Bilattice-Valued Predicate Fibrations and the Geometry of Closure

Proposed subtitle: Symmetry Breaking, Orientation, and Quasicrystalline Order in Belnapian Knowledge Structures

Thesis

When you fiber Belnap’s FOUR over a typed predicate vocabulary and close under inference, the bilattice’s Z₂ × Z₂ symmetry breaks in a predicate-dependent way. Directed predicates acquire directed truth chains; symmetric predicates retain their reflection symmetry. The resulting oriented geometry — not the bilattice itself — is the paper’s real mathematical object. Under generic conditions on graph topology, this geometry has quasicrystalline character: long-range algebraic order without translational periodicity.

Arc

The paper is NOT “here is a construction, here are its properties.” It is:

1. FOUR is a geometry (2D modal space, not a truth-value set)
2. Predicates reduce to two adjoint generators (extends/restricts)
3. Fibering FOUR over predicates yields a fiber functor (the construction)
4. Closing under inference breaks FOUR’s symmetry (the discovery)
5. The broken symmetry produces oriented, quasicrystalline geometry (the punchline)

Everything before step 4 is scaffolding. Step 4 is the theorem. Step 5 is what the theorem means.

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Section 1. Introduction (~1000 words)

Content

Open from three independent formalisms that knowledge organization requires but that have never been unified:

• Typed predication: Frege (1879) → Lawvere (1969). Predicates are unsaturated functions. A knowledge structure needs typed relations, not flat labels.
• Incomplete and contradictory evidence: Belnap (1977) → Fitting (1991, 2002). Real knowledge admits gaps (⊥) and conflicts (⊤). Two-valued logic is too coarse.
• Part-whole decomposition: Leśniewski (1916) → Cotnoir & Varzi (2021). Knowledge units compose mereologically. Parts inherit properties from wholes.

Each pair has been studied (bilattice + DL, mereology + ontology, predicates + graphs). The triple has not.

State the construction: Predicate-Fibered Belnapian Knowledge Structure (PFBKS) — a knowledge graph whose edges carry typed predicates valued in Belnap’s FOUR, with vertices ordered by a mereological preorder.

State the discovery: closing a PFBKS under inference eliminates ⊥ (the “no information” state), collapsing FOUR’s diamond to a directed triangle {f, ⊤, t}. This collapse breaks the bilattice’s Z₂ × Z₂ symmetry. The specific way it breaks depends on the predicate: directed predicates get directed truth chains (f → ⊤ → t), symmetric predicates retain Z₂. The result is an oriented geometry over the knowledge graph whose properties encode semantic content invisible to graph theory alone.

State contributions:

1. The PFBKS construction with Fiber Functor theorem
2. The two-generator thesis: all predicates decompose into extends (generative) and restricts (constraining), forming an adjunction
3. The Symmetry-Breaking theorem: closure on {t, f, ⊤} produces predicate-dependent directedness
4. Characterization of quasicrystalline order in the closed product geometry

Why these citations here

• Frege (1879): the origin of typed predication as function application. Cited for the intellectual lineage, not for technique.
• Lawvere (1969): hyperdoctrines formalize Frege’s predicates categorically. The fiber functor theorem generalizes his construction.
• Belnap (1977): the epistemic motivation for FOUR. Knowledge is what a computer has been told — it can be told nothing, truth, falsity, or both.
• Fitting (1991, 2002): bilattice theory and its logic programming applications. The fixpoint technique we adapt.
• Cotnoir & Varzi (2021): the mereological framework we use. Deliberately weaker than classical extensional mereology.

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

Content

This section reframes known material with a geometric thesis: FOUR is not “a four-valued logic” but a 2D modal space with a symmetry group.

2.1 Bilattices and FOUR. Define bilattice: a set L with two bounded lattice orderings (L, ≤_t) and (L, ≤_k), plus negation ~ and conflation -. Define FOUR = {⊥, t, f, ⊤}. State Fitting’s result: FOUR is the free bounded distributive bilattice on one generator.

2.2 The Twist Product. FOUR ≅ 2 ⊗ 2 — the twist product of the two-element Boolean algebra with itself. Each value is a pair (a, b) ∈ {0,1}²:

Value	Pair	Truth	Knowledge
⊥	(0,0)	none	none
t	(1,0)	yes	partial
f	(0,1)	no	partial
⊤	(1,1)	both	full

The two components are two modalities: a truth dimension and a knowledge dimension. FOUR is the modal algebra of a 2-point Kripke frame.

