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Interval Orders, Biorders and Credibility-limited Belief Revision

This paper introduces and axiomatically characterizes new families of belief revision operators based on interval orders and biorders, demonstrating how the latter can model dissonance and lead to credibility-limited revision strategies where agents may initially reject inconsistent information but accept it upon further explanation.

Original authors: Richard Booth, Ivan Varzinczak

Published 2026-05-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Richard Booth, Ivan Varzinczak

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

The Big Picture: Updating Your Beliefs

Imagine your mind is a library of books (your beliefs). Every day, you get new information (a new book) that you want to add to the library.

In the standard, "perfect" way of doing this (called AGM revision), you are assumed to be a robot that:

Always accepts the new book immediately (Success).
Never allows the library to become a mess where two books contradict each other (Consistency).

If the new book says "The sky is green" and your library says "The sky is blue," the robot instantly throws out the old "blue" book to make room for the new one. It never hesitates, and it never ends up with a broken library.

This paper argues that real humans aren't robots. Sometimes, new information is so weird or shocking that we shouldn't just blindly accept it. Sometimes, accepting it might break our entire worldview. The authors explore what happens when we relax the rules of "always accepting" and "always staying consistent."

The Tools: Ranking Your World

To decide which books to keep and which to throw away, we need a way to rank how "plausible" different scenarios are.

The Old Way (Total Preorders): Imagine a single, straight ladder. Every possible world is on a rung. If World A is higher than World B, A is better. This is the standard AGM approach.
The New Way 1: Interval Orders (The "Range" Analogy):
Instead of a single rung, imagine every scenario has a range (like a temperature range).
- Scenario A: "It's between 60°F and 70°F."
- Scenario B: "It's between 65°F and 75°F."
- Scenario C: "It's between 80°F and 90°F."
Here, A and B overlap, so they are "close" in plausibility. But C is clearly hotter. This allows for more nuance than a single ladder. The paper shows how to use these "ranges" to update beliefs.
The New Way 2: Biorders (The "Negative Space" Analogy):
This is the paper's main innovation. Imagine that some scenarios are so weird that their "range" has negative length.
- Normal Scenario: "It's between 60°F and 70°F" (Length = 10).
- Dissonant Scenario: "It's between 70°F and 60°F" (Length = -10).
A "negative length" doesn't mean the temperature is cold; it means the scenario is unstable or dissonant. It's a world that feels "wrong" or "impossible" to the agent. If you try to force your beliefs into this negative space, your library might collapse.

The Problem: When New Info Breaks Everything

The authors show that if you use these "Biorders" (with negative lengths) to update your beliefs, you run into a problem: Consistency breaks.

The Scenario: You believe "My nephew is 5 years old."
The New Info: "My nephew had lunch with King Charles today."
The Result: In the "Biorder" model, this new info might push your beliefs into a "negative length" zone. The result is a belief set that is logically broken (inconsistent).

The paper calls these "Destabilizing Sentences." They are inputs that, if accepted, cause the agent's belief system to crash.

The Solution: The "Credibility Filter"

Since we can't always guarantee consistency with these fancy new tools, the authors propose a safety valve: Non-Prioritised Revision.

Instead of blindly accepting the new book, the agent has a Credibility Filter.

If the new info is "Credible" (it fits within the safe, positive ranges), the agent accepts it and updates the library.
If the new info is "Incredible" (it lands in the negative, dissonant zone), the agent rejects it entirely and keeps the old library exactly as it was.

The Trade-off:

Old Rule: Always accept new info (Success), even if it breaks consistency.
New Rule: Always keep consistency, even if it means rejecting new info.

The "Context" Twist

The paper makes a very interesting point about why we might reject something.

Imagine you hear: "My 5-year-old nephew had lunch with King Charles."

Reaction: "That's impossible! I reject it." (It's incredible).

But then, you hear a follow-up: "King Charles visited a local school today."

Reaction: "Oh! Now I believe it. The nephew was at the school."

The paper argues that the first sentence wasn't "false"; it was just incredible on its own. It needed context (the second sentence) to become credible.

