PhilSci Archive

On the Common Logical Structure of Classical and Quantum Mechanics

Oldofredi, Andrea and Carcassi, Gabriele and Aidala, Christine A (2022) On the Common Logical Structure of Classical and Quantum Mechanics. [Preprint]

[img]
Preview
Text
quantumlogic_final.pdf

Download (551kB) | Preview

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 noncommutativity 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.


Export/Citation: EndNote | BibTeX | Dublin Core | ASCII/Text Citation (Chicago) | HTML Citation | OpenURL
Social Networking:
Share |

Item Type: Preprint
Creators:
CreatorsEmailORCID
Oldofredi, AndreaAndrea.Oldofredi@unil.ch
Carcassi, Gabrielecarcassi@umich.edu0000-0002-1071-6251
Aidala, Christine Acaidala@umich.edu0000-0001-9540-4988
Keywords: Quantum Mechanics; Quantum Logic; Classical Logic; Borel Algebra; Distributivity; Distributive Lattice
Subjects: Specific Sciences > Mathematics > Logic
Specific Sciences > Physics > Quantum Mechanics
Depositing User: Dr. Andrea Oldofredi
Date Deposited: 22 Jun 2022 16:25
Last Modified: 22 Jun 2022 16:25
Item ID: 20779
Subjects: Specific Sciences > Mathematics > Logic
Specific Sciences > Physics > Quantum Mechanics
Date: 21 June 2022
URI: https://philsci-archive-dev.library.pitt.edu/id/eprint/20779

Monthly Views for the past 3 years

Monthly Downloads for the past 3 years

Plum Analytics

Actions (login required)

View Item View Item