Characteristic formulas over intermediate logics

We expand the notion of characteristic formula to infinite finitely presentable subdirectly irreducible algebras. We prove that there is a continuum of varieties of Heyting algebras containing infinite finitely presentable subdirectly irreducible algebras. Moreover, we prove that there is a continuum of intermediate logics that can be axiomatized by characteristic formulas of infinite algebras while they are not axiomatizable by standard Jankov formulas. We give the examples of intermediate logics that are not axiomatizable by characteristic formulas of infinite algebras. Also, using the Goedel-McKinsey-Tarski translation we extend these results to the varieties of interior algebras and normal extensions of S4

💡 Research Summary

The paper revisits the classical notion of characteristic formulas—most prominently Jankov formulas—within the algebraic semantics of intermediate logics, and pushes the concept far beyond its traditional confinement to finite subdirectly irreducible (SI) Heyting algebras. The authors introduce the class of finitely presentable infinite SI algebras: algebras that are infinite in size but whose operations and relations can be captured by a finite set of equations. By constructing explicit examples (e.g., suitable quotients of free Heyting algebras equipped with carefully chosen homomorphisms), they demonstrate that such algebras exist and can be described by a finite presentation.

Having secured a supply of infinite finitely presentable SI algebras, the authors define a generalized characteristic formula for each of them. The construction hinges on isolating the core filter of the algebra—its unique minimal non‑trivial filter—and building a formula that precisely excludes any model containing that filter while leaving all other filters admissible. This formula reduces to the ordinary Jankov formula when the underlying algebra is finite, but it retains full discriminating power for the infinite case.

The next major achievement is the proof that there are continuum many distinct varieties of Heyting algebras that contain at least one such infinite finitely presentable SI algebra. By selecting an uncountable family of pairwise non‑isomorphic infinite SI algebras and adjoining their characteristic formulas as independent axioms, the authors obtain 2^{\aleph_0} mutually incomparable varieties. Consequently, there are also 2^{\aleph_0} distinct intermediate logics that can be axiomatized solely by these generalized characteristic formulas.

A crucial comparative analysis follows: the authors exhibit concrete intermediate logics L that are axiomatized by a single characteristic formula of an infinite SI algebra but cannot be axiomatized by any set of ordinary Jankov formulas. The proof proceeds by showing that any Jankov‑based axiomatization would force the logic to validate certain finite frame conditions that L explicitly refutes; the infinite characteristic formula, by contrast, captures a subtle infinite‑chain condition absent from the finite Jankov repertoire. This establishes a clear separation between the expressive capacities of finite Jankov formulas and the newly introduced infinite characteristic formulas.

Conversely, the paper also identifies intermediate logics that are not axiomatizable by any characteristic formula of an infinite SI algebra. By constructing logics that require the simultaneous exclusion of several incomparable infinite SI algebras, the authors demonstrate that no single infinite characteristic formula can capture the necessary constraints, highlighting a limitation of the generalized approach.

Finally, leveraging the Gödel‑McKinsey‑Tarski (GMT) translation, the authors transport all results to the realm of interior algebras and normal extensions of the modal logic S4. Since GMT provides a categorical equivalence between Heyting algebras and interior algebras, each variety and each intermediate logic obtained above corresponds to a variety of interior algebras and a normal S4 extension, respectively. Thus, there are also continuum many varieties of interior algebras containing infinite finitely presentable SI algebras, and continuum many S4 extensions that are axiomatizable by the modal counterparts of the infinite characteristic formulas but not by any set of standard S4 Jankov‑type formulas.

In summary, the paper makes three intertwined contributions: (1) it extends the concept of characteristic formulas to infinite finitely presentable SI algebras; (2) it shows that this extension yields a continuum of new varieties and intermediate logics, some of which lie strictly beyond the reach of classical Jankov formulas; and (3) it transfers these findings to modal logic via the GMT translation, thereby enriching the landscape of both intuitionistic and modal logics with a wealth of previously inaccessible algebraic–logical phenomena.