Classical foundations of quantum logic

We construct a languageL for a classical first-order predicate calculus with monadic predicates only, extended by means of a family of statistical quantifiers. Then, a formal semantic model is put forward forL which is compatible with a physical interpretation and embodies a truth theory which provides the statistical quantifiers with properties that fit their interpretation; in this framework, the truth mode of physical laws is suitably characterized and a probability-frequency correlation principle is established. By making use ofL and, a set of basic physical laws is stated that hold both in classical physics (CP) and in quantum physics (QP), which allow the selection of suitable subsets of primitive predicates ofL (the set_P of pure states; the sets_o and_E of operational and exact effects, respectively) and the introduction on these subsets of binary relations (a preclusion relation # on_P, an order relation < on_E. By assuming further physical laws, (_E, <) turns out to be a complete orthocomplemented lattice [mixtures and atomicity of (_E, <) also can be introduced by means of suitable physical assumptions]. Two languagesL_E^x andL_E^S are constructed that can be mapped intoL; the mapping induces on them mathematical structures, some kind of truth function, an interpretation. The formulas ofL_E^x can be interpreted as statements about properties of a physical object, and the truth function onL_E^x is two valued. The formulas ofL_E^S can be endowed with two different interpretations as statements about the frequency of some physical property in some class (state) of physical objects; consequently, a two-valued truth function and a multivalued fuzzy-truth function are defined onL_E^S. In all cases the algebras of propositions of these logics are complete orthocomplemented lattices isomorphic to (_E, <). These results hold both in CP and in QP; further physical assumptions endow the lattice (_E, <), henceL_E^x andL_E/S, with further properties, such as distributivity in CP and weak modularity and covering law in QP. In the latter case,L_E^S andL_E^S, together with their interpretations, can be considered different models of the same basic mathematical structure, and can be identified with standard (elementary) quantum logics. These are therefore founded on the classical extended languageL with semantic model.