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Belief Contraction in Dynamic Epistemic Logic

This paper addresses the limitations of existing dynamic epistemic logic approaches to belief contraction by introducing a new mechanism defined directly on standard Kripke models that accommodates complex scenarios like hedged announcements and private events, while providing a sound and complete axiomatization for both the specific and generalized logics.

Original authors: Gaia Belardinelli (Stanford University), Snow Zhang (University of Berkeley, California)

Published 2026-07-01

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CC BY 4.0

Original authors: Gaia Belardinelli (Stanford University), Snow Zhang (University of Berkeley, California)

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine your mind as a library filled with books about how the world works. Some books are "facts" you are absolutely sure of (like "the sky is blue"), while others are just "possibilities" you keep on the shelves just in case (like "maybe it will rain tomorrow").

In the world of logic, scientists have long tried to build a perfect system to explain how we update these libraries when we get new information. This paper, titled "Belief Contraction in Dynamic Epistemic Logic," by Gaia Belardinelli and Snow Zhang, tackles a specific problem: What happens when someone tells you that something you are 100% sure of might be wrong?

Here is the story of their discovery, explained simply.

1. The Old Way: The "Hard" Library

For a long time, the standard way to model belief change was like a security guard at a library.

How it worked: If you told the guard, "The sky is not blue," the guard would immediately throw away every book in the library that said "The sky is blue."
The Problem: This works great for expanding your knowledge (adding new facts). But it fails miserably at contraction (letting go of a belief).
The Scenario: Imagine Alice is 100% sure she locked her office door. Bob walks up and says, "Hey, your door might be open."
- In the old "Hard" system, Bob's statement is confusing. It doesn't give a hard fact to throw books away, but it also doesn't let Alice keep being 100% sure. The old system couldn't model Alice saying, "Okay, I'm no longer 100% sure, but I don't know for sure it's open either." It just broke.

2. The "Soft" Attempt: The Plausibility Ranking

To fix this, other logicians tried a different approach. Instead of a security guard throwing books away, they imagined a ranking system.

How it worked: Every book in the library had a star rating. The books with 5 stars were the "most plausible" (what you believe). If Bob said, "The door might be open," the system would just take the "Door is open" book and move it up to the 5-star shelf, right next to the "Door is closed" book.
The Flaw: The authors found two big holes in this "Soft" system:
1. The "Know-It-All" Problem: This system forces you to always know exactly what you believe. If you believe something, the system assumes you know you believe it. But in real life, people can be wrong about their own beliefs (e.g., "I think I'm fair, but I actually have a bias"). The old system couldn't model this mistake.
2. The "Maybe" Problem: When Bob says, "It might be open," he isn't just moving a book up a shelf. He is opening a whole new section of the library that was previously locked. The "Soft" system couldn't handle this specific type of "hedged" announcement (where you aren't sure, but you're worried).

3. The New Solution: The "Expansion" Mechanism

Belardinelli and Zhang propose a brand new way to handle this. Instead of throwing books away or just re-ranking them, they suggest adding new books to the shelf.

The Metaphor: Imagine Alice's belief that the door is locked is a fortress.
- Old Logic: If you tell her the door is open, you smash the fortress.
- New Logic: When Bob says, "It might be open," Alice doesn't smash her fortress. Instead, she builds a new wing onto her library. She adds a "Maybe Open" section.
- The Rule: If she was already 100% sure the door was locked, she now adds every single possibility where the door is open into her library. She doesn't delete her old belief; she just admits, "Okay, I can't rule out the 'open' possibilities anymore."

4. What They Discovered

The authors built a new mathematical logic (a set of rules) to describe this "New Wing" approach. They called it HPAL (Hedged Public Announcement Logic).

Here are the key findings from their new system:

It Works for "Maybe": It perfectly models what happens when someone says, "This might be false." The agent stops being 100% sure and starts considering the opposite.
It Breaks Some Old Rules: In the old "Soft" systems, there were strict rules about how beliefs must behave (like "If I believe X, I must know I believe X"). The authors show that their new system breaks these rules. And that's actually good! It means their system can model real humans who are sometimes confused about their own beliefs.
The "Moore" Paradox: They looked at tricky sentences like "The door is open, but I don't believe it." In their new system, if you hear a "maybe" announcement about this, your belief state changes in a way that makes the sentence false. They figured out exactly which sentences survive this change and which ones don't.

5. The "General" Upgrade

Finally, they realized that life isn't just about public announcements. Sometimes, you tell a secret to one person, or a group of people hears something differently.

They created an even bigger system called GDEL (Generalized Dynamic Epistemic Logic).
Think of this as the Master Key. It can handle public announcements, private whispers, and "semi-private" hints.
They proved that this Master Key can simulate almost any kind of belief change, including the "Soft" ranking systems, but with much more flexibility.

The Bottom Line

This paper is about fixing a broken tool. The old tools for modeling belief change were too rigid—they could only handle "definite facts" or "re-ranking." The authors built a new tool that handles "maybe".

They showed that when we hear "It might be false," we don't just swap our beliefs; we expand our library to include new possibilities. Their new logic explains exactly how that expansion happens, even when it makes us a little less certain about what we know we know.

