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It is widely accepted that the logic of quantum mechanics is based on orthomodular posets. However, such a logic is not dynamic in the sense that it does not incorporate time dimension. To fill this gap, we introduce certain tense operators on such a logic in an inexact way, but still satisfying requirements asked on tense operators in the classical logic based on Boolean algebras or in various non-classical logics. Our construction of tense operators works perfectly when the orthomodular poset in question is finite. We investigate the behaviour of these tense operators, e.g. we show that some of them form a dynamic pair. Moreover, we prove that if the tense operators preserve one of the inexact connectives conjunction or implication as defined by the authors recently in another paper, then they also preserve the other one. Finally, we show how to construct the binary relation of time preference on a given time set provided the tense operators are given, up to equivalence induced by natural quasi-orders. [ABSTRACT FROM AUTHOR]

Orthomodular lattices were introduced to get an algebraic description of the propositional logic of quantum mechanics. In this paper, we set up axiomatization of this logic as a Hilbert style implicational logical system L, i.e., we present a set of axioms and derivation rules formulated in the signature {→,0}. The other logical operations ⋁, ⋀, ¬ are expressed in terms of implication (which is the so-called Dishkant implication) and falsum. We further show that the system L is algebraizable in the sense of Blok and Pigozzi, and that orthomodular lattices provide an equivalent algebraic semantics for it. [ABSTRACT FROM AUTHOR]