Substitution and Truth in Quantum Logic

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)