Methods of Logic - W. O. V. Quine

METHODSOF LOGIC

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”“I'i°°5 stOF LOGICFOURTH EDITIONW. V. QUINEHsrvartl University PrsssEatttbndge, Massachusetts

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PREFACE This hct-c|l~: undertaltes h-nth tct cunvey a precise understanding nf thefennel cenccpts ef rnedern legic and tc: tievelctp ccmvenient techniques nfftrrmal reasening. ln each successive edititm the heel: has grnwn. Thep-urtiuns that can he passed nver withcrut detriment tn an understanding ctfhasic legical thecrry amnunt new tt: ahuut a third Inf the httctlt. i havehraclteted them sci that they can ccrnveniently he shipped, entirely ctrselectively. fur rturpn-ses nf a shcrrter ccrurse. Hut in my -awn teaching 1have always cnvered the whcrle hcrelt. in its past editictns, in a sernesterceurse. ln the secend editiun. I959“. the main imp-rctvernents were twu: a rn-erereadahtc prctctf nf the scrundness uf my systcrn ef natural deducti-an and. inan appcndits, prcsnfs ctf cnmpleteness and l_i:iwenheim's thenrem. The third editinn, IQTE, was a mute drastic departure: a new heels hyhalf. Precrf precedures and decisiun |:-recedurcs prctliferated. nffeting asheaf nf alternatives. Appreaching lcrgical structure thus frnm a pluralityctf angles cctnduced, I felt. tu depth uf understanding. l‘v'ly main trrethttd. as I called it. in quantificatien thenry was ren1arlta~hly simple and easiiyjustified and it was strategic as a hase frnm which tndevelnp the array ef alternative methc-ds. Ftlse it admitted ctf a relativelycasy cumpleteness prctctf. Accordingly my system nf natural deduction,which had dctminated the first twu cditinns, was demctted in the thirdeditinn and accctrded t:-niy hrief treatment in the c-rnissihie sectnr. He-spttnding new tu teachers whn tilted that system and regretted its demu-tien, l have in this new editien restered it tcr full treatment, th-nugh stillmatting it nrnissihte. rttnnthcr majnr departure in that third cditinn was the pestp-unement efquantitiers hy dcveleping mnnadic lugic first alctng Enelean lines, theughwitheut assuming classes. ln this new editicrn I have pressed the ideafurther. hy eapleiting alsn the set-thectretic ncttatien ctf class ahstractienwithtnut assuming classes. .-flthstracts emerge with all the innn-cence efrelative clauses. The device pays its way in the treatment af suhstituticrn,it clarifies the prctneminal character cit’ huund variables. and it eventuatesin a philnsnphically agreeable pcrslltietive -nu classes. Further illuminatinn vii

v:||1 Prefaceof the hound variahle is undertalsen in an umissihle chapter on oomhina-tory functors. There have hcen patches where points explained in my earlier ho-oltsneeded to he explained again in essentially the same old way. At thesepoints I have adapted examples and expository passages from Math-entntlcnl Logic, Elernentnry Logic, and U -5'e.rttt'dtt do nova ldgicn, prefer-ring not to obscure genuine points of contact hy net!‘ hoe shifts of examplesor of phrasing- But the places are few. Chapter t draws in part on so of titSenrttfo and Chapter 5 draws in part on §2 of Mnthentntienl Logic-. lnChaptela 4, E, l4, and 3 l, examples are borrowed from Elementary llogichut are handled differently. Uut of loyalty to loyal teachers of this hoolt, l have been reluctant todepart from the notations of truth functions and quantifiers that linheritedfrom Whitehead and Russell. In this edition at last I do so. l switch toother symbols that are increasingly in vogue for the conditional, thehiconditional, an-d universal quantification. 1 ntalte the move not just as aresponse to fashion. hut for technical reasons that will he noted as thesymhols emerge. Partly because of the changed notation, the copytediting of this editionwas unusually exacting. I am grateful to lt-Latarina F-lice of Harvard Lini-versity Press for a perceptive and meticulous joh.

CONTENTSlntrodnction FPART I. TRUTH FUHCTIOHS Negation, Conjunction, and Alternatiun ii‘ Truth Functions Hi The Conditional El -lit-5.4-|'l“-.'tI—' Grouping L‘? Tmth-‘v'alue Analysis 3.? II- Consistency and \"v’alidity to lmplication ttj i!\"-.l-t:l\"t_.I‘l| Words into Symhols 53 ‘El Equivalence fit?ll] Alternational hlormal Schemata esll Simplification Pa‘II Duality Ft?-'l3 Axioms S5PAIIT ll. GENERAL TERMS AND OUAHTIFIEISltl Categorical Statements P3 H2’15 ‘v’enn‘s Diagrams 5-‘Sts Syilogisms H12ll’ Limits of These lvlctho-ds ltltit1S Boolean Schemata l Hll-ll Tests of ‘v'alidity l 21\"It]- Some Boolean lncitlentals IE3El. The Hound ‘v'ariah|e l.-ll’11- Quantification llitiE3- Rules of Passage- lvlonadic SchemataE4. Prenexity and Purity I-=15E5. ‘Validity Again i'.‘5=tEh- Suhstitution rso

Eran ten rs‘PART Ill. GENERAL THEORY OF OUAHTIFIEATIOH 2?. Schemata Extended tsr ES. Substitution Extended l'?e' Z9. FureExistentia1s I32 SD. The lvlain lvlethod 1'91‘) Lil. Application FPS 32- Completeness EH3 33. Lrlwenheimls Theorem Ell? 34. Decisions and the Llndecidahle El} 35. Functional Normal Forms ZIP 3-Ei. Herbrand‘:-2 lvlethod £'I.'st ST. Other Methods for Validity 23.? SS. Deduction E3? 39. Soundness .E'=t'.5 -tit. Deductive Strategy L‘-13PART l\"l|f. GLIMFSES BEYOND4t. SingutarTerrns 2.59 2?-S42. Identity 2153-13. Descriptions 2?-t-t4. Elimination ofSingularTerms45. Elimination of ‘variables BS3st-E. Classes E‘-SS47. I‘-lumber zsvtst-S. Axiornatic Set Theory JEDPartial answers to Exercises .i't’.t5Bibliography 325index 5'2?

INTRODUCTION Logic, iilte any science, has as its business tI'te pursuit of truth. Whatare hue are certain statements; and the pursuit of truth is the endeavor tosort out the true statements from the others, which are false. Truths are as plentiful as falsehoods, since each falsehood admits of anegation which is true. But scientific activity is not the indiscriminateamassing of truths; science is selective and seeits the truths that count formost, either in point of intrinsic interest or as instruments for coping withthe world. Strictly spealting, what admit of truth and falsity are not statements asrepeatable patterns of utterance, but individual events of statement utter-ance. For. utterances that sound alilte can vary in meaning with theoccasion of the utterance. This is due not only to careless am biguities, hutto systematic ambiguities which are essential to the nature of language.The pronoun ‘l’ changes its reference with every change of spealter;‘here’ changes its reference with every significant movement throughspace; and ‘no-w‘ changes its reference every time it is uttered. The crucial point of contact between description and reality is to besought in the utterance of a statement on the occasion of a stimulation towhich that string of words ha.s become associated. l‘-lot that the statementwill refer to stimulation or sensation; it is apt to refer to physical objects.Language is a social institution, serving, within its limitations, the socialend of communication; so it is not to be wondered that the objects of ourfirst and commonest utterances are socially shared physical objects ratherthan private experiences. Physical objects, if they did not exist, would {totransplant lv'oltaire\"s epigram] have had to he invented. They are indis-pensa ble as the public common denominators of private sense experience. The latest scientific pronouncement about positrons and the statementthat my pen is in my hand are equally statements about physical objects;and physical objects are ltnown to us only as parts of a systematic con-ceptual structure which, talten as a whole, impinges at its edges uponobservation. We have a networlt of statements that are variously Iinltedwith one another and some of which, out at the periphery of the network.are associated more or less strongly with sensory stimulation. Even theperipheral ones are mostly about physical objects: examples are ‘lvly penis in my hand‘, ‘The mercury is at Sfl’. l

2 Methods of Logic A. sensory stimulation elicits some closely associated statement and theassociations then rev erbcrate through the system of statements. activatingat length another peripheral statement whose sensory associations malteus expect some panicular further stimulation. Such, schematically, is themechanism of prediction. when prediction fails, we question the inter-vening networlt of statements. We retain a wide choioe as to what state-ments of the systcrn to preserve artd what ones to revise; any one of manyrevisions will suflice to un matte the particular implication that brought thesystem to grief. Normally the peripheral statements, closely associated to stimulations,are to be preserved from revision once the appropriate stimulations haveoccurred. If revision of the system should become necessary, other state-ments than these are to suffer. lt is only by such an allocation of prioritythat we can hope to claim any empirical content or objective reference forthe system as a whole. There is also, however, another and somewhat opposite priority: themore fundamental a law is to our conceptual scheme, the less liltely weare to choose it for revision. When some revision of our system ofstatements is called for, we prefer, other things being equal, a revisionwhich disturbs the system least. Where the two priorities come into conflict, either is capable of pre-vailing. Statements close to experience and seemingly verified by theappropriate experiences may occasionally be given up. even by pleadinghallucination, in the extreme case where their retention would entail acataclysmic revision of fundamental laws. But to overrule a multiplicityof such statements, if they reinforce one another and are sustained bydifferent observers, would invite criticism. The priority on law, considered now apart from any competition withthe priority on statements verified by experience, admits of many grada-tions. Conjectures of history and economics will be revised more will-ingly than laws of physics, and these more willingly than laws of math-ematics and logic. Ctur systcrn of statements has such a thiclt cushion ofindeterminacy, in relation to experience, that vast domains of law caneasily be held immune to revision on principle. We can always turn toother quarters of the system when revisions are called for by unexpectedexperiences. lvlathematics and logic, central as they are to the conceptualscheme, tend to be accorded such immunity, in view of our conservativepreference for revisions which disturb the system least; and herein,perhaps, lies the “neccssity“ which the laws of mathematics and logicare felt to enjoy.

