E2. =[]ttrii'itiJ|'it\"rtt:'r:u-i itll H|5TU'li|'l:Ftl-. HUTE: Eluanlificatien, in a different but structurallyequivalent nutatiun, was introduced by Frege in I579. The quantifierswere rcntteretl as tr)‘ anti ‘(Hr)’ by Russell in IQDE, and this netatiencame dawn thrnugh Friatiuiu Mathematics and subsequent literature intnthe previuus editiens ui this buelt- Meanwhile the mure syrnrnetrical pair“liI\".=r' and ‘1r' has gained gruund. i‘-‘tn advantage is that the \"V\" can heused separately alung with ‘H’, as in Chapter ID. Same writers, wishingta stress the anaiugy uf the twe quantiliers tn cnnjunctinn and alte-rnatien.have used enlarged variants *i\"'t‘ and “st” uf their cnnjunctinn and alterna-titin signs instead cif \"\"IiI\"' and ‘ 3'.EXERCISES1- Rewrite these with help cif quantitiers: Juhn cannnt nutrun any man un the teatn, Inhn cannet nutrun every man tin the tearn.E- Rewrite the premises and cenclusien abeut the slnvenly and the pa-er {Chapter IT} with help uf quantifiers- Similarly fur the nther enam- ples in Chapter I? and in Eitercise I uf Chapter I? and Exercises E and -il uf Chapter 19.3. Supp-using the universe tn cumprise just a, b, . . . , ii, enprcss these truth-functienally:1rlF.r v III}, \"tI\",r[F.r v Ilr], \"tI\"',r[F,r . GI},1r Fr v‘ Elr tlr, \"lit Fr v \"tI\",r Ex, \"tilt Fir . ‘tflr E-Zr.which cnme nut equivalent tn nne annther?
Id! H- General Terms arid‘ Quunrifiers 23 RULES OF PASSAGE. MONADIC SCHEMATA in the schemata in which we have illustrated quautificatieri thus tar. thescape at\" each eccurrence nf a quantifier has been a truth tnnetinn at\" *F,r'.‘Gs’, etc- Hut we can be mere liberal. allewing same cumpunents uf thescnpe ta be devnid nf ‘Jr’ and represented simply as ‘pl c-r ‘if. It makes gnnd sense, e-g., tn write ‘Esta . F11’. This sheuld cemenut uue under just these interpretatiens uf ‘p’ and ‘F ’ that rnaite ‘p . Fs”true uf at least tine thing ,1“. These, uf cu-urse, are just the interpretatiensthat malte ‘p’ true and ‘F‘ true cif semething; sci the twe schemata:[1] Hrip . Fir}, p . HrF.rare equivalent. Ely similar reasening, sci are the twn schemata:[E] Help . Fr}, p .\"li\".r F,r. The same rnay be especled nf the pair:[3] Hxtp vFJr]=, p v H1 F:and again at the pair:{4} lIiI\".r[p vF.-:1, p v'P'IFx.These espectatitins can he cantirnietl by direct reasening rather lilte whatled tn if ll and [2]. But alsu, and perhaps inure instructively, we can derivethese equivalences frum the ethers. Talte (3)- ‘ §,r[_u v Fri’ ameunts ta‘—\"liI\":r —tp vFJ:}’ and hence. by D‘el'v'lc.=rgan's law, tu ‘—\"'iI\"J:|[,|5 . —Fs‘-Hut the ‘Vale . —F'.r]I' in this is equivalent by 1'1} tti ‘fl .\"i\"'.t —Fx'; suthe whele '— \"tI\"Jrt'f' - —F.t}’ becnmes ‘- [p . \"tI\":r —F,t}', er, by EIe!'vInr-
ll‘. Rules\" qfFrr.ts'age. i'rt'enri.|:i'ir .‘iTt\"iie'r-nete ll‘-l3gan's law, ‘p v —\"I\",r —Fs:’, which is tn say, ‘p v it Fir’. Preciselyparallel steps carry us frem ll} te {=1}- The equivalences:{5} Hxtp —1- F1}. pi —-I 3.1“ Flt\",{I5} llillirtp —-1- F1], pl —1- H.-I Fatpreceed immediately frem {3} and {-1}, respectively, since ‘pl —1-‘ ameuntste ‘p v‘. But the twe funher equivalences that the reader is new perhapsen the peint el eitpeeting are nnt ferthceming- What are right are ratherthe fellewing twe; nete the curieus twist-{Tl ElstF.=t —1- pl. ‘tilt F.r -—* _u.{El liI\"Jr{Fr —1- pl, 11' FI -I p-Fer all its eddity, {T} is easily grit frem {3}. *3.r{F.r —r,ujt‘ ameunts te‘3.r{-Fnr vpl‘, hence te ' Els —Fx vii‘, hence te '— ‘tit Far vp‘, hencete \"it\"s: F.t —1- p‘. {ti} preeeeds similarly frem {-4}. in future applicatiens el t|']—{'-‘ll a habit already acquired in Part [sheuld be ltept: that ef ignering erder in cenjunctiens and in altematiens.Thus, in the argument abeve l equated ‘3.r{'—FJt vp]t' te ‘Hr —-Fir vp‘en the strength eftfift, ignering the tact that {Ill shews ‘p’ rather te the leftef ‘v‘ and 'F.r' te the right. Cluantifiers are subject. in peint ef greuping, te the same cenventienwhich geverns the negatien sign {Chapter -4}; a quantifier applies te theshnrtest pessible ensuing sentence er schema. In ‘H,r{F,r —1- pl‘ thequantifier is required by the parentheses te apply te the whele cenditienal;in ‘Hi: Fr —1- p’, en the ether hand, the quantifier is understeed as lyingwithin the antecedent and applying enly te it. The distinctien hetween‘ E|.r{F.t --1 pl‘ and ‘El.r Fnr —1- p‘ can be repreduced in werds thus: There is semething st such that il'F.r thenp, If there is semethings such that Fir then p.Similarly fer \"i\".r{F.r —t,tJl' and ‘llit',r F1 —1*p':Everything,r is such that iiF.r thenp.lf everythingx is such that F.r thenp-Te appreciate the difference ell meaning in this latter pair, talte ‘F1’ as
l~'-‘l-='l H- General Terms anal Qaantifiers'.r centributes' and 'p' as ‘I'll be surprised‘- Then *‘tt\"r{F,r -1-_a}' says efeach persen, even the must genereus, that if he cu-ntributes I'll be sur-prised. It says that I feresee ne centrihutien at all. fin the ether hand‘ids Fir —*~_n' reflects ne such cynicism; it says enly that I'll be surprised{as whe weuldn't\"l] in the estraerdinary event that they all cnntribute. Inerdinary language the difference is drawn by eppesing ‘any‘ te 'every';'liI\",t{F.1: —*pl' says that if anyene centributes I'll be surprised, while\"'iI\".r F.-'1' —s+ _a' says enly that if everyune cuntributes I'll he surprised. In general the difference in English usage between ‘any' and ‘every’may have struclt many ef us as unsystetnatic and even rnysterieus. Scupeef the universal quantifier is the ltey te it. ‘Where distinctien is neededbetween breader and narrewer scepe, as between \"'i\",rt_'F_r -s pl-' and'l'l'.r F: —*p', the English speaIter's nncenscieus understanding is that‘any‘ calls fer the breader scepe and ‘every' fer the narrewer. This rulewerl-ts net enly in cennectien with the cenditienal but alse elsewhere,nutably in cennectien with negatien. Thus talte the universe ef discuurseas censisting ef all peems. ‘I de net ltnew any peein', then, and ‘I de netltnew every peem', call respectively fer the breader and the narrewerscepe: his —=[I ltnew Jr], —\"tI\",r{l ltnewn]-. The equivalences {l]—{'El are rules qt” passage. They shew hew temeve a quantifier acress a cenjunctien sign, an alternatien sign, er acenditienal sign. Rules ef passage fer muving quantifiers acne-ss a nega-fien sig were neted in the preceding chapter. in these twe equivalences: {R} Hr —FJt, -r 31.\" Fa.{ID} Hr —Fs, —\"i\".r F.r. The 'p' in the rules uf passage schematiees a sentence deveid ef ',r',such as ‘I'll be surprised‘. But alse the sentence in this pesitien ceuld besemething lilte ‘y will be surprised‘, such as te invite the eventualsuperimpesitien ef an eutside quantifier \"tl\"y' er 'Hy'. Te see hew such asituatien ceuld arise, suppese we are given this eitample fer analysis: There is semeene [seme cynic] whe, if artyene cantributes, will be surprised.
E3- Rates aj\"F'tt:-'stt_t,-e- |lr:t'tuttrtft't' St‘-ltett-ttttttt M5Lising the letter ‘y’ in eur quantifier this time instead ef '.r' {just fer achange], we can transcribe eur esample thus as a first step;Hytif anyenc centributes. y will be surprised].Eiut the part ‘if anyeue centributes, _v will be surprised’ has the ferm“c\".t{F.t —t _a]', er \"'Iit\".r{'F.t —1- t_?_'|-\"J- The whele thus becemes:t I ll 3_v‘tiI\".r{.r centrihutes —1- _v will be surprised].an instance ef the schema ' Hy\"P'.r{F.r —t- Cy)‘. The alphabetical eheice af the letter af quantificatian is ebvieuslyarbitrary; ‘Ht H.t' and ‘Hy Hy' beth say simply that semething is an H-The abeve esampie ef nested quantificatien shews, hewever, why wewant a variety ef letters te cheese frem; we need te lteep eur cress-references straight. We want it te be clear in '3y\"t|\",r{F.r —-1- G_v]' that thefirst quantifier relates te the censequent 't'_'t_v' while the secend relates tathe antecedent ‘F.r'. Espressiens such as ‘.r centrihutes’, ‘_v will he surprised', and ‘.r een-tributes —t- y will be surprised‘, which are like statements escept fercentaining ‘.r' er ‘y' witheut a quantifier. are epen setttences; see Chapter21. [tpen sentences are neither true ner false, but they may, like terms, besaid te he true af and false qt\" varieus ebjects. The epen sentence ‘.r is abet:-1-:' may, like the tenn 'beek' itself, be said te be true ef each beelt andFalse ef everything else; and it is a beat: . .r is bering' may be said te betrue ef each bering heel: and false ef everything else. ‘.r = it‘ and it is aman —1-: is mertal' are true ef everything. In general, te say that an epensentence is true ef a given ebject is tu say that the epen sentence becemesa true statement when *s' is reinterpreted as a name ef that ebject. Thenetien ef the estensiaa ef a tertn {Chapter ltl-‘jt liltewise carries ever teepen sentences: the e:-ttensien ef an epen sentence is the class ef all theebjects ef which the npen sentence is true. Since the schematic letters '_u', 'q', etc. have ceme new te stand aswelt fnr epen sentences, e.g., 'y will be snrprised', as fer statements, Ishall talte te calling them sentence letters. Nate that these and theschematic term letters ‘F ', '5', etc. differ basically in functien frem ‘it’,'y', etc. Wliereas ‘.r' can appear in sentences—even in clased sentences, with help ef a quantifier—en the ether hand '_t:t', 'q', etc. and ‘F‘, '5',
lttti H. General Term at-tel Qmsrmtyiersetc. cannet appear in sentences at all;\"‘ they are tuerely dummy sentencesand dummy terms, used in schemata which depict eutward funns efsentences- The letters '.r', ‘y’, ‘.3’, and ethers se used, are variables. Care must betalten, hewever. tn diveree this traditienal werd ef mathematic-s fmm itsarchaic cennetatiuns- The variable is net best theught ef as semehewvarying threugh time, and causing the sentence in which it eccurs te varywith it- Neither is it te be theught ef as an unknewn quantity, discnver-able by selving equatiens. The variables remain me1'e preneuns fercress-reference; just as '.r' in its recurrences can usually be rendered 'lt' inverbal translatiens, se the distinctive variables '-r', 'y', 't', etc-, cerre-spend te the distinctive prennuns *fnrmer' and ‘latter’, er ‘first’. ‘sec-end’, and 'third', etc. The statement: 3-r[Sadie stele x at the Htnperium . 3y{'Sadie exchanged -r fer yllctnrespends fairly literally te the werds; There is semething such that Sadie stele it at the Empntium and such that there is semething such that Sadie exchanged the farmer fer the latter. In the sentence:llll H-r{y is uncle ef tr]the eccurrence ef 'y' is free, there being ne \"tt'y' er ' Hy” present; but thee-ccurrences ef ‘x' are beund, because ef \" Ex’. Clue and the same eccur-rence ef *-r' may be beund in a whele sentence and flee in a part; the finaleccuirence ef '-r' in {I2}, e.g., is beund in {I11} but free in the part ‘y isuncle ef:‘. In ene and the same sentence, mnreever, ene eccurrence at‘-r' may be free and ethers beund; this happens in the cenjunctien:{ I3] I is red . lI\".r{.t has mass},in which the quantifier has te de enly with the secend clause. {I3} means‘J is red and everything has mass’, and ceuld just as well be written withdistinct variables:_ 3 Escepliert: they may appear within quetatien martts in sentences. E-ut even ,1 mean-tngless mark may appear within quetatiun marl-ts in a sentence- The quetatien as a whele isa meaningful name ef the meaningless mark.
