A Modal de Finetti Theorem: Exchangeability under S4 and S5

A Modal de Finetti Theorem: Exchangeability under S4 and S5 Daniel Zantedeschi * Abstract W e introduce modal exc hangeability , a symmetry principle for probability measures on Kripke frames: in v ariance under those automorphisms of the frame that preserv e the accessibility relation and fix a designated world. This principle characterizes when an agent’ s uncertainty ov er possible-world v alua- tions respects the modal structure. W e establish representation theorems that determine the probabilistic consequences of modal exchangeability for S4 and S5 frames. Under S5 , where accessibility is an equiv alence relation, the classical de Finetti theorem is reco vered: 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 b ut forced by symmetry; it is a theorem, not a modeling choice. The proofs are constructiv e, requiring only dependent choice (ZF + DC), and yield computable repre- sentations for recursi vely 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 frame work 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. K eywords: modal exchangeability , de Finetti theorem, Kripk e frames, S4, S5, rigidity , probabilistic repre- sentation, hyperintensionality , rational learning 1 Intr oduction 1.1 Probability fr om logical symmetry Cox’ s theorem [ 1 ] establishes that any system of plausible reasoning satisfying basic consistenc y require- ments must be isomorphic to probability theory . The theorem assumes a Boolean algebra of propositions, * Email: danielz@usf.edu. School of Information Systems, Uni versity of South Florida. 1 where logical equi v alence is the fundamental symmetry: propositions true under the same conditions must recei ve the same plausibility assignment. Under these assumptions (classical propositional or first-order logic with a single, flat uni verse of discourse), rational belief reduces to Bayesian probability . This classical deri v ation presupposes a structureless index set. Propositions liv e in a single logical space; symmetry means permutation of logically equiv alent propositions. But modern reasoning operates over structur ed logical spaces: beliefs are index ed by states, contexts, agents, time steps, or information sets. Once logical space acquires structure, the relev ant symmetry is no longer unrestricted permutation b ut structur e-pr eserving transformation. The question becomes: what constraints does structured symmetry impose on permissible probability measures? 1.2 Modal structure and accessibility-pr eserving symmetry Modal logics pro vide the canonical setting for structured logical space. In Kripke semantics, a modal frame ⟨ W , R ⟩ consists of a set W of possible worlds and an accessibility relation R ⊆ W × W encoding which worlds are reachable from which. V aluations may v ary across worlds; the accessibility relation determines the interpretation of modal operators. T wo systems are fundamental. In S5 , accessibility is an equiv alence relation: knowledge is symmetric, and what is known is known to be kno wn. In S4 , accessibility is a preorder: kno wledge may be directed, modeling realistic information gro wth, temporal reasoning, and partial observability . The critical observation is that in Kripke semantics, the fundamental symmetry is not permutation of propo- sitions but automorphism of the frame : bijections of W that preserve the accessibility relation. These accessibility-preserving automorphisms are the natural analogues of logical equiv alence in the modal set- ting. A rational agent whose uncertainty respects the modal structure should assign equal probability to configurations that dif fer only by such a symmetry . 1.3 The repr esentation pr oblem This symmetry principle leads directly to a representation-theoretic question: Given a pr obability measur e over valuations on a Kripke fr ame, in variant under accessibility- pr eserving automorphisms that fix a designated world, what form must it take? This is the modal analogue of de Finetti’ s classical question [ 4 ]. De Finetti’ s theorem characterizes dis- tributions in variant under permutations of an index set as mixtures of i.i.d. processes. The classical result implicitly assumes that all indices are mutually interchangeable, forming a single equi v alence class where e very element is accessible from e very other . This is precisely the S5 situation. But what if accessibility is merely a preorder? 