Methods of Logic - W. O. V. Quine

5. tffnnststenry and Faltdhy -11it cemes eut t|11e when ‘p’ is interpreted as true and ‘-3' as false; but it isnet valid, since there are ether iaterpretatiens ef ‘p’ and ‘i;-‘ which malte itceme eut faise. The way te test a truth-functienal schema fer validity aad eensisteneyis ehvieust we carry eut a t|1_|th-value analysis and see whether we get ‘T’in every case [shewing validity) er ‘.l.’ in every case {shewing incensis—tencyft er neither- Twe examples ef validity and ineensistency are respec-t:ive1y ‘p —‘* pi and ‘p_fi':T-1-T P'—*P' l.—1-.l. FF‘ Tl. l_TTT l_ J.‘valid schemata were already espleited at ene peint in the argument efChapter 1, where '— Queers)’ was cited. — taflvrst— [Tlqrs] - Ll. Tqrs} T T In general, ebvieusly, a schema is vaiid if and enly if its negatien isincensistent, and a schema is incensistent if and enly if its negatien isvalid. Thus the negatiens ‘—{p_a]‘ and ‘—{Pfla'rs]‘ ef the incensistentschemata 'p_fi\" and ‘ppq-rs’ are valid, and the negatien '— [p --1 p)’ ef thevalid schema ‘p -1- p’ is ilteensistent. Ft. test ef validity may he stepped shert, with negative eutceme, asseen as we ceme te a case yielding ‘J.’; and a test ef censistency may hestepped, with affirmative eutceme, as seen as we ceme te a case yielding‘T’. Thus the analysis ef ‘pg var .—‘~. q -H r* in Chapter 5 might, if wehad been interested enly in censistency and validity, have heen discet1tit1-ued in this fragmentary state: pg var .—t-.qr-Hr Tqvir .—t-.q1—1-r '-Ei\"_”'-\"-3\" HFT—1.T+t—‘1-r r TLThis much already sulfiees te shew heth that ‘pi; var .—r. q ~t—t- r‘ iscensistent and that it is net valid-

41 I . Truth Fanetfens *\"v'alidity\" is net te be theught ef as a tertn ef praise. When a schema isvalid, any statement whese ferm that schemata depicts is beund te be, inseme sense. trivial. It will be trivial in the sense that it cenveys ne realinfermatien regarding the subject matter whereef its cempenent clausesspeak. The statement:ii} if the Bruins win then the Bruins win,whese ferm is depicted in the valid schema ‘p -1-,t:t\", gives us ne infer~matien abeut the eutceme ef the game: indeed, any ether clause, en artyether subject matter, ceuld be used here in place ef ‘the El-ruins win* withas much and as little effect. Valid schemata are impertant net as an endbut as a means. We shall see in anedter page er twe that simple cases efvalidity afferd shertcuts in the u'utl'|~value analysis ef ether schemata; andwe shall see in the nettt chapter that the determinatien ef certain cemplettcases ef validity is tantatneu nt te determining relatiens ef equivalence andimplicatien between ether schemata. Theugh the statements illustrative ef valid schemata are always trivialin the sense neted abeve, they are net always trivial in the sense ef being,like {I}. recegnisable en sight. vats schemata may run te any length andany degree ef eemple:-tity; and seme even ef mederate length cannet berecc-gnised as valid witheut sttbstantial cernptttatien. The same is true efcensistency. Belew is a schema ef quite mederate cetttplesity which,theugh by ne means reeegnicable as valid en sight, is feund te be valid bymath-value analysis. ' tag.-vprvpr v,|'trsvqrrvr.tTqvTrvJ.rv.i_.vvarvrs Lg.-v.LrvTrvTsvtj,v-vrsqvrvqrvrs rvsvarvrsqv.t.vaTvJ.s qvTval.v'I'.t TvsvaTvJ_s .t.vsvg.l.vT.t- qvq T T sitsTv.i. .l.vT Tvl. J.vT TT TTThe test shews validity, but unimplemented inspectien weuld haveavailed little. Tal-ting advantage ef the really evident cases ef validity arid incensis-tency, hewever, we may speed up eur truth-valtte analyses hereafter.Schemata like ‘q va’ and ‘s it s’, which emerged in the ceurse ef theanalysis abeve, are new lcnet-vn te be valid, and hence te reduce te \"T‘ in

ti- Eetaristenry rtrtrf Validity‘ 43all cases: hereafter, therefere, we may as well agree te reduce any suchresult directly tn ‘T’, witheut further ceremeny. Thus, in place ef thecenfiguratiens: ave .s'lt'.i'TvJ. .l.vT Tv.l. J.vT T T T Twhich appeared in the lewer centers ef the analysis abeve, we shall infuture write simply: ave svs TTIn general, any such patently valid schema may be reduced itrtmediatelyte ‘T’ whenever it eccurs in a truth-value analysis, whether in iselatien eras cempenent ef a lenger fermula. Similarly any patently incensistentschema such as ‘pyi-I‘, ‘pears’, ‘t§pqr‘, etc. may be reduced immediately te'.l.‘ whenever it turns up in an analysis. ‘With these shertcuts in mind, let us analyze a really cemples schema: pvt; .pvq .v,aq :-r—t-qr .:—r.,|sr v,-isTvq.Tvq.vJ.q:H.y.:-v.TrvTr .l.'I|t:;I..l.'th:§||».1|tTt:;|III—'*e'-I—l*-lr'It'.l.F' v-t-PW 'iti'=l‘~\"*=i'-\"“'-Ill q—t-T —il.vqr.r—1~t;-l T \"-is'—*-ti J.The reductien ef the left side te ‘q -1-. r v F‘ preceeded by varieus stepsef reselutien, as usual. Hut in the nest step, using eur new shertcut, weput 'T‘ fer the patently valid ‘r v F‘ and get ‘.1; -1 T‘, which then reselvedinte ‘T‘. Lin the right-hand side ef the analysis, the reductien te'— {ea v .-qr .H- aft’ preceeded by reselutien as usual; then, using eur newshertcut, we put ‘l.' fer the patently incensistent ‘qty’, and get'-t[.l. vq .H q)‘, which reselved in turrt inte '—lq H ell’. Here. usingeur new shertcut again, we put ‘T’ fer the patently valid ‘qt He‘ andcenclude eur wnrlt. Where te draw the line between what is patently valid er incensistentand what is net patently se is quite arbitrary. The ‘— [ea v at .+-1* ql‘ in theright-hand part ef the analysis abeve is itself an incensistent schema, andse might have been supplanted immediately by ‘.l.‘ if its ittcensistency

-1.4 I, Trash Fancrianshad been felt te be sufficiently ebvieus. Similarly the ‘a ——r. r v F’ in theleft-hand part ef the analysis might have heen supplanted directly by ‘T’.Fer unifermity ef classreem werlt, we might limit the eategery ef “pat-ently incensistent” schemata te these twe ltinds: fa] cenjunctiens such as‘apt;-r’, ‘p vq .r . —-tp 'tt'a']’, etc-. in which seme part appeals bethplain and negated as cempene nt ef the cenju nctien, and [bl bicenditienalslil-:e ‘p Hp‘ er ‘—l{arl H qr’. But beware ef’p -115’; it is censistent. We might limit the eategery ef “patently valid” schemata te these tweltirsllst {a] altematiens such as *a vp v a v r’, ‘pa tr‘ r v -tlpql’. etc., inwhich seme part appears beth plain and negated as cempenent ef thealternatien; lb] cenditientds er bicenditienals whese twe sides are alilte,e.g., ‘q Ha’, ‘qr -H-ar’, ‘p vq p it a’. It is enly te such schemata, then that ettr shertcttt is te be applied;these, and eniy these, will be reduced apprepriately te ’.L’ er ‘T’ en sight. Frem the validity ef a schema we may infer. witheut separate test, thevalidity ef any schema which is fermed frem it by saitstinrtian. Frem thevalidity, e.g., ef ‘p v ,tt' we may infer the validity ef the schema‘qr v —|[qr]' which is fermed frem ‘_a vp‘ by substituting ‘qr’ fer ‘p’.This is apparent frem the definitien ef validity. validity ef ‘p v p’ meansthat ‘p it p’ is beund te ceme eut true ne matter what statement be put fer‘p’; se it fellews, as a special case, that ‘ar v — tar)’ will ceme eut t:rttene matter what statement ‘qr’ be made te represent—hence rte matterwhat statements be put fer ‘a’ and ‘r’. Saltstitarien sy‘ schemata fer letterspreserves validity. But it is clearly essential dtat ‘substitutien fer a letter’be censu-ued as meaning unife-mt substitutien fer every eccurrence ef theletter. Frem the validity ef ‘p it ti’. e.g., we are net entitled te infervalidity ef ‘qr v pt’ uer ef ‘qr v — tar)‘. It is permissible te put the sameer different schemata fer different letters, but we must always put thesame schema fer recurrences ef the saute letter. Since inconsistency ef a schema is simply validity ef its negatien. wemay cenclude further that substitutien cf schemata far letters preservesittcertsistertcy. But nete en the ether hand that substitutien cannet bedepended upen te preserve censistency. The mere fact that the meregeneral schema has seme true instances [which is what censistencymeans} gives us ne reasen te suppese that the special case will share anyef the true instances. The schema ‘p v pa’, e.g-, is censistent {as may beverified by truth-value analysis}, hut substitutien ef ‘rr’ fer ‘p’ thereinyields an incensistent schema ‘rt’ vrrt;-'. Similarly, substitutien fer aletter in a nenvalid schema cannet he depended upen te yield a nenvalidschema; it may yield a valid er nenvalid ene.

?. .fmplt't\"att'-art 45EIER CISESl. Test each ef these fer validity by tt'ath~value analysis, e:-tpleiting the new shertcut regarding patently valid and patently ineensistent clauses: s—*-tr -it-e—rsv sitar‘-‘-tl -tl‘tt- s-are -it-s tee. stare -it-e err-it-s tet-2- in each ef the feur schemata abeve, substitute ‘p v a‘ fer ‘p’. This is chiefly an ettercise in adjusting dets te preserve preper greuping.3- “lf a schema is censistent but net valid, thert by ene set ef substitu- tiens we can get a valid schema frem it, and by anether set ef substitutiens an incensistent schema.” ls this tn.|e’l' Justify yeur an- swer.-l-. By negating a censistent schema can yeu get a valid ene‘? a censistent ene‘? an incensistent ene? Illustrate year affirrnative answers. 7 IMPLICATION The mest censpicueus purpc-se ef legic. in its applicatiens te scienceand everyday disceurse, is the justitieatlen and criticism ct‘ inference.Legic is largely cencemed with devising techniques fer shewing that agiven statement dees. er dees net, “fellew legically\" frem anether. Thestatement ‘hie drepped freshman is eligible fer the Hnwdelrt Prise‘, e.g-.fellews legically frem ‘hle freshman is eligible fer the Hewdnin erBechtel Price‘; and l.l'l=E statement ‘Cassius is net beth lean and hungry‘fellews legically frem ‘Cassius is net hungry‘. New the first ef these eveettarnples lies beyend the scepe ef the truth-functienal part ef legic withwhich we are cencerned in Part I, but the secend e:-tample can already betreated here. Frem the peint ef view ef legical theery, the fact that the statement‘Cassius is net heth lean and hungry’ fellews frem ‘Cassius is net hungry’

415 I . Trtrtlt Ftrnctiertsis cenveniently analysed inte these twe circumstances: [alt the twe state-ments have the respective legical ferms ‘— fpal’ and ‘a’ {with ‘Cassius islean’ and ‘Cassius is hungry’ supplanting ‘p’ and ‘a’l: and ibl there are netwe statements which, put respectively fer ‘p‘ and ‘q‘, malte ‘a‘ tme and‘— fpal’ false. Circumstance lb} will hereafter he phrased in this way: '4\"implies ‘— ipal’. In general. ene trttth-functienal schema is said te implyanether if there is ne way ef se interpreting the letters as te malte the firstschema true and the secend false. Wltether a tmth-functienal schema .57, implies anether, 5,, can be de-cided always by talting S, as antecedent and S2 as censequent ef a cendi-tienal, and testing the cenditienal fer validity. Fer, aceerding te eurdefinitien, S, implies .57, if and enly if ne interpretatien ma]tes.‘i', true andSt false, hence if and enly if ne interpretatien falsities the material cendi-tienal whese antecedent is 5, and whese censeqaent is Sty. ln a werd,implicatien is validity cf the cenditienal- Te determine dtat ‘rjr’ implies‘— l.‘_p'tjI'l’, e.g., we checlt the validity ef the cerrespending cenditienal:J-—t—ts'li -=t—=:—l.='-=-at T—=-—lsJ-l T T hlettt let us nete an esample which turns eut negatively. That ‘p v a’dees net‘ imply ‘pa’ is feund thus: ii\"\"\"ii'-_\"l7\"»l'T‘-ta .—-I-Ta ti‘ T J.Dnce having ceme eut with a ‘J.’, we discentinue eur test in the l.r.newl-edge that ‘p ‘via .—s pa’ is net valid: l.e-. that ‘p vq' decs net imply‘pa’. T'l1is result dues net mean dtat ‘p v q .—s pq‘ dees net ceme eut tmeuttder seme interpretatiens ef ‘p’ and ‘a’, net\" dees it mean that ‘p v q’and ‘pa’ themselves de net ceme eut simultarteeusly tn.|e under serneinterpretatiens ef ‘p’ and ‘qt’. The failure ef implicatien means merelythat seme interpretatiens which malte ‘p v gt’ trtlc malte ‘pa’ false; er,what cemes te the same thing, that seme interpretatiens malte‘p v q _~—-+ pa-’ false- Hy reflecting briefly en eur rnetlteds ef testing fer implicatien. valid-ity, and incensistency, ene sees that these feur general laws held:

