Quantum Logic and Non-Commutative Geometry

We propose a general scheme for the “logic” of elementary propositions of physical systems, encompassing both classical and quantum cases, in the framework given by Non-Commutative Geometry. It involves Baire^*-algebras, the non-commutative version of measurable functions, arising as envelope of the C ^*-algebras identifying the topology of the (non-commutative) phase space. We outline some consequences of this proposal in different physical systems. This approach in particular avoids some problematic features appearing in the definition of physical states in the standard (W ^*-)algebraic approach to classical mechanics.