Most previous theories said: "If you reject a sentence, you reject it forever."
This paper says: "You can reject a sentence today, but if someone gives you a better explanation tomorrow, you might accept it."

They call this Credibility-Limited Revision. It's like a bouncer at a club:

If you walk in alone (the sentence), you get rejected.
If you walk in with a VIP friend (the context/explanation), the bouncer lets you in.

Summary of the Authors' Claims

We can model "weird" beliefs: By using "Biorders" (which allow for negative intervals), we can mathematically represent beliefs that feel "unstable" or "dissonant."
Consistency isn't guaranteed: If you use these models to update beliefs, you might end up with a broken, inconsistent set of beliefs.
The Fix is Rejection: To fix this, we can create a system where the agent simply ignores (rejects) any new information that would cause a crash.
Context Matters: This system allows for a sentence to be rejected initially but accepted later if more context is provided. This breaks the old rule that "if you believe A, you must believe A combined with anything else."

In short: The paper provides a mathematical toolkit for agents who are smart enough to say, "Wait, that sounds too crazy to be true right now," but are also smart enough to say, "Oh, now that you explain the context, I believe it."

1. Problem Statement

The paper addresses limitations in the standard AGM (Alchourrón, Gärdenfors, and Makinson) belief revision framework. The AGM approach relies on total preorders over possible worlds to model an agent's beliefs. This framework enforces two core postulates:

Success: The new information is always accepted.
Consistency: The resulting belief set is always consistent (provided the input is consistent).

The authors argue that these postulates are too idealized for realistic agents. Real agents may encounter information that is destabilizing (causing a collapse of current beliefs) or incredible (initially rejected but potentially accepted with further context). The paper seeks to explore belief revision operators based on more general ordering structures that allow for:

Instability/Dissonance: Scenarios where accepting new information leads to an empty set of optimal models (inconsistency).
Non-Prioritised Revision: Scenarios where the agent rejects new information to preserve consistency, or where the "Success" postulate is dropped.

2. Methodology

The authors generalize the underlying semantic structure of belief revision from total preorders to two broader classes of orderings studied in rational choice theory: Interval Orders and Biorders.

A. Interval Orders (IOB)

Definition: A binary relation satisfying reflexivity and the Ferrers condition.
Semantics: Represented by an Interval-Based Interpretation $\mathcal{I} = \langle L, U \rangle$ , where each world $v$ is assigned an interval $[L(v), U(v)]$ of non-negative integers.
Revision Rule: A world $v$ is preferred to $u$ if $L(v) \leq U(u)$ .
Behavior: These operators satisfy the Success postulate but fail Vacuity and Subexpansion (standard AGM postulates) in general. They maintain Consistency.

B. Biorders (BOB)

Definition: A generalization of interval orders that satisfies the Ferrers condition but does not require reflexivity.
Semantics: Represented by a Biorder-Based Interpretation $\mathcal{B} = \langle L, U \rangle$ $B = ⟨ L, U ⟩$ . Crucially, the interval length $U(v) - L(v)$ $U (v) - L (v)$ can be negative.
- A negative interval ( $U(v) < L(v)$ ) signifies a dissonant valuation, representing a form of "impossibility" or instability.
Revision Rule: $v \preceq_{bi} u$ iff $L(v) \leq U(u)$ .
Behavior: These operators satisfy Success but violate Consistency. If the new information $\alpha$ is "destabilizing" (i.e., the optimal set of models for $\alpha$ is empty because all models of $\alpha$ are dissonant), the revision yields an inconsistent (empty) belief set.

C. Subclasses and Transitivity

The authors define two subclasses of biorders to impose stronger structural properties:

ZTBOB (z-transitive): Satisfies a weakened form of transitivity involving dissonant elements. This ensures Consistent Cautious Monotony.
TBOB (Transitive): Satisfies full transitivity. This recovers the Subexpansion postulate (a key AGM supplementary postulate) but still allows for inconsistency.

D. Non-Prioritised Revision (NPR)

To address the inconsistency issue in Biorder-based revision, the authors define Non-Prioritised Revision operators ( $\circ$ ).

