Harnessing Vagueness with Productivity

Harnessing Vagueness with Productivity November 12, 2025 Abstract The sorites paradox, exemplified by the gradual loss of hair in the "bald man" scenario, illustrates the fundamental problem of vagueness in language and logic. In this work, I analyze this phenomenon through the lens of productive sets—sets for which a productive function f systematically generates elements outside any recursively enumerable subset. I model the concept of "bald" as a productive set, where f produces borderline cases (e.g., an individual with n +1 hairs), thereby undermining the possibility of precise, algorithmic determination of semantic thresholds. Classical forms of induction fail to capture the open-ended nature of such concepts, revealing structural limitations of formal systems in handling vagueness. I argue that vagueness is ontological rather than merely epistemic, arising from the inherent non-algorithmic character of certain predicates. This perspective calls for a reconsideration of classical approaches in logic and the philosophy of language and encourages further exploration of non-recursive models for understanding vague concepts. Notation and symbols Nthe set of natural numbers (including 0) A⊂B A is a subset of B(possibly A=B) A⊊B A is a proper subset of B(A⊂Band A=B) |A|cardinality of set A A′complement of Ain N Wxthe x-th recursively enumerable set (according to a fixed enumeration) χAthe characteristic function of the set A JPKthe denotation (extension) of the predicate P R(X)lower approximation of Xwith respect to relation R R(X)upper approximation of Xwith respect to relation R f, g, p (partial) recursive functions as defined in context 1 Introduction: Vagueness as a Problem of Computability The Sorites paradox reveals all the problems associated with vagueness.In this study of the issue, we will focus our attention on the bald man’s paradox. We do not provide any 1 argument, although it seems that the following conclusions regarding baldness can easily be transferred to other forms of the sorites paradox. The solution to the paradox that we have presented, from a philosophical point of view, does not negate traditional solutions, but rather complements and unifies them. Generally speaking, our solution reveals vagueness as a fundamental limitation of algorithmic classification. We analyze the predicate being bald in the following universes: •Universe U1: The set of people with Nhairs (main focus) •Universe U2: The set of hairs of a particular person •Universe U3: The set of all hairs of all people throughout history •Universe U4: The set of all predicates 1.1 Classical Formulations of the Paradox Standard induction (SP): •SP1: A person with 0 hairs is bald •SP2: If a person with nhairs is bald, then a person with n+ 1 hairs is bald •SP3: Therefore, every person is bald. Regression (SP’): •SP’1: A person with 100,000 hairs is not bald •SP’2: If a person with k+ 1 hairs is not bald, then a person with khairs is not bald •SP’3: Therefore, a person with 0 hairs is not bald. Both arguments employ two versions of the induction principle as formulated for the arithmetic of natural numbers. Mathematical induction is a dependable inference rule in the context of the natural numbers themselves. However, these arguments illustrate that extending induction beyond the natural numbers—particularly to sets that are merely well-ordered in a non-effective way—can result in paradoxes. 1.2 Mathematical Foundations of Induction Definition 1 (Mathematical Induction). (W(0) ∧ ∀n(W(n)→W(n+ 1))) =⇒ ∀n W(n) Definition 2 (Existential Regression). (∃x W(x)∧ ∀x(x= 0 ∧W(x)→ ∃y(y < x ∧W(y)))) =⇒W(0) Theorem 3 (Equivalence in Classical Logic).The principle of induction ⇐⇒ the principle of regression. The proof is based on the law of excluded middle (PEM): ∀x ( W ( x ) ∨¬W ( x )). The mathematical basis for the use of the aforementioned non-effective means is the following theorem: Theorem 4 (The Minimum (Well-Ordering) Principle in First-Order Logic). h∃n W(n)→ ∃mW(m)∧ ∀k(k < m → ¬W(k))i If there exists n such that W ( n )holds, then there exists the smallest m such that W ( m )holds, and for every k < m,W(k)does not hold. 2 1.3 Failure for Vague Predicates • Our considerations so far can be summarised as follows: Based on these paradoxical arguments, one can observe that mathematical induction does not work for the set of hairs, even after introducing equivalence classes (see Section 2). The source of the problem is ontological vagueness, not a reasoning error (false premise). 2 Numerical Explosion and Ontological Vagueness Combinatorial Problem Space Let H = {h1, . . . , h105} be the set of hairs on a head. The space of possible hair-removal sequences is: S= 105 [ k=0 Hk=⇒ |S| ≈ 10500000 As we can see the number of possible sequences ( ∼ 10 500000 ) vastly exceeds the number of atoms in the universe ( ∼ 10 80 ). From a philosophical point of view, this is surprising, because the number of hairs, which already seems enormous in relation to everyday life, explodes, as it were, to an unimaginable number. This observation alone should cause philosophers to question the validity of traditional constructions of vagueness Assumption of Homogeneity in Equivalence Classes For s, t ∈ S, define the equivalence relation: s∼t⇐⇒ length(s) = length(t) The equivalence class [k] = {s: length(s) = k}has cardinality: |[k]|= 105 k!