herbert keuth - tarski's definition of truth and the correspondence theory

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Tarski's Definition of Truth and the Correspondence TheoryAuthor(s): Herbert KeuthSource: Philosophy of Science, Vol. 45, No. 3 (Sep., 1978), pp. 420-430Published by: The University of Chicago Press on behalf of the Philosophy of ScienceAssociationStable URL: http://www.jstor.org/stable/187027

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DISCUSSION

TARSKI'S DEFINITION OF TRUTH AND THE

CORRESPONDENCE THEORY*

HERBERT KEUTH

University of Mannheim

I

The correspondence theory of truth is one of the most venerable

philosophical theses. Yet it was known almost from the beginningto entail the antinomy of the liar. This antinomy affects any use

of the word "true," not only the use which the correspondence theoristintends. A definition of truth for formalized languages which avoids

the antinomy was not formulated before 1931. Tarski's proposal of

that year uses an ingenious device. His definition concerns the truth

or falsity of the sentences of one language, the object language, but

it is given in a different language, the metalanguage. Adapting this

definition to natural languages poses some problems, but let us assume

they do not exist. Then we may freely apply the word "true" even

to sentences of natural languages. We need have no fear of getting

involved in antinomies. The use of the word "true" has been rehabili-tated. But what about the correspondence theory? Has it too been

rehabilitated? Many philosophers claim it has, in particularKarl Popperand his followers. But whether this is so depends of course on what

exactly the correspondence theory asserts.

Tarski's definition, or rather a definition of truthfor naturallanguages

along Tarski's lines, entails that the sentence "Schnee ist weiss"

is true if, and only if, snow is white, and so does the correspondence

theory. Here German is our object language and English is the

metalanguage. But for reasons which Tarski indicated himself, thecoherence theory and the pragmatist theory of truth entail the same

consequence. Hence the correspondence theory has to assert somethingmore if there is to be any difference between the theories of truth.

According to Popper it asserts "that truth is correspondence with

the facts (or with reality); or, more precisely, that a theory is true

if and only if it corresponds to the facts" ([6], p. 44). The last

*Received September, 1977; revised November, 1977.

Philosophy of Science, 45 (1978) pp. 420-430.

Copyright ? 1978 by the Philosophy of Science Association.

420



TARSKI'S DEFINITION OF TRUTH

clause sounds very much like "'Schnee ist weiss' is true if, and

only if, snow is white," but it is not as innocuous, for, according

to Popper, the correspondence theory deals "with the correspondencebetween statements and facts, and therefore with some relation

between statements and facts" ([5], p. 325).This is a rather strong interpretation of the correspondence theory.

There is also a weaker one, which is less vulnerable to criticism.

Both give descriptive sentences a realist interpretation. The sentence

"Schnee ist weiss" of the object language not only describes the

state of affairs that snow is white, but also asserts that the state

obtains, that it is a fact. Perhaps we should rather say "lawlike

generality" instead of "state of affairs," but this does not affectour present argument. According to the more careful interpretation,the correspondence theory simply claims that the sentence "Schnee

ist weiss" is true if, and only if, it is a fact that snow is white.

No relation of correspondence is postulated between the true sentence

"Schnee ist weiss" and the fact that snow is white. Yet "'Schnee

ist weiss' is true" is taken to make a proper assertion about the

sentence "Schnee ist weiss" rather than simply repeating what "Snow

is white" states. As "Schnee ist weiss" itself asserts that snow beingwhite is a fact, its translation "Snow is white" into the metalanguagemust state the same fact. Hence "Snow is white" gives the necessaryand sufficient conditions for "Schnee ist weiss" to be true, and

therefore we may write "'Schnee ist weiss' is true if, and only if,snow is white." The coherence theory and the pragmatist theoryof truth may imply the same string of signs, but according to them

it does not have the same meaning, as they do not presuppose that

"Schnee ist weiss" asserts a fact.

