Two Layers of Vagueness

1. Introduction

The Sorites paradox can be stated with deceptive simplicity. Consider a paradigmatic heap of sand constructed by adding grains one at a time. Formally, let H(n) abbreviate “n grains of sand make a heap.” The argument runs as follows:

Base premise: ¬H(1). A single grain of sand does not make a heap.

Tolerance premise: ∀n(¬H(n) → ¬H(n + 1)). If n grains do not make a heap, then n + 1 grains do not make a heap either; adding a single grain cannot transform a non-heap into a heap.

Conclusion: ∀n¬H(n). By mathematical induction from the two premises, no number of grains makes a heap.

Both premises appear compelling, and the inference is classically valid, yet a sufficiently developed state in the series is plainly a heap. The paradox therefore pressures us to reject the base premise, reject the universal tolerance premise, or revise the logic or semantics used for vague predicates.

Three families of response have dominated the literature.

Many-valued and degree-theoretic approaches revise classical semantics or logic. Some introduce one or more intermediate truth values between truth and falsity, while degree-theoretic approaches assign values on a scale, often the interval [0,1]. On some such accounts, adjacent predications may differ only slightly in truth value, allowing local tolerance judgments to remain highly acceptable without making the universal tolerance premise fully true. The exact treatment of conditionals, modus ponens, and logical consequence varies across theories. Such approaches consequently depart from classical bivalence or restrict some classical principles, although the extent of the revision depends on the particular system.

Supervaluationism evaluates vague discourse across admissible precisifications. A sentence is supertrue (or superfalse) when it is true (or false) on every admissible precisification. Each precisification of the fixed series places a sharp cutoff somewhere, so the universal tolerance premise fails on every precisification at that precisification’s cutoff, even though different precisifications locate the cutoff differently. Supervaluationism preserves classical theoremhood, but its treatment of validity and inference is not identical to classical semantics in every respect. It also faces the familiar problem of explaining admissibility and higher-order vagueness.

Epistemicism, most forcefully defended by Timothy Williamson, retains classical logic and bivalent semantics for vague language. For the fixed series, one transition changes the truth-value even though competent speakers cannot identify it. Borderline cases are therefore not truth-value gaps; they are cases of systematic ignorance. Williamson develops this diagnosis through inexact knowledge and margin-for-error reasoning: adjacent cases may be indiscriminable even though one adjacent pair differs in truth-value.

My position is a naturalistic variant of epistemicism. It retains classical logic, bivalence, and a sharp cutoff in the specified series, but it separates two questions that standard presentations often run together: what fixes the extension and why ordinary speakers cannot locate it. On Williamson’s picture, precise extensions supervene on the totality of meaning- and use-relevant facts, without a simple rule available to speakers. Later work reconstructs part of his epistemology through semantic plasticity and metalinguistic safety. The present proposal instead assigns linguistic practice the task of fixing a world-directed role and asks whether objective organization supplies a robust filler. The physical model is not a result already established by granular mechanics, but a candidate realization of this metasemantic architecture.

This places the view between several familiar options. Unlike supervaluationism, it posits one actual extension rather than a plurality of admissible sharpenings. Unlike contextualist accounts, it does not treat ordinary shifts in conversational standards as automatically shifting the truth-value in the fixed series. The distinctive proposal is thus role-relative naturalistic boundary-fixing within an epistemicist semantics.

More specifically, the account may be characterized as a form of normative metasemantic externalism. Linguistic practice contributes not merely patterns of use but standards of correct application, correction, deference, and world-directed answerability; objective structure then constrains which worldly state realizes the resulting coarse-grained role. The normative contribution helps determine what counts as relevant role-satisfaction without making the identity of the role-filler a matter of theoretical preference.

The proposal is developed for the predicate “is a heap of sand” as applied within a fixed construction series. It does not assume that all uses of “heap,” or all vague predicates, share the same grounding structure.

The paper’s central diagnosis is that Sorites reasoning projects a feature of pragmatic tolerance onto semantic truth-conditions. Its further, more contestable proposal is that a practice-fixed structural role may be filled by a physically distinguished transition in a sand-pile series. The first claim explains the appeal of tolerance; the second offers a naturalistic grounding for sharpness. Keeping them separate ensures that failure of the physical proposal would not by itself defeat the pragmatic diagnosis.

The rest of the paper goes on as following: Section 2 clarifies the agreement and disagreement with Williamson. Section 3 develops the pragmatic account of tolerance and the world-directed role inherited from use. Section 4 presents the naturalistic candidate: it identifies physically unequal transitions, states the role-filler conditions needed to connect physics with semantics, and proposes a multivariate operationalization. Section 5 explains the metasemantic dependence between practice and objective structure. Section 6 addresses logical alternatives, model non-uniqueness, the physics-to-semantics bridge, instability, higher-order vagueness, and domain restriction. Section 7 concludes.

2. Williamson’s Epistemicism: Agreement and Departure

Williamson’s epistemicism combines a semantic thesis with an epistemological explanation:

1. Classical logic and bivalent semantics apply to vague language. For every n, “n grains make a heap” is either true or false, including in borderline cases.

2. Vague predicates have sharp extensions. There is a natural number k such that k grains do not make a heap but k + 1 grains do; the universal tolerance premise is false at that transition.

3. The sharp boundary is characteristically unknowable. This is not merely an accidental absence of information: Williamson explains it through the inexactness of our discriminatory capacities and the margin-for-error conditions on knowledge.

I accept the first two claims as semantic commitments and the third as an accurate description of ordinary epistemic access. My disagreement concerns the explanatory framework: (i) how meaning-and-use facts relate to the sharp extension, (ii) why exact boundary knowledge fails, and (iii) how much of the characteristic appearance of vagueness should be assigned to pragmatics rather than semantics.

2.1 Where the Boundary Comes From

Williamson argues that vague predicates possess precise extensions despite ordinary speakers being unable to identify them. He does not posit a salient physical discontinuity or a consciously chosen numerical cutoff. Rather, the extension supervenes on the totality of facts relevant to meaning and use, including communal patterns of application and correction. The dependence is global and complex: ordinary speakers need not possess an explicit rule from which the cutoff can be calculated, and no individual disposition by itself determines it.

I agree that the extension is objective, but I deny that meaning- and use-relevant facts directly determine a numerical cutoff. On the present view, ordinary practice transmits a comparatively coarse, world-directed referential instruction: applications of “heap” are answerable to whether the grains form the kind of integrated aggregate represented by paradigmatic heaps. This instruction is inherited through social-historical practice, but it is not exhausted by speakers’ classificatory dispositions or tolerance judgments. Nor does it itself specify a cutoff, an admissible range of boundaries, or a particular physical metric. It instead directs inquiry toward relevant features of aggregate organization, such as mutual support, persistence, and collective behavior, while leaving their adequate scientific representation open. A theoretically justified physical model then investigates which transition, if any, best realizes that instruction under the specified construction conditions. Practice therefore supplies a world-directed target; objective structure and model-mediated inquiry constrain its physical realization.

