Bilattice-Valued Predicate Fibrations and the Geometry of Closure

Symmetry Breaking, Orientation, and Quasicrystalline Order in Belnapian Knowledge Structures

Thesis

Knowledge organization requires three things that have mature independent formalizations but have never been unified: typed predication (Frege → Lawvere), incomplete and contradictory evidence (Belnap → Fitting), and part-whole decomposition (Leśniewski → Cotnoir & Varzi).

This paper defines a Predicate-Fibered Belnapian Knowledge Structure (PFBKS) — a knowledge graph whose edges carry typed predicates valued in Belnap’s four-valued bilattice FOUR, with vertices ordered by a mereological preorder — and proves that closing it under inference produces a qualitative geometric phenomenon: FOUR’s Z₂ × Z₂ symmetry breaks in a predicate-dependent way, producing an oriented geometry whose structure encodes the knowledge system’s semantic content.

The paper is not “here is a construction, here are its properties.” It is: FOUR is a geometry → fiber it over typed predicates → close it → the symmetry breaks → the broken symmetry is the real mathematical object. Everything before closure is scaffolding. The symmetry-breaking theorem is the result. The oriented geometry is what the result means.

Background: Belnap’s FOUR

Nuel Belnap (1977) proposed a four-valued logic for reasoning about what a computer has been “told” by potentially unreliable sources. The four values are:

• ⊥ (neither): no information. Nobody has said anything.
• t (true): told true, not told false.
• f (false): told false, not told true.
• ⊤ (both): told both true and false. Contradictory evidence.

These four values carry two independent orderings:

• Truth ordering (≤_t): f ≤ ⊥ ≤ t, f ≤ ⊤ ≤ t, with ⊥ and ⊤ incomparable. A diamond with f at bottom, t at top.
• Knowledge ordering (≤_k): ⊥ ≤ t ≤ ⊤, ⊥ ≤ f ≤ ⊤, with t and f incomparable. A diamond with ⊥ at bottom, ⊤ at top.

The set FOUR with both orderings, plus negation (~ swaps t and f, fixes ⊥ and ⊤) and conflation (- swaps ⊥ and ⊤, fixes t and f), is a bilattice: a structure with two interacting lattice orderings. Ginsberg (1988) introduced bilattices to AI; Fitting (1991, 2002) developed their theory extensively and proved that FOUR is the free bounded distributive bilattice on one generator — the simplest non-trivial bilattice, and the universal one.

This paper’s central move is treating FOUR not as a set of truth values but as a geometry, fibering it over typed predicates, and studying what closure does to its symmetry.

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Section 1. Introduction (~1000 words)

Open from the three-formalism gap: typed predication, uncertain evidence, and mereological composition each have mature theories, but existing work only pairs them (bilattice + description logic, mereology + ontology, predicates + graphs). The triple is unstudied.

Introduce the PFBKS — a tuple K = (G, Σ, ν, ≤_m) combining a labeled directed graph G, a typed predicate vocabulary Σ, a Belnapian valuation ν, and a mereological preorder ≤_m. Explain in one paragraph what each component does and why all four are needed.

State the discovery: closing a PFBKS under inference eliminates ⊥, collapsing FOUR’s diamond to a directed triangle {f, ⊤, t}. This breaks the bilattice’s Z₂ × Z₂ symmetry. Directed predicates (part-of, requires, produces) acquire directed truth chains. Symmetric predicates (contrasts-with) retain Z₂. The result is an oriented geometry over the knowledge graph.

State four contributions:

1. The PFBKS construction with Fiber Functor theorem (a bilattice-valued generalization of Lawvere’s hyperdoctrine)
2. The two-generator thesis: all predicates decompose into extends (generative) and restricts (constraining), forming a Galois biconnection
3. The Symmetry-Breaking theorem: closure produces predicate-dependent directedness on the three-element residual {t, f, ⊤}
4. Characterization of quasicrystalline order in the closed product geometry under incommensurability conditions

Key citations

• Frege (1879): typed predication as function application — the intellectual origin. A predicate is an unsaturated function that takes a subject and returns a value.
• Lawvere (1969, “Adjointness in Foundations”): formalizes Frege’s predicates categorically as hyperdoctrines — functors from a base category of contexts to a fiber category of propositions. The Fiber Functor theorem generalizes this from Heyting algebra fibers to bilattice fibers.
• Belnap (1977, “A Useful Four-Valued Logic”): the epistemic motivation for FOUR. Not “there are four truth values” but “a knowledge system that receives information from multiple sources naturally occupies one of four epistemic states.”
• Fitting (1991, 2002): bilattice theory, fixpoint semantics for logic programming on bilattices. We adapt his fixpoint techniques.
• Cotnoir & Varzi (2021, Mereology): the mereological axiom system we adopt — deliberately weaker than classical extensional mereology, appropriate for knowledge structures.

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Section 2. FOUR as Modal Geometry (~1200 words)

This section establishes that FOUR is a 2D geometric object, not merely a set of truth values. All results here are known; the contribution is the geometric reading that the rest of the paper depends on.

2.1 Bilattices and FOUR. Define a bilattice formally: a set L with two bounded lattice structures (L, ∧_t, ∨_t, f, t) and (L, ∧_k, ∨_k, ⊥, ⊤), plus negation ~ (reverses ≤_t, preserves ≤_k) and conflation - (preserves ≤_t, reverses ≤_k). A bilattice is distributive if both lattice structures are distributive and the interlacing conditions hold (each meet distributes over the other’s join). A bilattice is interlaced if all four operations (∧_t, ∨_t, ∧_k, ∨_k) are monotone with respect to both orderings.

Define FOUR = {⊥, t, f, ⊤} with both orderings as given in the Background section. State Fitting’s (2002) result: FOUR is the free bounded distributive bilattice on one generator.

