Supervaluation-Style Truth Without Supervaluations
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Kripke’s theory of truth is arguably the most influential approach to self-referential truth and the semantic paradoxes. The use of a partial evaluation scheme is crucial to the theory and the most prominent schemes that are adopted are the strong Kleene and the supervaluation scheme. The strong Kleene scheme is attractive because it ensures the compositionality of the notion of truth. But under the strong Kleene scheme classical tautologies do not, in general, turn out to be true and, as a consequence, classical reasoning is no longer admissible once the notion of truth is involved. The supervaluation scheme adheres to classical reasoning but violates compositionality. Moreover, it turns Kripke’s theory into a rather complicated affair: to check whether a sentence is true we have to look at all admissible precisification of the interpretation of the truth predicate we are presented with. One consequence of this complicated evaluation condition is that under the supervaluation scheme a more proof-theoretic characterization of Kripke’s theory becomes inherently difficult, if not impossible. In this paper we explore the middle ground between the strong Kleene and the supervaluation scheme and provide an evaluation scheme that adheres to classical reasoning but retains many of the attractive features of the strong Kleene scheme. We supplement our semantic investigation with a novel axiomatic theory of truth that matches the semantic theory we have put forth.
KeywordsTruth Supervaluation Grounded truth Kripke’s theory of truth Axiomatic theories of truth Strong Kleene
This work was supported by the DFG-funded research project Syntactical Treatments of Interacting Modalities, the German BA, and by the European Commission through a Marie Sklodowska-Curie Individual Fellowship (TREPISTEME, Grant No. 703529). I wish to thank Catrin Campbell-Moore, Martin Fischer, Carlo Nicolai, Philip Welch and two anonymous referees for helpful comments on and discussion of my paper. Early versions of this paper were presented at Bristol, Bath, and Munich. I thank the audiences for their feedback.
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