Most General Assertions in Modal Logic

• Most General Assertion (MGA) is the least-committal geometric frame condition corresponding to a modal formula over Kripke frames.
• The MASSA algorithm extracts the MGA through defined syntactic phases, ensuring analytic, cut-admissible proofs via second-order quantifier elimination.
• MGA effectively connects modal semantics with first-order frame conditions, reinforcing classical Sahlqvist correspondence results in modal logic.

The Most General Assertion (MGA) of a modal formula is the first-order frame condition derived as the logical correspondent of the formula—formally, the frame condition that is both the least-committal geometric implication entailing and entailed by the formula over Kripke frames. The concept arises in the context of algorithmic correspondence theory and is operationalized via the MASSA algorithm as the first-order geometric axiom correspondent to a given (definite) analytic inductive modal formula. MGA thus internalizes the modal formula's frame-theoretic semantics in the analytic style of Gentzen-style proof theory and geometric logic (Domenico et al., 2022).

1. Definitions and Syntactic Foundations

The language of modal logic under consideration is the standard language of K:

$$\varphi ::= p \mid \bot \mid \neg\varphi \mid \varphi_1 \land \varphi_2 \mid \varphi_1 \lor \varphi_2 \mid \varphi_1 \to \varphi_2 \mid \Diamond\varphi \mid \Box\varphi$$

Modal formulas are assumed to be in negative normal form (NNF). Analytic inductive formulas are defined by the use of two key syntactic skeletons:

• Negative skeletons $\psi(x_1, ..., x_k; y_1, ..., y_l)$, constructed from variables, negated positive skeletons, conjunction, disjunction, implication from positive skeletons, and boxes, with monotonicity in the $x_i$ and antitonicity in the $y_j$.
• Positive skeletons (PIA) $\pi(x_1, ..., x_k; y_1, ..., y_l)$, constructed from variables, negated negative skeletons, conjunction, disjunction, and diamonds.

A formula is definite if, in its parse tree, neither conjunction (for negative skeletons) nor disjunction (for positive skeletons) is nested under implication or box (respectively, under diamond). Analytic-inductive formulas are constructed recursively by composing definite negative skeletons with definite negative PIAs as arguments, maintaining a well-founded critical-variable dependency relation. The Sahlqvist fragment is obtained when no critical variable of any PIA appears under another.

2. The MASSA Algorithm and MGA Extraction

The MASSA (Minimal Assumption Syntactic Schütte–Ackermann) algorithm takes as input a definite analytic-inductive modal formula $\varphi$ and outputs:

• An analytic geometric rule $r$ in the labelled sequent calculus G3K, extensionally equivalent to $\varphi$.
• The MGA of $\varphi$: the first-order frame correspondent of $\varphi$, produced through second-order quantifier elimination (via a SCAN-style Ackermann procedure).

The algorithm proceeds through five syntactic phases:

1. Logical Expansion in G3K: Color the root sequent, perform cut-free backward proof search with invertible rules, and prune to a multiplicative proof tree.
2. Atomic Cuts and Relational Harvesting: Cut on atomic formulas to obtain axioms with purely relational conclusions, reconstructing maximal PIA subproofs.
3. Skeleton-only Backward Search: Strip PIA components, translating sequents to frame-theoretic (relational) conditions.
4. Merging Points and Geometric Rule Formation: Identify merge points where PIA-proofs combine with the skeleton, yielding a geometric Gentzen rule whose axiom is the geometric frame condition.
5. SCAN-Style Second-Order Quantifier Elimination: Produce the first-order sentence (the MGA), potentially using Ackermann's method, yielding a Sahlqvist-style frame condition.

3. Termination and Scope of MGA Derivation

MASSA's termination is guaranteed for definite analytic-inductive formulas, including all Sahlqvist formulas. This is ensured by several measures: the finiteness of the formula's subformulas, bounded atomic cuts, strictly descending critical variable structures in PIAs, and the non-looping structure of the skeleton expansion. The SCAN-Ackermann elimination also terminates on Sahlqvist-shaped sequents. Outside this class, for instance with McKinsey's axiom $\Diamond\Box p \to \Box\Diamond p$, neither MASSA nor Ackermann elimination is guaranteed to succeed (Domenico et al., 2022).

4. Logical Equivalence and Analyticity

The geometric rule $r$ produced by MASSA has the following general form:

$$\frac{R_1(\vec{x}, \vec{y}_1), \Gamma_1 \vdash \Delta_1 \quad \cdots \quad R_n(\vec{x}, \vec{y}_n), \Gamma_n \vdash \Delta_n}{R_0(\vec{x}), \Gamma_0 \vdash \Delta_0} \quad r$$

The associated geometric axiom is:

$$\forall\vec{x} \left(R_0(\vec{x}) \rightarrow \bigvee_{i=1}^n \exists\vec{y}_i R_i(\vec{x}, \vec{y}_i)\right)$$

MASSA ensures that $r$ is analytic (cut-admissible), sound, and complete for the modal formula $\varphi$. The geometric axiom is logically equivalent over all Kripke frames to the standard Sahlqvist first-order condition for $\varphi$ and thus constitutes the MGA.

5. Paradigmatic Example

Consider the Sahlqvist formula:

$$\varphi_0 ::= \Box(p \rightarrow q) \rightarrow (\Box p \rightarrow \Box q)$$

Applying MASSA yields the geometric rule:

$$\frac{xRy,\, yRz \vdash xRy,\, xRz}{xRy,\, yRz \vdash}$$

whose geometric axiom is:

$$\forall x\, \forall y\, \forall z\, [(xRy \wedge yRz) \to xRz]$$

This is the standard frame condition of transitivity, and thus $\operatorname{MGA}(\varphi_0)$ is exactly the transitivity property.

6. Limitations and Extensions

MASSA only succeeds on definite analytic-inductive modal formulas. Counterexamples exist outside this fragment: e.g., for formulas analytic but not Sahlqvist, SCAN/Ackermann may fail. The methodology extends programmatically to broader systems, such as the modal $\mu$-calculus (with least fixpoints), hybrid logics with nominals, first-order modal logics, and substructural logics, provided the invertible/multiplicative decomposition and well-founded critical-variable ordering are maintained.

7. Significance Within Correspondence Theory

MGA, as constructed by MASSA, provides a syntactic pathway to internalize frame conditions as geometric, cut-free analytic rules in G3K, yielding an effective and constructive approach to correspondence for a large and prominent class of modal formulas. Within the tradition of algebraic and proof-theoretic correspondence, MGAs are recognized as the "most general" first-order assertions entailed by and entailing the original formula over Kripke frames, and correspond exactly to Sahlqvist's classical correspondence results (Domenico et al., 2022).