2.2 The Twist Product. Kalman (1958) showed that every bounded distributive lattice L generates a De Morgan algebra via the twist product L ⊗ L. Fitting (1991) extended this to bilattices: FOUR is isomorphic to 2 ⊗ 2, the twist product of the two-element Boolean algebra with itself. Concretely, each FOUR value is a pair (a, b) ∈ {0,1}²:

Value	Pair	Reading
⊥	(0,0)	No information in either dimension
t	(1,0)	Positive truth, no counter-evidence
f	(0,1)	Negative truth, no supporting evidence
⊤	(1,1)	Information in both dimensions

The first component is the truth dimension; the second is the knowledge dimension. FOUR is not a “four-valued logic” — it is a 2D modal space whose two axes are truth and knowledge.

2.3 The Symmetry Group. Negation ~ acts as (a,b) ↦ (1-a, b): it flips the truth axis and fixes the knowledge axis. Conflation - acts as (a,b) ↦ (a, 1-b): it flips the knowledge axis and fixes the truth axis. Together they generate Z₂ × Z₂, the Klein four-group — four elements: identity, negation, conflation, and their composition (180° rotation).

Z₂ × Z₂ is the symmetry group of the square. FOUR, plotted with ≤_t horizontal and ≤_k vertical, IS a square. Negation and conflation are reflections across the two axes.

This symmetry group is what closure will break.

2.4 Mereological Preorders. Define a mereological preorder on a set V: a reflexive, transitive relation ≤_m satisfying weak supplementation (if x <_m y, there exists z ≤_m y with z not overlapping x). Following Cotnoir & Varzi (2021, Ch. 3), we use this weaker axiom system rather than classical extensional mereology, because knowledge structures admit distinct units with identical parts (unlike physical objects under extensional mereology).

Key citations

• Kalman (1958, “Lattices with Involution”): the original twist product construction for De Morgan algebras. Priority for the algebraic technique.
• Fitting (1991): extended Kalman’s construction to bilattices; proved FOUR ≅ 2 ⊗ 2.
• Ginsberg (1988): introduced bilattices to AI reasoning.
• Dunn (2000, “Partiality and Its Dual”): treats negation as a modal operator. Supports reading FOUR as a modal space rather than merely a many-valued logic.
• Shramko & Wansing (2011, Truth and Falsehood): the philosophical framework for generalized truth values. Argues for treating FOUR’s values as genuine epistemic states, not technical conveniences.
• Arieli & Avron (1996, “Reasoning with Logical Bilattices”): establishes that bilattice structure supports genuine inference, not just truth-value assignment.

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Section 3. Predicate Primitives and the Adjunction (~1500 words)

First novel contribution. The claim: the surface vocabulary of predicates used in knowledge organization (part-of, requires, produces, extends, contrasts-with, governed-by, …) reduces to two irreducible generators that form an adjunction.

3.1 Predicate Signatures. Define a predicate signature as a tuple Σ = (P, F, τ, inv, comp) where:

• P is a set of predicate symbols (part-of, requires, extends, …)
• F: P → Family assigns each predicate to a semantic family (mereological, taxonomic, functional, deontic, oppositional, causal)
• τ: P → 2^{Properties} assigns algebraic properties (transitive, reflexive, symmetric, antisymmetric, …)
• inv: P ⇀ P is a partial involution giving inverses (part-of ↔ has-part, requires ↔ required-by)
• comp: P × P ⇀ P is a partial composition (component-of ∘ part-of = part-of, acts-on ∘ requires = produces)

Define coherence conditions: inv must be involutory (inv(inv(p)) = p) and respect properties (if p is transitive, inv(p) is transitive); comp must be associative where defined and respect families (composing two mereological predicates stays mereological).

A coherent predicate signature generates a predicate category C_Σ whose objects are predicate symbols and whose morphisms are generated by inv and comp. This category formalizes what BFO and DOLCE do informally: it gives the predicate vocabulary an algebraic structure.

3.2 The Two-Generator Thesis. Observation: every predicate in common knowledge-organization use decomposes as a mode of two operations:

• extends (ε): generative, expansive, forward. “X adds to Y.” Left-adjoint character.
• restricts (ρ): constraining, narrowing, backward. “X limits Y.” Right-adjoint character.

These are not inverses. Extending without restricting (adding capability without removing freedom) and restricting without extending (constraining without adding content) are both possible. They are orthogonal.

Predicate	Decomposition
part-of	ε (compositional): X extends Y’s structure
requires	ρ (dependency): Y restricts X’s existence
produces	ε (causal): X extends into Y
governed-by	ρ (deontic): Y restricts X’s behavior
extends (taxon)	ε (taxonomic): X extends Y’s abstraction
contrasts-with	mutual ρ: X and Y restrict each other
enables	weakened ε: X partially extends Y’s space
determines	strengthened ρ: X fully restricts Y

The symmetric predicate contrasts-with is algebraically distinguished: it decomposes as mutual restriction, making it self-inverse. This self-inverseness is the property that will preserve Z₂ symmetry through closure (Section 5).

3.3 The Extends-Restricts Adjunction. In the predicate category C_Σ generated by ε and ρ, there is a natural bijection:

Hom(ε(p), q) ≅ Hom(p, ρ(q))

“X falls within the extension of p relative to q” iff “p satisfies the restriction to X relative to q.” This is a Galois connection — and specifically, when lifted to bilattice-valued fibers (Section 4), it becomes a Galois biconnection in the sense of Jakl, Jung, & Rivieccio (2021): the bilattice analogue of a Galois connection, with compatible connections for both the truth and knowledge orderings.

The adjunction gives the predicate vocabulary orientation: the asymmetry between left adjoint (ε, generative) and right adjoint (ρ, constraining) is a structural asymmetry, not a notational convention. This connects to Lawvere’s (1996) reading of adjoint pairs as “unity and identity of opposites” — the extends/restricts pair is the unity of opposites for predication itself.

Also relevant: Busaniche & Cignoli (2021) study adjoint operations specifically within twist-product lattices, directly connecting FOUR’s twist-product structure to adjunction theory.

