Methods of Logic - W. O. V. Quine

llGENERAL TERMSAND QUANTIFIERS



14CATEGORICALSTATEMENTS There are many simple and legically se-and inferences fer which theferegeing techniques are inadequate. An example is this:hie philnsnphers are wicked. 5chenntricrtHy.' Ne ti are H .Same Greeks are philesephers; Sense F are G;Therefere seme Greeks are net wicked. Heme F are nut H.Hcte that 'F‘, '6‘, and ‘H’ here stand nntfcr statements, after the mannercf '_n* and 'q‘ in Part t, but fnr ccrnrnen nnuns—nr, in legical parlance,general terms.‘ Whether these nuuns be theught cf as substantive eradjective is an insignificant questien c-f phrasing. ‘G’ appears as a sub-stantive in the abeve example, via. ‘philcsuphers‘, and ‘H’ as an adjec-tive, \"wicked\"; but we ceuld rewrite the adjective as a substantive,‘wicked individuals\", if we liked. In the same spirit we can even treatintransitive verbs as terms, in effect, thus recltening ‘Frame fishes fiy‘ as aease bf ‘Scme F are G ’; fur the difference between ‘Seine fishes fiy* and‘Same fishes are flying things‘ is purely netatienal. The neuns ctr verbswhich figure as terms may alsn, cf ceurse, be cemplex phrases such as‘empl-eyed fnr ten years by Sunnyrinse', ‘wear brass rings in their nnses',etc. Whether terms be theught cf as in the singular er the plural is alsn alegically insignificant questien cf phrasing; thus there is nu need te dis-tinguish between ‘hlc philcsnphcr is wicked‘ and ‘l\"-lc philctsuphers arewicked*, ner between ‘All philnscphers are wise‘ and ‘Every pl-tilesnpheris wise’. There is nc need even tc distinguish between ‘Same Greek is aphilesepher' and ‘Same Greeks are philrtsuphers', prcvided dtat, as willbe eur practice here, we understand ‘sn-mel always te mean simply ‘atleast nne‘. But, fer all the latitude accerded tn the cencept cf term, it remainsclear that terms are never statements; and this is why the techniques cfFart l are inadequate te the inference exhibited abeve. Part l dealt with the ' W'|1at are spn-ltert nf as genera] terrns ur simply as terms in the present pages may, inview bf develuprnenls in Parts ll]—l\"v', be designated rnnre accurately as nun-t.rn':l't'r generalI'f'Fl\"l\"l'.T- \"93

it-=1 ll. General Terms nne‘ Qttttttttiliersstructures nf cnrnpnund statements relative nnly tn their enmpnnentstatements; statements remained the smallest units cf analysts. It is enlynnw, in Part ll, that we embark upen the analysis nf these cnmpnnentstatements in tum intn the still smaller parts, nnt statements at all butterms, nf which they are enmpnsed. Lngically snund inferences dependfnr their snundness nn the structures nf flte statements cnneemed, but therelevant stt't.1etures may be either the brnad nutward structures studied inPartl nr the finer substructures tn which we-are nnw taming. The exampleabnve is nne which depends nn shuctures nf the latter kind. lt is flte peculiarity nf a statement tn be true nr false. It is the peculiaritycf a term, nn the nther hand. tn betrtre sf many nbjccts. cr ene, er nnne.and false nf the rest. The term ‘Greek’ is true nf each Greek. and the term‘wicked' is true nf each wicked individual, and nnthing else. The temt‘natural satellite nf the earth‘ is tlue nf each natural satellite ef the earthand nnflting else, hence true nf but nne nbject, the mnnn. The term‘centaur‘ is true nf each centaur and nething else, hence true cf nething atall. there being nn centanrs. ln place nf the clumsy phrase ‘is true nf‘ we may alsn say ‘denntes‘, inthe best sense nf this rapidly deterinrati ng werd. Hut I prefer he-re tn resistthe temptatinn nf gnnd usage. ‘Denntes’ is sn current in the sense nf‘designates’, nr ‘names’. that its use in cennectien, say, with the werd‘wicked’ wnuld cause readers tn lnnk beynnd the wicked penple tn snmeunique entity, a quality cf wickedness nr a class cf the wicked, as namedabject. The phrase ‘is true cf‘ is less npen te misunderstanding; clearly‘wickedl is true nnt nf the quality cf wickedness, nnr nf the class ct‘wicked persens, hut nf each wicked persnn individually. Wl'ren we are minded tn speak nf classes, the class nf all the nbjccts nfwhich a term is nue may, in keeping with a lnng traditinn, be called theerrenslnn cf the term. The extensinn cf ‘wicked’ is thus the class cfwicked persnns; the extensinn nf ‘natural satellite nf the earth‘ is the classwhnse snle member is the mnnn; and the etttensinn nf ‘centaur' is theempty class. Tenns may be said tn have extensinns, just as statementshave tmth values; but there is nu need tn thinlt nf a term as snmehnw aname sf its extensinn, any mere than there is tn think nf a statement as aname nf its truth value. The use nf terms prnceeds smnnthly nn the whnlewithnut assumpticn nf any special eategery cf abstract nbjects calledclasses. It is erdinarily sufiicicnt tn knew that a given term is true cf thisand that individual and false nf the nther, withnut pnsiting any singlecnllective entity called the term's extensinn. Snme terms nf set thenry,indeed, cannnt have extensinns, nnt even the empty nne; see Chapter 45.

ld, Categarleal Statements‘ ‘-I-'5 Fnur ways nf jnining terms pairwise intn statemenm have been treatedas fundamental thrnughnut the lngical traditinn stemrrting frnm Firistntle:‘All F are El‘, ‘Hn F are G‘, ‘Same F are G‘, and ‘Same F are nnt G‘.Statements nf these fnur fnrms were called categarical. The fnur fnrrnswere distinguished by special nnmcnclature and by cntle letters ‘At’, ‘E’,‘I‘, and TII‘, as fnllnws.lit [Universal affirmative]: All F are GE [Universal negative]: bin F are GI [Particular affirmativel: Snme F are GEl‘ [Particular negative}: Snme F are nnt G The fnrm lit, ‘Fill F are G‘, may alsn be phrased ‘If anything is an F, itis a G‘; thus it is recngnieable as the \"generalised cenditienal“ whichwas tnuched nn in [ll—[3} nf Chapter 3. lvlany ether phrasings nf A alsncnme readily tn mind: ‘F are-G‘, ‘Each (Every, .tltny}F is at]F‘, ‘Whateveris an F is a -El‘, ‘F are ettelusively G‘, ‘Elnly G‘ are F‘. E likewise has many phrasings: ‘I\"'-In F is tare] G‘, ‘Hnthing is be-th anF and a G‘, ‘i\"'-lnthing that is an F is a El‘, and even ‘There is late] nn F-G‘{e.g., ‘There is nn black swan‘), ‘FG dn nnt exist‘. Efnrrespnndingly fnr I: ‘Same F is tare] El ‘, ‘Scrnething is bnth an Fand a G‘, ‘Samething that is anF is aft‘, ‘There is an F5‘. ‘There areFEE‘, ‘FEE exist‘. D, nf cnurse, has similar variants. Elften the tenns prnperly answering tn ‘F ‘ and ‘G’ are nnt directlyvisible at all in nrdinary phrasing nf statements. They may be partiallycnvered up by such usages as ‘nnwhcre‘, ‘anywhere‘, ‘always‘,‘everynne‘, ‘whnever‘, ‘whenever‘, etc. Thus the statement ‘l gnnnwhere by train that I can get tn by plane‘ is prnperly analy;-table as cfthe fnrm E, ‘l\"~lnF are C.-' ‘, but here we must understand ‘F as representing‘places I gn tn by train‘ and '5‘ as representing ‘places I can get tn byplane‘. The statement ‘Everynne in the rnnm speaks English‘ has the fnrtnA, ‘All F are G‘, where ‘F ‘ represents ‘persnns in this rnnm‘ and ‘tT‘represents ‘speakers nf English‘. In this last example the restrictinn tn persens implicit in ‘everynne‘ isessential, since there will be nnnpersnns in the rnnm that dc nnt speakEnglish. In such an example as ‘Everynne whe pays his dues receives theBulletin‘, nn the nther hand, ‘everynne‘ is used instead nf ‘everything‘nnly because nf a habit nf language, and nnt because the speaker feels anyneed nf hedging his statement against such absurd nbjccts as subhumanpayers nf dues. It wnuld be pedantic tn cnnstrue ‘F ‘ fnr this example as

9:5 ll- fieneml Tenns artei‘ Quantltiers‘persens whn pay their dues‘, and quite prnper tn censtrue it as ‘payers nfduesh In putting statements cf nrdinary language nver intn the fnrms lit, E, I,and {J we must be an the alert fnr irregularities cf idiem, and lcck beneaththem tn the intended sense. Etne such irregularity is cmissinn cf ‘-ever‘.as in ‘whn hesitates is lest‘, ‘t want tn gc where ynu gn‘ , \"tlllhen it rains itpnurs‘, ‘She gets what she gees after‘. Annther irregularity is the nnn-tempnral use cf ‘a|ways‘. ‘whcnever‘. ‘scmctimes‘, ‘ncver‘. E.g., thebl.-Ell¢f|‘ltl';\"»l'lll The sum nf the angles nf a triangle is always equal tn twn right anglesreally means: The sum nf the angles cf any triangle is equal tn twn right angles.and may he rendered ‘All F are G‘ where ‘F ‘ represents ‘sums nf anglesnf n'iangles‘ and ‘f]‘ represents ‘equal tn twe right angles‘. Frequently an I cnnsnuctinn having tn dc with time is implicit in theinfiectinn nf a verb; witness ‘We saw Strembeli and it was entpting‘,which cctnes nut as ‘Heme F are G‘ with ‘F ‘ eenstmed as ‘times we sawStrembeli‘ and ‘t]‘ as ‘times Strnmbnli was entpting‘. [See Chapter El.[5}.l Further examples nf tempnral idinms which call fnr a little reflec-tzinn, if the lngical structure is tn be prnperly extracted, are: I knew him befnre he lnst his fnrtune, I knew him while he was with Sunnyrinsc.‘Eefnre‘ and ‘while‘ here appear in the guise nf statement cennectives,like ‘and‘ ct ‘er‘ er ‘-1’. Hut the statements are better analyzed as nf thefnrm I, ‘E‘rnmeF are G‘, where ‘F ‘ represents ‘mnments in which I knewhim‘ and ‘G’ represents in the nne ease ‘mnments befnre he inst hisfnrtune‘ and in the nther case ‘mnments in which he was with Eun-nyrinse‘. Heflectinn, indeed, shntrld be the rule. Prnper interpretatinn is nntgenerally tn be achieved thrnugh slavish dependence upen a check-list nfidinms. ‘rklways‘ usually means ‘at all mnments‘. but it wnuld be unjusttn censtrue ‘Tai always eats with chnpsticks‘ as ‘Tai eats with chnpstieltsat all mnments‘. The prnper interpretat:inn nf this example is ‘All F are G‘

l 4- -Categarical Statements QTwhere ‘F ‘ represents ‘mnments at which Tai eats‘ {nnt simply ‘mnments‘}and ‘G ‘ represents ‘mnments at which Tai eats with chnpsticks‘. The impnrtance nf reflecting upnn cnntext and the cnmmnn sense nfthe cnncrete situatien. rather than lnnking tn any mere glnssary, is man-ifest even in sn basic a cnnstructinn as ‘An F is -ET‘- ‘lit lady is present‘ issurely nf the fnrm I, but ‘A Senut is reverent‘ is mere likely tn beintended in the fnrm ll. Cautinn is similarly needed in equating ‘any‘ with‘every‘; fnr, whereas the statements: Inhn can nutrun every man nn the team, Jchn can cutrun any man en the teamneed nc distinguishing, a divergence appears as sncn as ‘nnt‘ is applied: Inhn cannnt nutrun every man en the team, .lnhn cannnt nutrun any man nn the team.The first twn statements are indistinguishably ‘Fill F are G‘ 1,‘ where ‘F ‘ is‘man nn the team‘ and ‘G‘ is ‘wbnm Inhn can nutrun‘); the third, hnw-ever. is ‘SnmeF are nnt-El‘, while the fnurth is ‘l'~lnF are E.-\"-EXERCISE Classify the fnllewing statements as between lt, E, I, and Cl, andspecify in each case what terms answer tn ‘F ‘ and ‘t_‘iF‘.Blessed are the meek. We shnuld all be as happy as kings.tl‘ll.ll that glisters is nnt geld. A pn|icerrran‘s Int is nnt an ‘appy nne.There is nn gnd but Allah. There are smiles that make ynu blue.Hnpe springs etemal. I jnurneyed hither a Enentian rnad.The rtrle applies tn everynnc. I was stepped by the dnnr cf a tnmb.

