From semantics to syntax: Quantum logic of observables

An attempt of syntactical reconstruction of a logical counterpart of algebraic formulation of quantum theory is done (cf. [11]). It includes R. Goldblatt's orthologic [6] as the two-valued case. An axiomatization of such quantum logic, of observables (QLO) along with the set-theoretical semantics is proposed and soundness and completeness are proved. Also a normed quantum logic of observables (NQLO) is yielded. Exploiting von Wright's temporal operation two systems of temporal quantum logic of observables (TQLO and TNQLO respectively) are obtained for which soundness and completeness is proved. In conclusion, some perspectives for such an approach are considered.