Proposition (Two-Generator Sufficiency): Every morphism in C_Σ factors as a composition of instances of ε and ρ. The derived predicates (part-of, requires, etc.) are specific factorizations.

Key citations

• Lawvere (1969, 1996): adjointness as foundational structure. The 1969 paper shows quantifiers are adjoint to substitution. The 1996 paper makes the dialectical reading explicit: left adjoints are generative/free, right adjoints are constraining/forgetful.
• Ganter & Wille (1999, Formal Concept Analysis): extent and intent as a Galois connection — the prototype for extends/restricts.
• Jakl, Jung, & Rivieccio (2021, “Galois Connections for Bilattices”): Galois biconnections. Their bidirectional/ unidirectional distinction maps onto our pre-closure/post-closure distinction. Closest existing framework to extends-restricts on bilattice fibers.
• Busaniche & Cignoli (2021, “Adjoint Operations in Twist-Products of Lattices”): adjoint pairs within twist-product lattices, connecting FOUR’s algebraic structure directly to adjunction theory.
• Arp, Smith, & Spear (2015, Building Ontology with BFO): treats predicate families as distinguished categories but doesn’t axiomatize their algebra. We provide the axiomatization.
• Borgo et al. (2022, DOLCE): formal ontology with mereological relations. Same gap: rich vocabulary, no algebraic theory of it.
• Shinavier, Wisnesky, & Meyers (2019/2022, “Algebraic Property Graphs”): they give graphs algebraic structure; we give predicates algebraic structure. Complementary.
• Baader et al. (2003, The Description Logic Handbook): DL role hierarchies are a weaker form of what predicate signatures capture.

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Section 4. Belnapian Predicate Fibers (~1800 words)

The central construction. Every piece of the PFBKS is assembled and the main structural theorems are stated.

4.1 Labeled Directed Graphs. Define G = (V, E, L, src, tgt, lab) where V is a set of vertices (knowledge units — documents, concepts, entities), E is a set of directed edges (predicate assertions), L is a label set, and src, tgt: E → V and lab: E → L are source, target, and labeling functions. Standard definition following Hogan et al. (2021).

4.2 The PFBKS Definition. A Predicate-Fibered Belnapian Knowledge Structure is a tuple K = (G, Σ, ν, ≤_m) where:

• G = (V, E, L, src, tgt, lab) is a labeled directed graph with L = P (every edge label is a predicate from the signature)
• Σ = (P, F, τ, inv, comp) is a coherent predicate signature (§3.1)
• ν: E → FOUR is a Belnapian valuation: every edge gets a value from {⊥, t, f, ⊤} representing the epistemic status of that predicate assertion
• ≤_m is a mereological preorder on V (§2.4)

with compatibility: the mereological preorder ≤_m agrees with mereological predicates in Σ (if part-of(v, w) has ν = t, then v ≤_m w), and predicate composition in Σ respects edge composition in G.

4.3 Fiber Algebras. For each predicate p ∈ P, the fiber is the set of all edges labeled p:

Fib(p) = { e ∈ E : lab(e) = p }

The restriction of ν to Fib(p) gives a function Fib(p) → FOUR. The pointwise bilattice operations on this function space define the fiber algebra A_p: for two edges e₁, e₂ in Fib(p), their truth-meet ν(e₁) ∧_t ν(e₂), truth-join ν(e₁) ∨_t ν(e₂), knowledge-meet ν(e₁) ∧_k ν(e₂), and knowledge-join ν(e₁) ∨_k ν(e₂) are all well-defined because FOUR is a bilattice.

Proposition (Fiber Distributivity): Each fiber algebra A_p is a distributive bilattice. (FOUR is distributive; pointwise products of distributive bilattices are distributive.)

4.4 Fiber Composition. When predicates compose via comp in Σ (e.g., component-of ∘ part-of = part-of), their fibers compose too. If edge e₁ witnesses p and edge e₂ witnesses q with tgt(e₁) = src(e₂), and p ∘ q = r in Σ, then the composed edge witnesses r with value:

ν(e₁ ∘ e₂) = ν(e₁) ∧_k ν(e₂)

Why knowledge-meet ∧_k, not truth-meet ∧_t? Because composition combines evidential support: ∧_k preserves whatever both assertions agree on as information. If one says t and the other says f, ∧_k gives ⊤ (both pieces of evidence present, contradicting) — which is correct: a composed chain passing through contradictory evidence should surface the contradiction, not silently resolve it. This follows Fitting’s (1991) approach to composition in bilattice logic programming.

4.5 The Fiber Functor.

Theorem (Fiber Functor): The assignment p ↦ A_p extends to a contravariant functor

F: C_Σ^op → BiLat

from the predicate category (§3.4) to the category of distributive bilattices. Morphisms in C_Σ (predicate specialization, inversion, composition) map to bilattice homomorphisms between fiber algebras.

This theorem says: the algebraic structure of the predicate vocabulary maps coherently into the algebraic structure of the fibers. Predicate relations (p specializes q, p is the inverse of q, p composes with q to give r) are reflected as bilattice homomorphisms (A_q → A_p, A_p → A_{inv(p)}, A_p ⊗ A_q → A_r). The structure is already there.

The functor is contravariant: more specific predicates (further from the generators ε and ρ in C_Σ) have fibers that receive homomorphisms from less specific predicates. Specialization enriches.

Relation to Lawvere’s hyperdoctrines. Lawvere (1969, 1970) defined a hyperdoctrine as a functor from a base category to a fiber category (Heyting algebras for intuitionistic logic, Boolean algebras for classical logic). The Fiber Functor says a PFBKS is a hyperdoctrine with BiLat as the fiber category — bilattice fibers instead of Heyting algebra fibers. This is the non-degenerate version of what the Fregean frontmatter fibration paper identified as a “degenerate hyperdoctrine” (bare-set fibers, no algebraic structure within each fiber).

