Methods of Logic - W. O. V. Quine

33. Dsdarrlna EdiFrem the readiness cif this adjustment we see that the alphabetical restric-tien is really tighter than necessary. its virtue is that it is simpler than itsmere liberal alternatives, and mereciver the cases that it needlesslyexcludes ai'e easily adjusted. lit real fallacy that is excluded by the alphabetical restrictinn, andcannet be patched up by relettering, is this:eitssttss: ‘slx Hy F.r.'_'ytiitrirscrteiv: Hy Fwy {Lit} Five {El} ‘slx Fxe {U13} {wrung} Hy ‘s‘.r Fey {EU}The free variable ‘e‘ in the line grit by LIE is alphabetically later than theinstantial variable ‘iv‘ tit‘ that U13 step. ilind if we rectify this by using ‘ellinstead ef ‘iv’, we thereupen vielate the alphabetical restrictien in the Elstep. ln erder te facilitate cenfermily tn these restrictiens. and alse fer afurther purpese that will be neted presently, it is cenvenient tri_,|\"ilag eachstep cf El and LlCi by recnrding the instantial variable eff te the right efflie line which that step yields. ilitlsri it is cenvenient, and increasingly sein cemplex deductiens, te number the lines at the left and then refer bacltte them, as seurces, at the right. indeed the citatien ef rules {UL TF, ete.}is less te the peint; we recegniee the rule easily eneugh, given the sciuree.iltlsri, in preparaticin fer a certain later departure, let us signal the premisene lenger by name but raflier by starting a celumn cif stars there andrunning the celumn dewn the length ef the deduetien. Each star meansthat the starred line is affirmed enly ceriditienally en the premise. Here, then, rewritten in the new style and rectified in the alphabeticaldetail, is the leng deduetien last previeusly exhibited. *{l} ‘sly H.r{Fyy . Fyx .v Fry} till] 3.-'t{Fee t Fee .'vl Fee} {1} *i3l Fee - Fee‘ -vFe‘e ill E‘ *{=-1-} Fee v Felt {3} *{5} Ht Fxe v Fele {=1} =+={fi} Eli: Fxe v Ex Fm {5} ‘Fill 3.1: Fxe ll-5} *l3l ‘ill’ 31 Fri‘ {ill e

It-til lll‘- General Thrary af {Qiiaarliirarlaa The earlier little deduetien which used twe premises weuld I1-ew appearas fellews.=i{ I} ‘sl.r{t.'l.r —1-, Fr v Hr} {I} l-‘ ill=lvl={2} El.r{t'Ii.r . —Hx} l3ll4ltrill} Gr - —HrHt-1} E-‘r —r- Fr rllr {.5}\"{5} Fr - —-‘tr=i==i={fi} 3.r{F.r . —H.r} By reference te the flagging device, eur restrictiens en El and LTG cannew be phrased mere succinctly: ne variable is te be flagged twice, andthe flag is te be alphabetically later than all free variables ef the generic. The status claimed fer the last line in a deduetien, er indeed any line, isthat it is implied by its premises. Reservatiens are needed, hewever, incennectien with flagged variables. This implicatien claim is in erderprrivided that ne {lagged variable is free in the line that is said te beimplied, ner in its pretnises. Thus the last deduetien abeve shews that {1}and {2} therein imply {ti}, but net that they imply {S}. iltnd the reader cansatisfy himself that they in fact de net. Again, talte the deduetien beferelast and imagine line {1} thcreef as its premise, with {l} dropped; thisdeduetien weuld net claim te shew that {2} implies {E}, sinee lel is free inthat premise {Z}. Fl. deductirin is called finished if ne flagged variable isfree in its last line ner in any premise en which that last line depends; andthen the peint cf a finished deduetien is that its last line is implied by itspremises. There is a device that has had seme currency, in deduetien, under thename at‘ eaadliiaaal preef. Its adeptieri has the effect tif dispensing withthe clausal ferms ef UG and E123 after all. it has alse the mere impertantelfect ef dispensing whelly, in deduetic-ns, with the requirement ef prenexferm. Alse a certain naturalness can be claimed fer it. It eensists inadapting, ternperarily, an additlenal premise, and eventually dischargingit by caaalitlanalleariaa {Cd}: by incerperating it inte the cenclusien asantecedent ef a cenditienal. The pattern is this: en the assumptien dtat p,we shew that a; sri we cenclude that}: -1 an The precedure requires seme beeltlteeping, tn lteep truclt ef what linesdepend en the temperary premise. This is where eur netatien ef stars,rather peintless thus far, cemes inte its ewn. We star the ternperarypremise and each line that depends en it. if yet annther tetriperary premiseis adepted while ene such is already in ferce, we begin a further celumncif stars yet farther te the light-

eld. Drdartien I43 As an example let us return enee mere te the deduetien befere last andsee hew it might have heen managed with help ef cenditienal preef. =I={l} 's'yElx{Fyy . F_v.r .v Fry} {I} {3} e‘ rill 5|IiFtt - Fer -v Fret #113} Fee . Feel ,v Felt {4} **l*ll Fee *{3} 'l\"I'={5} ax Fxe 113} 'I={E} Fee —1~ 3.1.’ Fee l_‘3}[l3}{_\"'§l} **l7‘l Fe‘e {ltl-} 3 =l==t={E} alt Fee rill} Fele —i Elx Fre =I={]'[l} as FIE *{l1} sly 3.rF.1ry The celumn ef stars belnnging te a premise is finished fer geed whenintemipted. The little right-hand celumn ef twe stars at {4} and {5} be-lengs tei the temperary premise ‘Fee‘ and has netliing te dc with thesimilar celumn at {T} and {E}, which belengs te the temperary premise‘Fe le‘. The deduetien abeve ceuld be extended still by adding an unstarredline:{ll} ‘s‘yHx{Fyy-Fyx.vF.r_v}—i-‘sly 3xF.1r_v =l={l1}thus abserbing the eriginal premise itself inte the cenclusien. When thelast line ef a finished deduetien is unstarred it claims validity eutrightt it isne mere censequenee af a premise. A final example illustrates the utility rif temperary premises and cen-ditienalieatien in prebing schemata that are net prenex. It is the ene againabeut paintings and critics, which was dene at the beginning ef thepresent chapter. blew, hewever, we render the prerriise and cenclusiendirectly in the ferms that they received in Chapter 27. We neither putthem inte prenex fnrtn ner negate the cenclusien: =I={l} 3y‘[Fy . ‘slx{t1r —i H‘.ry}] rte} Fe - \"slx{t.i‘x —-i Her} {1} e =l={3} \"s‘x{[i‘x —-I Hxe} {3} =l={='-l} Giv -—i- Hive {3} =l\"l={.\".i} fiiiv

I-ll-'-‘l illlf. General Theary tif,Il'-[;ltratttt:,{‘it‘trl'ltirr**lb} Fe , Hive {2}{-ll}[5}\"I'll H}llFJl - Hi‘-llll ll-‘ll=r{fi} -Eliv —1- 3y{Fy . Hivy} \"'lT}=r={9} slr[Gx —i- 3y{Fy . Hxy}} {B} ‘H-' H l5TCl Rlclil. HDTE: The methed set ferdr in the present pages is cfa type ltnewn as natural deduetien, and stems, in its breadest eutlines,frem Ciertteen and lasltewslti {I934}. The rttle ef crinditienalieatien,which is the crux rif natirral deduetien, appeared as an explicit fcirrnal rtilesemewhat earlier, having been derived by Herbrand {l\"il3t}} and alse ineffect by Tarslti {I929} frem systems tif their ewn rif a type ether thannatural deduetien- The derivatien crinsisted in shewing that whenever nnestatement ceuld be deduced frnm anether in the cencemed system, thecenditienal fermed ef the twe statements ceuld alse be preved as atheerem by the eriginal rules ef that system. In this status ef derived rulerelative ta ene system er anether. the mle cf cenditienalieatien has cemete be ltnewn in the literature as the theerem cfdedatrrlaa. IasltewsIti‘s system cf natural deduetien is eenspieueesly unlilte that cfthe present pages; fer Jasltewslti dispenses with ECi arid El, and getsaleng with milder restrictiens an l_I-Cr, by the expedient cf treating lax‘,‘E-ly‘, etc., as abbreviatiens cf ‘-‘sl.r—‘, ‘—‘sly—‘, etc. This ceurse iseeenemieal in mics, but greatly increases the difficulty and cemplexity tit‘the deductiens themselves. The crucial difference between t3enteen‘ssystem and that er‘ the present pages is in El; he had a mere devieus rule. Because ef the presence ef El, the present system differs fremGentecnls and Jas ltewslti‘s cerisiderably en the scere ef restri ctiens upenthe rules. In particular the device ef flagging is a nevelty. Cienteen andlasltriwsld had restrictiens tee, but gave them different ferms. Ceeley, inpp. IZIS-—l-=1-I] cf his Primer a_}'\"‘Farnraf Legic {I942}, made use ef naturaldeduetien in a ferm which included substantially El. but witheut exactfermulatien cf restrictiens. Explicitiy ferrnulated reles and reshietiens,resembling the present system except far wide variatiens in the restric-tiens, were set ferth by Russet and independently by me in mimee-graphed lecture netes frem I‘.-lilfi en.El°lEltCl5E5I. Cheelt each cif the fellewing tci see whether it is a cerrect deduetien aceerding te the rules, and a finished ene-

3?, .\"i'nunrl'rrcss 1-15*tll F.'+' *i ll ‘tl\"Jt{FJr —I+ Gs}:12} \"tI'Jr Far -[ll r =|=f1] Fy —- Ei_v t |J'{3} FJF —!* \"Ira: F1 *t3l El.1r|{Fy -* GI] ill *t4l \"tI\"w EI.Ir{Fw -1- fie]: {3} gr*\"[ I l \"t|\"1.-|{F;r v G]-']I I11} Jr‘ =\"\"[ 1 l 'iiir.I[.F.: . -l!'_iF.l.'] H]'l=i'2,l Fy v G}- “[2! Fy . G_v ‘[31*[3'J ‘IiI\",r[Fy v Gr} ='=[3l Fy {'3} y ‘£41 \"I\".I.' F: {1}*ill Fa vfiy {5} it *{51 Ba f4]l'[5}Iefl} \"Ii'_t'{FJr vG__1»'}'l ill 1 e[fi.] ‘II’: Git*{f+] HJr\"|iI\"_y{F_:r v i'_,'F_i-]| {2} *|fT]| ‘II’; Fir . \"tI\".r Gs2. lf any bf the dcductiens abeve is cerrect but unfinished, append further lines se as tn preduce a finished deduetien. If any is incnrrect but can be revised intn a cerrect deduetien tn the same effect, se revise it.3. De the eitarnples and varieus eitercises bf Chapters 19 and 3| again by deduetien.I31} SOUNDNESS The deductive methnd last set ferth deniinated the first twe editiens dfthis beck. lts rules, we see, take lenger tn eaplain than these cf whatl amnew calling the main methed, and its seundness, we shall see, talteslenger tc preve. Once established, the methed has its virtues. It requiresnu prenexing, it is directed en implicatien rather than incnnsistency ereven validity, the steps are intuitive en the whele, and the seareh ferprecds tends tn gr: smnnthly. This last is nnt always true: semetimes an ebvieus implicatien, quicklypreved by the rnain methed, is stubbnrn under deduetien. Fer efficieney,en the average, the methed ef functienal nermal fnrms is perhaps best. Asfur the main methnd, it is eften effdflless, as we icnnw; it lends itself tn theeasiest cf cempleteness prcets; and it npens the way, which we havefellewed, tn a half deaen alternative metlmds that are easily derived fremit and are efficient er illuminating in their several ways. Thus it was that in

145 Hi’ . General Thenry sf I|Qndntfl|'it‘trtidnlthe third editinn I demnted deduetien, limiting my tt'eatment Inf it tn whatwe saw in the preceding chapter. The meve was deplnred by teachers whn were accustemed tn the earlyeditiens and wedded tn my rnethnd nf deduetien- Henee I am adding thepresent nniissible chapter and the near, in which I sketch the seundnessprnnf and discass deductive techniques. The seundness ni’ the methnd means that the last line t‘:-f a finisheddeduetien is implied by its premises {er is valid, if free cf prernises). Tnsee this we need pause nnly nver Cd, UG, and EI, sinee UI, EU, and TFare implicative. What needs tn he shnwn in the case ef Cd is that if a linein e deduetien is implied by its premises, and then we prefis its lastpremise tn it by ‘—#', the cenditienal thus fnrrned is implied by theremaining premises. New the ltey tn this is the equivalence cf 'pq —a- r* tc‘p —1-. q —1- r’. Fer, thin!-I nf ‘e —t- r’ as representing the inferred cnndi-tinnal and fin‘ the remaining premises. The line represented as \"r’ wasimplied by its premises, represented as ‘psi; EIJ ‘pq -—* r‘ represents avalid cenditienal. Sc then dees ‘p —s. q -s r’. Thus the inferred cnndi-tinna] Fe —s r'] is implied by the remaining premises |['p’}. It remains then tn censider HG and El. What is clear thus far is that if adeduetien laclts U‘-'3 and El, in nther werds if it has ne flagged variables,then it is snnnd: its last line is implied by its premises. H-aw let us assumethat finished deduetinns with just n flagged variables are s-mind, and enthat assumptinn let us preceed tn preve that finished deductinns withrt + 1 flagged variables are liltewise smtnd. Cnnsider then a finished deduetien with n + l flagged variables nfwhich the alphabetieally earliest is 'y*. Suppose it flags a step ef UG.Thus the flagged line has the Farm *‘li\"..t{. . ..r. . .1‘ and is infefled frnm aline ‘. . . y. . .'. lvlndify the deduetien by prefixing a whnle new celumn efstars and the new premise: =l't[U} ...y...—+'1I\",r[...,r...).Since the line ‘liI\".r{. . .,r. . .1‘ is derivahle frem the lines *. . . y. . .‘ and (Ellby TF, we can drnp the fiag ‘y’. Hy the alphabetical rule, ‘y’ is thealphabetically latest free variable in ‘. . .y. . F; yet it was the alphabeticallyearliest flagged variable in the deduetien; an nn flagged variables are freein {El} new that ‘y‘ is unfl agged. But neither are any ef them free in the lastline cf the deduetien er in its ether premises, since the eriginal deduetienwas a finished nne. Therefere the revised deduetien is a finished nne. Hutit has nnly rt flagged variables, and we assumed that stteh deductinns are

