On the Leibniz congruences

Josep Font

Banach Center Publications (1993)

  • Volume: 28, Issue: 1, page 17-36
  • ISSN: 0137-6934

Abstract

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The aim of this paper is to discuss the motivation for a new general algebraic semantics for deductive systems, to introduce it, and to present an outline of its main features. Some tools from the theory of abstract logics are also introduced, and two classifications of deductive systems are analysed: one is based on the behaviour of the Leibniz congruence (the maximum congruence of a logical matrix) and the other on the behaviour of the Frege operator (which associates to every theory the interderivability relation modulo the theory). For protoalgebraic deductive systems the class of algebras associated in general turns out to be the class of algebra reducts of reduced matrices, which is the algebraic counterpart usually considered for this large class of deductive systems; but in the general case the new class of algebras shows a better behaviour.

How to cite

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Font, Josep. "On the Leibniz congruences." Banach Center Publications 28.1 (1993): 17-36. <http://eudml.org/doc/262564>.

@article{Font1993,
abstract = {The aim of this paper is to discuss the motivation for a new general algebraic semantics for deductive systems, to introduce it, and to present an outline of its main features. Some tools from the theory of abstract logics are also introduced, and two classifications of deductive systems are analysed: one is based on the behaviour of the Leibniz congruence (the maximum congruence of a logical matrix) and the other on the behaviour of the Frege operator (which associates to every theory the interderivability relation modulo the theory). For protoalgebraic deductive systems the class of algebras associated in general turns out to be the class of algebra reducts of reduced matrices, which is the algebraic counterpart usually considered for this large class of deductive systems; but in the general case the new class of algebras shows a better behaviour.},
author = {Font, Josep},
journal = {Banach Center Publications},
keywords = {deductive system; protoalgebraic logic; Gentzen calculus; closure operator; abstract logic; algebraizable logic; Leibniz congruence; selfextensional logic; logical matrices; algebraic logic; self extensional logic; algebraic semantics; deductive systems; abstract logics; maximum congruence of a logical matrix; Frege operator; interderivability relation; protoalgebraic deductive systems; algebra reducts of reduced matrices},
language = {eng},
number = {1},
pages = {17-36},
title = {On the Leibniz congruences},
url = {http://eudml.org/doc/262564},
volume = {28},
year = {1993},
}

TY - JOUR
AU - Font, Josep
TI - On the Leibniz congruences
JO - Banach Center Publications
PY - 1993
VL - 28
IS - 1
SP - 17
EP - 36
AB - The aim of this paper is to discuss the motivation for a new general algebraic semantics for deductive systems, to introduce it, and to present an outline of its main features. Some tools from the theory of abstract logics are also introduced, and two classifications of deductive systems are analysed: one is based on the behaviour of the Leibniz congruence (the maximum congruence of a logical matrix) and the other on the behaviour of the Frege operator (which associates to every theory the interderivability relation modulo the theory). For protoalgebraic deductive systems the class of algebras associated in general turns out to be the class of algebra reducts of reduced matrices, which is the algebraic counterpart usually considered for this large class of deductive systems; but in the general case the new class of algebras shows a better behaviour.
LA - eng
KW - deductive system; protoalgebraic logic; Gentzen calculus; closure operator; abstract logic; algebraizable logic; Leibniz congruence; selfextensional logic; logical matrices; algebraic logic; self extensional logic; algebraic semantics; deductive systems; abstract logics; maximum congruence of a logical matrix; Frege operator; interderivability relation; protoalgebraic deductive systems; algebra reducts of reduced matrices
UR - http://eudml.org/doc/262564
ER -

References

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