The Semantic Conception of Truth

8/13/2019 The Semantic Conception of Truth

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THE SEMANTIC CONCEPTION OF TRUTHAND THE FOUNDATIONS OF SEMANTICS

Alfred TarskiUniversity of California, erkeley

P!"lis#ed inPhilosophy and Phenomenological Research$ %&'$$()Trans*ri"ed into #y+ertet "y Andre- C#r!*ky, O*t) ., &''/)

#is +a+er *onsists of t-o +arts0 t#e first #as an e+ository *#ara*ter, and t#e se*ond is rat#er+ole1i*al)

In t#e first +art I -ant to s!11ari2e in an infor1al -ay t#e 1ain res!lts of 1y investi3ations*on*ernin3 t#e definition of tr!t# and t#e 1ore 3eneral +ro"le1 of t#e fo!ndations of se1anti*s)T#ese res!lts #ave "een e1"odied in a -ork -#i*# a++eared in +rint several years a3o)&Alt#o!3# 1y investi3ations *on*ern *on*e+ts dealt -it# is *lassi*al +#iloso+#y, t#ey #a++en to

"e *o1+aratively little kno-n in +#iloso+#i*al *ir*les, +er#a+s "e*a!se of t#eir stri*tly te*#ni*al*#ara*ter) For t#is reason I #o+e I s#all "e e*!sed for takin3 !+ t#e 1atter on*e a3ain)4

Sin*e 1y -ork -as +!"lis#ed, vario!s o"5e*tions, of !ne6!al val!e, #ave "een raised to 1yinvesti3ations0 so1e of t#ese a++eared in +rint, and ot#ers -ere 1ade in +!"li* and +rivatedis*!ssions in -#i*# I took +art)7In t#e se*ond +art of t#e +a+er I s#o!ld like to e+ress 1yvie-s re3ardin3 t#ese o"5e*tions) I #o+e t#at t#e re1arks -#i*# -ill "e 1ade in t#is *ontet -illnot "e *onsidered as +!rely +ole1i*al in *#ara*ter, "!t -ill "e fo!nd to *ontain so1e*onstr!*tive *ontri"!tions to t#e s!"5e*t)

In t#e se*ond +art of t#e +a+er I #ave 1ade etensive !se of 1aterial 3ra*io!sly +!t at 1y

dis+osal "y Dr) Mar5a 8okos2ynska %University of 9-o-() I a1 es+e*ially inde"ted and 3ratef!lto Professors Ernest Na3el %Col!1"ia University( and David Rynin %University of California,erkeley( for t#eir #el+ in +re+arin3 t#e final tet and for vario!s *riti*al re1arks)

I. EXPOSITION

&) THE MAIN PRO9EM :: A SATISFACTOR; DEFINITION OF TRUTH)

O!r dis*!ssion -ill "e *entered aro!nd t#e notion$of truth) T#e 1ain +ro"le1 is t#at of3ivin3 asatisfactory definitionof t#is notion, i)e), a definition -#i*# is materially adequateand

formally correct) !t s!*# a for1!lation of t#e +ro"le1, "e*a!se of its 3enerality, *annot "e*onsidered !ne6!ivo*al, and re6!ires so1e f!rt#er *o11ents)

In order to avoid any a1"i3!ity, -e 1!st first s+e*ify t#e *onditions !nder -#i*# t#edefinition of tr!t# -ill "e *onsidered ade6!ate fro1 t#e 1aterial +oint of vie-) T#e desireddefinition does not ai1 to s+e*ify t#e 1eanin3 of a fa1iliar -ord !sed to denote a novel notion0on t#e *ontrary, it ai1s to *at*# #old of t#e a*t!al 1eanin3 of an old notion)


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3enerally, -e 1!st des*ri"e t#e for1al str!*t!re of t#e lan3!a3e in -#i*# t#e definition -ill "e3iven)

T#e dis*!ssion of t#ese +oints -ill o**!+y a *onsidera"le +ortion of t#e first +art of t#e+a+er)

4) THE E=TENSION OF THE TERM >TRUE)>

de*larative senten*e>0 asre3ards t#e ter1 >+ro+osition,> its 1eanin3 is notorio!sly a s!"5e*t of len3t#y dis+!tations "yvario!s +#iloso+#ers and lo3i*ians, and it see1s never to #ave "een 1ade 6!ite *lear and

!na1"i3!o!s) For several reasons it a++ears 1ost *onvenient to apply the term "true" tosentences, and -e s#all follo- t#is *o!rse)?

Conse6!ently, -e 1!st al-ays relate t#e notion of tr!t#, like t#at of a senten*e, to a s+e*ifi*lan3!a3e0 for it is o"vio!s t#at t#e sa1e e+ression -#i*# is a tr!e senten*e in one lan3!a3e *an

"e false or 1eanin3less in anot#er)

Of *o!rse, t#e fa*t t#at -e are interested #ere +ri1arily in t#e notion of tr!t# for senten*esdoes not e*l!de t#e +ossi"ility of a s!"se6!ent etension of t#is notion to ot#er kinds of o"5e*ts)

7) THE MEANIN@ OF THE TERM >TRUE)>

M!*# 1ore serio!s diffi*!lties are *onne*ted -it# t#e +ro"le1 of t#e 1eanin3 %or t#eintension( of t#e *on*e+t of tr!t#)

T#e -ord >true,> like ot#er -ords fro1 o!r everyday lan3!a3e, is *ertainly not !na1"i3!o!s)And it does not see1 to 1e t#at t#e +#iloso+#ers -#o #ave dis*!ssed t#is *on*e+t #ave #el+ed todi1inis# its a1"i3!ity) In -orks and dis*!ssions of +#iloso+#ers -e 1eet 1any different*on*e+tions of tr!t# and falsity, and -e 1!st indi*ate -#i*# *on*e+tion -ill "e t#e "asis of o!rdis*!ssion)


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If, on t#e ot#er "and, -e s#o!ld de*ide to etend t#e +o+!lar !sa3e of t#e ter1 >designate>"y a++lyin3 it not only to na1es, "!t also to senten*es, and if -e a3reed to s+eak of t#e desi3nataof senten*es as >states of affairs,> -e *o!ld +ossi"ly !se for t#e sa1e +!r+ose t#e follo-in3

+#raseB

A sentence is true if it designates an eisting state of affairs..

Ho-ever, all t#ese for1!lations *an lead to vario!s 1is!nderstandin3s, for none of t#e1 iss!ffi*iently +re*ise and *lear %t#o!3# t#is a++lies 1!*# less to t#e ori3inal Aristotelianfor1!lation t#an to eit#er of t#e ot#ers(0 at any rate, none of t#e1 *an "e *onsidered asatisfa*tory definition of tr!t#) It is !+ to !s to look for a 1ore +re*ise e+ression of o!rint!itions)

$) A CRITERION FOR THE MATERIA9 ADEUAC; OF THE DEFINITION)/

9et !s start -it# a *on*rete ea1+le) Consider t#e senten*e >snow is white)> snow is white>B

the sentence constituted y three words, the first of which consists of the #$th, #%th, #&th, and

'rd letters, the second of the $th and #$th letters, and the third of the 'rd, th, $th, '*th, and

&th letters of the +nglish alphaet.


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%T(! is true if, and only if, p.equialence of the form%T()>

No- at last -e are a"le to +!t into a +re*ise for1 t#e *onditions !nder -#i*# -e -ill

*onsider t#e !sa3e and t#e definition of t#e ter1 >true> as ade6!ate fro1 t#e 1aterial +oint ofvie-B -e -is# to !se t#e ter1 >true> in s!*# a -ay t#at all e6!ivalen*es of t#e for1 %T( *an "easserted, and we shall call a definition of truth "adequate" if all these equialences follow fromit.

It s#o!ld "e e1+#asi2ed t#at neit#er t#e e+ression %T( itself %-#i*# is not a senten*e, "!tonly a s*#e1a of a senten*e( nor any +arti*!lar instan*e of t#e for1 %T( *an "e re3arded as adefinition of tr!t#) states of affairs>( "referred to" y thoseepressions) As ty+i*al ea1+les of se1anti* *on*e+ts -e 1ay 1ention t#e *on*e+ts ofdesignation, satisfaction,and definitionas t#ese o**!r in t#e follo-in3 ea1+lesB

the epression "the father of his country" designates (denotes) /eorge 0ashington1

snow satisfies the sentential function (the condition) "' is white"1

the equation "' . 2 #" defines (uniquely determines) the numer&4)

designates,> >satisfies,> and >defines> e+ress relations %"et-een *ertaine+ressions and t#e o"5e*ts >referred to> "y t#ese e+ressions(, t#e -ord >true> is of a differentlo3i*al nat!reB it e+resses a +ro+erty %or denotes a *lass( of *ertain e+ressions, vi2, ofsenten*es) Ho-ever, it is easily seen t#at all t#e for1!lations -#i*# -ere 3iven earlier and -#i*#ai1ed to e+lain t#e 1eanin3 of t#is -ord %*f) Se*tions 7and $( referred not only to senten*est#e1selves, "!t also to o"5e*ts >talked a"o!t> "y t#ese senten*es, or +ossi"ly to >states of affairs>des*ri"ed "y t#e1) And, 1oreover, it t!rns o!t t#at t#e si1+lest and t#e 1ost nat!ral -ay ofo"tainin3 an ea*t definition of tr!t# is one -#i*# involves t#e !se of ot#er se1anti* notions,

e)3), t#e notion of satisfa*tion) It is for t#ese reasons t#at -e *o!nt t#e *on*e+t of tr!t# -#i*# isdis*!ssed #ere a1on3 t#e *on*e+ts of se1anti*s, and t#e +ro"le1 of definin3 tr!t# +roves to "e
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*losely related to t#e 1ore 3eneral +ro"le1 of settin3 !+ t#e fo!ndations of t#eoreti*alse1anti*s)

