Hello Aurino,
I'm glad that you decided to go forward with your reasoning. I often feel that you hold out as a form of a game  which can be frustrating since I don't know if I should always participate in that 'game'.
***H: the axiom as needfully true, but that would be demonstrate a proof of an axiom (which there is none). A: While the sentence is semantically valid, I maintain that it's not logical. The crux of the problem lies in the meaning of the word "proof", which in my opinion you are failing to fully take into account. It's my contention that, whatever it is that "proof" means, the word can only have meaning if the basic axioms of logic are accepted. That is, without logic there is no proof of anything!***
I agree, in a recent response to Dick I mentioned this very thing by saying that proof is contingent on a formal system of logic.
***So it's not that you can't demonstrate a proof of an axiom of logic, it's rather than any such exercise would be completely futile and meaningless. The axiom is not needfully true, it's needed for the concept of truth to have any meaning.***
Well, we need a category of 'is true' for the axiom to have any meaning. For example, (P=P is true) is the identity axiom and the axiom requires that 'is true' is part of the axiom. As Paul and I discussed, the 'is true' is meaningless from standpoint of the formal system. Truth is not being defined in terms of either a physical or abstract interpretation, it is merely an undefined term that applies for all usages in the formal system where P=P.
When we talk about reality (as Alan as mentioned), then we are seeking an interpretation for the LNC axiom. The term 'is true' must have some form of meaning as it relates to the world. Therefore, you need a concept of truth (or, better, an interpretation of truth) for the axiom. The most popular interpretation of truth involves identity (Tarski's interpretation of truth). This interpretation can be stated as:
(*) "snow is white" if and only if snow is white
There is an identity relationship with proposition P ("snow is white") and the fact of matter P* (snow is white).
So, in formal systems the undefined term 'is true' is used to define the indentity axiom, and in Tarski's interpretation of truth the identity axiom is used to give a physical interpretation of truth.
We can see the order of our reasoning as follows:
(1) Define a formal system of logic
(2) Introduce an undefined term 'is true' which holds no meaning but will be used to set a value (truth value) for a defined [identity] relationship.
(3) Define the identity axiom.
(4) Use the identity axiom to interpret a physical definition of truth as it applies to the world.
(5) Interpret physical truth in terms of the identity axiom.
(C) The term 'is true' in (2) takes on physical meaning from (5)
The conclusion (C) isn't circular since the 'is true' in (2) is undefined, whereas the 'is true' in (C) is a physical interpretation. We should only be precise what we mean by 'is true' (either as a formal system undefined term or as the result of an interpretation).
Now, if we consider your text again:
"The axiom is not needfully true, it's needed for the concept of truth to have any meaning."
We see that the formal axiom itself does not need a concept [i.e., interpretation] of truth, it merely needs to have a designated undefined term 'is true'.
***For instance, I said the statement A = A is valid in any conceivable universe. Notice, I didn't say "any universe", which is quite a different thing. There might in fact be universes in which A = A is false. There is one problem with those universes, though  they can't be conceived! Most important of all, since we can conceive the universe in which we exist, we can say with all certainty that A = A must necessarily be true in our universe. Taking it a bit further, the fact that we think logically is proof that logic is a valid means of thinking, and no other proof is required. But of course there are other means of thinking.***
When we are talking strictly about logic as a formal system, the LNC axiom "(P & P) is true" cannot be false by definition. We are talking in terms of a formal system where we define the rules and where we set a value for a certain relationship that exists (in this case, (P & P).
However, the problem is with the interpretation of truth. The interpretation of LNC is "(P & P)" is true" is subject to error. Maybe we cannot conceive of a world where it could be in error, but that is beside the point. There may be a world (or aspect of reality) where (4) and (5) is either false or meaningless. For example,
(I) "The basic constituent of matter is a particle" if and only if the basic constituent of matter is a particle.
The proposition (I) may be meaningless with respect to the way reality is. For example, the following may actually be the case:
(I*) "The basic constituent of matter is a particle and wave" if and only if the basic constituent of matter is a particle and wave.
In the case of (I*), the LNC does not hold. An object might be both P and not P. No violation of formal logic has occurred since the interpretation (I) was not valid. Nothing prevents (I*) from being true.
***I'm not sure this can be easily understood. Probably not. The subject is full of subtleties and unfortunately I have no time to further elaborate. That's all I can offer, I hope it helps explain my position. And if it doesn't, that's OK too.***
I'm here if you want to discuss it further.
Warm regards, Harv
