BINARY ORTHOLOGIC WITH MODUS PONENS IS EITHER ORTHOMODULAR OR DISTRIBUTIVE

Mladen Pavicic^1,2 and Norman D. Megill³

¹Atominstitute of the Austrian Universities, Schüttelstraße 115, A-1020 Wien, Austria; pavicic@ati.ac.at

² Department of Mathematics, University of Zagreb, GF, Kaciceva 26, POB-217, HR-10001 Zagreb, Croatia; mpavicic@faust.irb.hr; http://m3k.grad.hr/pavicic

³ Locke Lane, Lexington, MA 02173, U. S. A.; nm@alum.mit.edu

Abstract. We show that binary orthologic becomes either quantum or classical logic when nothing but modus ponens rule is added to it, depending on the kind of the operation of implication used. We also show that in the usual approach the rule characterizes neither quantum nor classical logic. The difference turns out to stem from the chosen valuation on a model of a logic. Thus algebraic mappings of axioms of standard quantum logics would fail to yield an orthomodular lattice if a unary - as opposed to binary - valuation were used. Instead, non-orthomodular nontrivial varieties of orthologic are obtained. We also discuss the computational efficiency of the binary quantum logic and stress its importance for quantum computation and related algorithms.

PACS numbers: 03.65.Bz, 02.10.By, 02.10.Gd

Keywords: quantum logic, orthomodular lattices, quantum computation, modus ponens, distributivity, classical logic.