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.