Difference between revisions of "On the infinite-valued Lukasiewicz logic that preserves degrees of truth"

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(New page: {{Paper| author=Josep Maria Font|author2=Àngel J. Gil|author3=Antoni Torrens|author4=Ventura Verdú| title=On the infinite-valued Lukasiewicz logic that preserves degrees of truth| journa...)
 
 
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== Abstract ==
 
== Abstract ==
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Lukasiewicz's infinite-valued logic is commonly defined as the set of formulas that take the value 1 under all evaluations in the algebra on the unit real interval. In the literature a deductive system axiomatized in a Hilbert style was associated to it, and was later shown to be semantically defined from Lukasiewicz's algebra by using a "truth- preserving" scheme. This deductive system is algebraizable, non-selfextensional and does not satisfy the deduction theorem. In addition, there exists no Gentzen calculus fully adequate for it. Another presentation of the same deductive system can be obtained from a substructural Gentzen calculus. In this paper we use the framework of abstract algebraic logic to study a different deductive system which uses the aforementioned algebra under a scheme of "preservation of degrees of truth". We characterize the resulting deductive system in a natural way by using the lattice filters of Wajsberg algebras, and also by using a structural Gentzen calculus, which is shown to be fully adequate for it. This logic is an interesting example for the general theory: it is selfextensional, non-protoalgebraic, and satisfies a "graded" deduction theorem. Moreover, the Gentzen system is algebraizable. The first deductive system mentioned turns out to be the extension of the second by the rule of Modus Ponens.

Latest revision as of 08:06, 27 April 2016

Authors:
Josep Maria Font
Àngel J. Gil
Antoni Torrens
Ventura Verdú
Title: On the infinite-valued Lukasiewicz logic that preserves degrees of truth
Journal: Archive for Mathematical Logic
Volume 45
Number 7
Pages: 839-868
Year: 2006




Abstract

Lukasiewicz's infinite-valued logic is commonly defined as the set of formulas that take the value 1 under all evaluations in the algebra on the unit real interval. In the literature a deductive system axiomatized in a Hilbert style was associated to it, and was later shown to be semantically defined from Lukasiewicz's algebra by using a "truth- preserving" scheme. This deductive system is algebraizable, non-selfextensional and does not satisfy the deduction theorem. In addition, there exists no Gentzen calculus fully adequate for it. Another presentation of the same deductive system can be obtained from a substructural Gentzen calculus. In this paper we use the framework of abstract algebraic logic to study a different deductive system which uses the aforementioned algebra under a scheme of "preservation of degrees of truth". We characterize the resulting deductive system in a natural way by using the lattice filters of Wajsberg algebras, and also by using a structural Gentzen calculus, which is shown to be fully adequate for it. This logic is an interesting example for the general theory: it is selfextensional, non-protoalgebraic, and satisfies a "graded" deduction theorem. Moreover, the Gentzen system is algebraizable. The first deductive system mentioned turns out to be the extension of the second by the rule of Modus Ponens.