{"id":"1810c3d4-47d2-47b9-8ad9-a110a4f65819","arxiv_id":"2411.18849","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Material consequence is always classical, {1}-preservation is supervaluationist, (0,1]-preservation is subvaluationist, and symmetric consequence becomes classical only at threshold 1.","lead":"This paper analyzes logical consequence defined by probabilities, comparing material, preservation, and symmetric consequence over classical propositional logic. It shows that certainty preservation equals supervaluationism, positive-probability preservation equals subvaluationism, and only a new symmetric consequence gradually becomes classical as the threshold rises to 1.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Fact 49's proof asserts CI2 is valid for all α with 0.5∉α, but CI2 is invalid for α=(0.6,1]; nontransitivity unproved for thresholds below 2/3.","rationale":"I read the paper in good faith and independently checked the central characterizations. Facts 11, 12, 40, 46, and 47 appear correct: the proofs are sound up to minor presentational issues (e.g., the singleton-measurability concern in Fact 11 is fixable via Dirac measures, and the finite-scope caveat is explicitly stated). The counting results in Section 6 and 7 are well-supported by the supplied lemmas and by the cited results of Adams and Knight, though those external theorems are not proved within the paper. The reader's weakest assumption—finite arguments—is real but explicitly scoped, and it does not undermine the stated theorems. However, the proof of Fact 49 contains a false intermediate claim: it asserts CI2 is α-symmetric valid for every α with 0.5∉α and α≠{1}, but CI2 is invalid for α=(0.6,1], as shown. This leaves the nontransitivity claim for thresholds between 0.5 and 2/3 unsupported as written. The central claims of the paper are not threatened, but the structural results in Section 7.4 require a repaired proof or a restricted statement, so a conditional acceptance is appropriate.","tokens_in":25932,"tokens_out":61467,"duration_ms":491887,"concrete_test":"Run a linear-programming search over small finite languages (e.g., up to 3 atoms) for a counterexample to transitivity of α-symmetric consequence with α=[0.6,1]: find finite Γ, ∆, ϕ such that Γ⊢∆ is invalid, Γ⊢∆,ϕ is valid, and ϕ,Γ⊢∆ is valid. If such a counterexample exists, Fact 49's statement is true and only its proof needs repair; if none exists for all small exhaustive searches, Fact 49's statement is false for this α, requiring a revision of the theorem.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In the proof of Fact 49 (§7.4), the authors claim: 'By our assumptions on α, we know from Fact 46 that CI2 is α-symmetric valid and that there is some k such that CIk is not α-symmetric valid.' The first conjunct is false. CI2 is p,q ⊢ p∧q, of size 3, and by Fact 46 it is (2/3,1]-symmetric valid and [2/3,1]-symmetric invalid. For α=(0.6,1] (which satisfies 0.5∉α and α≠{1}), the upset is wider than (2/3,1], so CI2 is α-symmetric invalid: take a model with p(p)=p(q)=0.65 and p(p∧q)=0.3, giving p(p),p(q)∈(0.6,1] and p(p∧q)∈[0,0.4), the mirror of (0.6,1]. Thus the proof does not establish nontransitivity for thresholds in (0.5, 2/3]. This is a concrete gap in a stated theorem about Tarskian properties of symmetric consequence; it does not affect the paper's main characterization results (Facts 11, 12, 40, 46, 47), which appear correct after independent verification.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops three probabilistic accounts of logical consequence over finite Set-Set arguments in a classical propositional language, using upsets α ⊆ [0,1] as sets of 'good' probabilities. Material consequence is shown to coincide with classical validity for every upset (Fact 9). Preservation consequence is characterized at the extreme upsets: {1}-preservation is supervaluationist validity and (0,1]-preservation is subvaluationist validity (Facts 11–12). For intermediate upsets, the paper gives sufficient conditions for invalidity, proves that distinct preservation relations are usually incomparable, and establishes a continuum of distinct relations (Theorem 32, Corollary 33). Symmetric consequence is then introduced: it is monotone, becomes classical exactly at {1}, and satisfies sharp threshold results for minimally classically valid arguments (Facts 40, 46–47). The paper closes with structural properties of symmetric consequence, including reflexivity and transitivity claims (Facts 48–49), and states an open conjecture about preservation consequence (Conjecture 24).","tokens_in":26170,"tokens_out":56671,"duration_ms":474732,"significance":"If the main results stand, this is a valuable contribution to probability logic and