhether its

simple constituent ""t"'"'it ^" t'ut or'false' However' truth tables tell

";:'í;;;;;u"ri-i¡" t*i "t falsirv of thesimple constitue¡rt statements'

in a simila¡ manner, t;;';;i;t';itíe probabiliw calculus tcll us hou' the

probabili§ of . "o*pi"t'

lütÁ""t is relatcd to the probability of its

simple constituent tttit'""it' U't they do not tcll us horv to clctcrminc

the probabiüties of sitpi"''i""-tntt' The problem of detcrrrrining the

orobability of simplc ti^itttnt' (or properties or arguments) is a problenr

of inducrire logic, but tt'l' ^ f'"üf"'n t¡ot is not solved by thc probability

'Jr'"'ü;,,,,, values assigned to co-nrplex statenre¡rts range frorn 0 to l'

.A,Iüough üe p,obabliiiy-"ultulu' does,not tell us horv to dcter¡ninc thc

orobabilities of ,lnlpl"

"tottnt"nt''

it cloes assign the cxtrcme ^values

o[

ü']iái';;;0."*l íira, of complex statemcntsl In Section I\/.3 rve dis'

cussed complex statements that arc truc no matter rvhat the facts are'

These statem"nt, tu"'" "ulled

tautologies' Since a tautology is gtraranteed

tobetrue,nomarter*h.tth"factsaie,itisassigncclt}rchighestpossiblcprobabilitY value'

Rulcl:iIastatementisatarrtology,thenitsprolla}lilityisequaltoL

Thus just as the complex statcmerlt sv-s is true no nratter rvhcther its

simple constibuent "i'"rn"'t' s' is true or false' so its probability is 1

resardless of tt',. p'ot'bility oi the s'mple constitucnt statement'

''i;":i;;i;;;rJ; ;ihá, type of sratement that is fnlse no nrattcr

rvhatthefactsare.Thistl'peofstatemcnt'calleclthesel[-contrac]iction'isassigned the lowest possible probabilit¡' value'

Rule 2: lf a statement is a self-contradiction' then its probability

is equal to 0,

Thus just as the complex statcnrent.s&-s is [a]se no nrattel rvhether its

simple constituent statement, s, is true or false' so its probability is 0

,egardless of the simple constituent statement'

V. 2 pnon.tsILITY, AncuMENTS, sTATEMENTS 13I

When two statements make the same factual claim' that is' when they

arebueinexactlythesamecircumstances,tlreyarelogicallyequiv:rlent.Ñ"* U a stateme;t that makes a factual claim has a ce¡tain probability'

unrtt,", statement that makes exactly the same claim in different rvords

;i;;ii be equally probable' The statement "My next throw of..-the dice

wil] come up a narural,, should have the same probabiliry as ..It is noÚ

the case that my next throw of the dice will nof come up a natural"'

This fact is reflected in the follorving rule:

Rule3:Iftwostatementsarelogicallyequivalent,thentheyhavethe same ProbabilitY'

By the truth table method it ís eas¡, to shorv that the simple statcment 2 is

ioglcotty equivalent to the complex statement that is its double negation'

,- -p, since they are true in exactly thc samc cases'

_-/p

Case 1:

Case 2:

TF

FT

TF

Thus the simple statement 'lMy next throw of the dice r"'ill come up a

natural" has,iecording to Ri" 3, the same probability as-jts double

;;;;;;., "It is no, thc case that m)' ¡rext throu' of the dice u'ill nof come

up a natural."'Th" 6.rt two rules cover certain special cases. T',he1, tell us the- prob-

"bilit of a complex statement if it is either a tatrtologl' or a contradiction'

f1]" ti]ira rule tells us how to fincl the probabilit)' of a c.omplex contingent

statementfromitssimpleconstituentstatemcnts,ifthatcomplexstate.m"ntislogicallyequivalenttooneofitssimpleconstituentstatements.Buttherearemanycomplexcontingcntstatementsthatarenotlogically

"qrr"ri.ra io .ny tt their simplc corrstitucnt statentcnts, and morc nrles

shall be int¡oduced to cover ihcm. The next two scctions present rtrlcs

for each of the logical connectives'

Exercises:

Instead of writing "The probability of p is i," we shall write, for short "Pr(p) -l." Now suppose that pr(pl = { ancl'Pr(q) = ¡' Find the probabilities of thc fol-

lorving compl"x stotements, using Rules 1 through 3 and the method o[ tnrt]t

tables:

I',r')

l. 1tvp.

2 q&q.

3. q&*q.a. -@e--q).

V, rrtn PRoBABTLITY cALCULUS

5. -(pv-P)'6. - -(pu-P)'7. pv@e.-q).B. q&(Pv-P).

\'.3.DISJUNCTIONANDNEGATIONRULES'Theprobabilitvof a d"jsjunctionpvq is most easill'calculated when its disjuncts, p and q'

at" *uíuolly exclusitse or inconsistent rvith each other. In such a case the

probabüqv Lf th. dlt¡unctjon can be caiculated from the probabilities of

üe disjuncts by mearu oL the special disiunction rule'We shall use the

notation introcluced in the cxcrcises on p. 132, writing "The probability

of p is r" as: "Pr(P) = 1."

Rule 4: If p arrd g are mutuall¡' exclusive, then Pr(pvq) : Pr(p) +

P'(q).

For example, the statements "socrates is both hald and rvise" and "socra-

tes is neither ba)d nor rvise" arc muhrally exclusive. Thus if the probabilify

that Socrates is both balcl ancl §'ise is f a¡rd thc probability that Socrates

is neither balcl nor rvise is .:., thcD thc probabilitl' that Socratcs fs cither

both bald and tvise or neither bald nor rvist' isi -i- 1, orl'\Ve can do e Iittlc ¡16¡¡ rvith the spccial altcirrrtron rulc i¡r tlrt'[<tl]o*'irrg

case: Suppose you are about to throi' a single six-sided clie ancl that cach

of the six outcomes is equally probable; that is:

Pr(thc clic rvill corne uP a 1) : iPr(the dic will come uP a 2) : iPr(the ilie u'ill conre uP a 3) = fPr(the die rvill come uP a 4) : iPr(the die rvill come uP a 5) : fPr(the die will come uP a 6) : i

Since the dic can show onl¡' one face at a time, these sü statements rnay

be treated as being mutually exclusive.' Thus the probability of getting

I Actuall)' thc statements are not muhtally exclusive in the logical sense' We

cannot shou' üat they are inconsistent with each other by the method of truth

tables, ancl it is logicalil' possible that thc dic might change shape upon being

thro*n so as to rlisplal' §r'o faces simultaneously. To trcat this case rigorously' we

*'ould havc to usc the general disjunction rule, along u'ith a battery of assump-

tions: Pr(l&2):0, Pr(2&3):0, Pr(1&3):0, etc' However' we shall see

that the ¡esult is the same as rvhen we use the special disjunction rule, and

treat these stxtemcnts as if the¡' rvcre mutually exclusil'e'

V.3 ors¡uNcrroN AND NEcATIoN nuLES 133

a I or a 6 may be calculated by the special disiunction rule as follows:

Pr(1v6) = Pr(I) + Pr(6) :+ + *: iThe probability of getting an even number may be calculated as

Pr(even) : Pr(2v4v6) : Pr(2) + Pr(4) + Pr(6)

--l--lt tJ--I-r;rü2

The probabili§ of getting an even number that is greater than 3 may be

calculated as

P(evenand greaterüan3) = Pr(4v6) = Pr(1) + Pr(6) =* + * = i

The probabiliry of getting an cven number or a 3 may be calculated as

Pr(even or 3) = Pr(2v4v6v3) : * = iFinally, calculating the probability of getting either a I' 2' 3' 4' 5' or 6

1,fr", u, the probatilitl' that the dic rvill shorv one facc or another) gives

j. or I.T'üi. *,ilt norv apply the special disjunction rule to a case of more general

interest. lt can úe. ,ho*n, by the method of truth tables, that anlt t¡"t"'ment p is i¡rconsistent rviü its negátion, -p' Since p and -p are there-

forc rnutuall¡, exclusive, the spcciai disjunction rule permits the conclusion

that

Pt(Pv-P): P¡(P) + Pr(-P)

But the staten)cnt 2tt'-p ís t tautolog¡', so b¡' Ilulc 1'

Pr(Pv-P) = L

Putting ¡|¡s5s frvo conclusions together gives

Pr(p) +Pr(-P):¡ ,

IfthequantityPr(p)issubtractedfrombothsidesoftheequatlon'thesides rT'ill renlain equal, so we nlay conclude that

P'(-P)=t-Pr(P)This conciusion holds good for any statement' since any statenrent is

inconslst.nt with its nelation, ancl for any statement p its disjunction

rvith its negation, puJp, is a tautolog¡'' This therefore establishes a

g"á...f neg;tion ruie, which allorvs us to calculate the probability of a

iegation fÁm the probability of its constituent statement:

Rule5: Pt(-P) = t - Pr(P)'

LY V' TIrr PRoBABILTTY cALCULUS

Suppose io the example using the die rve wanted to know the probability

of not getting a 3:

Pr(-3) =i-Pr(3) :t-t=*

liote tlat we get the same answer as we wouid i'[ we took the Iong road to

solring üe problem "nJ "on6n"d ourselves to using the special disjunction

rule:

Pr(-3) = Pr(1v2v4v5v6)

= Pr(l) + Pr(2) + Pr(a) + Pr(S) + Pr(6)

= + +-i -¡ *-:- *-;'* : *\Ve shall apply the special disjunction rule one more time in order to

estab[sh another g"t't'itty '''sefi rule' For any two statements' p' q' we

can show by the tt"t¡"iJ¡ü -"thod that üe complex statements p&q'

*.-o. and -p&q "'" inton'i'tcnt rvith each othár' As shown below'

fu"r"'¡ no casc in rvhich tu'o of them are true:

-p -q p&q P&-q -?&q

Case l:Case 2l

Case 3:Case 4:

FTFFTFTFFFFTTFFF

TTFTFFFTTFFT

a- Pr[(p&q)v(pa^.q)] = Pr(p&q) + Pr(p&-q)

o. ,'i[*nj,i-etq)] = Pr(p&q) + Pr(-p&q)

c. pr[(p&q)v(¡r&-q)v(-rri?ll'= pr(puq) + Pr(p&-q) + Pr(-p&g)

But üe complex statement (p&q)v(p&-q.).: 1:gt"il'."1:il:1"" to tn"

Since üey are mutudry excrusive, we can appr1, the special disjunction

n¡le and concludc:

,,ut u¡e uur¡¡PrL^ "-"";; :It;;n'by the iolJowing truü table:simPle statement P, as

p&q p&-q (P&q)v(Pa*q)

Case l:Case 2:

Case 3:

Case 4:

TTFF

TTFTFTFTFTFTFFFFFTFF

V. 3 ols¡uNcrro¡i AND NEc'{TIoN nL'I-ES 135

since according to Rule 3 logically equivalent statements have the same

;;;;;;i;ü, ;qi'tio'' (a) maY be rcwritten as

a'' Pr(P) = Pr(P&q) + Pr(P&-q)

A simiiar truth tabie rvill show that the complex statement (p&q)v(-p&q)

is logically eguivalent to 1i" 'i*pt" 'tut"t"''i q-' ih"'"fo" equation (b)

maY be rew¡itten as

b'' Pr(q) = Pr(P&q) + Pr(-P&q)

Finallv, a truth table rvill shorv that the complex statenrent iptq)v(p&-a)

,,t -m&o\is

logicalty equivalent to the "";;;t' statement pvq' u'hich

";bi;;lt t" re"'vrite equation (c) as

c'. Pr(pvq) : p¡(p&r¡) + Pr(p&-q) + Pr(*p&q)

Norv let us acld equations (a') ancl (b') togcthcr to get

cl' Pr(p) * f'(o) :2 Pr(p&q) + Pr(p&.*q) -{ Pr(-p&q)

l[ rve subt¡act the quantity Pr(p&q) from both sic]es of the preceding

c<1uation, u'c gct

d'' Pr(p) + Pr(q) - l'(poq) = Pr(p&q) * Pr(p&-q) * Pr(-p&q)

r f equ at i on i d,) i, co",r, ar;i)ü,,...i11;'ir l,1l rü).' ?"rl: i:üir,,"fl,':

"q"j to the same

:]],T,,;,:;';á"a io' "¡ ai'¡"'"tiáns' rvliethcr thc dis-

.".cral disiunction

il;;;; ;" átrturllv exclusivc or not:

Rule 6: Pr( pvq) = Pr(p) + Pr(q) - Pr(p&q)'

II some of the algebra used to,est"Y]Ltn" general <]islunction rule lras

lef t you bchind, tr,i rorto*,irg cliagram -^y l"*lp to nrakc the reasoning

cl car:

Pr(p)

P'(q)

Pr(p&-q)

Pr(P&q)

Pr(-p&q)

Pr(2vq)

), r q,r, u, Jt olill o I li, :"lJ';*'i,'irl ri" :, : :',li,l

ffiTrt;'í',;'"xfi*': J;.,l 11;'lli,''" )i;: il;;." gct Pr( pvq )' *''l a d < r

I :l(; V. rtl¡ PRoBABILITY cArcuLUS

I'r(¡r) ;rrrrl I'r(17) and then subtract Pr(p&q) to make up for having counted

rt trvic.c. In thc case in which p and q are mutually exclusive, this makes

r¡o rlificrcnce, because when p and q are mutually exclusive, Pr (p&g) =0. Nc¡ matter how many times 0 is counted, we will ahvays get the same

rcsult. For example, by the general disjunction rule, Pr(pv- p) = Pr(p) +

Pr(--O) - Pt(p&-p). But the statement p&-p is a self-contradiction, so

its profabilit¡,is zero. Thus we get the same result as if we had used the

rp".i"t disjunction rule. Counting Pr(p&q) twice does make a difrerence

,rlh"n p .ná q "r"

nof mutually exclusive. Suppose we use the general dis-

junction rule to calculate the probability of the complex statement pvp:

Pr(pvP) = Pr(P) -i Pr(P) - Pr(P&P)

But since the complex statement p&p is logically equivalent to the simple

statement p, Pr(p&P) = Pr(P).So we get

Pr(pvp) = Pr(p) + Pr(p) - Pt(p) : Pr(p)

\\¡e know this is the correct answer, because the complex statement ¡'vpis also logically cquivalcnt to the simple statement p'

Th" "á.pi" wlth the die shall be used to give one more iilust¡ation of

the use of the general disjunction rule Suppose that we \\'ant to know the

probability thl't the die iill come I p an even numbe¡ or a nunrbcr less

ti¡an 3. There is a §'a)1 to calculate this probability using onl¡' the spccial

disjunction ruie:

Pr(even v less than 3) : Pr(lv2v4v6)

= Pr(l) + Pr(2) + Pr(4) + Pr(6) =* = f

\ve may, use the special clisiunction rule because the outcomes l, 2, 4,

and 6 are mutually exclusive. However, the outcomes "even" and "less

than 3- are not mutually exclusive, since the die might come up 2' Thus

we mav apply the general disiunction rule as foliows:

Pr(even v less than 3)

= Pr(even) * Pr(less than 3) - Pr(even&less than 3)

Now we may calculate Pr(even) as Pr(2v4v6) by the special disjunction

rule; it is equal to:. We may calculate Pr(less than 3) as Pr(1v2) by the

,p..i"t disjunction'rule; it is equai to ':-' And we may calculate Pr(even&

less üan 3j as Pr(2), which is equal to r' So, by this method'

Pr(even vless than 3) = i + +- * : +

V.3 osIuNcnoN AND NEGATIoN RuLEs 137

The role of the subtraction term.can be seen clearly in this cxan:plc. What

*" ¡"r" done is to calculate Pr(even v less than 3) ns

Pr(2v4v6) + Pr(iv2) - Pr(2)

so the subtraction term compensates for adding in Pr(2) ¡u,igg \,vhg¡ rve

;J Pri;;"") and Pr(less than 3)' In this cxan.rple use of the general dis-

ir""U"" ,rl. *^, the long way of solving the problcm' But i¡r son¡e cases

it is necessary to use the general disjunction n-rle' Suppose you are told

that

P,(P) = )P.(q) = tPr(P&q) : I

You are asked to calculate Pr(pvq)' Now you cannot use the spccial dis-

junction rule since you know it'"i p and g arc not mutually exclusive' If

they were, Pr(p&q) would be 0, and you are told that it is ¿' Therefore

yo, murt use the general disiunction rule in the following way:

Pr(Pvq) : Pr(P) + Pr(q) - Pr(P&q)

=++ +-+=*ln Section V2, we compared üe rules of the probability calculus to

üe way in rvhich th" iruti., tables for the logical connectives relate the

truth or falsity of " "";;l; statement to the iruth or falsity of its simple

constituent statements' W" "" now at the point where we must qualify

this comparison. We t"n tt'u'y' determine the truth or falsity of a complex

statementifweknowwhetheritssimpleconstituentstatementsaretrueor false. But we

""nnot-ti*ors calciate the probability of a complex

statement from tbe probabilities of its ímple constituent statements'

Somerimes, as in the J-Ñ" "U"'e' inorderlo calculate the probability

of üe compl"* ,tutt"ni''puq,, *" need not only knorv üe probabilities

of its simple cor,rtitr.nisi.te.entr, p and q, we also need to know the

probability of *otl''"' "omplex

statement' p&q' We shall discuss the rules

i;;;;;r"; the probabiliües of such conjunctions in the next section'

However, we shall Áná tf'"t it is not dways possible to .calculate

the

probability of " "on¡*"tion

simply from the probabilities of its constituent

statemenB.

138 V. rgp PnoBABILITY CALCULUS

Exercises:

L suppose you have an ordinar¡' deck of 52 playing cards' You are to.,draw

on. "ord.

Arr.rme that each card has a probabiLi§ of Iof being drarvn' what is

üe probabilir.v that you will draw:

a. The ace of sPades?

b. The gueen of hearts?

c. The ace of spades or tlre queen o[ hearts?

d. An ace?

e. A heart?f. A face card (king, queen, or jack)?

g. A card that is not a face card?

h. An ace or a, sPade?

i. A queen or a heart?

j. A queen or a non-sPade?

2. Pr(p) = r, Pr(q) : r, Pr(p&q) = .].' What is Pr(¡rvg)?

3. Pr(r) = !, Pr(s) = 'l-, Pr(rvs) : s ' \\¡hat is Pr(r&s)?

4. Pr(u) =i, pr(¿) = J, Pr(u&-r¡ = 'l-. \Vlrat is Pr(trv-t)?

\¡.4, CONJUNCTION RULES AND CONDITIONAL PROBA-

BILIT"I'. Before ih" ,rle, that govern the probabili§ of conjunctions are

discussed, it is necessary to int;duce üe notion of conditional probabil-

ify. \\'e nia), ..'ritc Pr(q givcn p) as thc probability of q on tlte. co-nditiott

ilio, p This probability ,ia,v ot -oy not be- different from Pr(g)' We shall

deal rvith the concept of conclitional probabiiity on the intuitive level

before a precise definition for it is introduced'

In the Lxample rvith the die, rve found that -the probability of throwing

an even nu*b", rvas ]. Hou'ever, the probability of getting an even.num-

ber gioen tlwt a 2 o, o 4 i, throrvn is not i^but 1' And the probability of

"*ti?g an even number gioenlhat a 1 or a'3 is thrown is 0' To take a little

more compücated examfrle, suppose that the die ¡emains unchanged and

i:r, "* i" U"t on rvhethlr it wili come uP even' rvith a special agreement

,üatifitcomesup5allbetsrvillbeoflanditwillbeth¡ownagain,In

zucb a situaUon you would be interestqd in the probability that it wrU

;;;p even giaenrh¿, it will be either a I' 2' 3' 4' or 6' This probabiliW

.¡orl¿'b" gr"^I", than .| since the condition excludes one of the rvays in

which üe die could "ori",p odd' lt is, in fact' * ThY-tht probabilities

of 'even," given üree diffeient conditions' are tach di'serent from the

probabüty cf 'even" bY itseU:

a. Pr(even) = *

V. 4 coNluNCTroN nr.rt-Es 139

b. Pr(even liocn2v4) = !c. Pr(even giaenlv3) : Q

d. Pr(even gioen lv2v3v4v6) = -:-

conditional probabiiities allow for the fact that if a certain statement,p, is known to be tn¡e,lhis may affect the probability to be assigned toanother statement, g, The most striking *r., o""r, *,hen there is adeductively valid argument from p to q:

p = The next throw of the die will come up 2V

tle next ürow of the die will come up 4.

In this case, Pr(q gir:en p) : I:,Pr(even giocn 2v4) : I

Suppose thcrc is a deductively.valid argumcnt from p to *g:

p : The neit.th¡ow of the die rvill come up 1

vthe nerl th¡ow of ,the die wiü come up 3.

-g = The next throrv of the die rvill noú conre up even.

In this case, Pr(g giaen p) : g.

Pr(even giucn lv3) = Q.r

T'here are, ho*'ever, important cases rvhere neither the argument fromp to q nor the argument from p to -g is deductively ,alia and yetPr(q gioen p) differs from Pr(g), as in the previous exampre with the die:

Pr(even gioen Lv2v3v4v6) : IPr(even) : ]

I We must make one quali.6cation to üjs statement. When p is a sclf-contradiction, then for any statement q üere is a decructivcly valid argun,c.rfrom p to q ond a deductively valid argument fronr p to -q. ln such r c:rsr:,Pr(q giocn p) has no value.3. We must make one qualification to this statement. When ¡r is a sr.lf -contradiction, then for any statement q there is a deductively valid argrrrrrcntfrom p to q and a deductively valid argument from p to *r7. In such a <.,,sr,,Pr(q gioen p) has no value.

V. rrrr PnoaABILITY CALCSLUS140

Thcre are other cases where üe knowledge that^p is true may be com-

pl"t;ü lrt.ilt'rr, to the probab]riry ¡o be aisigncd to q For cxample' it

u'as said that the probabili§ that üe next. tñrow of the die will comc

;;";;'+. w" ""riJ ttl''tr'"t th" p'ouobilitv that the next throw. of

üe die rvill come ,p ",.n,'giu"n

that ihe President of the United States

;*;; simultaneouisly *iifou' throw, is stilla' The.President's sneeze

is irrelevant to the probability assigned to "even"' Thus the two state-

ments-Thenextthrowoftherliewillconreupeven''and..ThePresidentoftheUnitedStnt",*il-lsneezesimultaneouslywiththenextth¡owof the die" are indePentlent''

Wecannolvgivesubstancetotheintuitivcnotionsofconditionalprob-

"UiUty-"nd indJpendence by defining lhem

in terms of pure statement

probábüti"r' Fiist we wili define conditional probabiliry:

Deff nition L2: C ond itional pr ob ab ilíty :'

Pr(s given d: WlLet us see how this definition works out in the exrimple of the die:

b. pr(e'cn givcn 2r,4) = ESffi'gI = ffi = r

c. Pr(even givc' lr'3) = :li# = 9: o

d. Pr(c't'n given Iv2v3v4v6) = t'';i1t!'r'r"'rtt'ó"u"

P'(Z'¿vO) -*:'=T¡rE5;a'o¡-r-TNotice that the conditionar probabirities computed by using the definition

accord with the intuitive juigments as to conditional probabilities-in the

die example. We may t",t t¡t definition in another way' Consider the

,p..id ""s"

of Pr(g given p), where p is a tautology and q is a contingent

statement. Since a tautology't"kes no factual claim' we would not expect

Irnrr,rr" of independence is calle«l probabilistic or stochastic independence' lt

should not be confused *iií'-iil"1'"t'al logical independence discussed in

á"ár"rit" logic. Stochastic independence of two.statements is neither a necessary

""r " trm"iá, condition fo' üei' mutual logical independence'

sWhen Pr(p):0 the quotient is not deñned' In this case there is no

tu(q giwn P).

V. 4 cox¡ulrcr¡oN RriLEs 14I

knowledge of its truth to influence the probability üat we would assignto the contingeDt statemen! g. The probabilify that the die will come upeven given üat it will come up either even or odd should be simply theprobabilis that it will come up ¡evea In general, if we let ? stand for anarbitrary tautology, we should expect Pr(g given T) to be equal to pr(g).Let us work óut Pr(g given ?), using the definition of conditional prob-abili§:

Pr(T&q)Pr(g given r) = ffi

But the probability of a tautology is always equal to l. This gives

Pr(g given I) : Pr(T&C)

When I is a tautology and g is a¡y statement whatsoever, tlie complexstatement ?&q is logically equivalent to the simple statement g. This canalways be shown by truth tables. Since logically equivalent statementshave the same probability, Pr(g given ?) : Pr(g).u Again the de§nitionof conditional probability gives the expected result.

Now that conditional probability has been de§ned, that concept can beused to deEne independence:

Definition 13; Ind.ependence: Two statements p and q are inde-pendent if and only if Pr(q given p) : pr(g).

We talk of two statements p and g being independent, raüer than p beingindependent of q and g being independent of p. We ca¡ do üis becausewe can prove tirat Pr(g given p) = Pr(q) if and only if Pr(p given g) -Pr(p). If Pr(g given p) : Pr(q), then, by üe deEnition of conditionalprobability,

Pr/r&o)+='= Pr(q)I I\'P )

Multiplying boü sides of the equation by Pr(p) and dividing both sidesby Pr(q), we have

6 We muld have conshr:cted üe probability calculus by taking condiüona.lprobabil.ities as basic, and üen defining pure statarent probabihties as follows:The probability of a statement is de6ned as its probabiliry gioen a tautology.Instead we have taken statement probabilities as basic, and deEned condiüonalprobabüties. The choice of starting point makes no di-ffe¡ence to üe system as awhole. The s)Ttems are equivalen!,

742 V. r¡rr PRoBABILITY cArcl.ll,us

7 The e¡ception is when at lest one of the statements is a self<ontradiction and

üu has probability 0.

