A new approach to probabilities in mechanics

Abstract

In this paper a general theory of probabilities is sketched, suitable for any system of mechanics. I show that the classical probabilities of classical mechanics arise as a special case, and that the general non-classical probabilities display the striking features of quantum statistics. In particular, the rules for conditionalising in quantum mechanics turn out to be instances of the general conditionalising rules developed here. According to this analysis the probabilities of mechanics are generated from the propositional structure of a mechanical theory. A general account of mechanical propositions and of the logic used to express complex mechanical descriptions, is given in the first part of the paper. I argue that the logic is essentially classical, except that it has a generalised negation. It is important to stress that the same propositional logic is used by any mechanical theory. What distinguishes a classical theory of mechanics is the structure of its states. A mechanical state is analysed here in logical terms, and is crucial to the account of probabilities. The special properties of classical states give rise to classical probabilities, while in general 'peculiarities' arise. These turn out to be familiar features of quantum statistics. So I suggest that the odd features of quantum probabilities are in fact common to any non-classical theory, and that these peculiarities should be understood as rising from the structure of the theory, not from the peculiar nature of subatomic reality. It is the structure of states used by a theory which gives rise to non-classical probabilities. This is just to suggest that quantum theories are inadequate in a well-defined sense, and that the peculiarities of quantum descriptions are a symptom of this inadequacy. The paper is divided into four sections, entitled Propositions and the Logic LEt, Mechanical States, Probabilities, and Conclusions for Mechanics, respectively.