Methods of Logic - W. O. V. Quine

dd- Cl'es.tes 19 I Use tit‘ ‘=‘ rather than ‘E‘ frem the start, in Chapter Ell, weuld havebeen mere in keeping with my treatment ef ‘e‘. ‘l;‘, and ‘C’. l ceuldhave gene en te ettplain that i was there using ‘= ‘, like ‘e‘, ‘§‘, and ‘C‘,as a cepula rather than as a dyadic general term true ef ebjects- This,hewever, weuld have been an escess ef pedantic teal and a hazard teclear thinking. lt is safer te reserve ‘=‘ censpicueusly te the identifyingef ebjects and leave the equating ef general terms te ‘E'- Llp te a peint the admitting ef classes as values ef variables can heesplained away in tertns merely ef the censistency and validity efquantificatienal schemata. Te affirm that \"'tl'_v[\"IiI\"-rl.r ey] H H-rt-1: e_y,il, 3).-'[ H-rt-r cyl . — \"tI\";r[.r Eyl]ameunts merely te affimiing the validity ef the schema “lib: F-r —s Hr Fr‘and the censistency ef ‘ 3:: Fr . — ‘til-r F-r‘. Such an aeceunt is interestingin that, within its limits, it esplains statements abeut classes witheutpresuppesing classes. l-lse is made merely ef the cencepts ef quantifica-tienal validity and censistency- in Chapter 33 we feund that these cen-cepts ceuld be specified in terms ef deductive rules, er again in terms efsubstitutien, witheut dependence en the class cencept. ilt statement abeut classes can be thus esplained as leng as its classquantifiers, as l shall call them, are all prenes and furthermere all itsqua ntifiers, thmugb its last class quantifier, are universal er all ettistential-(By a class quantifier here I mean merely any quantifier whese variablehappens te recur after an epsilen.l if they are universal, truth ef thestatement ameunts te validity ef the cerrespending quantificatienalschema; if they are eitistential, it ameunts te censistency ef the schema.E.g-, truth ef the statements:\"If-r\"tft[-t er .-I Elyty eel]. at Hwltil-rt: Er -H- -t e iv]ef set theery ameunts te validity and censistency ef the respectiveschemata: F.t—+ 3yFy, ltt';r{F-t Htltlef quantificatiun theery- This expedient can he pushed a bit by bringing intemal quantifiers inteprenes pesitien in the usual way- Thus the statement:

Z91 fir’- Gtinqpses Heyerid 3e[‘ttl'w‘if:rt:r is w .-—t-. tr E cl ——l- Ely 3-star ey . .1: is al],with buried class quantifiers “'tt\"w‘ and ‘ Ely‘. can be transfnrmed inte: 3,; Elw3y[\"t\".rl-r ew .-s. .r es} —t El.1:i_'.r ey . .r eel],which can be captained as merely shinning in effect the censistency efthe quantificatienal schema: ‘it\".i:[t._?-r —-1-F-1:} -I 3-rtjt-is - Fr]. Laws ef classes cannet be explained away in the abeve fashien, hew-ever. when their prenes quantifiers are mittedly universal arid existentialas in the esamples: ‘lt\"a3w\"It'.tr{lr cc -*+- .t cw}, \"I\"-i:\"it'y3el.i cc -H. y cal.it is in such statements that the irreducible substance ef set theery, er thetheery ef classes, is te be seught- Similarly fer statements centainingburied class quantifiers which, when breught inte prenes pesitien by thet1.lles ef passage, beceme mittedly universal and esistential; e.g.; li|\".t'lt'y[\"il'rlI.t ea .--I-t y eel--1 ltfwty cw .—t~. -t ewll. Elut this pewer ef ettpressing irrcducibly new laws weuld ef itselfjustify little interest in class tlteery, were it net accnmpanied by a cerre-spending increase et pewer en the side ef applicatien. A geed example efthis effect may be seen in the definitien cf flie dyadic term ‘ancester‘ enthe basis ef ‘parent’. Tu simplify the situatien let us understand 'ancestet‘in a slightly breadened sense, thereby ceunting as a persen‘s ancesters neteniy his parents, grandparents, and se en, but alse the persen himself. Letus represent ‘parent’ by ‘F ‘, se that ‘F.ty‘ mcarts ‘.r is a parent efy‘. l'~lewthe preblem is te write ‘-t is an ancester ef y‘ using enly ‘F ‘ and eurvarieus symbels- hn impertant feature et the class ef y‘s ancesters is that all parents efmembers ef the class are members ef it in turn. Anether feature ef it isthat y himself belengs te it. But these twe features de net yet fit: the classcify ‘s ancesters uniquely; there are larger classes which alse centain y andcentain all parents ef members- tltne such class is the class ef the ances-ters ef y‘s gra|1dsens- Anether such class is the cembined class ef y‘s

dd. if-‘let-sses 193ancesters and neckties; fer, neckties being parentless, their inclusien deesnet disturb the fact that all parents ef members are members- But clearlyevery class which centains_v and all parents ef members will have at leastre centain all y‘s ancesters, ne matter what estra things it may happen tecentain. lvlereever, ene ef these elasses centains nething but y‘s ances-ters. Hence te be an ancester ef y it is necessary and suflicient te heleng teevery class which centains y and all parents ef members. 'l‘l1erefere ‘-r isan ancester ef y‘ can be written thus:-t belengs te every class which centains y and all parents ef membersand schematiaed th-us:ll} ‘tt\"n[veu.ltt'a‘li\"n-(wen.F.:w.—+.aea].—t~..sea]- This ingenieus censu-uctien admits ef many appticatiens besides thisgenealegical ene- An applicatien te number will be seen in the nestchapter. What is significant abeut the censtmctien fer present purpeses isthat it depends en quantifying ever classes. -anethet esample ef the pewer gained by quantifying ever classes wasprnpesed by Geach and perfected by David ltaplan, in private eerre-spendcnce:{2} Seme penple admire enly ene anether.hlaplan has preved that we cannet ettptess this in terms ef just ‘persen’,‘admire‘, identity, truth functiens, and quantificatiun. {2} dees net implythat there are twe peeple whe admire enly each ether. It might heldbecause ef seme irreducible greup ef say eleven mutual admirers. invek-ing classes and membership, we can de justice te {Z}:3at3xt_-t es) . ‘tI\".r[-t: ea .—t-. Hyl-1: admiresy} . \"'It'yl.t admiresy .—t-. s at y . y ctlll. A simpler illustmtien ef ear new access ef pewer may be seen in thefact that the sign ‘=‘ ef identity can new be defined; fer things areidentical just in ease they beleng te the same classes. -tr = y .H ‘ital-t ea .—t-. y es].

29¢ it’, Glimpses tleyentiWe can then preceed te define alse the universe, die empty class, me classwhese sele member is-1:, and the class whese members are .=r and y. l\"i|,,I\"= -r:-r =1}, A i l,1’:-t=.#;c}.l-rl= ls:-r = -tit lrtrl '- {a:a=-t'.v.a= Hisreiacsi. I~ttZtTtE= The censuuctien illustrated in the definitienef ancester was intrerluced by Frege in lE'l\"':i' fer applicatien te number. ltwas rediscevered independently a few years later by Peirce, and again byDedekind, whe prepeunded it in 135? under the name ef the methed efcltititts-l 47 NUMBER We say the Apestles are twelve, but net in the sense in which we saythey are pieus; fer we attribute piety, but net twelveness, te each. ‘TheApestles are pietts‘ has the ferm “til-rlF-t --t- Gt)‘, with ‘F ‘ fer ‘Apestle‘and ‘G‘ fer ‘pleas’; but \"lhe Apestlcs are twelve‘ has ne such ferm, andis mere nearly cemparable te the mere ettistential quantificatiun '3-tr Fr‘-'I'his familiar quantificatiun may be read ‘The Apostles are at least ene‘;and we might analegeusly think ef ‘The Apostles are twelve‘ as written‘ Hus Fr‘, using a numerically definite quantifier. These can be intredu ced en the basis purely ef the thcery ef quantifica-tien and identity as ef Ch apter -'-l-E; there is ne need here te assume classes-We can begin by explaining .3411 Fr‘: 3,],-t F.‘-t.‘ e-i-— H:-rF-t-Then we can ettplain each succeeding numerical quantifier in terms ef itspredeccsser in a uniferm way:

1! 5\". Number 295U} 3,1 Fa: H 3.r{FI . El,,y|_'Fy . y =.F=..t]i, 351.5\"; H H-:c{F,r . §1yt‘Fy . y #=..r]j,and se nn- Fer anything tti be true el’ rt + 1 things is fnr it tn be true pfsemething ether than whieh it is true ef rt things. By a deaen steps efespansien aeeel-rding te these definitiens, ‘H111 Fx‘ gees ever inte asehema nf identity theery. Similarly fnr any ether numerieatly definitequantifier- Hewever, we still have nn eitpansien el ‘Ht: Fir‘ with vari?able ‘n‘. We ean say there are twelve Ftp-nstles and twelve Muses, in thefnrni ‘Hurt Fa: . Ems GI‘, but we find diftieuity if we merely want tnsay tl1at there are just as many tltpnstles as l'vluses witheut saying hewmany. He definititins are at hand fnr expanding: §rr{ ing: FI . 3,1,1.‘ GI}. All high-sehe-nl students appreeiate the persistence, and seme the util-ity, ef number variables in algebra. The esample abeve and sueh relate-tlenes as ‘There are twiee as many eyes as faees' suggest that quantifiablenumber variables have a plaee alse in the analysis ef virtually unmath-ematieal diseeurse. If we are te have quantified variables fer numbers we must find entitiesin eur universe te view as numb-ers—er else eitpand nur universe teinelude sueh entities. As a step tewarti a reasenable theery ef numbers,censider the atljeetive ‘twelvefeld‘. If we are ta reeegniae sueh a terrrt,and net just the eerrespending quantifier ‘ 3,2:-1', we must reeegrtiae it asa term that is true he-t ef persens, e.g, Apestles, but el’ elasses, e.g. theelass e-F Apestles. Rendered as a term abstra-::t. twelvefeld I {y: 3,;.r{.r ey}}.The remaining step te the number llas an nbjeet is a shert ene: talte I2 asthe estensien ef ‘twelvefeid*, henee as the elass ef all twelvefeltl elasses.Se, new that absttaets and ether general terms are detibling as names eftheir estensiens, IE ={}'I3]+;.-'l.\"|:.1' eyl}.Wliere 3 is a number, set ennstrued, tu say that a elass_v has .1 members issimply te say that y E r.

E915 .i'l\". Glimpses Hrjnnnd The first ef the numbers, I], is then {fit}: the elass whese sele memberis the empty elass. I is then definable as the sueeesser ef ti, and 2 as diesueeessnr nf 1, and sn en, enee we define sueeessnr. Hnw tn de that isevident frem {I}: the sueeesser ef e, eall it Se, is the elass ef all theseelasses whieh, when deprived tit‘ a member, beeeme members ef e. Thus 5:5 = {yt Hrrtlr ey .y—{,r} eel}.Here —{:-r} is the elass ef everything but .t, se the eenjunetien y —- {1} isthe elass ef all members but I efy- Ftdtiitien bf numbers is easily defined in view ef this eireutnstanee: aelass has y + e members if and enly if it ean be brelten inte twe partshaving y ands members- But te say that a elass iv has e members is just tnsay that iv e 3; st] y+,-5={,t:Hu3iv|'_s'=uviv.aw=t\"‘t-uey.wei;]}- lvlultiplieatzien is definable in view ef a similar eireumstanee: a elasshas y - e members if it eatt be brnlten intn y pans having :5 members eaeh.And nnly if‘? Her quite; there is an e.‘-teeplibn where s = U, fnr we eannethave _v empty elasses if y 1'-'= I.” The ferms] definitien bf ynt is still areutine maner, but it runs leng and we may pass ever it. 'Wl1at new ef a fermat definitien el’ ';-; is a number’, 'l'~lz'\"i’ Numbers inthe present sense are just D, l, 1, . . .--that is, ti and the pesitive integers.Negative numbers, fraetinns, irratinnals, and imaginaries are net ef thesert used in measuring elass sine. Se eur present preblem is te- define ‘1\"'-l,-;‘in sueh a way that it will entrie eut true when and nnly when: is ti er l erE, ete. A means bf aeeemplishing this is suggested by the treatment bf‘aneestnr’ in the preeeding ehapter. Just as ‘aneester ef y‘ means ‘y erparent efy er parent el’ parent el’ y er . . .', se ‘number’ means ‘ll’ er Sill urSSE! nr . . In elt:-se anslegy te l l} pf the preeeding ehapter, aeenrd-inglyt we ean define thus:{2} I“-l:s—1-\"i\"u[{leu.'lfy{yeu .-1-. 5;-tett}.-r.et=tr].Tu be a number is tn belting tn every elass tn whieh El belengs and thesueeesser ef eaeh member belengs. \" Here I am indebted te Peter Cieaeh-

sli’- Number Eli? This definitien supparts rrrnthematicni iiiducrinn, a preef precedureeentral te number theery- We preve that Va [Na —* FE} by preving that Fl]and \"t|\"y[.Fy —1- Ftfiyjlfi- The reader sheuld ebserve hew {E} justifies thisinference. We are new in a pesitien te say that there are just as many ntpestles aslvluses. 3,:|[l\"'-lg . Fipestle ea . lvluse E.Tl,'UHere the iise, here again, ef general terrns as names ef their eittensicins. Besides letting menadic general terms mtinnlight as names, and thuspnsiting an abstract realm tif elasses, we tnight dd the same fur dyadicgeneral terms. This weuld mean pnsiting a secnnd abstract realm, that eidyadic relatiens, Pit this peint, he-wever, we can ectinerniae; the classesalready suffice fer generating a full cemplement ef relatiens tit theirreasenable facsimiles- A relatien can be eenstmed as a class ei erderedpairs. The uncle relatien is the class nf ali the uncle-niece and uncle-nephew pairs. Find what then is an tirtieretl pair? iltay arbitrary netinn ef a pair cifebjects I anti y will serve eur purpeses perfectly sin leng as these cendi-tiuns are met: [ii fnr any ebjects: and y there is such a pair; [ii] as seen asa pair is given, its first ebject :t is thereby uniquely deterrnirt-ed, and sci isits secend ebject y. [Thus the pair ef: and y is different frem the pair plyand ,:r, unless ,1.‘ = y,] it happens that these canditicins are met when thepair cif ,1: and y, written ‘{,r. J-'i\", is defined arbitrarily thus:till i1ti'i=iiIi.{Iii'll-—henee as the elass cif classes that has as members just the elasses {1}and {,r, y}. Dyadic terrn abstractinn thereuptin reduces te menadic, thus:{,ry:...,t...y...}!{1: Hxfiyte = {Ly}Lilte any menadic term abstract, it ccintinues tn functien as a general terl1'ialeng with dning deuble duty as a name within such limits as may beimptised tin class ertistence by the antincimies, In snme cases '\" Heatim less gifted than yciurself may feel at this peint that the lvluses are beingtreated mnre ttilerantly than Cerberus -[Chapter ill}, thus failing In appreciate that citireitample is nene the wtirse lclr being false.

