BELIEVING CHANGE AND CHANGING BELIEF

We present a first-order logic of time, chance, and probability that i s capable of expressing the four types of higher-order probability sen tences relating subjective probability and objective chance at differe nt times. We define a causal notion of objective chance and show how i t can be used in conjunction with subjective probability to distinguis h between causal and evidential correlation by distinguishing between conditions, events, and actions that 1) influence the agent's belief i n chance and 2) the agent believes to influence chance. Furthermore, t he semantics of the logic captures some commonsense inferences concern ing objective chance and causality. We show that an agent's subjective probability is the expected value of its beliefs concerning objective chance. We also prove that an agent using this representation believe s with certainty that the past cannot be causally influenced.