5?. Sen-trttinesx 147seund. Therefere its last line is implied by its premises, including [El]. Sethe cenditienal is valid whese censequent is that last line and wheseantecedent is the cenjunctien ef {ll} and the ether prentises- Apply ‘lily’ tethat cenditienal; the result is still valid- Hut ‘y\" is net free in tl1e ceu?sequent ner anywhere in the antecedent except in {El}, since the eriginaldeduetien was a finished ene- Se, by the rules el’ passage {El and [ll et'Chapter E3, we can change “'tt'.t‘ te ‘Hr’ and malte it gevern just {U}. Theresulting quantificatiun: Hy[...y...—*h\"x[....1t.-.]]is valid by if I3} ef Chapter 25 and se can be drnpped frem the antecedent,leaving the rest ef the cenditienal still valid. Se the last line el’ thededuetien is implied by the eld premises- We. assumed that the step using the alphabetically earliest flaggedvariable was by U5. The argument is parallel fer El. [ill is then‘art. . .1. . .1 —1- . . .y. . .’, and the pertinent law in Cbapterleistl-=1). Seehew ciesc the reasening is te the seundness preef in Chapter ED. We began by ehserving that deductinns witheut iiagged variables areseund. hlext, assuming that finished enes with just n flagged variables areseund, I preved that these with n + l fiagged variables are setlnd. Wecan new cenclude that all finished deductinns are seund. This way efreasening is called nintiten-tnticel indncrien.EXERCISE wherein dees the preef ef seundness depend en the requirement thatne variable be flagged twice? Hew de we l-utew that eur appeal just newte [l Ii] and {I4} cemplies with the last three lines ef page llll]?

Edit] HI- General Titeetfit effllndnrflireridni 40 DEDUCTIVE STRATEGY The fellewing six simple deductinns. establishing again three el’ eurfamiliar niles ef passage, will be cenvenient l‘er reference in illustratingstrategies ef deduetien-*If ll Dednctien I .' tll 1+’ *i ll Deducfien E‘: til It*1?) Hxtp 'vF.rl {3} =|==t=[Z] p v 11' Fr *II3l=|ni:[3] =I=t[1'-l) tllitll P ‘-‘Fr t[2I|t[5Il eefi} 3IF.r l5}=te|=[d} Fr *i4l Fr*i5i *1‘5}*tl5l 3.1‘ Fr *l5l 11: F1: —1- Fy Fy —1- Ex Far F\"-\"Fl p v 3.: Fx 1t[p v Fa} Dedncrien 3: Dedtrctien 4':i tn 3-rtr-' - Fall te:- tll P‘lie F in} I ‘lei 3.rF.r {El I F1: [3] i ta F.=r tlIlt3li ea Ex Fx t_'2]|{-fl-It =-= t-ii _l':t. F1: it-lls {5} p - Ex Fx * i5] Hxtp . Fr}*i ll Dednctien 5: tll I *t ll Deductien d: til I*t1l Hx[Fx —-r pl **l1II lillx Fx —1- p If 1 lliil**t3_l F:r —1- p [3] FJ: lit’; F_t a=t[3] lllx Fx std]ee[-:1.) tilts) Fx e#|[-\"-l] F l5]=|ee|f5} *[5]|- F.1:—s_n a #115} 3..t:l_'F.1: A-pl*i'5l *tI5l lilx Fx A p The ebvieus eperatien ef separating cenjunctiens by T'F may cenve-niently be left tacit by stating cenjunctive lines at will as bracketed pairsef lines. This is dene in Deductiens 3 and 4- Fer the discevery ef a desired deduetien there are seme simplestrategies. When as in [leductien I we are starting with a quantiticatien,

dill‘. Dee'nt'-rive 1'itrnreg_r 149the ebvieus strategy is te begin by drepping the quantifier by LII er EI[with er witheut change el’ variable}- Ct‘:-nversely. when as in Deductien Iwe are heping te get a quantificatiun as end 1'esttlt, the strategy is te try tededuce die desired schema witheut its quantifier {and with er witheutchange ef variable); afterward the quantifier may be supplied by LTC erEl]- If we were trying te diseever Ilteductiens l and 1, these twestrategies weuld afferd us just this much; [l]—[1l ef Deductien l and,werldng backward, (5)-{I5} el Eteductien 1. Se, if we were trying te devise Clcductien 1. the initial strategy efdrepping the quantifier weuld leave us with the preblem ef getting frem[El ef D-eduetinn l te {I5-l. Next, since [El and {I5} are alike te the extent ef‘p v’. it weuld he natural te wender whether a cenditienal jeining theremaining parts ef [2] and [ti], vie, ‘Fy -1- Hr Fir‘, might cembine withti} te imply [ti] truth-functienally. at fell sweep cenfinns the netien:‘p v q’ and *e —1- r‘ in cenjunctien imply ‘p v r‘. Se we new knew thatthe desired [I5] can be subj~eined by TF te {E} and [S]; the preblem remainsmerely ef getting (5). Here an ebvieus strategy ef the cenditienal cernesinte play: assume the desired antecedent as additienal temperary premise,try te deduce the censequent, and then get the cenditienal by Cd. Se weassume [3], frem which t4] happens te preceed witheut difficulty; andthus Deductien l has been created in full. If we were uying te devise Ilteductiea 1, the strategy ef quantifiersweuld have left es with the preblem ef getting frem ti] ef Deduetien 2 te{S}. The reasening which selves this preblem is exactly parallel te thatdetailed in the preceding paragraph. Truth-functienal implicatiens are checked easily eneugh enee they aretheught ef. But when we are building a deduetien, the implicatien has tebe theught up befere it can be tested. The way in which Deductien 1 wasachieved illustrates a ceurse which must eften be fellewed. There wecensulted cemmen sense fnr a suggestien ef ubtainable lines frem whichthe desired result {-5} might fellew. (2) was at hand, and cemmen sensesuggested {5} as an adequate suppiementatien. Se then we checked thesuggestien and feund that {1} and {5} de in cenjunctien imply [bl tmth-functienally. iltccerdingly we underteek te get [5]. The strategy behind the discevery -nf Deductien 3 is evident. The usualstrategy ef quantifiers leads frem ti} te {2} and [3], leaving us with thepreblem ef genlng fmm {2} and {Ii} te {S}; and this preblem effers linlechallenge te ingenuity. in Deductien 4 the strategy ef quantifiers leaves uswith the preblem ef getting {4} frem fl] and [Z], which again is the wctrkef a mement.

355' ill. Generrrf T.irecu\"y sf Qnenrgficnrinn The strategy behind the discevery ef Deductien 5 is as fellews. Thestrategy cf quantifiers leads us frem [ll te [2], leaving us with the preb-lem ef getting [ti] frem {E}. aceerding te the strategy ef the cenditienal,in erder te get {Er} we assume its antecedent as {3} and try te deduce itscensequent ‘p’. The strategy ef quantifiers leads us frem [3] te [4], se thatall that remains te he dene is get ‘p’ semehew frem the lines [l}—[-1}which are new at eur dispesal- Clbviettsiy {E} and {4} serve the purpese,via TF- The strategy behind Deduetien ti is rather as fellews. Since we wantts), the strategy ef quantifiers directs us te everleels its quantifier and aimfer t_'5]. aceerding te the strategy ef the cenditienal, in erder te get {5} weassume its antecedent [ll and try te deduce its censequent ‘p’. Se all thatrentains te be dune is get ‘pd serrrehew frnm [ll and [1]- The interveningline [3] quiet-dy suggests itself. it must he remembered that eur rules ef ded uctien apply enly te whelelines. LII and El serve te remcve a quantifier enly if the quantifier is initialte a line and cevers the line as a whele; and UG and EC: serve te intreducea quantifier enly inte such a pesitien. lt weuld be fallacieus, e.g-, tepreceed te the last line ‘Fr Fx —1- p’ ef Deduetien 5 frem the earlier line‘Fr —1- p‘ hy l.iC:, and it weuld he fallaeleus te preceed frem the first line“'tt'.r F.r: -4-pi cf Deductien ti te the subsequent line \"Fr -sp‘ by Ul.‘ltfx Fr —1- p’ is net a quantificatiun, but a cenditienal centaining aquantificatiun ‘lib: Fr\". What issues frem ‘Fr -1-p‘ by U-G, and yields‘FI -—* _n' by Lil, is nut \"'lI\".r Fr —r p’ but “'i\"x{F.r -1 pl-'. hlcxt let us undertake the inference abeut sleveniy persens at the be-ginning ef Chapter l'i\". Here eur premises and cenclusien are‘. \"li\".r[t'.T.=r —+. Fr v HI}, Elxffir - —H.r]t, H.r[F.r . —H.r}.The strategy ef quantifiers reduces the preblem te that ef getting frem‘Gr -1-.F.r vH.r’ and ‘G.-r - —H.r’ te ‘Fr . —H.r'. If by luck the ccn-jttnctien ef ‘Gr ~—-1-. F.-r v Hr‘ and ‘Gr . —H.rr‘ truth-functienally implies‘Fr - —HI'. then eur deduetien is cnmplete. Se we submit: p—s. q vr : pt‘ :-r-qt‘te a tn.|tlr-value analysis, and find that luclt is with us. ln full, then, fl} \"'i\"xt[Gr -1-.F.r v Hr}“lira; Elites . —H.rl* i3} Gr —r. Fr vH.r [ll

dill’. Detirtt't‘t'1-‘e .'i't‘t\"rtte,tt_t' E51 e -[ti] Gr . —H.r {1} r rt: {5} Fr . —H.r {3}t}tl-\",l =t= {til Exit‘-it . —.H.t} [5]- tlr mere substantial example is the traditienai ene abeut drawing cir-cles- The premise and cenclusien were sehematiaed in Chapter IT:\"v\".rtfF.r —a Gr}, \"tt\"_v[H.r{F.r . .l'1'_y.r]=—1- §.r[G.r . H'y.t}]New the steps ef deduetien frem the ene te the ether are dictated almestautematically by the snategies ef quantifiers and the cenditienal. Thedesired cenclusien being a universal quantiticatien, we first aim fer thisexpressien minus its \"\"liI\"y’. Hut this is a cenditienal; se we assume itsantecedent ‘3.r[F.1: . Hyxl‘ and try fer its censequent '5.r{G.'r . Hyxl‘.But in erder te get ‘ H.rt't'].t . Hy.r}' the strategy is te try fer ‘Gr . Hyr‘fer ‘Ct . H_vt', ete.). What we have te deduce this frem are\"'lil\".rt'F.1t —1- tlrjt‘ and ‘H.r[F.r - Hy.=r]\"; se the strategy ef dreppingquantifiers is breught te hear en these, and little prnves te be left te theimaginatinn. ln full the deduetien is as fellews.ell} \"Il\"';r[F.r —-1 Gr] {El t**tEl 3.r[F.r . Hyx} tl]=|==t=|[3l Ft . Hyt (3-]{-1}sele] Ft A Gt {5}r=s=|f5] Gt; . Hyt; std)sele] H.tt_'t'_'F.r . Hyx} [T] yat?) E.r[F'.t . Hyr} -r H=.rt'tlr . Hyx}std] 'tl\"y[Hxt[Fx . Hyxl A 3.r{G.=r . Hyxl]l'\"-lete that the shift frem ‘.r‘ te ‘t’, in line {El}, was necessitated by thealphabetical stipuiatien in El. [We ceuld emit mest such shifts by lettingthe erder cf the alphabet vary frem deduetien te deduetien.) Next there is the example abeut paintings and critics in Chapter 2?. Wealready went threugh the deduetien twice in Chapter 33. All the steps cfdeduetien are dictated by the strategies ef quantifiers and the cenditienal.Let us recerd the deduetien anew and review its genesis.=I= {ll 3y[Fy . lI'.rt?G.r -—r+H.ryl| 1 ll“ Jr lZl Fy*l[3l \"v\".rt_'t‘I.'Fr —1-Hryl [3]-=t= til} [Fr —rH.r_v

