A substructural Gentzen calculus for orthomodular quantum logic

Davide Fazio;Antonio Ledda;Francesco Paoli

;Gavin St. John

2023-01-01

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Abstract

We introduce a sequent system which is Gentzen algebraisable with orthomodular lattices as equivalent algebraic semantics, and therefore can be viewed as a calculus for orthomodular quantum logic. Its sequents are pairs of non-associative structures, formed via a structural connective whose algebraic interpretation is the Sasaki product on the left-hand side and its De Morgan dual on the right-hand side. It is a substructural calculus, because some of the standard structural sequent rules are restricted - by lifting all such restrictions, one recovers a calculus for classical logic.