On the Common Logical Structure of Classical and Quantum Mechanics

TL;DR

This paper demonstrates that quantum mechanics shares a common logical structure with classical mechanics, challenging the idea that quantum logic fundamentally rejects classical logic principles.

Contribution

It shows that quantum theory satisfies classical distributivity when quantum propositions are fully understood, and that classical and quantum logics are more similar than previously thought.

Findings

Quantum mechanics obeys classical distributivity with proper interpretation.

Quantum propositions can form a distributive lattice including expectation values.

Classical mechanics' statistical propositions follow a similar lattice structure to quantum propositions.

Abstract

At the onset of quantum mechanics, it was argued that the new theory would entail a rejection of classical logic. The main arguments to support this claim come from the non-commutativity of quantum observables, which allegedly would generate a non-distributive lattice of propositions, and from quantum superpositions, which would entail new rules for quantum disjunctions. While the quantum logic program is not as popular as it once was, a crucial question remains unsettled: what is the relationship between the logical structures of classical and quantum mechanics? In this essay we answer this question by showing that the original arguments promoting quantum logic contain serious flaws, and that quantum theory does satisfy the classical distributivity law once the full meaning of quantum propositions is properly taken into account. Moreover, we show that quantum mechanics can generate a distributive lattice of propositions, which, unlike the one of quantum logic, includes statements about expectation values which are of undoubtable physical interest. Lastly, we show that the lattice of statistical propositions in classical mechanics follows the same structure, yielding an analogue non-commutative sublattice of classical propositions. This fact entails that the purported difference between classical and quantum logic stems from a misconstructed parallel between the two theories.