Orthologic is non-classical logic and has been studied as a part of quantumlogic. OL is based on an ortholattice and is also called minimal quantum logic. Sequent calculus is used as a tool for proof in logic and has been examinedfor several decades. Although there are many studies on sequent calculus forOL, these sequent calculi have some problems. In particular, they do not includeimplication connective and they are mostly incompatible with the cut-eliminationtheorem. In this paper, we introduce new labeled sequent (...) calculus called LGOI, and show that this sequent calculus solve the above problems. It is alreadyknown that OL is decidable. We prove that decidability is preserved when theimplication connective is added to OL. (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)