The intention of this paper is to show that neither interpretationof the correspondence theory is compatible with Tarski's definitionof truth. If the correspondence theory is taken to assert that a particular

relation, the relation of correspondence, holds between any true

sentence and only true sentences on the one hand, and somethingreal on the other, then Tarski's definition cannot free it from its

antinomies. The same must be true even if the correspondence theorymakes any proper assertion about sentences, i.e., if there is anydifference in meaning between asserting a sentence and calling it

true. This affects the coherence theory and the pragmatist theoryof truth as well. There is probably no living philosopher who stressesthe philosophical importance of the correspondence theory more than

Karl Popper does. On the other hand, Popper's critical rationalism

gets much attention, particularly in the social sciences. Finally, his

is the stronger version of the correspondence theory. Therefore we

421



HERBERT KEUTH

shall first discuss Popper's exposition.We have been talking about sentences and statements. There are

even more kinds of entities, which are said to be true, e.g. propositions,but we cannot here resume the discussion about which of them may

properly be called true. (For more details see [1] and [2] .) We simply

adopt Tarski's terminology and call sentences true, and when we

say "sentence," we mean a class of strings of signs, which (the

strings of signs, not necessarily the signs) have the same meaning.If we dealt only with formalized languages, we could mean a class

of strings of signs, which have the same form (cf. [8], p. 370, note

5). Popper also really means "sentences," when he says "statements."

He only "gives" the word "statement" to some critics of Tarski

([5], p. 319, note 1).

II

According to Popper, Tarski was the first to interpret the corre-

spondence theory adequately because he "makes clear just whatfacta statement P will correspond to if it corresponds to any fact (i.e.,if it is true, H.K.): namely the fact that p" ([6], pp. 45-46). An

example will illustrate what Popper means. As we already said, a

definition along Tarski's lines indicates in a metalanguage the conditions

by which a sentence of the object language is true. Again, let German

be our object language and English the metalanguage, and let

(1) Der Mond ist rund

be the sentence of the object language for which the conditions of

truth are given. Then a definition of truth along Tarski's lines implies

(2) "Der Mond ist rund" is true if, and only if, the moon isround.

This already gives us a partial definition of the concept of truth or,more precisely, of the predicate "X is true" ([7], pp. 188-189). Now

Popper assumes that, according to this definition, the sentence (1)"Der Mond ist rund" corresponds to thefact that the moon is round

if, and only if, the sentence is true. But what does definition (2)

actually assert? Here the sentence

(3) The moon is round

is the definiens, while

(4) "Der Mond ist rund" is true

is the definiendum. Hence (3) and (4) have ex definitione the same

422



HERBERT KEUTH

is true if, and only if, the moon is round. In view of the role which

the concept of satisfaction plays in Tarski's definition of truth, we

need not discuss it in more detail. We can even answer our epistemo-

logical question without mentioning the concept any more. It will

be sufficient for our purposes to examine the partial defintions of

truth.

III

According to Popper, a definition of truthwhich involves the scheme

(5) asserts among other things that the sentence "Der Mond ist rund"

corresponds to the facts if, and only if, the moon is round. AndTarski introduces this scheme as a "criterionfor thematerial adequacy"

of his definition of truth ([8], p. 343). This criterion is meant to

insure that the definition does "justice to the intuitions which adhere

to the classical Aristotelian conception of truth-intuitions which find

their expression in the well-known words of Aristotle's Metaphysics:

(6) To say of what is that it is not, or of what is not that it

is, is false, while to say of what is that it is, or of what

is not that it is not, is true"

([8], pp. 342-343, his italics). According to the criterion a definition

of truth is materially adequate if, and only if, it entails all sentences

of the form (5), so, e.g. (2). From this we might conclude that any

theory of truth which satisfies the criterion must be a version of

the correspondence theory. This, however, cannot be true, for the

adherents of the pragmatist and the coherence theories of truth acceptthe scheme (5) as well, and Tarski himself shows why they do so.

Whoever rejects the scheme has to accept its negation

(7) X is true if, and only if, not p

instead ([8], p. 362). The only consequence of this move is that

for him the words "true" and "false" exchange their meanings. If

somebody rejected both (5) and (7), he would have to admit that

"Schnee ist weiss" might be false, though snow is white, while "Der

Mond ist rund" is true and the moon is round. Rejecting the scheme

(5) has unacceptable consequences. Hence, if its acceptance involved

an epistemological commitment to a particular theory of truth, there

could be but this theory. However, Tarski himself stresses that his

definition of truth is epistemologically neutral. It cannot take an

epistemological position, for it "implies nothing regarding the condi-

tions under which a sentence like 'snow is white' can be asserted.

It implies only that, whenever we assert or reject this sentence, we

424




must be ready to assert or reject the correlated sentence "'Schnee

ist weiss' is true" "([8], pp. 361-362, the choice of the German

objectlanguage is mine).

