{"id":"8abfc523-aab7-490e-937f-acfa41109a7a","arxiv_id":"2507.06042","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A probability threshold belief set is deductively closed exactly when the distribution has a step, and the new revision rule is the minimal Kullback-Leibler change that raises the new statement to the threshold.","lead":"A Lockean belief set is the list of statements whose subjective probability exceeds a threshold, and such lists are usually not closed under logical deduction. This paper proves exactly when such a set is deductively closed and gives a minimal-change revision rule that adds a new belief by adjusting probabilities as little as possible.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4 is false as stated: when P(psi) > lambda, R^lambda_psi equals P and violates the constraint P'(psi) = lambda, so the KL-minimality claim needs qualification to P(psi) <= lambda.","rationale":"I independently checked the main characterization. Theorem 2 is correct: the if-direction constructs lambda = P(Phi_omega) and verifies Theorem 1's condition; the only-if direction uses Theorem 1's generator psi and shows psi = Phi_omega for an atom omega attaining the minimum in JpsiK, yielding the omega-step. Positivity and finiteness are explicit assumptions, so no hidden inconsistency there. The reader's two gaps in Theorem 3 are real but repairable: in (i), when P(psi) > lambda the proof's assertion R^lambda_psi(psi) = lambda is false, though the conclusion can be recovered by noting the hypothesis forces tau > 1 - lambda and hence B_lambda_P is already the principal filter uparrow JpsiK; in (ii), one must first observe that Lemma 2 makes lambda = lambda_M for the revised probability before applying Theorem 1. The more serious issue I find is Proposition 4. As stated it claims R^lambda_psi minimizes KL among distributions with P'(psi) = lambda, but when P(psi) > lambda the definition makes R^lambda_psi = P, so R^lambda_psi(psi) = P(psi) != lambda and R^lambda_psi is outside the feasible set. The Appendix's Lagrange computation actually solves for the Jeffrey update with target lambda, which equals R^lambda_psi only when P(psi) <= lambda. This is a concrete false statement, not merely wording. It affects the paper's second central aim, minimal change, though not Theorem 2. The fix is to restrict Proposition 4 to the case P(psi) <= lambda and state separately that when psi is already believed, P itself is the minimal change under P'(psi) >= lambda. Since the main characterization survives and the revision operator is well-defined, the appropriate disposition remains conditional acceptance pending this correction, matching the reader's verdict.","tokens_in":15117,"tokens_out":17557,"duration_ms":170611,"concrete_test":"Recompute the KL minimizer in Appendix A for the case P(psi) > lambda, e.g., P(omega1) = 0.6, P(omega2) = 0.4, lambda = 0.55, psi = omega1. The Lagrange solution is P'(omega1) = 0.55 and P'(omega2) = 0.45, whereas R^lambda_psi = P has P'(omega1) = 0.6, so R^lambda_psi is not the minimizer and is absent from the constraint set; this confirms Proposition 4 requires the qualification P(psi) <= lambda.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's second advertised contribution, minimal revision, is formally supported by Proposition 4, which states that R^lambda_psi minimizes KL divergence among distributions P' with P'(psi) = lambda. This is false without a side condition. If P(psi) > lambda, Definition 2 gives R^lambda_psi(omega) = P(omega) for all omega (both max and min factors are 1), so R^lambda_psi = P and R^lambda_psi(psi) = P(psi) != lambda. Hence R^lambda_psi is not even a member of the feasible set {P' : P'(psi) = lambda}; it cannot be the minimizer. Example: Omega = {omega1, omega2}, P(omega1) = 0.6, P(omega2) = 0.4, lambda = 0.55, psi = omega1. Then R^lambda_psi = P with R^lambda_psi(psi) = 0.6, while the KL projection onto {P'(omega1) = 0.55} is P'(omega1) = 0.55, a different distribution. The Lagrange proof in Appendix A derives the Jeffrey update lambda P(. | psi) + (1 - lambda) P(. | not psi), which coincides with R^lambda_psi only when P(psi) <= lambda. Thus the minimality result, and the abstract's 'fewest possible changes' claim, need an explicit restriction to the case psi not in B_lambda_P (P(psi) < lambda); when P(psi) > lambda, psi is already believed and the minimal change is P itself under a constraint P'(psi) >= lambda. This is a localized mathematical error, not a flaw in Theorem 2, but it is load-bearing for the paper's minimal-change contribution.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":15380,"tokens_out":14063,"duration_ms":148666,"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":[{"comment":"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.","section":"§5, Proposition 4"},{"comment":"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.","section":"§5, Theorem 3(i) proof"},{"comment":"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.","section":"§5, Theorem 3(ii) proof"}],"minor_comments":[{"comment":"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.","section":"§2, last paragraph"},{"comment":"The name 'Hannson' in the first section should be 'Hansson'.","section":"§1"},{"comment":"The sentence 'Also in this in section we characterize...' contains a typo and should be rewritten.","section":"§5, paragraph after Definition 3"},{"comment":"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)).","section":"Appendix A, Fact 2 proof"},{"comment":"When P(ψ)=λ, the two cases in Proposition 3 coincide; stating this explicitly would avoid a small ambiguity in the piecewise definition.","section":"§5, Proposition 3"}],"recommendation":"major_revision","confidential_remarks":"The deductive-closure half of the paper is solid and novel; the problems are concentrated in the minimal-change section, and the fixes are clear and local. I would not reject the paper on these grounds, but the claims of KL-minimality and the proof of Theorem 3 need to be corrected before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe 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.