THE CONCEPTION OF SEMANTIC TRUTH
by
Hoke ROBINSON
In this paper I would like to examine the sense in which Tarski's
"Semantic Conception of Tru t h
-(
shall
first give my interpretation of Tarski's argument in the article of
that
1 title (supplemented by his T h e Concept of Truth in Formalized
Languages"
) c o n t (a i n s
2a
discussing,
and then follow this with observations on certain problems
)c ao nby
raised
nd Tarski's
c e pnotion
t i ofo truth which I am unable to clear up by
reference
on t h eto that text alone.
ro Tarski begins
his attempt to "catch the meaning o f an old nof
tion"
(
w
o
r
t
r
u
t
h
3
k. s :)the, definition must be materially adequate, in the sense that it
satisfy
)iI
must
be applicable with o u t violating unnecessarily o u r intuitions
about
b
n y truth ; and it must be formally correct, i.e. it must not violate
so t r criteria of correct language-usage. The question, however, is :
formal
a
what
d t are
e the conditions o f material adequacy ; and what are the
grounds
o f the criteria (concepts, rules, etc.) of formal correctness.
ir n
g
t Tarski begins by stipulating that the term " t ru e
applied
"propositions" nor to any other
to - w itol lsentences,
i n i t not
i a to
l l either
y
objects
though
the
possibility
of
extension
to other objects remains
h
b
e
m
open.
As
restricted
for
the
moment
to
sentences,
however, truth will
e
a
be
relative
to
the
language
of
which
the
sentence
is a sentence. As a
g
k
starting-point
in setting conditions o f material adequacy, Ta rski
e
appeals
to
Aristotle's
f o rmu la -" ... to say of what is that it is , is
n
c
e
l
re (
aSemantics,"
1
in Readings in Philosophical Analysis (Herbert Feigl and Wilfrid Sellars,
lreds.).
) Ne w York : Appleton-Century-Crofts, Inc., 1949, pp. 52-84. Reprinted f rom
Philosophy
and Phenomenological Research, Vol. IV, 1944. Hereinafter referred to as
A
cw
SCT.
l
rh
(f
iaLogic,
2
Semantics, Metamathernatics. Oxford: Clarendon Press, 1957. Translated by
r
tJ. H.
)e Woodger from De
Vol. 1, 1937. Hereinafter referred to as C I T E
e
A
iPhilosophica,
dr
(
W
a
h
r
h
e
i
t s b
l
rt T
e 3
g r i f f
f
A
i )
i rR
n
a
sd S
e
e
S
a
In C
d
K
d
af TTI o r m a l
i ,A
e
,s i e r t
m
Tn
fe pR
S .S
p
r
ia h5 c
h
K
e
ne 3IS n
,
i" .,e
t
tS -m
a T
a
d
94
H
.
ROBINSON
true" — and refines this into a restatement o f the correspondencetheory : a sentence is true if it agrees with / corresponds to / designates
an existent state-of-affairs / reality. In order to avoid the vagueries of
the slash-connected locutions, however, he starts with a concrete
example: "The sentence 'snow is white' is true if, and only if, snow is
white." (
4 The p rima facie trivia lity o f this definition is vitiated b y th e
)
explanation
that the quoted expression is the name of the expression
unquoted, and not that expression itself. I f , then, this statement
expresses the material conditions o f the truth o f the quoted expression, then a definition of truth is materially adequate only if from it all
expressions o f the same form as that statement are derivable. That
form, isolated, is: i f p is any sentence, and X is the name o f that
sentence, then :
(T)
x
is true if, and only if, p.
