NON-BOOLEAN DERIVED LOGICS FOR CLASSICAL-SYSTEMS

Birkhoff and von Neumann [Ann. Math. 37, 823 (1936)] proposed a nonsta ndard logic to describe quantum mechanics, in which the distributive l aws of Boolean logic do not hold. In this paper we develop two algebra s of propositions for classical mechanics that, like the quantum logic al algebra, are based on a measurement theory. We adopt a simple class ical measurement theory that allows the determination of any continuou s phase-space function to any finite precision. Surprisingly, the resu lting ''classical logics'' are non-Boolean, though the distributive la ws hold. It appears that any physical theory with a mathematical space of physical states and an adequate description of measurement natural ly 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.