Possibility operators over n-valued Gödel logic

Document Type : Original Article


Departamento de Matematica, Universidad Nacional del Sur (UNS), Bahia Blanca, Argentina


In the area of fuzzy logic, expansions of these logics by $\Delta$ operator have been intensively studied; the interest of $\Delta$ operator is due to the fact that it presents a fuzzy behavior, the associated systems were studied in propositional and first-order level.  On the other hand, the possibility operators that define Łukasiewicz-Moisil algebras have been studied over different classes of algebras; these operators are known as Moisil's operators in the literature. One of these operators coincides with $\Delta$, showing there are other operators with fuzzy behavior.   In this paper, we present the study of Moisil's operators over an extension of a fuzzy logic; namely, n-valued Gödel logic, thus opening the possibility to explore more fuzzy operators.


[1] F. Almin̆ana, G. Pelaitay, Monadic k × j-rough Heyting algebras, Archive for Mathematical
Logic, (2021), DOI:10.1007/s00153-021-00802-6.
[2] M. Baaz, Infinite-valued Gödel logics with 0-1-projections and relativizations. In GÖDEL 96,
LNL 6, Hájek P. (Ed.), Springer-Velag, (1996), 23-33.
[3] V. Boicescu, A. Filipoiu, G. Georgescu, S. Rudeanu, Łukasiewicz - Moisil algebras, Annals of
Discrete Mathematics, 49, North - Holland, 1991.
[4] M. Canals Frau, A.V. Figallo, (n + 1)-valued Hilbert modal algebras, Notas de la Sociedad
Matematica de Chile, X(1) (1991), 143–149.
[5] M. Canals Frau, A.V. Figallo, (n + 1)-valued modal implicative semilattices, 1992 Proceedings
The 22nd International Symposium on Multiple-Valued Logic, 190–196. IEEE Computer
Society, 1992.
[6] M.C. Canals Frau, A.V. Figallo, G. Pelaitay, Congruences on bounded Hilbert algebras with
Moisil possibility operators, South American Journal of Logic, (2021), URL: http://www.salogic.
[7] R. Cignoli, Estudio algebraico de lógicas polivalentes. Algebras de Moisil de orden n, Ph. D.
thesis, Universidad Nacional del Sur, Bahia Blanca, 1969.
[8] R. Cignoli, Proper n-valued Łukasiewicz algebras as S-algebras of Łukasiewicz n-valued propositional
calculi, Studia Logica, 41(1) (1982), 3–16.
[9] M. Coniglio, A. Figallo-Orellano, A. Hernández-Tello, M. Pérez-Gaspar, G’3 as the logic of
modal 3-valued Heyting algebras, Journal of Applied Logic, 9(1) (2022), 175–197.
[10] F. Esteva, L. Godo. Monoidal t-norm based logic: Towards a logic for left-continuous t-norms,
Fuzzy Sets and Systems, 124 (2001), 271–288.
[11] F. Esteva, L. Godo, P. Hájek, M. Navara, Residuated fuzzy logics with an involutive negation,
Archive for Mathematical Logic, 39 (2000), 103–124.
[12] F. Esteva, L. Godo, F. Montagna, The ŁΠ and the ŁΠ1
2 logics: Two complete fuzzy systems
joining Łukasiewicz and product logics, Archive for Mathematical Logic, 40(1) (2001), 39–67.
[13] A. V. Figallo, G. Pelaitay, J. Sarmiento, Cn algebras with Moisil possibility operators, Logic
Journal of the IGPL, 28(6) (2020), 1141–1154.
[14] A. Figallo Orellano, Notes on Hilbert lattices, International Mathematical Forum, 6(68) (2011),
[15] A. Figallo Orellano, A preliminary study of MV-algebras with two quantifiers which commute,
Studio Logica, 104 (2016), 931–956.
[16] A. Figallo Orellano, I. Pascual, On monadic operators on modal pseudo-complemented De
Morgan algebras and tetravalent modal algebras, Studio Logica, 107 (2019), 591–611.
[17] A. Figallo-Orellano, J. Slagter, An algebraic study of the first order version of some implicational
fragments of the three-valued Łukasiewicz logic, Computación y Sistemas, (2022), URL:
[18] A. Figallo-Orellano, J. Slagter, Monteiro’s algebraic notion of maximal consistent theory for
Tarskian logics, Fuzzy Sets and Systems, (2022), DOI:10.1016/j.fss.2022.04.007.
[19] C. Gallardo, A. Ziliani, A generalization of monadic n-valued Łukasiewicz algebras, Studio
Logica, (2021), DOI: 10.1007/s11225-021-09968-9.
[20] P. Hájek, Metamathematics of fuzzy logic, volume 4 of Trends in Logic. Kluwer, Dordrecht,
[21] P. Hájek, P. Cintula. On theories and models in fuzzy predicate logics, Journal of Symbolic
Logic, 71(3) (2006), 863–880.
[22] L. Iturrioz, Łukasiewicz and symmetrical Heyting algebras, Zeitschrift für Mathematische
Logik und Grundlagen der Mathematik, 23 (1997), 131–136.
[23] R. Wójcicki, Lectures on propositional calculi, Ossolineum, Warsaw, 1984.