E3- R't.ties tty|\"Frt.v.ttt,ge- McvtatIit' St-hemntn I4? -r is red . \"t|\"_v{y has mass}- Ely starting with ',tt', 'a', ‘F-r', 'Ct',r', 'F_'t\", etc., and applyingquantifiers and the truth-fnnctienal netatiens, we ebtain the rttttrtstct'ft'qattnttjicatienal scltenmta. The re strictien 'menadic' is due te the fact thatwe have net yet admitted elements lilte “Fry”. Chtantificatienal schemata,lilte sentences, will be called epen if they centain ene er mere freevariables, and etherwise clesed. Thus the schema: \"tt\".r[F-r —r Ely{G_t' . H-ti]is clescd, but its parts: Fr é Hylfiy . H-rl, Hyttly . H-rl, Cly . H-rare all epen. Tmth-lunctienal schemata. e.g., ‘p ~—t- a', ceunt as clesedquantificatienal schemata. tt quantificatienal schema, epen er clesed, is valid if it cemes eut trueunder all interpretatiens in all nenempty universes- Ft term tenet is inter-preted, as befere. by settling what things in the universe it is tn be true ef.tit sentence letter is interpreted, as befere, by assigning it a tmth value. titfree variable, finally, is interpreted by assigning it seme ebject in theuniverse. The cencepts ef censistency, implicatien, and equivalence ex-tend in similar fashien. HISTURICHL HUTE: The rules ef passage were explicit in\"w'hite.head and Russell, I915. The term is frem Herbrand- The term‘epen sentence‘ has been used by Carnap and ethers. The elder term ferthe purpese is 'prepesitienal functien‘, but this can he misleading, since afunctien in the mathematical sense is best seen as a certain type ef relatienrather than as a netatien.EIER CISESl. Suppesing the universe limited tea, h,, . . , it, expand {T} and {S} by expanding the quantiticatiens inte alternatien and cenjunctien.
Id-E H, General‘ Terrier and {Qaaattfiers2. There are nu rules ef passage fer 's—r'. Shaw, en the cuntrary, that nu twe ef the schemata: \"til\"-r{p HFI]1 p -t—t- lit’-r Fr, _a 1-1 1r Fr, Help tt-1-Fit] are equivalent. lvlethed: See what cemes ef reselutien when ‘T‘ is put fer ',n' thruugheut, and again when ‘J.’ is put fur it.3. What implicatiens held between the feur schemata in Exercise 2? lvlethud: Same. 24 PRENEXITY AND PURITY The rules uf passage werlt in either et' twe directiens: te widen thescepe ef a quantifier er te narrew it. The ene directien ends by bringingthe ferrnula inte prenex fenn, where all quantifiers stand in an initial ruwguverning all the rest ef the fermala. The ether directien ends by parity-itrg the furmula, by causing the scepe uf each quantifier tu be a truthfunctien enly ef cempenents each ef which shews free eccurrences af thevariable ef the quantifier. Beth et' these uses ef the rules ef passage depend en preparaterytransfermatiens uf ether kinds. lt' the ch esen directien is the ene that leadste prenex ferm. then certainly we must paraphrase away any eccurrenceef '-t—x' that jeins furmulas having quantifiers in either ef them. Fer wesaw in an exercise that there are ne rules uf passage fur 't—t~'. tltnyquantifier that is trapped in either ef the furmulas jeined by ‘H’ willremain trapped and incapable uf being breught inte prenex pesitien untilwe get rid uf ‘H’ by paraphrase. Ftnether transferrnatien preparatery te applying the rules uf passage ina prenexing way is the relettering uf beund variables. ‘Fr . Hr Gr’ is allright as it stands, simply as a cenjunctien ef the upen schema ‘Fr’ and theclesed schema ‘dtere are C‘ {which ceuld as well be written ‘Hy fly‘)-
24. l\"rene.tt'ty anti‘ Pttrtrv ataThe whele says that .r is an F and dtere are G. Hut it will net du tn applythe rule uf passage {I} ef the preceding chapter te 'F.r . 3,1: Gr‘ tu ubtain‘H-t{F.t . tIF.t}\". This last ceuld mean that semething is rettnd and square,which is false, while the furmer means that.r is reund and semething issquare, which is true ef any baseball .r. The thing tu du befere applyingthe rule ef passage {'l} tu 'F.t . El.r tI.F.r' is tu rewrite this as ‘Fr . Hy -Elly’;then we can preceed tu '3ylF.t . Gy}'. We must remember that {I}simply dues net directly apply te ‘Fr . Ex Gs’: the ‘p' in {I} standsnecessarily fer a furmula witheut free ‘.r’. Ctnce these prcparatiuns are made, we cenvert a schema inte prenexfenn simply by a successiun ef transfermatiens accurding te the mles efpassage {I}-{lfl}. Example: p H lIl\".r[Fs -I 3_v{F_v . t_T.r}|.Paraphrasing the ‘-H-' intu a cenjunctien ef cenditienals, we get: p —1- ‘tiI\".r|F_r —1- H_v{Fy . t?-r}] . lt\".rl.F;r —1- 3y{'Fy . G-r}] —ttp'.lie-lettering,p -+ \"tI\".r[F.r —“=t 3y{F_v . t?.r}l , ‘tiI\"slFs -1 an-*{F1-v . tltll —‘*p.The remaining transfermatiens are by the r1.tles uf passage {t5}, {T}, {Z},{I}. {Ii}. and {S}, twe steps at a time.\"lit\"-t[p —x, Fx —1- El;-'{F_v . tl'.:r}] . Hs[Fs —1- 3n.={'Fn- , [lg] _-1-p].\"li\".r Help -1. Fr -1 3_vtFy . C\"-r} : Ft: —1- 3tv{Fvt- , tflg} _-vp].\"tt'x Help —1- Hy{'F.r —-\"1-. Fy , fir} . En-{Fa —1-. Fit‘ . Gs} -—1~p].lit'.t Halaylp it-I Fr —t. Fy . tlr} . ‘tl\"n-{Fa —1-. Fit\" , Ga :—tp}].\"Ii\".rHsHyltt\"n={p A-; F-r -1-. F_v . Gs 1. Fa —1-. Fnr . Gs t-—>p}.ltlutc that at almest every stage there is seme freedem ef eheice as tewhich quantifier te bring lurward next. The distributivity laws til} and { ill} uf Chapter Z2 can cenduce te speeditt prenexing and te brevity in the result. Cunsider the example:{I} \"tf.rl H_v{F.r s—* =tS}‘.l v Hy Fy] . \"'tI\".r{F.r v Git}.
lfifl H. Gerterrti‘ Terttts rtasf Q!-l'|’.ll'l'l‘i:,iI‘It\"l\".lt'tltccerding tu the plan just new laid, we weuld begin tu prenex thisexample by relettering thus:I2} 'Iit'.r[ 3y{F.t H Ely} v 3: Fa] . \"iI'iv{Fw v Clw}.Frem this result by six appticatiens ef the rates ef passage tt}—t3} efChapter 23 we weuld meve finally re the prenex schema:iii} ll‘-r H_v3s\"tI\"t-v{F.r H-Gy .v Fr : Fw vt‘3‘w}-But if instead uf relettering tf I} as {Z} we apply the distributivity laws {it}and llfl} ef Chapter 11 te { I}, we get this: 'lt‘.r[H,v[F,r -H-Gy .v Fy} . .F.r vtS.t\"].tlt single applicalien uf the rule ef passage { l} uf Chapter E3 tums this intethe prenex schema:t-='l} \"II\"-r HylF-r H-t'}y .v Fy 1 F1‘ vtlt},which is simpler than the previeus prenex result IE3} and mere quicklyreached as well. Fer the salte ef such benefits we can be well advised eventu reletter quantificatiens te create duplicatiens rather than ta eliminatethem. If we had been given {2} te begin with, we weuld have gained byrelettering it as {I} and then preceeding as just new ebserved, Here,hewever, that this strategy af distributivity is the eeunsel enly ei‘ effl-ciency; schemata can always be prenexed witheut it. What we strictlyneed re cle is just paraphrase any bicenditienals that centain quantifica-tiens, diversify all variables ef quantificatiun, and apply the rules ufpassage- In relettering it is important always te aveid cellisiuns ef variables. It isall right te change ‘Hyl_‘F..r it Gy}' te ‘3.tt,‘F.r it Gel‘, but decidedly wrungtu change it te ‘H,t{F,t vGx}‘; fur this, far fmm being a superficialalphabetical change ef netatien, is a cmciat change ef structure. ‘W‘l'ten aquantificatiun is relettered, the schema that stands as flte scepe ef thequantifier rnust ceme te exhibit free eccurrences ef the new variable at allatttzl ettly the places where it had exhibited free eccurrences uf the eldvariable. The ‘Fx v Gs‘ ef the example abeve has free ‘s‘ where and ertlywhere ‘Fx v‘ Gy' had free ‘y‘; the ‘F1 v Cbr‘, en the ether hand, has free'-r‘ where but net enly where ‘Fr v Gy‘ had free ‘y'-
Ed, Fret:-e.rltjv anal Parit_'r I5I ‘Whereas prenexing brings quantifiers eut, purificatien drives them in-Furiticatien, litre prenexing, rests en preparatery transfermatiens. Thuscensider the impure quantificatiun ' 3.r{F.r . p v t-\"i.r}‘, whese impurity is‘p', lacking ‘.r‘. As a preparatery transferlnatiun we cenvert the scepe‘Fr . p v t.?.r' nf the quantifier inte altematinnal nutrnal ferm and thendistribute the quantifier threugh the alternatien, thus: Hx{F-r . p ,v, Fit . Git}, §l.r{F.r . p} v 3x{F,r . tlr}.blew we can apply the mle ef passage ll} et Chapter 23, cempleting thepurificatien. Ex Fx . p .'v' 3x{'Fx . Cr}. Te purify a universal quantificatiun whese impurity was similarlyburied, we weuld begin by cenverting the scepe uf the quantifier tecenjunctienal nermal ferm; fer a universal quantifier distributes threughcenjunctien. E-g-, by steps dual te these abeve, the impure quantificatiun“'iI\"xI[F.t v. p . t_'?.r}' fares as fellews:\"lt\".r[F.r vp . Fx v Ex},\"tf.rt_‘F.r v p} . \"tt'.r[F.r v tlrl.\"tt'.t .F.r vp .'tt\".r{F‘.r vtlix}. Ely a cemhinatien uf these metheds we can rid any quantificatienalschema ef all impurities. The general reutine is as fellews. 'ilt'hereverthere is an impure quantificatiun, put the scepe uf the quantifier intealtematinnal er cenjunctienal nermal ferm {aceerding as the quantifierwas existential er universal}. Hewever, prier te this and at every stage efthe precess, apply any applicable ene at the mles ef passage {l}—tE} efChapter E3 en sight—always su as te drive quantifiers inward. Centinueuntil all impurities are gene- E.g., let as purify {I}. By the t'ttle ef passage {-ft] ef Chapter E3, {I}becemes:{5} llllx §it'l_F.t 1-?-*Gy} v Hy Fy . \"tt\".tl‘F.r vllt}.hlext we turn the ‘Fr -Ht Cy‘ uf the impure quantificatiun ‘Hyt_Fx -Ht t]Fy}'inte alteinatiunal nermal ferm and distribute the ‘3y' threugh it. Thewhele ef {5} becemes:
I52 ll. General Tertns and {§!unnt.{tiers- \"i\"‘-r[Hyt_‘F.r . Gy} v Hy{—F.r . —t.?y}l ‘It’ Hy Fy . 'tiI\"..tt}f‘.r v Gr}-l\"~luw we apply the rule uf passage {I} ef Chapter 23 twice, getting: \"tt\".r[F.r- Hy Cy .v. —F-r . Hy —[ly] v Hy Fy . 'tt'.r{.F'.r vtlr}.l\"~lext we ntrn the scepe ef the initial \"tl'.r‘ inte cnnj unctienal nermal ferm.The whele becemes: vet.-at v Hy —t.'ly . Ely Ely v —i-'x . Hyfiyv Hy —t3yl v Hy Fy .\"tI'.rtF‘.rvt-Fixl.l\"~Iuw we can apply the mle uf passage {'1} uf Chapter E3. \"tI\".r[F.r v Hy —-Ely . Hy Gy v —F.r] . Hytiflyv Hy —Ct‘y ,v Hy Fy ';'tl‘xtF.rvtSx}.l\"-lext we distribute the ‘li\".r'. ‘I\".r{F.r v Hy —Gy} . \"I'.r{Hy Gy v —F.r} . Hy Gy v Hy —-Ciy .v Hy Fy :lI\":r{F.r vtlr}.blew we can apply the rule ef passage {4} ef Chapter 23 twice.lb} \"liI‘.xFx v Hy -—C-‘y . HyGyv\"tt'x —F.r . Hyfiyv Hy —-Gy ,v Hy Fy:b':{F,rv-Gr}.This at last is pare. If we transcribe \"ll'.r‘ as ‘-— Hr-—‘ and then drep allvariables, we can cendense the whele inte the Heulean schema:tn -aFvstt\"t.stuv-aF.Etevs|ts,v ElF:—3—{Fvt5'}.Thus the eriginal schema {i}, the prenex schemata {Ii} and {-1}, the pureschema {ti}, and the Heulean schema {T} all are equivalent. An alternative methed et purificatien is alse wnnh nuting. Here thequantifier may be universal er existential indifterentty; let us represent itas ‘fl.r‘. The impurity. let as say, is ‘p’: se the quantificatiun may bepictured as ‘t,','_t.r{. . .p . . .}'. New this is equivalent te:{S} p.fl.r{‘..,T,.§t..v.p.fix-|[...l.-.}
E-\"i- Pretre.rt'qr atra’ Parity I 53—|et the reader reasen why. Finally we resulve ‘T’ and ‘J.’ eut uf ‘, . .T- - .’ and '. - --I. . . -‘ by u'uth-functien legic- The instructiuns, we see, areadmirably brief- The reader weuld de well te run seme cemplex examplesthreugh by heth metheds and cempare them fer efficiency. tt menadic quantificatienal schema has impurities as lung as it hasstacltcd quantifiers, i.e., as leng as any quantifier has a quantifier in itsscepe. Fer, if a schema has any stacked quantifiers, then it will have atleast twe innermest stacked quantitiers—say ‘Hr’ and \"IiI\"y‘, By this Imean that the scepe ef ' Hr’ is a truth functien uf schemata ene ef whichis the \"'tt’y‘ quantificatiun, and the \"'ll\"y’ quantificatiun cuntains nu furtherquantifiers- Eat then either the ‘H-r’ quantificatiun er the ”'iI\"y’ quantifica-tiun is beund tu be impure- Fer, if the \"'Iity’ quantificatiun laclts free ‘.r’, itrenders the ‘ H-r‘ quantifi catien impure- if en the ether hand it has free ‘x’ ,then the scepe ef \"'iI\"y' is a truth functien uf schemata ene ef which is ‘Fr’er ‘-5.1:’ er the like; and this renders the \"'li|\"y‘ quantificatiun impure. We have seen hew tu purify any menadic quantificatienal schema, andnew we have seen further that the result will be deveid ef staclredquantifiers. Each quantificatiun will have as its scepe a truth functienmerely uf ‘Fx’, ‘Cir’, etc-, if ’.r’ is the variable uf the quanfifier; ‘Fy’, ‘p’,etc- are driven eut by the purificatien, and likewise all staclred quantifiers.Then, if we transcribe \"tt'.r’ again as '- H.r—' and then drep the vari-ables, as we did in passing frem {=5} tu {'7}, all quantificatiens will disap-pear in faver ef He-ulean existence schemata. We cannet quite say that every me nadic quantificatienal schema is thusreducible te a Eealean schema, such as {T}. {+5} was thus reducible be-cause it centained nu sentence letters ‘p’, ‘a’, etc., and ne unquantified‘Ft’ er the like. But we can say that every menadic schema that is clased{and hence deveid ef unquantified ’Ft‘ and the like} is reducible in theebserved fashien te a truth functien ef E-eeiean existence schemata andsentence letters. The methed cunsists in purifying and finally transcribinguniversal quantifiers and drepping variables. It thus emerges that the nutatiunal apparatus ef menadic quantifica-tienal schemata, with its variables el’ quantificatiun and all its stacltedquantifiers and impurities, is needlessly inflated. The letters and truthfunctiens nf Part t and the Eteelean existence schemata weuld have beenapparatus eneugh. Still, there is an impertant reasen fer talting up theinfiated apparatus: it prepares us fur the advent in Part Ill ef pulyadicschemata, centaining ‘Fry’ and the Iilte. t‘-‘it that peint the ltind uf reduci-bility here ebserved will lapse.
