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Vagueness and the Connectives

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Pith's one-line read The paper establishes that the Sorites premises—extremes plus denial of sharp cutoffs—are jointly consistent in orthologic and in fundamental logic, so the paradox does not force classical or intuitionistic logic, and fundamental logic is…

desk verdict Solid new models for the Sorites in orthologic, compatibility logic, and fundamental logic; the central consistency results hold, and Fact 10 has a one-character typo that should be fixed before publication. read the letter →

arxiv 2412.00356 v2 pith:SFLW4XZY submitted 2024-11-30 cs.LO

Wesley H. Holliday This is my paper · ORCID
classification cs.LO MSC 03B2003B4503B60
keywords Soritesparadoxvaguenessorthologicfundamentallogiccompatibilityepistemicmodalsnon-classicalexcludedmiddle
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

Vague predicates like 'young' generate a paradox when we assert the extremes and deny any sharp cutoff. The paper asks what logic we should retreat to if we also accept that epistemic modals like 'might' invalidate the distributive laws. It tries to establish that these Sorites premises are jointly satisfiable in orthologic (logic without distributivity) and in fundamental logic (which further drops excluded middle), and that the latter offers the most natural common base. The paper works by building explicit relational models with partial states, one symmetric and one pseudosymmetric, in which the offending conjunction is forced at a distinguished state.

What carries the argument

The central object is a relational model with an 'openness' relation $\mathrel{\rhd}$, read as 'x does not reject any proposition that y accepts.' A valuation is required to be a fixed point of the closure operator $c_\rhd(A)=\{x : \forall x'\rhd x\ \exists x''\lhd x' \text{ with } x''\in A\}$, which makes every formula's truth set a fixed point. Negation is interpreted as rejection along open states, and disjunction as the closure of the union, so a disjunction can be forced without either disjunct already being forced. In the Sorites models, states are pairs $(i,j)$ with $i+\delta<j$; the reflexive-symmetric symmetric model validates orthologic, and the pseudosymmetric model adds states $k$ for which $p_k\vee\neg p_k$ is rejected, validating fundamental logic. The spacing constraint $i+\delta<j$ is the workhorse: it forces $\neg(p_k\wedge\neg p_{k+\ell})$ for every $\ell\le\delta$, yielding the no-sharp-cutoff conjuncts.

What would settle it

A robust experimental demonstration that ordinary speakers accept sentences of the form '$φ$ and might not $φ$' as assertable without sarcasm or irony would falsify the paper's central motivation, since the rejection of distributivity—the ground for orthologic and fundamental logic—depends on those sentences being contradictions.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the formula $p_0 \wedge \neg p_{n-1} \wedge \bigwedge \neg(p_k \wedge \neg p_{k+\ell})$ is satisfiable, not only in classical or intuitionistic semantics, but in orthologic and in fundamental logic. The symmetric Sorites model uses states $(i,j)$ with $i+\delta<j$, letting $i$ mark the last predicate that holds and $j$ the first that is negated, so the gap $\delta$ blocks sharp cutoffs of length up to $\delta$. Since the model is reflexive and symmetric, it is a model of orthologic; with no constraints on the valuation beyond the fixed-point property, it also serves the compatibility semantics for weak pseudocomplementation. The pseudosymmetric model adds one state per $k$ that rejects $p_k\vee\neg p_k$, so the same formula stays consistent while excluded middle acquires expressive content, marking exactly where the series has a fact of the matter. The paper concludes that fundamental logic—obtained by dropping both distributivity and double-negation elimination—accommodates the non-classicality of epistemic modals and of vagueness together.

Load-bearing premise

The paper takes for granted the arguments that 'It's raining and it might not be raining' is a contradiction, and that this undermines the distributive laws; if those natural-language data are wrong, the case for orthologic and fundamental logic as the base for vagueness loses its footing.

Editorial extensions

If this is right

  • The Sorites premises are satisfiable in orthologic and compatibility logic, so denying distributivity is a sufficient logical response to the paradox.
  • The same premises are satisfiable in fundamental logic, so the paradox does not require accepting excluded middle.
  • In the fundamental approach, each instance of excluded middle $p_k\vee\neg p_k$ is forced exactly at states that settle the $k$-th item, giving a formal way to express 'there is a fact of the matter' versus 'no fact of the matter.'
  • The conjunctive and disjunctive versions of the Sorites behave differently in the fundamental treatment: $\neg(p_k\wedge\neg p_{k+1})$ is forced everywhere, but $\neg p_k\vee p_{k+1}$ is not, so one can accept the former without the latter.
  • The Gödel–Gentzen translation embeds orthologic faithfully into fundamental logic, so moving to fundamental logic loses no orthological reasoning.

