RETRACTED ARTICLE: A Completeness Theorem for a Functionally Complete Łukasiewicz Logic
This article was retracted on April 9, 2026
DOI:
https://doi.org/10.12775/LLP.2025.014Słowa kluczowe
many-valued logics, three-valued logics, propositional logic, non-classical logics, fuzzy logicAbstrakt
This article was retracted on April 9, 2026
The Editors-in-Chief have retracted this article because, after publication, it became apparent that it cites Mateusz Radzki's paper (2017), which contains an error in the proof of the claim that Słupecki's three-valued system is incomplete. Therefore, based on Radzki's paper, it cannot be concluded that Słupecki's system is really incomplete. Although the results contained in Víctor Aranda's LLP paper are not formally dependent on the content of Radzki's text, we have decided to ask the author to withdraw the original submission and prepare its corrected version. The corrected version is available at https://doi.org/10.12775/LLP.2026.010.
We want to express our profound gratitude to Professor Jan Zygmunt for his consultations and for pointing out the reference to Radzki's erroneous proof of claim that Słupecki's system is incomplete.
Supplementary Information
Below is the link to the electronic supplementary material:
Bibliografia
Avron, A., 1991, “Natural 3-valued logics: Characterization and proof theory”, The Journal of Symbolic Logic, 56(1): 276–294. DOI: https://doi.org/10.2307/2274919
Baaz, M., 1996, “Infinite-valued Gödel logics with 0-1-projections and relativizations”, pages 23–33 in P. Hájek (ed.), GÖDEL’96 – Logical Foundations of Mathematics, Computer Science and Physics, Lecture Notes in Logic, 6, Springer Verlag.
Hájek, P., 1998, Metamathematics of Fuzzy Logic, Vol. 4, Springer Science & Business Media. DOI: https://doi.org/10.1007/978-94-011-5300-3
Malinowski, G., 2007, “Many-valued logic and its philosophy”, pages 13–94 in D. M. Gabbay and J. Woods (eds.) Handbook of the History of Logic, Vol. 8, North-Holland, Amsterdam. DOI: https://doi.org/10.1016/S1874-5857(07)80004-5
Novak, V., 2005, “On fuzzy type theory”, Fuzzy Sets and Systems, 149(2): 235–273. DOI: https://doi.org/10.1016/j.fss.2004.03.027
Radzki, M., 2017, “On axiom systems of Słupecki for the functionally complete three-valued logic”, Axiomathes, 27(4): 403–415. DOI: https://doi.org/10.1007/s10516-016-9319-x
Słupecki, J., 1946, “The complete three-valued propositional calculus”, Annales Universitatis Mariae Curie-Skłodowska, 1: 193–209.
Słupecki, J., 1967, “The full three-valued propositional calculus”, pages 335–337 in: S. McCall (ed.), Polish Logic 1920–1939, Clarendon Press, Oxford.
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