Agreement and Updating For Self-Locating Belief

Suppose we have two ideally rational, perfectly introspective, individuals, who want to pool their information. Since both individuals are perfectly introspective, they are both in a position to characterize to the other exactly what they know. Suppose, then, that each tells the other what they know and each correspondingly incorporates this information into what they know. Given that such individuals are perfectly rational, we should expect that, having communicated in this manner, their resulting epistemic states should be such that any further communication would be otiose. Each will know exactly what the other knows, and will know that each knows exactly what the other knows and so on. Let us call an individual at a time an agent. And let us call a group of agents, whose epistemic states are of the sort that might result from this type of idealized information pooling, epistemic confidants. In what follows, we will focus on agents whose epistemic and credal states are defined over finite algebras of intensional propositions. We’ll also assume that any agent is such that there is a strongest proposition φ that they know, and that they know all and only those propositions entailed by φ. Obviously not every possible agent need satisfy these assumptions. However, even restricting our attention to such agents, the arguments that follow should be of interest. Moreover, we should, I think, expect similar considerations to apply to agents who don’t satisfy these constraints. Restricting our attention in this manner, though, will help to simplify and streamline our discussion. In what follows, I’ll argue that a very plausible principle about how epistemic confidants may rationally update their credal states has some surprising consequences for how rational agents may update their credal states in light of self-locating information. Let us call a proposition that may only vary its truth-value across worlds uncentered, and a proposition that may its truth-value, not only across worlds, but also across individuals or times centered. We can represent an agent whose epistemic and credal states are defined over an algebra that includes only uncentered propositions as having epistemic and credal states that are