Probability and logic in belief systems

This paper seeks to develop a formal (mathematical) model of belief systems based on the axioms of probability theory and propositional logic. By a belief system we mean a set of propositions along with an actor's objective probability assignments to (beliefs in) them, together with the relationships among and between propositions and beliefs. Belief systems are regarded as being comprised of interrelated elements.In the paper are developed measures of the distance between sets of beliefs; of the congruence, coherence, and consistency of belief systems; and of the degree of polarization of belief systems-which are derived from one basic operation, symmetric difference.We show that these measures possess a number of useful and powerful mathematical properties. Also, a model is set forth by which, from an actor's subjective probability assignment to propositions and pairwise conjunctions of propositions, we may then impute to the actor subjectively perceived truth functional relationships between proposition.The potential uses and practical difficulties with the approach taken in the paper are also discussed, and the assertion is made that the measures developed enable us to simply distinguish between certain notions (e.g., congruence, consistency, coherence) too often and easily confused, provide us with the possibility of interval (or at least, quasi-interval level) measurement of certain properties of individual belief systems, and also allows us to make comparisons between the structures of different actors' belief systems.

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