Interval Orders, Biorders and Credibility-limited Belief Revision

Ivan Varzinczak; Richard Booth

arxiv: 2604.27156 · v1 · submitted 2026-04-29 · 💻 cs.AI

Interval Orders, Biorders and Credibility-limited Belief Revision

Richard Booth , Ivan Varzinczak This is my paper

Pith reviewed 2026-05-07 08:23 UTC · model grok-4.3

classification 💻 cs.AI

keywords belief revisioninterval ordersbiorderscredibility-limited revisionnon-prioritised revisionpossible worldsrational choice theoryAGM postulates

0 comments

The pith

Interval orders and biorders from rational choice theory can serve as plausibility orderings to define new families of belief revision operators with full axiomatic characterisations.

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

The paper establishes that belief revision, usually based on total preorders over possible worlds, can instead use interval orders and biorders, which assign intervals of plausibility to each world. It supplies axiomatic characterisations for the resulting operator families and two families lying between them. Biorder-based revisions satisfy the Success postulate but can produce inconsistent outputs, so the authors modify the definition to discard inputs leading to inconsistency as incredible. This produces non-prioritised revision operators that satisfy the Consistency postulate but not Success, and that relate to credibility-limited revision without the single-sentence closure condition. The approach fits agents that may initially reject new information yet accept it once further explanation is supplied.

Core claim

Interval orders associate each possible world with a nonnegative interval of plausibility, while biorders generalise this by allowing intervals of negative length to capture dissonance or instability. Belief revision operators are defined by selecting the most preferred models of the incoming information under these orderings. The paper gives axiomatic characterisations of the four families and shows that biorder-based revisions satisfy Success yet do not always yield consistent belief sets. By treating inputs that would produce inconsistency as incredible and discarding them, the authors obtain non-prioritised operators satisfying Consistency but not Success; these are linked to credibility

What carries the argument

Interval orders and biorders as plausibility orderings over possible worlds, where each world is assigned an interval of plausibility (nonnegative for interval orders, possibly negative for biorders).

Load-bearing premise

The standard definitions of interval orders and biorders from rational choice theory can be directly lifted to define plausible-world orderings for belief revision without introducing inconsistencies or losing the intended semantics of the postulates.

What would settle it

A concrete set of worlds, a prior belief set, and an input sentence for which an operator constructed from interval orders or biorders violates one of the provided axiomatic characterisations, or for which the modified non-prioritised version fails to discard only the inputs that produce inconsistency.

Figures

Figures reproduced from arXiv: 2604.27156 by Ivan Varzinczak, Richard Booth.

**Figure 1.** Figure 1: provides an easier to visualise representation of the interval-based interpretation in Example 1. 0 1 2 3 L pq, p¯q¯ pq¯ pq¯ U pq, pq¯ pq, ¯ p¯q¯ view at source ↗

**Figure 3.** Figure 3: A compressed biorder-based interpretation. view at source ↗

**Figure 2.** Figure 2: provides an easier to visualise representation of the biorder-based interpretation in Example 3. 0 1 2 3 4 L p¯q¯ pq pq¯ pq¯ U pq¯ pq¯ pq p¯q¯ view at source ↗

**Figure 4.** Figure 4: System-of-spheres style representation of a TBOB revi view at source ↗

read the original abstract

Rational belief revision is commonly viewed as being based on a preference order between possible worlds, with the resulting new belief set being those sentences true in all the most preferred models of the incoming new information. Usually, such a preference order is taken to be a total preorder. Nevertheless, there are other, more general classes of ordering that can also be employed. In this paper, we explore two such classes that have been studied within the theory of rational choice but have seen limited or no application in belief revision. We begin with interval orders, introduced by Fishburn in the '80s, which associate with each possible world a nonnegative `interval' of plausibility. We then move on to biorders, studied by Aleskerov, Bouyssou, and Monjardet, which generalise interval orders by allowing the intervals to have negative lengths, a feature that can be used to capture a notion of dissonance or instability. We provide axiomatic characterisations of these two resulting families of belief revision operators, as well as of two further families of interest that lie between interval orders and biorders. We show that while biorder-based revisions satisfy the Success postulate, they do not always yield consistent outputs. By modifying their definition to discard inputs that lead to inconsistency as `incredible', we derive new families of so-called non-prioritised revision that satisfy the Consistency postulate, but not the Success one. These families are linked to credibility-limited revision operators of Hansson et al., but for which the set of credible sentences does not satisfy the single-sentence closure condition. We argue that the biorder-based approach is well-suited for scenarios where an agent might initially reject new information, but may accept it when presented with additional explanation.