95 H , General Tern-ts and Qunntrjfiers l5 VENN'5 DIAGRAMS In \"i=\"enn's diagrammatic methed I IEEIJ}, averlapping circles are usedtn represent l:l1e twn terms nf a categ-nrical statement. The regi-nrt in whichthe twe circles twerlap represents the ehjeets which are heth F and G.This reginn, called alens in genmetry, is shaded in Diagram Z. whele ‘F’is tat-Len as ‘French’ and ‘t'.\"i‘ as ‘generals’. this regien represents theFrench generals. Cerrespentlingljr the part ef the Frcirele which lies eut-sitle the G-eireie represents the ehjeets which are F hut net G: the Frenchnengenerals, in the e:-tatnple. This reginn, called a htne in genmetrgr, isshaded in Diagram l. The significance nf shading is emptiness; thusDiagram I affirms that nn.F are Ci, while Diagram I affirms that ncl F areeither than G’ , er in nther werds that all F are G.FE F -GH:-fillFareE E:HeFa|-eta Diiflml Diagram? FE FGi:5u-tr-eFan=E‘ El:5nrr|eFarern:|t-E Di-l§'l1'I'l-3 lIl'i-lgrar-til whiteness nf a reginn in a Venn diagram means nething hut lack nfinfnrmatinn. In Diagram I the twn lanes are left unshaded nnt hecause wethinlt there areF which are anti? and G which are nnt F , hut because ‘HeF aretT gives us nn inf-nrmatinn nn the suhject. All that ‘l‘~ltI:F are G' saysis that tl'|e lens is empty, and this is all the infnrmatien that Diagram Irecnrds- Similarly the lens and the right-hand lune in Diagram l are left

I 5 . l-\"crtn's Diagrams 9'5‘unshaded merely because ‘Fill F are G’ gives us ner inferrnatien cencem-ing these further regiens. The great reginn eutside beth circles represents the nbjccts, if any,which are neither F ner G. lt is left blanlc in Diagrams l and 1 because‘All F are G’ threws ne light en such ehjects, and neither decs ‘l'JeF areG’. Se, whereas shading means emptiness. nenshading decs nnt assurenenemptiness. Fer nenemptiness anether symbel is usedp via, a cress.Thus ‘Heme F are ti’, which aflirms nenemptiness ef the lens, is es-presfi by putting a cness in the lens as in Diagram 3. Here again theblan ltness ef the ether areas implies neither emptiness ner nenemptincss.but represents mere laclt ef infermatin-n. ‘Same F are net G’, finally, aflirms ne mere ner less titan that the partnf the F-circle which lies eutside the G-circle has semething in it; se it isrepresented lay putting a cress in that lune as in Diagram 4- Certain simple laws el’ categerical statements are graphically reflectedin the diagrams. The symmetry ef Diagram 1. and ef Diagram Ii, reflectsthe fact that in E and I the erder nf terms is inessential: ’l‘-le F are '5\"ameunts tn ’l\"~let'] are F’. and ‘Same F are ti\" te ‘Heme G are F’- Suchswitching ef tertns was ltnewn traditienally as simple cnnvcrsinrt. Thelepsidedness ef Diagrams l and 4 reflects the tact that simple cnnyersinnis net in general applicable tn A er D: ‘Fill Greeks are men’ is net tn becenfused with ‘All men are Grecirs’, ner ‘Same men are nnt Greeics’ with‘Same Greeks are net men’. A and D are mutual cc-ntradicterics. er ncgatinns: A is true if and enlyif D is false. This relatienship is reflected in the diagrams by the fact thatDiagram l shews shading where, and nnly where. Diagram It shews acress. ‘With respect te blanltness, er laclt ef infnrmatien, Diagrams I and4 are alilre; with respect te infermatinn they simply and directly deny eachether. Similarly E andl are mutual centradicteries: E is true if and enly if I is false. A and E. ‘All F are G’ and ‘He F are G’, may alse he felt te he semehew eppesite te each ether; hnwever. their eppesitien is ne maner ef mutual negatien. fer we cannet say in general that A is true if and enly if E is false. Dn the centrary. esamples chesen at randem at'e as likely as nnt tn cause A and E tn ceme eut he-tn false; this happens in particular when ‘F’ is taltcn as ‘French’ and ‘I1?’ as ‘generals’. Similarly] and U are quite cemmenly heth true, ef ceurse. as in this same esample. But A and D are never heth n-ne ner heth false, and similarly fer E and I; here are the pairs ef centradictnries. nr mutual ncgatinns.

|lj[l H- General Terms and -Jnnnttjicrs ‘Whereas A and E are very cemmenly beth false and I and ID are verycemmenly beth h'ue, it is less cemreen fnr I and D te ceme eut be-lh false,er fnr A and E in cnme eut beth true; but these things will happen wherethere are ne F. Clearly, where there are ne F, ‘5nmeF are t'.'.-\" and ‘HemeF are net G’ will heth be false. Alse, where there are ne F, ’l‘~le-F are CF’will ehvieusly he tme; and yet ‘All F are ti\".-\" will likewise be true, in dtatthere will he ne F which is net G. These pr:-ints are breught eut dia-grammatically by shading the F -circle in its entirety, as in Diagram 5, temean that there are ne F- This diagram verifies he-tlt A and E, fnr it shewsbeth ef the areas shaded which are shaded in Diagrams i and 2; and itfalsifles hetlt I and I], fer it shews shading in place nf beth creases efDiagrams 3 and 4. .F -G There are ne F Dingam 5 A, ‘All F are G’, weuld seem at flrst glance te he stmnger than I,‘SemcF are G’. and le imply it; bet it decs net, because ef the pessihilityef there being ne F. Diagram 5 depicts the very situatien where, theugh Ahelds, I fails. it may happen that all my dimes are shiny [in dtat I have ne-dime te the centraryl, and yet be false that seme ef my dimes are shiny.simply because I have nn dimes at all. The mest we can say is dtat if all Fare G and there are F then seme F are G. If the reader thinks it edd te say that all et ene’s dimes are shiny whenene has ne dimes, he is perhaps interpreting ‘All F are G’ tn mean, netsimply ‘There is neF dtat is net G’, but ‘There areF and each nf them isG ’. This, hewever, even if it he ene ef several defensible interpretatiensef art ambigueus idiem, is clearly nnt the interpretatien which weuldmake A the simple centradictery, er negatien, ef I]: ‘Same F are net C-\".It is the general legical practice, and a cnnvenient -nne, in understand ‘AllF arc G’ simply as the centradictery el’ D. Diagram 5 as it stands was seen in reflect the fact that I decs net fnllnwfrem A. Hut ‘Venn diagrams can alsn be used fer cnnstrttctive ends, as inshewing thatl fnllnws frem A supplemented with ‘There areF’. Te shew

I5, Penn’ s Diagrams Ii] Ithis we set dewn the diagram fer .+'t, via. Diagram l , and then enter ‘Thereare F ’ intn the diagram hy putting a cress in the F -circle. The cress mustge in the unshaded part ef the F -circle, since the shaded part is ltnewn tehe empty. Se the result, shewing a cress in the lens as it dees, verifies I- What has been said el‘ the relatienship between A and I applies equallytr: E and U: frem E, ‘He F are G’, we may infer U, ‘5nmeF are net G’,enly if we make the further assumptien that there are F. Diagram 5 shewsthe situatien where, theugh E helds, I] fails. But we can shew that {llfellews frem E and ‘There are F’, by putting a cress in the F -circle efDiagram 2 and ehserving that we have verified -El. Finally let us nbserve a ceuple ef simple inferences in which a cenclu-sien is drawn frem just a single premise, er assumptien, instead nf fremtwn:Seme F are G, There are ne F. There are G. l\"-le F are G.These inferences are justified respectively by Diagrams 3 and 5\", ferDiagram 3 shews a eress in the G-circle in suppert ef the cenclnsinn‘There are G’, and Diagram 5 shews a shaded lens in suppert ef thecenclusien ’l‘\"~leF are G’.EIEIICi5 E5l. Dues A, E, I erfifellew frem ‘Thereare ne-G\"? [lees dt, E, I. ert] cenflict with ‘There are ne G’? Appeal te diagrams.1. lviake a diagram fer ‘All F are U and all G are F’. Is this cempatihle with ‘hie F are G’? Esplain.

ll]? H- General Terms and Qaanrrfiers ‘I6 SYLLOGISMS ‘Hhat are speken ef traditienally as syllagistnsi are arguments whereina categerical statement is derived as cenclusien frem twe categericalstatements as premises, the three statements being se related dtat there arealtegetiter just three terms, each ef which appears in twe ef the state-ments. Silt e:-tamples fnllnw; ene ef them was already nnted in Chapterlr-l.All men are mertal, MIG areH,All Greeks are men; .allFare G; All Greeks are mertal. .~!t|lFareH.Ne men are perfect, Net? areH.All Greeks are men: .allFare G; It-le Greeks are perfect. l‘~leFare H.All philesephers are wise, .liillt’_i'areH,Seme Greeks are philesephers; SemeF arefi; Seme Greeks are wise. Se-meF areH.He philesepbers are wicked. Net? are H.Seme Greeks are philesephers; SemeF are G; Seme Greeks are net wicked. Seme F are nut Hall Greeks are men. A||HareG.Seme mertals are net men; Seme F are net G; Seme mertals are net Greeks. Seme F are net HSeme men are net Greeks. Seme G are net H,All men are mertal; All G are F; Seme mertals are net Greeks Seme F are net H A “valid” syllegism, erdinarily se-called, is a syllegisrn et’ such fermas te be incapable ef leading frem true premises te a false cenclusien. .-'-‘tn =t';‘atrgrnv'cat syllegistns, mere specifically, te distinguish them [rent ltyperlretirntsylln-gisms, which are certairi truth-fnnctienal arguments manageable by the melhclds efPart l.

l' t5- -Syl'lagt'.rrrr.r I U3easy test ef validity ef syllegisms is afie-rded by ‘v‘crtn’s diagrams. Threeeverlapping circles are used, as in Diagrams 45 and T, tn represent thethree terms ‘F’, ‘G’, and ‘H’ ef the syllegism. We inscribe the centcnt efll”F - Diagram E Diagam T-'the twe premises inte the diagram by the methed explained in cennectienwith Diagrams t—-4, and then we inspect the diagram te sec whether thecentent ef the cenclusien has autematically appeared in the diagram as aresult. Thus, let us test the secend syllegism ef the list abeve. ‘He recerdits iirst premise, ‘hie G are H ’, by shading the lens cemmerl te theG-circle and the H—circle; then we recerd the secend premise, ‘Fill F areG ’. by shading the lune which lies in the F -circle eutside the Gscircle.The result is Diagram I5. It bears eut the desired cenclusien ‘l\"'-leF are H’,since flte lens cc-mmen te the F -circle attd the H -circle is fully shaded. Let us nest test the feurth ef eur six examples. We recerd the firstpremise, ‘hleG are H’, as befere, and then we recerd the secend premise,‘Seme F are G ', by putting a cress in what remains ef the lens cemmen tethe F -circle and the G-circle; the result is Diagram '.l. It bears eut thecenclusien ‘Sc-me F are net H’, there being a cress in the F -circle eutsidethe H-circle. It is left te the reader te censtruct diagrams verifying the remainingfeur ef the abeve six syllegisms. lfln the last ef them, the secend premisesheuld be handled flrst; nete why.) The diagrammatic methed can he used te determine net merelywhethera given cenclusien fellews frem given premises, but whether anycenclusien at all [ef a syllegistic kind] is capable ef fnllewing fmm givenpremises. Fer, the cenclusien—in erder te he the cenclusien ef a se-called syllegism at all-—must be ‘All F are H’, ‘l‘~leF are H’, ‘SemeF areH’, er ‘Seme F are net H '; hence, unless the twe bettem circles ef thefinished diagram exhibit ene ef the feur pattems shewn in Diagrams l-4,there is nn cenclusien. In Diagram S, e. g- , the bettem circles exhibit nenenf the feur pattems ef Diagrams I-=1; and this shews fltat the premises