2.2 Why We Are Ignorant

Williamson’s explanation of ignorance begins from the idea that knowledge can be inexact. To know that a case falls on one side of a boundary, one must not be too close to an indiscriminable case on the other side. This is captured by a margin-for-error constraint: knowledge at a point requires truth throughout a sufficiently similar range of cases. Since adjacent members of a Sorites series are indiscriminable even though one adjacent pair differs in truth-value, speakers cannot know the exact cutoff. Formally, let Kφ represent knowledge that φ, and let sₙ ∼ sₙ₊₁ represent indiscriminability between adjacent cases. A simplified representation of the positive margin-for-error constraint is:

K(H(sn)) → ∀sm(sm ∼ sn → H(sm))

Similarly, knowledge that a case is not a heap requires that nearby indistinguishable cases also fail to be heaps:

K(¬H(sn)) → ∀sm(sm ∼ sn → ¬H(sm))

At the cutoff k, however:

¬H(sk) ∧ H(sk+1), sk ∼ sk+1

Therefore, neither H(sₖ₊₁) nor ¬H(sₖ) can be known with sufficient margin for error.

Williamson formulates the argument primarily in terms of inexact knowledge and margins for error; later authors often reconstruct the metasemantic component in terms of semantic plasticity and the epistemological component in terms of metalinguistic safety.

I accept that both phenomena identify genuine features of ordinary language, but I reject their semantic role and location.

First, what later authors call semantic plasticity concerns counterfactual variation in meaning: small changes in the total pattern of linguistic use may have fixed a slightly different precise extension. The actual extension is sharp, but nearby possible use-facts can correspond to nearby alternative cutoffs. On later semantic-plasticity reconstructions of Williamson’s framework, this sensitivity, together with speakers’ inability to discriminate the relevant use-situations, explains why exact metalinguistic beliefs about the cutoff are epistemically fragile. My proposal relocates much of the observed instability in speakers’ judgments to the pragmatic tolerance range, while allowing linguistic practice to help specify the referential role.

Second, I do not reject margin-for-error reasoning as a general epistemic constraint. My disagreement is with later attempts to connect the characteristic fragility of exact metalinguistic belief primarily to semantic plasticity in use. On the present account, nearby changes in speakers’ classificatory dispositions do not automatically move the semantic boundary, because use fixes a role rather than directly selecting its filler. Whether a belief about the boundary is safe or possesses an adequate margin for error depends instead on the stability of the proposed physical role-filler and on the reliability of the model used to identify it. I return to this issue in Objection 4.

On this account, ordinary ignorance need not be treated as proof of in-principle unknowability. It may partly reflect cognitive economy: everyday speakers have no practical reason to measure the structural dynamics of a controlled pile. That point explains ordinary non-inquiry, not the absence of knowledge in every possible epistemic situation. Whether expert knowledge is possible depends on the stability of the physical candidate, the quality of the measurements, and the adequacy of the metasemantic bridge.

2.3 Where Vagueness Resides

Williamson treats vagueness as compatible with semantic precision: borderline sentences have determinate truth-values, while our knowledge of their truth-values is blocked by the inexactness of our discriminatory capacities and the margin-for-error requirements on knowledge. Vagueness is therefore not identified with mere subjective uncertainty; it is a systematic form of ignorance generated by the interaction between precise extensions, fine-grained variation, and limited discrimination.

I instead distinguish two explanatory levels. Semantic precision concerns the truth-conditions and extension of the predicate. The characteristic appearance of vagueness in the heap case—hesitation, disagreement, and tolerance of adjacent cases—belongs largely to speakers’ pragmatic and cognitive capacities. What makes the Sorites premise intuitively compelling is our inability to discriminate adjacent cases within the tolerance range established during reference fixing. This supports a pragmatic diagnosis of the appearance of vagueness without implying that every phenomenon called “vagueness” is wholly pragmatic.

These disagreements reflect a methodological division of explanatory labor. Williamson combines classical semantics with an epistemology of inexact knowledge: meaning-and-use facts fix a precise extension, while margins for error explain why the cutoff cannot be known. The present framework preserves the semantic sharpness but redistributes the explanatory tasks. Pragmatics explains tolerance judgments and helps fix the referential role associated with the predicate; objective structure constrains which candidate can fill that role; semantics concerns the resulting truth-conditions; and empirical robustness, rather than semantic plasticity alone, bears on whether the boundary can be known.

3. The Pragmatic Layer: Tolerance as Cognitive Indiscriminability

How do speakers grasp the word “heap”? Not through an explicit definition. No one is told, upon learning the word, that a heap begins at exactly 137 grains. Speakers acquire the term through ordinary reference-fixing practice: they learn how to use it from paradigmatic cases, corrections, contrasts, and shared applications within a linguistic community.

This process should not be understood as the public stipulation of sharp truth-conditions. A vague term cannot be introduced by giving ordinary speakers a boundary that they are able to identify case by case; if such a boundary were available in ordinary practice, the term would not function as vague. Instead, the term is introduced through cases that are sufficiently clear for practical use. A single grain is a paradigmatic non-heap; a towering mound of sand is a paradigmatic heap. These examples anchor the term in uncontroversial applications, but they do not by themselves determine where the transition between heap and non-heap occurs. This fits a broader psychological picture on which ordinary category acquisition often begins from salient prototypes or family resemblances rather than from explicit necessary-and-sufficient definitions.

Paradigms alone therefore underdetermine extension. A term applied only to paradigms would have no determinate bearing on the indefinitely many cases lying between them. Speakers bridge this gap by acquiring a tolerance disposition: if two cases differ only by a sufficiently small amount, then they are not treated as differing with respect to heaphood. In the canonical Sorites series, the relevant difference is one grain. This disposition is not a semantic axiom but a pragmatic feature of ordinary classification, reflecting the limits of human perceptual and cognitive resolution. Our visual and practical capacities are not designed to track single-grain differences across the range of quantities relevant to heap classification.

From paradigms and tolerance, the result is not a unique cutoff. Rather, ordinary practice generates a tolerance range within which speakers may competently hesitate, disagree, or shift their judgments without violating their mastery of the term. This range concerns permissible classificatory variation; it does not constitute or delimit a set of admissible semantic boundaries. Speakers possess practical command of a tolerance range, not knowledge of a precise semantic boundary.

This tolerance range is not rigid. It displays context-sensitive and speaker-sensitive instability, but this should not simply be identified with Williamson’s semantic plasticity. Semantic plasticity concerns how small counterfactual changes in use may shift a term’s precise extension; the present tolerance range concerns fluctuations in actual classificatory practice, hesitation, and standards of application. Across contexts, speakers, and occasions, the tolerated range may shift. These are pragmatic fluctuations in classification, not by themselves changes in semantic extension.

Why do speakers acquire tolerance in the first place? The connection between cognitive indiscriminability and tolerance is not one of logical entailment. It is causal, practical, and evolutionary. Human perceptual systems evolved to track ecologically significant differences: predators, prey, obstacles, food, faces, and other practically salient features of the environment. They did not evolve to register every single-grain difference in a pile of sand. A linguistic practice that required strict single-grain sensitivity in the use of “heap” would be unusable for ordinary speakers. It would collapse into persistent and irresolvable disagreement. Tolerance is therefore not a semantic truth; it is a practical adaptation to finite discriminatory capacities. It is a trace of our cognitive architecture within linguistic practice.

