On Lockean beliefs that are deductively closed and minimal change

Lluis Godo; Lluis Subirana; Ram\'on Pino P\'erez; Tommaso Flaminio

REVIEW 3 major objections 5 minor 24 references

On Lockean beliefs that are deductively closed and minimal change

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Lockean belief sets close under logical deduction exactly when some world outweighs all strictly less probable worlds combined.

desk verdict Theorem 2 is a solid, citable result; the Section 5 minimal-change claims rest on a false KL-minimality statement and the Theorem 3 proofs need repair. read the letter →

arxiv 2507.06042 v1 pith:YMOZWE6Z submitted 2025-07-08 cs.AI

Tommaso Flaminio , Lluis Godo , Ramón Pino Pérez , Lluis Subirana This is my paper · ORCID

classification cs.AI MSC 03B4803B42

keywords Lockeanthesisdeductiveclosurebeliefrevisionminimalchangeprobabilitythresholdstepprincipalfilterrelativeentropy

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

According to the Lockean thesis, a rational agent believes a proposition when its subjective probability clears a threshold. Such Lockean belief sets are known to be awkward companions for classical belief revision because they are not generally closed under conjunction and deduction. This paper pins down exactly when they are: for a positive probability on a finite language, a non-trivial threshold belief set is deductively closed if and only if the distribution has a step at some world — one world whose probability exceeds the combined probability of all strictly less probable worlds. In that case the closed belief set is the principal filter generated by the worlds whose probability meets or exceeds that world's probability. The paper also introduces a minimal-change revision operator, closely tied to parameter-weighted conditionalization, that revises the probability so the incoming proposition is believed at the threshold, and characterizes when the revised belief set is again deductively closed.

What carries the argument

The $\omega$-step condition on a probability $P$: a world $\omega$ with $P(\omega) > \sum_{\omega': P(\omega') < P(\omega)} P(\omega') > 0$. This single dominance inequality is the exact condition that makes the set of formulas with probability at least $P(\Phi_\omega)$ a principal filter, hence deductively closed. The revision side is carried by the operator $R^\lambda_\psi$ defined by equation (4), which rescales the prior inside and outside $J\psi K$ to force posterior probability $\lambda$ on $\psi$; it is a form of parameter-weighted conditionalization and, when $\psi$ was initially disbelieved, it is the minimal relative-entropy update subject to $P'(\psi) = \lambda$.

What would settle it

Construct a positive probability on a finite set of worlds with no $\omega$-step and a threshold $\lambda \in (1/2,1)$ such that $B_{\lambda,P}$ is deductively closed and non-trivial; Theorem 2 says this is impossible. A familiar check is the uniform distribution on three worlds at $\lambda = 0.6$: the paper predicts the belief set is not closed, and indeed $\{w_1,w_2\}$ and $\{w_2,w_3\}$ are believed while $\{w_2\}$ is not, so any departure from that predicted failure would be a counterexample.

Watch

Extended reading notes

Core claim

Formally, the first characterization (Theorem 1) states that for a positive probability $P$ and threshold $\lambda > 1/2$, the set $B_{\lambda,P} = \{\varphi : P(\varphi) \geq \lambda\}$ is deductively closed if and only if some $\psi$ in the set has $P(\psi) = \lambda_M$ (the largest threshold value that gives this same set) and every model of $\psi$ has probability greater than $1 - \lambda_M$; in that case $B_{\lambda,P}$ is the principal filter $\uparrow\! J\psi K$. The sharper result (Theorem 2) is that such a non-trivial closed belief set exists for some $\lambda \in (1/2, 1)$ exactly when $P$ has an $\omega$-step: $P(\omega) > \sum_{\omega': P(\omega') < P(\omega)} P(\omega') > 0$. The belief set is then $\uparrow\! J\Phi_\omega K$, where $\Phi_\omega$ collects the worlds whose probability is at least $P(\omega)$. Building on this, the paper defines a revision operator $B_{\lambda,P} *_{ml} \psi = B_{\lambda, R^\lambda_\psi}$ that scales the probability of worlds inside and outside $\psi$ so that $\psi$ gets posterior probability $\lambda$ whenever it was below threshold; this operator coincides with parameter-weighted conditionalization on $\psi$ and is the unique minimal update in relative entropy subject to $P'(\psi) = \lambda$. Theorem 3 then characterizes exactly when this revised belief set is deductively closed: if $P(\psi) < \tau \lambda/(1-\lambda)$ with $\tau = \min\{P(\omega) : \omega \in J\psi K\}$, the revised set is closed and generated by $\psi$.

Load-bearing premise

Every world must have positive probability and the language must have finitely many variables; the characterizations and the revision rule's normalization both depend on these two assumptions.

