On the infinite-valued Lukasiewicz logic that preserves degrees of truth
Authors: |
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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.