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Note of the full generalized models of the extensions of a logic

Albuquerque H., Font J. M., Jansana R.

Reports on Mathematical Logic

2017

Vol. 52

57--68

In this short note we show that the full generalized models of any extension of a logic can be determined from the full generalized models of the base logic in a simple way. The result is a consequence of two central theorems of the theory of full generalized models of sentential logics. As applications we investigate when the full generalized models of an extension can also be full generalized models of the base logic, and we prove that each Suszko lter of a logic determines a Suszko lter of each of its extensions, also in a simple way.

On everywhere strongly logifiable algebras

Moraschini T.

Reports on Mathematical Logic

2015

Vol. 50

83--107

We introduce the notion of an everywhere strongly logifiable algebra: a finite non-trivial algebra A such that for every F ∈ P(A) \ {∅,A} the logic determined by the matrix [A, F] is a strongly algebraizable logic with equivalent algebraic semantics the variety generated by A. Then we show that everywhere strongly logifiable algebras belong to the field of universal algebra as well as to the one of logic by characterizing them as the finite non-trivial simple algebras that are constantive and generate a congruence distributive and n-permutable variety for some n ≥ 2. This result sets everywhere strongly logifiable algebras surprisingly close to primal algebras. Nevertheless we shall provide examples of everywhere strongly logifiable algebras that are not primal. Finally, some conclusion on the problem of determining whether the equivalent algebraic semantics of an algebraizable logic is a variety is obtained.

Fully adequate Gentzen systems and the deduction theorem

Font J., Jansana R., Pigozzi D.

Reports on Mathematical Logic

2001

No. 35

115-165

A deductive system over an arbitrary language type A is a finitary and substitution-invariant consequence relation over the formulas of A. A Gentzen system is a finitary and substitution-invariant consequence relation over the sequents of A. A matrix model of a deductive system S is a pair (A,F) where A is A-algebra and F is an S-filter on A, i.e., a subset of A closed under all interpretations of the consequence relation of S in A. A generalized matrix is a pair (A,C) where C is an algebraic closed-set system over A; it is a model of a Gentzen system G if C is closed under all interpretations of the consequence relation of G in A. A Gentzen system G is fully adequate for a deductive system S if (roughly speaking) every reduced generalized matrix model of G is of the form (A, FisA), where FisA is the set of all S-filters on A. The existence of a fully adequate Gentzen system for a given protoalgebraic deductive system S is completely characterized in terms of the following variant of the standard deduction theorem of classical and intuitionistic logic. The main result of the paper is the following: Theorem. A protoalgebraic deductive system has a fully adequate Gentzen system if and only if it has a Leibniz-generating PGDD (a parameterized graded deduction-detachment) system over all Leibniz theories. Two corollaries: (I) A weekly algebraizable deductive system has a fully adequate Gentzen system iff it has the multiterm deduction-detachment theorem. (II) A finitely equivalential deductive system has a fully adequate Gentzen system iff it has a finite Leibniz-generating system for over all Leibniz S-filters. Several different variants of the deduction theorem arise in the course of the paper showing that this familiar notion is only one manifestation of a surprisingly complex phenomenon.