The Full Import of Paradoxes
It appears that, from a cursory reading of the Wikipedia article on
fuzzy logic, that there are some/many aspects of reality that are spectral in character in terms of how we interact with them. The spectrum in question is divided up into manageable chunks, the boundaries between them are not precise values. This vagueness is what fuzzy logic was developed to tackle. I suppose we could say that fuzzy logic is
grey zone logic.
As for
the law of noncontradiction (the LNC) and fuzzy logic, my view is that the former holds in the latter. Yes that truth value can be somewhere in between 0 (false) and 1 (true) but it can never be 1 & 0 at the same time, nor is it that a truth value is (say) 0.7 and not 0.7.
Paraconsistent logics, on the other hand, are, as you rightly pointed out, systems in which a proposition has the combined truth value of 0 and 1 (at the same time and in the same respect) i.e. that proposition is a bona fide contradiction.
So, as per my analysis, fuzzy logic isn't a paraconsitent logic. I could be wrong of course, I do hope I'm not though.
Your
BothAnd system feels more like paraconsistent logic than fuzzy logic to me. Maybe it's the words "both" and "and" which indicates you want to reconcile thesis (yes/1)
AND antithesis (no/0) by approving
BOTH.
It could be that I'm getting mixed up between
the principle of bivalence and
the law of noncontradiction.