Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2410.12471

Axiomatic theories of supervaluational truth: completing the picture

Pablo Dopico·Daichi Hayashi
View original source ↗Read source document ↗Export metadata ↓

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 $φ_20$, thus standing halfway between VFM and VF.

Explore related subjects

math.LO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Pablo Dopico, Daichi Hayashi. 2024-10-16. Axiomatic theories of supervaluational truth: completing the picture. https://arxiv.org/abs/2410.12471

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Bluebirds and mockingbirds cannot produce a fixed-point combinator

Let $B$ be the bluebird combinator with reduction rule $Bxyz \to_{w} x\left(yz\right)$, let $M$ be the mockingbird combinator with reduction rule $Mx \to_{w} xx$, and let $I$ be the identity bird combinator with reduction rule $Ix \to_{w} x$. A fixed-point combinator, called a sage bird by Smullyan, is a closed term $Y$ such that, for a fresh variable $x$, $Yx$ is equivalent to $x\left(Yx\right)$ under these reduction rules. For a fixed variable $x$, we construct an invariant $\mathrm{Tr}_{x}\left(u\right)$ of a $BMI$-term $u$ with respect to $\to_{w}$. This invariant traces the occurrences of $x$ in the leftmost-innermost reduction sequence of $u$. We then prove that $\mathrm{Tr}_{x}\left(Yx\right) \neq \mathrm{Tr}_{x}\left(x^{r}\left( Yx \right)\right)$ for every $x$-free $BMI$-term $Y$ and every $r\geq 1$. Consequently, there exists no fixed-point combinator in $BMI$-combinatory logic. This provides a negative answer to the problem posed by Smullyan in 1985.

math.LO↗
arXiv2026

Pointwise provable equality and the failure of composition

Montagna (1989) and Di Paola--Montagna (1991) claim that the algebraic systems $S'$ and $S'_T$, respectively, are categories. We show that the proposed composition is not independent of the choice of representatives. For every consistent recursively enumerable extension $T$ of Peano arithmetic ($\mathrm{PA}$), we exhibit two program indices that are pointwise provably equal in $T$ but yield inequivalent composites when each is run after the same program. Montagna's $S'$ is the case $T=\mathrm{PA}$. The failure already occurs for partial maps from $ω$ to itself. Weak totality and the proposed range assignment also depend on the choice of representatives. More generally, for consistent $T\supseteq\mathrm{PA}$, pointwise provable equality is a composition congruence exactly when $T$ proves every true $Π^0_1$ sentence, in which case it is extensional equality. This completeness condition fails for every consistent recursively enumerable $T\supseteq\mathrm{PA}$ by Gödel's second incompleteness theorem. For every extension $T\supseteq\mathrm{PA}$, the least composition congruence containing pointwise provable equality is extensional equality if $T$ is $Σ^0_1$-sound and the universal relation otherwise.

math.LO↗
arXiv2026

Compactness via Consistency Properties

We will use consistency properties to characterize strongly compact cardinals, first showing an adequate Model Existence Theorem for larger fragments.

math.LO↗