.[rrtrneftrr'tion 3 ln the end it is perhaps the same to say, as one often does, that the lawsof mathematics and logic are true simply by virtue of our conceptualscheme- For. it is certainly by virtue of that scheme that those laws arecentral to it: and it is by viroie of being flies central that the laws arepreserved from revision at the expense of statements less strategicallysituated. But it mu st now be remarlted that our conservative preference for thoserevisions which disturb the system least is opposed by a significant con-trary force, a force for simplification. Far-reaching revision of the funda-mental laws of physics was elected in recent decades, by considerationsof simplicity, in preference to the welter of orf hoe subsidiary laws whichwould otherwise have been needed to accommodate the wayward experi-ences of Michelson and It-'lorley and other experimenters. Continued ex-periment “confirmed“ the fundamental revisions, in the sense of in-creasing the simplicity differential. lvlathematical and logical laws themselves are not immune to revisionif it is found that essential simplifications of our whole conceptual schemewill ensue. There have been suggestions, stimulated largely by quan-daries of modern physics, that we revise the true-false dichotomy ofcurrent logic in favor of some sort of tri- or rt-chotomy. Logical laws arethe most central and crucial statements of our conceptual scheme. and forthis reason the most protected from revision by the force of conservatism;but, because again of their crucial position, they are the laws an aptrevision of which might offer the most sweeping simplification of ourwhole system of lrnowledge. Thus the laws of mathematics and logic may, despite all “necessity,”be abrogated. But this is not to deny l.l'lElI such laws are true by virtue ofthe concepntal scheme, or by virtue of meanings. Because these laws areso central, any revision of them is felt to be the adoption of a newconceptual scheme, the imposition of new_meanings on old words. l\"-losuch revolution, by the way, is envisaged in this boolc; there will benovelties of approach and technique in these pages. but at bottom logicwill remain unchanged. l have been stressing that in large part our statements are linlted onlyremotely to observation. It is only by way of the relations of one state-ment to another that the statements in the interior of the system can figureat all in the prediction of experience, and can be found deserving ofrevision when prediction fails. blow of these relations of statements tostatements, one of conspicuous importance is the relation of logical im-plication: the relation of any statement to any that follows logically from

4 Methods of Logicit. If one statement is to be held as uue, each statement implied by it mustalso be held as true: and thus it is that statements intemal to the systemhave their effects on statements at the periphery. But for implication, our system of statements would for the most partbe meaningless; nothing but the periphery would rnalte sense. ‘fer impli-cation is not really an added factor; for, to say that one statement logicallyimplies a second is the same as saying that a third statement of the system.an ‘if-then‘ compound formed from the other two, is logically true or\"valid.\" Logical truths are statements on a par with the rest, but verycentrally situated; they are statements of such forms as ‘p or notp‘. ‘If pthenp’. ‘lfp andq thcnt;t‘, *lfeverything is thus and so then something isthus and so’, and others more complex and less quickly recognizable.Their characteristic is that they not only are true but stay true even whenwe malre substitutions upon their component words artd phrases as weplease, provided merely that the so-called “logical*' words '=’, ‘or’,‘not’, ‘if-then’, ‘everything, ‘something’, etc., stay undisturbed. Wemay write any statements in the ‘p' and ‘qr’ positions and any terms in the‘thus and so’ positions, in the forrrts cited above, without fear of falsity.All that counts, when a statement is logically true, is its structure in termsof logical words. Thus it is that logical truths are commonly said to betrue by virtue merely of the meanings of the logical words. The chief importance of logic lies in implication, which, therefore,will be the main theme of this be-ol-t. Techniques are wanted for showing.given two statements, that the one implies the other; herein lies logicaldeduction. Such techniques will be developed, for increasingly inclusiveportions of logic, as the boolt proceeds. The objects of deduction, thethings related by implication, are statements; so statements will constitutenot merely the medium of this hoelt [as of most}, but the primary subjectmatter. Strictly spealting, as urged earlier, what admit of trt:tth and falsity arenot the statements but the individual events of their unerancc. However, itis a great source of simplification in logical theory to tallr of statements inabstraction from the individual occasions of their utterance; and thisabstraction, if made in full awareness and subject to a certain precaution,offers no difficulty. The precaution is merely that we must not apply ourlogical techniques to examples in which one and the same statementrecurs several times with changed meanings, owing to variations in im-mediate context. But such examples are easily enough adjusted to thepurposes of logic by some preliminary paraphrasing, by way of bringingthe implicit shifts of meaning into explicit form. [See Chapter S.)

i'rttrc-=rilttt'tt'cm 5 Legie and mathematics were cnupled, in earlier remarks, as jcintiyenjcrying a central pusitinn within the tc-tai system cf disccurse- Lugic ascummenly presented, and in particular as it will be presented in this be-cit,seems tn differ frem mathematics in that in lngic we talit abcatt statementsand their interrelatic-nships, netahty impticatinn, whereas in mathematicswe tall-t about abstract ucnlinguistic things: numbers, functiens, and theliite. This ccntrast is in large part misleading. Lcgicai truths, e.g., state-ments nf the farm ‘ifp and q then t;-\"', are net abc-ut statements; they maybe about anything, depending nn what statements we put in the blanks ‘p’and \"qr'. When we tail-t nbnnt such iugical truths, and when we ettpnundimplicatinns, we are indeed tallting al:-nut statements; but se are we whenwe talit nbnnt mathematical truths. But it is indeed dte case that the truths cf mathematics treat espticitlyef abstract nunlinguistic things, e.g., numbers and functjens, whereas thetruths cf lngic, in a reascnahiy iimited sense cf the werd ‘]ngic*, have nusuch entities as specific subject matter. This is an impurtant diFtereace.Despite this difference, hcrwetrer, lngic in its higher teaches is fcund tcbring us by natural stages intu mathematics- Fer, it happens that certainunebtrusiye eatensiuns uf legical thecry carry us inte a realm, snmetimesalsn called ‘lugic' in a bread sense cf the wnrd, which dc-es have abstractentities nf a special itind as subject matter. These entities are classes; andthe lngical thenry cf classes, er set thenry, prcwes tn be the basic disci-pline nf pure mathematics. Frnm it, as first came te be ltnewn thrcrugh thewnrlt cf Frege, Dedekind, ‘Weierstrass, and their successcirs in the latenineteenth century and after, the whele cf classical mathematics can begenerated. Eiefcrre the end cf the bucllt we shall have ascended thrnughfdur grades bf legic in the narrewer sense, and emerged intc set thenry;and here we shall sec, as ettamples cf the deriyatiun cf classical math-ematics, h-nw the cnncept cf number and yarieus related nntinns can bedefined.



ITRUTH FUNCTIONS g



l NEGATION, CONJUNCTION, END ALTERNATION The peculiarity ef statements which sets them apart frem ether linguis-tic ferms is that they admit ei truth and falsity, and may hence besignificantly affirrned and denied. Te deny a statement is te affirm anetherstatement, itnewn as the negatien er centre.-iicterjtt cf the first. Te deny‘The Taj lvlahal is white‘ is tn afiirm ‘The Taj lvlahal is net white‘. l\"~lntethat this negatien is eppesed te the eriginai net as biactt tn white, but asnnnwhite tn white; it ceunts as true in every case except the case efwhiteness. The cemmenest methed ef ferming the negatien ef statements in erdi-nary language is by attaching ‘net‘ {er ‘dc-es net‘, etc.) te the main verb,as in the feregning e:-tarnple. But if the verb is gevemed by tsemetimes‘er ‘always’, the negatien is farmed rather by substituting ‘never‘, er ‘netalways‘. If the statement is cempc-und and thus has ne main verb, itsnegatien has te be phrased mere elaberately; e.g., ‘it is net the case beththat. . .and that. . Bat, despite such irregulatities cf crdinary language,a little care suffices fer censtracting a clear negatien ef any given state-ment, the guiding censideratien being simply this: the negatien is te ceuntas false if the given statement is true. and the negatien is te ceunt as trueunder any and all circumstances under which the given statement is false. In legical studies it is cenvenient te adept a single sign ef negatien,censisting ef the p-refit ‘ —‘, applied tn statements as wheles. Thus'- tlenes is away)‘ means tlenes is net away‘; the parentheses here servete greup, as a single whele, the statement te which ‘—-‘ is applied. Thesign ‘—‘ might be translated inte werds as ‘it is net the case that‘; briefiyit may be prcneunccd *nnt‘. when a statement is represented as a singleletter ‘p‘, as is cemmnnly dene in legical discussien, the sign ef negatienwill be placed abeve instead ef in frent; thus we shall write ‘,t'-P‘ instead ef‘—p‘ fer the negatien ef ‘p’. instead ef afiirnting each el‘ several statements we can, equivalently.affirrn a single statement which is ltnewn te legicians [in centrast tngrammarians} as the cenjnnctinn ef the given statements. The cenjunctienef twe er mere statements is cc-mmnniy ettpressed in English by linlting 9

ft] I, Truth Fttttcticlrtsthe statements by ‘and’, er cemmas, er a cembinatien ef [ll-tt twe: ‘Semeare hem great, scme achieve greatness, and seme have greatness flttustupen them.‘ ln legical studies it is cenvenient te ettpress the cenjunctiensimply by writing the cempenent statements in ju:-ttapesitien; e-. g. , ‘lfsemeare hem great] [same achieve greatness] tseme have greamess thmst upcntl1em}’—where again the parentheses serve mainly te ntarlt eff the cem-punent statements as wheles. if we thinlt ef ‘_t;I\", ‘qt ‘ , and ‘r‘ as statements,their cenjunctien is represented as ‘pqr‘. The meanings ef negatien and cenjunctien are summed up in theselaws: The negatien cf a true statentent is false; the negatien cf a falsesttttemertt is true,\" a cenjunctien cf statements aft cf which are trae is true,\"and a cenjunctien cf statements net all cf which are trae irfaise- We see immediateiy that *,-5‘, die negatien ef ‘ti’, will he true if andeniy if \"_h‘ is false, hence if and enly if ‘p\" is tme; se there is ne p-eint inwriting a deubie negatien ‘;5‘. ameunting as it decs simply te ‘_a‘. it isequally evident that the cenjunctien ‘pp-\" ameunts simply te ‘p’. Censider new ‘pit;-r)‘. This, being the cenjunctien ef ‘_e‘ and ‘qr’, is tebe true ifand eniy if ‘,n‘ and ‘qr’ are beth trite; and ‘qt-‘ in tum is tn be tt1.teit‘ and nnly it‘ ‘qt’ and 'r‘ are beth tme. Hence ‘_tt{t;-‘t\"l‘ is true if and eniy if‘yr-‘, ‘q’, and ‘r‘ are all true; in ether werds, ‘,t1[t_;'r}‘ ameunts simply te thethree-way cenjunctien ‘par’. ln the same way it may be seen that ‘t[pt_;t}r‘ameunts simply te ‘pt;-r‘. We may therefere drep parentheses and alwayswrite ‘_sIqr‘, viewing this at will as the cenjunctien ef ‘pq‘ and ‘r‘, as thecenjunctien ef ‘p‘ and *.-:_rr‘, and as the cenjunctien ef ‘p‘. ‘e’, and ‘r’.Cenjunctien is, in mathematical jargen, assnciative: internal greuping isimmaterial in tpqr‘, just as in the sum ‘.1: + y + c‘ er prnduct ‘.ryz‘ efarithmetic. Cenjunctien centrasts in this respect with the aritluneticaiup-eratien ef divisien; fer nete that the parentheses in ‘ll -1- {ti + 1)‘ and‘[12 -t- ti] + Z‘ matte all the difference between 4 attd l. While the assnciative law fer additien, multiplicatien, and cenjunctienjustifies suppressing parentheses, their suppressien sheultl net malte usferget that the law is still there and deing im werlt. We may best regardthe netatien ‘pqr‘ efthree-way cenjunctien as an abbreviatien specificallyand arbitrarily ef [say] ‘tpqi.-*‘; then, when in practice we treat ‘par’ alseas ‘F-‘ll-t_l't'l‘-. what is strictly afeet rather is a legical transferrnatien ef‘|[pq]r‘ inte its equivalent ‘gift;-rt‘. nnether respect in which cenjunctien resembles additicn and multipli-catien, and differs frem divisien, is that it is cetnttttttatfve,' i.e.. erder isimmaterial, there being ne need te distinguish between ‘pg-\" and ‘qp‘.

