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

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Abstract

Łukasiewicz’s infinite-valued logic is commonly defined as the set of formulas that take the value 1 under all evaluations in the Łukasiewicz 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 Łukasiewicz 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.

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Correspondence to Josep Maria Font.

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While writing this paper, the authors were partially supported by grants MTM2004-03101 and TIN2004-07933-C03-02 of the Spanish Ministry of Education and Science, including FEDER funds of the European Union.

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Font, J.M., Gil, À.J., Torrens, A. et al. On the infinite-valued Łukasiewicz logic that preserves degrees of truth. Arch. Math. Logic 45, 839–868 (2006). https://doi.org/10.1007/s00153-006-0001-7

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