Relation to Maruyama (2021). Maruyama’s fibered algebraic semantics provides a general framework: any non-classical logic with an algebraic semantics can be fibered over a base category. Our Fiber Functor instantiates his framework for FOUR-valued logic over predicate- structured knowledge graphs with the extends-restricts adjunction. The instantiation yields results (symmetry breaking, orientation) that the general framework does not reach because it works at full generality.

4.6 Mereological Inheritance. Parts inherit properties of wholes with evidence attenuated along the part-of chain. If v ≤_m w (v is part of w) and w has an edge e with lab(e) = p and ν(e) = x, then v inherits an assertion for p with value determined by ∧_k along the mereological chain.

Proposition (Inheritance Monotonicity): Inheritance is monotone in ≤_k. More information at the whole means at least as much inherited information at the part. (Follows from ∧_k being monotone in ≤_k.)

Theorem (Inheritance Coherence): The inheritance operation is a natural transformation η: F(−) ∘ whole → F(−) ∘ part, commuting with the Fiber Functor. Inheriting first and then applying predicate structure gives the same result as applying predicate structure first and then inheriting. They commute.

This interaction between mereological decomposition and bilattice-valued fibers has no precedent in the literature. BFO and DOLCE assume parts inherit properties from wholes but don’t formalize this inheritance bilattice-theoretically.

Key citations

• Hogan et al. (2021, “Knowledge Graphs”): standard definition of labeled directed graphs as the substrate for knowledge representation.
• Lawvere (1969, 1970): the hyperdoctrine as direct ancestor.
• Maruyama (2021, “Fibred Algebraic Semantics for a Variety of Non- Classical First-Order Logics and Topological Logical Translation”, JSL 86(3)): the general framework we instantiate.
• Fitting (1991): the ∧_k composition rule for bilattice-valued inference chains.
• Jacobs (1999, Categorical Logic and Type Theory): general reference for fibered category theory.

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Section 5. Closure and Symmetry Breaking (~2000 words)

THE CENTRAL SECTION. Everything before this is scaffolding for the construction. This section contains the paper’s main theorem.

5.1 Inference Rules. Define four rules that derive new FOUR-valued assertions from existing ones:

1. Fiber composition: if p ∘ q = r in C_Σ and there exist edges e₁ (for p, value x) and e₂ (for q, value y) with tgt(e₁) = src(e₂), derive an edge for r from src(e₁) to tgt(e₂) with value x ∧_k y.

2. Mereological inheritance: if v ≤_m w and w has edge e (predicate p, value x), derive an edge at v for p with value x ∧_k (the ∧_k product along the mereological chain from v to w).

3. Inverse: if inv(p) = p’ in Σ and there is an edge e for p from v to w with value x, derive an edge for p’ from w to v with value ~x (truth-negation: t ↔ f, ⊥ and ⊤ fixed).

4. Transitivity: if p is transitive (τ(p) ∋ transitive) and there are edges v→w (value x) and w→u (value y) both for p, derive edge v→u for p with value x ∧_k y.

Proposition (Soundness): All four rules preserve PFBKS structure — applying any rule to a valid PFBKS yields a valid PFBKS.

Proposition (Monotonicity): All four rules are monotone in the knowledge ordering ≤_k. No rule ever decreases the knowledge content of any edge. (This adapts Fitting’s 1991 monotonicity result for bilattice logic programming operators.)

5.2 The Fixpoint.

Theorem (Fixpoint): For any finite PFBKS K, iterative application of the four inference rules reaches a least fixpoint K* in finitely many steps. K* is the closure of K — the PFBKS with all derivable knowledge made explicit.

Proof strategy: The rules define a monotone operator T on the complete lattice FOUR^|E*| (where E* is the set of all potential edges), ordered by pointwise ≤_k. The Knaster-Tarski theorem gives the existence of a least fixpoint. Finiteness of V (hence of the potential edge set) gives finite convergence: each step pushes at least one edge strictly up in ≤_k, and ≤_k has height 2.

5.3 Elimination of ⊥. Under a scoped closed-world assumption — every edge within the scope of some inference rule either has direct evidence or is reachable by inference — the fixpoint K* has no ⊥-valued edges within scope.

The mechanism: edges start at ⊥ (no information). Each rule pushes edges upward in ≤_k: from ⊥ to t, f, or ⊤. The fixpoint is reached when no rule can push further. Within scope, every edge has been reached.

The closure acts as an order-theoretic filter on FOUR: it removes the bottom element of the knowledge ordering. The effective fiber algebra after closure operates on the three-element set {t, f, ⊤}, not on FOUR.

5.4 The Symmetry-Breaking Theorem.

Theorem (Symmetry Breaking): Let K* be the closure of a PFBKS K under the scoped closed-world assumption. The effective fiber algebra on {t, f, ⊤} = FOUR \ {⊥} has the following properties:

(i) Knowledge ordering survives as a V-shape. {t, f, ⊤} under ≤_k: ⊤ >_k t, ⊤ >_k f, and t and f are incomparable. This is FOUR’s knowledge diamond with the bottom removed — a V (or “fan”) with ⊤ at the apex.

(ii) Truth ordering collapses to a chain. {t, f, ⊤} under ≤_t is the total order f <_t ⊤ <_t t. In full FOUR, the truth ordering is a diamond with f at bottom, t at top, and ⊥ and ⊤ as incomparable middle elements. Removing ⊥ eliminates one of the two middle elements, and the remaining structure f < ⊤ < t is a chain. The partial order has become a total order. The diamond’s four-fold symmetry is broken.

(iii) Symmetry group reduces. The symmetry group of FOUR is Z₂ × Z₂ (Klein four-group), generated by negation and conflation. On {t, f, ⊤}:

• Negation (~) maps t ↔ f and fixes ⊤. This is still an involution on {t, f, ⊤}. It swaps the truth chain’s endpoints. Survives.
• Conflation (-) would need to swap ⊥ and ⊤ while fixing t and f. But ⊥ is gone. Conflation has no well-defined action on {t, f, ⊤}. Does not survive.