2 1.4 Main results The answer depends on the modal structure. W e introduce modal e xchang eability , defined as inv ariance of a probability measure under the stabilizer of a reference w orld within the automorphism group of the frame, and establish representation theorems for both S4 and S5 frames. Under S5 , the stabilizer acts transiti vely on the accessible cluster , yielding a single orbit. The representation reduces to classical de Finetti: valuations are conditionally i.i.d. gi ven a single directing measure. Under S4 , the preorder structure may fragment the accessible cluster into multiple orbits of the stabilizer group. The representation becomes orbit-wise : each orbit has its o wn directing measure, and cross-orbit dependence is not constrained by in v ariance. A rigidity condition emerges: the directing measure for each orbit must be constant across that orbit. Rigidity is not assumed but forced by the group action. If an agent treats structurally indistinguishable worlds as probabilistically equiv alent, then an y latent parameter gov erning the distribution must tak e the same value at all such w orlds. This condition deserves careful statement. For each orbit O of the stabilizer group, the directing measure Λ O (the latent probability la w go verning valuations within O ) is constant: it tak es the same v alue at ev ery w orld in O . This is not an assumption imposed for tractability but a necessary consequence of modal exchange- ability . As shown in Section 4 , Λ O is constructed from the G -in v ariant σ -field, and G -inv ariance forces constancy on orbits. Rigidity thus stands as the central structural consequence of combining exchangeabil- ity with accessibility structure. 1.5 Contributions The principal contribution is the concept of modal e xchangeability : in v ariance of a probability measure under the stabilizer of a reference world within the frame’ s automorphism group (Definition 2.9 ). W e show that this principle determines the form of permissible probability measures: • Under S5 , modal exchangeability recov ers the classical de Finetti representation with a single directing measure (Theorem 3.1 ). • Under S4 , it yields an orbit-wise decomposition in which each orbit carries its own rigid directing measure (Theorem 3.2 ). • Rigidity—constancy of the directing measure across an orbit—is forced by symmetry , not assumed. It connects to Kripke’ s rigid designators and to hyperintensional semantics (Section 5 ). The S4/S5 distinction thus directly controls whether the probabilistic representation is homogeneous or frag- mented. The main theorems are formally specializations of Crane and T o wsner’ s general relati ve e xchange- ability [ 3 ]; the contrib ution is the modal interpretation, which identifies rigidity as a named phenomenon and connects probabilistic representation to modal distinctions not visible at the le vel of general relational structures. The proofs require only dependent choice (ZF + DC), not full A C. 3 1.6 Paper outline Section 2 establishes the formal framework. Section 3 states the main theorems. Section 4 provides proofs. Section 5 interprets rigidity , connects to hyperintensionality , and giv es examples. Section 7 concludes. 2 F ormal Framework W e establish the technical setting: modal frames, their automorphisms, v aluations, and modal exchangeabil- ity . 2.1 Modal frames Definition 2.1 (Modal frame) . A modal frame is a pair ⟨ W , R ⟩ where W is a non-empty set of worlds and R ⊆ W × W is an accessibility relation . W e focus on two classes of frames: Definition 2.2 (S4 and S5 frames) . An S4 frame is a modal frame where R is reflexi ve and transitive (a preorder). An S5 frame is a modal frame where R is reflexi v e, symmetric, and transitive (an equi v alence relation). Throughout, we assume W is countable. This ensures the space of valuations is a standard Borel space, enabling measure-theoretic arguments without strong choice principles. Definition 2.3 (Accessible cluster) . For w 0 ∈ W , the accessible cluster from w 0 is: R cl ( w 0 ) : = { w ∈ W : R ( w 0 , w ) } In an S5 frame, accessible clusters are equi valence classes. In an S4 frame, they are upward-closed sets in the preorder . 