I am currently in a Logic class and was assigned this question.
In class we define complete as a set of connectives that could generate every truth function.
My answer to this question is no, the set of logical connectives is not complete in fuzzy logic. My reasoning behind this answer is the fact that .5 is .5-preserving. You can never get anything different from a .5. More clearly,
.5 and .5 = .5
.5 or .5 = .5
Not.5 = .5
.5 implies .5 = .5
.5 if and only if .5 = .5
Is this justification enough?
I'm assuming that fuzzy logic means some sort of infinite-valued logic. So, I don't know what the vast majority of outputs for the connectives you have for all inputs in the infinite truth set. George J. Klir and Bo Yuan in Fuzzy Sets and Fuzzy Logic write
"The insufficiency of any single infinite-valued logic (and therefore the desirability of a variety of these logics) is connected with the notion of a complete set of logic primitives It is known that there exists no finite complete set of logic primitives for any one infinite-valued logic. Hence, using a finite set of primitives that defines an infinte-valued logic, we can obtain only a subset of all the logic functions of the given primary logic variables."
This might get contrasted with Lukasiewicz-Wajsberg-Slupecki three-valued logic which has functionally complete semantics, and a complete and independent axiom set.