Quantum modal logic

TL;DR

This paper formalizes a quantum modal logic framework with semantics and proof calculus, laying the groundwork for various quantum modal logics like epistemic, temporal, and dynamic logic.

Contribution

It introduces the first formalization of quantum modal logic with relational semantics and a sequent calculus, establishing soundness and completeness.

Findings

Provided a relational semantics for quantum modal logic

Developed a sequent calculus for the logic

Proved soundness and completeness theorems

Abstract

A modal logic based on quantum logic is formalized in its simplest possible form. Specifically, a relational semantics and a sequent calculus are provided, and the soundness and the completeness theorems connecting both notions are demonstrated. This framework is intended to serve as a basis for formalizing various modal logics over quantum logic, such as quantum alethic logic, quantum temporal logic, quantum epistemic logic, and quantum dynamic logic.