The Adjoint Structure of Predication on Bilattice-Valued Fibers

¶Abstract

We name the mathematical object formed by a system of subject-predicate-object triples carrying values in the free bounded distributive bilattice $\mathbf{2} \otimes \mathbf{2}$ a predication modal geometry, and prove it is a Chu space. Lambek (1958) proved that predicate-argument combination in categorial grammar is a residuation; Lawvere (1969) established a related adjunction pattern $\exists \dashv f^* \dashv \forall$ in categorical logic. We instantiate residuation on $\mathbf{2} \otimes \mathbf{2}$ via the twist-product residuation of Busaniche & Cignoli (2014), obtaining operations $\odot$ (extending) and $\Rightarrow$ (restricting) with a verified adjunction $a \odot p \leq_t q$ iff $p \leq_t (a \Rightarrow q)$. This adjunction lifts to a fiber functor over any bilattice-valued SPO system, producing a bilattice-valued hyperdoctrine in the sense of Lawvere (1970) and Maruyama (2021). Under Jakl’s (2025) identification of the Chu construction with the twist construction — $\mathbf{2} \otimes \mathbf{2} \cong \mathrm{Chu}(\mathbf{2}, 0)$ — we prove every bilattice-valued SPO system is a predication modal geometry, i.e. a predicate-indexed Chu space over $(\mathbf{2}, 0)$, with $\odot$ and $\Rightarrow$ the Chu tensor and linear implication. The modal calculus is decidable (Rivieccio, Jung, & Jansana 2017); the operations and adjunction restrict to the post-closure set $\{t, f, \top\}$; and the dynamics of iterated extension and restriction on the four-element algebra are characterized by exhaustive computation.

¶1. Predication as residuation

Frege (1879) established that a predicate is an unsaturated function: it takes a subject and returns a value. In a subject-predicate-object triple $\langle S, P, O \rangle$, the predicate $P$ combines $S$ and $O$ into an assertion.

Lambek (1958) formalized how this combination works. In the Lambek calculus, a product operation $\cdot$ on expressions admits exactly two residuals:

$$a \cdot b \leq c \quad\text{iff}\quad b \leq a \backslash c \quad\text{iff}\quad a \leq c / b$$

The left residual $a \backslash c$ asks what must combine with $a$ on the right to yield something below $c$. The right residual $c / b$ asks what must combine with $b$ on the left. These two residuals exhaust the modes of functor-argument interaction. This is a theorem: the product has two argument positions, so there are exactly two residuals.

Galatos, Jipsen, Kowalski, & Ono (2007) proved that the Lambek calculus is the logic of residuated lattices. The residuals $/$ and $\backslash$ are residuals in the algebraic sense, and the full theory of residuated lattices applies.

Lawvere (1969) established a related adjunction pattern in categorical logic. The fundamental adjunction on a hyperdoctrine is $\exists_f \dashv f^* \dashv \forall_f$, where $f^*$ is substitution along a morphism $f$, $\exists$ is the left adjoint (generative: extends a predicate to a larger context), and $\forall$ is the right adjoint (constraining: restricts a predicate to where it holds universally). The Lambek and Lawvere adjunctions are not literally the same theorem — Lambek’s is a pair of residuals for a non-commutative monoidal product on a poset, and Lawvere’s is a chain of adjoints to base-change along a morphism — but the two exhibit a common pattern in which the generative/combining side is left-adjoint to a substitution/restriction operation. It is this pattern that the twist-product residuation of §2 instantiates in the algebraic case.

In the commutative case ($a \cdot b = b \cdot a$), the two Lambek residuals coincide into a single implication $\Rightarrow$. The adjunction retains its two sides: the product side (combining, generating, extending) and the implication side (testing, constraining, restricting).

¶2. Instantiation on the four-valued bilattice

¶2.1 The twist product $\mathbf{2} \otimes \mathbf{2}$

Belnap (1977) defined four epistemic states for a knowledge system receiving information from multiple sources: $\bot$ (no information), $t$ (told true), $f$ (told false), $\top$ (told both). Kalman (1958) introduced the twist product construction on bounded distributive lattices; Fitting (1991) extended it to bilattices and proved $\{\bot, t, f, \top\} \cong \mathbf{2} \otimes \mathbf{2}$. Fitting (2002) proved this is the free bounded distributive bilattice on one generator.