I’- irt-nsrlicatian -'-IT [ii tltny schema implies itself. til} if ene schema implies a secend and the secend a third then thefirst implies the third- {iii} An incensistent schc ma implies every schema and is implied byincensistent enes eniy. -|[iv’l tit. valid schema is irnplied by every schema and implies validenes eniy- We are telti in the secend half ef {iv} that implicatien, lilte substittttien,trttnsrrtits validity. Substittttien dees se, we saw, because all interpreta-tiens ef the secend schema are interpretatiens ef the first. implicatiendecs se fer a rather eppesite reasen: because all interpretatiens that maltethe first schema true mal-te the secend true. an easy familiarity with simple cases ef implicatien between truth-functienal schemata will be feund te facilitate censtmctien ef preefs ateven as advanced a level ef legic as Chapters SS ff. tltt that stage it willnet be eneugh te be able te answer raised questiens ef implicatien, whichwe can de by truth-value analysis as abeve; we must alse be able te raisethe questiens. We must be able te thinlt up schemata which imply er areimplied by a given schema and premise well as linlts in a prepesed chainef argument. Such pt'educts ef imaginatinn can be checked mechanicallyby n-uth-value analysis, but thiniting thern up is an unmechanical activity.Facility in it depends en grasping the sense ef simple schemata clearlyeneugh te be able, given a schema, te cenjure up quite an array ef fairlysimple variants which imply er are implied by it. Given ‘p v t;-‘, it sheuldeccur te us immediately that ‘p’ and ‘at’ and ‘pa’ and ‘pr -1- a’ imply it anddtat ‘p v a v r’ and ‘p —r q’ are implied by it. C-riven ‘p —1~ a’, it sheuldeccur te us immediately that each ef:tit s- sr- Prat -l—=*tit ttrrstv silt-—rsimplies it and that each ef: tl‘-’s- ti—t.s+ tt—r-st-tr.» ems-‘tris implied by it. Such flashes need net he highly accurate, fer we cancheclt each hunch afterwa-rd by truth-value analysis. Wltat is impertant isthat they be prelilic, and accurate eneugh te spare ettcessive lest metien. He deuht reperteire is an aid te virtuesity in centriving implieatiens.but understanding is the principal thing. Wlten simple schemata are

-=tS l. Trtttlt Ftmctienssufficiently trartmarent te us, we can see thteugh them by the light ef purereasen te ether schemata which must certie eut true if these de. er whichcan net ceme eut tme unless these de. it is well te reflect upen the abeveettamples and succeeding enes until it becemes ebvieus frem the sheermeanings ef signs tltat the implicatiens must held. lilleadirat-ss with implicatiens is aided alse, ne deuht, by ease ef checl-t-ing. tltccerdingly a quiclt implicatien test called the fell sweep will newhe er-tplained which, theugh net general, wetl-ts fer an impertant range efsimple cases. Setne schemata are visibly verifiable by ene and erdy ene interpreta-tien ef their letters. E.g., ‘pill’ cemes eut true when and enly when ‘T’ isput fer ‘p’ and ‘-l_’ fer ‘q’. l‘-lew when .5’ is such a schema, the questienwhether .57 implies a schema S‘ can be settled simply by supplanting ‘p’,‘a’. etc., in .57‘ by the values which malte fer truth ef S , and reselving. Ifwe ceme eut with ‘T’ er a valid schema, then S implies S ’; etherwise net.E.g., te determine dtat ‘pa’ implies ‘p —:-q .—t- r’ we put ‘T’ fer ‘p’ and‘l.’ fer ‘lg’ in ‘p -1- qr .—-1- r‘ and reselve the result ‘T —t- L .—t- r’, getting‘T’- ln particular a fell sweep will settle arty questien ef implicatien en thepart ef ‘p’ er ‘p’. Te find that ‘p’ implies ‘qr -1-pt’ we put ‘T’ fer ‘p’ in‘a —t- p’ and reselve the result ‘a -1- T’ te ‘T’. Te find that ‘p’ implies‘p -s q .-s qr‘ we put ‘T’ fer ‘p’ in ‘p —i- a .—t- q’ and reselve the result‘T —- qr .—\"=- a‘, ceming eut with the valid schema ‘q —-- a’. Te find dtat‘ts’ implies ‘rt ilisltl’. which was the esample ef Cassius, we ceuld havesimply put ‘J.’ fer ‘t;-‘ in ‘— tjpal’ and reselved the result ‘— lp.|.,‘|’ te ‘T’. Seine schemata, en the ether hand, are visibly falsifiable by ene andenly ene interpretatien ef their leuers. E_g-. ‘—{prl‘ cemes eut falsewhen and enly when ‘T’ is put fer ‘p’ and ‘r’; ‘p -1- r’ cemes eut falsewhen and enly when ‘T’ is put fer ‘p’ and ‘l’ fer ‘r’; ‘p vr’ cetttes eutfalse when and enly when ‘.l.‘ is put fer ‘p’ and ‘r’; ‘pr —i- s’ cemes eutfalse when and eniy when ‘T’ is put fer ‘p’ and ‘r’ and ‘L’ fer ‘.t’:, and‘p —-1-. r it s’ cemes eut false when and enly when ‘T’ is put fer ‘p’ and‘-l-’ fer ‘r’ and ‘-t‘. l\"-lew when S ’ is a schema thus falsifiable by ene andenly ene interpretatien, the questien whether a schema S implies 3‘ can besettled simply by supplanting ‘p’, ‘a’, etc. in S by the values which maltefer falsity ef.‘-‘I ’, and reselving. if we ceme eut with ‘J. ’ er an incensistentschema, then S implies S’; etherwise net. Fer, the implicatien can faileniy thteugb tmth ef S where 5 ‘ is false. E.g., te find that ‘p —t- a . a —i-r’ implies ‘p —i-r’ we put ‘T’ fer‘p’ and ‘I.’ fer ‘r’ in ‘p —t- a . q —t- r‘ and reselve the re-

F’- implicatien 4'9sult ‘T -—:-a . a —1- l.‘, getting the incensistent schema ‘ata’. Te findthat ‘p ‘v‘ a . a -1- r‘ implies ‘p it‘ r’ we put ‘l.’ fer ‘p’ and ‘r’in ‘p v a . a —1- r’ and reselve, getting ‘aa’ again. Te find that‘p —1-a . ar —1-s‘ implies ‘pr —1- s’ we put ‘T’ fer ‘p’ and ‘r’ and ‘l.’ fer‘s’ in ‘p —1- a . ar -—1-s’ and reselve. in particular this bacltward variety ef the fell sweep is cenvenientwhen we want tn ltnew whether a schema S implies ‘p’, er ‘p’. Te findthat ‘pa v pa’ implies ‘p’ we put ‘J.’ fer ‘p’ in ‘pa it pa’ and reselve theresult ‘la v la‘, getting ‘J.’. Te find that ‘p va .p v a‘ implies ‘p’ weput ‘J.’ fer ‘p’ in ‘p ‘via .p v a’ and reselve the result ‘l. ‘v‘ a . ll. v a’,getting the incensistent schema ‘ata’- Fell sweeps are pessible enly where the schema which is te de theimplying clearly cemes eut tme under ene and enly ene interpretatien, erelse the schema which is te be implied cemes eut false under ene and enlyene interpretatien. The general test ef implicatien, applicable in everycase, is rruth~value analysis ef die cenditienal; thefall .-tw-eep as eppesedte the fell sweep. implicatien may be made te relate statements as well as schemata.Wlten ene schema implies anether. and a pair ef statements are ebtainedfrem the schemata by interpretatien, we may say by etttensien that the enestatement implies the ether. Thus, besides saying that ‘a’ implies‘-— tpal’, we may malte interpretatiens and say that ‘Cassius is net hun-gry‘ implies ‘Cassius is net lean and hungry‘. Hut it is well here te saymere ettplicltly that the ene statement implies the ether trtrtit-jirnrtienaliy.adding the adverb as a reminder that the schemata which breught the twestatements inte an implicatien relatienship were truth-fu nctienal scbem atarather than schemata ef ltinds which have yet te be talten up in Part ll andbeyend. Truth-functienal implicatien is, in ether werds, the relatienwhich ene statement bears te anether when the secend fellews frem thefirst by legical censideratiens within the scepe ef the legic ef truth func-tiens. The terrns ‘truth-functienally valid’ and ‘truth-functienally incen-sistent’ may be applied te statements in similar fashien.| implicatien, as we have seen, is intimately related te the cenditienal. implicatien helds when and enly when the cenditienal is valid. This impertant cennectien has engendered a tendency ameng writers en legic te adept ‘implies’, eenfusingly, as a reading ef the cenditienal sign ‘-1-’ itself. Then, since ‘p —1- a’ has been esplained as ceming eut trtte whenever ‘p’ is interpreted as false er ‘a’ as true, it is cencluded with an air ef parades that every falsehettd implies every statement and dtat every

Stl i. Tratlt Ftrrt-t\"t!t’a1tt.rt:nttlt is implied by every statement. It is net perceived that ‘—1-’ is at bestan apprettimatien te ‘if-then’, net te ‘implies’. ln erder fully te appreciate the distinctien which I intend between ‘-1-’,er ‘if-flten’, and ‘implies’, it is necessary te becerne clearly aware ef thedifference between use and mentien. When we say that Cambridge ad-jeins Besten we mentien Cambridge and Besren, but use the names‘Cambridge’ and ‘Elesren’; we write the verb ‘adjeins’ net between Cam-bridge and Eesten, b-ut between their names. When the mentiened ebjectsare cities, as here, use and mentien are unliltely te be cenfused. Hut thesame distinctien helds when the mentiened er:-jects are themselves lin-guistic ettptessiens. when we write: The fifth werd ef “The Raven” rhymes with the elevenfltwe me ntien the werds ‘dreary’ and ‘weary’, hut what we use are names efthem. We write ‘rhymes with’ net between the rhyming werds but be-tween their names- We may alse write: ‘dreary’ rhymes with ‘weary’.but here again we are using names ef the rhyming werds in c_|uestien—tltcnames being in this case fertned by adding single quetatien marks. Itweuld be net ttterely untt'uc but ungrammatical and meaningless te write: Dreary rhymes with weary- l\"-lew when we say that ene statement er schema implies anetlter,similarly, we are net te write ‘implies’ between the statements erschemata ceneerned, but between their narnes. In this way we mentienthe schemata er statements, we talit aiseat them, but use their names.These names are usually fermed by adding single quetatien marks.\"\"v’alidity and censistency are in this respect en the sarne feeting withimplicatien; we say that a schema er statement is valid er censistent byappending ‘is valid’ er ‘is censistent’ net te the schema er statement inquestien hut te a name ef it. When en the ether hand we cempeund a statement er schema frem tweethers by means ef ‘if-then’, er ‘-1-’, we use the statements er schematathemselves and net their names. Here we de net mentien the statements er “ When the er.pressien re be named is displayed in an iselatcd line er lines, l malte acelen de the werlt ef single queratien marlts; abeve-

F. i’t't'rpl‘i-t‘-rII’li-t‘-l'l't 5Ischemata. There is ne reference te them; they merely eccur as parts ef alenger statement er schema- The cenditienal: lf Cassius is net hungry then he is net lean and hungrymentiens Cassius, and says semetlting quite trivial abeut him, but itmentiens ne statements at all. The situatien here is the same as withcenjunctien, altematien, and negatien. We have made a peint ef handling ‘if-then’ tmth-functienally. titrnengeur tepics et’ legical analysis, indeed, ne place has been made fer nen-truth-functienal ways ef cempeunding statements. But the fact remainsthat implicatien, as a relatien between statements, imputes intimatestrucmral cennectiens; it invelves far mere than the mere truth values efthe twe statements. This fact cenflicts in ne way with a strict adherence tetmth-functienal ways ef cempeunding statements and schemata, insefaras statements er schemata are re be cempeunded at all. The verbs ‘im-plies’, ‘is lenger titan’ , ‘is clearer than‘, and ‘thy mes with‘ are all en a parse far as the present centrasts are cencemed: they cennect, net statementste ferm cempeund statements, but names ef statements te ferm state-ments abeut statements. H ISTDRICAI. HCl’|’E: The distinctien that was suessed ju st new wasweefully neglected by ‘itihitehead and Russell, whe accerded ‘p —1- a‘ thereadings ‘ifp then a‘ and ‘p implies a’ indifferently. The eld centreversyever the material cenditienal [Chapter S} was, in censeqcence, aggra-vated. The truth functien ‘— tjpajt‘ meets seme eppesitien as a renderingef ‘if-then’; it meets mere, and rightly, as a rendering ef ‘implies’.Cenainly implicatien must be preserved as a streng relatien, dependent upen the suncture ef the related statements and netjust the truth values. The habit ef preneuncing ‘-1-’ as ‘implies’ still persists, and is te be depleted. Partly it is enceuraged by the trivial circumstance that ‘if’ hrealts the werd erder, while ‘implies’ falls inte the pesitien ef ‘—1-’.gatnyene thus tempted sheuld ebserve that ‘-1-’ can be read witheut change ef pesitien as ‘enly if’; see the nest chapter-EIEHCISESl. Determine by truth-value analysis whether ‘p --1-. a s—=1- r’ implies ‘r -1-1-. a —-1-p’ er vice versa-

51 I , Truth Firneriens1. De the same fer these: P Hat -—* se, F —*sr Has-3- Determine whieh ef the feur sehemata: F’-F\"*'?> '1i'-F\"—*\"¢i'> P-i5_\"*i'+ F'“\"'*Fi' imply ‘q' and whieh imply ‘,t3‘. This means eight fell sweeps-4. Determine what implicatiens held between these: _e—1-q, _ettq.—t-r, p—+.qrttr. hlnte that ‘pr —1-. q tt r’, lilte ‘p —+ q’, heeernes false under just ene interpretatien ef its letters.5. Determine what implicatiens held between these: F45 F\"H\"'5.I'= Pl\"?--5. Find as many sehelflata as yeu ean. eentaining ene eeeurrenee eaeh ef ‘p’ and ‘q’ and ne further letters, sueh that eaeh implies ‘pi Alsu find as many as yeu ean which are implied by ‘ti’.T. Determine whether either e-f these statements implies fl'l|E! ether\". The eempany is respensihie if and enly if the unit was an Interpleit and installed sinee January. if the unit was an Interples, then it was installed sinee January and the eempany is respensihle; and if the unit was net an Interple:-t, then it was nut installed sinee January and the eempany is net respensihie. Menlie.-:a'.\" Uhtain schemata representing the tugiea] ferms ef these statements by using ‘pl’, ‘er‘, and ‘r’ fer the enmpenent statements; then test the seherriata fer implicatien. He sure te use ‘p’ fer ene and the same eempenent threagheut heth eemp-tainds and similarly fnr ‘er‘ and ‘r’. Be sure alse te lteep the preper greupings.

