A Completeness Theorem for a Functionally Complete Łukasiewicz Logic
DOI:
https://doi.org/10.12775/LLP.2025.014Keywords
many-valued logics, three-valued logics, propositional logic, non-classical logics, fuzzy logicAbstract
Radzki has recently claimed the incompleteness of the axioms given by Słupecki for the functionally complete Ł3: some of its tautologies are not provable. In this paper, we provide a new axiom system for this logic (choosing a variant with two propositional constants and the Łukasiewicz implication as primitive symbols) and prove a Completeness Theorem.
References
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.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2025 Víctor Aranda
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 422
Number of citations: 0
