Theory of propensity

Satosi Watanabe

pp. 283-308

in: Robert S. Cohen, Mark W. Wartofsky (eds), Language, logic and method, Berlin, Springer, 1983

Abstract

Boolean logic and set theory are so closely bound up that it is hard to say which is the foundation of which. Yet, we may say that the backbone of this theoretical complex is the Axiom of Comprehension. But, this Axiom, at the same time, is precisely the most vulnerable assumption in the whole of logical philosophy. While the binary-valued characteristic function of predicates is instrumental in expressing the Axiom of Comprehension, our analysis of scientific obervation, in particular in psychology, suggests that we have to introduce what we call a "propensity function." This is, roughly speaking, a continuous-valued generalization of the characteristic function. Whereas the characteristic function leads to a distributive logic, we can derive a non-distributive logic from the propensity function. If we further introduce the assumption of compatibility (experimental results of two predicates do not depend on the order of the two observations), we can derive distributive logic, the usual laws of probability, set theory and the usual notion of extension of a predicate (Law of Comprehension). If we do not adopt the assumption of compatibility, our theory provides the framework for quantum mechanics, and hopefully, for future psychology. This way of deriving logic from a probability-like propensity function seems to be more desirable than the usual way of positing logical laws without unifying justification, because we can derive first a more general logic and then therefrom a more restrictive logic as a special case using a small number of clearly stated postulates. The necessity for modification of the Quinian definition of ontological commitment will be indicated.