Reading between the lines

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

  • The $\delta$ gap that limits sharp cutoffs is a free parameter; fitting it to psychophysical just-noticeable differences could turn the model into a quantitative account of granularity in vague predicates.
  • The paper's dual treatment suggests a unified picture: both 'might' contradictions and sharp cutoffs are states that block common refinement, so one semantic notion of openness handles modal and vagueness incompatibility.
  • If fundamental logic becomes the base, then $p\vee\neg p$ becomes an expressive assertion about determinacy; this could give a formal language for debates about whether certain mathematical or metaphysical questions have an objective answer.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 4 minor

Summary. The paper investigates which non-classical base logic for conjunction, disjunction, and negation can accommodate two sources of non-classicality: epistemic modals, which according to earlier work motivate dropping distributivity and moving to orthologic, and vagueness, which has motivated weakening classical logic along intuitionistic or Finean lines. It recalls the systems of fundamental logic, orthologic, compatibility logic, intuitionistic logic, and classical logic, together with relational fixed-point models. The paper's main formal contribution is a symmetric Sorites model S_{n,δ} in which the extremes p0 and ¬p_{n−1} together with all no-sharp-cutoff claims ¬(p_k∧¬p_{k+ℓ}) are jointly forced (Fact 4), yielding consistency in orthologic and compatibility logic (Fact 5). It then defines a pseudosymmetric model S_{n,δ} in which excluded middle instances are no longer tautologies but express settledness, and Facts 6–10 show the corresponding consistency in fundamental logic. The paper argues that fundamental logic is a natural common weakening and includes a discussion of costs and benefits relative to orthologic, including the Gödel–Gentzen embedding of orthologic into fundamental logic.

Significance. If the main model-theoretic results stand, the paper gives a clear and explicit demonstration that the Sorites constellation 'extremes plus no sharp cutoffs' does not force classical or intuitionistic logic, and that the retreat to orthologic or fundamental logic is coherent. The constructions are transparent, the main facts are proved directly, and the paper is candid about its reliance on the epistemic-modal data of [24], which it explicitly takes as a premise rather than re-deriving. The paper therefore makes a solid contribution to the debate on the right non-classical base logic for vagueness and epistemic modals, even though the philosophical conclusions are appropriately conditional and the completeness theorems for compatibility logic are only sketched.

major comments (1)
  1. [§6, Fact 10] Fact 10 as printed is false. By Fact 9, the extension of p0∨¬p0 is {(i,j)∈S | 0≤i or j≤0}. For n≥2 the state (−∞,n−1) satisfies neither disjunct, so it does not force p0∨¬p0 and hence cannot belong to the extension of the conjunction of all instances of excluded middle. The correct second state is (−∞,0), since j=0≤k for every k∈n. With that correction, the second equality—that the negation of the conjunction is forced exactly at the states with finite coordinates—does hold; as printed, that equality also fails, because (−∞,0) is related to (−∞,n−1) and would therefore not lie in the indicated set. Because Fact 10 is cited in §7 as the model-theoretic witness of Fine's 'global indeterminacy' point, it should be corrected before publication. This is a local one-symbol error: Facts 4–9 and the consistency results for orthologic, compatibility logic, and fundamental logic are unaffected.
minor comments (4)
  1. [§2, after Definition 2] The rule of proof-by-cases with side assumptions is misprinted: the conclusion should be α∧(ϕ∨ψ)⊢χ, not α∧(ϕ∨ψ)⊢ψ. As written, the rule does not match the informal description or the intended notion of compatibility logic.
  2. [§2, after Definition 2] The word 'psuedocomplementation' should be 'pseudocomplementation'.
  3. [§6, proof of Fact 7] In the second equation, the forward direction only considers successors (i′,j′) in S, but the frame also contains the integer states ℓ∈n, and Definition 12.4 makes (i,j)⊳ℓ for every ℓ. The proof should explicitly note that no integer state forces ¬pk (because (k,∞)⊳ℓ and (k,∞)⊩pk); the claim is true, but the proof as written is incomplete.
  4. [§4, Theorems 4 and 5] The completeness proof for compatibility logic is only a sketch; since these theorems are not needed for the central satisfiability results, this is acceptable, but a little more detail on why the Stone representation preserves weak pseudocomplementation would improve self-containedness.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the consistency results are direct model-theoretic constructions, with self-citations confined to background definitions and explicitly assumed motivations.

