Logic Journal of IGPL

Strict implication in quantum modal logic

Kenji Tokuo

Published Sep 4, 2026• 0 citations

Abstract

Abstract Quantum logic has no canonical implication connective, and polynomial candidates differ in the inferential principles they satisfy. We study a relational strict implication in quantum modal logic and interpret it through the modal accessibility relation. On reflexive modal frames, a formula formed with this connective is valid exactly when its antecedent semantically entails its consequent. The connective therefore internalizes semantic consequence. The forcing condition links modal accessibility to non-orthogonality. Consequently, any formula whose principal connective is this strict implication has the same truth value throughout each component determined by non-orthogonality. The truth set of each such formula therefore belongs to the Boolean algebra of unions of these components. We also establish further semantic laws and analyze quantum disjunction, contraposition, and nested implications.

DOI: 10.1093/jigpal/jzag052

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