Abstract. 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 $\LOM $, i.e., we present a set of axioms and derivation rules formulated in the signature $\{\to,0\}$. The other logical operations $\vee, \wedge, \neg $ are expressed in terms of implication (which is the so-called Dishkant implication) and falsum. We further show that the system $\LOM $ is algebraizable in the sense of Blok and Pigozzi, and that orthomodular lattices provide an equivalent algebraic semantics for it.
DOI: 10.14232/actasm-015-813-6
AMS Subject Classification
(1991): 06C15, 03G12
Keyword(s):
algebraizable logic,
axiom system,
derivation rule,
Dishkant implication,
logic of quantum mechanics,
orthomodular implication algebra,
orthomodular lattice,
semi-orthomodular lattice,
weak BCK-algebra
Received August 16, 2015, and in final form January 16, 2016. (Registered under 63/2015.)
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