The twist product goes back to Kalman (1958) for De Morgan algebras. Fitting (1991) extended it to bilattices. This is not new mathematics. What matters is the reading: the two orderings are not “two ways to compare” — they are two geometric axes.

2.3 The Symmetry Group. Negation ~ flips truth: (a,b) ↦ (1-a, b). Conflation - flips knowledge: (a,b) ↦ (a, 1-b). Together they generate Z₂ × Z₂, the Klein four-group — the symmetry group of the square.

The square has four symmetries: identity, horizontal reflection (negation), vertical reflection (conflation), and 180° rotation (negation ∘ conflation). These are the four elements of Z₂ × Z₂.

This symmetry is what closure will break.

2.4 Mereological Preorders. Reflexive, transitive, with weak supplementation. Following Cotnoir & Varzi (2021, Ch. 3), deliberately weaker than classical extensional mereology — appropriate for knowledge structures where distinct units may have identical parts.

Why these citations here

• Kalman (1958): the original twist product for De Morgan algebras. Priority for the construction.
• Fitting (1991): extended twist to bilattices; proved FOUR is the free bounded distributive bilattice. Both the algebra and the freeness result.
• Ginsberg (1988): introduced bilattices to AI. Cited for the multivalued-logic motivation.
• Dunn (2000), “Partiality and its Dual”: negation as a modal operator. Supports the reading of FOUR as modal rather than merely many-valued.
• Shramko & Wansing (2011): generalized truth values, the philosophical framework for treating FOUR geometrically.
• Arieli & Avron (1996): reasoning with logical bilattices, establishing that bilattice structure supports genuine inference.

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

Content

First novel contribution. The claim: the surface vocabulary of predicates (part-of, requires, produces, extends, contrasts-with, …) has two irreducible generators that form an adjunction.

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

• P = set of predicate symbols
• F: P → Family (mereological, taxonomic, functional, deontic, oppositional, causal)
• τ: P → algebraic properties (transitive, reflexive, symmetric, …)
• inv: partial involution on P (part-of ↔ has-part)
• comp: partial composition on P (component-of ∘ part-of = part-of)

Define coherence conditions (inv respects τ, comp respects families, etc.). This formalizes what ontology engineering does informally.

3.2 The Two-Generator Thesis. Every predicate in common knowledge- organization use decomposes as a mode of two primitive 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. “A extends B” does not entail “B restricts A.” Extending without restricting (adding capability without removing freedom) and restricting without extending (constraining without adding content) are both possible. The two operations are orthogonal.

Decomposition of common predicates:

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

The symmetric predicate contrasts-with is special: it decomposes as mutual restriction, making it self-inverse. This algebraic property (self-inverseness) is what will preserve Z₂ symmetry through closure.

3.3 The Extends-Restricts Adjunction. In the predicate category C_Σ generated by ε and ρ:

Hom(ε(p), q) ≅ Hom(p, ρ(q))

“X falls within the extension of p relative to q” if and only if “p satisfies the restriction to X relative to q.”

This is a Galois connection in the sense of formal concept analysis (Ganter & Wille 1999): ε plays the role of extent (what objects a concept covers), ρ plays the role of intent (what attributes define a concept). The adjunction is not a convenience — it is what gives the predicate vocabulary orientation. Left ≠ right. Generation ≠ constraint. This asymmetry will become the orientation of the product manifold.

Connection to Lawvere’s program: the 1969 “Adjointness in Foundations” paper argues that fundamental logical operations (quantifiers, substitution) are adjoint functors. Lawvere (1996, “Unity and Identity of Opposites in Calculus and Physics”) makes the dialectical reading explicit: adjoint pairs embody the “unity of opposites,” with left adjoints as the generative/free direction and right adjoints as the constraining/forgetful direction. The extends/restricts pair is this unity of opposites for predication.

Connection to bilattice-specific Galois theory: Jakl, Jung, & Rivieccio (2021, “Galois Connections for Bilattices”) introduce Galois biconnections — the bilattice analogue of Galois connections, distinguishing bidirectional biconnections (compatible pair, one per ordering) from unidirectional ones (single connection with bilattice structure). The extends-restricts adjunction on bilattice-valued fibers is a Galois biconnection. The symmetry-breaking theorem (Section 5) can be read as: closure collapses a bidirectional biconnection to a unidirectional one for directed predicates.