Editorial extensions

If this is right

A probability with a step at $\omega$ yields, for threshold $\lambda = P(\Phi_\omega)$, a consistent and deductively closed belief set generated by $\Phi_\omega$.
No non-trivial deductively closed Lockean belief set exists for any $\lambda$ in $(1/2,1)$ when the distribution has no step, for instance when probability is spread uniformly over two or more worlds.
The revision operator always satisfies the Success and Extensionality postulates, and it preserves deductive closure whenever the incoming formula $\psi$ satisfies $P(\psi) < \tau\lambda/(1-\lambda)$, in which case the revised belief set is generated by $\psi$.
Among all distributions that assign probability $\lambda$ to $\psi$, the revised distribution is the closest to the prior in relative entropy, and it is closer in total-variation distance than plain conditionalization on $\psi$ when $\psi$ was initially below threshold.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The step condition can be read as the distribution-level culprit behind the lottery paradox: conjunctive closure fails precisely when no world dominates the cumulative mass of all less probable worlds.
Because the $\omega$-step condition is strictly weaker than the total-ordering 'big-step' requirement, the characterization draws a precise line between coherent probabilistic acceptance and acceptance that requires an additional ordering structure.
The revision rule is defined for one-shot input; iterating it would require choosing a threshold at each stage, and the paper leaves open which iterated-revision postulates survive, so those postulates can be checked directly.
The results rely on every world having positive probability; allowing zero-probability worlds would likely replace the strict dominance inequality with a limiting version, and the revision identity would need a separate treatment of impossible worlds.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The short version: Theorem 2 is the real result — a clean, checkable characterization of when a Lockean belief set can be deductively closed and non-trivial: exactly when P has an ω-step, i.e. P(ω) > sum over strictly less probable worlds. That is a genuine weakening of big-stepped probabilities, it is not in the earlier literature, and it deserves to be cited.

What the paper does well: the proof of Theorem 2 is short and essentially correct. The step-probability condition is novel, and the correspondence between steps and principal filters generated by Φω is nice. Proposition 2 and Lemma 3 are fine. The paper is also honest about scope: finite language, positive probabilities, and the limitations are stated up front.

Where it breaks: Proposition 4 says R^λ_ψ minimizes Kullback-Leibler divergence among distributions P' with P'(ψ)=λ. That is false as stated. When P(ψ)>λ, Definition 2 sets both factors to 1, so R^λ_ψ = P and R^λ_ψ(ψ) = P(ψ) ≠ λ; the candidate is not even in the feasible set. The KL-minimality claim needs an explicit side condition, namely P(ψ) ≤ λ, equivalently ψ ∉ Bλ,P. This is not cosmetic: the abstract's "fewest possible changes" promise rides on it.

Theorem 3 has related proof gaps. In part (i), the proof says R^λ_ψ(ψ)=λ "by definition," which fails in the P(ψ)>λ subcase that the hypotheses allow; the conclusion may still be true there because ψ is already believed, but the argument needs a case split. In part (ii), the proof invokes Theorem 1 on the revised probability and writes 1−λ < τ; it should be 1−λ < λτ/P(ψ). Both are fixable, but the printed text does not establish the theorem as written.

The positivity and finite-language assumptions are explicit and reasonable for this line of work; the results just do not extend automatically, and the paper does not claim otherwise.

Net: the first half is solid and novel; the second half is under-cooked. This deserves a serious referee, not a desk reject. I would send it out and push the authors to repair Proposition 4 and the Theorem 3 proofs before publication.

Referee Report

3 major / 5 minor

Summary. The paper studies Lockean belief sets Bλ,P={φ:P(φ)≥λ} in finite classical propositional logic under the standing assumption of positive probabilities. Its first contribution is two characterizations of when some threshold λ>1/2 makes Bλ,P deductively closed: Theorem 1 gives a local condition in terms of a generator ψ and the maximal threshold λM, and Theorem 2 identifies the exact condition with P having an ω-step, i.e., P(ω)>Σ_{P(ω')<P(ω)}P(ω')>0, in which case the closed belief set is the principal filter ↑Φω. The second contribution defines a revision operator Rλψ, observes that it coincides with Jeffrey conditionalization when P(ψ)≤λ, characterizes when the revised belief set is deductively closed (Theorem 3), and claims a Kullback-Leibler minimality property (Proposition 4). The paper also verifies several AGM postulates for the proposed revision and compares the notion with big-stepped probabilities and P-stable sets.