·k!(e.g., |[104]| ≈ 102568) Problem: The assumption that all sequences in [ k ]have the same “baldness” status (i.e., ∀s, t ∈[k] : L(s)⇔ L(t)) is false. Heterogeneity of Classes Observation 2.1 (Counterexample to Homogeneity).For every k≥ 1, there exist s1, s2∈ [ k ] such that: L(s1)∧ ¬L(s2) For the same number of hairs k: •Configuration s1: hairs concentrated on the sides (pattern baldness) =⇒bald •Configuration s2: hairs evenly distributed =⇒not bald This prevents the use of induction even within equivalence classes. 3 Philosophical Conclusion At this point, we will summarise that we have three disturbing facts that require clarification. • Numerical explosion (10 500000 configurations) makes it impossible to define a sharp boundary for “baldness.” • Heterogeneity of equivalence classes demonstrates that the sheer number of hairs does not determine baldness status (hair distribution, thickness, cultural factors). • Non-extensionality of the predicate “Bald”: the truth of L ( x )is not a function of the mere number of hairs. For the same k , different configurations (distribution, thickness, length) and conventional factors can change the truth value of L ( x ), which undermines the extensional criterion and prevents a straightforward induction over k . We will not elaborate on this thread as it plays a secondary role to the main idea. 2.1 Sharp Predicates – Necessary Conditions? Already on the basis of a preliminary analysis, it intuitively appears that both the predicate to be a sharp predicate and the predicate to be a vague predicate are themselves subject to similar structural constraints. Somewhat jokingly, it is commonly accepted that sharp predicates are good, while vague ones are troublesome, due to difficulties in their use. Therefore, we first address those well-behaved sharp predicates for which we can provide necessary conditions for sharpness—i.e., conditions implied by it. A predicate Pis sharp only if it satisfies the following two necessary conditions: 1. Binarity Condition (BC): ∀x∈U:P(x)∨ ¬P(x),¬∃x∈U:P(x)∧ ¬P(x). 2. Recursive Decidability Condition (RC): χP(x) =    1if P(x), 0if ¬P(x), must be a total computable function. Claim 2.2 (RC implies BC).If the Recursive Decidability Condition (RC) holds for a predicate P, then the Binarity Condition (BC) also holds. Proof. If χP is a total function with range { 0 , 1 } , then for every x∈U exactly one of P ( x ) or ¬P(x)holds. This is precisely BC. Observation 2.3 (Logical asymmetry between RC and BC).RC entails BC, but not conversely. Hence the failure of BC necessarily entails the failure of RC. The contrapositive shows that whenever RC holds, BC must hold as well. This relation will be crucial for characterizing productive sets and, consequently, vague predicates. 4 2.2 Sentential and Predicate Negation for Sharp and Vague Predicates We can now formulate the correct definition of a vague predicate in light of the relation between RC, BC, and productivity. Definition 5 (Vague predicate).A predicate P is called vague if its denotation JPK⊆U is a productive set. Equivalently, P is vague when for every recursively enumerable subset We⊆JPK , there exists an effectively computable element x∗∈JPK\We . In particular, such a predicate fails the Recursive Decidability Condition (RC), although the Binarity Condition (BC) continues to hold. Intuitive reading. Vagueness thus corresponds not to a breakdown of classical bivalence, but to a deeper form of computational intractability: productive sets lie beyond recursive decidability while still maintaining sharp logical boundaries. A vague predicate in this sense is one whose extension is noncomputably generative—it cannot be exhaustively listed or decided, yet remains fully bivalent. Note 6 (Reformulated relationship between vagueness, BC, and RC).The logical relation among the three notions can be summarized as follows: RC ⇒BC,¬BC ⇒ ¬RC,Vague ⇒ ¬RC ∧BC. That is, every vague predicate fails RC but still satisfies BC. Vagueness does not imply a failure of bivalence, but rather a failure of effective computability. This interpretation aligns with the notion that the apparent “borderline cases” reflect the productive nature of JPK , not any genuine semantic indeterminacy. Erratum / Clarification In an earlier version of this framework, the definition of a vague predicate was stated as follows: a predicate P is vague if it fails to satisfy at least one of the necessary conditions BC (Binarity Condition) or RC (Recursive Decidability Condition) for sharp predicates. Upon further analysis, we realized that this formulation is logically redundant. Indeed, as shown the Recursive Decidability Condition (RC) implies the Binarity Condition (BC). Consequently, any predicate failing BC necessarily also fails RC. Therefore, the failure of BC is not an independent indicator of vagueness; the essential feature of a vague predicate is that its denotation JPK constitutes a productive set, i.e., a set whose elements cannot be exhaustively enumerated by any computable procedure. This clarification does not affect the subsequent analysis, as the main discussion correctly focuses on productive sets as the defining property of vague predicates. We thank the careful reader for noting this subtlety. Historical