I - l'lt\"t,t?__t,'-ffli'f-t\".I'l'l, Cenjanctfen, aarl sllterriatica ll ‘But cenjunctien was lately ebserved te enjey alse a third cenvenientpreperty, nrtt shared by addilien and multiplicatien; via., ‘p',t.t‘ reduces te‘_ti‘. Cenjunctittn is icletrtpvttent, tn persist in the jargen- Talten tegether,these three preperties ef cenjunctien ceme simply te this: nnce we havean inventery ef all the distinct cempenents efa centinued cenjunctien, nefunher details nf l‘l']E! cunstituticn ef the cenjunctien need eencem us. Having teuched en negatien and cenjunctien, which cerrespend te‘1tet‘ and ‘and‘, we turn new te a third way ef ferming statements ftemstatements. lt is called alterttatien, and eerrespends te the cennective‘er‘, er ‘either-er‘. This eennective is subject in erdinary discnurse tecenfiicting usages. Due sense is the ac-ne.rclnslve,' accerding te which thecempeund is tme sn lnng as at least ene ef the cemprtnents is true. Underthis usage the statement:{Either} lenes is ill er Smith is awayis true if Jenes is ill and Sn1ith is away, true again if lenes is net ill butSmith is away, tme again if lenes is ill but Smith is net away, and falsennly in case lenes is neither ill ner Smith away. The ether sense in which'nr‘ is semetimes used, called the exclusive, cnnstntes the cemp-eund astrue just in case exactly ene ef the cempenents is true. In this sense ef‘er‘, the cempeund becnmes false net enly when the cempenents are bathfalse {lenes neither ill ner Smith away] but alse when the cetnp-nnents arebeth true [Ienes ill and Smith away]. The ambiguity ef ‘er‘ is cemmenly reselved, in rtrdinary usage, byadding the werds ‘er bnth‘ er ‘hut net b-nth‘. Thus the nenexclusive senseis expressible in the unambigueus fashien:lenes is ill er Smith is away er beth,and the eitelusive sense thus:Jenes is ill er Smith is away but net bath. When we are cenfrnnted with ‘p er q‘ by itself, we de net in generalltnew which interpretatien tn assign te it. Elftett the eheice is immaterial,in that either sense weuld serve equally. Fer example, censider the eit- ' l fnllnw Ceeley in preferring this awkward tcI'm tn the mare usual but setttewhatmisleading ‘inclu sive'.

I1 i. Truth Futtcrienrpressien ‘.r Ey‘, i.e-, ‘.r sf.’ y erx = y‘. it makes ne dilference whether‘er‘ here is underste-ed in the nenexclusive er the exclusive sense. Thennly difference between the twe senses eceurs in the case where bed-tcempenents are true; but when the cempenents cencern-ed are ‘.t \"ii y‘ and‘.r = y‘, the case ef jeint truth -tines net atise either in fact er in the mindcf the spealter. lt is a cemmen errer te believe that examples lilte ‘.r -=11 y crx =-= y‘ areclear cases ef the use ef ‘er‘ in the exclusive sense, and in eetnsequence efthis errer there is a tendency te everestimate the rele which the exclusivesense ef ‘er‘ plays in everyday language. The clauses ‘.r st y‘ and ‘x = y‘are, ef themselves, mutually exclusive er incempatihle clauses; but thisincempatihility, far frem establishing that dte eentext ‘it st y ctr x = y‘uses ‘er‘ in the esclusive sense, deprives us ef the ene case in which wemight hep-e tn distinguish between the exclusive and nenexclusive senses-Since the clauses ‘.r sf. y‘ and ‘.r = y‘ are already ef such nature as teexclude each ether, it is immaterial whether we understand ‘er‘ as re-peating this exclusien er net. lf we want te establish indisputable instances ef the esclusive use ef‘er‘, we must imagine circumstances in which the persen whe uses ‘er‘has a p-nsitive purpese ef denying, explicitly within the given statement.the jeint uuth ef lil'i-E cnrnpettents. Such examples are rare, hut they exist.In an example given by Tarslti it is suppesed that a child aslts his father tntalte him te the beach and afterwards te the mevie. The father replies, in atune ef refusal, “We will ge either tn the beach er te the mevie.“ Herethe exclusive use is clear; the father means simultaneeusly te premise andte refuse. But it is much easier te find cases in which the nenexclusiveinterpretatien is ebligatery. Fer example, when it is decreed thatpassperts will be issued enly te persens whe were bern in the ceuntry erwhe are married te natives ef the ceunt|'y, this decs net mean thatpassperts will be refused tn persens whe were bem in the ceuntry and aremarried te natives. lvlest use cf ‘er‘ in everyday language is either cf thistype which admits eniy ef the nenexclusive interpretatien, er ef the typeef ‘x =1: y er.r = y‘, which admits built interpretatiens indifferently. Latin has distinct werds fer the twe senses ef ‘er‘: vcl fer the nenexclu-sive and nut fer the exclusive. In medern legic it is custetnary te write‘v‘, reminiscent ef ‘vel‘, fer ‘er‘ in the nenexclusive sense: ‘p v qt‘. it isthis mede ef cempeunding statements, and enly this, that is called alter-aatlen. when the ambigueus ‘er‘ ef erdinat'y language appears hereafterin the heelt, let us agree te censtrue it in this nenexclusive sense. If

l, lilegatlen, Cenjanctietn. arttl sllterrtatlen l3eceasiens arise where the exclusive sense ef ‘er‘ is really wanted, it iseasy eneugh te express it explicitly: p er q but net beth,er equivalently: Either p and net it urn and net p.i-e., in symbels: arrest The meaning nf alterrtatien, then, is given by this rt.tle: Arr alternatienis true if at least ene e-_,l\" the certtpenents is true, and etlterwlse false.‘Wrtereas a cenju nctien is true if and enly if its cempenents are all true, analternatien is false if and enly if its cempenents are all false. ln ametapher frem genetics. cenjunctien and alternatien may be centrastedthus: in cenjunctien, truth is recessive and falsity derrtinant: in alterna-tien, trttth is dentinant and falsity recessive. Because the explanatien ef alternatien is just the sarue as that ef cen-junctien except fer interchanging the reles ef tt't1tl1 and falseheed, it isevident that the ferrnal preperties ef cenjunctien must reappear as preper-fies ef alternatien; thus alternatien, lilte cenjunctien, is asseciative, cem-mutative, and idemp-t:-tent. We can render ‘t_,u v qt] v r‘ and ‘_c v {qt v r]‘indiflerently as ‘p v q v r‘; we can interchange ‘p v q-‘ with ‘q v pl‘; andwe can reduce ‘_t:r v p‘ te ‘,e‘. All that matters in a centinued alternatien.as in a ccrntinueti cenjunctien, is an inventery ef the distinct cempenents. Theugh the greuping ef cempenents is irrelevant within a centinuedcenjunctien and within a centinued alternatien, it is impertant wherecenjunctien and alternatien are mitted; we must distinguish, e.g., be-tween ‘pq v r‘ and ‘ptq v r)‘. In Cltapter 5 a systematic technique willappear whereby all eemplexes nf cenjunctien, alternatien, and negatiencan cenveniently be analyzed: meanwhile, hewever, it is easy te see inadvance that ‘pq v r‘ and ‘pin v ri‘ are beund te behave in quite unlilteways. Cine clear peinl ef divergence is this: ‘pie v ri‘, being a cenjunc-tien with ‘_rr‘ as a cempenent, cannet be true unless ‘_p‘ is true, whereasii .-fltfter Chapter 3-, this maybe alse wTitteI't ‘P H ti\"-

[.4 l , Tnah Ftrrtcticrns‘pp v r‘, being an alternatien with ‘r‘ as ene cempenettt, will be rrtre snleng as ‘r‘ is true. even if ‘pl’ be false. Greuping is liitewise impctrtam when negatien eccurs in cemliinatienwith cenjunctien er alternatien. We are net liltely, indeed, le cenfuse ‘,|!lt;-\"with ‘— incl‘, net ‘ii v q‘ with ‘— tp v ell‘. fer in the ene case eniy ‘_n‘ isnegated while in the ether case the whele cempeund is negated. Hut whatis less evident is l.l'lEll. we must distinguish alse hetween ‘ - incl‘ and ‘pa’,and between ‘— (pt v qt‘ and ‘ji v q‘- Let us see what these distinctiensare, talting ‘,t:-‘ as ‘penicillin was fiewn in‘ and ‘r;-‘ as ‘a quarantine wasimp-used‘. There are feur pessible situatiens:pry: Penicillin was flnwn in and a qutuantine was impesed.' I Penicillin was net fiewn in but a quarantine was impesed. Penicillin was fiewn in and rte quarantine was impesed.33% Penicillin was net fiewn in ner was a quarantine impcsed.l\"'-lew ‘— t'_,eqtI|‘ denies just the first ef the feur situatiens, and se cemes eutinte in the secnnd, tltird, and feurth. Thus ‘-— Len)‘ is quite different frem‘,fl.-:i‘, which helds in the feurth case unly. his fer ‘ti v ts‘, this heldswhenever ene er beth ef ‘p’ and ‘e’ held; hence in the secend, third, andfeurth cases. We can therefere equate ‘p vs’ with \" Lpei‘. Finally'—t'_p v ql‘ helds in the ene case where ‘pr v q‘ fails--hence in the feurthcase alen-e; se we may equate ‘ -- {_p v t;-fl‘ with ‘git;-“. Se ‘—{pt;-'1‘ decs net ameunt te Tie‘, but te ‘fl vii‘; and ‘—[p v ql‘decs net ameunt te ‘ti v e‘, but te ‘flat’. We may distribute the negatiensign nf ‘— [pq]‘ and ‘— [p v pi‘ ever ‘pt’ and ‘-:3‘ individually eniy en painef changing cenjunctien te alternatien and vice versa.“ at liule refiectien reveals the same relatienship in erdinary language.Clearly jerj‘, er ‘Help and nete‘, may be phrased ‘hleitherp aura‘; andit is scarcely surprising that ‘Helmet p ner t;-‘ sheuld ameunt te‘— tfp v ql‘, the negatien ef ‘Eitherp er t;-\". Again ‘- tpql‘ may be read‘l*-let beth p and qr‘, and frem this it is ne leap te ‘Either netp er nete‘. If we read the negatien sign as ‘it is net the case that‘, the distinctiensef greuping beceme autematic.— [pelt It is net the case that beth _n and q.ea: It is net the case thatp and it is net the case that t;-'.— [p v qt: it is net the case that either pr er a.it v if: it is net the case that p er it is net the case that qr.it T\"|!'|e.-,s.e equival-er'|t:e;5. are called [II-ellt-tnrgan's laws. See Chapter iii.

l, Negatlea, Cenjttacrlen. anti‘ .-tlrerruttiert l5Elf these feur the first and last, we have seen, ceme te the same thing; andsimilarly fer the sccend and third. l'|lSTUli'.lC.fil.. HUTE: The use ef the dash te express negatien. andef jutttapesitien te express cenjunctien, cemes dawn frem Peane [fl-ltlSS—l'ilt.lll and earlier writers- The bar ever the letter was used fernegatien by C. S. Peirce frem lST\"f.'l. The tilde ‘c-‘, a medified ‘n‘, sawseme use as a negatien sign in the nineteenth century and was revived by‘Whitehead and Russell. it has cctnsiderahle currency teday. alse in writ-ings ef my ewn. Because in the present heel: there is much negating efsingle letters, hewever. I have here favered Peirce‘s bar fer its cempact-rtess and pcrspieuity; and, given that, the dash is mere in lteeping than thetilde as negatien sign fer lenger expressiens- The alternatien sign ‘v‘dates frem \"w‘l1itehead and Russell and is new used by almest all legi-cians. Hut the term fer altematien is less uniferm; seme writers, fnllew-ing ‘Whitehead and Russell, use the less suggestive term ‘disjunctien‘.Custem varies in the netatien fer cenjunctien; seme invert the altematiensign fer the purpese, and seme, fnllewing Hilbert [ISIS], use ‘St’- Fernegatien there is a deviant sign “—|‘ that has gained seme currency.EIE RCISESl. Which cf the feur cases: lenes ill, Smith away, Jenes net ill. Smith away. I-rmes ill, Smith net away, lanes net ill, Smith net away malte the statement: lenes is net ill er Smith is net away ceme eut true when ‘er‘ is censtrued exclusively‘? nenexclusively?E. E‘-nnstruing ‘er‘ exclusively er rteneselusively is indifferent te the tmth er falsity ef the statement abeve in certain ef the feur cases- Which cases are these‘?Ii. Cunstruittg ‘,n‘ as ‘penicillin was flewn in‘ and ‘q‘ as ‘the t|t1arantine was lifted‘, distinguish in phrasing between ‘—[p val‘ and ‘p v qr‘.

