On witnessed models in fuzzy logic II

First the expansion of the Łukasiewicz (propositional and predicate) logic by the unary connectives of dividing by any natural number (Rational Łukasiewicz logic) is studied; it is shown that in the predicate case the expansion is conservative w. r. t. witnessed standard 1-tautologies. This result is used to prove that the set of witnessed standard 1-tautologies of the predicate product logic is Π2-hard.