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A complete many-valued logic with product-conjunction. (English) Zbl 0848.03005
Studies in the field of fuzzy sets related to triangular norms have shown that the infinitely many-valued logic over the real unit interval with 1 as the only designated truth degree and a product-based conjunction connective has not been discussed much till now, despite some representation theorems of $$t$$-norms which refer to the product as a kind of basic $$t$$-norm.
Choosing the product conjunction, a corresponding implication connective via residuation (i.e., both of them as an adjoint pair), and defining negation formally from implication and the truth degree constant 0 like in intuitionistic logic, the authors constitute the system they investigate.
The completeness proof is given via algebraic studies essentially like the corresponding proof for Łukasiewicz’s infinite-valued logic, but now introducing and investigating product algebras for this product logic instead of the MV-algebras for the Łukasiewicz case.

##### MSC:
 03B50 Many-valued logic 03B52 Fuzzy logic; logic of vagueness 03G25 Other algebras related to logic
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##### References:
  Alsina, C., Trillas, E., Valverde L.: On some logical connectives for fuzzy set theory. J. Math. Anal. Appl.93, 15–26 (1983) · Zbl 0522.03012  Balbes, R., Dwinger, P.: Distributive lattices. Missouri: Univ. Missouri Press 1974 · Zbl 0321.06012  Birkhoff, G.: Lattice theory, vol. 25. New York: 1948, Amer. Math. Soc. Colloquium Publ. · Zbl 0033.10103  Bouchon, B.: Fuzzy inferences and conditional possibility distributions Fuzzy Sets Syst.23, 33–41 (1978) · Zbl 0633.68100  Fuchs, L.: Partially ordered algebraical systems. New York: Pergamon Press 1963 · Zbl 0137.02001  Gödel, K.: Zum intuitionistischen Aussagenkalkül. Anz. Akad. Wissensch. Wien, Math.-naturwissensch. Klasse69, 65–66 (1932). Erg. math. Kolloqu.4, 40 (1933) · JFM 58.1001.03  Gottwald, S.: Mehrwertige Logik. Berlin: Akademie-Verlag 1988  Grätzer, G.: Universal Algebra. Berlin Heidelberg New York: Springer 1979 · Zbl 0412.08001  Gurevich, Y., Kokorin, A.I.: Universal equivalence of ordered Abelian groups (in Russian). Algebra i Logika2.1, 37–39 (1963)  Hájek, P.: Fuzzy logic and arithmetical hierarchy. Fuzzy Sets Syst.73, 359–363 (1995) · Zbl 0857.03011  Hájek, P.: Fuzzy logic and arithmetical hierarchy, vol. II. Submitted · Zbl 0857.03011  Hájek, P., Havránek, T., Jiroušek, R.: Uncertain information processing in expert systems. CRC Press 1992  Hájek, P., Valdés, J.J.: Algebraic foundations of uncertainty processing in rule-based expert systems I. Comp. Artif. Intell.9, 325–334 (1990)  Höhle, U.: Commutative residuated monoids. In: Höhle, U., Klement, P., (eds) Non-classical logics and their applications to fuzzy subsets (A handbook of the mathematical foundations of the fuzzy set theory). Dordrecht. Kluwer 1995  Ling, C.H.: Representation of associative functions Publ. Math. Debrecen12, 182–212 (1965)  Łukasiewicz, J.: Selected works. Amsterdam: North-Holland 1970 · Zbl 0212.00902  Novák, V.: On the syntactico-semantical completeness of first-order fuzzy logic I, II Kybernetika26, 47–26, 134–152 (1990) · Zbl 0705.03009  Paris, J.B.: The uncertain reasoner’s companion – a mathematical perspective. Cambridge: Cambridge University Press 1994 · Zbl 0838.68104  Pavelka, J.: On fuzzy logic I, II, III. Z. Math. Logik Grundl. Math.25, 45–52, 119–134, 447–464 (1979) · Zbl 0435.03020  Rose, A., Rosser, J.B.: Fragments of many-valued statement calculi. Trans. A.M.S.87, 1–53 (1958) · Zbl 0085.24303  Scarpelini, B.: Die Nichtaxiomatisierbarkeit des unendlichwertigen Prädikatenkalküls von łukasiewicz. J. Symb. Log.27, 159–170 (1962) · Zbl 0112.24503  Schweizer, B., Sklar, A.: Associative functions and abstract semi-groups. Publ. Math. Debrecen10, 69–81 (1963) · Zbl 0119.14001  Schweizer, B., Sklar, A.: Probabilistic metric spaces. Amsterdam: North Holland 1983 · Zbl 0546.60010  Takeuti, G., Titani S.: Fuzzy Logic and fuzzy set theory. Anal. Math. Logic32, 1–32 (1992) · Zbl 0786.03039  Zadeh, L.: Fuzzy logic. IEEE Comput.1, 83 (1988) · Zbl 0634.03020
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