lfl-='l H . General Terlttt end flaanrtjiers % Diagarrt El ‘Fill H are G’ and ‘Fill F are G’ cannnt be the premises nf any validsyllegism at all. as a funher example censider the premises ‘All G areH’ and ‘rill Gare F’. These are recerded in Diagram '5\". and we see that the diagramjustifies ne categerical cenclusien in ‘F ‘ and ‘H’. Eat this pair efpremises is interesting in that it afmest justifies a categerical cenclusien in GFH Diagram 5‘F ‘ and ‘H’, via., ‘Seme F are H‘. It’ we add just the further premise‘There are G’, te allnw us te put a cress in the -ene part ef the G-circle thatremains unshaded, we then find the cenclusien ‘Seme F are H ‘ justifiedby a cress cnmmen in the F-circle and the H-circle. Thus the reinfnrcedsyllegism:an Spartans are brave, All G‘ are H,All Spartans are Greeks, All G are F,There are Spartans; There are G; Seme Greeks are brave Seme F are H is valid.| In the lraditi-nnal terminelngy the term which plays the rele ef ‘F ’ in ‘ll.llF are G‘, ‘He F are G‘, ‘SemeF are G’, er ‘SemeF are nnt G’ is

i ti. S_vi'iegirnt.r IDScalled the subject ef the statement. The ether term, playing the rele ef‘G’, is called the predicate. The predicate ef the cenclusien is called themajer term ef the syllegism, and the subject cif the cenclusien is calledthe miner term nf the syllegism. The remaining term, eccurring in bethpremises but net in the cenclusien, is called the middle term ef thesyllegism. Thus all the feregeing examples have been lettered in such away as te make ‘F ‘ the miner term, ‘G’ the middle term, and ‘H’ themajer tetTn. The premise which centains the middle and majer tenns is called themajer premise ef the syllegism. The nther premise, centaining the middleand miner terms. is called the miner prerni.re. Thus all the feregeingexamples have been stated with the majer premise first and the minerpremise secend. lvtedieval legicians had a scheme fer ceding the varieus ferms efsyllegisms. They stipulated the respective ferms ef the premises andcenclusien {as ameng A, E, I, and G] by a triple cf letters; thus ‘EAG’meant that the majer premise was el’ ferm E, the miner premise A, andthe cenclusien D. This much was said te indicate the mead ef a syl-legism. But. even given the mend cf a syllcgism, there remains thequestien whether the majer premise has the majer term as subject and themiddle term as predicate, er vice versa; and cerrespnndingly fer the minerpremise. The feur pessibilities el’ arrangement which thus atise are calledfigures. and referred te by number as fellews: t sr and iian era lvlajer premise: GH HG GH HG Miner premise: FG FG GF GF Cenclusien: FH FH FH FHSpecificatiert el’ mend and figure determines the ferm ef a syllegismcemplelely. Thus the six examples at the beginning ef the present chapterare respectively AAA. in the first figure, EAE in the first figure, i-‘III in thefirst figure, EID in the first figure, ADD in the secend figure, and GAGin the third figure. The feurth ef the six abeve, via. EII] in the first figure, can be givenvariant ferms by simple cenversien {see preceding chapter} cf ene er hethpremises. We thus gel:l'~leHareG, l'~leGareH, l'~leHareG,SemeF are Gt Seme G are F; Seme G are F; Seme F are net H. Seme F are net H. Seme F are net H.

|{]Ei H- Generni Terms and QaanttjlfersThese are Ell} in the secend, third, and feurth figures. Similarly EAEand All in the first figure {the secend and third ef the examples at thebeginning rtf the ehapter] are eatried by simple ennyersinn inte EAE inthe see-and figure and All in die third. The fnur syllegisms last mentiened, ttiz. EAE in the first and seeendfigures and All in the first and third, ean he earried ever inte fear furthersylldgisrns by simple eemtersien el’ eaeh pf their eenelusietns. If we dethis, theugh, we must afterward reletter 'F* as ‘H * and ‘H‘ as ‘F‘tiudughnut the results in erder that ‘F’ may eentinue tn represent theminer term and ‘H’ the majer term; alse we must switeh the erder et\"the premises. se that the majer premise may eeatinue te appear first. Theresults, whieh the reader will de well te repreduee. are AEE in the feurthand seen-nd figures and IAI in the luurth and third- Altegetlfter. then, we haye feund fifteen yalid ferms: Ftasr HGURE seeetae Fit:-uasAAA, EAE+ AIL Eli] EAE, AEE, Eli], ADD rattte t-\"tauas Feuart-t Flt] uatsL!t.I, All, DAD, Eli) AEE, IAI, EIEIHate that ne twe uf these fifteen have the same premises, whenditferenees pf figure are talten inte aeeeu nt. We have here fifteen differentpairs ef premises. eaeh with its apprepriate cenclusien. And it is readilyverified hy inspeetien ef diagrams that nnne nf these fifteen pairs rd’premises justifies any fttrther syllegistie eenelusien in additien tn the enehere indicated fer it. ‘flawed in terms merely el’ ee-mhinatiens and witheut regard te theesistenee nf a yalid eenelusien, there are siaty-fettr pessibilities fer thepremises ef a syllegism. They may he AA. er AE, er Al, er AU, er EA,er EE, etc., te sisteea pessibilities, and eaeh ef these siateen may eeeurin any ef feur figures. In additien te the fifteen pairs ef premises whiehhave heen feund te yield valid syllegisms. therefere, there are ferty-ninefurther pairs te eensider. New we saw in eenneetien with Diagrams E and'5‘ hew as eheelt whether a given pair ef premises justifies any syllegistieeenelusien at all. If the reader se tests these t\"e-rty-nine pairs {art he=t.tr‘spastime}, he will find that nene ef them justifies a syllegistie eenelnsien.The fifteen ferms ef syllegism listed aheye are the enly valid enes. In additien, hetweyer, nine fnrrns eeme in fnr henerahle mentien.

id- .\"i'yilr:tgf.rm.t lU'l'These nine are ferms whieh. litre the ah-ave eaarnple uf the Spartans, needa small reinfurcing premise. ‘There are F’ fills the bill fer five ef them,‘There are El’ fer three. and ‘There are H‘ fer ene. Let me simply recerdthe nine in tabular fashien: Fl HST SE-FUND THl RD FEIU HT!-[ A DI]-ED Fl-ISl_|'RE Fl ['5 Ll RE. FIGU RE FIG-U RE FEE H I SEAA], EAU AEU. EA-[J AA], EAIIJ AED There are F EAt\"It There are G AAI There are H lnferences invelving se-called singular statements such as ‘Hectares isa maul. e.g.:All men are mertal. 5-iterates is a man; Sucrates is mertal,were traditiunally fitted intn the syllegistic meld by treating the singularstatements as etf the fe-rm A. This prucedure is artificial but net inebrreet;we can censtrue '5eerates is a man’ as ‘All-G are H ‘ where ‘G’ represents‘things identical with Se-crates’. The abeve inference thus was classifiedas AAA in the first figure. But we shall end up. in Fart I‘-I, with a differenttreatment bf singular inference. ln traditie nal Ingic it was eustumary ta pm-pt:-und varieus rules wherebyta test the validity uf a syllegism. Examples: every valid syllegism has auniversal premise {A nr E]; every valid syllegism has an affirmativepremise {A er I}; every valid syllegism with a particular premise [I er {llhas a particular cunelusi-nn; every valid syllegism with a negative premise[E er {ll has a negative cenclusien. There are further rules whese furrrtu-latien depends ea a cencept ef \"distributinn\" which has been emittedfrem the present esp-esitien. As a practical methed nf appraising syl-legisms., rules are less ennvenient than the methnd bf diagrams. lndeed.the very netiens ef syllegism and m-and and figure need never have beenteuched en in these pages. escept nut nf ennsideratien fur their premi-nenee in legic during twe theusand years; fer we can apply the diagramtest tn a given argument eut nf hand, witheut pausing te censider wherethe argument may fit in the tasenemy ef syllegisms. The diagram test isequally available fnr many arguments which db net fit any uf |]'te arbitrar-ily delimited set ef ferms ltnewn as syllegisms.

ltlh H. General Terms and Qttanttjtiers HIETURICHL HUTE: The terms *categerica|* and ‘syllegism’ de- rive frctm Aristutle. Se du the feurfuld classificatien bf categerical state- ments, and the nemenclature ef the pans cf a syllegism, and the classificatien inte me-ads and figures. except that the feurth figure is‘ attributed rather tu his pupil Theephrastus. The nutiun ef distributinn and the rules that invulve it are medieval- EIERCISES l - Censtruct diagrams velifying the remaining fnur cf the six syltegisms at the beginning ef the chapter- E. Determine hy diagrams what syllegistic cenclusien, if any, fellews frem each ef the fnllewing pairs rtf premises. All whe blaspheme are wiclt-ed; hie saint blasphemes. He snakes fiy; Seme snaltes lay eggs. Hething that lays eggs has feathers; Seme fishes have feathers. Whatever interests me bctres Geurge; Whatever interests Mabel bbres Eiearge. Whatever interests me beres G-enrge; \"Whatever interests Gee-rge be-res lvlabel. 3. Fer each ctf the pairs in Exercise I which failed tn yield a syllegistic cenclusien, determine by diagram whether a supplementary premise ef l.l'1-E ferm ‘There are F‘, er ‘There are G’, c-r ‘There are H‘. weuld suffice tu bring furth a syllegistic cenclusien.

1?. L-!'rr-tit,t' rgf T.lrc.ve ll-s|'ct|lte-ids l[l'El 'l7 LIMITS OF THESE METHODS The inferences te which we have thus far been applying ‘v'enn‘s dia-grams have all been made up ef categnricals A, E, I, ur U plus ane-ccasienal auxiliary ef the type ‘There are F ‘. Actually the diagrams canbe used semewhat mere widely; e.g., in arguing frem the Pltlssrrses: Everyene east ef the traelts is either sievenly er peer, l\"-let eve-ryene east ef the tracl-ts is peerte the Seme slevenly persens are nnt peer. certctustett:We set up a three-circle diagram as usual, wherein ‘F ‘ means ‘slnvenlypersens‘, ‘G’ means ‘persens east ef the traelts‘, and ‘H ‘ means ‘peerpersens‘. Then the first premise is entered in the diagram by shading asempty just that cempartment ef die circle G which lies eutside heth F andH L see Diagram lfi. The cempartment thus shaded is neither a lune ner alens, but a third shape. lt represents persens east ef the traclts whe areneither slnvenly ner peer; and just such persens are denied existence bythe first premise. New the secend premise, which says in effect ‘St:-me Gare net H‘, is rccerded as usual by putting a cress in what remains ef Geutside H. The result is seen te substantiate the cenclusien ‘Seme F arenet H ‘, since there is a cress in F eutside H. An innnvatien due te C. l. Lewis (1913) is the use ef a leng bar insteadef the cress in Venn‘ s diagrams. The adv arttage ef the bar is dtat it can be G‘ ,t' H Elia-germ ID

l It] H- General Tetvnr and tQt-tr-:t.I'tJ't'_,t‘ie't\".'.\"made te lie acress a beundary and thus indicate nenemptiness ef a cem-peund regien. This innevatien is useful in reasening, e.g., frem the easrvttsss: All ef the witnesses whe held steclt in the firm are empleyees,te the eeatctustert: Seme ef the witnesses are empleyees er held steclt. in the firm Seme ef the witnesses are empleyees.We set up a three-circle diagram in which ‘F‘ means ‘witnesses’, ‘G’means ‘steelthelders in the firm‘, and ‘H’ means ‘empleyees’. The lenscemmen teF and G, then, stands fer the witnesses whe held steel: in thefirm; se, en the basis ef the first premise. we shade the pan ef dtat lenswhich lies eutside H. [See Diagram l 1.} Next, ea the basis ef the etherpremise. we run a bar threugh as much ef the unshadedF as lies withinHerG. The mearting ef the bar is fltat ene er anether part ef the tetal regienmttrlted by the bar has semetlting in it. But the bar lies whnlly wifltin Fand H; se the cenclusien is sustained. G FH Di-Igr-amt! The utility and versatility c-f ‘iFenn‘s diagrams are particularly evidentfrem these last twe examples. A shertceming ef the diagrams, hewever,is that they lend themselves less readily te arguments invelving feur ermere temts. A diagram ef everlapping ellipses can be censtructed ferfeur-term arguments, but it calls fer careful drawing; and there is ne wayef censtructing a diagram which will exhibit five er mere reasenablysimple regiens in all cembinatiens ef everlapping. Where many terms areinvelved we may, hewever, u-y te breal: the argument dewn inte pansinvelving manageably few terms. The fellewing example is frem LewisCan-ell:

fl’, f.irrtr't.r hf Tltese ll-i'eti'ted:-\" lllPrtestrstts: ll} The enly animals in this heuse are cats;eet~tct.usretv: {Z} Every animal is suitable fer a pet, dtat leves te gaae at the meen; {3} ‘When l detest an animal, l aveid it: {4} He animals are carnivereus, unless they ptewl at night: [5] hie cat fails te ltill mice; lfil He animals ever talte te me. except what are in this heuse; [Tl F-langarees are net suitable fer pets; {til} l\"-lune but carnivera ltill mice; [Ell I detest animals that de net talte te me; if til} Animals that ptewl at night always ievc re gate at the me-en. l always aveid a ltangaree.This argument can be brelten dewn as fellews. Frem {1} and {5} we getthe lernrrtn e1‘ intermediate cenclusien:if l l‘,t All animals in this heuse ltill mice;this is the sert ef step te which a simple three-temt diagram is adequate.Frem {El and ll I], similarly. we get:[ill All animals in this heuse are carnivera.Frem id} and {ll} we get:till-It All animals in this heuse prewl at night.Frem [15] an-d {I3} we get:t[l=‘-ll All animals that talte te me prewl at night.Step by step in this fashien we can preceed re eur desired cenclusien.never using mere than a three-term diagram fer any ene step. {lf thereader cares tn carry this th reugh in detail, he sheuld thinlt ef the universeas limited fer purpeses ef the argument tn animals—thus never betiteringwith ‘animals‘ itself as a terrri.)