It is +er#a+s -ort# -#ile sayin3 t#at se1anti*s as it is *on*eived in t#is +a+er %and in for1er+a+ers of t#e a!t#or( is a so"er and 1odest dis*i+line -#i*# #as no +retensions of "ein3 a!niversal +atent:1edi*ine for all t#e ills and diseases of 1ankind, -#et#er i1a3inary or real)

;o! -ill not find in se1anti*s any re1edy for de*ayed teet# or ill!sions of 3rande!r or *lass*onfli*ts) Nor is se1anti*s a devi*e for esta"lis#in3 t#at everyone e*e+t t#e s+eaker and #isfriends is s+eakin3 nonsense)

Fro1 anti6!ity to t#e +resent day t#e *on*e+ts of se1anti*s #ave +layed an i1+ortant role int#e dis*!ssions of +#iloso+#ers, lo3i*ians, and +#ilolo3ists) Nevert#eless, t#ese *on*e+ts #ave

"een treated for a lon3 ti1e -it# a *ertain a1o!nt of s!s+i*ion) Fro1 a #istori*al stand+oint, t#iss!s+i*ion is to "e re3arded as *o1+letely 5!stified) For alt#o!3# t#e 1eanin3 of se1anti**on*e+ts as t#ey are !sed in everyday lan3!a3e see1s to "e rat#er *lear and !nderstanda"le, stillall atte1+ts to *#ara*teri2e t#is 1eanin3 in a 3eneral and ea*t -ay 1is*arried) And -#at is-orse, vario!s ar3!1ents in -#i*# t#ese *on*e+ts -ere involved, and -#i*# see1ed ot#er-ise

6!ite *orre*t and "ased !+on a++arently o"vio!s +re1ises, led fre6!ently to +aradoes andantino1ies) It is s!ffi*ient to 1ention #ere t#e antinomy of the liar, Ri*#ards antinomy ofdefinaility%"y 1eans of a finite n!1"er of -ords(, and @rellin3:Nelsons antinomy ofheterological terms)'

I "elieve t#at t#e 1et#od -#i*# is o!tlined in t#is +a+er #el+s to over*o1e t#ese diffi*!ltiesand ass!res t#e +ossi"ility of a *onsistent !se of se1anti* *on*e+ts)

.) 9AN@UA@ES undefined%orprimitie( terms>0 and -e 1!st 3ive t#e so:*alled rules ofdefinitionfor introd!*in3 ne- or defined terms) F!rt#er1ore, -e 1!st set !+ *riteria fordistin3!is#in3 -it#in t#e *lass of e+ressions t#ose -#i*# -e *all >sentences)> Finally, -e 1!st

for1!late t#e *onditions !nder -#i*# a senten*e of t#e lan3!a3e *an "e asserted) In +arti*!lar,-e 1!st indi*ate all aioms%orprimitie sentences(, i)e), t#ose senten*es -#i*# -e de*ide toassert -it#o!t +roof0 and -e 1!st 3ive t#e so:*alled rules of inference%or rules of proof( "y1eans of -#i*# -e *an ded!*e ne- asserted senten*es fro1 ot#er senten*es -#i*# #ave "een

+revio!sly asserted) Aio1s, as -ell as senten*es ded!*ed fro1 t#e1 "y 1eans of r!les ofinferen*e, are referred to as >theorems> or >proale sentences)>

If in s+e*ifyin3 t#e str!*t!re of a lan3!a3e -e refer e*l!sively to t#e for1 of t#ee+ressions involved, t#e lan3!a3e is said to "eformali3ed) In s!*# a lan3!a3e t#eore1s are t#eonly senten*es -#i*# *an "e asserted)

At t#e +resent ti1e t#e only lan3!a3es -it# a s+e*ified str!*t!re are t#e for1ali2edlan3!a3es of vario!s syste1s of ded!*tive lo3i*, +ossi"ly enri*#ed "y t#e introd!*tion of *ertainnon:lo3i*al ter1s) Ho-ever, t#e field of a++li*ation of t#ese lan3!a3es is rat#er *o1+re#ensive0
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-e are a"le, t#eoreti*ally, to develo+ in t#e1 vario!s "ran*#es of s*ien*e, for instan*e,1at#e1ati*s and t#eoreti*al +#ysi*s)

%On t#e ot#er #and, -e *an i1a3ine t#e *onstr!*tion of lan3!a3es -#i*# #ave an ea*tlys+e*ified str!*t!re -it#o!t "ein3 for1ali2ed) In s!*# a lan3!a3e t#e asserta"ility of senten*es,for instan*e, 1ay de+end not al-ays on t#eir for1, "!t so1eti1es on ot#er, non:lin3!isti*

fa*tors) It -o!ld "e interestin3 and i1+ortant a*t!ally to *onstr!*t a lan3!a3e of t#is ty+e, ands+e*ifi*ally one -#i*# -o!ld +rove to "e s!ffi*ient for t#e develo+1ent of a *o1+re#ensive"ran*# of e1+iri*al s*ien*e0 for t#is -o!ld 5!stify t#e #o+e t#at lan3!a3es -it# s+e*ifiedstr!*t!re *o!ld finally re+la*e everyday lan3!a3e in s*ientifi* dis*o!rse)(

The prolem of the definition of truth otains a precise meaning and can e soled in a

rigorous way only for those languages whose structure has een eactly specified. For ot#erlan3!a3es :: t#!s, for all nat!ral, >s+oken> lan3!a3es :: t#e 1eanin3 of t#e +ro"le1 is 1ore orless va3!e, and its sol!tion *an #ave only an a++roi1ate *#ara*ter) Ro!3#ly s+eakin3, t#ea++roi1ation *onsists in re+la*in3 a nat!ral lan3!a3e %or a +ortion of it in -#i*# -e areinterested( "y one -#ose str!*t!re is ea*tly s+e*ified, and -#i*# diver3es fro1 t#e 3iven

lan3!a3e >as little as +ossi"le)>

/) THE ANTINOM; OF THE 9IAR)

In order to dis*over so1e of t#e 1ore s+e*ifi* *onditions -#i*# 1!st "e satisfied "ylan3!a3es in -#i*# %or for -#i*#( t#e definition of tr!t# is to "e 3iven, it -ill "e advisa"le to

"e3in -it# a dis*!ssion of t#at antino1y -#i*# dire*tly involves t#e notion of tr!t#, na1ely, t#eantino1y of t#e liar)

To o"tain t#is antino1y in a +ers+i*!o!s for1,&*onsider t#e follo-in3 senten*eB

The sentence printed in this paper on p. %4, l. #, is not true.

For "revity -e s#all re+la*e t#e senten*e 5!st stated "y t#e letter s)

A**ordin3 to o!r *onvention *on*ernin3 t#e ade6!ate !sa3e of t#e ter1 >true,> -e assert t#efollo-in3 e6!ivalen*e of t#e for1 %T(B

%&( 5s5 is true if, and only if, the sentence printed in this paper on p. %4, l. #, is not true )

On t#e ot#er #and, kee+in3 in 1ind t#e 1eanin3 of t#e sy1"ol s, -e esta"lis# e1+iri*allyt#e follo-in3 fa*tB

%4( 5s5 is identical with the sentence printed in this paper on p. %4, l. #.

