Disbelief as the Dual of Belief

The duality of truth and falsity in a Boolean algebra of propositions is used to generate a duality of belief and disbelief. To each additive probability measure that represents belief there corresponds a dual additive measure that represents disbelief. The dual measure has its own peculiar calculus, in which, for example, measures are added when propositions are combined under conjunction. A Venn diagram of the measure has the contradiction as its total space. While additive measures are not self‐dual, the epistemic state of complete ignorance is represented by the unique, monotonic, non‐additive measure that is self‐dual in its contingent propositions. Convex sets of additive measures fail to represent complete ignorance since they are not self‐dual.