l ll H- General Terri-t.r and Qnentffierr lie we see that the purely mechanical medied uf diagrams pr-evesinadequate when an argument turns en a ferge ttattttber cf rerr.rr.s,' asupplementary technique has te be invelted, such as dtat ef reselving theargument inte parts. New anether place where the unaided methnd efdiagrams begs dewn is where there is an trrirntrrure sf truth functiens, asin the fellewing example?Pttlssttsesz if all applicants whe received the secend anneunce-cet-ter.ustet~t: ment are ef the class ef ‘llii, then seme applicants did net receive the secend anneuncement- Either all applicants received the secend anneunce- mertt er all applicants are ef the class cif ‘fill. If all applicants ef the class ef ‘tlitt received the secend anneuncement then seme applicants net ef the class ef Till‘ received the secend anneuncement.If we assign ‘F‘, Ti\", and ‘H ‘ in ebvieus fashien, the inference taltes enthe ferm: AllF whe areG areH —-s semeF are netti. AllFareGvallFareH, AllF whe areH are-G —1- semeF whe are netH aret}.Diagrams are suited te handling the cempenents ‘AllF whe are-G areH ‘,‘Sernel-\" are net E-\", etc., and the metheds ef Part 1 are suited tn *-1-‘ and‘v‘; but just hew may we splice the twe techniques in erder te handle acembined inference ef the abeve l-rind? Se it is time te address eurselves te a mere cemprehensive theery. Thethree ferrnulas last set ferth, centaining ‘F ', ‘G’, and ‘H ‘, are schemataef a sen, but differ frem the schemata ef Part I in centaining ‘.F ‘, ‘G ‘ , etc.and such werds as ‘a|l‘, ‘seme’, ‘whe are‘, etc., te the exclusien nf ‘p’,‘r,-t‘, etc. ln the ensuing chapters such schemata will be reduced tu alegical netatien subject re a “decisie-n ptecedure“-—i.e., a mechanicalreutine fer deciding validity, implicatien, censistency, etc. Such a deci-sien ptecedure exists fer truth-functiens] schemata in truth-valueanalysis; fer the new class ef schemata, hewever, the ptecedure will havete be mere elaberate. Once it is at hand, all inferences ef the sert we havebeen censidering in the present Part-—including the stubbern lastexample—-—can be adjudged mechanically by an implicatien test en prem-ises and cenclusien.

f I5‘, .i.irnit.'t qt“ T'lte..re ll-.'t‘r=‘Iftt;It1'.'F l l3EIERCIEEEI- Checlt the seundness ef each ef these inferences by diagram: All ef the witnesses whe held steelt. in the firm are empleyees. All ef the witnesses are empleyees er held ste-clt in the firm; All ef the witnesses at'e empleyees. Ftverynne whe ltnews beth Geerge and lvlabel admires Mabel- Eeme whe ltnew Mabel de net admire her. Eieme whe ltnew Mabel de net ltnew Eieerge. Ne Eurepean swans are blaclt. All blaclt swans are Eumpean. Ne swims are black- Clnly high-scheel graduates whe can read French are eligible- Seme can read French whe are net high-scheel graduates- 5-nme whe are net eligible can read French. Everything is either a substance er an anribute. lvledes are net substances. Mndes are attributes Hint: Be prepared te shade the limitless regien eutside all circles.2. Finish the ltangaren argument and supply a dtree-temt diagram te justify each step-fl- Using ‘F ', ‘G’, and ‘H ' in the ebvieus way, malte a Venn diagram fer each ef the fellewitig twe statements [Lillian]: A thing is flexible if and enly if it is either granulated er heavy- lf nething flexible is heavy then net everything granulated is heavy.

ll-=1 H . Ger-terni Terms arid [Euanrtjiers 13 BOO LEAN SCH EHIATA Te say efart ebject-tr dtat it is an F, we write ‘Fr’. Here then is a newsert ef sentence schema: ‘Fr’, ‘tlr’, etc. These may be cempeunded bytruth functiens; e.g.[ll F.r.—G.r.v.G.r-sH.r.We may cenveniently abbreviate such cempeunds by extracting the ‘.r’everywhere and putting it at the end, thus: [F-[T v‘t[tIF ——rH}]-Ir.We arrive in this way at schematic representatiens ef certain cemplexterms: schemata such as ‘fl’, ‘Fill’, ‘G -+H’, ‘FG v [G —sH]’, ete-They will be called Heeleen term scltetra:rrct- liithere ‘F’ is interpreted as ‘blaclt’ and ‘ti’ as ‘swan’. ‘G’ is‘nenswan’; ‘Ft?’ is ‘blaclt swan’; ‘FG’ is true ef just the ether blacltthings; and ‘F v G‘ is true ef swans and blaclt things generally. ‘F —i- til‘,‘if blacl: then a swan’, is true el’ everything that is a swan if blacl-t—henceall swans and all that is net blaclt; it is equivalent te ‘F v -[F ’, er ‘— {FG‘]‘. when ‘and’ and ‘er’ are applied tu terms, there is a curieus escillatienre allew fer. The things that are swans er blaelt (F v G] are the swans andthe blaclt things. The things that are swans and hiaclt {FIG} are net theswans and the blaclt things, but just the blaclt swans. The cencepts ef validity, censistency, and implicatien carry evernaturally te Heelean tertn schemata frem the eld tnttb-functienalschemata. A term schema is called valid if, when the ‘.r’ is restered, theresulting sentence schema is truth-functienally valid in the eld sense.l:i.g., since ‘F1 v —F.r’ er ‘{F vF_].r’ is valid, we call ‘F vi?“ itselfvalid. Cerrespendingly fer censistency and implicatien; thus ‘FF’ is ea]-led incensistent and ‘Fifi’ is said te imply ‘F v G’. The significance ef such validity en the part ef a term schema is dtatevery interpretatien ef its letters tums it inte a term that is tme ef everyebject .r. But interpretatien in what sense‘? ln tall-zing ef interpretatiens ef

IE. Hr:-tztlettn i'i'c.lterrtcrttt I l5term letters, we de best te allew free-dem in cheesing the universe efttisceurse-—the range ef ebjects: relevant le the legical argument we areplanning tn carry threugh. such freedem will cemmenly diminish by enethe required number ef terms. as we saw a few pages baclt in the ltangareeexample. We. interpret a term letter ‘F’. then, by senling which enes efthese ebjects ‘F’ is te be eenstmed as true ef. A valid term schema, then.is ene that will ceme eut true ctf all ebjects e-f any chesen universe underell‘ interpretatiens, within dtat universe, ef its term letters. A censistent term schema is true ef settle ebjects ef seme universesunder st:-the interpretatiens ef its term letters. Elne term schema impliesanether when, in every universe ef disceurse, cvet'y interpretatien maltesthe secend schema ceme eut true ef everything that the first cemes euttrue ef. Since fleelean term schemata have just the structure ef the eld truth-functienal schemata, we can test them fer validity, censistency, andimplicatien by truth-value analysis as if the ‘F‘, ‘G’, etc. were ‘p’, ‘q’,etc. We can even use fell sweep where apprepriate. lf it seems ndd tesubstimte ‘T’ and ‘J.’ thus fer term leners, then thinlt ef these tests asapplying in reality net tci the term schemata themselves but te the sentenceschemata that weuld be get frem the term schemata by restering ‘I’ asin t,l'i. We shall malte gnnd use ef the fnllewing law- [il lf a B-eelean tertn schema is censistent, then in any universecentaining a given ebject there is an interpretatien ef term letters thatmaltes the schema ceme eut true ef that ebject-Fe-r, being censistent, the schema resulvcs te ‘T’ under seme substitu-tien ef ‘T’ and ‘J.’ fer the term letters. We have then merely te interpreteach term letter as true er false nf the given ebject aceerding as itssubstitute was ‘T’ er ‘.l.’. Since censistency is invalidity ef the negatien, til can alse be put thus: {ii} If a Eeelean term schema is net valid, then in any universe centaining a given ebject there is an interpretatien ef term letters that makes the schema ceme eut false ef that ebject. Sincc failure ef implicatien means censistency ef the ene schema withthe negatien ef the ether, [ij has alse this further cerellary:

llfi H. General Tenns and fluaruifiers {iii} lf ene Eeelean tertn schema fails te imply anether, then in any universe centaining a given ebject there is an interpretatien ef term letters dtat makes the enc schema true ef that ebject and the ether false ef it. We preceed new te a genetaliaatien ef ii]: {iv} If.-11, . . . , tflik are Heelean term schemata and each ef them separately is censistent, then in arty universe centaining distinct ebjects .r1, - . - . rt, there is an interpretatien ef term letters dtat simultaneeusly maltes all true efxl, and .-*1; true 'li}f.I5, and se en.The reasening is as fellews. As in til, we interpret the term letters as trueer false ef .r, in such a way as te malte A 1 ceme eut tt1.te ef..t 1. Hut thisleaves us free still te interpret dtese same term letters as u't.te er false, aswe please. ef rs; and we tie se in such a way as te malte.-la ceme eut trueefrg. We centinue thus- Example: Talte .-11, A3, and sly as ‘F-ET’, ‘SH’, and ‘FH’, and talte 1:1,If, and .1\"; as Tem, Diclt, and Harry. Substitutien ef ‘T’ fer ‘F ‘ and ‘.l.’fer ‘G’ maltes ‘Fff’ reselve te ‘T’: se interpret ‘F ‘ and ‘H’ as true ef Tem,and ‘G’ net. Similarly, in view ef ‘EH’, interpret ‘F’ and ‘G’ as true efDiclt, and ‘H’ net. Similarly, in view ef ‘FH’, interpret ‘G’ and ‘H’ asnne ef Harry, and ‘F’ net. Sc ‘F ‘ is interpreted in the universe ef Tern,Diclt, and Harry as true ef Tem and Eliclt; ‘G’ is interpreted as true efDiclt and Harry; and ‘H’ is interpreted as u'ue ef Tem and Harry. Having recerded fer future reference these feur laws ef Beelean termschemata, we tum te E-eelean schemata ef a furtlter ltind. Let us write ‘ 3 ’fer ‘thcrc are’, in the sense ‘there is at least ene’. It is a prefix which,applied te a general temt. preduces a statement; thus ‘There are blacltswans’. schematically ‘ HF-G’. A schema censisting, lilte ‘HFG’, ef ‘ 3’fellewed by a E-eelean tenn schema, will be called a Haaleatt existencercltctua. The schematism ef the example abeut the class ef ’[l[l, at the end ef thepreceding chapter, gees ever new intn this:tn — area —1- see. — are v — Eire. - at-we -i area.Schemata such as these three, which are truth functiens ef Beelean exis-tettce schemata, will be called Baalean rtrtrerrrerrr .rc.lrerrtat‘a- Such schemata are ceunted valid when they ceme eut true under allinterpretatiens in all rtatrernpry universes- it is cenvenient new te malte

id. ftaelean Schemnta I ITthis exceptien ef the empty universe because there are ameng the Beeleanstatement schemata. unlilte the term schemata, certain enes such as‘HF v HF’ which fail fer the empty universe but are valid and usefuletherwise- Usually the universe relative te which an argument is beingcarried eut is already ltnewn er cenfidently believed net te be empty, sethat the failure ef a schema in the sele case ef the empty universe isusually nething against it frem a practical peint ef view. Usually, indeed, the universe wanted in arguments wnrthy ef Heelefijlschemata is ltnewn er believed tn have net merely seme members, butmany. tn the definitien ef validity, then, instead ef saying ‘everynenempty cheiee ef universe’ why net say ‘every eheice ef universe efmere than eleven members\"? The reasen fer net deing se is that na newschemata weuld be added te the categery ef valid schemata by sueh aliberaliaatien ef the definitien. If a schema can be falsified by semeinterpretatien ef ‘F’. ‘G’, etc. in seme small but net empty universe, itcan he falsified alse in bigger universes witheut end. The reasening is as fellews. We are suppesing a small but net emptyuniverse; let-I be an ebject in it. We are suppesing, further, an interpreta-tien ef ‘F’, ‘G’, etc- in that universe that falsifies a given schema. Add,then, as many new ebjects te the universe as yeu lilte. and interpret ‘F’.‘G’, etc. in the enlarged universe by letting all the new ebjects ge alengwith -r. That is. talie ‘F ‘ as true ef all er nene ef the new ebjects aceerdingas ‘F‘ was interpreted as true er false ef 1:; and similarly fer ‘G’ andfunher leners. Then the new ebjects will be indistinguishable frem _lt' sefar as the interpretatien ef eur schema is cencemed; the schema will befalsified new as befere- Happily, thus, the enly eaceptinn dtat there is any eccasien te malte indefining validity is that ef the empty universe. But we must net under-estimate this exceptien. Occasienally an argument may mest cenve-niently be carried threugh under a eheice ef universe whese nenempti nessis epen te questien. Still there is ne need te cever the empty universe ineur general tbeery ef validity, because the questien whether a schemahelds fer the empty universe is easily handled as a separate questien. Tedecide whether a schema cemes eut true fer the empty universe we havemerely te put ‘.l.’ fer the existential clauses and reselve. lust as a El-eelean statement schema ceunts as valid when uue under allinterpretatiens in all nenempty universes, se it ceunts as censistent whentrue under seme interpretatien in seme nenempty universe. Dne schemaimplies anether if, in every nenempty universe. every interpretatien thatmalres die ene true maltes the ether true. As befere. thus, incnnsistency isvalidity ef the negatien and implicatien is validity ef the cenditienal.