In the twist-product representation, each value is a pair $(a, b) \in \{0,1\}^2$: $\bot = (0,0)$, $t = (1,0)$, $f = (0,1)$, $\top = (1,1)$. The first component tracks positive evidence; the second tracks negative evidence. The bilattice carries two orderings: truth ($\leq_t$, with $f$ at bottom and $t$ at top) and knowledge ($\leq_k$, with $\bot$ at bottom and $\top$ at top).

¶2.2 The residuated pair

Theorem 1 (Busaniche & Cignoli 2014; Chajda & Langer 2021). On the twist product $L \otimes L$ of any integral commutative residuated lattice $L$, the operations

$$(x,y) \odot (z,v) = (x \cdot z,\ (x \to v) \wedge (z \to y))$$

$$(x,y) \Rightarrow (z,v) = ((x \to z) \wedge (v \to y),\ x \cdot v)$$

form a residuated pair with respect to the twist-product order (corresponding to $\leq_t$):

$$a \odot p \leq_t q \quad\text{if and only if}\quad p \leq_t (a \Rightarrow q)$$

This is the Lambek residuation instantiated on $\mathbf{2} \otimes \mathbf{2}$. The product $\odot$ is extending: it combines two epistemic states by fusing their positive evidence and cross-referencing each side’s positive evidence against the other’s negative evidence. The implication $\Rightarrow$ is restricting: it tests implication in both truth directions and combines negative evidence.

We verified this adjunction by exhaustive computation on all 64 triples $(a, p, q) \in \{\bot, t, f, \top\}^3$. Zero violations.

¶2.3 Algebraic properties

The following properties hold on $\{\bot, t, f, \top\}$ (verified computationally):

Proposition 1. $\top$ is the identity element for $\odot$: $\top \odot a = a$ for all $a$.

Proposition 2. $f$ is the absorbing element for $\odot$: $f \odot a = f$ for all $a$.

Proposition 3. $t$, $f$, and $\top$ are idempotent under $\odot$. $\bot$ is not: $\bot \odot \bot = f$.

Proposition 4. $a \Rightarrow a = t$ for $a \in \{\bot, t, f\}$, and $\top \Rightarrow \top = \top$.

Chajda & Langer (2021) also establish a second adjoint pair on twist products — the Nelson implication $x \Rightarrow_N y = (x \supset y) \wedge (\mathord{\sim} y \supset \mathord{\sim} x)$, incorporating contrapositive symmetry.

Proposition 5 (Nelson identity). For all $x, y \in \{\bot, t, f, \top\}$:

$$(x \Rightarrow (x \Rightarrow y)) \wedge_k (\mathord{\sim} y \Rightarrow (\mathord{\sim} y \Rightarrow \mathord{\sim} x)) = x \Rightarrow y$$

Proposition 6. The simple contrapositive fails: $\mathord{\sim}(a \odot b) \neq \mathord{\sim} b \Rightarrow \mathord{\sim} a$ in general. Neither $\odot$ nor $\Rightarrow$ commutes with negation $\sim$ or conflation $-$.

Propositions 1–6 are verifiable by inspection of the operation tables (a finite computation on 4 elements). The Nelson identity (Proposition 5) means negation-aware iterated implication is sound even though naive contrapositive is not.

¶3. The fiber functor

¶3.1 Predicate fibers

Let $\mathcal{S}$ be a system of subject-predicate-object triples where each triple carries a value in $\{\bot, t, f, \top\}$. For each predicate $p$ occurring in $\mathcal{S}$, the fiber of $p$ is the collection of all triples using $p$:

$$\mathrm{Fib}(p) = \{ \langle s, o, v \rangle : \langle s, p, o \rangle \in \mathcal{S} \text{ with value } v \}$$

The pointwise bilattice operations on $\mathrm{Fib}(p) \to \{\bot, t, f, \top\}$ define a fiber algebra $A_p$.

Proposition 7. Each fiber algebra $A_p$ is a distributive bilattice. (Pointwise products of distributive bilattices are distributive; Fitting 2002.)

¶3.2 The functor

Let $\mathcal{C}$ be the category whose objects are predicates occurring in $\mathcal{S}$ and whose morphisms are generated by three classes of predicate relationships in the vocabulary: specialization (one predicate restricts to another), inversion (one predicate is the converse-and-negation of another), and composition (one predicate is the pointwise conjunction of two others). These are the three classes of relationships between predicates that induce bilattice homomorphisms between fiber algebras; they are exhaustive within the homomorphism constraint that defines $\mathcal{F}$.

