Kripke-style Semantic for Modal Orthomodular Logic

In their 1936 seminal paper, Birkhoff and von Neumann made the proposal of a non-classical logic for quantum mechanics founded on the basic lattice-order properties of all closed subspaces of a Hilbert space. These lattice-order properties are captured in the orthomodular lattice structure, characterized by a weak form of distributivity called orthomodular law. This “weak distributivity”, which is the essential difference with the Boolean structure, makes it extremely intractable in certain aspects. In fact, a general representation theorem for a class of algebras, which has as particular instances the representation theorems as algebras of sets for Boolean algebras and distributive lattices, allows in many cases and in a uniform way the choice of a Kripke-style model and to establish a direct relationship with the algebraic model. In this procedure the distributive law plays a very important role. In absence of distributivity this general technique is not applicable, consequently to obtain Kripke-style semantics may be complicated. Such is the case for the orthomodular logic. Indeed, Goldblatt gave a Kripke-style semantic for the orthomodular logic based on an imposed restriction on the Kripke-style semantic for the orthologic, but this restriction is not first order expressible, making the obtained semantic not very attractive. Later, Miyazaki introduced another approach to the Kripke-style semantic for the orthomodular logic based on the representation theorem by Baer semigroups. In this way a Kripke-style model is obtained whose universe is given by semigroups with additional operations. Several authors added modal enrichments to the orthomodular structure based on generalizations of classic modal systems, or generalization of quantifiers in the sense of Halmos. Recently we have introduced an orthomodular structure enriched with a modal operator called Boolean saturated orthomodular lattice. This structure has a rigorous physical motivation and allows to establish algebraic-type versions of the Born rule and the well known Kochen-Specker (KS) theorem. The aim of our contribution is to study this structure from a logic-algebraic perspective. We first introduce the class of Boolean saturated orthomodular lattices OML ̄ and we prove that this class conforms a discriminator variety. Then, a Hilbert-style calculus is introduced obtaining a strong completeness theorem for the variety OML ̄. Finally, we give a representation theorem by means of a sub-class of Baer -semigroups for OML ̄. This allows to develop a Kripke-style semantic for the calculus. A strong completeness theorem for these Kripke-style models is also obtained.