152 HI- General Tiitectry elf flnnnrijleerinn=t==t~tf5\",l Gr tlltsltf-l tfilrrttil Fr - fits rtlltrill 3:-=tFr - Hrrl tsl 1still GI -t 3]-'lF}* - Hullstilt ‘ttaltlr —* ElrtFr - H-ellLines {2}-{4} issue autematically frem eur strategy ef dreppingquantifiers. lvlereever. since we want £9], the backward strategy nfquantifiers directs us te aim fer {ti}; and in erder te get {Bl the strategy efthe cenditienal directs us te assume t5] and aim fer tll. ln erder te get {Tlwe n'y fer U5}. aceerding te the backward strategy ef quantifiers. Se newthe deduetien is eemplete if in fact tn] happens te fellew frem its pred—ecessers by TF. Trttth-value analysis er inspeetien shews that it dees.Thus the truth-functienal implicatien leading te {til frem {E}, [4-}, and{5} did net need te be theught up; it autematically presented itself ferappraisal. In the example ef Chapter IT abeut philesephers, the prernise andcenclusien have the respective ferms:Hyllfy . ‘tt'.'rt[F.r —t- fi'1}|}]+ 3.rt[F.r . EH}.Uur strategies ef quantifiers prempt us tn derive ‘Fy’ and\"'I\".t:t'F.r --r Gayl’ frc-m the premise. and te aim fer ‘Fr’ and *Gx.r'—-ersay ‘Fy’ and ‘Gyy‘. Se all that remains is te get 'Gyy* frem ‘Fy’ atrd\"'tt\".r[F.r -r 1'1?-t'Jtl', which is easy. The full deduetien, then, is this:-= tn 3y[Fy . \"tt\".rt_‘F.r —1- G.ry]] ii“ 3“ til Fr l3] l3ll'il‘liter veri -i can {5}* l4] Fl’ “-\" GP?* f5] FF - 53?)’1' {6} 3.rt[F.tr . Gar]nlete that flre changes ef variables between [3] and en and between {5}and tel are net, as in previeus examples, prempted by the resuictiensanending UG and El. They are simply steps in the ebvieus reute frem [2]and {3} te rs}. ills an example cf a deduetien calling fer mere ingenuity in the ma-nipttlatien ef variables, let us shew that symmetry and transitivity te-gether imply reflexivity. Ctur preblem is te deduce “tt\".rlt'ytfF.r;y —r. Fxx .

dill. Dedtrcrt've .‘.'i'rrnregy 153Fyyl‘ frem the symtnen-y premise \"\"tt\".r\"i\"ylF.r\"y —1- Fyr)‘ and the transitiv-ity premise ‘h\"xh'y'v'a[F.ty . Fyr; .-+ Frat‘. Backward strategy tells us teaitn fer ‘Fry —1-. F.r;r . Fyy‘, er perhaps ‘Feiv —t-. Fun . Fiviv‘, after-ward getting the desired cenclusien by twe applicatiens ef UG. ln deduc-tinns which premise te invelve serieus shuffling ef variables, we caneften ebviate awkward cenflicts ef variables by immediately shifting tewhelly new variables in eur backward strategy; se let us aim fer‘Ftnv —1-. Fun . Fiviv'. Te get this, the strategy is te assume ‘Fttiv‘ andtry fer heth ‘Ftut‘ and ‘Fiviv‘. Se the preblem reduces te that ef getting‘Fun’ and ‘Fiviv‘ frem ‘Fniv‘ and the twe eriginal premises ef symmetryand transitivity. Turning then te fetward strategy, we censider dmppingthe quantifiers frem the premises; but there remains the preblem ef pick-ing new variables in suitable ways. We are well advised te pick them as‘tr’ and ‘iv’ exclusively, since enly these appear in the desired results‘Fan’ and ‘Fiviv' and the intermediate premise ‘Feiv’. ln the symmetrypremise the relettering ‘Feiv -—-i- Fivn‘ is mere premising titan ‘Fivu —rFeiv‘, since eur intermediate premise ‘Ftriv‘ will eembine with‘Feiv ee Fivtr‘ te yield semething mere, \"Fivtt‘, with which te werk. Senew we have *Faw‘ nttd ‘Fivt-t‘ te ge en. Therefere the twe reletterings efthe transitivity premise which we seem te have tn cheese be-tween, v'1.t.,‘Fnr-e _ Freer ,—a Ftrttt‘ and ‘Ft-vtr . Few .—tr Fiviv‘, will beth be useful:ene will yield eur desired result ‘Fen‘, and the ether will yield eur etherdesired result ‘Fit-tiv‘. Se eur deduetien uses the transitivity premisetwice, and r|.tns as fellews {subject te subsequent refinement}:*{-[ll \"v\".r\"tt\"y[F.ry —1- Fyxl ill {E} \"ll\".r‘tfy\"s\"atF.rj_v - Fye .—ti Fre] {Ii-It {2}if 'l3l \"‘}\"lF\"}' _*F.l'\"ls= {4} Feiv —1- Five K5]s= {5} \"tt\"y\"tI\".rt[Ftry . Fye .—i+ Fus] {fr}* tfil ‘v‘-ttFviv - Fivt -—> Fvs] {2} ill}=t= {T} Ftriv . Fr-vtr .--I Fun tillis {ii}ti {9} \"I\"_v\"tt'rI_'Fivy . Fya .—i- Five] tlllfsll=t=tIl-ill {I l}[l2]{T'] \"v\".t{Fivn . Fut .—r Fivalwill ll Fivtr . Ftriv .-ii Fiviv ttaitttiitieissritlfl} Fnw“\"i“‘i Fivu Fun l.-riw.-

Eli-it Ht . General Ttteery qt\" Qnnntttierrtlen-ritjlrl-It Few —1-. Fun . Fir-iv iii-{I3}=t={l5] \"I\"y{Ftty —+. Fna . Fyy] {l-ii} iv=t={lfiIt h\".r\"tt\"y{Fry —i-. Fxx - Fyyl {l5} u Ctnce a deduetien is diseevered, it is easily eneugh revised se as teeliminate unnecessary intervening steps. rt stretch ef unquantified dedue-tien such as appears in lines {-1}, {T}, {ltltl-{till is beund te emhedy asingle nuth-functienal implicatien. In the precess ef discevery we builtup the implicatien piecemeal, but new that its end peints are visible wecan verify meebarrically that {Isl is truth-functienally implied directly bythe cenjunctien ef {st}, {T}, and { ltltjt. Se in retrespect we can refine eurdeduetien by deleting {l l]|-{l3_l and justifying {[4} directly by the cita-tien ‘{4}{T]{ll]}‘. Incidentally, fer cendensatien we might emit {5]l— [9]and just write ‘similarly‘ after { ltlll. The advantage ef having aimed fer {Id} in the ferm ‘Few —1-.Fun . Fwtv‘. rather than in the ferm ‘Fry -1. F.r.t - Fyy‘, may be ap-preciated by rewriting the abeve deduetien with ‘.r’ attd ‘_v‘ in place ef ‘tr‘and ‘iv‘ everywhere. Difficulty will be feund te arise in {E}. The strategy ef reductie ad absurdum, which underlay the mainmethed, can usefully be revived where ether strategies fail. lt eensists inassuming the centradictery ef what is te be preved and then leeking fertreuble. It may be illustrated by deducing ‘—~ Elr Fr‘ frem *\"v'.r- —F.r': stti vi - Fr t ll stilt - Fir {3} .1: **~{3] Hr Fx *{-I‘-l} =t==r{-fl} Fr l3ll5l *t[5] 31 Fx —r Fr rtfil — 31 F1Here the usual strategy ef quantifiers leads frem{l]te{1}, leaving us withthe preblem ef preceeding thence te {I5}. Resetting te reductie ad absur-dum, we assume the centradictery ef the desired {til as {3}. Thence by thestrategy el‘ quantifiers we meve te {4}, and find the treuble we wereleeking fer; fer {4} cenflicts with {Z}. Cd and TF tlten lead threugh {5}te {ti-II. A supplementary strategy werth remarking is that ef tire dilemma,which is useful in getting a cenclusien frem an alternatien. First deducethe desired cenclusien separately frem each cempenent ef the alternatien,and derive a cenditienal in each case by Cd; then frem these cenditienalsand the eriginal altematien infer the cenclusien by TF. See page 243.

-ill Dedut'ti'-'e Err-elegy 155EHERCISEEI. ln striet analpgy tn the deduetien last presented. deduee '— V: Fr‘ frnm ‘H2: — Fx’-2. Establish the rest pf the rtties pf passage. namely {Z}. if-4}. {5]. [5]. and [E] pf Chapter 23. by pairs pf deduetipns.3. Establish by detluetinn the syllpgisms in Chapter lb. ineludirig the reinfereed nne abeut Spartans. Model: the abpye deduetipn abeut sitwetily persens.4- Similarly fpr the inferenee abeut witnesses. middle e-ti Chapter I7.5- Deduee ‘Hit —F.t' frem '-‘tt\"s Fx’. Plan pn haying this inter- mediate line: =l={4] F1 —1- \"liI\"Jr Fit *[3'JFtnalngnusiy. deduee \"\"iI\".t —F.r‘ frnm ‘ — 3.1: Flt‘-I5. Establish the equiyalertee tif ‘ltfxtp . Fir —+ F1)’ tn ‘p' by mutual deduetien. Hint: ‘p’ trtith—fiiaetinnally implies ‘p . F1 —1- Fit’.T\". Establish the eqtiiyaienee pf ‘Brig: . FA.‘ -1 Fr)‘ [Cl ‘p’.3. The fnllbwing are eseerpts frnm a deduetiye sdltttitm nf Esereise 4 t:-f Chapter E5. Ctirnplete it.=I=-E3\"; GI —1- H: {1}\"If-='r-} Hr|[F.r . G1} ts) IHts) Fr . ta: {E} y=t=ee{'l} \"a\".rI[H:: —* *.-'.r,'|- =I=t[9jl t3][5_',h[1fi]=l=*{llIi Hy —1- —-Jy .-1. Fl —i- —H.1:=l==l==I.'l 1} Hy . Jy 9. Establish the infetenee abeut the elass erf Till. end bf Chapter 17. Warning: This deduetien is a herliday yenture. Hdne pf the tmth- funetipnal implieatibns iiwelyetl is as ftlrrmitlahle as that whieh leads tp {I 1] in the preeeding esereise. but the deduetien rues td 1E lines [in my versien anyway]. seme pf whieh are ndt easily eprne hy. It tends tp heighten nne's appreeiati-an pf the snlutiein in Chapter I9.ill]. Deduee 'Fyy‘ fmm V1 Fity it ‘ix F‘y.t'. fellewing the strategy pf the dilemma



|vGLIMPSES BEY_OiD



41SINGULAR TERMS The Idgic rif mrth functiens and quantificatiun is new under ccntrcl.snme simple serts cf inference. hewever. still want discussien—-eietahlythese turning upen singular terms sueh as ‘Secrates‘:All men are mertal. Secrates is a Greeit.Sdcrates is a man; Scerates is wise: Seer-ates is mertal. Seme Greelts are wise.Furthertndre the thcery til’ identity. including such evident laws as ‘I = it‘and ‘rt = y .-H-. y = it‘. remains untnuched. Ft few chapters will sufficctc dc justice te singular terms and identity. There will be an epticinaleitcursiun inte semething called term fnncters. and in cenclusien we shallhave a brief glimpse cf set theery. tit the theery nf classes—a disciplineintimately linked tc legic but better regarded as the basic discipline cfclassical mathematics. What were called \"terms\" in Chapters l4 and ET and represented by‘F‘. ‘G’. ete.. are general terms. as eppesed tn singular terms. Hutgenerality is net tn be cenfused with ambiguity. The singular term ‘.lenes'is ambigueus in that it might be used in different cpnteitts tc name any cfvarieus persens. but it is still a singular term in that it purpcrrts in anyparticular cnntest tn name cine and enly ene persen. The same is true evencf pruncuns such as ‘l‘ and ‘thuu‘; these again are singular terms. butmerely happen tu he highly ambigueus pending determinatien threughthe cnnte:-tt err nther circumstances attending any given use nf fl1em. Thesame may he said cf ‘the man‘. pr mere clearly ‘the President‘. ‘thecellar‘; these phrases {unliltc ‘man'. ‘president’. and ‘cellar’ themselves}are singular tenns. but the unc and enly ene ebject tc which they ]Ill.ll'|J'EJ-11td refer in any given use depends tin attendant circumstances fur itsdeterminatien. Besides the classificaticn cf terms inte singular and general. there is acress ciassificatien intn cnncrete and elistrnet. Cencrctc terms are thdsewhich purpcrt tc refer tn individuals. physical ebjects. events; abstractternns are these which purpert tu refer te abstract ebjects. c.g.. tc num-bers. classes. attributes- Thus seme singular terms. c.g.. ‘Sucrates‘. 259

see IF- Gllinpses li‘eyan-rl‘Cerberus‘. ‘earth‘. ‘the anther ef Waverley‘. are cenerete. while ethersingular terms. c-g.. ‘T‘. '3 + rt‘. 'piety‘. are abstract. i'-'i.gain seme gen-eral terrns. e.g., ‘man’. ‘heuse’. ‘red heuse‘. are cenerete [since eachman er heuse is a cenerete individual}. while ethers. e.g.. ‘primenumber‘. 'aeelegical species‘. ‘virtue‘. are abstract [since each number isitself an abstract ebject. if anything. and similarly fer each species andeach virtue]- Cautinn then cemes te be needed in eur tallt ef term abstracts. te whichwe have been accusteming eurselves: fer a term abstract may er may netbe an abstract term. rt term abstract is a general term. and an abstract eneenly if the several ebjects ef which it is true happen te be abstract. lnChapter i-tti we shalt indeed eentemplate the beld step et’ letting generaltenns de deuble duty as names ef their estensiens and thus letting termabstracts serve as abstract singular terms. but enly en a par with etl1ergeneral terms. The divisien ef terms inte cenerete and abstract is a distinctien enly inthe ltinds ef ebjects referred te. The distinctien between singular andgeneral terms is mere vital frem a legical peint ef view. Thus farl havedrawn it enly in a vague way: a term is singular if it purperts te name anebject {ene and enly ene}. and etherwise general. l\"-lete the ltey werd‘purperts‘: it separates the questien eff frem such questiens ef fact as theeitistence ef Sc-crates and Cerberus. ‘Whether a werd purperts tu name eneand enly ene ebject is a questien ef language. and is net ctintingent enfacts er esistence. In terms cf legical structure. what it means te say that the singular term“purperts te natne ene and enly ene ebject\" is just this: The singularterrn belengs in parlrlanr aftlte l:lna’ in tvlilclt lt tvealrl alse lie calrerenr tause varlaliler ‘.i:‘. ‘y’. etc. (er. in erdinary language. preneuns]. Centeittslilte:Secrates is wise. Piety is s virtue.Cerhertts guards the gate. T = 3 + 4,etc-. are parallel in ferm te epen sentences:I is wise. I guards the gate. .r is a virtue. .1’ = 3 + 4such as m.ay eccur in clesed statements having the ferm cf quantificatiens:‘§.t't_‘.r is wise)‘. etc. The terms 'Secrates‘. ‘Cerberus‘. ‘piety‘. and ‘T

