Kripke-style semantic for modal orthomodular logic

G. DOMENECH, H. FREYTES AND C. DE RONDE

Gdansk, Polonia

Congreso; 9th Biennial IQSA Meeting "Quantum Structures Brussels-Gdansk '08"; 2008

International Quantum Structures Association

In their 1936 seminal paper, Birkhoff and von Neumann made the proposalof a non-classical logic for quantum mechanics founded on the basic lattice-orderproperties of all closed subspaces of a Hilbert space. These lattice-order propertiesare captured in the orthomodular lattice structure, characterized by a weak formof distributivity called orthomodular law. This “weak distributivity”, which is theessential difference with the Boolean structure, makes it extremely intractable incertain aspects. In fact, a general representation theorem for a class of algebras,which has as particular instances the representation theorems as algebras of sets forBoolean algebras and distributive lattices, allows in many cases and in a uniformway the choice of a Kripke-style model and to establish a direct relationship withthe algebraic model. In this procedure the distributive law plays a very importantrole. In absence of distributivity this general technique is not applicable, consequentlyto obtain Kripke-style semantics may be complicated. Such is the case forthe orthomodular logic. Indeed, Goldblatt gave a Kripke-style semantic for theorthomodular logic based on an imposed restriction on the Kripke-style semanticfor the orthologic, but this restriction is not first order expressible, making the obtainedsemantic not very attractive. Later, Miyazaki introduced another approachto the Kripke-style semantic for the orthomodular logic based on the representationtheorem by Baer semigroups. In this way a Kripke-style model is obtained whoseuniverse is given by semigroups with additional operations.Several authors added modal enrichments to the orthomodular structure basedon generalizations of classic modal systems, or generalization of quantifiers in thesense of Halmos. Recently we have introduced an orthomodular structure enrichedwith a modal operator called Boolean saturated orthomodular lattice. This structurehas a rigorous physical motivation and allows to establish algebraic-type versionsof the Born rule and the well known Kochen-Specker (KS) theorem. The aimof our contribution is to study this structure from a logic-algebraic perspective.We first introduce the class of Boolean saturated orthomodular lattices OML2and we prove that this class conforms a discriminator variety. Then, a Hilbertstylecalculus is introduced obtaining a strong completeness theorem for the varietyOML2. Finally, we give a representation theorem by means of a sub-class ofBaer ?-semigroups for OML2. This allows to develop a Kripke-style semantic forthe calculus. A strong completeness theorem for these Kripke-style models is alsoobtained.(trabajo completo enviado para publicación)