The symmetry group reduces from Z₂ × Z₂ to at most Z₂ (generated by negation alone). One of the two independent symmetries is destroyed.

(iv) Directed predicates acquire directed fibers. For predicates with non-self-inverse character (part-of, requires, extends, produces — all directed from source to target), the surviving Z₂ (negation) acts as an anti-automorphism of the truth chain: it reverses direction (t ↦ f, f ↦ t, ⊤ ↦ ⊤). The fiber has a preferred direction f → ⊤ → t, inherited from the predicate’s semantics: denial, then contested evidence, then affirmation.

(v) Symmetric predicates retain Z₂. For self-inverse predicates (contrasts-with, dual-of), negation acts as an automorphism: the predicate doesn’t distinguish source from target, so swapping t and f preserves the structure. The Z₂ symmetry survives fully. The fiber has no preferred direction.

5.5 Contradiction as Waypoint. The truth chain f <_t ⊤ <_t t gives contradiction (⊤) a role that FOUR’s diamond hides: it is a waypoint between denial and affirmation, not an error state. Evidence accumulates along the chain:

• part-of: “not a part” → “contested parthood” → “confirmed part”
• requires: “no dependency” → “contested dependency” → “confirmed need”
• produces: “no output” → “contested production” → “confirmed product”

The direction belongs to the predicate, not to the evidence. Each predicate’s semantics determines which end of the chain (t or f) counts as resolution.

5.6 The Closed Fiber Is Not a Bilattice.

Corollary: The effective fiber algebra on {t, f, ⊤} is not a bilattice. (Bilattices require two lattice orderings satisfying interlacing conditions. On {t, f, ⊤}, ≤_t is a 3-element chain and ≤_k is a V-shape — this does not satisfy interlacing.)

The two orderings interact asymmetrically on specific pairs:

Pair	≤_k	≤_t	Relationship
(t, ⊤)	⊤ > t	t > ⊤	Opposite directions
(f, ⊤)	⊤ > f	⊤ > f	Same direction
(t, f)	incomp.	t > f	Only truth speaks

The opposite-direction pair (t, ⊤) is the source of orientation: gaining knowledge (moving toward ⊤ in ≤_k) can move you away from truth (away from t in ≤_t). The predicate’s resolution direction navigates this tension.

5.7 Consistency and Gap Detection.

Proposition (Consistency Detection): Local inconsistency at a vertex v = existence of edges at v whose truth-meet yields ⊤. Formally: v is inconsistent for predicate p if there exist edges e₁, e₂ ∈ Fib(p) with src(e₁) = src(e₂) = v and ν(e₁) ∧_t ν(e₂) = ⊤. Decidable for finite graphs (enumerate edges at each vertex).

Proposition (Gap Detection): Under the scoped closed-world assumption, missing information at a vertex v = expected edges (predicted by predicate structure in Σ) whose closure value is ⊥. After closure, within-scope edges with value ⊥ indicate information the structure expects but does not have. Computable for finite graphs.

Proposition (Mereological Coherence): Three equivalent conditions for when inheritance does not contradict direct assertion at a vertex: (a) no vertex has both a direct edge and an inherited edge for the same predicate with ν(direct) ∧_t ν(inherited) = ⊤; (b) the inheritance natural transformation η (§4.6) preserves truth- consistency; (c) the mereological preorder is compatible with the fiber functor in the sense that F ∘ ≤_m is a subfunctor of F.

Key citations

• Fitting (1991, 2002): fixpoint theorem for bilattice logic programming. We adapt his monotone-operator/Knaster-Tarski approach.
• Arieli & Avron (1996, 1998): bilattice-based reasoning calculi. Our inference rules generalize theirs with mereological and compositional rules.
• Dunn (2000): negation as modal operator. The symmetry-breaking theorem gives this a geometric sharpening: after closure, negation doesn’t just flip truth — it reverses a direction on a chain.
• Shramko & Wansing (2011): what FOUR’s values mean as epistemic states. The theorem says what happens to those meanings under closure.

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Section 6. The Oriented Product Geometry (~1500 words)

What happens when multiple predicates interact in the closed algebra.

6.1 Product Fibers. With n predicates valued in FOUR, the combined epistemic state at a vertex is a point in FOUR^n ≅ 2^{2n} (a 2n- dimensional Boolean hypercube, via the twist product). Coherence conditions from Σ (composition rules, transitivity, inversion) cut out a sub-bilattice of FOUR^n — not all combinations of truth values across predicates are consistent.

After closure: the effective product operates on {f, ⊤, t}^n, constrained by coherence. This is a subset of a 3^n-point lattice.

6.2 Two Generators, Two Dimensions. The two-generator thesis (§3.2) means the product fiber is fundamentally 2-dimensional. Derived predicates are not independent axes — they are specific positions within the extends × restricts product space:

• part-of: high ε, moderate ρ (extends Y’s composition, constrained by containment)
• requires: low ε, high ρ (pure constraint)
• produces: high ε, low ρ (pure generation)
• contrasts-with: symmetric ρ (mutual restriction, on the diagonal)

The product {f, ⊤, t}^n projects down to a 2D space indexed by an extends-coordinate and a restricts-coordinate.

6.3 Orientation.

Proposition (Orientation): The product fiber geometry is oriented if and only if the predicate vocabulary contains at least one non-self- inverse generator.

The extends-restricts adjunction provides orientation: ε defines the positive direction along one axis, ρ along the other. The unit and counit of the adjunction (the canonical maps η: id → ρ∘ε and ε: ε∘ρ → id) are the two canonical directions. Their asymmetry (the adjunction is not an equivalence, so η ≠ ε⁻¹) is the asymmetry of the geometry.

Orientation is load-bearing: an oriented discrete structure admits well-defined summation over configurations. The partition function (§7.6) requires orientation to exist.

6.4 Curvature from Coupling. In an uncoupled product (no coherence conditions between ε-derived and ρ-derived predicates), the two dimensions are independent — a rectangular lattice, flat geometry.