Yet tolerance cannot be the whole story. If it were, “heap” would amount to no more than a record of speakers’ classificatory dispositions: to call something a heap would be merely to report that we are inclined, under ordinary conditions, to treat it as one. But this does not fit ordinary use. Speakers distinguish between something’s seeming to be a heap and its being a heap. They allow that a person may misclassify a case because of bad lighting, misleading perspective, poor attention, or ignorance of the relevant arrangement. They also allow that further inspection could improve judgment. These features show that “heap” is not treated merely as a projection of our classificatory habits. It is treated as answerable to the object.

These features are normative in a metasemantic rather than moral sense. A linguistic practice does not merely record how speakers classify cases; it distinguishes correct from incorrect application through patterns of correction, deference, error attribution, and improved access to the object. Those standards help constitute the world-directed role inherited by the predicate. Their contribution is therefore not to select a cutoff by preference, but to specify what any adequate worldly realizer must answer to.

This answerability is what I call the predicate’s referential instruction. By an instruction, I do not mean a rule consciously represented by ordinary speakers. Speakers who use “heap” competently need not think that they are trying to locate a physical ground of heaphood. They may not even be able to formulate such a question. The instruction is embedded in the social-historical practice through which the term is acquired and transmitted. In this respect, the account borrows a limited externalist insight from Kripke and Putnam: competent speakers may successfully use a term without possessing a complete descriptive specification of its reference. Putnam’s water/H₂O case illustrates the point. Ordinary speakers can refer to water without knowing its chemical structure; their reference is fixed partly by samples, environment, and communal practice rather than by an explicit definition in each speaker’s mind. “Heap” is not analogous to “water” in every respect, but it shares this weaker feature: competent use need not include knowledge of the condition that ultimately fixes correct application.

The referential instruction is not that speakers already know a numerical cutoff or consciously intend a scientific measurement. It is a structured constraint inherited from their practice: in the sand-pile case, correct application is answerable to whether the grains form the kind of spatially and mechanically integrated pile represented by paradigmatic heaps. Relevant features include mound-like organization, mutual support through a contact network, persistence under ordinary perturbation, and the capacity to behave as an aggregate rather than as isolated grains. This role is richer than bare “objectivity,” yet it does not prejudge which physical variable best realizes it.

This is a limited structural externalism, not the claim that “heap” is a natural-kind term. “Heap” does not name a substance or a hidden essence, and different materials may realize heap-like organization differently. The useful analogy with externalism is only that correct application can outrun speakers’ explicit descriptions and immediate judgments. Slater’s cluster-based account of natural kindness is relevant at this limited point: objective organization may be explanatory without reducing to one essential microscopic property. However, Slater does not discuss heaps, and nothing here attributes the present metasemantics to him.

This point clarifies the status of the referential instruction within the two-layer framework. The instruction belongs to the pragmatic layer insofar as it is inherited through reference-fixing practice. It is part of how speakers acquire and transmit the term. But it is semantically directed insofar as it points beyond that practice toward the truth-condition of the predicate. The instruction is therefore pragmatically inherited but semantically directed. The pragmatic layer fixes how speakers manage the predicate; it does not exhaust what the predicate is answerable to.

The pragmatic layer therefore establishes two claims. First, finite discrimination and ordinary training generate a tolerance range that explains the appeal of the Sorites premise. Second, the practice also supplies a fallible but contentful structural role that is not exhausted by that tolerance range. It does not yet establish which physical transition, if any, realizes that role. That further bridge requires explanatory, counterfactual, and empirical support, which Section 4 makes explicit.

4. The Semantic Proposal: A Naturalistic Candidate for Boundary-Fixing

Section 3 located the characteristic tolerance of the fixed series in the pragmatic layer: speakers are disposed to treat adjacent states alike because their discriminatory capacities are coarse-grained. That account explains why the tolerance premise feels compelling. It does not yet explain what makes H(n) true or false. That is the task of the semantic proposal.

Semantics concerns truth-conditions: what the world must be like for H(n) to be true. Truth cannot be exhausted by speakers’ dispositions to classify n grains, because those dispositions can be mistaken and revised. The semantic question is therefore which worldly condition, if any, satisfies the structural role inherited from use.

The proposal in this section is conditional: within a fixed sand-pile construction series, a transition that robustly realizes the world-directed role inherited from linguistic practice is a candidate semantic cutoff. Granular mechanics is relevant because it provides theoretically informed ways of representing the organization, mutual support, persistence, and collective behavior associated with paradigmatic heaps—not merely because it produces numerical maxima. These physical variables are proposed operationalizations of the role rather than dimensions already specified by ordinary use. A transition becomes semantically relevant only if the model validly tracks that role, correctly distinguishes clear paradigms, and remains robust across reasonable variations in measurement, aggregation, and experimental protocol.

The argument proceeds in four steps. Section 4.1 shows that transitions in a fixed granular series can be physically unequal. Section 4.2 states the conditions under which such a transition would fill the semantic role. Section 4.3 presents a proposed multivariate operationalization and its limitations. Section 4.4 explains how a uniquely fixed cutoff supports the paper’s bivalent semantics without making empirical identifiability a condition of bivalence.

4.1 What the Physical World Provides: Structurally Unequal Transitions

The first step is to deny that all transitions in the series are physically equal. If the transition from n to n + 1 grains involved only a change in cardinality, every possible cutoff would appear equally arbitrary. But a pile is not merely a number of grains. It is a granular system in which contacts transmit load through a heterogeneous, history-dependent network. The same number of grains can realize different arrangements and stability profiles; successive additions can therefore have structurally unequal consequences.

This matters because adding one grain need not have the same physical significance at every point in the series. Many additions will make only local or negligible changes. A grain may settle into an already stable region, slightly alter a contact, or produce a small rearrangement. Other additions may disturb a marginally stable configuration. A local contact may fail; a force chain may reorganize; a small perturbation may trigger a larger cascade of rearrangements. The transition from n to n+1 is therefore not always physically equivalent. Some increments are structurally shallow; others are structurally deep.

The relevant physical contrast is not between a perfectly smooth continuum and a miraculous discontinuity. Granular systems often display intermittent behavior: under slow driving or loading, they may remain locally stable for some time and then respond through sudden rearrangements or avalanche-like events. Such events can vary greatly in size. In many crackling or avalanche-like systems, event sizes are broadly distributed, often with power-law-like statistics over relevant ranges. Such distributions are often approximated by:

P(s) ∝ s-α

where P(s) represents the probability density or relative frequency of events of size s, and α is a system-dependent exponent. The distribution represents the occurrence of many small events together with fewer large events.

The important point for the present argument is not that every pile must undergo an absolute phase transition. It is that transitions in the physical series may be structurally unequal and may therefore differ in their capacity to realize the practice-fixed role. One transition may prove better than its competitors under an independently justified multivariate model and experimental protocol.