and Philosophical Context The distinction between sentential and predicate negation, and their relation to existential presupposition, was already recognized by Aristotle (in Categories, Cat. 13 b 17-35) and later discussed by Russell and Strawson. Thus, the 5 sentential negation (“It is not true that Socrates is sick”) is true when Socrates does not exist, while the predicate negation (“Socrates is not-sick”) is false. This illustrates that the equivalence ¬P(x)⇐⇒ ˜ P(x) holds only for existing objects, and that in natural language and philosophical logic, the distinction between these two types of negation is crucial—especially in the context of vagueness and existential presuppositions. This relies on an existential presupposition for predicate application in natural language; in classical first-order semantics with total domains the contrast is modeled differently. Note 7. The binarity condition (BC) is fundamentally relative to the choice of universe U : •Transfers to subsets: If V⊂Uand Pis sharp on U, then P|Vis sharp on V. • Does not transfer to supersets: If U⊂W , sharpness on U does not imply sharpness on W. Remark. The intuitionistic perspective on binarity and decidability: For a predicate P, the binarity condition (BC) ∀x∈U:P(x)∨ ¬P(x) holds constructively if and only if P is decidable (RC)—that is, there exists a uniform algorithm which, for every x, outputs either P(x)or ¬P(x). •Finite sets: All predicates are trivially decidable, so BC and RC coincide. • Infinite sets: Constructively, BC coincides with RC; if RC fails (for instance, for the Halting set K ), BC cannot be verified constructively—there is no algorithmic guarantee. Classically, BC may still hold even if RC fails, but no effective procedure exists to determine it. This highlights the limits of what can be known or computed. Productive sets exemplify this situation: in classical recursion theory, they are c.e. sets such that for any c.e. attempt to enumerate their elements, a productive function produces a fresh outsider—hence the set resists full computable capture. Intuitionistically, existence claims require witnesses; the productivity of a set is meaningful only if the productive function is given explicitly and can be executed step by step. Relation to Vague Predicates Under the present framework, a predicate P is considered vague if its denotation JPK is a productive set. Such predicates fail RC (no total computable characteristic function exists), while BC remains valid in the classical sense and coincides with RC in the constructive sense. This ensures that the concept of vagueness corresponds to a form of computational intractability rather than a failure of bivalence, and aligns with the intuition that so-called “borderline cases” emerge from the generative nature of productive sets rather than any semantic indeterminacy. 6 3 The Formal Framework of Productive Sets Definition 8 (Productive Set [Pos44], [Soa16]).A set P⊂N is called productive if there exists a total recursive function fsuch that ∀e(We⊂P=⇒f(e)∈P\We), where Wedenotes the e-th recursively enumerable (r.e.) set. Definition 9 (Effective Inseparability, cf. Soare [Soa16], pp. 41).A disjoint pair ( A, B )of c.e. sets is effectively inseparable if there exists a partial computable function ψ (called a productive function for (A, B)) such that for all xand y, [A⊂Wx&B⊂Wy&Wx∩Wy=∅] =⇒[ψ(⟨x, y⟩)↓&ψ(⟨x, y⟩)/∈Wx∪Wy]. Effective inseparability shows that even among computably enumerable sets, there exist disjoint pairs that cannot be separated by any recursive set. Moreover, for certain such pairs, a productive function always produces an element outside any attempted separation. This provides a direct link to productivity and highlights the limitations of algorithmic classification in computability theory. Epistemic Limitations for Vague Predicates Theorem 10. Let JPK⊂Nbe a productive set with productive function f. Then: 1. JPKis not recursively enumerable (r.e.). 2. Consequently, P is undecidable: there is no total algorithm deciding membership for all inputs. 3. For any r.e. approximation We⊂JPK , f ( e )generates a boundary case escaping classification. Proof. (1) Follows directly from the definition of productivity: if P were r.e., there would exist ewith We=P, but then f(e)∈P\We=∅, a contradiction. (2) Immediate from (1). (3) By productivity: f ( e ) ∈P\We is a boundary case not captured by the approximation. From Epistemic Barriers to Ontological Productivity While classical approaches often focus on the epistemic symptoms of vagueness—semantic or contextual indeterminacy—the productivity model reveals its ontological source: Impossibility of algorithmic control =⇒the generative nature of boundary cases. This shifts the discussion from epistemology to the ontology of computable concepts: vagueness is a structural property of certain predicates, rooted in the generative (productive) character of their extension. 