Coherence conditions couple the axes: extending can trigger restriction (e.g., adding a component to a system introduces new dependencies); restricting can trigger extension (e.g., imposing a constraint creates new structure to manage it). Each advance along ε shifts position in the ρ-fiber, and vice versa.

A trajectory advancing monotonically in multiple non-parallel coupled directions curves. The curvature is determined by the coupling strength: the number and nature of coherence conditions between ε and ρ.

6.5 Directed and Symmetric Dimensions. Directed predicates (from ε, ρ, or their non-self-inverse compositions) contribute dimensions with a preferred truth-chain direction. The symmetric predicate (contrasts-with = mutual ρ) contributes a Z₂-invariant dimension with no preferred direction.

The interaction between directed and symmetric dimensions in the product is qualitatively different from either alone — this is the structural basis for the quasicrystalline claim in Section 7.

Key citations

• Craig, Davey, & Haviar (2020, “Expanding Belnap”): dualities for bilattice families beyond FOUR. Their duality framework provides tools for the topological reading of product geometry.
• Jung & Rivieccio (2012, “Priestley Duality for Bilattices”): the dual of a bilattice is a Priestley space; our product has a dual Priestley space encoding the oriented structure.
• Priestley (1970): Priestley duality for distributive lattices. General framework for the geometric/topological reading of lattice algebra.

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Section 7. Quasicrystalline Order (~1500 words)

The most speculative section. All claims are clearly marked as conditional on explicit conditions or as structural analogies.

7.1 The Product Lattice. The full product {f, ⊤, t}^n is a lattice with 3^n points. It is periodic: translating by any basis vector gives an isomorphic copy. This is a “crystal” in the lattice-theoretic sense.

Coherence conditions define a sublattice — not all 3^n configurations are consistent with the predicate signature’s composition and transitivity rules. This sublattice is a “cut” through the product.

7.2 The Cut-and-Project Analogy. Quasicrystals (Shechtman et al. 1984) have long-range order but no translational periodicity — a third kind of order between crystalline (periodic) and amorphous (random). The mathematical framework (de Bruijn 1981, Meyer 1972, systematized in Baake & Grimm 2013) shows that projecting a higher-dimensional periodic lattice along an irrational slope produces a lower-dimensional quasicrystalline structure.

The analogy: {f, ⊤, t}^n is the higher-dimensional periodic lattice. The coherence conditions define the projection. If the projection direction is incommensurate with the lattice periods, the result has quasicrystalline character.

7.3 Incommensurability.

Definition (Incommensurability Condition): A PFBKS K satisfies the incommensurability condition if, for at least two predicate families derived from ε and ρ respectively, the depth distributions of their subgraphs (the subgraph of G restricted to edges from each family) have irrational ratio in the large-graph limit.

Concretely: the mereological tree (part-of edges) has one depth distribution and the dependency DAG (requires edges) has a different one. These depths are determined by different structural constraints and are generically incommensurate.

7.4 The Quasicrystalline Theorem.

Theorem (Quasicrystalline Order, conditional): Let K* be the closure of a finite PFBKS K satisfying the incommensurability condition. The closed product fiber, viewed as a subset of {f, ⊤, t}^n, has aperiodic long-range order: its autocorrelation function has dense Bragg-like peaks but no translational period.

Proof strategy: adapt the cut-and-project framework of Baake & Grimm (2013) from Euclidean space to the discrete bilattice product. The incommensurability condition provides the “irrational slope.”

7.5 The Predicate Thresholds.

Predicates	Type	Product geometry
1 (ε only)	directed	1D chain. Trivial.
2 (ε + ρ)	both directed	2D lattice. Crystalline.
3 (+ self-inv)	dir + sym	2D + Z₂. Quasicrystalline threshold.
5 (full vocab)	4 dir + 1 sym	Rich product. Non-trivial partition fn.

The 3-predicate threshold — adding the first symmetric predicate — is where the geometry changes qualitatively. Two directed predicates give a crystal. The symmetric predicate’s Z₂ interacts with the directed chains to break periodicity.

7.6 Toward the Partition Function. (Sketch, not a theorem.)

Z(K*) = Σ_{σ ∈ consistent configs} w(σ)

where w(σ) is a weight determined by knowledge content (e.g., product of ∧_k values across edges). Z counts/weights consistent configurations of the quasicrystalline structure.

Structural parallel to statistical mechanics: G is the spatial lattice, {f, ⊤, t} is the spin space, coherence conditions are the interaction Hamiltonian, Z is the partition function. Full development is future work.

Key citations

• Shechtman et al. (1984): experimental discovery of quasicrystals.
• de Bruijn (1981): algebraic theory of Penrose tilings via cut-and-project. The mathematical technique we adapt.
• Meyer (1972): Meyer sets — the mathematical foundation for aperiodic point sets with long-range order.
• Baake & Grimm (2013, Aperiodic Order): the modern mathematical treatise. Reference for cut-and-project and Bragg-like diffraction.

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Section 8. Examples (~800 words)

Three abstract, domain-independent examples demonstrating specific structural phenomena.

8.1 The 2-Predicate Case. A small graph (8 vertices, 12 edges) with only extends (ε) and restricts (ρ). Walk through:

• Initial PFBKS: assign FOUR values to each edge (some ⊥, some t/f/⊤)
• Closure: show specific ⊥ edges being pushed to t, f, or ⊤ by the four inference rules
• Symmetry breaking: exhibit the directed chain f < ⊤ < t on each fiber
• Product: {f, ⊤, t}² — a 3×3 grid. Crystalline (periodic).
• Consistency check: identify any ⊤-valued edges (contradictions)

8.2 The 3-Predicate Threshold. Same graph, add contrasts-with edges.

• Show how ⊤ values arise naturally: two vertices contrasting where one has evidence for t and the other for f → ⊤ (both present)
• Z₂ symmetry on the symmetric fiber: no preferred direction
• Product {f, ⊤, t}³ with coherence constraints: exhibit that the constrained configurations form a pattern that does not repeat periodically
• Contrast with the crystalline 2-predicate case

8.3 Full Vocabulary. Larger graph (20+ vertices), all five base predicates active. Sketch:

• Derived predicates from the two generators (governed-by = specific ρ, enables = weakened ε)
• Conflict detection via ⊤ values across interacting predicates
• Gap detection via the scoped closed-world assumption
• Gesture at quasicrystalline character of the full product

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Section 9. Related Work and Discussion (~1200 words)

Position against six lines of prior work.