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

3.4 The Predicate Category. C_Σ is finitely generated by ε and ρ with coherence conditions from Σ.

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.

Why these citations here

• Lawvere (1969, 1996): adjointness as foundational structure. The 1969 paper provides the framework; the 1996 “Unity and Identity of Opposites” paper makes the dialectical reading of adjoint pairs explicit. Direct intellectual ancestor of the extends/restricts adjunction.
• Ganter & Wille (1999): Formal Concept Analysis. The extent/intent Galois connection is the prototype for extends/restricts.
• Jakl, Jung, & Rivieccio (2021): Galois biconnections for bilattices. Their bidirectional/unidirectional distinction maps onto our pre-closure/post-closure distinction. Closest existing algebraic framework to the extends-restricts adjunction on bilattice fibers.
• Busaniche & Cignoli (2021): adjoint operations in twist-product lattices. Directly connects FOUR’s twist structure to adjunctions.
• Arp, Smith, & Spear (2015), BFO: the Basic Formal Ontology treats predicate families as distinguished categories but does not axiomatize their algebra. We provide the axiomatization.
• Borgo et al. (2022), DOLCE: formal ontology with mereological relations. Same gap: rich predicate vocabulary, no algebraic theory of that vocabulary.
• Shinavier, Wisnesky, & Meyers (2019/2022): algebraic property graphs. They give graphs algebraic structure; we give the predicates algebraic structure. Complementary.
• Baader et al. (2003): The Description Logic Handbook. Role hierarchies in DLs are a weaker form of what predicate signatures capture. We cite to position against.

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Section 4. Belnapian Predicate Fibers (~1800 words)

Content

The central construction. Fiber FOUR over the predicate vocabulary.

4.1 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
• Σ is a predicate signature (Section 3)
• ν: E → FOUR is a Belnapian valuation
• ≤_m is a mereological preorder on V (Section 2.4)

with compatibility conditions: lab(e) ∈ P for all edges e, and ≤_m respects compositional predicates.

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

Fib(p) = { e ∈ E : lab(e) = p }

The restriction of ν to Fib(p) gives a FOUR-valued function on the fiber. The pointwise bilattice operations on this function space define the fiber algebra A_p.

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

4.3 Fiber Composition. When predicates compose via comp in Σ (e.g., component-of ∘ part-of = part-of), the fibers compose with truth values combined by knowledge-meet ∧_k.

Why ∧_k (not ∧_t)? Because composing two assertions means combining their evidential support. Knowledge-meet preserves whatever both assertions agree on — it’s the conservative composition. If one assertion says t and another says f, knowledge-meet gives ⊤ (both pieces of evidence are present, contradicting). This is correct: a composed inference chain that passes through contradictory evidence should surface the contradiction, not resolve it.

4.4 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 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. It is the formal version of “the structure is already there.”

The functor is contravariant: more specific predicates (deeper in the category, further from the generators ε and ρ) 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 the category of Heyting algebras (or Boolean algebras, or lattices). The Fiber Functor theorem says a PFBKS is a hyperdoctrine with BiLat replacing HeytAlg as the fiber category. This is the non-degenerate version of the “degenerate hyperdoctrine” identified in the frontmatter fibration paper (which had bare-set fibers, no algebraic structure).

Relation to Maruyama (2021). Maruyama’s fibered algebraic semantics for non-classical logics provides a general framework: any non-classical logic with an algebraic semantics can be fibered over a base category. Our Fiber Functor theorem instantiates Maruyama’s framework for the specific case of FOUR-valued bilattice logic fibered over a predicate category with the extends-restricts adjunction. The instantiation yields specific results (the symmetry-breaking theorem in Section 5) that the general framework does not reach.

4.5 Mereological Inheritance. Parts inherit properties of wholes. If v ≤_m w (v is part of w) and w has an edge e with ν(e) = x, then v inherits an attenuated version: ν_inherited = ν(e) ∧_k (propagation along the part-of chain, combining via knowledge-meet).

Proposition (Inheritance Monotonicity): Inheritance is monotone in the knowledge ordering ≤_k. More information at the whole means at least as much inherited information at the part.