el . Singular l\"erni.r 2151are. in shed. substitutable fer variables in epen sentences witheut vie-lence re grammar; and it is this that mal-res them singular terms. Whetherthere is in fact such an ebject as Secrates (which. tenselessly. there is} erCerberus {which there is net] er piety er T {en whieh philesephers dis-agree] is ef cniirse a separate questien. General terms. in centrast te singular enes. de net eccur in pesitiensapprepriate te variables. Typical pesitiens ef the general term ‘man’ aresecn in ‘Secrates is a man‘. ‘All men are mertal‘; it weuld net malte sensete write:[ll Se-crates is an .-1'. All I are medal.er te imbed such ertpressiens in qtiantilieatiens in the fashien:[2] Hhrtfiecrates is an .r]|=.-[3] ‘til‘.r{all .1: are mertal —+ Secrates is mertal]-.The ‘.r‘ ef an epen sentence may refer te ebjects ef any tdnd. but it issuppesed te refer te them ene at a time; and then applicatien ef \"lt|l.r‘ er‘‘ Hr‘ means that what the epen sentence says cf .1: is true ef all er semeebjects talten thus ene at a time. There are indeed legitimate epen sentences semewhat resembling [1]but phrased in terms ef class membership. thus:{4} Secrates is a member ef .r. All members efir are mertal.But these de net. lilte til. shew Zr‘ in place ef a general term such as‘man‘: rather they shew ‘.r’ in place ef an abstract singular term. ‘man-ltind‘ (‘class cif all men‘). as in ‘Secrates is a member -ef manltind‘. ‘iltllmembers cf manltind are mertal‘. The epen sentences [4] may quiteprnperly appear in quantificatiens:[ii irtffiecrates is a member efr].{ti} \"tt‘.r[all members ef .r are mertal -—i- Secrates is mertal).incidentally [til can be further analyaed:{T} li\".r[\"tt'yt_fy is a member ef it —1- y is mertalfi —=- Secrates is mertall-

Etiil IF. Gllinpres Beyanrlllis an alternative te {4} we might alse appeal te attributes instead efclasses. thus:[ii] Secrates has I. Everything that hasr: is mertal.Here ‘.r’ appears in the pesitien ef an abstract singular term such as‘humanity‘ which purperts te narne an attribute- Cluantifieatiens anale-geus te l_'5}—t_”l} can then be built en [ti]- lust as the sentence letters in a schema stand as dummy sentences andthe terrn letters as dummy general terms. se die free variables may be seenas standing as dummy singular terms. Te represent the abeve syllegismsabeut Sncrates schematically. then. we may simply use a free ‘y' lerepresent ‘Secrates‘. The inferences are then justified directly by the twevalid cenditienals: \"i\"‘.r|'.'Fr: -Pfiirl . Fy .—r tfly. Fy . Gy .——r~ 3.:t|[F.r . Gr]. rltnetber example: eiisiuisss: rltldrich bribed every member ef the cernrriittee. Barr is a member cf flie cemmittee; C{}HELLl5IEIl\"~lt Semeene bribed Barr.This is justified by the validity ef the cenditienal: ‘lt\".t[F.r A-Gar] - Fw -1 Hr: Girw.which gees inte a pure eitistential and is quicltly checlted. These valid cenditienals shew that the cenclusieas will ceme eut trueif the premises de. ne matter what ebjects ef the universe we cheese ininterpretatien ef ‘y’. ‘s‘. and ‘w‘. ln particular diereiere we may cheeseSecrates. Aldrich. and Barr--previded merely that the universe centainssueh things. But this last pi-avian is essential te the intended applicatien ef eurdeductive results. .4. singular term may er may net name an ebject. llisingular term always purparts te name an ebject. but is p-ewerless teguarantee that the alleged ebject be ferthcerrring; witness ‘Cerb-ertis‘. Thedeductive techniques cf quantilicatien theery with free variablesservevery well fer inferences depending en singular terms when we are assuredthat there are ebjects such as these terrns purpert te name: se this questien

4 l‘ - Singular Terms 153cf existence then becemes the central questien where singular terms arecencemed. l shall find ne use fer the narrew sense which seme philesephers havegiven tn ‘existence’. as against ‘being’; vi:-.'-. cencreteness in space-time-lf any such special cennetatien threatens in the present pages. imagine‘exists’ replaced by ‘is’. When the Parthenen and the number T are said tebe. ne distinctien in the sense cf ‘be’ need be intended. The Parthenen isindeed a placed and dated ebject in space—time while the number T (ifsuch there be] is anether sert ef thing; but this is a difference between theebjects cencemed and net between senses ef ‘be’. ln centrast te T and the Parthenen. there is ne such thing as Cerberus;and there is ne such number as tlltlt. Clearly these repudiatiens de net efthemselves depend en any limitatien ef existence te space-time. Themeaning ef the particular werd ‘Cerberus’ merely happens te be suchthat. if the werd did name an ebject. that ebject weuld be a physicalebject in space and time. The werd ‘Cerberus’ is lilte ‘Parthenen’ and‘Hucephalus’ in this respect. and unlilte ‘T’ and ‘[tl'[l‘. But the werd‘Cerberus’ differs frem ‘F‘arthenen‘ and ‘Hucephalus‘ in that whereasthere is semething in space-time such as the werd ‘Parthennn’ purperts tename t[vi:'.:-. at Athens fer seme deitens ef centuries including part er all el’the twentieth}. and whereas there is ftenselesslyjt presumably semethingin space and time such as the werd ‘Hucephalus’ purperts te name {via-.at a successien ef pesitiens in the hlear and lvliddle East in the feurthcentury rt-c-fl. en the ether hand there happens te be nething such as thewerd ‘Cerberus’ purperts te name. near er remete. past. present. erfuture. lt is surely a cemmenplace that seme singular terms may. theughpurperting te name. flatly fail te name anything at all. ‘Cerberus’ is eneexample. and ‘t-it'll’ is anether. But. cemmenplace theugh this be. experi-ence shews that recegnitien ef it is beset with persistent cenfusiens. te thedetriment ef a clear understanding ef the legic ef singular terms. Let usmalte it eur business in the remainder ef this chapter te dispel certain efthese cenfusiens. There is a tendency te try te preserve seme shatlewy entity under thewerd ‘=I.\"_Ierbetus’. fer example. lest the werd lese its meaning. ll’ ‘Cer-berus’ were tneaningless. net enly weuld peetry sufl'er. but even certainblunt statements ef fact. such as that there is ne such thing as Cerbenis.weuld lapse inte meaninglessness. Thus we may hear it said. c.g.. that-Cerberiis exists as an idea in the mind- But this verbal maneuver cendueesenly te cenfusien. Ctf a tangible ebject such as the Parthenen. te change

2154 lit’- Glimpses lieyaiidthe subject fer a mement. it weuld be wantnn ebscurantism te affirm sriaulrle existence: in llithens and in the mind- Far mere straightferward teadmit twe {er marry] ebjects: the tangible Parthenen in Athens. and thePardienen-idea in the mind {er the Parthenen-ideas in many minds}. ‘Parthenen’ names the Parthenett and enly the Patthenen. whereas ‘thePaithenen-idea‘ names the Parthenen—ldea. Similarly net ‘Cerberus’. but‘the Cerbertis-idea’. narnes the Cerb-erus-iclea; whereas ‘Cerberus’. as ithappens. names nething. This is nnt the place te try te say what art idea is. er what existence inthe mind means- Perhaps frem the peint ef view ef experimental psychet-egy an idea sheuld be explained semehew as a prepensity te certainpatterns ef reactien tn werds er ether stimuli ef specified itinds; andperhaps “existing in the mind” then means simply “being an idea.” Butne mauer: the idea ef “idea” is entertained here enly as a ceneessien tethe edter party. The peint is dtat theugh we be as liberal abeut eeun-tenancing ideas. and ether nenphysical ebjects as anyenc may aslt. still teidentify the Partltenen with the Parthenen-idea is simply te cenfuse enething with anether; and te try te assure there being such a thing asCerberus by identifying it with the Cerberus-idea is te malte a sintitarcenfusien. The elfert te preserve meaning fer ‘Cerberus’ by presenting semeshadewy entity fer ‘Cerberus’ te name is misdirected: ‘Cerberus’ remainsmeaningful despite net naming- lvlest werds. like ‘and’ er ‘s-alte‘. ai‘equite meaningful witheut even put'petting te be names at all. Even when awerd is a name ef semething. its meatting weuld appear net te be identifi-able with the thing named.1 lvleunt Everest has been ltnewn. frem eppe-site peints el’ view. beth as Everest and as Chemelungma;* here thename-d ebject was always ene. yet the names can scarcely be viewed ashaving been stilte in meaning er synenymeus; fer ne insight inte thecembined minds ef all users ef ‘Everest’ and ‘Chemelungma’ ceuldreveal that these named the same thing. pending a strenueus investigatienel’ nature. Again there is Frege’s example ef ‘Evening Star’ and ‘lvtemingStar’; the named planet is ene. but it teelt astrenemy and net mereanalysis ef meanings tc establish the fact. Precise and satjsfactery fermulatien ef the netien cf meaning is anunselved preblem ef semantics- Perhaps the meaning ef a werd is bestcenstrued as the asse-ciated idea. in seme sense ef ‘idea’ which needs te l This much neglected peint was well urged by Frege. “l_lber Flinn urtd H-edcutung. ' ‘ * Erwin Schriii:tinger. lit‘-liar is .Lijt’eF’. last paragraph. nearly enmigb-

ri l. Singular Tet-ins 255be made precise in turn: er perhaps as the system ef implicit rules incenfermity with which the werd is used. suppesing that a criterien ef“implicit rule” can be devised which is selective eneugh te allew same-ness ef meaning nn the part ef distinct expressiens. Perhaps. indeed. thebest treatment ef the matter will preve tn censist in abandening all netienef se-called meanings as entities: thus such phrases as ‘having meaning’and ‘same in meaning‘ might be tlrepped in faver ef ‘significant’ and‘synenymeus‘. in hepes eventually ef devising adequate criteria efsignificance and synenymy invelving ne exciirsien threugh a realm cfintermediary entities called meanings- Perhaps it will even be feund thatcf these enly significance admits ef a satisfactery criterien. and that alleffert te malte sense ef ‘synenymy’ must be abandened aleng with thenctien ef meaning.\" l-lewever all this may be. the impertant peint ferpresent put-prises is that significance ef a werd. even efa werd which [Iii-te‘Cerbertts‘} purperts te he a name. is in ne way centingent upen itsnaming anything; and even if a werd dne_s name an ebject. and even if weceuntenance entities called meanings. there is still ne call fer the namedebject te be the meaning. The mistalten view dtat the werd ‘Cerberus’ must name semething inerder te mean anything turns. it has just new been suggested. en cenfu-sicn cf naming with meaning. But the view is enceuraged alse by anetherfacter. vii-'.. eur habit ef thinking in terms el’ the misleading werd ‘abeut’.If there is ne such thing as Cerberus. then. it is asl-red. what are yeutallting airriur when yeu use the werd ‘Cerbems’ {even te say that there isne such thing}? Actually this pretest ceuld be made with the same ce-gency (via. nene] in ceuntless cases where ne weuld-he name such as‘Cerberus’ eccurs at all: ‘tli-‘hat are yeu tallting abeut when yeu say thatthere are ne Enlivian battleships‘? The remedy here is simply te give upthe unwarranted netien that tallting sense always necessitates there beingthings tallied abeut. ’l’lte nctien springs. ne deuht. frem essentially thesame cenfusien which was just previeusly railed against; flten it wascenfusicn between meanings and ebjects named. and new it is cenfusi-enbetween meanings and things tallied abeut. This mistahen view that ‘Cerberus’ must name semething has beenseen te evelte. as ene lame effert tn supply a named ebject. the netinn thatCerberus is semething in the mind. Ctther expedients te the sante end arecemmenly enceuntered- There is. c.g.. the relativistic dectrine accerdingte which Cerberus exists in the wnrld ef Crreelt mythelegy and net in thell Sci: my Freer a l'.r.t_glr'.-:.rl Frrlttt tri’ l\"t'i.*I-1-'- Essays ll and lll-

