Updating as Communication*

Abstract

On many traditional theories of belief, your belief state is represented by an assignment of credences to propositions, or sets of possible worlds. If you are rational, your credence distribution will be a probability measure. Traditional theories of belief fit with a standard Bayesian theory of rational belief change: on learning a proposition, you must update your belief state by conditionalizing your credence distribution on the proposition you learn. That is, you must update by assigning 0 credence to those worlds incompatible with what you learn, and re-normalizing your credence distribution over the remaining worlds. Following Quine 1969, Lewis 1979 argues that we should instead represent your belief state by an assignment of credences to sets of centered worlds: world-timeindividual triples. For instance, if you have .5 credence that it is 3:00pm, your belief state should be represented by a measure that assigns .5 to the set of centered worlds with that time coordinate. Unlike traditional theories of belief, Lewis’s theory does not fit with a standard Bayesian theory of rational belief change. For instance, Bayesian conditionalization preserves certainties. If you update by conditionalizing on the set of centered worlds you learn, it follows that if you are ever certain that it is 3:00pm, you must always remain certain that it is 3:00pm. But clearly this is not what rationality requires. If we agree with Lewis about how to represent belief states, we must develop another set of principles governing rational belief change.