This possibility is conditional rather than dogmatic. The semantic account does not require every granular sequence to contain a metaphysically dramatic phase transition or a single largest avalanche. It requires only that, in the relevant series, one transition may be comparatively privileged relative to the antecedently specified role. Granular mechanics makes this possibility plausible because force transmission in granular matter is heterogeneous, history-dependent, and often organized through marginally stable force-chain networks. A small perturbation need not produce a merely proportional response; in fragile or slowly driven disordered systems, local changes can trigger plastic rearrangements or avalanche-like events of very different magnitudes.

For each addition transition n → n + 1 in a specified construction series, let the multivariate addition profile be whose components measure changes in force-network organization, grain displacement, acoustic emission, and contact-network topology:

xadd(n) = (ΔFn, ΔDn, ΔAn, ΔCn)

A theoretically justified aggregation rule A maps this profile to a score measuring realization of the practice-fixed role:

Sadd(n) = A(xadd(n))

Within the specified experimental window, the model’s leading addition candidate is:

kadd = arg maxn Sadd(n)

This formalization does not make the mathematical maximum semantically privileged or alter the standard Sorites formalization. It identifies the model’s best-supported candidate only if the variables, normalization, and aggregation rule independently and validly operationalize the role, and if the result remains robust across reasonable variations in model and experimental protocol. These specifications remain scientifically revisable.

Model mediation neither entails conventional invention nor guarantees non-arbitrariness. The model must earn representational authority by tracking the practice-fixed role. At minimum, it should classify clear paradigms correctly, respond to arrangement and stability rather than grain count alone, support counterfactual predictions about perturbation and reorganization, and yield comparable results across reasonable repetitions and protocols. Sensitivity analysis can test the last requirement; it cannot supply the semantic relevance of the chosen variables.

One transition is “joint-carving” only relative to this antecedently specified role. Nature does not contain an intrinsic label “heap,” and naturalness by itself cannot select which physical dimensions matter. If a model satisfying the stated adequacy conditions identifies one transition as robustly realizing the role better than neighboring transitions, that transition is a better-supported candidate filler. If the conditions are not met, the paper has not identified the semantic boundary, even if the scoring function returns a unique numerical maximum.

This claim remains compatible with gradualism. The physical world need not contain an absolute phase boundary between heap and non-heap. There may be many rearrangements of different magnitudes, and a role-sensitive aggregation model may rank one transition as the best realizer. Comparative privilege relative to the role could be sufficient for semantic boundary-fixing if the role, model, and result are independently justified. The account therefore does not infer sharpness merely from the existence of a numerical maximum; it treats the model ranking as defeasible evidence within a broader metasemantic argument.

4.2 From a Physical Candidate to a Semantic Boundary

A physical transition is not a semantic boundary merely because it is large, measurable, or unique. The bridge must begin from the practice-fixed role identified in Section 3: the predicate concerns an integrated sand pile whose grains form a persisting, mutually supporting aggregate. Granular variables are relevant only insofar as they measure the organization, stability, and causal behavior constitutive of that role.

The proposed sufficiency claim can be stated as follows:

If (i) linguistic practice fixes the sand-heap role just described, (ii) the selected physical variables validly measure realization of that role, (iii) the aggregation rule is independently justified rather than chosen to manufacture a cutoff, and (iv) one transition in the finite series robustly realizes the role better than its competitors under reasonable variations of model and protocol, then that transition is eligible to fix the extension in the specified series.

The externalist analogy explains only why speakers need not know the realizing science; it does not establish that force-network dynamics are the correct realization. That conclusion must be earned by the bridge conditions. Putnam’s and Kripke’s cases show that reference can outrun individual description, but “heap” lacks the stable microstructural essence associated with paradigmatic natural-kind terms. The analogy therefore licenses epistemic deference, not an automatic identification.

The aggregate-rearrangement model is relevant because it aims to measure several dimensions of pile integration at once. It gains semantic support only if it improves on rivals such as grain count, angle of repose, visible shape, mechanical stability, or energy release. Comparative explanatory success matters: the preferred model should better track paradigms, explain why rearrangement affects pile persistence, and remain invariant under changes that ordinary practice treats as irrelevant.

The relation is one of conditional role satisfaction. Use specifies what kind of difference matters; physics and modeling test which transition best realizes it. When the bridge conditions are met, the cutoff lies between k and k + 1 grains: H(k) is false and H(k + 1) is true. When the conditions are not met, the model is inconclusive. Bivalence may remain a semantic commitment, but the paper has not thereby located its boundary.

This conditional structure avoids both simple conventionalism and crude physicalism. Classificatory habits do not directly select k, yet physical salience alone cannot determine semantic relevance. Linguistic practice fixes a role; a scientifically justified model may identify its worldly filler. Both contributions are necessary.

4.3 Constitutive Role-Filling and Experimental Operationalization

The third step is to clarify the relation between the physical model and the predicate’s extension. The claim is not that heaphood is metaphysically identical with any single physical variable, nor that one measurement directly determines the meaning of “heap.” Rather, linguistic practice fixes a structural role, and a physically grounded model attempts to identify which transition, if any, realizes it. Force-chain rearrangement, acoustic emission, displacement fields, contact-network topology, and related variables should therefore be understood as components of a multivariate model for identifying candidate role-fillers, not as independently sufficient definitions of heaphood. Future scientific refinement may revise which physical features best operationalize the role or whether a different composite model provides the most adequate representation.

The proposal is constitutive only in a conditional role-functional sense: if the role is adequately specified and a worldly transition uniquely satisfies it, no further boundary fact is required.

First, the semantic boundary is not an additional entity over and above satisfaction of the referential role. To ask for the boundary of “heap” is to ask which worldly transition, if any, fulfills the boundary-role partly fixed by the practice that introduced the predicate. Second, that role is externalist: it directs the predicate toward objective organization rather than merely toward speakers’ current classificatory dispositions. Third, the multivariate model operationalizes the practice-fixed role by comparing candidate transitions across theoretically relevant dimensions of pile integration. It does not treat the largest rearrangement as independently semantically privileged. If the role is adequately specified, the variables and aggregation rule are independently justified, and one transition robustly realizes the role better than its competitors, then that transition’s satisfaction of the role fixes the semantic boundary; the model supplies defeasible evidence for identifying it.

On this constitutive proposal, no additional semantic boundary is required once the practice-fixed role has been adequately specified and a worldly transition has been shown to satisfy it. A further fact would be needed only if the physical transition merely caused or indicated a boundary fixed elsewhere. The satisfaction relation is intended to complete the semantic work. The argumentative burden therefore lies in justifying both the role and the claim that the worldly transition fills it; the model cannot discharge that burden merely by producing a maximum.

