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
7. The Antinomy of the Liar
Brief Summary:
Tarksi has provided the (T) scheme for designating the truth of sentences: (T) X is true if, and only if, p. In this formulation, p is the sentence in question, and X is the name for it, often times being that same sentence with quotations around it. We encounter the liar paradox, however, when X refers to its own self and is predicated as not true. For example: This sentence is not true. The formulation would then read “This sentence is not true” is true, if and only if, this sentence is not true, or, after symbolic substitution, ‘s’ is true if and only if ‘s’ is not true. [The substitution is clearer when considering the more precise formulation that refers to where the sentence is on the page. See below.] Tarski thinks it is important to deal with this paradox, as its solution could play a central role in the foundation of theoretical semantics.
Summary
Previously Tarsky noted that in order to give a definition of truth, we cannot use any natural language, as they structurally speaking will cause too many difficulties [on account of ambiguities for example]. Instead we need to make a formalized language that approximates a natural language as close as possible. Part of this project of creating a suitable formalized language is determining the parts and generative operations of that language. We need primitive or undefined terms along with rules for defining new terms. We also need axioms (primitive sentences) as well as rules for inferring new sentences from them. Now in this section, Tarski says that
In order to discover some of the more specific conditions which must be satisfied by languages in which (or for which) the definition of truth is to be given, it will be advisable to begin with a discussion of that antinomy which directly involves the notion of truth, namely, the antinomy of the liar.
(339)
Tarski then creates a liar paradox using the (T) formulation: (T) X is true if, and only if, p. In this case, he will make p be an actual sentence on the page. In this publication, the page this text is on is 339, and the stated sentence is on line 11. So Tarksi formulates it like this
To obtain this antinomy in a perspicuous form, consider the following sentence:
The sentence printed in this paper on p. 339, l. 11, is not true.
(339)
Tarski abbreviates this above sentence to the letter ‘s’. Then he places this sentence and its name into his (T) scheme.
According to our convention concerning the adequate usage of the term “true,” we assert the following equivalence of the form (T):
(1) ‘s’ is true if, and only if, the sentence printed in this paper on p. 339, l. 11, is not true.
(339)
Yet, as we know, a name is identified with what it is a name of. So
On the other hand, keeping in mind the meaning of the symbol 's,' we establish empirically the following fact:
(2) 's' is identical with the sentence printed in this paper on p. 339, l. 11.
(339)
So since they are identical, we can substitute one for the other, which will produce a contradiction.
Now, by a familiar law from the theory of identity (Leibniz's law), it follows from (2) that we may replace in (1) the expression “the sentence printed in this paper on p. 339, l. 11” by the symbol “‘s.’” We thus obtain what follows:
(3) 's' is true if, and only if, 's' is not true.
In this way we have arrived at an obvious contradiction.
(339)
Tarski thinks that this is more than a joke. Since our structure produces it, we must take it seriously and deal with it. Specifically
We must 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)
In fact, this antinomy of the liar can play a central role in semantics just as other antinomies have played central roles in other areas of philosophy.
It should be emphasized that antinomies have played a preeminent role in establishing the foundations of modern deductive sciences. And just as class-theoretical antinomies, and in particular Russell's antinomy (of the class of all classes that are not members of themselves), were the starting point for the successful attempts at a consistent formalization of logic and mathematics, so the antinomy of the liar and other semantic antinomies give rise to the construction of theoretical semantics.
(340)
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
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