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Rational Pavelka predicate logic is a conservative extension of Łukasiewicz predicate logic. (English) Zbl 0971.03025
Rational Pavelka logic extends Łukasiewicz infinitely valued logic by adding truth constants \(\overline{r}\) for rationals in \([0,1]\). It is shown that this is a conservative extension which solves an open problem raised by the first author [Metamathematics of fuzzy logic, Kluwer, Dordrecht (1998; Zbl 0937.03030)]. The crux is proving the existence of the sups and infs giving the truth values of quantified formulae involving the new constants. It is noted that this would be unnecessary if one used a theorem of L. P. Belluce and C. C. Chang [J. Symb. Log. 28, 43-50 (1963; Zbl 0121.01203)] stating that every theorem of Łukasiewicz predicate logic is true in all evaluations for which its truth value is defined. However the authors present a counterexample that shows that this theorem is false, not only for Łukasiewicz logic, but also for product logic. It is shown as a consequence of the conservative extension theorem that results on partial truth can be expressed in Łukasiewicz logic itself, without the extension to rational Pavelka logic.

MSC:
03B50 Many-valued logic
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