Quantum Logic in Intuitionistic Perspective

TL;DR

This paper develops a framework combining quantum logic and intuitionistic logic by introducing disjunctions into property lattices, resulting in a Heyting algebra that better captures the logical structure of physical properties.

Contribution

It introduces a complete Heyting algebra of propositions for physical properties, refining the mathematical structure of quantum logic with an intuitionistic perspective.

Findings

Establishes a bijective correspondence between property lattices and propositional lattices with operational resolution.

Refines the correspondence to include orthocomplementation with operational complementation.

Derives an intuitionistic functional implication on property lattices.

Abstract

In their seminal paper Birkhoff and von Neumann revealed the following dilemma: "... whereas for logicians the orthocomplementation properties of negation were the ones least able to withstand a critical analysis, the study of mechanics points to the distributive identities as the weakest link in the algebra of logic." In this paper we eliminate this dilemma, providing a way for maintaining both. Via the introduction of the "missing" disjunctions in the lattice of properties of a physical system while inheriting the meet as a conjunction we obtain a complete Heyting algebra of propositions on physical properties. In particular there is a bijective correspondence between property lattices and propositional lattices equipped with a so called operational resolution, an operation that exposes the properties on the level of the propositions. If the property lattice goes equipped with an orthocomplementation, then this bijective correspondence can be refined to one with propositional lattices equipped with an operational complementation, as such establishing the claim made above. Formally one rediscovers via physical and logical considerations as such respectively a specification and a refinement of the purely mathematical result by Bruns and Lakser (1970) on injective hulls of meet-semilattices. From our representation we can derive a truly intuitionistic functional implication on property lattices, as such confronting claims made in previous writings on the matter. We also make a detailed analysis of disjunctivity vs. distributivity and finitary vs. infinitary conjunctivity, we briefly review the Bruns-Lakser construction and indicate some questions which are left open.