Pr(¿&o)

ffi=Pr(P)But b1' the de6nition of conditional probabilify, this mcans Pr(p given g)

: pr(p).

This proof only works if neither of the t"vo statements has 0 probability.

oüenvise one of the relevant quotients would not be defined. To take

ca¡e of this eventuality, rve may add an additional clause to the de6nition

and say tlat trvo statements are also independent if at least one of them

has probability 0. It is important to realize the difierence between inde-

p.ná"oa" and mufual exclusiveness. The statcment about the outcome of

the th¡ow of the die and the statement about the Prcsident's snecze are

independent, but they are not mutually exclusive. They can very rvell be

truelogether. On the other hand, the statements "The ncxt throu'of thc

die will come up an cven number" and "The next thro$' of the díc rvill

come up a 5" are mutually exclusive, but they arc not indepenclent'

Pr(cvcnj - l, lrrrt PI(t'r't'n givcn 5) : 0. Pr(5) :r, but Pr(S givcn cven)

= 0. In genáral, if p and g are mutually cxclusive thcy arc not indcpend-

ent, and if they are independent they are not mutually exclusive'?

Having specified the definitions of conditional probability and inde-

p.nd"nc!, the rules for conjunctions can nou, be introduced. The general

Loniunction rulc follorvs clircctly from the dcfinition of conditional prob-

abiliry:

Rute 7: Pr(p&q) : Pr(p) X Pr(q given P),

The proof is simple. Take the definition of conditional probabili§:

Pr(g given p¡ = P'=(49)Pt(p)

\lultiply both sides of the equation by Pr(p) to get

Pt(p) X Pr(g given p) : Pr(P&q)

rvhich is üe general conjuncrion rule. since, u,hen p and q are independ-

ent, Pr(g given p) = P¡(g), we may substitute Pr(g) for Pr(g given p) in

the general conjunction rule, thus obtaining

Pr(p) XP'(s) =Pr(p&q)

V.4 cox¡uNcTroN nULES

Of course, the substitution may only be made in the spccial case whenp and q are independent. This result constitutes the special coniunctionrule:

Rule 8: If p and q are independent, then Pr(p&q) =Pr(p) x Pr(q).

The general conjunction rule is more basic tlian the special conjuuctionrule. But since the spccial conjunction rule is simpler, its application u,ilIbe illustrated first. Suppose that trvo dice arc throrvn simultaneousll,. Thebasic probabilities are as follorvs:

Die A Die B

Pr(i) : -:-

Pr(2) : rPr(3) : -:-

Pr(4) : rPr(S) : :Pr(6) = :-

Since the face shown by die A presumably does not influence the faceshorvn by die B, or vice versa, it shall be assumed that all statementsclaiming various outcomes for die A are independent of all the statementsclaiming va¡jous outcomes for die B. That is, the statements "Die A rviUcome up a 3" and "Die B will come up a 5" are independent, as are thestatements "Die A will come up a 6" and "Die B wil.l come up a 6," Thestatements "Die A rvill come up a 5" and "Die A u,ill come up a 3" are

not independent; they are mutually exclusive (u,hen made in regard tothe same throw).

Nou, suppose we rvish to calculate the probability of throrving a I ondie A and a 6 on die B. The special conjunction rrrle can nou,be used:

Pr(I on A & 6 on B) : Pr(I on A) ¡ Pr(6 on B)

:fx+:*In tlie same u,ay, the probabilify of each of the 36 possible combinationsof results of die A and die B may be calculated as f , as shown in Tablc l.Note that each of the cases in the table is mutuallv exclusive of cach othcr

I43

Pr(l) = -.t-

Pr(2) : ;Pr(3) = -:-

Pr(4) : rPr(5) = rPr(6) : r

t,t4 V. nrr PRoBABTLITY cAl.ct.aus

r'ust'. 'l'l¡u.s by' tbe special alternation rule, t]¡e probabiJity of case I v case 3

is ec¡ual to tie protability of case I plus the probability of case 3'

Table IPossible results when throwing two dice

Case Die A Die B Case Die A Die B-I2J

456I<)

J4J6IIJ456

4444445D

55))66666b

t9202loooa

24%26

2829303l.to

JJ343536

1.)

J4561

2J4b61

2J456

IIIII1

2ó

L

2o

2JJJ3.tJ

Io

.)4b6

8I

l01lt2t3l{

1516T7l8

Suppose now üat we wish to calcuiate the probability that the dice will

*rn" r.,p showing a I and a 6. Thereare two ways this canhap-pen: a l on

die A and a 6 on die B (case 6), ora6 on dieA and a I on die B (case 3i)'

Th" prob"bility of this combination appearing may be calculated as

follows:

Pr(l & 6) = Pr[(t onA& 6 on B) v (l on B & 6 on A)]

since the cases are mutually exclusive, the special disjunction rule may be

used to get

Pr[(l on A & 6 on B) v (I on B & 6 on A)]' = Pt(t onA&6onB) * Pr(l on 8&6onA)

But it has already been showl, by the special coniunction rule' that

Pr(lonA&6onB) = *Pr(IonB&6onA) =*

so the answer is fi + *, or* .

The same ro.tÜf ,"L-o-,rir.ig'c"r, b" used to solve more complicated prob-

Iems. Suppose we want to -know üe probabili§ that the sum of spots

V, 4 coN¡uNcrroN nul-Es t45

showing on both üce will equal 7. This happers onl1, in cases 6, 11, 16,21, 2Á, and 31. Therefore

Pr(total of 7) = Pr[ (1 on A & 6 on B)v (2onA&5onB)v (3onA&4onB)v (4onA&3onB)v (5onA&2onB)1, (6onA&lonB)l

Using the special disjunction rule and the special conjunction rule Pr(totaiof 7) = -o,ot*.In solving a particular problem, there are often several rvays to applythe rules. Suppose we wanted to calculate the probábiIis that both dicervill come up even, We could determine in rvhich cases both dice areshowing even numbers, and proceed as before, but this is the long rvay tosolve the problem. Instead we can calculate the probability of getting aneven number on die A as r by the special disjunction rule:

Pr(even ""

o'

:: i:l_", x; iT,áJ.liiÍ',*u "" o,

and calculate tf,. prob"U'ltity lf g"ftlng an even number on die B as i bythe same method. Then, by the special conjuncbion rule,s

Pr(cven on A & even on B) = Pr(even on A) X Pr(even on B)

=+x+:+We apply the general coniunction rule when two statements are ¡rot

independent. Such is the case in the following example. Suppose you arepresented rvith a bag containing l0 gumdrops, 5 ¡ed and 5 black. You areto shake the bag, close your eyes and draw out a gumdrop, look at it, eatit, and then repeat the process once more. We shall assume that, at thebime of each draw, cach gurndrop in thc bag has an cqual probabilityof being drarvn. The problem is to find the probability of drarving trvored gumdrops.

6 It can be shown that the statements "Die A will come up even" and "Die Bwill come up even" are independent, on the basis of tlre independenceassumptions made in setting up üis example.

146 V. rru PRoBA¡ILITY CALCILUS

To solve üis problem we must Gnd the probability of the conjunction

Pr(red on I & red on 2). We will fust 6nd Pr(red on 1). We rvill designate

each of the gumdrops by a letter: A, B, C, D, E,F, C, H,l, /. We know

tl¡at we rvill drarv one of tlese on the 6rst draw, so

Pr(A on lvB on I v C on lv...v,lon 1) : 1

\ow, by the special disjunction rule,

Pr(Aon t) + Pr(B onl) * Pr(Con 1) + ... + (Pr(/on l) = 1

Since each of the gumdrops has an equal chance of bcing drarvn, and

tlere are l0 gumdrops, therefore

Pr(A on l) = *,'(, "i 1) = *

Pr(/ on I) : *\Ve said that there rvere 6ve red ones. We will use the letters A, B,C, D,

and E to designate the red gumdrops and thc remaining letters to desig-

nate the black ones. By the special disjunction rule, the probability of

gening a red gumdroP on drau' I is

Pr(A on 1 v B on I vCon I v D on I vE on I): Pr(A on l) + Pr(B on 1) + Pr(C on I) * Pr(D on 1) + Pr(E on 1)

=*=i\\,e shall have to use the general conjunction rule to 6nd Pr(red on 1 & rcd

on 2), since the statements "A red gumdrop u,ill be drarvn the first ti¡ne"

"nd :,A red gumdrop will be drawn the second time" are not independent.

If a red gur,clrop ii drarvn the first time, this rvill leave four red and 6ve

black gumdrops in the bag u,ith equal chanccs of being drarvn on the sec-

ond dÁ','r'. Buiif a black gumdrop is dra§'n the first time, this will leave five

red and four black gumdrops arvaiting the second drarv. Thus the knorvl-

eclge that a recl one is drarvn the 6rst time rvill influence the probability

,ve assign to a red one being drarvn the second time, and thc trvo state-

ment§ are not independent. Applying the general conjunction rule, rvc get

Pr(red on 1&red on 2) = Pr(red on I) X Pr(red on 2 given red on 1)

\Ve have alreadl,found Pr(recl on 1). Norv "ve

must calculate Pr(red on 2

given red on 1). Civen that we drarv a red gumdrop on the fir'st drarv'

ihere r,'ill be nine gumdrops remnining: four red and 6ve black' We nlust

V. 4 coNluNCTroN Rr.rLEs r47

draw one of them, and they each have an equal chance of being d¡awn,By reasoning simila¡ to that used above, each has a probabiñry of {of being drawn, and the probabirity of drawing u ,ed onJ is a. Therefore

Pr(red on 2 given red on l) : +We can now complete our calculations:

Pr(red on I &redon 2) : i X * = *We can calculate Pr(black on I & red on 2) in the same way:

Pr(black on 1) - i-Pr(red on 2 given'black on l) = r

Therefore by the general conjunction rule,

Pr(blackon I &red on 2) : i X * = *-

At this point the question arises as to what the pr(red on 2) is. we knowPr(red on 2 given red on t) : * . We know pr(reá on Z gve,n black on1) =,*. But what we want to knów now is the probabiliry oT getting a redgumdrop on the second draw befofe we have made üe 6rrt dr"ir. wecan get the ansrver if we reaüze that red on 2 is logically equivalent to

(red on I & red on 2) v (not-red on I & red on 2)

Remember that the simple statement g is logically equivalent to the com-plex statement (pAq)v(-p&q). Therefore

Pr(red on 2) : Prl(red on ] & red on 2) v (not-red on I & red on 2)]By the special disjunction rule,

Pr(red on 2) : Pr(red on I & red on 2) *Pr(not-red on I & red on 2)

we have calculated Pr(red on I & red on 2) as f . we have arso carculatecl

Pr (not-red on I & red on 2) = Pr(black on I & red on 2) : *Therefore

Pr(redon2) = * * * = * + * = * =iThe same sort of applications of cpnditionai probabirity and üe generrrl

conjunction rule would apply to card games *¡ere the ca¡ds that havr:been p-layed are placed rn a discard pireiather than being retu¡ned to thedeck. such considerations are treatá very carefuü¡,in rnanuars on pokr:r

V. nm PRoB^BILITY cAlcuLUSI48

and blackjack. In fact, some gambling houses have resorted to using a new

deck for each hand "f

Uft"t¡?tt in oid"' to keep astute students of prob-

ability from gaining an advantage ovcr the house'

Exercises:

f. Hp) =*, Pr(q) : {, p and q are independent'

a. What is Pr(¡r&q)?

b. Are P and q mutuallY exclusive?

c. What is Pr(Pvg)?

2.Supposetrvodicearerolled'asintheexampleabove.a. What i, tt" p,oúlüilii, "f

U"t¡ dice showing a 1 ('tnake-eves")?

[. ri'¡", l, tt," p.oU"iiiiti of Uott' dice showing a 6 ("boxcars")?

c. \Vhat is the probability that the total n-umbJ' of spots showing on both

dice will b" "ith"t 7 or 1 1 ("a natural" ) ?

3. A coin is flipped three times' Assume that on each toss Pr(heads)= { and

Pr(iaik) = ¡. Assume that the tosses are independent'

a. \\'hat is Pr(3 heads)?

b. What is Pr(2 heads and I tail)?

c. What is Pr(l head and 2 tails)?

á. W¡", ls Pr(heaJon toss I & tail on toss 2 & head on toss 3)?

e. What is Pr(at least )' tail)?

f. What is Pr(no heads)?

g. What is Pr(either 3 heads or 3 tails)?

4. Suppose you have an ordinary deck of 52 cards A card is drawn and is

not replaced, then another card is drawn' Assume that on each draw each of

il. .rla, then in the deck has an equ:rl chance of being drawn'

x. What is Pr(ace on draw l)?

b. What is Pr(I0 on draw 2 given ace on draw l)?

c. What is Pr(ace on draw I & i0 on draw 2)?

d. What is Pr(10 on draw I & ace on draw 2)?

e. What is Pr(an ace and a 10)?

f. What is Pr(2 aces)?

5. The probability that Ceorge will study for the test is *' The probability that

he rvillpass the test gt';; il;;i; ti"ai"t íti The probaüi,tv that he will pass

,¡" ,",,'Ñ* that hJ ,1;;t ;;i "udv

i' : ' r'i¡hat is the probabilitv that Ceorge

wil) pass the test? Hint: The 'i-pit 'i't"-ent."Ceorge will pass the test" is

logicalll' equivalent t" if'"-t"tpltx statement."EitheJGeorge will study and

pittfrá test or Ceorge will not study and pass the test"'

V.5' EXPECTED VALUE OF A GAMBLE' The attractiveness of

a wager depends no, *t, ot' tht probabilities involved' but also on the

odds given. The probability of g"tting a head and a tail on tw'o independ-

V. 5 rxpscrED vl¡-uE or A GAMBLE 149

ent tosses of a fair coin is ,!, whi-le üe probability of getting two headsis onlyf. But if someone were to ofrer either to bet me even money thatI will not get a head and a tail or give 100 to I odds against my gettingtwo heads, I would be well advised to take the second wager. The prob-ability üat I will win the second wager is less, but this is more thancompensated for by the fact t]¡at if I win, I will win a great deal, andif I lose, I wiLl lose much less. The attracliveness of a wager can be mea-sured by calculating its expec-ted oalue. To calculate the expected valueof a gamble, 6¡st list ail tle possible outcomes, along with their prob-abilities and the amount won in each case. A loss is listed as a negativeamount. Then for each outcome multiply the probability by the amountwon or lost. Finally, add these products to obtain the expected value. Toillustrate, suppose someone bets me 10 dollars that I will not get a head anda tail on two tosses of a fair coin. Ttre expected value of this wager forme can be calculated as follows:

Possible outcomesToss I Toss 2

Probability Gain Probability X Cain

- $10

t010

-10Expected value: $0.00

Thus üe expected value of the wager for me is 90, and since my opponentwins what I lose and loses what I win, the expected value for him is also

$0. Such a wager is caüed a fair ba. Now let us calculate the expectedvalue for me of a wager where my opponent will give me 100 dollars ifI get two heads, and I will give him one dollar if I do not,

Possibleoutcomes ProbabilityToss I Toss 2

Cain Probability X Cain

¡.

IIL

II

H

THT

HHTT

- $2.50

2.fl2.fl

- 2.s0

I

ItI

II

H

T

H

T

H

H

TT

$100

-1-l

$25.00

- 0.25

-0j5-0.25-1.

E*p"ct"dvail* §24.25

r50 V. rsr PnoBABILITY cALCULUS

Ttre expected value of this wager for me is $24.25. Since my opPonent

le5s5 u'tat I rvin, the expected value for him is -$24'25' This is not a fai¡

bet, since it is favorable to me and unfavorablc to him'The procedure for calculating expected value and the rationale behind

it are ciear, but let us try to attach some meaning to the numerical answer.

This can be done in the following way. Suppose tlat I make the foregoing

Eager many times. And suppose that over these many times the distribu-

tion'of results corresponds to the probabilities; that is, I get trvo heads

one-fou¡ü of üe time; a head and then a tail one-fourth of the time; a

tai.l and then a head one-fourth of the time; and hvo tails one-fourth of

üe time. Then the cxpected value u,ill be equal to my average rvinnings

on a \r'ager (that is, my total rvinnings divided by the number of rvagers I

have made).I said that expected value was a measure of the attractiveness of a

wager. Cenerally, it seems reasonable to accept a wager rvith a¡ositiveexp-ected gain and reject a wager with a negative expected gain. Furüer-móre, if yóu

"r" offered a choice of wagers, it seéms reasonable to choose

üe u,ager with the highest expected value. These conclusions, horvever,

are oversimplifications. They assume that there is no positive or negative

ualue associated with risk itself, and that gains or losses of equal amounts

of monel, represent gains or losses of equal amount of value to the indi'r"idual involved. Let us cxamine thc first assumption.

Suppose that you are compclled to choose an even-money wager either

for I dollar or for I00 dollars. The expected yalue of both rvagers is 0.

But if you rvish to avoid risks as much as possible, you rvould choose the

s-.llei wager. You would, then, assign a negative value to risk itself.

Horvever, ii you enioy taking larger risks for their own sake, you would

choose üe larger *rg"r. Thus although expected value is a major factor

i¡ determini"g th" attractiveness of rvagers, it is not the onll, factor. Thc

positive or negative values assigned to the magnifude of the risk itseU

must also be taken into account.

we make a second assumpbion when we calculate expected value in

terms of money, we assume that gains or ]osses of equal amounts of

money represenl gains or losses of equal amounts of vaiue to the individ-

ual involved. In -üe

language of the economist this is said to be the

assumption üat money has a constant marginal utility' This assumption

is quite often false. For a poor man, the loss of 1000 doila¡s might nrean

be would starve, rvhile ttre gain of 1000 dollars might mean he would

merely live somewhat more comfortably, In this situation, thc real loss

accompanying a monetary loss of 1000 dollars is much greater than thc

,eul gái, accómpanying á mouetary gain of 1000 dollars' A nran in these

V. 5 sxpecrED VALUE oF ^

GAMBLE

cücumstances would be foolish to accept an even money bet of 1000dollars on the 0ip of a coin. In terms of money, the wager has an expectedvalue of 0. But in terms of real

'alue, the u'ager has a negative expectedvalue.

suppose you are in a part of t]¡e city far from home. you have lostyour rvallet and only have a quarter in change. Since the bus fare homeis 35 cents, it looks as though you rviil have to walk. Now someone offersto flip you for a dime. If you win, you can ride home. If you losc, youare hardly any worse off than before. Thus although the expected valueof the wager in monetary terms is 0, in terms of real value, the wager hasa positive expected value. In assessing the attractiveness of rvagers bycalculating their expected value, we must always be careful to see whetherthe monetary gains and losses accurately mirror the real gains and lossesto thc individual involved.

Exercisesr

I. what is the expected value of the follorving gambre? you are to roll a pairof dice. If the dice come up a natural, 7 or Il, you win l0 dollars. If üe dice come upmake-eyes,2, or boxcars, 12, you lose 20 dollars. Otherwise the bet is off.

2' what is the expected value of the following gambre? you are to flip a faircoin. if it comes up heads you win one dollar', and the rr,¡gs¡ is over. If it comes uptails you ]ose one dollar, but you flip again for two doilars. If the coin comes upheads this time you r,in trvo dollars. If it comes up tails you lose two dollars, bulflip again for four dollars. If it comes up heads you rvin four dollars, If it comesup tails you Iose four dollars, But in either case the u,ager is over.

Hint: The possible outcomes are:

Toss l Toss 2 Toss 3

None NoneH NoneTHTT

3, suppose you extended the doubling strategy of Exercise 2 to four tosses.Would this change the expected value?

4. supposc that vou triplcd your stakes instead of doubling them. woulcl tlrischange the expected va)ue?

5. 'suppose that you have jr,rst enough money to buy needed meclici¡¡e [c.¡ryourself or your family. someone offers to bet you on üe flíp of a fair coin. If itcomes up heads, he will give you l0 dollars. If it comes up tails, you rvill givr.him five dollars. In these circumst4nces it would be unreasor¡able for yorr t,accept this normally lttractive wager. \\¡hy?

t5I

HTTT

t;12

.

V. r¡rr PRoB^-BILITY cArct¡Lus

v,6. GAMBLING AND TIIE PROBABILITY CAIIULUS.If proba-

lrilities are to be r.rsed to calculate erpected value of gambles as in üe last

s¿ctior\ üere are elementary practical reasons for üe maüematical rules of

the probability calcu_lus. kt us assume for simplicity t}at money has con^stant

marginal utilify and that there is no value, positive or negative, attached to risk

itself. Then the erpected value that you attach to a wager üat pays you 1 dollarif p and such üat you lose noüing if -p, is just $P(p).

$1 x P(p) + $0 x P4-p) = $P(p)

Then a tautolngy should have probability I because the wager: "You get Idolla¡ if a tautology is tn:e, noüing otheru"ise" should have a value of Idoila¡-a payoff that you are certain to get because tautologies are tnre in all

cases. Likewise, a conttadichon should get probability 0 because üe wager"You get 1 dollar if a contradiction is true, nothing oüerwise" is obviouslyworü nothing.

Nothing should get a probabiliry greater üan I or less üan 0. If a state-

ment p got a probability greater than l, the wager "You get I dollar if p,

noüing otherwise" would get an expected value greater than I dollar, an

erpected value greater üan an¡thing you could possibly win. If a statement P

got a probabiliry less than 0, then üe wager "You get I dollar if p, nothing

otherr+'ise," which you can win but not lose, would get negative expected

value; and the wager "You get -1 dollar (that is, you lose I dollar) if p, nothing

oüeru'ise," which you can lose but not win, would get positive expected value.

We can also give a gambling rationale for the special disjunction mle.

Suppose that p;{ are mutually exclusive. Then a bet "l dollar if p, nothingoüerq'ise," taken together with a bet "1 dollar if g, nothing otherwise" gives a

total payofi of I dolla¡ if p is true or if g is true, nothing if boü are false (they

can't boü be true because they are incompalible)' That is, üe bet on p and on

g, taken together, is tantamount to üe bet: "1 dollar if P or q, noüingotherwise." If I'm willing to pay $P(p) for the first bet and üen $P(g) for the

second one, I will have paid $P(p) + P(g) for üe lot. If my evaluations are

consistent, I should üen give that same value to üe bet "l dollar f p or q,

nothing otherwise." But the erpected value of üis bet is P(pvq), so I have üespecial disjunction nrle:

P(prc)=P(p)+P(q)(when p;q are mutually excluive)

All üe rules of the probability calculus given in this chapter can be shown to

follow from those jrxtified in üis section. A more detailed development of üis

sort of decision-theoretic approach to probability is contained in Chapter \¡II.

V.7 neyes'THEoREMI53

V.7. BAYES' THEqREM. you may wonder what the relation isbetween a conditional probability pr(g given p) and, its converse pr(pgiven g), They need nor be equal. The p.obrbirrty that Ezekial i; ;; ,p",given tlat he is a gorilla, is

J. Bu-t the piobability that Ezekial i, " fo.iu",given that he-is an ape, is Iess tlan r-, The value of ^;;;i,i;;io."u-abüty is not determined by the varue of its converse arone. But the varueof a conditional probabiüty can be calculated from the varue of its con-verse, together wiü certln other probabiüty values. The basis o] ,f,i,calcu.larion is set forth in Bayes' tlworem. a simplified *;;;;;."proofof Bayes' üeorem is presented in Tabre 2. step 4 of üis tabre statesüe simplified version áf n"y"r' ü"o;:; ñ"," ,¡", it arows us to com_pute conditional probabilities- going in one direction_that ir, pr(f givenp)-from conditional probabirities loing in the opposite direction-that

Table 2

Step Justi6cation

1. Pr(s given p) : PiÍP&Í)rr\p )

Pr(s given ,) = ¡rtlraffiil--3. Pr(q givcn p) : ,, , Pr(P&g)

rr(ptq)-+T(p&=q)4. Pr(q given p) =

De6nition of conditionalprobabiliry

p is Iogically equivalentto (p&q)v(p&-q)

Special disjunction rule

Ceneral conjunction rulepr(q) X pr(p given s)tn

,.The general form of Bayes' tlreorem arises as folrows: suppose that instead ofsimply üe two statements g and -g *. "onri,l"r

r r"t o¡ n.rtrrlfy "_"lrrir"statements, qy q2,.,. en, which is ethauslioe. That is, the complex st;;;;;qtvq2v "'ven, b a tautolog'y. Then it can be proven tl¡at the ri-pr" ,,","-"ri'pis logically equivalent to the complex statemenr (Ái-)r(pdqrir. . .;ü;: íThis subsütution is made in step 2, and üe rest oi the'proof fo,ows the modelof the proof given. The ¡esult is

Pr(q,givenp):,,=,,, = . Pt(9,) XPr(pgir.r g,)t(pr(q¡) X pr (p r,r." Or)1ffi

154 V. rur PnoBABrLrrY clrcu¡.us

is, Pr(p given g) and Pr(p given -g)-together with certain statcment

probrbüties-tlat is, Pr(q) and Pt(-q). Let us see how üis theorem

is applied in a conc¡ete examPle.

