Pre-Stable Canonical Rules

• Pre-stable canonical rules are syntactic and semantic constructs that preserve modal and implicational operations on bounded critical domains within finite algebras.
• They optimize countermodel constructions and underpin duality-based completeness proofs in non-canonical systems like KM and GL.
• Their local, pre-filtration approach relaxes global preservation requirements, enabling finite axiomatization and more efficient embedding conditions.

A pre-stable canonical rule is a syntactic and semantic construct arising in the algebraic and Kripke semantics of modal and intuitionistic modal logics. These rules generalize the notion of stable canonical rules, relaxing the global preservation requirements of algebraic or relational structures to more local, domain-restricted conditions. Their development is motivated by proof-theoretic limitations in certain modal logics, notably those (like the Kuznetsov-Muravitsky system, $\mathsf{KM}$, and Gödel-Löb logic, $\mathsf{GL}$) where full filtration does not guarantee the preservation of all modal or implication operations on entire finite algebras. Pre-stable canonical rules, calibrated to "critical" domains, enable algebraic and dual topological descriptions of axiom systems, optimization of countermodel constructions, and new duality-based completeness proofs.

1. Definition and Formal Structure

A pre-stable canonical rule encodes, for a finite algebra $A$ and a designated finite set (domain) $D$ of elements or pairs, the preservation of certain critical operations. It takes the form of a single-conclusion rule whose premises enforce the algebraic structure on propositional variables indexed by elements of $A$, and only require the preservation of modal or non-truth-functional operations on the specified domain $D$.

In the intuitionistic modal ("sim") signature, for a finite frontal Heyting algebra $H$ and $D = (D^\to, D^{\boxtimes})$ (where $D^\to \subseteq H \times H$, $D^{\boxtimes} \subseteq H$ are finite), the pre-stable canonical rule $\operatorname{Can}(H,D)$ is

$\operatorname{Can}(H, D):\quad \Gamma/\Delta$

with

$$\begin{aligned} \Gamma &= \{\,p_0 \leftrightarrow \bot,\; p_1 \leftrightarrow \top\,\} \ &\;\;\cup\, \{\,p_{a\wedge b} \leftrightarrow p_a \wedge p_b : a,b\in H\,\} \ &\;\;\cup\, \{\,p_{a\vee b} \leftrightarrow p_a \vee p_b : a,b\in H\,\} \ &\;\;\cup\, \{\,p_{a\to b} \leftrightarrow p_a \to p_b : (a,b)\in D^\to\,\} \ &\;\;\cup\, \{\,p_{\boxtimes a} \leftrightarrow \boxtimes p_a : a\in D^{\boxtimes}\,\} \ \Delta &= \{\,p_a \leftrightarrow p_b : a\neq b \in H\,\} \end{aligned}$$

The classical modal ("clm") case, for a finite $\mathsf{K4}$-algebra $M$ and domain $D\subseteq M$, yields:

$\operatorname{Can}(M, D): \Gamma/\Delta$

where

$$\begin{aligned} \Gamma &= \{\,p_0\leftrightarrow\bot,\;p_1\leftrightarrow\top\,\} \ &\;\;\cup\,\{\,p_{a\wedge b}\leftrightarrow p_a\wedge p_b : a,b\in M\,\} \ &\;\;\cup\,\{\,p_{\neg a}\leftrightarrow\neg p_a : a\in M\,\} \ &\;\;\cup\,\{\,p_{\square^+ a}\to\square^+ p_a : a\in M\,\} \ &\;\;\cup\,\{\,p_{\square a}\leftrightarrow\square p_a : a\in D\,\} \ \Delta &= \{\,p_a : a\in M\setminus\{1\}\,\} \end{aligned}$$

Premises enforce the Boolean/Heyting structure and only record the modal structure on the critical domain $D$; the conclusion forces all $p_a$ to be pairwise distinct or non-top, as appropriate. These rules are calibrated to the limitations of algebraic embeddings and Kripke morphisms in non-canonical logics (Bezhanishvili et al., 13 Nov 2025).

2. Motivation: Pre-filtrations and Embedding Conditions

Pre-stable canonical rules emerge from the observation that, in systems like $\mathsf{KM}$ and $\mathsf{GL}$, standard filtration—collapsing Kripke models or algebras to finite quotients preserving formulas in some set $\Theta$—does not guarantee the preservation of modal/implicational structure globally. Instead, only a bounded "domain" of critical subformulae (e.g., those occurring in the rule under consideration) are preserved. This leads to the concept of pre-filtration: a finite model $(\mathcal{K}, V')$ is a pre-filtration of $(\mathcal{H},V)$ through $\Theta$ if the $(\boxtimes, \to)$-free reduct is the sublattice generated by $V[\Theta]$, $V'$ agrees with $V$ on $\Theta$, and the inclusion is a pre-stable embedding satisfying bounded domain conditions on $D$ (the set of images of critical connectives within $\Theta$):

$$D^\to = \{ (V(\varphi), V(\psi)): \varphi \to \psi \in \Theta \}, \qquad D^{\boxtimes} = \{ V(\varphi) : \boxtimes \varphi \in \Theta \},$$

with analogous conditions in the classical case. This conceptual link directly connects algebraic refutation of a pre-stable canonical rule to the existence of such a pre-stable embedding preserving the structure only on the critical domain (Bezhanishvili et al., 13 Nov 2025).