Ififi ll-\". Glimpses Heyartirl werld cf medern science. This is a perverse way ef saying merely dtattireclts believed Cerbenis te exist and that [if we may tiust medemscience thus far] they were wt‘eng. lvlyths which aflirm the existence efCerberus have estltetic value and antlrrepelegical significance; mereeverthey have internal structures upen which eur regular legical techniquescan be breught te beat‘; but it dees happen that the myths are literallyfalse. and it is sheer ebscurantism te phrase the matter etherwi se. There isreally enly ene werld. and there is net. never was. and never will be artysueh thing as Cerberus- Anethcr such expedient which had best net detain us leng. if enlybecause ef the mazes el’ metaphysical centreversy in which it weuldinvelve us if we were te tarry. is the view that cenerete individuals are el’twe ltinds: these which are actuali:-red and these which are pessible but netactua]iaed- Cerberus is ef the latter ltind. aceerding te this view; se dtatthere is such a dting as Cerberus. and the preper centent ef the vulgardenial ef Cerberus is mere cerrectly expressed in the fashien ‘Cerberus isnet actualieed’. The universe in the breader sense becemes badly ever-pepulated under this view. but the cemfert ef it is suppescd te he dtatthere cemes te lie semetlting Cerberus abeut which we may be said te betallting when we rightly say [in lieu ef ‘There is ne such thing as Cer-berus’] ‘Cerberus is net actualiaed‘. But this device depends en the pussibility ef Cerberus; it ne lengerapplies when we shift eur example frem ‘Cerberus’ te seme weuld-bename nf cemplex fenrt invelving an eut-and-eut impnssibility. e.g., ‘thecylindrical pyramid ef Cepilce‘. Having already cluttered the universewith an implausible let ef unactualiaed pessibles. are we te ge en and adda realm cf unactualiaable impessihlesl‘ The tendency at this peint is techeese the ether hem cf the suppescd dilemma. aitd rule that expressiensinvelving iinpessibility are meaningless. Thence. pedtaps. the net un-cemmen netien that a statement legically incensistent in ferm must bereclassified as a nenstatement: net false but meaningless. This nctien.besides being unnatural en the face ef it. is impractical in that it rules eutthe pessibility ef tests ef meaningfttlness; fer legical censistency admitsef ne general test. ner even ef a eemplete preef precedure [see ChapterS4]. All this piling ef expedients en expedients is. insefar as prempted by anctien that expressiens must name tn be meaningful. quite uncalled fer.There need be ne mystery abeut attributing nenexistence where there isnething te attribute it tn. and there need be ne misgivings ever the recan-

til .i . Singular Tenns Edi‘ingftilness el’ werds which purpert te name and fail- Te purpert te nameand fail is already preef ef a full share ef meaning. lf fer ether reasens the recegnitien ef unactuali-ted pessibles is felt tuhe desirable. there is nething in the ensuing legical theery that needcenllict with it as leng as essential distinctiens are preserved. Se-calledmental ideas and sn-called meanings were previsienally telerated abeve;“ideas” ef the Platenic stripe. including unactualiecd pessibles. ceuld beaccemmedated as well- What must be insisted en is merely that suchshadewy entities. if admitted. be named in seme distinguishing fashien:‘the Cerberus-pussibility’. er ‘the Cerbertts-idea’. er ‘the meaning ef‘Cerberus’ ’. If we can get tegelher en this much by way cf cenventien.flten everyune can be left te his faverite metaphysics se far as anythingfurther is cencemed. The essential message te be carried ever frem thischapter inte succeeding enes is simply this: Seme meaningful werdswhich are preper names frem a grammatical peint ef view. netably ‘Cer-berus’. de net name anything.‘EIERCEE5I. Shew dtat the premises: -I Hart and Williams did net bedt centribute. lf Elalte centrihuted then se did everybedy imply the cenclusien: Elalte did net centribute. allewing free variables te stand fer the preper names.1. Shew that the premises: Edith envies everyune mere fertunate than she. l-terbcn is ne mere fertunate than any whe envy him imply the cenclusien: Herbert is ne ntere fertunate than Edith- * Fit-r nrrrrc en this theme 5-Ee Russell. “Eta derieting.” and my Frrnn rt Lngical Paint-r.1_|\" l-\"’t’E\"|-‘I-’. Essaysl and \"v’l-

263 IF- Giirnpres £i‘eyent:t' 42 IDENTITY identity is such a simple and fundamental idea that it is hard te explainetherwise than threugh mere synenyms. Te say that x and y are identicalis te say that they are the same thing. Everything is identieal with itselfand with nething else. But despite its simplicity, identity invites cenfu-sien. E.g., it may he aslted: Elf what use is the netien ef identity ifidentifying an ebject witl1 itself is trivial and identifying it with anythingelse is false’? This particular cenfusien is cie ared up hy reflecting that there are reallynet just twe ltinds ef cases te censider, ene trivial and the ether false, hutthree: Eicere = Cicere. Cicere = -Catiiine, Eieere = Tully.The first ef these is trivial and the secend false, hut the third is neithertrivial ner false. The third is infermative, hecause it jeins twe differentterms; and at the same time it is true, hecause the twe terms are names efthe same ebject. Fer trufli ef a statement ef identity it is necessary enlythat ‘=’ appear hetween names ef the same ehjeet; the names may, and inuseful cases will, themselves he different. Fer it is net the names that areaflirmed te he identicai, it is the things named. Cicere is identical withTully [same man}, even theugh the name ‘Cicere' is different frem thename ‘Tully’. Te say anything abeut given ebjects we adjein apprepriatewerds te names ef the ebjects; hut there is ne reasen te espect that what istherehy said ef the ebjects will he true alse ef the names themselves. TheHile. e.g., is lenger than the Tusceleesahatehie. but the names are eppe-sitely related. Since the useful statements ef identity are these in which the namedebjects are the same and the names are different, it is enly because ef apeculiarity ef language dtat the netinn ef identity is needed.‘ Still, nelinguistic investigatien ef the names in a statement ef identity will su ffice,erdinarily, te determine whether the identity helds er fails. The identities: 5 T|‘l|l-ls it was that ‘Hume had treuble aeeeunting fer the erlgin ef the identity idea inexperience. See Tre-zttise e_t'Hurnnn Hntnre, Eh. l. Pt. I‘-i’, Sec. Il.

-rs. r'rfe'fi'l'li\"}' _ tat-1'Everest = Chemelungma [see Chapter -ti},Evening Star = lt-tlerning Sitar,25th President ef l_l-S. = first President ef l.l-S. inaugurated at =1-2,lvlean temperature at Tustla = €'t3'“Fall depend fer their suhstantiatien upen inquiry inte eittra-linguistic mat-ters ef fact. A pepuiar riddle, cemmenly asseciated with identity. is this: Hew cana thing that changes its suhstance he said te remain identical with itself?Hew is it, e.g-, that enels hedy may he spelten ef as the same hedy ever aperied ei years? The prehiern dates frem Heraclitus, whe said “Yencannet step inte the same river twice, fer fresh waters are ever flewing inupen yeu.\" Actually the key te this difficulty is te he seught net in theidea ef identity hut in the ideas ef thing and time- A physical thing—whether a river er a human hedy er a stene—is at any ene meme nt a sumef simultaneeus mementary stages ef spatially scattered atems er ethersmall physical censtituents. New just as the thing at a meme nt is a sum efthese spatially small parts, se we may thinlt ef the thing ever a perled as asum ef the temperally small pa_rts which are its successive mementarystages. Eemhining these cenccptinns, we see the thing as extended intime and in space alike; the thing becemes a sum ef mementary stages efparticles. er hriefly particle-mements. scattered ever a stretch ef time aswell as space. All this applies as well te the river er human h-edy as te thestene. There is enly a difference ef detail in the twe cases: in the case efthe stene the censtituent particle-mements pair eff fairly cempletely fremene date te anether as mementary stages ef the same particles, whereas inthe case ef the rivet er human h-edy there is mere heteregeneity in thisrespect. The river er human hedy will regularly centain seme mementarystages ef a particle and eitclude ether mementary stages ef the sameparticle, whereas with the stene. barring small peripheral changes erultimate destruetien, this is net the case. Here we have a distinctienreminiscent ef the distinctien in traditienai phitesephy hetween “medes\"and “su I:-stance s\". But things ef heth ltinds are physical things in ene andthe same sense: sums ef particle-mements. And each thing is identicalwith itself; we can step inte the same river twice. What we cannet de isstep inte the same tempnral part ef the river twice, where the part istemperally sherter than a stepping-while. Diversity amettg the parts ef awhele must net he allewed te ehscure the identity ef the whele, ner efeach part. with itself-

ETD H-\"'. tfiliatprer Heytim-ti‘ Thus far we have been thinking ef statetnents ef identity cetttpcrsed ef *=‘ flanked by singular terms. But ‘=’ is an erdinary relative term, and se may be flartked as well by variables; e.g.: 's'r'tt'yp:isaged. y isag-ed .—tt. .1: =y), El.r{.rlsaged.‘I'y{yisaged —r..r ==y]], Q0-'3y{.tisaged.yisaged..r qty], 'tt'.r‘tt\"y’tt'g{risaged.yisaged-eisagni;t.—r-:.r=y.v..r=.r.v.y =1}. t_\"Ihe netatien ‘J: at y’ is a cenvenient abbreviatien fer ‘-— tr = y}‘.} As the reader can verity en a linle reflectien, these feur statements ameunt respectively te the fellewing: There is ene ged at mest,“ There is esactly ene ged, There are at least twe grids, There are twe geds at mest. Statements cf identity cenaisting ef '=’ flanked by siugtdar terms areneeded because twe singular terms may uarne ene thing. Hut the need ef‘=* flanked by variables arises frem a diflerent peculiarity cf language;v'ia., frem its use ef multiple varlahles ef quantificatiun ter their pre-neminal aualegues in erdinary language}. Twe variahles are ailewed terefer te the same ebject, and drey are alsn ailewed te refer tu differentebjects; and thus the sign cf identity eemes te be needed when, as in theabeve feur esatnples, there arises the questien ef sameness er differenceef reference en the part ef the variables. Frem a legical peint ef view it isthe use ef the identity sign between variables, rather than between singu-larterrrts, that is fundamental. We shall see, indeed [Chapter 44], tlratthewhele eategery ef singular terms is theoretically supetflueus, and thatthere are legical advantages in thinking ef it as theeretieally cleared away. The legic ef identity is a branch net reducible te the legic ef quan-tifieatien. its netatien may be theught ef as cemprising dtat ef the legic efquantificatien plus the ene additienal sign '='. Thus the sehemata ef thelegic ef identity are the same as the quantificatienal sehemata eitcept dtatthey may centain, aleng with the clauses *p*, ‘a*, ‘Fr’, ‘Gay’, etc.,additienal clauses ef the ferm el’ identities: ‘J: = _v‘. ‘.r = r‘, etc. \"'v’atiditymay be defined fer schemata ef the legic ef identity precisely as it was ‘ This, aceerding tea quip ef'W'ltitehead's, is theereed t;rffl'|¢ [J|qj[g|-ig|-y;_

='l.'il- Identity‘ I'lldefined fer quantificatienal schemata {Chapter 21'}- Thus there are validschemata sttch as: iris =1-*-Fr.'t—* Elrts =-rlwhich are like valid quantificatienal schemata except that ‘.r = .r',‘s = y‘, etc., tum up instead ef schematic clauses ‘t.'i.r.t’, ‘Gs,-a‘, ete.These are valid still by virtue simply ef their quantificatienal structure.and independently ef any peculiarities ef identity. Hut there are furthervalid schemata whese validity depends specifically en the meaning efidentity; ene such is;ill F.t..r=y.—“i Fy-Fer. censider any eheice ef universe. any interpretafien ef ‘Fl therein,and any assignment ef ebjects te the free variables ‘x’ and 'y'. lf theebject assigned te Zr’ is the same as that assigned te ‘y’, and is an ebjectnf which ‘F l is interpreted as true, then {ll cemes eut true threugh trudt efits censequent ‘Fy’; and in any ether case {ll cemes eut true threughfalsity ef its antecedent. Threugh having added the identity sign te eur legical netatien we findeurselves able, fer the first time, te write genuine sentences witheutstraying frem eur legical netatien. Hitherte, schemata were the h-est weceuld get; estralegical materials had tn b-e imperted frem erdinary lan-guage when. fer the sake ef illustraticn, a genuine sentence was wanted.ln ‘x = .:t' and *'h\".rtf.r = .=tl*, hewever, we have sentences—the first epen,the secend clesed and true. Any schemata ebtained by putting identities it = .:t', ‘I = y’, etc., inplace ef say ‘Get’. Tir_t\", etc., in valid quantificatienal schemata are. ithas been esplained, valid schemata ef the legic ef identity. Hut suchputting ef identities fer schematic clauses can, when thereugh, yield asentence instead ef a schema, e.g.: .1‘ =.t--I Hyls =yl.The cencept ef validity rnay he applied te such sentences aleng with theschemata, fellewing the precedent ef Chapter Eh- This e:-tamplc is even quantificatienally valid, in the sense ef thatchapter. Ctf these whese validity depends specifically en the meaning efidentity. en the ether hand, the simplest is:

ZTE ll\"- Glimpses Eeyend{II} I = I.t'-'i.nether, truth-functienally implied by the twe feregeing sentences, is‘iris = rt’- Eubstitutiens may be made fer ‘F‘ in {ll just as esplained in Chapter23, cscept that the substituted ettpressiuns may new centain identity signsinstead cf term leuers- lteflectien en the general mechanics ef substitutien[Chapter 215} reveals that, in {ll in particular. ‘Fr’ and ‘Fy’ may in effectbe directly supplanted respectively by any schemata that differ enly in thatthe ene has free Zr‘ in seme places where the ether has free ‘y’- Fer,picturethe twe schemata as ‘. . ..'r....1r...y- - .‘ and ‘- . -r- - .y. _ .y.-.‘;then the abstract theereticalty substituted fer ‘F‘ weuld be ‘ls: . . .1 . - .3. . -y. . Thus ene result ef substitutien in {ll is: w=.1r..t;=_y ..-—!-.1-v=_y—the law ef u-ansitivity at identity (cf. Chapter Iii]. The universal clesures ef {ll and {ll}, via: 'fl'.th\"y[F.r - .t = y -1 Fy}, \"tt\".t{.r =1],tegether with the universal clesures ef any results cf substituting fer ‘F ’ intil, will be called urinals‘ cf identity. New the technique ef preef inquantificatiun theery carries ever te the legic ef identity; we simply treatthe axierns as premises. E.g., the law ‘.1: = y .—r. y = .r' ef the sym-metry ef identity fellews truth-functienally frem the case: .:t=.t-.t=y.—t.y=.ref [I] tegelher with ill}. Fer further illustraticn let us shew \"\"it'y[.t = y .—i- Fyl’ equivalent te‘F.r'. by establishing implicatien fervvard and backward in the style ef theend ef Chapter 3?. [ll --1.], hlylr = y .-it Fy1I—i- Fr. {h\"_v I -—-1-.] Fir -i- h\"y|‘,r = y .—1- Fy}.Each ef these full lines is menadically valid. It must be remembered that in these preefs the identity sign behavesmerely as an inert predicate letter, as if ‘I = .r' and ‘.1: = y‘ were ‘GM’