293 ll’. Glimpses Btrycind Hale = {iry:.,..r, . .y...}],in enters net- Liite any menadie terms, dyadic enes are subject autematically te theliieelean functers, These rnay be theught cif as defined by abstractinn as intfE]—{ I 2} ef Chapter El- alse the dyadic terms usher in a useful assert-ment ef farther funetets- Ubvieus enes are these ef inversien, er cenver-sien, and ef reliectien; they simply repeat the ‘inv Fl‘ and ‘Ref F“ elChapter 45. Fur\"-lher fliere are the Cartesian pratlact, the resultant, and theimage, defined thus: F I-it G E lry: Fir , Gyi, Flt? E {,ry: Hr[F:nt . G-ell. F“G E {,r: Hy[F.ry . Gyil-Thus ‘cat it deg* relates every cat te every deg; ‘mntherl parent‘ isequivalent te ‘grandmether'; and the veteran“war are the veterans cifwars, .t'-'lt carrelatlert is a relatien that ne ene ebjeet bears tn mere than ene,and ne twe ebjects hear tn any nne. Again we may as well adjust thedefinitien te dyadic general terms witheut regard ta whether they arenames ef relatiens. EFIIIF s—1\"l'.tlI'y\"G';-_'{F,I.t t Fyg .'|tt'. F31 . Fgy :—ii, I = _p}_This afferds a see-end way el’ saying that there are just as many Fipestlesas lvluses: they are in cerrelatien, Fer saying that the F and the G are incerrelatiee there is an eld netatien, 'sm' fer *similar‘: F smG .1-i 3,r{Crln,t . F E=,i:“G . G It {inv,t]|“F}, l\"~lurnbers were treated ab-eve as sizes ef classes. Being enly U and thepesitive integers, hewever, they were the sizes eniy ef finite elasses, Thecardinal nurnbers, se called in set theery, are mere inclusive; they mea-sure alse infinite classes. New just as the finite cardinal numbers werecenstrued as classes nf litre-sized classes, se may the infinite enes. Ingeneral then the cardinal number ef a class I, finite er infinite, is{y: y sms}, Dne might suppese that all infinite classes are aiilte in sireand henee that there is at mest ene infinite number, and this weuld be tnsuppese that any twe infinite elasses are in cerrelatien, Far frem being

4' F. Number 299ebvieus, this can fer genernus universes be disprev-ed; see Set Tfisatry andlts Laglrr, ‘HE, ‘Ways ef censtruing numbers are available that are very ualtlte the enewe have been studying, An elegant alternative, mere widely fellewednewadays, taltes H simply as ilt and Se as it v {ti Each number thusbecemes the class ef all emlier enes. The definitien {2} el’ ‘H3’ thencarries ever, using the new versiens el’ ‘ll’ and '5'- This versien tilnumber alse eittends neatly inte the infinite, but I shall net pursue it, Te say that y has: members was simply te say, in the earlier versien cifnumber, that y ts 3, Hew weuld we say it new? The answer is evidentwhen we reflect that each number e has :5 predecessers, ceunting El, andhenee, in the new versien, r members. Te say that y hast: members is thente say that y sin s, Htsreitucat hl1DTE= The definitien cf numerically assistsquantificatiun, in effect, is in Frege {I354}. Se is the interpretatien cifnumbers as elasses ef apprepriately sized elasses, and se is the definitien{2} ef *l'\"-Lt’. The elegant alternative versien cif number neted in the lastparagraphs is due te vcin l‘-leumann (I923)- The thetiry el’ infinite numbersis due te Canter (13911). It was he whn equated sizes ef elasses byeerrelatitin and preved, en certain reasenable assumptiens in set theery,that there is ne highest infinite. The eld algebra ef relatiens stems frem Delviergart and Peirce and inpart frem Cayley‘s grtiup theery {I554}. See abeve, end ef Chapter 2?.Wiener in llilltl was the first te efier a generally applicable censtmctien efthe erdered pair. The versien used abeve is a variant due te ltiuratewslti.EIE HCISES1 . Where F I {,ty:,r is a mart . y is a mement . r is alive in y}, -G ‘E {y:y is a mement in 1955} put these feur terms fairly naturally inte werds: F“tF, F\"d, -tjt-‘rm, —tF\"aji.

EDI] ll-\". Glimpses Beyand I. ‘Where ‘F’ and ‘G’ stand fer ‘admires’ and ‘brether ef’ , put these seven terrns fairly naturally intn werds: — [F llii}. linv Fl ll], F l [inv G}. {inv F} l {inv G], invffinv F] ltil}, [inv Fllti. —[[irtv F} l [inv Gil.3. Find interpretatiens ef ‘F‘, ‘G’, ‘H*, and ‘Kl such that - ltFfi'l“H E tF“llltt3“Hll. — s-“tare E ts\"Hitr\"s;-1. Justify yeur answers.it. Shaw that {3} fulfills the requirement {ii} en erdered pairs.5- Spell eut the justificatien by [2,] ef mathematical inductien- AXIOMATIC SET THEORY We have been seeing samples cif what can be eitpressed in set dreary.The main sample eemprised the finite cardinal numbers, One can ge en tedefine the raties, the irratienal numbers, the imaginary numbers, funcatiens, indeed the whele apparatus ef elassieal mathematics, withtiut es-eeeding the frugal netatien ef set theery. This netatien, mnrenver, can bevery frugal indeed. All that is needed besides quantificatiun and the truthfunctiens is the epsilen ef membership. Abstraetien is net needed. It canbe defined by deseriptien thus:[I] {II Fr} = fty}\"lif.rt,r ey ,-i—iF,rland descriptien is eliminable as in Chapter ll-4, This definitien ceversabstraetien enly in its use as a elass name and net in its breader use as ageneral term; but that breader use is heuristic and dispensable. Schematic term leners tum up in eur disceurse abeut the netatiens tilbeth legic and set thent‘_‘r'. as just new, and tertns ef all sens tum up insentences that eetnbine set theery with ether tepies in applying set theery

dd. itrlamatic Set Tlteary 3[l'itn the wnrld- ln actual sentences cif set theery, hewever, there are neschematic letters—sentences never centain t'nem—-and furthermere thereis an need ef abstractinn, thanlts te t_' ll, ner any call fer any general termeitcept epsilen itself. As ntited in Chapter -tlti, epsilen figures in thesecenteitts as a dyadic general term. The reducibllity ef the whele vast cenceptual apparatus ef classicalmathematics te this minuscule capsule is a reasen fer interest in settheery. Specialists are drawn alse by an interest in the pure mathematicsnf the higher infinite-s, witheut regard te the translating er medeiing efclassical mathematics. ln these pages en set theery l have been eccupied with questiens el’definitien and net cif theerems and preef, but l did mentien that there areantinemies and that they stand in the way ef treating all general terms asnames cif their estensiens. The simplest and best ltnewn et these an-tinemies is fi‘itsssll’s parades. By cencretien fill} ef Chapter 2]},ill is irr —is till -e'r—tr til.but there is ne such class; fer, talting y as that class, we ceuld infer frem{ti that r tr -at —ti til—a cnntradictinn. fl] is indeed true fer every ebject y, butill -* 3:vt.i' = {II r is trill-Here, fer all its negativity, is a firm little theerem ef set theery. The antinemy admits ef variants and embellishments, ltindred invarying degrees, and there is ne way ef neatly segregating them. Thereare pairs ef abstracts such that either ceuld cnnsistently be talten tn name aclass, but net beth. We cannet eltcise the lethal classes witheut eltcisingin necueus enes as well, in ene quarter er anether. There are censequentlymany alternative aitiem systems er preef prncedures, differing fmm eneanether in respect cif classes admitted and excluded. Cine system mayexcel in simplicity in seme respects, anether in naturalness in seme re-spects. tllae system may be belder and richer in its yield ef theerems.anether pri-erer and thus less liltely tn turn eut incensistent. The ene inwidest use by specialists teday is Zermelels er nest ef ltin. l shall sltetchZertnele's.

HUI lit’. Glimpses Eryand its mest characteristic assumptien ef class ertistence is the artiemsehema: Elsi-t = 'll’=l’ H - Fill.which afiirms the ettistence ef any specifiable subclass ef any given class,r- Since it yields elasses enly frem prier classes, further eitistence asiemsare Iieietied te prime the pump. We are given these: 3:1‘.-t = {:t.:-rll. are = lr=r E-ill. Elsi: = ly: Hit-'l_'y eiv . iv e,r}}}.A n'end ef the system is that a elass eitists unless it is tee big. Cen-sequently Eermele cannet use the Fregean versien cif number studied inthe preeeding chapter; each number sci censtrued, escept ll, weuld be teebig a class te ettist. But the ven hleumann versien at the end tri thatchapter werlts very well. Zermeie dees need infinite classes--just net tee eiteessively infinite.The feregeing aitiems being tee medest in that regard, he adds an artiempreviding an infinite class. Further attiems trail eff inte the increasinglyetIlttjet2ttll't1lI the asiem ef eheice and the centinuum hypethesis, te givenames te twn. There is ne clear stepping place, since a cnmplete system isnet te be aspired te. That much is clear fmm Eiiidel's preef ef the incem-pletahility ef number theery [Chapter 34}, since number theery can befnrmulated in set theery. ‘tien Heumann ctrntributed alse a set theery ef his ewn. It departedfrem Zermelcfs by admitting seme additienal classes, large enes, in amarginal status: they were declared incapable nf being members ef furtherclasses. Here it is that a use was feund fer the superfluity ef terms. ‘set’arir:l ‘ciassl: 'set* is reserved fer classes that can be members. That I is aset can be rendered ‘Hair rt.‘-1]‘. lt seems that classes cart be assume-dwitheut restrictien as leng as their rnembers are restricted te sets, theughvan Neumann did net gct dtat far. The burden ef further esistence aitiemsthen cemes in saying what elasses are te qualify as sets. ‘lien hleumanntaltes as sets Zermelels classes. Classes are censequently sets fer vtinl‘~leumanrr if they are net tee big. ln the ceurse ef these last three chapters we have ineenspicueusly butdecisively talten leave ef legic prnperly se called and meved inte anetherdepartment tif mathematics, set theery. The transitien eccurs when the

-ti-5’. .rl,rr'c-vrtatlc ,\"i'et Tlterrry 3-[iiiquestien is answered, er even fnrmulated, as te what classes there are:which enes ameng eur general terms are le succeed in deubling as names,Because ef the cemmen failure tri grasp the very distinctien betweengeneral terms and class names, the cressing ever frem legic inte settheery has eften gene unheeded dewn the years. lvtuch ef set theery hasbeen thnught ef vaguely as legic and cenvetsely. Yet the distinctien issharp and significant. Set theery, lilte se much else in mathematics andlilte natural science as well, treats cif its peculiar demain ef ebjects- Thelegical trttths, in centrast, are merely the sentences ebtainable by sub-stitutien in valid legical sche mata- They treat ef ne distinctive ebj ects andindiscriminately ef all. Hl5TClfi.lC.#r.L HQTE: The reductien ef ratitrs tn set theery is clue tn Peane and that ef the irratinnals is due chiefly te Dedekind {lS'l'2]. The eitecutien ef die whele reductien pregram in detail came with Whitehead and Russell’s menumental Frlnclpia Mathematics [ii-'llIl— I 3], eseept fer w'iener‘s step cif reducing relatiens te classes. [See preceding histnrical nete} Russell discevered his parades in lihill. He prepeunded in l'ilt]S his fameus theery ef types, a brand ef set theery that c-rrmmanded much faver fer fifty years but weuld require ertcessive ertpesitien fer present pur- peses. Zertrtele-ls system dates liltewise frem lijifill, and ven l‘~leumann’s| frem 1925. Fer fuller esp-tisitien and cemparisen el these systems and ethers see my Set Thee-ry and its Legic.