No-, "y a fa1iliar la- fro1 t#e t#eory of identity %9ei"ni2s la-(, it follo-s fro1 %4( t#at -e1ay re+la*e in %&( t#e e+ression >the sentence printed in this paper on p. %4, l. #> "y t#esy1"ol > s) >


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*o1+elled to assert a false senten*e %sin*e %7(, as an e6!ivalen*e "et-een t-o *ontradi*torysenten*es, is ne*essarily false() If -e take o!r -ork serio!sly, -e *annot "e re*on*iled -it# t#isfa*t) referrin3 to senten*es of t#is lan3!a3e0 -e #avealso ass!1ed t#at all senten*es -#i*# deter1ine t#e ade6!ate !sa3e of t#is ter1 *an "easserted in t#e lan3!a3e) A lan3!a3e -it# t#ese +ro+erties -ill "e *alled >semanticallyclosed)>

II) 3en!ine> lan3!a3e %or, at least, t#at all >3en!ine>lan3!a3es are 1!t!ally translata"le() Ho-ever, t#is restri*tion does not affe*t t#e needs orinterests of s*ien*e in any essential -ay) T#e lan3!a3es %eit#er t#e for1ali2ed lan3!a3es or ::-#at is 1ore fre6!ently t#e *ase :: t#e +ortions of everyday lan3!a3e( -#i*# are !sed ins*ientifi* dis*o!rse do not #ave to "e se1anti*ally *losed) T#is is o"vio!s in *ase lin3!isti*

+#eno1ena and, in +arti*!lar, se1anti* notions do not enter in any -ay into t#e s!"5e*t:1atter ofa s*ien*e0 for in s!*# a *ase t#e lan3!a3e of t#is s*ien*e does not #ave to "e +rovided -it# anyse1anti* ter1s at all) Ho-ever, -e s#all see in t#e net se*tion #o- se1anti*ally *losedlan3!a3es *an "e dis+ensed -it# even in t#ose s*ientifi* dis*!ssions in -#i*# se1anti* notionsare essentially involved)

T#e +ro"le1 arises as to t#e +osition of everyday lan3!a3e -it# re3ard to t#is +oint) At first"l!s# it -o!ld see1 t#at t#is lan3!a3e satisfies "ot# ass!1+tions %I( and %II(, and t#at t#erefore it
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1!st "e in*onsistent) !t a*t!ally t#e *ase is not so si1+le) O!r everyday lan3!a3e is *ertainlynot one -it# an ea*tly s+e*ified str!*t!re)


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It is desira"le for t#e 1eta:lan3!a3e not to *ontain any !ndefined ter1s e*e+t s!*# as areinvolved e+li*itly or i1+li*itly in t#e re1arks a"ove, i)e)B ter1s of t#e o"5e*t:lan3!a3e0 ter1sreferrin3 to t#e for1 of t#e e+ressions of t#e o"5e*t:lan3!a3e, and !sed in "!ildin3 na1es fort#ese e+ressions0 and ter1s of lo3i*) In +arti*!lar, -e desiresemantic terms%referrin3 to t#eo"5e*t:lan3!a3e( to e introduced into the meta6language only y definition) For, if t#is +ost!lateis satisfied, t#e definition of tr!t#, or of any ot#er se1anti* *on*e+t, -ill f!lfill -#at -e

int!itively e+e*t fro1 every definition0 t#at is, it -ill e+lain t#e 1eanin3 of t#e ter1 "ein3defined in ter1s -#ose 1eanin3 a++ears to "e *o1+letely *lear and !ne6!ivo*al) And,1oreover, -e #ave t#en a kind of 3!arantee t#at t#e !se of se1anti* *on*e+ts -ill not involve !sin any *ontradi*tions)

essentialri*#ness)> If -e restri*t o!rselves to lan3!a3es "ased on t#e lo3i*al t#eory of ty+es, t#e *onditionfor t#e 1eta:lan3!a3e to "e >essentially ri*#er> t#an t#e o"5e*t:lan3!a3e is t#at it *ontainvaria"les of a #i3#er lo3i*al ty+e t#an t#ose of t#e o"5e*t:lan3!a3e)

If t#e *ondition of >essential ri*#ness> is not satisfied, it *an !s!ally "e s#o-n t#at aninter+retation of t#e 1eta:lan3!a3e in t#e o"5e*t:lan3!a3e is +ossi"le0 t#at is to say, -it# any

3iven ter1 of t#e 1eta:lan3!a3e a -ell deter1ined ter1 of t#e o"5e*t:lan3!a3e *an "e *orrelatedin s!*# a -ay t#at t#e asserti"le senten*es of t#e one lan3!a3e t!rn o!t to "e *orrelated -it#asserti"le senten*es of t#e ot#er) As a res!lt of t#is inter+retation, t#e #y+ot#esis t#at asatisfa*tory definition of tr!t# #as "een for1!lated in t#e 1eta:lan3!a3e t!rns o!t to i1+ly t#e

+ossi"ility of re*onstr!*tin3 in t#at lan3!a3e t#e antino1y of t#e liar0 and t#is in t!rn for*es !s tore5e*t t#e #y+ot#esis in 6!estion)

%T#e fa*t t#at t#e 1eta:lan3!a3e, in its non:lo3i*al +art, is ordinarily 1ore *o1+re#ensivet#an t#e o"5e*t:lan3!a3e does not affe*t t#e +ossi"ility of inter+retin3 t#e for1er in t#e latter)For ea1+le, t#e na1es of e+ressions of t#e o"5e*t:lan3!a3e o**!r in t#e 1eta:lan3!a3e,t#o!3# for t#e 1ost +art t#ey do not o**!r in t#e o"5e*t:lan3!a3e itself0 "!t, nevert#eless, it 1ay

"e +ossi"le to inter+ret t#ese na1es in ter1 of t#e o"5e*t:lan3!a3e)(


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T#!s -e see t#at t#e *ondition of >essential ri*#ness> is ne*essary for t#e +ossi"ility of asatisfa*tory definition of tr!t# in t#e 1eta:lan3!a3e) If -e -ant to develo+ t#e t#eory of tr!t# ina 1eta:lan3!a3e -#i*# does not satisfy t#is *ondition, -e 1!st 3ive !+ t#e idea of definin3 tr!t#-it# t#e e*l!sive #el+ of t#ose ter1s -#i*# -ere indi*ated a"ove %inSe*tion () true,> or so1e ot#er se1anti* ter1, in t#e list of !ndefined ter1s of t#e1eta:lan3!a3e, and to e+ress f!nda1ental +ro+erties of t#e notion of tr!t# in a series of aio1s)

T#ere is not#in3 essentially -ron3 in s!*# an aio1ati* +ro*ed!re, and it 1ay +rove !sef!l forvario!s +!r+oses)&7

It t!rns o!t, #o-ever, t#at t#is +ro*ed!re *an "e avoided) For the condition of the "essentialrichness" of the meta6language proes to e, not only necessary, ut also sufficient for the

construction of a satisfactory definition of truth1i)e), if t#e 1eta:lan3!a3e satisfies t#is *ondition,t#e notion of tr!t# *an "e defined in it) sententialfunctions)> T#ese are e+ressions like > is white,> > is greater than y,> et*) T#eir for1alstr!*t!re is analo3o!s to t#at of senten*es0 #o-ever, t#ey 1ay *ontain t#e so:*alled freevaria"les %like and y in > is greater than y>(, -#i*# *annot o**!r in senten*es)

In definin3 t#e notion of a sentential f!n*tion in for1ali2ed lan3!a3es, -e !s!ally a++ly -#atis *alled a >re*!rsive +ro*ed!re>0 i)e), -e first des*ri"e sentential f!n*tions of t#e si1+leststr!*t!re %-#i*# ordinarily +resents no diffi*!lty(, and t#en -e indi*ate t#e o+erations "y 1eansof -#i*# *o1+o!nd f!n*tions *an "e *onstr!*ted fro1 si1+ler ones) S!*# an o+eration 1ay*onsist, for instan*e, in for1in3 t#e lo3i*al dis5!n*tion or *on5!n*tion of t-o 3iven f!n*tions,i)e), "y *o1"inin3 t#e1 "y t#e -ord >or> or >and)> A senten*e *an no- "e defined si1+ly as asentential f!n*tion -#i*# *ontains no free varia"les

As re3ards t#e notion of satisfa*tion, -e 1i3#t try to define it "y sayin3 t#at 3iven o"5e*tssatisfy a 3iven f!n*tion if t#e latter "e*o1es a tr!e senten*e -#en -e re+la*e in it free varia"les

"y na1es of 3iven o"5e*ts) In t#is sense, for ea1+le, sno- satisfies t#e sentential f!n*tion > iswhite> sin*e t#e senten*e >snow is white> is tr!e) Ho-ever, a+art fro1 ot#er diffi*!lties, t#is1et#od is not availa"le to !s, for -e -ant to !se t#e notion of satisfa*tion in definin3 tr!t#)

To o"tain a definition of satisfa*tion -e #ave rat#er to a++ly a3ain a re*!rsive +ro*ed!re)


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%It 1ay see1 stran3e t#at -e #ave *#osen a ro!nda"o!t -ay of definin3 t#e tr!t# of asenten*e, instead of tryin3 to a++ly, for instan*e, a dire*t re*!rsive +ro*ed!re) T#e reason is t#at*o1+o!nd senten*es are *onstr!*ted fro1 si1+ler sentential f!n*tions, "!t not al-ays fro1si1+ler senten*es0 #en*e no 3eneral re*!rsive 1et#od is kno-n -#i*# a++lies s+e*ifi*ally tosenten*es)(

Fro1 t#is ro!3# o!tline it is not *lear -#ere and #o- t#e ass!1+tion of t#e >essentialri*#ness> of t#e 1eta:lan3!a3e is involved in t#e dis*!ssion0 t#is "e*o1es *lear only -#en t#e*onstr!*tion is *arried t#ro!3# in a detailed and for1al -ay)&.

