Quasiary Predicate Logic

• Quasiary predicates are n-ary relations that generalize monadic predicates by employing versatile quantifier rules and encompassing both ancient deduction practices and modern computational applications.
• This logic framework integrates natural deduction, model theory, and duality theory to establish sound and complete systems for analyzing arbitrary-arity relational structures.
• Applications range from expressing complex numerical properties in formal languages to modeling coinductive predicates in transition systems, highlighting its enhanced expressiveness and computational relevance.

The logic of quasiary predicates concerns the behavior, expressiveness, and formal manipulation of predicates with fixed but arbitrary arity—generalizing beyond the monadic case to the polyadic (n-ary) regime. Quasiary predicate logic is distinguished by its focus on quantifier rules, model-theoretic properties, and computational expressiveness when applied to formal languages, type theory, and coalgebraic systems. Fundamental developments span from ancient deduction practices in polyadic contexts to modern work on the duality theory of existential first-order fragments indexed by predicate arity and the design of modal logics for coinductive, arbitrary-arity relational properties.

1. Core Natural Deduction Rules for Quasiary Predicates

In systems that accommodate relations of arbitrary arity, the standard natural deduction rules for quantifiers—universal introduction (∀I), universal elimination (∀E), existential introduction (∃I), and existential elimination (∃E)—are generalized to apply to predicates $P(\vec{x})$ where $\vec{x}$ is an $n$-tuple, $n \geq 1$ (Protin, 2022). The sequent calculus presentation is as follows, with all components traced directly to polyadic settings:

• Universal Elimination (∀E): From $$\Gamma \vdash \forall x_1 \dots x_n \, P(x_1,\dots,x_n)$$, infer $\Gamma \vdash P(t_1,\dots,t_n)$, for arbitrary terms $t_i$ free for $x_i$.
• Universal Introduction (∀I): From $\Gamma \vdash P(x_1,\dots,x_n)$, infer $$\Gamma \vdash \forall x_1\dots x_n \, P(x_1,\dots,x_n)$$, provided $x_1,...,x_n$ are not free in any undischarged assumption.
• Existential Introduction (∃I): From $\Gamma \vdash P(t_1, ..., t_n)$, infer $$\Gamma \vdash \exists x_1 ... x_n \, P(x_1,...,x_n)$$.
• Existential Elimination (∃E): From $$\Gamma \vdash \exists x_1\dots x_n \, P(x_1,\dots,x_n)$$ and $[P(c_1, ..., c_n)] \vdash Q$, infer $\Gamma \vdash Q$ for fresh constants $c_1,...,c_n$.

Nested quantification and mixed quantifier blocks over polyadic predicates are central to both ancient and contemporary proof-theoretic approaches, allowing the direct encoding of multiple generality (Protin, 2022).

2. Quasiary Predicates and Expressiveness in Logic on Words

In computational logic, quasiary (fixed-arity) numerical predicates are formalized as families $R^k$ indexed by word length $n$ such that $R^k(n) \subseteq \{0,\dots,n-1\}^k$ (Zaïdi, 2022). Uniformity requires $R^k(n) = Q \cap \{0,...,n-1\}^k$ for a fixed $Q \subset \mathbb{N}^k$.

The fragment $\Sigma_1[\mathbb{N}^u_k]$ consists of sentences with a single block of $k$ existential quantifiers over quantifier-free formulas involving only letter predicates and uniform $k$-ary numerical predicates. The Boolean closure $\mathcal{B} \Sigma_1[\mathbb{N}^u_k]$ enables arbitrary Boolean combinations of such sentences, yielding a robust family of languages that strictly extends the regular $\Sigma_1[<]$ languages even for $k \geq 1$, thus facilitating global numerical properties not available in the strictly regular case (Zaïdi, 2022).

3. Topological Duality and Ultrafilter Equations for Quasiary Fragments

Duality theory reveals a topological structure underlying quasiary existential fragments. The Boolean algebra $\mathcal{B} \Sigma_1[\mathbb{N}^u_k]$ has as its Stone dual a subspace $X_k$ of the Vietoris hyperspace $V(\beta(\mathbb{N}^k))^{A^k}$, with points characterized combinatorially via colorings of $\mathbb{N}^k$ (Zaïdi, 2022). In the case $k=1$, explicit ultrafilter equations—such as $E_{ab=ba}, E_{aab=abb}, E_{a=a.a}$—completely characterize $\mathcal{B} \Sigma_1[\mathbb{N}^u_1]$, with soundness and completeness obtained through topological methods.

