Modal Exchangeability: Centered Symmetry and the Credal Architecture of Kripke Frames

TL;DR

This paper explores how modal structures in Kripke frames influence probabilistic exchangeability, leading to new representation theorems and applications in learning and credal modeling.

Contribution

It introduces modal exchangeability, characterizes it via orbit decomposition, and derives representation theorems for countable frames with applications in learning and credal structures.

Findings

Worlds in the same orbit are conditionally identically distributed.

On certain frames, worlds are conditionally i.i.d. given an orbit-specific measure.

Orbit decomposition determines whether learning pools globally or remains local.

Abstract

We ask what happens when the index set carries modal structure, with possibilities organized into a Kripke frame. We define modal exchangeability as invariance under accessibility-preserving automorphisms that fix a designated base world, and derive a representation theorem for countable frames. The orbit decomposition of the centered symmetry group governs the within-orbit structure: worlds in the same orbit are conditionally identically distributed, and on orbits satisfying a richness condition and countable infinitude they are conditionally i.i.d. given a rigid orbit-specific directing measure. Point-homogeneous S5 frames yield a single de Finetti parameter; S4 frames may admit multiple orbits, with the richer orbits carrying rigid directing measures and the remainder carrying only weaker invariant structure. Two applications follow. First, the orbit decomposition determines whether learning pools globally or remains orbit-local. Second, it supplies a mechanism for structural credal fine-graining indexed to orbit regions, distinct from hyperintensionality in the strict sense of distinguishing coextensive propositions.