What is the implication of quantum intuitionism?

Because of their relevance to intuitionistic logic and quantum logic, respectively, Hilbert algebras alias positive implication algebras (PIAs) and quasi-implication algebras (QIAs) are important generalizations of Abbott's implication algebras (IAs). We try to find the join of the variety of all PIAs and the variety of all QIAs; in other words, we search for all equations (in the language of implication algebras) which are valid in intuitionistic logic as well as in quantum logic. We present five simple equations which axiomatize an upper bound of these two varieties such that the most important common properties of PIAs and QIAs are inherited, and one gets an axiomatization of PIAs as well as of QIAs by just adding one additional equation. Studying negations, we show that in our common generalizations of PIAs and QIAs the negated elements always form an orthomodular lattice. (orig.)SIGLEAvailable from TIB Hannover: RN 3109(288) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman