A SUBSTRUCTURAL GENTZEN CALCULUS FOR ORTHOMODULAR QUANTUM LOGIC
Davide Fazio, Antonio Ledda, Francesco Paoli, Gavin St. John · The Review of Symbolic Logic · 2022
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.