The Objective Indefiniteness Interpretation of Quantum Mechanics

TL;DR

This paper proposes that quantum mechanics can be understood through a dual logical framework called partition logic, which models objective indefiniteness in micro-reality, offering an alternative to classical Boolean logic.

Contribution

It introduces partition logic as a dual to Boolean logic, develops its mathematical foundation, and applies it to interpret quantum mechanics via the objective indefiniteness perspective.

Findings

Partition logic is dual to Boolean subset logic.

Lifting set-based partition logic to vector spaces reproduces quantum mechanics.

Partition logic provides a coherent interpretative framework for quantum indeterminacy.

Abstract

The common-sense view of reality is expressed logically in Boolean subset logic (each element is either definitely in or not in a subset, i.e., either definitely has or does not have a property). But quantum mechanics does not agree with this "properties all the way down" picture of micro-reality. Are there other coherent alternative views of reality? A logic of partitions, dual to the Boolean logic of subsets (partitions are dual to subsets), was recently developed along with a logical version of information theory. In view of the subset-partition duality, partition logic is the alternative to Boolean subset logic and thus it abstractly describes the alternative dual view of micro-reality. Perhaps QM is compatible with this dual view? Indeed, when the mathematics of partitions using sets is "lifted" from sets to vector spaces, then it yields the mathematics and relations of quantum mechanics. Thus the vision of micro-reality abstractly characterized by partition logic matches that described by quantum mechanics. The key concept explicated by partition logic is the old idea of "objective indefiniteness" (emphasized by Shimony). Thus partition logic, logical information theory, and the lifting program provide the back story so that the old idea then yields the objective indefiniteness interpretation of quantum mechanics.