Gluts and Gaps

I can't think of any law deeper than the Law of Non-contradiction; I consider questioning it to be more fundamental than the question of God's existence.

Does rationality (any and all rationality) /presuppose/ that the Law of Non-Contradiction is always going to be there? Can there, alternatively, be "true contradictions"?: I'm referring to logic gluts (a proposition that is both true and false) and gaps (a proposition that is neither true nor false) -- or ought we consider the consideration of these nothing more than the fanciful experimentation of an unrestrained imagination?

(n.b. The common strategy to attack the Law of Non-Contradiction is via. a challenge to The Law of the Excluded Middle, which states that a proposition is either "a" or "not-a", but not "both" nor "neither".)

It's hard to think of many good examples (paraconsistent/non-Classical logic is a very new development, so is not yet furnished with a wealth of palatable metaphors), but self-referential sentences may serve:

"This sentence is a lie" might be an example of a gappy proposition. How legitimate of a challenge is the Liar's Paradox to Classical (Aristotelian) logic? Does it break the shiny, perfect toy that we call the Law of Non-Contradiction? Are we left only with tautologies like "A=A"?

If you want more context, please refer to:

https://plato.stanford.edu/entries/dialetheism/


(I've mentioned this in earlier threads ("Paraconsistency" and "Fuzzy Logic"), but I think the phraseology of this OP is different enough to warrant another treatment.)

Logic is fundamentally flawed! Gödel proved that any logical system necessarily includes inconsistencies, but the thrust of your argument is correct — "logical gluts" exist in the void of inconsistency that Gödel posits.

Not usually a responder to these high-order topics, but this one is close to the heart.

Have you read “Gödel, Escher, Bach” by Hofstadter?