Janusz Czelakowski

Equivalential logics (II)

In the first section logics with an algebraic semantics are investigated. Section 2 is devoted to subdirect products of matrices. There, among others we give the matrix counterpart of a theorem of Jónsson from universal algebra. Some positive results concerning logics with, finite degrees of maximality are presented in Section 3.

Weakly algebraizable logics

In the paper we study the class of weakly algebraizable logics, characterized by the monotonicity and injectivity of the Leibniz operator on the theories of the logic. This class forms a new level in the non-linear hierarchy of protoalgebraic logics.

Algebraic aspects of deduction theorems

The first known statements of the deduction theorems for the first-order predicate calculus and the classical sentential logic are due to Herbrand [8] and Tarski [14], respectively. The present paper contains an analysis of closure spaces associated with those sentential logics which admit various deduction theorems. For purely algebraic reasons it is convenient to view deduction theorems in a more general form: given a sentential logic C (identified with a structural consequence operation) in a sentential language I, a quite arbitrary set P of formulas of I built up with at most two distinct sentential variables p and q is called a uniform deduction theorem scheme for C if it satisfies the following condition: for every set X of formulas of I and for any formulas α and β, βεC(X∪{{a}}) iff P(α, β) AC(X). [P(α, β) denotes the set of formulas which result by the simultaneous substitution of α for p and β for q in all formulas in P]. The above definition encompasses many particular formulations of theorems considered in the literature to be deduction theorems. Theorem 1.3 gives necessary and sufficient conditions for a logic to have a uniform deduction theorem scheme. Then, given a sentential logic C with a uniform deduction theorem scheme, the lattices of deductive filters on the algebras A similar to the language of C are investigated. It is shown that the join-semilattice of finitely generated (= compact) deductive filters on each algebra A is dually Brouwerian.

Congruence distributive quasivarieties whose finitely subdirectly irreducible members form a universal class

By a congruence distributive quasivariety we mean any quasivarietyK of algebras having the property that the lattices of those congruences of members ofK which determine quotient algebras belonging toK are distributive. This paper is an attempt to study congruence distributive quasivarieties with the additional property that their classes of relatively finitely subdirectly irreducible members are axiomatized by sets of universal sentences. We deal with the problem of characterizing such quasivarieties and the problem of their finite axiomatizability.

Logical matrices and the amalgamation property

The main result of the present paper — Theorem 3 — establishes the equivalence of the interpolation and amalgamation properties for a large family of logics and their associated classes of matrices.

Reduced products of logical matrices

The class Matr(C) of all matrices for a prepositional logic (ℒ, C) is investigated. The paper contains general results with no special reference to particular logics. The main theorem (Th. (5.1)) which gives the algebraic characterization of the class Matr(C) states the following. Assume C to be the consequence operation on a prepositional language induced by a class K of matrices. Let m be a regular cardinal not less than the cardinality of C. Then Matr (C) is the least class of matrices containing K and closed under m-reduced products, submatrices, matrix homomorphisms, and matrix homomorphic counter-images.

Local deductions theorems

The notion of local deduction theorem (which generalizes on the known instances of indeterminate deduction theorems, e.g. for the infinitely-valued Łukasiewicz logic C∞) is defined. It is then shown that a given finitary non-pathological logic C admits the local deduction theorem iff the class Matr(C) of all matrices validating C has the C-filter extension property (Theorem II.1).

Fregean logics with the multiterm deduction theorem and their algebraization

AbstractA deductive system

The parameterized local deduction theorem for quasivarieties of algebras and its application

\mathcal{S}