PARTIAL ANSWERSTO EXERCISESChapter 1. E.r:.2- The first three. Er. tl- ‘fies, when ‘p’ and ‘er‘ are b-nth talten as false-Chapter 1- Es, t. The indicated reutine gjves: —is-srl -tnerl —tr‘i'tti‘l —terltl —l.flt?tl- But this is equivalent; -it='ttrl—l.Ii*tll rldfl —'[.ill‘l-Chapter 3. Es. 2. —te val v —tP ‘till-Chapter -fl. Er. i, Jehn will play er sing and lvtary will sing. Jcihn will play er Jehn and lvlary will sing. Er. 3, secnndparr. —ln ‘-‘cl -rvs -—* -ta\"-‘slr- Er. .5, last part. ECpl=Z.-lltr;-'ri'ltpl§lasCi'lil\Il\lal*lrll{l‘lp Fthlplllslllp.Chapter 5. Er. I-l. ,l,.l. v T .—i TJ. T —1- 1. ,1, Er. 3, third part. eemes T——t~.T—t-ty .l.—1-..l.—\"*q lT—\"*ql T if T J. dlllfi

3t'.|t5 Partial ..~larivers ta iitercisesChapter ti. Er. l,secarrri_t:1arr. e era -ll-F Ht-‘ TH-q.v.Tt—irt l.-i—i-it;-.v..l.—i-it;.= tvt tri TT Er- E’, first part. pvt} .—iitj|' tlti r:_]' —i-.p‘Ii|'.-.y. Er. ti. ta} l‘~ln. [b] Yes, the negatien nf ‘p v rg’ is censistent. (cl Yes, the negatien nf ‘p vp* is incensistent.Chapter T. Es. i. p~H-.a-H-rI—+ir+i-.qr—ip TH-.r;,-t—ir;—i;ri—'i.rj'——iT .q'{—ll',F\",—'l'f' ty-t-ii-T,-r-T grit-ti-_]__—ri_]_ T t;- Se‘p -i—1~.q H-r‘ dees T J. netimply*r-i-i.q—+p‘. r~H.q—rpI—Hpt—r.qs—rr rt-1-.ty-—iT:—i:Tt-i.t;rr—ir r—i,q-r—ir- T—i-.a-t-1-T ll Sn nnt vice versa. T J, Ex. it. Twe fell sweeps shnw that neither ‘pr it q .-i r’ ner ‘p —1-. ry v r’ implies ‘p -—i- qr’. l pvq.-sir p—t-.r;-'vr -I 'v‘.l..-—i~r T—i-.J.vr _|'?J 1\"\"-er r r Twe fell sweeps shew that ‘p —r qr‘ and ‘p v a .—> rl each imply ‘p -'1-. g v r’. i II.‘ p'Itq.—t-r _|'t:t l'--e-. ]—\"'r _|'t: |_*-Er Tv.l..—i.I. l— lt-l Trt.tth-value analysis shews ‘p —i+ q -.—-1*: p v q ,--ii r’ nnt valid. Sn ‘p —i-al dees nnt imply ‘p vq .—ii r’. But it

Partial sbisivers ta iilrercilrer 3'37 weuld, by [ii], if ‘p —+, q it r‘ did, since ‘_n —1- qt‘ implies ‘F -1-. q v r‘; se ‘pi --i.q vr‘ dees nntimply ‘p v q .—i- r‘ either. Er. F’, partial answer. The schemata are ‘p -H qr‘ and ‘qr ——i- rp . ts] —1- tip‘,Chapter it E,t- .-Ti‘- The ‘then’ clause erttends threugh ‘Barbuda‘, since the ‘l‘1l‘ has tn gnverrr ‘fiy‘. Se the whele is a cenditienal: Either the Giants er the Bt'uins win and the Jacltals talte secend place --I l‘ll recever past lnsses and either buy a clavicherd er fly te Barbuda. The antecedent nf this cenditienal is a cenjunctien and net an alternatien, since ‘the Ciiants‘ and ‘the Hnrins‘ are ceerdinated by the shared verb ‘win’. The Giants er the Bruins win . the Iacltals talte secend place .—i l‘ll recever past lesses and either buy a clavicherd er fiy tn Barbuda. The censequent is liltewise a cenjunctien, since its ‘either‘ stands after the ‘and’ rather than befnre the ‘I‘ll‘. Final result: The Criants win v the Bruins win . the Jacltals talte secend place .—.=-. i‘ll recever past less-es . I'll buy a clavicherd v I'll fly tn Barbuda.Chapter 9 Es. l. ‘p —ri r . q --i- r‘ is equivalent tn ‘p v qr .—> r‘. and the ether three are equivalent tn ‘pq —i r‘. But the reader sheuld preduce the truth-value analyses.Chapter ll]- Er, l , first sclrerrra. p .qv—[rv —iiy vpl]. lift I -tjl ‘ti. F . {II ‘v‘ ltl, sir isr-i vars- This simplies tn 'pq v prq‘, and indeed re ‘pq.-‘. Te see the advantage nf werlting fmm the eutside inward, try deing this ettereise in the eppesite way—first changing the in- nermest pnrtinn ‘— la v pl‘ tn ‘pp’ and then wnrlting nut- ward.

3lJS Partial .rlr-rsivers re Exercises Er- .-', last trait‘. The last three ef ti\"re sis schemata end up thus, after ebvi- eus simplificatiens: par vpqf, par vper vpr]-‘F vpqr. par vpr var. But the reader sheuld fill in all steps. Er, s, last its-.y. Twe nf these last three have already gene inte dcvelnped ferm. as seen abeve- The last ene becemes: par vpal‘ vper vpar with nmissinn nf a repeated clause. Es. 3. Yes. The reader sheuld still justify it, Er. -ti, last part {cerrespending re Est. 4 eftfhapter .7]. The three sche mata ge inte altematinnal nnrmal fnrm thus: pvq, pavr, pvt;-vr. Develnped, they eventually becnme respectively; that is-tr ‘test ‘-‘ti-Ill‘ veer rear. par vpr}? vpqr vpar vpdr, per vpal‘ vptp vper vpqrr lrljlI'r_:_]IF vp-gr, Rearranged fnr matching, they appear thus; est rear it at-v ‘rest that \"Pct. est rest “Pct istr it Pct. pqr vpqr vprjr vper vprjr vpqs vper. Cnntainment then shews that the third schema is implied by the first and by the secnnd, and that there are ne nther implicatiens. Clearly the methnd is peer.Chapter I l Es. .i,_t'irstpart. We see by a fell sweep that ‘qr’ implies the rest nf the altematien and sn is redundant. [Ilrepping it, we have ‘par v _flqr v pr‘. By a funher fell sweep we see that this whele is implied by 'pq‘ and hence that the ‘F‘ is still redundant.

Partial Ansivers ta Ererclsss lllltiiChapter l I Ex. 5, last The last three schemata ebtained in Es. l ef Chapter ll]Chapter l3 yield ne cnnsensuses, since ne twe clauses are eppesed inCh apter I5 eitactly ene letter. Sn there are nn further prime implicants. Er. F’, last iarlf. These three schemata have ne sherter altematinnal nnrmal equivalents, since all prime implicants are there and nene is redundant. Er- l- The first is dual tn each nf the ether three, these three being equivalent. Fer, by the secnnd law ef duality, ‘p t—+ q‘ is dual te ‘—l,n re ill‘. Er. 3, clausal part- The clause is redundant if a fell swn-np shews it tn be implied by the rest nf the schema. Er, 4, first srirerna. s-sv—trv—ts\"-*all. _e:av.r..;-vp, p.avr.qvt;-vp. Simplifying, p . t'.jl' vr . qt vp p.qvr-r;-'. se- .E.il.'. l. Erilltll-'1-1—'=lP—*P-—‘-F‘ —>t-- Eir. 2. \"r\"es- There might be nn F {and hence nn C]. ti ~ iii tiChapter l'i\" Er. i, .recrinr:i inference- ‘iltirite ‘F ‘, Ti‘, and ‘H‘ fnr ‘ltnnws Creerge‘, ‘ltrrews lvlatiel‘, and ‘admires lvtabel‘.

3 lfi Partial .-*tn.rnver.r tri Exercises G FH Er. 3, secend part. Hint: Estend the bar device nf Diagram llChapter IS, Er- l, j‘irst example. — EIFGH. — HFHG, -- HFH. Es- i, last example. — 3 —lG viii. — HFG, -— EIFH. Es. E- It dees nnt- {Reader sheuld still eitplain, er, if baffled, read t'.'Il'l.]Chapter ill. Er. 1‘. ‘F‘ is ‘talies legic‘, 't'_'I‘ is ‘taltes Latin‘, and seen. 3FGFl‘,-7 . — B[I[C vii} -{F.-ll]. l\"-legateti: —tEtrar1t-‘. — Eltta v tri — {F.l}]}. Cnnverted tn cenjunctienal nermal ferm: - Ett-\"atl.l v atte v tr] - trii]. Cnnverted tn an er-tistential cenditienal: 3FGi:l.f——t- Hlftfi vii} — t_‘F.l‘}]‘. Fell sweep tn test whether ‘FIGHT’ implies ‘ta vtr] —t[F,l‘t‘: it-7 vi-\"II -tF-ll tlvttt -{TJ.} T Se the negatien was valid- Sn the eriginal was incnnsis- ttiltll.

Partial flamers re Exereise.r 3! 1 Ex. El‘, first pan‘- — HFGH . — HFHG .—1- — HF!-i‘. Cerijunetiena] net1T1a| ferm: HFGH tr HF.-‘it? tr - 3.-“H. E:-tistetitisl eenditienalr HF]? —1- 3=[FGF' '-.-'Ft'i'G}. Valid; fart, fell sweep shews that ‘FH* implies ‘FEE’ vFE'G‘.Chapter it}. Ear. J‘, seearrdfarmuta- it = -[FvtT]|-G tr —[Fw§]'IG .—+. it = FE? HFG, —H$wfiflv—EwGFL+-Hfiflvflfl see vFl'I¥l —1- Ems vi?)-G tr —-{F i-G151. ‘Jerify by truth-value analysis that ‘PG '-.-‘ Ft?‘ implies: {F v-S16 tr -tsveya.Chapter 11. Ex. I, first e..tam,a:'e. ‘t[r[,r is a man an the team -—1- — Jahn ean eutrun 1]. Ex. 2, first example. \"I'.r{x is eest at t|'ae]is —=-..r is slnvenly six is pear], — ‘ti'.t{.t is east ef tr:-ieks -1-.1: is peer}, Hit: is sleveniy . — I is peer]. Ex. _i', partial answer. The twe in the middle are net equivalent.Chapter 23 Ex. 2, partial answer. “tLr[_p -H-Fir)’ and ‘p 1+ ‘ix Er’ reselve respectively te \"tilt —FJr’ and ‘—\"Ii|\"x Ff when ‘J.’ is put fer ‘_a*, se they ere net equivalent- Ex. 3- The first impiies eeeh end eaeh implies the last. The reader sheuld shew this.Chapter I4. Ex. I. Expanding ‘H’ and relettering, we have: '||i!',1;F,1'-—s.\"li|FyG}* --1-\"'|I\"EHE -Ht‘ H1.'—1-‘tI'wGw.

312 Partial .r\"ll'iI..'|-‘III?-E'l\".!‘ Ia Erereises New we are free te cheese the errler ef espertatian ef quantifiers. fine at the final results is: 11:3} 31=\"I'z\"I'w{F.:r —+. Gy -—1~ Hz . Hr —1~ Gm]. Hate alse the fellewing refinement. We might prneeetl as Far as the interrnetiiate stage: §;[F1 -1-. \"I\";i[‘U\"3,r G}: —1- Hg} . \"li\"w[\"tI\"'v Hr —* Gwll and then reletter thus: 3.:t[FJ; —1-. \"fl\"::[\"i|\"}' {Ty --1 He} - ll\":-[lIi|\"t* Ha --1- Gal] with a view ta esplniting the tlistrihutiirity law [I'D] pf Chapter II, thus: H:[F:r —1- ‘tI':[\"\"i\"Jr G11 —1- Hz . \"IiI\"v Hv —i Gel]. The en-rl result then has enly feur quantifiers. Half: Hy Hv{.FJ: —i-. G} —1- Hr: . H1-P —1- Ge}. Ex. 3. 3.r'fl\"_1|.r{Fx vfiy vHg.: . -F.1r '-.-' ~—t._'i_v '-.-'Hy}. §.1'[li'_f-ri[F.r tr-G]; VHF] . \"4i|\"y[—Fir '-.-' — Gy ‘u'H'y]]. H.1'[F.:r tr Vjrtfi-‘y ‘|rHy] . —FJr tr \"lit\";-,r[— G]; vH;|.-J], H.t[F.1r . ‘til\"_'|r[--G_!.= ‘I|I'H}=] Jr. -Fnr - Vylffiy 'ttHy] .'-J. ‘lI'ytG_1r tr Hy] . \"i\"_ir{—t.¥3r 1rHy]]. H.=r[FJr . ‘tt'j,=[—G_j-,r ‘IJH;-,r] N. —F;r . \"i'}={G;|r ‘u'H;|r‘,I] tr. \"IiI\"}'l_'t_T}' 'trHj.=] . lIiI\"yt_'—{F}' tr Hy], 1Ir[F1 . \"I'y{—Gy '-.-' H_ir}]1uI' H.r[—FJr . 'U'Jr[Gy '-.-' H_p}] tr. ‘U']r[G}= ‘trH3r] . ‘tI\"J-'[—-G}: '-.-' Hy}, 3.1:FAr -\"I'}r{—Gy ‘It-'Hy} .v. 11' —F:: .'1I'_}={[?]|r vH;1:] .v. \"fir-|[t.?_j-.* tr Hy} . ‘IQ-.={—Gy trHy}-Chapter Z5. Ex. 4'. Hint: Represent the eenelusian as: H_1:=|[FJr . Girl —* 3:r{H:r . J1].Chapter 1+5. Ex. 2. {ezys = EJ.‘ + 3}.Chapter Z7. Ex. 4 . lIiI\"..1:[.1r was a finger ef hers -—+ Hyty was a ring .3: was an :1].