I IS H. tlcnerai Terms arm‘ ggaaadfiersEquivalence is; as befere, mutual implicatien; alse validity ef the bicen-diticnai- We preceed with a few laws abeut Beelean existence schemata. lvl A Heelean existence schema is valid if and enly if its term schema is valid.Fer, if the term schema is valid, then every interpretatien rnaltes it trueef everything, in any chesen universe, and therefere true ef semething inany nenempty universe. lfcn the ether hand the term schema is net valid;flten by [ii] there is art interpretatien that maltes the term schema ceme eutfalse ef the sele ebject in seme universe, and die existence schema underthat interpretatien cemes eut false- tjvijt A Be-elean existence schema is censistent if and eniy if its tcmt schema is censistent-Fer, if the tertn schema is censistent. then by [ii there is an interpreta-tien that maltes the term schema ceme eut true ef semething, and thusmaltes the existence schema cnme eut u-ue. If en the ether hand the termschema is incensistent, every interpretatien maltes it false ef all ebjectsand fltus falsifies the existence schema- rvii] Cine Heelean existence schema implies anedter if and enly if the ene term schema implies the ether.Fer, suppese the ene term schema implies the ether. Then any interpreta-tien that maltes the ene tenn schema tnre ef semething will malte theether true ef the same thing. Then any interpretatien dtat maltes the eneexistence schema true will malte the ether true; se the ene existenceschema implies the ether. l'~lext suppese dtat the ene term schema decs netimply the ether. Then. by (iii), fltere is an interpretatien dtat maites theene temr schema ceme eut true cf the sele member cf scme universe andmakes the ether ceme eut false ef it. This interpretatien maltes the eneeitistence schema ceme eut true and the ether false. Se the ene existencescherua dues net imply the ether. Anetber remarlrable thing ab-cut Be-clean existence schemata is thatthey cannet cenflict with ene anether. The cenjunctien ‘ HFG .EIGH . HFH‘ is censistent despite the ccnflicts amnng ‘FG ‘, ‘DH’, and‘Fri’. We have this:

IE. Beeieun Sriteiuate I 1'5‘ {viii} A ccnjuncticn cf E-cc-iean existence schemata is censistent scleng as each cf them separately is censistent-Fcr, whcn each cf the existence schemata is selficcnsistent, the termschema in each is self-censistent hv [vi]. In view cf [iv], then, there is insrrrrie universe an interpretatien tl1at simultaneeusly rrialtes each {if thesetern1 schemata ceme eut true cf ene ur annther chj-ect, and this is aninterpretatien that malres the existence schemata ali true tegether. Important use will he made in the next chapter cf this scmewhat merecemplex lavv: [ix] A Bceleart existence schema is implied hv a cnnjunctinn cfBrieleasi existence schemata enly if it is implied by ene ttf thern inisciaticn.Fer, imagine E-erilean term schemata in place cf ‘A’, ‘xi 1*, . . . , 51,,\"and suppcse that nu cue cf the existence schemata *3.-=l1*, ‘El.-t,_\", etc.separately implies ‘El.-t‘; then we want tti shew that the cenjunctien ‘HA.. Hat; ..... 3:1,,’ dues nut imply ‘HA’ either. Ev [vii], ‘xi,’ ric-es netimply ‘i-1‘; sc ‘A115’ is censistent. Sc is ‘i-1,.-ii, sc is ‘A3.-i*, etc. Sc, by{iv}, there is in a universe eftt cl:-jcctsx,,- . . ,x,, an interpretatien cf termletters that sirnultancuuslv makes '21,.-ii true -sf I1, and ‘.112;-i* hue cf xi,and sc rm. U11 this interpretatien ‘A ‘ is true cf nnne cf the c-hjectsx,,. . .,x,, in the universe, sc ' HA |’, ‘3.-4;\", etc. all cnme cut trtte and \" H.-t’ false.Therefere ‘HA, . HA2 .. . . . 3.-t,,* dces nut imply ‘EA’. We turn finaliv tc an ebvieus hut impertant law tc the efl‘ect that ‘H’ isdistrihutive nver alternatien. This is valid:[3] 3{F 'v'G] H. HF v HG. Clearly, scrne persens are either saints cr geniuses if and cnlv if snrneare saints cr st:-me are geniuses. lt is nct exclude-d, cf ceurse, that there heheth saints and geniuses, ner is it excluded that seme cf the saints hegeniuses. The chviuttsness cf {3} rnust nut he allewed tu encourage the theughtthat *3‘ is distributive alsn river cenjunctien. {vii} assures us rightlyett-rnugh that ‘EFF impties ‘HF . HG’, hut the ccnverse faiis. There arercund things and square, after all, but ner reund squares. if this waming is unneeded, perhaps the next is rncrc tc the purpese:the cemhinatien ‘— H‘ is nut distributive even river alternatien. Thus talre

I21] H. General Tenns and fleeettjiers‘F’ as ‘hers-el and ‘G’ as ‘unicern’- Then ‘HG’ is false and therefere‘— HF v — HIT is true; en the ether hand '— 3=[F VG)’ is false, since‘F v G’ is true ttf the herses. Certainly {3} allews us te de seme distributinn in ‘— 3{F '-.-‘ 5}’. Itgive!-I -3-[F 'li|'G}-t—t+—[3F'v' 3G] 1-1. -3.7: . -'-HG. HISTDIICAI. HDTE: The idea ct changeahle universes gees baclrtc Delvtergan if 13415], as riees the plirase ‘universe ef disccurse’. Exclu-siert et\" the empty universe was implicit already in Frege's lugie ef 1319.The argument that was set fcrth abeve, fer excluding the empry universe.is in Hilbert and ittcltermann (i933, p. 92} and tie deubt in earlier werhs.The sign * H‘ and its distributinn law [3] are frem Peane, 1395.EIERCISES1. Schematiae, in the style ef {Z}, the five examples in Exercise l ef Chapter 1?.1. In view ef {ix} and {vii}, yeu can decide whether the schema: are e-es . EIFG . arise; implies EIFG‘. Hew? De se.

I 5|’- Tests cif lfrrlfrlitjv ill 19 TESTS OF VALIDITY We are new in a pesitien te test any Heelean statement schema fervalidity. l shall state and justify the tesfi case by case, aceerding te theferm ef the schema. In all cases the tests reduce tu truth-value analyses efEl-etileatt term schemata. ti) at Beelean existence schema is valid if and enly if its term schema is valid.This was {vft cf the preceding chapter. {iii The negatien efa Eieclean existence schema is valid if and enly if the term schema is incensistent.Fer, the negatien is valid if and enly if the existence schema itself isincensistent, and se, hy [vi] cf the preceding chapter, if and uuly if theterm schema is incensistent. {iii} Fin alternatien cf negatiens ef Beeleart existence schemata is valid if and eniy if ene cf these negatiens meets the abeve validity test.Fer, the altematien rif negatiens ameunts te the negatien ef the cenjunc-tien ef the existence schemata; and this is valid just in case the ceri-junctien is incensistent. Hut, by {viii} cf the preceding chapter, thiscenjunctien is incensistent just in case seme ene ef the existenceschemata is incensistent, and hence just in case seme ene cf the negatiensef existence schemata is valid. The next test cericerns the e.t:t'stentt'ni' cenditienal, by which I mean acenditienal whese antecedent is a Beelean existence schema er a cen-junctien cf sueh and whese censequent is a Eteeiean existence schema. [iv] .-\"tn existential cenditienal is valid if and enly if the Beelean term schema in ene ef the existence schemata in the antecedent impiies the Beelean term schema in the ccnsequent.

IE1 H. General Terms and fi'tttItt.tt§,l'it='t‘sFer, the existential cenditienal is valid if and eniy if the antecedentimplies the censequent. Hy [ix} ef the preceding chap-ter, this will happenif and enly if ene ef the existence schemata in the antecedent implies thecenseqnent. Hy [vii] ef the preceding chapter, this will happen if and enlyif die ene term schema implies the ether- The next ene gees witheut saying. tv} it cenjunctien cf Eeelean statement schemata ef any ef the ferms cevered in [ii—{iv]t is valid if and enly if each cf them cemes eur valid under the tests ti]-(iv). Fer ccmpletcness cf uur test ef validity cf Hcclean statement sch ematait new remains enly te shew that every Eli-ettlettn statement schema can begrit ever inte ene ef the five ferms cevered in t_'i}—t_'v], by pcrfermingequivalence transfermatiens upen its parts. This receurse te interchange-ahle parts depends upen a law ef interchange similar tc what we saw inChapter it. We saw there that any cempenent ef a truth-functien al schemacan be supplanted by any equivalent witheut disturhing the validity ercensistency ef the whele and witheut atfecting its relatiens ef implicatienand equivalence tu ether schemata. Such interchange werlts equally ferBeelean statement schemata, and fer much the same reasen. It werlted ferpure u'uth functiens because the truth value cf the cempeund statementdepends en ne features ef the cempenent statements except their truthvalues. Find it wtttlts new fur truth functiens and * 3' because the truthvalue ef die cempeund statement depends en ne features cf the cempe-nent statements and terms except their being true er false er true efsemething er ef nething. Tc transferrn any given Heelean statement schema S inte ene cf thelive ferms, we begin hy transfermingS inte cenjunctienal nermal furm asef Chapter 12. Fer this purpese we treat each existence schema in S as if itwere a single statement ietter ‘p*, Hy‘ , etc. This hrings S inte the ferm ct acenjunctien ct\" altematiens. Each ef these aiternariens is an altematien efexistence schemata er cf negatiens uf existence schemata er beth. Hew-ever, there is never any need tn rest with an alternatien ef existenceschemata, thanlts te the distrihutivity law, [3] ef the preceding chapter.Transfermed hy this law, each ef eur altematiens will he an altematien ef{ene er mere} negatiens ef existence schemata and at mest ene existenceschema. if it is HI] alternatien ef ene such negatien. it falls under [ii]; if efseveral, {iii}- If it is an alternatien ef ne negatiens and ene aflirrnativeexistence schema, it falls under lfi]. if finally it is an alternatien ef ene er

ll}- Tests cf ttnlidity 113mere negatiens ef existence schemata and ene existence schema, itameunts te an existential cenditienal and falls under {iv}; fer, ‘it, v pg v.--tt,|E|,, l||ll|!_|l'1 is equivalent te ‘plpvy. . .p,, —t q\"- Fer a simple illustralien let us apply the test te the third syllegism inChapter lb- Rendered as liieeleau statement schemata, the premises efthis syllegism are ‘— 355\" and ‘ HFET and the cenclusien is '3FH’. Sethe inclusive Beclean statement schema te test fer validity is: — 3-IJH . HF-5 .—-r HFH.Tewatd cenverting this te cenjunctienal nermal ferm, we translate thecenditienal inte alternatien and negatien. —{— HG!-T . HFG} v HFH. |.Cenverslen inte cenjunctienal nermal ferm taltes ene mere step: Elt3Hv — Eli-\"ti v HFH.Then we fuse the twe aflirmativc existence schemata. HFG it HUSH vFH}.Rendered as an existential cenditienal, this becemes: EIFG -i- H|[Gll' vFH}.By l.l't-E criterien {iv}, finally. this is valid; fer a fell sweep shews that ‘Ft?implies ‘CH ‘v‘ FH'. Let us apply the test te the argument regarding the class ef ‘fill. It get asfar as the fermulatien {1} midway in the preceding chapter. Te cstahlishthis implicatien we want te shew the validity ef the cenditienal: - EIFGH —=- are . - are v - st.-sis .-s. - sane -s sane..-appreciable laber can cemmenly he saved if, te begin with, we permutethe letters ef each term inte alphabetical erder. This enhances the match-ing ef terms and, therewith, the chances ef simpliiicatien. We get this: ~ ElFcH -s HFG . -- 3.-as v — EIFH .-.-. - EIFGH -s E|Ft3-‘H.