I54 if . General‘ Terms atta’ fllaantrjfiers HISTUIICAI \"UTE: Twe names are particularly asseciated re-spectively with prenexing and purifying: these ef Skelem and Behmann.Skelem depended heavily en the prenex ferm fur his preef precedure inpulyadic quantificatiun theery, much as we shall de. Cln the ether handBebmann's decisien precedure ef I922 {see end uf Chapter Iii‘ abeve}depended en driving quantifiers in.EIEIICISESl. Put the schema \"Itt’.t Fir —-it, \"tt\",r tlr -H ’Il\",r Hx’ inte prenex fenn-Il. Likewise ’Hr Fr -H Hit G-r .—I- H.r Hr‘.3. Purify ’H.r\"t\"y{F.t H lily .—r H_v}' by the first methed.4. Likewise \"tf:r Hyilfx —t~ Cy .~H- H'y}’,ii. Repeat It and ti by the secend methed-I5. Establish by infermal argument the equivalence er {B} te ’flr{. . .p- . .}’- 25 VALIDITY AGAIN A universal quantifier is indifferent tu the validity ef the schema that itgevems. This is evident frem the meaning ef universal quantificatiun. Tusay e.g. that “tt\"y{F'y —1- Hr F.r}' cemes eut true under all interpretatiensef ‘F’ is the same as saying that ‘Fy —1- HI Fx‘ cemes eut tme under allinterpretatiens uf ‘F’ and all assignments uf ebjects tu ‘y’; and sueh is themeaning ef validity ef ‘Fy —tt Hx Fx’. {See Chapter 23.} Curre-spendingly an existential quantifier is indifferent tu the censistency ut’ aschema that it gevems. We can test an epen schema fer validity, then, by testing its universal’citxture. This is a clesed schema that we get frem it by prefixing auniversal quantifier fur each free variable. In erder tu test a clesed schema
25 . lt‘ait'r.Ir'l‘_v .~t,cat'n I 55fer validity, se leng as it has nu sentence letters, we reduce it tu a Eeeleanschema and then apply the methed uf existential cenditienals er the cel-lular mcthe-d {Chapter I'll} as we please. Hew tu reduce it te a Heeleanschema was illustrated in Chapter 2-1 in the transfurmatien et{1ipreeres-sively inte {S}, {I5}, and finally {T}, Let us test the epen schemata: ’Iil\"xFx-I-Fy, Fy—*H.-'tFxfur validity. lltfl remarked, this means testing the clesed schemata:\"Il\"y{\"tt\"x Fx —r Fy}, ’IiI\"_v{Fy —r Hr Fx}.Purifying them by means uf the rules ef passage {bland {ll} ef Chapter Z3,we get: \"tl’.rl\"‘x-i-\"c\"_vFy, HyF'y-—r H-r.Fx.If despite their patent validity we persist in reducing these te Buuleanschemata, we get: —HE-1-—HF, HF-1-HF.Fer a less trivial example, censider the inference:Fa, \"Iit’.rl Hy Fy —-I Gx}, Hy{F'y . Gy}.Te check the implicatien we test die epen cenditienal:Ft: . ’tl'.r{ Hy Fy —1- Cx} ,—t~ Hy{Fy . Ely}fer validity. This means testing the clesed schema:’Il\"i;[Fr, . ’tl\".r{ Hy Fy —r flr} .—-It Hy{Fy . t'_Ty}l-Pruceeding te purity it, we apply successively the mics uf passage {S},{I}, and {I5} ef Chapter E3.Hell-\".2 - \"t\".t{Hr Fr —tt?xl] —* HyI.F_r - Gr}.He F: .\"I’x{HyFy —1-C-'x} .—-I Hy{Fy . Cy},Ht Fa . Hy Fy -1 \"tI\".r Gx .—1- Hy{Fy . {fly}-
lfiti H. Gerterrrl Tr'rrh.'t rrflrf E-'tttt-'ttt'_.I'it=*t‘s'Then we transcriii-e the universal quantifier and drep the variables. HF.§F—-1-—3G.—1- HFCF.Eheesing new the methed ef existential cenditienals, we tam this intueeiijunctienal nermal ferm and simplify. lt hctils cluwn te: —3Fv Hfiv HFG.Cellecting the afiirmative existeatial schemata as ‘ alt? v FE)’, we endup with the existential cenditienal: HF -s Hit? vi-\"G1.We checlt its validity by criteritm (iv) pf Chapter I9. That is, we citeclt byfell swqctp that ‘F’ implies '5' v FIE’. Hete that this technique, again, depends fer its snundness rm a law efinterchange pf equivalents. Again, hnwever, as in the case tlf truth-fuactienal schemata {Chapter 9} and Bepiean schemata {Chapter 19], allis in erder; fur it is evident here again that the truth value nf a cempeundstatement depends cm ne features at the cempenent sentences and termsexeept their heing true er false er true el’ semething er pt‘ nething. We turn finally ttr the questien cif a validity test fer clesed menadicquantificatienal schemata centaining sentence letters. Such a schema willbe valid, clearly, just in case it resctives te ‘T‘ er te a valid schema undereach substitutien cif ‘T’ and ‘J.’ fer its sentence letters. Se the test istruth-vaiue analysis. If the analysis issues in ‘J.’ at any peint, -cif ceursethe schema was net valid. U'the1\"wise we fclllclw up with validity tests pfany quantificatienal schemata issuing frem the truth-value analysis. Example:\"li\".tI[F.t HP} - HIFI 1-1-q .—1-. p Hq h.=_'fisidr'.'‘li'.t'{F.r=-HT] . EIFJ Ha .—1+. T1—\-q \"|i|'.rF.t.3J.'F.t'-i—1r_|r.—!~qr‘tl\".tF.t.El;t'F.tt—1~T.—+'T \"t|\".tF.r.H.xF.tH_L .—1J_ T —f\"II\".r F::.—H.tF.t\"} rig.irtsr'd'r'.''lfJ:l[F.=|.' Hi} . EJFI -Hr; .—-I-. 1. Hq\"fir —F.r . H.xFJrs-as .—>a‘li|\".1r -rF.t+3.tFx-1-I-T.—1-J. \"I\".t —FJr.El.xF.ts—I~J..--1-T—t_\"i';r --Fnr . 1rFx_} T
25- l\"rtir'ri't't'_'t' ..-'=l,t.1'|:I.'t'I'I I 5TI’The questien ef validity here reduces te the questien ei validity ei ‘ — {\"'t\"'.tFr . — 31 Fri‘ and ‘-{\"tl'.t —Fx . Hr Fri’. These ge ever inte Eeeleanschemata as ‘—t[— HF . — HF)‘ and ‘—~tj—- HF . EFT. The latter has thevalid ferm ‘—{pp]|*- The ether gees inte cenjunctienal nermal ferm as‘HF v HF‘, and se, by cellectien ef aftirmative existence schemata.'3 [F it F1’. This is vaiid by criterien ti] ef Chapter I9. Dismissal ef ' -— [— HF . EFT as ef the ferm '— |[p_t:Fj' was an ebvieusshertcut, falling eutside the mechanical reutine thus far set dewn. Thatreutine weuld prescribe cenverting '- I[- HF . EFT te cenjunctienalnermai ferm as ‘HF v — HF ’, then rendering this as an existential cen-ditienal ‘ HF —1~ HF‘, and finaily ehserving that the term schema in theantecedent implies the ene in the censequent by fell sweep. it is werthneting that the mechanical reutine cevers such cases, but it weuld be sillyte fellew it- When truth-value analysis is applied te quantificatienal schemata, asabeve, feur supplementary rules ef reselutien are semetimes needed inatlditien te these in Chapter 5. Hamely, \"'tt'.tT'. ‘ H.rT’, \"\"If.t.l.'. and‘Elsi’ reselve tn ‘T‘, ‘T’, '1‘, and ‘J.’. Une ef these is used when wetest the fellewing rule ef passage, {E} ef Chapter 23. \"tI\"x[F.r Apt t-1-. 1rFx ap\"lit'.r[Fr—rT]| H. 3.1? Ft -1-T ‘tf.t{F.t~—1-.l.] -H-. 1rF.r—t-1. \"li\".rT \"li\".t —FJr -Hr — 1rF.t TThe remainder ef this test ceitsists in testing the leftever bicenditienal\"\"tI\",r —F.t H — Elsi F'.:r' fer validity. It gees ever inte the truth-functienaliy valid Eeelean schema ‘~— HF H — HF‘. We cenclude with a fuil—dress example, starting in werds.Passttses: If the Bissages repert is te be trusted tiien the chnrgécenctttsiert; d’n_fi\"'nires is a mere teei ef the sisal interests and nene ef the natives really favered the ceupen plan. If the charge d’n,[i'ut're.s is a mere teel ef the sisal interests then seme ef the natives either really fa- vered the ceupen plan er were actuated by a per- senal anintesity against the deputy resident. If the Eissages repc-rt is te be trusted then seme whe were actuated by a persens] animesity against the deputy resident did net really faver the ceupen pian.