Any substitution-instance of this equivalence is an equivalence of the
form ( T ) (
5 This "semantic" conception o f truth allies it with other semantic
)notions,
.
such as "satisfaction,
— brings it within range of the family of antinomies known as the
thus
"semantic
d e s iparadoxes.
g n a t i oAsnthese
"
can be dealt with only insofar as the
alanguage
n used
d is unambiguous in the relevant respects, and a s
" d e language
ordinary
f i n is i generally
t i
not thus unambiguous, Tarski begins
his
discussion
of
the
formal
correctness
o f the semantic definition of
o n , "
truth
by detailing
the notion of a language with a specified structure.
a
n
dSuch a specification must determine (I) what expressions are to be
considered meaningful ; (2) what expressions are primitive (undefined); (3) what rules introduce defined terms ; (4) which expressions
are sentences; (5) when sentences are to be a sse rte d (
sentences
are primitive (axioms); and (7) what the rules of inference
6
or
a language whose structure is thus
); proof
( 6 ) arew t oh be.
i Granted
c h
(
(4
5
that
) sentence; the definition of truth in general would be the totality of these.
()C
language
(see below).
T
4
S
)h
T
u
N
,
sp
o
te
.
a
e
5
tc4
h
.
i
a
tn
,s
t
h
THE CONCEPTION OF SEMANTIC TRUTH
9
5
specified, Tarski attacks the paradox of Epimenides the Liar.
Epimenides the Cretan said, " A ll Cretans are liars." I f we represent the quoted expression as c, then to suppose c true is to deny what
Epimenides said, and hence to suppose c false. On the other hand, to
suppose c as false is to affirm what Epimenides said, and hence to
suppose c true. We should scarcely call a definition of truth "formally
correct" which allowed the inference of the denial of a sentence from
its affirmation, and conversely. Tarski finds two essential() presuppositions leading to the paradox, one of which must thus be altered if
the paradox is to be avoided. One of these is the assumption that the
laws of logic hold ; this is clearly to be abandoned only under utmost
duress. The other is that the (now, specified-structure) language in
question is "semantically closed."
To assume that the language producing the antinomy is semantically closed is to assume that it contains its expressions, the names of
each expression, the term " tru e " , and all expressions determining the
use of t r uparadox-producing
this
e"
assumption, and avoid languages which are
.
semantically
closed. This can be done by using two languages, an
I"object-language
f
the
- object-language. The definition of truth will obviously be in this
w
meta-language,
and hence so will all the equivalences o f form (T)
e, a n d
which
a
it
must
imply
(i.e., in order to be materially adequate). Yet the
a
of
(T)
is
obviously
fro m the object-language. Hence the metar" m e t a l a n gmust
u acontain every sentence of the object-language, together
elanguage
ng e "
o( (
t8
Cretan"
are possible - is, he holds (in footnote I I to SCT), inessential here. since the
7
)
can be constructed from the first t wo premises alone, as follows: S is any
t)paradox
"
S * is the sentence
wT
ho f the form, "E v e ry sentence
osentence
correlated
with each sentence S so that, where S is "Ev ery sentence is 0", S* is "The
a
risentencec'every sentence is 0' is 0 . " S is "self-applicable" if and only if S* is true,
r
hs
enon-self-applicable
if and only if S* is false. Then let 0 be "non-self-applicable". Then
t is false (i.e., the sentence "The sentence 'Every sentence is non-self-applijif" Sk is o, S*
is false), and hence S is not O. Contrariwise, if S is not 0,
a i is non-self-applicable"
l
ecable'
S*
is true,
and hence S is O. See also the Grelling-Nelson paradox of heterological
'
k
s
cwords, which also appears to require no empirical statements.
b
ta s(
o t8
umeta-language, relative to which the original meta-language is the objectthigher-level
h
t )i
"
hlanguage.
e Tr
l hd
i
a as
w sd
s
s iu
o sm
f tp
i
l tn
i
96
H
.