I14 H . General Terms and QnuntrfiersThe stages ef cenversien inte cenjunctienal nermal ferm are left te thereader. If patently valid clauses ef the type ‘ti ‘it q ‘v‘ pl are deleted as theyarise, this is what the reader is apt te end up with: — 3Ft’.i v EIFH v ElFt'iH v EIFGH.R-\"lien we fuse the three affirmative existence schemata, this becemes: — EtFr\";' v state veers veem.Rendered as an existential cenditienal, this becemes: HFG -1» H{Fi‘? vFGH vFGl?}-Hy [l't'E criterien {iv}, finally, this is valid; fer a fell sweep shews that ‘FE’implies ‘FH it FC-‘H it FCFF-l\". New ene mere example: — HFGH. HF—1- HG’ .v HFC-Paraphrasing the cenditienal inte alternatien and negatien, we get: - area . - as v Etc .v sac,which distributes te cenjunctienal nermal ferm thus: — 3Ft?Hv HFG . — HF v HG v EFF.Rendering this as a cenjunctien ct existential cenditienals, we have: seen —v Else . HF -v are vac;-.Here {v} is applicable; we have te see that beth cenditienals are valid, ifthe whele is te qualify as valid. Beth cenditienals tie, by {iv}, preve valid;fer we find by fell sweep that ‘FC-‘H* implies ‘FIT and ‘F’ implies‘ts v PG’. A decisien ptecedure fer validity is cf ceurse a decisien ptecedure atthe same time fer censistency, implicatien, and equivalence, since incen-sistency, implicatien, and equivalence are validity ef the negatien, thecenditienal, and the bicenditienal.

llil. Tests cf l=’aliifity l15 The test cf validity which is new befere us always preceeds, we mightsay, by transferming a Be-elean statement schema inte a cenjunctien efgenuine er defective existential cenditienals; fer the schemata cevered itt{i}, {ii}, and {iii} ameunt te defective existential cenditienals, lttcltingantecedent er censequent. tic the methed may be called the rnetlicsl e_,fexistential centlltierntls.‘ A different validity test fer Beeleau statement schemata, due te Her- brand, I shall call the cellular methed. In practice it tends te he mere laberieus than that ef existential cenditienals, but it is werth eeting be- cause it is se simply fermulated. It eperates with cellular existence schemata- lit cellular existence schema in rt given letters ‘F’, ‘G’, etc. has as its term schema just a cenjunctien ef n literals in alphabetical erder. each being ene ef these letters er its negatien. Thus the cellular existence schemata in ‘F‘ are ‘HF’ and ‘HF’; these in ‘F’ and ‘G’ are ‘HFtT, ‘El’-‘Ell, ‘EIF-5*, and 'HF't§'; these in ‘F‘, “G”, and ‘H’ are *H.Ft‘l}'H’, ‘HFGH ‘ , and six mere; and se en. Each cerrespends thus te an uncut cell ef the Tenn diagram and says there is semething in it. ltlete next that any Heelean existence schema. say in rt leners, can be transfermed inte an alternatien cf cellular existence schemata in tltese rt letters. The methed is as fellews. Crivcn an existence schema, put its term inte develep-ed alternatienal nermal ferm {see Chapter lfl} and flten dis- tribute ‘H' threugh the alternatien. Fer instance, given '3'[F{tI'.'r' 'v'H}]', we first put its term inte altematinnal nermal feitn. getting ‘H{F't_T v F'H}’. and then we develep it, getting: El{Ft3H vF-GE‘ vF\"tI-‘H vFt?H},er, drepping the duplieatien, secaveeevaeatFinally, distributing ‘ 3'. we have: EIFUH v EIFGH v E|Ff.iH. it fellews that any E-eelean statement schema S , say in rt letters, can berendered as a truth functien ef cellular existence schemata in tltese rtleners. First, by a trivial meve, each existence schema appearing inS canbe made te centain all n letters. Fer, if an existence schema lacits ‘H ’, we

I215 ll. General Terms earl Qanntllierscan just append ‘H v H\" re its term schema by cenjunctien. Thereupen,by the metlted ef the preceding paragraph. we preceed te render eachexistence schema as an altematien ef cellular existence schemata in the rtletters. New the validity test that was geing te be se simply ferrnulable is justthis: {vi} A truth functien ef cellular existence schemata in tt given letters is a valid Fieelcan statement schema if and enly if it reselves te ‘T‘ under all assignments ef ‘T’ and '.L‘ te these cellular schemata {disre- garding the assignment ef ‘l.* re all E\" cellular schemata}. I shall net pause fer a full preef ef {vi}, fer it can quicltly be madeplausible by the fellewing reflectien upen ‘tlenn diagrams- when weundertake te interpret a E-eelean statement schema, the enly infemtatienthat is geing te matter te its truth er falsity is infettnatien as te which cellscf the diagram ate eccupied and which are empty. lvtereever, interpreta-tien ef any ene cell as eccupied er empty is independent ef interpretatienef ether cells as eccupied er empty. Se the enly thing that can malte fervalidity ef a Heelean statement schema, enee the schema is reselved intethe cellular existence schemata, is the extemal truth-fnnctienal structureby which it is built ef these cellular existence schemata. {Exceptien: sincewe chese te define validity widteut respensibility fer the empty universe,the parenthetical exemptien is appended te {vi}.} Example: HFG ‘v‘ HFH -—1- 3[F[G vH’}].First we supply missing letters te all its existence schemata. 5[FG{H vl-ll] it §[.FHtfG' 'v‘|§}] .-t- H[F{t.'i' v‘H'].Then we expand the terms inte nermal ferm, helding te alphabeticalerder. El{Ft.\"lH vFG.'?} v El{FtFH vF‘t']l-l} .—-t- HIIFG vFl-l}.Then we develep the terms, helding te alphabetical erder and drepping aduplicate.

lil. Terra r.!,t\" Faltdlty illEltl-‘mil’ v Ft?Fl'} v streets v Fem .—t- arrest veers v Fete.Then we distribute ‘ 3 ‘-HFGH v HFGH v HFGH it HFCH .—-. EFGH it HFGFT v HF-CH.The truth-functienal validity is visible. The stipuiatien ef alphabetical erder in the definitien ef cellular exis-tence schema is ne mere laber saver new; it is essential te the seundnessef {vi}. if in this example we had written '3F't_?H' at seme peints and‘EFHEF ’ at ethers, the trutlt-functienal validity ef the eutward structureweuld have been lest; the parts ‘EFGH’ and ‘ElFHt]‘ weuld have faredlitre independent sentence letters ‘p’ and Hy‘- l\"|l5T'l:l'RlC..fil. HDTE: The first decisien precedure fer this much ef legic was l.iiwcnheim*s { l'Ell5}, but the first rcasenably manageable ene was Eehmann‘s {I912}. The methed cf existential cenditienals is seme- what reminiscent ef Eehmann‘s. The cellular methed stems rather frem Herbrantl { l lilfifi}..|.j—|—\.—iI—-I-I- EIERCISES l. Test each ef the fellewing schemata fer validity by the methed ef existential cenditienals. EIJF -v 3tFe vle v en}, HG! . ses .—t- El{,Ft]H vF.t vF.-‘xii. 1. Schematiae the fellewing statement and test fer censistency {by testing the negatien fer validity} aceerding te the methed ef existen- tial cenditienals. Seme whc talte legic and Latin take neither physics ner Ctreelt, but all whe taltc either Latin er chemistry talre beth legic and tfireelr- 3. Five inferences were sehematieed in Exercise l ef Chapter 13. Checlr

I15 ll‘. General Terms and fluantifiers them all, by testing the apprepriate cenditienals fer validity accerd- ittg te the methed ef existential cenditienals- 4. Checit this inference by the same methed- hlebedy attended but parents cf the cast- If all parents ef the cast were bered, then he parents ef the cast have a taste fer high-class entertainment. if any attended whe have a taste fer high-class entertainment, then seme parents ef the cast were net hm-ed.| 5. Repeat all feur exercises using the cellular methed.20SOME BOOLEANINCIDENTALS Eeelean term schemata are trt.|dt fiinctiens ef term lerters—fltus ‘F‘,‘Ft.\"l‘, ‘F vtfi‘, ‘F —t-ti\", ‘F -t-1-I5‘, and upward by iteratien. Eeeleanexistence schemata are term schemata with ‘H‘ prefixed, and Eeeleanstatement schemata are these and the trudt functiens ef them. This meagerinventery is already redundant, as we ltnew; ‘—l§Ft‘§'}‘ weuld serve fer‘F vti\", and ‘—{FG}‘ fer ‘F —1- G’, and ‘—{F|‘:} —[FG}‘ fer ‘F H El‘.Cenversely, cenveniencc can be served by further redundancy. Alengwith ‘ HF \" fer ‘there are F ' we may cenveniently adept “lt\"F ‘ fer ‘all is F ‘,‘dtere are nething but F ‘. Cenvenient funher netatiens are that cf lacta-sien, ‘F I; G‘. ‘all F are G’; that efccextenslveness, ‘F E G‘, ‘all andenlyF areG'; and that efpreper lttclttrtcn, ‘F C CF‘, ‘all F are G but netvice vcrsa‘. an these are easily reducible te the previeus netatiens, dtus: vre--at-‘. F;e.<-sve\"-its}F ee .t-1-vi.-'-\" -=-set ~1-t—3{FG}. v-s.F;e.egF -e-. — stare} . - street. t~‘t:e.v-v.Fge.—{tst;F} H. — stats} . Eltlas].

Ell- Seine -llaelenrt lrtr'lrdcrttrtl.s [Ill in {S} ef Chapter ill we neted that ‘ 3‘ is distributive threugh alterna-tien. Similarly “lt\"‘ is distributive threugh cenjunctien:{I} ‘il‘Ft_? -t-t-.\"t\"F.‘lt\"fi'.If the validity ef { l} were less ebvieus, ene might checlt it by translating itinte the previeus netatien, as just explained, and then applying themethed ef existential cenditienals. Similarly fer the validity ef this cen-ditienal:{1} ‘l'l‘F -r HF.Universality, ubiquity, implies existence. lt is natural and cenvenient tc recerd equivalences and implicatiensbetween statement schemata by setting dewn valid bicenditienals andcenditienals as abeve. These dc net themselves afiinn equivalence erimplicatien, but their validity censtitutes equivalence and implicatien.l\"~lew the signs ‘E’ and ‘t_;' ef ceextensiveness and inclusien pcrferm thatsame service fer term schemata. That ‘FG‘ is equivalent te ‘-— {F v Cl‘,and implies ‘F vtl‘ is reflected in the validity ef the cerrespendingstatement schemata nf ceextensiveness and inclusien: Fe e -tFv-sh. Fe t_._:Fvc. The werd fitncter, grammatical in impert but legical in habitat, be-cemes useful at this peint. A functer is a sign that attaches te ene er mereexpressiens ef given grammatical ltind er ltinds te preduce an expressienef a given grammatical itind. The negatien sign is a functer that attacheste a statement te predu ce a statement and tn a term te preduce a term. Thealternatien sign, cenditienal sign, and bicenditienal sign are functers thatjein statements in pairs te preduce statements and jein tenns in pairs tepreduce term s. ‘ 3‘ is a fu ncter that attaches te a term te ferm a statement;se is “tt\"‘. Finally ‘Q, ‘C’, and ‘E’ are functers that jein terms te fermstatements. Functers ef this last ltind are called cepnlas; in particular ‘I;‘means ‘are’. Clearly then ‘F Q G‘ and ‘F E G‘ de net cerrespend te ‘p —1- a‘ and‘p -H q‘ in the way that ‘Ft? and ‘F v G‘ cettespend te ‘pq‘ and ‘p v q‘.What cerrespend tc ‘p —1- a‘ and ‘p H q‘ in the way that the termschemata ‘FG\" and ‘F v G‘ cettespend tn ‘pq‘ and ‘_a v q‘ are rather theterm schemata ‘F ea G‘ and ‘F -H fl‘. The sentence schemata ‘F 1; -G‘and ‘F E C\" are net tn he cenfused with the term schemata ‘F —1- Ci‘ and