I55 H. General‘ Terms and QtuantijiersPutting ‘p’ fer ‘the Hissagns rep-ert is te be trusted’, ‘rs’ fer ‘the charged‘u_fi'uires is a mere teel ef the sisal interests‘ , \"F.r‘ fer \"Lt is a native‘, ‘Gt’Fer ‘.r really favered the ceupen plan‘, and ‘HI’ fer ‘.1’ was actuated by apersenal animesity against the deputy resident‘, we can represent thepremises and cenclusien by the mixed schemata: ,a -a. q .\"I'.r{F.r —a ~t'_\"=':r], q —a §_1't_'F.I . tilt’ '|tI'HJ.'], ,t=- —+ 3.r[Hs . —t?x}-rltceerdingiy we submit the cenditienal:pr —-r. g - 'lI\"xtF.x —-P --G1] 2 qr —1- 3:ci[F.t . C-Lt 'vH.r',l :—I~. p —* H;tl_'H_r . —-Git}te truth-value analysis. Let us save space by ahbreviatiltg the threequantificatiens as Z-1‘, ‘B’, and ‘C‘- P-red -s-*5-—r-r-'—r-ITT-—rr§n-‘I-q—rH.-—r.T—*C J.—1-t]|vl.qr—rB.—+..l.—tCuat . q —~i-E’ .—rC TTat .T—-iii‘ .——a-C L4 .1.-1.5‘ .—i-CAH -1 C‘ TSe eur prehiern reduces tu a validity test tn‘ ‘AB —1- E’, which is:\"lI'.t|[FJr —1- —~EF.x] . 3.xt.F;t . Gs 'v'H.tf| .—r E.rt'Hx . —Gir‘,l.Transcribed as a Heelean schema, this becemes: - 3- {F—\"=> G1 . 3|.-'~tt;\" vat; .-1 Had.er, in cenjunctienal nermal ferm, 3- tr at-ct v - state vH\",|] v sue.When we eellect the afiirmative existence schemata, this becemes: - Butte vH]1 v at-tr -set vHG]and se the existential cenditienal: '
ZE. lr'etr'n\"itjv rlgttirt! I59‘ 3|F{Ci vH}]-1» H[—tF—tG]vHt'T?].Finally we checlt the validity ef this, aceerding te the criterien [iv] elChapter I9, by verifying that ‘Fit; vHj:‘ implies '-[F -—-1-C] vHt\"_'i‘.This again is dune by truth-value analysis- F{'GvH}—1-l—{F—+-GII vans]F{TvH'}—r|—--[F—-I-L}vHJ.] FtivH}-—>[—-[F—sT}vHT] F—rF FH—aH T TEHERCISESI. Test these six schemata fer validity- \"tI'y=[Fy —t- \"Iii\"-it Fr], §ly[Fy -t ‘I’-tr F-tr], \"'t\"yt[ H-t F-t —+ Fy}, Elyt it F-‘t’ —1- Fy}, 3_vt\"tI\"-1' F-I —-+ Fy}, H_ytFy -r H-t F-Ir}.1. Checlt the bicenditienals cerrespending te tl'_|—t'l'] ef Chapter E3-3- Test each ef these pairs fer equivale nce, by testing the bicenditienal. ‘ti’-t F-r, \"tI\"riF-t vtlt] . \"lt\".t{F-r v —t11r]-; V-r F-t, \"til-i*[F.t . tilt] v \"tIbr{F-I . -5-ti; 3.1: Ft. 3.r{F.r ‘v‘ tilt} . 3.1-“(F-t v —t'_F-ti; H-t Fr, H-tI[F.r . G-1:} v 3r|['F.t . —t']-r}.4- Checlt the fellewing argument.Palstvttsss: The persens respensible fer the recent kidnap-cetvcttrstetv: pings are experimental psycbelegists. - if ne experimental psychelegists are ltnewn te the police, then nene ef the fermer besses ef the beetleg ting are experimental psychelegists. If any ef the ferrner besses ef the beetleg ring are respensible fer the recent ltidnappings then seme experimental psychelegists are ltnewn te the pelica-
IE1\"! H . General Terms rind‘ |Qlt|'.|5|'i'l'l'!:_|lf-E\"|\"'.'!-' 26 SUBSTITUTION In legical theery the schema is what is in fecus, but still it is thesentence finally that matters. The purpese ef schemata is simply te facili-tate the legical study ef sentences by tlepicting dieir legical ferms. Whatrelates the schemata te the sentences dtat they schematiae is substitutien.We get the sentence by substituting in the schema- It is time te examinethis cennectien. Fer truth-fnnctienal schemata the relevant nctien cf substitutien wasclear and simple: just substitute sentences fer sentence leuers. and alwaysthe same sentence fer the same letter. Fer Beeiean schemata the nctienwas equally simple\". just substitute general terms fer term letters, and thesame term fer the same letter. The meve te quantitieatien cemplicatcs substitutien by allewing ‘F-r’as a whele tn stand fer an unanalyted sentence. Clearly we cannet spealtsimply ef substitutien ef arbitrary sentences fer ‘Fir’, 'Fy*, etc-; a certaincerrespendence must be preserved between the substitutes fer ‘Fx’ and‘.Fy*. What cerrespendence‘? First appreximatien: the substitutes must he sentences that are alilteexcept that the ene has free eccurrences ef ‘.i:' in all and enly the placeswhere the ether has free eccurrences ef ‘y’. Examples: Fa Fy{1} -r is preud ef the team, y is preud ef the team-[ZI .t is preud ef t’, y is preud cf .t-[3] .t is preud efx, y is preud efy-[4] 32-'I[.r is preud ef .1], Hail’ is preud nf .3)-In each pair the ene sentence says ef.t what the ether says ef y: that he ispreud ef the learn, preud ef t, preud cf himself, preud ef semething.Still, the requirern-ent that the free eccurrences ef ‘.t' whelly match theseef ‘y’ is tee restrictive. Fer, aleng with pride cf self, pride ef semething,pride ef flie team. pride efc, anether alternative that needs tn be ailewedis pride ef::[5] ;t is preud ef-r, y is preud ef-r.
Ed, Errb.rrt'ttrtt'ert l I5 IThis is net te disallew the pair {fl-II; it is additienal. Beth pairs need te helecegnired as legitimate and different pairs ef substitutiens fer ‘Fir’ and._;-Ft; Thus the desired requirement, vaguely spealting, is that the sentencessubstituted fer 'F.t' and ‘Fy‘ be sentences that say respectively ef..t and ysum.-;=' ene same thing—-be it pride ef self er pride ef.-t. The right device lies ready te hand, baclt in Chapter 1|: term abstrac-tinn- We have merely te recegnite ‘Fl separately as a term letter again.Substitutien fer ‘F * in ‘F-Ir‘ eensists in twe steps: in putting a suitable termabstract fer *F‘ in ‘F-t’ and then perferrning cencretien {id} er I113} efChapter El)- Similarly fer substitutien in '.F'y'- Fer ebtaining the pair efresults fl} the apprepriate abstract is ‘lw: w is preud ef the team} '; fer {Z}it is ‘lw: iv is preud eftl‘; fer {3} it is *{w: w is preud efw]-’; fer {4} it is‘lw: Hztw is preud ef all‘: and fer {5} it is *{w: w is preud ef.t}'- Termabstracts figure net at all in quantificatienal schemata ner in the tests efvalidity, but they de yeem.an service when we want te apply theseschemata and ceme eut with sentences. They are wanted te gevern sub-stitutien. Certain restraints are needed. Substitutien cf the abstract ‘{w: Hxtw ispretid ef .r]|l’ fer ‘F‘ in ‘F.r' and ‘Fy’ might seem te give the pair efsentences:{ti} 3.1.1’,-r is preud cf 1:], H.r{y is preud cf .r].This pair, unlilte t'll—t5]I, is ill matched. There is ne ene “same thing“that the left sentence says abeut-r and the right abeut y. The sentence atthe right says that y is preud ef semething, but the ene at the left saysnething at all abeut-rt it says enly that semeene is preud ef himself, and itceuld as well be rendered ‘ Etta is preud ef:)‘. It has ne free ‘I'- Hence this first restrictinn.‘ Fin abstract is net te be predicated ef avariable that weuld be captured by a quantifier inside the abstract. Suppese next that we are substituting fer ‘F‘ net just in ‘F-1:’ and ‘Fj-1‘as heads ef twe celumns [ l}-{S}, but in the valid schema ‘ls’-t Fa: —1- Fy‘.Substitutien ef ‘{w: iv is preud ef.x}‘ fer ‘F’ in ‘Fr’ and ‘Fy\" had beenlegitimate, yielding the pair ef sentences £5]; but it is nnt legitimate in theeentcxt “'tf-r F-x —r Fyl, fer it weuld yield:{T} \"ti\"-tt-r is preud efx} -1-.y is preud ef-t,
lei H. General Terms and t}‘uann]i.srswhich fails te inherit the legical structure tn which “tfx F-r —a Fy‘ ewedits validity. This last says that if everything is.F then y is; (T) says that ifeveryerte is preud ef himself then, irrelevar:rtly, this fellnw is preud ef thatene. Brnadly speaking, the treuble in (T) is the same as in [Er]: an eccurrenceef ‘.t‘ has been captured by a quantifier. The causes, hewever, are eppe-site. ln [ti] a quantifier that had been lurlting in the substituted abstract'{w: H-rfw is preud ef.r:]}‘ captured the free *-rl that was waiting in ‘Fir’.in [T], cenversely, a quantifier lurlting irr the centext ef ‘F,:r' capmre-r;l thefree ‘.r’ ef the substituted abstract '{iv: w is preud ef;r}‘- Se we Freed stillthis sccend resrrirrritrrr-* A variable free in the abstract must ner be such asre be captured by a quantifier in the schema inte which the abstract issubstituted. We may sum up the twn restrictiens symmetrically thus: Quantifiets efthe substituted abstract must net capture variables ef the schema in whichthe substitutien taltes place, and variables ef the substituted abstract mustnet be captured by quantifiers ef the schema in which the substitutientaltes place. These restrictiens simply ward eff cenfusiens ef variableswhich, if allewed, weuld cause substitutien te deviate frem its intendedpurpese cf interpreting tertn leners. In turning quantificatienal schemata inte sentences we have net enlyterm letters but alse sentence letters te reclten with. Fer sentences theapprepriate eperatien ef su bstitutinn is simpler, there being ne questien efabstracts and eencretien. Substitutien efa sentence fer a sentence letter ina schema eensists as usual in putting the sentence fer all eccurrences nfthe letter. The first ef the twe restrictiens set ferth abeve. mereever, nelenger has a place, there being ne predicating ef abstracts. But the secendrestrictinn curries ever: variables free in the substituted sentence mustnet be such as te be captured by quantifiers in the schema inte which thesentence is substituted. This restrictien merely maltes explicit the under-standing that geverned ‘p’ in Chapter 23: ‘p’ represented, in these cen-texts. a sentence deveid ef free \".t’.ENE ltCl5E5 {air-praisedatey}, {:2 y praised a te Jr}l. Substitute each ef the abstracts: ltzx praised: tey}, {.t: y praised r te 2:},
Ed, ,'i'rrl'r.-..rr'r'trtir:-rt lb} fer ‘F‘ in ‘F-I‘- Cnrnpatibly with the restrictiens en substitutien, which ef these abstracts can be substituted fer ‘F' in ‘Eli: Fir‘? Wltat de the resulting sentences mean?1. Find an abstract which. substituted fer ‘F ‘ in ‘Fa: —1~ Fy‘, yields: ya‘ = -r-r +-r ya‘ =y.t +_v-
rrrGoErNQEuRAAr~Lrrrrrr=-rrcrtjAorr5z>v_r§r
27 SCHEMATA EXTENDED ln the legical traditinn terms are distinguished inte twe kinds, relativeand abselute- The characteristic ef a relative term is that it describesthings enly relatively te further things which have afterward te bespecilied in turn- Thus 'father‘ as in ‘father ef isaac‘, and ‘nerth‘ as in'nerth ef Beaten‘, are relative terms. What were spelten ef as terms inChapter I4, en the nther hand, are abselute terms. Wards capable efbehaving as relative terms can regularly be used alsa as abselute terms,threugh what ameunts te a tacit existential quantificatiun in the cente-xt;thus we may say abselutely that Abraham is a father, meaning that there issemething ef which Abraham is a father. in English a cenvenient earmark ef the relative use ef a term is theacljeining ef an ‘ef‘—phrase er pessessive mediiier whese sense is net thatef ewnership- Thus ‘father ef Isaac‘ . er ‘lsaac‘s father‘, has nething te dewith preprietership en lsaac‘s part, but means merely ‘that which bearsthe father-relatien in lsaac‘. We can appreciate the distinctien betweenthe pessessive ‘my‘ and the relative ‘my‘ by recalling what Dienysnderussaid te Ctesippus with refe rence te the latter‘s deg: ‘. . . he is a father, andhe is yeurs: therefere he is yeur father‘ (Plate, Eurhydemus]. It will be better hereafter tn speak ef terms net as abselute and relative,btrt rather as rnt.utau‘r‘c and pulyadic. mere particularly a'yad'r‘c. The changein terminelcgy reflects a change in peint ef view. The term 'father‘ asrelative is theught ef as refen'ing still te Abraham. even theugh relativelysemehew te lsaac. The term ‘father‘ er ‘father ef‘ as dyadic is theught efrather as true ef twn men ceerdinated as an erdered pair, Abraham andlsaac. A dyadic terrn. like a menadic ene, may eccur indifferently as sub-stantive, adjective, ur verb. tn ‘.1: is a helper ef y‘ we use the substantive.in ‘.r is helpful teward y‘ the adjective, and in ‘I helps y‘ the verb: butlegically there is ne need te distinguish the three. Lngicaily the impertantthing is that whereas the menadic terms ‘man’, ‘wallts‘, etc. are true efCaesar, Secrates, etc. ene by ene, en the ether hand the dyadic term'belps‘ is true ef Jesus and Lazarus as tr pair, and ef Uncle Tem and LittleEva as a pair. lf as in the feregeing pages we write ‘F-t‘ fer ‘.r is an F‘, id?
IE3 Hi‘. Generai Theerry t.r__||\"1,',;_r\"r-rarrtrj_iI‘it'rrtittrtthen the analegeus netatien in cennectien with relative terms sheuld be‘F-ry‘, ‘.r isF tey‘. The erder ef ‘.I‘ and ‘y‘ in ‘-1: helps y‘ is in nne respect accidental: ‘.rhelps y‘ can as well be phrased ‘_'r' is helped by I‘. Eut in anether respectthe erder is essential: ‘.r helps y‘. e-g-. ‘Jesus helps Lacan1s', is netequivalent te ‘y helps .:r:‘. Se the sentence ‘I helps y‘ may be describedequally as ef the ferm ‘F.ty‘ and as ef the ferm ‘Fy.r‘, but the interpreta-tiens thus successively impesed en ‘F‘ are then distinct frem eachether-—-as distinct as ‘helps‘ and ‘is helped by‘. ‘F-ry‘ cannet in general beequated with ‘Fy-t‘. Besides dyadic tertns we may recegnire triadic enes, rerradic, and senn; e.g., 't'I=‘.ryt‘ may In-ean ‘.r gives y te r‘, and ‘H;-:y.rw‘ may mean ‘-tpays}: te t fnr w‘. There are fertns cf inference, legically ne less seund than these dealtwith in Part ll, which are insusceptible te the metheds ef Part ll simplybecause their analysis calls fnr recegnitinn ef dyadic terms. An examplefrem -fungius ffi. I-54-D] is:All circles are figures: All whn draw circles draw figures.The premise can be represented in nur previeus netatien as\"'t\"-1rt_‘Fir -+ Gr)‘, but the cenclusien presents difficulties. We can indeedrepresent the cenclusien as 'ltt‘.rLf1'.r —1- Jxl‘, interprefing ‘Hr’ as ‘.r drawsa circle‘ and ‘.f.rr‘ as ‘.r draws a figure‘, but then the schemata\"tf.r[F.r —1- GI)‘ and “'tt‘.tt_‘.H‘x —1- Jxl‘ bear ne visible intercennectienwhich ceuld justify inference ef flie ene frem the ether. ‘What we must deis extend eur eategery ef quantificatienal schemata te admit such ferms as‘Hyx‘ fer ‘y draws .r‘. Then ‘y draws a circle‘ can be represented as‘Hx|[Fr . Hyxl‘, and ‘y draws a figure‘ as ‘ E.ti_‘G.t . Hy.r}‘:,tl1ereupen eurcenclusien as a whele, ‘All whe draw circles draw figures‘, becemes:-[ll \"tl\"y[H.rl_‘F.r . Hyr] A Elrftir . .Hy.r}].Quantificatien theery needs te be extended in such a way as te enable uste shew, amnng ether things, that ‘la’-r[F-t —s Gr)‘ implies {lt- Anether example ef the need cf thus extending quantificatiun theery isthis:PltEMlsE: There is a painting that all critics admire;cesrctusren: Every critic admires seme painting er nther.