Suipor" we have tlvo urns. Urn 1 contains eight red balls and two

blaci ta¡s. Urn 2 contains two red balls and eight black balls. Someone

l,rs selected an urn by fljpping a fair coin' He then has drawn a ball

from the urn he selected. Árru." üat each bali in t}e urn he selected

had an equal chance of being draull]. \\4-rat is the probability that he

selected urn I, given that he drerv a red ball? Bayes' tlreorem tells us

the Pr(urn I given red) is equal to

Pr(urn 1) X Pr(red given urn l);1) X Pr(red given -urn l)l

The probabilities needed may be calcuiated from the information given

in the problem:

Pr(urn t) = +Pr(-urn l) : Pr(urn 2) = +Pr(red given urn I) =-!-Pr(red given -urn 1) : Pr(red given urn 2) = *

If these values are substituted into üe formula, they give

:r(-Pr(urn I given red) - /t \1g)+1lY:)\: ,. to/ \. ,- ¡o/

A simi.lar calculation rvill shou'that Pr(urn 2 given red):r Thus the

application of Bayes' theorem conffrms our intuition that a rcd ball is

more likely to have come from urn I üan urn 2, and it tcUs us horv

mucb more [kely.It is important to emphasize the importance of the pure statemcnt

probabüties Pr(q) and Pt(-q) in Bayes'üeorem. If rve had not knorm

üat the urn to be d¡awn from had been selected by flipping a fair coin,

iJ we had just been told üat it was selected some way or other, we

could not úave computed Pr(urn I given red), Indeed if Pr(urn 1) and

Pr(-s, 1) had been diferent, then our answer would have been di-f-

feient. Srrppose üat the urn had been selected by throwing a pair of

dice. If the dice came up "snake-eyes" (a I on each die), urn I rvould be

selected; otherwise urn 2 would be selected. If this rvere the case, then

-:-10

tll 5

l0 ¡0

V. 7 nayps'THEonEM

Pr(urn l) : * and Pr(-¡¡¡ l) : Pr(urn 2) :the example the same, Bayes'theorem gives

Pr(urn I given red) =I r, 8

s6 , \ rñ8

360 8 a

8 J_ ?o -78-sDseo ' seo

(* x;i;) + (ii x *)This is quite a different answer from the one we got wben urns 1 and 2

had an equal chance of being selected. In each ""r"

prlrr* r given red) ishigher,than Pr (urn 1). This can be interpreted as saying that in bothcases t}le additional information üat a red ball was drawn would raiseconfdence üat urn I was selected. But the initiai level of confidencethat urn 1 was selected is di-fferent in the two cases, and consequentlythe 6nal level is also.

Exercises.:

l. The probability that George will study for the test is {. The probabilityüat he will pass, given that he studies, is -!-. The probrbiiii that he passes,given that he does not study, is -!-. what is ite probability that he has studied,gíven that he passes?

2. suppose there are three ums. urn I contains six red balls and four blackballs. urn two contains nine red balls and one black ball. um 3 contains five redballs and ffve black balls. A ball is drawn at random from um 1. If it is black asecond ball is drawn at random from urn 2, but if it is red the second ball is drawnat random from urn 3.

a. What is the probability of the second ball being drarvn from um 2?b. What is the probabiUfy of the second ball being drawn from urn 3?c. what is the probability üat the second ball drarvn is black, given that it

is drawn from um 2?

d' \14rat is the probability thet the second ball drawn is blaclq given that itis drawn from urn 3?

e. What is üe probability that the second ball is black?f. what is the probability that üe second ball was drawn from urn 2, given

t}at it is black?

g. what is the probabilitv that the second ball was d¡awn from um 3, giventhat it is black?

h. what is the probabiury that the second ball drawn was drawn fromurn 2, given that it is red?

i' what is the probability that the second baü d¡awn was drawn fromurn 3, given that it is red?

36.

155

Keeping üe rest of

V. rrte PRoBA.BILITY cArcuLUSli-l{i

¡. \\/hat is the probabil.ity that the ñrst brll dr¡rvn rvas red' giverr that the

second ball drawn is black?

k. \Vhat i,,t," proU"i;liy tf"t tlre first ball is black' given that the second

ball is bl¿¡ck?

l. What is the probabilit¡' that both balls drarvn are black?

3. A fair coin is flipped twice' The rwo tos.ses are independent' What is the

probability of a heads ", i¡. ntu toss given a heads on the second toss?

4. Their caPtors fr.r" i""ia"¿ that two of .üree prisoners-Smiü' Jones' and

Fitcb-will l.¡e executed;;;;;.-ñ" choice has beán made at random' but the

identi§ of the unfortun"," *i""i"tt ft to be kept from tl¡e prisonen until-the final hour'

Tbe prisoners, who a¡e h"li;;;;;" cells' t'table to communicate wiü each other'

t¡ow this. Fitch asks , g"";";;;'l';e name o[ one of the oüer prisoners who will be

executed. Regardiess oi *n"tll"' fitch was chosen or not' one of the others will be

executed, so the guard t"*;;";;;-;; is not giving- Fitch any illicit information by

a¡swering truthfully. A" *yl,;¡á"., will be exJcuteá." Fitch is heartened by t-he news

for he re¿sons tlrat hi, p,ouJii"y 'iU"ttg the one u'ho escaPes execution has risen from

i;ü:H; ñ';;;-;;i" , '";;íJ H; l''e s""d? use Baves'theorem to anarvze the

reasoning involved. (Hi,t, á^i"tlote the p'áb'bility that Fitch will not be executed

given üat the guard telts him ttrat Jot'"' t'iil be-executed' not the probability that Fitch

will not be executed gi'"n tf'"t foáts will be' What assumptions are.possible 'abof

the

;r;b"bilt ,h^r ü"'g.*,J;"iú-rit"t' üat Jones will be executed given that Fitch

escap€s execution?)

V.8. PROBABILIfi AND CAUSALITY' What is meant when it is

said that smoking causes lung cancer? Not that-smoki ngis a -suficien'

condition

for contraction of lung "tnt?',

for many people smoke and never contract the

d*;- Not üat ,*o-king is a necessar',1-"ondition for lung cancer' for some

*ho n"u., s.ok" n""'tñeless develop hurg cancer' What is meant is some-

,¡mg pt"U"Uilistic: üat smoking increases one's chances of getting lung

*fr""t*,rn, say üat smoking has a tendency in üe direction of sufñcientness if

f.f "*.-igiteá smoking) is lreater than Pi(cancer given -smoking)-üat is'

iiH;;;g"it po§ticely staíxti,olty rcbaant to cÁcer' we might say that

;;ñii* " t".,d"n"y in the diráction of necessary¡ness for lung cancer if

;ñ;;S smoked given cancer) is greater than Pr(having smoktd giv.en no

*),""ri-",nr, is, if ácer is posiiivel! statistically relevant to smoking.But we

; .il* hom üe probabiliry "al"uí,,,

that for any tv"o statements' P'Q'o I is

Jr,i'r"fy statistically ,.I.u,ít to Q if an¿ only if Q is positively statistically

^relcvant to P. 81'Baves' üeorem:

--l wi,¡ positive probabiliry.

V.8 pnosABrl¡Ty A¡¡D cAUSALrr,l¿157

P(p given P) = PII-ql:'' ?)l(P)- -EPI--_

So: P(Q given p) * p(p given p)-TTp)- -T(n-P is positively relevant to p just in case the ,eft-hand side of the equation isgreater tJran one; Q is positivery ¡erevant to p just in case the right-haiá side oftlre equation is greater.than one. so the frobabiristic not;;-;i';"-g

"tendency toward a sufHcient condition, and having a te-ndency ,oo,^raLr.rg "necessary condition come,to. the same thingl c-orxideratioís

"pp"* t" u"simpler. in üis way in a probabiristic setting tñan in a determinisutone.But there is a complication that we musr-now disc,us. Srpp;;;;^;'r_.,.rg

itself did not cause the cancer, but üat desire to smoke and cancer were boüeffects of som.e-una9r]rrng genetically determinecr biorogicar "oná,io-n. r¡"nsmoking would still be positively siatisticalry ."r"urnt"to

"r""*,-úr, * "symptom of having the bad gene rather thrn ,s a cause of cancer. If üishypothesis were correct, we wáuld not say that smoking raised one,s chances ofgetting lung cancer. If someone, say gou, ha<) üe bad g?r"r, ,f,", y;;-:;.".",of contracting cancer would be 1i.árdy high and ,,ioki.,i *oiJ*nJi'*"¡."them worse; if you didn't.have.the b"á gá"., you, chan-ces of contractingca¡cer would be lowe¡ and smoking *orrla,t ,,rk" th"* *o^". noa ir, ,¡.positive statistical relevance of smoking to cancer wourd disapp"", if *.looked at probabilities con.ditional on háving the bad g"n"r; tik'"'rir" u ,r"looked at probabiliües conditional on not having ü" bnjg"n"s:

P(cancer given smoking anci bad genes) _ p(cancer given bacl genes)P(cancer given smoking and good genes) _ p(cancer given goocl genes)

- To support the claim that smoking is a probabilistic cause of Iung cancer, theforegoing h¡pothesis (and others lik"e it) iust be rured out. r"riffi]"ntr""rtwins ca¡¡ be found such üat one of each pair is a rong-time ,,not"l,

"na ,no..of the smokers deveron cancer. perhaps surrjects who don,t want to smokebut a¡e forced to inháre smoke "n¡*ry 1""'ioi, raboratory mice, cocktailwaitresses, and so on) have a higher incidence of Iung "*""r.'u we believe t}¡at a certain constelration of factors determi¡res üe chance ofgetting lung cancer, then we consider smoking a probabilistic carlse ;ii,-gcancer if, when we hold all the other p."e*isúng frcto.s lxerl, sm"li"g ,"-creases tle chance of lung cancer. That is, if:

Pr(cancer given background factors ancl smoking) is greater üanPr(cancer given background factors and no ,rnot-ingi

1t3 V. mr PRoBABILITY CALCULUS

\%et}er X is a probabilistic caue of )'for inüvidual a may depend on just

r*üat constellation of backgrouncl factors is present for o. some luc§ people

have a biochemistry such üat for üem, contact with poison oak is not'a

probabilistic *rrr" áf skin enrptions and intense itching, b,t for most of u itunlortunately is.

Exercises:

l. Dlscuss üe following argument: lv{ost hcroin users have previorsly smoked

marijua-na Therefore, marijuana use causes heroin use'

2. Hou' would vou go about finding out whether/or you exPosure to ragweed pollen

is a causc of a stuffed-up nose' runny eyes, and so on?

3. Some studies have found that, on average, convicted criminals exhibit vitamin

deociencies. This suggests to some researchers that ütamin deñciencies might affect

penonaliq,and leadi criminal behaüor. An alternative hypothesis might be üat, in

,n,"y ".'.', there is a common cause_heroin addiction_that leads to bot}¡ criminal

behaüor and malnutrition with its attendant vitamin deñciencies. can you think of any

oJ", potslble hypotheses? \\trat sort of data worrld you gather to decide which is

correct?

Suggested readings

I. J. Coo<I, C'ood Thinking(Minncapolis: [J¡iversity of Minnesota Press' I983)'

chaps. 21, 22.

Patrick Sup¡:es, A lrol¡ol¡ílistic Thcory of Causulíty (Amsterdam: North Hol-

land, 1970).

Appendix to Chapter V:Sampling and Statistics

A.l.INTRODUCTION. The word "sampling" most readily brings

to mind public opinion polls, quality control procedures in industry, and

ratings of tel.üsion programs. But the use of sampling is far more uni-

,".r"'Í üun is suggesied by üese immediate associations. When you take

a sip of wine to ascertain üe qualify of the contents of the bottle, you

are iampling, The same is true when you judge the general reliabiliry

of a rn"lie oiautomobile on the basis of the experiences of several people

who have owned one, or when you decide how honest a man is on the

basis of ümited observation of his behavior. Whenever you attempt to

gain information about a group of things (persons, events, physical obiects,

ictions, etc.) by examining a portion of that group, you are sampling-'

The group that is the object of the investigation is called the population

and thJ portion that is examined is called tbe sample. When we draw a

conclusián as to the compositioí of üe population on üe basis of

information on the composition of the sample, we shall say that we are

making a statistical inlerence.

No ,ery interesting statistical inference is deductively valid. A deduc'

tively vaiid "rgu,n"nt

whose premises simply describe the sample and

say that it is included in the population can only conclude rvhat is true

of the population by virtue of the sanlple being a part of it; for example:

I have examined 10 lions and half of them have had tooth decay'

The t0 lions in my sample *"t" "! fto. A'"tit"*

@erican zoos at the time of

investigarion included at least five with tooü decay'

The evaluaüon of Iess trivial §pes of statistical inference is üe business

of inductive logic, for if üe arguments involved are to have any merit they

must be induclively strong. Such inferences are studied in a branch of

statisbics knorvn as proiectitse sfaÚrsfics. Projective statistics is based on t}¡e

üeory of probabili§ and on anoüer branch of statistics: descriptiae statis'

t;cs. bes&iptive statistics analyzes the various rvays in which the com-

position of , g.orrp of üings (sample or population) may be succinctly

cbaracterized, .

4.2. DESCruPTIVE STATISTICS. Suppose we wish to character-

ize üe composition of a group of pcople in terms of tt¡c scorcs that its

my

159

I (X) APPEI.{I)D( TO CHAPTI]II V: SAMPLINC AND S'I'ATIS'I-ICS

nrembcrs have nradc on a stancla¡d intelligencc tcst. The most complete

charactcrization of the composition of the group would consist of a

tal¡ulation of the number of pcople who rcccivcd cach sct¡re. Such a

tabulation is known o.s a frequency distribution. Ilut just because the

frequency dist¡ibution is so complete a characte¡ization, it runs the risk

of overwhelming us with detail. Gene¡al tendencies and patterns may be

difficult to distinguish in a forest of fgurcs. For this reason, descriptivestatistics has developed various ways of characterizing the compositionof a group which are less complete but more succinct.

Probably the most familia¡ concepts of descriptive statistics are those

associated with the word "average." Tliere are many dillerent types ofaverage, the most common being the mean, median, and mode.

Suppose we have a group of l0 gumdrops and wish to characterize the

composition of the group with respect to color. If we characterize the

composition of a group simply in terrns r¡f c¡ualitatiae categories (such as

color) the only type of average that can be applicd is the mode. Suppose

that the frecluency distribution of colors within the group of gumdrops is

as follows:

Category Fre clue ncy

RedGreentavenderBlack

5

3

II

The modal c'olor fo¡ üis group is red. The mode is simply that categorywhich has more membe¡s of the group in it than any other.

If, however, our categories are not merely qualitative, but fali in some

natural order, wc can apply not only the mode but also thc median. Sup'pose we wish to categorize nine cups of collee with regard to crudetemperature categories, as shown in the following frequency distribution:

Category Frequcncy

Boiling hotComfortably hotTepidCold

2o

I4

The median temperature in this group is tepid. The medbn category is

thc c¡¡¡e in üe middle, that is, the one such üat üere are a§ many üings in

A. 2 o¡sclup'rtvE srATrs-rrcs 161

higher catcgories as i, lowcr categories. it dicl not ,lakc scnsc to talkabout a mcdian in the gu,rdrop c.ramplc bec^use it docs not nrakc sc.scto talk about higher and lorvcr colors, alth,ugh it crocs ¡nakc scnsc totalk about higlier and lowcr tc,npcratu¡es. We can, howcvcr, talk aboutüe mode as wcü as thc median iil thc cxample of the cups of coilec.Note that whiie the "average" cup of coffee, in the scnse of median, istepid, the "avcrage" cup of coflee, in thc sensc of nrodc, is cold.

The third kirrd of average, thc mcan, is applicablc only whcn thecategories are quantitative. Suppose we wish to cha¡acterizc a groupof 100 students in terms of their scores on a lO-point quiz. Thc frequcncydistribution of thei¡ sco¡es is as follows:

Category (scorc)Frcquency (nurrrbcr

of studcnts)

t0I8

7

6

5

1

.J

o

I

I2620

LI

8

J

I0I

We may calculate the mean score by adding up all the individual scores(5 tens, 9 nines,26 eights, etc.) and dividing the total by the nurnber ofindividuals (100). The rnean score fo¡ this group is 6.g7. 'I'he nrcun clt,¿.rr-tity for a group is thc su¡¡r of the quantitics for cach ¡ncnrbcr of thegroup divided by the nurnber of membcrs. It did not makc sense to talkabout the mean in eiüer the gumdrop or the cofice examplc becausethe categories were not quautiües and could not be added or divided.But it does make sense to talk about both the rncdian arrd rnoclc, as r.vcll astle mean, rvhen the categories are quantitative. A chcck of the frequcncydistribution of test sco¡cs shorvs that thc median scorc is 7 and tl¡c rnod¿rlscore is 6.

We have surveyed only a small nunrbcr of thc conccpts of dcscriptivcstatistics. A more thorough treatment may be founrl i¡r somc of the sug-gested rcadings lístod at the cnd of this scction. Howcvcr, onc point that

ll

162 API'L]NDIX TO CIIAI''TER V: SAMPLINC AND STATISTICS

wc have made in conncctio¡t with v¡rious types o[ avcragc holds in geDera]

for clcscriptive statisticsr The types of dcscriptive statistics wlrich are

applicablc depend on the types of categories, uscd to characterize the

members of the group,

Excrciscs:

l. Supposc you liave a bag containing 100 mineral specimens. You ascertain

l,<.,* mairy c¡f tl¡crn fall i¡rt<.¡ cacl¡ of tlrc following catcgorics: lrard cnor,rgh to

scratch glass; hard cngrrgh to scr¡tch talc l¡ut not hard eno¡glt to scratclr glass;

not har<I enouglr to scratch talc. Civen only this information, which typc or types

of average would be applicable? WhY?

2. Nine str¡dc¡rts takc a lOO-poirrt qtriz. thc scores arc as follows: 79,78' 17'

75,7t, anr|70, a¡¡d thrce zeros. Calculato thc mcart, uredian, and modal scorcs'

Suggested rcudings

Darrell HufI, Hou to Líe With Sraúisrícs (New York: W, W. Norton and

Comparry, Inc., 1954).

I¡or a <liscussio¡l r¡[ ll¡c rclationslrip [)etrvec¡t tlrc typcs of categorics tlsed

to clr:rractcrizc thc mr¡rnllers of a grorrp al¡d tlre types of descriptive

statistics w)iicli are rpplicablc, the advanced student should see the

fr-rllowing:

S. S. Stevens, "OIr thc Theory of Sc¿lcs of lrleasuremettt," in Plrilo'sopñy

ol Sciercc, Arthur I)r¡rto and Sidney lvforgenbesser, eds' (New York:

Nlerielialr Boo|s, Inc., 1960), pp' l'11+9.

A.3, SANÍPLINC AND PROJECTIVE STATISTICS' When lve

lc¡row that a ccrtai¡t <Jcscriptivc stxtistic corlcctly charactcrizcs ¿r salnple

(for example, that the mean weight of a samplc of 100 "one-pound" boxes

of Sti"t y bookies is 14 ounccs) and wc wish to conclude somcthing about

the rlcscriptiv<: statistics which co¡rcctly charactcrizes the population (for

example, üat the mcan weight of all "onc-pourld" boxcs of sricky cookies

is beüw one pound) thcn wc arc involved in problcms o[ statisticai in-

ference. The great majority of the descriptive statistics with which we are

presented arJ arrived at in üis way, rather üa, by an exami¡ration of

ihe entire population. The Sunday supplements regularly print statem-enls

as to the or.rrg" (mean) weight of American women between 30 and 40'

the average (mean) number áf hui* on an adult male Caucasian's head,

and the froportion o[ Americans who are suflering from u,diagnosed

mental ilit es.. lt should be rather obvious that none of these figures were

A, 3 S ¡TTT'¡-I¡'¡G AND PROJI,CTIVE STATISTICS 163

obtaincd frorn a complete survcy of thc population, but r¿rthcr by infcr-cnce f¡'o¡n sanrplcs.

In gcncral, the infcrence from sanrplc statistics to population statisticsis, at bcst, inductively strong. Conrmon sensc tclls us that somc typcs ofsarnples provide a lirmer basis for such infcrences than others, Sornconewho claims that the mean weight of Japanese ¡nc¡r is ovr:r 200 pouncls

on the basis of a sample cornposed solely of Sumo wrcstlcrs would notconvirrcc rna»y pco¡rlc. Such ¡ sar¡¡>lt: is said to l¡c l¡ioscd by viltuc o[thc fact that onc nrust be quitc hcavy to l¡c a Sr:mo wrcstlcr. Althoughtlrc Sur¡o wrcstlc¡ san-rple is so blatantly biascd that it would ¡nislcad ¡ro

onc, sarnplcs can be biased in more subtlc ways. Considcr, for cxanrplc,a poll takcn on ¿r Ncrv York st¡cctcorner i¡.t whiclt cvcry tcrtth pcrsottwho passes thc st¡cctcol'nc¡ is intcrvicivcd. This may appcar to bc a

rvay of obtainirrg an unbi¿ist:tl sartt¡>lc o[ thc opirrio¡ts t¡[ Nt:w Yt¡tkcrs.

But suppose that thc poli dcals with attitudes toward thc wclfare statear¡d that it is takcn on strcctcor¡rcrs in thc ltcart of the financial distlictduring lunch hour on wcekdays. Wc rvoulcl ccrtainly not cxPCct sttclt a

sanrplc to bt: rcplcscutativc o[ tlrc r:r,tirc population of Ncw York. It is

clcar thcn that for thc infcrcncc frorn thc samplc to the populatio¡r tr.r

bc inductivcly strong, the sarnplc nrust bc u¡ll¡iasccl,Ilut cvcn ¡n unbiased satnplc tnay not bc ntuclr good if tltc saruplc is totr

snlall. An u¡tbi¡scd sarnplc c¡f ouc Ncrv Yo¡'kcr "vould

not bc of ntuclr

hclp in tclling us thc pcrccntage of Ncw Yorkcrs s'ho arc f<:nl¿rlc. A ne*'

drug woulcl hardly bc juclged mcdically safc if it htd l¡cc¡r tricd o¡r ¡sarnple of only l0 ¡rcople. For thc infcrc¡.tcc fronr thc san',plc to thc

population to be inductrvely strong, thc slrrrrplt: ¡¡rust bc sLrfficicntl¡' largc,

and tl¡e largcl thc bcttcr.Conrmon sense has given us somc rules of thunlb for cvaluating statis-

tical i¡rfere¡rces. If wc arc to go farthcr wc ¡r'lust vcnturc into thc donrain

of projcctive statistics. Projcctivc statistics blings thc thcory of prob-

ability to bcar o¡r thc problcnr of statistical i¡rfcrcncc by rctluiring that

a sample be ra¡rdon. Saying that a satrtplc is ra¡ldo¡rl docs ¡rot refcr toany characteristics of tltc sample itself, but lathcr to thc sclcctiou proccss

used to pick the sam¡rlc out Irom tlre populatio¡r. A ¡a¡rdom-sclectio¡r

process has two main virfues. Thc first is that it clintinatcs L¡ias fro¡¡r sanl-

pling. The second is that if we know that a sample had becn picked by a

randonr-sclection process and if we know a few things about the popula-

tion from which it was selected, wc can somotimes calculatc thc prob-

ability that the sample will resemblc the population.

rl i,;l

l(i,l AI,I,I.JNI)¡X 'I.O CHAPTIIR V: SAMI,LING AND STATISTICS

'l'lrcrc rtrc various typcs of ra¡tclonr-sclcction proccsses' We shall exam-

ir¡c four of them. Lct us suppose that we have a herd of 10 elephants as

our population, and wish to Ánd out something about üc average weight

of tl," "l"ph^.,ts,

but that because of the diflicu-lties involved we can afiord

to weigh only 4 of them' One way to eliminate bias would be to select

o,.,..oñpl" by th" follo*ing processr We number thc elephants I tlrrough

10. Wc ihen'roll a l6-sidá, weli-balanced die and weigh the elephant

whose number colres up. Wc roll the die three more tilncs and weigh

the indicatcd elcphants ln or.l"r to come up with a list of four weights'

e .r*pf" selectá in this way is called a sitnple tondom sample uith-r-epla"i*rrt.

The phrase "witi replacement" alludes to the fact that if

the sample is taken i¡r üis ma¡rnei, the sa'ne elephant may be included

more than once in the sample, since his number may come up on more

thanonethrowofthec]ie.Wemightsaythatafterthe§rstelephantisselected to be a mcmber of the sailplc , ít is replnced in the population

and is thus eligible to be selccted again' A selcction procedure gives us

a simple random sample with replacemcnt if:

I. Each membe¡ of the population has an cqual probabili§ of

being selectetl as thc 6rst member of the sample'

2. Each ¡le¡nber of the population, ilrcluding those selectcd as

previous members of the samplc, has atr equal cirance of being

ielected as the nth nre¡nber of thc salrrplc'

A sinrplc ralrdot¡t saniplc rvith rcplaccrnent, although unbiascd' is typ-

icalty an inerflicie¡rt *"y uf gaining infornraüon abr¡ut a popuhtion' lt*ori¿ tr" silly to *cigl, tl'e '*""'" "l"pl'""t four tir¡rcs Thc natural thing

to do wouid be to require that the sa"tple be cornposed of fou¡.difiercnt

;i";ilit. We could use the "rn" '"tpling procedure exccpt Ürat if on

Á"-t"*"¿, third, or fourth throw of the die the number of arr elephant

already selectcd carne up, we would roll again until wc got a diflerent

;rntr;t. A sarnplc ubtoi,'"i by such a ¡rrntttdt"e is called a shnple random

,orinit" *¡rnutt're¡tlucetr*nt' i 'cl"ctitli' proce<lure givcs us a sirnplc r¡r¡dom

sample without rePlaccrncnt if:

l. Each member of the population has an equal probability of

being selectecl as the first member of the sample'

2. Eaü member of üe population, excluding those selected. as

previous ¡nembers of the^simple, has an equal chance of being

ielected as the nth member of the samplc'

A. 3 s¡.l,rp¡-tuc AND PI{oJECTrVE STATISTICS 165

In other worcls, iti a sirnplc ra¡¡donr sarnplc without replaccnrcnt c¡f size r¡'

each group of lr distinct mentbers of thc population has an equal chance o[

constiuting the sample. Thus this type of samplc is just as unbiascd as a

si,,',ple ranáon, ,^.p1" with replacemcnt, but .it avoids tlic risk of loss of

infonnration through duplication in thc samplc' If thc only information

that we have about the population is a list of its rncnrbers, thcn a simple

ranclom sample without rcplaccmcnt is thc bcst bet, and thc larger the

sample the bettcr.Hlrrevcr, if wc hllvc so¡nc a<]clitionul inft¡r'rnation itbout tlic popul;rtiorr,

we may be iu a position to nrakc good usc of a .diffcrcnt typc of ra¡rd-onr

,o.nft". Suppose^we know that the l¡erd of elephants is conrposcd of five

^,lrit "t"ptrmts ancl fivc baby clcphants Using eithcr of thc typcs of

,,,rpt" .*ao,n sampling *" iun tlie risk of gctting a sanrple cornposed

of rll brbi., or all aáults. In either case, the avcragc rvcight ofthe sanrple

rvould surcly not bc reprcsentativc o[ thc avcragc wcight of the.hcrd'

In orcler to cxcludc sucir a possibility, rve ntay sclcct our samplc of four

elephants by taking a simple random sample of two from the adults and

" ,i.pl. ton,lo,n ,lu,rpl" of t*o fro'n the babies' Such a sanrpling pro-

.",1*^" i, calleri stratr¡f ed raulom sanryling' The adults and the babies

coristitute thc two strata. We have stralificd ranclom sampling with

,"fir."*"nt or rvithout replaccmetrt dcpcnding on whether .the cotr-

stituent simple ran<lon, ,"rnpl", are takcn with or without replaccmcnt'

Noticc that thc fcature of this cxaniplc ü¡at rnade stratified random

sanipling dosirul¡lo was our knowlcdgc thut buby oloplrunts tc¡ld to havc

, ,,r"lr'ioru", wcight than aclult elqrl'a"ts A stratificd random saniple

" irr"i".rúr" to a siDplc raDtlo¡tr '""'pl" if ancl o,ly if rvt: caD divjdc tlrc

populatiorr illto sLrata suclr tl¡at n'"'itbtt" of olrc stritttlrn arc relativcly

alike, and men]bers of different strata are relatively cli[I''rt:nt' with

r"rp""t to the property in which we arc intcrested'- ft; "on""¡,

i,rtrád,,c"cl hcre fornr only a small part of thc. topics

dcalt with in proicctive statistics' A morc dctailed discussio¡r is bcyoncl

the scope of this book.

