On Structural Completeness of Tabular Superintuitionistic Logics

As usual, the superintuitionistic (propositional) logics (that is, logics extending intuitionistic logic) are being studied “modulo derivability”, meaning such logics are viewed extensionally — they are identified with the set of formulae that are valid (derivable in the corresponding calculus) in this logic. Under this approach, the lattice of all superintuitionistic logics ordered by set-inclusion is dually isomorphic to the lattice of all varieties of pseudo-Boolean algebras. If a logic is defined by a calculus, we introduce a notion of derivability of a formulae from a collection of formulae. The notion of derivability can be generalized to not finitely axiomatizable logics (for instance, as a consequence operator [1]). Sometimes, in such cases the consequences relation can be defined constructively (for instance, it can be defined by a finite matrix [1]). In the present paper, we study precisely these consequence relations: the ones that are defined by finite pseudo-Boolean algebras (regarded as matrices with a unique designated element — the greatest element of the algebra)