arXiv subjects

Guillaume Massas

Publications and source records attributed to Guillaume Massas.

6 recordsLinked to original sources

arXiv2026

Fundamental Logic Through the Lens of Modality

Fundamental logic is a non-classical logic based only on the introduction and elimination rules for conjunction, disjunction, negation, and the quantifiers in a Fitch-style natural deduction system. In this paper, we attempt to obtain a better understanding of fundamental logic and its semantics through the lens of modality. Using modal logic, we develop means of mutual understanding between the fundamental logician, on the one hand, and the orthologician and intuitionistic logician, on the other: we prove that the Gödel-McKinsey-Tarski (GMT) translation of intuitionistic logic into the classical modal logic $\mathsf{S4}$ is a full and faithful embedding of fundamental logic into the orthological version of $\mathsf{S4}$; that the Goldblatt translation of orthologic into the classical modal logic $\mathsf{KTB}$ is a full and faithful embedding of fundamental logic into an intuitionistic version of $\mathsf{KTB}$; and that the GMT translation is a full and faithful embedding of intuitionistic logic into a modal extension of fundamental logic.

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arXiv2026

Conditionals and Modalities in Constructive Quantum Logics

We investigate logics that generalize both intuitionistic logic and quantum logic. In earlier work, we introduced Ex-logic, an extension of Holliday's fundamental logic that coincides with the intersection of orthologic and the implication-free fragment of intuitionistic logic. In this paper, we add an implication connective to Ex-logic and axiomatize iEx-logic, the intersection of full intuitionistic logic and orthomodular logic with the implication connective interpreted as the Sasaki hook. As a consequence, we obtain a characterization of the lattice of logics extending iEx-logic as the product of the lattice of intermediate logics and the lattice of orthomodular logics. We also explore the robustness of our algebraic approach by briefly discussing extensions of iEx-logic with modal operators.

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arXiv2025

Constructive Quantum Logics

Following a suggestion of Birkhoff and Von Neumann [Ann. Math. 37 (1936), 23-32], we pursue a joint study of quantum logic and intuitionistic logic. We exhibit a linear-time translation which for each quantum logic $Q$ and each superintuitionistic logic $I$ yields an axiomatization of $Q\cap I$ from axiomatizations of $Q$ and $I$. The translation is centered around a certain axiom (Ex) which (together with introduction and elimination rules for connectives) is shown to axiomatize the intersection of orthologic and intuitionistic logic, solving a problem of Holliday [Logics 1 (2023), pp. 36-79]. We prove that the lattice of all super-Ex logics is isomorphic to the product of the lattices of quantum logics and superintuitionistic logics in the signature $\{\land,\lor,\neg\}$. We prove that there are infinitely many sub-Ex logics extending Holliday's fundamental logic.

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arXiv2024

Goldblat-Thomason Theorems for Fundamental (Modal) Logic

Holliday recently introduced a non-classical logic called Fundamental Logic, which intends to capture exactly those properties of the connectives "and", "or" and "not" that hold in virtue of their introduction and elimination rules in Fitch's natural deduction system for propositional logic. Holliday provides an intuitive relational semantics for fundamental logic which generalizes both Goldblatt's semantics for orthologic and Kripke semantics for intuitionistic logic. In this paper, we further the analysis of this semantics by providing a Goldblatt-Thomason theorem for Fundamental Logic. We identify necessary and sufficient conditions on a class K of fundamental frames for it to be axiomatic, i.e., to be the class of frames satisfying some logic extending Fundamental Logic. As a straightforward application of our main result, we also obtain a Goldblatt-Thomason theorem for Fundamental Modal Logic, which extends Fundamental Logic with standard Box and Diamond operators.

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arXiv2022

Choice-Free de Vries Duality

De Vries Duality generalizes Stone duality between Boolean algebras and Stone spaces to a duality between de Vries algebras (complete Boolean algebras equipped with a subordination relation satisfying some axioms) and compact Hausdorff spaces. This duality allows for an algebraic approach to region-based theories of space that differs from point-free topology. Building on the recent choice-free version of Stone duality developed by Bezhanishvili and Holliday, this paper establishes a choice-free duality between de Vries algebras and a category of de Vries spaces. We also investigate connections with the Vietoris functor on the category of compact Hausdorff spaces and with the category of compact regular frames in point-free topology, and we provide an alternative, choice-free topological semantics for the Symmetric Strict Implication Calculus of Bezhanishvili et al.

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arXiv2022

A Semi-Constructive Approach to the Hyperreal Line

Using a recent alternative to Tarskian semantics for first-order logic, known as $\textit{possibility semantics}$, I introduce an alternative approach to nonstandard analysis that remains within the bounds of \textit{semi-constructive} mathematics, i.e., does not assume any fragment of the Axiom of Choice beyond the Axiom of Dependent Choices. I define the $F\it{-hyperreal}$ $\it{line}$ ${^\dagger\!\mathcal{R}}$ as a possibility structure and show that it shares many fundamental properties of the classical hyperreal line, such as a Transfer Principle and a Saturation Principle. I discuss the technical advantages of ${^\dagger\!\mathcal{R}}$ over some other alternative approaches to nonstandard analysis and argue that it is well-suited to address some of the philosophical and methodological concerns that have been raised against the application of nonstandard methods to ordinary mathematics.

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