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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 \(B\).

MSC:
03B52 Fuzzy logic; logic of vagueness
03B50 Many-valued logic
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