Bilattice reasoning (Fitting 1991, 2002; Arieli & Avron 1996, 1998; Bou & Rivieccio 2013; Rivieccio, Jung, & Jansana 2017). They develop bilattice algebra, logic, and four-valued modal semantics. We fiber bilattice algebra over typed predicate vocabularies and study the geometry of closure. The symmetry-breaking result has no precedent in this literature.

Paraconsistent description logics (Bienvenu, Bourgaux, & Kozhemiachenko, KR 2024). Four-valued querying in DL frameworks. We share the FOUR-valued motivation but work with fibered predicate structures (not DL TBoxes), and the geometric/quasicrystalline analysis is entirely novel.

Categorical knowledge representation (Spivak 2014; Shinavier, Wisnesky, & Meyers 2019/2022). Algebraic property graphs, ologs, functorial data migration. We share the categorical approach; our contribution is bilattice-valued fibers with the extends-restricts adjunction and the symmetry-breaking result. They don’t use bilattice values or study closure phenomena.

Fibered algebraic semantics (Lawvere 1969, 1970; Maruyama 2021; Jacobs 1999). Maruyama provides the general framework for fibered non-classical logic semantics. We instantiate for FOUR over predicate- structured knowledge graphs. The instantiation yields symmetry breaking, orientation, and quasicrystalline order — results the general framework does not reach. This is the closest related work technically.

Mereological ontology (Cotnoir & Varzi 2021; Simons 1987; BFO; DOLCE). They axiomatize part-whole. We take mereological preorders as given and study their bilattice-theoretic interaction with predicate fibers. The Inheritance Coherence theorem is novel in this setting.

Belnap-Dunn logic (Omori & Wansing eds. 2019; Shramko & Wansing 2011; Dunn 2000). Comprehensive four-valued/many-valued logic. Our contribution is not a new logic but a geometric reading of FOUR and a structural result about what closure does to it.

Future directions:

• Temporal evolution: dynamic PFBKS with time-varying valuations, where the symmetry-breaking structure provides directionality
• Higher-order bilattices: Craig, Davey, & Haviar (2020) construct bilattice families beyond FOUR. The symmetry-breaking analysis should extend.
• DL complexity: connect closed PFBKS to Bienvenu et al.’s complexity results
• Full partition function: make the statistical mechanics analogy rigorous
• Manifold interpretation: in the large-graph limit, does the discrete product fiber converge to a continuous manifold? What are its topological invariants?
• Implementation: algorithms for closure, symmetry-breaking detection, and quasicrystalline structure identification

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Section 10. Conclusion (~500 words)

FOUR is a 2D modal geometry with Z₂ × Z₂ symmetry. Predicates reduce to two adjoint generators (extends and restricts). Fibering FOUR over the predicate vocabulary gives a fiber functor — a bilattice-valued hyperdoctrine. Closing under inference eliminates ⊥ and breaks the symmetry: the diamond collapses to a directed triangle, and the direction depends on whether the predicate is directed or symmetric. Under generic incommensurability conditions, the closed product has quasicrystalline character.

This is a new mathematical object. Not a bilattice, not a knowledge graph, not a mereological structure, but something that emerges from their interaction under closure: the oriented geometry of typed predication under uncertainty.

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Formal Results Summary

#	Type	Name	Section	Status
1	Definition	Predicate signature Σ	3.1	Novel
2	Definition	Extends-restricts adjunction	3.3	Novel*
3	Definition	PFBKS tuple K	4.2	Novel
4	Definition	Fiber algebra A_p	4.3	Novel
5	Definition	Incommensurability condition	7.3	Novel
6	Proposition	Two-generator sufficiency	3.4	Novel
7	Proposition	Fiber distributivity	4.3	Standard
8	Proposition	Inheritance monotonicity	4.6	Novel
9	Proposition	Soundness of inference	5.1	Adaptation
10	Proposition	Monotonicity of inference	5.1	Adaptation
11	Proposition	Orientation criterion	6.3	Novel
12	Proposition	Consistency detection	5.7	Novel
13	Proposition	Gap detection	5.7	Novel
14	Proposition	Mereological coherence	5.7	Novel
15	Theorem	Fiber Functor	4.5	Novel
16	Theorem	Inheritance Coherence	4.6	Novel
17	Theorem	Fixpoint	5.2	Adaptation
18	Theorem	Symmetry Breaking	5.4	Novel
19	Theorem	Quasicrystalline Order	7.4	Novel†

*Confirmed novel by literature search: extends/restricts as adjoint pair in KR has no precedent. Ingredients exist independently (Lawvere’s quantifier adjoints, FCA Galois connections, Spivak’s data migration adjoints, Jakl-Jung-Rivieccio’s Galois biconnections) but the synthesis is new.

†Conditional on incommensurability; proof adapts cut-and-project to discrete bilattice products.

Novelty Assessment

Component	Novel?	Builds on
Two-generator thesis (ε/ρ adjunction)	Yes*	Lawvere, Ganter-Wille, Jakl+
Predicate signatures as algebra	Yes	BFO/DOLCE, DL roles
Bilattice-valued fibers over predicates	Yes	Lawvere, Maruyama
Fiber Functor theorem	Yes	Maruyama’s techniques
Symmetry-Breaking theorem	Yes	New result
Contradiction-as-waypoint reading	Yes	Dunn, Shramko-Wansing
Oriented product geometry	Yes	Priestley, Craig et al
Quasicrystalline characterization	Yes	Baake-Grimm framework
Mereological inheritance in bilattice	Yes	Cotnoir-Varzi, BFO
Consistency/gap/coherence detection	Yes	Standard techniques
Bilattice theory itself	Known	Belnap, Fitting, A&A
Labeled directed graphs	Known	Hogan et al.
Mereological preorders	Known	Cotnoir-Varzi
Cut-and-project for quasicrystals	Known	de Bruijn, Meyer
Galois biconnections for bilattices	Known	Jakl, Jung, Rivieccio

~65% novel construction/results, ~35% known/adapted foundations.

