Partial quantum logics and Łukasiewicz’ quantum logic

Abstract

In Chapter 5, we have considered examples of partial algebraic structures, where the basic operations are not always defined. How does one give a semantic characterization for the different forms of quantum logic corresponding, respectively, to the class of all effect algebras, of all orthoalgebras and of all orthomodular posets? We will call these logics: unsharp partial quantum logic (UPaQL), weak partial quantum logic (WPaQL) and strong partial quantum logic (SPaQL), respectively. By PaQL we will denote any instance of our three logics. Since PaQL as well as partial classical logic PaCL (studied in Chapter 13) are examples of partial logics, one could expect some natural semantic connections between the two different cases. However, we will see that the basic idea of the semantic characterization of PaQL is somewhat different with respect to the PaCL-semantics.