The fact that it is possible to define three different material conditionals in orthomodular lattices suggests that there exist three different orthomodular logics whose conditionals are material conditionals and whose models are orthomodular lattices. The purpose of this paper is to provide equationally definable implication algebras for each of these material conditionals.

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)