Theorem (Inheritance Coherence): The inheritance operation is a natural transformation η: F(−) ∘ whole → F(−) ∘ part, commuting with the fiber functor. That is: it doesn’t matter whether you first inherit and then apply predicate structure, or first apply predicate structure and then inherit. They commute.

Why these citations here

• Lawvere (1969, 1970): the hyperdoctrine is the direct ancestor. The Fiber Functor generalizes it to bilattice fibers.
• Maruyama (2021): fibered algebraic semantics. Our construction instantiates his general framework with specific gains from the instantiation (symmetry breaking, orientation).
• Fitting (1991): the bilattice-valued semantics we fiber over the predicate vocabulary. Also the source of the ∧_k composition rule.
• Jacobs (1999): Categorical Logic and Type Theory. General reference for fibered category theory and its logical interpretation.
• Hogan et al. (2021): knowledge graph formalism. Standard definition of labeled directed graphs.

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

Content

THE CENTRAL SECTION. Everything before this is scaffolding. Everything after is consequences.

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

1. Fiber composition: if p ∘ q = r in C_Σ, then edges for p and q compose to give an edge for r, with value ν(p) ∧_k ν(q).
2. Mereological inheritance: if v ≤_m w, edges at w propagate to v with value attenuated by ∧_k along the chain.
3. Inverse: if inv(p) = p’, then an edge for p with value x gives an edge for p’ with value ~x (truth-negation).
4. Transitivity: if p is transitive and there are edges v→w, w→x for p, derive v→x with value ν(v→w) ∧_k ν(w→x).

Proposition (Soundness): All four rules preserve PFBKS structure (the output is still a valid PFBKS).

Proposition (Monotonicity): All four rules are monotone in the knowledge ordering ≤_k. Applying a rule never decreases the knowledge content of any edge.

Adapts Fitting’s (1991) techniques for bilattice logic programming.

5.2 The Fixpoint.

Theorem (Fixpoint): For any finite PFBKS K, iterative application of the four inference rules starting from K reaches a least fixpoint K* in finitely many steps. K* is the closure of K.

Proof strategy: the rules are monotone operators on the finite lattice FOUR^|E|, ordered by pointwise ≤_k. Knaster-Tarski gives the fixpoint. Finiteness of the graph gives finite convergence.

5.3 Elimination of ⊥.

Under the 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.

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

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

5.4 The Symmetry-Breaking Theorem.

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

(i) Knowledge ordering: {t, f, ⊤} under ≤_k is a V-shape (or “fan”): ⊤ >_k t, ⊤ >_k f, and t and f are incomparable. The knowledge ordering survives intact from FOUR, with only the bottom removed.

(ii) Truth ordering collapses to a chain: {t, f, ⊤} under ≤_t is the total order f <_t ⊤ <_t t. The diamond (partial order with two incomparable middle elements ⊥, ⊤) becomes a chain (total order with one middle element ⊤). The four-fold symmetry of the diamond is broken.

(iii) Symmetry group reduction: The symmetry group of FOUR (Z₂ × Z₂) reduces on {t, f, ⊤} to at most Z₂. Negation survives as a map t ↔ f (swapping the chain’s endpoints). Conflation does not survive as an automorphism (it would need to fix ⊤ while swapping t and f, but ⊤ is no longer symmetric — it is strictly between f and t in truth). Only one generator of Z₂ × Z₂ survives.

(iv) Directed predicates: 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. The fiber has a preferred direction: f → ⊤ → t. The direction is inherited from the predicate’s semantics.

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

5.5 Contradiction as Waypoint.

The truth chain f <_t ⊤ <_t t places contradiction (⊤) between denial (f) and affirmation (t). This is not an error state. It is a stage in evidence accumulation:

For a directed predicate, the resolution direction f → ⊤ → t reads:

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

Contradiction is the epistemic state where evidence has arrived from both directions and has not yet been resolved. The chain’s direction says: resolution, when it comes, goes toward affirmation (for the predicate as stated) or toward denial (for the negated predicate). The direction belongs to the predicate, not to the evidence.

5.6 The Asymmetric Interaction.

Corollary: The closed fiber at a directed predicate is not a bilattice (bilattices require two lattice orderings that interact via the interlacing conditions; {t, f, ⊤} with ≤_t and ≤_k does not satisfy interlacing because ≤_t is a chain and ≤_k is a V-shape on a 3-element set).

