Towards transfinite type theory: rereading Tarski’s Wahrheitsbegriff

Synthese 191 (10):2281-2299 (2014)
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Abstract

In his famous paper Der Wahrheitsbegriff in den formalisierten Sprachen (Polish edition: Nakładem/Prace Towarzystwa Naukowego Warszawskiego, wydzial, III, 1933), Alfred Tarski constructs a materially adequate and formally correct definition of the term “true sentence” for certain kinds of formalised languages. In the case of other formalised languages, he shows that such a construction is impossible but that the term “true sentence” can nevertheless be consistently postulated. In the Postscript that Tarski added to a later version of this paper (Studia Philosophica, 1, 1935), he does not explicitly include limits for the kinds of language for which such a construction is possible. This absence of such limits has been interpreted as an implied claim that such a definition of the term “true sentence” can be constructed for every language. This has far-reaching consequences, not least for the widely held belief that Tarski changed from an universalistic to an anti-universalistic standpoint. We will claim that the consequence of anti-universalism is unwarranted, given that it can be argued that the Postscript is not in conflict with the existence of limits outside of which a definition of “true sentence” cannot be constructed. Moreover, by a discussion of transfinite type theory, we will also be able to accommodate other of the changes made in Tarski’s Postscript within a type-theoretical framework. The awareness of transfinite type theory afforded by this discussion will lead, in turn, to an account of Tarski’s Postscript that shows a gradual change in his logical work, rather than any of the more radical transitions which the Postscript has been claimed to reflect

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Iris Loeb
VU University Amsterdam

References found in this work

Saving Truth From Paradox.Hartry Field - 2008 - Oxford, England: Oxford University Press.
The Logical Syntax of Language.Rudolf Carnap - 1937 - London: K. Paul, Trench, Trubner & Co..
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What Are Logical Notions?Alfred Tarski - 1986 - History and Philosophy of Logic 7 (2):143-154.
Principia Mathematica.A. N. Whitehead - 1926 - Mind 35 (137):130.

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