The Ramsey Test and the Indexicality of Conditionals: A Proposed Resolution of Gärdenfors' Paradox

Working within the AGM approach, Peter Gärdenfors has proved that — given certain auxiliary assumptions — there are no non-trivial belief revision systems that satisfy both the Ramsey test and the Preservation condition. There are various ways of reacting to Gärdenfors' paradoxical result. Gärdenfors himself has described his result as a dilemma: we must give up either the Ramsey test or the Preservation condition. Isaac Levi has pointed to an implicit assumption in Gärdenfors' approach: conditionals express truth-value bearing propositions and are therefore eligible as members of belief sets. Here, we shall focus on another implicit assumption: a conditional sentence A > B expresses one and the same proposition relative to every belief state. We shall argue that contrary to the this assumption, epistemic conditionals are naturally interpreted as context-sensitive: an epistemic conditional expresses a proposition, but only relative to a belief state. According to the approach advocated here, a sharp distinction is made between the semantic level containing propositions and belief states and the linguistic level containing sentences and sets of sentences. Belief revision is viewed as an operation on belief states; and it is primarily propositions rather than sentences that are accepted relative to belief states. At the semantic level, Gärdenfors' result applies to the Ramsey test in the form: (P-R) The proposition P ⇒ Q is accepted in the belief state S iff Q is accepted in the state S∗P which is the result of revising S with the proposition P. Thinking of the connective > as corresponding to a binary operation ⇒ on propositions is, however, tantamount to assuming that epistemic conditionals are context-independent. Once, we give up this assumption, we see that the proper way of formulating the Ramsey test for propositions is not (P-R) but rather: (P-Ramsey) P ⇒S Q is accepted in the belief state S iff Q is accepted in S∗P. But, as we shall see, (P-Ramsey) is compatible with Gärdenfors' postulates for belief revision. We also show how to accommodate the full Ramsey test for conditional sentences, thus allowing for unlimited iteration of conditionals. Our conclusion is that there is no real conflict between Preservation and the Ramsey test — once we take the context dependency of epistemic conditionals into account.