The two orderings interact asymmetrically on specific pairs:

• (t, ⊤): ≤_k says ⊤ > t. ≤_t says t > ⊤. Opposite directions.
• (f, ⊤): ≤_k says ⊤ > f. ≤_t says ⊤ > f. Same direction.
• (t, f): ≤_k says incomparable. ≤_t says t > f. Only truth speaks.

The disagreement between truth and knowledge on the (t, ⊤) pair is the source of orientation: gaining knowledge (moving toward ⊤) can take you away from truth (moving away from t). The predicate’s resolution direction navigates this tension.

Why these citations here

• Fitting (1991, 2002): the fixpoint theorem for bilattice logic programs. We adapt his techniques for our inference rules. Also the monotonicity result.
• Arieli & Avron (1996, 1998): bilattice-based reasoning. Our inference rules generalize their logical calculus with mereological and compositional rules.
• Dunn (2000): negation as modal operator. The symmetry-breaking result gives Dunn’s observation a geometric sharpening: negation doesn’t just flip truth — after closure, it reverses a direction.
• Shramko & Wansing (2011): generalized truth values. They study what FOUR’s values mean philosophically. The symmetry-breaking theorem says what happens to those meanings under closure.

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

Content

What happens when multiple predicates interact in the closed algebra.

6.1 Product Fibers. With n predicates, each valued in FOUR, the combined state at a vertex is a point in FOUR^n. Coherence conditions (from Σ) cut out a sub-bilattice. After closure: the effective product operates on {f, ⊤, t}^n, constrained by coherence.

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

• part-of: high ε (extends Y’s composition), moderate ρ (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 fiber {f, ⊤, t}^n projects down to a 2D space indexed by the extends-coordinate and the 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 are the canonical maps between the two directions. Their asymmetry (the adjunction is not an equivalence) is the asymmetry of the manifold.

Orientation is load-bearing. An oriented manifold admits integration (in the discrete case: well-defined summation over configurations). The partition function (Section 7) requires orientation to exist.

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

Coherence conditions couple the axes: extending can trigger restriction (building something new constrains the builder). Restricting can trigger extension (removing possibilities creates new structure). Each advance along ε shifts your position in the ρ-fiber, and vice versa.

A trajectory that advances 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 directed dimensions to the product: dimensions with a preferred truth-chain direction. Symmetric predicates (contrasts-with = mutual ρ) contribute Z₂-invariant dimensions: dimensions with no preferred direction.

The product of directed and symmetric dimensions is qualitatively different from either alone. This is the structural basis for the quasicrystalline claim in Section 7.

Why these citations here

• Craig, Davey, & Haviar (2020): “Expanding Belnap” — dualities for default bilattices. Extends the duality theory beyond FOUR to richer bilattice families. Their duality framework provides tools for the topological reading of our product geometry.
• Jung & Rivieccio (2012): Priestley duality for bilattices specifically. The dual of a bilattice is a Priestley space; our product geometry has a dual Priestley space whose topology encodes the oriented structure.
• Priestley (1970, 1972): Priestley duality for distributive lattices. The general framework underlying the geometric/topological reading of lattice algebra.
• Lawvere (1969): the adjunction providing orientation. Already cited in Section 3, but its geometric consequences appear here.

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Section 7. Quasicrystalline Order (~1500 words)

Content

The most speculative section. Claims are conditional on incommensu- rability conditions. The section is honest about what is proven vs. what is conjectured.

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

Coherence conditions define a sublattice — not all 3^n configurations are consistent. The sublattice is a “cut” through the full product.

7.2 The Cut-and-Project Analogy. The mathematical theory of quasicrystals (Shechtman et al. 1984, mathematicized by de Bruijn 1981, Meyer 1972, Moody 1997, and systematized in Baake & Grimm 2013) shows that projecting a higher-dimensional periodic lattice along an “irrational slope” — a direction incommensurate with the lattice periods — produces a lower-dimensional structure with long-range order but no translational periodicity.

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

7.3 Incommensurability. Each predicate’s subgraph (the subgraph of G restricted to edges labeled with predicates derived from ε or ρ) has its own topology. The mereological tree has one depth distribution, the dependency DAG another, the taxonomic hierarchy another.

