Agreement theorems for Self-Locating Belief

In this paper, I first outline Aumann’s famous “no agreeing to disagree" theorem, and a second related theorem. These results show that if two or more agents, who have epistemic and credal states that are defined over algebras that do not include any self-locating propositions, have certain information about one another’s epistemic and credal states, then such agents must assign the same credence to certain propositions. I show, however, that both of these theorems fail when we consider agents who have epistemic and credal states that are defined over algebras that do include self-locating propositions. Importantly, these theorems fail for such agents even when we restrict our attention to the credences that such agents have in non-self-locating propositions. Having established this negative result, I then outline and prove three agreement theorems that hold for such agents. Call a centered proposition one that may vary its truth-value not only across worlds, but also across individuals or times, and call an uncentered proposition one that may only vary its truth-value across worlds.1 It is well-known that, for agents whose epistemic and credal states are defined over algebras of uncentered propositions, there are strict formal constraints on the extent to which such agents may rationally assign distinct credences to propositions, given knowledge of one another’s epistemic and credal states. Interestingly, though, if one adds to such algebras centered propositions, then such formal constraints on rational disagreement, even concerning uncentered propositions, no longer obtain.2 The purpose of this paper is to explore the extent to which there are formal constraints on disagreement amongst agents whose epistemic and credal states are defined over algebras that include centered propositions, given knowledge of one another’s epistemic and credal states. Let us say, roughly, that a set of agents are epistemic confidants just in case (i) it is common knowledge amongst them what each agent knows, and (ii) it is common knowledge Received: November 10, 2015. 1 Examples of centered propositions include the proposition that I am hungry at time t , and the proposition that it is now raining in Matera. Examples of uncentered propositions include the proposition that NN is hungry at time t , and the proposition that it is raining in Matera at time t . One might notice that centered propositions are often expressed by sentences that contain indexicals like ‘I’ and ‘now’. Given this, one might naturally wonder whether we should also allow propositions to vary their truth-values over locations. Such propositions may be expressed by sentences involving the indexical ‘here’. The answer is that, while we can allow for this sort of variation, we need not take it as basic. Instead, we can take such variation in truth-value over location to be deriviative from variation over times and individuals. As we’ll see, we can represent centered propositions by sets of world, individual, time triples. Given this, we can then, for example, take the proposition that it is raining here at time t ′ to be the set of world, individual, time triples, ⟨w, i, t⟩ such that t = t ′ and it is raining at the location of i in w at t . 2 One can see this sort of phenomenon in cases discussed in Leslie (1997), Bostrom (2000), Bostrom (2002), and Lewis (2004). The formal underpinnings of these cases, though, are left unexplored by these authors. c ⃝ Association for Symbolic Logic, 2016 1 doi:10.1017/S1755020316000101