In this paper we enrich the orthomodular structure by adding a modal operator, following a physical motivation. A logical system is developed, obtaining algebraic completeness and completeness with respect to a Kripkestyle semantic founded on Baer*-semigroups as in [22].

Topos quantum theory represents a whole new approach to the formalization of non-relativistic quantum theory. It is well known that TQT replaces the orthomodular quantum logic of the traditional Hilbert space formalism with a new intuitionistic logic that arises naturally from the topos theoretic structure of the theory. However, it is less well known that TQT also has a dual logical structure that is paraconsistent. In this paper, we investigate the relationship between these two logical structures and study the implications (...) of this relationship for the definition of modal operators in TQT. (shrink)

An effect algebra is a partial algebraic structure, originally formulated as an algebraic base for unsharp quantum measurements. In this article we present an approach to the study of lattice effect algebras (LEAs) that emphasizes their structure as algebraic models for the semantics of (possibly) non-standard symbolic logics. This is accomplished by focusing on the interplay among conjunction, implication, and negation connectives on LEAs, where the conjunction and implication connectives are related by a residuation law. Special cases of LEAs are (...) MV-algebras and orthomodular lattices. The main result of the paper is a characterization of LEAs in terms of so-called Sasaki algebras. Also, we compare and contrast LEAs, Hájek's BL-algebras, and the basic algebras of Chajda, Halaš, and Kühr. (shrink)

Logical matrices for orthomodular logic are introduced. The underlying algebraic structures are orthomodular lattices, where the conditional connective is the Sasaki arrow. An axiomatic calculusOMC is proposed for the orthomodular-valid formulas.OMC is based on two primitive connectives — the conditional, and the falsity constant. Of the five axiom schemata and two rules, only one pertains to the falsity constant. Soundness is routine. Completeness is demonstrated using standard algebraic techniques. The Lindenbaum-Tarski algebra ofOMC is constructed, and it is shown to be (...) an orthomodular lattice whose unit element is the equivalence class of theses ofOMC. (shrink)