Significance. Conditional on the revision-section errors being repaired, the paper makes a clean and publishable contribution. The deductive-closure half is the strongest part: Theorem 2 gives a crisp, non-obvious equivalence between an epistemic property (deductive closure of threshold beliefs) and a purely probabilistic structural condition (existence of an ω-step), and the proof is elementary, self-contained, and free of fitted parameters beyond the threshold λ. The related-work discussion connecting the result to big-stepped probabilities and to Leitgeb's P-stability is informative. The minimal-change half is currently overstated: Proposition 4 is false as stated and the proof of Theorem 3 contains incorrect applications of Theorem 1. These are local mathematical errors with clear repairs, not failures of the main characterization, but they are load-bearing for the paper's second advertised contribution and therefore require a careful revision.

major comments (3)

[§5, Proposition 4] Proposition 4 (Section 5, with proof in Appendix A) is false as stated. When P(ψ)>λ, Definition 2 gives Rλψ(ω)=P(ω) for every ω, so Rλψ(ψ)=P(ψ)≠λ; hence Rλψ does not even belong to the feasible set {P' : P'(ψ)=λ}. For example, take Ω={ω1,ω2}, P(ω1)=0.6, P(ω2)=0.4, λ=0.55 and ψ=ω1; then Rλψ=P, while the KL projection onto {P'(ψ)=0.55} is a different distribution. The Lagrange-multiplier proof derives the Jeffrey update λP(·|ψ)+(1−λ)P(·|¬ψ), which equals Rλψ only when P(ψ)≤λ. The proposition, and the abstract's 'fewest possible changes' claim, must be restricted to P(ψ)≤λ or reformulated as minimization over {P' : P'(ψ)≥λ}; as printed, the minimality claim is incorrect.
[§5, Theorem 3(i) proof] The proof of Theorem 3(i) claims that 'Rλψ(ψ)=λ holds by definition' and then applies Theorem 1 with χ=ψ. This is not valid under the theorem's hypothesis, because the max factor in Definition 2 makes Rλψ(ψ)=P(ψ) when P(ψ)>λ, and the hypothesis P(ψ)<τλ/(1−λ) does not exclude this case (e.g., P(ω1)=0.7, λ=0.6, ψ=ω1). The conclusion of (i) may still be true, but the proof needs a case split for P(ψ)<λ and P(ψ)≥λ, and in the latter case it must work with the appropriate λM for the revised probability; the printed argument leaves a gap in a stated characterization.
[§5, Theorem 3(ii) proof] In the proof of Theorem 3(ii), after Lemma 2 identifies JψK as the generator of Bλ,P∗mlψ, Theorem 1 must be applied to the revised probability Rλψ, not to P. The correct inequality is 1−λ < min_{ω∈JψK} Rλψ(ω) = λτ/P(ψ), which yields P(ψ)<λτ/(1−λ). The printed line 'by Theorem 1, 1−λ<τ' is too strong and does not follow; it should read '1−λ<λτ/P(ψ)'. The subsequent algebra should be adjusted accordingly.

minor comments (5)

[§2, last paragraph] The blanket positivity assumption P(ω)>0 is introduced in a single sentence at the end of Section 2; since Theorem 1, Lemma 2, and the normalization in Definition 2 all rely on it, it would help readers if this were stated as a standing assumption in the introduction or abstract.
[§1] The name 'Hannson' in the first section should be 'Hansson'.
[§5, paragraph after Definition 3] The sentence 'Also in this in section we characterize...' contains a typo and should be rewritten.
[Appendix A, Fact 2 proof] The proof of Fact 2 writes Rψ instead of Rλψ, and the computation of d(P(·|ψ),P) would be easier to follow with one intermediate step showing d(P(·|ψ),P)=2(1−P(ψ)).
[§5, Proposition 3] When P(ψ)=λ, the two cases in Proposition 3 coincide; stating this explicitly would avoid a small ambiguity in the piecewise definition.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the step-probability characterization and the revision-minimality result are genuine derivations, with only background self-citations.

full rationale

The central result, Theorem 2, is a genuine equivalence rather than a definitional reduction. The Lockean set Bλ,P = {φ : P(φ) ≥ λ} is defined by thresholding, while the ω-step condition (Definition 1) is an independent probabilistic condition. The proof derives closure from the step condition via Proposition 2 and Theorem 1, and conversely derives the step inequality from closure plus non-triviality; neither side is assumed in the other. The revision operator Rλψ is constructed from max/min rescaling, and Fact 2 and Proposition 4 compare it to external benchmarks (total variation distance and KL divergence), so the minimal-change claim is not obtained by renaming the construction. The self-citations in the paper ([4], [21]) are pointers to prior discussion or to a framing epistemic-space formalism, not premises of the proofs. One non-circular caveat: Proposition 4 as stated misses the case P(ψ) > λ, where Rλψ = P and hence does not satisfy the constraint P′(ψ) = λ; this is a localized mathematical correctness issue, not a circularity. Since no step reduces by definition to its input, the circularity score is 0.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper relies on standard finite Boolean algebra, classical logic, and probability theory; the only domain-level assumptions are positivity of probabilities and a finite language. The threshold λ is a model input, not a fitted constant. No invented entities such as new forces, particles, or conserved quantities are introduced; the ω-step is a defined condition on the distribution.