multiple-conclusion logic. The paper extends Paris's and Knight's Set-Fmla analyses to the Set-Set setting in a way that reveals new identifications: supervaluationism and subvaluationism appear as probability-preservation at the extreme upsets, and symmetric consequence approaches classical logic with an exact threshold (n−1)/n for minimally valid arguments of size n. The paper is commendably explicit about its assumptions: all arguments are finite, the probability notion is classical, and the three cited external results (Adams's Theorem 20, Knight's Theorems 29 and 46) are clearly flagged rather than hidden. Several proofs, notably Fact 18 and Lemma 28, are substantial and appear correct, and the open Conjecture 24 is honestly labelled as unproved and is not used to derive other results. These strengths make the paper worthy of publication if the issues below are addressed.","major_comments":[{"comment":"The proof asserts that 'By our assumptions on α, we know from Fact 46 that CI2 is α-symmetric valid' whenever .5∉α and α≠{1}. This is false. CI2 is p,q ⊢ p∧q, which has size 3; by Fact 46 it is ((2/3),1]-symmetric valid and [2/3,1]-symmetric invalid. For α=(0.6,1], which satisfies .5∉α and α≠{1}, CI2 is α-symmetric invalid: a model with p(p)=p(q)=0.65 and p(p∧q)=0.3 gives p(p),p(q)∈(0.6,1] and p(p∧q)∈[0,0.4), the mirror of (0.6,1]. Thus the displayed proof does not establish nontransitivity for open thresholds in (0.5, 2/3]. Since Fact 49 is the basis for the paper's claim that symmetric consequence is non-Tarskian except at {1}, this gap is load-bearing and must be repaired.","section":"§7.4, proof of Fact 49"},{"comment":"The biconditional in Fact 49 as stated is false for α=(0.5,1]. For this α, suppose Γ⊨Δ,φ and φ,Γ⊨Δ are both α-symmetric valid, and put S=Γ∪¬Δ. The two assumptions say that S∪{¬φ} and S∪{φ} are both α-unsatisfiable. If S were α-satisfiable, the set P={p: p(s)>0.5 for all s∈S} would be a nonempty relatively open subset of the finite probability simplex. The two unsatisfiability conditions force p(φ)=0.5 for every p∈P, which is impossible for a nonempty open set. Hence S is α-unsatisfiable and Γ⊨Δ, so the relation is transitive. Therefore the conclusion drawn after Fact 49 that only {1}-symmetric consequence is fully Tarskian is incorrect: (0.5,1]-symmetric consequence is reflexive (Fact 48), monotone (Fact 38), and transitive, although it is not classical (CI2 is invalid). The corrected classification should exclude (0.5,1] from the nontransitive case; nontransitivity occurs when the threshold of α lies strictly between 0.5 and 1.","section":"§7.4, statement of Fact 49"}],"minor_comments":[{"comment":"In the left-to-right direction, the sentence 'To see that ⟨W,A,JK,p⟩ is a {1}-preservation counterexample' should refer to the newly defined probability function p′ rather than the original p; as written it attributes the counterexample to the wrong probability function.","section":"§4.2, proof of Fact 11"},{"comment":"The displayed tuple in Fact 4 contains a typo ('pΓ p )⟩'), and the notation 'JpKΓ p' is cumbersome; please clean up the superscripts and add a missing parenthesis.","section":"§2.1, Fact 4"},{"comment":"In the six-sided die example, the letter p is used both for the proposition 'the die comes up > 1' and for the probability function p(·); even given the paper's convention of treating p ambiguously, this makes the example harder to read. A different letter for the proposition would help.","section":"§4.1, die example"}],"recommendation":"major_revision","confidential_remarks":"The main characterization results (Facts 11, 12, 40, 46, 47, and the counting results for preservation consequence) appear correct and are substantial. The Fact 49 problem is localized but affects the paper's advertised conclusion about Tarskianness: the statement needs correction and the proof needs a genuine construction for thresholds above 0.5. This is fixable within the paper's scope, so I do not recommend rejection, but the manuscript should not be accepted until Fact 49 and the surrounding discussion are revised."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the core results are right and worth engaging. In Set-Set, {1}-preservation is supervaluationism (Fact 11), (0,1]-preservation is subvaluationism (Fact 12), {1}-symmetric consequence is classical (Fact 40), and Facts 46–47 give exact size thresholds for minimally valid arguments. That is a clean extension of Paris and Knight, and the open-vs-closed threshold distinction at rationals is handled carefully (Lemma 28, Theorems 31–32). The paper is also honest: Conjecture 24 is labeled open and not used to prove anything else, the three cited external results (Adams's Theorem 20, Knight's Theorems 3.5 and 4.14) are flagged as unproved here, and the finite-argument assumption is explicit and genuinely load-bearing.