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Algebraic and Proof-theoretic Aspects of Non-classical Logics pp 247–274Cite as

Notes on Strong Completeness in Łukasiewicz, Product and BL Logics and in Their First-Order Extensions

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 4460))

Abstract

In this paper we investigate the problem of characterizing infinite consequence relation in standard BL-algebras by the adding of new rules. First of all, we note that finitary rules do not help, therefore we need at least one infinitary rule. In fact we show that one infinitary rule is sufficient to obtain strong standard completeness, also in the first-order case. Similar results are obtained for product logic and for Łukasiewicz logic. Finally, we show some applications of our results to probabilistic logic over many-valued events and to first-order many-valued logic. In particular, we show a tight bound to the complexity of BL first-order formulas which are valid in the standard semantics.

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Authors and Affiliations

  1. University of Siena, Department of Mathematics and Computer Science, Pian dei Mantellini 44, 53100 Siena, Italy

    Franco Montagna

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  1. Franco Montagna
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Stefano Aguzzoli Agata Ciabattoni Brunella Gerla Corrado Manara Vincenzo Marra

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Montagna, F. (2007). Notes on Strong Completeness in Łukasiewicz, Product and BL Logics and in Their First-Order Extensions. In: Aguzzoli, S., Ciabattoni, A., Gerla, B., Manara, C., Marra, V. (eds) Algebraic and Proof-theoretic Aspects of Non-classical Logics. Lecture Notes in Computer Science(), vol 4460. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75939-3_15

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  • DOI: https://doi.org/10.1007/978-3-540-75939-3_15

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-75938-6

  • Online ISBN: 978-3-540-75939-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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