ROBINSON
with the name o f each sentence (i.e., X ) (
9
semantic
terms, especially " tru e " . I f the semantic terms, especially
") tru
, le o
", g
arei cintroduced
a l
into the meta-language by definition (rather
than
consider them adequate.
o p asep rimitive
r a t )(n
o ,r we
s may
,
Note
that
the
meta-language
definition's formal correctness re a
n
d
quires that the meta-language be "essentially richer" than the objectlanguage, i.e. it must contain every term in the object-language,(
and
u some the object-language does not contain (e.g., the names o f
object-language
sentences X, and " tru e " , etc.). I f it were not essen)
tially richer, i t could b e interpreted in the object-language, and
Epimenides' Paradox could be reconstructed at the me ta -le ve l.(
12
)
(
a variety
9
of the calculus of classes symbolism for the translation of an object-language
sentence
into the meta-language (so that the meta-language "contains" the object-lan)
guage
in the sense that it includes a translation of every object-language sentence), and
I
another
notation (perhaps based on Boole) for the names of these sentences. Thus the
n
sentence
"given any two classes, one contains the other" is in the object-language:
C
T f lx f lx A I x x lx x
F
in Lits meta-language translation:
,
for any classes a and b we have a b or b c a
T
and
a has as its name in the meta-language:
r
n
s
i
(See
n
p. 187.)
k CTFL,
,
i(
(
Logic,
u Semantics,
Metamathematics, p. 405, and elsewhere, introduction of the term
L
1
"t ru
s0 e " , by axiom has two disadvantages: there is always the "accidental" or contrived
,
character
e
)
2 that attaches to stipulated axioms, violating our intuitions of the material
adequacy
sA + of the term; and there is the possibility that further axioms could render the
L
meta-language
inconsistent. Both these problems are avoided if "true" is introduced by
sa I
definition.
vT ,
( " ) See
note 8.
a
a 2)
(rr
•
object-language
remains "c los ed", that is, everything that can be said in the meta-lan1is
guage
c an be said, mutalis mutandis, i n t he object-language, and the semantic
2e
k
paradoxes
are reconstructable. See especially CTFL, pp. 247-251, in which it is shown
)ti
that
where
a richer meta-language is not possible (in this case, because the object-lanTyp
guage
is of infinite order, and transfinite ordinals are not considered), both a sentence
ho
andao
fi its negation are derivable (following Godet). See also CTFL, p. 274.
tP
n
io
t
sls
tio
osu
sh
t
al(
yo
i
,g
n
ui"
THE CONCEPTION OF SEMANTIC TRUTH
9
7
Within th is framework o f language and meta-language, Ta rski
defines truth via the semantic notion of satisfaction. This is defined as
a relation sometimes obtaining between objects and sentential functions. Sentential functions are defined "recursively", that is, the
simplest cases are described, together with rules which ma y b e
applied recursively to these simple cases to generate all and only
sentential functions. The notion o f satisfaction, according to Tarski,
may also be defined recursively : " We indicate which objects satisfy
the simplest sentential functions ; and then we state the conditions
under which given objects satisfy a compound function — assuming
that we know which objects satisfy the simpler functions from which
the compound one has been constructed." (
satisfaction
and o f sentential function, Ta rski defines " t ru t h " as
13
applying
functions with no free varia) G i vtoethose
n sentences
t h i (sentential
s
bles)
d e which
f i are
n satisfied
i t i oby nall objects, all other sentences being false.
o f
1944
T ; in 1951 R. M. Martin, noting that "after reading the relevant
passage
the reader may feel that he knows no more about the truth
h
concept
e
than he did before," attempted to clarify this p a ssa g e (
(CTFL
had not yet appeared in English). Wh y is it the case, asks
1 4S
Martin,
that " fo r a sentence only two cases are possible: a sentence is
) e
either
m satisfied by all objects, or by no objects" ? (
1 5a o f functions being satisfied, not by objects one-at-a-time, but
talks
) nT a by
rather
r sinfinite
k i , sequences
h
e f of objects f
function
is satisfied by sequence f if f
s t a yof one
1
s argument
,
ltwo-argument
e s
i satisfied
t
, sif a t i s f i function
is
by f ; if f
it
2 ;caetc.
la
n Then
d
f the partial definition of satisfaction — e.g., "satisfaction
for
, C2-place
2
r I e s nfunction
p e c t(relation)
t i vh e il Ry "s— is the correlating o f it t o a
sequence
sw o a a tf such
i y that
s ff
y
ta Martin
R
f
then
2
•
quotes
Lamma
A of CTFL:
n
c
LEMMA
A. I f the sequence f satisfies the sentential function x, and
e
thep infinite sequence g s a t i s f i e s the following condition : for every
t
(i
(1o
Philosophy
3n
1
and Phenomenological Research, Vol.