2.2 A utomorphisms and stabilizers Definition 2.4 (Frame automorphism) . An automorphism of ⟨ W , R ⟩ is a bijection π : W → W such that: R ( w , v ) ⇐ ⇒ R ( π ( w ) , π ( v )) for all w , v ∈ W Let Aut ⟨ W , R ⟩ denote the group of automorphisms under composition. Automorphisms are the “symmetries” of the frame: permutations that preserve the modal structure. For S5 frames, automorphisms permute within equi valence classes. For S4 frames, they preserve the preorder . 4 Definition 2.5 (Stabilizer) . The stabilizer of w 0 ∈ W is: G : = Stab w 0 = { π ∈ Aut ⟨ W , R ⟩ : π ( w 0 ) = w 0 } The stabilizer G is a subgroup of Aut ⟨ W , R ⟩ consisting of automorphisms that fix the reference world. Note that G permutes R cl ( w 0 ) : if π ∈ G and w ∈ R cl ( w 0 ) , then R ( w 0 , w ) implies R ( π ( w 0 ) , π ( w )) = R ( w 0 , π ( w )) , so π ( w ) ∈ R cl ( w 0 ) . Definition 2.6 (Orbits) . The G-orbits in R cl ( w 0 ) are the equi v alence classes under the relation: v ∼ v ′ ⇐ ⇒ ∃ π ∈ G : π ( v ) = v ′ W e write O = { O 1 , O 2 , . . . } for the set of orbits. In an S5 frame where G acts transitiv ely on R cl ( w 0 ) , there is a single orbit. In general S4 frames, there may be multiple orbits, reflecting the finer structure imposed by directed accessibility . 2.3 V aluations and probability measur es Definition 2.7 (V aluation space) . Let L = { ℓ 1 , . . . , ℓ k } be a finite set of propositional atoms. A valuation on W is a function: V : W → { 0 , 1 } L The valuation space is Ω = ( { 0 , 1 } L ) W , equipped with the product σ -field F . Since W is countable and L is finite, ( Ω , F ) is a standard Borel space. Automorphisms act on v aluations by permuting worlds: Definition 2.8 (Automorphism action on v aluations) . For π ∈ Aut ⟨ W , R ⟩ and V ∈ Ω : ( π · V )( w ) : = V ( π − 1 ( w )) This action is measurable: π : Ω → Ω preserves the product σ -field. 2.4 Modal exchangeability W e now define the central symmetry condition. Definition 2.9 (Modal exchangeability) . A probability measure P on ( Ω , F ) is modally exc hangeable at w 0 if P is G -in v ariant: P ( π − 1 ( A )) = P ( A ) for all π ∈ G , A ∈ F Equi valently , for V ∼ P : the distrib utions of V and π · V coincide for all π ∈ G . 5 Modal exchangeability captures the principle that an agent has no probabilistic basis to distinguish between worlds that are “structurally identical” from w 0 ’ s perspecti ve. If π ∈ G , then π permutes accessible worlds while preserving the frame structure and fixing w 0 ; a rational agent should treat V and π · V as equiprobable. Remark (Connection to classical exchangeability) . When R is an equiv alence relation ( S5 ) and G contains all finite permutations of R cl ( w 0 ) , modal exchangeability reduces to classical exchangeability of the sequence { V ( w ) } w ∈ R cl ( w 0 ) . 2.5 Structural conditions W e identify conditions under which the representation takes a clean form. Definition 2.10 (Point-homogeneity) . The frame is point-homogeneous at w 0 if G acts transiti vely on R cl ( w 0 ) : for any v , v ′ ∈ R cl ( w 0 ) , there exists π ∈ G with π ( v ) = v ′ . Point-homogeneity means there is a single orbit. This holds for S5 frames when the equi v alence class is “structureless” beyond its cardinality . Definition 2.11 (Extension property) . The stabilizer G has the finite-support e xtension pr operty (Ext) if: for any finite, pairwise-disjoint subsets F 1 , . . . , F m ⊆ R cl ( w 0 ) and F ′ 1 , . . . , F ′ m ⊆ R cl ( w 0 ) with | F i | = | F ′ i | for all i , there exists π ∈ G such that π ( F i ) = F ′ i for all i . Remark (When (Ext) holds; link to ultrahomogeneity) . Property (Ext) is the modal-structure analogue of fi- nite ultrahomog eneity : any isomorphism between finite induced substructures of an orbit O (in the signature induced by the frame, e.g. the restriction of R to O ) extends to an automorphism of O . By Fraïssé’ s theorem, a countable relational structure is ultrahomogeneous if and only if it is the Fraïssé limit of an amalgamation class (equi valently , of its age) [ 15 – 17 ]. For modal frames, this yields a clean suf ficient criterion : an orbit O satisfies (Ext) whenever the relational structure induced on O by the accessibility relation is a Fraïssé limit. The simplest case arises when the induced subframe on O carries only reflexiv e accessibility (a discrete preorder), so that G | O = Sym ( O ) and (Ext) holds trivially . If instead O inherits a dense linear order without endpoints, then O ∼ = ( Q , < ) , the Fraïssé limit of finite linear orders, and ultrahomogeneity (hence (Ext)) follo ws [ 15 , 16 ]. More generally , whene ver the age of the induced structure on O satisfies the hereditary , joint embedding, and amalgamation properties, O is a Fraïssé limit and (Ext) holds [ 15 – 17 ]. The failure of (Ext) corresponds to the presence of additional rigidifying structure on O , that is, relations that distinguish locations within the orbit and obstruct amalgamation. In such cases, the representation theorem still applies at the lev el of orbit-wise mixtures, but the classical conclusion of conditional i.i.d. can weaken: we retain identical marginal distributions within the orbit (Lemma 4.2 ) without full exchangeability and hence without the conditional-independence conclusion that exchangeability underwrites. In this sense, (Ext) marks the boundary between statistical homogeneity (within-orbit i.i.d. giv en a directing law) and mere mar ginal indistinguishability induced by G -in v ariance. 6 For finite orbits, (Ext) reduces to G | O = Sym ( O ) ; the Fraïssé machinery is essential mainly in the countably infinite case, where amalgamation ensures suf ficiently many automorphisms e xist. Property (Ext) ensures that G contains suf ficiently many permutations to make modal exchangeability equiv- alent to classical exchangeability within orbits. It corresponds to the disjoint amalgamation property in model theory . 3 Main Repr esentation Theorems W e state the main results: representation theorems for modally e xchangeable measures under S5 and S4 . 3.1 The S5 case: Recov ery of classical de Finetti Theorem 3.1 (Modal de Finetti: S5) . Let ⟨ W , R ⟩ be an S5 fr ame with R an equivalence r elation. Let P be a modally exc hangeable pr obability measure on Ω . Assume: 1. The frame is point-homog eneous at w 0 (single orbit) 2. The stabilizer G has pr operty (Ext) Then ther e exists a pr obability measur e µ on the space M ( { 0 , 1 } L ) of pr obability measur es on { 0 , 1 } L such that: P ( V ∈ A ) = Z M P Λ ( V ∈ A ) µ ( d Λ ) wher e P Λ denotes the pr oduct measur e under which { V ( w ) } w ∈ R cl ( w 0 ) ar e i.i.d. with common law Λ . Theorem 3.1 shows that the modal framew ork recovers the classical de Finetti representation in the homoge- neous S5 case: when the frame imposes no asymmetric structure, the classical result carries ov er unchanged. The substanti vely new content arises under S4 , where the preorder structure breaks the single-orbit assump- tion. 3.2 The S4 case: Orbit-wise decomposition When the frame is S4 but not S5 , the accessible cluster may decompose into multiple orbits. This yields a fragmented structure: each orbit is go verned by its own rigid parameter , and observ ations do not pool across orbit boundaries without additional assumptions. Theorem 3.2 (Modal de Finetti: S4 / Orbit-wise decomposition) . Let ⟨ W , R ⟩ be an S4 frame and P a modally exc hangeable pr obability measur e on Ω . Let O = { O 1 , O 2 , . . . } denote the G-orbits in R cl ( w 0 ) . Assume that for each orbit O ∈ O , the r estricted action of G on O has pr operty (Ext) . Then ther e e xist random pr obability measures { Λ O : O ∈ O } on { 0 , 1 } L such that: 7 1. W ithin-orbit i.i.d.: F or each O ∈ O , conditional on Λ O , the valuations { V ( w ) } w ∈ O ar e i.i.d. with law Λ O . 2. Cross-orbit structure: Conditional on the G-in variant σ -field F G (equivalently , on Ξ ), the restric- tions V | O ar e i.i.d. within each orbit with dir ecting law Λ O ( Ξ ) . Cr oss-orbit dependence is not ruled out by G-in variance and may be mediated thr ough Ξ , equivalently thr ough the joint distribution of { Λ O } O ∈ O . 3. Rigidity: Each Λ O is F G -measurable: for any π ∈ G with π ( O ) = O, Λ O ( π · V ) = Λ O ( V ) P-a.s. In particular , Λ O does not depend on the choice of world within O. The orbit-wise decomposition captures the essential phenomenon: the accessible cluster fragments into regions, each with its o wn rigid governing parameter . W orlds within an orbit are probabilistically indistin- guishable, but worlds in dif ferent orbits can hav e entirely dif ferent statistical properties. The rigid parame- ters { Λ O } are the “local laws” governing each region; their joint distrib ution (mediated by Ξ ) may encode arbitrary cross-orbit dependencies. The rigidity condition (3) is the principal departure from the classical setting. It asserts that the directing measure for each orbit is G -inv ariant: it cannot depend on which world within the orbit we “observe from. ” This follo ws from the G -in v ariance of P , not from any additional assumption. Remark (Relation to