The class of weakly implicative fuzzy logic (WIFL) was introduced to encompass the existing class of the so-called fuzzy logics in one general framework. Roughly speaking, WIFL is the class of propositional logics (understood as a consequence relation) complete w.r.t. linearly ordered logical matrices of truth values.
In this talk we concentrate on first-order variant of logics in WIFL. In particular we deal with problems appearing in their very basic aspects. As WIFL is very broad class we run into problems with proper formulating of an axiomatic system of these predicate logics and then with proving the completeness theorem.
The former one lies in the fact that the connective of max-disjunction plays a crucial role in the predicate fuzzy logics known from the literature. However, in WIFL there are logics without this connective in the language. The later one is caused by fact that by the transition to the first-order we lose (in some cases) some important proof-theoretic meta-theorems of the propositional logics necessary for the completeness proof.
We try to overcome both problems and provide some partial results.
Generally, we can say that for good-behaving propositional logics
(with some form of deduction theorem, definable lattice structure,
etc.) the transition to the first-order is rather smooth and
painless and so we concentrate on those problematic ones.