(A
)o") SCT, p. 63.
)S
f
C
R
T
.T
,r
M
.pu
.t
M
6h
a
r3
"
t.
iw
a
XI (Mar. 1951), pp. 411-12.
98
H
.
ROBINSON
k [where k is some positive integer, and members o f sequences, as
well as variables in functions, are ordered — say, alphabetically — and
numbered], f
sequence
k =
gg also satisfies the function x . (
1
6
k
i f
)T
v his lemma presents very little problem as it stands; it is clear that we
k dealing wit h infinite sequences in order t o avoid having t o
are
construct
a different notion o f satisfaction fo r one-argument funci
s
tions,
two-argument
functions, etc.; but whether or not a sequence
a
satisfies
f
r a function depends only on those members of the sequence
corresponding
to variables in the functions ; all others are irrelevant.
e
e
Thus
v aonly
r f
satisfies
two-argument function R; f
li aa n bd
obviously,
if some other sequence g has members identical with those
3
fl e
o,2
o f f lin all the relevant places (i.e., those with indices corresponding to
the
ta
of the free variables of the function under consideration),
r
f indices
then
i
f
f
satisfies
the
so will g, no matter how much the
,e
e
t
c
.
a function,
r e
t
sequences
may
differ
in
the
irrelevant
places.
irh m
m
a
t
e
r
i
a
l
.
e l
Since
a
sentence
is
a
function
with
no
free variables, there are no
H
n
c
e
,
e v ea
members
of
any
sequence
which
are
relevant
to the satisfaction of this
n
f t u
function,
in
c and hence Lemma B follows trivially :
n
t
i
LEMMA
d
en B. I f x [is a sentence] and at least one infinite sequence
o
satisfies
cx
i the sentence x , then every infinite sequence s a t i s f i e s
xd . i
;
(n Thisg exposition does indeed clear up the sense of Tarski's remark
t
1w "satisfaction by all objects" ; but if Martin thinks that, by virtue of
on
h
7h exposition alone, the reader (or, to speak for myself, this reader)
his
e
)knows
any more about the truth-concept than he did before, he is
e
n
sadly
mistaken.
Th e feeling persists that a great deal o f logical
t
slight-of-hand
has
been performed, and that, rabbit to the contrary
h
notwithstanding,
there
is something funny about that hat.
e
To
begin
with,
what
can it mean to speak o f an "object" "satisr
fying"
a sentence? We do speak of terms, perhaps loosely of objects,
s
satisfying a sentential f u n
e
ct
q
i o( n ; b u t
u
i (1 n
P
M
e
a 61 n
d
n
e 7) l s e w h
c)
eP
r e
eC
.
a T1
f
f F9 u
n
c
L
t 8 i
o
n
i ,;
s pI
.u
1
THE CONCEPTION OF SEMANTIC TRUTH
9
9
a sentence with one o r more vacant places, and the function is
satisfied (gesdttigt) when these places are filled. Thus, it would seem,
all objects (terms) o f all sequences would neither satisfy nor fail to
satisfy all sentences, which would consequently be of indeterminate
truth-value. I f the truth o f, " We are now under nuclear attack"
depends upon the sequence "Hamster, yak, abominable snowman,
. . . " , we are in sad shape.
But Tarski clearly does not mean this. I n company with every
serious thinker since Kant, he holds that a criterion of material truth is
not possible:
In fact, the semantic definition of truth implies nothing regarding the
conditions under which a sentence like (1):
(1) s n o w is white
can be asserted. It implies only that, whenever we assert or reject this
sentence, we must be ready to assert or reject the correlated sentence
(2):
(2) t h e sentence "snow is white" is true.
