A logical account of quantum superpositions

TL;DR

This paper proposes a logical framework for understanding quantum superpositions, focusing on a syntactical approach at the propositional level, and introduces a new quantum logic with three negations to clarify the concept.

Contribution

It introduces a novel syntactical logical approach to quantum superpositions and develops a new quantum logic with three types of negation.

Findings

Superpositions can be modeled without contradictions.

A new quantum logic with three negations is proposed.

Misinterpretations of superpositions as contradictions are addressed.

Abstract

In this paper we consider the phenomenon of superpositions in quantum mechanics and suggest a way to deal with the idea in a logical setting from a syntactical point of view, that is, as subsumed in the language of the formalism, and not semantically. We restrict the discussion to the propositional level only. Then, after presenting the motivations and a possible world semantics, the formalism is outlined and we also consider within this schema the claim that superpositions may involve contradictions, as in the case of the Schr\"odinger's cat, which (it is usually said) is both alive and dead. We argue that this claim is a misreading of the quantum case. Finally, we sketch a new form of quantum logic that involves three kinds of negations and present the relationships among them. The paper is a first approach to the subject, introducing some main guidelines to be developed by a `syntactical' logical approach to quantum superpositions.