Exercises:

l. From an ordinaty <ieck of playing cat'ds, a sample o[ four cards is selected

¡yl;*pf" ."na"r" ,"ipfi,,g'*iti "ptl""'n""t' whai is the probability that all

four selections will be aces?

2. From an ordinary deck of playing cards, a sample of four calds is selected

by simple random ,"*pli,rg *ith"ui '"itn""rntnt' What is the pLobability that all

four selections will be aces?

166 API,ENDIX TO CIIAPTEN V: SAIIf PLING AND SI'ATISTICS

3. Pollsters trying to predict the outcomes of national electio¡ts always usest¡atified random samples. Why do you suppose that they do so? If you wereconducting such a poll, how would you choose your strata?

Suggested readings

Morris J. Slonim, Sanytlinpl in a N utshell (New York; Simon and Schuster,1960), pp, l-53.The following is lccornmcnclcd for thc advanccd stuclcnt:i{enna¡i Chcrnoff l¡nd Linc<.¡ln E. l\,foses, I)lementary Dccísiot Tltcory(Nerv York: John Wiley & Sons, Inc., lg59),

VI

Coherence

VI.l. INTRODUCTION. Thc concepts of c¡>istcmic and i¡¡tluctivcprobability werc introduccd in chapte. L., nu,r"iical ¡neasures gradingdcgrcc of r,tio¡ral l>clicf in a stato.rcnt a.<l dcgree oI srp¡rort thc p.,:,,,iscs ofalt argtttncttt givc its t'rtrtclusion. In Oha¡rtcr V wr: cn<:or¡¡¡tr:rc<l a r¡rathcr¡r;¡tic¿rlcharactcrization of probabilitics and corrcljtio¡ral probabilitics. why .shouldc¡>istcrrric atlcl ir¡dr¡ctive probalrilitics olrcy thc ¡natlrc¡natical rules Iajcl c¡;w¡rfor ¡rrol-,allilitios an<l cor¡<litiorral prolralrilitics? O¡rc reaso¡r that r:arr Io givc¡ i.sthat these nlathematical rules arc ret¡uiretl by the rolc that episternic'p.oba-bility ¡rlays i¡r ratiorr¿l dccision.

Ttris sort of argume,t can be ¡racle at vari<¡r¡s lcvels, clepe.cling on whatsinrplifying a.sstrrn¡rtior.rs are rnacle. 'I'hc rli.sc.s.sio. ca¡r bc clcrr¡Át¿rry ¿¡rdt¡ar¡sp¿re¡it, in tl¡c sort of cl¿rssical ganrbling sitrration clisc.ssecl in V.6. I-lcrewe a*§urnc that all that is at issue in the decisio¡l problern is rnoncy. we also¿ssune that rnoney has constant nrargir)al utility across thc rangc áf stakes atissue; that is, a¡r cxtra dollar courts for as nruch whcthcr it is aclcicd to bigwi*ings or big losses. Finally, we assurne that if üe bettor takes ,"u".ulindiüdual bets as fair (favorable, unfavorable), he t¿kes the rcsult of makingthem all together as fair (favorable, unfavorablc). under these assrrrnptions wecan sho\v that if ¿ bcttor violates tha rulas of thc probubility calculus, hc canhave a Dutch.Booft made against him; that is, a clever bookiecan make a seriesof bets wiü him, all of which he considers fair or favorable, such that he suffersa net los no nlatter what happerc regarding the propositiors hc is lrctting on.

The assunrptions for a Dutch Book argument can prausibry be held to b"e tnre(or approximately true) for.typical rnonetary g,mbles witir small stakcs, lrrteach of thcm l¡re¡ks down when we co¡rsidcr thc problcrn of rutio.¿l dcci.sro¡rmore globally. These consideratio¡rs lead to a deeper level, which involves thetheory of utility and cuhninates i¡r a¡r analysis that shows how cot¡erent systcm.sof preference can always be represented as havi,g cr.¡me from prob,bility andutility, with preference going by expected utility. In this chapter we wili startwith the simpler situation and end with a sketch of the lcaáing iclcas of thcdeepcr results.

VI.2. THE PROBABILITY CALCULUS IN A NUTSIIULL. Indiscussing these questions it rvill be uscful to have ¿rs coricisc a clrrr-

r67

rl¡r

ti

VI. corruncNcr

rulttriz:rtit¡¡r «¡f tlic mathc¡natical conccption of a probabiliry as is possible.I lt:l: is tlre classic one, due to Kolmogorov:

De[inition 14: A probability (on statemc¡rts) is a rule assigningcach state¡ncnt, S, a unique p¡obability, Pr(S), such that:

^' '? a. No probabiliry is luss than zero.

b. lf T is a tautology, Pr(?') : 1

c. If P; Q are rnutually exclusive, then Pr(I\Q) = Pr(I') + P(Q).

Lct us see horv tllis l¡¡icf chalactcrizatio¡r yiclds the longcl ottc of Chap-ter V. Rules I and 4 of Chapter V are explicitly contained in Definition 14

as 14b and l4c. We saw in Chapter V that tirese two rules yield üe negationrule. Since Pv-P is a tautology and P;-P are mutually exclusive,

Pr(P) r- Pr(.-P) : I

RulcS: Pr(-P) = I -l'r'(P)

Si¡ce the deniul of a cc¡ntradictio¡r is a ttutology, Iab and Iiule 5 show

that if C is a cont¡adiction:

l'(c) : r-lRule 2: Pr(C) : 6

It is ¡rot s0 ollvious that Ilulo 3 ("if two stltcmcnt§ ;rrc logicllly cqt¡iva.lent, üey have the same probability") is a conse(lr¡encc of Delinition 14,

but it is. Supposc P is logically equivalent to Q. Then P; ---Q arc nrutuallyexclusive for -Q is true when P is falsc a¡rd falsc rvhe¡t l'is true. (lfthe foregoing statement is not obvious, revicw what logical equivalenccmeans a¡rd prove it.) By üe sarne token, I\-Q is a tautology. So by l4b¿urd I4c: Pr(I') + I(*Q) - l. Using thc ttcgatiori rule:

Pr(P) +I-Pr(Q) :1Pr(P) = P(Q)

Rule3: If I'and Q arc logically c<¡uivalent, then Pr(I) = l']r(Q)

We have now only to show üat Definitio¡¡ 14 restricts the possible

probability vaiues to tlre range from 0 to l, for everyüting else in Chap-

ter V was shown to follorv from what wc have here developed. We willdemonstrate this by shorvilrg sorncthing mucl¡ Inorc gencrai and i¡rter-

esting. First, a fcw prelimi¡¡aries.

VI. 2 p¡ro¡esrl-rrlr cALCULUS rN A NUTSuELL 169

Definition 15: Q is a logical consequence of p just if e is truein every case in which P is true.

So, for example, R is a logical consequence of It & S, as is S; and Rv.S isa logical consequence of R and of S.

tt S R&S Rv.S

Case 1:

Casc 2:Case 3:Case 4:

Notice that a tautology is a logical consequcnce of cvcrydring, since a

tautology is true i¡r all cases. And evcrything is a logical consequence ofa contradiction, since a contradiction is ¡rever t¡ue. Wc will norv showthc L o gicul C o ns e q u e nc e P riruiple t

If Q is a logical conscquencc c[ P, thc¡r Pr(Q) nrust be at least as

great as Pr(P).

Thcn, of course, cvery probability nrust fall in the irrtcrval fro¡n 0 to 1.

If Q is a logical consequence of P then either P and Q are true inexactly the same cases or Q is true in the cases where P is, plus some

ertra eascs, In the former instanes P and Q aro logieally cquivalcnt andthus havc the same probability. Lct us, then, look at thc latter. Forexanrple:

(Q&-P)

TTTTTFFTFTFTFFFF

rii

Casc l:Case 2:Case 3:

Casr: 4:LASC D:

Case 6:Case 7:Case B:

TFT1'FFTF

TFT.I'

TFTT

FFFt'T¡-FT

Here Q is a logical consequence of P. It is true in the cases wl¡cre P is

[1,3,4,7] plus some cxtra ones [5,8]. At the right is üe statemcnt,

170VI. conpnrNcr

Q&-/', u,l¡iclr is truc in iusl tlrcsc extracnscs ancl falsc othcrwisc. Noticetlrat (Q&-p) aul p are mutually exclusiae. Next notice that pv(e&*p)is logically equü:alent ro e since (ea_fi

"¿¿s just the ."o",ria-1*oocascs to P. Thc¡r:

Pr(Q) = Pr[Pv(Q&-P)] = pr(p) + pr(e&-p)Since Pr(Q&*P) is at worst zcro, I)r(e) rrrust l¡c at .lcast as great as I,r(p).This result completes_ th" urg,r.n"'nt thai beñnitio. f<""a,"r*',,ir"probability calculus as .evelopá i, Ctr^pi". V. It is, however, of morctha¡¡ incidcntal importance. It

-assures tl'r.'protouitirtic r:aridity of aeduc-tirse argument. If

Pl

;^C

is a. (truth-furrctionally) dccluctivcry varicr argu,rent, thcn trrc co¡rclusion,C, is a logical conscquence.of tl".on¡un"U'* of üe premises p, & p,' ' & P"' ou¡ result shows that our cpnclusio n mwt bi at reast as' ptrob-"bk,y,the c.oniunction ol the premises.

¡ n¡nx Qt the di§a§trou§ ean§equenee§ if this were n0, tru€r we egurdhave good rcason for believing.that "il

tr," p.".ies of an argument are huc,deduce the conclusion from ih"m, and *t ¡^r" equally good reason forthercby bclieving the co¡rclusio¡¡. under ,,.r"i "]."u-.tances it wourcr beha¡d to imagine a ratjonale f*-

"pplylng Jedir"tir" logl".The probabilistic validiry "r ¿"á""íi"J"rgu-"r,, provides a justi'ca-tion for applying deductive Iogic to ,it"rtioir

-*f,.r" *" ,r"

"rii,i"J i"assl83 .hiSh probablüty to. üe conjunction of tlre premises but are notentitled to be certain of their trutú. There o."1,.ong arguments to t,eeffect üat ti¡is covers almost all applications of á"a,l"tü ;;"';;empirical knowledge.

Exercises;

I. Show:

a. JI

P is.-a logical consequmce of e ancl e is a logical consequencc o[ p,then P is logicaüy cquivalent to 8. 'This scctio¡r may be omitted without los of continuity

VI.3 locrc,r.r- coNSEeuENcE prirNcrpLEt71

b. II P is a logical consequer)cc of e arrd e is a logic.rl uo¡)surlucr¡cc of Fl,then p is a logical consequence of Il.c. if P is iogically cqurvalent to e, thrrn p arid () are ]ogic¡t collscquer)cesof thc same stf,tcmcr)ts a¡rcl i,rre r. iogl"t consc(luences tl¡e.samcstatcrncr)ts,

2. Shorv rhrt if Ii is a Iogical cor¡sc(l.crco .[ 1, r¡¡cl ,f e arrrl Il llas as alogical co.scq.cncc Pve, thc, Jl i.s iogic,rlly "1,,ir,,,t.,,,, t, cithcr 1, or t, e.r t.PuQ.

3. Study thc followirrg truth trl)le:

PQR -P&g&R P&8&*It (-P&Q&R)v(P&e&- R)Case ICasc 2:Case 3:Case 4:Uase 5:Case 6:Case 7:Case 8:

TTTF-FTFFTTTFITTt' F-

l,

F-

F-

FTFIrF

Show th¡rt for rzny eiue in crry truth tahlo you san eonstruet 0 sentcneewhich is tn:e in that case andialse in olt otÁ", "*"r,

"

Show that for any s¿f of cases in any truth table you can construct aser)tcncc truc i¡r those cases alrd false in all other cascr_

TTTT¡-FFF-

FTL'

FTFF'

F

T.t,

r-FIrF'

F-

Then:

1,.ta1

vI.3. .THE LOGICAL CONSEQUENCE pnINCIpLE ALONEIS NOT ENOUCH. T'e.Logical Con.eq,i.,"e principle is both plausibieand powerful. It is hardly open to clispute tf,riif p i, ,.r" in "í.;;;;;i¡r wlricl¡ P is, Q must have at lcast as good a cl¡a¡rcc of bcing truc as p.

We,should.be pleased üat Definition 14 of the previous section leadsto this result; wc should have been clisrnayccl if it trad not.It is.inte¡esting to ask here how. far the Logicar corrsequcncc prirrciprewill take us toward mathematical probabirit] as characterizecl by Defi-

t7,.\ VI. conpnrNcr

rrrlrorr l,l. lt leads immediately to the principle that logically equivalent

st,¡tr.r¡[.r¡ts lravc the same probability since P and Q are logically ccluiva-

l.rrt jrrst in case each is a logical consequence of the other. It leads to

tl¡r, Iirt't that there must be a maximum probability value, shared by

irll trutologies, and a minimum probability value, shared by all con-

tr.lrlictions. But even if we arbitrarily choosc I as thc maximunr a¡rd 0

ls the minimum probability, the logical consequence principle does

not thcreby lead to the additivity of probabilities of mutually exclusive

scntences, that isr

14 c. If P; Q are mutually exclusive, then Pr(.PvQ) - Pr(I') + Pr (Q)

This can be most easily seen by considering a new quantity, plnusibility,

which is defined in terms of probability as on üe graph in Figure 4'

(To find a statement's plausibility from its probability, first find its prob-

obility o' the horizonial axis, go straight up to the curve and straight

over to its plausibility on the vertical scale.) Noticc that the curve

clefining plausibility is so d¡awn that greater probabilities lead to greater

plausibiliUes and greater plausibilities arise only from greater prob-

abilities. That is:

Probability (A) ) Probability (B)

if and only ifPlaustbtllty (A) > Pluustbllity (B)

The upshot of this is that if we were to arrange §ome stateme¡lts .inorder of lncreasing plausibility, we would place them in the sam¿ otd¿r

as we would if we were arranging them in order of increasing probabiliry'

A short way of saying this 1s to say üat probability ,and plausibility

are ordirully simiLar.It is now casy to see that plausibility must satisfy thc,Logical Con'

scquence Principle. If Q is a logical consequenc¿ of P then the probability

of Q must be ai least as great as the probability of P (since we showed,

in the last scction, that probability satisfies üc logical consequencc

principle). Sincc plausibility is ordinally similar to probability, plausibility

of Q must be at least as great as plausibili§ of P.

However, plausibility need not add for mutually exclusive statements.

Assume that sonre statemcnt, P, is as likely as not; so:

VI.3 LOCICA L CONStrQUENCE PItl¡-CtPLE

Pr(P) :Pr(--P) :rPlausibility

173

oY, 1

Probability

Figure 4

Referring to Figure 4, we see that the plausibility values of P and of

-P are r. We also sec that since Pr(Pv-P) = l, tlie plausibility of

ffr:P) =1 t. so plausibility of (Pv--P) is definitcly nor equal to plausi-

bility of (P) plus plausibility of (-P).Altlrough plausibility satisfies the logical co¡rsequence principle, and

has a maximum of 1 and a minimum of zcro, it is not a probability.To understand the nced for t]¡e extra property of additivity which

distinguishes a probability we must look to quantitative applications ofprobability. In the next few sections, we shail consider the oldest andmost general quantitative application of probabilities-that of a guidefo¡ the intelligent gambler.

Exercises for the advar¡ced student:

i. The following principle has been proposed for any grading of rationaldegree of belief.

L For any statements, R, S, T, if R and T are mutually exclusive and S and

I are mutually exclusive, then the degree of belief in R is greater thanüe degree of belief in S, if and only if the degree of belief in flvT sfould

be greater tha¡r the degree of belief in SvT.

174 VI. corrrrtBNc¡

Shorv that probability satisffes principle I and that any quantity

ordinally similar to probability does also.

b, Show that any quantity representing degree of belief which:

i. Satisfies principle I,ii. Cives logically equivalent se¡¡tences the same probability, and

iii. Requires that tautologies have the maximum ¡rrobability and con-

tradictions the minimum

must satisfy the logical consequence principle.

2. Consider a language containing only two sim¡:le statements, P, Q, togerher

with every complex statement which can be built out of them using &, v, -.a. Show that every statement in thc language is logically equivalettt to one

of the following sixteet¡ statements:

P&-Pi -P&*Q; - P&Q; -P; P&-Q; -Q; (P&-Q)v(- P&Q); - Pv

-Q; P&Q; (P«Q)v( -P&-Q); Qr - PvQ; P; pv-Q; PvQ; Pv*P.(Hint: Look at the truth ta[¡le for all these statements.)

lr. Assumirtg PI(P&Q) =.4 lr Pr(P&-Q) =.29; Pr(-P&Q) = '2; Pr(-P&

-Q) =.1; calcul¡te the probability values for each of thc folegoing

statemcnts.

c. Co¡rsider the quantity reprcsetrting a degree of belief rvhich takes as

values just the foregoing probrbilities cxcept that (-Pv-Q) and

(-P"Q) srvitch values. That is, for any statement R, D(R) = Pr(R)except that D( -Pr-Q) = .71and D(-fvq¡ = .69

i. Show that D satisfies the Logical Consequence Principle'(Hint; to velify that the Logical Corrsequence Principle is not vio-

luted you treed o¡rly verify thnt no sontcncc of whlch *PvQ is a

consequence has a prol-rability gre¡tcr than .69 and no staternent

which is a consequence of -P&-Q has a probability less than '71')ii. Show that D rriololes principle I.

(Hint: -PvQ is logically equivalent to (P&Q)v.-P and -Pv.-Q is

logically equivalent to (P&- Q)v-P.)3. Exercises I and 2 show that satisfying the Logical consequence Principle

is not a sufficient condítion for being ordinally similar to probability. Show that

satisfying principle l also fails to be sufficient for ordinal similarity to probability.

(This is iaign"tt problem. The required proof can be found in the first of the

suggested readin¡;s. Conditions wl¡ich are sufficient to guarantee oldinal similarity

to probability may be found in both of the suggested readings.)

Suggested readings

Kranz, Luce, Suppcs, and Tversky, Foundttiotts of MeasuremenÚ (New

Y<.¡rk: Academic Press, l97l).Richard Cox, The Algebra of Probable lnletence (Baltimore, N'fd.: The

Johns Hopkins Press, l96l).

VI. 4 ¡¿'rs L75

VI.4. IIETS. A l¡r:t on a statrrrne.nt, P, is ¿t¡r arrangcnrent by u,hichthe bcttor collccts ¿r cortain su¡r) d, if P .is truc a.d forfcits a certair.¡sum, b, if P is falsc, TIre situation can b,_r charactcrizcd in a payoff table:

P Nct Cain

-Fa

- l¡

Thc total an)ou¡tt involvcd, a lodtls.

ú, is tht, sl¿rl.¿ urJ tlrc r,rtio I i, t1,.,

TF

A bct o¡¡ onc st¿ltcn)ent rtray also constjtutc ¿ ltct o¡i ¿r¡totlrt,r sttIc-nreut. In the ¡lir.¡st h'ivial case, if hvo statc¡nt:nts arc logically equivalentthen a bct o¡r one is cqually a bct on thc othcr. Ir: thc ¡rc-rt r¡rost triviiilcase a negative l¡ct on -P is identical to a positivc bct on P.

-P Net Cain

1'I:

-b1-u

FT

If B, and I1, arc bctting arrangcnrc¡rts, tltcir surr rs iur Jrr.irr)g(.¡r¡er)tby which the bettor fulfills his obligations undcr. I¡oth IJ, arrd 13,. F'orexanrple: c"^yl,o

B,

Nct Cairr a Net Cain

't'F

a* l¡

Tt-

C

-d

Sunr of I3, ancl I).

D o Net Crin

I-J,

TTIrF-

't'

F1'

Ir

u+ca-dc*b

*(b F d)

It should be clear that thc sunl of trvo l¡etsgiven statenrcut. A bct ol1 a state¡lcltt, S, pays

¡tecd not bc l bt:t ol) a¡ty.r certa.in valuc if S is t¡ut:

I'itt VI. coxrnrNce

,rrr,l r r¡,,1r .r r r'¡l¡rur vllrrc if S is false. There is no dillcrcntiation aboutrr rrrTr llr.rt li r.ur lrt truc or oays that S can be false. So i{ in the payollt.¡l,l, lr,¡ llrr srurr lrt't, ürcre are at least ürrce different figures under nct

¡i.rrrr, tlrl sru¡r lrct cannot be interpreted as a bet on any statenrent.l,r't t¡s ¡¡rt¡vc to a r¡rore interesting case of a sum bet being a bet on

.r st,rlrr¡rt'nt. Supposc that P and Q ale mutually exclusive, that B, is

.r lx't or¡ I'f<¡r stakes (a + b) at odds ü "rrd

B, is a bet on Q for stakesel

Ir t l) at rxlcls :¡ :

PQ B, B" Sum of B, and B,

T!-I;TI.' F

o *d-b c

-b -d

a-dc-b

-(b + d)

!oL

Sirrcc P ar' i Q are mutually exclusive, there are only three possible

tt¡r¡¡I.¡i¡ratio¡¡s of truth values. If the sum bet has a different payofi valueir¡ each of the three cases, we know that it is not a bet on any staternent.llrrt what if thc payoff values in the lirst two cases are the sanre (thatis o d = c - b)? Then the bettor wins this value in either of these

crr-scs, fhat is Wbqeoer PvQ is true and loses b + d in the üird case,

tlwt b whenPvQ islaisa, So if ¿ - d = c - b, the sum bet is e bct on

I'v() with stakes (a - d) + (b + d) = (a * b) and o¿¿, L ai. und",

rvlrat conclitions does this interesting phenomenon occur? It doesn't takerrrrrclr algebra to show that¿- d= c- b just in case a -l- b=c) d,

tlrat is just in case üe stakes of our bets on P and Q are equal. Insun¡fnary,

ll P and Q are mutually exclusioe, tlrc sum of bas on P a,ld on Qot equal stakes is a bet or¡ PvQ at the same sbkes.

, ,.( 'l'lrcrc is a¡rother kind of betting arrangenlent which is o[ general

,, ¡¡rl('r'('st u¡rcl which is not a bet on any statement. This is the sort o[,' l)('t tlu¡t rs c¡llcd ofi if ccrtain conditions are not met; cali it a conditíorwl

/rr't. ll tl¡t: lrct is on Q and the conditions to be met are specified by P,

tl¡r'r¡ it is «'lllt'd, not surprisingly, a bct on Q conditional on P and SiyS:rirl to tl¡r'following sort o[ payoff table: A,:';,''":;':.: ::

'i YNrl_

VI. 4 nr.:-r.s 177

P Q l']ayofl

TTu1' Ii' -'ITTOF-FO

A little ¡eflcctic¡¡r should convi¡rcc you tl)rt n)xny o{ tlrc bctting situa-tions that wc get ourselves into are conditional bets. sc.¡rlctiÁes wernay wish that even nro¡e of üenr were. If so, it shoulcl come as good,crvs that we can always construct a betting arrangcnrent conditL,,,rlon P by a simple hedging strategy.

Consider the su-ñiñwo bels, the frst bcing a bct on p&e and thcsecond being a bet on -P.

PQ P&Q Bet I _P Bet 2 Sunr of Bcts 1

and 2

FFTT

T'I'TTFFFTITF¡'F

c)

-LL

-d_)

-f-i

e

e

^_toJ

-@1ne*de*d

If we arrange Ilet 2 so that our rvinnings on -p, c, crlual our losscsfrom Bet l, d, üe sum of Bets l and 2 will be a bet on p conclitionalon P, as follows:

Sum of Bcts I and 2

c-l*(d+D

0

0

In summary: The sum of two bets, the 6rst on P & Q and the seconclon --P with the winnings on the second being cqual to thc losscs onthe 6rst, is a bet on Q condiüonal on P.

