You are right that something I said sounded "preachy"; I thought so too, thought about changing it; but kept it rather than spend more time trying to figure out how to change it.
Quote: "we have never confirmed a clear case":
here you introduce a qualifier "clear".
I stand by the claim that it is true by definition that something cannot exist and not exist. If there is a LACK of clarity then the matter to be resolved is one of clarity and error margins. It does not alter the truth of the law of non-contradiction.
Any "contradictory" elements occuring in a not-fully-clarified scenario do not breach the law-of-non-contradiction, as those "contradictions" are pseudo and not real. Real contradictions are impossible by definition.
Page 700 of "LIGHT" by Ditchburn gives a clear explanation of wave-particle duality showing there is no mystery. Note that "frequency" is a characteristic of waves and particles; "mass" is frequency (as is "energy"). Waves and particles are considered to be "limiting theories" on energy and matter. High frequency wave quanta pack a punch like a particle (which itself is a high frequency wave bundle).
Whatever rules so-called paraconsistent logic has; the law of non-contradiction will still be implied somewhere even if not stated.
But (2) is not defeated! It cannot be by definition. Lack of clarity is lack of clarity; induction has nothing to do with the law of non-contradiction, it is impossible to exist and not exist! It is possible to get muddled though; or for an existent pattern to get diffused among other patterns so change its form of existence.
Things just ARE.
You are very open-minded and willing to face the activities of the logicians; but I contend that their attempts to circumvent the law of non-contradiction can be easily defeated!
Open-mindedness has nothing to fear, because by definition their attempts can only fail. Of course you cannot "prove" an ultimate explanation, as that would be contradictory as the "proof" would be more ultimate.
But I don't think you need to downgrade to "belief" status your acceptance of non-contradiction; I think you or I or anyone can defeat anyone who tries to refute that law (perhaps unless the logician one tackles gets so bogged-down in jargon that one cannot be bothered unravelling it all!).
Funny thing is; to defeat the logician one would have to show he was contradicting himself! He would then go round in circles saying he was allowed to do that. So he cannot be refuted. But he can die! That's why I said the earlier comment; if people want to fool themselves, it is they who do it, and they who bear the consequences. So the responsibility of dealing with that law is for each of us to face himself. But I don't mean to be preachy; its just that there is a responsibility for each one of us to get real and not kid ourselves, including those puzzling logicians!
Challenging post though. Thank you for that.
Hope I don't sound too terse or anything.
Sorry if I do.