A Modal de Finetti Theorem: Exchangeability under S4 and S5
Daniel Zantedeschi
Abstract
We introduce modal exchangeability, a symmetry principle for probability measures on Kripke frames: invariance under those automorphisms of the frame that preserve the accessibility relation and fix a designated world. This principle characterizes when an agent's uncertainty over possible-world valuations respects the modal structure. We establish representation theorems that determine the probabilistic consequences of modal exchangeability for S4 and S5 frames. Under S5, where accessibility is an equivalence relation, the classical de Finetti theorem is recovered: valuations are conditionally i.i.d. given a single directing measure. Under S4, where accessibility is a preorder, the accessible cluster decomposes into orbits of the stabilizer group, and valuations within each orbit are conditionally i.i.d. with an orbit-specific directing measure. A rigidity constraint emerges: each directing measure must be constant across its orbit. Rigidity is not assumed but forced by symmetry; it is a theorem, not a modeling choice. The proofs are constructive, requiring only dependent choice (ZF + DC), and yield computable representations for recursively presented frames. Rigidity has direct epistemic content: rational agents whose uncertainty respects modal structure cannot assign different latent parameters to worlds within the same orbit. The framework connects probabilistic representation theory to the S4/S5 distinction central to epistemic and temporal logic, with consequences for hyperintensional belief and rational learning under partial information.