TTTFFTFF

VL5. FAIR BETS. llcniernbcr fronr Chapter V tl¡at tl¡c cxl)cctcd

tsalue of a betting arrangcmcnt is the sum of the quantitics obtaincd

by multiplying thc payoff in a given case by thc probability of tlrat ciisc'

For example, the bet:

178

Exercise:

If you bet sornconc a dolllr at 2 to I odds thlt P

a. What is your payoff table for P? c

b. What is your payofi table for - P? d

Bet I

Payofl

a

-l)has an

mcnt:cxpccted value of oPr(P)

Bet 2

Vl. cousnsNce

What is /rrs payofl table for P?

What is his payoli trble for - P?

- bPr( -P) and the betting arrange-

TF

Payoff

o

ltc

-cl

has an expected value of cPr(P&Q) + lPr(P&-Q) l cPr( -P&Q) -dPr(- P&- Q).--ii t¡" .*pi"t"a value of a bct is positive, it is called tr loaorable ber;

if negative, it is an unfaoorabtebel;if '"'o, it is a fair bet' Whethcr a bet

i, fuii, f"rotuUl", o, uáf"uorable ¿epends on how the probabilitit's bala¡rcc

out the odds. Consider Bet I on P. It is fair just in case:

aPr(P) -bPr(:¡¡ =gaPr(P) -D[I-Pr(P)] :aoPr(P)-b1'bPr(P)=QaPr(P) +bPr(P) =12

Pr(P)[a+b)=6, ¡r_\pr(p) =(rnrr7

TTTFFTFF

VI. 5 r,rur ¡r:'rs 179

l,TIrt tlurntity if¡ it callecl the betting quotie¡Lt for P. So we can siry

that a bct on P is fair just in case the probability of P cquals tlrc bcttingquoticnt for P.

Sr-rpposc we have fai¡ bets on P and Q:

Payoll o I'ayofI

TF

TF

a

-bc

-d

Must the sum of these two bets be fair? The sum bet

Payofi

a* c

a-dc-b

-b-d

is fair if and only if :

Pr(P&Q)(c + c) + Pr(P&-Q)(a - r/) + Pr(-P&Q)(c -l;)+Pr(-P&-8X*b-d)=g

0r: '

aPr(P&Q) + cPr(P&Q) + aPr(P&-Q) - r/Pr(P&-8)"i cPL(-P&())

- ÜPr(-P&Q) - üPr(-Pu.-Q) - r/Pr(-P&-Q) :6or:

aIPr(P&Q) + Pr(P&-Q)] - L¡[Pr( -P&Q) + Pr(-P&-Q)]-l- c[Pr(-P&Q) + Pr(P&Q)] - d[Pr(P&-Q) + Pr(-P&-Q)] : 0

or:

aPr(P) -bPr(-P) +cPr(Q) -dPr(-Q) :0But sincc our bct on P is fair, aPr(P) - DPr( -P) = 0; and sincc ourbct on Q is fair, cPr(Q) - ¿P(-a) :0. So if bets o¡r two statementsarc fair their su¡n is fair.'

rNote that thls is not the oniy way tltat tlre sum bet can bc fair, If the erpectedvaluc of or¡e bct is 'f- c and tirat of thc otller is -e , thc¡r tlrc srrm l>ct will bc

f¡ir.

TTTFFTF!-

¡ ,ti( I VI. cor¡rn-aNcr

'l'lrr argument is summarized in Figure S. Each squar.e contains the¡'.ryoll for one caso of the sum bet multiplied by the probability of that,,r\('((:.9., in the upper left-hand square Pr(P&e)(a + c)..= apr(p&e) +, l'r(I'&Q)). Thus, the expected value of the sum bet is just the sum of.v.rything in all the squares. The squares are.divided intb triangles toru¡1gt'st a way of adding. The quantities in the lorver left triangles arcrr<l«L'd downward and summed at the bottom of the columns. The quan-tilics in the upper right triangles are added to the right. The suin ofllrr: r¡uantities at the bottom of the columns is the expected value of thc

PPayof.f a

-PPayoff .b

\ +cPr(P&Q)

\+aPr(P&Q)

\ ¡

\

\. +cpr(-p&e)

\JPr(-P&Q)\ \

1 dPr(Pa-Q)

\\

+aPr(P& -a) ' I

\ aar.1-ra-e¡

\-6Pr(-P&-Q)\a

-dPr(-Q)

aPr(P) -bPr(- P)

Figure 5

QE>\aú

'§

oñt

o.

cPr(Q)

^-o

bet on P; the sum of the quantities a the right of the rot+,s is the cxpccteclvalrre of the- bet on Q. So ue lruoe shoun tlut tlrc expected. ailue o¡tltt: sum of bets on P and on e rs the xtm of the erpected oalues o,ftltose bets. Of course, then, if two bets are fair, ih"i, ,r* bet is fai¡,

. So far, we have only talked about the sum of two fair beús, rathertlrarr two fair betting arrangements. Remember, a bet on a statenrent, p,adr¡rits of only two possibilities-p is true or p is false-and specifies au,ir¡ue payoff in each case. As we saw in the last section, the sum oftl¡c two bets (on statements) is a betting arrangement, which typically isr¡r»t a bet on any statement. Thus, we have still to ask whether thc su,nof arty two fair betting arÍangenlents is a fair betting arrangenient. The

VI. 5 rann ¡rrs181

answer, happily, is yes-by the same sort of argument we uscd i¡r the

ilTp]:t case. TI¡e argumenr is i¡rdicatcd in Figure 6. Again thc lowerrett trlangles are summed downwards and when they are addccl at thebotton: we find that their sum equals.-the expectecl value of t_,cttirigarrangement l' Likewise the surn of ail the cántents of a, thc uppeiright triangles equals the expected varue of bctting arrargemcnt 2. I3utthe sum of all the triangres ii just the expected ,"ri" of ,r-.,.' r,r, u.,,rüarrangement.

CasePayof(

c !! t

c2ü z

Betring Arrangemenc I

u¡pr (c )Figure 6

So we know in general that:The expected value of a sum of betting an.angements is the surnof the expected values of the individu"ib"tting arrangements.

and in particular that:

A sunr of fair betting an.angements is fair.Perhaps you may think-that this is not very surprising and that rvciave been, perhaps, belaboring the obvious. Well and golo,l! Think th"nhow surprising it would be if this were not true-in fact how disr¡slrousit

.would be for niaking decisions under uncertainty. We coulcl under-take a series of fair risks and yet havc no assurance that thc totalar¡angement was not unfair.

'j 'j

e.¡

Éq)

qJho

aú

óo

C)ro

cl c,- -^ cl', uz u;

,,¡t t 1c ¡A c'¡ )

v¡lr(c¡&c¡) \ 1

I fi2 VI. cor¡rnrNcr

What is surprising is not that probabilities, as measures of beltef,Iead to such well-behaved concepts o[ expected value and fair bets butthat pLobabilities are the only kinds of measures of belief which willdo so.

Inragine us using some neu measure of l¡clief, call it plausibility (Pl),to take üe place of probability, Again, we will call a bet on a state-ment S

Pavoff

a

-b

fair iust ín case the betting quotient fo, S, ;j7, equals the plausibility

of S,

Norv re¡nember that the foregoing bct on S is also a bet on -S withncgative pavoff (-/r) if -S is true a¡rd a negative loss (-a) i-f ,-S is false.'

-'s Payoff

TF

-b-(-o)

Thc betting quoticnt for' -.S is

-a o

-=-

(i +l-il -;+nNow suppose S has a ccrtain plausil>ility, Pl(S). If a bet on S is fair,

the bctting quoticnt for .§ nrust equal that plausibility:

TF

Pt(.s) :-,-u'l l)

Since this bet is also a bet on -S and since it is fair, the betting quo-tic¡¡t for ,-S must equal its plausibility:

2 Rcmenrber that ¿ bct on a state¡nent, P, is an arrangcnrcnt by »'hich thebcttor collccts a certai¡r sum, o, if P is true and forfcits a ce¡tain sum, b, if P isfalsc. Thcse qrrantitics rnay lrc nr:gativc. Tl¡rrs a lrcton P is literally aiso a bct on

-P. This argunrent thus depends on fairncss bcing a property of the arrange-nrcnt rat\er than on thc way it is rlescribr:d (as l¡ct on P or bet on -P, etc.).

VI. 5 rarn arrs

Pl(-S) : o =-a+ b

Notice that plausibility is

Pl(^*s) :l-PI(S)beginning to resernble probability since:

183

'Norv let's look at the case where we have two mutually exclusive

statements, P; Q. Suppose we fincl the proper betting qunU"nt, ;f7.dand ;i-¿ to Írssure fair bets. Keeping to thesc quotic»ts, wc ca¡.r

clroosc thc stakes so that they are equal on the bets (a + b : c + d). lnsection VI.4 we saw üat undcr such circumstances the sum of thcsc

bcts is a bct on pve with betting quoticnt 6=*++=4. (tf you

dorr't unclcrst¿rnd rvlicrc this carnc from, go back'to s'ection'VI.4 and

work it out.) This is ¡ust llj = =+ + -!- Sincc rvc assrrmcdo+t) a+b a+t)thc stakes rvclc crlual, a t- b : c * d. So the betting quoticnt for

PvQ is cr¡url to --l!- , *rtthat is, ro thc sum of tlrc bctting

quoticnts for P and for Q,At this ¡roint u,e need to adcl one ¡nodcst assumption about fairncss

to nrake any headrvay:

. lf a ba s o sum of fair bets ít is fair.

Civcn tltis a.ssrrrtrlttion it follorvs that:

If P arul Q are mutually exclusiae then P\PvQ) = P(P) + P(Q).

Takcn togr:thcr with thc for.cgoirrg fact about ncgation Ii.c., thatPl (^.-.S) : I - Pl(S)l this shorvs us the Pl(I'v.-P) : I ar¡d PI(P&-P) --0. Rcmernbcr now from VL4 that if two statements are logically equiva-It:nt, a lrct on ono is cqrrally a bct on thc othcr. Thus cvcry tautology¡nust l¡avc plrrusibility l and cvcry contritdiction plausibility 0. It rvouldbe hard to inraginc anything lcss rvorthy of belief than P&-P or more

"vorthy than Pv-P. If rvc r¡akc this final assumption,

A totttoLogy lms tlrc ntaxintunt. ¡tlausibility anrl. a co¡ttrarlictiotttlrc tninintu¡tt

wc havc iusured that all üc claucs of Dcfinition 14, section VI.2, havcl;ccn nlct ancl thus that plausibility urust, in fact, bc probabiliry.

I8,t

-I'o sum uP:

proltability:(r)

(ii)(iii)

VI. cosnnr¡lcr

lf pttusibility meets the lollouíng conditio¡ts' it is

A bet on a statement, S, is fair 'r and only if the

betting quotient for S equals the plausibili§ of S'

If two bets are fair, their sum bet is fair'

A tautology has lhe maximum plausibitity value and

a contradiction the minimum.

Exercises:

l l be t on P rvith you, witlr the betting quotient for P being ;fu ona

the stakes being a + b with o ancl b both positive quantities' Describc your bet

on -P with nre.

2. N'fore preciscly, when rve bet t¿i¿ñ someone he enters into an arrange-

ment wllerc ottr rvinnillgs are his losses, and uics o¿rsa That is' the entrics on

ii, p"yoff table are the negative of the corresponding payoffs in our table'

Call his bet lhe complanent of our bet'

Consider the principle that a b¿t ís foir if ond oaly if its conp.lement k fair' Us'

ing thcresults of Exercise l, show that this principle can replace üe use of negative

winnings and losses in the argument for;

Pl(s)+Pl(-s):l

3. Wtrat kinds of bets give rise to betting quotients greÉter than I or les

thln 0? (Hint: if o ancl b are both positive,;{7-*"tt be between 0 and l'a

If a is positive and b is z"ro,rJr-- l lt b is positive and a is Tero';:l-i

= 0. What if a an<l b are both negative?)

.1. Civc ntt intt¡itive argulnent as to why the ki¡rds of bcts that give rise to

lr.,ti,,,g quuti"¡lts less th,,u O or greater tlratt I could not reasonably be de-

..rib"á * f¡ir no matter how plausible or implausible the statement in ques'

tion. Remember that a bet that is favorable or unfavorable is not fair'

5. Suppose we allowecl some statements to have greater plausibility than

a tautoloiy, atrd some less than a contradiction' but kept to the other^restric-

,i,,nr. so]'io, example: Pl(P) =2 ancl Pl(-P) = I-Pl(P) :*l suppose

also that we calculate the expected value of a bet on P in the normal way as

aPl(P)-bPl(-.P).a. Show that these plausibility values wor'¡ld give an expected value of

0 to some of the iypes of bcts discussed in Exercises 3 and 4'

b. Show that these plausibility vr'lues would violate the following prin-

ciple,

VI.6 rur DUTcH BooK r85

If Ilet I a¡rcl Bet 2 difler only in that Bet 2 has a greatel payofl in one

c¿se thatt Bet l, then Bet 2 has at least as grelt an expected v¡luc lts

Bet l.6. When rve showed that if fair bets and exfected values are to rvork reason-

ably, plrrrrsibility must be probability, orrr clc¡no¡rstratiotr was b:tsctl otr rnilrimll

assumptions lbout fair bcts. If we assume more about fair l¡ets, the argumcnt

becomes very short. Assume that the expected value of a bett;nE otangament

is the sum of the products of the plausibility of a casc and the payofi in that

case. Wc will consider only unfair bcts.

a. Corlsider:

Bct I Bet 2

Payoff Pv - P Payofl

o

o

TF

a

-bTF

Shou, thlt they are the samc bet'

b. Assurrrc tlrc follorving sure'thing princi¡>le: tl t bct pays ofr o fit1da¡nount, a,, in eaerrl possible caie, then the cxpactt:d 'salua ol tlnt bct

ntust lta a. Ulrtler this assumption, show that the plrrrsibility of

tautology must be I and the plausibility of -S must be 1- l'l(S)'

". Suppnr.-" that P ancI Q are mutually exclusive Consider the betting

alrangeñcnt:

P Q Payofl

TFoFT a

FFa

Usingtheruleforcalcu[atingexpccteclvllueancltltestlre.t}rirlg¡lrin.ciplel relate Pl(P&-Q); pi(-raQ); Pl (-P&-Q) Assuming tlrat

loglcally equivalent statements have the same plausibility' shorv that

Pl(P)+Pl(Q) =Pl(PvQ)d. Show that logically equivalent stltenrents must lt¿rve the samc plausi-

bility.

VI.6. THE DUTCH BOOK. If y'ou are so foolish and your bookic

is sc clcvcr thnt you concludc i.t scrics of l¡cts rvith hir¡l such tlrat he

wins üe sum bet na matter uhat lnppens hc is said to )rave nlacle a

Dutch Book against you.

186 VI. corrrnrNce

The following striking fact is oftcn citcd as a iustification for the

assunrptiort tlrai epistt:mic problbilitics should obcy thc rules of the

probability calcult¡s :

If you count as fair any bct oIr S if the bctting quotient for S

equals the plausibility of S, and if you are rvilling to make any

serics o[ bcts cach of rvhich you rcgard as [air, thcn il your

¡ttawibility aalucs do not <tltcg tlw rules of the probabilittt cal-

cuhs a Dutclt Book can l¡e nnde against you'

Against thc background o[ thc prcvious trvo scctions, thc reasons for

this tht:ort:nr shoultl bc fairly tr¿rrrsll¡rcnt' Lct rrs t¿rkt: tlr<: conditions for

bcing r probal;ility in ortl<'r.

l4a: No probability is Iess than z-ero.

If you havt' donc tho t'xcrciscs ;'<ltt h:tvc lrlrt:ncl1' cliscol'<:rcd the u¡l-

plcasant rcsults of takirrg ¡rlausil¡ilitics lcss than 0. Suclr plarrsibilitics

rroulcl lcad rnc to rcgat<i a [rt:t on I'as fair, which rvot¡ltl rcsult in a loss

whcthcr P is trrrc or fllst'. l-or cxartrplt:, a pl;rtrsibility of - 10 rvould

lcad nrc to rcgard tht' folkrrving l;ct as frir'

-10

sirrcc tlrc Irt'ttirrg qttoti.rri i. -fl: '- , ,,10" " d.' l, (- lI0¡ r t{l - l0 lrr gen.rrl'

any plarrsibility vnlrrc, e, [or..s rvill jtrstify as fair a l¡ct on s rvith u,ir.r-

nings rz if S is true ancl losses I¡ if S is false jtrst in c"r" f, :!-]. ¡E'"'-

ciser shorv that this is truc.) 'I'lrus, iI e is nogrtivc, it rvi)l .justify a btrt "vitlr

negativc 'rvinnings (a) lnd positivc Iosscs (/r).

14b. If Iis a tautologY, Pr(I) = I.

Anr,, plausibility grcatt'r than I rvill gt't us into rvh:rt u'e iust cliscr.rsscd

bt'carrse if e is grcntcr th¡r I, ?t, n('8.tti'c. Strpl;ost:, on thc other

Ilancl, rvc undcrestint¡t(: a tarrtology and givc it plausibility' lcss tha¡r l.Tlrcn thcre will bc sor¡c ocl<ls at rvhicll .,r,e cor¡siclcr it fair to bct against

T (i.c., bct on ?' rvith ncgativc rvittnitrgs and ncgative losscs)"1'his is a

bct wc arc sure to regrct. For cxarnplc, supposc rve assign a plausibility

of .75 to a tautology. Thi.s justifics a l>ct rvhcrt'a = $-25 and ü : $-75'N{y bookic nccd ortly tlt¡ a truth taLlt: to collcct nr1' $25

VI. 6 urr DUTCIT BooK

l4c: ln P;p are mutually exclusive, tlien Pr(I\Q) = Pr(P) + P(8).

We havc alrcady leanrcd that if P and Q arc mutually cxclusivc, thesum of bets on P and on Q of cqual stakes is a bet on PvQ "such that thebetting quotient on PvQ is the betting quoticnt on P plus the betting quo-ticnt on Q. Since I am committed to acccpting any series of bcts, cach lnenr-ber of rvhich I consider as fair, my lrookic can alw,ays conrpcl rnc to ¡ct (1.s ifPl(PvQ) = I'l(P) J- Pl(Q) by placing separatc bcts at eqLral stakc.s onP and Q. [Ic makcs a bet on i'rvhich I cousiclcl fair. This nrcans thatthe bctting quotient for P equals what I take to bc PI(P). Likewise rvithQ at cqual stakcs. The sunr o[ thcsc l¡ct:- is a l¡ct on (Ii,Q) rvl¡ich Irvor¡lcl consirler fair if ancl only if I took Pl(Pv()) to cqual Pl(f') + PI(Q).Ilut lvc arr: assurning that I takc PI(PyQ) to ha,,,c onotlrcr valr¡c rvhichcstablishes a diffcrent bctting rluoticnt for',vhat I takc to l;e ¿r firir bct.I am oflr,ring nry bookit: t\\,o scpiuirtc sr:ts of odds on PvQl Obviously,the thing for him to do is bct on PvQ at one sct r¡f ot.lcls and againstPvQ at thc othcr, choosing thc rr.:ost lucrativc oclds and catching rnc inthc rniridlc. Iior cxanrplc, sr.rppos(,tllc cf[cctivc [¿rir l¡cttinc ratio orr Pr,()rcsrrlting frour scparntt: bct.s o¡r P atrrl Q ts.6 u,lrilc tlrc bctting ratio Ijudgc rlircctll,to l¡c fair is.5. TIrr:n I rvill jrrdgc to bc flir sc¡rarltc: lrctso¡r 1'a¡r<l Q *,hosc suln rvill pay rlrc $l if I'r'Q is truc uncl cost ¡lrc: $G ifPvQ is f:rlse. I will also judge a br:t to bc fair rvl¡ich costs nrc $5 if PvQis trt¡c irncl pays nrc $5 if PvQ is falsc. If rny Lror>kir: nrrki:s all thcst, lr<rts

he rvill u,in $l from me no ntnttcr \\,hat h¡rl)p('ns, Thc u,ltolc story is iuthc follorvrng table:

187

()Ilct on Bct or¡

P '... Q

[]ct o¡l Srrrn r¡f l]cts r>n

P,Q P; Q; PvQSur¡l r¡f

P;Q

1'f¡7Ii T *3F ir -3

7

4

4

-6

I

--i*l-5

+5

If you unclcrstand tlle principles at r'"'ork, ),ou should bc able to norvshorv for yoursclf horv this can bc rlonc irr gi'rrcral.

' Tlrc inoral of thc story is inrl.tortant, "l-hc Dr¡tch Rook b<'ing rnrrrlt:against nlc rcsults front nry Itaving trr,o di[Icrcltt clJcctivc bcttirrg quo-ticnts for I']r,Q. if rvc rcgerd thc odcls tltlt a pcrson is rvillinq to sivt'onP a ¡rc:asurc of his clcgrcr: of l¡t:licf on P, rny problr:nrs stcrn from myhaving trvo incompatible dcgrccs of bclicf, Tllc most cxtrc¡rc clsc ofthis diseasc u,oulcl occrir if I glvc a pro¡rosition di,grocs of lrclicf 0 lnrl 1,

tlius irr cl[«'ct [>clir:r,ing *,itlr ccrt:Lirrtv botlr P anr] .--1','fl¡rrs, lf clcr:ri,t:s

VI.7 coNprlloNALrzATroN

tim of a semi-l)utch Book. (This conccpt js due to Shiruony', ls arc tlre followrng

thcorcms, )

a l'tove that if you assign prob¡l)ilit)' I to uny stirtemerlt othcr than a

tautology, you Iay yourself opcrr to a sr:mi-L)utch Book.

lr, SIrorv th¡t iI you atlllerc to thc rtrlcs of thc llrol;lbilitl' calculrrs and

assign probability I only to tautologtcs, you arc not open Io a semi-

Dutch Book.

Suggested rcadings

Abrrcr Slri¡nony, "scicrrtific IIt[crcttcc," i¡r 7'l¡c Na¿¡rr¿: aul l;urrcti<ttr olSciutlific 7'hc<¡rics, ed. llobert Cokrclrrl' (Pittslrrrrgh; Urrivelsity oI Pitts-

burgh l'rcss, 1f170), pp.79-172.Richarcl Jcflrcy, TIc Logíc ot' Dccistott (Nov Yolk: McCra*'-l{ill Book

Conrprrnv, 1965).

VI.7. CONDITIONALIZATION. In thc prcceding scctiolrs we

hevc discusscd rvhy epistcnric probtbilitics sl¡orrl<1, in fact, bc 1;rob-

abilitics. Thc qtrcstion of iulttctiac probrlbilitics has be0tr lclt opt:n. lnthis scction, lct us appr.olch tlrc r$rcstion of intluctivc probllbilitics fronlr¿rthc'r a tlii[t'¡ t'nt arlgl(' than that of Clraptcr 1; that is, virt , tlrc rrrlc

of conditionalizatioll.Li't us assurttc, for thc nlolllcllt, that rvc arc opcratillS witl)in n cel'-

trinty nroclcl. We get to know rliore alrd nrorc things rvith ccrtainty

rrltd tllcsc ltorrrs pllo tlp, so to spcak, in rrtl cvcr'growlrrg stock of krlorvl-

cclgc. Sup¡rosc u'c liavc I ccrtaitr "initial" sct of cpistcrntic probtbilitit:s;l'r,l antl our s0lrscs toss n Itcrv itent of krlorvlcdgt:, P, i¡lto our stock o[

knorvlcdgt'. llr¡§.'arc we to nrovc to a "final" sct of cpistc¡nic probabilitics,

Pr1, u'hich acconr.noclirtc our ¡1ew itcr¡r of knorvlcclge in a rational

fashion'/ Thc ruic of conclitionalizrrtion givcs tlris lrrs\\'cr;

Ilulc C: For any statclttcnt Q, takt: its ncw prol)ill)ility to l¡t: its

old proLrlbility con(litior)al on thc tlcrv itt-'trl of knorvlcdgt', i t:,Pr¡(Q) - Pr,(Q givcrn i'),

NotC that Ilrrlt, C gir.cs P thc rrcrv valuc of 1, a st:rtlrs con)lllcllstrratcrvitll its ttt'u'-fortncl ccrtlirlt¡r. Rtrlc C, llorvcvcr, catr also cflcct e clrangt:

in thc cpistenric probabilit¡,of ncitrly cvcry othcr st¿rtcn)crrt. Wlrrt justi-

fic¡tioD is tircrc for ntaking thcsc chaDgt,s rccording to this rlrlc? l¡or

thc unsrt't'r \\,('n)Llst $o Lrlck to i¡cts ol)cc lllorc,

Ilcllrcurbcr. that tirc sunl o[ bcts or.r P&Q llncl orl -.-P is, if tlrc stakcs

Vl. coltrnrNcn I tjf)I,B¡.t

ol bclicf arc hcld to bc ticcl tr-r bctting <luoticnts and bctting l¡chavior'

in tlrc nrantrcr inclicatccl, the adclitivity rerluircmcnt for probability is

a kind of conststenctJ requirement for dcgrecs of belief ' The fact that we

do use our epistcmic probabilitics as rveiglits for determining what risks

to takc iIr u¡rcertain situatiotrs rnirktrs this thc strongcst argumcrlt to the

efiect that epistenric probabilities ore probabilitics'

The a.gument lacks onc step o[ being compiete We have shown that

if you violate the rules of the probability calculus 1'ou lay yourself open

to a Dutch Book. But rve have not shou'n that compliance with those

rules protects yorr agninst a Dutch Book' Docs it? Stop norv, if you do

not know, and think about thc arlswer'

The answer is, of cotrrse, implicit in section VI'5' When someot)e

rnakes a Dutch Bo<.¡k against yo., h" enticcs 1'ou into a sum l¡et such

thet it is a sum of incli.,iciual bcts *,hich yorr consirler to be fair, but

which itsclf gurraltt(..,,i r',trr a loss i¡t Cvcrl, casc. Norv if you arc tltralirlg

in gcnuinc proLrabilrtics (rathcr tlr:Úr sornc rvackl,plausil)ilitics with

values lcss than 0 or greetcr than l) 1,ou rl,ill considcr thc surrr bct tobe unfair. The expcctcd valuc of thc sum bct is thc surn of thc proclucts

of thc payoff, "nt th" corrcsponding probabilitics' Some of the prob-

abilitie; will be positive; none will be negativc. All the payoffs rvill bc

negativc; sr-¡ will tl.ic t:xllcctccl valuc'

ñorr *" provccl in scction Vl.5 that if rvc arc usir.rg gcnuine probablli-

ties to defiirc fairncss, if luo ltets arc ltir, tlrcír sum ltcl is fair, and iltvabettings.rÍao.gamants lra fair the¡ tr,,n is fair It follq\a:s that if you

are rrsing !"n,rin.' probtrbiliti<;s, no sc(luon('c of bcts you consiclcr fair

can constitr¡tc a Dutclt Llook against you.