Mechanism: If a biorder-based revision ( $\bullet$ ) of $\alpha$ results in an inconsistent set, the NPR operator rejects the input and returns the original belief set $KB$.
Credibility: An input $\alpha$ is deemed credible if $KB \bullet \alpha$ is consistent; otherwise, it is incredible.
Link to Credibility-Limited Revision (CLR): The authors map these NPR operators to the framework of Credibility-Limited Revision (Hansson et al.), but with a crucial modification: they reject the Single-Sentence Closure property.
- Standard CLR: If $\alpha$ is credible, any logical consequence is credible.
- Proposed CLR: An agent may reject $\alpha$ initially but accept $\alpha \land \beta$ if $\beta$ provides necessary context (e.g., "King Charles visited a school" makes "Nephew had lunch with King Charles" credible).

3. Key Contributions

Extension of Revision Semantics:
- Introduced Interval Order-Based (IOB) revision, completing the picture started by Rott (2014) for contraction.
- Introduced Biorder-Based (BOB) revision, utilizing negative interval lengths to model "dissonance" and instability, a concept previously unexplored in belief revision.
Axiomatic Characterizations:
- Provided representation theorems for IOB, ZTBOB, and TBOB operators.
- Identified that TBOB operators satisfy all AGM postulates except Consistency, plus Disjunctive Rationality.
- Characterized the resulting NPR operators (BOB-NPR, ZTBOB-NPR, TBOB-NPR) using a set of postulates including Relative Success (Success or no change) and modified Disjunctive Rationality.
Credibility-Limited Revision with Contextual Closure:
- Formulated NPR operators as Credibility-Limited Revision operators.
- Proposed a relaxation of the Single-Sentence Closure condition (replacing it with a weaker closure for disjunctions). This allows for context-dependent credibility, where a sentence is rejected in isolation but accepted when combined with explanatory context.
Handling Destabilizing Inputs:
- Defined Destabilizing, Irreconcilable, and Precarious sentences.
- Demonstrated that biorders can model scenarios where an agent enters "maintenance mode" (rejecting input) until further explanation stabilizes the belief state.

4. Results

Theorem 1-3: Established the correspondence between Total Preorders, Interval Orders, and Biorders and their respective revision operators.
Theorem 4 & 5: Showed that ZTBOB and TBOB operators satisfy specific strengthened sets of postulates (e.g., TBOB satisfies Subexpansion).
Theorem 6-8: Provided axiomatic characterizations for the Non-Prioritised versions of these operators, introducing postulates like P12+ (if a disjunction is credible and one disjunct is not, the result is equivalent to revising with the credible disjunct alone).
Theorem 9 & 10: Proved that ZTBOB-NPR and TBOB-NPR operators are equivalent to Credibility-Limited Revision operators where the underlying revision is based on Interval Orders and AGM (Total Preorders), respectively.
Counterexamples: Demonstrated that standard AGM postulates like Vacuity and Subexpansion fail for general biorders, and that Consistency fails for BOB revision unless inputs are filtered.

5. Significance

Realism in AI: The paper moves belief revision closer to realistic agent behavior by allowing for instability and conditional acceptance of information. It models the cognitive process of rejecting "too good to be true" or "too weird" information until it is contextualized.
Theoretical Novelty: It bridges the gap between Rational Choice Theory (specifically biorders and interval orders) and Belief Revision, introducing concepts like "negative intervals" and "dissonance" into the logic of belief change.
Flexibility: By relaxing the Single-Sentence Closure condition, the framework supports more nuanced reasoning where the credibility of a statement depends on its conjunction with other statements (context), rather than just its logical form.
Future Directions: The work opens avenues for studying iterated revision with biorders, defeasible reasoning, and normative reasoning, suggesting that the extra structure in biorders (beyond what is needed for a single revision step) may be crucial for handling sequences of updates.

In summary, the paper proposes a robust theoretical framework where belief revision is not a blind acceptance of truth, but a selective process governed by the structural properties of the agent's belief ordering, capable of handling instability and context-dependent credibility.