# 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.

## Bibliography

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