3 , H-\"en-is inte Sylmhefs 53 3 WORDS INTO SYMBOLS Legical inference leads frem prsmises—-statements assumed er he-lieved fer whatever reasen—te cenclusiens which ean he shewn enpurely legical greunds tn be true if the premises are true. Techniques tethis end are a primary business ef legic. and have already begun teeccupy eur attentien. But whereas the certncetien hetween premises andeenclusiens is thus greu nded in legic, erdirtarily the premises and cenclu-siens themselves are rtet; and herein precisely lies the npplicerien ef legicte fields ether than itself. The premises and eertclusiens may treat ef any tepics and are ceuched.te begin with, in erdinary language rather flian in the technical ideegraphyef medern legic. it is as an aid te establishing implicatiens that we thenpre-ceed te mutilate and distert the statements, intreducing schematicletters in erder te hring eut relevant slteletal suuctures, and translatingvaried werds inte a few fit-ted symbels such as ‘H-‘ and ‘v’ in erder te gaina manageable ecenemy ef stmcteral elements. The taslt ef thus suitablyparaphrasing a statement and iselating the relevant structure is just asessential te the applicatieri ef legic as is flie test er preef ef implicatien ferwhich that preliminary taslt prepares die way. ftn esample ef hew such paraphrasing reduces varied idierns te uni-fenuity has already heen neted in the netatien ef negatien [see Chapterll. The netatien ef cenjunctien has s similar efiect; fer in erdinary lan-geage cenjunctien is espressed net enly hy ‘aad‘ hut alse by ‘but*, by‘a|thet1gh‘, hy unspelteti punctuatien. and in varieus ether ways. Censid-eratien ef *but‘ and ‘altheugh’ is instructive, fer it brings eut a distiactienbetween what may be called the legical and the rheterical aspects eflanguage. We are liltely te say: lenes is here bur Smith is away,rather than? lenes is here and Smith is away,

5.4 J‘, Trirrh Fr-rrreti'cI.I't.t\"because ef the centrast between being here and being away; er, if thecentrast between ‘lenes is here‘ and ‘Smith is away‘ attains such preper~t:iens as te cause surprise. as it might. e-g., if lenes is net in the habit efceming eseept tn see Smith; we are liltely te say: lenes is he-re altheugh Smith is away-But the circumstances which render th-e cempeund tree are always thesame. via., jeint truth ef the twe cempenents, regardless ef whether‘and’; ‘but’; er ‘altheugh’ is used. Use ef ene ef these werds rather thananeflier may malte a difference in naturalness ef idiem and may alseprcivitle seme incidental evidence as te what is geing en in the speal-;er’smind; but it is incapable ef matting the difference between truth andfalseheed ef the cempeund. The difference in meaning between ‘and’,‘hut’. and ‘altheugh‘ is rheterical, net legical. Lngical netatien, uncen-eerned with rheterical distinctiens. espresses cenjunctien unifermly. Fer a further eitample ef the reductien ef manifeld idiems cf erdinarylanguage te uniferrnity in legical netatiens. censider the idinmatic var-iants ef ‘if-then‘: Ifp then q. p enly ifq, e ifp; -:_t previded thatp, q in casep.The netatien ‘p —1- q’, insefar as it may be admitted as a versien ef ‘if pthen q’ at all, is a versien at enee ef all these variant idiems. Hete that the antecedent ef a cenditienal, cerrespending te the ‘p’ ef‘p —1- q’, is net always the part which cemes first in the vemacular. it isthe part rather that is gevemed by ‘if’ [er by ‘in case’; ‘previdetl that’,erc.l. regardless ef whether it cemes early er late in the cenditienal. Thusit is that ‘p ifq' gees ever inte ‘q ~—1~p‘; net ‘p -I q‘. But whereas ‘if‘ isthus ertlinarily a sign ef the antecedent. the attachment ef *enly‘ reversesit; ‘enly if’ is a sign ef the censequent. Thus ‘p enly if q’ means; net ‘p ifq’, but ‘ifp then if; net ‘q -1-pl’; but ‘p —“=* if. E.g.. “feu will graduateenly if ynur bills have been paid‘ decs net mean ‘If yeur bills have beenpaid yeu will graduate‘: it means ‘If yeu will graduate. yeur bills twill]have been paid’. The reader may have feund *ifp then if awltward as a prenunciatien ef‘p —-sq’; because ef the separatien ef *if‘ frem 'then‘. If se, the abeveebservatien en ‘enly if‘ deserves special attentien; ‘-t-‘ may be read‘enly if’.

15. I-‘Ferns inte .'l’yrnhel,v 55 It is particularly te be neted that ‘enly if’ decs net have the sense ef‘H’, which is ‘ffnnti enly if’. sis the werds suggest, ‘p if and enly if qr‘ isa cenjunctien ef ‘p if q’ and ‘p enly if q’—hence ef ‘qr —-1-p’ andup _‘_ q._Ptmeng the linguistic variants ef ‘ifp then qr’ listed abeve, ene meremight have been included: ‘net p unless qr’. This variant leads tn thefellewing cnrinus reflectien: if ‘net p unless e’ means ‘p --I q‘, and‘p —1- q’ means ‘p v q‘, then ‘netp unless q’ must mean ‘p v qr’, whichmaltes ‘unless’ answer te ‘v’ and henee te ‘er’. Whatever strangenessthere may be in equating ‘unless’ te ‘er’ is precisely the strangeness el‘equating ‘if-then’ te lt is semetimes felt that ‘if-then’ suggests acausal cennectien, er the liltc; and, insefar as it tiees, se alse decs ‘un-less’- Hut when we distill a truth functien eut ef ‘if-then’ we have '—‘*’,and when we distill a truth functien eut ef ‘unless’ we have ‘v‘, ‘er’.The evident cemmutatlvity ef ‘er’, i-e-, the equivalence ef ‘p er q’with ‘qr nrp’, is less evident with ‘unless’. The statements:t[ I} Smith will sell unless he hears frem yeu,{2} Smith will hear frem yeu unless he sellsseem divergent in meaning. Hewever, this divergence may he attributedin part te a subtle tendency in ‘unless’ cempeunds te mentien the earlierevent last when time relatienshlps are impertant. Because ef this ten-dency, we are liltely te censtrue the vague ‘hears frem yeu’ in ll} asmeaning ‘hears frem yeu that he sheuld net sell‘, and in [2] as meaning‘hears frnm yeti that he sheuld have seld’. Hut if we are re cempare ll}and [Ii as genuine cempeunds ef statements, we must lirst render eachcempenent unambigueus and dnrahle in its meaning—if net ahselutely.at least sufllciently te esclude shifts ef meaning within the space ef thecemparisen. Thus we sheuld perhaps revise if I) and [2] te read:Smith will sell unless yeu restrain him,Smith will he reprimanded by yeu unless he sells,and se censider them te be related net as ‘p unless qr’ and ‘q unless p’, butmerely as ‘p unless q’ and ‘r unless p‘. Thus far we have been surveying in a cursery way that aspect efparaphrasing which turns en mere veeahulary- We have been cerrelatingcennective werds ef erdinary language with the cennective symhnls nf

lift l‘- Truth Fnnctienssymbelic legic. The last enample, hewever, has breught te light anetherand subtler aspect ef the task ef paraphrasing; en eccasien we must netenly uttnslate cennectives hut alse rephrase the cempenent clauses them-selves, te the estent anyway ef insuring them against material shifts efmeaning within the space ef the argument in hand. The necessity ef thiseperatien is seen mere simply and directly in the fnllewing enarnple. Thetwe cenjunctiens:['3] He went tn Pawcatuclt and I went aleng,{rt} He went te Sangatuclt but I did net ge alengmay beth he true: yet if we represent them as nf the ferms ‘pp’ and ‘rd’,as seems superficially te fit the case. we ceme eut with an incensistentcemhinatien ‘pi;-re‘. actually ef ceurse the ‘l went aleng’ in {3} must bedistinguished frem the ‘l went aleng’ whese negatien appears in (4); theene is ‘I went aleng te Pawcatuclt’ and the ether is ‘ll went aleng tnSaugatuclt’. When E3] and {4} are cempleted in this fashien they can nelenger he represented as related in the manner ef ’_fI|t|l\" and ‘ref, but enly inthe manner ef ‘pg’ and ‘rs’; and the apparent incensisrency disappears. Ingeneral, the trnstwnrthiness ef legical analysis and inference depends eneur net giving ene and the same espressien different interpretatiens in theceurse ef the reasening. ‘vlielatien ef this principle was ltnewn traditien-ally as the fallacy nfequivecnrien. lnsefar as the interpretatien cf ambigueus ettpresslens depends encircumstances ef the argument as a whele—spealter, hearer, scene, date,and underlying preblem and purpnse—the fallacy ef equivecatien is nette be feared: fer. these bacttgreund circumstances may be espected teinfluence the interpretatien ef an ambigueus espressien unifermly wher-ever the espressinn recurs in the ceurse ef the argument. This is whywerds ef ambigueus reference such as ‘I’, ‘yeu’, ‘here’, ‘Smith’, and‘Elm Street’ are erdinarily allnwable in legical arguments witheutqualilicatien; their interpretatien is indifferent te the legical seundness efan argument, previded merely that it stays the same thmugheut the spaceef the argument. The fallacy ef equivecatien arises rather when the interpretatien ef anambigueus espressien is influenced in varying ways by immediate cen-tettts, as in {3} and [4], se dtat the ettpressien undergces changes efmeaning within the limits ef the argument. in such cases we have terephrase befere preeee-ding; net rephrase tn the extent ef reselving all

ll- Wards inra Symhefs 5'1‘ambiguity, but te the etttent ef reselving such part ef the ambiguity asmight, if left standing, end up by being reselved in dissimilar ways bydifferent immediate centests within the prepesed legical argument. Thelegical cennectives by which cempenents are jeined in cempeunds mustbe theught el’ as insulating each cempenent frem whatever influences itsneighbers might have upen its meaning; each cempenent is te be whellyen its ewn, escept insefar as its meaning may depend en these breadercircumstances which cenditien the meanings ef werds in the cempeund asa whele er in the legical argument as a whele- lt eften becemes evident, when this waming is berne in mind, that acempeund which superficially seems analyaable in terms merely ef cen-junctien and negatien really calls fer legical devices ef a mere advancednature. The statement:{'5'} We saw Strembeli and it was eruptingis net adequately analyzed as a simple cenjunctien, fer the censtmctien‘was . . .-ing’ in the secend clause invelves an essential tempera] refer-ence haclt tn the first clause. A mere adequate analysis weuld censtrue E5}rather as:Eleme mement ef eur seeing Strembeli was a mement ef its erupting,which invelves legical structures talten up in Part ll- The general enterpiise ef paraphrasing statements se as te iselate theirlegical structures has, we have thus far seen, twe aspects: the directtranslating nf apprepriate werds inte legical symbels {cnmprising justtmth-functienal symbels at this level], and the rephrasing ef cempenentclauses te circumvent the fallacy ef equivecatien. blew a third aspect, efequal imp-nrtance with the ether twe when eur esamples are ef any cen-siderable cemplesity, is determinatien cf hew te erganiae paraphrasedfragments preperty inte a structured whele- Here we face the preblem efdetermining the intended greu ping. A few clues te greuping in statementsnf erdinary language have been neted {Chapter -'-ll, but in the main wemust rely en nur geed sense ef everyday idiem fer a sympathetic under-standing ef the statement and then re-thinlt the whele in legical symbels-When a statement is cemples, it is a geed plan te leelt fer the eaterrnesrstructure first and then paraphrase inward, step by step- This pnecedure

jfi I , Truth Funerfan.rhas the deuble advantage cf dividing the pmblem up inte manageableparts. and nf lteepiag the cemplesities ef greuping under centrel. E.g.,censider flte statement:{I5} lf-lenes is ill er Smith is away then neither will the Argus deal be cencluded ner will the directers meet and declare a divi- dend unless ltebinsen cemes te his senses and taltes matters inte his ewn hands.First we seelt the main cennective ef [6]. Reasening as in Chapter 4, wecan narrew the eheice dewn te ‘if-then’ and ‘unless’; suppese we decideen ‘if-then’. The eutward structure ef lb}, then, is that ef a cenditienal; selet us impese just this much structure espticitly up-en [I5], pestpeningminuter analysis. We have:{T} lenes is ill er Smith is away -—s neither will the Argus deal be cencluded ner will the directers meet and declare a dividend unless ltebinsen cemes re his senses and taltes matters inte his ewn hands.It-lest we may censider, as if it were a separate preblem remeved frem (T),just the leng cempeund ‘neither. . . hands’. We decide, let us suppese,fltat its main cennective is ‘unless’. Treating ‘unless’ as 'v', we tum {T} asa whele inte:{S} lenes is ill er Smith is away -tr. neither will the Argus deal he cencluded ner will the directers meet and declare a divi- dend v ltebinsen will ceme te his senses and taite matters inte his ewn hands.New we talte up, as if it were a separate preblem remeved frem {S}, thelengest cempenent net yet analyzed; vis., ‘neither - - . dividend‘. Themain cennective here is clearly ‘neither-ner’. Reflecting then that ‘neitherr ner r‘ in general gees inte symbels as vr’, we rewrite ‘neither . .-dividend’ accerdingly; [S] thus becemes:{S} lenes is ill er Smith is away -1. — [the Argus deal will be cencluded] —t'tlte directers will meet and declare a divi-

