The Probabilistic Argument for a Nonclassical Logic of Quantum Mechanics

Abstract

The aim of this paper is simple. I want to state as clearly as possible, without a long discursion into technical questions, what I consider to be the single most powerful argument for use of a nonclassical logic in quantum mechanics. There is a very large mathematical and philosophical literature on the logic of quantum mechanics, but almost without exception, this literature provides a very poor intuitive justification for considering a nonclassical logic in the first place. A classical example in the mathematical literature is the famous article by Birkhoff and von Neumann (1936). Although Birkhoff and von Neumann pursue in depth development of properties of lattices and projective geometries that are relevant to the logic of quantum mechanics, they devote less than a third of a page (p. 831) to the physical reasons for considering such lattices. Moreover, the few lines they do devote are far from clear. The philosophical literature is just as bad on this point. One of the better known philosophical discussions on these matters is that found in the last chapter of Reichenbach’s book (1944) on the foundations of quantum mechanics. Reichenbach offers a three-valued truth-functional logic which seems to have little relevance to quantum-mechanical statements of either a theoretical or experimental nature. What Reichenbach particularly fails to show is how the three-valued logic he proposes has any functional role in the theoretical development of quantum mechanics It is in fact fairly easy to show that the logic he proposes could not possibly be adequate for a systematic theoretical statement of the theory as it is ordinarily conceived. The reasons for this will become clear later on in the present paper.