Many-valued Logics and Quantum Mechanics

Jarosław Pykacz

Jarosław Pykacz

Hans Reichenbach

Jarosław Pykacz

A many–valued logic from an old neglected Lukasiewicz paper is adopted to describe results of physical (esp. quantum) experiments. Truth–values of propositions are interpreted as probabilities of their experimental confirmation. It is shown that the Lindenbaum algebra of the obtained logic is an orthomodular partially ordered set, i.e. a “quantum logic” in the Birkhoff–von Neumann sense. Short outline of the historical development of applications of many–valued logics in foudations of quantum mechanics is given.

Anthony Sudbery

According to quantum mechanics, statements about the future made by sentient beings like us are, in general, neither true nor false; they must satisfy a many-valued logic. I propose that the truth value of such a statement should be identified with the probability that the event it describes will occur. After reviewing the history of related ideas in logic, I argue that it gives an understanding of probability which is particularly satisfactory for use in quantum mechanics. I construct a lattice of future-tense propositions, with truth values in the interval [0, 1], and derive logical properties of these truth values given by the usual quantum-mechanical formula for the probability of a history.

M. Katz

The multiple valued logic of inexactness and approximation we proposed in the past as framework for scientific thought in general is in this paper considered as logic for quantum theory in particular. In this logic truth-values are viewed as degrees of error and a deduction is considered valid if the error in the conclusion can be minimized whenever the error in the premise is brought under control. Applications of this logic to quantum mechanics encompass notions from earlier (non-classical) formal-logical approaches (modal, intuitionistic and multi-valued) and enable rigorous formulations of ideas relating to uncertainty, to states of quantum mechanical systems and to the familiar Hilbert space presentation of quantum physics.