Sharp quantum logics

Abstract

We will first study two interesting examples of logics that represent a natural logical abstraction from the class of all Hilbert lattices. These are represented respectively by orthomodular quantum logic (OQL) and by the weaker orthologic (OL), which for a long time has been also termed minimal quantum logic. In fact, the name “minimal quantum logic” appears today quite inappropriate for two reasons. First, a number of, in a sense, weaker forms of quantum logic have recently attracted much attention; and second, the models for the “minimal quantum logic” do not provide for the possibility of an adequate modeling of the generalized probabilities that are induced by states of QT. However these probabilities do not usually play a fundamental role in the logical developments that follow. And the “minimal quantum logic” provides a “floor” for the other logics, so we include it. In the following we will use QL as an abbreviation for either OL or OQL.