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.

Desk Editor's Note private letter to a colleague

Booth and Varzinczak lift interval orders and biorders from choice theory into belief revision, deliver axiomatic characterizations for the operators and two intermediates, and fix the consistency problem in biorders via a credibility-limited non-prioritized variant.

read the letter

The main point is that this paper imports interval orders (Fishburn-style intervals of plausibility) and biorders (allowing negative lengths for dissonance) to select most-preferred models of new information, then supplies representation theorems for those families plus two that sit between them. They also modify the biorder operators by discarding inputs that would produce an empty set of maximal models, yielding non-prioritized revisions that keep consistency at the cost of success and that connect to Hansson-style credibility-limited operators without single-sentence closure. The argument that this setup suits agents who initially reject information but might accept it with more context is straightforward and fits the constructions. What the paper does cleanly is the translation itself and the explicit handling of the success-consistency trade-off; the characterizations appear to be new results not present in the cited prior work. The soft spots are modest. The direct lift of the order definitions works at the semantic level they describe, but one would still want to check whether the negative-length cases introduce any unintended side effects on the induced revision operators beyond the inconsistency they already flag and repair. The practical payoff over ordinary total preorders is not quantified, so the added generality stays mostly formal for now. No circularity or hidden fitting shows up in the high-level setup. This is for people already working on generalized or non-prioritized belief change in logic and knowledge representation. A reader who wants fresh axiomatizations or tools for modeling initial rejection will get concrete value; others outside that niche will not. It deserves a serious referee because the characterizations are formally grounded and the constructions are reproducible from the given definitions.

Referee Report

0 major / 4 minor

Summary. The manuscript generalizes standard AGM-style belief revision (based on total preorders over possible worlds) by lifting interval orders (Fishburn) and biorders (Aleskerov et al.) from rational choice theory to define new families of revision operators. It supplies representation theorems characterizing the operators induced by interval orders, biorders, and two intermediate classes lying between them. The paper shows that biorder-based operators satisfy Success but can produce inconsistent outputs; it then modifies the definition by discarding inputs that would yield inconsistency (treating them as 'incredible') to obtain non-prioritized revision families that satisfy Consistency but not Success. These are related to credibility-limited revision (Hansson et al.) but without the single-sentence closure condition on the credible set, and are argued to suit agents that may initially reject information yet accept it with further explanation.

Significance. If the representation theorems hold, the work meaningfully extends the formal foundations of belief revision beyond total preorders, incorporating structures that can model dissonance or instability via interval lengths (including negative ones). The explicit Success/Consistency trade-off and the resulting credibility-limited operators without single-sentence closure provide a clean, falsifiable framework for non-prioritized revision. This could support AI applications involving agents that handle conflicting or unstable information, and the paper correctly credits the source literature while positioning the contribution as a direct translation plus targeted modification.

minor comments (4)

[Preliminaries / §2] The preliminaries section would benefit from an explicit side-by-side comparison (perhaps as a table) of the four families of orderings (interval orders, the two intermediate classes, and biorders) together with the key properties each induces on the corresponding revision operators (e.g., Success, Consistency, and any closure conditions).
[§4] The definition of the biorder-based revision operator and the subsequent 'incredible input' filter should include a short worked example showing a concrete biorder with a negative-length interval that produces an empty set of maximal models, followed by the effect of the credibility-limited modification.
[Throughout] Notation for the interval assignment function (and the associated 'length' or dissonance measure) is introduced in the preliminaries but reused with slight variations in later sections; a single, consistently subscripted symbol throughout would improve readability.
[Discussion / §6] The connection to Hansson et al.'s credibility-limited revision is stated clearly in the abstract and conclusion, but the manuscript should add one paragraph in the discussion explicitly contrasting the credible-set closure properties (single-sentence vs. none) with a reference to the relevant theorem in Hansson et al.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of the manuscript and for recommending minor revision. We appreciate the recognition that the representation theorems extend AGM-style revision beyond total preorders and that the credibility-limited variants offer a clean framework for non-prioritized revision without single-sentence closure.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via external definitions

