Multiple Conclusion Rules in Logics with the Disjunction Property

We prove that for the intermediate logics with the disjunction property any basis of admissible rules can be reduced to a basis of admissible m-rules (multiple-conclusion rules), and every basis of admissible m-rules can be reduced to a basis of admissible rules. These results can be generalized to a broad class of logics including positive logic and its extensions, Johansson logic, normal extensions of S4, n-transitive logics and intuitionistic modal logics

A new proof of structural completeness Łukasiewicz's logics

The problem of structural completeness of the finite-valued Lukasiewicz's sentential calculi was investigated and solved in [4], [7], [6]. The present paper contains a new proof of all these results. I. Let (S; F1,...,Fn) be a propositional language. A matrix M = (|M |, |M |*; f1,...,fn) of this language is embeddable in a matrix N = (|N|, |N|*; gi,..., gn) (M C N) iff there exists a monomorphism h : M M N. The symbol N xM stands for the product of matrices and the structural consequence generated by M (cf. [1]) is denoted by M (Fragment tekstu)

Studies on Church's calculus

An extended version of this abstract will appear in Reports on Mathematical Logic. 1. In Church's calculus we establish classes of equivalent formulas built from only one propositional variable p in order to obtain the theorem on existence exactly one Lindenbaum's extension for Church's system. Moreover, we construct a class of finitely axiomatizable systems between Church's and Grzegorczyk's systems and we consider the problem of structural completeness of Church's calculus reduced to formulas formed from the variable p (Fragment tekstu)

On structural completeness of the infinite-valued Łukasiewicz's propositional calculus

This is an abstract of my paper “On structural Completeness of manyvalued logics” submitted to Studia Logica. I. In the paper the quasi-structural consequence generated by the infinite-valued Lukasiewicz's calculus (Ro»,A<») is examined. The notions of consequence operations Sb(X),Cn(R0,Sb(X)) and Cn(Ro*,X), where R0 = {r0} and R0* = {r0,r} (r0 is the modus ponens rule and r* is the substitution rule), are defined in [2]. Recall that a consequence Cn is quasi- structural (Cn e Sb — Struct) iff there exists a consequence Cn1 e Struct such that Cn = Cn1 Sb. The structural consequence generated by a matrix M is denoted by M, and the symbol E(M) stands for the set of all valid formulas in this matrix (E(M) = M(0)). For every matrix M we have (Fragment tekstu)

The program-substitution in algorithmic logic and algorithmic logic with non-deterministic programs

This note presents a point of view upon the notions of programsubstitution which are the tools for proving properties of programs of algorithmic logics [5], [3] being sufficiently strong and universal to comprise almost all previously introduced theories of programming, and the so-called extended algorithmic logic [1], [2] and algorithmic logic with non- deterministic programs [4]. It appears that the mentioned substitution rule allows us to examine more deeply algorithmic properties of terms, formulas and programs. Besides the problem of Post-completeness and structural completeness of algorithmic logics strengthened additionally by the rule of substitution is raised (Fragment tekstu)

A mind of a non-countable set of ideas

The paper is dedicated to the 80th birthday of the outstanding Russian logician A.V. Kuznetsov. It is addressing a history of the ideas and research conducted by him in non-classical and intermediate logics

On structural completeness versus almost structural completeness problem : a discriminator varieties case study

We study the following problem: determine which almost structurally complete quasivarieties are structurally complete. We propose a general solution to this problem and then a solution in the semisimple case. As a consequence, we obtain a characterization of structurally complete discriminator varieties. An interesting corollary in logic follows: Let L be a propositional logic/deductive system in the language with formulas for verum, which is a theorem, and falsum, which is not a theorem. Assume also that L has an adequate semantics given by a discriminator variety. Then L is structurally complete if and only if it is maximal. All such logics/deductive systems are almost structurally complete.submittedVersionFil: Campercholi, Miguel Alejandro Carlos. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.Fil: Stronkowski, Michal M. Warsaw University of Technology. Faculty of Mathematics and Information Sciences; Polonia.Fil: Vaggione, Diego José. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.Matemática Pur