A note on the first-order logic of complete BL-chains. (English) Zbl 1152.03019

The relationship of the set of predicate tautologies of all BL-chains and the set of predicate formulas valid in all standard BL-algebras (i.e., the set of all standard tautologies) is discussed. This paper is a continuation of the papers of F. Montagna and L. Sacchetti [ibid. 49, No. 6, 629–641 (2003; Zbl 1035.03010)] and [ibid. 50, No. 1, 104–107 (2004; Zbl 1039.03013)], and its main result shows that a coomplete BL-chain \(B\) satisfies all standard BL-tautologies if and only if for any transfinite sequence \((b_i: i\in I)\) in \(B\), there holds \(\bigwedge_{i\in I}(b^2_i)= (\bigwedge_{i\in I} b_i)^2\). Another equivalent condition is this one: the formula
\[ \forall x(\varphi(x)\&\varphi(x))\to ((\forall x\varphi(x))\&(\forall x\varphi(x))) \]
is valid in \(B\).


03B52 Fuzzy logic; logic of vagueness
03B50 Many-valued logic
Full Text: DOI


[1] Aglianó, Varieties of basic algebras I: general properties, J. Pure Appl. Algebra 181 pp 105– (2003)
[2] A. Glass, Partially Ordered Groups. Series in Algebra 7 (World Scientific, 1999). · Zbl 0933.06010
[3] P. Hájek, Metamathematics of Fuzzy Logic (Kluwer, 1998). · Zbl 0937.03030
[4] Hájek, Basic fuzzy logic and BL-algebras, Soft Computing 2 pp 124– (1998) · Zbl 05469956
[5] Hájek, Fuzzy logic and arithmetical hierarchy III, Studia Logica 68 pp 129– (2001) · Zbl 0988.03042
[6] Hájek, Basic fuzzy logic and BL-algebras II, Soft Computing 7 pp 179– (2003) · Zbl 1018.03021
[7] Hájek, Arithmetical complexity and fuzzy predicate logics: a survey, Soft Computing 30 pp 1– (2005) · Zbl 1093.03012
[8] Hájek, On theories and models in fuzzy predicate models, J. Symbolic Logic 71 pp 863– (2006)
[9] Montagna, Three complexity problems in quantified fuzzy logic, Studia Logica 68 pp 143– (2001) · Zbl 0985.03014
[10] Montagna, Kripke-style semantics for many-valued logic, Math. Log. Quart. 49 pp 629– (2003) · Zbl 1035.03010
[11] Montagna, Corrigendum to ”Kripke-style semantics for many-valued logic”, Math. Log. Quart. 50 pp 104– (2004)
[12] Mundici, Interpretations of AF C *-algebras in Łukasiewicz sentential calculus, J. Functional Analysis 65 pp 15– (1986) · Zbl 0597.46059
[13] M. E. Ragaz, Arithmetische Klassifikation von Formelmengen der unendlichwertigen Logik. Ph. D. thesis, ETH Zürich (1981). · Zbl 0516.03011
[14] Ward, Residuated lattices, Trans. Amer. Math. Soc. 45 pp 335– (1939)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.