Crane–T owsner) . Theorem 3.2 can be viewed as a specialization of Crane and T ows- ner’ s general representation theorem for relativ ely exchangeable structures [ 3 ]. In their framework, G - in v ariance on a relational structure yields a representation in terms of the structure’ s “definable closure. ” For modal frames, this specializes to the orbit decomposition, with rigidity corresponding to the directing measure being constant on definable equi valence classes. The present work interprets this general machinery in specifically modal terms, connecting the representa- tion to the S4/S5 distinction central to epistemic and temporal logic; pro vides constructi ve proofs using only ZF + DC, whereas Crane and T owsner work in full ZFC; and identifies rigidity as a named phenomenon with epistemic and hyperintensional significance (see Section 5 ). Corollary 3.3 (Bernoulli parameterization) . Under the conditions of Theor em 3.2 , if the L coordinates ar e conditionally independent within each world given Λ O , then: Λ O = k O ℓ = 1 Bernoulli ( Θ O ,ℓ ) for rigid parameter s Θ O = ( Θ O , 1 , . . . , Θ O , k ) ∈ [ 0 , 1 ] L . 3.3 Comparison: S5 vs. S4 The theorems re veal ho w modal structure af fects probabilistic representation: 8 S5 (Homogeneous) S4 (Orbit-wise) Orbits Single Multiple Parameters One Λ One Λ O per orbit Cross-orbit structure N/A Unconstrained (mediated by Ξ ) Rigidity Single parameter Per -orbit constraint Structure Unified Fragmented Under S5 , the group action is maximal: all accessible worlds are interchangeable, yielding a single directing measure as in classical de Finetti. Under S4 , the preorder structure fra gments the accessible cluster into distinct epistemic regions. Each orbit carries its own rigid parameter; dependence across orbits is permitted but not determined by in v ariance, and information transfer across orbit boundaries depends on the joint prior ov er { Λ O } . The orbit-wise decomposition is forced by the frame constraints; it is not a modeling choice b ut a consequence of the weaker group action a v ailable in S4 frames. 4 Pr oofs 4.1 Proof strategy Both theorems follow a common strate gy . W e first disinte grate P o ver the G -in variant σ -field F G to e xtract latent structure, then use the orbit decomposition to analyze the structure orbit-by-orbit, apply the classi- cal Hewitt–Sa vage exchangeability theorem within each orbit (using property (Ext)), and finally establish rigidity from F G -measurability . The proofs use only ZF + DC. Disintegration on standard Borel spaces requires dependent choice [ 7 ], or- bit enumeration requires only countable choice (at most | W | orbits), and the Hewitt–Sa v age theorem for countable sequences uses countable choice [ 10 ]. 4.2 Disintegration over the in variant σ -field Lemma 4.1 (Ergodic decomposition) . Let P be modally exchang eable. Ther e exists a F G -measurable random variable Ξ and a re gular conditional pr obability P ( ·| Ξ ) suc h that: 1. P = R P ( ·| Ξ = ξ ) P Ξ ( d ξ ) 2. F or each ξ , the measur e P ( ·| Ξ = ξ ) is G-er godic. Pr oof. Since ( Ω , F ) is a standard Borel space (countable W , finite L ), disinte gration ov er the sub- σ -field F G exists by [ 7 , Theorem 452I]. This requires only dependent choice. The ergodic decomposition is the conditional probability gi ven F G . Each fiber P ( ·| Ξ = ξ ) is G -ergodic: the only G -in v ariant e vents ha ve conditional probability 0 or 1. 9 4.3 Within-orbit structur e Lemma 4.2 (Identical distribution within orbits) . F or any v , v ′ in the same G-orbit: L ( V ( v ) | Ξ ) = L ( V ( v ′ ) | Ξ ) P-a.s. Pr oof. By definition of orbits, there e xists π ∈ G with π ( v ) = v ′ . By G -in variance of P : P ( V ( v ) ∈ B | Ξ ) = P ( V ( π − 1 ( v ′ )) ∈ B | Ξ ) = P (( π · V )( v ′ ) ∈ B | Ξ ) Since π · V d = V under P and Ξ is F G -measurable (hence π -in v ariant), this equals P ( V ( v ′ ) ∈ B | Ξ ) . Lemma 4.3 (Exchangeability within orbits) . F ix an orbit O ∈ O and assume G | O has pr operty (Ext) . Then, for P-a.e. r ealization of Ξ , the conditional distribution of { V ( w ) } w ∈ O is exc hangeable . Pr oof. Property (Ext) ensures that for any finite permutation σ of O , there exists π ∈ G with π | O = σ . By G -ergodicity of P ( ·| Ξ ) , the conditional distribution is in v ariant under all such permutations. This is exchangeability . 