An explicit basis for admissible inference rules in table modal logics of width 2

We construct an explicit finite basis for admissible inference rules in an arbitrary modal logic of width 2 extending the logic Grz. Translated from Algebra i Logika, Vol. 48, No. 1, pp. 122–148, January–February, 2009.     

Preservation of Admissibility of Inference Rules in the Logics Similar to S4.2

We show that an arbitrary finitely approximable logic extending S4.2(Grz.2,KC) preserves all admissible inference rules of the logic S4.2(Grz.2,KC) if and only if this logic possesses the so-called semantic cocovering property.     

Table admissible inference rules

A recursive basis of inference rules is described which are instantaneously admissible in all table (residually finite) logics extending one of the logics Int and Grz. A rather simple semantic criterion is derived to determine whether a given inference rule is admissible in all table superintuitionistic logics, and the relationship is established between admissibility of a rule in all table (residually finite) superintuitionistic logics and its truth values in Int. Translated from Algebra i Logika, Vol. 48, No. 3, pp. 400–414, May–June, 2009.     

Finite bases with respect to admissibility for modal logics of width 2

It is proved that every finitely approximable and residually finite modal logic of depth 2 over K4 has a finite basis of admissible inference rules. This, in particular, implies that every finitely approximable residually finite modal logic of depth at most 2 is finitely based w.r.t. admissibility. (Among logics in a larger depth or width, there are logics which do not have a finite, or even independent, basis of admissible rules of inference.) Translated fromAlgebra i Logika, Vol. 38, No. 4, pp. 436–455, July–August 1999.     

Bases of admissible rules for <Emphasis Type="Italic">K</Emphasis>-saturated logics

Admissible inference rules for table modal and superintuitionistic logics are investigated. K-saturated logics are defined semantically. Such logics are proved to have finite bases for admissible inference rules in finitely many variables. __________ Translated from Algebra i Logika, Vol. 47, No. 6, pp. 750–761, November–December, 2008.     

Bases of admissible inference rules for table modal logics of depth 2

In the present article, we prove the theorem which states that every table modal logic λ of depth 2 over S4 has a finite basis of admissible inference rules. In addition, it is established that a finite algebra ℒ belongs to F_ω(λ)^Q iff there exist numbers n₁…, n_k such that (Lemma 5). Let F be a λ-frame of depth 2 and b a cluster of the second layer in F. We show that for any n₁,…,n_k, there exist no p-morphisms from (F_n1⊔…⊔F_nk)⁺ a local component K (b) such that, for any n, there is no p-morphism from any local component of F_n onto K (b) (Lemma 6). Translated fromAlgebra i Logika, Vol. 35, pp. 612–622, September–October, 1996.     

Describing a Basis in Semireduced Form for Inference Rules of Intuitionistic Logic

It is shown that a set of all rules in semireduced form whose premises satisfy a collection of specific conditions form a basis for all rules admissible in IPC. The conditions specified are quite natural, and many of them show up as properties of maximal theories in the canonical Kripke model for IPC. Besides, a similar basis is constructed for rules admissible in the superintuitionistic logic KC, a logic of the weak law of the excluded middle.