Quasi-Heyting Algebras: A New Class of Lattices

Abstract

Quasi-Heyting algebras (QHAs) generalize boththe Heyting algebras (HAs) of intuitionistic logic andthe orthomodular lattices (OMLs) of quantum logic. As inHAs, negation is a Galois connection, which expresses abandonment of the law of theexcluded middle, and as in OMLs, incompatibility ofpropositions is expressed by departures fromdistributivity. Formulating an equational definition ofQHAs leads to generalizations of familiar operations. QHAsare the truthvalue objects of a generalization oftoposes. So far, this development has aimed to providefoundations of logic and model theory suitable for addressing computer science problems, but theyalso appear applicable as formulations of the logic ofsome types of scientific measurement. Many properties ofOMLs are likely to have generalizations to QHAs.