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An inconsistent theory is not terribly interesting because it proves every proposition, so there is no way to separate what is true from what is false.

But some contradictions take more steps to prove than others. And if you limit the theory to propositions that are within a certain "distance" from the axioms, you may still produce interesting results without proving every proposition.

Or, perhaps the theory proves every proposition, but if you measure distance in logical steps, the theory contains "bubbles" of consistency within which there is meaningful structure.

Is there any theory like this?

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    $\begingroup$ Note that inconsistent theories can be quite interesting when we weaken the ambient logic - see e.g. "paraconsistency." $\endgroup$ Jun 6, 2018 at 2:19

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