The account proposes a constitutive role-satisfaction relation rather than a merely causal correlation. Acoustic emission, displacement fields, contact-network analysis, and force-chain visualization are not treated as independent symptoms of a boundary fixed elsewhere. They provide different components of a multivariate event profile describing the structural consequences of each added grain. These components need not each reach an independent maximum at the same increment. Instead, they are combined by a theoretically specified aggregation rule into an overall measure of structural significance. If the resulting model robustly identifies one transition as realizing the practice-fixed role better than its competitors, this supports the claim that the role has a unique physical satisfier. If reasonable aggregation rules remain persistently divided, the current modeling framework fails to uniquely identify which candidate fills the role and must either be refined or treated as inconclusive.

The following is a proposed operationalization, not an established protocol in granular mechanics. Under controlled geometry, grain properties, and loading conditions, grains would be added one at a time. For each addition transition, the experiment would record grain displacements, photoelastic force-network changes, contact topology, and—where methodologically appropriate—acoustic or vibration signals. These measurements would form the addition profile, and a pre-specified aggregation rule would evaluate its role-realization score. The leading candidate would be the transition with the highest robust role-realization score, not the point at which any raw indicator independently peaks.

A complementary removal protocol can represent the transition n + 1 → n by a separate event profile:

xrem(n) = (ΔFn, ΔDn, ΔAn, ΔCn)

Using the same independently justified aggregation rule, the removal score and leading candidate are:

Srem(n) = A(xrem(n)), krem = arg maxn Srem(n)

Agreement between the leading addition and removal candidates would provide stronger cross-protocol evidence. Divergence would not automatically refute the addition-series candidate, because granular systems can be path-dependent; it would instead show that the result is protocol-relative and that generalization across directions requires further explanation.

Using several indicators reduces dependence on one instrument, but it also introduces choices about scaling, weighting, and aggregation. The evidential case therefore depends on preregistered or independently motivated specifications, out-of-sample or cross-protocol robustness checks, and sensitivity analysis. These procedures test whether the candidate is an artifact; they do not establish that the measured construct is semantically privileged.

Evidence for uniqueness is model-mediated and role-relative; it does not require every component of the raw event profile to peak at the same transition. Two distinct event vectors may receive the same aggregate score, while a nominally leading candidate may be too close to its competitors to remain distinguishable under measurement error or reasonable model variation. The account therefore requires more than a formal argmax: one transition must remain the best-supported realizer of the role, be separated from its competitors by more than the relevant measurement uncertainty, and retain that ranking under reasonable variations in normalization, weighting, and experimental protocol.

Experiments on granular piles and other crackling systems support the possibility of intermittent rearrangements and broad event-size distributions. They do not show that every grain-by-grain series contains one uniquely dominant structural event. Accordingly, uniqueness of a role-filler must be an empirical and model-mediated result rather than a default.

Experimental design can improve the reliability of the aggregate result. Vector-valued event profiles reduce dependence on a single noisy channel; repeated trials estimate variability; and sensitivity analysis tests whether the selected candidate transition remains stable under reasonable changes to normalization, weighting, and model specification. These procedures cannot guarantee uniqueness, but they can distinguish a robust candidate from an artifact of one measurement or aggregation choice.

Exact ties or persistent failure to identify a uniquely supported transition remain logically possible. Even after refined modeling, repeated trials, and multivariate measurement, such a result would not establish the Sorites tolerance premise, which requires every transition from nnn to n+1 to preserve non-heaphood. It would show only that the present measurements and aggregation rule do not identify a single transition as the best-supported realizer of the practice-inherited role. Where the model robustly does so, it supplies an empirically grounded candidate cutoff; where it does not, the identification procedure remains incomplete. This is a limitation of the method, not evidence that the predicate lacks a bivalent extension.

Moreover, failure of the model to locate a unique candidate should not be assimilated to reference failure. Reference failure concerns whether a linguistic practice succeeds in targeting the intended kind of worldly phenomenon; model underdetermination concerns whether our present evidence and theoretical apparatus identify the boundary. These questions can come apart. A practice may preserve ordinary reference to heaps even when available physical models disagree about the exact cut-point. Conversely, a radically deviant practice may involve misuse, reference failure, or semantic change; the present framework requires additional criteria to distinguish those possibilities.

Scientific revisability does not by itself undermine semantic precision. If future work identifies a better operationalization of the practice-fixed role, the model should be revised. That would be a revision in our account of the role’s physical realization, not evidence that ordinary tolerance judgments constitute the extension. The form of the proposal survives only if the revised model continues to satisfy the bridge conditions.

4.4 Classical Logic at the Semantic Layer

If the role-filler conditions fix the cutoff between k and k + 1 grains, the semantics remains classical: H(n) is false below the cutoff and true above it. The law of excluded middle holds for every n.

The tolerance premise is false at k: ¬H(k) is true while ¬H(k + 1) is false. Ordinary inability to discriminate the adjacent cases does not entail semantic invariance between them.

Sharpness and bivalence are semantic commitments of the account, not conclusions guaranteed by the experimental procedure. Exact or near ties, instability under reasonable specifications, or disagreement among justified models would show that the proposed route to the cutoff is inconclusive. Such results would neither establish the universal tolerance premise nor decide between epistemicism and its semantic rivals.

The epistemological consequence is more modest. Cognitive economy helps explain why ordinary speakers do not measure force-chain reorganization, acoustic emission, displacement fields, or avalanche energies when using “heap”: tolerance is cheap and practically sufficient. This explains ordinary non-inquiry, not a complete theory of ignorance. Whether experts could know the boundary depends on the stability of the physical phenomenon, the adequacy of the aggregation model, and the metasemantic argument connecting the model to the predicate.

Section 4 has therefore proposed a semantic and epistemic framework rather than established a completed physical reduction. The physical world supplies structurally unequal transitions; linguistic practice fixes the relevant boundary-role; and a theoretically constrained aggregation model attempts to identify which transition best realizes it. If the role, model, and candidate realizer are adequately justified, the resulting boundary is compatible with classical logic and bivalence. If the model is inconclusive, our identification of the boundary remains epistemically unresolved; this does not imply that the predicate lacks a determinate extension.

5. Relating the Two Layers: Metasemantic Dependence Without Reduction

Sections 3 and 4 distinguished two explanatory layers. The pragmatic layer explains how speakers manage a vague predicate in practice: they acquire a tolerance range through paradigms, correction, shared use, and finite discriminatory capacities. The semantic layer explains what makes applications of the predicate true or false: in successful cases, a referential role inherited from linguistic practice is filled by a structure that is privileged relative to that role. The layers are distinct, but they are not wholly independent. There is a metasemantic dependence between them: linguistic practice helps specify the role whose satisfier determines extension, while objective structure constrains which candidate can fill that role.

The pragmatic tolerance range and the semantic boundary should therefore be distinguished without being isolated. The tolerance range does not itself constitute the semantic boundary; it explains why speakers find a region of cases difficult to classify. Yet reference-fixing practice contributes to semantic determination by specifying the relevant role. Conversely, the physical structure does not directly cause speakers to possess exactly that tolerance range; ordinary speakers do not detect force-chain rearrangements or acoustic-emission profiles when using the word “heap.” Pragmatic tolerance arises from human cognitive economy and social practice, while semantic boundary-fixing depends on the satisfaction of a practice-fixed role by objective physical structure.

