***He said "may actually BE"; note this word "BE". As soon as you apply the verb "to BE": you have lost the case. LNC applies whenever the verb "to be" applies.***
Not necessarily. Logic that rejects the LNC hasn't lost the notion of something 'obtains'. Again, the formal system without LNC has an undefined notion of 'is true'. It simply means a category (or bucket if you will) that says that whatever satisfies the criteria of that axiom formula will be thrown into that bucket ('is true' bucket). For something 'TO BE' the case, merely means that the formal system that we are interpreting against reality cannot fail in that interpretation. For example, interpreting reality using the LNC we might see that it fails (e.g., particle-wave duality).
***Harv: "Nothing prevents (I*) from being true."
"BEING true"? Whatever pattern-matching (e.g. Tarski pattern matching) you use to give the relationship that is your meaning of "truth";***
We aren't throwing out all logic axioms when we disregard the LNC as a true interpretation. We can still have notions of Tarskian truth even without the LNC. For example, Tarski's approach to truth is:
(T) "Snow is white" is true if and only if snow is white
Notice that (T) can modified without a reference to LNC:
(T*) "Snow is white and gray" is true if and only if snow is white and gray
(T*) shows that Tarski's interpretation of truth can still apply. There needs to exist an isomorphic relationship (or some identity) between "snow is white and gray" and snow is white and gray.
***what prevents (I*) from being true is that (I*) is prohibited from BEING anything.***
The denial of LNC does not prohibit (I*) from being either true or false. It merely denies that (P and not-P) are not necessarily false. Again, the LNC does not prevent two-valued logic of something being either true or false. There is another logic axiom called the law or principle of bivalence that states that truth values can only be true or false, but not neither and not both (which is a principle that is also rejected by some logicians and philosophers, such as Michael Dummett - one of the most well-known philosopher today). Antirealism stems often from the denial of this principle.
***As (I*) negates BEING, (I*) cannot BE true, or BE at all. (I*) is (I*), isolated from all that exists in its non-existence, you might say.***
I'm not sure what you mean by BEING in terms of a pure formal system (i.e., a formal system is not concerned with reality per se). (I*) only negates ((P and not-P) is false). This is only a negation of the law of non-contradiction (LNC). That's it.
We should probably talk about the formal system and its axioms first, and then seek to apply an interpretation of these systems to reality (BEING as you call it). Otherwise it is easy to confuse the two.
Warm regards, Harv