13!} ll. General Teri-as ancl t',Qttentt}'iers‘F 1-1' G‘; they are related tn them rather, tve saw, as “'t\"{F —t G}‘ and“'t\"[F H G}‘- The Eteelean legic cf terms that is new befnre us is substantially whathas traditienally been cenceived ef as an algebra ef classes and calledEeelean algebra. The letters ‘F ‘, ‘G‘. etc., used just new as schematicletters standing fer general terms, weuld frem that peint nf view beceunted rather as standing fer class names, er as variables ranging everclasses. The cepula ‘F‘ gives way en that appreach te simple identity,‘=‘; fer classes are identical when they have the same members. Tradi-tiertally indeed equatiens predeminate in Eeelean algebra; an algebraicleelt prevails. Thus instead cf ‘ EIF‘ and “'t\"F‘ we get ‘F e l't‘ and‘F = \"v“, ill. being the empty class and ‘v‘ the universe. The intent is ne-lenger legic preper, but the substantive rlteery ef abstract ebjects ef aspecial sert: classes. This disseciatien frem tttith-functien legic is accen-tuated by further departures in netatien; tltus ‘F —+ G‘ and ‘F H G‘ arenet written, and instead ef the alternatien ‘F v G‘ and cenjunctien ‘FG‘ef terms we get ‘F Ll G‘ and ‘F t\"! G‘. saitl te designate the rtrtlert andinrersecrlan ef classes. Waiving tltls last departure, we may describe a Bceleart equatien as anequatien each side ef which is ene ef the letters ‘.-'1‘, \"‘v“, ‘F‘, ‘G’, etc.,er is cempeunded ef them by negatien, alternatien, cenjunctien. In the erdinary algebra ef numbers, it will be recalled, any equatiencan he cenverted te a ferm in which ene side is ‘IT. l\"-lew it happenslikewise dtat in Eteelean algebra any equatien can be cenverted re a ferntin which ene side is ‘.-'l.‘. Fer, ‘F = G‘ means that the classes F and Ghave the same members; hence that there is netlting in FG ner in FG; andhence that Ft‘? vF't'.'I = it. Equally, if we prefer, we can always malte ene side “-l‘; fer ‘F = G‘is equivalent alse tc ‘FG vFG = \"v\". The fermulas ef Eeelean algebra are the Beeleaa equatiens and thetruth functiens ef them. We can test any such fertnula fer validity bytranslating it late a Beelean schema as ef Chapter 13 and testing it as inChapter IS. Se let us see hew te translate. We saw hew te transfetttt anyequatien inte the fenn ‘ll. = . . . ‘; and then ‘ll. =‘ can be translated as‘— 3‘. If there are residual eccurrences nf ‘ll.’ er “'v“, they can bereselved eut truth-functienally in the manner ef ‘.l_‘ and ‘T‘ er, mereclumsily, rewritten as ‘FF’ and ‘F v F‘. any truth functien cf Beeleanequatiens gees ever thus inte a ttttth functien ef Eeelean existenceschemata, ready fer the mutine cl‘ Chapter 19. Is there mere than a netatienal difference between a Elenlean legic ef

El}. Se-me Bnnlenn Ineidentnls l3ltenns, as in Chapters 15* l El, and this algebra el’ elasses? l'-lnminally thereis the difference between ‘F‘ as depicting a general term and 'F* asdepicting a class name. There is the difference between the general term‘man’, er ‘is a man’, and the class name ‘rnanltind‘. [See Chapter l-1-}The general term is true el’ eaeh pf varieus individuals, men. The elassname is a narne ef nne abstract nbject, the class cf men- By means cf thegeneral term we may spealt generally ef men and nnt raise the philnsephi-eai questien whether, besides these varieus men, there is an additienalebject which is the class bf them- The class name dees raise that questien,anti it snlicits an affirtrtative answer. The questien is net ah empty ene. tn Chapters 45 and 4? we shall seethat the assumptinn bf elasses can add materially tn what can be l'ermu—lated in a theery, even when what we are fnrrnulating is nnt itself abntttclasses. Where we are interested in eccinc-my nf means, and cif lteepingaccaunt nf the assumpticns needed in prn-ving er defining semething, wemust talte nete pt\" whether classes figure in the apparatus\", semetimes theydn and snmetirnes net. In the case ef Eu-nlean algebra, still, we are free te disavew the as-sumptien at\" classes, representing it as a mere tnanner -nf spealting; fnr wehave seen haw tn translate Hnnlean algebra intn the rnnre innn-eertt idiempf a Heelean legic cf general terms. When the assumptien cif classesmaterially centributes in a thenry, and cannet be thus faeilely disavnwed,the classes are assumed amnng the very nbjeets that the general te11'nsthemselves arc true cf. They are assumed as values ef the variables-—tnanticipate a netinn which we shall talte up in the nest chapter. lvleanwhilewe may best in-nlt upen the En-nlean algebra ef elasses as enly a simulatediinie theery efelasses and a figurative rendering nf what is really nethingtnnre than the En-nlean legic nf general terms. Classes are alse called sets. A speculative branch nf mathematicsealled set theery stems frem Canter tfl. lllillft}. It is eccupied largely withprnbierns nf infinity, and it treats nf classes in substantial ways fltat are bynu means tc be dismissed like the Hnelenn class algebra as mere mannersbi speaking- We shall glimpse a little cif it in Chapters =1-ti-45. in thatrarefied air the term ‘set’ tends tn be preferred te ‘class’, er-teept in acertain technieal ennteitt [see last pages at beelt] where the deuble ter-minnlngy has been put tn use tn niarh a special distinctinn. ln the current “new mam“ cf the elementary schn-nls, the esaltedname nf set theery is cnnfusingly applied te the nther end ef the seale: tnthe Ennlean algebra nf classes, henee really the simple legie cif generaltertns.

I3.\"-1 ll’. General Terms and Qnnmtjiers H IETURFCHL HDTE: The inclusien signs ‘C’ and ‘ ';\", new currentin set theery, are derived frem Ctergnnne*s use in llillti nf ‘C’ fnr cnn-tainment. The Hnelean algebra cf classes was [banded by He-ale in IE4?and llTlp-1't:t1t'tl'=Cl by lltelvlcrgan and Jevctns in lbtitl and lab-1. It was antici-pated in seme measure by Lcibnia in the seventeenth century. Betweenthis legic and the legic cf truth functiens, whese antecedents are mereancient (see Chapter 1], there are cf cnurse censpicueus parallels; andthey were neted already by L-eibnia and He-ale. But the full relatienshipbetween the legic bf truth functiens and the Bee-lean algebra nf elasseswas nnt as clear as it later becarne, e.g. with Eehmann [I93]. Therelatienship is best reflected by the legic nf terms, Chapters IE and iii\"abuve, putting classes aside.EIE IICISESI. Translate the fnllewing fnrmulas intn Elnnlean schemata in the sense nf Chapters IE and iii‘ and test them fnr validity by the methed cf e:-tistential cenditienals. .F‘=Fvt3vH.—>.F=GvH. Fvt\".'t=t.IF.—t-.F=tT-'. F=FvG.s-1-.Ft3'=t'l-‘, Ft? =Fvt?.+s-.F=t';.2. Rcfcrmulate the methed bf esistential cenditienals as a decisien prn- cednre applicable directly tn truth functiens at‘ Hcclean equatiens. {A serieus esercisc.] 21 THE BOUND VARIABLE Ft. general terrn may, as remarked in Chapter ltl-, be theught nf in-dilferently as singular er plural and as substantive, adjective, er verb.This latitude is all very well, but we ceuld dn with mere. We weuld liltetn be able tn handle any statement abeut an ebject as predicating seme

If . The Retain’ Ferrable l33general term el’ that nbject, even when the sentence is cemples and shewsnn neatly segregated term tn the purpese. Already in the little esercise nfChapter I4 we saw that verbal cnntnrtiens were needed tn segregate theapprepriate terms- In rnany examples, e-g.:ll] Tnm used tn werit fnr the man whn murdered the secnnd husband ef Tttm’s ynungest sister,it can be rather a challenge tn find a tert't1 whieh, predicated nf Tem, hasthe same effect. We ceme nut with:(E) fnrmer empluyce nf ewn ynungest sister's secend husband’s murderer. Ctrdinary language already uffnrds a general espedient fnr such cases:the relative cla-use. The purpese ef III} is served, very nearly, by this:{3} whn used tn wnrlt. fnr the man whn murdered the secend hus- band ef his ynungest sister.The new trnuble is the cress-reference nt’ ‘his’; presimity steers it tn thewrnng ‘whn’. Prnblems nf cress-reference invelve us in vesatinus re-casting, as we an well ltnew. There is an inelegant quasi-mathematicalusage, hewever, that cuts thrnugh all that. lt is the ‘such that’ censtruc-tinn, with a suppnrting variable supplanting the prnneun:{4} Jr such that tr used tn wnri: fnr the man whe murdered the secend husband nl’r’s ynungest sister,er, tn be prnlis,[5] .t such thatr used tn wnrlt fur the man y such that y murdered the secend husband nfJr’s ynungest sister.The Zr’ and ‘y’ here are but-ind vnrt'ehl'es.' mere devices fnr prnneminalernss-reference. lvlultiple eheice cf letters is wanted fnr avniding am-biguity nf cress-reference when, as in {S}, relative clauses are nested. ln the relative clause, then, rendered with ‘whe’, ‘which’, ‘that’, er ’Isuch that‘, we have a general term that segregates what the eriginalsentence said abeut the ebject in questien. Pretiicating it nf the ebject isequivalent tn the nriginal sentence. ‘Tent is lfsnmennel whe enjnys music’

I34 ti’ . General Terms and fiannttfiersameunts tn ‘Tem enjnys music’; ‘Tem is {same-nneft .-’I.' sueh that: used tnwttrlt etc-‘ ameunts ttt {I}. The inscrtinrt cif ‘snmettnc’ here is just acenccssicn tn grammar, adjusting an adjective] clause tn substantive]pnsiticn. In fnrming a relative nr ‘such that’ clause, we segregate and weabstract. In -|[l}—{f-It what {I} says nf Tem is segregated frnm the name‘Tem’ anti abstracted t’rnm the sentence {I}. What we thus abstract is ageneral term; there is nn thnught here nf an abstract ebject- The clausethus abstracted may, like any general term, be tnte nf any number nfebjects nf whatever snrts- abstract nbjccts are as may be; we shall cnmetn direct grips with them nnly briefly tcward the end nt’ the bnnh. The cnnstructinn it such that. . -.r - - -‘ will hereafter be rendered ‘lit:. - ..t . . .}’ and called term nhstraennn. The cnmples general term thusfermed will be called an abstract. Just as predicating {4} -nt’ Tem simplyreaffirms ll). sn in general{E} {.1::....t.-.iySuch reductien nf a predicatinn ‘{.r:- . ..t:. . .ly‘ tn '- . . y. . .‘ l call cnncre-tintt. When the sentence that I have represented as ‘. . ..r . . .’ is just asimple predicatinn. say ‘.r is wise’, the abstract becnmes merely a redun-dant rendering cl’ ‘wise’ itself; {..r: I is wise} E wise.Thus {.r:F.r} HF, {.t::FJ:}y -t—1~Fy.[T] The ‘.r’ that appeared sn briefly in [ll nf Chapter lfl has new cnmebaclt tn stay. It is the beund variable nt’ term abstractinn; but let us bemere precise. It is beund in the abstract ‘{.:r: .1: is wisel’, beund by theprefitt ‘.r:’, but free in the cnmpnnent espressinn ‘.r is wise’ taken byitself. This ettpressien is called an npen sentence. It differs frnm aclasedsevttence, nr a statern-ent, in cnntaining a variable in place nl’ a name. It isneither tnte nnr false. The analngue nf a free variable in nrdinary lan-guage is a prnnnun fnr which nn grammatical antecedent is e:-tpressed erunderstand, and the analngue cf an epen sentence is a clause cnntainingsuch a dangling prnnnun.