IV

The adherents of all theories of truth state the conditions under

which they assert that a sentence is true according to the scheme

(5). They all agree that the sentence "Der Mond ist rund" is true

if, and only if, the moon is round. Hence, if there is to be anydifference between the theories of truth, the conditions, on which

their adherents are prepared to assert that the moon is round, mustdiffer. In other words, the sentence "The moon is round" must have

different meanings depending on the theory of truth, which we

presuppose, and so must all other sentences, which we may substitute

for the variable "p" in (5). When the correspondence theorist saysthat the moon is round, he must mean something different from what

the coherence theorist means, and neither of them can agree with

what the pragmatist means. Nevertheless their assertions will have

something in common. Each of them assigns a property, the property

of being round, to an object, the moon. None of them asks a question,nor does any of them give an order. Rather they all assert something,but possibly asserting something does not mean the same to all of

them.

1. Suppose when the correspondence theorist says "The moon is

round," he means "The moon is in fact round," and let the meaningof "in fact" be determined solely by the contrast to the followingsomewhat caricatured positions of the coherence theorist and the

pragmatist. We assume that, when the coherence theorist says "Themoon is round," he means "Perhaps the moon is round, perhapsnot. At any rate the assumption that it is round is consistent with

the rest of my assumptions," and we assume that the pragmatistmeans "Perhaps the moon is round, perhaps not. At any rate the

assumption that it is round is useful." If the correspondence theory

postulated only that statements should be interpreted as our corre-

spondence theorist does, then it would certainly be rehabilitated byTarski's definition of truth, provided of course that the definition

can be applied to natural languages.

2. Quite obviously this is not Popper's interpretation of the corre-

spondence theory. Nor does he interpret the sentence "The moon

is round" in this way. He likes profuse ontologies and when asserting

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HERBERT KEUTH

"The moon is round" he not only implies that the moon exists and

that it is in fact (in the above sense) round, but also that there is

the fact that the moon is round. To him this fact is an entity inits own right. It is as real as the moon. But even if the correspondence

theory postulated no more than such an interpretation of sentences,its use of the word "true" would be rehabilitated.

This is easily seen once we reconsider the above partial definition

of the predicate "X is true." If the sentence (1) "Der Mond ist

rund" of the object language asserts the existence of the fact that

the moon is round, then its translation (3) "The moon is round"

into the metalanguage asserts the same and, as (3) is the definiens

of (4) "'Der Mond ist rund' is true," the latter asserts the same

again. As opposed to this the equivalence (2) "'Der Mond ist rund'

is true if, and only if, the moon is round" neither asserts anythingon the moon, nor on the fact that it is round. A fortiori it does

not assert that the peculiar relation of correspondence holds between

(1) "Der Mond ist rund" and the fact that the moon is round. Rather

it renders the convention by which Tarski assigns the meaning of

(3) to (4). In view of this interpretation the correspondence theorycould not give rise to antinomies.

3. But even this is not Popper's interpretation of the correspondence

theory. He is not content with the assumption that sentences assert

the existence of facts. Rather he believes that "the peculiar relation

of correspondence to a fact" holds between true sentences and facts

([6], p. 46, his italics). And he believes that Tarski's definition finallymakes it possible to state the conditions, under which each sentence

of some language corresponds to the facts ([6], p. 46). His reasoningis as follows. Both the sentence (1) "Der Mond ist rund" and its

translation (3) "The moon is round" into the metalanguage state thefact that the moon is round. Hence, when we want to speak about

the correspondence of a sentence to a fact, the metalanguage allows

us to state the fact, or the alleged fact, about which the sentence

in question speaks. On the other hand the metalanguage contains

the name "Der Mond ist rund" of sentence (1). Hence the metalanguage

permits us to speak about the sentence and assert e.g. that it

corresponds to the facts ([6], p. 46). Up to this point Popper's reasoningis obviously correct, provided of course the metalanguage contains

a predicate which permits him to assert that a peculiar relation of

correspondence holds between some sentence and some fact. Popperintroduces the predicate "X corresponds to the facts," which Tarski

had not needed.

Now Popper thinks that the sentence (1) "Der Mond ist rund"

426




corresponds to the facts if, and only if, the moon is round. And

as this applies correspondingly to all other sentences Xi of the object

language and their translations "pi" into the metalanguage, he intro-

duces the scheme

(8) X corresponds to the facts if, and only if, p

([5], p. 326). He thinks that he can state in this way the conditions

under which each sentence of some object language corresponds to

the facts and that he may therefore introduce the definition

(9) X is true if, and only if, X corresponds to the facts

([6], p. 46). According to Popper, Tarski pointed out that this is"an objectivist or absolutist notion of truth," the correspondence

theory. This part of Popper's argument is not tenable and in (a) and

(b) we shall try to show why.