S- l-l\"ara‘.t' fl'Il't'II' .Syrrtl':Iaf.t' SS‘dend} v ftehinsen will ceme te his senses and talte mattersinte his ewn hands.Directing nur attentien finally te the varieus shert cempeunds whichremain unanalysetl in lit], we turn the whele inte:[ill] lnnes is ill v Smith is away .--1-: —[the Argus deal will he cencluded} —{the directers will meet . the directers wilt declare a dividend} v. Rehinsen will ceme te his senses . Rnbinsen will talte matters inte his ewn hands.Put schematically. the tetal structure is:llll p va .—r. F —t[.rt} vuv.EXERCISES lustify inference ef the cenclusien: If Smith is away and ltebinsen decs net cntne tn his senses then the Argus deal will net he cencluded frem U5}. lviethed: Find the schema which cerrespends te this cenclu- sien as {ll} dees te (hi; then shew that this schema is implied by til]-I. Etetermine which ef these statements implies which: lenes is nnt eligible unless he has resigned his cemmissien and signed a waiver. lenes is eligible if he has resigned his cemmissien er signed a waiver. lenes is eligible enly if he has signed a waiver. lviethed: Paraphrasc the statements, represent their smtcture schematically, and test the schemata. Shew all steps.3. Paraphrase inward, shewing and justifying each step: If either the Giants er the Bruins win and the Iacltals talte

tstft I. Truth Funerienr secend place, then l’ll recever past lessee and either buy a clavicherd er fly te Barbuda-4. Paraphrase inward, shewing and justifying each step: if the tree rings have been cerrectly identified and the mace is indigeneus, then the Ajn culture antedated the Tula if and enly if the Tula culture was centemperary with er derivative frem that ef the present er-tcavatien.9EQUIVALENCE Twe truth-fnnctienal schemata are called equivalent if they agree witheach ether in peint ef truflt value under every interpretatien ef theirletters, er in ether werds if they agree case hy case under truth-valueanalysis. in anticipatien, varieus cases ef equivalence were netetl inChapters l-Il:‘.c‘te ‘rs.’-ap’‘aasn’d, and ‘rt vr’, ‘rs - t‘ ts ‘r -ct‘.‘pa’ te ‘—t_'p vat‘, ‘p vi;-' .v r’ te ‘p v.a vr’,‘sits’ ts-‘s ‘tr-=\" and ‘-tr‘?--ti‘. '—trr-rt‘ tc '-t-vr’.‘s-rs’ts‘-tvct'sss‘svs‘- ‘—trvst‘ts‘rs‘-‘rt tars’ ts ‘rt -ts -tr -*r1’aad‘-\"ll1‘til*lef'l’- Te test twe schemata fer equivalence, we might malte truth-valueanalyses ef the twe schemata and see if they agree case by case. But thereis anether way which tends te be emier: we may ferm a bicenditienal efdie twe schemata, and test it fer validity. Fer, aceerding te eurdefinitien.twe schemata S, and 5, are equivalent if and enly if ne interpretatienmaltes.'i‘, and SE anlilte in truth value; hence if and enly if ne interpreta-tien falsilies the bicenditienal whese sides are S, and S2. Thus, just asimplicatien is validity cf the cenditienal, se equivalence is validity nf thebicenditienal-

S‘. Eauivrrfcnee ll l Te determine the equivalence ef ‘p . a v r’ te ‘pa vpr‘. e.g-, wecheclt the validity cf the cerrespending bicenditienal?“ p.r,tvr,-t—1-.pqvprT.avr,-i-_'fqvTr l..avr.~=—‘r.l..qv'_Lrqr\",tr,s—r,r;|\"Iv‘r T Tin similar fashien it may be checlted that ‘p’ is equivalent tn each ef:tli ii. rs. arr. stirs. it-sits. nsvrc. rrvs-.s‘~'c- lt has been said that the mest censpicueus purpese ef legic, in itsapplicalien tn erdinary disceurse, is the justificatien and criticism efinference. But a secend purpese, almest as impertant, is transfermatinnnf statements. it is eften desirable te transfettu ene statement inte anetherwhich “says the same thing” in a different ferm--a ferm which is sim-pler, perhaps, er mere cenvenient fer the particular purp-eses in hand.blew insefar as such transfermatiens are justifiable by censideratienspurely nf n\"uth-functienal structure {rather than tttrning upen ether serts eflegical structure which lie beyend the scepe nf Part ll, a technique fertheir justificatien is at hand in eur test nf equivalence ef u\"uth-functienalschemata. Transfermatien, e.g., nf:{El The admiral will speal; and either the dean er the president will intreduce himinte:{3} Either the admiral will speal: and the dean will intreduce him er the admiral will spcal: and the president will intreduce him,er vice versa, is justified by the equivalence el’ ‘p . a v r’ te ‘pp vpr‘,and this equivalence is veriiied by mechanical test as abeve. The state- \" in purse ancc cf the pelicy annnunccd in Chapter 5, all intenriediatc steps cf reselu-tien in this analysis arc left te the reader te fill in- The reasen an intcnnediatc stage‘a v r ,-t-. q v r‘ is shewn in the left-hand part ef the analysis is that the passage frem thisre ‘T’ is net hy reselutien but by the rule ef patently valid clauses, Cltaptct ft.

fifi I- Tnrtlt Feitctinrtsments {El and [3] may. by an erttensinn nf tertnjnnlngy similar tn fliatmade in Chapter Ti. be spnlten nf as truth-funetinnally equivalent. It is evident frnm nur defi nitinns and testing techniques that til Fquivalence is mutual implieatinn.Frem this lavv and ti}--t iv] nf Chapter T, these clearly fnllnw: {ii} Any seherna is equivalent tn itself. {iii} If nne schema is equivalent tn a seenntl and the sccend is equivalent tn a third then the first is equivalent tn the third. tfivl If nne schema is equivalent tn a secnntl then the secnnd is equivalent tn the first. t'l\"'int sn tint implicatinnlj [ti] Valid schemata are equivalent tn nne anether and tn nn nthers; and similarly fnr incensistent schemata. F-ubstitutinn was nbservecl in Chapter ti tn preserve validity. Sinceimplicatien and equivalence are merely validity nf a cnntlitinnal and abicnntlitinnal, it fnllnws that substitutien alsn preserves implicatien andequivalence. Frnrn the equivalence nf ‘p’ tn each nf the schemata in til,e.g.. we may infer by substitutien that 'r‘ is equivalent tn each nf ‘F, ‘rs’,‘F vF‘., \"F vfs‘, etc.; alsn that ‘er‘ is equivalent tn each nf ‘— —t{t:jr]t‘,‘p.-qt\", ‘er v at\". ctc.: and cnnespnndingly fnr any nther substitutienupnn ‘_n* and ‘q’ in f ll. The particular family nf equivalences fltus gener-ated will be used later as a means nf simplifying schemata. Appeal tn substitutien is helpful incidentally in justifying these twecenvenient ways ef describing implicatien in tcrrns nf equivalence: -[vi] 5 1 implies 51 if and nnly if 5 I is equivalent tn the cnnj unetinn nf .'i'1 and 5-_=. {vii} .5\", implies 5, ifand nnly il'Sy is equivalent tn the altematinn nf S, and S3.Tn justify [vi]. nbserve tn begin with that ‘p —1- qt‘ ancl ‘p Hpq‘ areequivalent by truth-value analysis. It fnllnws by substitutinn. then. thatthe cnnditinnal fnrmeti nf 5'1 and E2 is equivalent tn the bicnntlitinnalfnt*m=etl nfS| and the cnnju netinn nlS 1 and .572. Sn, by iv}, that cnnditienaiis valitl if and nnly if this bicnnclitinnal is valitl. Hut validity nf thecnntlitinttal is implicatinn, and validity nl' the bicenditienal is equiva-

P- Equivalence -E13lence; sn {vil fnllnws. The jttstificatinn nf {vii} prneeeds similarly frnmthe verifiable equivalence nf ‘p —+ a’ and ‘q -H-. p 'v' .-;-‘. The fnllewing twn laws have tn dn nnly with implieatinn, but I havedeferred them tn here because they are nbtainetl easily frnm [vi] and {vii}. [viii] .5‘ implies each nl’ 51 and 5'3 if and nnly it'll‘ implies the cnn-junctinn nf .57 , and SE. [is] 5'1 and .5’; each imply S if and nnly it the alternatinn nf 5 I and SEimplies S .Justificatinn nf {viii}: Supp-nse .5‘ implies 5, and SE. Then 5 is equivalent,by {vi}, tn the cnnjunctinn nf .5\" and 51, and alsn tn the cnnjunctinn nf Sand 5;. Sn S is equivalent tn the cnnj nnctinn nf all th tee. But. by [vi], thisls the same as saying that 5 implies the cnnjunctinn nf S1 and 5,. Sn wehave seen that, if .57 implies 5 | and alsn SE, it implies their cnnjunctinn.The cnnverse is trivial. at parallel argument fnr {is}, based nu [vii], is left tn the reader. The lavvs {viii} and tis] atlnrcl a cenvenient eatensinn nl' the methnd nffell swn-np. E.g., tn see whether ‘pa v pr v ail’ implies a given schema,we have nnly [thanlts tn tisl} tn see whether each nf ‘pi;-*, jar‘, and ‘qr’individually implies it: and this may be seen by three fell swnnps. Again,tn see whether a given schema implies ‘p v if . p —1- r . q 'v' r‘, we havennly [thanks tn tviiill tn see whether it implies each nf ‘_a v qr, ‘p -—-1» r’,and ‘q v F‘; and this again may be seen by three fell swnnps. We must nnt bnpe tn build {viii} and t_' is] nut intn a square by suppnsingthat 5 implies rm-e nr the nther nf 5'1 and .5‘, whenever 5 implies theiralternatinn; nnr, again, tbat 5, nr S; implies 5' whenever their cnnjunctinnimplies .5’. After all, ‘_n' implies ‘pa v pa‘ withnut implying either ‘lat;-‘ nr‘pa’; and ‘p’ is implied by ‘p v q . p vi?‘ but neither by \"p vq‘ nnr by‘p v qr‘. Substittltinn cnnsists always in putting schemata lnr single letters, andfnr all recurrences nf the letters. when these restrictinns are nnt met, theputting nf nne schema fnr annther will be called nnt substitutien butinterchange. Thus interchange cnnsists in putting nne schema fnr anntherwhich need nnt be a single letter, and which need nnt be supplanted in allits recurrences. What has been said nf substitutinn, that it preservesimplicatien, equivalence, and incnnsistency, cannnt nl' cnurse be said ingeneral ni interchange. But there are useful laws nf interchange, the leastnf which is this first law cf interchange: Thinlt nf ‘ . . .p . . . ‘ as any

fist I . Trash Funetiansschema enntaining ‘p', and nf ‘ . . .q . . . * as fermed frnm ‘ . - ._n . . .‘ byputting ‘a’ fnr nne nr mnre nccurrences nf ‘IF; then ‘p ~t—\"rqr* implies[Similarly fnr any nther letters instead nf ‘p‘ and ‘q*.} Let us see why thelaw hnlds. We want tn shnw that any interpretatien nf letters which maltes‘p -H-qr‘ nnme nuttme will mslce‘...p....~t—r-.. . .t;-...* cnme nut true.But tn malte ‘p -H a’ cnme nut true we must either put ‘Tl int beth ‘p'and ‘a’ nr else ‘J.’ fnr bntlt ‘p’ and la’; and in either case ‘ . . -p - - .' and‘ . . .1; . . . ’, which differed nnly in ‘p’ and ‘q‘. became indisunguishablefrnm each nther, sn that their biennditinnal reduces tn ‘T*. New we can establish a rnnre impertant secend law sf interchange.‘ IfSi and 5: are equivalent, and Si is fnrrned l'rnm.‘.iI by putting S; fnr nne nrmntc nccur|'enees nf 5' 1, then S] and SQ are equivalent. E.g., this lawenables us tn argue frnm the equivalence nl ‘p ——r+ 1;.-* and *— tjatill tn theequivalence nf ‘p -sq .v r‘ and *—{pa} vr’. The rnugh idea is, inschnnl jargnn, that putting equals fnr equals yields equals. This secnnd law nf interchange is established as fnllnws. Chnnse anytwn letters nnt appearing in 5 I nnr in Si. They are, let us imagine, ‘pl and‘al. Then put ‘pl fnr the nccurrences nf S I in questinn in 5 I; the result maybe represented as ' . . .p. . . ‘, and the result nf similarly using \"q* may berepresented as ‘ . . . a . . . ‘. By the first law nf interchange, ‘p H t;-\"implies '. - .p. . ..t-1-.. . .q. . By substitutien uf.5'; l'nr\"p\" and .5\"; l'nr ‘qr’in this implicatien, we may cnnclude that the biennditinnal nf .5’, and S;implies the biennditinnal nfS I and .5 Q. But the biennditinnal nl'S , and-52 isvalid, since S, and S, are equiv atent. Therefnrc the biennditinnal nf S 1 and55 is valid; see tfivl nf Chapter T. Therefnrefil and El are equivalent. This secend law assures us that we can interchange equivalents S, and52 in any schema E] withnut allecting the nutcnme nf a u1.tth+valueanalysis; fnr. El and the result El will be equivalent, and equivalentschemata are schemata that agree case by case under truth.-value analysis.There thus fnllnws this third law nf interchange: Interchange nf equiv-alents preserves validity, implicatinn, equivalence, and incnnsistency;and, unlilre substitutien fnr letters, it even preserves censistency, unn-validity. nnaintplicatinn. and nnnequivalence. Substitutinn fnr letters must, we saw, be censtrued as unlfnrtn andexhaustive; but there is nn such requirement in the case nf interchangingequivalents. If in the valid schema ‘p v _rl\" we substitute ‘qrl int ‘p’, wemay infer the validity nf ‘qr v -— tart’ and this nnly; but if in that same

