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Tarski demarcates the theoretical territory at stake in a 1944 paper:

Tarski, A., 1944, “The semantic conception of truth”, Philosophy and Phenomenological Research, 4 (3): 341–376. https://sites.ualberta.ca/~francisp/Phil426/TarskiTruth1944.pdf

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S. THE INCONSISTENCY OF SEMANTICALLY CLOSED LANGUAGES.7 If we now analyze the assumptions which lead to the antinomy of the liar, we notice the following:

(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.

It turns out that the assumption (III) is not essential, for it is possible to reconstruct the antinomy of the liar without its help.” But the assumptions (I) and (II) prove essential. Since every language which satisfies both of these assumptions is inconsistent, we must reject at least one of them.

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. 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.

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Following Tarsky, the solution to the Liar paradox must jettison either a recursively enumerable truth constant or Boolean algebra. The Liar’s revenge statement associated with incompleteness (of Boolean algebra) demonstrates that incompleteness isn’t a solution to a paradox.

The result is that logic continually destroys and recreates itself because both that logic disproves itself and that the disproof of logic also disproves itself.

September 24, 2021