Tarski (§8) of “The Semantic Conception of Truth and the Foundations of Semantics”, entitled ‘8. The Inconsistency of Semantically Closed Languages’

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

8. The Inconsistency of Semantically Closed Languages

 

 

Brief Summary:

The liar’s paradox arises because we considered the language in which it arose as being semantically closed, meaning that within it are expressible all its components and generative rules. Thus the languages we examine using Tarski’s method cannot be considered semantically closed, or else they will be inconsistent.

Summary

Previously Tarski explained how his (T) scheme for giving a semantic definition of truth – (T) X is true if, and only if, p, – brings about the liar’s paradox when the sentence in question refers to itself and is predicated as being not true:

‘s’ is true if and only if ‘s’ is not true.

He said that we need to address this problem, because it can help us uncover foundational principles in theoretical semantics, and to do so, we need to

discover its cause, that is, | to say, we must analyze premises upon which the antinomy is based; we must then reject at least one of these premises, and we must investigate the consequences which this has for the whole domain of our research.
(339-340, from the prior section)

Now Tarski will examine “the assumptions which lead to the antinomy of the liar.” (340, current section) In do so, we notice three things. Firstly, we assumed that the paradox occurs in a certain language. We assumed that everything in the formulation

‘s’ is true if and only if ‘s’ is not true.

are also in the same language in which the paradox arises. We also assumed that all the sentences that determine how ‘true’ can be used are as well found in this language. A system with these two features is called “semantically closed” [perhaps because all the semantic properties are described within in and do not need a metalanguage for that description.] Secondly, we assumed that this language operates according to the normal laws of logic. And thirdly we assumed this language is capable of saying such sentences as:

‘s’ is true if and only if ‘s’ is not true.

[Note, my interpretation of the third assumption below needs revision, as I seem to misunderstand the terminology, particularly the term ‘empirical premise’ and perhaps I do not know the usage here of ‘argument’]

I. We have implicitly assumed that the language in which the antinomy is constructed contains, in addition to its expressions, also the names of these expressions, as well as semantic terms such as the term “true” referring to sentences of this language; we have also assumed that all sentences which determine the adequate usage of this term can be asserted in the language. A language with these properties will be called “semantically closed.”

II. We have assumed that in this language the ordinary laws of logic hold.

III. We have assumed that we can formulate and assert in our language an empirical premise such as the statement (2) which has occurred in our argument.
(340)

Tarski says that the third assumption is not essential, because we can create the liar paradox by other means within the language. He writes in footnote 11:

This can roughly be done in the following way. Let S be any sentence beginning with the words “Every sentence.” We correlate with S a new sentence S* by subjecting S to the following two modifications: we replace in S the first word, “Every,” by “The”; and we insert after the second word, “sentence,” the whole sentence S enclosed in quotation marks. Let us agree to call the sentence S “(self-)applicable” or “non-(self-)applicable” dependent on whether the correlated sentence S* is true or false. Now consider the following sentence:

Every sentence is non-applicable.

It can easily be shown that the sentence just stated must be both applicable and non-applicable; hence a contradiction. It may not be quite clear in what sense this formulation of the antinomy does not involve an empirical premiss; however, I shall not elaborate on this point.
(358, endnote 11)

But although the third assumption is not essential, the first two are. [The second assumption is that the language follows normal laws of logic. The first is that the language is semantically closed, meaning that all its semantic and logical properties can be described within it.] Because the first two assumptions together would make a language inconsistent, we must reject at least one of them.

Consider if we said that the language does not follow the normal laws of logic. This would have gravely detrimental consequences [for example, we would be unable to formulate it consistently, perhaps. A paraconsistent logic might not have such problems, but Tarski here is not specific about them, since they are apparently too obvious.]

It would be superfluous to stress here the consequences of rejecting the assumption (II), that is, of changing our logic (supposing this were possible) even in its more elementary and fundamental parts.
(340)

Thus we must reject assumption 1 and say that we are not using a semantically closed system.

We thus consider only the possibility of rejecting the assumption (I). Accordingly, we decide not to use any language which is semantically closed in the sense given.
(340)

Some people might object saying that we cannot work with semantically closed languages. Such people might think that either there is one genuine language [and all natural languages are variations of it] or that all languages are mutually translatable [and thus there is no language that can be conceived of as standing above and beyond the others.] Tarski’s reply is that this objection is not a problem in science at least, which does not need semantically closed language. (341)

Another objection would be that our everyday language is closed [perhaps because it can say anything we want it too, and we do not need a separate language to describe the first one. English can be fully defined and described in English dictionaries and grammars. We do not need French for example to explain all the features of English.] But Tarski notes that everyday language does not have a specified structure. The problem of inconsistency only arises for languages with specified structures. We do our best to approximate everyday language in a formalized one, and we can only surmise that the inconsistency of the approximation is indicative of inconsistency in the original. (341)

 

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