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)

In this study, new sequent calculi for a minimal quantum logic ) are discussed that involve an implication. The sequent calculus \ for \ was established by Nishimura, and it is complete with respect to ortho-models. As \ does not contain implications, this study adopts the strict implication and constructs two new sequent calculi \ and \ as the expansions of \. Both \ and \ are complete with respect to the O-models. In this study, the completeness and decidability theorems (...) for these new systems are proven. Furthermore, some details pertaining to new rules and the strict implication are discussed. (shrink)

Unified quantum logic based on unified operations of implication is formulated as an axiomatic calculus. Soundness and completeness are demonstrated using standard algebraic techniques. An embedding of quantum logic into a new modal system is carried out and discussed.