5*- Equivalence tjfivalid schema ‘p v p‘ we elect rather tn put ‘pp’ fnr its equivalent ‘p’, weare entitled thereby tn infer the validity nnt merely nf ‘pp ‘if '\" fltJ',ttl\". butequally nf ‘pp vp‘ and ‘p v — tppl’. Since interchange nl equivalents dne-s nnt affect the nutcnme nf atruth-value analysis, it pmves tn be a cnnvenient adjunct tn the techniqueat trutl1-value analysis; fnr, if we supplant schemata by simpler equiv-alents in the cnurse nf such analyses, nur cnmputatinns are reduced. lnparticular, accntdihgly, whenever a cnrlfiguratinn nf any cf the sevenfnnns depicted in [ll maltes its appearance in the cnurse nf a tnJth—valueanalysis, let us immediately simplify it befere prnceeding. We are nntnnly tn put 'p' fnr its equivalents ‘p’, ‘pp‘, ‘p v p’, ‘p v pal, etc., butcnrrespnndingly vs fnr ‘F, ‘rt’. ‘r v r‘, ‘r v tr‘, etc-, and vp-* fnr‘— —td','r}‘, \"av-pr‘, ‘qr vqr', etc. with nur new pnlicy in mind let us talce annther tam at the lnng schemawhich was analyzed in Chapter t5: pvt;-.pva.vpq :s'—i-a .:—t-.prv'pr eras-*\"-r=-isT'v\".l.a.s—>.-:yt—rT .l.vTa.s—va:—>.l. T —ta He] .l.Here the eriginal schema is subjected tn sntne sirnplificatin-ns befere thesubstitutien nf signs fnr ‘p’ is even begun. The simplificatinns cnnsist inreducing hnth ‘p vq . p v cl and ‘pr vpr’ tn ‘p’; fnr ‘p vq . p v Q‘ isthe last nf the schemata in f ll, and ‘pr v pl\" is the ['lEIIti tn last with ‘r‘substituted fnr ‘q’. hlest let us turn haclt tn the first lnng truth-value analysis nf Chapter ti.Under the new prncedure it wnuld run rather thus: pa v_st vprvps vqrvrrTqvTrvJ.rv.l.rvrp~vrr l.qv.trvTrvTrvqrvrtqvrvervrr rvrvarvlttqvrvar rvrvrrTvrvlr .LvrvTr Tvrvl..t J.vrvTrT Pvr T svs TTIn this case nnne nf the seven fnrms listed in ll] is visible in the nriginalschema as it stands, but snme emerge as the analysis prnceeds. fin the leftside, ‘q v F v er v rs’ is reduced tn ‘q v r v at-* by puning ‘r’ fnr

frfi I , Truth F't.inctians ‘F vrs’. Cln the right side, similarly, ‘r vs var vrr' is reduced tn ‘r v s v ht’ by putting ‘r’ fnr ‘r v at-*. Hnth nf the simpliticatinns last nnted are based nu the equivalence nf ’p v pq’ tn ‘p‘; but they invnlve alsn a mental switching nf cnnjunctinns and alternatinns. The clause \"r v rs‘ which is tn give way tn ‘F’ is nnt even visible in ‘q vr v at vrs‘ until we diinlt cf the part ‘qr v rs‘ as switched tn read \"rs var’: nnt is the clause ‘r var‘ visible in ‘r vs var v rs‘ until we thinlt nf the part ‘s var‘ as switched tn read ‘ar v s’- Even when this clause ‘r v at-* has been isnlated, mnrenver, its equivalence tn ‘r’ is nnt inferred frnm the equivalence nf ‘p v pa’ tn ‘p’ merely by substitutinn; we have alsn, mentally, tn reread ‘r v qr’ as ‘r v rq’ by switching the cnnjunctinn- Such Preparatnry switching nf alternatinns and cnnjunctinns invnlves a tacit appeal tn further equiva- lences: the equivalence nf ‘p v qr’ tn ‘a v p’ and nf ‘pp’ tn ‘ap’. Hut these steps drnp nut nf cnnscinusness if we schnnl eurselves, as we well may, tn disregard typngraphical erder amnng the cempenents nf a cnnjunetinn and nf an altematinn. It is arbitrary tn single nut just these seven equivalences, via., the equivalence nf ‘p’ tn each nf the seven schemata in ti], as a basis fnr simplificatinns auxiliary tn truth-value analyses. tilt. further cnnvenient equivalence, which cnuld in fact have been eitplnited in beth nf the truth-value analyses last set fnrth, is the equivalence cf ‘p v pg’ tn ‘p v q’. Pitt-ntlter cnnvenient nne is the equivalence nf ‘p . p v q’ tn ‘pr;-\". The practical investigatnr will use any simplificatnry equivalences that nccur tn him- Fnr the standardizing cf esereises, a cnnvenient cnm- prnmise might be tn allnw use nf the seven equivalences singled nut in -If l] and the funher enes assembled at the beginning cf the present chapter.j Pare bit-enditiencis. that is. schemata built up nf statement letters by means snlely nf ‘-H’, admit nf a startlingly simple test cf equivalence. They are equivalent if and nnly if just the same letters nccur an odd number nt’ times in each. it. pure biennditinnal is valid if and nnly if nn letter eccurs in it an ndd number cf times. blegatinn signs, mnrenver. can be sprint-tied thrnugh a pure biennditinnal withnut any effect if they are even in number- These nbservatinns are due tn Lesniewslti tpp. Eb, Iii]. HISTURICEL HDTE: The cnnfusinn between implicatien and the cnnditinnal, depleted at the end cf Chapter ‘l. has canted with it a cnniu-

ii‘. Eqaivrtfent'e '-’I’t-\"sinn between equivalence and the biennditinnal- The deplnrable habit stillpersists nf prnnnuncing ‘t—t’ as ‘is equivalent tn’ instead nl’ as ‘if andnflqiflEIERCISESl. [letermine which ni’ these are equivalent tn ‘pq —1- r’ and which tn ‘p v if .-—r- r’:p-1-..q|-—1-r, -|f_,'—i'*.j!;I'—-tit‘. jt‘.t-+t*.if_|'—t*t', p—t>r.'|u\".q'—-Ir.E. Determine which nf these is equivalent tn ‘p —'r qr’ and which tn ‘p —r. q v r’: p—t-i;-.p—vr, p-1-.-:j.v.p—t-r.fl- Determine any equivalent pairs frnm amnng these: F-‘lift P-til» F'—lti'~ ti‘-‘*'F'= ti\"\"*il-\"= ti‘_\"'l5'~This means fifteen shnrt tests.rt. lvlalring full use nf the new simplificatinn prncedure, test each nf the fnllewing three pairs fnr equivalence by truth-value analysis nf bicnnditinnals:it-rvrtvsr. are-err-arr:par vpqtt vpr,tvq'rs, pvt;-vr.pvq-vs._svrvs.evrvs;pqrvpqrvparvpqr, pvs.-vr.,-:.-vqvr.pvevr._svevr.5. By the same methnd, checl-t the schemata: P-\"-ti‘“_\"l’+ Fri f-‘lift Pl’?-H‘-Pf’fnr equivalence each with each.ti. Checit these fnr equivalence by the same methnd: ii‘-l\"'Ii-1i‘“\"\"-”—\"l-7\"» F\"‘-*i’-ti\"H'l'-T. See if the biennditinnal is assnciative.ti. Justify [is]-

lib i‘. Truth Furrctiatu\" ll] ALTERNATIONAL NORMAL SCHEMATAT|-|,-; |1fl[3[j[|t|'|5 ‘1,r‘__ and ‘+1-‘ are superflunus, we ltnnw, in that alluse nf them can be paraphrased intn tertns nf cnnjunctinn and negatien.The sign ‘—*’. hnwever. has been seen tn have a special utility in thetesting nf implicatinu: fnr. tn test implicatien we fnrrn a cnnditinnai [withhelp cf ‘—s’} and test its validity. The sign ‘-H-‘ has been seen tn be nfsimilar use in the testing nf equivalence. fin there is gnnd reasnn fnrhaving added the strictly superfiunus signs '—-1-’ and ‘H’. hlnw the ad-vantages nf retaining ‘v‘ are cf quite a different ltind, and will bee-timeevident in the cnurse nf the present chapter and the nest twn. What are lmnwn as [Itelvlnrgan‘s laws affirm the equivalence nftit ‘—tpvqv...vs]|‘ tn 'pe.-.s‘and{ii} ’—[pa.-..rl’ tn 'pvev...vs‘.Fnr the case nf just ‘p’ and ‘q’, these laws were already nnted in Chapter1. The further cases fnllnw by substitutinn and interchange. E.g., frnmthe equivalence nf ‘ — [pt v el’ tn pp‘ we have, by substitutien nf ‘p v q‘fnr ‘p’ and ‘r’ fnr ‘qr’, the equivalence nf ‘-— tp v q v r’,I’ tn ‘ - {p v ,-;-].F‘;and thence, putting ‘pt;-‘ fnr its equivalent ‘—(p v qt‘, we nbtain theequivalence nf ‘—|[p v a v rl’ tn ’jl‘-r.'_',_-‘.i\"’. Delvlnrgan’s laws are usefttl in enabling us tn avnid negating cnnjunc-tinns and altematiens. We never need apply negatien tn the whnle nf analtcmatinn, since ‘—{p vq v . . .v.t}‘ is equivalent tn ‘pr? . . .r‘; and wenever need apply negatien tn the whnle nf a cnnjunctinn, since ‘— (pig. . .sl’ is equivalent tn ‘p v if v. . . v .i'. iltlsc nf cnurse we never need applynegatien tn a negatien, since ‘p‘ is equivalent tn ‘p’. Fnr that matter, wealsn never need apply negatinn tn a cnnditinnal nt biennditinnal: fer. bythe methnd nf the preceding chapter it is easy in veri fy the equivalence nf

lit- ttirernsttienai Nnrrtazrl-ieliematn bitlllll ’— ll? \"\"\"‘ Ill’ ill‘ Til\"and nl’{iv} ‘—{p Hal’ tn ‘p H-q’ andtn ‘p HQ’.Sn any truth—functinnal schema can he put nver intn an equivalent inwhich negatien rt-ever applies tn anything but individual letters. ‘ferns-fnrmatinn nf this tiind is generally cnnducivc tn easy intelligibility. E.g., cnnsider the fnrbidding schema:tlt ~-is-its -—r—tct—tvi = —l-:tr.st—tv-—rtil}-[it is nf seme help tn vary parentheses thus with bracltets and braces whenthey are deeply nested-) I\"-lnw since {ll has the nutward fnrm ‘— ital‘, itcan be transfnrrned by till tn read: — ts —r-is -—t —t-at —rrti1 v — — t—tcu —ttt -i-tilCannellatinn nf ‘— — ’ reduces this tn:till —tr-'-—rrc -—* -\"ire —rnllv—ttvl —tt='—‘-'-rl-Then by tiiifl we transfnrm the first half nf {1} intn: it er-‘-‘ti - — —t-at —*ttl.nr p —1- sq . sq —r ptsn that t2] becemes:{3} p—rsa.sq —-I-p.‘-.-'—lrp\"t—t[p —-1-st.By {ii} again. ‘—trpl‘ here becemes ‘r vp’. and, by tiii] again,‘— tp —1- sjt‘ becnmes ‘pi’ nr ‘pr’, sn that tit] cemes dnwn tn:{4} p—1-sq.sa—vp.v.rvp.ps,

jt[} i’- Trnrtlr Fnnctiararin which, finally, all negatinn signs are limited tn single letters- it-ll is fareasier tn grasp than fl)- Such is the advantage nf nnnfining negatien tn single letters. l\"lnw itwill be feund in general that still funher perspicuity can be gained bycnnfining cnnjunctinn tn single letters and ncgatinns cf letters; and it willbe feund alsn that such cnnfinement nf cnnjunctinn can, lilte the cnnfine-ment nf negatien tn single letters, always be accnmplished. The lavvwhich maltes this pessible is ltnnwn as the lat-v cf rfistrihntivity nf can-_,inrtctian thrrrttgh alternatien, and runs as fnllnws: 'p.qvr'-.-'.--vt’ isequivalenttn ‘pqvprv.-.vpt'.Regardless nf the number nf letters invnlved, the equivalence is readilyverified by the methnd nl’ the preceding chapter:p,q1itru-.-vt.Hr.pqvprv.--'v'plT,q,-vt-v.-.vi.-r-i-.TqvTrv-..vTr J. .evrv.-.vr.-—--.l.qv.l.rv...v.l..rq-vrv,-,vt,--.1;-vrv--.vr J.s—vJ.TTThis law, lilte the familiar identity: .rl_'y+s+...+tv]-=.*qv +-rs +...+.rtvnf algebra, authnriaes the cnnvenient nperatinn nf “multiplying nut.\"Thanlts tn it, we need never acquiesce in a cnnjunctinn which has analternatien as cnmpnnent; we can always distribute the nflier part nf thecnnjunctinn thrnugh the altematien, as abeve, sn as tn cnme nut with analtematien cf simpler cnnjunctinns. Since nrder is immaterial tn cnnjunctinn, such distributinn can bewnrlied equally well in reverse: nnt nnly is ‘p . q v r v. . . v t‘ equivalenttn ‘pa vpr v. . .vpt’, but alsn ‘a vr v...vr . p’ is equivalent tn‘qp v rp v. . . v tp’. These twn snrts nf disu'ibutinn are indeed nne andthe same, nnce we learn tn ignnre nrder nf cnnjunctinn. When we have a cnnjunctinn nf twn alternatinns, distributinn taltesthe fnrm nf the familiar “crass-multiplying“ nf algebra: e.g.,‘p vt . ry vr vs’ cnmesnut ‘pa vpr vps vtq vtr vts’. Fnr, we beginby handling ‘qt v r vs‘ as we might a single letter ‘ii’; thus, just as‘p vr . rt’ wnuld became ‘pa v tn’ by {reverse} distributinn, sn‘pvt . q vrvs‘ becemes ‘p . rj vrvs .v. r . gurus‘. Afterward,

fti. .-'llter'nrrt‘ittra:ti' Nnrntai .'i'citernata T1distributinn nf ‘p’ turns the part ‘p . qr v r v r’ intn ‘pq vpr vpr‘, anddistributinn nf ‘r’ turns the pat't ‘t . q v r v s‘ intn ‘try vtr vtr’- Let us new gn baclt tn [ti] and imprnve it by distributing. We therebychange the part ‘F v p . p-v’ nf tit-ll tn ‘Fps v pps’, sn that [ti] becnmes:t5} tt—rt=r-re-re -‘-‘t‘ttrvtittt-We can npen the way tn further distributinn if we get rid nf ‘—1-’, translat-ing ‘t —1~ rt’ in general as ‘i v ii’. Such translatinn tums ffil intn:p vrq . — {sq} vp .v rps vpps,which, when ‘— tlrql’ is changed tn ‘s vi?’ by {ii}, becnmes: pv.tqr..tva‘vp.vl‘p.rvpps.l\"~lnw the part ‘p v-ts . s v lg vp‘ can be “crnss-multiplied.” sn that thewhnle bccnrnes:[bl as vpa vpp vsqr v sat? v sap v fps vpps. We can quicltly shnrten this result by deleting the patently incnnsistentclauses ‘pp‘, ‘.tqa‘, and ‘pps‘. We then have:[T] pr vpr; v sqs v sap v fps.Fluch deletinn is a case nf the prnendure explained in Chapter ti: each nfthe patently incnnsistent clauses may be thnught nf as supplanted by ‘.l.‘,which afterward drnps by resnlutinn {iii} nf Chapter 5]- ittlsn we may dmp any duplicatinns frnrrt cnnjunctinns—thus reducing\"st;-s‘ in ‘sq’. This was the secnnd nf the seven fnrrns nf simplificatinnnnted in {1} nf the preceding chapter. Sn {Tl becemes:[ll] pr vpevtqvsepvnir.which wears its meaning nn its sleeve- This its equivalents {1} and {4}cuuld scarcely have been said tn dn- The fnrtrts t-'-i]—[E] share the fnllewing three nntewnrthy preperties:‘—-+’ and ‘s-1’ dn nnt nccur; negatinn is cnnfined tn single letters; andcnnjunctinn is cnnfined tn letters and ncgatinns nf letters- lichemata hav-

