Tarski’s Theorem and the Extensionality of Truth
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In his informal presentation, “The semantic conception of truth and the foundations of semantics”, Alfred Tarski (1944, 348) defines a language to be “semantically closed” if it 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” and if “all sentences which determine the adequate usage of this term can be asserted in the language”.
Tarski points out that every semantically closed language in which “the ordinary laws of logic hold” is inconsistent. Presumably, when he says that, in semantically closed languages, the sentences that determine the proper use of the semantic vocabulary (e.g., the T-sentences) “can be asserted”, he means that the rules of the language use warrant such assertion. No paradox arises from the fact that certain sentences are capable of being uttered in a certain mood.
Tarski concludes that to get a consistent truth predicate, we must either give...
Thanks to Kevin Scharp for encouraging me to develop the main point in this note, and to Carrie Jenkins for valuable discussion of the issues. I appreciate the spirit of collegiality. Thanks also to two anonymous referees for this journal, who made helpful suggestions for clarification.
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