7 Ontological vs. Epistemic Barriers Not all limits in mathematics are epistemic. Some boundaries are intrinsic to the structure of concepts or sets, independent of what we know. Whether a set is r.e. or decidable is a formal fact, not a matter of human knowledge. Productive sets, such as Fin , exemplify this: no algorithm can exhaustively decide membership. The impossibility of algorithmic classification is structural, not merely epistemic. Example 3.1. Consider a model for the predicate “being bald,” denoted L over universe U1 , with extension JLK . The predicate L behaves analogously to the productive set Fin (indices of Turing machines enumerating finite sets): •Fin is productive and belongs to Σ0 2.1 •Its complement is r.e. but not recursive. •Consequences for “baldness”: ∄algorithm to confirm baldness in all cases, ∄algorithm to refute baldness with certainty, Boundary cases (e.g., individuals with 100,001 hairs) escape every fixed rule. Remark. Mathematically, “being bald” is not literally a productive set; at any fixed time and in a finite population, extensions are finite. The analogy concerns complexity of classification, not cardinality. The vagueness of “bald” acts like productivity: whatever crisp, algorithmic rule you propose, boundary cases will always slip through. This generativity is a structural feature of the predicate itself, echoing the productive counterexample property from recursion theory. BC, RC, and Vague Predicates Recall that in this framework: •Productive sets fail RC (no total computable characteristic function exists). • BC remains valid in classical logic and coincides with RC in constructive (intuitionistic) contexts. • Thus, a predicate P is vague iff JPK is productive. Boundary cases correspond to elements generated by the productive function f. Remark. The Binarity Condition (BC)—that for every x , either P ( x )or ¬P ( x )holds—has a different status depending on the system: • In classical systems with the law of excluded middle and r.e. proof rules, BC can, in principle, be verified algorithmically for each formula. • In non-effective systems (like true arithmetic or systems with the omega-rule), BC may hold in the full model (“God’s view”), but there is no recursive way to verify all instances; truth in the model does not guarantee provability inside the system. Notes: •“Bald” is modeled after Fin (conceptual analogy; not a literal identification) 1 For reference, see [Soa16, p. 82]: x∈Fin iff Wx is finite, i.e., x∈Fin ⇐⇒ ( ∃s ) ( ∀t )[ t≤s∨Wx,t = Wx,s ]. 8 Predicate type BC RC Example Sharp ✓ ✓ Prime numbers Epistemically vague ✓×Halting set (K) Ontologically vague ✓דBald” (Fin) Extremely vague × × Higher-order vagueness: e.g., “borderline borderline bald” (where even being a borderline case gets fuzzy) Table 1: Hierarchy of vagueness based on computability and logic • Ontological barrier: In the productive case, vagueness comes straight from the mathematical structure—the extension isn’t r.e., so no algorithm pins it down. It’s not about missing information, but about how the set is built. • Dynamic boundaries: The productive function f spells out precisely why boundary cases won’t ever settle: for any attempt at full classification, you can always algorithmically toss in a new “close call” just outside. The dividing line keeps moving. 4 Productiveness and the Logic of Vagueness: The Case of “Bald” Remark. Productiveness formalizes "escaping algorithmic control": any attempt to enumerate a recursively enumerable subset P is ineffective—the function f always finds a new element outside the enumerated set. Recall that Fin —the set of indices of Turing machines enumerating finite sets—serves as the canonical productive set. In what follows, we model the vagueness of “bald” using the generative logic of Fin : no matter what r.e. rule is proposed, there is always a borderline case that escapes classification, mirroring the productive property. This is not a formal theorem, but a conceptual analogy that captures how boundaries in vague predicates behave. This set provides the canonical model for analyzing productive phenomena within the logic of vagueness. Note that the set of all finite subsets of the natural numbers is recursively enumerable (r.e.), whereas the set Fin (the set of indices of Turing machines enumerating finite sets) is not. The reason for this difference lies in a profound property of computability theory: while every finite subset of natural numbers can be effectively listed, there exists no algorithm to decide whether a given program (Turing machine index) generates a finite set. This stems from the undecidability of the halting problem—there is no general procedure to determine whether a machine will halt on all inputs and thus enumerate only finitely many elements; therefore, the set of such indices is not r.e., even though the finite sets themselves are. 4.1 Key Properties •Non-Recursively Enumerable: No productive set is recursively enumerable (r.e.). • Generator of Borderline Cases: For the predicate “bald”, the function f corresponds to adding a “grain” in the sorites paradox. 9 example, the set of "bald" individuals—without reference to an opposing class. This reveals that vagueness, and the perpetual generation of borderline cases, can be modeled entirely in terms of the open-endedness of a single concept, not a dichotomy. • Contextual theories (Shapiro): Boundaries depend on the context of use. Critique: Ignores the fundamental instability—the function f operates across contexts, generating universal exceptions.