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arxiv: 2606.31861 · v1 · pith:2J34RHN7new · submitted 2026-06-30 · 💻 cs.LO · cs.AI

Belief Contraction in Dynamic Epistemic Logic

Gaia Belardinelli (Stanford University) , Snow Zhang (University of Berkeley , California) This is my paper

Pith reviewed 2026-07-01 02:25 UTC · model grok-4.3

classification 💻 cs.LO cs.AI
keywords belief contractiondynamic epistemic logicKripke modelsreduction axiomshedged announcementsprivate announcementsdoxastic accessibility
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The pith

Belief contraction is modeled by a direct transformation on standard Kripke models without accessibility constraints.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Standard dynamic epistemic logic can expand beliefs by eliminating possibilities but lacks a way to contract beliefs in factual propositions. Adding plausibility orderings to models has expressive shortcomings, such as failing to represent beliefs that violate positive introspection or contractions triggered by hedged announcements. The authors therefore introduce a contraction mechanism that works directly on ordinary Kripke models by modifying the doxastic accessibility relation. They examine which contraction properties this mechanism preserves, identify cases of unsuccessful contraction, and supply reduction axioms that give a sound and complete logic. The work also extends to a general dynamic logic that handles private and semi-private contraction events with its own sound and complete axiomatization.

Core claim

We introduce a mechanism for belief contraction defined directly on standard Kripke models, without any constraints on the doxastic accessibility relation. This mechanism satisfies some of the standard properties of belief contraction but not others. We study the conditions under which contraction may be unsuccessful and provide a sound and complete axiomatization of the logic via reduction axioms. We also define a more general dynamic logic that is an extension of standard DEL and accommodates belief contractions due to events such as private or semi-private announcements, and provide a complete and sound axiomatization of the general logic.

What carries the argument

The contraction mechanism that directly transforms the doxastic accessibility relation on standard Kripke models to effect belief contraction.

Load-bearing premise

That the limitations of plausibility-enriched models are real and that contraction dynamics can be captured accurately by a transformation on unconstrained Kripke models.

What would settle it

Finding a specific Kripke model and hedged announcement where the defined contraction does not produce the expected belief change, or discovering a valid formula not derivable from the given reduction axioms.

Figures

Figures reproduced from arXiv: 2606.31861 by California), Gaia Belardinelli (Stanford University), Snow Zhang (University of Berkeley.

Figure 1
Figure 1. Figure 1: Left: The Kripke model M = (W,R,V) representing Alice’s initial belief state. An edge from a world v to a world w means w ∈ Ra(v). We have M ⊨ Ba¬p∧¬Ba p. Middle: The standard event model E = (E,Q, pre) representing Bob’s announcement. The event e contains its own precondition pre(e) = p. The loop on e means e ∈ Qa(e). Right: The Kripke model M ⊗E = (WE ,R E ,V E ) representing Alice’s belief state after t… view at source ↗
Figure 2
Figure 2. Figure 2: Left: The plausibility model M = (W,⪯,V) representing Alice’s initial belief state. An edge from v to w means w ⪯a v. We have M ⊨ Ba¬p∧ ¬Ba p. Middle: The plausibility event model E = (E,≤, pre) capturing a soft update that p holds. An edge from f to e means e ≤a f . Right: The plausibility model M ∗E representing Alice’s belief state after updating, with M ∗E ⊨ Ba p∧ ¬Ba¬p. 3.1 Limitations of plausibility… view at source ↗
Figure 3
Figure 3. Figure 3: Left: The plausibility model M = (W,⪯,V) representing Alice’s initial belief state, where M ⊨ Ba¬p∧¬Ba p. Conventions for edges are as in [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Left: The Kripke model M (same as [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Alice’s belief update as a contraction on [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Left: The Kripke model M representing Alice’s initial belief states. Middle: The generalized event model E = (E,Q,Q +, pre) representing Bob’s announcement to Alice that p might hold. An edge from an event f to an event h labeled by +a means that h ∈ Q + a (f). We omit the Q-edges since Qa(e) = E for all e ∈ E. Right: The Kripke model M ⊗E = (WE ,R E ,V E ) representing Alice’s belief states after Bob’s an… view at source ↗
Figure 7
Figure 7. Figure 7: Left: The Kripke model M representing Alice and Clark’s initial belief states. We have M ⊨ Ba¬p∧Bc¬p∧BcBa¬p. Middle: The generalized event model E = (E,Q,Q +, pre) representing Bob’s private announcement to Alice that p might hold. An edge from an event f to an event h labeled by +a means that h ∈ Q + a (f). If an edge’s label does not contain +, then the edge is a Q-relation, e.g. e ∈ Qc(f). Right: The Kr… view at source ↗
Figure 8
Figure 8. Figure 8: Left: The Kripke model M representing Alice and Clark’s initial belief states. We have M ⊨ Bc(Ba p∨Ba¬p)∧Bc(Baq∨Ba¬q). Right: The Kripke model N representing Alice and Clark’s belief states after Bob’s hedged announcement, where N ,w ⊨ ¬Bc(Ba p∨Ba¬p)∧Bc(Baq∨Ba¬q). Proposition 7.10. There is no formula ϕ ∈ L ÷ such that M÷ϕ ≡ N . Proof. Suppose towards a contradiction that M÷ϕ ≡ N for some ϕ ∈ L ÷. Since ev… view at source ↗
Figure 9
Figure 9. Figure 9: The generalized event model E = (E,Q,Q +, pre) representing Bob’s announcement to Alice and Clark. As before, we omit the Q-edges, since Qi(x) = E for all events x ∈ E and agents i ∈ {a, c}. Each event’s precondition is given by the conjunction of the literals listed inside the event. So for example event e has precondition p∧q [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
read the original abstract

