Axiomatic theories of supervaluationist truth: completing the picture
Pablo Dopico
In [1], Kripke proposed to carry out his fixed-point semantics for truth over supervaluationist schemes in the style of [2]. Three schemes stood out, corresponding to different admissibility criteria for extensions of the truth predicate that are considered in the supervaluationist satisfaction relations: (1) the scheme vb, which considers extensions consistent with the original one; (2) the scheme vc, which considers consistent extensions more generally, and; (3) the scheme mc, which considers maximally consistent extensions. In [3], Cantini presented the axiomatic theory of truth VF, which happened to be sound with respect to the fixed points of Kripke’s theory constructed over the scheme vc: that is, every classical fixed-point model for vc over the standard model of arithmetic \(\mathbb{N}\) is also a model of VF. As [4] shows, this is the best one could hope for in relation to any of the supervaluationist schemes mentioned above—there is no sound theory satisfying the reverse implication. Furthermore, Cantini proved that VF is a remarkably strong theory from a proof-theoretic point of view, matching the strength of the impredicative theory ID\(_1\).
In this paper, we complete the picture of axiomatic theories of supervaluationist truth, introducing two new theories that correspond—and are sound with respect—to the schemes vb and mc. In the case of the former scheme, we advance a theory that we call VF\(^-\), and establish its proof-theoretic strength, which equals that of VF.
The most substantial part of the paper, however, is dedicated to the theory which axiomatizes the fixed-point semantic theory over mc. We provide a proof-theoretic analysis of this theory, which we call VFM, in two stages. For the lower bound, we show that the theory can define Tarskian ramified truth predicates up to \(\varepsilon_0\) (RA\(_{<\varepsilon_0}\)). For the upper bound, we provide a cut elimination argument formalized within the theory ID\(^*_1\), which is known to be proof-theoretically equivalent to RA\(_{<\varepsilon_0}\). Hence, we conclude that VFM is proof-theoretically much weaker than the rest of theories of the supervaluationist family, and indeed on a par with the well-known theory KF.
Finally, we also introduce the schematic reflective closure of the theory VFM, as defined in [5]. We establish its consistency, and carry out the proof-theoretic analysis for this theory, which confirms that this schematic reflective closure as proof-theoretically strong as the theory RA\(_{<\Gamma_0}\).
This is joint work with Daichi Hayashi.
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