The “advance rate” along each predicate dimension — how many steps it takes to traverse the subgraph from root to leaf — is generically incommensurate across families. The mereological depth and the dependency depth are determined by different structural constraints.

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 have irrational ratio in the large-graph limit.

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 peaks (Bragg- like) but no translational period.

The proof uses the cut-and-project framework of Baake & Grimm (2013) adapted from Euclidean space to the discrete bilattice product.

7.5 The Predicate Thresholds. The qualitative character of the product geometry depends on the number and type of generators:

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)	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. Adding the symmetric predicate breaks the periodicity.

7.6 Toward the Partition Function. Define (as a sketch, not a theorem):

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

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

The structural similarity to lattice models in statistical mechanics (where the graph is the spatial lattice, {f, ⊤, t} is the spin space, coherence conditions are the Hamiltonian, and Z is the partition function) is suggestive. Full development is future work.

Why these citations here

• Shechtman et al. (1984): the experimental discovery of quasicrystals. Sets the historical context. Cited to acknowledge that quasicrystalline order was first a physical discovery.
• de Bruijn (1981): the algebraic theory of Penrose tilings as cut-and-project from higher-dimensional lattices. The mathematical technique we adapt.
• Meyer (1972): Meyer sets. The foundational mathematical concept for aperiodic point sets with long-range order.
• Moody (1997): model sets and their properties. Extends Meyer’s framework to the general setting we need.
• Baake & Grimm (2013): the modern mathematical treatise on aperiodic order. The reference for the cut-and-project framework and the definition of Bragg-like diffraction.

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Section 8. Examples (~800 words)

Content

Three examples, abstract and domain-independent. Each demonstrates a specific structural phenomenon.

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

• The PFBKS definition (ν assigns FOUR values to each edge)
• Closure (show ⊥ edges getting pushed to t, f, or ⊤)
• The symmetry breaking (show the directed chain f < ⊤ < t on each fiber)
• The product: {f, ⊤, t}² — a 3×3 grid. Crystalline (periodic).

8.2 The 3-Predicate Threshold. Same graph, add contrasts-with edges (symmetric). Walk through:

• How ⊤ values arise naturally from opposition (two vertices contrasting on a predicate where one says t and the other says f → both pieces of evidence present → ⊤)
• The Z₂ symmetry on the symmetric fiber (no preferred direction)
• The product {f, ⊤, t}³ with coherence constraints: show that the constraints create a pattern that does not repeat periodically
• (Informal: just exhibit the aperiodicity, don’t prove it here)

8.3 Full Vocabulary. A larger graph (20+ vertices). Sketch (not fully worked):

• All five base predicates active
• Show how derived predicates (governed-by = specific ρ, enables = weakened ε) arise from the two generators
• Show conflict detection via ⊤ values
• Show gap detection via the scoped closed-world assumption
• Gesture at the quasicrystalline character of the full product

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

Content

Position the paper against six lines of prior work. For each: what they do, what we add, how the contribution is distinct.

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, “Queries With Exact Truth Values in Paraconsistent Description Logics”; Odintsov & Wansing 2010). They handle four-valued querying in DL frameworks. We share the FOUR-valued motivation but work with fibered predicate structures rather than DL TBoxes. The geometric and quasicrystalline analysis is entirely novel.

Categorical knowledge representation (Spivak 2014; Patterson 2017; Shinavier, Wisnesky, & Meyers 2019/2022). They bring category theory to KR: algebraic property graphs, ologs, functorial data migration. We share the categorical sensibility. Our contribution is specific: bilattice- valued fibers with the extends-restricts adjunction and the symmetry- breaking result. Their work doesn’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 semantics of non-classical logics. We instantiate it for FOUR over predicate-structured knowledge graphs. The instantiation yields specific results (symmetry breaking, orientation, quasicrystalline order) that the general framework does not provide. This is the closest related work technically.

Mereological ontology (Cotnoir & Varzi 2021; Simons 1987; Arp, Smith, & Spear 2015, BFO; Borgo et al. 2022, DOLCE). They axiomatize part-whole relations. We take their mereological preorder as given and study its interaction with bilattice-valued predicate fibers. The inheritance coherence theorem (Section 4.5) is novel in the bilattice setting.