Technical Summary: Belief Contraction in Dynamic Epistemic Logic

Problem Statement
Standard Dynamic Epistemic Logic (DEL) models belief change primarily through the elimination of possibilities (hard updates), effectively capturing belief expansion but failing to model belief contraction regarding factual propositions. While a popular alternative enriches Kripke models with plausibility orderings to represent "soft" updates (promoting possibilities rather than eliminating them), this paper identifies two critical expressive limitations in the plausibility framework:

Positive Introspection: Because plausibility orderings are transitive, the framework validates axiom 4 ( $Ba\phi \to BaBa\phi$ ), rendering it incapable of modeling agents who are mistaken about their own beliefs.
Hedged Announcements: The framework cannot model belief contractions resulting from "hedged" public announcements (e.g., " $\phi$ might be false"). Such announcements intuitively cause an agent to suspend judgment on $\phi$ without necessarily adopting new factual beliefs. The paper demonstrates via Theorem 3.2 that no plausibility event model can transform a model where an agent believes $\neg p$ into a model where the agent suspends judgment on $p$ (neither believing $p$ nor $\neg p$ ) while maintaining modal equivalence.

Methodology
The authors propose a new framework for belief contraction defined directly on standard multi-agent Kripke models, without imposing constraints (such as transitivity or Euclideanness) on the doxastic accessibility relations.

Hedged Public Announcement Logic (HPAL): The core mechanism is a dynamic modality $[\div \phi]$ , representing a hedged public announcement that $\phi$ might be false. The update operation, denoted $M \div \phi$ , modifies the accessibility relations $R_a$ as follows:
- If an agent $a$ believes $\phi$ at world $w$ ( $M, w \models Ba\phi$ ), the agent's set of accessible worlds is expanded to include all worlds where $\neg \phi$ holds: $R^{\div \phi}_a(w) = R_a(w) \cup \{v \in W : M, v \models \neg \phi\}$ .
- If the agent does not believe $\phi$ , the accessibility relation remains unchanged.
  This approach treats contraction as an extension of the model (adding edges) rather than a restriction (removing worlds).
Generalized DEL (GDEL): To accommodate more complex scenarios, such as private or semi-private announcements, the authors introduce a generalized event model $E = (E, Q, Q^+, pre)$ . This extends standard event models by adding a relation $Q^+_a$ representing events that are introduced as possibilities by the event itself, distinct from $Q_a$ which represents possibilities not ruled out. The generalized product update $M \otimes E$ combines standard elimination (via $Q$ ) with the addition of new possibilities (via $Q^+$ ).

Key Contributions and Results

Axiomatization of HPAL: The paper provides a sound and complete axiomatization for the logic of hedged public announcements (Table 1). The system includes reduction axioms that allow dynamic formulas to be translated into static doxastic formulas. A key reduction axiom (A5) characterizes belief after contraction: an agent believes $\psi$ after contracting on $\phi$ if and only if they either did not believe $\phi$ initially and believe $\psi$ after the update, or they believed $\phi$ and $\psi$ holds in all newly added $\neg \phi$ -worlds.
Analysis of Preservation and Success:
- Preservation: The authors define a formula $\phi$ as preserved if $\phi \to [\div \neg \phi]\phi$ is valid. They show that while standard public announcements preserve universal formulas, hedged announcements preserve existential formulas (built from literals, $\land$ , $\lor$ , and $eBa$).
- Success: A formula is successful if it is not believed after a hedged announcement that it might be false ( $[\div \phi]\neg Ba\phi$ ). The paper proves that universal formulas are successful. Conversely, "Moorean" sentences like $p \land Ba\neg p$ are shown to be self-refuting and thus not preserved.
AGM Postulates: The paper evaluates standard AGM contraction postulates within this dynamic setting.
- Valid: Closure, Extensionality, and Consistency hold.
- Invalid: Success, Inclusion, and Vacuity do not hold unrestrictedly. For instance, an agent may acquire new beliefs about their own lack of belief (violating Inclusion) after a hedged announcement.
- Strong Vacuity: A strengthened version of Vacuity holds: if no agent believes $\phi$ anywhere in the model, the contraction has no effect.
Generalized DEL (GDEL) and Simulation: The paper establishes that GDEL is a conservative extension of standard DEL. A key theoretical result (Theorem 7.8) is that any generalized update produces a model that is a simulation of a refinement of the initial model. This formally captures the dual nature of the update: it can eliminate possibilities (refinement) and add possibilities (simulation).
Expressive Power: The authors demonstrate that GDEL can model dynamics that HPAL cannot. Specifically, Example 7.9 and Proposition 7.10 show that HPAL cannot model a scenario where a hedged announcement causes an agent to suspend judgment on one proposition ( $p$ ) while retaining a specific belief about another ( $q$ ). GDEL overcomes this by using $Q^+$ relations to selectively introduce new possibilities compatible with the agent's background knowledge.

Significance
The paper claims to offer a solution to the expressive limitations of plausibility-based approaches to belief contraction. By defining contraction directly on standard Kripke models, the framework successfully models agents who lack positive introspection and agents who suspend judgment in response to hedged announcements. The authors position their work as a natural extension of standard DEL that accommodates information loss (contraction) alongside information gain (expansion). They note that while their framework can emulate plausibility dynamics by adding edges, it diverges from AGM theory by allowing the truth values of epistemic formulas to change during contraction, a feature inherent to dynamic settings but absent in static AGM models. The work leaves open the characterization of all successful formulas in HPAL and the conditions under which introspection properties are preserved in GDEL.