T2 l. Truth Ftrnctiattring these three prnperties will be called altem-atirmal nrmrral sclrernara. This essentially negative characteriaatinn may be tefnrtnulated in mutepnsitive terms as fnllnws. Let us spealt nf single letters and ncgatinns nfsingle letters cnllectively as literals; thus ‘p’, ‘t;-‘, ‘p’, etc. are literals.Tbe alternatinnal nnrmal schemata, then, are the literals, the cnnjunctinnsnf literals. the alternatinns cf literals, the alternatinns cf cnnjunctinns nfliterals. and the altematinns nf literals with cnnjunctinns nf literals- Thiseharaeterieatinn can be put much mnre cnmpactly if we allnw eurselves tnspeak nf cnnjunctinns and altematiens nnt nnly nf twn nr mnrc cnmpn-ttents but alsn nf nne cnmpnnent—meaning thereby the cnmpnnent itself.Under this usage. ‘p‘, ‘pa’, and ‘pi_rr‘ are cnnjunctinns respectively nfnne, twe. and three literals; and cnrrespnndingly fnr alternatien. An alter-natinnal nnrmal schemata, then, is describable simply as any alternatienaf crrnjnnctinns sf literalr—that is, any alternatinn nf nne nr mnre cnn-junctinns nt’ nne nr mete literals. These cnnjunctinns, wherenf the alter-natinnal nnrtual schema is an alternatien, are called its ciatrses. The prneess whereby [ll was transfnrmed intn its altematinnal nnrrrtalequivalent {bl can be reprcduced fnr all schemata. Given any schema, wecan rid it nf ‘—t-’ and ‘H’ by fantiiiar translatinns: ‘p —+ qr’ becemes‘p v qr‘ and ‘p H qr‘ becnmes ‘pi;-' v pa’. We can cnnfine negatien tnsingle letters by ti}-{iv}, nr simply by ti}-{ii} having first gnt rid nf ‘-1-‘and ‘H’. Finally we can cnnfine cnnjunctinn tn literals by persistentdistributinn. The reader will find by esperiment that labnr is generallysaved. in these transfermatiens, by wnrlting frnm the nutside inward. This is all there is, su-ictly spe-airing, tn transfermatinn intn altema-tinnal nnrmal fnrm. Simpiificatinn. hewever, as in passing frnm an tnts], is always wetcnme tna. This gets rid, we saw. cf duplicatinns cf aliteral within a clause, and it gets rid nf any incnnsistent clause—unlessnf cnurse the whnle schema has bniled dnwn tn a single incnnsistentclause. say ‘en;-‘, whese nmissinn weuld leave us with nnthiag at all.Even in this esueme case a small simplificatinn cart still be tnade: we canwrite ‘pp’, since all incnnsistent schemata are equivalent. Tl1us in the altematinnal nnrmal fnrm we have an immediate test nfincensisrency: just simplify by drnpping incnnsistent clauses, in the abevefashinn, and see if nnthing but a visibly incnnsistent clause remains. atltematinnal nnrmal schemata are generally cnnvenient because theirnet impnrt is sn readily grasped: we can tell at a glance what interpreta-tinns will malte them true. E.g., an interpretatien will malte [E-It true if andnnly if it either interprets ‘p’ and ‘s’ as false tmalting the first clause nf {fill

l't\"l. -tllternariarral l\"t\"artriat’ .i’t‘ltett1r1ttI T3nne], nr interprets ‘p’ and ‘q’ as false [melting the secnnd clause true}, nrinterprets ‘s’ as false and ‘q’ as trite, nr etc- Elne nf the simplilicatinn laws that are cnllected in [ll nf Chapter ‘Elreduced ‘pq v pq’ tn ‘p’. lf perversely we apply this law baeltwards. as acnmplicatinn law, we can espand nne altematinnal nnnnal schema intnannther which has certain bnld traits wnnh nnlicing. This espansivetransfermatinn is called iievelapn-tent: ‘p‘ gnes intn ‘pi; v pq‘ by de-velnpment with respect tn ‘q’. l‘~lnw if we persistently develnp each clausecif an aiternatinnal nnrmal schema with respect tn each letter that is absentfrnm the clause, we arrive at the rievelapecl alternatianai nnrmal fnrm.Fnr instance, ‘sq v .Fp.t\"’ becnmes first: lqp v -tqp v rpsq v fprqand eventually: rqpr v sqpl“ v sqpr v rqpr v fpsq v rpsq,nr, alphabetized: pqrr v pqrs v pqrs v pqrs vpqrr v pqrs. This fnrtn, when thus alphabetiscd and freed nf repetitinns, is in effecta u'uth table; each nf its clauses depicts nne cf the ways nf assigning truthvalues tn letters that maltes the schema cnme nut u1.te. implicatien be-cnmes recngniaable nn sight: if the letters nf twn dcvelnped altematinnalnnnual schemata are alilte, then the nne schema implies the nther just incase all its clauses are amnng the clauses nf the nther. The marl-t nfvalidity, in dcvelnped alterrtatinnal nnrmal fnrm, is that all pessibleclauses are present: all 2\" nf them. where n is the number nf differentletters. The marlr. nf incnnsistency is disappearance; nn clauses. l'll5TCl|ilCtHl. HUTE: D-elvlnrgan’s laws were named fnr liuguslusEl-ehtlnrgan, whn llnurished in tats-st; but they were ltnnwn tn ‘Williamnf Ct-cldram five centuries earlier. (Cf. -Lultasiewics, “Eur Geschichte-”’,lThe idea nf tievelnprnent and its name gn baclt tn Iltelt-lnrgan’s centem-pnrary Ifienrge Binnie. The altematinnal nnrmal fnrm was familiar tn Ernst

'.|'-rt f- Truth FanclinrtrSchriider by IETT and is dnubtless nlder. This fnrtn is nften called, lesssaggestively. the disjunctive nnrmal fnrm.EIERCIEESI - Hy successive tran sfnrrnatinns, transfnrm each nf these schemata intn an altematinnal nnrmal schema. -trt it r ls it ctr‘-' -is \"villi. p——t-q .q—i-r.--1-.p—i-r, p -—r- q ,—r-gr :s—r-l|I|_r F’ “'l\"=.i' - ti‘ “\"\"f~ F’ “'*'i' -‘H’ fr pq it-1-r.2- Eieveinp each nf the results cif the preceding etrercise intn dcvelnped altematinnal nnrmal fnrm-3. Judge whether this is a snund general methnd nf testing alternatinnal nnrmal truth-functinnal schemata fnr equivalence: decide by fell sivanp whether each clause cf each schema itnplies the nther schetna. Justify ynur judgment-rt. Check the implicatiens in Esercises t-it nf Chapter T by the methnd nf dcvelnped altematinnal nnrmal fnrms. ll SIMPLIFICATION The perspicuity nf the altematinnal nnrmal fnrtn can be enhanced bypressing simplificatinn. In passing frnm (\"ll tn (E) in the preceding chapterwe used nne nf the seven fnrrns nf simplificatinn which were tinted incennectien with {ll nf Chapter ‘ii, via. ‘pp’ tn ‘p’. But nthers nf the sevenmay liltewise be used tn advantage. e-g-. that nf ’p vpq’ tn ‘p’. Thus,reducing ‘sq v sqp’ tn ‘sq’ in {E}, we get.‘

ll- 5t'rnpltjt'icatinn T5-ll] psvpqvsqvrpr.The result {ll itself is susceptible tn yet a funher simplificatinn, cnveredby nnne nf the seven. The initial clause nf [ll is in fact redundant; {ll isequivalent tn:[1] pqvrqvr-pr. There is a quick way -nf testing any clause nf an altematinnal nnrmalschema in see if it can be thus drnpped as redundant- The law {vii} nfChapter 9 tells us hnw: just check, by fell swnnp, whether the clauseimplies the rest nf the schema- The clause ‘pr’ nf ll} is feund by fellswnnp in imply the remainder nf rt), and this marks ‘pr’ as redundantin [il- The schema id} nf the preceding chapter was already an altematinnalnnrmal schema. fl} here is annther, and, we see, equivalent ta [ti].Simplificatinn can gn a lnng way. Snmetimes an alternatinnal nnrmal schema can be simplified by drnp-ping nnt a whnle clause but just a literal. This happened already in thepreceding chapter when we mnvcd frnm {T} tn [E-It en the strength nf l ll nfChapter El. It can happen alsn in cases untnuched by t‘ ll nf Chapter ii. Anettample is affnrded by 'pq vpqr v per‘, which prnves equivalent tn‘pq vpr v par‘. There is a quick way nf testing a literal in snme clause nf an altema-tinnal nnrtnal schema tn see if it can be dmpp-ed as redundant. Just see, byfell swnnp, whether the rest cf the clause implies the whnle schema. Thustake ‘pqr v pr v qr‘. Tn check that the ‘r‘ nf ‘pqr’ is redundant here. weneed merely check that ‘pq’ implies ‘pqr v pr v qr‘. Why this methnd always wnrks can be seen well ennugb by justanalysing the present esample- We want tn check that ‘pqr v pr v qr‘ isequivalent tn ‘pq v pr v qr‘. llifitat we have checked by fell swnnp is that‘pq’ implies ‘pqr v pr v qr‘, and this amnunts tn saying [by {vii}, Chap-ter ‘Ell that ‘pqr v pr v qr’ is equivalent tn the cnmbined altematien‘pq vpqr v pr vqr‘. Hut then we reduce the first twn clauses nf thisalternatien tn ‘pq’ [by ll}, Chapter 9}, getting ‘pq v pr v qr’- Snmetimes the eliminatinn nf a redundant literal can reward us dnubly,by engendering a clausal redundancy that we can drnp as well. Thuscnnsider again nur last ertample, ‘pqr vpr vqr‘. We see by three fellswnnps that nnne nf its three clauses is redundant. tlifter we drnp the

Tti l. Truth Frurcriens redundant ‘r’, hewever, the remainder ‘pq vpr var’ has a redundant clause. Find it. ltn altematinnal nnrmal schema is quickly checked fnr censistency. as nnted in the preceding chapter: each incnnsistent clause dreps as patently incensistent. at validity check nf altematinnal nnrmal schemata is harder. but it is prnvided nnw by eliminatinn nf redundant literals. Fer. recall the redundancy test fnr a literal: it is that the rest nf its clause implies the whnle schema- But if the whele schema is in fact valid, anything implies it; se then each literal will fall away by the redundancy test until we are left with nnly nne-letter clauses. This remainder will be nnt nnly valid but patently valid: snrnetlting like ‘p v q v q’. Twp gnnd ways are nnw befnre us fnr simplifying altematinnal nnrmal schemata. We cart test a clause fnr redundancy, and we cart test a literal fer redundancy, in each case by fell sweep. it-.n altematinnal nennal schema can, hewever, resist bnth redundancy tests and still admit nf simplificatinn in mere devinus ways. An ettarrrple is: [3] lid \"r’.5'ti‘ \"~\"tl‘l’ ‘t’tit'~ By twelve fell sweeps the reader can test each clause and each literal nf [3] fer redundancy and draw a blank every time. \"t’et [3] has a simpler equivalent, ‘pq v pr v qr‘. Simplificatiens by these fell-sweep techrtiques are always beneficial, even failing the assurance that we have feund a shnrtest equivalent. We may ferge such further reductinns as that cf [3] tn ‘pq v pr v qr’, nr we may press nn with the nest paragraphs.j— It is remarkable that ne quick and general methnd is kunwn fnr reduc- ing art altematinnal nnrmal schema tn a shnrtest equivalent. We have tn etthaust pessibilities. an adequate range nf pessibilities can be staked nut by the fnllewing censideratiens. an altematinnal nemtal schema is an alternatien ef clauses each nf which implies the schema [since ‘p’ itnplies ‘p v q’]. Further, if we have checked fnr redundant literals and deleted them, we may be sure that each clause is a prime implicant ef the schema; that is, when yeu drep any literal frem a clause, the remainder ef the clause ceases tn imply the schema. Sn, if we sntnehnw assemble all the prirne implicants nf a given schema, we may be sure that we have caught all the clauses nf any shnrtest altematinnal nnrmal equivalent nf eur schema. lttt nur clumsiest we may then simply try the varieus cemhina-

l’ l- t'i'irnplt]'ic‘alittn TTtinns nf prime implicants and select a shnrtest altematien that cnmes nutequivalent in the eriginal schema. There is a mechanical methed, due tn Samsnn and lvlills, fnr generat-ing all the prime implicants nf a schema. We start with an alternatinnalnnrmal schema that has been freed nf redundant literals. This schema isan altematien nf seme nf its prime implicants. hlnw if twn nf these clausesare eppesed in nne and nnly nne letter, say ‘p’ [sn that nne clause has ‘p’and the nther has ‘p’], talte the cnnjunctinn nf the rest nf their literals.This cnnjunctinn [minus any duplicatinns] is what I call the cnnsensus nfthe twn clauses. Fnr instance. the first and third clauses nf [3] have thecensensus ‘pr’; the secnrr-d and fnunh have ‘pr’- it can be prnved dtat thisprecess nfcnnsensus-taking will turn up every missing prime implicant.” Thus, tn find a shnrtest nnrmal equivalent nf [3], we begin by generat-ing its twn missing prime implicants as abeve. ittrrneaing them tn [3], wehave:[4] pq vpqvqrvqr vprvpr.This is equivalent still tn [3], by virtue nf [vii] ef Chapter 9; it is aredundant equivalent. ‘live knew that its last twn clauses can be drnpped;but if, instead, we keep ene er beth nf them, perhaps we can drep nthers,and mnre. Sn we make varieus esperimental fell swnnps. In the cnurse nfthem we find net nnly that ‘pq’ implies the rest nf [4] and sn can bedrnpped, but alse that ‘qr’ implies the rest nf what remains and an can bedrnpped in turn, and alse that ‘pr’ implies the rest el’ that. We end up thuswith ‘pq vpr vqr’, an impmvement en [3]. There is annther equallyshert versien that we ceuld get ten: the reader might like tn search it nut. Htsrnrucat stern The preblem nf simplifying uurlr-functienalschemata has interested industry, because ef art applicatien tn electr'iccircuits. Thus picture twn terminals and twn intervening switches. if theswitches are cennected in parallel, then the current is en just in ease thenne switch er the ether is clnsed. if they are cennected in series, thenthe current is en just in case the nne switch and the nther are clnsed. Suchare the reles ef altematien and cenjunctien; and as fnr negatien, it an-swers te the threwing ef a switch. There results, as Claude Shannen ll See my Selected Legic Papers, pp- lti-ti f.