This comparison also clarifies the relation to Lewisian reference magnetism. Like reference-magnetic views, the present account allows objective structure to constrain reference. However, the mechanism differs. Reference magnetism typically treats naturalness as selecting among a range of candidates already licensed by linguistic practice. The present account does not allow worldly structure to choose directly from the pragmatic tolerance range. Instead, linguistic practice first determines a referential role: it specifies what kind of organization or similarity is relevant. Objective structure then constrains which feature, if any, fills that role. Naturalness or structural privilege therefore cannot independently fix the boundary; it operates only after the semantic role has been specified. The proposal is neither pure use-theory nor unconstrained reference magnetism: use fixes the role, and worldly structure constrains its filler.

5.1 Contingent Alignment Between Tolerance and Worldly Structure

For the practice to be useful, its clear positive and negative paradigms must normally bracket the semantic cutoff, but the exact cutoff need not coincide with any conscious judgment boundary. This expected fit is contingent on the history of the practice, human discriminatory capacities, and the kinds of macroscopic organization encountered in ordinary life.

A historical and practical explanation of this rough fit is more defensible than a strong evolutionary claim. Communities retain low-cost classifications that support successful interaction with salient macroscopic structures, while correction and deference help stabilize paradigm use. This story may explain why the role and ordinary judgments overlap roughly, but it does not determine the exact content of the role or vindicate the proposed physical metric.

This makes the alignment between the two layers contingent, but not inexplicable. There is no logical guarantee that ordinary tolerance practices will roughly align with the structure that fills the practice-fixed role. Still, linguistic practices that systematically failed to track practically relevant worldly organization might be unstable or undergo correction. A community whose word “heap” was applied only to two or three grains would not straightforwardly preserve ordinary English reference to heaps. Its practice might involve persistent misuse, reference failure, or a semantic shift toward a different concept; the present framework does not by itself decide among these diagnoses. Successful reference-fixing practices generally persist because they point, roughly and fallibly, toward structures that matter in interaction with the world.

The referential role is inherited through a history of applications, corrections, and practical engagements with the world. The evolutionary and historical story offered here is suggestive rather than a complete teleosemantic theory. It helps explain why tolerance ranges and objective structures often align roughly, but it does not uniquely determine the content of the referential role or prove that the proposed physical metric is semantically correct.

5.2 Misalignment, Misuse, Reference Failure, and Semantic Change

Because alignment is contingent, linguistic practice and worldly structure can come apart in several ways. These possibilities must be distinguished: ordinary misclassification by an individual, systematic but corrigible misuse, failure to preserve ordinary reference, and a genuine semantic shift in which a community introduces or develops a different concept.

First, a tolerance range may be badly calibrated while the practice still preserves ordinary reference to heaps. Speakers may classify borderline cases poorly even though the referential role remains directed toward pile-like organization. The appropriate diagnosis is ignorance, error, or pragmatic miscalibration.

More radical divergence raises a metasemantic question rather than yielding an automatic diagnosis. If a community systematically applies “heap” only to two-to-five-grain arrangements, the practice may fail to preserve ordinary English reference to heaps. But it may instead constitute a semantic shift toward a different concept, or involve systematic misuse maintained by social convention. The absence of any particular physical feature does not by itself distinguish misuse, reference failure, and semantic change. The relevant question is whether the altered practice preserves the same world-directed role; physical evidence concerns whether that role is realized, while historical continuity, correction norms, intentions, and inferential role help determine whether the practice still expresses the same concept.

The framework can therefore identify a condition for concern: the practice no longer targets the kind of pile-like phenomenon represented by paradigmatic heaps. It cannot infer from that condition alone that the expression has an empty semantic value. Additional evidence about speaker intentions, correction practices, historical continuity, and inferential role is needed to distinguish misuse, failed reference, and semantic change.

Likewise, overlap between the pragmatic tolerance range and the identified role-filling transition is neither strictly necessary nor by itself sufficient for successful reference. A badly calibrated community may still preserve ordinary reference, while a superficially overlapping practice may target a different role. The relevant question is whether the historical and normative practice continues to fix the same boundary-role, not merely where speakers happen to place their borderline judgments.

5.3 Stability of the Boundary and the Illusion of Plasticity

One consequence of the two-layer picture is a partial reinterpretation of the phenomena used to motivate semantic plasticity. Later semantic-plasticity reconstructions of Williamson’s framework concern nearby possible changes in meaning-fixing use and corresponding shifts in precise extension. The present account does not deny that sufficiently substantial counterfactual changes in use could help fix a different referential role or extension. It denies that ordinary context-sensitive shifts in hesitation or classificatory strictness must themselves be semantic shifts. Such changes primarily alter the pragmatic tolerance range, while the boundary does not move merely because a speaker becomes more cautious, inattentive, strict, or permissive.

This does not mean that the semantic boundary is independent of all empirical conditions. If the material system changes, the physical boundary may change. Different grains, different shapes, different friction coefficients, different loading conditions, and different pile geometries may produce different structural transitions. But that is physical variation, not semantic plasticity. The boundary changes because the worldly system changes, not because the tolerance range has drifted.

The appearance of semantic plasticity arises when these two forms of variation are conflated. Since speakers experience only their tolerance practices, they naturally take the instability of judgment to reflect instability in the predicate’s truth-condition. But on the present account this is a projection error. The pragmatic envelope may move, while the boundary remains fixed by whichever worldly transition satisfies the practice-fixed role within the specified series. Sorites reasoning gains its intuitive force precisely by mistaking movement in classificatory practice for movement in semantic truth.

5.4 Distinction and Metasemantic Dependence

The two-layer relation can now be summarized. Pragmatic practices supply tolerance-based methods of classification and help fix the referential role. The semantic layer concerns the truth-condition obtained when objective structure fills that role. The layers are therefore distinct but metasemantically dependent. Their empirical alignment is contingent, and failures of alignment can produce ignorance, error, misuse, reference instability, or semantic change without automatically producing semantic indeterminacy.

The framework is neither straightforward conventionalism nor a claim that the world contains semantic labels independently of language. Linguistic practice contributes by fixing the relevant role; objective structure contributes by constraining its filler. The proposed physical model mediates our epistemic access to that filler and must itself be theoretically justified. Tolerance practices neither create the worldly structure nor become irrelevant to semantic determination.

The relationship between the layers is therefore one of distinction together with metasemantic dependence, rather than mere correlation or causation. Human cognition and linguistic practice supply both a rough, fallible mode of classification and a referential role; the physical world constrains which candidate can fill that role. When ordinary tolerance and the role-filling structure roughly align, vague predicates function successfully despite grain-by-grain indiscriminability. When they misalign, speakers may be ignorant or badly calibrated. In extreme cases, a linguistic practice may fail to preserve ordinary reference or may instead institute a semantic shift. In none of these cases does pragmatic tolerance by itself become a semantic truth-condition.

6. Objections and Replies

Objection 1. Why not many-valued logic?