Thus, we may accept the semantic conception o f truth without
giving up any epistemological attitude we may have had ; we may
remain naive realists, critica l realists o r idealists, empiricists o r
metaphysicians — whatever we were before. The semantic conception
is completely neutral toward all these issues.(')
We are not, therefore, using the semantic conception to determine the
assertability o f an object-language sentence such as (1) above. Bu t
then, what is the purpose of the semantic conception, if it is neutral
toward epistemological attitudes?
Notice that in Lemmas A and B quoted above there was a deletion
following the term "sequence". In Martin's version of these lemmas,
sequences are explained as sequences o f o b je ct s, (
19
)(a n d
t h e
k
e
y
(
1
8
1
)
9
)S
C
M
T
a
r,
tp
i.
7
n
,1
(
p
H. ROBINSON
sentence in SCT also says, " A sentence is true if it is satisfied by all
objects, and false otherwise." (
2°
appear
in the subsection o f CTFL entitled, " Th e Concept o f True
)Sentence
B u tin thet Language
h e of the Calculus of Classes," and the lemmas
o r speak
there
i g io fnsequences
a l
o f classes. Thus in this particular case, a
sentence
in
the
formalized
L
e
m
m
a language of the calculus of classes is true if
it
s is satisfied by all infinite sequences of classes, and false otherwise.
A We may, however, still ask : when is this the case ? A closer look at
athe notionn o f a formalized
d
language may help us here. Tarski first
excludes
ordinary
o
r
colloquial
language from consideration : since
B
this is semantically closed, no "formally correct" definition is possible: t h e antinomy o f the L ia r is always constructable i n such
languages.(
21
languages,
i.e. languages with a specified structure. (
I) I
above
(
)F 3oWr m
2
language
e isathat
n the
o conditions
t e d under which sentences in the language
)l assertable be specified. As it is obviously not possible to specify
are
the
under which material (empirical) sentences are to be
tc hoconditions
ar r
asserted
with
in
a formalized language, we are dealing either with
te c t n
equivalences
of the kind that, e.g., if sentence (I) above is assertable,
e s s
o
iso is sentence (2), and conversely ; o r we are dealing with "logical
n
stru th s
e
(axioms)
stipulated t o b e assertable without p ro o f o r evidence,
co— o
together
with
those sentences (theorems) the evidence fo r which
o
f
n d
consists
in the axioms and the "truth-preserving" inference-rules
ilt th
which
iye o produce these theorems from the axioms. It is clearly sentences
of
o osecond kind which Tarski refers to as "true if satisfied by every
pf the
n
infinite
ro
s m s sequence of objects."
l bus examine the formalized language of Section 3 of CTFL, the
fia Let
language
tli z e o f the calculus o f classes. The definition o f truth fo r this
language
which is held to result from Lemmas A and B is:
ie d
h
nl
e
DEFINITION
23. x is a true sentence — in symbols x T r — if and
sfa
o
rn
p
m
g
a (
e
l
ciu (2 z
iea (20 d
2(
fg 1)22
)S
ye )C3
ii S
T)
nn C
,P
F
gq TLp.
ou ,.9
p
64
fe .p3,
ts 5.(n
ht 71eu
i ;5m
m
THE CONCEPTION OF SEMANTIC TRUTH
1
0
1
only if x ES [i.e., " x is a sentence" ; see Def. 12, p. 1781 and every
infinite sequence of classes satisfies x . (
24
) The move to classes has not, o f course, solved our problem as to
how a sentence can either be satisfied or fail to be satisfied by classes
or by any other "objects". The notion of "satisfaction" is defined in
Def. 22, but contains the term "function", the definition for which is :
DEFINITION 10. x is a sentential junction i f and only if x is an
expression which satisfies one o f the four following conditions : (a )
there exists natural numbers k and I such that x = Lk,1 [lk, 1 reads, "the
k-th class-variable is included in the 1-th class-variable,
— o r class
" c kl . "a1 (s s
I
includes
2 5 x = y [y reads --y.] ( y ) there exist sentential functions y and z
that
such
) ; ( that
0 ) x = y + z [ y + z read y v z.] ( 6 ) there exists a natural
number
t h e krand
e a sentential function y such that x = nk y [read, "the
universal
of the k-th variable over the function y .] (26)
e x i quantification
s t
((I),
s (y) and (45) all presuppose the notion of sentential function: it is (a)
which
mu st b e examined. I n the calculus o f classes, th e mo st
a
elementary
s e n form
t eof function is that in which one variable is related to
another
and other functions must be defined in terms of
n t iby ainclusion,
l
this.