Farrier .-tln.r|r|:er's ta .Exerer'se.r 313 Ex. Ji, iasr example. Hint: As a first step put just the inside quantifieatien inte wards. retaining ‘x’. Ex. ti, last example. False- Ifx and e were as deserihed. then hp taiting 3,» as: we euuld ennelude that x was between e and a. Ex- F, first example. ‘-— Ex — Hytx = 3;)‘, er,er|uiiraientl1r. “tfx —‘t\"y -1}: = 3.=}*. er, equivalently, \"tfx 3;-rtx = gr)‘. True.Chapter IE Ex- I, left rainrrtn. The first twe may he suhstituted; the third nnt. The results nf the first twe substitutiens rnean that 1' praised himself tn semeene and that y praised semeene te his faee. Ear. 1?. partial answer. Uni)» the third ahstraet ean he substituted in \"'tl'x Fx —i- F;-.\" er in ‘Fy —1- Ex Fx'; enly the third and feurth in (4); nene in[l3ler[l-1]. Ex. 3, partial answer. Feur ef the eight are thus ahtainahle. The secend an the leit anti the three an the right are net, altheugh twn ef these epuld he inferred in an additienal step er twe.Chapter 29 Ex- I. rliirrl parr. Stages hetween {E1 and [E-1'}: \"tiI\"z{Fg -1 Ge} —-1 \"IiI\"}*[Hxl_'Fx - Hyx] —1- Hwtffiw . Hywl], \"i\"}'[‘li'a{Fe -+-Ga} -I-. 3x[Fx - H_1rx}-i- Hwtfi-w . H_pw]], lil'y|[‘i';;-{Fe —1- Gel —*- 'lI\"x[Fx . Hjrx -—1~ Hwlfiw . Hpwlll, \"il'}\"lI\"x['tI\".:{Fz —1- Gr} —-It Fx . Hpx .—-a Hwtl-Gw . H_].'w]], liI\"y\"lI\"x §z[F: -115: -—i-2 Fx . Hyx -1 Hw[t_?w . Hpwlj, \"IiI\"j,r\"|i\"'x Ha[F.=: ~—rt]: .—> Hw{Fx . Hp: .—-I-. Gw . Hywll, l'ill}'lli'r.1' Hz 31+-'I[Fa -+ Ci: -—-H Fx - H’;-ix .—i-. Gw . Hyw]. Ex. F’. \"tI\"x'tl'_1.\"liI\"E[Fxy - Fya --1-FIE} - llllx —F.tJ.' .—1 \"i|\"x'li\"}\"{Fx}' —1- —F]|rx], ‘tI'xh\"3.r‘tI'e[Fx_i* . F)‘: .-r FE} . ‘tiw -—Fww .—I* li\"a\"i\"u{Fau —1- --Fvul.

314 Farriai Answers re Exerciser \"tiI\"uli\"'|»* it Ely 3: 3wtF.::y . F}: .—1- Fxx I —Fww :—1-. Fun —-a- —.F1-raft. E-lxgy 3: 3HFI:F.!t'_‘f,l' . F3-re -—r FJII I —F'ww I—-In Fan —1- —F=.'a}, Fuu . F1-Pu .—r Fur; t —-Fan I-1. Fm: -s —Fp1¢_Chapter 3|]. Ex. i'. In ‘Fygf and '—Fay’ the instantial variable is net new. ‘— Ex Fxx‘ is net-prenex. but malte it se and the preef gnes threugh, with premises \"ll\"; -—Fr;r’ and ‘Ty FJr;-.=‘ and instances as befnre. Ex. .2, first pan‘. See beginning ef Chapter 33.Chapter 31- Ex. 3, anrirring same steps. \"i\"x‘lI'_}=[Fx . Gary .—i- Hetia;-.r . HI.‘-51]., \"liI\"x[Fx - Hy Gxj-9 .—+ Hy Hxjn]. PREHEI PREMISES FDR IHICUHEIF-TEHCTi VxltI'}* 3a[Fx . Gary .-i-. Jay . Hxe] Ex Hp Va —-[Fx . tltjr .-+ Hxz] IHSTAHCES1 5': ‘-iFa . Gun .—1- Hue] [deuble step) Half!-\"rt . Gun .—i-. Jen . Hue] [deuble step} Fir . Gav .—-in Jwv . Hint: -{Fa - Gan .—1 Haw] Ex. 4. Fer the results t:-f paraphrasing, see answers tn Chapter 35. Ex. 5. r-aesnses: ‘lfxtx laughs at x —1- I like x] 'ti'x['lil'yt';r is friend efx —i+x laughs at pl -1 I detest x] PLLTITUDEI 'il\"x[I detest x —1- — 1 like x} 'CC|'l\"i|CLU5]Cll'~iI Hx‘t|\"y[;r is friend ef x —-I-x laughs at gr] —\"1- 3:: —xis friend efx FREHEK PREHIEEE FUR. [H'CC|'i\"'i5T5TEl\"'i'Ell\"I llllxtfixzi: A LI] lil\"x 3}'[F}'x —1- Exp .—1- Dx]

Parn1aiAeswer.r re Exerrises 3I5 \"tl'x{Dx -I —Lr} E11 lily ‘Ilia —{F}'.1t —\'-[i'.1t'y .—1- —FZEl' IHETAHEEEI up ‘ti’: — {F-rw —r Gwy .—i- — Feel 3y{Fyw -Pr Gwy .—:- [Jwl Ft:-.1: -—t- Gwv .-+ [iw frt-s'1»s' —i~Lt-|-=‘ Dw —* -Lw —[Fw|-* —1- Gwv .—i- ——F'ww‘,t {deuble step} — [Fww -1 t]u=w .—1 —Fww} {deuble step}Chapter 32 Ex- 2- The ndd nnes are T and the even enes .L-Chapter -\"id. Ex- i'. There ean be nn eemplete dispre-ef precedure here, sinee dispreefs ei negatiens afferd prnnfs. There ean be ne deei- sien precedure missing nnly finitely many at the tmths, sinee that preeedure tagether with a list ef the etnined truths wnuld afferd a eemplete pre-eedure- Ex. E’. Because etherwise yeu enuld always preve a sehema tn he finitely valid by pre-ying that its negatien was infl.Chapter 35 Ex. i. Make substitutiens tn mateh these that were made in the first preef in this ehapter. exeept fnr using ‘r.,.,_._‘ in {.5}. Ex. E. litewnrlting Ex. 4 at‘ Chapter 3| in Etreben*s way: Pnesuses: 3xtfF.:t .\"ti\"y[Gy . 3.t[Fa - Hjtal -1-Hyxll \"tI'y[tZiy —1- 3-tiFa - H-1-'-Ill — 3-r'[Fx . ‘tt\"y{Gy —-i Hyxl] tfeenelusiea negated] autusrstenrsz 3-r{Fx .\"tt'y[—Gy '-.-' ‘ti’: ——[.Fa - Hya} trHyx]l ‘t\"yi*t'-Ty tr Hail’: - H.'i'tll \"lI\"xI—~ Fr ‘ti Hy — {Cy -1 Hyx]-] FUl~ttITTtJt'-l.-'|.l- naastxt Feasts: Fx . ‘iI'y[— Gy tr ‘tie — [Fe - Hya] ‘I-*Hyxl Vrtrfis ‘-*- F:-as - Hrin h\"x{—F'x it —~ (Cy, —-I Hy_._.x_l]

Elfi Feniei rln.rwer.t' re Exerei.-resChapter 36 ll\"'-|'5T.=\l\"-iCE5I Fr ' -6.?-1' if _i-Tier r H.l\".r3u.1-ii l\"rH}'x-T _ GT1 V‘ F3:-I-r ' Hliriis; “F-1’ \"I\" \" iGJ\"x_i'H}\"r1l Ex. 2- Preef ef {4} ef Chapter 29: Fxx —1'F.1}'|‘H\"-F.I-'_}’ —1- Fyy EZLFIZ -i FEE} 1|-r‘ 3.Ti:FJ:E -1* Fey] 3at'.Fxe —i-Fe_vlChapter Ii? Ex. 2'. Ftenf ef{ I] nf Chapter 2'? by last methnd el’ Chapter 3?: [1] Fy . ‘tl'x[Fx -1-Eixy} —1-. Fy . -Gyy i\"P\":r I -1:-l 5‘.viFr - \"II\"stF1 -1\"-'lr.vll —=~ iii-rlf.Fx . -CF-tar] ‘What valid menadie sehema cevers [l|’l Fy . liix-l[Fx -1-Hx] .—t-. Fy - Hy. Preef ef {4} at’ Chapter 1'3: [ll 5'2 FIE -r Fxy [l -1] 1t[Fx-E! -1-Frgyl The valid menadie sehema that envers this last line is as fellews. Test it far validity. ‘II’: Ge -—-I Hx .—i- Hattie —-—=- He}. Pre-ef ef {5} ef Chapter 29:Chapter 33 [I] \"liI\"xIfFx -1 Cix} —1-. 3x{Fx . Hyx] —1- Hxlfix . Hyx} [\"t\"'y I -1.] h\"'x[Fx --I fix} —-1- \"iI\"'yf3x|f.Fx . Hy-tr} —1- Hxlfix . H_‘|t.t}] Preef ef [I I} ef Chapter 29: [ll \"t\"::{Fxz --i-Fxe], [1 -rl 3Jt\"Ii\"a{Fn —*FJ-‘al- Ex. i. The first and third en the left are eerreet hut unfinished. The seeend en the left is ineerreet [vialating the alphabeti-

Partial Answers ta Exerciser 3 I \"l eal rule] and unfinished- The ene at the upper right is enrreet and finished. The last ene is finished but inenrreet, ‘y’ he-ing flagged twiee. Ex. 2 in part. Revise the last ene by bringing in a new line ‘Fa . Ca‘. Ex. 3. Rewerlting Es. 5 el Chapter ill {whereaf see answer abeve}: rt ll \"li|\"x|[G-r_r —-is LI} 1et=[1] liI\"x[‘liI'yi[Fyx —* Cry] —t- D-rl =t==|=1=[]] \"tftt{[-ix —* —L.I} 4==|et=a|:4} Hr\"'I\"y{Fyx —1- Ely] is-l iv -t=x=-#115} \"tI\"yt'F_-pie —s- Gtvy] its-|-=1-t_'fi] Crww -—1* Lw [ll] ****t\"l'} eeeelfij \"t|I'yi_'Fyw —s C-wy] —r Dw {III [lie —a- —L1.-|.l' [3] ***'I\"{Eil sweet ll-II] Fww -t- Gan-v [5] —F1-vw i[5]tI[-li-]Il'T]t'_E]t[5ll ven-tl |_l Elx —F.tn: {tut Ivl-I=I[l2l Hx\"lI\"y[Fyx -I Gxyft —1- Ex —Fxr sill]Chapter 41 Ex. I by methed afpare existentials. —i[Fy .Fz] . Fw-—>h\"xFx .—1- -Fw, 3x[— [Fy . Fe} - Fw —1-Fx .-!- —Fw], —{Fy . Fa) . Fw -1-F_v ---r —Fw :v: —i_'Fy . Fe} . F-.1: -1-F; .—v -—Fyy. Ex. 2- \"lil\"x[Fx_y —1- Gyx} . lfxtfixa —+ —Fa1r] .—1~ —Fzy, 11rHwtFxy —rGyx . C-‘we -1 —F:rw .A —F.:y]|, Fay -1-Cy: . Cy: —P r-Fey .—-I —Fay. This ean be shertened by use ef i til] ef Chapter 21.Chapter 42 Ex. l- 'Criven that \"tI\"x{Fx —1-: -r = y .v. x = .1}, 5x{Fx . Gx], and given ‘ltI'xll'w{tf.?x . x = w .—=-r Gw]-' as axiem ef identity, te shew that Cy v Gr. lvlain methnd:

315 Partial elnrwerr ta Exerciser Fll'.Eh'l15E51 \"I\"x{.Fx -1-: x = y .v. x = all tnsrxnees: §x{Fx . Gx} h\"xli|\"w[Gx . x = Hr .-i- Cw] —{Gy vile] Fa . t-in Ftt——=rt rt =y.'tl-t-t =-I Ga - it = y -—+ Cy Cu . I-t = at .-1-G.-; The last five lines are truth-funetienally ineensistent. The reader might try repeating the exereise by the methed ef pure existentials- Ex- .2. The ferward eenditinnal is a variant ef fr}. by a rule at passage. The reverse implicatien, when buttressed with the ease ‘y = y‘ efill], helds by E13.Chapter 43 Ex. l. {-.-x'j|[x is a man in tewn . \"ll'y['y is a man in tnwn . x st‘-‘y .—t-x is taller than y}]- Ex. .2 by methed‘ efpare existential-t. h\"xl_'xwrntelillr—i-.x =y}.y =y.ywretel .—1- Hxtr wmte W . x wrete I], 3x3t':i_xwrete\"t\"tis—1-.x =y:y =y.ywretel :-1. e wrete ‘ill . 5 wrete I}, ywrnteW'-t-i-.y =yty=y -ywretel :—s. y wrete ‘W . y wrete l.Chapter 44 H{rx}{Fx . fix} - .l'i[tx]i[Fx . Cixl, Hyllzly . h\"x{Fx . -Ex .+-1. x = y]] . 3y[.ly .\"tI'xI[Fx - [ix .+t—t-. x = yftl.Chapter 4'? Ex. l. last half. — -[F“t?} eemprises everything. human and etherwise. ex- eept the men whe were alive in 195i]. - [F“{}'] eensists ef the men whe were barn befere l95{i and survived that year. Ex. .2. feurth relatien. The relatien efx ta y wherex is admired hy semeene [male er female] whn has y as a brnther-

Partial Answers re Exerr-ises 319'Ex. 3, .teeeari' hall.Talte F as the henefaetien rel alien, H as the elass ef Anne-nians, and it\" as the elass ef erphans- Then F ‘*|fHlt.'It eem-prises just the benefaeters ef .|'i'l|.I'l'l'lIl2t‘l=iEll1 erphans, whereas{F“H]li'°\"\"li;'] ineludes alse anynne else whe benefits hethan Armenian and an nrphan.

BIBLIOGRAPHY This list includes enly such legical and nearly legical werlrs as happentn have been alluded ta, by title er etherwise, in the enurse ef the bnetlt.Fer a cemprehensive register nf the literat1.tre af mathematieai lngie tn theend nf 1935 see Churehis Bihiiagraphy. That werlt, whieh is helpfullyanaetated and thereughly indexed by subjects, is invaluable tn legicians-Subsequent literature is cevered by the Reviews seetien et‘ tl1e .lea'rrtal efSytnhelie Legic. which is indexed by subjects every five years and byanthers every twe.li't.Ill'.El\"'-I, Heward H., and ethers. Eymhesls ef Elertrenlc Crinurntlag and Carttral Clreaits. Cambridge, lvlass.: Harvard University Press. 1951.Eleniaatvtv, Heinrich. lllierinvnaril: and Lnglir. Leipzig, 192?.———— “Beittige eur iltlgehra der Legilr, insbesendere aura Ent- scheidungsprnhletrt.\" Hatlternatisrlte Anitalen. val. Eli [I922], pp- ll53-E29.Benrvavs, Paul. “Uher eine natiirliche Erweiterung des Relatie-nenltal- lriils,\" in a. Heyting, etliter, Canstrttetlvlty in lidathemaries [Amster- darn: Herth-Hnlland, l959]. pp. l-l4.—-——— and ll-\"leses Schizinfinltel. “Eum Entscheidungspreblem der mathematisehen Le-gilt. ” Mathematlsehe Annalee, val. 99 {I913}, pp- 342-371-Be-EILE, Genrge. The lslathematieal elnalyrlr ef Legic. Lentlnn: Eieerge Hell, and Cambridge, Eng.: tvtacmillan, 154?. Reprinted in Calleetea’ Lngical Waritr. Ctxferd: Blackwell, and Hew ‘lerlt: Fhilnsephieal Li- hnary, I943-—-—-- xlrt lnvertigatian ef the Laws ef Tltetrgltt- Lende-n: Walten and lvlaberiy, I354. Reprinted Hew ‘ferlt: De-ver. l95l.CAHTDR, Genrg. “Cher eine elementare Frege der l‘vIaru1igfrilt:iglteits~ lehre.“ Jehresireticltte der deutschen lvlathematilter-‘llereinigungen, val. l llfllltll-‘ill, pp. T\"5—'l'E. Reprinted in Gexammelte ell:-ltandi lungen. Berlin: Springer, 1932.Caatwsr, Rudelf. The iiegieal Syntax ef Language. Lnndnn and New Yarlt: Harceurt Brace, 1'93?-CARRDLL, Lewis. Symbalir Legic. Lnndnn: Macmillan, 139?.310

Hibiidgrapfiy satCav1-Ev, Arthur. “fin the theery ef g;|'ettps as depending an flte symbnli- ca] equatien 19\" = I.\" PhiIa.tapltiral Magaaitte. vel. T H354], pp. 4l]—4'l, 4-I'.]3—4i39. Reprinted in Calleeterl ll1IatltenaarieaiPapers. Cam- bridge, Eng.: Cambridge University Press, I395-CHLHte|-t, Alenae. .4 liibiiagraphy ef Symheiir Legit‘. Prnvidence, R.I.: Asseeiatinn fnr Symhelie Legie, I933- Reprinted frem Jenmal sf Syrnhaiir.‘ Legic lI93ti, I933].-———- “A nete en the Entscheidungspr-:1:-blem-\" leurnei ef Syrniielie Lrigie, vet- I £19315], pp. 41344]. Cerrectien en pp- ltll—tt}2-Caatev, iehn C. .4 Primer ef Farmei Legic. I\"-lew \"t'erlr.: lvlacmillan, I942.Ctttntv, H. 13., and Re-bert Feys- Cami:-inetery Legic. Amsterdam: l\"-la-rth-Hnlland, I953.Dxvts, hlartin. Cempataliiliry arai l.ieselvairiiiry- Hew \"t'e1-It: I'v!lct3raw- Hill, I953-—-———-, ed. The llrtti‘er:idalJle- Hewlett, I'~i.\"t’.: Raven Press, 1955-Denesnve, Richard. Stengireit and irratianale Eaitlert. Braunschweig: liieweg, 1322. Later editiens I392, 19125, I912.—--~—- Was sinei aria‘ was salien rile Ealtien? Braunschweig: ‘Iiieweg, I33?-l3'El'v‘ldltear~t, Augustus. Farntal Legic. Landen: Taylnr, I342.—-—-- “Cin the syllegism.\" Trartsartiens ef the Cambridge Philesephi- cal Seeiety, val. 3 III349], pp. 329-433; val. 9 [I356], pp. T9-I22‘; val. ll-l [I3-54], pp. I’i3-233, 331-353, 423-43?. These had been read at earlier meetings: I343, lsfiti, tsse, I363.Etassen, Eurten. “Cln the cempleteness ef quantificatiun theery.\" Pra- eeetiings cif the Hatienal Aeademy ef Sciences, val. 33 [I952], pp. l34'l-52-Ft-I|t.u., Herbert, and ‘vllilfrid Sellars, eds. Readings in Pitilesaphieal .-=laaly.rt's. New ‘ferlt: Appleten-Century-Crafts, I949.FltEt3E, Cettleb. Begnjiltsr-hri_fl'- Halle: Hebert, I379. Translated in van Heijenenrt.—-— [lie Gran.-zllagen tier rlrithrrtetiiz. Hreslau: It-laebner, 1334. Re- printed with English translatinn, Clttfnrdt Blaeirwell, and New Yerlt: Phi lesephical Library, I951]--—-— t]rtrna'ge-terse trier rlririnrteriiiz. ‘llel. 1, I393; vel. 2, I933. .Iena: Pnhle.-—~— “lilher Sinn und Eedeutung.“ Eeitseitriii _,iiir Piiilasepitie anti pitilasepitlseite ifririk, val. I33 [I392], pp. 25-53. Translated in Feigl and 3ellars.

322 SihliegrapiiyFrttnsHitt., R. Abstract in Stimlrtaries af Talks at rite Summer institute ef Synthalie Legic. lvlirneegraphed. It-haea, l\"~i.'t\"-1 Cernell University, I952, pp. 2ll—2l2.CEHTIEH. E:iEl‘I'IEIIt3- “Untersuchungen iiber das legisehe Schliessen-\" ll-latltematisrlre .E.'eitsrl:rr]\"i‘, vel. 39 [I934—35il1 PP. l\"ll5-2lI], 4-I35- 4-3i.l3E.atser-IHE. l. D. “Essai de dialectique ratienelle.“ .4mta.les tie rrtanhtimaiiattes pares er appliauées. vel. T [I3lIi-—I'l], pp. i39—223.Cnxexts, l'vI. I. \"In-edundant disjunctive and cenjunctive ferms ef a Beeiean t'un.-ctien.\" t.a..tr. Jearuei ef iieseareit aml ilievelepm-em. vel. I [I952], pp. ITI—I?t5.l3|ii[JEL, Kurt. The Censisteaey ef lite Caarinrtttm Hypetitesis. Prineelen, l’~l.J.: Prineeten University Press, 19413.4 “Die llellstiindiglteit der Axieme des legischen Funlrtienenlra|- lriils.“ Manatshefie fiir irlatltematilr mid‘ Plty.rii:, vel- 3'l [I933], pp. 349-3193. Translated in van Heijeneen.----—- “Ilber fertnal unentscheidbare Slit-“fie der Principia Ivlathematics und verwandter Systeme.\" ilfanarshefle flir ihlatlterrtarilr tmri Pltyshi. vel. 33 [I93 I], pp. l'l3—l93- Translated in van Heijeneert.—-Z-— “Eum Entseheidungspreblem des legisehen Funlttienenltallriils. ‘ ‘ il»i'ertatsi:e_,iie_i'iir ll-rlailiematii: ttmal Pltysiir, vel. 413 [ I 93 3]. P13. 43 3-443.—-——— \"(In undecidable prepesitiens ef ferrnal mathematical systems.\" Ivlimeegraphed. Princeten, l\"~l..l-, I934. Printed in Davis, The Unde- eiriaitle.Cieenstart, Nelsen. “The preblem ef ceunterfaetual cenditienals.\" Jearnai af Pitilaseplty, vel- 44 [I942], pp. II3—I23- Reprinted in Fact, Fietien, and Fareeast, Indianapolis, Ind.: Hebbs-Ivlerrill, I955.Hxtt-tes, Paul. “Algebraic legic [II].“ Fttmtiameata ll-iaritemarieae, vel. 43 [19515]-. pp. 255-325-Heaaaase, .lact|,ues- Recitercites stir la tltiiarie tie la démertstrarien. ‘Warsaw: Seciéte des Sciences et des Lettres, I939. Translated in Her- brand, Legieal Hlrirings, Cambridge, I'vIass.: Harvard University Press, l9TI.Httaettr, David, and ‘ill. Acltermann. Gmndstige tier titearetlseiten Legile Berlin: Springer, I923. 2d ed., I933. 3d ed., I949.i and P- Etemays. Cirandlagen sler Mathemarik. ‘v\"el. I. I934; vei. 2, I939. Berlin: Springer.Jxsxewstn, Stanislaw- \"Dn the mles ef suppesitiens in ferrnal legie-” Erttelia Lagiea, ne- I [‘l‘larsaw, I934].ilevens, W- Stanley. Pare Legic. Lenden: Stanferd, I354.