\n\nWhat 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.\n\nWhere 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.\n\nTheorem 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.\n\nThe 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.\n\nNet: 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.","headline":"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.","tokens_in":16045,"tokens_out":4345,"would_cite":true,"duration_ms":44841,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["03B48","03B42"],"pacs":[],"model":"deepseek-v4-flash","headline":"Lockean belief sets close under logical deduction exactly when some world outweighs all strictly less probable worlds combined.","keywords":["Lockean thesis","deductive closure","belief revision","minimal change","probability threshold","step probability","principal filter","relative entropy"],"falsifier":"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.","tokens_in":14804,"feed_emoji":"🎲","tokens_out":19047,"duration_ms":169835,"temperature":0.7,"pith_summary":"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.","feed_headline":"Dominant worlds make Lockean beliefs deductively closed","feed_subtitle":"Threshold belief is deductively closed exactly when one world dominates, with a minimal-change update to keep it closed.","key_machinery":"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 $P'(\\psi) = \\lambda$.","core_discovery":"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 $\\psi$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Originates the Lockean thesis that defines beliefs by probability thresholds.","marker":"[9]"},{"why":"Sets out the belief-revision postulates that motivate and test deductive closure.","marker":"[1]"},{"why":"Introduces big-stepped probabilities, the step notion that the paper weakens to its omega-step condition.","marker":"[3]"},{"why":"Presents the closest prior account of closed probabilistic belief sets and the stability condition the paper compares against.","marker":"[17]"},{"why":"Provides the conditionalization rule that the revision operator reduces to when the input formula is disbelieved.","marker":"[14]"},{"why":"Applies a variant of that conditionalization to belief revision, the approach the paper builds on for minimal change.","marker":"[5]"}],"fun_headline_variants":["Dominant worlds ensure deductively closed Lockean beliefs","Minimal revision keeps Lockean beliefs deductively closed","Lockean closure iff a world dominates in probability","Fewest changes to close Lockean beliefs under logic","A dominant world gives deductive closure for Lockean sets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Dominant worlds ensure deductively closed Lockean beliefs","Minimal revision keeps Lockean beliefs deductively closed","Lockean closure iff a world dominates in probability","Fewest changes to close Lockean beliefs under logic","A dominant world gives deductive closure for Lockean sets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00022,"raw_usage":{"total_tokens":1513,"prompt_tokens":1079,"completion_tokens":434,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":695,"completion_tokens_details":{"reasoning_tokens":358}},"tokens_in":695,"tokens_out":434,"duration_ms":5028,"temperature":1.0,"reasoning_tokens":358,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:15:04.589355+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Oxford University Press, Oxford (1992)","cited_arxiv_id":null,"evidence_quote":"Originates the Lockean thesis that defines beliefs by probability thresholds."},{"cited_title":"Journal of Symbolic Logic 50, 510–530 (1985) On Lockean beliefs that are deductively closed and minimal change 15","cited_arxiv_id":null,"evidence_quote":"Sets out the belief-revision postulates that motivate and test deductive closure."},{"cited_title":"Journal of Logic and Computation9(6), 873–895 (1999)","cited_arxiv_id":null,"evidence_quote":"Introduces big-stepped probabilities, the step notion that the paper weakens to its omega-step condition."},{"cited_title":"Annals of Pure and Applied Logic 164, 1338–1389 (2013)","cited_arxiv_id":null,"evidence_quote":"Presents the closest prior account of closed probabilistic belief sets and the stability condition the paper compares against."},{"cited_title":"University of Chicago Press, Chicago, 2nd edn","cited_arxiv_id":null,"evidence_quote":"Provides the conditionalization rule that the revision operator reduces to when the input formula is disbelieved."},{"cited_title":"Annals of Mathematics and Artificial Intelligence87, 259–291 (2019)","cited_arxiv_id":null,"evidence_quote":"Applies a variant of that conditionalization to belief revision, the approach the paper builds on for minimal change."}],"review_version":1}