If the physical world is gradual, model-mediated, and structurally continuous, why preserve bivalence at all? A many-valued or degree-theoretic account may seem more faithful to the phenomenon. Heaphood does not appear all at once. As grains accumulate, the system becomes increasingly pile-like: contact networks become richer, stability changes, force chains develop, and the aggregate gradually acquires the physical features associated with heaps. On this picture, degrees of truth may correspond not merely to speakers’ degrees of confidence or assertability, but to real degrees of physical organization. If the world itself comes in degrees, perhaps semantics should come in degrees too.

Reply. This is a serious alternative, and the present account need not deny its scientific motivation. Granular systems do exhibit graded physical organization. There may be increasing stability, increasing structural complexity, increasing contact-network density, and increasing heap-likeness across the series. A degree-theoretic semantics can naturally represent this continuous physical profile. In that respect, many-valued logic may offer a useful model of the physical spectrum underlying heap formation.

The disagreement concerns what semantic role that spectrum should play. A graded physical basis does not by itself entail graded truth. One may accept that the underlying physical phenomenon is continuous while still holding that a predicate’s extension is fixed when one transition uniquely realizes a practice-fixed role within that continuum. Many scientific classifications work this way: a system can vary continuously along many dimensions while a theory identifies a threshold or transition point as explanatorily privileged. The question is therefore not whether the physical basis is graded; it is whether semantic truth should mirror that gradation directly, or whether the practice-fixed role is uniquely realized by one transition within it.

The present account chooses the second option. It treats degrees of physical organization as the worldly domain within which the practice-fixed role may have a unique realizer, not as degrees of semantic truth. The multivariate model is relevant because its variables bear on mutual support, persistence, and aggregate behavior, not merely because it produces a numerical ranking. Many-valued semantics remains a legitimate rival, especially if vague predicates are taken to report physical degree directly. The dispute is therefore metasemantic: whether graded structure enters truth-values or instead helps realize a role that fixes a sharp extension.

The choice between these frameworks is therefore not settled simply by pointing out that the physical world is gradual. Both accounts can acknowledge that. The real dispute is metasemantic: whether vague predicates inherit the gradation of their physical basis into their truth-values, or whether a practice-fixed role can be uniquely realized within that graded basis. This paper defends the latter option because it preserves classical logic while still explaining why the Sorites premise feels compelling and why the underlying physical phenomenon appears continuous.

Objection 2. What if there is no unique physical role-filler?

The account requires the aggregation model to identify a unique and robust candidate for the physical role-filler. Yet granular systems are noisy, path-dependent, and multidimensional. Different event vectors can receive equal aggregate scores; a nominal maximum may be unstable under small changes of weighting or normalization; and different reasonable aggregation rules may rank the increments differently. If so, the physical evidence may fail to locate a unique candidate even though each sub-model is informative.

Reply. The aggregation model does not require the separate indicators to peak at the same increment. They are components of an event vector evaluated by a theoretically specified scoring rule. What the account requires is one transition that remains the best-supported realizer of the practice-fixed role across reasonable specifications, measurement error, and repeated trials. Sensitivity analysis, cross-validation across experimental conditions, and independent physical justification for the selected variables and aggregation rule can strengthen the case for a candidate. They cannot guarantee uniqueness in advance.

If reasonable aggregation rules remain divided, the result neither vindicates the tolerance premise nor undermines bivalence. It shows that the proposed physical identification procedure is inconclusive. The theory must then remain agnostic about the boundary’s location or improve the model. The physical proposal is conditional as an epistemic method, while bivalence remains a separate semantic commitment.

Objection 3. Why should a model-ranked transition be a semantic boundary?

Even if the model ranks one transition above all others, it does not follow that this transition is the boundary of the predicate “heap.” Physical facts do not arrive with semantic labels attached. A model ranking is a claim about granular organization under specified measurements and assumptions, not yet a fact about language. The account therefore seems to require an illicit jump from physics to semantics.

Reply. The objection is decisive against any inference from physical prominence alone. The proposed bridge instead has three stages. First, ordinary practice must fix a sufficiently determinate structural-functional role for a sand heap. Second, the physical variables must validly measure realization of that role rather than merely correlate with grain addition. Third, the preferred aggregation model must outperform plausible rivals in paradigm fit, explanatory relevance, counterfactual behavior, and robustness. Only then is the transition that best realizes the role eligible to fix the extension.

The missing link is therefore not a further physical fact. The role has semantic authority because the practice through which the predicate is acquired is norm-governed: speakers distinguish correct from incorrect applications, defer to better-situated judgments, revise errors under improved access, and treat the predicate as answerable to its objects. Those norms help constitute the world-directed role rather than merely summarizing classificatory frequencies. Externalism enters at the next stage: once that role is fixed, whether a candidate transition satisfies it is constrained by the organization of the worldly system, not by speaker preference. The account is thus normative at the metasemantic level and externalist at the level of realization; the model mediates evidence about the filler but does not create its semantic relevance.

Externalist examples show that speakers need not possess the scientific theory that describes a role-filler, but they do not prove that the proposed filler is correct. Nor does Lewisian naturalness select a cutoff without a prior account of which respects of similarity matter. The paper’s positive claim is therefore conditional and comparative: the multivariate transition profile is a serious operational candidate because its components bear directly on mutual support, persistence, and aggregate behavior, but the resulting ranking earns semantic relevance only if the bridge conditions are met. A numerical maximum by itself is insufficient.

Objection 4. Doesn’t physical instability restore principled ignorance?

Even if ordinary instability in hesitation and classification is located primarily at the pragmatic layer rather than treated as semantic plasticity, nearby physical perturbations might still alter which transition best realizes the role. A slight breeze, a change in humidity, or a minor rearrangement of surface grains could change the event profiles and their comparative role-realization ranking. If so, beliefs about the precise boundary might remain unsafe. The account would then reintroduce a margin-for-error problem at the level of the physical phenomenon and its model-mediated identification.

Reply. The objection identifies an empirical constraint on knowability rather than a direct challenge to bivalence. The proposed candidate is identified only relative to specified materials, construction conditions, measurements, and modeling assumptions. Small perturbations may alter the event profiles or the comparative ranking of candidate transitions. Whether the identification of a best role-realizer is stable must therefore be established through robustness analysis; it is not guaranteed by describing any feature as “global” or structurally significant.

The contrast with Williamson is consequently limited but substantive. Williamson’s original margin-for-error account explains why exact boundary knowledge may fail when nearby cases are indiscriminable despite differing in truth-value. Later semantic-plasticity reconstructions add that nearby changes in meaning-fixing use may shift the extension and undermine the safety of metalinguistic beliefs. The present account accepts the general safety constraint but denies that ordinary fluctuations in classificatory dispositions automatically alter either the referential role or its worldly realization. If the physical phenomenon is stable, the measurements and model are reliable, and the metasemantic bridge is adequately justified, expert knowledge may be possible in principle. If these conditions fail, knowledge may remain practically or systematically unavailable. That would be a contingent, empirical, and model-dependent limitation—not by itself a demonstration of principled unknowability, and not evidence against bivalence.

Objection 5. Does higher-order vagueness return at the pragmatic layer?