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

Future directions:

• Temporal evolution: dynamic PFBKS where valuations change over time, with the symmetry-breaking structure providing directionality for the dynamics
• Higher-order bilattices: Craig, Davey, & Haviar (2020) construct bilattice families beyond FOUR with richer Priestley dualities. The symmetry-breaking analysis should extend.
• DL query complexity: connect closed PFBKS to the complexity results of Bienvenu et al. (2024)
• Full partition function development: make the statistical mechanics analogy rigorous
• The 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 computing closure, detecting symmetry breaking, and identifying quasicrystalline structure in real knowledge graphs

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Section 10. Conclusion (~500 words)

Restate the arc: FOUR is a 2D modal geometry with Z₂ × Z₂ symmetry. Predicates reduce to two adjoint generators. Fibering FOUR over the predicate vocabulary gives a fiber functor (a bilattice hyperdoctrine). Closing under inference eliminates ⊥ and breaks the symmetry: the diamond collapses to a directed triangle, and the direction depends on the predicate. The resulting oriented geometry, under generic incommensurability conditions, 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 geometry of typed predication under uncertainty.

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Formal Results Summary

#	Type	Name	Section	Status
1	Definition	Predicate signature	3.1	Novel
2	Definition	Extends-restricts adjunction	3.3	Novel
3	Definition	PFBKS	4.1	Novel
4	Definition	Fiber algebra	4.2	Novel
5	Definition	Incommensurability condition	7.3	Novel
6	Proposition	Fiber distributivity	4.2	Standard
7	Proposition	Two-generator sufficiency	3.4	Novel
8	Proposition	Inheritance monotonicity	4.5	Novel
9	Proposition	Soundness of inference	5.1	Adaptation
10	Proposition	Orientation criterion	6.3	Novel
11	Theorem	Fiber Functor	4.4	Novel
12	Theorem	Inheritance Coherence	4.5	Novel
13	Theorem	Fixpoint	5.2	Adaptation
14	Theorem	Symmetry Breaking	5.4	Novel
15	Theorem	Quasicrystalline Order	7.4	Novel*

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

Novelty Assessment (revised)

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
Oriented product geometry	Yes	Priestley, Craig et al
Quasicrystalline characterization	Yes	Baake-Grimm framework
Mereological inheritance in bilattice	Yes	Cotnoir-Varzi, BFO
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

*The specific framing of extends/restricts as an adjoint pair in knowledge representation is confirmed novel by literature search. The ingredients exist independently (Lawvere’s quantifier adjoints, FCA Galois connections, Spivak’s data migration adjoints, Jakl-Jung- Rivieccio’s Galois biconnections) but nobody has synthesized them into a two-generator predicate adjunction for KR.

~65% novel construction, ~35% known/adapted. Higher novelty ratio than v1 because the symmetry-breaking result and the geometric analysis are entirely new.

Key References (Complete)

Foundational (known results we build on)

• Belnap, N. D. (1977). “A Useful Four-Valued Logic.” In Dunn & Epstein (eds.), Modern Uses of Multiple-Valued Logic. Reidel.

• Fitting, M. (1991). “Bilattices and the Semantics of Logic Programming.” Journal of 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 Artificial Intelligence.” Computational Intelligence, 4(3), 265–316.

• Arieli, O. & Avron, A. (1996). “Reasoning with Logical Bilattices.” Journal of Logic, Language and Information, 5(1), 25–63.

• Kalman, J. A. (1958). “Lattices with Involution.” Transactions of the 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.” Proceedings of the AMS Symposium on Pure Mathematics, 17, 1–14.

• Cotnoir, A. J. & Varzi, A. C. (2021). Mereology. Oxford University Press.

• Jacobs, B. (1999). Categorical Logic and Type Theory. Elsevier.

• Priestley, H. A. (1970). “Representation of Distributive Lattices by Means of Ordered Stone Spaces.” Bull. London Math. Soc., 2, 186–90.

• Lawvere, F. W. (1996). “Unity and Identity of Opposites in Calculus and Physics.” Applied Categorical Structures, 4, 167–174.