&4) CONSEUENCES OF THE DEFINITION)

T#e definition of tr!t# -#i*# -as o!tlined a"ove #as 1any interestin3 *onse6!en*es)

In t#e first +la*e, t#e definition +roves to "e not only for1ally *orre*t, "!t also 1ateriallyade6!ate %in t#e sense esta"lis#ed in Se*tion $(0 in ot#er -ords, it i1+lies all e6!ivalen*es of t#efor1 %T() In t#is *onne*tion it is i1+ortant to noti*e t#at t#e *onditions for t#e 1aterial ade6!a*y

of t#e definition deter1ine !ni6!ely t#e etension of t#e ter1 >true)>&/T#erefore, everydefinition of tr!t# -#i*# is 1aterially ade6!ate -o!ld ne*essarily "e e6!ivalent to t#at a*t!ally*onstr!*ted) T#e se1anti* *on*e+tion of tr!t# 3ives !s, so to s+eak, no +ossi"ility of *#oi*e

"et-een vario!s non:e6!ivalent definitions of t#is notion)

Moreover, -e *an ded!*e fro1 o!r definition vario!s la-s of a 3eneral nat!re) In +arti*!lar,-e *an +rove -it# its #el+ t#e laws of contradiction and of ecluded middle, -#i*# are so*#ara*teristi* of t#e Aristotelian *on*e+tion of tr!t#0 i)e), -e *an s#o- t#at one and only one ofany t-o *ontradi*tory senten*es is tr!e) T#ese se1anti* la-s s#o!ld not "e identified -it# t#erelated lo3i*al la-s of *ontradi*tion and e*l!ded 1iddle0 t#e latter "elon3 to t#e sentential*al*!l!s, i)e), to t#e 1ost ele1entary +art of lo3i* and do not involve t#e ter1 >true> at all)

F!rt#er i1+ortant res!lts *an "e o"tained "y a++lyin3 t#e t#eory of tr!t# to for1ali2edlan3!a3es of a *ertain very *o1+re#ensive *lass of 1at#e1ati*al dis*i+lines0 only dis*i+lines ofan ele1entary *#ara*ter and a very ele1entary lo3i*al str!*t!re are e*l!ded fro1 t#is *lass) Itt!rns o!t t#at for a dis*i+line of t#is *lass the notion of truth neer coincides with that of

proaility0 for all +rova"le senten*es are tr!e, "!t t#ere are tr!e senten*es -#i*# are not+rova"le)&/Hen*e it follo-s f!rt#er t#at every s!*# dis*i+line is *onsistent "!t in*o1+lete0 t#at isto say, of t-o *ontradi*tory senten*es at 1ost one is +rova"le, and :: -#at is 1ore :: t#ere eistsa +air of *ontradi*tory senten*es neit#er of -#i*# is +rova"le)&

&7) E=TENSION OF THE RESU9TS TO OTHER SEMANTIC NOTIONS)

Most of t#e res!lts at -#i*# -e arrived in t#e +re*edin3 se*tions in dis*!ssin3 t#e notion oftr!t# *an "e etended -it# a++ro+riate *#an3es to ot#er se1anti* notions, for instan*e, to t#enotion of satisfa*tion %involved in o!r +revio!s dis*!ssion(, and to t#ose of designationanddefinition)

Ea*# of t#ese notions *an "e analy2ed alon3 t#e lines follo-ed in t#e analysis of tr!t#) T#!s,*riteria for an ade6!ate !sa3e of t#ese notions *an "e esta"lis#ed0 it *an "e s#o-n t#at ea*# oft#ese notions, -#en !sed in a se1anti*ally *losed lan3!a3e a**ordin3 to t#ose *riteria, leadsne*essarily to a *ontradi*tion0&'a distin*tion "et-een t#e o"5e*t:lan3!a3e and t#e 1eta:lan3!a3e

"e*o1es a3ain indis+ensa"le0 and t#e >essential ri*#ness> of t#e 1eta:lan3!a3e +roves in ea*#*ase to "e a ne*essary and s!ffi*ient *ondition for a satisfa*tory definition of t#e notion
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involved) Hen*e t#e res!lts o"tained in dis*!ssin3 one +arti*!lar se1anti* notion a++ly to t#e3eneral +ro"le1 of t#e fo!ndations of t#eoreti*al se1anti*s)

one) I do not #ave t#e sli3#test intention to*ontri"!te in any -ay to t#ose endless, often violent dis*!ssions on t#e s!"5e*tB >44I 1!st *onfess I do not !nderstand -#at is at stake in s!*# dis+!tes0for t#e +ro"le1 itself is so va3!e t#at no definite sol!tion is +ossi"le) In fa*t, it see1s to 1e t#at

t#e sense in -#i*# t#e +#rase >t#e ri3#t *on*e+tion> is !sed #as never "een 1ade *lear) In 1ost*ases one 3ets t#e i1+ression t#at t#e +#rase is !sed in an al1ost 1ysti*al sense "ased !+on t#e"elief t#at every -ord #as only one >real> 1eanin3 %a kind of Platoni* or Aristotelian idea(, andt#at all t#e *o1+etin3 *on*e+tions really atte1+t to *at*# #old of t#is one 1eanin30 sin*e,#o-ever, t#ey *ontradi*t ea*# ot#er, only one atte1+t *an "e s!**essf!l, and #en*e only one*on*e+tion is t#e >ri3#t> one)

Dis+!tes of t#is ty+e are "y no 1eans restri*ted to t#e notion of tr!t#) T#ey o**!r in alldo1ains -#ere :: instead of an ea*t, s*ientifi* ter1inolo3y :: *o11on lan3!a3e -it# itsva3!eness and a1"i3!ity is !sed0 and t#ey are al-ays 1eanin3less, and t#erefore in vain)

It see1s to 1e o"vio!s t#at t#e only rational a++roa*# to s!*# +ro"le1s -o!ld "e t#efollo-in3B


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It see1s to 1e t#at none of t#ese *on*e+tions #ave "een +!t so far in an intelli3i"le and!ne6!ivo*al for1) T#is 1ay *#an3e, #o-ever0 a ti1e 1ay *o1e -#en -e find o!rselves*onfronted -it# several in*o1+ati"le, "!t e6!ally *lear and +re*ise, *on*e+tions of tr!t#) It -illt#en "e*o1e ne*essary to a"andon t#e a1"i3!o!s !sa3e of t#e -ord >true,> and to introd!*eseveral ter1s instead, ea*# to denote a different notion) Personally, I s#o!ld not feel #!rt if af!t!re -orld *on3ress of t#e >t#eoreti*ians of tr!t#> s#o!ld de*ide :: "y a 1a5ority of votes :: to

reserve t#e -ord >true> for one of t#e non:*lassi*al *on*e+tions, and s#o!ld s!33est anot#er-ord, say, >frue,> for t#e *on*e+tion *onsidered #ere) !t I *annot i1a3ine t#at any"ody *o!ld+resent *o3ent ar3!1ents to t#e effe*t t#at t#e se1anti* *on*e+tion is >-ron3> and s#o!ld "eentirely a"andoned)

&?) FORMA9 CORRECTNESS OF THE SU@@ESTED DEFINITION OF TRUTH)

T#e s+e*ifi* o"5e*tions -#i*# #ave "een raised to 1y investi3ations *an "e divided intoseveral 3ro!+s0 ea*# of t#ese -ill "e dis*!ssed se+arately)

I t#ink t#at +ra*ti*ally all t#ese o"5e*tions a++ly, not to t#e s+e*ial definition I #ave 3iven,

"!t to t#e se1anti* *on*e+tion of tr!t# in 3eneral) Even t#ose -#i*# -ere leveled a3ainst t#edefinition a*t!ally *onstr!*ted *o!ld "e related to any ot#er definition -#i*# *onfor1s to t#is*on*e+tion)

T#is #olds, in +arti*!lar, for t#ose o"5e*tions -#i*# *on*ern t#e for1al *orre*tness of t#edefinition) I #ave #eard a fe- o"5e*tions of t#is kind0 #o-ever, I do!"t very 1!*# -#et#eranyone of t#e1 *an "e treated serio!sly )

As a ty+i*al ea1+le let 1e 6!ote in s!"stan*e s!*# an o"5e*tion)47In for1!latin3 t#edefinition -e !se ne*essarily sentential *onne*tives, i)e), e+ressions like >if) ) ), then,> >or,> et*)T#ey o**!r in t#e definiens0 and one of t#e1, na1ely, t#e +#rase >if, and only if> is !s!allye1+loyed to *o1"ine t#e definiend!1 -it# t#e definiens) Ho-ever, it is -ell kno-n t#at t#e1eanin3 of sentential *onne*tives is e+lained in lo3i* -it# t#e #el+ of t#e -ords >true> and>false>0 for instan*e, -e say t#at an e6!ivalen*e, i)e), a senten*e of t#e for1 >p if, and only if, q,>is tr!e if eit#er "ot# of its 1e1"ers, i)e), t#e senten*es re+resented "y p and q, are tr!e or "ot#are false) Hen*e t#e definition of tr!t# involves a vi*io!s *ir*le)

If t#is o"5e*tion -ere valid, no for1ally *orre*t definition of tr!t# -o!ld "e +ossi"le0 for -eare !na"le to for1!late any *o1+o!nd senten*e -it#o!t !sin3 sentential *onne*tives, or ot#erlo3i*al ter1s defined -it# t#eir #el+) Fort!nately, t#e sit!ation is not so "ad)