4.? . identity ET3and ‘Gay’- The peculiar preperties ef identity are inveked esplicitly.when wanted, by citing an asient ef identity. It is cenvenient semetimeste cite the asiem with its quantifier er quantifiers and semetimes wiflteut,as the abeve esamples shew- Htstctlttcat r-tens east shewed in rtea lTl'teercm vtn eathese attiems ef identity are eemplete. That is, every valid schema ersentence ef the legic ef identity can he preved fmm them by the legic efquantificatien.EIERCISEEl. Writing \"y\" fer ‘Harri, ‘;;‘ fer ‘the cashier‘. and ‘F‘ fer ‘had a key‘, put the statentcnt: l'~lene but Barr and the cashier had a key inte legical netatien with help ef identity. F-hew that this and: Semeene whe had a key teek the briefcase imply the cenclusien: Barr er the cashier teek the briefcase with the help ef an attiem ef identity.E. Preve the equivalence ef: 3.r[F.'t . Jr = y]-, Fy by preving the twe intplicatiens with help ef asiems ef identity-

ET4 IF. Glimpses Beyunrt 43 DESCRIPTIONS It is usual in legic te write ‘[11]’, with an inverted ieta, te mean ‘theebject I such that‘. Thus the cemples singular tenns ‘the anther efltlfnverley‘ and ‘the prime number between 5 and ll‘ hcceme:lrrltir wrete l-Fuuerleyl, {'.u:][.r is prime . 5 =-“Ix si ll}.Singular terlns are called descriptiertt when written in this ferm. Thesingular terms ef erdinary language which may be represented thus asdescriptiens begin typically with the singular ‘the’, but by ne meansnecessarily se, as these esamples shew:what he went after, trxltfhe went after Jtl;where he was hem. t[r:t]t[he was b-ern at it};_lehn‘s mether, {salts bere Jehnl;Smith's heuse. {trite is a heuse . .1‘ is Smith‘s]. In general a singular term purperts te name ene and enly ene ebject.and in particular a singular term ef the ferm ‘t1.t}F.r‘ purperts te name theene and enly ebject ef which the general term represented by ‘F‘ is true.Thus, if y is the ebject {'.-.r]F.r. then y must be such thatFy . F nething-but-y-This cenjunctien ameunts te saying dtat, fer each things. ‘F‘ is true ef .rifs = y, and false efr etherwise. ln shert:[ll ‘liI\"..t[F'.r -t-t-. .1: = yl.E.g., te say that Scett is [1.t}t_'.t wrete Waverley} is te say that‘ll'.t{.t wrete Waverley H. .1‘ = Eeett].If ‘F‘ is true ef nething er ef many things, then there is ne such thingas t[r:t]F.1:. Actually the term appearing in the rele ef the ‘F ‘ ef ‘ital Fr‘ in

-tl'_i|'- Best-riptians 175verbal examples frem erdinary disceurse very frequently needssupplementary clauses te narrew it dewn tn the peint ef being t|'ue ef enlyene ebject, but this situatien can cemmenly be viewed merely as a ease efthe familiar practice ef depending en ctrntcxt er situatien te t'eselve am-biguities ef erdinary language. Meteever, it weuld be unnatural te cen-strue all use ef the singular ‘lhe‘ in this way; eften a better aeceunt issimply preneminal. Cemmenly ‘the buy‘, ‘the car‘. serves merely as apreneun whese grammatical antecedent is seme name er descriptien erperhaps a general term and quantifier. in legical netatien such a preneunmight ceme threugh simply as a beund variable. we saw in Chapter 4] that arguments invelving a singular term can becarried threugh by straight quantificatiun theery with a free variable, say‘_v‘. fer the singular term, but that the applicatien ef the results depends encenstruing y as the ebject named by the singular term, and hence iscentingent en existence ef such an ebject. This censtruing ef y, and theexistence assumptien en which it rests, figured newhere in theschematism ef the preef, but enly in the infermal step ef applicatien.blew the beauty ef dcscriptiens is that here the censtruing ef y as thenamed ebject can itself he schematiaed quite explicitly as an additienalprerni se, ef the ferm [ll abeve. Se eur technique fer arguments invelvingdcscriptiens is as fellews: we use free variables fer the dcscriptiens as ferany singular terms, but we alse add a ulcseriptiururt premise ef the ferm[ll fer each descriptien. An axiem ef identity is alse usually called fer,because ef the ‘=' in the descriptienal premise. Thus let us u'y thisexample:PREMISEZ The breker whe hired .luhn hired enly henerser:-r~trr|_LJstctt~t: gradu ates, Iehn was an heners graduate.Here we have twe singular terms, the simple ene ‘Iehn‘ altd the cemplexene ‘the breker whe hired lehn‘. Let us represent them by free variables‘w‘ and ‘y‘ respectively. Writing ‘F ' fer ‘hreker‘ and ‘G‘ fer ‘hired‘,we may alse render the cemplex singular term as a descriptien‘{r.r}t[F.r . tlevl‘; se the cerrespending descriptienal premise is: \"tt‘xtF.r . tlrtv .-H. .t = yl.Se frem this and the eriginal premise, which is ‘ldxttfiyx -I Hrl‘ where‘H’ represents ‘hnners graduate‘, and perhaps an axiem ef identity in

2?t5 IF- Glimpses Baymtd adtlitiefl, we want te shew that ‘Hie’ fellews. We cle se by transfertrting the cenditienal: lI\".1'=[Fr . first .~t—s-. ,1: = y] .\"i\".tlI}'y.r -a-Hr] . \"I\".rt_'.r = Jr} .-+ Hwintn a pure existential and finding it valid. 'w*hether a prnpesed deduetien is te enjuy the benefits c-1‘ a descriptienaipremise depends, evidently, en whether a given singular term can fairlybe translated inte the thrm efa descriptien. New fairness ef translatinn isa vague matter, hinging as it dees en the cencept ef synenymy which wasse dimly regarded in Chapter 41. ‘The auther ef Waverley‘ seems fairlytranslatable as *-|[1,r]t}r wrete Waverley)‘, hut '5cett’ and ‘the auther eflvunlims‘ du nut, despite the fact that all uf these name the same ebject;fer it is felt that ‘the anther ef Waverley\" is cennected with ‘[1..r]l'l_',r wreteWavarlayl‘ by sheer meaning, whereas '5cett' and ‘the autlier ef lranltec‘are cennected with it threugh accidental matters ef fact. At the same time it seems that singular terms can depart widely in farmfrem the singular ‘the’ idiem and still be fairly deemed translatable intntleseriptiens: witness 'lehn’s mether'. Indeed, even se simple a term as‘Se-crates‘, Russell has argued,\" is fer each ef us synnnymeus with semedescriptien, perhaps ‘(tilts was a philesepher . 1 dranlt hemleeltft‘ erperhaps anether depending en hew each el’ us first learned ef Sn-crates.are then all singular terms te he censidered capahle c-f fair translatinn intedescriptiens. escept fer these very few names which we may be suppes-edte have learned by direct ct:-nfrcmtatien with name and ebject’? lvlust aseparate categery then be ltept epen fer these few hypothetical encep-tiens? Happily, we can iselate such epistemelugical censideratiens frem thelegic ef singular tenns by a very simple expedient: by insisting c-n theprimacy ef general terms. We may insist that what are ieamed by eaten-sien, er direct cnnfrentatien, he never names but suleiy general terms.This we may insist en at the level strictly ef legical grammar, witheutprejudice te epistemnlngy er entelegy. witheut prejudice te epist;emel-egy because we may grant the epistemelegist any el’ the werds which hetraces te estensien; we merely parse them differently. Instead uf treatingthe ustensively teamed werd as artnmc cf the shewn ebject te begin with,we treat it te begin with as a general tenn traa exclusively ut\" the shewnebject; then we censtrue the name, as such, as ameunting te ‘[s.r}F.I’ ' E.g., in “llin-uwledge by acquaintance-“

#3- {lest-riptfelts 1??where ‘F ‘ represents that primitive general term. l\"~le matter te epistemel-egy, but much tn the smclethness nf lngical theery. Sn there is ne lenger an ebstaele tn treating all singular terms asdcscriptiens- Given any singular term uf erdinary language, mnrenver,say ‘Secrates‘ er ‘tL‘.erberus‘ er ‘the auther uf Waverley‘, the prepereheice nf ‘F‘ fer translatinn ef the term inte ‘|;s,1tlF,r‘ need in practicenever detain us. If a pat translatinn such as ‘[i,r]-[st wrete Waverley)‘ liesready te hand, very well; if net, we need net hesitate tn admit a versien efthe type nf‘ti,1t]|[,1: is-Sucratesl‘ er ‘t_'1;t]=t,t is-Cerberus)‘, since any lesslame versien weuld, if admissible as a translatinn at all, differ at mest inettpesitnry value and net in meaning- Prnefs ef the type ef the breker e:-tamplc set ferth abeve are in ne wayfacilitated by thus trivially lransferming simple singular terms inte de-scriptinns. Eenstruing ‘lehn‘ in thc breker eitampie as a descriptien'{1r]-Lt is-.lehn]'. er ‘tt,t].l,1t‘, weuld entitle us tn a further descriptienalpremise “'|iI\",tt.l,t -H-. .1‘ = wl‘, but this is neither necessary ner useful ferthe pregress ef the deductien- The advantage ef treating all singular termsas descriptinns is nf a mere theeretical kind: that ef sparing us having teadmit intn the frame-werk ef ntir technical theery a distinctien between acategery ef dcscriptiens and a eategery ef netidescriptive singular terms.It is theereticaily impertant net tn have tn admit diis distinctien because,as we have seen, the questien nf there being essentially nnndescriptivesingular terms at all, and if se what, was shruuded in the theery efknawledge and meaning. We have segregated that issue frem c-ur een-cerns, by shifting it frem the realm ef singular tertns inte that uf generaltenns. Every singular term can nnw, trivially if nnt etherwise, be handledas a descriptien; what had been an issue ever names leamcd estensivelyversus names learned discursively new becnmes an issue ever generalterms learned estensively versus general terms learned discursively. inthis ferm the issue ceases te cut acress any uf eur schematism c-f legicalfnrms anti eategeries, and can be left tn ether minds. There is a yet mere striking benefit tn be gained frem treating allsingular terms as descriptinns, but it must await the nest chapter.EXERCISESl- Express ‘the tallest man in tel-vn' in the ferm ‘tzutlt. - -1- . .}‘, using ‘taller than‘ but net ‘tallest'.

TF3 lF. tliirnpses Beyaad1. Shew dist a descriptienal premise, an axiem ef identity. and the premise: The auther ef Waverley wrete lvanhae teged-ier imply: Semeene wrete beth Waverley and lvanhae- 44 ELIMINATION OF SING-ULAR TERMS Let us next take up the preblem, which has been teeming fer semetime. ef the truth value ef such statetnents as ‘Cerberus harks‘. Falsity, asa sweeping answer cevering all statements centaining ‘Eerberus‘, weuldbe ever-hasty: first. because the statement ‘There is ue sueh thing asCerberus‘, at least, is true; and secend. because whatever statements weadjudge false must admit ef cempeunds, e.g., their negatiens, which willhe true. Truth, as a sweeping answer, weuld enceunter parallel difficuI-ties. Uur deductive metheds fer singular terms threw ne light en the ques-tien; fer we already assume that the singular term names an ebject whenwe represent the singular term by El free variable. and we make the sameassumptien again when we adept a descriptive premise fer a descriptien.Failing a named ebject, eur metheds shew nefliing, fer what they purpertte shew rests dien en a centrary-te-fact assumptien. Cemmen usage,mnrenver, likewise leaves us in the dark; fer, excepting such centexts as‘There is ne such thing as Cerbems'. a singular term is erdinarily usedenly when the speaker believes er cares te pretend that the ebject exists. Under erdinary usage, we saw [Chapter 3], truth values attach net teindicative cenditienals as wheles but eniy tn the eensequents cenditien-aliy upen truth ef the antecedents. analegeusly, under erdinary usagetruth values attach te centexts ef singular tenns fer the tnest part enly