ttt l’- Truth Fancttans Under what circumstances weuld ene ef these eemp-nunds ceme eut inte and the ether false‘?-‘ll. titre there circumstances in which ‘p vq‘, ‘p vs‘, ‘p vt;l‘, and ‘—l[_e val‘ all ceme eut true‘?2TRUTH FUNCTIONS All that is strictly needed fer a precise understanding ef negatien,cenjunctien, and altematien is stated in these laws:‘,|ft‘ is tme if and eniy if ‘_t:r‘ is false.‘pq- . ..s‘ is tme if and enly if all ef tr I ‘sI -I are tme, J‘_rr va v. . .v s‘ is true if and eniy if ‘p‘, q ,...,--I] -IIi ‘s‘ are net all false.New it is evident frem these laws that negatien, cenjunctien, and altema-tien share the fnllewing impertant preperty: in erder te he able te deter-mine the trtlth er falsity ef a negatien, cenjunctien, er alternatien, it issuflicient te ltnew the tmth er falsity ef the cempenent parts. lt is cc-nvenient te spealt ef truth and falsity as truth values; thus thetruth value ef a statement is said te be truth er falsity accnrding as thestatement is true er false. ‘illltat we have just ebserved, then. is thatthe truth value ef a negatien, cenjunctien, er altematien is deterrrtined bythe truth values ef its cempenents. This state ef affairs is expressed byspealting ttf negatien, cenjunctien, and alternatien as truth jisnctiens. lngeneral, a cempeund is called a truth fartctian cf its cempenents if itstnrth value is determined in all cases by the truth values ef the cempe-nents. lvlere precisely: a way ef ferming cempeund statements fremeempenent statements is truth-fanctiarmi if the cempe-unds thus fermedalways have matching tmth values as leng as their cempenents havematching tr1.tth values.‘‘ l thanlt James rt- Tltnmas fer reperting an errer in my eld phrasing ef this-

Il. Truth lfant-ttans l i‘ The preperty ef truth-furtctienality which is thus enjnyed by negatien.cenjunctien, and alternatien may be better appreciated if fer centrast weexamine a neu-truth-functinnal cempeund;lenes died because he ate fish with ice cream.Even agreeing that the cempenents ‘lenes died‘ and ‘lenes ate fish withice cream‘ are tr1.|e, we may still dispute ever the tmth value cf thiscempeund- The tmth value ef the cempeund is net determined simply bythe truth values nf the cempenent statements, but by these in cempatlywith further censideratiens; and very ebscure these further censideratiensare. fin the ether hand the tmth value cf the cenjunctien: lenes ate fish with ice cream and diedctr ef the altem atien: lenes ate fish with ice cream er dieder ef the negatien: lenes did net dieadmits ef ne dispute whatever encc the truth values ef ‘lenes ate fish withice cream‘ and ‘lenes died‘ are l-tnewn individually. The cempeund ‘pr because q‘ is shewn net te be a truth functien ef ‘p‘and ‘e‘ by the fact that it cemes eut tme when seme truths are put fer ‘_e‘and ‘t;-\" and false when ether truths are put fer ‘_u‘ and ‘t;-‘. In the case ef‘p v 1;-\", ‘pa‘, and ‘pl’, en the ether hand, ene tnre cempenent is as geedas anether and ene false cempenent is as bad as anether se far as the truther falsity cf the cempeund is cencerrted. Any particular tmth functien can be adequately described by present-ing a schedule shewing what n'uth values the cempeund will talte en fereach eheice ef truth values fer the cempenents. Our three basic truthfunctiens themselves, indeed, were summarily se described in the epen-ing lines ef the present chapter. Any unfamiliar feurth t|'uth-ftrnctienalsymbel ceuld liltewise be intreduced and adequately explained simply bysaying what truth values en the part ef the cempenents are te maite thenew cempeund true and what enes are te matte it false. it symbel

lit l . Trrttlt Fattctiens‘excl-er‘ fer the exclusive ‘er‘, e.g., weuld he fully explained by astipularien that ‘pl excl-er q\" is te be false when ‘p‘ and ‘a’ are talten asbeth tme er beth false, and true in the remaining twe cases {‘p‘ true and‘t;r‘ false er vice versa}. This questien new arises: de eur negatien, cenjunctien, and alternatiencenstitutc a sufficie nt language fer all truth-functieaal pnrpeses? -Given anexplartatien ef a new u-uth-funetienal symbel [e.g., ‘excl-er‘), can wealways be sure that the new symbel will be translatable inte eur existingnetatien? The answer is that negatien and cenjunctien are alwayssufficient, witheut even alternatien! E.g., censider again ‘p excl-er t;-‘. This has been explained as false injust the case ta] where ‘p’ and ‘qr‘ are beth tme and the case [bi where ‘pt’and ‘t;-\" are beth false. Therefere ‘p excl-er t;-‘ ameunts simply tn denying,simultaneeusly, ‘pa’ and ‘p-Q‘; fer ‘pt;-‘ helds in case ta] and fltere alene,and ‘_tiej‘ helds in case {bl and there a|ene- Therefere ‘p excl-er q‘ameunts te: - teal — tse-the cenjunctien ef ‘—{pa}‘ and ‘- tpqi‘; fer this cenjunctien simultane-eusly denies ‘,ea‘ and ‘pa’ and nething mere. The cempeund ‘p excl-era‘is false in the twe cases where ‘— Lea] —{p.-1]‘ is false, and true in the twecases where ‘—-{pet - fpqjl‘ is true. Se the symbel ‘excl-er‘ is super-ftueus; cenjunctien and negatien suffice. ln the same way the symbel ‘v‘ ef alternatien itself can be seen te besuperfiueus. The ene case where ‘_tr v a‘ is te be false is the case where‘_ri‘ and ‘a’ are beth false—i-e., tlte case where ‘,r'r-tj‘ helds. Se instead efwriting ‘p v a‘ we may simply deny ‘,t-la‘, writing ‘— tpal‘. These twe simple examples ef translating tmth functiens inte negatienand cenjunctien illustrate a genera] methnd which werl.-rs fer almest anytruth functien. Given a descriptien ef a truth fI.tnctien—i-e-, given simplya schedule shewing what truth values flte cempeund is te talte en fer eacheheice ef truth values fer the cempenents—we can certstruct a truthfunctien eut ef negatien and cenjunctien which answers the descriptien-Tl-te general methnd will beceme evident if illustrated ence mere, thistime with a less simple and mere arbitrary example than ‘excl-er‘ and ‘v‘.This time a certain truth functien ef ‘_e‘, ‘t;-‘, and ‘r‘ is described asfellews, let us say. lt is tn ceme eut true in the five eases: ‘p’ false. ‘ti\" true. ‘r‘ true. ‘p‘ tme, ‘t;-\" false, ‘r‘ tme,

2 , Trtrth Frtrtr'tiens l'i‘ EI-ls -1'“ ‘.1;-‘ true, ‘r‘ false, Hy‘ true, ‘r‘ false, ‘ false, q|, It false, r|. ll false g‘\"1:_s'eft‘ false,and false in the remaining three cases: E PI-II ‘a‘ true, ‘r‘ true, ‘ ‘ false, ‘i;-‘ false, ‘r‘ true. \"‘'es:\"te: true, ‘.;;=‘ false, ‘r‘ false-Hew these three latter cases are the cases respectively where ‘pt;-'r‘ is true.where ‘par’ is true. and where ‘p.-F‘ is true; se the cempeund which weare seeiting is ebtained simply by simultarteeusly negating these threeunwanted cases, in a cenjunctien thus: —tt-\"itl —tti'tlt\"l r (Fifi-tflur cempeund thus denies, explicitly, just these cases in which it was tnceme eut false; in all ether cases it cemes ettt true. Clearly this same methed will werlt fer any example se leng as thereare -seme cases, ene er mere, in which the desired cempeund is te cemeeut false. We thus have a reutine whereby almest any described trtrthfunctien can he written eut in terms ef negatien and cenjunctien. Theenly truth functiens which eur reutine fails te taite care ef are the eneswhich are te be true in all cases, regardless ef the truth values ef thecempenents. These trivial exceptiens call, then, fer separate treatment;and a treatment is straightway ferthceming which is cerrespendinglytrivial. lfeur preblcm is te express a truth functien ef ‘p‘, ‘a‘, ‘r‘, and ‘s‘[say] which will ceme eut true regardless ef what truth values are as-signed te ‘p‘, ‘a’, ‘r‘, and ‘s‘. we may snlve it simply by writing: ' l_pptyrs]-.Clearly ‘ppt;-rs‘ will ceme eut false in all cases, en acceurtt ef ‘pp‘;therefere ‘— fpparsl‘ will ceme eut true in all cases. Se it is new clear that negatien and cenjunctien censtitutc a sufficientlanguage fer all truth-functienal purpeses- Far frem needing ever te addfurther netatiens fnr hitherlu incxprcssible truth functiens, we can evendrep die netatien ‘v‘ which is already at hand. But we shall net drep it,fer it facilitates certain technical manipulatiens [cf Chapters lll—lZ]-

ID I. Truth Furrtrtiarts It shuuld be remarltetl that eunjunetian is really nu less superfluousthan altematien: fur the faet is that an adequate netatien far tmth fune-tiens is ettnstittttetl net nnly by negatien anti eunjunetien, but equally hynegatinn and alternatien. Tu see this it is sttffieient tti ebserye that theeunjttttetittn \"pg\" itself is translatable intn terms bf negatien and altema-tien, as ‘-1.35 ti it)’. This espressien is equivalent ta tat\", in the sensethat it enmes nut true where ‘,1:-*' and ‘q’ are beth true. ant] etherwise false.Fer. ‘— [,5 '-.-' ajl‘ is true if and eniy if ‘p t-' it‘ is false, henee if and ttnly if‘gt’ and ‘ts’ are beth False. aritl henee if arttl eniy if flu‘ atttl ‘is-‘ are hathIFIJB. ln liett nf negatien antl cenjunctien, er negatien antl alternatien, asingle eunneetire ean he tttarle tu suffire—yie., *|‘, eunstruetl as fellaws:‘p Ia’ is ta be true if and ualy if ‘p’ and ‘q’ are nut beth true. ‘p Ia‘arn-tatnts tn what wuultl be ettpresseti itt terms nf er.-njunetinn and negatienas ‘— =l[_ttq]*; but, if we start rather with * ‘ as basic, we ean express ‘ft’ interms nf ‘|‘ as ‘_t:r |p', anti ‘pq* as ‘lfp la} | {pi |q)'. An-ather eentteetiyewhich wuuitl sufliee by itself is ' .1, ’, at ‘neitlter-rte-r*. ‘pi ,|, q’ ameunts tewhat weuld he expressed in terms nf euttjunetitttt anti negatien as *pt‘,l‘;hut, if we start rather with ‘ i ' as haste, we eart ettpress ‘ti’ as ‘ti 1 P’an-t‘se‘as‘ta -l at i ts 1 sit‘-Hl5TURlCHL HUTE: The It:-git: uf alternatien, eettjuneti-an, and nega-tien was in vestigatett systematieai]y in aneient times by the Stttiesfi in theit-'litlt:lie Ages by Perms Hispauus. Duns Seuttts, and ethers,\"* and in mati-ern times mainly by He-ale if iii-1? enwartil] and Sehriitler if I ET? unward].The terms ‘truth value‘ antl ‘truth functien’ are respeetiyely frem Frege[I392] and Whitehead and Russell |[l9]l]}. The reduetiens tn ‘I’ atttl ' ,|. *are clue ta Shelter { l9i3].EIEIICISESI. Cibtain a eurrtpuunti ef ‘pl’. ‘q’, anti ‘r’, using uttiy ettnjuttetittn and negatien, which wiil erime nut true whenever ettaetly twe nf ‘p’, ‘Q’, atttl *r* are true, antl atberwise false.2- Du the same using eniy alternatien and negatien. time way weuld he tu translate the previuus answer. 5 See Lultasiew let, “.T1.\"1.tr Geschiehte,“ eitetl in the Bibti-ugtaphy.