[Sha06] 5.2 Limitations of Classical Approximation Models Before fully justifying the productivity-based approach, it is instructive to critically examine the two most influential classes of models for vagueness: rough set theory (Pawlak) and fuzzy set theory. Both attempt to capture the lack of precise boundaries by introducing gradations of membership or dual approximation. However, as will be shown in the following subsections, these methods are inherently tied to the partitioning of the universe into a concept and its complement ("bald" vs. "not bald"), and their boundary-blurring is limited to this dichotomy. In a sense, this is a presupposition of their approach and, unlike our approach, it undermines their theoretical position from the outset. From a linguistic point of view, their approach concerns so-called complementary antonyms, while ours additionally concerns contrary antonyms. By closely analyzing their construction, it becomes clear that neither approach accounts for the generative aspect of vagueness modeled by productive sets—namely, the perpetual possibility of escaping any enumeration or explicit definition of the predicate itself. This key limitation undermines their ability to capture the dynamic, one-sided open-endedness that lies at the heart of vague concepts. 5.3 Pawlak’s Rough Set Theory in a Nutshell We briefly present Pawlak’s theory2which shows certain similarities with our approach. Definition 18 (Approximation Space).Let A= (U, R)be an information space, where: •U — as introduced earlier as U1 , denotes the set of all individuals, each with a certain number Nof hairs, •R– an indiscernibility relation (usually equivalence). For x∈U , the equivalence class [ x ] R = {y∈U : xRy} represents objects indistinguishable with respect to the available knowledge. Definition 19 (Approximations).For a set X⊂U: Lower approximation :RX={x: [x]R⊂X} Upper approximation :RX={x: [x]R∩X=∅} Boundary region : BND(X) = RX\RX 2[Paw82] 16 Example 5.1 (Modeling the "bald" predicate in Pawlak’s framework).Let U = { 0 , 1 ,..., 10 5} , where the element nrepresents the number of hairs. Define: •X={0}— the set of "bald" individuals (exactly 0 hairs), •U\X={1,2,...,105}— not bald (including the extreme case of 105hairs). Then: RX={0}(definitely bald) RX={0,1,...,105−1}(potentially bald) BND(X) = {1,2,...,105−1}(region of vagueness) Here, the boundary region consists of individuals with 1to 10 5− 1hairs — these are the borderline, indeterminate cases between clearly bald and clearly not bald. Remark. It is important to note that, in Pawlak’s theory, the width of the boundary region depends entirely on the choice of the indiscernibility relation. By defining broader equivalence classes, one can generate a wider region of vagueness. This is a modeling strategy: the source of vagueness is not explained, but rather shifted to the definition of the relation itself. Note 20 (Heterogeneity of abstraction classes: limitation of Pawlak’s model).As discussed earlier in this article, abstraction classes for the predicate “bald” are empirically heterogeneous: individuals with the same hair count may differ significantly in the distribution or density of hair or other features relevant to baldness. Pawlak’s rough set model assumes that equivalence classes defined by indiscernibility are homogeneous, which rarely holds for natural concepts. This limitation means that Pawlak’s approach, while effective for mathematical objects, does not fully capture the ontological vagueness present in real-world phenomena. Note 21 (Critique of the static nature of abstraction classes).Pawlak’s rough set model is based on static abstraction classes [ x ] R , defined a priori by an indiscernibility relation R (e.g., “having the same number of hairs”). The boundary region BND ( X )is a closed set with no mechanism for generating new borderline cases. In contrast, our productivity-based model (see Def. 13) reveals a fundamental limitation: Static classes cannot capture the ontological dynamics of the productive function f. The function f ( e )systematically generates new borderline cases (e.g., a person with n + 1 hairs), which: 1. Belong to JPK(are “bald”), 2. Escape all existing approximations RXand RX. Note 22 (Key limitation).Pawlak’s model does not account for the dynamics of borderline cases: •The boundary region BND(X)is static, •There is no mechanism for generating new exceptions (e.g., adding one hair), •The model is incompatible with productive denotations (see Theorem 4.3 in this work). 17 Pawlak’s Model Productivity Model [x]R={y∈U: hairs(y) = hairs(x)}f(e)∈JPK\We Fixed boundary: Dynamic generativity: Equivalence classes define fixed (predefined) vagueness intervals. The productive function f always generates new borderline cases outside any algorithmic approximation. Table 2: Comparison of vagueness modeling: Pawlak’s static approach vs. dynamic productivity-based model. Concept Mechanism Ontological status Pawlak’s lower approximation R(X) = {x: [x]R⊂X} Static set of definite elements Productive set (PR) f ( e ) ∈P\We , for We⊂P , Wer.e. Dynamic generation of definite cases outside r.e. subsets Pawlak’s upper approximation R(X) = {x: [x]R∩X=∅} Static set of potential elements Counterproductive set (CPR) g ( e ) ∈B\A , for B⊃A , B r.e. Dynamic generation of gaps in potential supersets Table 3: Relationships between Pawlak’s approximations and productive/counterproductive sets. 18 5.4 Comparison of Pawlak Approximations with Productive and Counterproductive Sets Definition 23 (Contraproductive set [Dek55], [Odi89]).A set A⊆N is contraproductive if there exists a partial recursive function p(x)such that for all x, A⊂Wx=⇒   p(x)is defined, p(x)∈Wx\A, where Wxis the x-th recursively enumerable set. If such p ( x )exists, we say that A is contraproductive relative to p , and that p is a contraproductive function for A. Theorem 24 (Dynamic irreducibility).There does not exist a relation R such that for a productive set Pand a counterproductive set A: R(P) = {f(e) : We⊂P}, R(A) = {g(e) : B⊃A, where Bis r.e.