The concept of a proposition in classical and quantum physics

Ernst-Walther Stachow

Bernard d’Espagnat

Knowledge of the classical and nonclassical theories of logical propositions is not required for an understanding of the analysis of the nonseparability of systems. Some very elementary notions about that matter may help to place the general problem in appropriate perspective. Classically, physical propositions bear upon physical systems. A proposition a bearing on a system S is a qualitative statement about S. The quantum calculus of propositions has been successfully used as an alternative and more general basis for the foundations of quantum mechanics. It is a truism that the advent of the modern physical theories—relativity, quantum mechanics, quantum electrodynamics, S matrix theory, and so on—has induced us to abandon many familiar intuitive concepts. All the successful theories mentioned above are built on elaborate sets of axioms that are justified only a posteriori, that is, by agreement between some of their consequences and observed facts.

Peter Mittelstaedt

Vassilios Karakostas

Abstract The logic of a physical theory reflects the structure of the propositions referring to the behaviour of a physical system in the domain of the relevant theory. It is argued in relation to classical mechanics that the propositional structure of the theory allows truth-value assignment in conformity with the traditional conception of a correspondence theory of truth. Every proposition in classical mechanics is assigned a definite truth value, either ‘true ’ or ‘false’, describing what is actually the case at a certain moment of time. Truth-value assignment in quantum mechanics, however, differs; it is known, by means of a variety of ‘no go ’ theorems, that it is not possible to assign definite truth values to all propositions pertaining to a quantum system without generating a Kochen-Specker contradiction. In this respect, the Bub-Clifton ‘uniqueness theorem ’ is utilized for arguing that truth-value definiteness is consistently restored with respect to a determinate sublattice of propositions defined by the state of the quantum system concerned and a particular observable to be measured. An account of truth of contextual correspondence is thereby provided that is appropriate to the quantum domain of discourse. The conceptual implications of the resulting account are traced down and analyzed at length. In this light, the traditional conception of correspondence truth may be viewed as a species or as a limit case of the more generic proposed scheme of contextual correspondence when the non-explicit specification of a context of discourse poses no further consequences.

Peter Mittelstaedt

Harald Atmanspacher A, Peter beim Graben

The concept of complementarity in combination with a non-Boolean calculus of propositions refers to a pivotal feature of quantum systems which has long been regarded as a key to their distinction from classical systems. But a non-Boolean logic of complementary features may also apply to classical systems, if their states and observables are defined by partitions of a classical state space. If these partitions do not satisfy certain stability criteria, complementary observables and non-Boolean propositional lattices may be the consequence. This is especially the case for non-generating partitions of nonlinear dynamical systems. We show how this can be understood in more detail and indicate some challenging consequences for systems outside quantum physics, including mental processes.