TE I . Truth Functinnr ehservecl in 1933, a eerrespcndenee hetvveen circuits and schemata. Pt practical technique fer rerltleing a schema te a shnrtest equivalent schema weuld enahle the engineer he reduce a circuit te a simplest equivalent circuit. l was still unaware cf the eiusiveness cf such a techaiq ue as late as 1943, when I was werlting at the present heel: and hnping te hase my whele treatment nf tnttli-fttnctien legic upen an easy simplificatinn reutine. lvly first article en the ptehlern appeared in I951. Hy then the eemputatien laheratery at Harvard had feund it wc-rthwhile actually te tahulate and publish the simplest equivalents cf the 55,53-'5 tI1.1t]1 functiens pf feur distinct letters. [See Aiken.) Because autc-mats require cempli- caterl circuits, rnany studies ef the simpliticatien preblem have been published in engineering media. it was in ene such in I954 that samsen and lvltlls presented the cnnsensus inethett, as 1 am calling it, fer getting all the prime implicants as in [4]. ln anrather such paper, in 195?, Eihazaia sh-awed a way cf expediting the rest cf the jch, that ef selecting an adequate minimum cf prime implicants as in ‘pa v pr v qr‘- Meanwhile anether engineer, Rnlf ii. lvliiller, sh-nwed me i|1 1955 that the majer technical treuhle lies in the surprising multitude cf prime implicants, in the case ef schemata with six er a dexen difierent letters. Fridshal cites a nine-letter schema which, he claims, has 1.693 prime implicants. En- gineers have heen pregrantming eernpttters te find simplilicatiens ef such schemata, hut even se the tasl: can he ferhidding. A simplificatinn| technique that did net depend en any exhaustive survey ef the prime implicants weuld he a been. EIERCIEES l. Eheci: the equivalence ef If l} te {2} hy truth-value analysis. 2. Similarly fer [3] and pa vpr v qt\". 3. Investigate these schemata fer redundant clauses and redundant liter- als: tall‘ vita var van as vtr-alt vast- 4. De the same fer the six altematinnal nnrmal schemata ebtained in Exercise 1 cf Chapter lit.i 5. Find anether equivalent nf t-1]. as shert as ‘pa vpr v qr’. Shaw steps.

ts. Duality \"ta I5. Find all prime implicants cf each cf the six altematinnal nnrmal schemata ebtained in Exercise l cf Chapter It]. [First drep any re- dundancies that were revealed just nnw by Exercise -1.]l T- Find a shnrtest altematinnal nnrmal equivalent cf each ef the six. 12 DUALITY All legical cnmptttaticn at the truth-fimetienal level is essentially cnm-ptttatirtn with ‘T’ and ‘J. '. Hence it is tn he expected that twe schematawill he quite parallel in their behavier if they are just aiilte under truth-valtle analysis except fnr a there-ughgeing interchange cf ‘T’ and ‘J.’.Schemata se related are called duals cf each ether. They behave in rela-tien te each ether aceerding te laws which, fer their theeretical interestand -nceasinnal cnnvenience, warrant settle netiee. Theugh duals are eppesed snrnewhat in the manner cf ‘T’ tn ‘l.', theyare net te be cenfuscd with mere cnntradicteries er mutual ncgatinns. Theprime example cf duality. rather, is cnnjunctinn versus altematien- Cen-junctien and alternatien are alilte except fnr a thereugltgeing interchangecf ‘T* and ‘J.', in the fnllewing sense. Cenjuttclic-n, te begin with, isdescrihahle thus: lsr ecnnenxxr-tr Etaecextretvtstr ttssutrr T TT l'\"-l'-'- l—l——l l—l—l—New te interchange ‘T’ and ‘.l.* merely in the last celumn weuld indeedprnduce a truth functien which is the negatien cf cenjunctien. interchange‘T’ and 1* threugheut all three celumns, hewever. and what yeu get isprecisely a descriptien cf alternatien:

at] I . Truth Functtnnxlat\" tI'tJhIF'Cltl\"~l'EHT EH11‘! CDHFDHEHT REEU LT J. J. .L —-l-t -l—l\"\" —l—l—lSuch is the sense in which ‘pq’ and ‘p v q‘ are said ta be duals. In generalthe relatienship herween dual schemata S and S ' is this: whenever each cf‘p‘. ‘e’. etc- is interpreted eppesitely fnr 5 and .5“, the tmth values cf Sand 3' turn nut eppesitely tn each ether. Trivially, by this standard, ‘ti’ is dual net tc ‘pi but te ‘,1?’ itself; fer,give eppesite valaes tn 'p‘ and yeu get values fnr *p‘ which are eppesitetn each nther. Instead cf inverting 'T' and '1.‘ individually threugheut the table, aswas dene abeve, we ceuld get an equivalent tabulatieu cf the dual byleaving the celumns fnr cempenents intact and just inverting the resultcelumn bndily [sci that a tap ‘T’ becemes a bettern 1*]. This can beveri fied in the abeve case, and it can be seen en reflectien te werlt always.In particular then the marl: ef a self-dual truth functien is that its resultcelumn leelts the same upside dewn. The duality el’ ‘pq’ te ‘p v q’ is evident witheut resert tn die abevetabulatinn if we simply cempare the eriginal descriptiens ef cenjunctienand alternatien- at cenjunctien is mte when its cempenents are all tine.and etherwise false; whereas an alternatien is false when its cempenentsare all false, and etherwise nne. These twe deseriptiens are alilte exceptfer interchange cf ‘tme’ and ‘false’; hence flee‘ and ‘p v l;-' are beund tnbehave alilte except fnr a therettghgeing interchange ef the reles cf ‘T’and '1.‘--which is what duality means. The self-duality cf ‘p‘ is evidentsimilarly frem the general descriptien cf a negatien as “tme er falseaceerding as its cempenent is false e-r true\"; fnr, switch the werds *true*and ‘false’ in this descriptien and yeu simply have the descriptien efnegatien ever again. lvlere generally new, censider any schema 5 huitt up cf letters bymeans exclusively at negatien. cenjunctien, and altematien thence de-veitl ef ‘—1-’ and ‘H-’}. Suppcse a secend schema S ' is lilteS except that ithas altematien wherever 5 has cenjunctien and vice versa. Then truth-value analyses ef S and S’ we beund tc match except fer interchange nf‘T’ and ‘J.’ threugheut; fer, we just saw that the explanaticns cf cenjunc-tien and alternatien are alilte except fnr switching ‘true’ with ‘false’, and

L? . Dttelt’l‘}' 5]that the explanatien ef negatien is unchanged by switching ‘tme’ with‘false’. Sell and 3' are duals. What has just been established will be called the first law cf ci'unlt'ty.''Where.S is any truth-fimctienal schema deveid ef ‘—1-’ and ‘+-I-’. the resultcf changing alternatien tn cenjunctien and vice vetsa threugheut S is dualtn 5' . This law aflirms immediately the duality ef ‘pq’ tn ‘p v tj’, and thesell’-duality nf ‘p’ and cf ‘p’. It affirms alse the duality el’ ‘p . q v r‘ in‘pr vet’, the duality pl ‘pa vpr’ tn ‘p ’|\"-Ejl . ll vr’, the duality cf\"_nqvqrvpr’tu‘,u ‘vi; . evr .p vr’,etc. Se we new have a quick and graphic way cf farming a dual el’ aschema: interchange cenjunctien and alternatien. This prc-cedure de-pends, he it nnted, nn absence cf ‘—-1-’ and ‘H’; but we can get rid cf ‘—*’and ‘-H-’ in advance, since ‘p —+q’ may he rendered as ‘p vq‘ and‘p ~t-sq‘ as‘_l1-vtj . tl'vp’er‘pq vpa‘. In interchanging cenjunctien and alternatien tn get duals, special caremust be taken tn preserve greuping. in case ef deubt, think cf full paren-theses as rcstered in lieu cf the det cenventiens. Thus ‘p . ty v r’ has asdual net ‘p vtj . r‘ but ‘p v qr’. Fer, ‘p . q vr’ means ‘pen v r)’, and‘p v qr‘ means ‘_n v tart‘, in which the same pattem nf greuping is pre-served; ‘_e vq . r’, en the ether hand, means ‘tp v qlr’, and is dualrather tn ‘pq v r’. Given any schemata 5 and S ’, new, we can test whether5 ’ is a dual el’S by fc-rming an explicit dual uf.5’ aceerding te the abeve methed and thenchecking it fer equivalence te S‘. ln particular we can thus determinewhether a given schema .57 is a dual cf itself; we have merely tn ferm theexplicit dual e-f5 by switching cenjunctien with altematien, as explained.and then te test this result fer equivalence te .5‘ . Apart frem trivial cases,such as ‘p’, self-duality is rather rare; but it decs eccur. E.g.,‘pq v pr v qr’ is dual te itself, since it is equivalent te its ewn explicitdual ‘p vs . p v r . q v r’. {See Chapter 9, Exercise 4.] Switching alternatien with cenjunctien is net the enly cenvenient wayef farming a dual. .-anether way, which dues nnt even require a prepara-tcry eliminatinn cf ‘wt’ and ‘H’. is prcvicted by the secend law cfduality.\" if in any schema yeu negate all letters and alse the whele, yeu geta dual. This law is evident frem the eriginal definitien cf duality; fernegating the letters has the same effect as reversing all interpretatiens ufletters, and negating the whe-lc reverses the truth value el’ the nutcnme. [lel'vlergan’s laws themselves {Clmpter Ill] are essentially duality plin-ciples, as may be seen by rearguing them in the present centext- l‘-its dual

E1 l . Truth Funetiensef ‘pq . - -:-r’ the first law ef duality cites ‘p vq v. . .vs’, whereas thesecend cites rather ‘— [pd . . ..tl’\", these twe must tlten be equivalent teeach ether, and thus Deli-\"lergan’s first law, ii] ef Chapter lll, helds- {ii}admits ef a parallel argument. A third fctw cf duality is this: A schema is valid if and enly if its dual isincensistent. Fer. if twe truth-value analyses differ te the extent ef aLhereughgeing interchange ef ‘T’ and ‘J.’, clearly the ene will shewvalidity if and eniy if the ether shews incnnsistency. Fe-urth tttw efd‘ttniity.' A schema S, implies a schema Es if and enly ifthe dual ef .572 implies the dual ef 5 1. This is seen as fellews. The duals ef-5'1 and 5; behave like 5 1, and Es, under truth-value analysis, except fer aswitching ef ‘true’ with ‘false’ threugheut. Hence te say that ne interpre-tatien ef letters makes S, true and S, false is the same as saying that neintetpretatien maltes the dual ef.5’ , false and the dual el’S, trtte. Fifth htvv ejfelhnltty: Schemata are equivalent if and enly if their dualsare equivalent. This fellews frem the feurth law, sinee equivalence ismutual implicatien. The third, feurth, and fifth laws enable us, having established enevalidity er incensistency er implicatien er equivalence by truth-valueanalysis er etherwise, te infer an additienal incnnsistency er validity erimplicatien er equivalence witheut funher analysis. E.g., having verifiedthat ‘p vs . q v r . r vs’ implies ‘p vs’ {as maybedene by the methedef the fell sweep, Chapter T}, we may cenclude by the feurtlt law efduality that ‘pr’ implies ‘pt;-‘ v qr v rs’. This eperatien may, in centradis-tinctien te the first. be speken ef as the full swap. Either ef Del'vlergan’s laws, ti]-—|[ii} ef Chapter IIJ, fellews frem theether by the fifth law ef duality. Again, frem the law ef distributjvity efcenjunctien threugh alternatien {Chapter ltl] we can, by the fifth law nfduality, infer a law nf chstrihutivity tr-_,felternutt'nn thrnugh cenjunctien: ‘pvqr..-t’isequivalentte’_ttvtj .pvr....._ttvt’.This law shews that cenjunctien and altematien are in still mere cenge-nial relatiens te each ether than are multiplicatien and atlditinn. In arith-metic we can multiply eut, thus: .t[y+.t +...+w]l=,ty +13 -I-...+.tw.but we cannet “add eut” flies:

tlfltmlity E3 .t +m.-.w =t[x +y]{.r +,3]l-.-t'.r + iv].ln the case ef alternatien and cenjunctien, en the ether hand, distributinnwnrks betli ways. Indeed, since by the tifth law et’ duality all equivalences centinue teheld when cenjunctien and alternatien are switched, we may cenclude atence that the technique el’ reducing a truth-fnnctienal schema te a nermalschema may be repru-duced entire with alternatien and cenjunctienswitched- This switched ptecedure issues in cenjunctienal nermalschemata such as: pvevr.,evs.t;.-vrvs—i.e, in cenjunctiens ef alternatiens ef literals. The clauses ef a een-junetienal nermal schema are alterttatiens, net cenjunctiens; and theschema is a cenjunctien ef its clauses. Because ef duality, any ptecedure fer simplifying altematinnal nermalschemata has an exact parallel fer cenjunctienal nermal schemata. tnparticular, just as any incensistent clause such as ‘ere’ dreps frem analtematienal nermal schema [as leng as seme clause remains}. se anyvalid clause such as ‘tit vs vty’ dreps frem a cenjunctienal nermalschema. It was netcd in Chapter lll that the altematinnal nertnal ferm afferds animmediate test ef ineensistency. The cenjunctienal nermal fnrm afferds acerrespending test ef validity: just simplify by dn::-pping valid clauses inthe abeve fashien and see if nething but a visibly valid clause remains. Aleng with the dcvelnped altematinnal nermal fnrm that cenfmnted usin Chapter ill, a develep-ed cenjunctienal nermal ferm is assured byduality. l-lere the apprepriate eperatien el’ dcvelepment is the ene thatcarries ‘_n’ inte ‘p v q . p v Q’. The cenjunctienal nermal ferm‘s v q . l‘ v p v s’ has the tleveleped cenjunctienal nermal fenn:,svqvrvs.pvqvrvs.svs-vrvs .,evqvrvs.pvqvrvs.pvnvrvs,the dual ef what we saw at the end ef Chapter ill. Fer twe develepedcenjunctienal nermal ferms having the same letters. the test ef implica-tien is the reverse ef what it was fer develeped altematinnal nermalferms; all the clauses ef the implied schema are ameng these ef theimplying schema. The mark -nf incnnsistency in devele-ped cenjunctienal

