Minimal Change in Modal Logic S5

We extend belief revision theory from propositional logic to the modal logic S5. Our first contribution takes the form of three new postulates (M1-M3) that go beyond the AGM ones and capture the idea of minimal change in the presence of modalities. Concerning the construction of modal revision operations, we work with set pseudo-distances, i.e., distances between sets of points that may violate the triangle-inequality. Our second contribution is the identification of three axioms (A3-A5) that go beyond the standard axioms of metrics. Loosely speaking, our main result states the following: if a pseudo-distance satisfies certain axioms, then the induced revision operation satisfies (M1-M3). We investigate three pseudo-distances from the literature (Dhaus, Dinj, Dsum), and the three induced revision operations (*Haus, *Inj, *Sum). Using our main result, we show that only *Sum satisfies (M1-M3) all together. As a last contribution, we revisit a major criticism of AGM operations, namely that the revisions of (p ∧ q) and (p ∧ (p → q)) are identical. We show that the problem disappears if instead of material implication we use the modal operator of strict implication that can be defined in S5.