free parameters (1)

threshold λ = > 1/2 (model input, not fitted)
The Lockean threshold is chosen by the agent and is an input to Bλ,P and to the revision rule Rλ_ψ; the paper characterizes for which P such a λ exists but does not fit λ to data.

assumptions (5)

standard math Classical propositional logic over finitely many variables, with formulas identified with subsets of the finite set Ω of valuations.
Section 2 sets the language L over x1,...,xn and identifies formulas with their model sets; all results are stated in the finite Boolean algebra 2^Ω.
domain assumption Probability functions are positive: P(ω) > 0 for all ω in Ω.
Stated at the end of Section 2; used in Lemma 2, Theorem 1, and in the normalization of Equation (4).
domain assumption The threshold λ is strictly greater than 1/2, and in the revision section 1/2 < λ < 1.
Section 1 argues rationality forces λ > 1/2; revision uses both λ and 1-λ as renormalization factors.
standard math Lagrange multipliers locate the global minimum of the Kullback-Leibler divergence under the linear constraint P'(ψ)=λ.
Used in the proof of Proposition 4; the objective is convex, so the stationary point is the minimizer.
standard math Filters of 2^Ω correspond exactly to consistent deductively closed theories.
Section 2 establishes the identification used throughout to reason about deductive closure via filters.

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Cite this review

Pith. "Pith review of On Lockean beliefs that are deductively closed and minimal change." pith.science (2026). https://pith.science/paper/YMOZWE6Z

@misc{pith2026250706042,
  author       = {Pith},
  title        = {Pith review of: On Lockean beliefs that are deductively closed and minimal change},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YMOZWE6Z}},
  note         = {Machine review of arXiv:2507.06042}
}

read the original abstract

Within the formal setting of the Lockean thesis, an agent belief set is defined in terms of degrees of confidence and these are described in probabilistic terms. This approach is of established interest, notwithstanding some limitations that make its use troublesome in some contexts, like, for instance, in belief change theory. Precisely, Lockean belief sets are not generally closed under (classical) logical deduction. The aim of the present paper is twofold: on one side we provide two characterizations of those belief sets that are closed under classical logic deduction, and on the other we propose an approach to probabilistic update that allows us for a minimal revision of those beliefs, i.e., a revision obtained by making the fewest possible changes to the existing belief set while still accommodating the new information. In particular, we show how we can deductively close a belief set via a minimal revision.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Journal of Symbolic Logic 50, 510–530 (1985) On Lockean beliefs that are deductively closed and minimal change 15

Alchourrón, C.E., Gärdenfors, P., Makinson, D.: On the logic of theory change: partial meet contraction and revision functions. Journal of Symbolic Logic 50, 510–530 (1985) On Lockean beliefs that are deductively closed and minimal change 15

work page 1985

[2] [2]

The British Journal for the Philosophy of Science69(2), 509–552 (2016)

Benci, V., Horsten, L., Wenmackers, S.: Infinitesimal probabilities. The British Journal for the Philosophy of Science69(2), 509–552 (2016)

work page 2016

[3] [3]

Journal of Logic and Computation9(6), 873–895 (1999)

Benferhat, S., Dubois, D., Prade, H.: Possibilistic and standard probabilistic se- mantics of conditional knowledge bases. Journal of Logic and Computation9(6), 873–895 (1999)

work page 1999

[4] [4]

Erkenntnis88(3), 965–990 (2023)

Bonzio, S., Cevolani, G., Flaminio, T.: How to believe long conjunctions of beliefs: probability, quasi-dogmatism and contextualism. Erkenntnis88(3), 965–990 (2023)

work page 2023

[5] [5]

Annals of Mathematics and Artificial Intelligence87, 259–291 (2019)

Cantwell, J., Rott, H.: Probability, coherent belief and coherent belief change. Annals of Mathematics and Artificial Intelligence87, 259–291 (2019)

work page 2019

[6] [6]

Oxford Uni- versity Press, New York (2000)

Cori, R., Lascar, D.: Mathematical Logic: A Course with Exercises. Oxford Uni- versity Press, New York (2000)

work page 2000

[7] [7]

Darwiche,A.,Pearl,J.:Onthelogicofiteratedbeliefrevision.ArtificialIntelligence 89(1), 1–29 (1997)

work page 1997

[8] [8]

In: Proceedings of the11thConferenceonUncertaintyinArtificialIntelligence(UAI’95).pp.149–156

Dubois, D., Prade, H.: Numerical representation of acceptance. In: Proceedings of the11thConferenceonUncertaintyinArtificialIntelligence(UAI’95).pp.149–156. Montreal (1995)

work page 1995