On the failure of standard completeness in ΠMTL for infinite theories

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 ΠMTL (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.