3. Semantic and Algebraic Characterization

The algebraic semantics of pre-stable canonical rules are governed by the existence of pre-stable embeddings between finite algebras. Specifically, for a finite algebra $B$ and a domain $D$, the failure of $A$ to validate $\operatorname{Can}(B,D)$ is equivalent to the existence of a pre-stable embedding of $B$ into $A$ that preserves non-truth-functional operations (like implication or box) on $D$:

$A \not\models \operatorname{Can}(B, D) \iff \text{there is a pre-stable embedding } B \to A \text{ preserving %%%%36%%%%}.$

On the topological/Kripke side, the dual is the existence of a pre-stable surjection of spaces (or frames), satisfying bounded domain conditions mirroring the algebraic scenario. This precise correspondence is essential in recent duality-theoretic completeness proofs, such as the new proof of the Kuznetsov–Muravitsky isomorphism (Bezhanishvili et al., 13 Nov 2025).

4. Comparison with Stable Canonical Rules

Stable canonical rules require global preservation of all modal and implication operators across the finite algebra $A$, suitable for logics admitting full stable filtrations (e.g., certain modal logics above $\mathsf{K4}$ that admit definable filtration). In contrast, pre-stable canonical rules are inherently local, requiring only domain-limited preservation. Formally, stable rules enforce, for all $a$ and $b$:

$$p_{\boxtimes a} \leftrightarrow \boxtimes p_a \quad \text{for all } a \in A$$

$$p_{a\to b} \leftrightarrow p_a \to p_b \quad \text{for all } (a,b) \in A^2$$

which is only possible in systems where the filtration supports such uniformity (e.g., for logics with definable filtration in the style of Lemmon or Gabbay). In logics like $\mathsf{KM}$ or $\mathsf{GL}$, only the pre-stable variant matches what pre-filtration guarantees (Bezhanishvili et al., 13 Nov 2025).

5. Applications: Axiomatization, Duality, and Preservation Results

Pre-stable canonical rules enable finite axiomatization of universal classes above non-canonical logics. In the context of the Kuznetsov–Muravitsky isomorphism, the entire proof is recast in terms of the refutation and transfer of pre-stable rules under cluster-collapse and skeleton-reconstitution maps in the duality between modal algebras and Esakia-type spaces. The key steps are:

• Any rule system above $\mathsf{KM}$ is canonically reducible to a finite family of pre-stable canonical rules (Theorem 4.8).
• Refutations of these rules commute with the dual maps $\rho$ (collapsing clusters) and $\sigma$ (reverting to stricter orderings), crucial for the lattice isomorphism between extensions (Corollary 5.11).
• Kripke completeness and the finite model property are preserved under these transfers (Theorem 6.11), yielding robust preservation theorems for extensions axiomatized via pre-stable canonical rules (Bezhanishvili et al., 13 Nov 2025).

This framework replaces more complex, non-constructive, or case-by-case arguments from classical modal completeness theory.

6. Examples and Explicit Constructions

A representative example in the intuitionistic modal case is for $H = \{0 < a < 1\}$ with $D^\to = \{(a,a)\}$ and $D^{\boxtimes} = \{a\}$:

$$\Gamma = \{p_0 \leftrightarrow \bot,\, p_1 \leftrightarrow \top,\, p_{x\wedge y} \leftrightarrow p_x \wedge p_y,\, p_{x\vee y} \leftrightarrow p_x \vee p_y,\, p_{a\to a} \leftrightarrow (p_a \to p_a),\, p_{\boxtimes a} \leftrightarrow \boxtimes p_a\}$$

$$\Delta = \{p_u \leftrightarrow p_v : u \neq v \in \{0, a, 1\}\}$$

A pre-stable canonical rule thus encodes the explicit requirement that embeddings preserve only the specified connectives on the domain elements, dictated by the logical structure of the formulas at hand (Bezhanishvili et al., 13 Nov 2025).

7. Significance and Outlook

Pre-stable canonical rules have transformed the algebraic and duality-theoretic analysis of modal and intuitionistic modal logics where standard stable methods are too rigid. Their flexible "bounded domain" approach not only accommodates the technical limitations of pre-filtrations, but also yields conceptual and efficient proofs of isomorphism results, preservation properties, and finite-model axiomatizability for broad classes of logics. Their algebra-space alignment fits seamlessly into duality theories for modal logics, promising further generalizations in non-classical and substructural settings. The systematic use of pre-stable rules has improved both the abstraction level and the uniformity of completeness and correspondence arguments in contemporary modal logic research (Bezhanishvili et al., 13 Nov 2025).