Ta
f
u rskin goes on to define "sentence" (function without free
variables,
Def.
1 2 )(
c
t
i
2
rules
7
for
producing
theorems from the a xio ms. (
o
n
),
2
the
y 9 conditions under which sentences in this formalized language (i.e.,
of
(condition (5), p. 94
)ss tthe
Ti h
pcalculus
u
e ls aet oef cclasses)
o n saret to
i tbeuasserted
t e
above),
and thus contribute to specifying its structure.
su
of satisfaction itself, then, is as follows:
tc The
h definition
e
a
x
i
o
h
DEFINITION
m
s
(22. T h e sequence f satisfies the sentential function x
if
2 f is 8an infinite sequence of classes and x is a sentential function and
these
)
, satisfy one o f the following four conditions : (a ) there exist
a
n
d (
(2
p (24 r
o
v (2s) i
d
e (2b)C s
T
t (27)C
F
T
h 21)C
9
T
C
3
e FL)TF
,L
C
L
F
,p
T
,p
L
.
F
,p
.1
L
.1
p
9
,.1
5
7
p
7
1
.
5
.7
.
1
102
H
.
ROBINSON
natural number k and a sentential function y such that x = n
xk = 'the k-th class variable is included in the 1-th class-variable, and
the
o f sequence f is included in the 1-th member o f
y k-th
a n member
d
sequence f "1 ; ((3) there is a sentential function y such that x = y and f
does not satisfy the function y (y ) there are sentential functions y and
z such that x = y + z and f either satisfies y or satisfies z; (8) there is a
neutral number k and a sentential function y such that x = n
every
of classes which differs from f in at most the
k y infinite
a n sequence
d
k-th place satisfies the function y . (
30
Once
)
again, (0), (y) and (8) contain the term "satisfies" already, so it
is to (a) that we confine our interest. I f we restrict our consideration to
functions of one and two variables only, we may talk, not of infinite
sequences of objects (classes), but of individual classes and pairs o f
them (since, in this case, other members o f the sequence would be
irrelevant to the satisfaction o f the functions being considered). We
then read (a): " x is satisfied b y f in case x is a two-place function
stating that the first free variable is included in the second, and f is a
pair of objects (classes) of which the first is included in the second.
— universally-quantify
we
I f
th e second variable, we get a one-place
function, which states : f satisfies x, where x = f l
i2 L s
includedi n a llclasses"],i n casef isa n object(class)such
that
it is included in all classes, i.e. where f = the null class. Note that
i
in
, both
2 cases,
[ r whether
e a d or, not x is" satisfied
x depends upon properties of
f.
The
question,
then,
in
both
cases
is
:
is there such a pair, or such an
=
object (class) f, such that it satisfies the function x ? The answer to this
question is found in the symbols, axioms and rules of the calculus of
classes, the formalized language under consideration here: according
to these rules, there is a pair o f classes such that one includes the
other, and there is a null class. Hence satisfaction here is determined
by which "objects" " e xist" in the language under consideration.
Now i t is clear fro m Lemmas A and B that either a ll objects
"satisfy" a sentence, or none do ; the question was, what is —
s a t i for
tion"
s f asentences.
cTarski gives an e xa mp le (
3l
)( o f
a
t r u e
s" e n t e n c e
symbolism.)
i ()
n
3
C
1
T
)F
C
L
T
,
F
p
L
.
