An anhomomorphic logic for quantum mechanics

TL;DR

This paper explores an anhomomorphic logic framework for quantum mechanics, focusing on quadratic coevents, their properties, and the structure of associated projections, introducing the concept of precluding coevents and their duality with preclusive coevents.

Contribution

It develops the theory of quadratic coevents in an anhomomorphic logic, introduces precluding coevents, and establishes their duality with preclusive coevents in quantum logic.

Findings

Projections form an orthomodular poset

Precluding coevents are stronger than preclusive coevents

Duality between precluding and preclusive coevents

Abstract

Although various schemes for anhomomorphic logics for quantum mechanics have been considered in the past we shall mainly concentrate on the quadratic or grade-2 scheme. In this scheme, the grade-2 truth functions are called coevents. We discuss properties of coevents, projections on the space of coevents and the master observable. We show that the set of projections forms an orthomodular poset. We introduce the concept of precluding coevents and show that this is stronger than the previously studied concept of preclusive coevents. Precluding coevents are defined naturally in terms of the master observable. A result that exhibits a duality between preclusive and precluding coevents is given. Some simple examples are presented.