4.4 Proof of Theor em 3.1 Pr oof. Under point-homogeneity , R cl ( w 0 ) is a single G -orbit. By Lemma 4.3 , the conditional distrib ution of { V ( w ) } w ∈ R cl ( w 0 ) gi ven Ξ is e xchangeable. By the He witt–Sav age theorem [ 9 ], there exists a directing measure Λ ( Ξ ) on { 0 , 1 } L such that, conditional on Ξ , the v aluations are i.i.d. with law Λ ( Ξ ) . Define µ as the distribution of Λ ( Ξ ) . Then: P ( V ∈ A ) = Z P Λ ( ξ ) ( V ∈ A ) P Ξ ( d ξ ) = Z P Λ ( V ∈ A ) µ ( d Λ ) 4.5 Proof of Theor em 3.2 Pr oof. Step 1: Orbit-wise r epr esentation. F or each orbit O ∈ O , Lemma 4.3 gi ves exchangeability of { V ( w ) } w ∈ O conditional on Ξ . By He witt–Sa v age, there e xists a directing measure Λ O ( Ξ ) on { 0 , 1 } L such that { V ( w ) } w ∈ O are conditionally i.i.d. with law Λ O ( Ξ ) . Step 2: Cr oss-orbit structur e. No automorphism in G maps one orbit to another (by definition of orbits). Hence G -in v ariance yields exchangeability within each orbit separately , but imposes no constraint forcing independence across orbits. An y cross-orbit coupling is encoded in Ξ (equiv alently , in the joint distribution 10 of { Λ O } O ∈ O ). The orbit-wise i.i.d. structure holds conditional on Ξ ; the marginal joint distribution of { Λ O } may exhibit arbitrary dependence. Step 3: Rigidity . Each Λ O is constructed from Ξ , which is F G -measurable. Hence Λ O is F G -measurable. By definition of F G , for any π ∈ G : Λ O ( π · V ) = Λ O ( V ) P -a.s. In particular , Λ O does not depend on which world w ∈ O we “observe from”; it is constant across the orbit. This is rigidity . 5 The Rigidity Phenomenon Rigidity is the central structural consequence of modal exchangeability on S4 frames. This section e xplains why it arises, what it rules out, and ho w it connects to hyperintensional semantics. 5.1 The logical structure of rigidity Rigidity is forced by in v ariance through a short chain of observ ations. The G -in variance of P is the sole premise, and the directing measure Λ O is determined by the distrib ution of { V ( w ) } w ∈ O . For an y π ∈ G with π ( O ) = O , G -in variance implies that Law ( { V ( π ( w )) } w ∈ O ) = Law ( { V ( w ) } w ∈ O ) , so Λ O is the same whether computed from { V ( w ) } or from { V ( π ( w )) } . Formally , Λ O is F G -measurable: Λ O ( π · V ) = Λ O ( V ) P -a.s. for all π ∈ G . The directing measure is thus a G -in variant of the orbit, constant across all worlds in O . 5.2 What rigidity rules out W ithout the in v ariance requirement, one could assign world-r elative parameters: a different Λ w for each w ∈ R cl ( w 0 ) . Rigidity eliminates this within orbits: probabilistically indistinguishable worlds must share a common governing parameter . Across orbits, parameters can dif fer: if O 1 , O 2 are distinct orbits, no π ∈ G maps one to the other , so the group action imposes no constraint relating Λ O 1 to Λ O 2 . This yields a precise sense in which the orbit-wise factorization of Theorem 3.2 is maximal : the directing measure Λ O is an intrinsic property of the orbit (analogous to a geometric in variant), and further factorization into a single parameter go verning all orbits is impossible without assumptions be yond modal e xchangeabil- ity . 5.3 Rigidity and hyperintensionality Rigidity connects naturally to hyperintensional semantics, that is, frame works that distinguish necessarily equi valent propositions based on structural or conceptual dif ferences [ 12 , 13 ]. In standard possible-worlds 11 semantics, a proposition is identified with the set of worlds where it is true, creating the granularity pr oblem : necessarily equi v alent propositions cannot be distinguished, yet agents manifestly hold different attitudes to- ward them. Hyperintensional semantics addresses this by enriching the semantic framew ork with impossible worlds, topic structures, or subject-matter constraints [ 12 , 13 ]. The orbit decomposition provides a pr obabilistic implementation of hyperintensional distinctions. Proposition 5.1 (Hyperintensional discrimination) . Let O 1 , O 2 be distinct orbits in R cl ( w 0 ) . Even if pr opo- sitions concerning O 1 and O 2 ar e necessarily equivalent (hold at the same worlds within each orbit), an agent can maintain dif fer ent cr edences about them, captur ed by distinct rigid parameters Λ O 1  = Λ O 2 . Pr oof. By Theorem 3.2 , no element of G maps O 1 to O 2 , so G -inv ariance imposes no constraint relating Λ O 1 to Λ O 2 . The joint prior ov er ( Λ O 1 , Λ O 2 ) may assign arbitrary (and distinct) mar