}, where fand gare the productive and counterproductive functions, respectively. Proof. Suppose, for contradiction, that such a relation R exists. Then the set {f ( e ) : We⊂P} would coincide with a lower approximation R ( P ), which is always ∆ 0 2 . By Dekker’s theorem, productive sets have strictly higher arithmetical complexity and cannot be captured this way. Similarly for counterproductive sets and upper approximations. This contradiction implies that no such Rcan exist. Remark. This result highlights a fundamental mismatch between static approximation frameworks (such as Pawlak’s lower and upper approximations) and the inherently dynamic nature of productive and counterproductive sets. No choice of indiscernibility relation can bridge this gap: the generative mechanisms defining these sets cannot be captured by any fixed relation. Implications for the “bald” model Let P= Fin (a productive set) and A=K(a counterproductive set, cf. Dekker 1955): •Lower approximations cannot capture the generative nature of P: For any recursively enumerable approximation R , there exists a computable f ( e )such that f(e)∈Fin \R(Fin), i.e., every effective lower approximation omits some new element generated by f. • Upper approximations fail to model the “escaping” behavior of counterproductive sets: For any upper approximation R(K), there exists a function gsuch that g(e)∈K\R(K), i.e., every attempt to overapproximate Kmisses some new member of K. 19 •Conclusion: Static approximations (as in rough or fuzzy set theory) are ontologically insufficient for truly generative vagueness: productive (or counterproductive) sets evade any static, effective boundary by continuously generating counterexamples. 5.5 Critique of Fuzzy Logics and an Alternative Proposal In this section, we argue for the superiority of the emergent model for vague predicates over the approach based on fuzzy logics. The argument is that fuzzy logics are both ontologically and computationally inadequate for productive denotations. Vagueness is not an "approximable degree of truth" but a fundamental impossibility of algorithmic closure. 5.5.1 Computability in the sense of rational numbers Q Definition 25 (Densely computable function on rationals).Let D = Q∩ [0 , 1]. A function µ : D→Q∩ [0 , 1] is called densely computable if there exists a total recursive function F:N→Nsuch that: • Every rational number q∈D is assigned its canonical code ⌜q⌝∈N (for example, by means of a pair (a, b), where q=a/b,0≤a≤b,b > 0), •For every q∈Dwe have F(⌜q⌝) = ⌜µ(q)⌝, where ⌜µ ( q ) ⌝ denotes the code of the rational value µ ( q ) ∈ [0 , 1] (using the same coding method as above). • For arguments that do not code an element of D , the value F may be arbitrary (e.g., F(n)=0). 5.6 Computability on a Dense Set (the Rationals) Theorem 26 (Mismatch of Fuzzy Logics for Dense Functions).There exist disjoint, recursively enumerable, recursively inseparable sets A, B ⊆N such that there is no total computable function µ:Q→[0,1] (computable on rational inputs, e.g., outputting a rational in reduced form) that satisfies µ(x)=1 for x∈A, µ(x)=0 for x∈B, µ(x)∈(0,1) for x∈N\(A∪B), with µdefined on the dense domain Qand computable on (canonical) codes of rationals. Proof sketch. By Post’s theorem, there exist disjoint r.e., recursively inseparable sets A, B ⊆ N . Suppose, for contradiction, that such a µ exists. Evaluate µ on integers via their canonical encoding as rationals x7→ x/1, and define D={x∈N|µ(x)>1 2}. 20 Since µ is total computable on rational inputs and returns rational outputs, the predicate “ µ ( x ) >1 2 ” is decidable; hence D is recursive. Moreover, A⊆D (because µ = 1 on A ) and B∩D = ∅ (because µ = 0 on B ), so D separates A and B , contradicting their recursive inseparability. Therefore, no such µexists. Example 5.2. The function µ:Q→[0,1] defined by µ(q) =            1, q ≤10, 0, q ≥20, 1−q−10 10 ,10 < q < 20 is a total computable function on the dense domain Q : for any rational input q = a/b (with b = 0) in canonical form, µ ( q )is effectively computable as a rational. The theorem shows that no such computable function can realize the required A/B/ (0 , 1) pattern on integers when A, B are r.e. and recursively inseparable. 5.7 Three Pillars of Critique 1. Impossibility of Computability (see Thm. 26): For productive sets (e.g., Fin —the set of indices of programs generating finite sets—or vague concepts such as X = “bald”), there does not exist a computable function µ that satisfies the requirements of fuzzy logics. Here, we mean densely computable functions on Q (i.e., functions for which there is an algorithm to compute µ ( q )for every rational q ). Computability of µ is understood in the classical sense, on codes of pairs of natural numbers. Even though µ formally maps N into the closed interval [0 , 1], the assumption that a computable version exists for productive denotations leads to contradiction. Moreover, attempts to approximate µ result in algorithmic contradiction. 