Counterfactuals and updates as inverse modalities (Q1357671)

Assign sentences in the usual way to the elements of the set \(W\) of possible worlds. For each sentence \(A\), let \(R_A\) be the accessibility relation such that \(R_A(x,y)\) holds if and only if \(x\) is among the \(A\)-worlds that are closest to \(y\). Then the Katsuno-Mendelzon update of \(B\) by \(A\) consists of the \(A\)-worlds that are closest to some \(B\)-world, i.e. of the worlds \(x\) such that \(R_A(x,y)\) for some \(B\)-world \(y\). Furthermore, the Stalnaker counterfactual ``if \(A\) was the case then \(B\) would be the case'' can be understood as holding in the worlds such that in all their closest \(A\)-worlds, \(B\) holds, i.e. in the worlds \(x\) such that if \(R_A(y,x)\) then \(y\) is a \(B\)-world. Hence, counterfactuals can be understood as a necessity modality whose accessibility relation is the inverse of the accessibility relation of a possibility modality that corresponds to updates. This elegant connection is competently developed in this paper, and in particular it is shown how properties of updates and properties of Stalnaker counterfactuals correspond to each other.