3 . The Cuttditieaui El 3 THE CONDITIONAL Besides ‘arid’ and ‘er‘, anether eutttteetive uf statements which playsan itnpurtaut part in everyday language is ‘if-then‘. Ft statement pf theferm *ifp thet1q' is ealletl arerrtelititmtti. The erimpenent in the pusitiein e-f‘F’ here is railed the antecedent pf the euntlitit:-rial, and the ettmpenertt inthe pesitintt ttf ‘is’ is ealled the etinsequent. A eenjuneti-an ef twe statements is true, we l-tntiw, just in ease hetheurnputtents are truei and an alternatien is true just in ease ene tlr h-lathcempenent-s are true- hiriw under what eireumstartees is a eenditienaltrue? Even tu raise this questien is tu depart frem every-tlay attitudes. Anafli rmatiun hf the furrn ‘ifp then if is eummunly felt less as an atfi rtttatitrnuf a eentlitiuttal than as a euhditiunal aflirntatinn ef the eenset|ttertt.5 If.after we have made sueh att affirmatien, the anteeedertt turns eut true,then we eensid-er eurselves eemmitted te the eunsequent, atttl are ready teael-tnuwiedge errer if it pruves false. If an the nther hand the anteeetlentturns eut te have been false, uur eenditienal aflirmatinn is as if it hadnever been made. Departing frem this usual attitude, hewever, let us thinlt ef eentiitien-als simply as eernpttund statements whieh, lilte e-unjunetiutts and alterna-tinns, adntit as whules ef truth and falsity. Under what eireumstattees,then, shuuld a euuditiunal as a whele be regarded as trtte, anti under whateireutttstanees false‘? ‘Where the anteeedent is true. the ah-rive aeeuunt ufeurrtmtitt attitudes suggests equating the truth value uf the tenditie-nalwith that ttf the eunsequent: thus a eunditiuttal with true anteeedent andtrue eensequent will eeuttt as true, and a ennditienal with tn.le artteeedentanti false enn sequent will eeunt as false. W'l'tere the anteeedettt is false, uttthe ether hand, the adupliun ef a truth value ft:-r the eenditinttal heettrnesrather mare arbitrary; but the deeisitin which pruves must eenvenient is tttregard all eeaditiunals with false antecedents as true. The eandititmat *ifpthen qt‘, su er:-nstrued, is wriltett ‘p -1 q’ and ealled the material rt:-rttt'i~tirinui- lt is ennstrtted as true where ‘p’ and ‘ql are true, alsu where ‘p’ isfalse art-rl ‘is’ true, and alse where ‘p' and ‘if’ are beth false; anti it is ti I am inigiehterl here ta Dr. Philip Fthirielarider- Elsewhere irl this ehapler l draw tl|II-artSeetiun I tif my H-'fr1'F.lI-rt-rtt.rt'it'ttt' Legit\" by |:|errnissien ef the Harvard University Press.

22 i. Truth Furtetienseenstrued as false enly in the remaining case, via., where ‘p‘ is true and‘ts’ false. The sign ‘--i-‘, like ‘v\", is superfluuus- We knew frem Chapter I hewte eensmtct, by means ef cenjunctien and negatien alene, a cempeundwhich shall be false in just the ene case where *p* is true and ‘i;-'* false;vit.. '~ [,nqi}‘- We ceuld dispense with '—t-‘ altegethet, always writing‘— tjpal‘ instead ef ‘p -1- q‘- Tet anether rentteiing, readily seen te eentete the same thing, is ‘ti liq‘- I-lewever, the superflueus sign '—1’ willpreve eventually te facilitate teehnieal manipulatiens. New censider the statement:ll] if anything is a vertebrate, it has a heart.This, te begin with, is net a cenditienal in the sense with which we havebeen eeneemed abeve, fer it is net really a cempeund ef twe statements‘anything is a vertebrate’ and ‘it has a heart’. The ferrn ef werds ‘it has aheart‘ is net a statement, true er false, which can he entertained in its ewnright, and be meeted te be true in case there are vertebrates. Rather, ll]must be viewed as affirming a bundle ef individual cenditienals: lf a is avertebrate, a has a heart: ifh is a vertebrate, it has a heart; and se en. Inshert:[E] He matter what I may be, if .1: is a vertebrate then .r has a heart.But it is irnpertaut te nete that, ef the bundle ef cenditienals which {1}all’-irrns. each individual cenditienal can quite suitably be interpreted as amaterial cenditienal. Fer, if we reflect that the material cenditienal‘p -1- q’ ameunts te ‘-— [pg]!-’, and then rewrite {2} a-ccertlingly, we have: He matter what .1: may he, it is net the ease that .r beth is a vertebrate and tie-es net have tt heart.er briefly:[3] hletlung is a vertebrate and yet dees net have a heart—which dees full justice te the eriginal ll}. Se a geiteraiised eentiitientti,such as (ll, can in full aecerdance with cemmen usage be eenstmed asaffinning a handle ef material cenditienals. Talten as a whele, thegeneralized cenditienal is a tepic fer Pan ll: it lies heyend the present

.:i‘. The Cmniitiettei 23phase ef analysis, which eenecms enty the cetnpnunding ef statementsexplicitly frem hleeltliltc cempenents which are self-centain-ed statementsin tum. .i5t.neth-er use ef ‘if-then’ whieh is certainly net te be censtrued in thefashien ef ‘p -+ q‘ is the cnntrttfztcnmi cenditienal; e.g.:{4} lf Walhurg had attended, the measure weuld have lest.Wlteever aflirms a cenditienal thus in the subjunctive meed is alreadyprepared in advance te maintain alse, uncenditienally, the falsehe-ed efthe antecedent, hut still he thinlts the cenditienal adds serne infermatien.Surely, then, he dees net censider that such a cenditienal is autematicallyverified [lilte ‘p -1- q‘) simply by the falsity ef the antecedent. This ltindef ennditienal is net suhjeet te the earlier retnartt te the efiect that inerdinary usage a cenditienal is drepped frem eensideratien, as empty anduninteresting. enee its antecedent preves false. The eentrafactual cenditienal is best disseeiated frem the erdinarycenditienal in the indicative mend. ‘Whatever the preper analysis ef theeentrafactttal cenditienal may he, we may he sure in atlvanee that itcatmet he truth-functienal; fer, ehvieusly erdinary usage demands thatseme eentrafaetual cenditienals with false antecedents and false een-sequents be tnte and that ether centtafactual cenditienals with false an-tecedents and false eensequents he false. itmy adequate analysis ef theeentrafaetual cenditienal must ge beyend mere t|'uth values and censidercausal eennectiens, er ltinttred relatiettsltips, between matters speltest efin the antecedent ef the cenditienal and matters spelten ef in the cen-sequent. It may he wandered, indeed, whether any realty eehercnt theeryef the centrafactual cenditienal ef erdinary usage is pessihte at all, par-ticu larly when we imagine trying te adjudicate between such examples asthese:If Hizet and verdi had heen cempatriets, Biaet weuld have been Italian;If Bieet and Verdi had heen cetnpatriets, ‘tterdi weuld have been French.The prehlem ef centrafactual cenditienals is in any case a perplexingene,’ and it belengs net te pure legic but te the flteery ef meaning erpessihly the philesephy ef scienee. We shall net recur te it here.i See l\"'iels-en Ciuedtnan, “The prebletn ef eeunterfactual cenditienals. \"

Id l‘- Truth Fttrtr'tt'.en.v Se the material cenditienal ‘p ~+ qr‘ is put ferward net as art analysis efgeneral cenditienals such as ll], ner as an analysis ef centrafactual cen-ditienals such as £4], but, at must, as an analysis er the erdinary singularcenditienal in the indicative meed. Even as an analysis ef such cenditien-als the versien ‘p —t- q‘ [er *— ll:-gt‘) is semetimes felt te be unnatural, ferit directs us te censtrue a cenditienal as true ne matter hew irrelevant itsantecedent may be te its censequent, se leng as it is net the case that theantecedent is true and the censequent false. The fellewing cenditienals,e.g., qualify as true:{5} if France is in Eurepe then the sea is salt.{ti} Elf France is in Australia then the sea is salt,[T] If France is in Australia then flte sea is sweet.l'~le deuht this result seems strange: hut I de net thinlt it weuld be any lessstrange te censtrue |,5]-['1'] as false- The strangeness is intrinsic rather tethe statements £5]-{Tl themselves, regardless ef their tmth er falsity; ferit is net usual in practice te ferm cenditienals eut ef cempenent state-ments whese truth er falsity is already lrnewn uncenditienally. The reasentltis is net usual is readily seen: Wlty affirm a leng statement litre t 5] er ts)when we are in pesitien te affirm the sherter and strenger statement ‘Thesea is salt‘? And why aflirm a leng statement lilte id) er ti] when we arein pesitien te affirm the shertcr and strenger statement ‘France is net inAustralia‘? In practice, ene whe affirms ‘If p then q’ is erdinarily uncertain as tethe truth er falseheerl individually ef ‘p* and ef ‘q’ but has seme reasenmerely fer disbelicving the cemhinatien ‘_n and net qr‘ as a whele. We say: If lenes has malaria then he needs quinine,because we ltnew aheut malaria but are in deubt beth ef lenes’ ailmentand ef his need ef quinine. Only these cenditienals are are werth affinit-ing which fellew frem seme manner ef relevance between antecedent andcensequent—-seme law, perhaps, cennecting the matters which these twecempenent statements describe. But such cennectien underlies the usefulapplicatien ef the cenditienal witheut needing te participate in its mean-ing. Such cennectien underlies the useful applicatien ef the cenditienaleven theugh the meaning ef the cenditienal be understeed precisely as‘ - err

i. The ti?‘-anditteuui 15 The situatj en is quitc similar, indeed, in the case ef the cennective ‘er‘.The statement:France is in Eurepe er the sea is sweetis as little werth affirming as l5}—{'.ll and fer the same reasen: we can savebreath and yet cenvey mere iefermatien by affirming simply ‘France is inEurepe‘. In practice ene whe affirms ‘p er q‘ is erdinarily uncertain as tethe truth er falsehe-ed individually at yr and ef -st, but lJttllB‘v'E5 merelythat at least ene ef the twe is true because ef a law er seme ether manneref relevance cennecting the matters which the twe cempenent statementsdescribe. Yet clearly ne meaning nced be imputed te ‘er‘ itselfbeyend thepurely truth-functienal meaning “net beth false“. The questien hew well ‘p —t- q‘ cenferms te the erdinary indicative‘if-then‘ is in any case ene ef linguistic analysis, and ef little eenscqucncefer eut purpeses. What is impertant te nete is that ‘p -—‘-'- tj‘. the se-calledmaterial cenditienal, is te have precisely the meaning ‘— (pt;-‘It’ {er‘ft v q‘}:, and it will beceme evident eneugh, as we preceed, hew welladapted this cencept is te purpeses fer which the idiem ‘if-then‘ naturallysuggests itself. In particular, as already neted, the material cenditienal isprecisely what is wanted fer the individual instances cevered by a generalcenditienal ef the type [1]. The idiem ‘p if and eniy if q‘, called the hiceitditiettel, ameuntsebvieusly te thc cenjunctien ef twe cenditienals, ‘if p then lg‘ and ‘liqthen p‘. All that has been said regarding the interpretatien nf the cendi-tienal applies atutatis mutundis tn the bicenditienal; whatever use ‘if-then‘ may be put te, and wh atevcr meaning it may be cenceived te have, acerrespending use and a cerrespending meaning must accrue te ‘if andenly if’. when in particular the cenditienal is eenstmed as the materialcenditienal ‘p —-I qr‘, the cerrespending bicenditienal is called the mete-riui hicenditinnal and written ‘p H tj‘. Since ‘p H e‘ may be regardedsimply as an abbreviatien ef ‘lp -1- all tfq -1-pi‘, er ‘- [prjl — leflli. it isevidently false in twe and eniy twe cases: in the case where ‘pl is true and‘q‘ false. and in the case where ‘q’ is tme and ‘p’ false. ln ether werds, amaterial bicenditienal is tme if the cempenents are alilte in truth value{beth true er beth false}, and it is false if the cempenents differ in tmthvalue. The sign ‘Hi, lilte ‘—-i-' and ‘v‘, is dispensable; indeed, we havealready seen that ‘p H ql may be expressed in terms ef cenjunctien and

