Quantum logical calculi and lattice structures

In a preceding paper [1] it was shown that quantum logic, given by the tableaux-calculus T_eff, is complete and consistent with respect to the dialogic foundation of logics. Since in formal dialogs the special property of the value-definiteness of propositions is not postulated, the calculus T_eff represents a calculus of effective (intuitionistic) quantum logic.Beginning with the tableaux-calculus the equivalence of T_eff to calculi which use more familiar figures such as sequents and implications can be investigated. In this paper we present a sequents-calculus of Gentzen-type and a propositional calculus of Brouwer-type which are shown to be equivalent to T_eff. The effective propositional calculus provides an interpretation for a lattice structure, called quasi-implicative lattice. If, in addition, the value-definiteness of quantum mechanical propositions is postulated, a propositional calculus is obtained which provides an interpretation for a quasi-modular orthocomplemented lattice which, as is well-known, has as a model the lattice of subspaces of a Hilbert space.