On real and rational completeness of some predicate fuzzy logics with truth-constants

Moreover the term predicate fuzzy logic L∀ will denote the predicate calculus obtained from the propositional core fuzzy logic L by adding the “classical” quantifiers (∀ and ∃) together with the same axioms and rule of generalization used by Hàjek in his book [7] when defining BL∀. On the other hand, given a left-continuous t-norm ∗, if L∗ denotes the (propositional) fuzzy logic which is standard complete with respect to the MTL-algebra [0, 1]∗ defined by ∗ and its residuum, L∗(C) (resp. L∗∀(C)) will denote the core (predicate) fuzzy logic which is the expansion of L∗ (resp. of L∗∀) with a truth-constant for each element of a countable subalgebra C of [0, 1]∗ and the corresponding book-keeping axioms (see also Hájek’s book). The aim of this paper is to investigate at large conservativeness results for the expanded logics L∗(C) and L∗∀(C), as well as real and rational completeness results in the sense defined in [2]. To this end, we take advantage of a number of already available results in the literature, namely