El’. The Hat-tire‘ Farinble I35 With abstractinn at band, the relatien et the truth functiens el’ terms tnthese ef sentences can be rendered vividly anew: ts} F ={.e —Fx}. ['5'] FE E ix: Fx . Ex},{ID} F vt? E {x: F1: vt'Jx},{ll} F—1~GE{x:Fr—t-GI},{l2} F 1-1-til E ix: Fx -I-1-{Ex}. Instead ef expressing predicatinn by juxtapctsitien, as in ‘Fx’, I shallsemetimes use an epsilen as cepula—particularly in cennectien withabstracts. Thus ts} becemes:llitjt ye{.r:...r...}.~t+...y-... ‘Whereas the cepulas ‘Q’, ‘C’, and ‘E’ jein general terms, the cepula‘is’ jeins a singular term te a general term; fer the variable behavesgrammatically like a singular term. The cepulas ‘e’ and ‘t_;’ answer tn ‘is’and ‘are’. The abstract and its epsilen are asseciated with set theery. The class efall ebjects x such that . . ..r . . . is called {x: . . .x . . .} in set theery, andmembership is expressed by epsilen. My adeptien nf these netatiens ferthe nntelegically innecent relative clause and cepula is ef a piece with myttntelngically innecent use ef the Bnelean fermalism that has beengratuiteusly saddled with the burden ef classes dewn the years. Seme readers will feel that I weuld de better here te render termabstractinn by a distinctive sign ‘st’ fer ‘such that’, as I and ethers havesemetimes dene, and leave ‘~{x: . . .x . . and ‘tr’ tn unregenerate settheery. I new find it mere instructive philesephically and mere elegantlegically te use the same netatien fer beth. W't1en the time cemes tebreach set theery and recegnine classes as ebjects {Chapters 4e--43},general terms will be enlisted fnr deuble duty as names ef classes. Termabstractien will then resume intermittently its accustemed effice ef classabstractinn- HIETURICEL HDTE: Being the initial letter el’ the Crreelt cepula,epsilen was adnpted fer the singular cepula by Peane in I339. He in-verted it fer the ‘such that’ ef term abstractien because nf hew they cancel

lfitii ' ll’- General Terms anal Qnanrfiicrseach nther; see {I3}. Peartn’s ‘e’ is in universal use tnday fer class mem-bership, but his ‘a’ is used newadays rather fnr the nencemmittal ‘suchthat’ nf term abstractinn. usually lnfnrrnally. Peane used ‘e’ fer beth thecepula and membership. and ‘st’ fnr beth term abstractinn and classabstractinn, much as I am venturing tn dc-——tht1ugh his was a case efgenuine ceniusinn- A netatien using circumflex accents and serving un-ecjuivecally fer class abstractinn was used by Russell in 1903 and per-sisted thrnugh Principle lttlnrltenantica and much subsequent literature,including early editiens ef this benlt. The netatien ef curly braces that Iam new using is usual tnday in set theery. In fermulatiug set flieery in taae Ciiidel appealed infenrtally tn ficti-tinus supplementary ebjects that he called nniin-vs. They were lilte classesbut were net values ef variables. They were just a manner ef spcalting,aveidable by circumlecutien. In my Brazil lectures ef I942 I pressed thisfictien. I presented what I called the virtual theery nf classes and rela-tiens, in which I used the netatien nf set theery as far as I ceuld witheutassuming classes. See El Scntida tin neva lcigicn, §5l. lvlartin was urgingthe idea enneurrently. In Set Titeary artrl its Legic {l'§l'b3. l9~b'l\",'t= I madeextensive use ef the virtual theery ef classes as an auxiliary tn the realtheery ef classes. It prnved te be a valuable auxiliary, but I persisted inpresenting it as a mere prelegemenen tn set theery and as a sly partialsimulatien therenf, failing, as we all did, tn appreciate that it sheuld standsquarely in elementary legic as a regimentatien ef a fundamental featureef language, the relative clause. 22 QUANTIFICATION Te say that all men are mertal, and that seme bnelts are bnring, we cannew write:{ll \"tl’{x: mart x -1 mnrtal-1'},[2] 3{-tr: be-eltx .bnring Jr}-

22'- {lnrtnrificatiria I 3'?applicalien nf the functers “ti” and ‘ El‘ tn term abstracts in this fashienprnves strategic. Legic can be pursued withnut using the abstracts ether-wise than after “ti” and ‘ El’. The cenverse is alsn true: “ti” and ‘E’ arenever needed etherwise than in applicatien tn term abstracts, sinee, by {Tlef the preceding chapter, \"1ifF’ and ‘HF’ can always be rendered \"'tt'{-t:Fri’ and ‘H-[-t: F:t}’- In shnrt, the cnmbinatinns \"Ill-t:’ and ‘Hist’, alsn\"'tt'{v:’, '3 {_v;’, etc., can be treated hereafter as simple prefixes. I shall dcse, and shenen them te ‘lit’-t’, ‘Hr’, \"'tt'_v', ‘Hy’, etc. ll} and [It becemeanantt]‘icntinn.t\":{3} \"til\"-t:|[man x —1- mertal -Il-[4] ltlbnnlt -t .bnring x].Tbe prefixes ‘la’-t’, ‘ Ex’, \"‘tI\"y’, etc-, called anantiliers, are the fecal peintcf nceclassical legic. Relatinns nnted in Chapter Ell between “F\" and '3’ reappear newbetween quantiticatiens:\"H\"-r —F-rs—i-—HxF-x, 31 —F-t H—’v’;rFx, \"t|\"’irl'-’-r-—i- H-t.F-x. when we switch tn quantificatienal style, we drep the E-eelean cnm-peunds ‘F’, ‘PG’, ‘F ‘JG’, ‘F —t-G’, and ‘F H--G’; they ge ever intcterm abstracts as in lEIt—{lE} ef the preceding chapter and disappear intnquantilicatiens. Term letters thus cease te eccur etherwise than with avari able appended: ‘F-t\", ‘Gr’, ‘Fy’. Thencefnrward we can thinlt ef the sccnmbinatinns as standing fer sentences as unanalyaed wheles. ‘F-t‘ standsfer any epen sentence cnntaining free ‘.r’ buried anywhere within it.perhaps in several places, as ‘Tem’ was in ll} ef Chapter 2] , and we needhave an theught ef segregating a term represented by ‘F‘ itself. If we dnhave the theught, we can always reactivate term abstractinn; the term ists} ef Chapter El . at ef new the meve te quantificatien is ne imprnvement. Cln thecentrary, Elenlean schemata as nf Chapters IE and 1’-J are simpler andmere perspicunus, unencumbered by either ternt abstracts er quantifiers.We have nur test ef validity by existential cenditienals. and linle is left tebe desired--until we meve frem abselute er tnenatfic terms lilte ‘bnelt’ tnrelative er dyadic terms liltc ‘uncle’. This is the meve that cemplicatcslegic and maltes fer its stature as a serieus subject. It is enly in prepara-tien fer it that we new switch tn quantiliets. Elut we shall have five

I35 ll’- General Tcrtns anal Qttanrijierschapters en them befnre we tacltle the dyadic terms fer which they areultimately wanted. The qusntiticatienal renderings et’ the categerical fnrms it and I el’Chapter Id are illustrated in (3) and {4} abeve; and E and Ct are equallyevident- Te sum up:A: ihllFarcG E: ltlnFaret.-'3‘ b\"-rt{F-t —r G-1'} ‘till’-r{F-t —t ——t‘fF-1']I: Seme F are G El: Seme F are net G 3-t{F-t . Gs] 3-t[F’-r . —tixl Tn discnurage the erreneeus reading ‘3xt{Fx —t~ G-ti‘ nf I euce and ferall. let us step tn see what ‘H.r{F-r —rt'_T-r]‘ really says. ‘F-r —t- Gr’ isequivalent tn ‘ —-Fx ‘v‘ Gr’, sn ‘3.r=[Fx —1- G-r)’ says nnly that at least nneebject is nen-F er ET; and this is beund tn be true, regardless ef hew ‘F‘and ‘ti?’ are intetpreted, except in the ene extreme case where ‘F ‘ is truenf everything in the universe and ‘G’ is true nf nething. The fnrm‘ 3-rt_'F-t: —1- t']-ti‘ is sn rarely false as tn be seldem werth affirming. tlin ndd feature ef nrdinary language is the grammar ef the werds‘everything’ and ‘semething’, which behave lilte names except at crucialpeints. The statements:Hand is a beett and is bering,Maud is a benlt and Maud is beringare clearly interchangeable, as are: Maud is in prese nr in verse, Maud is in prnse nr Maud is in verse;but the statements:{5} Semething is square and reund,tn] Semethiag is square and semething is mendhave eppesite truth values, as dn these:{T} Everything is visible er invisible,

.?-\".’l'- Qttrtnrt:,l‘it'ttt't’t-in I 1-'3'[fl] Everything is visible er everything is invisible.Such is the impnrtance nf scepe in quantificatien. In '3-ttF;r . t‘_T-rjt‘ and\"'t\"x{F-r v C-rlt’ the scepe nf the initial quantifier extends tn the end; in‘H-r F-r . H-r t}'-tr’ and “s‘-r F-r v \"if-r fit\" itsteps halfway. in these cases itmatters- tn the eppesite cases it is indeed immaterial: existentialquantificatinn is distributive threugh alternatien and universal quantifica-tinn is distributive thrnugh cnnjunctinn. liil H-tilt’-t v G-ti *-r- Hr F-r v H-t C-t.tllill ls\"-rlF.r - Gr} H. ll\"-It F-r . ’lil\".t l-’t’.r.Tn say this is just tn reiterate [3] nf Chapter IE and ll} nf Chapter Ell inthe light nf [it] and t tut nf Chapter 2 I- Statements nf nrdinary language which at first glance seem tn be cnn-junctiens er cenditienals eften demand intetpretatien as quantificatinns efcnnjunctinns er cnnditienals. Examples arc:[I I] Sadie stele semething at the Empntium and exchanged it fer a blnuse,['12] If Sadie wants anything she manages tn get it.These must be interpreted as quantificatinus:{ill Hxrjfladie stele-t at the Emperiutrt . Sadie exchanged -t: fnr a bleuse},til-ill \"It\"-rtffiadie wants .r —r Sadie manages te get -ti,rather than as a cnnjunctinn and cenditienal:[I51 Hrlfiadic stnlc -r at the Empnrium] . Sadie exchanged it fnr a blnuse,{lb} 3-ttfSadie wants rl —-it Sadie manages tn get it.Fnr, the ‘it’ ef if l ll clearly refers baclt acress ‘and’ te ‘semething’, andcnrrespendingly the ‘it’ nf [ll] refers bacit tn ‘anything’. The quantifiersmust he made te cever the whnle cempeund as in [I3] and ll-ill, ratherthan just the first clause as in tftfit and {tn}, sn as tn reach nut tn a laggingrecurrence nf ‘-r’ in the pesitien nf ‘it’.

I-lit] if . General Terms and Quumijiers If we thinlt c-f the universe as limited ta a finite set uf ubjectsa, .!t,. . .,li, we can eitpatlcl eitistential quantifieati-ans intn alternati-ans and univer-sal quantificatie-ns intu cenjunctiens; ‘Q1 F1’ arid \"'lil'J: Fir’ became re-spectively: FavFliv---vF.ti_ Fa.Fb.-.-.Fh-The distitictiun between {5} and [fill then cnmes nut quite clearly; 'E'ur[FJ: . Girl‘ becurnes: Fa . t_?a.v. Fit . Gil .v..-..v.Fh .511.whereas ‘ 31 F1: . H1 Gs‘ becnmes: Fa vFs v. . .vFli . Ga vfis v. - .vGa.The distinctian between {T} and [H] cemes nut equally clearly. Further-mure the interchangeability nf —\"i\"x' with ‘H.=r—' and ef \"— 31' with'li\"x—‘ tums nut tn be a mere applicatiun uf DelvIargan’s laws {Chapter lfi]; fur, ‘— \"IiI\"x Fr’ and ‘H: —F.r‘ becutne respectively: —-I[Fn.Fb.....Fn], —Fuv—Fbv...v—F.li,and ‘— 1|: F1\" and ‘Vs — Fir’ becurne respectively: —{FavF.!.iv...v.Flij|+ —Fa. *Fl:.-.....-Fa. lt tl1us appears that qu antificatitin ceuld be dispensed with altugether infavur e-f truth functiens if we were willing ta agree fer all purpnses en afitted and finite and listed universes, b,. . . ., .li. Hewever, we are unwill-ing; it is cenvenient tu allnw fer variatiuns in the cbuiee at universe. Thisis cnnvenient nnt nnly because phiinsaphers disagree regarding the limitsat reality, but alse because—as already nute~tl—saaie lngical argumentscan be simplified by deliberately limiting the universe el’ disce-arse teanituals er ta persens at ta the empleyees at a given firm tar the space atthe preblem in hand. Fe-r must pr-ablems, merenver. the 1'elevant universecu-mprises ebjects which we are in nu pesitien tu list in the manner ufa,s,. . .,fi- Fnr many prablems the universe even eemprises infinitely manyebjects; e.g., the integers. Thus it is that quantiiicatian is here tu stay.