Tarski (§4) of “The Semantic Conception of Truth and the Foundations of Semantics”, entitled ‘4. A Criterion for the Material Adequacy of the Definition’

by Corry Shores

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[The following is summary. All boldface, underlying and bracketed commentary are my own.]

Alfred Tarski

“The Semantic Conception of Truth and the Foundations of Semantics”

Part I. Exposition

4. A Criterion for the Material Adequacy of the Definition

 

 

Brief Summary:

The conditions for our definition of truth are that it must be formally correct and materially adequate. It is formally correct if takes the form of the (T) scheme:

(T) X is true if, and only if, p.

Here, X is the name for a sentence, conventionally placed within quotation marks, and p is the sentence expressed in some language (given without quotations). For example:

The sentence “snow is white” is true if, and only if, snow is white.

The definition then is materially adequate if this equivalence is fulfilled [that is to say, if the content or material of the left side of the equation and on the right side are fully adequate to one another.]

Summary

Previously Tarski examined some correspondence theory definitions of truth, and he found them to be too unclear and imprecise. So we must formulate the definition in a different way. Tarski will do so here first by beginning with an example. He has us consider the sentence:

“snow is white”
(334)

We wonder, under what conditions would this sentence be true or false? If we stick with the classic correspondence theory, we would say that

the sentence is true if snow is white, and that it is false if snow is not white. Thus, if the definition of truth is to conform to our conception, it must imply the following equivalence:

The sentence “snow is white” is true if, and only if, snow is white.

(334)

[We above have an equivalence, since we have a biconditional, if and only if.] Tarski notes that “snow is white” on the left side of the equivalence has quotation marks around it, while on the right side it does not.

On the right side we have the sentence itself, and on the left the name of the sentence. Employing the medieval logical terminology we could also say that on the right side the words “snow is white” occur in suppositio formalis, and on the left in suppositio materialis.
(334)

Tarski provides two reasons for why we need to have the name for the sentence on the left and the sentence itself on the right.  Recall again our formulation.

The sentence “snow is white” is true if, and only if, snow is white.

The first part,

The sentence “snow is white” is true

has the form “X is true”. In this grammatical structure, we need to replace X with a name, for otherwise it would not be meaningful, “since the subject of a sentence may be only a noun or an expression functioning like a noun.” (334) The second reason is that “the fundamental conventions regarding the use of any language require that in any utterance we make about an object it is the name of the object which must be employed, and not the object itself. In consequence, if we wish to say something about a sentence, for example, that it is true, we must use the name of this sentence, and not the sentence itself” (334).

Here we are using quotations around a sentence to indicate its name, but we can use other methods as well. For example, we could arbitrarily assign it some letter symbol. Or we can be more mechanical and

use the following expression as the name (the description) of the sentence “snow is white”:

the sentence constituted by three words, the first of which consists of the 19th, 14th, 15th, and 23rd letters, the second of the 9th and 19th letters, and the third of the 23rd, 8th, 9th, 20th, and 5th letters of the English alphabet.
(335)

Tarski will now generalize this procedure. Consider some sentence, and call it ‘p’. We then form the name of this sentence (in the above we did so with quotation marks), and we name it with another letter, for example, X. Now we have two sentences. We have p, which is the sentence in question. (Above the example was: snow is white). And we also form this sentence: X is true. (Above it was: “snow is white” is true). We now want to know, what is the logical relation between X is true and p? (Or as above, what is the logical relation between “Snow is white” is true and snow is white?) Tarski says that given the way we conceive of their truth, they are equivalent.

We shall now generalize the procedure which we have applied above. Let us consider an arbitrary sentence; we shall replace it by the letter 'p.' We form the name of this sentence and we replace it by another letter, say 'X.' We ask now what is the logical relation between the two sentences “X is true” and 'p.' It is clear that from the point of view of our basic conception of truth these sentences are equivalent. In other words, the following equivalence holds:

(T) X is true if, and only if, p.