3.7. 5t'.lterrrrrr'a .l:.'-ttertrferf lft5|'\"'i\"r‘itli ‘r'_'r‘.r‘ interpreted as ‘.1 is a critic‘. and ‘Hay’ as ‘.r admires jr‘, we mayrepresent ‘all critics adruirey‘ as \"'tt\"-rt_'I5‘-r —1- H-ry}'. Se, interpreting ‘Fy‘as ‘y is a painting‘, we may represent the abeve premise as:{2} 3y{Fy . \"it\"-til-?.r —t H-ry]].Further. since ‘.r admires seme painting er ether‘ becemes ‘ 3ytFy. H-ry]r‘- the cenclusien as a whele takes en the ferm:[Fl] \"ti\"-r[t?.r ——1r Hj!rl‘F}‘ . H-1'y‘,l].Cine mere example:easarrse: There is a philesepher whem all philesnphers cen- tradiet.eet~rt\"LLtster~r: There is a philesepher whe centradicts himself.The premise here has a ferm clesely similar te {I}.|[='-i‘,l Ey[Fy . \"tfx{F-r —t t3‘a'}'ll.The cenclusien is simply ‘H.rr[F.r . Ext)‘- We saw in Chapter 22 hew differences in greuping ceuld affect themeaning ef a quantificatiun; \"'t\"'.t=[F.r v Ex)‘ had te be distinguished frem‘Fr Fr v ‘tilt Cr‘, and ‘3-t'lF.t' . Gr)‘ frem ‘Eb: Fx . it C1‘. Censider-atiens cf this ltind ceme tn lnnm larger new that we need quantificatienswithin quantificatiens. Thus, let us refiect next en the expressien:[5] \"Iit‘.t[F-I —s HytfFy . tlryl].if we interpret ‘F-r‘ as ‘.r is a number‘ and ‘G.ry‘ as ‘.r is less thany‘, then{5} cemes re mean:Every number is such that seme number exceeds it,er briefly ‘Every number is exceeded by seme number‘. This mightcarelessly be rcphrased ‘Seme number exceeds every number‘ and thenbe put back inte symbels as: iris is a number . y exceeds every number}.
l'i'[l' Hf. t‘_,?ener'a.f i\"‘.freary cf Quanrrjficariani.e., If-ti}. Hut actually there is all the difference between [5] and -|[-1] thatthere is between truth and falsity. -[5] says dtat fer every number th.-ere is alarger, which is true, whereas {4} says there is seine great number which,at encc, exceeds every number. This last is false en twe ceunts: fer thereis ne greatest number. and even if there were it weuld net exceed itself. ‘l‘he distinctien in fnrm between [3] and fl] is the same as just newstressed between [5] and Ht. The warding cf the premise and cenclusienabeut paintings illustrates again the awkwardness ef erdinary language inkeeping the distinctien clear. The netatien ef quantifi-catien is handler inthis respect. The mathematical cencept ef limit prevides, fer readers familiar withit, an apt further illustraticn cf the abeve distincfien. A functien fir} issaid te appreach a limit ft, as .r appreacheslr, if fer every pnsitive numbere there is a pesitive number 5 such that fix] is within s ef tr fer every .r[ask] within 5 ef fiI. In terms ef quantifiers this cenditien appears asfellews: ‘tt\"c{e:=r'[t.-1- eats ;=--tt.‘ti\".rttlI-r: I.-1' - s| c a.-1-.tft.rIr —s|c e}]]-.As textbeelis rightly emphasize, we must thinlt ef is as chesen first; fereach eheice cf c a suitable s can be chesen. This warning is. in effect. awarning against cenfusing the abeve ferrnula with the essentially differentene: 3-s{s :-=- n . \"t|\"s:[e :=~ n --s ‘tr’-rte --=: |.r — ri;| -c s -i. [gs] - ri| c ti];-.The distinctien between these twe fermulas will be recegnixed as identi-cal with that between {5} and {ti}, and between {3} and {E}. The essential centrast between [5] and |[-'-ll, and between E3] and [2],becemes simpler and mere striking when we cempare:[Er] ‘six H-jr Fry, Hy ‘ifs Fry.Suppese we interpret ‘F.ry‘ as ‘.r and y are the same-thing‘. se that theschemata tfi} beceme:{Tl ‘itfx Eytx and y are the same thing},{E} 3y‘tI\".rr[;t and y are the same thing}.Fer each chesen ebject-1:, clearly there will be an ebject which is the sarnetviz. , me chesen ebject .=: itself]. {if each ebject .r, therefere, the sentence:
EF‘. Schemrrrrr .-‘ittenrferf li‘l Hyrf-r and y are the same thing}is true. Se {Tl is true. Cln the ether hand. as leng as there are mere ebjectsthan ene in the universe. ne ene ebject can be the same as each; i-e., neene ebject y can he such that ‘iI\".rI[.t and _tr are the same thing].Se {fit is false. ln general \"ls\"-1: Hy Fry‘ says that cnce any ebject whatever -r is fixedupen, an ebject y is ferthceming such that Fry. [Itiffere nt cbeices ef rr maybring ferth different cbeices ef y. Cln the ether hand ‘Hy \"fir F.ry‘ saysthat seme ebjecty cart be fixed upen such that, fer this same fixed y, ‘F.ry'will held fer all cemers .r. Supp-esing a limited universe ef ebjects a, it, . . . , fr, let us see hew“tiI\".t Hy Fxy‘ and ‘Hy ‘tiI\".r Fxy‘ cempare when the quantificatiens areexpanded inte cenjunctiens and altematiens {see end ef Chapter 22].“tt\";t Hy F.r_v‘ becemes first: 5yFay . 3yFhy .. - .. HyFhyand then: Fca 'v'Fa.i:r v. - .vFah . Fen vFbb v. . .v.\"-\"trh .. . .. Fna vFirb v. . .vFtrlt.Urn the ether hand ‘Ey \"tt\".t F‘.ry‘ becemes first: 'ifxF.xa v 'tt‘.t Fxhv. . .'-.-\"'s\".t Frrhand then:Faa . Fba Fha .'ir\". Fab .Fbb .....Fhb .v.-...v. Fab . Fhh .....Fhh- It was remarked in Chapter 12 that theugh in erdinary language thewerds ‘semething’ and ‘everything‘ masquerade as substantives, theirbehavier deviates frem that ef genuine substantives. Further examples efsuch deviatien are previded by (‘Ir’) and rs;-. Fer, (‘it might be put intewerds as ‘Everything is identical with semething‘, and rs] as 'Sen'rethingis identical with everything‘. if ‘everything‘ and ‘F‘-ttniething‘ really be-
I'll Ill- General i\"lrear_v cf r,1'n-aurrjricrrrlanhaved like narnes, we sheuld expect these twe statements te beequivalent-—and in fact we sheuld expect beth te be false. Eur actually,as seen. {‘l] is true and [E] false. Further, if ‘tie-thing‘ and ‘everything‘were genuine names we sheuld certainly expect ‘Hething is identical witheverything‘ te be false; actually, hewever, this statement simply denies{E} and hence is true. Alse we might expect ‘Everything is identical witheverything‘ te be equivalent te the truth ‘Everything is identical withitself‘, whereas actually it expresses the faiseheed:[El] ‘s\"-r‘tiI\"ytr is identical with y‘,l.Cine reasen why quantificatienal analysis aids clear drinking is simply thattbe spurieus substantives ‘samething‘, ‘everything‘, and ‘nething‘ {andtheir variants ‘sc-mebedy‘. ‘ever-ybedy‘. ‘nebedyjr give way re a lessdeceptive idiem. The cemhinatien ‘ltfr \"I\"y‘ in (9) is net te be theught ef as semehew adeuble quantifier; “slit lily Fry‘ is simply a quantificatiun ef “tf_v Fry‘ as awhele ‘alhereas t ljl-(fit shew existential quantificatiens wirlun universalenes and vice versa. til] shews universal within universal. Part lll brings ne new legical symbels fer use in sentences. Alreadytcward the end ef Chapter 23 we bad an example abeut 5adie‘s sheplift-ing that was the equal ef any sentence fermulable in Part III. Clur newgains are net in sentences but in schemata. The schemata new befere usare the aaantrficariarml sclrerrrara generally, net just the menadic enes.They ccmprisc ‘tr’. ‘c‘. ‘F.r‘. “Fr”. :c.r*, ‘Hr,v‘. ‘lr‘yr‘, ‘r‘-frr‘, ‘Fryr:‘,etc., and everything thence censtructible by truth-functien netatiens andquantifiers. The pelyadic ingredients—‘H-ty‘ and its suite—are what arenew. {lt is custemary net te use the same term letter with different num-bers ef variables—thus ‘Fr‘ and ‘Fryz‘—--within ene schema er withinthe schemata ef ene preblem. Cenventiens ceuld be devised tn accum-medate such use.) Tbeugh the enrichment teuches enly the schemata, it is a crucial ene.As remarked, it enables as te establish new implicatiens between sen-tences. The breadened demain ef schemata sustains a breadened cenceptef validity, and hence ef implicatien, which se exceeds the beunds ef Partll as te resist any cemparable neatment. Fer validity ef quantificatienalschemata in general it is impessible te devise a decisien precedure. [SeeChapter 34.) The same applies at ceurse te censistency, implicatien, andequivalence. The definitien ef validity is as befere: truth under all inter pretatrens in
2?. .'i'r\"hr-mtrrtl E.t'tent.lt=tl H3all rt-anemptv universes. The defittitierts afee-nsis1;ette1r. implicatien, andequivalenee lblltiw suit. interpretatien bf a terrri letter in the menadic easerneans settling what things in the universe the letter is tn be true tit‘. lnpulvadie eases it means settling what pairs ef things in the universe, erwhat triples, ete., it is ta be true ef- The impdssihility af a decisien praeedure fer validity will net deter asfrem develdping pre-eedures far p-raving validity‘. fits neted in Chapter l3,the difference is that a deeisidn precedure assures an atfirmative er nega-tive answer every time, while a preef preeeelure assttt'es at best an even-tual afiirmative answer where an affirmative answer is in erder. Praefpreeedures far general quantifieatien theery will eeeupv mast nf Part lll. H|5TU'H|Cltl HDTE: An algebra nf relatiens was prepeanderi by[lelvlrirgan in 1354 and mueh impre-ved in |5'l't]' by Peiree. lt is related tepnlvadie quantificatiun theery mueh as the Eeelean algebra df elasses isrelated tn rnenadie quantitieatien theery. theught as iterselt sheweti inabeut till-=1,‘ its see-pe is nnt quite sn bread. Ev lliltl, thanlts te Frege.full quantifieatitin theery was at hand.EIERCISESI. Supp-esing the universe limited ta n, tr-,. . . , Ft, eapand the quantifies- tierts inte alternatien and eenjunetien in eaeh ef the fellewing easin- pies: \"'tl\"x 'lil\";|: Fry, H1 3_v Fa’;-.=, by VJ: F111. Hy H.rF.rv. De eaeh in twn stages.1. Rewrite these with help ti-f quantifieatian: Every snlid is seluble in same liquid er ether. There is a liquid in whieh every stilid is st:-luhle-3. Rewrite this eaample tllelvlerganlsl with help ti-f quantifieatienr lf all hnrses are animals then all heads ef berses are heads n-t“ aninlals. '5-ee vari I-leijenn-art, pp. EH, 233-
I'M Ill. General‘ l\".'t|t'r:+rjt't'-*_,l‘l1j',_'|ltitsntil'ie|srti.t.rrt4. E:-tpress, with help af quantilieatitin, the lilteliest interpretatien bf tl1e statement: She hart a ring en every finger.5. Where ‘F * means ‘harms’, and the universe is mankind, put these unarribigutnisly and itlie-matiealiy intn werds: HI by Fyx, \"li|\"'.t't' Hy Fry —1- F.r_r}, ‘lI\".t[li\"y[Fy.r -* Frey] —-it Fat].5. Supposing the universe ta euntprise just the paints an an endless line, judge eaeh at\" these statements as te treth value and eaplain ynur l'-E!-El5[l'fllI'lE,. 3.1:'lfy3.tl[.t is between y and sl. ‘lfy HI Hats is between y and til. 3.: Halfytx is betweeny and it].T. Eapress eaeh af these statements with help bf quantificatiun and indieale its truth value. Nething is identieal with nething, Something is identieal with stirnething, Everything is identieal with nething, Hething is identieal with anything. 23 SUBSTITUTION EXTENDED In Chapter 16 we saw that the tierivatinn af sentenees by substitutien inmenadic quantificatienal sehemata is a ratber eemples and subtle affair.and we saw that it ean be smnnthly managed threugh substitutien efabstraets fer the term letters. New that we have meved tn pniyadiesehemata, werse enrttplertity is ta be eap-eetetl. Haw sheuld seatenees
Ed‘. 1'i'ti'r'tl.'trt'ttrl‘r'tr-It lL'.t'tett-tilted\" ITSrepresented by ‘.F:ty’, *Fy.t*, and ‘F;t.t’ be related tn ene annther’? Wemust he able ta tell e.g- that the sentenee:t’ 1} Barty amuses .1: mere than y amuses yfi v Ext: amuses y mere than y amuses xi .—-1 El.rt_'.t arnuses x mere than y amuses xiis a etrrreet result ttf substitutien in the schema:{2} 3.tF.tyv 3.1: Fy.t .—t- H.tF.t;:t. Happily the salntitm is mueh the same as befere. The apprepriateauxiliary is again the term absttaet. but ptliyadie: thus '{.ty; - . ..t . . .y.. ‘{.1yz:. . .:r. . -_v- . -1. . .}', ete. Te get ill. the absttaet tn substitutefar ‘F’ in {2} is:{sine w amuses :5 mere than y am uses ivWe have nnly ta put it fnr ‘F‘ in all three plaees and then prn-eeed witheaneretian in ebvieus fashien: in the first at the three plaees the beundvariables ‘.3’ and *1-v‘ nf abstraetitnt give way tn ‘.r’ and ‘y’, in the s-eenndtu ‘y’ and Zr‘, and in the third tr: ‘.1r* and '1‘. The twn restrietinns in Chapter Eti enntinue t-a apply. lnspeetinneentirms that the present ettatiiple is in erder en that seere. ln the mnnadie ease, term abstraets are a regimentatien ef erdinaryrelative elauses. ln the ptllyadie ease they are analt-gtlus: they are ape-Iyadie eatensitrn til the relative elause. Clut erdinary language lael-tspelyatlie relative elauses, and ene ean see in part why. Relative elauseshave mest nf their utility in eentettts equivalent tn qu antitieatinn: enntestsin whieh they are gevemed in effeet by the funettir ‘H’ er “ll”. Hut apelyadie relative elause effers ne advantage when gevemed by ‘H’ er\"li\"’. beeause the same warlt ean be aeedmplished with nested rnanadierelative elauses sttbjeet separately te ' H’ ttr ‘V’. Fnr instanee: E {.1-_v: .t saved y irntn ttrewning}ameunts in menadic elauses ttr ‘There is semen-ne whe saved sdmett-newhn was tlrewning‘, er: His: 3{v: it savetly frem drewningii.