Complete Bibliography

Foundational

• Arieli, O. & Avron, A. (1996). “Reasoning with Logical Bilattices.” J. Logic, Language and Information, 5(1), 25–63.
• Belnap, N. D. (1977). “A Useful Four-Valued Logic.” In Dunn & Epstein (eds.), Modern Uses of Multiple-Valued Logic, Reidel, 5–37.
• Cotnoir, A. J. & Varzi, A. C. (2021). Mereology. OUP.
• Fitting, M. (1991). “Bilattices and the Semantics of Logic Programming.” J. Logic Programming, 11(2), 91–116.
• Fitting, M. (2002). “Bilattices Are Nice Things.” In Bolander et al. (eds.), Self-Reference, CSLI Publications.
• Ginsberg, M. L. (1988). “Multivalued Logics: A Uniform Approach to Inference in AI.” Computational Intelligence, 4(3), 265–316.
• Jacobs, B. (1999). Categorical Logic and Type Theory. Elsevier.
• Kalman, J. A. (1958). “Lattices with Involution.” Trans. AMS, 87(2), 485–491.
• Lawvere, F. W. (1969). “Adjointness in Foundations.” Dialectica, 23(3–4), 281–296.
• Lawvere, F. W. (1970). “Equality in Hyperdoctrines and Comprehension Schema as an Adjoint Functor.” Proc. AMS Symp. Pure Math., 17, 1–14.
• Lawvere, F. W. (1996). “Unity and Identity of Opposites in Calculus and Physics.” Applied Categorical Structures, 4, 167–174.
• Priestley, H. A. (1970). “Representation of Distributive Lattices by Means of Ordered Stone Spaces.” Bull. London Math. Soc., 2, 186–90.

Contemporary (closest related work)

• Bienvenu, M., Bourgaux, C., & Kozhemiachenko, D. (2024). “Queries With Exact Truth Values in Paraconsistent Description Logics.” Proc. KR 2024.
• Bou, F. & Rivieccio, U. (2013). “Bilattices with Implications.” Studia Logica, 101(4), 651–675.
• Busaniche, M. & Cignoli, R. (2021). “Adjoint Operations in Twist- Products of Lattices.” Symmetry, 13(2), Article 253.
• Craig, A. P. K., Davey, B. A., & Haviar, M. (2020). “Expanding Belnap: Dualities for a New Class of Default Bilattices.” Algebra Universalis, 81, Article 50.
• Jakl, T., Jung, A., & Rivieccio, U. (2021). “Galois Connections for Bilattices.” Algebra Universalis, 82, Article 37.
• Jung, A. & Rivieccio, U. (2012). “Priestley Duality for Bilattices.” Studia Logica, 100(1–2), 223–252.
• Maruyama, Y. (2021). “Fibred Algebraic Semantics for a Variety of Non-Classical First-Order Logics and Topological Logical Translation.” J. Symbolic Logic, 86(3), 1189–1213.
• Rivieccio, U., Jung, A., & Jansana, R. (2017). “Four-Valued Modal Logic: Kripke Semantics and Duality.” J. Logic and Computation, 27(1), 155–199.
• Shinavier, J., Wisnesky, R., & Meyers, J. G. (2019/2022). “Algebraic Property Graphs.” arXiv:1909.04881.

Philosophical and interpretive

• Dunn, J. M. (2000). “Partiality and Its Dual.” Studia Logica, 66(1), 5–40.
• Omori, H. & Wansing, H. (eds.) (2019). New Essays on Belnap-Dunn Logic. Synthese Library, Springer.
• Shramko, Y. & Wansing, H. (2011). Truth and Falsehood: An Inquiry into Generalized Logical Values. Springer.

Ontology and KR

• Arp, R., Smith, B., & Spear, A. D. (2015). Building Ontology with Basic Formal Ontology. MIT Press.
• Baader, F. et al. (eds.) (2003). The Description Logic Handbook. Cambridge University Press.
• Borgo, S. et al. (2022). “DOLCE: A Descriptive Ontology for Linguistic and Cognitive Engineering.” Applied Ontology, 17(1).
• Ganter, B. & Wille, R. (1999). Formal Concept Analysis: Mathematical Foundations. Springer.
• Hogan, A. et al. (2021). “Knowledge Graphs.” ACM Computing Surveys, 54(4), 1–37.
• Spivak, D. I. (2014). Category Theory for the Sciences. MIT Press.

Quasicrystal mathematics

• Baake, M. & Grimm, U. (2013). Aperiodic Order, Vol. 1. CUP.
• de Bruijn, N. G. (1981). “Algebraic Theory of Penrose’s Non-Periodic Tilings.” Proc. KNAW, A84(1–2), 39–66.
• Shechtman, D. et al. (1984). “Metallic Phase with Long-Range Orientational Order and No Translational Symmetry.” Phys. Rev. Lett., 53(20), 1951–1953.

Estimated Scope

~12,000–13,000 words. 5 definitions, 9 propositions, 5 theorems.

Target venues (in order of fit):

1. Journal of Logic and Computation — algebraic logic + computation
2. Studia Logica — philosophical logic with mathematical substance
3. Journal of Applied Logic — applied formal logic
4. KR proceedings — if shortened to 8-page version focusing on the Symmetry-Breaking theorem
5. Algebra Universalis — if algebraic content (fiber functor, symmetry breaking) is foregrounded over KR motivation