It is !ndo!"tedly t#e *ase t#at a stri*tly ded!*tive develo+1ent of lo3i* is often +re*eded "y*ertain state1ents e+lainin3 t#e *onditions !nder -#i*# senten*es of t#e for1 >if p, then q,>et*), are *onsidered tr!e or false) %S!*# e+lanations are often 3iven s*#e1ati*ally, "y 1eans oft#e so:*alled tr!t#:ta"les)( Ho-ever, t#ese state1ents are o!tside of t#e syste1 of lo3i*, ands#o!ld not "e re3arded as definitions of t#e ter1s involved) T#ey are not for1!lated in t#elan3!a3e of t#e syste1, "!t *onstit!te rat#er s+e*ial *onse6!en*es of t#e definition of tr!t# 3ivenin t#e 1eta:lan3!a3e) Moreover, t#ese state1ents do not infl!en*e t#e ded!*tive develo+1ent oflo3i* in any -ay) For in s!*# a develo+1ent -e do not dis*!ss t#e 6!estion -#et#er a 3ivensenten*e is tr!e, -e are only interested in t#e +ro"le1 -#et#er it is +rova"le)4$

On t#e ot#er #and, t#e 1o1ent -e find o!rselves -it#in t#e ded!*tive syste1 of lo3i* :: or

of any dis*i+line "ased !+on lo3i*, e)3), of se1anti*s :: -e eit#er treat sentential *onne*tives as!ndefined ter1s, or else -e define t#e1 "y 1eans of ot#er sentential *onne*tives, "!t never "y1eans of se1anti* ter1s like >true> or >false)> For instan*e, if -e a3ree to re3ard t#e e+ressions
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>not> and >if) ) ), then> %and +ossi"ly also >if, and only if>( as !ndefined ter1s, -e *an define t#eter1 >or> "y statin3 t#at a senten*e of t#e for1 >p or q> is e6!ivalent to t#e *orres+ondin3senten*e of t#e for1 >if not p, then q)> T#e definition *an "e for1!lated, e)3), in t#e follo-in3-ayB

(p or q) if, and only if, (if not p, then q).

T#is definition o"vio!sly *ontains no se1anti* ter1s)

Ho-ever, a vi*io!s *ir*le in definition arises only -#en t#e definiens *ontains eit#er t#e ter1to "e defined itself, or ot#er ter1s defined -it# its #el+) T#!s -e *learly see t#at t#e !se ofsentential *onne*tives in definin3 t#e se1anti* ter1 >true> does not involve any *ir*le)

I s#o!ld like to 1ention a f!rt#er o"5e*tion -#i*# I #ave fo!nd in t#e literat!re and -#i*#see1s also to *on*ern t#e for1al *orre*tness, if not of t#e definition of tr!t# itself, t#en at leastof t#e ar3!1ents -#i*# lead to t#is definition)4?

T#e a!t#or of t#is o"5e*tion 1istakenly re3ards s*#e1e %T( %fro1 Se*tion $( as a definition

of tr!t#) He *#ar3es t#is alle3ed definition -it# >inad1issi"le "revity, i)e), in*o1+leteness,>-#i*# >does not 3ive !s t#e 1eans of de*idin3 -#et#er "y e6!ivalen*e is 1eant a lo3i*al:for1al, or a non:lo3i*al and also str!*t!rally non: des*ri"a"le relation)> To re1ove t#is >defe*t>#e s!33ests s!++le1entin3 %T( in one of t#e t-o follo-in3 -aysB

%T(! is true if, and only if, p is true,or%T(! is true if, and only if, p is the case (i.e., if what p states is the case).

T#en #e dis*!sses t#ese t-o ne- >definitions,> -#i*# are s!++osedly free fro1 t#e old,for1al >defe*t,> "!t -#i*# t!rn o!t to "e !nsatisfa*tory for ot#er, non:for1al reasons)

T#is ne- o"5e*tion see1s to arise fro1 a 1is!nderstandin3 *on*ernin3 t#e nat!re ofsentential *onne*tives %and t#!s to "e so1e#o- related to t#at +revio!sly dis*!ssed() T#e a!t#orof t#e o"5e*tion does not see1 to reali2e t#at t#e +#rase >if, and only if> %in o++osition to s!*#

+#rases as >are equialent> or >is equialent to>( e+resses no relation "et-een senten*es at allsin*e it does not *o1"ine na1es of senten*es)

In 3eneral, t#e -#ole ar3!1ent is "ased !+on an o"vio!s *onf!sion "et-een senten*es andt#eir na1es) It s!ffi*es to +oint o!t t#at :: in *ontradistin*tion to %T( :: s*#e1ata %T( and %T( donot 3ive any 1eanin3f!l e+ressions if -e re+la*e in t#e1 + "y a senten*e0 for t#e +#rases >p is

true> and >p is the case> %i)e), >what p states is the case>( "e*o1e 1eanin3less if + is re+la*ed"y a senten*e, and not "y t#e na1e of a senten*e %*f) Se*tion $()4.

And I t#ink even t#at I *anri3oro!sly +rove t#is state1ent on t#e "asis of t#e follo-in3 definitionB An e+ression is said to

"e >inad1issi"ly lon3> if %i( it is 1eanin3less, and %ii( it #as "een o"tained fro1 a 1eanin3f!le+ression "y insertin3 s!+erfl!o!s -ords)

&.) REDUNDANC; OF SEMANTIC TERMS :: THEIR POSSI9E E9IMINATION)

T#e o"5e*tion I a1 3oin3 to dis*!ss no- no lon3er *on*erns for1al *orre*tness of t#edefinition, "!t is still *on*erned -it# *ertain for1al feat!res of t#e se1anti* *on*e+tion of tr!t#)
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! is true> ase6!ivalent to t#e senten*e denoted "y = %-#ere = stands for a na1e of a senten*e of t#e o"5e*tlan3!a3e() Conse6!ently, t#e ter1 >true> -#en o**!rrin3 in a si1+le senten*e of t#e for1 >! istrue> *an easily "e eli1inated, and t#e senten*e itself, -#i*# "elon3s to t#e 1eta:lan3!a3e, *an

"e re+la*ed "y an e6!ivalent senten*e of t#e o"5e*t:lan3!a3e0 and t#e sa1e a++lies to *o1+o!ndsenten*es +rovided t#e ter1 >tr!e> o**!rs in t#e1 e*l!sively as a +art of t#e e+ressions of t#e

for1 >! is true)>

So1e +eo+le #ave t#erefore !r3ed t#at t#e ter1 >true> in t#e se1anti* sense *an al-ays "eeli1inated, and t#at for t#is reason t#e se1anti* *on*e+tion of tr!t# is alto3et#er sterile and!seless) And sin*e t#e sa1e *onsiderations a++ly to ot#er se1anti* notions, t#e *on*l!sion #as

"een dra-n t#at se1anti*s as a -#ole is a +!rely ver"al 3a1e and at "est only a #ar1less #o""y)

!t t#e 1atter is not 6!ite so si1+le)4/T#e sort of eli1ination #ere dis*!ssed *annot al-ays"e 1ade) It *annot "e done in t#e *ase of !niversal state1ents -#i*# e+ress t#e fa*t t#at allsenten*es of a *ertain ty+e are tr!e, or t#at all tr!e senten*es #ave a *ertain +ro+erty) Forinstan*e, -e *an +rove in t#e t#eory of tr!t# t#e follo-in3 state1entB

All consequences of true sentences are true.Ho-ever, -e *annot 3et rid #ere of t#e -ord >true> in t#e si1+le 1anner *onte1+lated)

A3ain, even in t#e *ase of +arti*!lar senten*es #avin3 t#e for1 >! is true> s!*# a si1+leeli1ination *annot al-ays "e 1ade) In fa*t, t#e eli1ination is +ossi"le only in t#ose *ases in-#i*# t#e na1e of t#e senten*e -#i*# is said to "e tr!e o**!rs in a for1 t#at ena"les !s tore*onstr!*t t#e senten*e itself) For ea1+le, o!r +resent #istori*al kno-led3e does not 3ive !sany +ossi"ility of eli1inatin3 t#e -ord >true> fro1 t#e follo-in3 senten*eB

The first sentence written y Plato is true.

