Canonical interpretation of propositions as events

Abstract

This paper establishes conditions under which Savage's (1954) informal interpretation of subjective probabilities as measures of confidence in the truth of propositions can be formally justified. For this purpose we construct, for any given propositional language, a canonical state space such that each proposition a of the language is associated with a unique event A defined on this state space. As our main result we establish a one–one onto correspondence between the canonical state space and the set of all truth conditions for the propositional logic such that proposition a is exactly true at every truth condition that corresponds to some state in A. According to our approach, an agent's degree of confidence in the truth of a proposition can, therefore, be interpreted as his or her subjective probability that some truth condition holds at which the proposition is true. Such an interpretation, however, is only valid for agents with unlimited powers of logical reasoning.