.Eiii-riiegraplty 32.3Rt-EErv£, S. C. lnrrealueri.-an re ilrleramarlrematiet. Amsterdam and Hew \"t’erlt: ‘v'an hlestrand Reinheld. I952.i “Recursive predicates and quantifiers.“ Traesarrtiarr-s ef the Ameriean It-ilathematical Seciety. vel. 53 [I943], pp, 41-“l3.l'{t.‘|t.i.1'e1.vstt|, Casimir. “Stir la netien de l’er'dre daus Ia theerie des ensembles.\"Funtlamerrrallrlethemarieae, vel. 2 [I921], pp. I-fil-IT].Lrzsrvtewsttt, Stanislaw- “Crrundriige eines neuen Systems der Ii.ir‘t1nd- Iagen der l'v'lathematilt.“ Fnntiamenla llllathematieae, vel- I4 [I929], pp. I-3|.Lewis. C. I- .4 .3urv\"ey agf .'-iymhelir .l.agir:- Eerlteley: University ef Califernia Press, I913.Ldvvt=.rvHEl1vI, Leepeld- “Uber lvliigliehlteiten im Relativlr.aIiti.'1I.\" lllariremariseire rlnrtalen. vel. \"lb iI9l5], pp- 442-4'lII. Translatien in van Heijeneert-Li_|tt.=.srav.-res. Jan. “Cl legice trejw'artesciewej\" [Cln three-valued legic]. itueir Filrrrafirrny, vel. 5 [I929], pp. Id9-l?I-4 “Uwagi e altsyemacie lilie-ed‘a i n dednltcyi uegdlniajacej” [Re- marlts en I\"~lieed's axiem and en generalizing by deduetien]. ilsiega Pamiatl:.-nva Pal-rltiegrr T-a1var.:yst‘wa Filezafirenega we Lwawie, I.\"-.vt-iiw, |93I.i “Eur Ccschichte der Aussagenlegilt.\" Frireantnis, vel. 5 [I935-3'5], pp. I I I-I3].l'vlanrl.‘s, R- rvt. “A hemegeneeus system fer fermal legic.\" .lattrrtai tzf Symlralir I-agirr, vel. 3 [I943], pp. I-23.I\"-Ir-t1s1.i.:v|v, J. ven- “Eur Einftihrung der transfiniten Eahlen.“ .-it-ta Lir- terarnm er: .5'r.\"lenriarnm Regine l-lniversitaiis Htrngarieae Francisca- .lesepirinae [sect. scient. math.], vel. I [Saeged. I923], pp. I99-293. Translatinn in van Heijeneert.-—— “Eine Axiematisierung tier Ivlengenlehre. ’ ’ Jrrarnalfrir reine and angewamzire lvlatlrematiiz, vel. I54 [I925], pp- 2l9-2413; vel- I55 [I923], p. I23. Translatien in van Heijeneert-hlreeu, lean. \"A reductien in the number ef primitive prepesitiens ef legic-\" Preeeeriings ef the Cambritlge Philnsephieal Seciety, vel. 19 rrstr-rtn. pp. as-er-Peane, Giuseppe. elrirlrmeriees Prineipia. Turin, I339.—--—-Farmnlaire sle i'rlariternatiaues'. lntreductien, I394; vel. I, I395; vel. 2, I39'l-99- Turin- ‘llel. 3, I931, Paris. ‘Riel- 4. I932-3; vel. 5 [_s.v. Fnrmularia lrfatematiea], I935-3. Turin: Becca.Perltt-TE, C. S. Cellerterl Papers. vels. 2-4- Cambridge, lvlass-: Harvard University Press. I932-33.

liiirliagrapiry 323HLEEHE, S. C. lnrrerittctian ta ile'etamatlrenraiics- Amsterdam and New Yerlc llan l\"-Iestrand Reinheld, I952--—-— “Recursive predicates and quantifiers.“ Tran-ractians ef flre American Ivlathematieal Seciety, vel- 53 [I943], pp. 4]-‘l3.ltELntr.Tewsrtt, Casimir. “Sur la netien dc I’erdre daus la theerie des ensembles.“ Fundamenta Mathenmticae, vel. 2[I921], pp. It'll-I'll.Lesntrzwsttt, Stanislaw- “Cirund-tiige eines neuen Systems tler t'.irand- Iagen der lv'lathemariIr-“ Fttnclamertta Harhenmtieae, vel. I4 [I929], pp. I-3|.Lrswts, C. I. .4 Survey ef Symbelic Legic. Herlreley: University nf Califernia Press, I913.Lljwet~tHr.1rvr, Leepeld- “I3her lvliigliehlreiten im Relativltallciil.“ lhlarnematisclre rlnnalert. vel. Tb [I915], pp- 44?-423. Translatien in van Heijeneert.Les.-.s1swrca, Jan. “Ct legice trdjwarter-iciewe_i“ [Dn three-valued legic]. llnclr Filarajicsny, vel. 5 [I923], pp. I69-I'll.4 “Uwagi e aitsyemacie l'~lieed'a i e dedultcyi uegdlniajacej“ [Re- marlrs en I'~licnd‘s axiem and en generalizing by deduetien]. ilsiega Parrtiatirewa Pelslriega Tewaraysnva Fileacjicsnege we i'.wewie, Lwd-w, I931.-i— “Eur Gesehichte der Aussagcnlegilt.\" Erlzenrrtnis, vel. 5 [I935-315], pp. I I I-I3].Ivlxrtrnv, It. M. “A hemegeneeus system fer ferrnal legic.“ .larrrr|al ef Symbalic Legic, vel. 3 [I943], pp- I-23.l\"~lEtrr.1r.n|r~t, J. ven. “Eur Einfiihrtrng der transfiniten Eahlen-“ .4eta Lir- terarnm ac Scienriarttm ilegiae llrtiversitaris Htmgaricae Francisca- lrrsepninae [sect. seient- math.], vel. I [Sr-eged, I923], pp. 199-2{l3. Translatien in van Heijeneert.-—-— “Eine Axiematisierung der hllengenlehre.“ J’-aarrtalfiir reins und angewamzlte Matnematik, vel. I54 [I925], pp. 219-2411}; vel. I55 [I923], p. I23. Translatien in van Heij-eneerLIt-lrceu, Jean. “A reductien in the number ef primitive prepesitiens ef legic.\" Praceerlings ef flre Cambridge Philesephical Seciety, vel. I9 [1912-2'3], pp. 32-44-Pexne, Giuseppe. rlritlrmerices Principle. Turin, I339--———Farrnttlaire rle Marhenrarianes. Intreductien, I394; vel. I, I395; vel. 2, I392-99- Turin. ‘lei. 3, 19131, Paris. ‘rial- 4, 19132-3; vel. 5 [s-v. Farmularia ll-iatemarire], 19135-3. Turin: Becca.PEIRCE, C. S. Callectetl Papers, vels. 2-4- Cambridge, I'vIass.: Harvard University Press, I932-33.

324 Biirliegrapiry Pest. E. L. “lntre-duetitrn te a general flreery ef elementary prepesi- tiens.“ American .i'et.trrtal cf lldathentatics, vel. 43 [I921], pp- I33- 135- Reprinted in van Heijeneert. PRESEURCER, IvI- “Uber die Itlellsfandiglteit eines gewissen Systems der Arithmetih.“ Sprmttasdanie I I Iilengresu lvlaternmyltfiw Rrajdw Slewansltych Hilarsaw, I933], pp. 92- l3l, 395.Qutns, ‘Iii. ll. ilrlatirentatical Legic. Hew ‘lerlt, I943. 2d printing, Cam- bridge, lvlass.: Harvard University Press, I942. Rev. ed., I951--—-—Elementary Legic. Besten, I941. Rev. ed., Cambridge, Iv'lass.: Harvard University Press, I933.itii Sentide ria nave lagica. She Paula: Ivlartins, I944.-—-— Fram a Lngical Paint‘ a_,i' I-\"iew. Cambridge, lvlas-s-: Harvard Uni- versity Press, I953--i Set Tlteery and lts Legic. Cambridge, lvIass.: Harvard University Press, I933, I939.——-—-Selected Legic Papers. New 't\"erit: Rattdem Heuse, I933.-—-— The I-I-\"ays cf FIIF-lIl£lt2.I and‘ tlitlter Essays, enlarged editien- Cam- bridge, lvIa.ss.: Harvard University Press, I923.L “llariahles explained away.“ Praceerlings ef the Ameriean Philesephical Seciety, vel. III4 [I933], pp. 343-342. Reprinted in Selected Legic Papers.Reeetts, Hartley, Jr. Tl.-eery r.y\"ilecersive Fttttctians and Efective Cem- pt-tta.lrility. New ‘ferlt: lvlclf-iraw-Hill, I932-RtissELL, Bertrand. The Principles cf Mathematics. Cambridge, Eng.: Cambridge University Press, I933. 2d ed., blew ‘ierlr. I933.i “Cln tleneting-“ ll-find. vel. I4 [I935]. pp. 479-493. Reprinted in Feigl and S-ellars.-——- “lvlathematical legic as based en the theery ef types.“ altnerican Jettrnal afllfathemutics, vel. 33 [I933], pp. 222-232.i “Rnewledge by acquaintance and ltnewledge by descriptien.“ Preceedings ef the Aristetelian Seciety, vel. I I [l9I I], pp. I33-I23. Repri nted in Tire Preirletns af Pltilasapiry. Lenden: Williams and hier- gate, and New 't'c-rlt: Hell, I912, and in Mysticism anti Legic, New ‘ferlt: Lc-ngmans, 1913.Srtatsetv, E. I-li., and B. E. Ivlills. “Circuit minimiaatien: algebra and algerithms fer new Bee-lean eaneaical expressiens.” .-IRCRC Techni- cal liepert 21, I954.Scttiftnatnttet, Ivleses. “Ueher die Bausteinc der mathematisehen Legilt.“ iriatlrenretiscire .-innalen, vel. 92 [I924], pp. 335-313. Translatien in van Heijcneert.

Bilrliegrapiry 325Sat-I rr.l'inE1t, Ernst. Der tjiperarianslrrels ties Lagilrilialiriilt. Leipzig: Teuhner, I377 [37 pp-].i Farlesnngen iiber clie .-Ilgelrra cler l.agi.i:-I \"v'eI- I, I393; vel. 2, I391-I935; vel. 3, 1393- Leipzig: Teubner-Sr-t.t.t~tr~tt:-rt, Claude E. “A symbelic analysis ef relay and switching cir- euits.“ Transactien-r ef the American Institute ef Electrical Engineers. vel- 57 [I933], pp. 713-723.SneFFes, H- Iv1- “A set ef five independent pesrnlares fer Beelean algebras-“ Transactian-s ef the American lvlathematical Seciety, vel. |-=t [I913], pp. 4-S1-433.Sneentnate, Jeseph R. lhlatirenratical Legic- Reading, lvlass-I Addisen-Wesley, I93?-Suetxst, Thnralf. “Legiseh-lternhinaterisehe I-Intetsuchungea iibcr die Erliillbarlteit n-der Beweisharlteit mathernatiseher Siitze.“ Sl;rt]'i’er ut- gin av Der Iverslre Ilidensltaps-Altademi i Clsle, I. Ivlat.-naturv. R1. I923, ne. 4. Translatien in van I-Ieijeneert-i- “Uber die mathematische Legilt. \" Alarsl: it-rlatematisir Tirlsslrrifi, vel- I3 [I923], pp- 125-142. Translatien in van Heijeneert.i— “Uber einige Satzfunlttienen in der Arithmetilt. “ S.l.\"r[li‘er utgitt av Der hlerslte Iiidenshaps-Akademi i Dsle, I. Ivlat.-naturv. RI. 1933, ne- ‘i-Sstuttvan, Raymend Ivl. Titeary ef Farnral Systems. Princeten, 1*-i.l.: Princeten University Press, I931.Ttittssr, Alfred. inrrerluctian re Legic. New Terlt; Clxienll University Press, 1941.i .4 Decisien il=i'eti‘rea'fer Elementary rllgehra and Geametry. Santa Ivlenicaz Rand, I943. Rev. ed., Berkeley: University ef Califernia Press, 1951.4 Legic. Semantics. i‘:-ietanrariternatics. CI-xferd: Clarenden Press. I953.-——- “Remarques sur les netiens fendamentales de Ia méthedelegie des mathématiques.“ Race-nil; Pelsltiege Tewarzystwa Ivtatemstycn nege, vel. 7 [I929], pp. 273-272-----, A. lvlestewslrzi, and R. lvl. Rebinsen. Undecirlable Tltearies. Amsterdam: lrierth-Helland, I953--i and F. S. Thempsen. Abstracts in Bulletin et‘ Am-erican Math- ematical Seciety, vel. 53 [I952], pp. 35f.Ttt.ttTt—|TE|~ta|r.err, B- A- “lmpessibility ef an algerithm fer the decisien preblem in finite elasses“ [Russian]- iJelrlarly Alrademii l'~|anlt SSSR, vel. 73 [I953]. pp- 539-572. Reviewed in Jaurnal e_i'.'5'yntl.ralic Legic, vel. I5 [I953]. p. 229.

323 EibllagraphyTURIHC. Alan l'vl- “Cln cetnputahle numbers, with an applicatien te the Entseheidungspreblem-“ Jaurnal ef the Lenden lvlatlrernatical Seci- ety, vel. 42 [I933-37], pp. 233-235. Cerrectinn in vel- 43, pp- 544-543.\"v\".tr~t Hsustveenr, lean, ed. Frem Frege re t3t1itiei.' A Seurce Baal: in ihietitentetieei legic- Cambridge, IvIass.: Harvard Ilniversity Press, I937.vertn, lehn. Symbelic l.aglr.'. Lenden: Ivlacrtiillan, 1331. 2d ed., 1394.-i “I3-n the diagrammatic and mechanical representatiens nf prepesitiens and reasening.“ The Lanclan Edinburgh, anti Dublin Plrilasapirlcal rllagazitre and Jaurnal cf Science, vel. I3 [I333], pp. I-I3.ven hleernsnn. see l'\"~ieu1naru1.‘iintrenenn, A- ll-, and H. Russell. Principle Mathematics, vel. I, 1913; vel- 2, l9I2; vel. 3, I913. Cambridge, Eng.: Cambridge Uni- versity Press, 2d ed.. 192-5-27-‘I-litt-;r~tE1t. Herbert. “A simplificatinn ef the Iegie ef relatiens.“ Praceerl- ings ef the Cambridge Philesephical Seciety, vel. I7 [1912-I4], pp- 337-393. Reprinted in van Heijeneert.‘lIl1TreEHsTEthl, Ludwig. Tractatus Lagica-Pitileseplticus. Lenden: Regan Paul, I922. Reprint el’ “l..egisch-philesepltische Abhandlnag“ [rtnrralen der Alarurpltilesapitie, I921] with English translatien in parallel-Flsrtt-tete, Ernst. “Untersuehungen ilbet die Crrundlagen der IvIengen- lehre-“ ltlatltentatisclte .4nnelen, vel. 35 [I933], pp. 231-231- Translatien in van Heijeneert.

