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On the failure of standard completeness in \(\Pi\)MTL for infinite theories. (English) Zbl 1117.03032

Summary: It is well-known that Hájek’s basic fuzzy logic (BL), Łukasiewicz logic, and product logic are not strongly standard complete. On the other hand, Esteva and Godo’s monoidal t-norm-based logic (MTL) and its involutive extension IMTL are strongly standard complete. In this paper we show that \(\Pi\)MTL (an extension of MTL by the axioms characteristic of product logic) does not enjoy the strong standard completeness theorem like BL, Łukasiewicz, and product logic.

MSC:

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