(a) The scheme (8) resembles (5) very much and it might serve

a similar purpose. Then (8) would be the criterion for the material

adequacy of Popper's definition of correspondence. According to this

criterion a definition of "X corresponds to the facts" would be

materially adequate if, and only if, it implied all sentences of theform (8), hence e.g.

(10) "Der Mond ist rund" corresponds to the facts if, and onlyif, the moon is round.

Each of these sentences would be a partial definition of "X correspondsto the facts," while (10) would be a complete definition of

(11) "Der Mond ist rund" corresponds to the facts.

In (10) the clause (3) "The moon is round" would be the definiens

and (11) the definiendum. Hence ex definitione (3) and (11) would

have the same meaning. On the other hand, as we already mentioned,

(3) is as well thkedefiniens of (4) "'Der Mond ist rund' is true,"if we follow Tarski. Hence

(12) "Der Mond ist rund" is true if, and only if, "Der Mond

ist rund" corresponds to the facts

says no more than

(13) The moon is round if, and only if, the moon is round.

Correspondingly (9) says no more than

(14) p if, and only if, p.

427



HERBERT KEUTH

Who could not call such a tautology correct? But in view of this

interpretationof

(8), Popper'sdefinition

(9)can

hardlyrender the

correspondence theory.

(b) But Popper does not intend to define the predicate "X corre-

sponds to the facts" by means of (8). Rather he presupposes thatit "relate(s) expressions to the world of facts" ([5], p. 326). In thiscase Popper's definition (9) contradicts Tarski's definition of truth.

Tarski assigns (4) the meaning of (3). On the other hand (4) and

(11) must have the same meaning according to Popper's definition

(9). Obviously (11)is the

definiens,while

(4)is the definiendum.

Hence there are two definitions, both of which assign a meaningto (4). If we now assume with Popper that "X corresponds to the

facts" relates expressions to the world of facts, then the compatibilityof Tarski's and Popper's definitions depends exclusively on the

interpretation of (3). We can choose either of two possible interpreta-tions, both of which make the definitions incompatible.

(i) Let (3) "The moon is round" simply assert that the moon is

round,as Tarski

assumes,whereas

(11)"'Der Mond ist rund' corre-

sponds to the facts" asserts that the peculiar relation of correspondenceholds between (1) "Der Mond ist rund" and the facts. Then Popper'sdefinition (9) assigns a different meaning to (4) "'Der Mond ist rund'

is true" than a definition along Tarski's lines, which follows the scheme

(5). Hence the definitions are incompatible.

(ii) Now let (3) "The moon is round" assert that the peculiar relation

of correspondence holds between (1) "Der Mond ist rund" and the

facts. Then(3)

and(11) have the same meaning. Only in this casethere is no formal contradiction between Popper's definition (9) and

Tarski's scheme (5). Together they imply Popper's scheme (8), which,

according to Popper, entails that (3) indicates the necessary and

sufficient conditions for the peculiar relation of correspondence to

hold between (1) and the facts. To me this seems to be the essence

of Popper's reasoning, which tries to show that Tarski has rehabilitated

the correspondence theory. At first sight the semantical antinomies

seem to have been avoided in this case. But the impression is deceptive,

for, accordingto

Tarski, (3)is

the translation of (1) into the metalan-guage. Hence both must have the same meaning. On the other hand

we assumed that (3) and (11) have the same meaning and consequentlythis must hold for (1) and (11) as well. In other words, given this

interpretation of (3), the sentence (1) "Der Mond ist rund" must

428




assert that itself is related in the peculiar way of correspondenceto the facts. Hence it is self-reflexive and

accomplishes tasks whichTarski assigned to its name in the metalanguage and to the semantical

predicates of the metalanguage. Against Tarski's intention-and prob-

ably against Popper's as well-the object language is "semanticallyclosed," i.e., along with its sentences it contains the linguistic means

to speak about them. Therefore it admits semantical antinomies.

Popper's definition (9) is formally compatible with Tarski's scheme

(5), but the interpretation of (3) cancels the separation of object

language and metalanguage, which Tarski's definition of truth pre-

supposes. Hence Popper's and Tarski's definitions are incompatiblein this case as well.

