On the infinite-valued Lukasiewicz logic that preserves degrees of truth

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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


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.