,1
p
9
.2
1
THE CONCEPTION OF SEMANTIC TRUTH
1
0
3
the sense o f Def. 23. First , the function n•2 11,2 [read, " t h e f irst
class-variable is not included in any class"' is false for all the first
class-variables (and hence it s negation, 1.1
second
class including the first"] is true for all first class-variables) in
2 t
case
there is some class including all classes. Tarski's calculus o f
1
classes
assures us that this is the case, and that furthermore we may
,
universally
quantify
2 [ r e a
d , the" first
t class-variable.
h e r e Hence we arrive at a true
sentence
the calculus of classes) of form (T):
i
s (in the language of a
n
ti ru e
—
a
Uc b. (
]2 2
3
)iThus
1 f the truth of A
a1
b
l including
U 2 1 2 all
, classes a, which existence (together with the existence
n
of
, any classes
2
i s at all) is guaranteed by the stipulated axioms, rules, etc.
dg
of
2 the
u calculus
a r a of
n classes (including such existential assumptions as
otC eaxiom
the
e dof infinity (
nT
3
truth
b 3 o f aysentence might be, " x is a true sentence if and only if it
)ltr) •
asserts
Asomething
already assumed by the axioms and rules o f the
h
sye
formalized
l e r
in question." (We ignore undecidable sentences
[ i m p language
for
i"r the
w
e amoment.)
x y i Should the axioms and rules allow the construction
fof
o
e
s objects
t fe (here,
n classes) which could compose sequences not "satisfying"
some
(i.e ., incompatible wit h it ), then (modus
fa
p
u
t
tsentence
i
c e
"
tollens
on
Lemma
B),
no
sequence "satisfies" (is compatible with) it.
oo
n
d g
this
trAnd
h resulti is strange only if we forget that we are dealing with
f,
languages here, and thus that there are no "contingensaformalized
"a
only
truths" and "logical falsities".
lcies",
d
e
f
i
'c
l "logical
It
seems,
then,
that
a true sentence is satisfied by all objects in the
la
n
i
t
i
T
sense
that
no
object
(as
stipulated in the specified structure o f the
csh
o
n s
lelanguage under consideration) may be other than the sentence deo
farscribes it as being. If, in the language o f the calculus o f classes, a
tsesentence says that there is a null class, then this sentence is "satissified" by all sequences of —
h
no sequence will contain any object
o bspecified
j e c t s "by the
i language,
n
esas
sac a s e
h CTFL
e , p.r 196.e
act (32)
itl (
s
haa 3
3
l
l
esn ) u
c
l
a
s
rs S
s e
ew
i
n
ih( e s
,
sic r e
e
ac m
ch a
li r
104
H
.
ROBINSON
incompatible with there being a null class). And this same situation
would obtain in any other language whose structure could be specifi
e specification stipulates as existent are obtained. The philosophithis
cal
d question o f the " t ru t h " o f statements is transformed in to the
question
of the existence of certain objects of certain kinds in certain
,
languages
; and even if this transformation is helpful logically, it is
p
scarcely
illuminating epistemologically. Thus we see the sense o f
r
Tarski's
note that this conception of truth is epistemologically neutral,
i
that
it
solves
none of the issues between realists, idealists, etc. ; it is a
n
formal
notion.
c
i
p One is tempted to ask : then what good is it? As a formal notion,
aTarski has used his conception of truth to provide a solution to the
lage-old semantic paradoxes ; and in so doing has arrived at logical
lresults paralleling those o f Gödel, but in a more perspicuous form.
yThese results are far from trivial. Tarski is able to show that, since the
bmeta-language must be "essentially rich e r
- ( h eofr aehigher
r elogical
a dtype")
,
than the object-language, then when
eobjects
the
"
object-language
m
u
s
t
is
o
f
infinite
order, n o definition o f truth is
c
c o
n
tfromawhich
i ancontradiction is not d e riva b le (
aconstructable
3
there
4
are
true
sentences
which are not decidable. (
u
),
3
5
a
n
d
t
h
a
t
s
Hoke ROBINSON
Memphis
State University
e)
Dept.
o
f
Philosophy
i
nMemphis, TN 38152
USA
s
p
e
c
i
f
y
i
n
g
t (
(") CTFL, pp. 274-76.
h 3
e 4
s )
t C
r T
u F
c L
t ,
u p
p