Eh l‘- Tntth Functic-asnegatiett as ‘— [pd] — lap)’. But, as will appear in due eettrse, each efthese three dispensable signs plays a special pan in facilitating thetechniques ef leglc. Htsteltlettt I'~lt.'JTE= The material cenditienal gees hact-t te Pldle eflvlegara- It was revived in medem legic by Frege [iE'lFt} mid Peircelllidfill. The apprepriateness ef the material versien was vigereusly de-hated in ancient times {see Peirce, 3.4-at ff; Lultasiewice, “Zur Cre-sehichte,\" p. ltd], and has heceme a current tepic uf centreversy aswell. The issue has heen cleuded, hewever, by failure te distinguishclearly between the cenditienal and implieatien [see Chapter T). The sign ‘Ii’ was used by Gergenne fer the cenditienal as early aslltlti, ttm-‘ugh net in the material sense, and was revived hy Peane andused by lhlltitehead and Russell in the material sense. Its use has beenwirlespread in subsequent literature. including previeus ediriens ef thisbe-elt. It had flte disadvantage, hewever, ef leelting lilte a cenverse ef theinclusien sign ‘E‘ that will emerge in Chapter II] and is standard usage inset theery. At last l have switched te l-lilbert’s ‘—+’, which is new widelyused and is mere suggestive- Cenfermably I have switched te ‘H’ fer thehieenditienal. lv'ly sign fer this in previeus editiens, fellewing ‘Whiteheadand Russell, was ‘r-a‘, which will be put te a different use in Chapter lift.EIERCISESl. It was said in a feetnete in Chapter l dtat ‘p er .-zj‘ in the exclusive sense eeuld be written ‘p H d’. Explain why.E. Translate ‘p H q’ inte terms purely ef alternatien and negatien.

4 . Grasping E? 4 GROUPING A censpicueus type ef ambiguity in erdinary language is ambiguity efgreuping. The statement: Rutgers will get the pennant and Hebart will be runner-up if titaymslri is disqualified,e.g., is hepelessly ambigueus in peint ef greuping: there is ne tellingwhether Rutgers’ getting the pennant is suppesed te be centingent upenRaymslti‘s being disqualified. lf se the legical fnrm is ‘p —i-qr’, andetherwise it is ‘qtp -t- rl’, where ‘p’ represents ‘Rxymslti is disqualified’,‘q’ represents ‘Rutgers gets the pennant‘, and ‘r’ represents ‘l-lebart isrunner-up‘. In cemplex statements ef erdinary language the intended greupingsemetimes has te be guessed, as abeve, and semetimes has te be inferredfrem unsystcmatic cucs, as illustrated by the fnllewing example:-[I] lf the new mail-erder campaign dues net brealt the Dripsweet menepely and restere freednm ef cempctitien then lenes will sell his car and mertgagc his heme.The werds ‘if’ and ‘then’ here are helpful in determining the greuping, ferthey frame the cemplex antecedent ef the cenditienal just as clearly as ifthey were parenthcscs- But they de net shew hew much text is intendedfer the censcqucnt ef the cenditienal. Sheuld we step the censequent efthe cenditienal at the last ‘and’, er censtrue it as rttn ning clear te the end’?The prnpet answer is evident at a glance; hewever, let us nete explicitlywhy. The clauses ‘lenes will sell his car’ and ‘lenes will mertgagc hisheme’ have been tclesceped by emitting the rcpetitien ef ‘Jenes will’; andthis afferds cnnclusivc evidence that the ‘and’ here is intended te ceerdi-nate just these twe clauses, rather than reaching farther baclt te include awhele cenditienal as ene cempenent ef the cenjunctien. Se we ltnew thatt‘ ll is tn he censtrued as a cenditienal. having as antecedent:

IE I . Tmtfi Fu.I'tcti'crrts the new rnailscrrtlet campaign tines net brealt the illripsweet munepely and restere freednm ef cempctitienand as cnnsequent: lenes will sell his car and mcngage his heme.But there remains a questien bf greuping within the antecedent: is the‘nnt’ tn gbvem the whele, t:-r is it te gevern just the part preceding ‘and’?Uhvieusly the whele. An-d nete that the ebviuusness bf diis eheice is duetn much the same telesceping device as was ebserved hefere: the werds‘restere freedem bf cernpetitien’ which fnllnw the ‘and’ tnust. becatise bftheir fragmentary character, be eenstmed as cnnrdinate with ‘break the[Jripsweet meneptily’. Sn {ll is a cenditienal nf the ferm ‘— lfpq} —1- rs’,where ‘p’ means ‘the new mail-erder campaign hrealts the Dfipsweetmenepbly’, ‘qr’ means ‘the new mail-erder campaign reslbres ffeedetn bfcbmpetitibn’, ‘r’ means ‘lenes will sell his car’, and ‘s’ means ‘Jnnes willmertgagc his heme’. We all have an eittracmdinary finesse at erdinary language; and t_hus it isthat the eerreetness ct’ the abeve interpretatien at tl] is beund te havebeen mere immediately evident tn all nf us than the reasens why. But anesamjnatien bf the reasbns afferds seme nbtibn ef the serts cut’ unsystem-atic devices whereby erdinary language succeeds in its indicatiens bfgreuping, such times as it succeeds at all. We neted the effectiveness bf ‘if’ anti ‘then’ in marking the bnundariesef the antecedent ef a cenditienal. in similar fashien ‘either’ and ‘er’ maybe used tn marl-t the bcrnndaries bf the first cempenent bf an alternatien;and similarly ‘beth’ and ‘and’ may be used te rtiarlt the beuttdaties nf thefirst cempenent sf a cenjunctien. Thns the ambiguity nf: lenes came and Smith stayed er Rebinsen leftcan, by inserting ‘either’ at the app-repriate peint, be reselved in favbr bf‘pq v r’ nr ‘pin vs)’ at will: Either Jenes came and Smith stayed nr Rnbinsen left, lenes came and either Smith stayed er Rebinsen left. Grasping may alse be indicated in erdinary language by inserting avacueus phrase such as ‘it is the case lhat’, balanced with anether ‘that’ tn

-i‘- Grasping I'llshew ceerdinatien ef clauses. A further device is the insertien ef em-phatic particles such as ‘else’ aftcr ‘er’, er ‘alse’ er ‘furthermere’ after‘and’; such reinferccment ef a cennective has the effect ef suggesting thatit is a majer ene-It is evident by new that the artificial netatinns nf legic and math-ematics enjey a great advantage ever erdinary language. in their use efparentheses te indicate greuping- Parentheses shew greupings unfail-ingly. and arc simple te use- They have the further virtue ef allewingcernple:-t clauses te be tirepped mechanically inte place witheut distertienef clause er ef centcitt- This particular viI1ue has heen ef incalculableimpertance; witheut it mathematics ceuld never have develeped beyend arudimentary stage.Even se. parentheses can be a nuisance. Unless cenventiens areadapted fer emitting seme ef them, eur lenger ferrnulas tend te bristlewith them and we find eurselves having te ceunt them in erder te pairthem eff. Actually twe cenv entiens fer minimizing parentheses have beentacitly in use new fer seme pages; it is time they were stated- [line is this:the cennectives ‘v’. and ‘H-’ are treated as martting a greater brealtthan cenjunctien. Thus ‘pa v r’ is unde-rsteed as having the greuping‘tpql v r’, and net ‘pin v r_i’—as is well suggested by the typ-egraphicalpattern itself. Similarly ‘p v qr’ means ‘p ‘--' iqrl’. ‘pt; -s r’ means‘(pal —-1- r’, etc. The ether cenventien tn which we have been tacitlyadhering is this: the negatien sign is understeed as geverning as littleas pessible ef what fellews it. Thus ‘—[pq}r’ means ‘[— tpqllr’.net ‘—tt_nnlrl’; similarly ‘--tp '--‘air’ means ‘t-\"Ir-1' Vellr’. net’\"* its v i;-‘lei’: and se en. :il|.l'I ausiliary netatien ef dets will new be adepted which will have thectfect ef eliminating nit parentheses. se far as Part l is cencerned. esceptthese directly cennected with negatien. Perhaps this expedient will seemte reduce parentheses beyend the peint ef diminishing returns; actually itstnain value lies in clearing the way fer a new influs ef parentheses in Partll and beyend.[lets are reinfnrcements. They may be thaught ef as a systematicceu nterpart ef the practice in erdinary iangaage. netted abeve. nf inserting‘else’. ‘alse’, etc. Te begin with. if we want tn cenvey the meaning‘pin v r]’ and thus create a greater brealt at the peint ef cenjunctien thanat the peint ef altematien, we shall insert a det at the peint ef cenjunctienthus: ‘p .q v r’. Fer ‘t_’p vteflr’ similarly we shall write ‘p v‘ ta . r’. fer‘pin —-—‘- rl’ we shall write ’i-I - at -1 r’. etc.

