arXiv subjects

Vicent Navarro Arroyo

Publications and source records attributed to Vicent Navarro Arroyo.

3 recordsLinked to original sources

arXiv2026

Ultrafilter Extensions for Veltman Semantics

In this paper, we present a first-order frame condition for interpretability logic and show that the condition is not modally definable. Yet, the frame condition holds both on ILM and on ILP frames and, hence, is of potential importance for the long-standing open problem about the interpretability logic of all reasonable arithmetical theories. In the light of the Goldblatt-Thomason Theorem, the modally inexpressible frame condition serves as motivation to develop ultrafilter extensions for interpretability logic. We develop the necessary algebraic tools to define these ultrafilter extensions and prove the main properties about both the tools and the ultrafilter extensions.

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arXiv2026

Gödel-Dummett and $\mathsf{BD_2}$: Linearity and Depth-Two Branching in Kripke Semantics

We study the semantic relationship between Gödel-Dummett logic $\mathsf{GL}$ and bounded-depth-2 logic $\mathsf{BD_2}$, two well-known intermediate logics. While $\mathsf{GL}$ imposes linearity on Kripke frames, $\mathsf{BD_2}$ bounds their depth to two. We prove these logics are incomparable (neither contains the other) through minimal frame conditions. Notably, their combination $\mathsf{GL+BD_2}$ collapses to the logic of one or two world frames, bringing it remarkably close to classical logic. This illustrates how controlling breadth and depth in intuitionistic semantics leads to mutually exclusive structural constraints. Finally, we give a conceptual and philosophical interpretation of the previous results. This is an extended abstract of work in progress. Comments and suggestions welcome at: vicent.navarro@ub.edu

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arXiv2025

On the Contingency of Logic in Possible World Semantics

This paper investigates the contingency of logic within the framework of possible world semantics. Possible world semantics captures the meaning of necessitation, i.e., a statement is necessarily true if it holds in all possible worlds. Standard Kripkean semantics assumes that all possible worlds are governed by one single logic. We relax this assumption and introduce mixed models, in which different worlds may obey different logical systems. The paper provides a first case study where we mix classical propositional logic ($\mathsf{CPC}$) and intuitionistic propositional logic ($\mathsf{IPC}$) in the possible world semantics. We define the class of mixed models $\mathcal{M}\mathcal{M}(\mathsf{CPC}, \mathsf{IPC})$, together with a subclass of concrete mixed models ($\mathcal{CMM}$), and establish their semantic properties. Our main result shows that the set of formulas valid in $\mathcal{M}\mathcal{M}(\mathsf{CPC}, \mathsf{IPC})$ corresponds exactly to the intuitionistic modal logic $\mathsf{iK}$ extended with the Box Excluded Middle axiom ($\mathsf{iK} + \mathsf{bem}$). To demonstrate this, we prove soundness and completeness results linking $\mathcal{M}\mathcal{M}(\mathsf{CPC}, \mathsf{IPC})$ and $\mathcal{CMM}$, and birelational models for $\mathsf{iK} + \mathsf{bem}$.

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