\n\nWhere it is soft: the proof of Fact 49 (transitivity of symmetric consequence) has a real gap. The text says that from the assumptions on α we know via Fact 46 that CI2 is α-symmetric valid. That only follows when α is contained in (2/3,1], i.e. when the threshold is at least 2/3 and the upset is open at that threshold. For α=(0.6,1], CI2 is not valid: take p(p)=p(q)=0.65 and p(p∧q)=0.3, giving an α-symmetric counterexample. So the nontransitivity half of Fact 49 is unproved as written for thresholds in (0.5,2/3]. The theorem may well be true—a different witness argument or a direct counterexample construction would likely fix it—but the proof as given does not cover those thresholds. This is a local gap in a side structural theorem, not in the main characterization results; I independently checked Facts 11, 12, 40, 46, and 47 and they hold.\n\nThe citation pattern is appropriate: the self-citations are to connected work on tolerance and degrees of truth, not padding. The paper is mathematically careful, clearly written, and advances the subfield. The main caveats are the three cited theorems taken on trust and the finite-argument restriction, both of which the authors openly acknowledge.\n\nWho this is for: anyone working in probability logic, multiple-conclusion logic, or the formal semantics of super- and subvaluationism. It deserves a serious referee, not a desk reject. I would send it out, ask for a fix to the Fact 49 proof, and otherwise expect minor revisions.","headline":"A genuinely useful paper: it moves Paris/Knight preservation results to Set-Set, identifies supervaluationism/subvaluationism at the extreme upsets, and maps symmetric consequence's gradual approach to classical logic, with one local proof gap in Fact 49.","tokens_in":26743,"tokens_out":5806,"would_cite":true,"duration_ms":50211,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["03B48","03B60"],"pacs":[],"model":"deepseek-v4-flash","headline":"Preserving probability 1 is supervaluationism, and symmetric consequence reaches classical logic only at certainty.","keywords":["probabilistic consequence","multiple-conclusion logic","supervaluationism","subvaluationism","probability preservation","symmetric consequence","threshold logic","classical validity"],"falsifier":"Consider the three-ticket lottery argument: let $p$ say ticket 1 loses, $q$ say ticket 2 loses, and take the argument $p, q \\vdash p \\land q$, which has size 3 and no classically valid proper subargument. Fact 46 predicts that at threshold $2/3$ the closed upset $[2/3,1]$ admits a symmetric counterexample (the uniform lottery gives $p(p)=p(q)=2/3$ and $p(p\\land q)=1/3$), while the open upset $(2/3,1]$ does not. So test whether any probability distribution has both $p(p)>2/3$ and $p(q)>2/3$ with $p(p\\land q)<1/3$: if one exists, the paper's exact-threshold claim is false, and if none exists, the standard lower bound on conjunction probabilities confirms the threshold.","tokens_in":25696,"feed_emoji":"🎲","tokens_out":10760,"duration_ms":87045,"temperature":0.7,"pith_summary":"This paper asks what logical validity becomes when it is defined by preservation of probability instead of preservation of truth, and when arguments may have multiple conclusions as well as multiple premises. In this multiple-conclusion setting, preserving certainty from premises to at least one conclusion is not classical validity: it is exactly supervaluationist validity, and preserving merely positive probability is exactly subvaluationist validity. The paper's third notion, symmetric consequence, compares premises against an upper threshold and conclusions against the mirror-image lower threshold, so that as the threshold rises toward certainty the logic gradually strengthens; at the extreme threshold $\\{1\\}$ it coincides exactly with classical validity. The paper also shows that a classically valid argument of size $n$ first becomes symmetric-valid at threshold $(n-1)/n$, and that there are continuum many distinct probabilistic consequence relations.","feed_headline":"Certainty preservation is supervaluationism, not classical logic","feed_subtitle":"Probability-1 preservation gives supervaluationism; symmetric consequence approaches classical logic gradually.","key_machinery":"The central object is an upset $\\alpha \\subseteq [0,1]$ of probabilities treated as good: it must contain 1, exclude 0, and be upward closed, so it is determined by its threshold $\\inf \\alpha$ together with openness or closedness. A probabilistic model is a finite-additive probability space whose atoms are worlds carrying classical truth values, and every finite probability distribution