full rationale

The paper imports the definitions of interval orders (Fishburn) and biorders (Aleskerov et al.) directly from the independent rational-choice literature and lifts them to define revision operators that select maximal models of the input sentence. Representation theorems are proved from these imported axioms rather than being presupposed. The non-prioritised modification is introduced by an explicit, non-equivalent redefinition that discards inputs yielding empty maximal sets; this change is motivated by the observed failure of Consistency and is not forced by any prior equation or self-citation. No load-bearing step reduces to a fitted parameter, a self-referential equation, or a uniqueness theorem supplied only by the authors' own prior work. The Success/Consistency trade-off follows directly from the semantic properties of the lifted orders and is stated explicitly rather than smuggled in.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard mathematical definitions of interval orders (nonnegative intervals) and biorders (allowing negative lengths) taken from the rational choice literature; no new free parameters, invented entities, or ad-hoc axioms are introduced beyond those definitions and the usual AGM-style postulates.

axioms (2)

domain assumption Interval orders associate each world with a nonnegative real interval representing its plausibility
Taken directly from Fishburn's definition and used as the semantic basis for the revision operators.
domain assumption Biorders generalize interval orders by permitting intervals of negative length to capture dissonance
Taken from Aleskerov, Bouyssou, and Monjardet; used to define the more general revision family.

pith-pipeline@v0.9.0 · 5613 in / 1595 out tokens · 63613 ms · 2026-05-07T08:23:03.519891+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

For Disjunctive Overlap, we consider two cases •(a)α∨β /∈ C KB

Closure, Relative Success, Inclusion, Extensionality and Consistency all straightforwardly follow using the fact that IOB revision operators satisfy the corresponding postulates. For Disjunctive Overlap, we consider two cases •(a)α∨β /∈ C KB. ThenKB ◦ ∗,C (α∨β) =KB. FromC4 ′ we haveα /∈ CKB orβ /∈ C KB. HenceKB ◦ ∗,C α=KB orKB ◦ ∗,C β=KB. In either case w...

work page
[2]

Since◦is also a ZTBOB NPR operator, we already know from the proof of that theorem that◦=◦ ∗′,C′ where∗ ′ =∗ ◦ andC ′ =C ◦, and that⪯ int KB is an interval order, where⪯ int KB is defined from the biorder⪯ bi KB that◦associates toKB. It suffices to show that the extra assumption that⪯ bi KB is fully transitive is enough to show that⪯ int KB is not just an...

work page

[1] [1]

For Disjunctive Overlap, we consider two cases •(a)α∨β /∈ C KB

Closure, Relative Success, Inclusion, Extensionality and Consistency all straightforwardly follow using the fact that IOB revision operators satisfy the corresponding postulates. For Disjunctive Overlap, we consider two cases •(a)α∨β /∈ C KB. ThenKB ◦ ∗,C (α∨β) =KB. FromC4 ′ we haveα /∈ CKB orβ /∈ C KB. HenceKB ◦ ∗,C α=KB orKB ◦ ∗,C β=KB. In either case w...

work page

[2] [2]

Since◦is also a ZTBOB NPR operator, we already know from the proof of that theorem that◦=◦ ∗′,C′ where∗ ′ =∗ ◦ andC ′ =C ◦, and that⪯ int KB is an interval order, where⪯ int KB is defined from the biorder⪯ bi KB that◦associates toKB. It suffices to show that the extra assumption that⪯ bi KB is fully transitive is enough to show that⪯ int KB is not just an...

work page