Of *o!rse, sin*e -e #ave a definition for tr!t# and sin*e every definition ena"les !s tore+la*e t#e definiend!1 "y its definiens, an eli1ination of t#e ter1 >true> in its se1anti* sense isal-ays t#eoreti*ally +ossi"le) !t t#is -o!ld not "e t#e kind of si1+le eli1ination dis*!sseda"ove, and it -o!ld not res!lt in t#e re+la*e1ent of a senten*e in t#e 1eta:lan3!a3e "y asenten*e in t#e o"5e*t:lan3!a3e)

If, #o-ever, anyone *ontin!es to !r3e t#at :: "e*a!se of t#e t#eoreti*al +ossi"ility ofeli1inatin3 t#e -ord >true> on t#e "asis of its definition :: t#e *on*e+t of tr!t# is sterile, #e 1!sta**e+t t#e f!rt#er *on*l!sion t#at all defined notions are sterile) !t t#is o!t*o1e is so a"s!rd

and so !nso!nd #istori*ally t#at any *o11ent on it is !nne*essary) In fa*t, I a1 rat#er in*lined toa3ree -it# t#ose -#o 1aintain t#at t#e 1o1ents of 3reatest *reative advan*e1ent in s*ien*efre6!ently *oin*ide -it# t#e introd!*tion of ne- notions "y 1eans of definition)

&/) CONFORMIT; OF THE SEMANTIC CONCEPTION OF TRUTH


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state1ents -it# o!r ne- for1!lation, and to ask t#e1 -#et#er it a3rees -it# t#eir intentions)Unfort!nately, t#is 1et#od is i1+ra*ti*al sin*e t#ey died 6!ite so1e ti1e a3o)

As far as 1y o-n o+inion is *on*erned, I do not #ave any do!"ts t#at o!r for1!lation does*onfor1 to t#e int!itive *ontent of t#at of Aristotle) I a1 less *ertain re3ardin3 t#e laterfor1!lations of t#e *lassi*al *on*e+tion, for t#ey are very va3!e indeed)4

F!rt#er1ore, so1e do!"ts #ave "een e+ressed -#et#er t#e se1anti* *on*e+tion does refle*tt#e notion of tr!t# in its *o11on:sense and everyday !sa3e) I *learly reali2e %as I alreadyindi*ated( t#at t#e *o11on 1eanin3 of t#e -ord >true> :: as t#at of any ot#er -ord of everydaylan3!a3e :: is to so1e etent va3!e, and t#at its !sa3e 1ore or less fl!*t!ates) Hen*e t#e +ro"le1of assi3nin3 to t#is -ord a fied and ea*t 1eanin3 is relatively !ns+e*ified, and every sol!tionof t#is +ro"le1 i1+lies ne*essarily a *ertain deviation fro1 t#e +ra*ti*e of everyday lan3!a3e)

In s+ite of all t#is, I #a++en to "elieve t#at t#e se1anti* *on*e+tion does *onfor1 to a very*onsidera"le etent -it# t#e *o11on:sense !sa3e :: alt#o!3# I readily ad1it I 1ay "e 1istaken)it is snowing> *o!ld "e tr!ealt#o!3# it is not sno-in3, or *o!ld "e false alt#o!3# it is sno-in3, -o!ld nat!rally "e verysi3nifi*ant for o!r +ro"le1)

T#erefore, I -as "y no 1eans s!r+rised to learn %in a dis*!ssion devoted to t#ese +ro"le1s(t#at in a 3ro!+ of +eo+le -#o -ere 6!estioned only &?J a3reed t#at >true> 1eans for t#e1>agreeing with reality,> -#ile 'J a3reed t#at a senten*e s!*# as >it is snowing> is tr!e if, andonly if, it is sno-in3) T#!s, a 3reat 1a5ority of t#ese +eo+le see1ed to re5e*t t#e *lassi*al*on*e+tion of tr!t# in its >+#iloso+#i*al> for1!lation, -#ile a**e+tin3 t#e sa1e *on*e+tion -#enfor1!lated in +lain -ords %-aivin3 t#e 6!estion -#et#er t#e !se of t#e +#rase >t#e sa1e*on*e+tion> is #ere 5!stified()

&) THE DEFINITION IN ITS RE9ATION TO >THE PHI9OSOPHICA9 PRO9EM OFTRUTH> AND TO ARIOUS EPISTEMO9O@ICA9 TRENDS)

I #ave #eard it re1arked t#at t#e for1al definition of tr!t# #as not#in3 to do -it# >t#e+#iloso+#i*al +ro"le1 of tr!t#)>7Ho-ever, no"ody #as ever +ointed o!t to 1e in an intelli3i"le-ay 5!st -#at t#is +ro"le1 is) I #ave "een infor1ed in t#is *onne*tion t#at 1y definition, t#o!3#it states ne*essary and s!ffi*ient *onditions for a senten*e to "e tr!e, does not really 3ras+ t#e>essen*e> of t#is *on*e+t) Sin*e I #ave never "een a"le to !nderstand -#at t#e >essen*e> of a*on*e+t is, I 1!st "e e*!sed fro1 dis*!ssin3 t#is +oint any lon3er)

In 3eneral, I do not "elieve t#at t#ere is s!*# a t#in3 as >t#e +#iloso+#i*al +ro"le1 of tr!t#)> I

do "elieve t#at t#ere are vario!s intelli3i"le and interestin3 %"!t not ne*essarily +#iloso+#i*al(+ro"le1s *on*ernin3 t#e notion of tr!t#, "!t I also "elieve t#at t#ey *an "e ea*tly for1!latedand +ossi"ly solved only on t#e "asis of a +re*ise *on*e+tion of t#is notion)
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sno- is -#ite> is taken to "e

se1anti*ally tr!e if sno- is in fact-#ite %itali*s "y t#e *riti*( :: lo3i* finds itself involved in a1ost !n*riti*al realis1)7&

If t#ere -ere an o++ort!nity to dis*!ss t#e o"5e*tion -it# its a!t#or, I s#o!ld raise t-o +oints)First, I s#o!ld ask #i1 to dro+ t#e -ords >in fact,> -#i*# do not o**!r in t#e ori3inal for1!lationand -#i*# are 1isleadin3, even if t#ey do not affe*t t#e *ontent) For t#ese -ords *onvey t#ei1+ression t#at t#e se1anti* *on*e+tion of tr!t# is intended to esta"lis# t#e *onditions !nder-#i*# -e are -arranted in assertin3 any 3iven senten*e, and in +arti*!lar any e1+iri*alsenten*e) Ho-ever, a 1o1ents refle*tion s#o-s t#at t#is i1+ression is 1erely an ill!sion0 and It#ink t#at t#e a!t#or of t#e o"5e*tion falls vi*ti1 to t#e ill!sion -#i*# #e #i1self *reated)

In fa*t, t#e se1anti* definition of tr!t# i1+lies not#in3 re3ardin3 t#e *onditions !nder -#i*#a senten*e like %&(B

%I(snow is white*an "e asserted) It i1+lies only t#at, -#enever -e assert or re5e*t t#is senten*e, -e 1!st "e readyto assert or re5e*t t#e *orrelated senten*e %4(B%4( the sentence "snow is white" is true

T#!s, -e 1ay a**e+t t#e se1anti* *on*e+tion of tr!t# -it#o!t 3ivin3 !+ any e+iste1olo3i*alattit!de -e 1ay #ave #ad0 -e 1ay re1ain naive realists, *riti*al realists or idealists, e1+iri*istsor 1eta+#ysi*ians :: -#atever -e -ere "efore) T#e se1anti* *on*e+tion is *o1+letely ne!tralto-ard all t#ese iss!es)

In t#e se*ond +la*e, I s#o!ld try to 3et so1e infor1ation re3ardin3 t#e *on*e+tion of tr!t#-#i*# %in t#e o+inion of t#e a!t#or of t#e o"5e*tion( does not involve lo3i* in a 1ost naiverealis1) I -o!ld 3at#er t#at t#is *on*e+tion 1!st "e in*o1+ati"le -it# t#e se1anti* one) T#!s,t#ere 1!st "e senten*es -#i*# are tr!e in one of t#ese *on*e+tions -it#o!t "ein3 tr!e in t#eot#er) Ass!1e, e)3), t#e senten*e %&( to "e of t#is kind) T#e tr!t# of t#is senten*e in t#e se1anti**on*e+tion is deter1ined "y an e6!ivalen*e of t#e for1 %T(B

The sentence "snow is white" is true if, and only if, snow is white.

Hen*e in t#e ne- *on*e+tion -e 1!st re5e*t t#is e6!ivalen*e, and *onse6!ently -e 1!st ass!1eits denialBThe sentence "snow is white" is true if, and only if, snow is not white %or +er#a+sBsnow, in fact,is not white).

T#is so!nds so1e-#at +aradoi*al) I do not re3ard s!*# a *onse6!en*e of t#e ne-*on*e+tion as a"s!rd0 "!t I a1 a little fearf!l t#at so1eone in t#e f!t!re 1ay *#ar3e t#is*on*e+tion -it# involvin3 lo3i* in a >1ost so+#isti*ated kind of irrealis1)> At any rate, it see1sto 1e i1+ortant to reali2e t#at every *on*e+tion of tr!t# -#i*# is in*o1+ati"le -it# t#e se1anti*one *arries -it# it *onse6!en*es of t#is ty+e)

I #ave d-elt a little on t#is -#ole 6!estion, not "e*a!se t#e o"5e*tion dis*!ssed see1s to 1every si3nifi*ant, "!t "e*a!se *ertain +oints -#i*# #ave arisen in t#e dis*!ssion s#o!ld "e taken
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into a**o!nt "y all t#ose -#o for vario!s e+iste1olo3i*al reasons are in*lined to re5e*t t#ese1anti* *on*e+tion of tr!t#)

&') A99E@ED METAPH;SICA9 E9EMENTS IN SEMANTICS)

T#e se1anti* *on*e+tion of tr!t# #as "een *#ar3ed several ti1es -it# involvin3 *ertain

1eta+#ysi*al ele1ents) O"5e*tions of t#is sort #ave "een 1ade to a++ly not only to t#e t#eory oftr!t#, "!t to t#e -#ole do1ain of t#eoreti*al se1anti*s)74

I do not intend to dis*!ss t#e 3eneral +ro"le1 -#et#er t#e introd!*tion of a 1eta+#ysi*alele1ent into a s*ien*e is at all o"5e*tiona"le) T#e only +oint -#i*# -ill interest 1e #ere is-#et#er and in -#at sense 1eta+#ysi*s is involved in t#e s!"5e*t of o!r +resent dis*!ssion)