2. Ontological Error of Reductionism: Fuzzy logics ignore the generative role of the function f , which continually creates new borderline cases (e.g., f(e) = 105+ 1 hairs). 3. Static Illusion: Modeling vagueness as “degrees of membership” overlooks its inherently dynamic nature as an open process, by analogy with Brouwer’s intuitionism [Bro81]. Remark. In classical literature, Fin denotes the set of indices of programs generating finite sets, i.e., Fin = {x : Wxis finite} . In this article, Fin is not identified with the concept of “baldness,” but serves as an example of a productive set: precisely defined, yet exhibiting a vague boundary. The “bald” example illustrates vagueness in the everyday sense. 5.8 Paradigm Comparison Theorem 27 (A Productivity Barrier for r.e. Fuzzy Catalogues).Let D be a countable domain and let values lie in Q∩ (0 , 1). Let W be any recursively enumerable family of total 21 Criterion Fuzzy Logics Emergent Models (Productivity) Computability of µ Constrained; computable profiles exist, but r.e. catalogues are not exhaustive Not required; focus on the productive mechanism f Generation of cases Catalogue-driven (passive enumeration) Active generation (productive f) Ontology Static, predefined membership repertoire Dynamic, generative (cf. Brouwer) Relation to productivity Fails under r.e. limitations Directly satisfied (built-in productivity) Table 4: Comparison of approaches to modeling vagueness. recursive fuzzy classifiers ( µi ) i∈W , where each µi : D→Q∩ (0 , 1) is total and computable. Assume there exists a point q∗∈Dsuch that the pointwise image S(q∗) := {µi(q∗) : i∈W} is a proper subset of Q∩ (0 , 1) (e.g., finite, or otherwise not exhausting Q∩ (0 , 1)). Then there exists a total recursive classifier µ∗:D→Q∩(0,1) with index e∗/∈Wsuch that µ∗(q∗)∈Q∩(0,1)\S(q∗)and hence µ∗=µifor all i∈W. Proof sketch. Since S ( q∗ ) ⊊ Q∩ (0 , 1) and Q∩ (0 , 1) is countable and r.e., pick r∈ ( Q∩ (0 , 1)) \ S ( q∗ ). Define µ∗ by setting µ∗ ( q∗ ) = r and, for q = q∗ , let µ∗ ( q ) = 1 2 (or any fixed rational in (0 , 1)). This µ∗ is total and computable. For every i∈W , we have µi ( q∗ ) ∈S ( q∗ )  = r , so µ∗=µi. Therefore the index e∗of µ∗does not belong to W. Remark (On a stronger single-point version).A strictly stronger statement is possible: one can aim for a single point q∗ at which a newly constructed total computable classifier disagrees with all members of an r.e. catalogue, without first assuming a witnessed gap. However, such a result requires additional structural assumptions on the catalogue (e.g., a global bound on the pointwise range, a fixed finite value alphabet at each argument, or other uniform saturation constraints). Absent these extra hypotheses, an r.e. family may be arranged to exhaust Q∩ (0 , 1) at every fixed point, making unconditional single-point separation impossible. Note 28 (Interpretation).The theorem shows that whenever an r.e. catalogue leaves a pointwise gap at some q∗ , one can construct a total computable classifier that falls outside the catalogue yet differs from every listed classifier already at that same single point. Conceptually, this captures the idea that any fixed, effectively generated catalogue leaves room for new cases—precisely at locations where its pointwise range is not saturated.To put it more simply, this means that the set of fuzzy logic classifiers is a productive and vague set. 22 Note 29 (Crucial observation on 0 and 1 in fuzzy logic).In fuzzy logics and fuzzy sets, it is assumed that some elements of the universe have membership value 1, some have 0, and the rest take intermediate values. The problem is that the assignment of elements to values 0 or 1 must be fixed in advance, typically as a recursive or at best recursively enumerable set. In other words, the base is static and predetermined. However, for productive concepts this static assumption fails — borderline cases keep emerging continuously and cannot be ultimately assigned to either 0 or 1, as new cases always arise demanding consideration. Fuzzy logic cannot capture this, since it presupposes a stable partition into 0 and 1 values, which does not reflect the dynamic, open-ended nature of genuine vagueness. Therefore, fuzzy logic does not fully address the problem of vagueness — it overlooks the essential generative and open character required for adequate modeling of inherently productive concepts. From a philosophical point of view, this means that the fuzzy logic approach only seemingly solves the problem of uncertainty. 