Excrciscs:

1. Suppose someone assigns a plausibility to P v Q diflerent from the -sunr

of the plausibilities hc assigns to P rnrl to Q. civc cxplicit instructit¡trs for nrrking

a Dutch Book against hinr.

2. Show that if I,c¡u are usiDg gentriltc ¡rrObabilitics, t)o sequer)cc of bcts ¡,ou

consider flir or favorablc can cotistittttc a I)tltch Book agrinst vou

3. lf you conclude u series of bcts suc)¡ tllirt tlrcrc is rtrr l.rossil;lc cit<tll¡stlllce

rrncler rvhich you cilrt rvi¡t tln tllt'srtlll bct an<l tli<:rc is sr¡ntc ll0ssilllc circttllr-

stance ulttler rvhich vou calr lose o¡ thc sulll bct, rvc rvill say t[at yotr are tlrc vic-

3 That a serics of bets nray bc ¡tlltde if tlrc btts irxlividrral)y arc fair

VI. coxrnr:Ncs190

arc right, a bct orr Q corrclitional on P rvitlr a payofl tablo as rcprcscntcd

bclow:

f Q I'avoll

a

- l)

0

0

Thi5 bctting arrangement is Iail iust in casc its cxpccted valuc is zero'

that is:

aPr(P&Q) -DPr(P&-Q):0or:

PrlP&O) , l.¡

Pr(P& - Q) a

Note that sevcral scts oI probability valucs rvill rcn<ler such a l¡ct [air'

For cxamplc, if a : $l ancl i¡ : $2, ticn Pr(P&Q) = { antl Pr(P&-Q) .=

-'-- rcnrlcrs tlic bct t^ir' "s cioes Pr(P&Q) - f antl Pr(P&--_Q) :t. T\"

il';'"r "^r,*t t"l"' r'1r'¡ = l'nncl thc sácond sct ¡nakes Pr(P) = r'

I¡r [act, any value of P is co'npatiblc with our conditional bct so lon¡;

as it if <livided up into P(l'&g) and P(f&-Q) in the same ri¡tio as

btoa.In othcr worcls, thc bct is fair jtrst in case:s

hPr(P&Q) = -:-rPr(P)

n- l'r(P)Pr(P&-(l) =o+ b

t{0,1?) : Pr(Q givcn P) : i+,Pr(P)

Pr(|t=o) = pr(-Q given P) = r+nPr(P)

-- t ,t rrot 7,cro

sOrPr(P):0o¡¿:b:o'

TTTFFTFF

r9t

If wc call L , ,h" bating qtrctient on Q corulitío¡nl on P' rvhicha* lt

scems reasonable, we can now say that a conditional bet is fair *'hen

the conditional betting quotients equal the corresponding conditional

probabilities.' ñ;; the interesting thing to notice about all this is the connection

between a conditional be(l remainíng fair undcr a belicf change and

,h^, .hnng. taking place by con<iitionalization' Suppose a bet on Q

conditional on P with condibional bctting quotient *, * fair on a

set of initial probabilities Pr¡. Thcn +ffiP - Pr'(Q givcn P) =

l' .. Su',,rnse ¡low that a clrangc to a ncrv sct of plobabilitics is

a-ll¡*ra,,-try conclitionalizing on P. Then Prr(P&Q) = Pr'(P&Q givr:n P) =

Pr,(Q givcn P) and Pr1(P) = Pr,(P give' P) = I rtt"' }ffiP =

Pr'(Q givtn P) = -l+ so, íl Itelk:f.s art: r:lttr,rgci l'r¡ condilionoliza'

I orlttio¡t o¡r L', lair ltets conditíonal on P remain lair'

Ve,ry rric'c. Rrrt wliat's nict:r is tlrat coutlitionaliz'atiorl is thc only

rnctlrorl for t:hanging bclicf.s undcr tllesc circunlstanct:s" u'hich has

this propcrty. Supposc a bct on Q conditional on P is fair l¡cforc and

after a bclief change fr.o¡i Pr¡ to I',r'r. rr',.,' ffi = #-=I'rr(P&Q) If this bclicf is a result of P bccorning certoitt, tlr.n Pr,(P): l.

Prr(l)Furthermorc, Pr¡(P&Q) must equal Pr1(Q) for Prl(Q) = Prl(P&Q) -l-

Prr(.*P&Q) and Prr(.*P&Q) musi bc 0, sinct: Pr'(P) '= I So Pr1(Q) =-

Pr-.(P§9) ancl the change has takcn placc by conditto¡r¡liz.atiorr' The

Pr,(P)onht tttetltod ol clvnging belíels suclr tlnt P becomes ccrlain aruI l¡ets

'"Jn',itrr,á-,*1 n,,'p,. l,;71, "arc

lui'r rc¡nai¡t /air rs c,rrrlili,ri¿rli¡¡¡f it'rr¡ orr l)''

If this is true, a Dutch Book argurncnt canrtot be fal away Wt: havc

shorvn tl.¡at if sonrconc docs noÚ Jhnng" his bclicfs accorclirrg to Rt¡lc,C'

thc corrditional bctting quoticnt 'uhich hc regards as assuring a fair

ü"t nn p conclitional ,ln i' *ill clwnge upon thc acqtrisition o[ P as irr¡

VI. 7 covolnoN^LlzATlol\j

6 CcrtaintY modcl.i Erccpting s¿5s5 *hcre Pr(P) :0 or o: b:0

VI. co¡rrrui¡.¡crr92

itt.ril of knorvlcclgc. If thc L¡ookic knorvs /¿¿.rr¿r thc bcttoI will changc

his bctting quotiÁts he is clcarly in a position to guarantec a profit if. P

occurs. Bf ,nakiug con¿itio¡ral [>t'ts bt'fore a.cl after thnt occurrc'ncc' ht:

is es.scntially betting on Q at trvo diflcrcnt scts of odds' We havc alrcrcly

secn how a bookic carl assurc hirnsclf a plofit irl such a situation lfPr,(Q given P) is less than PL1(Q givcn I'), hc will bct initially on Q

-nd'itlá.,ul on P and ffnally againsr Q co,ditional on P if Pr,(Q given P)

is grcatcr than Pr1(Q gir",, l'i, hc will bct i.itially agtti',st Q conditional

on P and finally on Q conclitional on P. choosirrg thc stakes corrcctly,

he guarantees himscll a profit i[ P occurs, Iiurthcrnrorc, hc brcaks evcn

if p does not occur., since all bets are co¡rditional ori..P. only one mole

srnall stcp is requircd to nchicvc a propcr Dutch Book. The bookic con-

.sidcrs thc "-nl,nt,

a, that he has guarantecr'l hc r'vill rvih if P occurs, and

makes a sicle bct of ]a on .- P, guararrtccing hinrself liet winl)ings,

come wlrat rttay.

Thc virtues o[ conclitionalization lraving been firmly cstab]ishcd, norv

let us look a littlc rnorc closcly at thc rvorkings of the certainty n-rodcl

rvith the rtrlc of conclitioniilizatiolr. As rvc travel through life, wiÜl

Oul eycs opcn, wc contcr to k¡row, lltorc ltlltl llrorc t[rirlgs, Tlris grOrvt]t

of krrorvlcdgc is represcnted by'the adding of statemcnts (O,, O,, O")

to our stock of knorvlt.clgc. llpon tllc atlclition oI a tlcrv itcl¡l r¡[ knorvl-

cclgc, O,, to otrr stock of klt«,)r.r'lcclgt:, u'c rcvisc our bt'licf 'structtrrc iry

pntri,rg lrorn olci cPistcrnic probabilitics, Pr,, to trcrv cpistt'rrric prob-

iUilitiÁ prrr. il by conditionrrlizrrtion 0n O,,," S0 for alrv statcmct)t Q,

Pr,^ . ,,(Q): Pr,(Q givcn O,), and

Pr,, - ,,(Q) : Pr,,, ,(p given O" . ,)

Ixtt

Pr,, ,(Q givcr.r O,. ,) :Pr". ,(Q&O, , ') --?.*(o;;f -

Pr,,(Q&O". ' givcn O,) -Pr,(O". r givcn O,,)

Pr"(Q&O",,&O,)/Pr,,(O,,) __

Pr,,(O.,,&O,)/I']r,,(O,,)

8We are assuming that I'r, (O,,) > 0, so tlllt thc conditior;rl probabilitics arc

well dclined.

VI. 7 co¡¡ornoN^LrzATroN

Pr,(Q&O..,&O.) _Pr"(O" - ,&O,)

Pr,,((.t givcn O,&O,,. ,).

So two stcps can bc comprcssed into onc. First co¡lditionalizing on O,,

and nioving from thc rcsulting distril¡ution by conditionalizing on O,,,is equivalurrt to rnoving frorn tllc original tlistributio¡r by conditional-izing on thc conjunction O,&O",,. It follows thlt u'e can conlprcss

any finite numbel of steps into onc.

Tlrc sct of episteníc probabilities, P4",,XQ) orrit¡ccl at by succcs-

sítse conditionalízations on itcms in a ,stock of knowledge (O,, O" . . . O")i.s identícaL to tlte set ot' proltalillities u¡lúcl¿ u;ould bc artiocd at lrycorulitionalization on tlrc cofiunction ol ulL tltose itc¡ns ol knortlcrlgc,Pr,(Q giaen O,&O,&. . .&O").

Pr,(Q givcn O,&O.& . . . &O,) is a nrcast¡rc of thc firmness with rvhich

O,&O,& . . . &O, supports Q. Sincc Pr, is not thc lcsult of a conclition-

alization, it clocs not dcpcnd upon thc contcnts oI otrr stock of krrorvl-

cclgc, TIris sr.rggcsts that u,c might idcntif)' it as tlrc indttctiac prob.rllilityof thc argurnent:

o,o,

o,,

o'Ihis iclcntification is vouchsafcd by Dcfitrition 5 of Chaptcl l:

In tlrt ccrtainty rnodel thc cpistcrrric probal.,ility of ¿r stltcrt.¡c¡tt

is tlrc inductivc probability oI tlrat argunrcnt rvhicl¡ i)r(s tl):tt statc-

ment as its co¡rclusion ancl whosc prcmiscs co¡lsist of all thcoL¡scrvation rcports rvlrich comprisc our stock of knowlcdgc.

ancl it arrsn'crs tlrc <lucstion w'ith rvhich tvc bcgan thir scr:tiotr. "l¡ulttctiac

proltaltilitics" tnrtst, ín Iact, bc co¡ulitio¡wl. proltabilitics.In Chaptcr I, rvc startcd rvith i¡lciuctivc probat)ilitics ancl, in thc

ccrtainty n-ro<lel, clclincd cpi.stcrnic probabilitics in tcrnts o[ tlrcril. Inthrs section rve startcd with e pistcnric probabilitics atrcl, rvithin thc

assuntlttioris ol tllt: ccrtainty rttoclcl, rt'c,ovt'rt:cl itttlttctivt: probabilitics.

r93

VI.8 rn¡-¡.mu-trY

Payoff

195194

-l'lrc approaclr of Chaptcr I is that of Carnap; that of this section is

,..sociati,cl with thc Baycsian school, T¡at thr:y coincidc to suc¡ an

cxtc¡rt is a plcasarlt atld irlforrnlttivc fact.

Exercisc:

Shorv that if I movc fronl an initr¡l sct of ¡;robabi)ities, first by conditional-

ii.ing o. p, th<:rr orr (), tlrcrr ,. Il, to a fi¡r¡l sct ,f pro¡abilitics, thcn for any state-

rnents S rnrl T:

' Prr(P&Q&n&S) Pr¡(P&Q&R&S)

F P&oaF&TI(¡1,'ovidr:tl tlrc i¡ritirl prol.rlbilititrs arc positivc)'

V1.8. FALLIBÍLITY.'A man would be rash indeed if his acts of

observation all rcsultcd in certainty in an associated observafion state'

rncnt. In [act, therc are reasons to believc that it is never rational to be

ccrtain (in the sense of assigning cpistcrnic probabilitv of 1) o[ any

observational stat'e ment. Thc first rcason is Shirnony's argurnent that

assigning probability I or 0 to any statement not a logical truth or a

.o,rlr"aiition ..rp..tir.ly lays rrs open to a quasi-Dutch Book' Thc

othcr rcasons lrarc ,,,nerged fronr nluch thrcshirrg about by cpistemolo'

gists in this ccntury. Thc thrcshing is pcrhaps not yct ovcr, and no bricf

,r-*r.y of its results is likely to be regarded as fair by all sides' Ncver-

theles.s, what I ttke to bc thc herrt o[ tltc rnntter is thisr rto lllatt0r what

language wc use to describe our ol¡servations, the act of obsérvation

"uJ th" act of believing a scntcncc attribrrting a certain character to

that obscrvabion are diitinct.o Doing onc does not entnil doing the

othcr. The link betrvecn them is cau-sal, rrot logical' If I am of sound

nrind anci bocly, adopt ¿r modcst obscrvirtion language, and am proficient

in its usc, this causal process may bc highly reliable as a means for

gcnerating truc bclie[s. But thcrc is no rcason rvhatsoevcr to believe that

it is,infallible.In such circurnstances it is lrard to sce horv it rvor¡ld bc rcasonat;le

to be certain. Remember that certain§ for us means an cpistcmic prob-

ability equal to 1. Ancl if Pr(P) : 1, thc l¡ct

VL cot¡rneNcs

.This scction <lcals with an adva¡¡ced topic and nray lrc omittcd withortt loss oI continrriiy.

e N.B. "Distinct" trrcans "not identical." It docs not mcan "disjoint"; it docs not

mcan "ttnrclatctl,"

Tr

0

-b

is fair no matter how great b is. It is common folk knowlcdgc that someonc

who says he is ccrtain and who cvcn f ecl,s certain, may shrink f rom putting

his money where his mouth is. ccrtainty of the sort in rvhich $'e arc in-

terested involves the willingness to risk eoen¡tlúng if you are rvrong

over against no gain if you are right'If all this has not convinced you that certainty is never warranted

for contingcnt sratcmcnts, I hopc it has at lcast convincccl yotr thltthcre arc some times when we wish to changc our bclicfs undcr thc

pressurc of ncw evide¡rcc rvhcre thc certaillty modcl is ina¡rpropriatc'iVc ¡,eecl, thcn, a rvay of clranging our cpistenlic probabilitics rvhen an

obscrvrttiotr r.aiscs our dcgrcc oI br:licf in a statcnlcl)t, \vithout rrrisirrg

it all thr¡ rvay to I.Strppost: tilat an obstlrvation causcs Lls to chirngc otrr tlcgrcc oI bclit'I irl

p from I,r,(p) to I,r1(P). lVc nright hope that orrr rulc for changing

belicfs in srrch a situation r.vould bc such tliat llcts conclitionrl on P arld

I¡crts con<litional on -P rvlrich arc fair l>cfor<l thr: clrarlgc rc¡rlnirl fltir.

Wc sllr.v in section VI.7 that bets conditioual orr I' rctnait.r [aiI irrst irr

casc thc ratio oI Irr(P&Q) to Pr(P&-Q) rc¡naitrs constant. And this ratio

r0r)taiDs c0nst0nt just in ensc tho conrlitional probrhilitÍcs Pr(Q givcn P)

and Pr(-Q given P) remaiu constant. By tlrc samc tokcn, fair bcts

co¡rditional on -P remairi,fair ir.rst in casc thc conclitional probabiliticspr(Q givcn --P) and Ir.(-Q given .-P) rer¡air.r constant. Thtrs if fair'

bcts conditio¡ral on P and on -P are to renlain fair:

Pr,(P&Q) : Prr(P)Prr(Q given P)

Pr,(P&-Q) : Pr/(P)Pr¡(^-Q given P)

Pr¡(-P&Q) = Prr(-P)Pri(Q given -P)Pr¡(-I'&--Q) : Irrr(-P)Pri (-Q giaen-P)

rncl

I']r1(Q) '= Pr7(P&Q) -l- I',r/(--P&Q)

Putting thcsc togethcr rvc have:

l{Xj VI. conrn-s¡¡cr

Jeffrey's Rule: If our nel\' information is representecl as a changein dcgrce of belicf in P fronr Pr,(P) to Pr¡(P), then for any state-

nrcnt Q, takc:

Prr(Q) = Prl(P)I']r,(() givcn P) + Pr1(-P)Pr,(Q givcn -P)Noticc that Jeffrey's ¡-ule is e gcneralization of Rule C. ln tlre special

case rvhcre Pr7(P) = 1, Jcflrcy's rulc rcclt¡ccs to Rulr: C. Noticc also thatjcffrcy's rule can be vicrvcd as a wcightcd avcrage of Rulc C to both P

ancl to -P. Conditionalizing on P, Pr,(Q) wor¡ld be Pr;(Q given P).

Conditio¡ralizing on -P, Pr,(Q) rvould be Pr,(Q given -P). Averagingthcse rcsults, wcighting the average by Pr¡(P) and Pr¡(,.-P), gives us

Jeffrcy's rulc.IVe have, tlrcn, a viablc fallibility ruodcl for changing frorrt otrc st:t

of epistcmic probabilitics to arrotlrcr. But nou' it is not so casl' as itr.r,as in the certainty moclcl to represcnt an epistemic'probability as theresult of an inductivc probability opcmting on a stock of kno*,lcclgr:.What observation givcs us now is not 'a st't of certain scntertces

0,,0r, .. ., but rathe¡ a set of observational probabilities, Prr(0,); Prr(0r);. . . . The obscrvational probabilitics ¿re to bc thc outcornc solely of thcobscrvation, rrot of incluctivc rcasottitrg, for tlrc point is to scpirr¿tc outthc factors of ol¡scrvation and induction.

In the ccrtaintv nroclcl rvc shorvcd that conditiorraliziug first on 0,,

thcn on 02, etc,, gavc thc san.¡e ¡csult as co¡ldiüo¡lalízing on their co¡r-

junction 0r & 0,. Hcnco the possibility o[ "factoring" our cpisternic prob-:rbility into a stock of knorvlcdgt: iurd ¿r sct of rrrtluctivi: prol¡abilitics.In gr:ncral thcrc is no long corrjunction ancl associatccl probability torvhicll rve can apply Jcllrcy's rulc and gct tlrc s:trnc st't of c¡risternicprobabilitics as wc wotrlcl Iravc gottc'n frorl succcssivc applicatiorrs ofthat rrrle.

Suppose we attempt to define our cpistcmic probability as üe resultof applying Jeffrey's rule successivcly to cach itcm in our stock of knowl-edgc, taking inductive probabilitics as the conclitional probabilities inthe ffrst stcp, üe resulting epistemic conditional probabil:ties as con-clitional probabilities for thc ncxt stcp, and so on. This rvill not clo, forseverai reasons. The first is that the final rcsult diffcrs dcpending onthe order in u,hich thc itenrs in or:r stock of knou,ledgc are taken inthis process, This will not rvork, since the sane data coupled with thesanrc inductive probabilities should generate the same episternic prob-alrilitv. Thc second rcason is thnt at cxch stagr: in this proccss thtr

VL 8 r'au-mu-trr

observational probability is taken ", th" finrl probability. In the cer-

tainty model if P is observed, it bccomes certain. Well and good. Itsfinal probability becomes 1. But if we are sophisticated enough torcalize tlrat observations may f all sl¡ort of ccrtainty, we should be

sophisticated enough to realize that observational probability need notbe the only factor inSuencing final probability. Final probability is

rather the result of the interaction of observational probability withtheories which rve may hold on the basis o[ previous observations.

l-ct me illustrate. Suppose I see a bird at twilight which I clearlyidentify as a raven. Because the light is not so good, the probability Ican assign to him being black on the basis of that ol¡servation is only.8.Suppose further that I hold the theory that all ravens are black and

that this theory is buttressed by massive numbers of previous observa-tions. In such a situation the final probability I assign to the statementthat thc ravcn is black rvill be highcr than the observational probability,and quitc propcrly so. Othcrrvisc I could disconfirrn lots of theorics just

by running around at night.All right, my, tlreory (rvlrich is really tlrc couclr¡it of thc force of pre-

vious observations) pulls up the observational probability in this case.

It is ir¡st as eesy to think of cases where it pulls it doutn, say where I think I sec

a water buflalo on the Sa¡l IJenlardi¡ro Irrceway at 3 ¡,.r'r.

Can wc have an analysis o[ thc interaction of thcory ancl obscrvatiort along

thcsc lines? Is there a valid Dutch llook argumcnt for Jcflrcy's rulp?'I'hcse are

crintrovcrsial questions under currcnt investigation, If you find them ofr:orn¡rclling intcrcst, you rnay want to follow trp thc srrggcstcd rcadings.

Exercises:

l. Start with initial probabilities Pr(P&Q) = 173 Pr(P&-Q) : .001;

Pr(-P*Q) = l,/3; P¡(-Pa-Q): l/3. Applv Jeffrev's rule taking Prl(P) = 99'

Calculate Pr¡ (P&Q); Pr1 (Pt-Q); Prl (-P&Q); Prl (-P&-Q)' Now takingthisset of probabilities as initial probabilities, apply Jefirey's rule taking Ptr(Q) :.99. Calculatc the final probabilities of all tl¡e same statements.

Now repeat the process in opposite order; that is, first apply Jeffrey's rule on

Q at PrlQ) = .99, then on P at Pr(P) = .99. Compare this set of ñnal proba-

biliües with the previous one.

2. Suppose we move from Pr, to Pr1 by applying Jellrey's rule to P, takingPrr(P) to have some value between 0 and l. Suppose also that Prr(S) = I onlyif S is a tautology and Pr1(S) = 0 only if S is a contr¡diction.

a. Shorv thatPr¡(S):1 ifS isa tautology and Prl(S):0onlv ifS is a

contradiction.

197

I {t9VL 9 urrlrrYlg8 VI. cotlrnrNcs

b. Show that for any contirrgcnt slatcments S and 7',

Pr,(P&S) Pr/(P&S)

li?7&nand

Prr(^-P&S) _ Prr(-P&.S)Pr¡(--P&I) Pr¡(-P&I)

c. If Pr,(I') - r¡ and Pr7(i') = Ü, slrow that first applyirrg Jcflrcy's rrrlc to

Pr¡ witlr Pr1(P) = D ar¡tl tlro¡r applying it to that set of probatrilitics

with Pri(P) " ¿, g(rts y,rtt lrrtck to llrt: irritr¡l sct ol prol;iririlitics'

Suggested readings

Brad Amcndt, "ls'l'hcrc a Dtrtch Ilook Ar¡Srmcnt for Probability Kincma-

tics?" /'Iiioso¡rh t¡ o.[ Scfu:ncr 47 (l9l]0): 5&3-588.

Ilrr«lolI Oarnal;, "l¡rrlrrr:tivc l-ogic anrl Ilational f)ccisions," in Slr¡¿ics ín

I¡uluctíae I'ogic rtn,l t'roltabilitr¡ I, c<1. Carnap and Jeflrcy (Los Angeles:

University of Califomi¿ Prcss, I971), pp.5-31

IIartry Field,"^ Notc on Jcflrey Conrlitionalization," l'hilosophy of Scicnci:45

(1978): pp.361-367.

Ilicharrl lclfrey, 't'ltc I'ogir: o.f I)«;isio¡t (2ntl ed.) (Chicago and London:

Univcrsity of Clricago I'ross, l1)&j), cha¡r. I l.Brian Skyrms, "ltighcr Ortlcr Dcgrccs of Bclicf," in lrospccls lor Prognrotísn:

Iissays irr llttnor of I| I'. Ilarrrst'r7, cd. I). II. N'f cllor (Carnbri<lgc; Canrbridgc

University Press, 1980).

Psul Tellcr, "Conrlitiorlaliziition itn(l Ollscrvttion," 5r¡nl/rrr'ra ?0 (1073):

218-258.

VI.9. UTILITY. We havc been operating so far within a set of as-

sr¡mptions üat often a1.»proxinrate the tnrlh for monetary garnbles at smitll

stakes. It ís time to take a ¡norc global viewpoint and question these a*ssurnp-

tions.An extra hundred dollars mcans lcss to a ¡nillionairc than to an ordinary

person. But if I win a nlillion, I'rn a rilillio¡r¡ire, so the differencc bctrvcer¡

winning a million + l0O <loll¿rs anrl rvinnirrg ¿r million nrc¿rns lcss to mc tli¿n

thc dilfcrcncc bctwccrr wirrning l(D dollars and winning notlríng. In tlrcterminology of cconol¡rics, moncy has decrcasing rathcr than constant rnar-

ginlrl rrtility for me.

The idea of rrtility was irttrr¡clttced into the litcrature on gambling in this

conncction by Darricl Rcrnoulli iD 1738. IJenroulli w:rs r:onr:erncd with the st.