T#e -#ole 6!estion o"vio!sly de+ends !+on -#at one !nderstands "y >1eta+#ysi*s)>Unfort!nately, t#is notion is etre1ely va3!e and e6!ivo*al) true> is of t#is kind, sin*e no !niversal*riterion of tr!t# follo-s i11ediately fro1 t#e definition of t#is ter1, and sin*e it is 3enerally

"elieved %and in a *ertain sense *an even "e +roved( t#at s!*# a *riterion -ill never "e fo!nd)
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T#is *o11ent on t#e a*t!al *#ara*ter of t#e notion of tr!t# see1s to "e +erfe*tly 5!st) Ho-ever,it s#o!ld #e noti*ed t#at t#e notion of tr!t# does not differ in t#is res+e*t fro1 1any notions inlo3i*, 1at#e1ati*s, and t#eoreti*al +arts of vario!s e1+iri*al s*ien*es, e)3), in t#eoreti*al

+#ysi*s)

In 3eneral, it 1!st "e said t#at if t#e ter1 >1eta+#ysi*al> is e1+loyed in so -ide a sense as

to e1"ra*e *ertain notions %or 1et#ods( of lo3i*, 1at#e1ati*s, or e1+iri*al s*ien*es, it -illa++ly a fortiorito t#ose of se1anti*s) In fa*t, as -e kno- fro1 Part I of t#e +a+er, in develo+in3t#e se1anti*s of a lan3!a3e -e !se all t#e notions of t#is lan3!a3e, and -e a++ly even a stron3erlo3i*al a++arat!s t#an t#at -#i*# is !sed in t#e lan3!a3e itself) On t#e ot#er #and, #o-ever, I *ans!11ari2e t#e ar3!1ents 3iven a"ove "y statin3 t#at in no inter+retation of t#e ter1>1eta+#ysi*al> -#i*# is fa1iliar and 1ore or less intelli3i"le to 1e does se1anti*s involve any1eta+#ysi*al ele1ents +e*!liar to itself)

I s#o!ld like to 1ake one final re1ark in *onne*tion -it# t#is 3ro!+ of o"5e*tions) T#e#istory of s*ien*e s#o-s 1any instan*es of *on*e+ts -#i*# -ere 5!d3ed 1eta+#ysi*al %in aloose, "!t in any *ase dero3atory sense of t#is ter1( "efore t#eir 1eanin3 -as 1ade +re*ise0

#o-ever, on*e t#ey re*eived a ri3oro!s, for1al definition, t#e distr!st in t#e1 eva+orated) Asty+i*al ea1+les -e 1ay 1ention t#e *on*e+ts of ne3ative and i1a3inary n!1"ers in1at#e1ati*s) I #o+e a si1ilar fate a-aits t#e *on*e+t of tr!t# and ot#er se1anti* *on*e+ts0 and itsee1s to 1e, t#erefore, t#at t#ose -#o #ave distr!sted t#e1 "e*a!se of t#eir alle3ed1eta+#ysi*al i1+li*ations s#o!ld -el*o1e t#e fa*t t#at +re*ise definitions of t#ese *on*e+ts areno- availa"le) If in *onse6!en*e se1anti* *on*e+ts lose +#iloso+#i*al interest, t#ey -ill onlys#are t#e fate of 1any ot#er *on*e+ts of s*ien*e, and t#is need 3ive rise to no re3ret)

4) APP9ICAI9IT; OF SEMANTICS TO SPECIA9 EMPIRICA9 SCIENCES)


20/25

Se1anti* notions are !ndo!"tedly involved, to a lar3er or s1aller de3ree, in +sy*#olo3y,so*iolo3y, and in +ra*ti*ally all t#e #!1anities) T#!s, a +sy*#olo3ist defines t#e so:*alledintelli3en*e 6!otient in ter1s of t#e n!1"ers of true%ri3#t( andfalse%-ron3( ans-ers 3iven "y a

+erson to *ertain 6!estions0 for a #istorian of *!lt!re t#e ran3e of o"5e*ts for -#i*# a #!1an ra*ein s!**essive sta3es of its develo+1ent +ossesses ade6!ate designations1ay "e a to+i* of 3reatsi3nifi*an*e0 a st!dent of literat!re 1ay "e stron3ly interested in t#e +ro"le1 -#et#er a 3iven

a!t#or al-ays !ses t-o 3iven -ords -it# t#e sa1e meaning) Ea1+les of t#is kind *an "e1!lti+lied indefinitely)

T#e 1ost nat!ral and +ro1isin3 do1ain for t#e a++li*ations of t#eoreti*al se1anti*s is*learly lin3!isti*s :: t#e e1+iri*al st!dy of nat!ral lan3!a3es) Certain +arts of t#is s*ien*e areeven referred to as >se1anti*s,> so1eti1es -it# an additional 6!alifi*ation) T#!s, t#is na1e iso**asional 3iven to t#at +ortion of 3ra11ar -#i*# atte1+ts to *lassify all -ords of a lan3!a3einto +arts of s+ee*#, a**ordin3 to -#at t#e -ords 1ean or desi3nate) T#e st!dy of t#e evol!tionof 1eanin3s in t#e #istori*al develo+1ent of a lan3!a3e is so1eti1es *alled >#istori*alse1anti*s)> In 3eneral, t#e totality of investi3ations on se1anti* relations -#i*# o**!r in anat!ral lan3!a3e is referred to as >des*ri+tive se1anti*s)> T#e relation "et-een t#eoreti*al and

des*ri+tive se1anti*s is analo3o!s to t#at "et-een +!re and a++lied 1at#e1ati*s, or +er#a+s tot#at "et-een t#eoreti*al and e1+iri*al +#ysi*s0 t#e role of for1ali2ed lan3!a3es in se1anti*s *an

"e ro!3#ly *o1+ared to t#at of isolated syste1s in +#ysi*s)

It is +er#a+s !nne*essary to say t#at se1anti*s *annot find any dire*t a++li*ations in nat!rals*ien*es s!*# as +#ysi*s, "iolo3y, et*)0 for in none of t#ese s*ien*es are -e *on*erned -it#lin3!isti* +#eno1ena, and even less -it# se1anti* relations "et-een lin3!isti* e+ressions ando"5e*ts to -#i*# t#ese e+ressions refer)


21/25

T#e 6!estion arises -#et#er se1anti*s *an "e #el+f!l in solvin3 3eneral and, so to s+eak,*lassi*al +ro"le1s of 1et#odolo3y) I s#o!ld like to dis*!ss #ere -it# so1e detail a s+e*ial,t#o!3# very i1+ortant, as+e*t of t#!s 6!estion)

One of t#e 1ain +ro"le1s of t#e 1et#odolo3y of e1+iri*al s*ien*e *onsists in esta"lis#in3*onditions !nder -#i*# an e1+iri*al t#eory or #y+ot#esis s#o!ld "e re3arded as a**e+ta"le) T#is

notion of a**e+ta"ility 1!st "e relativi2ed to a 3iven sta3e of t#e develo+1ent of a s*ien*e %or toa 3iven a1o!nt of +res!++osed kno-led3e() In ot#er -ords, -e 1ay *onsider it as +rovided -it#a ti1e *oeffi*ient0 for a t#eory -#i*# is a**e+ta"le today 1ay "e*o1e !ntena"le to1orro- as ares!lt of ne- s*ientifi* dis*overies)

It see1s a priorivery +la!si"le t#at t#e a**e+ta"ility of a t#eory so1e#o- de+ends on t#etr!t# of its senten*es, and t#at *onse6!ently a 1et#odolo3ist in #is %so far rat#er !ns!**essf!l(atte1+ts at 1akin3 t#e notion of a**e+ta"ility +re*ise, *an e+e*t so1e #el+ fro1 t#e se1anti*t#eory of tr!t#) Hen*e -e ask t#e 6!estionB Are t#ere any +ost!lates -#i*# *an "e reasona"lyi1+osed on a**e+ta"le t#eories and -#i*# involve t#e notion of tr!t# And, in +arti*!lar, -e ask-#et#er t#e follo-in3 +ost!late is a reasona"le oneB

An acceptale theory cannot contain (or imply) any false sentences.