44 . Eltmittatien a,t’.5't'n_gnlar Terms ITEIcnnditienally upen existence ef the ebjects. But if we are te have asmeeth legical theery we must fill such gaps, even theugh arbitrarily, insuch a way that every statement cemes te have a truth value. Thus it wasthat we cenventienaily extended the cencept ef the cenditienal, in Chap-ter 3, se as te allew truth values generally te cenditienals as wheles- Anextensinn in the same spirit is needed new en the scere ef singular termsthat de net name- We cannet, we have seen, aecemplish this extensien by any blanketdecisien that all centexts nf a term such as ‘Cerberus’ are te he false, erall true- We can, hewever, decide the simple centexts and then let thetmth values ef the cempeunds fellew frern these decisiens. Let us speakef a general term as simple, fer eur purpeses, when it dees net explicitlyhave ll‘!-E! ferm el’ an abstract er a negatien, cenjunctien, alternatien,cenditienal, er bicenditienal ef sherter cempenents. ‘i'1\"l'ien any such sim-ple general tertn is applied te a singular term which fails te name, let usclassify the resulting sentence as false [fer all values ef any free variablesit may have}. Thus ‘tllerberus barks‘, fermed as it is by applying thesimple term ‘barks‘ te ‘tlerh-ems‘, is adjudged false. This rule is suited fer use enly upen sentences which are censidered tebe fully analyzed in peint ef legical structure. If the sentences are still tebe subject te further paraphrasing ef werds inte symbels, we must bewary ef treating a predicate as “simple\" in the sense abeve and thenparaphrasing it inte a cemplex ene. Fer illustratien let us reexamine the statement:{ll The breker whe hired lehn hired enly heners graduates.If we use ‘F‘ fer ‘breker’ and ‘E’ fer ‘hired’, then ‘the breker whe hiredlehn‘ may be rendered ‘[tx}[Fx . G x lehnl‘. Te say that diis allegedpersen hired tr is then te say: ‘[ll G {'-'x_l{F.r - G I Jehn} it,se that [ll becemes:{3} ‘ti-“alt? [1.r}l_'Fx . C-'x Iehn] n -1-Hit]where ‘H‘ means ‘heners graduate‘. Hew let us suppese that ne breker erseveral hired Iehn, se that there is ne such thing as the bmker whe hiredlehn. aceerding te the decisien which we have newly adeptecl te cever

Elli] ll-\". Glimpses Beyenelsuch cases, the simple eentext {Z} ef ‘t[t.x]{F,t: - G ;t Jehnl‘ is then te beclassified as false fer all cheices ef a. Thereupen die eenditienal in til]becemes true fer all cheices ef it threugh falsity ef antecedent, se [Illbecemes true. The eutceme is therefere that {ll becemes tme. indepen-dently et any censideratiun ef heners graduates. in ease lehn was hiredby ne breker er several. This particular eutceme is the merest curiesity,neither welceme ner unwelceme, sinee erdinary usage leaves cases suchas this undecided. Even when a singular tenn fails te name, hewever, we de have verypreper precencepttiens abeut the truth value ef the special centext ‘Thereis Ier: is netl‘ such a thing as . . But statements ef this ferm call fer aseparate analysis, aleng lines which are already pretty evident frem thesepast ebservatiens: ta} We may take ‘ttrlF.r‘ as the general ferm fer singular terms. {bl ‘t'tt,rlFr‘ purperts te name the ene and enly ebject ef which ‘F ‘ is true [suppesing any general term fer ‘F ‘ here]. {cl \"'IiI\",rt[Fr -H-. x = y]' ameunts te saying that y is the ene and enly ebject ef which ‘F‘ is true.Te say that there is sueh a thing as ti-xll-‘x is te say, in view t:-fthjt, thatthere is ene an-d enly ene ebject el which ‘F ‘ is ttue: and this may. in viewef tel. be said as fellews:(4) 3ylI'xtF.1: H-. x = yl.Here, then, is an adequate fermulatien ef ‘There is such a thing ast1.t:}Fx’; and ne mere can be wanted, in view ef ta], in fermulatien ef thegeneral idiem ‘There is such a thing as . . Curieusly eneugh, the translatinn st} ef ‘There is such a dting as[tr] Fr‘ is deveid ef the singular term ‘[11t]F.r‘. New eliminatien ef‘|ji.:t].F.r‘ frem ether centexts can alse be accemplished. Fer, think ef ‘G’as representing any general term which is “simplc“ in the recentlydefined sense. Then ‘G [*Lr}Fr', which attributes ‘G’ te ttx]F.r, may beparaphrased as:{5} Hy[tI¥y . \"t|\".r[Fr -t-1-. x = y]].This is seen as fellews. First suppese (Case 1] that there is such a thing as[1.1t}F.t:. Then the clause 'ltt\",r{.F.r -H. ..l.‘ = y]' identities y with t[t..r}Fx. and

std. Ellmlnarlan af.,'i'lng'a'lar Terms Eli]accerdingly [51 as a whele becemes true er false aceerding as ‘G’ is trueer false ef ttx]F,r. Hext suppese [Case El that diere is ne such thing ast'.ut'lF,r- Then \"\"i\"x{Fx H. x = yl‘ becemes false fer all cheices ef y; se{5} becemes false. Hut ‘C-' t1,r}F'.r‘ likewise is te be false in this case.aceerding te eur recent agreement abeut simple general terms in applica-tien te singular terms that de net name. We ai'e new in a pesitien te eliminate singular tetrrts everywhere.Given any sentence invelving singular terms, we begin by paraphrasingthe sentence inte the esplicit netatien ef quantificatiun and tmth functiensas fully as we can, leaving the singular terms undisturbed as cempenentsbut putting each in the ferm ef a descriptien. Then we supplant eachsimple centext ef each descriptien by its equivalent ef the ferm t5l—erby {4} if it happens te have the ferln ‘there is such a thing as tar] Fr‘- Fer simplicity we have been imagining always clesed singular terms.as eppesed te epen enes such as ‘x + 5‘ er ‘the eldest sen efx‘ er ‘tsxltrwrete el’. Clearly, hewever, the epen enes admit ef eliminatien by thesame precedure; the fact that a free variable is being carried aleng altersnething essential te the reasening- Te see hew the eliminatien ef singular terms preceeds in practice, letus nettirn te {ll and eliminate the singular terms ‘Inhn’ and ‘the brekerwhe hired lehn‘. its a first step we may eliminate the descriptien frem thesimple centext {Z}. The general methed ef deing this was seen in thetranslatien ef ‘G l_\".ut}Fx‘ inte (5); but what we have te deal with new inplace ef ‘G |[tx}F.r‘ is {Z}, which is ef the ferm ef ‘G [s.r}Fx‘ with‘it: Fe - G\" e Jehn]-‘ fer ‘F‘ and ‘la: Gen]-‘ fer ‘GI These same substi-tutiens in {5} give:[ti] H=y[t]F_vu . \"iI\".rtFx . Gr lehn .H. x = y]].then, as the translatien ef [2]. But ‘.lehn‘ has yet te be dealt with. Writing‘J’ fer ‘is-Jehn‘, we render ‘Inhn’ as a descriptien ‘italic’. se that‘G .r lehn' in [bl becemes ‘G.r{trlJ‘r'. This clause is ef the ferm‘Gt'.'x]I.Fx’ with '3' fer ‘.r’, ‘J’ fer ‘F‘, and ‘{.:tt'_'lx.:}’ fer ‘G’. Cet're—spending changes in {5} give:ll] Hiv[Grw . Vail: H. .1 = wj|],then, as translatien ef ‘t1t{sr}.lr‘. New we have eliminated beth singularterlrls. It remains enly te assemble the pieces, by putting {Tl fer‘G I Inhn‘ in {til and then putting the result fer {E} in [3]- We thus get:

IE1 ll-’. Glimpses Beyertsl \"'l\"at[HyICrya . \"'t||'.r[F.r . Hw[t_T-rw . \"'ifat‘,l.t; -t-s, g = wj] .H. x = y'_t| —t~Ha]tas eur final paraphrase ef fl}. Hut by new it begins te appear that theeliminatien sf dcscriptiens is ef essentially theeretical interest, and fliat inpractice the alternative handling ef the breker preblem which was netedin the preceding chapter recemmends itself highly. Nevertheless, the theeretical eliminability ef singular terms—-the dis-pensability ef all uatnes—is se startling that its impnrtance scarcely needsdwelling upen except in the negative fashien ef peinting eut what it deesnet mean. ti dees nnt mean dtat eur language leses all means ef talkingabeut ebjects; en the centrary, the feregeing censideratiens shew that theextrusien ef singular tenns is unaccempanied by any dirrtintttien in thepewer ef the language. Wliat the disappearance ef singular terms deesmean is that all reference te ebjects ef any kind, cenerete er abstract, ischanneled threugh general terms and beund variables. We can still sayanything we like abeut any ene ebject er all ebjects. but we say it alwaysthreugh the idiems ef quantificatiun: ‘There is an ebjectx such that . . .‘and ‘Every ebjects: is such that . . -‘. The ebjects whese existence isimplied in eur disceurse are finally just the ebjects which must, fer thetmth ef nur assertiens. be acknewledged as “values ef variables\"-i.e.,be reckened inte the tetality ef ebjects ever which eur variables efquantificatien range. Te be is te be a value ef a variable. There are neultimate philesephical preblems cenceming singular terms and dieir ref-erences, but enly cencerning variables and their values; and there are neultimate philesephical preblems cnnccrning existence except insefar asexistence is expressed by the quantifier ‘ Hr‘. Except when we are cen-cemed with philesephical issues ef linguistic reference and existence, enthe ether hand, there is ne peint in depriving eurselves ef the cenvenienccef singular terms; and accerdingly the techniques ef irtfenmce liidiertedevel-eped fer singular terms are net te be theught el’ as abandened. H lsreitlcat nete Frege ata a netatien fer the d-escriptien prefixin lb‘.3i3. The present netatien was adapted later frem Peane by Russell.The impertant idea ef eliminating dcscriptiens by paraphrasing the cen-text, substantially as abeve, was eriginal with Russell, liltlfi; and it is inallusien t\" Russell‘s examples that the auther ef Waverley centinucs te bementiene-.l in subsequent writings en the subject. Russell did net take thefurther step ef treating all names as dcscriptiens and thus eliminating

all-f. Elimittatlen tiff k'art'alrl'e-t IE3them tee. He preferred te preserve art cpistcmeiegical distinctien betweennames that were shert fer dcscriptiens and names that were irrcduciblypreper. learned by acquaintance-EllEllCl5E Put the statement: The weman whe lives abeve us is Cnerman and leves flewers inte symbels, using ‘Fr’ fer ‘x is a weman‘, ‘E}'.r‘ fer ‘x lives abeve us‘, ‘H.r‘ fer ‘x is German‘, and ‘.lx‘ fer ‘r leves flewers‘- Then transferm the whele se as te eliminate use sf descriptien.HST orttrmvmanamrtcamtts We acquired in Chapters ill and iii an efficient Heelean legic efmenadic terms. Terms were cempeunded te ferm lhrther terms witheuthelp ef abstractinn, beund variables, er quantifiers. Terms were fermedfrem terms by negatien and cenjunctien and, dispensably, the furtherfuncters ‘v‘, ‘—r‘, and ‘H‘. Sentences were fermed frem terms by theexistence functer '3' and, dispensably, the further functers ‘ltI\"', ‘Q’,‘E’, and ‘E’- tltbstracts and beund variables were added afterward in preparatien ferquantificatiun, which was wanted enly in preparatien fer a legic efpelyadic terms. lt will new be shewn that by generalizing negatien,cenjunctien, and the existence functer te apply te all pelyadic terms. andthen adding feur mere functers, we can previde fer a legic ef pelyadicterms that is just as pewerful as the legic ef abstracts er quantificatiun butretains the spirit at eur little Eta-tlean legic sf menadic terms. By this lmean that it has ne beund variables. ne abstracts, ne quantifiers. Thescheme will be explained by explaining hew te translate baclt and ferth

EB-=t IF. Glimpses Beyeneibetween it and the legic ef variables and abstracts nr, ultimately. el’quantiiicatien. These translatiens serve e:-tplicitly tn depict and analyzetl1e cembinaterial werl: that beund variables pcrferm. lt is this deepenedunderstanding ef tl1e variable itself, rather than any practical advantagesel’ a legic withnut variables, that maltes the present escarsien werthwhile. .Teens can be menadic er pelyadic. an n -adic term will he said in be efdegree n- In quantilicaiien theery, where the schematic term letters 'F*,“G”, etc. appear enly with variables attached, we can tell the degree ef aterm letter by ceunting the variables. Herc, hewever, where variables areeventually te disappear, we shall still need tn ltnew the degrees ef tennletters; se let us adept distinctive styles el’ term letters fer difierent de-gtees. I shall use ‘F1’. ‘Ur’, etc. as menadic letters, ‘FE’, ‘GT, ete- asdyadic, and se en. Alse fer unifenniiy I shall use ‘F“‘, ‘GT, ete. insteadef *p‘, Hg‘, etc. as sentence letters: fur sentences fit smeethly inte thescheme as terms ef degree U.This netatien is net te be viewed as making legic newly dependent ennumber. The censtant espenents are te be seen merely as parts ef Fancyletters. The numerical inferrnatien that they cenvey is nething mnre thanwas already available in quantificatien theery by ceunting the variablesattached te a term letter.Hegatien and cenjunctien, lamiliar thus far as applied te menadicterrns and in sentences {hence terms ef degree {l'], are new te begeneralized in ebvieus fashien te terms ef all degrees. Restiscitating thepelyadic terrn abstract {Chapter 23} fer espesitery purpc-ses, we may putthe general case thus:{1} F\" —{-I-'l.I'2...I1.|:_.F.I|_.I:....-3l.'1||l',,{El F\"-U\" 5 l-I113. . -I,|IF“.I|.Ig. . ..r,, . Gnlllg . . .1,,}.Incidentally the ceextensiveness functer ‘E’ itself has undergene gen-eraliaatien at this peint, jeining net just menadic terms but ii-adic enes ferany n. Lilte the beund variables ‘If, ‘Is’, etc. and the term abstracts.hewever, ‘E‘ enters here enly as an espesitery device and net. likenegatien and cenjunctien, as part ef the final netatien. The numerical variable ‘rt’ in -[ll and {Z} will seem te have invekednumerical ideas unequivecally. Se it has, hut enly, again, at the especi-tery level and net within the netatien thatl am explaining. {ii and {1} aregeneral esplanatiens ef \"F'\"', \"F“G'i\", ‘F\", ‘F151’, and se en. Theseexpressiens are indeed schematic in turn‘, actual examples ef ‘Fm,

