Supervaluation-Style Truth Revisited
Pablo Dopico, Carlo Nicolai, Johannes Stern
TL;DR
This work reevaluates Stern's supervaluational style truth by showing that the original SSK program does not fully realize its aims but that several coherent first-order realizations do yield $\mathbb{N}$-categorical axiomatizations. It introduces and analyzes multiple fixed-point frameworks, notably $\mathrm{SSK}$, $\Theta$, and $\Theta^*$, and provides axiom systems $\mathrm{PK}$, $\mathrm{PK}^+$, $\mathrm{IT}^-$, $\mathrm{IT}^*$, and variants that achieve $\mathbb{N}$-categoricity under different conditions, including $\omega$-logic closure and disquotation principles. The paper clarifies how these realizations relate to standard supervaluation fixed points such as $\mathrm{VB}$, showing that while the least fixed points align in important respects, they diverge in others, and that no uniform single axiomatization captures all aspects. The results present a nuanced landscape in which the essential penumbral truths of classical logic and arithmetic can be captured by several well-mscoped, $\mathbb{N}$-categorical theories, each trading off semantic self-scrutiny, compositionality, and logical strength.
Abstract
Supervaluational fixed-point semantics for truth cannot be axiomatized because of its recursion-theoretic complexity. Johannes Stern (\emph{Supervaluation-Style Truth Without Supervaluations}, Journal of Philosophical Logic, 2018) proposed a new strategy (supervaluational-style truth) to capture the essential aspects of the supervaluational evaluation schema whilst limiting its recursion-theoretic complexity, hence resulting in ($\nat$-categorical) axiomatizations. Unfortunately, as we show in the paper, this strategy was not fully realized in Stern's original work: in fact, we provide counterexamples to some of Stern's key claims. However, we also vindicate Stern's project by providing different semantic incarnations of the idea and corresponding $\nat$-categorical axiomatizations. The results provide a deeper picture of the relationships between standard supervaluationism and supervaluational-style truth.