6 Philosophical Justification for the Shift Vagueness is not a flaw in language or logic, but a fundamental feature of certain mathematical concepts: the impossibility of algorithmically closing their extension. It’s not about eliminating vagueness, but embracing it as a constructive aspect of formal systems. Recursive transparency: The function fis explicit and present, which means: •the illusion of sharp, computable membership degrees disappears, •the necessary undecidability of binary decisions is justified, • vagueness is shown not as a problem, but as a source of conceptual growth and adaptability. Classical approach Productivity model Static boundaries (e.g., “threshold of 1000 hairs”) Dynamic generation of counterexamples (function f) Subjective membership degrees Objective non-algorithmizability (cf. section 4.1) No explanation of the sorites regress Unification of regress through recursive f No account of borderline contradictions Explains contradictions as computational instability Epilogue: Beyond the “Safe World” of Recursive Sets: Two Reflections 1. The boundaries of computability are closer than we think In the foundations of mathematics and theoretical computer science, we often assume that we operate within the “safe” world of recursive and recursively enumerable (r.e.) sets, where our intuitions are clear and the boundaries of concepts are well understood. However, both 23 theory and practice reveal that, even unconsciously, we frequently encounter much subtler notions: productive, creative, and non-constructive sets—whose properties transcend classical intuitions. • Productive sets and other “non-structural” sets arise naturally in algorithm analysis, proofs of undecidability, and investigations into the limits of computability. • Many concepts we use daily—such as “all programs that do not halt” or “problems that cannot be algorithmically decided”—are, in fact, examples of sets beyond the recursive and r.e. classes. • Our cognitive sense of security is thus illusory: the boundary between the “constructive” and “non-constructive” worlds is subtler than it appears. 2. Everyday mastery of vague predicates: taming the wild It is also worth emphasizing that our everyday ease in using vague predicates (such as “bald,” “tall,” “old”) is not only a linguistic phenomenon, but also a profound cognitive process. This ability allows us to domesticate and utilize concepts that, in logic or mathematics, are formally very difficult, non-constructive, or even productive. • Everyday practice: Every person, regardless of education, intuitively handles vague concepts, even though their formal definition leads to paradoxes or undecidability. • Taming complexity: Using vague predicates in practice shows that we can “tame” complexity and non-constructiveness, making them useful tools in communication and thought. • A bridge between daily life and theory: Analyzing these processes can help us understand how the human mind copes with non-constructiveness, and how we might transfer intuitions from everyday life to advanced mathematics or logic. Conclusion. The reality of computability theory is much richer and more unsettling than the classical world of recursive sets suggests. We regularly employ concepts—sometimes without realizing it—that force us to revise our intuitions and recognize that the boundary between the constructive and the non-constructive is fluid. At the same time, our everyday cognitive strategies for handling vagueness allow us to domesticate even the wildest theoretical notions, making them accessible and practical for everyone. These reflections suggest that the true challenge for mathematics, logic, and AI is not merely to formalize the boundaries of the computable, but to understand and harness the generative, open-ended nature of vagueness and non-constructiveness that pervades both theory and everyday life. Note on the Use of AI Assistance In preparing and editing this article, I have made use of AI-based tools and consultation. Artificial intelligence assisted me in clarifying the formulation of certain technical passages, synthesising critical perspectives on alternative models of predicate vagueness, and optimising the language of the article. All substantive decisions, the main idea of the paper, the structure of the argumentation, and the final editorial shape of the text, however, are entirely my own. The role of AI in this process is best characterised as analogous to expert consultation or editorial support. The final version of the work fully reflects my own scholarly standpoint and approach to the subject matter. 24 Bibliografia [Bro81] Luitzen Egbertus Jan Brouwer. Brouwer’s Cambridge Lectures on Intuitionism. Ed. by D. van Dalen. Cambridge: Cambridge University Press, 1981. isbn: 0521234417. [Dek55] J. C. E. Dekker. “Productive sets”. In: Transactions of the American Mathematical Society 78.1 (1955), pp. 129–149. doi: 10.1090/S0002-9947-1955-0067049-X. [Fin75] Kit Fine. “Vagueness, Truth and Logic”. In: Synthese 30 (1975), pp. 265–300. [Łuk20] Jan Łukasiewicz. “O logice trójwartościowej”. In: Ruch Filozoficzny 5 (1920). Tłumaczenie angielskie: On Three-Valued Logic wSelected Works, pp. 169–171. [Odi89] Piergiorgio Odifreddi. Classical Recursion Theory: The Theory of Functions and Sets of Natural Numbers. Sole distributors for the USA and Canada, Elsevier Science Pub. Co., 1989. [Paw82] Zdzisław Pawlak. “Rough Sets”. In: International Journal of Computer & Information Sciences 11.5 (1982), pp. 341–356. doi: 10.1007/BF01001956. [Pos44] Emil Post. “Recursively enumerable sets of positive integers and their decision problems”. In: Bulletin of the American Mathematical Society 50 (1944), pp. 284– 316. [Sha06] Stewart Shapiro. Vagueness in Context. Oxford University Press, 2006. [Soa16] Robert I. Soare. Turing Computability: Theory and Applications. Springer, 2016. [Wil94] Timothy Williamson. Vagueness. Routledge, 1994. isbn: 9780415139809. 25