ginals. Hence an agent can be (say) nearly certain about the bias in O 1 while remaining maximally uncertain about O 2 , without violating modal exchangeability . This connects to Holliday’ s work on hyperintensional belief [ 12 ] and Berto’ s analysis of topic-sensiti ve credence [ 14 ]. In their framew orks, credences can differ across “subject matters” ev en when the propositions are logically equi v alent. The orbit decomposition provides the formal structure: orbits correspond to topics, and rigid parameters to topic-specific credences. Rigidity ensures coherence within a topic; the absence of symmetry constraints across orbits ensures flexibility acr oss topics. 6 Examples W e illustrate the theorems with explicit examples. 6.1 Example 1: S5 with symmetric accessibility Example 6.1 (Classical exchangeability reco vered) . Let W = { w 0 } ∪ N , with R the uni v ersal relation (all worlds mutually accessible). This is an S5 frame. Let L = { p } (single atom). The stabilizer G = Stab w 0 consists of all permutations of N (fixing w 0 ). The accessible cluster is R cl ( w 0 ) = N , and G acts transiti vely with property (Ext). By Theorem 3.1 , any modally e xchangeable P has the form: P ( V ( 1 ) = x 1 , . . . , V ( n ) = x n ) = Z 1 0 θ k ( 1 − θ ) n − k µ ( d θ ) where k = ∑ n i = 1 x i and µ is the de Finetti measure on [ 0 , 1 ] . This is the classical binary de Finetti representation. 12 6.2 Example 2: S4 with two clusters Example 6.2 (T wo-orbit structure) . Let W = { w 0 , a 1 , a 2 , . . . , b 1 , b 2 , . . . } . The accessibility relation is defined as follows: w 0 accesses itself and all other worlds; each a i and b j accesses itself; the a -worlds are mutually accessible ( R ( a i , a j ) for all i , j ); and there is no accessibility from any a i to any b j or vice versa. This is an S4 frame. The accessible cluster is R cl ( w 0 ) = { a 1 , a 2 , . . . } ∪ { b 1 , b 2 , . . . } . The stabilizer G consists of automorphisms fixing w 0 . Since a -w orlds are mutually accessible b ut not acces- sible to b -worlds, an y π ∈ G must map { a i } to { a i } and { b j } to { b j } . Hence: G ∼ = Sym ( { a i } ) × Sym ( { b j } ) There are two orbits: O a = { a 1 , a 2 , . . . } and O b = { b 1 , b 2 , . . . } . By Theorem 3.2 , any modally e xchangeable P decomposes as: P = Z P Λ a , Λ b µ ( d Λ a , d Λ b ) where, conditional on ( Λ a , Λ b ) , the v aluations { V ( a i ) } i ≥ 1 are i.i.d. with law Λ a and { V ( b j ) } j ≥ 1 are i.i.d. with la w Λ b . Both Λ a and Λ b are rigid on their respecti v e orbits. Their joint distribution is unconstrained by symmetry; dependence between Λ a and Λ b is permitted through Ξ . The agent has two biases: one for a -w orlds, one for b -worlds. Observations at a -worlds inform Λ a ; whether they also inform Λ b depends on the joint prior ov er ( Λ a , Λ b ) , which is not constrained by symmetry . 6.3 Example 3: T emporal structur e Example 6.3 (P ast-future asymmetry) . Consider a temporal frame: W = Z (discrete time), with R ( t , t ′ ) ⇐ ⇒ t ≤ t ′ (the future is accessible from the past). Fix w 0 = 0. Then R cl ( 0 ) = { 0 , 1 , 2 , . . . } = N . The stabilizer G = Stab 0 consists of order -preserving permutations of N fixing 0. If we require automor - phisms to also preserve the full structure (including w orlds before 0), then G is sev erely restricted. Suppose G consists only of the identity (a rigid frame). Then every world is its own orbit: O = {{ 0 } , { 1 } , { 2 } , . . . } . By Theorem 3.2 , each time t has its own rigid parameter Λ t ; no nontrivial symmetry constraints apply , and no pooling across time steps is forced. Modal exchangeability imposes no constraint beyond consistency of the measure; this is the degenerate case where symmetry is tri vial. This illustrates that modal structure matters: more symmetry (larger G ) yields stronger representation theo- rems. 13 6.4 Example: The Principal Principle as orbit-wise rigidity of chance Le wis’ Principal Principle (PP , 11 ) is often stated informally as: an agent’ s credence in an event should match the event’ s objectiv e chance, conditional on admissible information. In a minimal form, for an event A and objecti ve chance function Ch , P ( A | Ch ( A ) = p , I ) = p , where I is admissible background information (roughly: information that does not itself “screen of f ” the chancy setup). Our representation theorems pro vide a natural implementation of PP in modal settings. Fix