Constructive Quantum Logics

Juan P. Aguilera; Guillaume Massas

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arXiv · 2503.15292

Constructive Quantum Logics

Juan P. Aguilera·Guillaume Massas

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Abstract

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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BibTeX RIS

Juan P. Aguilera, Guillaume Massas. 2025-03-19. Constructive Quantum Logics. https://arxiv.org/abs/2503.15292

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