Pcterslrtrrg garnc. In this garnc, yorr flip a [air coin rrntil it con¡cs rrp Ilca<l.s. If it

conres up hcacls on the first toss, yorr get $2; if on thc second toss, $41 if on the

üircltoss,$B;ifonthenthtoss,$2"Thcexpectedrlollarvalrrcofthisgamcisi.,ñnit". (Eterci.sc, check thisl) How ¡nuch wáulcl you Pay to get into this gamc?

Á"*nutii,, idea wa^s üat if the marginal utility of money dccrcascd in Üre right

*uy " ,trc St. Petersburg gu*" "otiltl

have a reasonablc finitc exPectcd u'i¿i'y

evón though üe monetary expectation is infinite'

When we consider decisions whose payoffs are in real goo¿s rathcr than

moncy'thcreisanothercomPlicationwemusttakeintoaccollnt.Thatis,thevalue of having two goods iogether may not be

-equal to Üle stt¡n of their

inclividr¡al values because of interactions betwecn thc goods. If a man wants to

start a pig farm, anrl getting a sow has valuc b for him and getting¿ l>oar has

,rt,r" ., tlien getting Ütth alow ancl a boar may have valuc grcatcr th¿r¡r I¡ * c'

Tlrc sorv an,i ilc boirr are, for him, crnn¡tlemaúarry goocls lnteraction bctween

goocls can also be negative, as in the case of thc prospective chicken farmer

who wins two roosters in tu'o lotteries' The Presence of an active market

recluccs, lrut <.locs not eliminate thc eflect of complementaritics The sccond

rooster ;s still of more value to a prospective chicken fa,ner in Kans¿us than to

Robinson Crusoe. The farrner "un,

toi"*utnple, swap it for a hcn; or at least sell

it an<l prrt the moncv tow'ard a hcn Rec¿ruse of complerncntaritics'^wc.cannot

i" o".irof &ss.r.c i¡at if a Scttor makcs a scries o[ Scts cach of u'hic¡ he

considcrs to bc fair, he rvill juclge thc rt:sult of making tlrc¡n all togctltcr as ftrir'

\\¡hcrc payoffs intcract, tti".i.-gtrt hancl may nccd t<¡ know rvhat thc lcft is

doing.Tlrc prccecling points about how utility works are intuitively casy to grasP'

But it ii hardcr to say just what utility i§' We know how to count money' pigs'

ttnd chickens; ltut how do wc mcmurc utility? Von Nctlmann tnd Morgcnstem

showe<lhowtousethecxpectedutilityprinciplctomeasurcutilityifwehavesomecl¡a¡rce<levice(suchasawheelofforttrne,afaircoin,a.lottery)forwhichwe know the chanccs. We pick the l¡est payoff in our dccision problem and

give it (by convcntion) utiliiy I; likewjse, we. givc the u'orst payofl utility 0'

?h"n *" ,r,"orure üe r¡tility ;f a Payofi, P, in bctween by jrrdging what sort of

a gamble with the worst and thgbest payoffs as possible outcomes has valuc

"q,rol to P. For instance, far¡ner Joncs wants a horse' a pig' a chicken' and a

ht,sban<].[Ierctlrrentdecisionsituationisstructrrredsothatshewillgetcxactly one of thcsc. She ranks the payoffs:

I. Horse2. Ilu-sbanrl

3 Pig,1. Chicken

ro Utility = log NloncY.

201:l(x) VI. colrnnrNcr

slrt: is incliflcrcnt bctwcett (l) a lottcri' that gives I clran"e of a horsc andIr:llancc of a chickcn, ancl onc that gives ¿ husl¡and for srrrc; and (2) a lottery that

givcs I chance of a horse and I chance of a chickcn, and one that gives a pig for

srrrc. Thus, hcr rrtility scale looks like this:

UtilitrJI Iorse Illtrsband .u

l'ig .5

Chickcn 0

If hcr <lccision sitrral.ion wcrc structrrrcd so tliat shc rDight cnd up w'ith all thcse

goocls, ahd if they didn't irrtcrfr:rc with onc atrothcr, then her utility scalc Inighth¿rve a differcnt to¡r: Florsc and llusband and Pig and Chicken. lf it were

stnrctured so Ürat shc might end trp gctting nonc of these goocls, aftcr going to

somc expcnse, there nright be a different bottonr, which would have utility 0'

Utility, i:-s mea-sured by the von Nctrrnantr-Nlorgcnstern r¡ethod, is xrbiec-

tioc utility, detcrmined by the dccision ¡naker's o$'n prefercnces. Therc are, no

doubt, various scttses in which a decisio¡r maker can Lrc wrong about what is

good for him. However, such questions ¿rc not add¡essed by this theory'

From a decision maker's utilities wc can infer his degrees of bclief. Farmer

Srnith has bought two tickcts to win for a race at thc cotrtitv [ait', olre on

Stewball and one on Molly. lf he holds a ticket on a winning horsc, he wins a

pig; otherwise he gets nothing. We assur:ie that he does not care abotrt the

outcome of thc horse race per sc; it is irnportant to hinr only insofar as it docs or

does not rvin him a pig. tle is incliflcrent to keeping his tickct on Stervhrll or

exchanging it for an objective lottery ticket with a known I0 lrcr ccnt cha¡rce of

winning; likewisc for Molly or irn objcctive lottcry ticket with a l5 ¡rer cctrt

chance o[ winning.Farmer Smith's utility scalc looks likc this:

Li tilitrlPig I

f icket on lVlolly .I5l'ickct r¡n Stcu'ball .10

Nothing 0

If he maximizes cxpectctl utilitv, his cx¡rcctcrl utility for lris lrct (tickct) on

Nlolly is:

Degree of llelicf (tvf olly Wins) Utilitv (Pi¡j) +Dcgrec of Relicf (N{olly Loscs) Utilit¡'(Notlring)

This is just equal to his degree of belief that Molly wins. Tlren his degree olbetief úat Mály wins is .15; in t1.,e same way, his degree of belief that Stewball

wins is .10. Suújective degrees of belief are here recovered from subjective

utilities in an obvious andiimple way. (Things would be more complicated i[farrner Smiü cared about the outcome o[ the race over and above the question

of üc pig, but ¿us we shall see in üe next section, his subjective degrees of belicf

could still be found.)

Exercises:

l. A decision maker with declining marginal utility of money is rhk aaerse. in.

monetary terms. fle will prefer $50 for sure to a wager that gives a chance of t of

winningit00 and a chrnce of * of winning nothing, because the initial $50 has more

utility ór him üan üe second $5O. Suppose that winning $ 100 is the best thing that can

happen to him and winning noüing is t}re wo¡st'

a. What is his utility for winning $ 100?

b. What is his utility for winning noüing?c. What is his utility fo, u *ug""' that iives a known objective chance of I of

winning $100 and t of winning nothing?

d. What ."n *" ,^y about his utility for getting $5O?

e. Draw a graph of utility as against money for a decision maker who is generally

risk aversc.

2. Suppose farmer Smith has one ticket on cach horse mnning at the county fair' and

üus wili win a pig no matter which horse wins l-et U(pig) = I and U(nothing) - 0-

§uppose farmer Smith's preferences go by expcctcd utility'

o. Fnrnrer Srniü Úelicves t¡at utt tis tickets takcn togethor 0r0 worth on. pig lor

rure.Whatdoesthistellyouabouthisdegrcesofbeliefal¡outtherace?ll.Str¡lposethatfarmerSmiüalsobelievesthateachofhisticketshasequal

utiiity. What does this tell you about his degrees of belief ¡bout the race?

3. (Advanced) Letussaythatthephysícalsunrof twobets, I)ti82'?'¿ysof['atea"h

"*", t,oth the pirysical goáds tl,at A, p,ys oll and the physical goocls that B' pays ofl'

Ho*"u"r, Iet,rss"ythaiabet Bristhimatlrcntoticalsumof B,andB'if itpaysoff'at

"o"h ""r", a physlcal goocl whosc utility is equal to the sunt o/llrc uÚiliti¿s of the physical

payolls of B,and B, irr that crse. Show the following:

a. Iior someonc who carcs only about golcl, and whose marginirl utility for.goki is

constant, the physical sum of two bcts (with payoffs in gold) is the mathcmat-

ical sum.

b. Iior someone who cares only abotrt gold but whose marginal utility for gold is

cleclirring, thc physical ,unt of t*o bcts need not ctlttal thcir nrathematical

surn. (llint: sce Excrcise 1.)

c. F,,, puynffs in arbitrary physical goods, the physical sum of two bcts may fail

to cc¡ral tlleir m¿thcmatical sum as a rcsttlt of colnple rnentaritics'

VI. 9 urrllrv

202 VI. cor¡rn¿¡rc;s

d, Suppose that propositions p antl r¡ are incompatible and that one has a bettingarrantcmcnt th¡t l¡as ¡ ¡rayofl o[ util¡ty r if ¡r is tnre (and g falsc), anrl again a

payoff of utility r if r¡ is tnrc (and ¡r false), and a utility 0 if p and g are lnth false(¡ > 0). This be tting arrangcmc¡rt can be correctly described in two ways; (i)as a bet with payofl of utility ¡ if p or q is tnre, utility 0 otherwisc; (ii) as thenollxnnatical sum of two l;cts, onc of which yiclds a payoff oI utility r if p is

true, utility 0 otheruisc; thc othcr of which yields a payoll of r¡tility r if q is

true, utility 0 otherwise. Reconsidcr the Dutch Book arguments in Iight of theforegoing.

Suggcstcd rcaclings

R. D. [,r¡ce and FL R¿ifla, Ou¡nt.s tttul Dcci^sitnrs (New York: Wilcy, 1957),

chap.2.

J. von Ncunrann and O. lvlorgcnstern, 'l'hcory ol Crnncs otul Econunic Be-lruabr (2nd cd.) (Princctorr: Princctr¡n Univcrsity Prcss, 19.17).

VI.10, IIAMSEY. Thc von Ncrrrn¿rn¡¡-Morgcnstcrn theory of utility isre¡lly a rediscovery of ideas containcd i¡r ¿r remarkable essay, "Tnrth andProbability," written by F. P. Rarnscy in t926. In the essay, Ramsey goes cvendecper into the foundatio¡rs of utility and protrability. The von Neumann-Morgerrstern nlethod requircs that the decision maker know somc objectívechances, which are then rrscd to scale his subjective r¡tilities. From his subjec-tivc r¡tilities and prcfcrences, infor¡nation al>out his subjective probabilitiescan be recovered. Ramsey .starts withor¡t the assumption of knowledge of somechancs§, and \yith only thc dceision n)aker's prcfaroneos,

Ramsey starts by identifying propositions that, like the coin flips, lotterics,and horse races of the previous scction, have ¡ro value to tlre decision maker ina¡rd o[ üemselvcs, brrt only insofar as ccrtain payofls hang on thcm. tlc callssuch propositions "ethically ¡rcutr¿r1." A proposition, p, is ethically ner¡tral for a

collcction of payofls B if it makes no rtiflerence to the agent's prefcrcnces, thatis, if hc is i¡rdifferent betrvccn B with ¡r tnrc ¿rnd B wiü p false. A ¡rroposition, p,is ethically neutral if p is cthically ncr¡tral for nraximal collection.s of payoflsrelevant to the decision problcrn. Thc nicc thing about ethícally neutralpropositions is that thc cxpcctcd utility of gamblcs on thcm depcnds only onüeir probability and the rrtility of tlrcir outcornes. Their own utility is not a

complicating factor.Now we can identify an cthically ncutral proposition, II, a-s having proba-

lrility l for thc decisio¡r rnakcr if thcrc arc two payoffs, A;8, such that heprefers A to B lx¡t is indiffcrent bctwcr:n the two garnbles: (I) Cet A if Il is tnre,B if 1/ is false; (2) get 13 if I/ is tn¡c, A if /I is falsc. (If hc thought ,Il wa*s rnorelikcly than -/I, hc worrkl ¡rrcfcr garnlrlc l; if hc thorrght -]/ rvas rlorc likcly

VI. l0 n¡usav

üan l/, hc would prefcr uamblc 2. Fr¡r the purpose of scaling thc decisionnrakcr's utilitics, srrclr a Pro¡rosition is jrr.st as good as tlrc pro¡rositir¡n that a faircoin comes rrp heads.

The same procedure works in general to identify surrogatcs for fair lotteries.Srrp¡rosc üere arc 100 ethically ncutral propositions, II,;f1r.,.. . ;/Ir*, which arepairwise incompatible and jointly exhaustive. Suppose there are 100 payoffs,CiCz; , . .;C¡m, such that C, is ¡rrefcrred to Cr, G2 is preferrecl to G., and soforth up to (i,*. Strpposc thc dccision makcr is indiflcrcnt betrvccn thccomplex gamble;

If /1, get G, &If H, get C. &

If a, gei c, er

If II,oo get G,*

and every other complex gaml)le you can get from it by moving ú-re G,saround.Then cach of the fl,s gets probability .001, and togcthcr thev are jrrst as flood as

a [air lottery with 100 tickets for scaling the dccision maker's rrtilities.A rich enough pre[crcnce ordcring ha^s enough cthically ncr¡tral propositions

forming equiprobable partitions of the kind just discussed to carry out the vonNctrmann-lvlorgenstern type of scaling of utilities described in thc last sectionto any desired dcgrce of precision. Once the utilitics havc l¡ccn determined, thedegree of belief probabilities of the rerrraining ethically ncr¡tral proPositionsc¿n be determined in the sinrplc way we havc secn bcfore. Thc decisionmaker's dcgree of belief in thc ethically neutral ¡rroposition, ¡r, is just thc utilityhe ¿ttachcs to the tlamble: Cú C í.[ p, B othcnoL¡c, whcrc C has trtility I and Bhu-s utility 0.

With utilities in hand, rve can also solve for the decision m¡kcr's dcgrees ofbelicf irr non-cthically ncutral irropositions, althorrgh things arc not r¡rrite sosimplc herc. Suppose that farmer Smith owned Stewball and wantcd his horseto win, as wcll ¿u wanting to win a pig. Thcn "stewball wi¡rs" a¡rcl " Molly wins"are not cthically neutr¿l for him. Now suppose we want to dctermine hisdegrec of lrelief in the proposition that Molly wins. Givcn orrr convcntions, wecan't sct rrp a gamblc üiat gives utility I if Molly wins bcc¿ruse what far¡nerSnriü desircs most and gives rrtilitv I is: "Cet a pig and Stewball wins." But weknow that thc expectcd utility of thc rvagcr "Pig if tr{olly wi¡rs, nr¡ prizc if sheIoses" is er¡ual to:

I'(i\lollr, *,ins) (r(rct pig and lrlollv rvins) +1 - Pr(lv'lolly wins) (/(no yrrizc anrl lvf ollv losr:s)

on r

904 VI. coHensNce

If we know the utility of the wager, of "Cet pig and Molly wins," and of "No

¡rnzc arrd Molly loses," we can solve for Pr(Molly wins).

For a rich and cohcrcnt prefercnce ordering over gambles, Ramsey has

con¡ured up both a sul:jectivc utility assignment and a degree of belief proba-

l,iliiy assignáent such that preference goes by expected utility. This sort o[

reprcsentation thcorem shows how deeply the probability concePt is rootcd in

practical reasoning.

Exerciscs:

l. Suppose t}¡at the four proPositions, IIII;LtT;TH;TT, are pairuise incompatible (at

most one of them can be true) and ¡ointly exhaustive (at least one must be tnre)'

Describe t-he preferences yorr worrld need to find to conclude that they are ethically

nerrtral and equiprobable.

2. Suppose that farmer Smith owns Stewball and that "Molly wins" is not ethically

neutral. His most preferred outcome is "Cet pig and Stewball wins"; his least prefcrred

is "No pig and Stewball loses"; therefore, these get utility I and 0, respectively'

Propositions, HH;ItT;Ttt;TT tre as in Exercise I. Farmer Smith is indifferent between

"C"t pig and Molly wins" and a hypothetical gamble üat would ensure that he would

get the pig and Stewball would win if HH or HIor IH a¡¡d üat he would get no pig and

§t"*lr"il would lose if TT. (What does this tell you about his utility for "Cet pig and

Molly wins"?) Hc is indiEerent between "No pig and Molly loscs" and the hypothetical

gamble that would ensure üat he would get the pig and Stewball would win if HH and

ihat he would get no pig and stewball would lose if IIT or TI! or IIH. He is in<lifferent

between the gamble "Pig if Molly wins; no pig if she loses" and üe gamble "Cet pig

and Stewha¡¡ wins if HH or HI, but no Pig and §tewball loses if IH or IT"'a. What are his utilities for "Cet pig and Molly wins"¡ "No pig and Molly loses";

üe gamble "Pig if Molly wins; no pig if Molly lcses"?

b. What is his degree of belief probal;ility that Molly will win?

Suggested readings

F. P. Ramsey, "Truth a¡¡d Probability," in The Fountlations of lvlothenntics orul

Other lttgical Essoys, ed. R. B. Braithwaite (London: Routledge and Kegan

Paul, l93l) and in Slldí¿s in Subicctioe Probability, ed. I{. Kyburg and H'

Smokler (lluntington, N.Y.: Krieger, l98O).

For üe advanced student:

Peter Fishbum, "subjective Expected Utility, a Rcview of Normative

Theories" The.on¡ ond DccLsion 13 (l98l), pp. 139-199.

Terrence Fine, I/r¿o¡i¿s ol Probobility (New York md l-ondon: Academic

Press, 1973), chap. VIII.

VII

Kinds of Probability

VII.1. INTRODUCTION. Historically, a number of distinct but relat-ed concepts have been associated with the word ¡trobabilify. These fall intothree families: rational degree of bclief, relative frequency, and chance. Eachof üe probability concepts can be thotrght of as conforming to the ¡nathemat-ical mles of probability calculus, but each carries a different meaning. WeIiave, in one way or another, met each of these probability concepts already inthis lrook. A biased coin has a certain objcctive clnnce oÍ coming up heads. lfwe are uncertain as to how üe coin is biased and what the objectivc chancereally is, wc ¡nay have a rational degree of belief that thc coin will comc rrpheacls that is rrnequal to the tme chance. If we flip the coin a nrrml.ler of timcs,a certain percentage of the tosses will come up headsi that is, thc rckúitc

fre.quotct¡ of heads in the class o[ tosscs will be a nu¡nber in thc intcrval from 0to l. The relative frequency of heads may well differ from both our dcgree ofbelief that tle coin will come up heads and the objective chance that üe coincomcs up heacls. 'Itre concepts are distinct, but they are closcly related.Olrserved relatiae frequencies are important evidence that infltrences our ro-tk»tal dcgrec.s of belicf aboul obiectiae t)ta¡tces. If, initially, wc arc u¡rsurewhetlcr the coin is biased 2 to I in favor of heads or 2 to I in favor of tails(dcgrec of belief |)l and then we flip the coin 1000 timcs and gct 670 heads, wcrüll have gotten strong evidence indecd that thc coin is biascd toward hcads.

AJont jrrst üese lines, Cicero evaltrated divination as a statistical thcory andforrncl it rrnworthy of a high degree of bclief. In our own tinrc, rnicro¡;hysicsconsists of thcories that postrrlatc clranccs, ¿rr¡rl that arc largcly tcstccl againstfrcquentist evidence. This final chapter is clevotcd to ¡ r«:vien'of these con-ceptions of probability and a sketch of their interrelation.

VII2. RATIONAL DEGREE OF BELIEF. Belicf is not really an all ornothing affair; it adnrits of degrees. You might be rcasonably sure that thepresitlent was guilty without lreing absolutcly certain. You mi{ht bc extrcmclyrlubiotrs about thc ¡rlaintifl's "^trpposccl

whiplash injury without lrcing certaintJrat hc is mlrlingering. You might thi¡rk of it as only slightly rnorc likcly thanrlot that üc carrse of a sore throat is n vinrs. Degrees of helief can be rcprc-sentccl nrrrncrically, with larger numbers corresponclinS to stror)ger trclicfs.\Vhat shorrld thc mathernatics of thesc numbers [¡c for a rational agent?

Clralrtcr Vl irrtroclucccl sorne o[ tirc rcaso¡rs wlry it has becn hcld that rutionul<lcttrccs of lrclicf shoultl adrnit ir nrrnrcrical rcprescntl.ttiorr that obcys thc

205

VII. xtNos otr PnollAlllLll'Y VII. 2 rr¡,r'¡oN^L DECRtiti ot¡ IlIil,lIiIr 207

<lctcr¡ninccl original clegrees of belicI uni<iuely fr¡r such age nts*that is, clegrccs

of belief prior to encountering any cvidcnce-thcn thc only disagreemcnt

possible for rational agents would be due to thcir having different cvidence.

Strch is thc conccption o[ indrrctive logic that we find in Carnap's carly work.

Carna¡r then thought that thc probabilities a rational agent would have on

no eviclence was a matter of rationality-a matter of logic. The probabilities

th¿rt a rational agent worrld havc on a body of cviclc¡rcc wotrld thcn bc tlctcr-

rninccl by coDrlition¿rlizntiorr on thc evidcncc. I-:rtcr, Carna¡l ca¡nc to thc vicw

that the constraints of rationality deter¡nined not one, but rather a lalrgc class of

pcrmissible original degrees of belief, tle sct forth candidatcs for this class in

Tlrc C¡»ttí¡tttu¡i of lntluctioc Metlnds. One peculiarity of Carnap's rnetho¿s

rvas that all of thern had the conscquencc that no ttniversal gcneralization (for

exalnplc, AII ¡¡¡cn arc nrortal) cotrld achicve non-zcro proltattility orr any fiIitearlrorrnt oI cvirlcncc. Carnirp tlcfcr¡<lc<l this skcptictl concltrsi«rr¡, lrrrt othcrs

argrrc<l th;rt it shorvccl that his rncthorls cor¡lci not givc an aclcr|ratc accottnt oI

thc confirn¡ation of scicntific laws. Ilintikkir showcrl how Cartrit¡r's class o[

priors corrlcl ltc ttxpancleil so as to allorv a ration¿rl agcnt to assign to a tt¡riversal

gencralization high protrahility on finitc cviclc¡rcc. Such invcstigatio¡¡s of

c.orrfir¡n¡tiolr tlrt:ory in thc Carna¡rian tradition providc trscftll inforlnation

abotrt thc irlrluctivc bchirvior o[ variorts liriors. Thcre is, howcvcr, nothing in

crrn:rpilrn co¡rfir¡natir¡n thcory likc thc cohcrencc ¡r[ltlrnel]ts to explain why

tIe constraints oI rationality should linrit one to thc prior probabilitvassi g'nrncnts irt r¡ucstion.

Dc Finctti, on thc othar lrand, bclicycs that thc nrla§ 0f tlrc probahilitycalculus constitute the onlry constraints on rrtional lrclicf. In acldition, he

crnphasizcs both tltc oPcrr-crrrlcclncss antl cvolvir)fl natrrre of Lhc s¡racc of

propositions about which wc have degrccs of bclicf, as rvcll as thc fact üatwhat wc leam may not be strfficiently surnmarizcd bv propositio¡rs in otrr

probability spacc. For dc Finctti, "indrrctivc rcasoning" is ¿ ¡nuch more wild

inrl frec-swinging allerir, with tlrc ct.r¡rstrlints of ratiorrality lcaving il grcat dcal

oI ro6r¡ [¡r inrlivi<hurl <lillcrcnccs arrrl itliosyricr;rsics. It shorr]rl not lrtr thorrght,

howcvcr, that r:omnron cvirlcn<.c in this vicw hns no forcc wltatcver to compcl

agreclnc¡t. F9r a wiclcly tlivc,rsc clirss of priors, it catr l¡c sllorvtr thllt tnorc

a¡<l ¡tr¡l.c cvirlcrrcc [orr:cs convcrgclrce, in tlic li¡nit, tr:¡ tltc sttl¡tc ¡lostcritlrprolrrrlrility u.¡ortvtll o.[llu:nrrtt.ltnnatiutlslntt:tttrcoI tlrt:prrtltttltililry r',r1r;ttlt¿s.

'l'lris rlrrrcll ilrtlr¡ctive Lrgit.is lrrrilt into thc prolr;rlrilit¡,calr:rrlrrs itsr:l[.

Excrcise:

Wr: l¡9tll lxrlicvc t[¿t a t:pirr is cithcr (A): lriast:tl 2 tr¡ I in favor of lrcrrtls or (B): bitscrl

2 to I i¡r [¡vr¡r of tails, lrrrt u,r-. tlillcr a.s to horv our initial tlcgrecs of ltclicf a¡e distribrrtcd

lrctwct:n (A) an(l (l]). )'otr bclicvc (A) to rlcgrcc .8 arr<l (li) to tlcgrcc .2. I lrclicvt: (i\) to

rlcgrec .3 ancl (R) to clegrcc.T. \\¡c flip thc coin fivc tirilcs antl gct onc ltcacl r¡rtl fotrr lails.