E4 l . Truth Functiensnermal fertn is presence ef all the 2\" pessible clauses, and the marlt efvalidity is disappearance. A cencluding werd el’ cautien: the altematinnal and cenjunctienalferms ef a schema are erdinarily net duals ef each nther. Duals, alter all,are net equivalent, except in the case ef self-duality. Htsretttcltt were The essence ef duality is Delvlergan‘s laws,which, as neted, ge back six centuries te William ef Elcltliam. ivtercdeliberate treatment ef duality dates frem Schrilidet {lSTl‘].EXERCISESl- Which el’ the schemata: P \"rte. P stilt F res. —tt-t Hui are dual tn which‘? Justify-2. Write the duals ef these: a—ts- s —*a- *—tr-—*-st. —ts —>tu.3. We saw in Chapter l l a test ef redundancy fer a clause er a literal ef an altematinnal nermal schema. What, by censideratiens ef duality, sheuld be the tests el’ redundancy fer a clause er a literal ef a cen- junctienal nennal schema‘?4. Hy successive transferrnatiens, transierm each er the schemata ef Exercise l ef Chapter ll} inte a cenjunctienal nermal schema. Simplify where yeu can.5. Expand the results ef Exercise 4 inte develeped cenj unctienal nermal ferm.I5. Test the feur schemata ef Exercise l el’ Chapter t5 fer validity by putting them inte cenjunctienal nennal ferm.

.i;i’ . .-l-stems E5 13 AXIOMS The applicalien ef legic te ene er anether scientific theery, such asarithmetic er seme branch ef physics, is semetimes made explicit bysetting up what is called an axiem system- Certain statements ef thethcery are chesen as a starting peint under the name ef axiems. and thenfunher statements, called theerems, are generated by shewing that theyare legically implied by the axiems. The implicatien cencemed here geesbeyend the kind ef implicatien that we have thus far censidered, namelytruth-fnnctienal implicatien, and expleits alse the further reseurces el”legic that we shall tum te in later parts ef the beelt... A variant ef the axiematic methed has eften been used within legicitself. and even within trttth-functien legic, te generate valid schemata.Here the generating relatien can ne lenger be cited simply as implicatien,since implicatien here is ubiquiteus: any valid schema is implied by anyand every schema. instead. specific fermal mics efinference are given. Ausual ene is me.-::lus pensns.' if a theerem [er axiem; axinms ceunt asthcercms] is a cenditienal whese antecedent is alse a theerem, then putdewn the censequent as a theercm- Aneflier usual ene is suhstt’t‘utt'ert.'substitute any schema fer all eccurrences ef a letter in a theercm- lf westart with valid axiems, ebvieusly these twe rules will lead enly te validtbeerems- Cine interesting eheice ef axiems, due tn Lukasiewicx, eemprises thesethree:[ll ju—:-q-.—1*Iqr—-tr.-'-i'.,t;t—t-r,th spreefit s—w=*sLet us generate seme theerems by the twe rules. Substitutien el’ ‘p’ fer ‘t;-‘in [1] gives:Hi enemaSubstitutien ef ‘p’ fer ‘r’ in ll] gives:

ss i‘. Truth Fuuctiensfii r*s=Hsea=ea*r-Substitutien ef ‘_s- —1- p’ fer ‘q’ in {5} gives:fl5J' ti-1'-.-\"T\"—tt1' =—“\"=-t7'—’-\"F -—‘-‘ti’ i“\"-li\"'*F'-lvledus pencns, applied te lb] and {ti}, gives:ill .5\"“'P -tit =-t-li'—*P-lvledus pencns, applied te {Tl and t3}. gives:th spa This sequence ef steps is called a precf ef ‘pr -1 p’. ln a cendensednetatien the whele preef can be put thus: Br til. l3 tel-ti —*= 3 -1:-ls -'*t-='-The nutncral ‘Ii’ here stands as an abbreviatien ef the schema (3), hence\"p —il- F _—} p‘. The expressien ‘I [tjhi]-’ stands fer the schema I4}, thisbeing what {2} becemes when ‘q’ is supplanted by ‘p’. Thus the whelestring ef symbels ameunts te {I5}. The mentien ef {ll means that thiswhele schema ts; can be get frem tjl} by certain substitutiens. A readerpresented merely with this cendensed preet\" weuld necnver lb), cempareits structure with that ef l[ l }, and diseever fer himself that ‘p -1 p’ and ‘pr’were the required substitutes fnr ‘q’ and ‘r’ in {I}. The square brackets,finally, indicate excisien by medus penens—twice ever, in this case. This system visibly treats enly ef negatien and the cenditienal. Hew-ever, any truth-functienal schema can be translated inte these tenns, since‘p vs‘ and ‘pq’ are equivalent te *,s —1- a‘ and_‘—[,e —1- cl‘. There is aprnef that every valid btttli-funetienal schema, er its translatien, can beget frem the axiems {l’,t—-[3] by the twe rttles- The system is in this sensetemplate. Axiematists are naturally cencemed that their axiems be indeperufent:that nene be derivable as a theerem frem the rest, and hence dispensable-The independence ef an axiem is neatly established it‘ we can se reinter-pret the syrnbels as tn falsify that axiem while still preserving the validityef the ether axiems and the seundness ef the rules ef inference. Ferinstance, we can sh-ew that the axiem [3] is independent, in this-system,

i' 3 . .1l.rt'nm.r E?by reinterpreting the negatien sign as simply yielding a falsehn-ed wher-ever applied. We keep the interpretatien df Henee the rules pfmedus pnnens and substitutien remain snund and ll) remains valid. iltlse[El cemes but valid; fnr if ‘ft’ cemes nut always false, ‘p —s q‘ ennses nutalways true- Hut {Ii}, thus reinterpreted, ennies nut false fnr false ‘p’.Again, tn shnw {1} independent, reinterpret tl1e negatien sign as yield~ing a truth wherever applied.varieus nther eemplete systems ef independent axinms have beeneuntrived fnr tmth-functien lngie. hint all nf them are based en negatienand the cenditienal- Seme use negatien and alternatien. Cine, due tnHiend and impre-ved by Lultasiewiee, uses nnly the tmth-fiinetipnal enn-nective ‘I’, which, we saw [Chapter 2}, suffices fnr espressing all the truthfunctiens. This system has just nne asinm: p|.qir:1::s|..t|s:iI.siq .|:p Is-|.p |.r.its rules bf inference are substitutien and a variant nf medus punens whichsays that if a thedretn has the fnrm ‘A B i tf.\" and has a thenrern in theU1‘ place, then the schema in the ‘E’ place is a l.l'l'lE!1lIl!t‘EI‘l'l. If we define‘p —-I qr‘ as ‘p q lq’, the asitnn ab-uve beentnes less bewildering: F‘ I--sir =l=-i'—*-<-= -l=-sis -—*-is IL ilisiematie legic is tme thing; the applieatidn at full-bluwn legic tnestralegical nsinms is anether. The centrast between them is the centrastbetween what F-heifer ealledfr.-nndntinnni and pestfannddtiaitel systems.Feundatidnal systems have their autnnntnnus rules pf inference; pest-fnundatinna] systems just infer their thenrerns frem their asidrns by lugiseal itnplieatinn, deferring tn lngie fnr the analysis and the teehnnldgy -nfthat relatien. Still, fnundatinnal systems can he fashinned alsn fer estralngica] sub-jeet matter. An esample is the fnllewing asinmatie algebra bf subtrac-tien. \" There are twe aitidtns: 1 =s — ti\" —:Pl, I — I:-1 —al=.: —ti' -11-Suhstitutinn is again nne nf the rules nf inference, and a secnnd rule nfinferenee allnws us tn put the left side bf a thenreni fnr the right in anythenrem. Suhstitutinn in the secnnd asicm, e.g., gives the thenrem: ‘\"1 I-'r-nnt my Eeieetert f.rigit\" Papers. pp. ,5-=i—fit'lI.

BE; I- Trtrrh Fr-tne.*r'r:-vtr E—lI\"“lJr\"“J’il=lJ’ r.'tI*—tI-cl.and frem this and the lirst asinm by means cf the secnnd mlc nt\" inference-we get the thecrrent: t—I=t1'\"\".vl—Lt—t}- A qneerer esample: the secnnd t't|le nt' inference allnws us tn put theleft side ‘.Ir' uf the first at-titttn fer the right side in that axiem itself; we getthe thenrem ‘J: = Jr’. We can get l‘l\"tt]-t'e- The system can be pnnved ccmplete, in this sense:every equatinn that is cnnstmctcd cf this subtraetinn netatien and is valid.er true fur all values nf its variables, is derivable frem the twn asinrns bythe twn rules. Mnrepver, the netatien is str-ringer than it lctctlts; Jr + y canbe espressed in it, asr — Hy — y} — _v]I. Usually. hnwcver, asinmatiaatinns nf e:-ttralngical subjects are pust-fnundatinnal. There is a gnnd reasen: mass pruductien. Once fer all wedevelnp legical techniques fnr establishing implicatien; afterward we canuse these fer deriving thectrerns frnrn axinms in pestfeundaticnai systemsan any subject. fin the ether hand an asinm system fer legic is necessarily fctunc|a-ticna], and I weuld in cenclusien remark further that it is cf dubidusvalue—-esp-ecially in the legic cf truth functiens. This dnmain, after all,enjdys the lusury cf a decisirtrr prct'er:n-ire fer validity-—that is, amechanical test. Truth-value analysis afl’ttrds ene such test bf validity;truth tables atfnrd anether. transfermatinn inte cenjunctienal nerrna] fnrmaffnrds a third. Thus blessed, we sheuld be unwise tn tnalce practical usecf the axiematic mefltrtd in this dc-main. it is inferinr in that it affnrds nugeneral way pf reaching a verdict -bf inv alidity; failure tc diseever a prnnffer a scherna can ntean either invalidity cr mere bad luclt. Lultasiewics swcre by axiematic truth-fttnctiu-n legic as a traininggreund fer attiernatic methed in tnpre demanding dctrnains. I swear ratherby the sufliciency, untc the day, nf the evil thcre-bf. axiematic legic, withits schemata and its specific rules pf inference, is very unlilte pctstfc|unda-tipnal aztictrn systetns. Training in the latter, if wanted, is best snught assuch. its main cempenent is training in the recngnitinn er prnnf bf impli-catien, since implicatien is what relates pestfcundatidnal axinms tn thethenrems; and such training is a pervading purpese cf this btiplt. Therewill be seme specific netice bf pnstfctundatin-na] aaticnns in Chapters 42and 43. Meanwhile 1 have included this brief acc-aunt pf axiematic

Li‘. Axiums 59truth-functien legic partly fnr centrast and partly because nf the premi-nence cf the subject in earlier literature. HISTDRICEL HDTE: Frege, in 1li’t\"§t, was the first tn aaiematiac the legic nf truth functiens and tn state fnrtnal t1.1les r:-f inference. Frege*s system, lilte the -Lultasiewice system t[l',l—i_'3}, was cuuched in negatien and the crtnditiunai and preceeded by trtndus p-unens and substitutiutt‘. but the axinms were mere cutTtb-ersctme, and the rule uf substitutien vvas nut made explicit- Whitehead and Russell cuuched their system, liilll, in negatien and alternatien. The Luhasiewica system dates frem i*5i.?9.“’ The blicnd-Lultasiewica ttttium dates frem l\"5'l’i‘ and I931. The attiumatic appreach cnntinued tn dcminate truth-functien legic thrnugh the thirties, and a scere pf systems are in print. The first pttauf pf cnmpleteness ctf such a system is due tn Fest, I921.“ EIERCIS-E5 l. Preve ‘p —1- p .—1-. ,tTI —* qr‘ frem t[l]—[3} using the cendensed prnef netatien. 2. Shttw the independence cf {2} in detail. 3. In the hlicud-Lultasiewic: system, prdve: r|:s i..r'|r.:i::,n [.q Ir :|r.:|:.,n |.q Ir ' [which says ‘r |.s -P r !—t-:.p |.q Ir tit’ if we define ‘—1-’ as be~| fete] . if See Tamlti‘s,[.ng=ir-, .‘i'emu.utt'cs, it-fcteerdthernerics, p. 43. 1“ Fur a cc-mplclcness prncf shrtrter than m-nsl, see my Scfcrtert Legit? F'c,|!iIv!'t'.t. pp-155'—ltiTi.