l7{t HI- General Th.-.=v:rq' nf Quttntificnrinnin ether wards. the cemhinatien '3{.ty:' ameunts merely ta the iteratedquantifiers ‘HI Hy’; and similarly fer \"li\"{.t'y:‘. Hut if pcllyadic relativeelauses after little tu erdinary disecrurse. they are a gtidsend te the theeryef substitutien. Here. as in rnenadie terrn abstractinn, l have drawn an the netatien at‘set theery. The netatien . . ..t . . -y - . is standard fnr relatiens.censtrued as elasses at ardered pairs- Remarlts parallel te these at the endef Chapter 1| apply again here.lt was remarlted at the end cif Chapter \"l that validity may be ascribednnt nnly te truth-funetienal sehemata but alse, by etttensi-en, ttt the sen-tences whrise fnrms these sehemata depict; but that it is well then ttl addthe qualifier ‘truth-functinnal’. Ceirespendingly a sentence ehtainable bysubstitutien in a valid quantificatienal schema is qnnnlificntierially valid.Such a sentence is true, er true fer all values ef its free variables. Hut itmay er may net be truth-functienally valid; its truth may depend se-lely DI‘!its truth-functinnal strueture. clr it may depend partly en hew thequantifiers are arranged.We may alse nete an intermediate grade, mnnad’t't.- validity. Ft sentenceis quantiticatic-nally valid if it can be get by substitutien in a validquantificatienal schema; it is menadieally valid, me-re particularly, if itcan be gtit by substitutien in a valid quantificatienal schema which ismenadic; and it is truth-funeti-anally valid if it cart be gt:-t by substitutienin a valid truth-functinnal schema.Similar remarlts apply te implicatien an the part cif sentences: it ean bequantificatienal, and mc-re particularly menadic, and still mere particu-larly trutl't~functiettal. Similarly fnr inetrnsistency.Substitutien in schemata alse taltes anether directien and servesanether purpese: substitutien ttf sehemata in schemata tn preduce mereschemata. A natable feature cif such substitutien was remarited up-an atthe t|'uth~funetic-nal level in Chapters ti and 13: substitutien in validschemata gives valid schemata. lust new we nnted that a quantificatlenally valid sentence might ermight net be. mare particularly, mdnadicatly valid er even truth-functirinally valid. New similar distinctiens apply tn quantificatienalsehemata themselves. A valid pulyadic schema may be mtivntdicrtllyval't'd‘. in the sense ef issuing by substitutien frem a valid mnnadie ene.An example is ‘VI Gay -P Gyy’, whieh issues frem \"'tl'.r F1 —s+ Fy'. .-andelf ceurse a quantificatienal schema may be truth~funetittnally valid.When we were getting sentences by substitutien. what we substitutedwere sentences fer sentence letters and term abstracts fnr term letters. Tn
ES. ,§'nEt.vtt'rurinn Etrentled’ l T7get schemata by substitutien, what we substitute are quantificatienalschemata fur sentence letters and again term abstracts fer term letters; butthe term abstracts are new schcmatic—e.g. ‘{w: Cw v Hi; Hawl’. Sub-stitutien cf this fnr ‘F’ in \"lil'.t Fr ~—t- Fy’ pnticeeds in the way and underthe restrictiens already neted, and yields:{3} \"tI'.t=[C.t v Ht Hat} —*. Cy v Es Hay.Ptnuther cttample: substitutien tit {w: Cw v —Hw}’ itit ‘F’ and ‘Cy’ fnr‘p’ in:{4} \"li\".t[.F.t —l*yt1':l='l'-l'- 3.: F1’ -—a+pyields:{S} \"i\".i:t_'C.r v —H;t .—+ Cy] -H-. E.t{C.t v —H.t]- -1 Cy.The utility cf substitutien here, as in Part l, is as a means cf generatingvalid schemata frem valid schemata. E.g., since [3] and {5} were get bysubstitutien in schemata which were seen in Chapter 15 te be valid, wecenclude that {3} and {5} are valid. Substitutien can he depended upen tn transmit validity t’er essentiallythe reasttns already neted in Chapter I5. But it will be well tc review thematter new in the new setting. Tc begin with let us see why it is that theabeve substitutien in til-’,| yields a valid result. ‘validity cif the result {5}means truth under all interpretatiens cf ‘C’ and ‘H’ and the free variable‘y’, within any nenempty universe. Supp-use the universe fitted, then, andcensider any particular eheice Si cf such interpretatiens; what we want tesee is that {5} cemes cut true under 53. Te see this we derive frem 53 thefnllewing interpretatiens fnr the schematic letters -cl’ {-1}: we interpret ‘F’as true cf just the things that ‘{w: Cw v — Hw}’ cemes tn be true ct’ underill, and we interpret ‘p’ as having the truth value which ‘Cy’ cemes tuhave under ill. Being valid, t=t-ft must ceme nut true under these interpreta-tiens; hence {S}, which simply repeats [4] under these interpretatiens.cemes cut true ttiti. lvlcre generally, suppcse a sentence schema 5 ’ ebtained by substitutieninS . Each free variable and each schematic letter in S has a cnrrespcndentamnng the materials cif S ’:, this currespendent is in each case either thesame letter ever again, ct else a substituted sentence schema er term letterat abstract- hlnw given any eheice iii cf interpretatieins fer the free vari-
l'l'E til . Cenernl Thariry cf t_,§Innntilicntlcn ables and schc matic letters ttf.1'i\" , let us adept as interpretatien cif each free variable cir schematic letter ut’ S the interpretatien that has already accrued tn its ccirres-pendent thrnugh El. S, sci interpreted, matches S’ as inter- preted by H. -Since this wcirits far each chciice cf E, we see that$’ is validtar true fur all interpretatiens] if 5 is. The functien cit’ the twe restrictiens un substitutien in Chapter 26 is tuassure that the eurrespcndents just new speiten cf really ccrrespcnd- Letus new have same examples shciwi ng haw su bstitutinn can fail tti transmitvalidity when the restrictiens are vitilated. Substitutien cf ‘ fa: Ely Cty}' fnr ‘F‘ in \"\"'|il’.1r F1‘ —# Fy’, in virilatiriu ttfthe first restrictien. weuld yield:[ti] bx Hy Cxy —1- Hy Cyy- [invalid]That this is net valid. despite the validity nf \"'il'.t F.r —1- Fy‘, is seen bycunfining the universe tc numbers and intil-‘|‘lI\"‘¢ting ‘C’ as ‘is less than‘;thereupon the antecedent cf {-5} becnmes true (‘fnr every number there is agreater‘) and the ccnsequent false. Substitutien ct’ ‘la: Cxcl’ fer ‘F‘ in \"'I\".t Fx —1- Fy’, in viclatidn at‘ thesecend restrictien, weuld yield:[T] ’lil\"sr C.t;t —1- Cry. {invalid}That this is net valid may be seen by talting ‘C’ as ‘identical with’; then{T} says ‘lf everything is identical with itself then I is identical with y‘,and this is clearly net true fnr every cheiee trfx and y. Cr, tti restate thisrcfutatien in mere explicit relatien ttr the definitien cf validity: when weadept a universe cf twe -tn‘ mere ebjects, and talte DUE-' cf these ebjects asintetpretatien ef the free ‘.r’ el’ {Tl and a different cine as interpretatien cf‘y’. and interpret ‘C’ as ‘is identical with’, thereuptin {Tl becemes false. In {ii}. the expressiens ‘Cr v He Hat’ and ‘Cy v Ht Hay‘ whichsupplanted the ‘Fx’ and ‘Fy‘ tit’ \"'tl';t Fr —-it Fy‘ are symmetrical in ‘x’ and‘y’: the ene expressien has ‘.t’ where, and nnly where, the ether has ‘y’.In the invalid substitutien which led te {T}, en the ether hand, the ex-pressiens ‘Car’ and ‘Cxy’ which supplanted ‘Fx’ and ‘Fy’ fail tn shewthis symmetry; ‘Cay’ dues net have ‘y’ everywhere dtat ‘Cat’ has ‘x’.The reader must be warned that this asymmetry has nti-thing in de with theinvalidity ct‘ {T}. It is unnecessary, in general, fcr the expressiensupplanting ‘Fx’ tu have ‘.r’ where and eniy where the expressien
I-id. Sahstitniitin Ettettdied JTSsupplanting ‘Fy’ has ‘y’- lt is quite preper, e.g., tc substitute ‘it: Cayi’fer ‘F’ in \"ll'.r F_=r —a Fy’ and infer the validity cf:{E} ’l:l'.t C.t_v —1- Cy_y{e.g., ‘lfeveryene hates y then y hates himself’}. Despite the asymmetryelf ‘Cay’ and ‘Cyy’ with respect te ‘.r’ and ‘y’, [El is a genuine special caseut \"il':r Fr —1~F_v‘. as a verbal cetnparisen immediately reflects: ‘lf ev-erydring is an F then y is an F’; ‘lteverytbing is a C cl-fy flten y is a C cfy’; ‘lf evcryenc is a Herbert-hater then Herbert is a Herbelt-hater‘- Te be assured cf the cettectness cf a substitutien, we need lee-lt enly tethese peints: we must be able, en demand, te specify the actual sentenceschema er term letter er n-adic abstract that is substituted fnr the sentenceletter nr n-adically eccurring term letter; we must be sure that it has beenintrcduccd at each eccurrence cf the letter; we must be sure that at eachpt:-int -nt‘ intreducing the abstract the particular variables there appended tethe term lener have been put fer the variables cf abstractinn; and finallywe must he sure that the substitutien has net led te new capturing ctvariables by quantifiers, centrary tn the twe restrictiens. Let us new shift frem ‘h\".t Fr -s Fy‘ te anether ebvieusly validschema. ‘F_v -1 Elx Fr’- Frem this we may preceed te:isi Cyy —1- Ex er;..~by the legitimate substitutien cf ‘la: Cr-:y}‘ fer ‘F’;, but it weuld be il-legitimate tci substitute ‘it: Cxtl’ fer ‘F’ and thus preceed te:till} Cxy —+ it Cxx. {invalid}an example el’ {9} is ‘If Herbert hates himself then seme-ene hates Her-bert‘, which is quite uncxceptienable; but an example el’ iltl} is ‘lf Amt:-sis uncle cf Herbert then sc-merine is uncle cif himself’. l\"~lttte that theughthe same principles bf substitutien are uperative here as befere. the pairtil} anti till} is rather eppesite in appearance tn [E] and if'l'}. The valid {E}had unlilte variables in the antecedent, but in the valid {9} the eppesite isthe ease. as tar as substitutien in eur particular examples \"il'.t Fx -—1- Fy’ and‘Fy —t- Hr Fr’ is cencerned, nete that the net effect ct the twe restric-tiens is just this: the schemata that ceme te supplant ‘Fy’ and ‘Fx’ mu st be
