Tarski (§10) of “The Semantic Conception of Truth and the Foundations of Semantics”, entitled ‘10. Conditions for a Positive Solution of the Main Problem’

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

10. Conditions for a Positive Solution of the Main Problem

 

 

Brief Summary:

We are semantically defining truth by using a meta-language to talk about an object-language. We use the term ‘true’ in that meta-language in order to define parts of the object language. But we do not want to resort to an additional higher meta-language to define semantically this term ‘true’. We want instead to define it within that meta-language itself. We will be able to, so long as the meta-language is essentially richer than the object-language (meaning that the object language cannot say everything in the meta-language). In the next section we learn how that is so.

 

Summary

Previously Tarski provided

a clear idea both of the conditions of material adequacy to which the definition of truth is subjected, and of the formal structure of the language in which this definition is to be constructed.  Under these circumstances the problem of the definition of truth acquires the character of a definite problem of a purely deductive nature.
(343, from this current section).

Solving the problem of how to define truth in detail would require “the whole machinery of contemporary logic”. So here Tarski will just give a rough outline of the solution. (343)

The solution may be positive or negative, depending on whether or not the meta-language is “essentially richer” than the object-language. (434d). Although it is not easy to define essential richness, Tarski says one way is to say that the meta-language is essentially richer if it is of a higher logical type. (343-344)

So, if the object language can say everything in the meta-language, then the meta-language is not essentially richer than the object-language. This can result in the liar paradox.

If the condition of “essential richness” is not satisfied, it can usually be shown that an interpretation of the meta-language in the object-language is possible; that is to say, with any given term of the meta-language a well determined term of the object-language can be correlated in such a way that the assertible sentences of the one language turn out to be correlated with assertible sentences of the other. As a result of this interpretation, the hypothesis that a satisfactory definition of truth has been formulated in the meta-language turns out to imply the possibility of reconstructing in that language the antinomy of the liar; and this in turn forces us to reject the hypothesis in question.

(The fact that the meta-language, in its non-logical part, is ordinarily more comprehensive than the object-language does not affect the possibility of interpreting the former in the latter. For example, the names of expressions of the object-language occur in the meta-language, though for the most part they do not occur in the object-language itself; but, nevertheless, it may be possible to interpret these names in term of the object-language.)
(344)

Without the essential richness of the meta-language, we encounter inconsistency, hence it is “necessary for the possibility of a satisfactory definition of truth in the meta-language” (344). [Semantically defining a term  of a metalanguage may require a higher order meta-language to explain that to which this term is equated. Thus even the word ‘true’ cannot have a semantic meaning for the meta-language.]

If we want to develop the theory of truth in a meta-language which does not satisfy this condition, we must give up the idea of defining truth with the exclusive help of those terms which were indicated above (in Section 8). We have then to include the term “true,” or some other semantic term, in the list of undefined terms of the meta-language, and to express fundamental properties of the notion of truth in a series of axioms. There is nothing essentially wrong in such an axiomatic procedure, and it may prove useful for various purposes.
(344)

But, so long as the meta-language is essentially richer than the object-language, it will be able to construct a satisfactory definition of truth. Tarski will show how in the next section.

It turns out, however, that this procedure can be avoided. For the condition of the "essential richness" of the meta-language proves to be, not only necessary, but also sufficient for the construction of a satisfactory definition of truth; i.e., if the meta-language satisfies this condition, the notion of truth can be defined in it. We shall now indicate in general terms how this construction can be carried through.
(344)

 

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