Believing change and changing belief

Abstract

We present a first-order logic of time, chance, and probability that is capable of expressing the four types of higher-order probability sentences relating subjective probability and objective chance at different times. We define a causal notion of objective chance and show how it can be used in conjunction with subjective probability to distinguish between causal and evidential correlation by distinguishing between conditions, events, and actions that: 1) influence the agent's belief in chance; and 2) the agent believes to influence chance. Furthermore, the semantics of the logic captures some common sense inferences concerning 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 believes with certainty that the past cannot be causally influenced.