On when a semantics is not a good semantics: The algebraisation of orthomodular logic

KOWALSKI, TOMASZ STANISLAW;PAOLI, FRANCESCO;GIUNTINI, ROBERTO

2016-01-01

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Abstract

It has been taken for granted for a long time that orthomodular lattices are the "algebraic counterpart" of orthomodular quantum logic. Pavicic and Megill have questioned this claim by pointing out that orthomodular quantum logic is sound and complete with respect to a proper supervariety of the variety OML of orthomodular lattices (the so-called weakly orthomodular lattices). The same authors conclude that "in the syntactical structure of quantum logic there is nothing orthomodular". After reviewing in a certain detail some concepts from Abstract Algebraic Logic, especially Blok and Pigozzi's theory of algebraisable logics, we argue that the weakly orthomodular semantics introduced by Pavicic and Megill is not a good semantics, and that the role of OML as an algebraic counterpart of quantum logic is unaected by their allegations.