EQ-algebras. (English) Zbl 1184.03067

Novák, Vilém; De Baets, Bernard

Fuzzy Sets Syst. 160, No. 20, 2956-2978 (2009).

In this paper, the authors introduce the concept of an EQ-algebra, which should become the algebra of truth values for a higher-order fuzzy logic (a fuzzy type theory, FTT). The motivation stems from the fact that, until now, the truth values in FTT were assumed to form either an MTL-, BL-, or MV-algebra, all of them being special kinds of residuated lattices in which the basic operations are the monoidal operation (multiplication) and its residuum.
The main operation of an EQ-algebra is a fuzzy equality \(\sim\) (a natural interpretation of the main connective in fuzzy type theory) accompanied by the binary operations of meet and multiplication (\(\otimes\)). The essential difference between residuated lattices and EQ-algebras lies in the definition of implication \(\rightarrow\). Unlike residuated lattices, where \(\rightarrow\) is joined with \(\otimes\) through the adjointness property, in EQ-algebras \(\rightarrow\) is defined directly from the fuzzy equality \(\sim\). Hence, the adjointness property, which strictly couples \(\rightarrow\) and \(\otimes\), is relaxed. This has as consequence that the multiplication can be non-commutative without forcing two kinds of implication. In this paper, the authors study basic properties of commutative EQ-algebras (i.e. the multiplication \(\otimes\) is commutative) and present several special kinds of EQ-algebras.
The relation between EQ-algebras and residuated lattices is quite intricate and it seems that the former open the door to another look at the latter. When considering implication only, it can be shown that the corresponding reducts of EQ-algebras are BCK-algebras, and so, residuated lattices are “hidden” inside. On the other hand, EQ-algebras form a variety and they are not equivalent with residuated lattices; in fact, EQ-algebras generalize residuated lattices because they relax the tie between multiplication and residuation (i.e. between conjunction and implication in logic).
The EQ-algebras open the door also to introducing a class of EQ-logics, which might cast a different light on fuzzy logics studied until now only from the point of view of the properties of implication.

Reviewer: Florentina Chirteş (Craiova)

MSC:

03G25	Other algebras related to logic
03B52	Fuzzy logic; logic of vagueness

Keywords:

EQ-algebra; fuzzy equality; fuzzy logic; fuzzy type theory; higher-order fuzzy logic; residuated lattice

Software:

ETPS

References:

[1]	Andrews, P., An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof (2002), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 1002.03002
[2]	Blyth, T., Lattices and Ordered Algebraic Structures (2005), Springer: Springer London · Zbl 1073.06001
[3]	Bodenhofer, U., A similarity-based generalization of fuzzy orderings preserving the classical axioms, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 8, 593-610 (2000) · Zbl 1113.03333
[4]	Castañeda, H., Leibniz’s syllogistico-propositional calculus, Notre Dame Journal of Formal Logic, XVII, 4, 338-384 (1976) · Zbl 0314.02004
[5]	Cignoli, R. L.O.; D’ottaviano, I. M.L.; Mundici, D., Algebraic Foundations of Many-valued Reasoning (2000), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0937.06009
[6]	Cintula, P., Advances in the \(Ł \Pi\) and \(Ł \Pi \frac{1}{2}\) logics, Archive for Mathematical Logic, 42, 449-468 (2003) · Zbl 1026.03017
[7]	Cintula, P.; Hájek, P.; Horčík, R., Formal systems of fuzzy logic and their fragments, Annals of Pure and Applied Logic, 150, 40-65 (2007) · Zbl 1140.03010
[10]	Esteva, F.; Godo, L., Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy Sets and Systems, 124, 271-288 (2001) · Zbl 0994.03017
[11]	Esteva, F.; Godo, L.; Montagna, F., The Ł \(\Pi\) and Ł \(\Pi \frac{1}{2}\) logics: two complete fuzzy systems joining Łukasiewicz and product logics, Archive for Mathematical Logic, 40, 39-67 (2001) · Zbl 0966.03022
[12]	Galatos, N.; Jipsen, P.; Kowalski, T.; Ono, H., Residuated Lattices. An Algebraic Glimpse At Substructural Logics (2007), Elsevier: Elsevier Amsterdam · Zbl 1171.03001
[13]	Gerla, B.; Leuştean, I., Similarity in MV-algebras, Fundamenta Informaticae, 69, 287-300 (2006) · Zbl 1098.06008
[14]	Gispert, J.; Torrens, A., Bounded BCK-algebras and their generated variety, Mathematical Logic Quarterly, 53, 206-213 (2007) · Zbl 1123.06010
[15]	Goguen, J. A., The logic of inexact concepts, Synthese, 19, 325-373 (1968-69) · Zbl 0184.00903
[16]	Hájek, P., Metamathematics of Fuzzy Logic (1998), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0937.03030
[17]	Hájek, P., Fuzzy logics with noncommutative conjunctions, Journal of Logic and Computation, 13, 469-479 (2003) · Zbl 1036.03018
[18]	Hájek, P., What is mathematical fuzzy logic, Fuzzy Sets and Systems, 157, 597-603 (2006) · Zbl 1108.03028
[19]	Henkin, L., A theory of propositional types, Fundamenta Mathematicae, 52, 323-344 (1963) · Zbl 0127.00609
[20]	Höhle, U., Quotients with respect to similarity relations, Fuzzy Sets and Systems, 27, 31-34 (1988) · Zbl 0666.18002
[21]	Höhle, U., Many-valued equalities, singletons and fuzzy partitions, Soft Computing, 2, 3, 134-140 (1998)
[22]	Höhle, U., Fuzzy sets and sheaves. Part I: basic concepts, Fuzzy Sets and Systems, 158, 11, 1143-1174 (2007) · Zbl 1111.18003
[23]	Iseki, K., An introduction of the theory of BCK-algebras, Mathematica Japonica, 23, 1-26 (1978) · Zbl 0385.03051
[24]	Kabzinński, J.; Wroński, A., On equivalential algebras, (Proc. Internat. Symp. Multiple-Valued Logic (1975), Indiana University: Indiana University Bloomington, USA)
[27]	Novák, V., On fuzzy type theory, Fuzzy Sets and Systems, 149, 235-273 (2005) · Zbl 1068.03019
[29]	Novák, V., Which logic is the real fuzzy logic?, Fuzzy Sets and Systems, 157, 635-641 (2006) · Zbl 1100.03013
[30]	Novák, V., EQ-algebras in progress, (Castillo, O., Theoretical Advances and Applications of Fuzzy Logic and Soft Computing (2007), Springer: Springer Berlin), 876-884
[33]	Novák, V.; Perfilieva, I.; Močkoř, J., Mathematical Principles of Fuzzy Logic (1999), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0940.03028
[34]	Pavelka, J., On fuzzy logic I, II, III, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 25, 45-52 (1979), 119-134, 447-464 · Zbl 0435.03020
[35]	Ramsey, F., The foundations of mathematics, Proceedings of the London Mathematical Society, 25, 338-384 (1926) · JFM 52.0046.01
[36]	Slomczyńska, K., Equivalential algebras. Part I: representation, Algebra Universalis, 35, 524-547 (1996) · Zbl 0855.03039
[37]	Slomczyńska, K., Equivalential algebras. Part II: independence, Algebra Universalis, 37, 426-439 (1997) · Zbl 0903.03041
[38]	Slomczyńska, K., Free spectra of linear equivalential algebras, The Journal of Symbolic Logic, 70, 1341-1358 (2005) · Zbl 1106.08004
[39]	Tourlakis, G., Mathematical Logic (2008), Wiley: Wiley New York · Zbl 1155.03001