a base world w 0 and let O be the set of G = Stab w 0 -orbits in the accessible cluster R cl ( w 0 ) . Under modal exchange- ability , Theorem 3.2 yields orbit-wise directing laws { Λ O } O ∈ O and, in the Bernoulli parameterization, rigid parameters Θ O = ( Θ O ,ℓ ) ℓ ∈ L such that conditional on Θ O the v aluations { V ( w ) } w ∈ O are i.i.d. Interpret Θ O ,ℓ as the objective chance of atom ℓ within orbit O . Then PP becomes an immediate consequence of the conditional i.i.d. structure: for any w ∈ O and any atom ℓ ∈ L , P  V ( w ) ℓ = 1   Θ O ,ℓ = p , Ξ  = p , where Ξ is the G -in variant latent variable from Lemma 4.1 . In words: conditional on the orbit-wise chance parameter , an agent’ s credence matches that chance. This e xample also clarifies the role of modal structure. In the S5 (homogeneous) case, there is a single orbit and hence a single rigid chance parameter gov erning the entire accessible cluster, reco vering the f amiliar “global” reading of PP . In general S4 frames, the orbit decomposition yields local chance parameters Θ O that are rigid within orbits but may dif fer across orbits, corresponding to regime-dependent objectiv e chances not identified by modal expressibility alone. 7 Conclusion 7.1 Summary W e hav e established representation theorems for modally exchangeable probability measures on Kripke frames. Under S5 , modal exchangeability recovers the classical de Finetti representation with a single di- recting measure (Theorem 3.1 ). Under S4 , it yields orbit-wise i.i.d. representations with rigid directing measures, where cross-orbit dependence is unconstrained by symmetry (Theorem 3.2 ). Rigidity , the con- stancy of directing measures on orbits, is the central structural consequence. It emerges from symmetry alone and provides a representation-theoretic foundation for hyperintensional probability , connecting the S4/S5 distinction to topic-sensiti ve credence and fine-grained belief. 14 7.2 Implications f or learning The orbit structure determines what can be inferred from observ ation. Under modal exchangeability , ob- serv ations within orbit O update the directing measure Λ O but are uninformativ e about Λ O ′ for O ′  = O ; cross-orbit transfer depends on the joint prior o ver orbit-specific parameters, which symmetry alone does not determine. Under S5 (single orbit), all observations pool; under S4 (multiple orbits), learning is local by default. Hierarchical priors coupling orbit parameters along the accessibility relation can encode directed information transfer , b ut this is a modeling choice orthogonal to the representation theorem. 7.3 Scope and limitations This work addresses representation, not decision-making or dynamic learning. W e characterize the structure of modally exchangeable measures but do not dev elop decision procedures or analyze con ver gence rates in detail. The assumption of countable W and finite L ensures standard Borel structure, enabling constructi ve proofs. Extensions to uncountable settings would require stronger choice principles and dif ferent techniques. 7.4 Future dir ections Se veral extensions merit in vestigation. In dynamic modal logic, one may ask ho w representation theo- rems interact with belief revision and dynamic epistemic updates, a nd whether there exist “exchangeability- preserving” update rules. Our results concern measures on v aluations rather than modal formulas directly; de veloping a probabilistic semantics in which rigidity appears as a semantic constraint on formula probabil- ities is a natural ne xt step. On the algebraic side, relaxing full in variance under the stabilizer to approximate or partial exchangeability (inv ariance under a subgroup) would yield weaker representations, potentially with approximate rigidity . Finally , formal analysis of sample complexity and con v ergence under modal exchangeability w ould connect the present framew ork to P A C learning and statistical learning theory . 7.5 Concluding remarks De Finetti’ s theorem shows that permutation in variance forces a mixture-of-i.i.d. structure. W e have sho wn that modal in v ariance, under accessibility-preserving automorphisms, forces a finer structure: orbit-wise mixtures with rigid directing measures. The central lesson is that the accessibility structur e of a Kripke fr ame determines the pr obabilistic struc- tur e of exc hangeable measur es defined on it . 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