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Axiomatic theories of supervaluational truth: completing the picture

Pablo Dopico, Daichi Hayashi

Abstract

Supervaluational fixed-point theories of formal truth aim to amend an important shortcoming of fixed-point theories based on the Strong Kleene logic, namely, accounting for the truth of classical validities. In a celebrated paper, Andrea Cantini proposed an axiomatization of one such supervaluational theory of truth, which he called VF, and which proved to be incredibly strong proof-theoretically speaking. However, VF only axiomatizes one in a collection of several supervaluational schemes, namely the scheme which requires truth to be consistent. In this paper, we provide axiomatic theories for the remaining supervaluational schemes, labelling these systems VF$^-$ (for the theory which drops the consistency requirement), and VFM (for the theory which requires not only consistency but also completeness, i.e., maximal consistency). We then carry out proof-theoretic analyses of both theories. Our results show that VF$^-$ is as strong as VF, but that VFM's strength decreases significantly, being only as strong as the well-known theory KF. Furthermore, we introduce and analyse proof-theoretically two variants of these theories: the schematic extension, in the sense of Feferman, of VFM; and a theory in-between VFM and VF, that we call VFW, and which drops the assumption of maximal consistency. The former is shown to match the strength of predicative analysis; for the latter, we show its proof-theoretical equivalence with ramified analysis up to the ordinal $\varphi_20$, thus standing halfway between VFM and VF.

Axiomatic theories of supervaluational truth: completing the picture

Abstract

Supervaluational fixed-point theories of formal truth aim to amend an important shortcoming of fixed-point theories based on the Strong Kleene logic, namely, accounting for the truth of classical validities. In a celebrated paper, Andrea Cantini proposed an axiomatization of one such supervaluational theory of truth, which he called VF, and which proved to be incredibly strong proof-theoretically speaking. However, VF only axiomatizes one in a collection of several supervaluational schemes, namely the scheme which requires truth to be consistent. In this paper, we provide axiomatic theories for the remaining supervaluational schemes, labelling these systems VF (for the theory which drops the consistency requirement), and VFM (for the theory which requires not only consistency but also completeness, i.e., maximal consistency). We then carry out proof-theoretic analyses of both theories. Our results show that VF is as strong as VF, but that VFM's strength decreases significantly, being only as strong as the well-known theory KF. Furthermore, we introduce and analyse proof-theoretically two variants of these theories: the schematic extension, in the sense of Feferman, of VFM; and a theory in-between VFM and VF, that we call VFW, and which drops the assumption of maximal consistency. The former is shown to match the strength of predicative analysis; for the latter, we show its proof-theoretical equivalence with ramified analysis up to the ordinal , thus standing halfway between VFM and VF.

Paper Structure

This paper contains 18 sections, 74 theorems, 51 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notational preliminaries
  4. Supervaluational theories of truth
  5. The theory $\mathrm{VF}^-$
  6. The theories VFM$^-$ and VFM
  7. Presentation
  8. Proof-theoretic analysis of VFM: lower-bound
  9. Proof-theoretic analysis of VFM: upper-bound
  10. The system VFM$^{\infty}$
  11. Truth-as-provability interpretation for VFM
  12. Proof of the Soundness Theorem
  13. Formalising the consistency proof
  14. The Theories VFW$^-$ and VFW
  15. The schematic extension of VFM
  16. ...and 3 more sections

Key Result

Proposition 1

$\mathrm{VF}$ is sound with respect to $\mathrm{vc}$ fixed-point models, i.e., if $X=\mathcal{J}_\mathrm{vc}(X)$, then $(\mathbb{N}, X)\vDash \mathrm{VF}$.

Theorems & Definitions (144)

  • Definition 1: $\mathrm{VF}$
  • Proposition 1: Cantini
  • Proposition 2: Cantini
  • Definition 2: $\mathrm{VF}^-$
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Definition 3
  • Remark 1
  • ...and 134 more