The tolerance range is itself vague. There is no sharp boundary between competent and incompetent use, nor between admissible and inadmissible applications of a tolerance disposition. This raises a higher-order question: can the pragmatic norms that help fix the referential role be vague without transmitting indeterminacy to the semantic boundary?

Reply. Higher-order vagueness in competence and application does not automatically propagate into the referential role. The objection assumes that the role-fixing function is exhausted by the distribution of ordinary classifications, but an externalist account of reference rejects this assumption. Linguistic practice helps determine what kind of worldly structure a predicate tracks, but the referential instruction need not encode every borderline judgment or hesitation among speakers. Therefore, small variations in classificatory dispositions may introduce uncertainty at the pragmatic level without substantially altering the role that guides reference.

More formally, let U represent the usage profile and let Cₕ represent the relevant communal, historical, and external reference-fixing conditions. The referential role is given by:

R = g(U, Ch)

Furthermore, the argument requires a stability premise rather than the function notation alone. For an ordinary ε-neighborhood of classificatory variation:

U' ∈ Nε(U) ∧ Ch' = Ch → g(U', Ch) = g(U, Ch)

Thus, small higher-order variations in application do not alter the referential role; only variation large enough to leave this stability neighborhood threatens role identity. The conclusion is conditional but deductive: if the role is stable under ordinary variation and a physical role-filler is identified, higher-order vagueness in ordinary application need not produce semantic indeterminacy.

Objection 6. Does the account generalize beyond “heap”?

The proposal is deliberately domain-restricted. It concerns “is a heap of sand” in a fixed construction series, not every use of “heap” and not all vague predicates. Laundry heaps, crowds, baldness, color, wealth, intelligence, and beauty may involve different physical, functional, perceptual, social, or evaluative roles. Some may lack a unique externalist role-filler.

Reply. The account should not be generalized crudely. Its strongest claim concerns the heap case: Sorites pressure can arise when pragmatic tolerance is treated as semantic invariance, and objective structure may constrain boundary-fixing. Different predicates may involve materially, biologically, perceptually, institutionally, or evaluatively constituted roles. Some may not fit an externalist role-filler model at all. The proposal is therefore a local framework with a possible structural lesson, not a complete taxonomy of vagueness.

Consider “bald.” One possible precisification takes complete absence of relevant scalp hair as a sharp physical condition, while treating ordinary applications to people with some hair as pragmatically extended. But this is not obviously the ordinary lexical semantics of “bald”; it may be a theoretically convenient sharpening rather than the discovered boundary of the existing predicate. The example therefore illustrates how the two-layer strategy might be applied, not that it has already identified the correct semantic role-filler.

A competing account may treat “bald” as essentially dependent on socially and contextually governed standards, without a single privileged physical role-filler. If so, the present externalist model would have limited application. This would not refute the heap proposal, but it would prevent the framework from being advertised as a general theory of vague predicates.

“Red” involves reflectance, illumination, perceptual systems, and color-space structure; “rich” depends partly on economic and institutional standards; and “beautiful” may lack any stable externalist role-filler. These examples suggest that the metasemantic architecture may vary by predicate. The modest conclusion is that, where a practice fixes a sufficiently determinate world-directed role and objective structure robustly supplies a filler, semantic sharpness can coexist with pragmatic tolerance. The paper does not establish that these conditions hold beyond the heap case.

The resulting position is a local case study and research program. It offers a pragmatic diagnosis of the fixed-series heap Sorites and a conditional naturalistic account of how its sharp boundary might be grounded. Its broader methodological lesson is not that every vague predicate has a hidden physical cutoff, but that tolerance, metasemantic role-fixing, empirical realization, and logical consequence should be evaluated separately.

7. Conclusion

The heap Sorites paradox begins with an apparently innocent thought: one grain cannot make the difference between a non-heap and a heap. This thought is compelling because it reflects something real about us. Our perceptual and classificatory capacities are tolerant. We do not ordinarily discriminate adjacent cases in a one-grain series, and our linguistic practices are not designed to require such discrimination. The mistake is to treat this pragmatic tolerance as a semantic truth-condition.

The two-layer framework developed in this paper diagnoses that mistake. At the pragmatic layer, speakers use vague predicates through tolerance ranges shaped by reference-fixing practices, paradigmatic cases, shared corrections, and finite discriminatory capacities. These ranges explain why competent speakers hesitate, disagree, and shift their judgments in borderline cases, and why the Sorites tolerance premise feels almost undeniable. They do not, by themselves, determine the predicate’s extension. This pragmatic diagnosis is strongest in the heap case and should not be assumed without argument to exhaust every form of vagueness.

At the semantic layer, linguistic practice fixes a world-directed boundary-role, while objective structure constrains its filler. For “heap,” the paper proposes a multivariate model that compares transitions across theoretically relevant dimensions of granular organization, treating a transition that robustly realizes the practice-fixed role better than its competitors as the candidate boundary. This proposal is model-mediated and scientifically revisable. It aims to track real structural inequalities among transitions, but its success depends on the justification and stability of the aggregation model and on the metasemantic bridge connecting structural significance to heaphood.

The account therefore diagnoses the fixed-series heap Sorites rather than dissolving every Sorites paradox. Under successful role-filling conditions, the base premise is true; the tolerance premise fails at the cutoff; and classical induction reveals the incompatibility of those premises with the existence of a clear heap state. The intuitive force of tolerance is explained pragmatically. The physical model then offers a fallible way to locate the cutoff, provided the role and bridge conditions can be defended.

The view retains epistemicist commitments to bivalence, sharpness, classical logic, and the falsity of universal tolerance, while changing the proposed grounding. Use fixes a structural role; objective organization constrains its filler; and modeling supplies defeasible evidence. If the model is inconclusive, the boundary remains unidentified. That result does not establish a truth-value gap, although the independent case for bivalence must still come from the epistemicist semantics rather than from the model’s success.

In this sense, the proposal is a form of normative metasemantic externalism: norm-governed linguistic practice helps determine the world-directed role, while objective structure constrains its realization. The normative component is not a license for stipulative sharpening; it supplies standards of correct application to which the worldly filler must answer.

The proposal is intentionally limited. It does not establish that every vague predicate has a physically grounded extension, nor that every materially anchored predicate has a uniquely privileged physical metric. Its broader lesson is methodological and conditional: before revising logic or inferring semantic indeterminacy from ordinary hesitation, we should ask whether pragmatic tolerance can be distinguished from a truth-condition constrained by objective structure. The answer may differ across predicates.

The heap Sorites gains much of its force when pragmatic tolerance is treated as semantic invariance. Separating the two weakens the authority of the tolerance premise: our inability to discriminate adjacent cases does not by itself show that no adjacent pair differs in truth-value. The physical account then supplies a substantive but incomplete proposal for locating that difference. Its value lies not in having finished a general dissolution of vagueness, but in making clear which pragmatic, metasemantic, physical, and epistemic claims must be defended separately.

Acknowledgement: AI-assisted tools were used for language editing and document preparation. The author takes full responsibility for the argument, sources, and final text.

Notes