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Substitution and Truth in Quantum Logic

Published online by Cambridge University Press:  01 April 2022

Itamar Pitowsky
Itamar Pitowsky*
Affiliation:
Department of Philosophy, University of Western Ontario
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Abstract

If p(x1, …, xn) and q(x1 …, xn) are two logically equivalent propositions then p(π(x1), …, π(xn)) and q(π(x1), …,π(xn)) are also logically equivalent where π is an arbitrary permutation of the elementary constituents x1, …, xn. In Quantum Logic the invariance of logical equivalences breaks down. It is proved that the distribution rules of classical logic are in fact equivalent to the meta-linguistic rule of universal substitution and that the more restrictive structure of the substitution group of Quantum Logic prevents us from defining truth in a classical fashion. These observations lead to a more profound understanding of the Logic of Quantum Mechanics and of the role that symmetry principles play in that theory.

… Its decisive difference in comparison to the Classical-Model is the fact that gratings in vector-space defy superposition.

—Hermann Weyl (1949)

Type
Research Article
Information
Philosophy of Science , Volume 49 , Issue 3 , September 1982 , pp. 380 - 401
Copyright
Copyright © Philosophy of Science Association 1982

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Footnotes

†

I wish to thank Professor J. Bub and Professor W. Demopoulos for their valuable suggestions and remarks.

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