can be represented on the finite set of worlds determined by the argument's atoms. Preservation consequence uses $\\alpha$ for premises and the complement of $\\alpha$ for conclusions; symmetric consequence uses $\\alpha$ for premises and its mirror image $\\overline{\\alpha} = \\{x : 1-x \\in \\alpha\\}$ for conclusions, which links the proof of Fact 51 to duality and makes Fact 37 turn symmetric validity into $\\alpha$-unsatisfiability of $\\Gamma \\cup \\lnot\\Delta$. Finiteness is what lets the proofs work: to build a $\\{1\\}$-preservation counterexample from a supervaluationist one, the paper spreads probability uniformly over one falsifying world per conclusion, and to extract a classical counterexample it uses $p(\\bigwedge\\Gamma)=1$ and $p(\\bigvee\\Delta)=0$ to locate a world outside the conclusion set. The precise threshold results rest on the size $|\\Gamma \\cup \\lnot\\Delta|$ of an argument and on minimally classically valid subarguments.","core_discovery":"The central discovery is that the move from single-conclusion to multiple-conclusion arguments changes the probabilistic meaning of classicality. For preservation consequence, an argument is valid when no model gives every premise a probability in the 'good' set $\\alpha$ and no conclusion a probability in $\\alpha$; at the smallest upset $\\{1\\}$ this relation is shown by Fact 11 to be supervaluationist validity, and at the largest upset $(0,1]$ it is shown by Fact 12 to be subvaluationist validity. Material consequence, which rolls the whole argument into one conditional sentence, remains classically valid for every $\\alpha$. Symmetric consequence instead counts an argument as valid when every model that puts all premises in $\\alpha$ also puts some conclusion outside the mirror image $\\overline{\\alpha} = \\{x : 1-x \\in \\alpha\\}$; Facts 40, 46 and 47 show this relation strengthens monotonically as $\\alpha$ narrows, reaches classical validity exactly at $\\{1\\}$, and first validates a minimally classically valid argument of size $n$ at threshold $(n-1)/n$. These results extend earlier Set-Fmla probabilistic entailment results to the Set-Set framework and to open thresholds.","pith_inferences":["The same two-threshold template could characterize other multiple-conclusion nonclassical logics: allowing the premise threshold and conclusion threshold to move independently generalizes symmetric consequence toward the tolerant and strict logics studied in fuzzy logic, and the paper's Facts 39 and 51 give a natural bisection of that space.","Because the threshold at which an argument becomes valid depends on its finite size $|\\Gamma \\cup \\lnot\\Delta|$, symmetric consequence suggests a resource reading of certainty: the more independent sentences an inference needs, the more probability the premises must carry, which could connect to quantitative proof complexity.","Fact 44 implies that closed thresholds are never the first point of validity for symmetric consequence, so the logic changes only when passing from a closed upset at a rational threshold to the adjacent open upset; this makes the set of rational numbers the exact discontinuity set, a fact that could be studied as a stability or learnability property of probabilistic consequence."],"forward_implications":["Preserving certainty does not recover full classical logic once conclusions are allowed to be multiple: the $\\{1\\}$-preservation relation is supervaluationist, so for instance $p \\lor \\lnot p$ does not entail the pair $p, \\lnot p$, and $p, \\lnot p$ does not entail $p \\land \\lnot p$.","Every $\\alpha$-preservation consequence relation is Tarskian, and each one is either weakly paracomplete or weakly paraconsistent according to whether $0.5$ is in $\\alpha$; none is self-dual.","Symmetric consequence relations form a linear chain as $\\alpha$ narrows, are never fully Tarskian except at $\\{1\\}$, and still number continuum many distinct relations; the only thresholds at which a closed and an open upset give different symmetric logics are the rational numbers.","Larger arguments need higher certainty: a classically valid argument of size $n$ is guaranteed to be symmetric-valid at every upset with threshold above $(n-1)/n$.","Material consequence is indifferent to the threshold: it coincides with classical validity for every upset, and so it is the only one of the three notions that never departs from classical logic."],"supporting_citations":[{"why":"Supplies Theorem 20, used to construct $\\alpha$-preservation counterexamples from minimally sufficient premise sets, and the Set-Fmla claim that probability-1 preservation coincides with classical entailment.","marker":"Adams 1998"},{"why":"Gives the earlier Set-Fmla sound and complete