INDEIIA, see Categerical E-ehrnann,H-, I27, I32, I54 Bemays, P., I31, III9, 213, 233; en L-3w-\"Ii\", I23-133. 137. 141Alasnlule term, I37, I37 enhe.itn‘s theerem, 213-212, 213Abstract nbject, 5, 94, 259-231. 239. See Eicunditienal , 25-23, 35, 55, 3I , 33-39; alse Class and equivalence. 33, 37; pure, 33; ef terms. 123, I311Abstractietr, 134- I 33, I31; pelyadic, 175-173, 234-. 297-293'. as descrip- Beale, IS-, 23, 73,132 tien, 333 Be-elcan algebra, II4, I33-I32 H-nelean schema, ll4—1I3,137,I53Aclrertnann, ‘Iii-'., 123, 131 Beund variable, I33, I34, I45. I43, 131 Hreuwcr, 1.- E. -1., 233Ailten, H. H., 73Algebra: el sublractien, 37-1l3;e:I' Cantnt, Cr., I31, 213. 212. 299 Capture, 131-132 classes, 114, 133-132; nfrelatierts. Carnap, R., 147, 2I7n I73; cf rest numbers, 213, 295 Carrell, L., I13Alphabetical erder, I27 Categerical, 95, I33, I95Alphabetical restrictien, 243-241 Causal cennectien. 23. 24, 55Allcrnati-un, il, 13, I5, I3. 211, 72; in Cayiey, A-, 299 English, I2, 25, 23-29; laws ef, 13, Cd, 242- 244, 243 13. 34,73. 31-32; el terms, I14. Cellular, I25-I27 129-I33 Church, A.. 214-213, 319Alternatienal nennal ferm, 72-73. 73 Circl-ed numeral. I31Ambiguity, I, 4, 27. 55-57 Circuit, electric, 77-73Ancester, 292-293 Circular reasening, 233And, I3, 23-29, 53-54 Class, I32, 231; es theme af rnatltetnatics,Antecedent. 2|, 54Antinemy. 239, 331, 333 5, 333; as extensinn, 94; empty, 94, 133;defer1'ed, 94, 131. I35; algebra ef.Any vs. every, 97, 144 I 14, I33-I32; as value ef variable, I31, 291; virtual, I33, 291-292; iden-AmmmJM tity ef, 239, 293-291; reified, 239- 293. 331-333; utility ef. 292-295,Applicatien el legic, 4-5, 45, 53. 195. 333- See alse Ahstractien Clausal generalizatien, 227-231 333; te circuits, 77-73; tn axiems, 35. Clause, 72, 33:relative,I33—l3-5,175, 37. 33. 271-273 293Aristntle. 1193 Ctnsed, 134, 147, I53Arithmetic, see NumberAsseciativity, 13. I3, 37Attribute, 94. 2t=.2, assAut, I2Axiem, 35-S9, 223, 234, 235; ef iden- tity. 271 --2'13; at set theery, 332 327

323 lttclexClesure, I54 214, 217; fer Beelean schemata, I22-Ceextcnsivencss, I23-I29, 233, 239- 127; fer I-I-eelean algebra, 133; fer menadic scI'ternata,154-159,137, 235; 291 fnr pure existentials. 134- 1 315. 214; ferCelltsien, I53, I31 pure univcrsals, 213--214; impnssibilityCelt-tn, 53n ef, 214-213; fer finite validity. 215; ferCembinater. 233-233 real numbers, 213Cemmutativity. III, I3. 55, 33- Set‘ alse Dedeltind, R., 5, 294, 333 Dcductien. 233-239, 245-243; natural, Symmetry 244; theerem ef, 24-4; seundness ef,Cernpactness, 233Cempleteness. 214; in truth. functiens, 343- 247'; strategies ef, 243-254 Degree, 234 I3-23, 33, 37; in subtractien, 33; in i3e!'vIergan, A.; 73, I23, 132,173, 2991 quantificatiun theery, 193, 237-233, 215-2l-3: in number thcery, 213-213 laws ef. l4n, 33, 73, 31-32, 34, I43 in identity theery, 273 [1-enetatien. 9'4. See alse Extensien:Cemputer. 73Cetrcrctien. I34 blaming; ReferenceCenditienal, 21-23, 34-35, 54-55: D-esc|'iptien, 274-233; eliminatien ef, generalized. 22. 95, ti-9; cenltafactu a1. 23; and implicatien, 43-47, 49-5|; 231-232 un-der negatien, 33-39: cf terrns, I I4 Designatien. 94 See alse I\"-lam-ing I29- I33: existential, I21, I25; El. Elcveleprnent. 73, 33 192; cenditienal preef. 242 Dilemma, 227, 254Cenditienalizatien, 242-244, 243 Disjunclien. I5, 74Cenfinement ef quantificatiun. 142- I43 Distributivity, 73, 32; el H. I19; ef Iii. 147. 151-154Cenjunctiun. 9-I I, 17-13. 23-29, 34. 129; ef quantifier. I39, 143, I53; ef 53-54; in English. I3. 23-29. 53-54 negatien. See 1'Je1'vlergan, laws ef: Pas- ef terms, 1 I4, 129-133; infinite, sage 233-235; pelyadic, 234. 235. 237 Det. 29-33Cenjunctienal nertnal ferrn. 33 Dreben, 13., 233. 222, 224, 233 Duality. 79-34, 224-225C|;1-1'|-sensus, 77,73 Duns Scetus. 23 Dyadic. I37, I37-133. I37-I33Censequent. 21Censistency, 39-41, 43, 47; tests ef, 4] E. see Categnrical 5‘. I13, I19, 235 73; patent, 43- 44; ef Heelean EC. see Crencralizatien sehemata, I I4-I I9: in quantificatiun El. see Instantiatien theery, I54. I93, 215-213 Einstein, A., I95Censlructivity, 235, 213 Empty class, 94, I33, 294Centrafactual, 23 Empty universe, I13-I [7, 123.123Cenversien, simple, 99. See alse Inver- English, see Paraphrasing sten Enlhymene, I99-232Ceeley, I. C-, Iln, 244 Epistemelegy, I-4, 273, 233Cnpula, I29. l35—l33. 293-29] Epsilen, I35-I33, 293, 333Cerrelatinn, 293 Equivalence, '31-32, 37, 74; and bicendi-Curry. H. B., 233 tienal. 33, 37; under duality, 32, 34; pfDavis, I\"-'l., 213 Beelean schmata, I13, 122.55-e elseDecisien precedure, 33, I12, I72-I73, Interchange

313 {odorl.31»o-sur-e, I54 114, 11?: for Boolean schemata, 111-Coextenaiyeness, 113-119, 133, 139- 111; for Boolean algebra, I311; for menadic schemata. 154- I 59, 111?, 135; 191 fer pure esistenliais. 134- IE5, 114; forCollision, I511, IE1 pure universalia, 113-114; impossibilityColon, 511n of. 114-1115; for flnile valiuiity, 115; forCombinator, 1315-133 real numbers, 115Commutatiyity, I11, 11, 55, 1515. See oiso Dedekind, R., 5, 19-4, 3113 Deduction, 133-139, 145-1415; natural, 3ymmetry 1-14; theorem of, 144; soundness of,Compactness, 1113 1415--141'] strategies oI', 143-15-4Completeness, 114; in u'utI1 functions, Degrae, 134 I3l\el1-Iorgan, A413, I111, I1-1,113, 199; 13-111, 33, 31'; in sl.!I1t1‘ac1ior|, E3; in laws of, I4n, I515, 1'3, 31-31, 34, I411 quantification theory, I911, 1111'-1113, Denotation, 9'4. See n.E.so Ezttetssion; 115-1115; io number theory, 1115-113 Naming; Reference in identity- tlteo-ty, 113 Description, 114-1311; eliminaliofl of,Computer, 13 131-131Conerefion, 134 Designalion, 94. See o-‘so HomingConditional, 11-115, 34-35, 54-55; Development, 13, 33 generalized, 11, 95, 139\", conlrafactuai, Dilemma, 111', 15-4 13; ind imp1icalion, 4ti-4'1‘, 49-5 I; Disjunelion, I5, 1'4 and-er negation, ~55-39; of temis, I I4, Distributiytly, ‘111, 31; of H, I19; of \"I1, 119-1311; e.tistentia1, 111 , 115; El, I191 of quantifier, 139', I411, I511; o1’ I91: conditional proof, 141 nega|iozn,,5ee [1-ell-'[oegan, laws of: Pas-Conditionalieation, 141-144, 1415Confinement of quantification, 141- 143 543\": I41, I51-I54 Dot, 19-31111onj=|.:nction, 9-I I. I1‘-I3, 13-19, 34, Dwell-e11, H., 1113, 111, 114, 133 53- 54: in English, 11], 13-19, 53-5-4 Duality, 19-34, 114-115 ofterlns, I14, I19-I311; infinite, Duns Scotus, 111 1113-1115: polyanie, 134, 135, 1111 Dyadic, I31, I151‘-I153, I31‘-I33Coojunetional oorrnal form, 33Consensus, 1'1, 13 E, see Categories}Consequent, 11 3,1115, II9, 135Consistency, 39-41 , 43, 41\"; tests of, 41 E13, see Generalization 1'3; patent, 43-44; of Boolean E1, see lnstantialion schemata. 114-119; in quantification Einstein, 4., I95 theory. 154, 1911,1I5-1115 Empty eiaaa, 94,1311, 194Conslluetiyity, 1115, 1111' Empty universe, I 115-111, I111, 1115Eontrafaerual, 13 Eng1isI'|,,ree ParaphraaingConversion, simple, 9'9. .I,i'e'e of,-:o [nver- Enlhymene, 199-1111 sion Epistemology, 1-4, 11-5, 133Cooley.1. E.‘-, |1n,144 Epsilon, 135- 1 315, 1911, 31111Copslla, 119, I35-1315, 191.1-191 Equivalence, -I51-~51, I51, 1'4: and 1:|icontli-Correlation. 1111Curry. H. H-, 133 liionai, 1511, I51; llI'b|III.','l' duality, 31, 34; ofDayis, H., 113 I3-oolean 5121211113111, I 13, I11. Egg alsoDecision proeedure,11ili, I11. I11-I13, Interchange

Index 319Equiyoeation. 4. 55-51 Hilbert. D-, I5.1111. 131, 113; on i_.i'iw-Eyery vs. any. 91'. I4-4 enheim‘s theorem. 1I11—1|1, 113Eyery't]'ting,133-139, I'1'I—l'1'1Eitclusiye, 11-I3. 13 Homogenizing. 133Esistenee, llfi, 1151-115-4. 131. See clto ltlea, 1113-1154, 1151' Ciarss Inempotenee, I1, 13Existential, pure, 133-134 Identity, I39. 133 -113: of elasses, 139,Existential conditional. i1| , 115Eslensiofl, 94, 145 1911-191E.ttensio-oalily. 139 If, see Bieontlitional; Contlitionai Image, 193Fell swoop, 43-49, 153, I11 Implicani. prime, 115-13Finished, 141 Impiication. 4. 415. 33. 115; ind 111\"-'l1'IIIi—Finite uniyetse, 1411.111. 1I5—1It5Flag. 141-141, 14-4 tionai, 43-43 , 49-51; laws of. 43.Foundational system, 3'1‘ E-1-33; tests of, 43-49; truth-Free yariable, I34, I45, I415. 131 functional, 49. 51--E13, 31: Boolean.Frege, G.,1111. 1315; on nutnher, 5, 194, II4, I13, l19;polyat1ie, I31-I33; proof procetlttre for. ses Deduction 199; on truth functions, 1|-1, 115. 3-9, 39 Inclusion. I13—I19, I31, 133, 1911-191 on quantification. 141. I13, I94 Ineontpleteness, see CompletenessFritlshal, 3.. 1'3 Inconsistency, see Consiate ncyFull swap, 31 lnti-epentlen-ce, 31i-33Full sweep. 49 Indirect proof, I94Function. 119-1111. 133, 31111 Induction. mathematical, 143, 191'Functional norma.I fortn. 119-114. 131- Inference. 45-43. 93;1'tt1es of, 35 . 31', 134 113-1311, 134, 135. 139; singular, 111?,Funeto-1'. I19. 133-133 159. 151; truth-functional, 133 Infinity, 1411: lemma, 1113; schema. 115-Geaeh, F., 193, 19I5n 115Generalization {in proofs], 111-131 . 11-4 Ir|F.lttl'hI;e. I311, 1911. 191 instantial variable. 1911-191 135. 133-141 lnstantiation {in proofs}. 1911. 1115-11.113,Generalized connitiottal. 11. 95 113. 1111, 133-1413; restraints on. 191.Genera! term, 93-94, I3!-I33 111. 113, 1311-131, 1411-141; multi-Generic. I91. 111 ple. 1111-111, 133-13-4. See alto In-Gentzen. C-. , 144 stance; Suisstittttion Interchange. I53-ti-5.111. I531]t:rgt,1-t'|:'n:, J.[1'., 115 Inlerpretatio-rt: of sentenoe letters. 33-34‘, of Iertn letters, I14—I E1, 113. 1119-Ghaxala. M- .1-, 1'3 111; bypassed. 111Giizlei. H... I345. 1113. 113-113, 113 Intersection. I311Goodman, H., 13n inyersion, 135-133. 193. 31.111. See alsoGrouping: ofsymlaols, I11, 13- I4, 19-31 Commulatiyity 143; of component sentences. I4. 13-19. 51-53;ofqt.tantiftcat1on, 139, I41-144.139, I93Haltnos, F'., 133 Iasltourslti, 3., 144Her1:n'ane1. 1., 144; on rnonattic logic, 11? Jeyotts, W. 3., 131 Jungius, 1,, II53 141; polyatiie. 139, I94, 113, 131

