Tarski (§9) of “The Semantic Conception of Truth and the Foundations of Semantics”, entitled ‘9. Object-Language and Meta-Language’

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

9. Object-Language and Meta-Language

 

 

Brief Summary:

We are semantically defining truth using the (T) scheme:

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

p is the sentence we are defining, while X is the name for that sentence, often given in quotation marks. For example,

“snow is white” if and only if snow is white.

The text in the quotation marks is in the language that the statement is talking about, and that language is called the object-language. The parts of that expression which are not in quotes and which are talking about the object-language are parts of what is called the meta-language.

Summary

Previously Tarski concluded that in order for our semantic theory to avoid the liar’s paradox, we cannot “employ semantically closed languages.” Instead, “we have to use two different languages in discussing the problem of the definition of truth and, more generally, any problems in the field of semantics” (341, both quotes from this section). The first language, called the object language, is the one we are talking about, while the second one, called the metalanguage, is doing that talking.

The first of these languages is the language which is “talked about” and which is the subject matter of the whole discussion; the definition of truth which we are seeking applies to the sentences of this language. The second is the language in which we “talk about” the first language, and in terms of which we wish, in particular, to construct the definition of | truth for the first language. We shall refer to the first language as “the object-language,” and to the second as “the meta-language.”
(341-342)

The terms ‘object-language’ and ‘meta-language’ have a relative sense, since if we in turn talk about some meta-language with another meta-language, then the first becomes the object-language.

It should be noticed that these terms “object-language” and “meta-language” have only a relative sense. If, for instance, we become interested in the notion of truth applying to sentences, not of our original object-language, but of its meta-language, the latter becomes automatically the object-language of our discussion; and in order to define truth for this language, we have to go to a new meta-language so to speak, to a meta-language of a higher level. In this way we arrive at a whole hierarchy of languages.
(342)

Tarski then explains how we obtain the vocabulary of the metalanguage. [It seems he is saying that the metalanguage is mostly what is contained in the (T) formulation. So it also includes everything as well in the object language, which will be stated in the (T) formulations. It is not explained yet what to do with the fact that ‘is true if and only if’ could easily be found in an object language like English or formal logic.]

The vocabulary of the meta-language is to a large extent determined by previously stated conditions under which a definition of truth will be considered materially adequate This definition, as we recall, has to imply all equivalences of the form (T):

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

The definition itself and all the equivalences implied by it are to be formulated in the meta-language. On the other hand, the symbol 'p' in (T) stands for an arbitrary sentence of our object-language. Hence it follows that every sentence which occurs in the object- language must also occur in the meta-language; in other words, the meta-language must contain the object-language as a part. This is at any rate necessary for the proof of the adequacy of the definition – even though the definition itself can sometimes be formulated in a less comprehensive meta-language which does not satisfy this requirement.

(The requirement in question can be somewhat modified, for it suffices to assume that the object-language can be translated into the meta-language; this necessitates a certain change in the interpretation of the symbol 'p' in (T). In all that follows we shall ignore the possibility of this modification.)
(342)

Because the (T) scheme needs names for sentences in the object-language, the meta-language should be rich enough to create names for all such sentences. As well, the meta-language needs basic logical terms like “if, and only if.”

Furthermore, the symbol 'X' in (T) represents the name of the sentence which 'p' stands for. We see therefore that the meta-language must be rich enough to provide possibilities of constructing a name for every sentence of the object-language.

In addition, the meta-language must obviously contain terms of a general logical character, such as the expression “if, and only if.”
(342)

The meta-language should not have undefined terms except ones in the object language, ones that refer to the object language expressions’ form, ones used to create names for those expressions, and logical ones.

It is desirable for the meta-language not to contain any undefined terms except such as are involved explicitly or implicitly in the remarks above, i.e.: terms of the object-language; terms referring to the form of the expressions of the object-language, and used in building names for these expressions; and terms of logic. In particular, we desire semantic terms (referring to the object-language) to be introduced into the meta-language only by definition. For, if this postulate is satisfied, the definition of truth, or of any other semantic concept, will fulfill what we intuitively expect from every definition; that is, it will explain the meaning of the term being defined in terms whose meaning appears to be completely clear and unequivocal. And, moreover, we have then a kind of guarantee that the use of semantic concepts will not involve us in any contradictions.
(343)

We do not need further requirements for the object and meta-languages. They should be like other known formalized languages.

We have no further requirements as to the formal structure of the object-language and the meta-language; we assume that it is similar to that of other formalized languages known at the present time. In particular, we assume that the usual formal rules of definition are observed in the meta-language.
(343)

 

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