T#e ans-er to t#e last 6!estion is *learly ne3ative) For, first of all, -e are +ra*ti*ally s!re, ont#e "asis of o!r #istori*al e+erien*e, t#at every e1+iri*al t#eory -#i*# is a**e+ted today -illsooner or later "e re5e*ted and re+la*ed "y anot#er t#eory) It is also very +ro"a"le t#at t#e ne-t#eory -ill "e in*o1+ati"le -it# t#e old one0 i)e), -ill i1+ly a senten*e -#i*# is *ontradi*tory toone of t#e senten*es *ontained in t#e old t#eory) Hen*e, at least one of t#e t-o t#eories 1!stin*l!de false senten*es, in s+ite of t#e fa*t t#at ea*# of t#e1 is a**e+ted at a *ertain ti1e)Se*ondly, t#e +ost!late in 6!estion *o!ld #ardly ever #e satisfied in +ra*ti*e0 for -e do notkno-, and are very !nlikely to find, any *riteria of tr!t# -#i*# ena"le !s to s#o- t#at nosenten*e of an e1+iri*al t#eory is false)

T#e +ost!late in 6!estion *o!ld "e at 1ost re3arded as t#e e+ression of an ideal li1it fors!**essively 1ore ade6!ate t#eories in a 3iven field of resear*#0 "!t t#is #ardly *an "e 3iven any

+re*ise 1eanin3)

Nevert#eless, it see1s to 1e t#at t#ere is an i1+ortant +ost!late -#i*# *an "e reasona"lyi1+osed on a**e+ta"le e1+iri*al t#eories and -#i*# involves t#e notion of tr!t#) It is *loselyrelated to t#e one 5!st dis*!ssed, "!t is essentially -eaker) Re1e1"erin3 t#at t#e notion ofa**e+ta"ility is +rovided -it# a ti1e *oeffi*ient, -e *an 3ive t#is +ost!late t#e follo-in3 for1B

As soon as we succeed in showing that an empirical theory contains (or implies) false sentences,

it cannot e any longer considered acceptale.

In s!++ort of t#is +ost!late, I s#o!ld like to 1ake t#e follo-in3 re1arks)

I "elieve every"ody a3rees t#at one of t#e reasons -#i*# 1ay *o1+el !s to re5e*t ane1+iri*al t#eory is t#e +roof of its in*onsisten*yB a t#eory "e*o1es !ntena"le if -e s!**eed inderivin3 fro1 it t-o *ontradi*tory senten*es) No- -e *an ask -#at are t#e !s!al 1otives forre5e*tin3 a t#eory on s!*# 3ro!nds) Persons -#o are a*6!ainted -it# 1odern lo3i* are in*linedto ans-er t#is 6!estion in t#e follo-in3 -ayB A -ell:kno-n lo3i*al la- s#o-s t#at a t#eory

-#i*# ena"les !s to derive t-o *ontradi*tory senten*es ena"les !s also to derive every senten*e0t#erefore, s!*# a t#eory is trivial and de+rived of any s*ientifi* interest)


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I #ave so1e do!"ts -#et#er t#is ans-er *ontains an ade6!ate analysis of t#e sit!ation) I t#inkt#at +eo+le -#o do not kno- 1odern lo3i* are as little in*lined to a**e+t an in*onsistent t#eoryas t#ose -#o are t#oro!3#ly fa1iliar -it# it0 and +ro"a"ly t#is a++lies even to t#ose -#o re3ard%as so1e still do( t#e lo3i*al la- on -#i*# t#e ar3!1ent is "ased as a #i3#ly *ontroversial iss!e,and al1ost as a +arado) I do not t#ink t#at o!r attit!de to-ard an in*onsistent t#eory -o!ld*#an3e even if -e de*ided for so1e reasons to -eaken o!r syste1 of lo3i* so as to de+rive

o!rselves of t#e +ossi"ility of derivin3 every senten*e fro1 any t-o *ontradi*tory senten*es)

It see1s to 1e t#at t#e real reason of o!r attit!de is a different oneB


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+ra*ti*al +oint of vie- t#ere is no *lear:*!t line "et-een 1eta: 1at#e1ati*s and 1at#e1ati*s+ro+er0 for instan*e, t#e investi3ations on defina"ility *o!ld "e in*l!ded in eit#er of t#esedo1ains)

47) FINA9 REMAR8S)

I s#o!ld like to *on*l!de t#is dis*!ssion -it# so1e 3eneral and rat#er loose re1arks*on*ernin3 t#e -#ole 6!estion of t#e eval!ation of s*ientifi* a*#ieve1ents in ter1s of t#eira++li*a"ility) I 1!st *onfess I #ave vario!s do!"ts in t#is *onne*tion)

ein3 a 1at#e1ati*ian %as -ell as a lo3i*ian, and +er#a+s a +#iloso+#er of a sort(, I #ave #adt#e o++ort!nity to attend 1any dis*!ssions "et-een s+e*ialists in 1at#e1ati*s, -#ere t#e

+ro"le1 of a++li*ations is es+e*ially a*!te, and I #ave noti*ed on several o**asions t#e follo-in3+#eno1enonB If a 1at#e1ati*ian -is#es to dis+ara3e t#e -ork of one of #is *ollea3!es, say, A,t#e 1ost effe*tive 1et#od #e finds for doin3 t#is is to ask -#ere t#e res!lts *an "e a++lied T#e#ard +ressed 1an, -it# #is "a*k a3ainst t#e -all, finally !neart#s t#e resear*#es of anot#er1at#e1ati*ian as t#e lo*!s of t#e a++li*ation of #is o-n res!lts) If net is +la3!ed -it# a

si1ilar 6!estion, #e -ill refer to anot#er 1at#e1ati*ian C) After a fe- ste+s of t#is kind -e findo!rselves referred "a*k to t#e resear*#es of A, and in t#is -ay t#e *#ain *loses)

S+eakin3 1ore serio!sly, I do not -is# to deny t#at t#e val!e of a 1ans -ork 1ay "ein*reased "y its i1+li*ations for t#e resear*# of ot#ers and for +ra*ti*e) !t I "elieve,nevert#eless, t#at it is ini1i*al to t#e +ro3ress of s*ien*e to 1eas!re t#e i1+ortan*e of anyresear*# e*l!sively or *#iefly in ter1s of its !sef!lness and a++li*a"ility) a++li*ation> is !sed in a

very -ide and li"eral sense) It is +er#a+s not less o"vio!s t#at not#in3 follo-s fro1 t#ese 3eneralre1arks *on*ernin3 t#e s+e*ifi* to+i*s -#i*# #ave "een dis*!ssed in t#is +a+er0 and I really donot kno- -#et#er resear*# in se1anti*s stands to 3ain or lose "y introd!*in3 t#e standard ofval!e I #ave s!33ested)

I9IO@RAPH;

Only t#e "ooks and arti*les a*t!ally referred to in t#e +a+er -ill "e listed #ere)

Aristotle) K&L)Metaphysica) %


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8ntroduction to -emantics) Ca1"rid3e, &'$4)@del, 8) K&L)

>"er for1al !nents*#eid"are St2e derPrincipia Mathematica!nd ver-andter Syste1e,I)>Monatshefte f9r Mathemati: und Physi:, vol) ===III, &'7&, ++) &/7:&')

@del, 8) K4L)>"er die 9n3e von e-eisen)>+rgenisse eines mathematischen ;olloquiums, vol)

II, &'7., ++) 47:4$)@onset#, F) K&L)>9e Con3rs Des*artes) !estions de P#iloso+#ie s*ientifi6!e)>Reue thomiste, vol)=9I, &'7, ++) &7:&'7)

@rellin3, 8), and Nelson, 9) K&L)>e1erk!n3en 2! den Paradoien von R!ssell !nd !rali:Forti)>Ahandlungen der

+r:enntnis, vol) I, &'7., ++) &$7:&.?)8okos2ynska, M) K4L)

>Synta, Se1antik !nd Actes du >ongr?s 8nternational dePhilosophie -cientifique, vol) III, Paris, &'7., ++) ':&$)

8otar"inski, T) K&L)+lementy teori po3nania, logi:i formalne i metodologi nau:) %+lements of+pistemology, "er die es*#rnkt#eit der A!sdr!*ks1ittel ded!ktiver T#eorien)>+rgenisse einesmathematischen ;olloquiums, vol) II, &'7., ++) &?:47)

Na3el, E) K&L)Revie- of Hofstadter K&L) The =ournal of -ymolic 7ogic, vol) III, &'7, +) ')

Na3el, E) K4L)Revie- of Carna+ K4L) The =ournal of Philosophy, vol) ===I=, &'$4, ++) $.:$/7)

Ness, A) K&L)> Tr!t# As Con*eived "y T#ose -:rifter utgitta @et ors:e Bidens:aps6A:ademi i Cslo, 88. Dist. +r:enntnis,vol) , &'7?, ++) 7//:$.)

R!ssell, ) K&L)An 8nquiry 8nto Meaning and Truth) Ne- ;ork, &'$)

S*#ol2, H) K&L)Revie- of -tudia philosophica, vol) I)@eutsche 7iteratur3eitung, vol) 9III, &'7/, ++)&'&$:&'&/)

Tarski, A) K&L)>S!r les ense1"les dfinissa"les de no1"res rels) I)>


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Tarski, A) K4L)>Der @r!ndle3!n3 der -issens*#aftli*#en Se1antik)>Actes du >ongr?s 8nternational de

Philosophie -cientifique, vol) III, Paris, &'7., ++) &:)

Tarski, A) K$L)>"er den e3riff der lo3is*#en Fol3er!n3)>Actes du >ongr?s 8nternational dePhilosophie -cientifique, vol) II, Paris, &'7/, ++) &:&&)

Tarski, A) K?L)>On Unde*ida"le State1ents in Enlar3ed Syste1s of 9o3i* and t#e Con*e+t of Tr!t#)>The =ournal of -ymolic 7ogic, vol) I, &'7', ++) &?:&&4)

Tarski, A) K.L)8ntroduction to 7ogic) Ne- ;ork, &'$&)

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The Semantic Conception of Truth