Predicate Type	Fragment	Dual Characterization
$k=1$ (unary)	$\mathcal{B} \Sigma_1[\mathbb{N}^u_1]$	Ultrafilter equations, finite colorings
$k>1$ (quasiary)	$\mathcal{B} \Sigma_1[\mathbb{N}^u_k]$	Vietoris subspaces, finite colorings of $\mathbb{N}^k$

A plausible implication is that, while k-ary fragments with $k \geq 1$ extend expressiveness, explicit equational/ultrafilter bases become increasingly intricate due to new combinatorial phenomena akin to Ramsey theory on $\mathbb{N}^k$ (Zaïdi, 2022).

4. Dependent-Type and Coalgebraic Perspectives

Dependent-type theory internalizes quantification over quasiary predicates as dependent products (Π-types) and sums (Σ-types), enabling fine-grained control over domains of quantification. For a binary relation $R(x,y)$, the Σ-type $\Sigma x:A. \Sigma t:B. R(x,t)$ expresses nested existence, preserving polyadic binding structure (Protin, 2022). Ancient deduction patterns like ekthesis correspond to Σ-elimination and pattern-matching in this type-theoretic context.

In coalgebraic logic, arbitrary-arity (quasiary) coinductive predicates are treated through fibrational semantics. A fibration $p:\mathbb{E}\to\mathbb{C}$ facilitates uniform treatment of n-ary predicates and enables liftings of functors $\overline{B}:\mathbb{E}\to\mathbb{E}$ inducing greatest fixed points, representing coinductive predicates (e.g., bisimilarity, similarity, behavioral metrics) (Kupke et al., 2020). The logical adequacy and expressiveness of modal logics for these coinductive quasiary predicates are established by verifying inequalities between the semantics of formulas and the greatest fixed points.

5. Soundness, Completeness, and Model-Theoretic Properties

The four generalized quantifier rules together with propositional deduction yield a sound and complete natural deduction system for full first-order logic with n-ary (polyadic) predicates (Protin, 2022). Soundness is shown inductively on derivations, and completeness entails that any valid sequent admits a natural deduction proof, corresponding to standard results for Gentzen systems and modern polyadic first-order logic.

In the duality-theoretic setting for logic on words, $\mathcal{B} \Sigma_1[\mathbb{N}^u_1]$ admits a complete ultrafilter axiomatization, and the proof utilizes explicit colorings of $\mathbb{N}$ and invariance under basic operations on marked words (Zaïdi, 2022). For $k>1$, model-theoretic completeness invokes combinatorial Ramsey-type arguments.

6. Illustrative Examples and Applications

Classical and contemporary examples illustrate the scope of quasiary predicate logic. In the ancient context, Aristotle formalizes syllogisms involving ternary and binary relations, with deduction steps that instantiate and generalize in accordance with the generalized quantifier rules (Protin, 2022). In computational applications:

• The unary predicate “is prime” specifies positions in a word; the sentence $\exists x\,\mathrm{prime}(x) \land a(x)$ picks out words containing the letter $a$ in a prime position (Zaïdi, 2022).
• A binary numerical predicate $\leq$ (interpreted as “$i \leq j$”) supports languages such as $A^*aA^*bA^*$, where a letter $a$ occurs before a letter $b$.

Coalgebraically, arbitrary-arity coinductive predicates characterize relations in transition systems and automata—ranging from bisimilarity to similarity and weighted behavioral metrics—by assigning modal logics whose formulas reflect the full quasiary structure (Kupke et al., 2020).

7. Complexity-Theoretic and Expressivity Considerations

The introduction of k-ary uniform numerical predicates into logical fragments introduces expressive resources that strictly transcend regular languages, especially for $k \geq 1$ (Zaïdi, 2022). Boolean closures remain “tame” in the sense that dual spaces and axiomatizations are still tractable. The Boolean algebra $\mathcal{B}\Sigma_1[\mathbb{N}^u_1]$ is closed under all Boolean operations and admits a concise equational axiomatization, yet can define non-regular languages (e.g., positions indexed by primality). For $k\geq2$, new complexity-theoretic phenomena emerge, suggesting connections with decision problem hardness and the potential role of number-theoretic oracles.

Quasiary predicate logic thus provides a unified framework for modeling, reasoning about, and analyzing relational structures of arbitrary arity in both ancient philosophical and modern computational settings, underpinning major results in proof theory, duality, expressiveness, and complexity (Protin, 2022, Zaïdi, 2022, Kupke et al., 2020).