Showing papers on "Linear logic published in 1970"

Quantum and classical logic: Their respective roles

Abstract: That quantum mechanics is different from classical mechanics, in what it says about the physical world and how it says it, needs no proof. How precisely to describe and explain these differences, and what significance to attach to them is being continually discussed. One of the claims that is being made is that the most significant difference between classical mechanics and quantum mechanics is that the latter uses or needs to use a non-classical kind of propositional logic, a logic that has been called a 'quantum logic'. The classical logic is often described as Aristotelian. More accurately, it is the propositional logic of two-valued truth-functional propositions, the logic of classes and the logic of quantification as, for example, these are developed in Russell and Whitehead's Principia Mathematica (PM). G. Birkhoff and J. von Neumann were the first in 1936 to put forward the view that the 'physical quantities' of quantum mechanics constitute an orthocomplemented non-distributive lattice, and not the Boolean algebra of classical PM logic. 1 Other proposals were made about the same time and later by Reichenbach, yon Weizs~icker, Heisenberg and others in favor of multi-valued logics in which the classical principle tertium non datur is violated. Bas C. van Fraasen has given a brief and systematic survey to the various quantum logics in his paper 'The Labyrinth of Quantum Logics'. I shall not be concerned in this paper with multi-valued quantum logics, but only with the view common to Birkhoff and von Neumann, Segal, Mackey, Finkelstein, Jauch, Putnam 2 and others, that quantum logic is a non-distributive lattice. This theory says that the basic empirical propositions of quantum mechanics obey, not the axioms and rules of classical P M logic, but those of a logic obtained from classical logic by dropping the distributive laws for 'and' and 'or' and replacing them by what looks like a weaker form of connection. In Jauch's version, the basic empirical propositions