Counterfactuals and Updates as Inverse Modalities

We point out a simple but hitherto ignored link between the theory of updates and counterfactuals and classical modal logic: update is a classical existential modality, counterfactual is a classical universal modality, and the link between the two (called the Ramsey rule) is simply the link between two inverse accessibility relations of a classical Kripke model. 1 I n t r o d u c t i o n B a c k g r o u n d . An intuitive connection between theory change and counterfactuals was observed by F. P. Ramsey [17], who proposed what has become known as the Ramsey Rule: To find out whether the counterfactual 'if A were true, then B would be true' is satisfied in a state S, change the state S minimally to include A, and test whether B is satisfied in the resulting state. 1 It was initially hoped that the AGM theory of belief revision [3, 13] would provide the right notion of minimal change. However, the intuitively acceptable AGM postulates for belief revision axe known to be incompatible with the Ramsey Rule [2, 3]. It turns out that the theory of updates proposed by Katsuno and Mendelzon [10] is compatible with the Ramsey Rule [6]. Updates, like revisions, axe a formalisation of theory change; but whereas revisions are intended to model changing knowledge about a fixed world, updates are intended to model a changing world. The difference between the formalisations of updates and revisions can be seen in terms of postulates; for example, the AGM postulate A • B = A A B if A A B is consistent 1Actually, Ramsey proposed the rule only for non-counterfactual conditionals, but the term 'Ramsey Rule' is now taken to refer to counterfactuals too.