Hereditarily Structurally Complete Superintuitionistic Deductive Systems

Abstract

The paper studies hereditarily complete superintuitionistic deductive systems, that is, the deductive system which logic is an extension of the intuitionistic propositional logic. It is proven that for deductive systems a criterion of hereditary structurality -similar to one that exists for logics -does not exists. Nevertheless, it is proven that many standard superintuitionistic logics (including Int) can be defined by a hereditarily structurally complete deductive system.

Varieties of Positive Modal Algebras and Structural Completeness

“…If 1 is not join-prime, then there are a, b < 1 such that a ∨ b = 1. Together with (14), this implies that 1 = (a ∨ b) a ∨ Qb = 0 ∨ b = b which contradicts the fact that b < 1. Hence we conclude that 1 is joinprime.…”

“…{1} and c/θ for every c ∈ L. The fact that 1 is join-prime and (14) imply that φ is congruence of A. Moreover, φ is different from the identity relation and a, b / ∈ φ.…”

“…Structural completeness and its variants have been the subject of intense research in the setting of modal and Heyting algebras [58], yielding an array of deep results. To name only a few of them, HSC varieties of K 4 -algebras, and of Heyting algebras have been fully described in [14,57]. Moreover, explicit bases for the admissible rules of intuitionistic 1 Observe that passive quasi-equations are vacuously admissible.…”

Hereditarily Structurally Complete Positive Logics

“…Cintula & Metcalfe (2010)). All hereditarily structurally complete intermediate logics were described by author in Citkin (1978). In (Rybakov, 1995, Theorem 4.5) Rybakov obtained a similar description for the extensions of normal modal logic K4, and as a consequence, he gave an alternative proof of the criterion on hereditary structural completeness for intermediate logics (cf.…”

Complexity of Admissible Rules in the Implication-Negation Fragment of Intuitionistic Logic

“…Kripke-frame based characterizations of hereditarily structurally complete (i.e., each extension of the logic is structurally complete) intermediate logics and transitive modal logics have been obtained by Citkin and Rybakov [3,16]. Bases have been provided for certain intermediate logics by Iemhoff [8] and for transitive modal logics by Jeřábek [11], and Gentzen-style proof systems have been developed for these logics by Iemhoff and Metcalfe [10,9].…”