Contemporary (closest related work)

• Maruyama, Y. (2021). “Fibred Algebraic Semantics for a Variety of Non-Classical First-Order Logics and Topological Logical Translation.” Journal of Symbolic Logic, 86(3), 1189–1213.
• 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.
• Jung, A. & Rivieccio, U. (2012). “Priestley Duality for Bilattices.” Studia Logica, 100(1–2), 223–252.
• Bienvenu, M., Bourgaux, C., & Kozhemiachenko, D. (2024). “Queries With Exact Truth Values in Paraconsistent Description Logics.” Proceedings of KR 2024.
• Shinavier, J., Wisnesky, R., & Meyers, J. G. (2019/2022). “Algebraic Property Graphs.” arXiv:1909.04881 (revised 2022; not yet formally published).
• Rivieccio, U., Jung, A., & Jansana, R. (2017). “Four-Valued Modal Logic: Kripke Semantics and Duality.” Journal of Logic and Computation, 27(1), 155–199.
• Bou, F. & Rivieccio, U. (2013). “Bilattices with Implications.” Studia Logica, 101(4), 651–675.
• Jakl, T., Jung, A., & Rivieccio, U. (2021). “Galois Connections for Bilattices.” Algebra Universalis, 82, Article 37.
• Busaniche, M. & Cignoli, R. (2021). “Adjoint Operations in Twist- Products of Lattices.” Symmetry, 13(2), Article 253.

Philosophical and interpretive

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

Ontology and KR

• Arp, R., Smith, B., & Spear, A. D. (2015). Building Ontology with Basic Formal Ontology. MIT Press.
• Borgo, S. et al. (2022). “DOLCE: A Descriptive Ontology for Linguistic and Cognitive Engineering.” Applied Ontology, 17(1), 45–69. [VERIFY: exact pagination]
• 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.
• Baader, F. et al. (eds.) (2003). The Description Logic Handbook. Cambridge University Press.
• Ganter, B. & Wille, R. (1999). Formal Concept Analysis: Mathematical Foundations. Springer.

Quasicrystal mathematics

• Shechtman, D. et al. (1984). “Metallic Phase with Long-Range Orientational Order and No Translational Symmetry.” Physical Review Letters, 53(20), 1951–1953.
• de Bruijn, N. G. (1981). “Algebraic Theory of Penrose’s Non-Periodic Tilings.” Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, A84(1-2), 39–66.
• Baake, M. & Grimm, U. (2013). Aperiodic Order, Volume 1: A Mathematical Invitation. Cambridge University Press.

Estimated Scope

~12,000-13,000 words. 5 definitions, 5 propositions, 5 theorems.

Target venues (in order of fit):

1. Journal of Logic and Computation — algebraic logic + computational aspects
2. Studia Logica — philosophical logic with mathematical substance
3. Journal of Applied Logic (if still active) — applied formal logic
4. KR 2025/2026 proceedings — if shortened to 8-page version focusing on the symmetry-breaking theorem
5. Algebra Universalis — if the algebraic content (fiber functor, symmetry breaking) is foregrounded over the KR motivation

What Changed from v1

The original outline (v1) was structured as: “here is a construction (PFBKS), here are its properties.” The construction was the contribution.

This outline (v2) is structured as: “here is a geometry (FOUR), here is what happens to it under closure (symmetry breaks), here is what the broken symmetry means (oriented quasicrystalline structure).” The discovery is the contribution. The construction is scaffolding.

Specific changes:

1. New Section 2: FOUR reframed as modal geometry, not just “a four-valued logic.” The twist product and symmetry group are now foundational, not background.
2. New Section 3: The two-generator thesis and extends-restricts adjunction. This was implicit in v1’s predicate signatures; now it’s the first novel contribution.
3. Section 5 is now central: The symmetry-breaking theorem. This was absent from v1 entirely. It is the paper’s main result.
4. New Section 6: Oriented product geometry. The geometric reading of the closed product fiber as an oriented 2D structure.
5. New Section 7: Quasicrystalline order. The characterization of the closed product under incommensurability. Speculative but precise.
6. Mereological inheritance demoted: From a central section (v1 §5) to a subsection of the fiber construction (v2 §4.5). Still present, but it’s a tool, not a thesis.
7. Inference calculus compressed: From its own section (v1 §7) to a subsection of the closure section (v2 §5.1-5.2). The inference rules are means to the fixpoint, not ends in themselves.
8. New citations: Kalman (1958), Dunn (2000), Ganter & Wille (1999), Shechtman et al. (1984), Baake & Grimm (2013), de Bruijn (1981). These reflect the paper’s expanded scope into geometry and quasicrystalline mathematics.