31;] I , Truth Ft-|nctt'r:rn.s l\"~ies.t. if at seme eccurrence ef ‘v‘ er ’-\"’ er '-H-’ we want Le create astill greater brealt than is e:-tpressed by the det ef cenjunctien. we shallinsert a det alengsidc ‘v‘ er ‘—1-‘ er ‘H’; thus ‘tp . qr v rl it-1 s’ becturtes‘p . qt vr .s—i- r’. lust as the underted ‘v‘ er ‘—i-’ er ‘H’ marlts a greaterbrealt than the undetted cenjunctien, se the detted ‘v‘ er ‘-1’ er ‘s—1-’marlts a greater brealt than the det ef cenjunctien. The det which is thusadded te reinferce ‘v’ er ‘—r’ er ‘H’ gees en the side where the rein-fereement is needed; thus ‘tp —t- is . r] vs’ becemes ‘p -1* qt - r -it 5’,but ‘p -1- in . r H-sl’ becemes ‘p --I-. at . r I-rs’. r\"!t_aeiI1’iP - ti‘ “Fl \"'1'tp —# qt . rl’. calling fer reinfercement en beth sides ef the central ‘v‘.hecemes‘p .q I-er .v.p --sq ..r’. When we want te create a still greater breait at seme peint ef cenjunc-tien than is espressed by a detted ‘v’ er ‘—'*’ er ‘H’ in the neighberheed,we shall put a deuble det ‘:’ fer the cenjunctien. Wlien we want te createa still greater break. than this at ‘v’ er ‘—i’ er ‘H’. we shall put a deubledet alengside ‘v’ er ‘—+’ er ‘H’; and se en te larger greups ef dets. Whatmight be written fully in terms ef parentheses as: r ‘t inte —* rl \"H in it elrlr.e.g., is written with help ef dets as fellews:ill sv!p.q—+r.++.pvq.r:t. in general thus the cennectives “v\". ‘—t-’. and ‘-H-‘ fare alilte. Anygreup ef dets alengside any at’ these cennectives represents a greaterbreait than is represented by the same number ef dets standing alene as asign ef cenjunctien. but a lesser brealt than is represented by any largergreap ef dets. Parenthcses will centinue te be used te enciese a cempeund gevemedby a negatien sign; the netatiens ‘er [pa-fl’, ‘-— tp v pl’. etc. thus persistunchanged. Dets have ne pewer. ef ceurse, te transcend parentheses; in‘— tp v q . rls’. e.g., the det is pewerless te greup the ‘s’ with the ‘r’. Ei We need it-e parentheses er ether indicaters ef greuping if. with-Lultasiewice, we write each cennective befere the espressiens that itcennects instead ef writing it between. His netatiens fer ‘pt;-‘, ‘p v q‘, ll Test marked eff by the right angles may cenveniently be emitted fer pttrpnses ef asherter ceurse-

4- Greaping 3l’F' —r ii’, ‘n rial ass ‘ti’ were ‘i*1r'a'- ‘deal \"':Ptl\"i ‘Est-\"1 and ‘Hit’-The altematien {E} abeve becemes this in his netatien:{3} nt.ritEl'tpEqrrl£F'tpqrt-The greuping is unique and inevitable. The initial ‘A’ shews that thewhele is an altematien whese first cempenent is ‘s’ and whese secendcempenent is the rest. The ensuing ‘it’ shews that the rest is a cenjunc-tien. The first cempenent ef that cenjunctien is, in view ef the ensuing‘E’. a bicenditienal- Centinuing thus, we elicit the intended structure infull. We have been reeegnizing cenjunctiens ef net just twe cempenents,but ef any number; and altematiens similarly- ln the -Lukasiewica neta-tien this freedem must be curtailed; ‘per’ may be rendered either‘l{pl'-‘Lair’, that is, ‘pl . qr’, er ‘liltipqr’, that is, ’_ttq . r’, the-se beingequivalent; but it must be rendered ene way er the ether, and net simply‘Kpi;-'r’. Cerrespe-ndingly fer ‘ti v qr v r’; it must be rendered ‘Ftp,-‘liar’ er‘Pttltptyr’. Te permit ‘Kpqr’ and ‘rtpqr’ weuld epen the way te ambiguityef the fnllewing sert: ‘lilitpqrs’ ceuld be taken either as ‘lt[ti'tpqr}.t’ thatis, ‘p v q vr . s’ er as ‘ttt_'n.pqr]rs‘, that is, ‘p vq . rs’. This eenfusianweuld matter: the reader can find an assignment ef truth values te ‘p’, ‘q’,‘r’, and ‘s’ that makes ‘p v q v r . s’ true and ‘p v q . rs’ false. =Lukasiewica’s netatien enjeys an arrestingly simple grammar. ‘Whatstrings ef letters censtitutc eeherent fermulas? ‘Epq’ yes; ‘.n.pl‘~lKpq’ yes;{3} yes; ‘,nl(,ul‘slt;-\" nu; ‘tltpliq’ ne. Here is a simple test, due alse te-Lultasiewice. Ceunting frem the beginning ef the fermula, lteep scere efthe eccurrences ef statement letters |[\"p’, ‘rt’, etc.) and ef twe-place cen~nectives t”C’, ‘Pt’, ‘hi’, ‘E’}. The eccurrences ef statement letters eut-number the eccurrences ef cennectives when yeu reach the end ef apreper lerrnula, and net befere. I leave the reader te satisfy himself ef thisrule by esperiment. This netatien eemmands theeretical interest, and it has prnved usefulin the pregramrning at cemputers; but dets and parentheses seem mereperspicueus. Students are tempted te tinker with the det cenventiens with a view teecenemy. They nete, e.g., that ‘p v. q -H-. r v s’ weuld de in lieu ef my‘p 'v: qt t-ti. r v s’; it admits ef eniy the desired interpretatien‘p via H-.r vs)’. since ‘lip v.a -t—1~]r vs’ makes ne sense. Suchecenemies are ill advised. lf an eittra det speeds up the reading, it pays itsway. When elegance is what we want. we can get it in full measure byswitching te the -Lukasiewicc netatien.

32 f_ Trm',f;|: ,FitiI'it|I'l'fE.Il'it.'.i Htsretacat. HDTE: During the fifteenth, sisteenth, and seven-teenth centuries the vincafam was cemmenly used in mathematical writ-ing te indicate greuping. It is an underline er everline. Thus‘p vtq i-at his\" might be rendered ‘p vg Hr_\n.r‘- The tints lites\" with Peane and were taken up by ‘Whitehead and Russell and later legi-|cians, subject te varying cenventiens. The l'_.ultasiewica netatien dates frem iil2il-\"EIERCISES l. Shaw hew the ambigueus statement: Jehn will play er Jehn will sing and Mary will sing ceuld be rendered unambigueus. in each ef twe senses. by tclescep- ing clauses. 2. indicate and justify the apprepriate greuping uf: if they either drain the swamp and reepen the read er dredge the harber. they will previde the uplanders with a market and themselves with a bustling trade.3. Rewrite these using dets: trite vrl —>rl H the —=--sltrv —=-st. —tr-* rele vii —i — ta reli--tt. Rewrite this using parentheses: p —+.qvr ._evqs :H:.arv.,a .av.t :—i-pt.l5. Rewrite the three abeve in Lultasiewica‘s netatien. \" See Tarstti, Legit-_ ii-:-mantles. i'cI'ctamntltemnric.t, p. hit.

5 - Train-False Analysts 33 5 TRUTH -VALUE ANALYSIS In Chapter E a cempeund was said te be a truth functien ef its cempe-nents when its truth value is determined by these el’ the cempenents; andit was ebserved that cenjunctien and negatien censtitutc an adequatenetatien fer truth functiens. In view ef this latter circumstance it is naturaland cenvenient hereafter te eenceive the nctien ef truth functien in apurely netatienal way: the tratlijiineriarrs sf given cempenents are all thecempeunds censtrueted frem them by means esclusively ef cenjunctienand negatien {and the dispensable further eennectives ‘v‘, ‘—r’, ‘-H’).Thus ‘fit’ is a truth functien bf ‘p’, and ’-—[_u v F .s—i- pry] —# r’ is a truthfunctien ef ‘p’, ‘a’, and ‘r’. We alse ceunt ‘p’ itself a tmth functien efP‘ A tmth functien ef letters pi’. ‘ty’, etc-, is strictly speaking net astatement, ef ceurse, since the letters are themselves net actual statementsbut mere dummies in place ef which any desired statements may beimagined. Hereafter the letters ’,n’, ‘a’, etc., and all truth functiens efthem will he called schemata (singular: schema}. lvlere specifically theywill be called truth-firnctianai schemata when it becumes necessary tedistinguish them frem schemata invelving legical devices ef ether thantmth-functienal kind. Schentata are legical diagrams ef statements; theletters ‘p’, ‘if’. etc., by supplanting the cempenent clauses ef a statement,serve te blet eut all the internal matter which is net germane te the breadeutward structures with which eur legical study is cencemed. Hy interpretatien ef the letter ‘p’ {er ‘a’, etc.) may be meant specifica-tien ef an actual statement which is te be imagined in place ef the letter.By interpretatien ef ‘p’ may alse be meant simply specificatien ef a truthvalue fer ‘p’. The twe senses ef ‘interpretatien’ can be used pretty inter-changeably because each actual statement 5 has a specific truth valuefknewn er unknewnl and that truth value is all that matters te the tmthvalue ef any tmth functien efS. at cenvenient graphic methed uf impesing interpretatiens, at the sec-end ef the abeve varieties, is simply te supplant the letters in a schema bythe mark ‘T’ fer truths and ‘.l.’ fer falseheeds. Cemputing then directlywith these marks, we can quickly determine what truth value the whele

34 I. Truth Functiansschema takes en under the impesed inteI‘prctatiens.\"‘ Thus, suppese eurpreblem is te determine the tmth value ef die schema ‘— [pg v pal’ ferthe case where ‘p’ is interpreted as true and ‘+5-\" as false. We simply put‘T’ fer ‘p’ and ‘J.’ fer ‘q’ in the schema, getting ‘—lT.l. v Tll’. But,since ‘T’ reduces te ‘J.’ and ‘.l.’ te ‘T’, this becemes ‘—[TJ. v J.TJ’.Further, since a cenjunctien with false cempenent is false. ’TJ.’ reduceste ‘J.’ and se decs ’.l.T’- Se the whele is new dewn te ’— t’,J. ti‘ .l.]’- But,an altematien ef falseheeds being false, ‘J. v J.‘ reduces ta ‘J.’; thewhele thus becemes ‘.l.’ er ‘T’. This eutcc-me means that eur eriginalschema ‘— Lea ‘v‘ pal’ cemes eut true when ‘p’ is interpreted as true and‘a’ as false. The precess whereby ‘— tfT.l. v T.l.]’ was reduced ta ‘T’ will be calledreselutien. The simple st ef the steps invelved in reselutien, via. neduetienef ‘T’ te ‘J.’ and ef ‘.l.’ te ‘T’, will always be tacit hereafter, we shallnever write ‘T’ ner ‘.l.’, but immediately ‘J.’ and ‘T’, as if the netatien efnegatien as applied te ‘T’ and ‘J.’ censistetl simply in inverting- Theether steps ef reselutien illustrated in the abeve e:-tample were reduetianef ‘T.l.’. ’.l.T’, and ‘.L v J.’ te ‘J. ’. These steps, and all further enes ferwhich there might be eccasien in ether eitamples, may cenveniently becedified in the ferm ef nine rules efreseiutien: ti] Delete ‘T’ as canrpenent afcenjunctfan. [Thus \"l'fT' reduces te ‘TT’ and thence te ‘T’; ‘J. T’ reduces te ‘L ’; etc. Reasenr a cenjunctien with a true cempenent is true er false aceerding as the rest ef it is true er false} fiil Delete ‘J.’ as cempenent ef alternatien. {Thus ‘J. v J. v J.’ reduces te ‘J. v J.’ and thence te ‘J.’; ‘J. v T’ reduces te ‘T’; etc. lteasen: an alternatien with a false cempenent is true er false aceerding as the rest ef it is true er false.) iiiil Reduce a cenjunctien with ‘J.’ as cempenent ta ‘J. ’. fivl Reduce an alternatien with ‘T’ as cutnpeuent ta ‘T’. [v] ilrep ‘T’ as antecedent cfa cenditienal. flieasen: a cenditienal with uue antecedent is true er false aceerding as the cunsequent is true er false.) {vi} Reduce a cenditienal with ‘J.’ as antecedent, at ‘T’ as can- \" We need net fumble fer a pie-riunciatien ef ‘J.’ cu-erdinate with the prenunciatiun‘tee’ ef ‘T’, fer the werds ‘true’ and ‘false’ themselves are shert eneugh te serve cenve-niemiy as prenuneiatiens ef the twe signs. Flefen: depleting my preference ef ‘J.’ tu theinitial ‘F’ trf ‘false’ , nete the urgent need ef ‘F ’ fur ether purpuses in Farts H—l’tf.