axiomatization of probabilistic preservation logics that this paper extends to the Set-Set framework and to open thresholds.","marker":"Paris 2004"},{"why":"Provides Theorem 29 on maximal rational satisfiability thresholds and Fact 46 on the exact threshold $(n-1)/n$ for minimally valid arguments.","marker":"Knight 2002"},{"why":"Provides Fact 18, which lets probabilities of premises be preserved while forcing the conclusion's probability to 0.","marker":"Adams and Levine 1975"},{"why":"Supplies the structural properties of supervaluationist and subvaluationist consequence that the paper transfers to $\\{1\\}$- and $(0,1]$-preservation.","marker":"Kremer and Kremer 2003"},{"why":"Defines the supervaluationist framework whose validity notion is identified with $\\{1\\}$-preservation in Fact 11.","marker":"Fine 1975"},{"why":"Defines subvaluationist validity and weak paraconsistency, which anchor the identification in Fact 12 and the structural discussion.","marker":"Hyde 1997"}],"fun_headline_variants":["Preservation at certainty gives supervaluationism, not classical logic","Symmetric consequence is classical only at threshold 1","Material consequence stays classical; preservation splits at extremes","At certainty, preservation logic is supervaluationism; at positive, subvaluationism","Threshold choice flips preservation logic between super and subvaluationism"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central identifications all assume arguments are finite sets of premises and conclusions, since the proofs spread probability over finitely many worlds and infer properties of finitely many conjunctions and disjunctions; allow infinite arguments and the match with supervaluationism, subvaluationism, and classical logic would need new proofs and might fail.","fun_headline_variants_meta":{"raw":{"variants":["Preservation at certainty gives supervaluationism, not classical logic","Symmetric consequence is classical only at threshold 1","Material consequence stays classical; preservation splits at extremes","At certainty, preservation logic is supervaluationism; at positive, subvaluationism","Threshold choice flips preservation logic between super and subvaluationism"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000751,"raw_usage":{"total_tokens":3322,"prompt_tokens":901,"completion_tokens":2421,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":517,"completion_tokens_details":{"reasoning_tokens":2336}},"tokens_in":517,"tokens_out":2421,"duration_ms":16467,"temperature":1.0,"reasoning_tokens":2336,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T10:50:23.892979+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Consider the three-ticket lottery argument: let $p$ say ticket 1 loses, $q$ say ticket 2 loses, and take the argument $p, q \\vdash p \\land q$, which has size 3 and no classically valid proper subargument. Fact 46 predicts that at threshold $2/3$ the closed upset $[2/3,1]$ admits a symmetric counterexample (the uniform lottery gives $p(p)=p(q)=2/3$ and $p(p\\land q)=1/3$), while the open upset $(2/3,1]$ does not. So test whether any probability distribution has both $p(p)>2/3$ and $p(q)>2/3$ with $p(p\\land q)<1/3$: if one exists, the paper's exact-threshold claim is false, and if none exists, the standard lower bound on conjunction probabilities confirms the threshold.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Theorem 20, used to construct $\\alpha$-preservation counterexamples from minimally sufficient premise sets, and the Set-Fmla claim that probability-1 preservation coincides with classical entailment."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the earlier Set-Fmla sound and complete axiomatization of probabilistic preservation logics that this paper extends to the Set-Set framework and to open thresholds."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides Theorem 29 on maximal rational satisfiability thresholds and Fact 46 on the exact threshold $(n-1)/n$ for minimally valid arguments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides Fact 18, which lets probabilities of premises be preserved while forcing the conclusion's probability to 0."},{"cited_title":"and Kremer, M","cited_arxiv_id":null,"evidence_quote":"Supplies the structural properties of supervaluationist and subvaluationist consequence that the paper transfers to $\\{1\\}$- and $(0,1]$-preservation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the supervaluationist framework whose validity notion is identified with $\\{1\\}$-preservation in Fact 11."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines subvaluationist validity and weak paraconsistency, which anchor the identification in Fact 12 and the structural discussion."}],"review_version":1}