206

mathematical nrles of thc probability calcr¡lus lf numerical dcgrees of belief

are used to calculate expected vaitre in üe standard way' then for üe

*p"*r,f"t to be well-Lehaved, numerical clegrecs of -bclief rnrrst be

miü"matic"l probabilities. This fact is dramatized by üe Drrtch .Book

"rg*"t,, If yáu üolate t}¡e rules of the probability calculus' your calcula-

if* "f "*p"át"d value will be incoherent in such a way that you can be

systematicallyvictimizcdby¡clevcrllcttor'Thc.argtlmentC¿lnlletakcnastcpfírth", when consiclcrations of utility arc introdrrced' Fronr a colicrent prcf-

erence ordering over a rich array of options, olle c¿ln extr¿rct nrtmerical

proUaUillty aniutility a-ssignmenti such that the original preferences go by

expected utilitY.bn" poprrlri.isconception of subjectivc probability shor.rld 5e corrccted at

tt,ispoint.rhatisüeideathatthctheorysaysthatcveryrationalpcrsonmttsthavJ a sha¡p nttmencol ortl¡rc for his clt:grec of bclicf in any proposition' This is

not really so. It is not a condition of rationality th¿rt one's prcferenccs lle over

,rch "

áh field of options a-s to allow thc Iiamsey mcthotl of Chaptcr VI to

converge to a uniqtie ¡rumerical proltaSility-rrtility rcprescntation. -But

it

might fiirly be held üat one's p."fcic,,c-es- bc ó-oherent and bc capablc of lreing

cofierently extendcd to arbitrarily rich ficlds o[ oPtions. This way of looking at

üe representation theorcnls allows that a rational system of prcferences adrnit

u ,"r,g" of numerical probability-utility rePresentütions, each of which does

üe jó of providing expected utilitics that correspond to the preference,s. In

""cú of these repreientations, thc probability nrrmSers oSey the laws o[ t]re

probability c"l.ulur. ln this way, *" "an

say that a rational agent's clcgrecs of

Lti.t ,"y not be ¿etermineá up to a unique nurnerical value' hut that

neverthelás the probability col.rilus givcs us nr'lcs for ratiotrirl tlcgrecs o[

belief.Suppose,then'thatrationaldegreesoflleliefnlaybcrepresentc<la.snttmllers

üat are mathematical probabiliiies. If a rational agent trpdatcs- lris proba-

bilities in üe light oI new cvirlcncc according to rr¡les of bclícl charrgr;, tlrcn

there are coherÁcc arg.rnc.ts [r-¡r ct:rt¿rin nrlcs. T¡c l)trtch I]ook arqttrtrcttt for

conütionalization discirsse¿ in Chapter VI is one srrch arg,ntent. According to

üis argrtment, if what one lcan¡s ii jrrst that a certain prol;osition' p' in orre's

probañlity space is true, thcn rnc slrorrl<l rr¡r<latc 6y co¡rditio,alizi.g .rt ¡r. Tltc

c¡restion oi the full scopc o[ s.ch cliachrr¡nic cohcrcr¡ce irr¡]nrrncnts is a nrtttcr o[

current research.

Forthemoment'letrrsrcstrictottrattcntjontothccascq.llcrccvi<lc¡ltle:comes nicely packaged as just lcarning the tnrth of sontc proposition in onc's

probability spa"". lir"n two agcnts rvlt. agrc<: o, tltcir degrccs '[ lrclicI l>c[.rr:

getting the eridence, anrl who arc both rationiil in thc scnsc of thc last

f,"rrgi"ptr will agrcc on thcir final dcgrccs of lr.licf. Initial <legrce of lrt:lit:f ,ltrs

"uiden"" will deicrminc final degrcc of bclicf. I[ thc constraints oI rationality

VlI. xlNps oF PnoB^BILITY

f we both conditionalize on this evidcnce, what are our final degrees of belief in (A) and

(B)? (Usc Bayes' theorem.)

Suggested readings

Rudolf Carnap, I'ogical Foundatíons of Probobitittl (2nd ed ) (Chicago: Uni-

versity of Chicago Press, 1962).

Rudou Camap, The Conlinuuln of lnductiac llctlnds (Chicago: Urriversity of

Chicago Prers, 1952).

Enrno de Finetti, P¡ol¡obility, lnduction and S'aÚis'ics (New York: Wilcy'

1972).

Bnrno tle Finetti, IAeorq of I'robability (2 vols ) (Ncw York: Wiley' 1975)'

Jaakko tlintikka,"AT*o Dimcnsional continuum of Intltrctive Methods." in

Aspects of Inductitsc Ingic, ed. tlintikka and suppes (Ncw York: IIt¡nranitics

Press, 1966).

Richard Jefr:ey,'f|," lttgir: of Decision (2nd ed') (Chicago and l'ondon: Uni-

versity of Chicago Press, l9&j).

Isaac l,evi, The Lnterprisc of Krcu;lcdgc (Cambridge, Mass': MIT' 1980)

F. P. Ramsey, "Tmth and Probability," in The Foundations of Mathetnatics and

Other LogiotÁrsoys, ed. Il. B. Braithwaite (London: Routledgc & Kegan Paul'

l93l) anJ in Srudi¿s in Subicctioe Probobility, ed l{ Kyburg and }l Snrokler

(Huntington, N.Y.: Krieger, I98O).

L, J. Savage, The Fotndotiottr ry' Stati.stics (2nd cd )(Ncw York: Dover' 1972)

VuS.REI.ATIVEFnEQUENCY.IfIllipac0inte¡ltirncsanrlitcomes up hcads six of those tirnes, thc observed relativc frcquency of heads is

.6; six heads out of ten trials' If our language contains the means of reporting

the outcomes of single trials and contains standard ProPositional logic, it

already contains üe m"rns for reporting the relative freqrrertcies in finite

numbárs of trials. For example, (supposing that Ht means "heads on trial 1,"

etc.) we can render "relative frequency of heads equals .1" as the disjunction o[

üose sequences of outcomes with exactly one head and nine tails:

Hl&T2&T3¡(T4&T5&T6&T7&TB&T9&T10Tl& H2&T3&T4&T5&T6&fi &T8&T9&Tl 0

T1&T2& ¡I3&T4&T5&T6&fi &T8&T9&T l0T1&T2&T3& H4ó¡ T5& T6&fi &T8& T9& T I 0

Tl&T2&T3&T4& HS&Tffi fi &T8d(T9¿{Tl 0

T1&T2&T3&T4&T5& H6&T7& T8&T9& T 1 0

TI&T2&T3&T4&T5&T6& Ifi &T8&T§& T 10

T l&T2&T3&T4&T5&T6&T7& H8&T9&Tl0T 1&T2&T3&T4&T5&T6&T7&TB& H9& Tl 0T l&T2&T3&T4&T5&T6&fi &T8&T9& ¡i i 0

ororororororororor

VII. 3 nrl¡rrrvE rrnEeuENcy

([ixerci:e: Write out üe comparable description for "relative frequency ofheads eqrrals .2" in ten trials.) Relative frequencies of outcome types obey thelaws of üe probability calculus: They are always nrrmbers in the interval from0 to l. The relative frequency of a tautological outcome type (for example,either heads or not) is I and üat of a contradictory outcome type (for e*r-p1",both heacls and not) is 0. The relative frequency of two mutually exclusiveoutcorne types is the surn of thcir individual relative frequencics.

Statenrents oI ol¡served rclativc frequencies constitute an espccially irnpor-tant kind of evidence for statements about chance. In certain familiar <:a-ses

they summarize everything that is salient to the chances in a series of r.¡l;scrvecltrials. Consider again the examplc of the coin with unk¡rown l>ias. Srrpposc thatthe coin is flipped ten times with the outcorne being:

I I 1&T2& H3& H4&T5& t 16&T7& H8& I I9& H l0

The rclative frequency of heads is .7. Now supposc that we wrnt to find thcprobability conditional on the outcome üat thc chuncc of lreads is-!. We trseBayes' thcorelrl, '

pr[chancc H -floutcorn.l - Prloutcome/chancel =j]-tltgylalI = il,Sunr, ['r[outcotnc/r'hant:c II = i] I'rIr'hrrncc II -, i]

The outcornc evidcnce figures in Bayes' theorv solely thror.rgh thc conditionalprobabilities of thc outcome given the chance hypothescs, Brrt for ¡ givcnclrance hvpotlicsis, cuery ottt.come sequcncc loith seDen lrcud,y out o.f tc,1 tosscs

IuLs the so¡rc probaülity conditional o¡ lhe chancc hypothe sis. (This is bccausethe trials are independent, conditional on the chance hypothcses.) Any otheror¡tcorne sequence of ten trials with the sar¡te relative frequency of heads

would lead to the same calculation (order is not important). With regard to orrrinlcrcnces about chances here, all thp relevant information in the outcomesequence is capfured by a report of relativc frequency. Rclativc frcqucncy is

s¿rirl here to be a xtlftcient statistic. (It is a sufilcient statistic whon we areconsidering outcorne sequcnces of fixcd lcngth. Thc pair consistirrg of thclcngth of thc outco¡¡rc sc(lucnce togcther with the relativc frcqrrcncy consti-trrtcs a suflicient statistic in general for all outcornc scr¡rrcnccs.) Wlicnevcr thctrials are inclcpenderrt, conditional on thc chance h,v¡rothcscs, relativc frt:-quency is a sulficicnt statistic. In this typical situation, reports of rclativcfrcqrrcncy captrrrc all the salicnt informati<¡n in thc cxpcrinrcntal rcstrlts.

llcports of rclative freqrrency l¡cco¡nc rnore corn¡rcllirrg as thc nrrrnbcr ofobserved trials becomes large. Seven out of tcn hcads rnight inclinc rrs tolrrrlicvc that thc coin is biased toward heads, brrt our inclirratiorr rvr¡rrld turrrinto sorncthing strongcr on thc cvidc¡rco of70 otrt of lOO hoacls, Ict alonc r¡n 7CX)

209208

210 VII. xrNos oF PRoIIABILITY

Suggested readings

Alonzo Church, "On the Concept of a Random Sequence," Bullettn of the

Amencan Mathenatical Society 44 (1938): 130-135.

William Feller, An lnt¡oduction to Probabillty Theory and lts Appllcatrons(2nd ed.) (New York: Wiley, l95O), chap. VII (for üe wcak law of large

numbers).

Terrencc Fine, Il¡eori¿s of Probability (New York and London: Academic

Pres,1973), chaps. lV and V.

VII.4 cuaNcr 2tl

L. J. Savage, The Foundatiorw of Statistics (2nd ed.) (Ncw York: Dover, 1972),

sections 3.6 and 3.7.

fuchard von Mises, Probability, Srdtisúic.r and Truth (2nd ed.) (New York: TheMacmillan Company, 1957).

I'lans Reichcnl¡a ch, Erpcricnce and Prediction (Ch icago: Un ivcrsity of ChicagoPres, 1938).

John Venn, The Logic of Chance (4th ed.) (New York: Chelsea, 1962.)

(The last three writers identify chance in one way or another with relativefreqrrency in the long run.)

YII.4. CHANCE. We have already discussed some aspects of chancc inr»¡rnection with degree of belief and relative frequency. The biased coin is

held to have some true chance of corning up heads, determinecl by its physicalcomposition (in conjunction wiü üe nature of üe flipping apparatus), In thelong nln we approach certainty tlat üe relative frcquency of heads willapproxirnate üe chance o[ heacls, but in the short nrn chance and relativefrequency may well diverge. We may be uncertain about the chances, in whichcase we have various degrces of bclief for the chance hy¡;othescs, I.Iere,

observcd relative frequencies can serve as cvidence-evidcnce that bcco¡nesmore and more conclrrsivc as the nrrmber of trials approachcs infinity*aboutthe chances Dia Bayes'theorem.

l,et us look at the use of Bayes' theorem a little nrore closely. The qtrantities

that arc crucial t0 tho way in wl¡icll üc ol¡scrvcd frcquoncics cut for and

against the chance hypoüeses are the conditional probahilities of outcomes

con<litional on the chance hypotheses. Now I am going to ask a qucstion thatrnay sccnr trivial. trtr4raf Ls thc proltability of gening lwad.s corulitional on tlrcclmru:e o.f lwad.s beingl ? The natur¿rl answcr is f , and it mav sotrncl tautolog-ical. Ilrrt rcrncmber, tlrc conditional probability in question is a clcgree o[lrclief. What we havc is a principle connecting degree of belief and chance thatis so ol>viou.s that it is constantly rrsed brtt rarely statcd. II otrr cvidcncc for an

outcornc corrrcs only from knowlcdgc of thc chances involvcd, thcn:

I)cgree of bclicf [O given chanct{O) - ¡] - ,It is üis principle üat allorvs us to use Bayes' tleorem in inductive rcasoningabor¡t chances.

It also allows us to determine i¡l a nahrral way degrees of belief aboutoutcomes from degrees of bclief aLrout chances. Suppose that you are certaint}at üe chance o[ heads is either { or -} and that you regard those chance

h¡>othcses as cr¡rallv Iikcly (that is, you give t-hem each degrcc of belief f ).Then what should yorrr clcgrce of bclief be that thc ncxt coin flip comes u¡r

out of 1000. These garnbler's intuitions are mathematically well foundcd.

Suppose that you ^rc

in th" coin tossing case, with a finitc number of chance

t ypott "t"t,

eách of which has positivc prior (dcgree of 6elief) probability and

that üe trials are independent, conditional on üe chance hypothescs. Thcn as

üe number of trials lr""o-", larger and larger, your probability üat the

relative frequency approximates the true chances to any tiny preassignederror

tolerance approaches l. This is a conseqrrence of a theorem of the probability

calculus known as the weak laa of latge nulntl¡ers. Ilere, you nrt¿sf lielieve

strongly that the relative freqtrencies will approximate the chances in the long

n:n if your degrees of belief are coherent.

We expect a large number of intlcpendent flips of a coin to produce an

outcome i"qu"n"" whose relative frequency of heads approximates the tnre

chance of heads given üe true bias o[ thc coin. We also expect üat the

outcome sequence wilt be disordered or random without any consistent

pattem. An outcome sequence wíth a simple pattern, such as:

TIHTHTITI.IHTI IHTHHTTITITI-I[ITI{HTIII{TI I I{TI.II{THI I . . .

would strain our belief that the trials really are independent. This intuition also

has a maüematical foundation in the probability calculus. using a srritablc

definition of randomness, it can be shown that a large number o[ irrdcpcndcnt

trials is likely to produce a highly random outcome seqr¡encc'

Situations in which wc are confronted with a large nurnber of trials that we

üink are independent (or approximately independent), conditional on the

shanqe hypeüqsesr are all around u.s, They are of intere§t not only in the thcory

of classicai games of chance, but also in statistical physics, social scienccs, and

medicine. In such sittrations we exPect (with high degree of bclief) to

encounter relatively random data whose relative frequency approximates the

chances. Some writcrs focrts ot.l this sort of casc to the extent of defining

probability (that is, chancc) a-s the li¡niting rclativc lrequcncy of random

áutcome sequences. But without going that far, we can still see the írnportance

of relative frequencies for anyonc who h¿ls cohercnt dcgrees of belief about

chances.

2t2 Vll. xrNus or l'no¡ABIr-r'rY VlI. 4 r;rrnNcr.; 213

üe seqrrence of trials be random; some versions (such as that o[ Rcichenbach)do not. Frequency views derive a certain motivation from limit theoremsof the probability calculu^s, such as üe law of large numbers. I[ we considerinfinite sequences of independent trials,t the law of large numbers can bestated in a strong way: The probability that the chance etluals the relativcfrequency is l. Why not just say üat chance ís lirniting relative frequency.

This view has its difficulties. The most obvious is that the appropriate infinitesequences may not exist. Suppose, with some plausibility, that my biased coindoes not last for an infinite number of flips. Still, we want to say that on eachflip there was a definite chance of üat tlip coming up heads. There is anotherproblem as well in finding üe appropriate infinite sequences. The physicalfactors dctcrmining chance may change. The coin may wear in sucli a way as tochange üe bias after a few million flips. Or, to vary the exanrple, thc chance ofa newborn baby living to age 50 may changc from deca<lc to dccade andcountry to country, and so on. For these reasons, sophisticated rclative fre-quency vicws from the time of Venn (1866) onward have had to talk abouthypothetical relntíoe frequencies: uhat the limiting relatiae frequency uouldhat¡e been if the expeñment had been repeated (independently?) an inf,nitenutil¡cr of tinrcs such that the factors dctcrnúning tlrc cln¡tccs dful ¡x¡t clumgc.Notice üiat üis definition o[ chance, as it stands, is circular. It uses both thenotion of thc factors deterrnining the chances, and, apparently, thc notion crf

independent trials. A noncircrrlar hypoüetical relative frequency definition ofchance worrld reqr.lire some way of elirninating üese references to cliancc.

Even if this were ilceomnlishsd, there worrkl ile some question as to thogrounding of the hypoüetical. Improbability, even probability 0, is not im-possibility. There is nothing in probability theory üat savs that it is impossiblefor a fair coin to come up heads every time in an infinitc se(¡rcnce of inde-penclent triais. What, üen, does üc frequentist takc as grounding üe truth ofhypotheticals about what the limiting relativc frequcncy would have been, insuch a way that they capture the conccpt o[ chancc? Thc idca o[ a relativefrer¡uency interpretation of chance offers to bring thc conccpt of chance downtr.¡ earth, but a-s üe relativc frequency interpretation becomes rnore sophisti-cated, it becornes more remote from the real events that constitute scquences

of actual trials.Personalists war)t to do without the concept of ch¿rncc. The primary

conccption of probability for the¡n is rational degree ofbclicf. But they need toprovide some cxplanation of what appears to be rational degrecs of belicfolnut cluutcc,:r-s in thc case of thc l¡iascrl coin.'I'hcy <Jo this by showing horv

ordinary degrces of belief about thc cxperimental setttp and thc outconrc

heads? A .atural answcr is to average üe possiblc chances 50/50, getting (+)

(+) + (+) (t) = i'.The answer is a consequence of üe foregoing principle and

üe probabili ty calculus:

p(H) =Pr[Ch(H) =+&II] +Pr[Ch(H) :+&H]

= Pr[Ch(H) =]1rr¡Itzcl,(Il) =+l + Prlch(rl) =]1rr¡uzctr(H) =il

= Pr[Ch(H) =f,] (+) + PrlCh(H) =+l ('i)

The proper degree of bclicf that the next flip conres up heads is an average o[

the portibl" ch"n""., wiü the weights of the average being the degrees of

belief in üe proper chance hypotheses. we have met such weighted avcrages

before when we discussed rational decision. They are called cxpectotions. fn

connection with rational dccision, we wcre interested in expccted utility. The

principle we lrave herc is that rutional degree of ltclicf is the expcctation ofchtnce.

The preceding two Paragraphs round out our quick survey of the connec-

tions between chance and the other conceptions of probability (although they

nrly lcave lirrgcring questir¡ns ¿rtrorrt thc crcdr:ntials of the "nattrral" principlcs

invokcd). But, what ts chonca in and of itsclf ? This qucstion is highly contro-

versial. I will attempt a rough and ready sketch of the main positions, t¡ut thc

reader shoulcl be warned that I will not l¡c able to d<-r justice hcrc to the subtle

and sophisticated variations on these positions to i¡e forrntl in thc literature.

Basieally, mstaphysicll vipws of chance fall into thrae mailr en¡egories; (l)

chance as primitire and irreducible, (ll) chance as reducible in some way to

relative frequency, (III) chance as reducible in son,e way to degree of belief.

Those who think o[ chance a-s an irreducible notic¡n of physical tendency or

propensity usually ürink o[ it as a theorctical conccPt on a Par with the concePt

of force in physics. Physical theorics stated in terms of chance permit

preüctions about üe cha¡lces. But all that we can observe are the relativc

irequencies in scquences of outcomcs. Thcse are inductively rather than de-

ductively linked to statcrncnts about chance. That is, our rational degrees of

belief about t-he chances are influenced by observed relative frcquencics.

Conversely, our l¡eliefs al¡out the cliances i¡rfluence our anticipations abotrt

relativc frequcncies. ln short, all üat we k¡iorv about chancc in gencral is its

connectioniwith rational degrec o[ belicf and relativc frequency. The main

shortcoming o[ üis view is that it has so little to say; its nrain strength is the

drortt'omings o[ colttl>r'tilrq vicrvs.

Frcqrrentists üink of chancc ¿ts rclative frc<¡tency it tlrc ktr44 /ün-ntore

precisely, a-s the limiting rclative frcr[rcncy in an infinite so(lucnce oI trials.

§nnr",r.rrio,,, of tlie theory lsuch s that o[ vorr I{iscs):xld the llrovisLr tltittrAntl strcrrgthcrr thc additivity nrle of tlrc probability crlcrrlrrs to allr¡s,infi¡¡itc rdditivitv

VII.4 c:1r,.'rvcs 2t5214 VII. xrNt¡s oF I'tt()l]ABILIl'Y

scqucncc cl¡tr lrxrk likr: tlcgrccs tlI lrclicf ¡tlrot¡t cllatlc'c'

P.cs,rmably therc arc ccrtain ovcrt Plrysical facts that yotr takc as dcter-

mining thc chanccs, for cxarnPlc, thc sh;rpc and distribtrtirln of dcnsity of thc

c<.¡in. Let trs say that yorrr' ltcrsorral chuttcc ("¡r-<'lrrrncc," for slrrlrt) of hcacls is

yorrr degrec of bclicf in hcurls contlitior¡al on thr: spccification of thcsc physical

iac,ts. Ifár cxirrn¡rlc, lt:t L(.llv.ll'l'l tlcscrilrc I coin with a lrtmp on thc hcacls side

such that yor¡r dcgrcc of bclicf that the coin will come uP hoads contlitional on

that physical asyrnrnetry rrf thc coin isf . Then wc will say that yotrr p-chance

of heads i: f in any sittrution in whiclr LI-IMPH is ¿r tnre descriPtion o[ the coin.

Now thc pcrsonalist can show tll¿tt ¡t-clranccs work jrrst thc way that chanccs

are sltpposcd to, and that this is a conseqr¡crlc€ of the probability calcrrlus.

I nus:

Pr[lrcrrcls/p-clrarrcc(hca<ls) - + ] . ifrorn tl¡c «lcfilritiorr oI p-chance, anil ration:rl tlcgrcc of lrelicf is t]tc cxPcctation

o[ p-chancc.Ño*, p-clrantcs ¿rc /)c,".sórlr,t/; thcy <:orrtc fro¡t¡ onc's dcgrccs of llclicf. Dif-

[ererrt rational agcnts can assigrr diflcrent 1;-chaDccs to thc sal]lc possiLrlc

circu¡nstanccs. This srrbjcctivc charactcr of ¡l-chanccs, howcvcr, givcs way to

an ol:rjective deter¡ni¡ration in ccrtai¡r s1'rccial circtrnrstallces. O¡rc sort of ovcrt

fact al>out thc world that otrc might appcal [o in constnt<:ting p-c]rances

consistsof therrrlr¡/itit: frrr¡ttencr¡of anotrtcolltctypc(fortrxlittrplc,hea<ls) inan

outcome seql¡cnce. If tw,o r:¡tional agcDts havc degrccs of l;clief suc:h that their

p-chanc€s const!'ue tcd in this way give irrrlcpenri0nt triills, thcn thcir p-chancos

intlst agree in the lirniting casc of .infinitc trials as a consc(}rencc of the law_of

largc numbers. Here oltjcctivc agreemcnt is a conscquence of a thcorer¡ of the

¡lrobal rility crtl<'ulus.

Thc analysis of the condition rurder which a clcgree of belief has thc sort of

stnlctur.c srch that coDclitioning on st¿ltc¡ilents o[ rclative freqrrcncy gives

in«.lependcnce in the linrit is dr¡c to dc Finctti. It is üat thc dcgrec of bclief

prolralrility makes the trials erclrdrrgeoble, which is to say that for any finite

length srrbse<¡trcncc, rcl:itivc frcqrrcncy is a sufllcicnt statistic. (For dctails, sec

thc strggcstcd readings.) I¡¡ this c¿rsc, Pcrsoualisrn cotttcs sttrprisingly closc to

freqrrcntism: Dcgree of lrclirtf is lhc cx¡tt:ctrtlittrr of linitittg rehlioe fraqueru\.This vcrsion o[ pcrsonalisnr inhcrits sr¡rnc oI tht: strcngths of [rc<¡rctrtisrn, suclr

zr-s thc olijective charactcr oI laws of largc tttrmlrcrs; llrt ít also irrl¡erits sonre of

the ¡troblents of fre<¡rcntism, strch as (ltrostions al)out thc i<lcalizcd character

of infinite sequcnccs o[ trills.The qrrestion of tlrc natrrrc oI c:hancc is a <liflicult und corttrovcrsial nrct¿t-

physic,ai tolli,:. Irro¡r«:rrsity, frcr¡trcncy, untl ¡t<:rs0rralist vir:rvs oI chattcc sccltt

initially to bc qtrite distinct. I3ut sophisticatcd vcrsions of c¿¡cl¡ of ücsc vicws

rnakc contact with the ccntral concepts ¿rnd methods of thc others, and there is

consiclcrablc common grorrnd betrvecn them'

Suggested readings

Bmno de Finetti, "Foresight: Its Logical Laws, Its Subjectivc Sources," in

Slurii¿s i¿ Subiectioe Probobility, ed. H. Kyburg and H. Smokler (Huntington,

N.Y.: Krieger, 1980).

Henry Kyburg, Jr., "Propensities and Probabilities," Brí,ir/r !ournal for tlwPhílosophy of Scícnce 25 (197a); 358J75.

Davicl Lewis, "A Subjectivist's Cuidc to Objcctive Chance"' i¡ Shtdies in

Iuluctíoe L,ogíc antl Probabilitg, Vol. II, ed. R' Jeffrey (Berkeley and Los

Angcles: Univcrsity of California Press, 1980) and in I/s, ed' W' Harper,

R. §talnr¡kcr, and G. Pcarce (Dortlrccht, Thc Ncthcrlands: D. Rcidcl, l98l)'

Karl Popper, "The Propensity Intcrpretation of Probability" British lournal fott)rc Phílosophy of Science l0 (1959):25-42.

Brian S§rms, "statistical I-aws and Pcrsonal Propcnsitics," in PSA 1978, Vol'

2, etl. P. Asqtrith ancl I. tlacking (East Lansing' Michigan: Philosophy of

Scicnce Association, I982).

Ilrian Skynns, Pxtgnulias axl Lnqtiricísit(New IIaven, Conn : Yalc Univcrsity

l)rcss, 1984), chap.3.

John Vcnn, 'flw Ltsgic ttf Chana: (4lh cd.) (New York: Chclsca, 1962)'