We generalize the (∧, ∨)-canonical formulas to (∧, ∨)-canonical rules, and prove that each intuitionistic multi-conclusion consequence relation is axiomatizable by (∧, ∨)-canonical rules. This yields a convenient characterization of stable superintuitionistic logics. The (∧, ∨)-canonical formulas are analogues of the (∧, →)-canonical formulas, which are the algebraic counterpart of Zakharyaschev's canonical formulas for superintuitionistic logics (si-logics for short). Consequently, stable si-logics are analogues of subframe si-logics. We introduce cofinal stable intuitionistic multi-conclusion consequence relations and cofinal stable si-logics, thus answering the question of what the analogues of cofinal subframe logics should be. This is done by utilizing the (∧, ∨, ¬)-reduct of Heyting algebras. We prove that every cofinal stable si-logic has the finite model property, and that there are continuum many cofinal stable si-logics that are not stable. We conclude with several examples showing the similarities and differences between the classes of stable, cofinal stable, subframe, and cofinal subframe si-logics.formulas that provide a uniform axiomatization for large classes of si-logics and transitive modal logics. Zakharyaschev [26,27] generalized these results by introducing canonical formulas, which axiomatize all si-logics and all transitive modal logics. Jeřábek [21] further generalized Zakharyaschev's approach by defining canonical multi-conclusion rules that axiomatize all intuitionistic multi-conclusion consequence relations and all transitive modal multi-conclusion consequence relations.The algebraic counterparts of Zakharyaschev's canonical formulas for silogics are the (∧, →)-canonical formulas of [4]. The (∧, →)-canonical formula of a finite subdirectly irreducible (s.i.) Heyting algebra A fully describes the ∨-free reduct of A, but describes the behavior of the missing connective ∨ only partially. In fact, the algebraic content of Zakharyaschev's closed domain condition (CDC) is encoded by D ⊆ A 2 , where the behavior of ∨ is described fully. The (∧, →)-canonical formulas, though syntactically quite different, serve the same purpose as Zakharyaschev's canonical formulas in providing a uniform axiomatization of all si-logics.One of the main technical tools in developing the theory of (∧, →)-canonical formulas is Diego's theorem [11] that the variety of implicative meet-semilattices is locally finite. Another locally finite variety closely related to the variety of Heyting algebras is that of bounded distributive lattices. This suggests a different approach to canonical formulas, which was developed in [5], where (∧, ∨)-canonical formulas were introduced. The (∧, ∨)-canonical formula of a finite s.i. Heyting algebra A fully describes the bounded lattice reduct of A, and only partially the behavior of the missing connective →. In this case the CDC is encoded by D ⊆ A 2 , where the behavior of → is described fully. As in the (∧, →)-case, the (∧, ∨)-canonical formulas provide a uniform axiomatization of a...

Stable logics are modal logics characterized by a class of frames closed under relation preserving images. These logics admit all filtrations. Since many basic modal systems such as K4 and S4 are not stable, we introduce the more general concept of an M-stable logic, where M is an arbitrary normal modal logic that admits some filtration. Of course, M can be chosen to be K4 or S4. We give several characterizations of M-stable logics. We prove that there are continuum many S4-stable logics and continuum many K4-stable logics between K4 and S4. We axiomatize K4-stable and S4-stable logics by means of stable formulas and discuss the connection between S4-stable logics and stable superintuitionistic logics. We conclude the article with many examples (and nonexamples) of stable, K4-stable, and S4-stable logics and provide their axiomatization in terms of stable rules and formulas.

With each superintuitionistic logic (si-logic for sort), we associate its downward and upward subframizations, and characterize them by means of Zakharyaschev's canonical formulas, as well as by embedding si-logics into the extensions of the propositional lax logic PLL. In an analogous fashion, with each si-logic, we associate its downward and upward stabilizations, and characterize them by means of stable canonical formulas, as well as by embedding si-logics into extensions of the intuitionistic S4.

NNIL-formulas, introduced by Visser in 1983–1984 in a study of $\varSigma _1$-subsitutions in Heyting arithmetic, are intuitionistic propositional formulas that do not allow nesting of implication to the left. The first results about these formulas were obtained in a paper of 1995 by Visser et al. In particular, it was shown that NNIL-formulas are exactly the formulas preserved under taking submodels of Kripke models. Recently, Bezhanishvili and de Jongh observed that NNIL-formulas are also reflected by the colour-preserving monotonic maps of Kripke models. In the present paper, we first show how this observation leads to the conclusion that NNIL-formulas are preserved by arbitrary substructures not necessarily satisfying the topo-subframe condition. Then, we apply it to construct universal models for NNIL. It follows from the properties of these universal models that NNIL-formulas are also exactly the formulas that are reflected by colour-preserving monotonic maps. By using the method developed in constructing the universal models, we give a new direct proof that the logics axiomatized by NNIL-axioms have the finite model property.

Abstract.A lattice P is transferable for a class of lattices K if whenever P can be embedded into the ideal lattice IK of some K ∈ K, then P can be embedded into K. There is a rich theory of transferability for lattices. Here we introduce the analogous notion of MacNeille transferability, replacing the ideal lattice IK with the MacNeille completion K. Basic properties of MacNeille transferability are developed. Particular attention is paid to MacNeille transferability in the class of Heyting algebras where it relates to stable classes of Heyting algebras, and hence to stable intermediate logics.Mathematics Subject Classification. 06D20, 06B23, 06E15.