Dynamic epistemic logic represents belief change via model transformations induced by epistemic events. Its standard formulation (Baltag, Moss, Solecki, 1998) provides a natural account of belief expansion through the elimination of possibilities, but it cannot model belief contraction about factual propositions. A classic response enriches Kripke models with plausibility orderings, representing contraction as an update that promotes certain possibilities over others. We show that this approach has expressive limitations. In particular, the approach cannot model belief that violates positive introspection and contraction dynamics in response to a hedged public announcement that phi might be false. Motivated by these considerations, we introduce a mechanism for belief contraction defined directly on standard Kripke models, without any constraints on the doxastic accessibility relation. We show that it satisfies some of the standard properties of belief contraction but not others, study the conditions under which contraction may be unsuccessful, and provide a sound and complete axiomatization of the logic via reduction axioms. We also define a more general dynamic logic that is an extension of standard DEL and accommodates belief contractions due to events such as private or semi-private announcements, and provide a complete and sound axiomatization of the general logic.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that plausibility-enriched models have expressive limitations for representing belief contraction in dynamic epistemic logic, specifically failing to capture beliefs that violate positive introspection or the effects of hedged public announcements. It introduces a contraction operator defined directly on standard Kripke models with no constraints on the doxastic accessibility relation, shows that the operator satisfies some but not all standard contraction properties, identifies conditions under which contraction is unsuccessful, and supplies sound and complete axiomatizations via reduction axioms both for the core logic and for a generalized dynamic logic extending DEL to handle private and semi-private contractions.

Significance. If the central claims hold, the work is significant for dynamic epistemic logic because it offers a framework for contraction that avoids additional plausibility structure while still delivering reduction axioms. The explicit treatment of unsuccessful contraction and the extension to private announcements are useful contributions, and the provision of complete axiomatizations via reduction axioms is a clear technical strength that supports syntactic reasoning about the dynamics.

major comments (2)
  1. [§3] §3, definition of the contraction operator on the accessibility relation: the transformation must be shown not to implicitly enforce seriality or other frame properties on arbitrary input relations; otherwise the claimed advantage over plausibility models (which already impose orderings) is lost and the soundness of the reduction axioms for the base language becomes fragile.
  2. [§5] §5, soundness argument for the general dynamic logic: the reduction axioms for semi-private announcements are stated to be complete, but it is unclear whether they continue to hold when the contraction operator is applied to models that lack positive introspection; a concrete counter-model or preservation lemma is needed to support the claim that the general logic remains a conservative extension of standard DEL.
minor comments (2)
  1. Notation for the updated accessibility relation after contraction should be introduced once and used consistently; occasional reuse of R for both original and updated relations reduces readability.
  2. The discussion of unsuccessful contraction would benefit from an explicit example showing a concrete Kripke model in which the operator returns the original model.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and detailed report. The comments raise important technical points about frame properties and the scope of the soundness results. We address each major comment point by point below.

read point-by-point responses

  1. Referee: [§3] §3, definition of the contraction operator on the accessibility relation: the transformation must be shown not to implicitly enforce seriality or other frame properties on arbitrary input relations; otherwise the claimed advantage over plausibility models (which already impose orderings) is lost and the soundness of the reduction axioms for the base language becomes fragile.

    Authors: The contraction operator is defined to act on arbitrary accessibility relations without imposing additional constraints. Specifically, the transformation removes certain worlds from the set of accessible worlds but does not add accessibility links or enforce properties like seriality (which would require at least one accessible world). We will add an explicit lemma in the revised §3 proving that the operator preserves the absence of seriality, transitivity, and other properties from the input model. This maintains the claimed advantage over plausibility-enriched models and ensures the soundness of the reduction axioms holds for the unrestricted class of models. revision: yes

  2. Referee: [§5] §5, soundness argument for the general dynamic logic: the reduction axioms for semi-private announcements are stated to be complete, but it is unclear whether they continue to hold when the contraction operator is applied to models that lack positive introspection; a concrete counter-model or preservation lemma is needed to support the claim that the general logic remains a conservative extension of standard DEL.

    Authors: We agree that a preservation result is necessary to confirm the reduction axioms apply even to models without positive introspection. Since our framework explicitly allows arbitrary accessibility relations (which may violate positive introspection), we will include a preservation lemma in the revised §5 showing that the semantics of the contraction operator and the reduction axioms for semi-private announcements are preserved under such models. This lemma will also establish that the general logic is a conservative extension of standard DEL. If the referee prefers, we can instead exhibit a counter-model, but our preliminary verification indicates the axioms hold. revision: yes

Circularity Check

0 steps flagged

No circularity: new contraction operator defined directly on Kripke models with independent reduction-axiom axiomatization.

full rationale

The paper defines a belief contraction mechanism directly on unconstrained Kripke models, demonstrates selected AGM-like properties via explicit model transformation, identifies failure conditions, and supplies reduction axioms whose soundness and completeness are established relative to the base DEL semantics. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or definitional renaming; the cited Baltag-Moss-Solecki framework is external and the new operator is introduced by explicit construction rather than by re-labeling prior results. The derivation therefore remains self-contained against the standard DEL semantics.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the work relies on standard Kripke semantics and reduction axioms from dynamic epistemic logic without introducing fitted parameters or new postulated entities.

axioms (2)
  • standard math Kripke models with accessibility relations represent epistemic states
    Base structure invoked for defining belief and its updates.
  • standard math Reduction axioms characterize dynamic operators
    Used to obtain sound and complete axiomatizations.

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Reference graph

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