Theorem 2 (Fiber Functor). The assignment $p \mapsto A_p$ extends to a contravariant functor $\mathcal{F}: \mathcal{C}^{\mathrm{op}} \to \mathbf{BiLat}$.

Proof. Each of the three morphism classes of $\mathcal{C}$ must map to a bilattice homomorphism in $\mathbf{BiLat}$. Specialization induces restriction of valuations, which preserves pointwise bilattice operations. Inversion applies negation $\sim$ pointwise, which is a bilattice homomorphism (Fitting 1991). Composition applies $\wedge_k$ pointwise, preserving bilattice structure by Proposition 7. For functoriality, $\mathcal{F}$ must satisfy $\mathcal{F}(\mathrm{id}_p) = \mathrm{id}_{A_p}$ and $\mathcal{F}(g \circ f) = \mathcal{F}(f) \circ \mathcal{F}(g)$ on composable generators. Identity is immediate: the identity morphism in $\mathcal{C}$ is the identity specialization, inducing the identity restriction on $A_p$. Composition: the composite of two specializations is the specialization along the composite fiber inclusion, whose induced restriction is the composite of the two restrictions; two successive inversions apply $\sim \circ \sim = \mathrm{id}$ to each valuation; and repeated $\wedge_k$-composition is associative and commutative by Proposition 7. $\square$

This makes $\mathcal{S}$ a bilattice-valued hyperdoctrine in the sense of Lawvere (1970), instantiating Maruyama’s (2021) fibered algebraic semantics for the specific case of $\mathbf{2} \otimes \mathbf{2}$ over predicate-structured data.

Corollary 1. The adjunction $\odot \dashv \Rightarrow$ lifts to fibers: for each predicate $p$, extending and restricting on $A_p$ are adjoint fiber-wise.

¶4. Predication modal geometries as Chu spaces

The object whose adjoint structure the previous sections describe is a Chu space. This identification is what gives the name predication modal geometry: predication from the SPO indexing, modal from the paired Priestley-dual geometry of $\mathbf{2} \otimes \mathbf{2}$ (Jung & Rivieccio 2012), geometry from the Chu-space structure we now establish.

¶4.1 The Chu construction

For a bounded distributive lattice $L$ with a distinguished element $\bot \in L$, the Chu construction $\mathrm{Chu}(L, \bot)$ is the category whose objects are triples $(X, A, r)$ with $X, A$ sets and $r: X \times A \to L$ a pairing, and whose morphisms $(X, A, r) \to (Y, B, s)$ are pairs of maps $(f: X \to Y, g: B \to A)$ satisfying $r(x, g(b)) = s(f(x), b)$. The construction is due to Barr; we use here only its object-level structure.

¶4.2 Jakl’s identification

Theorem 3 (Jakl 2025). For $H$ a Heyting algebra and $0 \in H$ its bottom, the Chu construction $\mathrm{Chu}(H, 0)$ is isomorphic to the twist-product construction: $(a, b) \in H \otimes H$ corresponds to the Chu object with the natural pairing into $H$. Under this isomorphism, the multiplicative tensor $\otimes$ of the Chu construction corresponds to the twist product $\odot$ of Theorem 1, and the linear implication $\multimap$ corresponds to $\Rightarrow$.

Specializing to $H = \mathbf{2}$ gives $\mathbf{2} \otimes \mathbf{2} \cong \mathrm{Chu}(\mathbf{2}, 0)$, and in particular $\mathbf{FOUR} \cong \mathrm{Chu}(\mathbf{2}, 0)$ as residuated structures.

¶4.3 Bilattice-valued SPO systems are predication modal geometries

Definition. A predication modal geometry is a bilattice-valued SPO system $\mathcal{S}$ regarded through its structure as a predicate-indexed Chu space: for each predicate $p$ occurring in $\mathcal{S}$, the triple $(E, E, v(-, p, -))$ with $v(-, p, -): E \times E \to \mathbf{2} \otimes \mathbf{2}$ is an object of $\mathrm{Chu}(\mathbf{2}, 0)$ under Theorem 3, and the family $\{(E, E, v(-, p, -))\}_{p \in \Pi}$ is indexed by the category $\mathcal{C}$ of §3.2.