IE1] Hi’. Cenerel T.lreer_y cf Qaentifieetirntaliite except fliat the ene has free ‘y’ wherever the ether has free ‘.r’. Bethmay have additienal free eccurrences ef ‘y’, as seen in {E} and til}. Thusthinlt ef‘. . .y. . .y. . .’ and ‘. . ..v. . .y. . .’ as the twe schemata; we can putthem fur ‘Fy’ and ‘Fx’ in \"tl'x F..t—‘*Fy’ and ‘Fy -—-ii HxFx’ by sub-stituting ‘{t: . . ..t . . . y . . .}‘ fer ‘F‘. Se. insefar as we are cencemedmerely with substitutien in \"'lI\"x F.1' —1- Fy’ and ‘Fy —t Ex Fx’, we mayemit all theught ef abstracts, instead directly supplanting ‘Fr’ as a wheleby any sentence schema S, centaining free ‘..t’, and ‘Fy’ by a schema 5,which is lilte S, except fer having free ‘y’ in place ef all free ‘.t’. Such, then, is the netatienal relatien between sentence schemata dtatare suited te the respective reles tif ‘ltfx Fx’ and ‘Fy’ in ‘hlx Fr: —i- Fy‘, eref ‘Ex Fx’ and ‘Fy’ in ‘Fy —t Ex Fx’. It has a name: ene schema iscalled an instance ef the ether- ‘Fy’ is an instance cf ‘hlx Fx’ and‘Ex Fx’. The tenninelegy applies alse, ef ceurse. where letters etherthan ‘I’ and ‘y’ are vied; thus ‘Fw’ is an instance ef \"tl\"y Fy’ and‘Ely Fy’. Furlhertnute, ‘C1-vw’ is an instance ef\"'I'y Cwy’, cif‘Hy Cwy’,cf \"'I\"y Cyy’. uf ‘ Hy Cyy’. ef \"lily Cyw’, and cif ‘Hy Cyw’. Ctn the etherhand ‘Cwy’ is net an instance cf \"ltl'y Cyy’ er ‘ Ely Cyy’. The general descriptien is this: an instance at a quantificatiun exactlymatches the eld epen schema that fellewed the quantifier, except that itmay shew a different variable in place ef the recurrences el’ the variable efthat quantifier. lf it dees shew a different variable. it must shew it in allthe places tat least} where the eld variable had been free in the eld epenschema that fellewed the quantifier. lvlerenver, it must shew it free inthese places. The reader will recegnize in these requirements the effectsef the restrictiens en su bstitutien fer term letters. A universal quantificatiun implies each el’ its instances. and an ex-istential quantificatien is implied by each cf its instances. Such arethe implicatiens that ceme ef substitutien in the valid schemata\"li\".t Fx -—-t- Fy’ and ‘Fy —1- Ex Fx’. When we substitute fer ‘F’ in the valid clesed schemata:ll ll b\"y{\"i\".t .F.t' -—i- Fy}, {I3} HytF_y —t+ \"tI\".t Fx},{ll} \"i\"y{Fy —-Ir 3.: F1}, {I4} Eyt HI FJ: -it Fy}tel‘ Chapter E5, especially Exercise I}, the effect ef the twe restrictiens ismere stringent: the schemata that supplant ‘Fy’ and ‘Fx’ here must bealike except that the ene has free eccurrences ef ‘y’ where and tint’y wherethe ether has free ‘.i:’. Fer, the secend restrictinn requires that the abstract
ES, Eultsrirnrien Etterttfetff IS-lsubstituted fnr ‘F ’ be deveid cf free ‘y’, in view ef the initial quanti fters in{l l}-{ it-l}. Thus, whereas it was allewable te substitute ‘{a: Cty}‘ fer ‘F ’ in\"lit\"-t Fr —i- Fy’ and ‘Fy —1- 3: Fx’ se as te nbtain {B} and {'9}. it is ferbid-den tti malte the same substitutien in {I l}—{ I4} se as tti nbtain:{I5} ’liI\"'y{’lil’.r C-ty -1: Clll'l~ {IT} 3y{Cyy —+ \"tl'.1t Cxy}, {invalid}{lb} \"lil\"yCiyy —1- 3.1: Cry}, {IS} Elyiir Cry —+Cyy}- {invalid}{I5} and {lti} happen te be valid anyway, because they are the universalclesures ef the valid epen schemata {S} and {'9}- {IT} is net valid, as maybe seen by adepti ng a universe cf twe ct mere ebjects and interpreting ‘C’as ‘is identical with’. ‘Cy_y thereupen becnmes true and ‘lfx C-ry’ false ferevery ebject y, and thus {IT} cemes eut false- That If lfl} is net valid maybe seen by interpreting ‘C’ as ‘is distinct frem‘. l\"|'i5TCl'R|’Cfii|.. HDTE: l ltnew cf nu full and cerrect fertnulatien efsubstitutien fer pulyadic tenn letters prier te dist cf I934 by Hilbert andEernays. The auxiliary rele played by abstracts in my present methed wasplayed in theirs by schemata called Nenrgfermen. In Eierrrenlery i’..t.tgir: Iused auxiliary fcirntulas tit‘ anether style fer that purpese. They eentainedcircled numerals and were called stencils, nr, in the secend editien cf thatheels and three editiens cif this nne, prttdienres. lt is a pleasure new teinvelte the versatile abstract and banish these intrusiens- The twe restrictiens tin substitutien date frem Elementary Legitr, i941;Hilbert and acltermann had met these needs by the mere drastic expe-dient ef using different parts cif the alphabet fer their beund and freevariables.EIERCISESl. Decide which cf the fellewing abstracts may be substituted fer ‘F’ in ‘ it Fxy’ cempatibly with the restrictiens en substitutien.law: tv praised w te -I i. {.7-jw: w praised y te I}.{rs-; w praised s tn a}. law: Hytw praisedy te all.isw: w praised -t te ti. -law: Hntw praised it te ti}-
ill’- Ceeerni T-ftettry sf fi|'trtrrtt'rj,|'t’t'ttrr'|:-*t'tPut the results ef these legitimate substitutiens inte werds, suppesingthe universe limited te manltind.List all the schemata whese validity can be shewn by legitimatesubstitutien ef ene er anether ef:it: C.r.t v Cay}. la: Cys v Cry},it: CI: v Car ls it: ‘tt’y{Cyc v Ctyllfer ‘F’ in ”'|I\".t Fx -s- Fy’ er ‘Fy —s HI F‘-r’ er {I1} er {iii} er {id}.Talting the universe as the members el’ the ceuncil and interpreting‘C’ as 'dennunced’ and ‘p’ as ‘steps must be talten‘, put the resultsinte erdinary language.Determine which cif the fellewing schemata are legitimately ebtaine-ble frem \"'tl'-ir F-r —> Fy‘ er ‘Fy —s+ Ex Fit’ er {4} er {I3} er {l-fl} bysubstitutien. Identify the substituted abstract in each case.’U‘.r F.t'x —1- F_‘t‘_1~'. Cy-It —I- it CI]-',E-'.rtt=-rs —1- as Cyy}, Ely{Cy - re .-+. ’li\".t' er . an.h'r{C_vx -1 Hrs} —t-. Cyy —+ Hye, City . Hya .-at E'lr{C.cr . H-ts},Cy . ‘b\".r{Cy —r Cr} .—t 3.t[C.t . ‘tt'r{C.t —i- Cr-:}I,‘r‘rt‘r‘.rttFe= -—r first —1- first H- 5|-it'll‘:-=t'5-ti»: —* first ~t fiss- 29 PURE EXISTENTIALS In general quantificatiun theery as elsewhere, implicatien is validity efthe cenditienal. Te justify an inference, then. such as the ene in Chapter2? abeut the philesepher, what we need te dc is preve the validity cf acenditienal. in this example the apprepriate cenditienal is: Hy[Fy .ltt'x{Fx —s-C-t'y}] —-P 1t'{F-r . Cat}{see {4} in Chapter ET}. Cc-averting it te prenex ferm, we get perhaps: \"lI\"yH.t3z{Fy . Fx —+C.ty .—t. Fa . Cat}.
2'jt- Pure Eti.trenrftli.t IE3Clther prenex equivalents ceuld be get, but this will de. in fact, sincevalidity is eur cencern, we may as well drep the \"'v\"_y‘; fer an initialuniversal quantifier is indifferent tn validity. ‘What we want te shew, then,is the validity ef:{2} 3.rHt{Fy . Fr —t- C-ry .—t-. Fa . Cra}.blew try this experiment: substitute the free ‘y’ fer the existential vari-ables. drepping the quantifiers. The result:{'3} Fy . Fy -+ Cyy --+. Fy . Cyyis trudt-functienally valid- This eutceme. preperty viewed, establishesthe validity ef {2} and therewith ef tl}. Fer {3} is an instance ef: 3elFy . Fy —1- Cyy .-1-. F:-: . C-teland accerdingly implies it. which in tum implies {Z}. The matrix ef a prenex schema is what cemes after the quantifiers. Apare existential is a prenex schema with nene but existential quantifiers.l\"-lew what the example abeve teaches us is that if a pure existential hasjust ene free variable, and substitutien cf that variable fer the existentialvariables mms the matrix truth-functienally valid, then the pure existen-tial was valid. We can claim mere: the pure existential is valid enly if dtat substitutienturns the matrix truth-functienally valid. The reasen may best be seenthreugh anether example. Let us medify {2} just eneugh te sp-nil it.{T} El.rHr,{Fy . Fx vC-try .—1-. Fr: . Cat}.Substitutien ef ‘y’ fer ‘.r’ and ‘.3’ in the matrix ef {T} gives:{Ii-’} Fy . Fy 'vCyy re. Fy . Cy_y,which, rather than being truth-functienally valid, is falsifiable by assign-ing T te ‘Fy’ and J. te ‘Cyy’. But {T} itself beils dewn te {Ii‘} in auniverse cf just a single ebject y. Se {T} cemes eut false in that universeif we interpret ‘F’ and ‘C’ se thatFy and —Cyy. Se {'2'} is net valid: nettrue under all interpretatiens ef ‘F’ and ‘C’ in all nenempty universes-
IE4 Hi . -Certerat‘ Titeer]-' t;I_,|f flaanttjfieatian We have arrived at a decisien precedure fer pure existentials in enefree variable. The scherna is valid if and enly if its tTIatt'ix becemestruth-functitrnally valid en substitutien cf the free variable fer the exis-tential variables. The test can be extended te pure existentials in tnere than ene freevariable. in such a case there are a multiplicity ef ways cf substituting thefree variables fer the existential enes: but then we talte the altematien trfall such results and checlt it fer trt.ttl'i-functienal validity- Example:{4} 5|t{F-rt\" —=~Fts}-There are twe ways ef substituting the free variables fer the existentialvariable. They give ‘Fstx —+F.ry‘ and ‘Fry —t-Fyy‘. The alternatien:{5} Fxx --1» Fry .v. Fxy —-1 Fyyis truth-functienally valid- Each half cf {5} implies {4}; se any interpreta-tien that maltes either half true nraites {ti} true. In shert, {5} implies {4}-Se {4} is valid- lt is strange but tme. legically true. fer any twe penpleltrhn and lvlary, that there is strmeene whe, if admired by lehn, admireslviary. Te see in general that the validity ef the altematien is net enlysufficient fer the validity uf the pure existential but alse necessary, let usagain reasen frem an example. Let us medify {4} just eneugh te speil it.{4'l 3ttFse -*F.rsl-The alternatien:{f-’} Fxx —t Fyx .v. Fxy -1-Fyyis falsifiable by assigning T tu ‘Fxx’ and ‘Fry’ and .l. te ‘Fyx’ and ‘Fyy’.But {-1'] itself bells dewn te {5’} if we talte a universe efjust-tr and y. Se{el’} cemes eut false in that universe under such an interpretatien trf ‘F ‘. at this peint the ferm tit‘ argument is evident fer this general theurem:A pure existential is valid ifanti enly if we get a truth-functienally val‘t'a‘schema by tailing the alternatien c-f the results tzf substituting the freevariables fur the existential tines in tire ntarri-.t. in hepes cf checlting validity aleng these lines, we dd well, whencenverting a schema te prenex ferm. te cheese eur steps se as l.'tIl giveprierity te universal quantifiers where pessible. If we can get them all in
Eil. Fttre Exi.stentiat's l S5frent, we can drep them and test the pure existential. if in canverting {l}we had bmught its final ‘ Ex’ eut first, this test weuld have been deniedus. Twe fringe cases need nntice- What if en cenversien te prenex fcrnr allthe quantifiers became universal, and thus are dreppedi‘ Then ef ceursewe test the schema as it stands fur truth-functinnal validity, there being nnquantifiers. and what if we end up rather with just existential quantifiersand ne free variables‘? Then we just put the arbitrary letter ‘x’ fer thevariables in the matrix and test the resulting ene-variable schema fertruth-functinnal validity. E-g-, the validity ef: Ex Ey{F-ry -—> Fyx}cemes dewn te that ef ‘Fax —-I F.r;t’. The criterien is suffieient, since then-ne-variable schema is an instance {ef an instance . . .} cf the existentialschema and se implies it. ass it is necessary, since, in the universe c-f asingle ebject-1:. the existential schema bails dewn te the ene-variableschema. ’it’alid pelyadic schemata may er may net he menadically valid {seeChapter I-lb}. The validity cf {l} and {4} is an irrcducibly pelyadic matter.The same is true ef:{ti} H-r \"'tI\"y F-ry —1- ltfy Ex Fxy,which tends ta be the first genuinely pelyadic example that ene thinlts ef.Let us try eur new methed en it. Cenverting it te prenex ferm withprierity en universal quantifiers, we get: ‘li’.r\"v'y Ha 3w{Fxr; --it Fwy}and se preceed te test this:{T} Ha Ew{F.t.t ->Fwy}.Cine way ef substinrting the free variables ef {T} fer ‘e’ and ‘w’ in‘F-tr —1- Fwy‘ gives ‘Fry —1- Fry’; and there are three edicts. The test cfvalidity ef {\"i},_ and hence cf {ti}, weuld cnnsist in farming the alternatiencf ‘Fry —1- Fxy’ with its three ccmpanicns and examining the result fertmtlr-functienal validity. Eut cf ceurse there is ne need el’ all this. since‘F-ry —> Fry’ is valid by itself; the alternatien has te be valid.
