Probabilistic consequence relations

Ellie Ripley; Paul \'Egr\'e

REVIEW 2 major objections 3 minor 38 references

Probabilistic consequence relations

T0 review · 2 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Preserving probability 1 is supervaluationism, and symmetric consequence reaches classical logic only at certainty.

desk verdict 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. read the letter →

arxiv 2411.18849 v2 pith:XKA6JZMR submitted 2024-11-28 math.LO cs.LOmath.PR

Paul Égré , Ellie Ripley This is my paper · ORCID

classification math.LOcs.LOmath.PR MSC 03B4803B60

keywords probabilisticconsequencemultiple-conclusionlogicsupervaluationismsubvaluationismprobabilitypreservationsymmetricthresholdclassicalvalidity

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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

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.

Where 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.

The 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.

Who 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.

Referee Report

2 major / 3 minor

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).

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 (2)

[§7.4, proof of Fact 49] 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.
[§7.4, statement of Fact 49] 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.

minor comments (3)

[§4.2, proof of Fact 11] 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.
[§2.1, Fact 4] 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.
[§4.1, die example] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: central results are proved from the definitions and from independent external cited theorems, not from their own targets.

full rationale

This paper contains no curve fitting and no derivation that assumes its own target. The characterization results, including Fact 11 ({1}-preservation validity iff supervaluationist validity), Fact 12 ((0,1]-preservation validity iff subvaluationist validity), Fact 40 ({1}-symmetric validity iff classical validity), and Fact 46 (Knight's threshold theorem), are each proved directly from the definitions of probabilistic models, upsets, and the three counterexample notions, or cited from independent external sources such as Adams and Levine (1975), Adams (1998), and Knight (2002). Fact 46 is explicitly cited to Knight (2002, Thm. 3.5), an external theorem with its own proof, and it is used as a lemma rather than as a restatement of the paper's conclusions. The open Conjecture 24 is explicitly labeled unproved and is not used to establish any later result. Self-citations such as Ripley (2013, 2017), Cobreros et al. (2024), and Egré et al. (2024) appear only as contextual pointers to related work and are not load-bearing in the proofs. The proof of Fact 49 in Section 7.4 appears to contain a genuine mathematical gap, since the claim that CI2 is α-symmetric valid for every α with 0.5 notin α fails for α = (0.6, 1]; however, a proof gap is a correctness issue, not a circularity issue, and it does not affect the main characterization results, which are independently verifiable from the definitions. No step reduces, by construction or by self-citation, to its own input, so the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper's results depend on the choice of finitely additive classical probability, the upset structure of thresholds, the restriction to finite arguments, and three cited theorems (Adams 1998, Knight 2002) that are used without proof in the text.

assumptions (6)

domain assumption Probabilities are real-valued, finitely additive measures on an algebra of worlds.
Definition 2; the entire paper works with this notion of classical probability rather than conditional or non-classical probability.
domain assumption The set alpha of 'good' probabilities is an upset containing 1 and excluding 0.
Section 2.3 restricts alpha to upsets; the open/closed threshold taxonomy and all classification results depend on this restriction.
domain assumption Arguments are finite sets of sentences.
Definition 1 restricts to finite Gamma and Delta; Fact 4, Fact 11, Fact 40, Fact 41 all rely on finiteness.
standard math Adams's Theorem 20 holds as cited.
Stated in Theorem 20 with proof deferred to Adams 1998; used in Fact 21 to build probability distributions, underpinning Corollaries 22 and 23 and Theorems 31 and 32.
standard math Knight's Theorem 29 (maximal satisfiability threshold is rational) holds as cited.
Stated in Theorem 29 with proof deferred to Knight 2002; used in Corollary 30, Theorems 31 and 42, and Fact 44.
standard math Knight's Fact 46 (minimal classically valid arguments become symmetrically valid at threshold (n-1)/n) holds as cited.
Stated as Fact 46 with no proof given; drives the size-based thresholds for symmetric consequence (Facts 46, 47) and the discussion of CIn and MPn.

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Cite this review

Pith. "Pith review of Probabilistic consequence relations." pith.science (2026). https://pith.science/paper/XKA6JZMR

@misc{pith2026241118849,
  author       = {Pith},
  title        = {Pith review of: Probabilistic consequence relations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XKA6JZMR}},
  note         = {Machine review of arXiv:2411.18849}
}

read the original abstract

This paper investigates logical consequence defined in terms of probability distributions, for a classical propositional language using a standard notion of probability. We examine three distinct probabilistic consequence notions, which we call material consequence, preservation consequence, and symmetric consequence. While material consequence is fully classical for any threshold, preservation consequence and symmetric consequence are subclassical, with only symmetric consequence gradually approaching classical logic at the limit threshold equal to 1. Our results extend earlier results obtained by J. Paris in a SET-FMLA setting to the SET-SET setting, and consider open thresholds beside closed ones. In the SET-SET setting, in particular, they reveal that probability 1 preservation does not yield classical logic, but supervaluationism, and conversely positive probability preservation yields subvaluationism.

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Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Adams, E. (1996). Four probability-preserving properties of inferences. Journal of Philosophical Logic , 25:1--24

work page 1996

[2] [2]

Adams, E. W. (1998). A Primer of Probability Logic . CSLI Publications, Stanford, California

work page 1998

[3] [3]

Adams, E. W. and Levine, H. P. (1975). On the uncertainties transmitted from premises to conclusions in deductive inferences. Synthese , 30(3/4):429--460

work page 1975

[4] [4]

Bergmann, M. (2008). An introduction to many-valued and fuzzy logic: semantics, algebras, and derivation systems . Cambridge University Press

work page 2008

[5] [5]

Biazzo, V., Gilio, A., Lukasiewicz, T., and Sanfilippo, G. (2005). Probabilistic logic under coherence: Complexity and algorithms. Annals of Mathematics and Artificial Intelligence , 45:35--81

work page 2005

[6] [6]

Blasio, C., Marcos, J., and Wansing, H. (2017). An inferentially many-valued two-dimensional notion of entailment. Bulletin of the Section of Logic , 46(3/4):233--262

work page 2017

[7] [7]

and \'Egr \'e , P

Chemla, E. and \'Egr \'e , P. (2019). Suszko's problem: Mixed consequence and compositionality. The Review of Symbolic Logic , 12(4):736--767

work page 2019

[8] [8]

Chemla, E., \'E gr \'e , P., and Spector, B. (2017). Characterizing logical consequence in many-valued logic. Journal of Logic and Computation , 27(7):2193--2226

work page 2017