Being incompatible with Tarski's definition is quite legitimate, but

because of the incompatibility the problem of the antinomies remains

to be solved for Popper's version of the correspondence theory. And

Popper has to find an answer to the question of what he means

when he calls a sentence true. According to Tarski "'Der Mond

ist rund' is true" simply means "The moon is round."

4. But Tarski might at least have rehabilitated the weak version of

the correspondence theory, which we mentioned in section I. Contraryto the fictitious version in 2 and to Popper's version, it does notconsider the fact that the moon is round to be as "real" an entityas the moon. Even if it did, it would not for this reason get into

conflict with Tarski's definition. And unlike Popper's version it does

not assume that a peculiar relation of correspondence holds between

true sentences and facts. But like Popper's version it takes (4) "'Der

Mond ist rund' is true" to make a proper assertion on the true sentence(1) "Der Mond ist rund" rather than simply repeat in different words

what (3) "The moon is round" states on the moon. Hence accordingto the weak version (3) and (4) cannot have the same meaning. If

they had, then either both would make the same statement on the

sentence (1), the statement that (1) is true, or both would be confined

to the same statement on the moon, the statement that the moon

is round. However, according to the weak version of the corre-

spondence theory, (4) makes a statement about the sentence (1),

whereas (3) is confined to a statement about the moon. As Tarskimakes (3) the definiens of (4), this version of the correspondence

theory is not compatible with Tarski's definition of truth either. Hence

it still faces the problem of the antinomies and has to answer the

question of what it means to call a sentence true.

429



HERBERT KEUTH

V

Tarski's definition of truth has rehabilitated the application of the

word "true" to sentences of formalized languages. Probably thedefinition can be adapted to specially prepared parts of natural

languages. So we need no longer fear getting involved in antinomies

when using the word "true." But a correspondence theory accordingto which a sentence is true if, and only if, it is related in the peculiar

way of correspondence to the facts, is incompatible with Tarski's

definition. This is easily overlooked, since Tarski's definition impliesall sentences of the form "X is true if, and only if, p" and since

the sentences substituted for "p" may be taken to assert facts.

Therefore Tarski's definition may be taken to assert that the sentencewhose name is substituted for "X," is true if, and only if, there

is the fact, which the sentence substituted for "p" states. From this,some authors seem to infer that Tarski's definition relates all true

sentences and only true sentences to facts and that this relation is

of the peculiar kind of correspondence. But for a simple reason this

view must be mistaken. According to Tarski the equivalence "X is

true if, and only if, p" expresses that the sentence which is substituted

for "pI," e.g., "The moon is round," and the sentence which results

from "X is true" on proper substitution, in this case "'Der Mondist rund' is true," have the same meaning. Hence Tarski's definition

cannot possibly relate sentences like "Der Mond ist rund" to facts

like the fact that the moon is round, for "'Der Mond ist rund' is

true" does not assert anything on "Der Mond ist rund." Rather it

asserts that the moon is round, just as "The moon is round" does.

For this reason no theory of truth which claims to make a properassertion about a sentence when calling it true, is compatible with

Tarski's definition of truth. Hence each of them has to find its own

solution to the problem of the semantical antinomies and each hasto explain what it means to say that a sentence is true.

REFERENCES

[1 Keuth, H. Realitit und Wahrheit.Zur Kritikdes kritischen Rationalismus. Tiibingen:J.C.B. Mohr (Paul Siebeck), 1978.

[2] Pitcher, G. (ed.) Truth. Englewood Cliffs: Prentice-Hall, 1964.

[3] Popper, K. R. The Logic of Scientific Discovery. New York: Basic Books, 1959.

[4] Popper, K. R. Objective Knowledge, An Evolutionary Approach. Oxford: Clarendon

Press, 1972.

[5] Popper, K. R. "Philosophical Comments on Tarski's Theory of Truth." In [4].

pp. 319-340.

[6] Popper, K. R. "Two Faces of Common Sense: An Argument for CommonsenseRealism and Against the Commonsense Theory of Knowledge." In [4]. pp. 32-105.

[7] Tarski, A. "The Concept of Truth in Formalized Languages." In Tarski, A. Logic,Semantics, Metamathematics. Oxford: Clarendon Press, 1956. pp. 152-278.

[8] Tarski, A. "The Semantic Conception of Truth and the Foundations of Semantics."

Philosophy and Phenomenological Research 4 (1943/4): 341-375.

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