334] IndexEqtllan. D-\".293 Hnrntai fumisz alte1‘|:|.atitrr1aI. T2-1'3, T15;llleene. S. C., 2l4n. 2lT. 2lB cnttjuncliunal. H3;fu:nc1i-e-nal. 2I9. 224IIli:ini1,, I., 2I]3Iitlts-eIt,.lt., I13 Hclt. see Heg.atianIturatewski. C.. 299 H-cltinn, I3-I5 Humher. 3?-BE. 2I19—2lfi;11:aI. 2113.Languagnt. see FaraphrasingLeibniz. G. W. 1|'nn.I32 2lE. 3IEl'[l. 393; elementary theery ef,Lemma. ll] 2] 1; tlelitteti, 295-299Lesuiewski . 5. . t'iI‘.'1=Letter, see Schematic; 5-enlen|:e D. see Calcgericail; Ecru1..ewis, ‘E- I., H39 lIl'|J-ject. see Abstract nh_'|e:ct; Existence;Lesic-c-tt. 221]-22]Limit, [T9 Dnlulegy; Physical ebject; PnssihleLiteral. T2La-gic. I-3, 5. 53. IIE 1fi=J'a=1Lngical t1'tIt1‘|. 4-5. 42L43-1HI:IlhE-‘iIl'l, L., l2'-I‘, 299-212. 2il5 Ehckhaln, W. nrf, T3. E4Lukasiewice, 1-, 2I]n, 215, 3i]-31, 3'9, T3; finiy if. 51. 55 Dntelagy, 215-3-2-I57, 235-, 152 azltierns ef, E5, 3?-B9, 2315 Dpen. 134. 145. I54 U=l'.see Ptltcmatien Urdcled pair. I151‘. 291\". 299 Clrtlinary ianguag-e.see Para|:|hrasi|'|g Dstcrrsien. 2T5Main methed. I90-I93, 295-2945 Padding. 21351'-*IarIi|1. H. It-'[., I3-ti Pair, I152, 29?, 299Mathematics, 2. 3. 5. 3III3, 3I]2-393 Parades. 2159. 3l]l . 3933I'[lI1'iI, IE5. -see m',1r||:| '[.]p||::|] Parapiuusing. I93; intn truth functiens,He-aliillg. I. 4, 2153-215-'.F, 2'J'I5—2T||' 53-55; inward. 51-59. 195-I97; ofMechanical test. see D-ccisitln precedure implicit IE!l'I']'|, 95-93. I33. I15}; of sin.-Membership. I35-I36. 299, 31]} gular term. I137; af Bee-lean equatitm.Mctarnathematica, 213 I313; inltl quantificatiun. 135. I39. 195Mills. B. E., TI, TB Parcnthescs. 9. ill. I3. 29-3|]. I43Ilrlinltawslti, H., I95 Particular, 95Hiscd schema, I42, 153. l5I-5—I59 Passage. I42-I4-4, I43, I43H-udtis |:|~t|nen-s. E5. ET. 234, 235 Pa1e|1t. 42-44. fil nIHI-unatlic, 9311. I33, 14?, I53. Il.I'.F; yaIid- Peane. I1, I5. 32. I29. l35—l3I5-. 292 Ity. implicatien. cnrlsistency, I115, 235 Peirce. C. 5.. I5. 25, 39, I33, 294Ihlfiller. Ft. E... TE Permutation, see [nversiun Fetitia principii. IUDHame, 25-3, 2?-ti-21? Pctrtls Hispanus. 29Haming. 94. 262-25? Phila cf Iulegara. 215H-ecessity. 2 Physicai abject. I. 259. 2&9Hegatian, 9-I'D. I5. 34; trf terms. H4. Plate. I-E? Palyadic, IE?-153. l'I'5+l'i'II5-. 233-235. I23; crf quantifica1ie-n, I37. 144; pelyadic. 234, 25.5. See arise D-cli-'Ic|r— 29?-293 gan. laws at Feasible abject. 2&6-25?Heurnan|1,.I. ven. 299. 3132-3133 Pest. E- 1... 39. E9Hew math, 13] Ptlstfnundatittnai system, 3?I'~ii-c.t|-tl, 1., ET, B9 Predicate, IIII5, IE1

flirts: 331Predicate 1'unclor, sec Term 1'llI'|t-‘I11! Relative clause. 133-1345. I25Predication, I32-135 Relative term, I32. 132Ptediction. 2 Eelettcring: indilferetlee ef. I45. I415; I11Pte1'nisc.53. |9'|l'l-l'91t!;I1-E!1t,1'.2\".'2—2122; avoid collision. [43, 1512. 193-194, [fl|T|];it;||1:\".a1'y, 242-244; des-criprrional, 231], 241]-241; for‘ distribution, 1511, 225 312Prene:t.143—i49. 154. 222. 243; I-tesolulion, 34-35, 32-33. ti-in. I13 strategies ef. 1513. 134-E35, 214 Resultant, 293Pi-esburger, 191., 212 Rhinelandcr. P- H-, 2InPrime implicant, 23-23 Right angle. 31]nFrnduct; arithmetical, 295; Cartesian. Rigid routine. 295-233, 213 293- 51-5 alse Cnnju action: Intersection Rogers, H-, .Ir.. 213Pronoun. 133. 1415 li1'.ost-er, H-, 24-4Pm-n1‘ procedure, 214: for truth-functienal Russell, H., I33. 141, 23-I5, 223, 232- validity, 33; for t|uar|tificationa1ya1itl- 2.33‘. and Wltitehe-H11. 15. 32, 5|, 39, ily, 123. 191]. 224-223, 232-235: for I-1| . 141', 3113; parades of, 33! , 3113 implication. 193; for incnnsistency. I91]-I93. 235 -233; fer censistenfiy. Samson. E- ‘H., 22, 23 215-215. See alse Cempleleness; De- Schema. 33, I12, I53. It'll]; normal, ductinnProof theory. 213 22-23. 215. 33. 219. 224; ’tt=rr|:1, I14;Property. 94. 239 Boclean. I14-I13-;esistenee. 11-I5;Frncpnsitinnalfunction. 142 tnised. 142, 153. 1515-159; quantifica-Pure: bicenditienal. I5-I5: quantification. titt-I1aI,142,I22:monadic. 142. 132. I51-I54; esistenlial, |33-135;ut1i1rer- sal. 213-214 Era [Ilsa EH1!-stilttlitiitlQuantification. I321 under negatien, 132. Schcmatic letter: fnr sentence, 33, 143; I4-tl-;distril:iutic|-n of, 139. 1412. 153': fclrterm. 93, [12, I32, 3'31]-3-111: scope of. I39, I42-I44: in finite uni- pelyadic. 133. 122, 234-235; ys. vari- serse.141].121;hisIuryof, I41; able, 239 confinementef, I42-I43. I42. 151-- 154; purificatien ol',151-E54tstac1red. Schcinlinltel. H., 139. 233 153,115-9, I32; numerically definite. Schr-Eider. E.. 213. 39. 24. 34 234. 239 Science. 1-3. 333 Scope efquantifier. I39. I42-I44, 169,lflluantificational: seherna, 142, 122; validity, implication. corlaistency. 1945- 122. I23 Scntence. I31. ltil1, I215 Sentence 1etter.9. 33. I45. 153. I53. IE2Qluimby. D. P.. 2-33 S-ct. 131 . 332- Sec also ClassUuetation. 53. I43, 215-3 Shannon. C. E., 2'2 Sheffer, H.1I-‘1., 21]. 32I-Tteductio ad absurdum, i912. E94. 254 Shocnfield, 1., 213Eedundancy, 25-2-E1, 34 Simplificalion. 3, 35-33; -til‘ normalReference. 232, 239Eefiecunn, 235 schemata. 22. 24-23Reflesiyily. ‘[32-I39 Singular term: in inferenee. I112. 25-9.3clalinri,1o2.12-ti. 292-293 252; and variable, 135, 233-2152; ‘rs. general. 259. 231. 233-239, 3133: laclting reference. 232- 233, 223-2312; as description. 223-222; eliminatinn of. 231-232

332 1rtel'eJ: 51tt:t1I:tTt, T-, 154, 194, 2133-21 1, 221 Translation. see ParaphrasingSmullyan, 11'.-, 213 True of. 94SnrnetI'i1r|.g,l33-I39. 121-122 Truth, 1, 9; logical. 4-5. 42;1rttth func-Suutv1ness:et' main method. 192- 193; of tioo, I3-2[I, 33, 114-115. 123; truth Herhrand‘s methnd, 223, 231; of de- value, 13. 213'; trutlt-value gap. 21. 29; ductive method. 243-242 1t't.tl1:t-1'1tnl;'1iot‘tal.'1-c11it\"1‘ttI. 33: tmth-value analysis, 32-33. 41, 43. 31, 35.115.Stat1tedquariti3ers,153,1t59,132 I52; nuth tattle, 33-39; trut:11- Star, 241-243- functions] validity. implication,Etatement. I-2. 9. 42. 49, E14. l1-t;.='.i1-.- equivalence. 49, 32-33. 123; truth- functienal inference. 239 gular. 1112 Turit'tg.11|..1h'1..214-213Stencil. I31Stoics. 211 UG, see GeneralizationStrategy: in truth functiens. 33. 411. 3132: U1. sec Instantiation L|l1iao,1- 3-. II3 in p-araplfirasing. 52-59. 193-192, 193; Llnien, I311 in prenexing, I51], 134-135. 214-;i|1 Universal; afli rmative and negati ve. 95; deduction, 243-25-4Subject, 1115 closure, 154; pure, 213-214Subscript. 2211-221. 224 L1niverse:oftlisco1.lrsc.111. 115.1211.Substantive. 93, 132; spurinus, I33-139, 121-122 193;e-mpty, 113-112. I211. I23; uni-Substitution: for '_a‘, '.-4‘. etc., 44. 4'1. 152. verse elass. I39; f|njte.1411. 121. 35, 32; for 'F.:r'. ete.. 1311. 132; for 215-213 ‘F:ry', etc.,125-I31 Unless, 55Suhstitutiviljv. 33-35. 122. 153. 221-223 Usage. sec ParaphrasingSui:-tractien. 32-33 Lise vs. mention, 511, 14-Etn, 215331.1-cit dtat. 133-133Sum. 293. .5'ee also fltlterrtltiun; Llnic-n '11\". 1'1, 1311, 141, 29\"-'1Eupcrstitution. 235 1.\"a1idity. 4: trut1'|-functional. 39-42, 49,Syllogism, 1112- 1113Symmetry, 132-133 I23‘. ptltent. 42-44, tiln; transmission ef. 44, 42. 32, 34, I22, 229-2311;Tarsli. 14... I2. 32n. 39n. 213, 244. 233 under duality. 32; of syllogism, 1112-Te1esr;.|;;-ping. 22-2-3 1113,1113-1112;13oo1ean. 114-113. 121 - 122; of npen schema, 154;Tense. 93, 195-193 monadie. 154-159. 123. 235;Term: general. 93-94. 131-133; major, quantificatienal. I22-123, I911, 212; sf pure csistcntial, 133-134, I39; middle, minor, 111-5; absolute vs, ]'g|g. proof procedures fur. I911. 224-223. 232-235; in identity theory. 221 11\"\"-‘*5. 132. |52;Po1yadic. 132. See airs- 'v'an 1-leijenourt, 1., 123n .4.l:|-stractzion; Singular term ‘hiarialrle, 133.134,132,143, 131; andTenn functor, 233- 232 singular terrn. 135, 2311-232; instan-Term scltcma.1|4,122 tial, 1911-192; elimination of. 234-TF. 233 233; vs. schematic lcttcr. 239. Ere alsoTlreophrastus, I113 FleletteringThomas. ll. A.. I-tin \"v'el, 12Time. 52, 93,195-1'93. 239 \"1-'enr|..1_. 93. 1113. 1119-1111Trachteniirot, 13. .41., 213. 213Transit] vily, 132-139; of implication, 42; of equivalence. 32

Indexllfineulurn, 32 ‘Wiener, H., 299. 3133\"v\"1rt1ta1. I33 ‘Wittgenstein, L., 39\"I.-\"on l\"~leu=mann. see Heumarm Wurfit. are Parstlhrasing'91-’ei-eratrass, 11'... 5 2'e:|1rte1-u, E., 313-1 -3132W1tite1tead..-1.-1*I-,22-11|r;and11.usae1l.15. Zero. 295, 2'-111 32, 51, 39,141,142. 393

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