-is. Eliminetinn cf raaasa-l ass'F‘ifF1*, and ‘F\" weuld be '— [it is raining)’, ‘blaclt deg’, ‘rain-uncle ef‘.Numbers are net invelved. hlegatien and cenjunctien, previeusly at hand fer terrns ef degree ll{sentences} and l, are generalized in [ll and {2} te apply tn terms ef anydegree tit- l\"'-lnw a generali:-tatien ef 'El‘ is likewise needed- Escept fer amementary escursien in Chapter E3, where ‘ H‘ was tentatively applied tedyadic abstracts, we have been applying it steadfastly te menadic termsand latterly enly tn menadic abstracts, rewriting the results as quantifica-tiens. The ebvieus generaliaatien is the nne suggested by that escursienin Chapter 13, whereby '3' when applied te an n-adic term weuld yield aterm ef degree ill {a sentence]. Censider, hewever, this dilierent generali-Jtatien: just as ‘ El‘ when applied te a term ef degree l yields a term efdegree ll, sn when applied tn a term ef degree n it yields a term ef degreeii — I . This is the versien that will serve us. -HF\" E‘[.-T5,.-In: 3.1].-F“.-T|.I'g...I1.|l'. Cine ef the feur further functers wanted is reflectien, the self-functer.Applied te the dyadic terrn ‘hates' , it yields the menadic term ‘hates sell\"- RelF“ E {.t: Filtxl.The generaliaatien is this:{til RefF\" H ~{.=r;- . . ..r,,: Faxary. . -.r,,}. anether is padding. which gees peerly inte werds. it inveltes gratui-teus ebjects. Applied te ‘deg‘, it preduces a dyadic tern1 ‘Pad deg‘ thatrelates anything and everything tn every deg. Pad F‘ E l_;cy: Fly}.The generaliaatien is this:-[5] PadF\" E-l.r,-,1:|...Jc,,:F'tt|....r,,l- The remaining twe are functers el pcrmntatien, er inversien. Theinevitable ene in the dyadic case cenverts ‘parent ef‘ te ‘elispring ef’ andactive veice te passive. lts estensien te rt-adic terms ceuld talte either cif

EH-ti ll-’. Giimpsesfileyendtwe natural lines. I shall need beth, and will call them majer and nu'.m';irinversien.lilll l|ll\"'F“ E l-Ti---Iai F“-'l'a-7'51---~Fn—i.l'+[Tl inv Fa E l.t|-....t,,1.F'\".ty.t|,ty---,t,,l- tteratien ef ‘lnv’ serves tn bring any variable te initial pesitien- E.g., F “awry; H [inv F5}l+'.1'_‘t-'21-‘ H {lnv lnv Fitlsyrttw.But this maneuver disturbs the erder ef the rest el’ the variables; it draggedthe eld terminal variables ‘yr’ ferward. New this diserder can be rectifiedby a secend maneuver, which banishes a variable frem secend place tethe far end. We apply 'inv' arid then ‘inv’, thus:lb} -l'}5.t'yavw H -[inte Gfilyxavw =1-I-llltvirtv G'E'}.tavu{y'.Repeating this maneuver, we can banish ‘r’ in turn, ending up with theseught erder ‘.rvwyr'. Summed up, Ffivwxys H [Inv inv lnv inv lnv lnv Fi=],ri=1-uya. What this shews is that repeated use ef ‘lav’ and ‘inv’ enables us temeve any variable te initial pesitien and still restere the erder ef the etliervariables. But this means that we can achieve any permutatien whatever,building backward frem right tu left. lest pull the desired last variable teinitial pesitien, then pull the desired penultimate variable tu initial pesi-tien in frent ul it, and se en. Invelcing ‘Ref’, then, we can eliminate any rcpetitien el’ variables.First we permute the rcpetitien tn initial pesitien and then we apply ‘Rel’;fnr .FH.1-1|IE- . ..t,, H l:R.Ef.F\"l.Ig. . .,t:,,.By ‘Pad’ we can threw in any new variable, hewever irrelevant; fer F111. . ..i:,, H [Pad F*l}.t¢..1:,. . .r,,-iliftenvard we can repesitien the new variable as desired, by permutatien. Thus it is that eur feur added functers aecemplish all the recembi-

-'15. Eliminetien ef Feriebles Ell?natery werlt ef variables. They suffice te hemegenitc any twepredicatiens-—that is, te endnw them with matching strings ef variables,and indeed in any erder, free ef repetitiens. Fer esample, the beieregeneeus predieatiens ‘F lwawry‘ and ‘tIF\"v.try;;‘ are verifiably equivalent tethese hemegeneeus enes:til] [Pad Ref lnv inv F l}vw,t:ya, [inv Pad G‘l)vw,irya- blew l shall shew that eur seven functers, applied in iteratien te termletters, are adequate te the whele legic ef quantificatiun. Beund variablesge by the beard. Functers and schematic term letters remain. The translatien preceeds as fellews- Translate universal quantificatieninte esistential quantificatiun and negatien in the familiar way, and alltmth functiens inte negatien and cenjunctien. Put espenents en the termletters aceerding te hew many variables are attached, and change anysentence letters ‘p‘, ‘er‘, etc- te ‘F “‘, ‘t']\"\", etc. ‘Wherever a predicatinn isnegated, apply the negatien rather te the term letter. ‘Wherever twe predi-catiens eccur in cenjunctien, hemegeniae them as abeve and then reducetheir cenjunctien te a predicatien ef a cenjunctive term; fer, by [E],F\"_r,. . .s,, .G*'=.r, . . .,r,, .H [F\"tfi*'l.r; . - -.=r,,.Centin ning thus, we reduce each innermest quantificatiun te an esistentialquantiticatien ef a single predicatien. lts term will be cemplex, usually,built up ef term letters by eur functers; and new we further cemplieate itby applying functers suflicient te bring the quantified variable inte initialpesitien witheut rcpetitien. The quantificatiun becemes ‘H'..rF,ry, . . .y,,‘,say, where ‘T\" stands fer seme cemples temi; and this quantlticatienreduces in turn, by [3], te ‘{ElT]y| . . .y,,‘. The beund variable ‘.r‘ is nemere. ll‘-\"is the translatien centinucs, quantificatiens that were net inner-mest becnme se and are rid el’ their quantified variables in turn. an clesedquantificatienal schemata, having ne free variables, thus reduce te eem-pleit terms ef degree ll built up ef term letters by eur seven functers. Further ecenemy is available. Geerge lvlyre shewed me in Itliil that ifinstead ef ‘inv’ and ‘inv‘ we adept just a single permutatien functer‘Perm‘ equivalent te the iteratien ‘inv inv‘ illustrated in {S}, then all thewerlt ef beth ‘lnv‘ and ‘inv‘ can be dune by ‘Perrn', '3‘, and ‘Pad‘.'l Uurseven functers thus reduce te sis. t‘ '|11is can he seen with help ei ll-\"rig-\"s t1i'Pared'a.r. isrs. p- 295. The cmp-ping functerthcrc is eur present ‘ H '.

IEE IF- -Glinrpses Beyend |'|lfiT'lJRlCH.l. HUTE: It was Echiinilnkcl in title whe first analysedthe cemhinalery werlc ef variables by eliminating them in faver ef cem-binatery functiens. t'f.‘enrbinerery legic, thus feunded, has since accreted asubstantial literature, primarily by H- E- Curry- That ap-preach differsfrem the present ene in that its cembinaters are net mere functers liltecurs, applicable te terms, but are terms in their ewn right, singular terms,names ef functiens. They are censequently applicable te themselves andene anether, whereas the present functers apply enly te terms te term newterrns, and never apply te themselves er te ene anether. As a result theSchcintlnkel-tfurry netatien is as pewerful as that ef higher set theery, andnet readily shielded frem the antinemies {q.v- lnfre}, whereas the presentnetatien is equivalent merely te that ef quantiiicatien theery- The present scheme dates frem my “lfariables esplained away\"[l'§l'l5[l), apart frem superficial details. lt has seme allinity with Ta1slci‘s“cylindrical algebras“ ll!-3'51} and Halmes‘s \"pelyadic algebra“ (1956),and mere with a medified cylindrical algebra by Eemays if I959].EIERCISES l- \"v\"erit‘y by eitplicit steps ef transfurmatien that ‘F awe‘-vry‘ and ‘t?‘s-ere‘ are equivalent te [El].2- Transfenn the schema “ti-rtiy Fry —r ‘lily Eta Gy-rel‘ step by step inte a cemples term built ef ‘F1‘ and ‘G3’ by the term functers.ltlb CLASSES Cenfusien between a general term, true ef each ef varieus cenereteebjects. and a singular term naming an abstract ebject, is neither new neruncemmen- The same werd is apt te play beth reles. ‘lted‘ is true ei‘ eachred ebject and may alse be said te name a preperty, a celer- ‘l'v'lan‘ ls trueef each man and may alse be said te name a class, mankind-

slt'i- {Tie-tses 159 Classes, attributes, and preperties, if such there be, are abstract eb-jects. Between attributes and preperties l malte ne distinctien, ner is thereany call te distinguish classes frem preperties escept en ene peint: classesare e-rtensieneih What this means is that they are ceunted identical whenthey have the same members- Preperties, such as that ef having a heartand that ef having kidneys, may be preperties ef all the same individualsand yet be distinguished. What dees make fer identity ef preperties is netwell defined; we are apt te be febbed eff with leese tallt ef essence versusaccident. necessity versus eentingency, er the like. Se, despite the factthat ‘preperty‘ er ‘attribute‘ is mere usual in everyday parlance than‘class‘. l faver the latter. There are similarities in the behavier ef general and singular terms thatenceurage cenfusien, and sueh cenfusien may well underlie the verybelief in these abstract ebjects, whether classes er preperties. lie that as itmay, we must price the eutceme; fer the assumptien ef preperties, erbetter ef classes, has implemented the theeretical reseurces ef eur lan-guage in ways indispensable te science and even te mueh unscientificcemmunicatien- We shall presently nete a few instances- Tbe cenfusien has shewn itself in a readiness en the part ef semelegicians te let the schematic term letters ‘F ‘, ‘-G‘, etc. inte quantifiers asif they were variables. What had been schematic letters fer terms werethus beund as variables admitting ebjects ef seme sert as values—perhaps prepcnies, better classes. at this peint twe cenflatiens eccur:general terms are cenflated with class names, and schematic letters arecenflatcd with variables. l prepese new te ge aleng with the fertner ef these cenflatiens. lt mayhave begun in cenfusien. but it can be fellewed witheut cenfusien. Wecan deliberately assign a term deuble duty, letting it be true ef each manand at the same time name a class. mankind. We can let general termsdeuble as names el their estensiens- We shall find in Chapter 43 that wecannet assign this additienal duty te every general term, because ef an-tinemies; btit we can assign it selectively. Cenllatien et schematic letters with variables, en the ether hand,pretlts us net at all and engenders enly ebscurity. lf there are te heclasses, let them be values ef regular variables ‘-r‘, ‘y‘, etc- like every-thing else. The schematic lettcr ‘F ‘ centinucs te stand fer general temisand, insefar as these terms are ailewed new te deuble as class names, itstands as a schematic letter alse fer class names; but this still dees netmake it n bindahlc variable. That there are men is written ‘E-rt man -rl‘- That there is man-—-here

Iilll flit’- Glimpses Eeyenetthe term deubles as a class name—is written ‘H-rt-r = man)‘- That thereare disembedied spirits is written: Hridisembedied J.’ -spirits)er, equivalently, Ell-r:diseml1edied-r . spirit-Ir}{see Chapter Z1}. That they cnnstirute a class, en the ether hand, iswritten: Hyty = {It disembedied-r. spirit-ill.The set-theeretic netatien ef abstractinn, apprepriated in earlier chaptersfer the relative clause er tenn abstract, has new ceme heme te reest- Likeether general terms, it deubles new as a class name. Epsilen was apprepriated, aleng with the abstractien netatien, as acepula alternative re justapesitien fer er-tpressing predicatinn: ‘Jr is F‘ fer‘F-r‘- Linlc use was made ef it then. blew that general terms are ceming rede deuble duty as class names, epsilen likewise cemes heme te reest: itcemes te espress membership as well as predicatinn. With classes asvalues ef variables, indeed, it really cemes te be needed: we can write‘-r e y‘ where readers weuld be puaalcd by the justapesitien ‘y-r‘ theughfamiliar with ‘F-t‘. When used thus befere class names er variables,epsilen changes its grammatical status frem that ef a cepula te that ef adyadic general term. In ‘-r is y‘ the general term epsilen is predicated ef-rand y. What l have said cf epsilen applies equally te the functers ‘~;‘ and‘C'. Traditienally they belenged te set theery, but when they emerged inChapter Ell it was rather as cepulas like epsilen. blew that general tertnsare ceming te de deuble duty as class names, ‘§‘ and ‘C‘ ceme likeepsilen te de deuble duty as dyadic general terms; thus Zr I; _'|-' ‘, ‘.r C y ‘- The functer ‘ E‘ received in Chapter El] the same status as ‘ Q‘ and ‘C ‘:that ef a cepula fer use between general terms. When general temis cemete deuble as class names, ‘E’ takes en a secend status cerrespendingly,ceming like ‘t_;‘ and ‘-E‘ te serve as a dyadic general term betweenvariables er names ef classes- But what dyadic general term? Precisely‘=='l iltccerdingly l write ‘-r = y‘, net ‘Jr E y‘; l write ‘=‘ ratherthan ‘E‘wherever we are unequivecally cencemed with names rather than withgeneral terms that may er may net deuble as class names.