Conditionalization and total knowledge

This paper employs epistemic logic to investigate the philosophical foundations of Bayesian updating in belief revision. By Bayesian updating, we understand the tenet that an agent’s degrees of belief—assumed to be encoded as a probability distribution—should be revised by conditionalization on the agent’s total knowledge up to that time. A familiar argument, based on the construction of a diachronic Dutch book, purports to show that Bayesian updating is the only rational belief-revision policy. We investigate the conditions under which of the premises of this argument might be satisfied. Specifically, we consider the case of an artificial agent whose language (of thought) features a modal operator TK, where TKψ has the interpretation “My total knowledge is ψ”. We show that every proposition of the form TKψ is epistemically categorical: it determines, for every proposition φ in the agent’s language, whether the agent knows that φ. We argue that, for certain artificial agents employing such a language, the diachronic Dutch book argument for Bayesian updating is on firm ground. In the days before their connection to fragments of first-order logic was fully understood, modal logics were conceived of primarily as tools for the philosophical analysis of various fundamental concepts: necessity, time, obligation, knowledge. The axioms of these logics were formalizations of philosophical intuition; the ability to determine the precise consequences of those axioms, and the philosophical perspective it provided were the fruits of that formalization. Two developments have since radically altered this situation. The first is the discovery of relational semantics for modal logic, which made it possible to conceive of modal logic primarily as a collection of formalisms for describing relational structures. The second is the increasing influence of Computer Science in logic, which has pushed issues such as decidability, computational complexity and efficient automation, where modal logics have turned out to exhibit striking behaviour, to the fore. Indeed, it is probably correct to say that modal logicians of the younger generation are predominantly motivated by the special modeltheoretic and complexity-theoretic characteristics of the formal languages they study. These days, modal logic just isn’t—well, modal. The present paper harks back to the older, philosophical tradition in modal logic. Specifically, we employ epistemic logic to investigate the philosophical foundations of Bayesian updating in belief revision. By Bayesian updating, we understand the tenet that an agent’s degrees of belief—assumed to be encoded