.5 . Truth-l-\"at’ue slnniysis 35sequent. ta ‘T’- [Thus ‘T H T’, ‘J. —* T’. and ‘J. -1- J-’ reduce tu‘T’ fl {vii} tjfa canditt'anai has ‘J. ’ as rranseqruent, reduce the whale ta thenegatien rgf the antecedent. [viii] [trap ‘T’ as cempenent cf a hicandirienai. [Thus ‘T H T’reduces te ‘T’, and ‘T H .l.‘ and ‘J. H T’ reduce tn ‘J.’.l flit} Drap ‘J. ’ as cempenent cf a bicenditienal‘ and negate the etherside. {Thus ‘J. H J.’ reduces tn ‘T’, and ‘T H J.‘ and ‘J. H T’ re-duce te ‘J. ’.l Set up aceerding te these rules, eur eriginal esample ef reselutienameunts te ne mere than this:— {TJ. v LT} [changing ‘TJ.’ and ‘J.T’ each te ‘J.’ by {ll er tiiil] {changing ’.l. v J.’ te ’.l.’ by [ii]} —f.l. v J.) T Tuming te a mere elaberate esample, let us determine the truth valueef ‘pa v pr .——i-. a H r’ fer the case where ‘p’ and ‘a’ are interpreted asfalse and ‘r’ as true.J.J. v T.l. .—:-. J. H T {changing ‘.t H T’ te ‘J.’ by {viiil er [i:|t]’,| J__l_ v TJ. .—s- 1 {by {viii} - t’.l..l. v TJ.} |[by [iii] twice} —{J. v J.) {changing ‘.l. v J.’ tn ‘J.‘ by [tit] TThus ‘pa v pr .—r. a H r’ cemes eut true when false statements are putfer ‘p’ and ‘qr’ and a true ene fer ‘r’. Let us feign centact with reality by censidering an actual statement efthe ferm ‘pa v pr .—r. a H r’;ll} if either the resident and the deputy resident beth resign er the resident neither resigns ner es prises the charge d’ajl\"aire.r. ifl either case the deputy resident will resign if and erdy if the resident eitpeses the charge d’afi\"aires.‘tit-\"l1at we have feund is that ill cemes eut true in the case where neitherthe resident ner the deputy resident resigns and the resident eitpeses thecharge d’afl\"aires.

3+5 .!’. Tnuh Funtctians We have evaluated the schema ‘pa v pr .H. a H r’ fer ene interpre-tatiuu: ‘p’ and ‘a’ as false and ‘r’ as true. There remain seven etheritttet'pretat.iens that might be censidered: ‘p’, ‘a’, and ‘r’ all u'ue. ‘p’ attd‘qr’ true and ‘r’ false, ‘p’ and ‘r’ true and ‘a’ false. and se en. The eightcases can be systematically eltplered, with evaluatien ef the schema fereach case, by the fellewing methnd. First we put ‘T’ fer ‘p’, leaving ‘qr’and ‘r’ unchanged, and make all pessible resalutiens by ti]-{is}:Ta v J.l‘ .—i-. a H r {changing ‘Ta’ te ‘q’ by [ijt] {changing ‘.l.i-‘ ta ‘J.’ by fiiiftlq v it .—1+_ q H r a v J. .—+. q H r {changing ’tjI' ‘ti’ .l.’ ll} ’-tf’ by fii]] lg -at-, q H rThen we put ‘T’ fer ‘qr’ in this result and reselve further:T H. T H r tlltt it'll THr {by tjviiiil rWe have new feu nd that whenever ‘p’ and ‘if’ are beth interpreted as true,eur eriginal schema reselves te ‘r‘—hence becemes true er false aecerd-ing as ‘r’ is true er false. This dispeses ef twe ef the eight cases. Heat wereturn te eur intermediate result ‘q H. i;-' H r’ and put ‘.l.’ fer ‘if’:.l. —-1. .L H r tbs II‘-‘ill TThis shews that eur eriginal schema cemes eut uue whenever ‘p’ isinterpreted as tme and ‘q’ as false. regardless ef ‘r’. This iiispeses ef twemere ef eur eight cases. New we ge all the way back te eur eriginalschema and put ‘J.’ fer ‘p’: .l.q v Ti\" .—t-. q H r .L v Tr .—-+. is H r {by [iii}} Tr -s. q H r {by inn r —-1-,p Hr {by [iiiPutting ‘T’ fer ‘r’ here and reselving further, we have:

5 . Trttt‘-h— l”aiae el n-alysis 3’l'J. H. a H T ti?-it trill TThis shews that eur eriginal schema cemes eut hue whenever ‘p’ isinterpreted as false and ‘r’ as true, regardless ef ‘a’. Twe mere cases aredispesed cf- Finally we ge back te ‘t —i-. q H r’ and put ‘.l.’ fer ‘r’:T —-t-. qr H J. its tvll -it H 1- flI\"l\" lilll llSir whenever ‘p’ and ‘r’ are beth interpreted as false, eur schema reselveste ‘ii’-hence becemes false er true aceerding as ‘qr’ is interpreted as treeer false. The feregeing analysis might cenveniently have been carried eut in asingle array as fellews: sever-—i-setTtyrtt,l.l‘.—!-.a-t—rr J.q'v'Tr.—s_,-,y+t-trc_|ttt.l,_.i‘_-it-,q-t—i-r _l_‘v'Tt\".-—'r.a=t-ltrqvl. .—r.q Hr Tr-—t-.qHraH.qHr i‘—r.aHrT—1-.THr .l.—-r..l.Hr J.—1-.qHT T—r.uH-J.THr T T qHJ.rrt tarThis is called a truth-value analyst's. The general methed may be summedup as fellews. We make a grand dicheterny ef cases by putting first ‘T’and then ‘J.’ fer seme chesen letter, say ‘p’. The espressiens l.l’ttl5- fermedare the respective headings ef a bipartite analysis- Then we reselve betheitpuessiens. by ti}-tisl. until we end up with ‘T’ er ‘J.’ er seme schema.lf a schema results, we then prnceed te develep, under that schema, a newbipartite analysis with respect te a chesen ene ef its letters. We centinuethus until all end results are single marl-;s—‘T’ er ‘J.’. Each end resultshews what tn.|th value the eriginal schema will take en when its lettersare interpreted aceerding te the marks which have there supplanted them. Actually all intermediate steps ef reselutien are se ebvieus, and sereadily recenstrueted at will. that they may hereafter be left te the imagi-

35 I, Tntrh Furtctt’.-\"hrsnatien. Thus the truth-value analysis abeve weuld in future be eendensedas fellews: pu'v'pi‘-.-s_q+arTr_ir'-.i'J..i‘.-s.t_tH'r lei-'ttTP.—1-.i;-Hr .g—s_,g--t-1-r .F—1-_.gr-HrT—-r.THr J.—1-.J.Hr .l.H.aHT T--I--i;-'-t—r.l. r T T ifT.l. J.T There is ne need always ta cheese ‘p’ as the first letter fer which te put‘T’ and ‘J. ’. it is better te cheese the letter which has the mest repetitiens.if repetitiens there be. and ta adhere te this plan alse at each later stage.Thus it was, indeed. that whereas in the sccend stage en the left side efthe abeve analysis ‘a’ was chesen fer replacement by ‘T’ and ‘J.’, en theether hand in the sccend stage en the right side ‘r’ was chesen. Thisstrategy tends te hasten the disappearance ef letters, and thus te minimisewerk.I A methed ef truth-value analysis that has leng been usual in the litera- tt.re is that ef tri-ah tables. Fer the schema ‘pa var .--1-. qr H r’ last analyst-ed. the truth table is this: pavpl‘ -it ‘1 pa-vpr.—s.qv-sr T T J.l—tl—it—|t—t’i= l——li—t—l-Iit I-rlI-—-tl—l\"t I-t-it-1»-i-|.’i§-', —l'-l—l-l-\"’\=I -|-|t-|—i-+ -ti-t-t-t-I‘ta -l-—l-t—i— -t—tl—l-—l-I—t-I1 -|it-|—-i|-tThe first three celumns e:-thaust the assignments er’ truth values te theletters. The succeeding celumns list the truth values ef pregressivelymere eemplcs cempenents ef the schema fer each assignment te theletters. Each at’ these succeeding celu mus is derived frem earlier celumnsby reasening tantameunt te what l have called reselutien.

5 . Tn.rth- l”a.fue .-tlrrttiysis The uuth table can be censtructed mere cempactly thusP‘ ti‘ fir F art-rTTT.l.J.T J.J..l. _l.J.TTJ.J. TJ..L J.J.TJ.J..L .Ll.J. TJ.J.TTT TJ.}. TJ.].J..i.T J.J.T .l.TJ.T,l.J. TTT.l.J..L ll.T —|—|t—|—|—|—|—|l. J.Tl.. —l—lIi—-l—I~c TTTWe begin this censtmctien by writing eut the schema and inscribing theapprepriate celumns under its single letters and their negatiens. Then wederive the celumn fer building it under the middle ef that cenjunc-tien. Similarly fer ‘pr’ and ‘q H r’. Then we derive the celumn fer thealtematien, building it under the ‘v‘; and finally we build the celumn ferthe whele schema under its main cennective, ‘H’.But the branching netatien ef truth-value analysis that we just previ-eusly arrived at is, we see, briefer still titan this cempacted truth table.The saving increases markedly, mereever, when the number ef distinctletters is increased, as the reader can verify by ertperiment. Fer feurletters. after all, the truth table runs te sir-tteen lines; fer five. thirty-twe.The saving increases further when. as in the nest chapter, the analysisspecifically seeks censistency er validity; fer these questiens are an-swered by truth tabies enly when the tables are nearly finished, whereasunder cur branching preeedure ef tnith-value analysis they eften areanswered early in the game. |\"|l5T'l:’l'RlCJltL HUTE: The pattem ef reasening that the truth table tabulates was Frege’s, Peirce’s, and Schri:ider’s by ISSEI. The tables have been preminent in the literature since lililfl tlsukasiewica, Pest,| \"tlt\"ittgenstein}. The cempact style last displayed abeve is frem my Math- eniancai Lagic. liitlfl.

tlt] I . Truth FanctiattrEllEltCl5E5l. Suppcse they drain the swamp but neither reepen the read ner dredge the harber ner previde the uplanders with a market; and suppesc nevertheless they dc previde dremselvcs with a hustling trade. [Je- termine, under these circumstances, the tmth value cf the statement in Exercise 1 ef the preceding chapter. lvlethed: represent the cem- penents as ‘p’, ‘a’, ‘r’, ‘s’, ‘t’; put ‘T’ and ‘.l.’ apprepriately fer the letters; reselve-1. Suppese they neither drain the swamp net recpen the read but dtat they beth dredge the harber and previde the uplanders with a market. and still de net previde themselves with a bustling trade. What then is the tmth value ef the statement in Exercise Z ef the preceding chapter’?3. lvlake a truth-value analysis cf each ef the schemata: P_“P*'?'» PT’-F\"\"\"fi'~ F\"\"-F\"\"fi'= F”_\"‘-F\"\"\"\"'ii- Present yeur wetk in full, shewing intermediate steps ef reselutien; afterward circle these intermediate lines fer emissien, te shew hew the werk weuld leek in the cendensed style.4. Fer cempatisen perferm twe tt‘t.lth-value analyses ef ‘p Hq . q H r’, first fnllewing and then fleeting the strategy cf eheesing the mest frequent letter. 6 CONSISTENCY END VALIDITY it truth-functicnal schema is called cr:-nsisteni if it cemes eut mic undersame interpretatien ef its leuers; etherwise incensistent. at tmth-fun-ctienal schema is called valid if it cemes eut tn.te under every in-terpretatien ef its letters. The schema ‘pd’, fer example, is censistent, fer