full rationale

The paper's central formal claims—that the Sorites premises are jointly satisfiable in orthologic, compatibility logic, and fundamental logic—are established by explicit model constructions in §§5–6. Facts 4, 5, 8, 9, and 10 are computed directly from Definitions 10–12 and the forcing clauses of Definition 6, not derived by restating prior conclusions. The soundness theorems invoked (Theorem 1 from Goldblatt [17], Theorem 5 proved in the paper, Theorem 2 from the author's [21]) are used only to connect these models to the target logics; they are independent published results rather than placeholders for the paper's conclusions. The paper explicitly flags that it is 'simply taking for granted here the arguments against distributivity involving epistemic modals from [24]' (§7), so that reliance is a stated conditional premise, not a hidden circular step. The same applies to fundamental logic, whose definition and proof theory come from [21]; the new contribution is the pseudosymmetric Sorites model showing consistency in that system. There are no fitted parameters renamed as predictions, no uniqueness theorem imported to force a conclusion, and no ansatz smuggled in via citation. The apparent error in Fact 10's printed extension—the second state should be (−∞,0) rather than (−∞,n−1)—is a mathematical correctness issue (likely a one-symbol typo) rather than a circularity, since it does not make the conclusion depend on its own input. Overall, the derivation chain is self-contained and non-circular.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claims rest on standard mathematical background and on philosophical premises about natural-language data and Fine's framework. No non-mathematical entities are introduced. The main new content is model-theoretic and is proved explicitly.

free parameters (1)
  • δ (tolerance width in Sorites models)
    A model parameter in Definitions 11 and 12; the satisfiability facts are proved for all δ in the range 1 ≤ δ < n-1, so no specific value is fitted to data.
assumptions (4)
  • standard math ZFC set theory and standard algebraic facts (Stone representation, prime filters) for model constructions
    Used implicitly throughout, e.g., in the proof sketch of Theorem 4.
  • domain assumption Epistemic-modal data from [24]: sentences like p ∧ ♦¬p are contradictions, so proof-by-cases with side assumptions and pseudocomplementation are invalid
    The paper states it takes these arguments for granted (§7, p.17); they motivate rejecting distributivity and thus moving to orthologic or fundamental logic.
  • domain assumption Fine's global approach to vagueness, including weak pseudocomplementation as the negation of compatibility logic
    The paper builds on Fine's framework and does not defend it against alternative approaches such as supervaluationism or epistemicism, noting in the conclusion that such comparisons remain future work.
  • domain assumption Soundness and completeness of fundamental logic with respect to reflexive pseudosymmetric fixpoint frames (Theorem 2 from [21])
    Used in Proposition 4 to infer soundness of fundamental logic for the pseudosymmetric Sorites model; accepted from prior work without reproof.

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Cite this review

Pith. "Pith review of Vagueness and the Connectives." pith.science (2026). https://pith.science/paper/SFLW4XZY

@misc{pith2026241200356,
  author       = {Pith},
  title        = {Pith review of: Vagueness and the Connectives},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SFLW4XZY}},
  note         = {Machine review of arXiv:2412.00356}
}
read the original abstract

Challenges to classical logic have emerged from several sources. According to recent work, the behavior of epistemic modals in natural language motivates weakening classical logic to orthologic, a logic originally discovered by Birkhoff and von Neumann in the study of quantum mechanics. In this paper, we consider a different tradition of thinking that the behavior of vague predicates in natural language motivates weakening classical logic to intuitionistic logic or even giving up some intuitionistic principles. We focus in particular on Fine's recent approach to vagueness. Our main question is: what is a natural non-classical base logic to which to retreat in light of both the non-classicality emerging from epistemic modals and the non-classicality emerging from vagueness? We first consider whether orthologic itself might be the answer. We then discuss whether accommodating the non-classicality emerging from epistemic modals and vagueness might point in the direction of a weaker system of fundamental logic.

Figures

Figures reproduced from arXiv: 2412.00356 by the authors.

Figure 1
Figure 1. The symmetric Sorites model for n = 4 and δ = 1. Proposition 2. Sn,δ is a relational fixpoint model. Proof. We must show that V (pk) is a fixpoint of c≬. Given (i, j) ∈ S, we have i < j by Definition 11.1. Now suppose (i, j) 6∈ V (pk), so i < k by Definition 11.3. Let (i ′ , j′ ) = (−∞, k), so (i ′ , j′ ) ∈ S by Definition 11.1. Then max(i, i′ ) = i < min(j, j′ ), so (i ′ , j′ ) ≬ (i, j) by Definition 11.2. Now for … view at source ↗
Figure 2
Figure 2. The pseudosymmetric Sorites model for n = 4 and δ = 1. We indicate x ⊳ y by an arrow from y to x. The arrow from k to the dashed rectangle indicates arrows from k to all states inside the rectangle. An edge with no arrow head indicates arrows in both direction. Proposition 3. Sn,δ is a fixpoint model [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

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