lllti Hi . Cenerttrl Theaty riyf Qaantificatien it happens cenveniently eften, in testing pure existentials. that a singlesuch result cf substitutien tums eut thus te be valid by itself. Thus taltethe example in Chapter 2? abeut drawing circles. l-lere the cenditienal tetest fer validity is:{S} lil'x{Fx —r C-t} —1- ‘lil\"y’[3x{F-1: . Hyx} A 3x{C-rt . Hy-r:}l{see Chapter IT, {1}}. Cenverting it te prenex ferm with prierity anuniversal quantifiers and then drepping the initial universal quantifiers,we get the pure existential:fill at Elvllf-t —rCt.' -—tt Fx . Hyx .—t-. Cw . Hyw}.New there are feur ways cf substituting ‘-rt’ andier ‘y’ fer ‘x’ atrd ‘w’ in thematrix cf {ii}. Cine way gives:{lil} F.r—s-C.t.-s: F-r. Hyr .-s. Cx.H‘yx,and there are three ethers. The test ef validity ef {H}. and hence cf {S},weuld censist in ferming the alternatien ef {ill} with the ether three andtesting the leng result fer truth-functinnal validity. But again we dc betterte teel: befere leaping; {ltl} itself is truth-functienally valid. se cf ceursethe altematien will be se iiltewise. The reader will have the same experience again if he tries the secendexample in Chapter Ti. the ene abeut the paintings; again a single resultef substitutien tums eut valid. Semetirnes. hewever, we have te press ente the alternatien cf twe er mere such results cf substitutien. This wastrue already cf the simple example {-1}; neither half cf {5} is valid by itself. Se we have a decisien precedure fer validity ef pure existentials. lnview cf the warning that there is nc general decisien precedure fer valid-ity ef quantificatienal schemata, then, we may be sure that universalquantifiers cannet always be maneuvered inte initial pesitien. Here is asimple valid schema that resists such treatment:{ll} Hyltf-t{Fx.t —=|-Fyr}.Tc preve the validity cf such schemata we shall have tti restrrt ttr meregeneral metheds. But it is a gee-ti ptrlicy, when trying te preve the validity cf a schema,te try first fer a pure existential. This eften succeeds, and when it deesthere are twe advantages. Cine is that a direct test is available. assuring a
,?§I_ I-‘are E,xi.rrentiat's l RTcenclusively negative answer er a preef. The ether is that the precesstends tn be relatively swift, since it se eften happens, as in all three ef theexamples that we have been examining frem Chapter 2?, that a singlesubstitutien cf universal variables fer the existential enes settles matters- lncidentally we new have anether general decisien precedure fer va-lidity ef menadic schemata, altemative tn the metheds in Part ll- Fer wesaw in Chapter 14 hew any menadic schema can be purified; and we sawthat, encc purified, it will have ne staclted quantifiers. But the t'ules cfpassage enable us re bring quantifiers inte prenex pesitien in any erder welilte if nene cf them are staclted- Thus any menadic schema can be testedfer validity by the methed ef pure existentials. E-y way ef epe-ning a field fer further examples invelving pelyadicschemata, let us acquaint eurselves with sytntnetry, tratrsitivt'ry. re_,t‘i’e.rt'v-ity. and related cencepts—the-se be-ing werth neting alse in their ewnright. it dyadic term is called symtnetricai, asymmetrical. transitive.intransitive, tetally reflexive, reflexive, er irreflexive aceerding as itfulfills:hl‘-r‘tt\"y{Fx_y —+ Fytr} {symmetry}\"it‘.r‘li\"y{F-x_'|t —+ — Fy-1:} {asymmetry}\"It\"-r\"rI\"y'b\"-=;{F.x-y . Fyr; .--1. Fxx} {transitivity}\"I\"-t\"tt\"y\"Itt\"r'{F-ty . Fyr .—+. —F.ra} {intransitivity}‘if-t Fscr {tetal reflexivity}‘lit’-t'lrl‘y{F.ry —1-. Fxx . Fyy} {reflexivity}’lit‘.r —Fxx {irreflexivity}The dyadic term ‘ccmpatriet’ is symmetrical, in that if x is a cempatri ct efy then y is a cempatriet cf x. It is alse transitive. if we disallew multiplenarienality; fur then if -r is a cttmpatriet cf y and y cf .t:. .t will alse be acempatriet nf .:. lt is alse reflexive, if we censider a persen a cumpatrictcf himself--as indeed we must if ‘cempatriet ef‘ means ‘having samenatienality as’. El-ut it is net tetally reflexive, if we thinlt ef eur universe ascentaining any things deveid cf natienality. Examples ct’ tetal reflexivityare rare and trivial; ‘identical’ and ‘ccexistent’ are twe such. The dyadic ternr ‘nerth’ is again transitive, but it is asymmetrical anditteflexive; ‘r is nerth efy’ excludes ‘y is ncrth efx’, and nething is nttrthcf itself. The dyadic term ‘metlrer’ is intransitive, asymmetrical, andineflexive. The dyadic term ‘leves’ laclts all seven preperties. ’1iit’l‘tere-t leves y, ymay er may net le-vex; thus ‘leves’ is neither symmetrical ner asymtnetri-cal. Where .r leves y and y leves s, x may er may net ltrve 2;; thus ‘leves’ is
lfifl ll’! . General Tneat? sf flaaafifieatianneither transitive ner inn-ansitive. And, sinee seme lave theniselves whileethers {even amnng thnse whe leve hr are lev-ed] dd net have themselves,‘leves’ is neither reflexive nur irreftexive. The reader may wander why. parallel tn the distinctien hetweenreflexivity and tetal reflexivity, a tiistinetina is nnt drawn between“irreflexivitv“ in the sense at: \"'I\"x'1I'jvl_'Fx_a —1-. —Fxr . —F}[s}and “tetal irreflexivitv” in the sense hf \"'I'x —FJ;x'. The reasen is thatthis latter distinctien is illusnrv; the twe sehemata are equivalent. Teshnvv this we estahlish the validity elf twe ennditianals. as fellews:-[Ill 'li'x‘tl\"jv|[Fx}' —+. — Fxx . —F;v_v} —* \"tI'x —Fx'x,[I3] ‘liI\"x —Fx.r —1- \"I\"x‘I'}'{FxJ.= —1-. —F.t:.t . —Fy_}~},Taming them inte prenex fertn and drepping initial universal quantifiers,we get:‘El-ill §x3}'l[F.1.jv A. —Fxx . —F]|.'].= 2-r —Fzz\",i_llfrl 321%?-\"is -1-:F;|:;|.~ —1-. —Fxx . —F;v_v}.The tine is sh-awn valid by this truth-funetinnallv valid result at substitu-tinn: Fez —*. —Fat . —Fzz :--v —-Fa:-:and the ether by this truth-funetihnallv valid alternatien hf twe: —Fxx —>:Fx_v —r. —F.‘-I11‘ . —Fj,=},= .:v:. —.F}t_v —=~: F1]: —1-. —FJ:x . --.F'_v_v. As anether example it will he prnved that E}'lTi[I'lEl‘.I‘}f and l.I'i3l15ltIl‘||'il‘.\"_*||‘tegether imply reflexivity. This means [srrevittg the validity hf the trendi-tinnal:[IE] ‘tl'x\"il\"j|r[F,t*y —1- Fyx] . \"i'x‘i'}='li\"z{Fxjt' . Fya .—i- F13} .-I 'll\"x'li\"'y[Fxjv —*. Fast . Fyy].Turning it intn prenex ferm and drepping initial universal quantifiers, weget:
25'. Pure .l;'.t|\"t|'t's‘tttt'ttl'.s 155'{IT} 3vHtvH.rHy 3a{Fvw —1~Fwv t Fxy . Fyz .—-1- Fx:-5 t—t-2 Fin -1. Fa . Fan}.This is shewn valid by the truth-funetienal validity hf the alternatien hftwe tif the results pf substituting ‘t’ and ‘a' far the existential variables-[13] Fta —a-Fat : Fm . Fat .--1-Fit I--rt Ftn —-t. Ftt . Fan .:v:. Fta --v Far : Fat . Fta .-I-Fi..m:~—1-:Ftn -—-1-. Ftt . Fl-thl.The reader is urged te- dri a painstaking truth-value analysis befnre agree-ing th the tt‘uth—fi1rtt:tirma] validity el’ this altematien. HISTDRICAI. HDTE: The existeiiee at a validity test tn this purpeseseems tti have been first ntited by Berrtays and Sehtinfinlsel, 1923. ln thepresent methed there are eehees pf Herbrand that will beepnie elearer inChapter 315.EIERCISESI. Delineate all the steps at passage, relettering, and ti-eletitin that ear-ry {I} inte {Z}; U5] intti {T}; [E] inte[':T.l;{l1}intti[l4}; [13] iata H5]; {H5} inte l'l'l]|.E. Cheelt the validity el’ {IE}.3. De the example in Chapter 1? abeut the paintings.4. Repeat the exereises nf Chapter 25 by the new method.5. Prrive that asymmetry implies irreflexivity.ti. Pt-eve that intransitivity implies irrettexivity.T. Prave that transitivity and irreflexivity tegether imply asymmetry.
|'ElI[] HI . Genera! Themy sf Qanntifierttien 30 THE MAIN METHOD We turn new tu a pruut\" pr-:3-eedure that will he feund tu he eumplete:adequate tu establishing the validity uf any valid quantifieatiunai sehemaand henee alsu tu establishing any implieatiun and any ineunsisteney_ tnfaet it will be eriented tu ineunsisteney pruuts. Tu preve a sehema validwe shall preve its negatien ineu-nsistent; and tu pru-ve irnplieatien we shallpreve that the ene sehema is iueunsistent with the negatien uf the nther. ln {I1} nf the preeeding chapter we saw a eandidale fur a validityprrsnf, inaeeessible tn the methnd uf pure existentials. Te pruve it valid bythe new methed we negate it. assume the prenex furm e-f the negatien aspremise, and prueeed tu generate instanees frum it as fulluws: ssssnss: \"tI\"y it — {Fat —> Fysll uvsratvess: Hr; — [Fxa —>.F.r.:} — {Fxu —r FIN]Preufs uf incnnsistency in this system always end up with a truth-funetiunally ineunsistent sehema, as here, er a truth-funetienally incan-sistent eumbinatien uf sehemata; and this melanehe-ly exhibit is meant teshuw that the premise was at fault and thus incensistent. Each sueeessiveline is an instanee uf an earlier line. We seem tn see here the pattern traditinnally ltnewn as redaettti aduhsurdum: dispreef by derivatiun ef a elear euntradietiun. But derivatihnhew? Du quautitieatiens imply their instances? Universal quantificatiens du indeed. The first line uf the preef abeveimplies the seeund. The eperatien hf miiversui instunrt'utt'un—LTl, as lshall eall it—whieh leads frum the first line tn the seenttd is implieative. Hut existential instantiatien, El, whieh leads fruni the seetind line letthe last, is nut implieative. ‘ Ex Fa\" dues nut imply ‘Fy’. Cunsequently itis nnt evident that the derivatiun hf an inennsisteney by prugressive in-stantiatien, as in the preef abeve, suffiees tn shew the premise inee-nsis-tent. It dues suffiee, but the ptiint will talte snme prtiving- Hefure prueeeding tn justify the methnd, let us get a firmer idea ttf whatthe methnd is. Fur nne thing, El is tn he hedged thus: The in.vtuntiatvurinbie, whieh is substituted fur the variable that was beund by the
Search
Read the Text Version
- 1
- 2
- 3
- 4
- 5
- 6
- 7
- 8
- 9
- 10
- 11
- 12
- 13
- 14
- 15
- 16
- 17
- 18
- 19
- 20
- 21
- 22
- 23
- 24
- 25
- 26
- 27
- 28
- 29
- 30
- 31
- 32
- 33
- 34
- 35
- 36
- 37
- 38
- 39
- 40
- 41
- 42
- 43
- 44
- 45
- 46
- 47
- 48
- 49
- 50
- 51
- 52
- 53
- 54
- 55
- 56
- 57
- 58
- 59
- 60
- 61
- 62
- 63
- 64
- 65
- 66
- 67
- 68
- 69
- 70
- 71
- 72
- 73
- 74
- 75
- 76
- 77
- 78
- 79
- 80
- 81
- 82
- 83
- 84
- 85
- 86
- 87
- 88
- 89
- 90
- 91
- 92
- 93
- 94
- 95
- 96
- 97
- 98
- 99
- 100
- 101
- 102
- 103
- 104
- 105
- 106
- 107
- 108
- 109
- 110
- 111
- 112
- 113
- 114
- 115
- 116
- 117
- 118
- 119
- 120
- 121
- 122
- 123
- 124
- 125
- 126
- 127
- 128
- 129
- 130
- 131
- 132
- 133
- 134
- 135
- 136
- 137
- 138
- 139
- 140
- 141
- 142
- 143
- 144
- 145
- 146
- 147
- 148
- 149
- 150
- 151
- 152
- 153
- 154
- 155
- 156
- 157
- 158
- 159
- 160
- 161
- 162
- 163
- 164
- 165
- 166
- 167
- 168
- 169
- 170
- 171
- 172
- 173
- 174
- 175
- 176
- 177
- 178
- 179
- 180
- 181
- 182
- 183
- 184
- 185
- 186
- 187
- 188
- 189
- 190
- 191
- 192
- 193
- 194
- 195
- 196
- 197
- 198
- 199
- 200
- 201
- 202
- 203
- 204
- 205
- 206
- 207
- 208
- 209
- 210
- 211
- 212
- 213
- 214
- 215
- 216
- 217
- 218
- 219
- 220
- 221
- 222
- 223
- 224
- 225
- 226
- 227
- 228
- 229
- 230
- 231
- 232
- 233
- 234
- 235
- 236
- 237
- 238
- 239
- 240
- 241
- 242
- 243
- 244
- 245
- 246
- 247
- 248
- 249
- 250
- 251
- 252
- 253
- 254
- 255
- 256
- 257
- 258
- 259
- 260
- 261
- 262
- 263
- 264
- 265
- 266
- 267
- 268
- 269
- 270
- 271
- 272
- 273
- 274
- 275
- 276
- 277
- 278
- 279
- 280
- 281
- 282
- 283
- 284
- 285
- 286
- 287
- 288
- 289
- 290
- 291
- 292
- 293
- 294
- 295
- 296
- 297
- 298
- 299
- 300
- 301
- 302
- 303
- 304
- 305
- 306
- 307
- 308
- 309
- 310
- 311
- 312
- 313
- 314
- 315
- 316
- 317
- 318
- 319
- 320
- 321
- 322
- 323
- 324
- 325
- 326
- 327
- 328
- 329
- 330
- 331
- 332
- 333
- 334
- 335
- 336
- 337
- 338
- 339
- 340
- 341
- 342
- 343
- 344
- 345
- 346
