Quantum Logic and Classical Logic: Their Respective Roles

Abstract

The paper analyses the claim made by Birkhoff and von Neumann, Finkelstein, Jauch, Putman and others that quantum mechanics implies the use of a non-classical ‘quantum logic’ (an orthocomplemented non-distributive lattice) on the level of factual sentences. It is shown that this claim is confused by the erroneous assumption that simple theoretical statements about the Hilbert space state vector of a system are logically equivalent to simple empirical statements about the outcome of Yes-no tests. It is shown that simple theoretical statements are equivalent to statements in a meta-context-language which assert the existence, not of events, but of constellations of invariant physical conditions within which events occur. The logic internal to the meta-context-language is a non-classical logic; the logic of quantum event language is or could be classical. The existence of a quantum logic is shown to be related to the general context-dependent character of statements and this claim is illustrated by examples taken from outside the domain of physics.

The original version of this paper was read at a meeting of the Boston Colloquim for the Philosophy of Science, 21 January 1969 under the title ‘Quantum Logic Does Not Have to Be Non-Classical’. The author wishes to thank Professor R. S. Cohen and the President of Boston University for the hospitality he enjoyed at Boston University as Visiting Associate Professor of Physics during which time he wrote this Paper. The present version has been much improved due to conversations with Professor R. S. Cohen and A. Shimony of Boston University and especially with my commentator for that occasion, Professor David Finkelstein of Yeshiva University.