No, fuzzy logic is not what I'm after. Nor even the truth or falsity of single statements. I'm interested in the degree to which an entire inconsistent system of logic is inconsistent.

Here's an equivalent-ish problem:

Given a programming language, what's the smallest program that's an infinite loop?

Some languages are designed to never let you write infinite loops (every program halts in them), but maybe there's a mistake and they accidentally let some non-halting programs slip through.

Seems like it wouldn't be Turing complete if you couldn't make an infinite loop

Yep. Still, it turns out there's a lot you can do with non-Turing-complete languages, and it's nice knowing that a program will definitely finish.