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Non-Boolean derived logics for classical systems.

Birkhoff and von Neumann [Ann. Math. 37, 823 (1936)] proposed a nonstandard logic to describe quantum mechanics, in which the distributive laws of Boolean logic do not hold. In this paper we develop two algebras of propositions for classical mechanics that, like the quantum logical algebra, are based on a measurement theory. We adopt a simple classical measurement theory that allows the determination of any continuous phase-space function to any finite precision. Surprisingly, the resulting ``classical logics'' are non-Boolean, though the distributive laws hold. It appears that any physical theory with a mathematical space of physical states and an adequate description of measurement naturally yields a logiclike structure of experimental propositions, and that this ``derived logic'' can be non-Boolean even for theories much less radical than quantum theory.