We shall call any such equivalence (with 'p' replaced by any sentence of the language to which the word “true” refers, and 'X' replaced by a name of this sentence) an “equivalence of the form (T).”
(335)

[Perhaps another way of saying the above sentence is that a sentence is true if its equivalent form in a certain language is true.] The above equivalence is the material adequacy for the truth definition. [I am not sure why we use the term ‘material’. Perhaps this is because it concerns the adequacy of the content of the right side of the formulation to the content of the left side.] [So our usage of term “true” is adequate from the material perspective when we use it apply in the above manner. In other words, it seems that the material conditions for the truth of a sentence are that a) it is true in some language and that b) we can declare that truth by predicating truth to its name while c) designating its expression in some language and d) affirming its truth in that language.]

Now at last we are able to put into a precise form the conditions under which we will consider the usage and the definition of the term "true" as adequate from the material point of view: we wish to use the term "true" in such a way that all equivalences of the form (T) can be asserted, and we shall call a definition of truth "adequate" if all these equivalences follow from it.
(335)

We note that expression (T) is not itself a sentence but rather it is a schema of a sentence. [Perhaps this is because it is  not filled in yet with content, or perhaps for some reason even if it is filled in with content it should not be considered a sentence.] Tarski claims that neither (T) itself nor any instantiation of it suffices for a definition of truth.

We can only say that every equivalence of the form (T) obtained by replacing 'p' by a particular sentence, and 'X' by a name of this sentence, may be considered a partial definition of truth, which explains wherein the truth of this one individual sentence consists. The general definition has to be, in a certain sense, a logical conjunction of all these partial definitions.
(335)

Tarski closes with the following parenthetical remark:

(The last remark calls for some comments. A language may admit the construction of infinitely many sentences; and thus the number of partial definitions of truth referring to sentences of such a language will also be infinite. Hence to give our remark a precise sense we should have to explain what is meant by a "logical conjunction of infinitely many sentences"; but this would lead us too far into technical problems of modern logic.)
(336)

 

Text:

Tarski, Alfred. “The Semantic Conception of Truth and the Foundations of Semantics”. In The Nature of Truth: Classic and Contemporary Perspectives. Michael P. Lynch, ed. Cambridge, Massachusetts / London: MIT, 2001, pp.331-363.

A hyperlinked online version can be found here:

http://www.ditext.com/tarski/tarski.html

The Lynch edited book writes this in the acknowledgments:

Alfred Tarski. “The Semantic Conception of Truth and the Foundations of Semantics.” Philosophy and Phenomenological Research 4 (1944). Copyright 1992 by the Estate of Alfred Tarski. Reprinted by permission of Jan Tarski.

Further bibliographical information from
http://dingo.sbs.arizona.edu/~hharley/courses/522/522/MPPLecture4.html:

Alfred Tarski (1944) The semantic conception of truth and the foundations of semantics (Reprinted as Chapter 4 of Martinich’s anthology). This is an abridged and updated version of his 1935 long paper Der Wahrheitsbegriff in den formalisierten Sprache (The concept of truth in formalized languages), itself a translation from his book in Polish of 1933.

And yet further bibliographical information from the German wiki page for Tarski

http://de.wikipedia.org/wiki/Alfred_Tarski:

Der Wahrheitsbegriff in den formalisierten Sprachen. In: Studia Philosophica. [Lemberg] 1 (1936), S. 261–405 (Vorabdruck datiert 1935).[4] Der Artikel ist eine deutsche Übersetzung der erstmals 1933 gedruckten polnischen Arbeit, die aber schon 1931 der Öffentlichkeit präsentiert wurde. Nachdruck in Karel Berka, Lothar Kreiser (Hrsg.): Logik-Texte. Kommentierte Auswahl zur Geschichte der modernen Logik. Akademie-Verlag, Berlin 1983, S. 445–546, in englischer Sprache in Tarski: Logic, Semantics and Metamathematics - papers from 1923 to 1938 by Alfred Tarski. Oxford 1956, 1983.

The German text can be found here:

http://www.ifispan.waw.pl/studialogica/s-p-f/volumina_i-iv/I-07-Tarski-small.pdf