Theorem 4. Every bilattice-valued SPO system is a predication modal geometry: the assignment $p \mapsto (E, E, v(-, p, -))$ factors the fiber functor $\mathcal{F}$ of Theorem 2 through the Chu category as $\mathcal{F} = \jmath \circ \mathcal{F}_{\mathrm{Chu}}$, where $\mathcal{F}_{\mathrm{Chu}}: \mathcal{C}^{\mathrm{op}} \to \mathrm{Chu}(\mathbf{2}, 0)$ and $\jmath: \mathrm{Chu}(\mathbf{2}, 0) \to \mathbf{BiLat}$ is the algebraic isomorphism of Theorem 3.

Proof. By Theorem 3, $\mathbf{FOUR} = \mathbf{2} \otimes \mathbf{2}$ is isomorphic as a residuated structure to $\mathrm{Chu}(\mathbf{2}, 0)$; let $\jmath$ denote this isomorphism. Each fiber algebra $A_p = \mathbf{FOUR}^{\mathrm{Fib}(p)}$ is therefore a $\mathrm{Chu}(\mathbf{2}, 0)$-valued function space, equivalently the Chu object $(\mathrm{Fib}(p), \mathrm{Fib}(p), v(-, p, -))$ in which both state and attribute sides are the subject-object-pair index and the pairing is the valuation. Define $\mathcal{F}_{\mathrm{Chu}}: \mathcal{C}^{\mathrm{op}} \to \mathrm{Chu}(\mathbf{2}, 0)$ on objects by $p \mapsto (\mathrm{Fib}(p), \mathrm{Fib}(p), v(-, p, -))$ and on the three generating morphism classes of §3.2 by the correspondents of the $\mathbf{BiLat}$-homomorphisms of Theorem 2 under $\jmath$: these are well-defined because $\jmath$ is a category isomorphism on the underlying algebras. Functoriality follows from functoriality of $\mathcal{F}$ (Theorem 2) and the isomorphism $\jmath$. By construction $\jmath \circ \mathcal{F}_{\mathrm{Chu}} = \mathcal{F}$. $\square$

Corollary. The adjunction $\odot \dashv \Rightarrow$ on predicate fibers is the $\otimes \dashv \multimap$ adjunction of linear logic at the algebraic level, via the Chu identification of Theorem 3.

This is the naming theorem: the adjoint structure of predication on bilattice-valued fibers is the Chu-multiplicative structure on a predicate-indexed Chu space over $(\mathbf{2}, 0)$.

¶5. Consequences

¶5.1 Decidability

Rivieccio, Jung, & Jansana (2017) proved that modal operators on the four-valued bilattice decompose into pairs of classical modal operators on the twist components:

$$\Box(a^+, a^-) = (\Box_1(a^+),\ \Diamond_2(a^-))$$

The resulting four-valued modal logic has the finite model property and is decidable.

Corollary 2. The modal calculus arising from the fiber functor $\mathcal{F}$ is decidable.

This follows because the fiber algebras are distributive bilattices (Proposition 7) over $\mathbf{2} \otimes \mathbf{2}$, and the modal logic on $\mathbf{2} \otimes \mathbf{2}$ is decidable by Rivieccio, Jung, & Jansana (2017).

¶5.2 Post-closure restriction

Under closure (iterating inference rules to a fixpoint in the sense of Fitting 1991), $\bot$-valued assertions within scope are pushed upward in $\leq_k$. The effective algebra operates on $\{t, f, \top\}$.

Theorem 5. The set $\{t, f, \top\}$ is closed under both $\odot$ and $\Rightarrow$. The identity element $\top$ is in $\{t, f, \top\}$. The adjunction $a \odot p \leq_t q$ iff $p \leq_t (a \Rightarrow q)$ holds on $\{t, f, \top\}$ with zero violations.

Proof. Exhaustive computation on all 27 triples. $\square$

The adjunction persists after closure.

¶5.3 Dynamics

The following results are computed exhaustively on $\{\bot, t, f, \top\}$.

Proposition 8. All orbits under iterated left-$\odot$ or left-$\Rightarrow$ reach fixed points in at most 2 steps.

Proposition 9. The only oscillatory behavior in the system: alternating $\odot$-by-$\bot$ and $\Rightarrow$-by-$\bot$ produces period-2 orbits ($\bot \to f \to \bot \to f \to \cdots$). All other alternating dynamics converge.

¶5.4 Geometry

The Priestley dual of $\mathbf{2} \otimes \mathbf{2}$ as a distributive bilattice is a discrete two-point space with a swap involution encoding negation (Jung & Rivieccio 2012). Bilattices are two-sorted structures whose duals are pairs of Priestley spaces linked by continuous order-preserving maps (Rivieccio & Jung 2012). The modal calculus of predication lives in these paired spaces.

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