Thanks for replies, Harv.
About arguments generally:
(Note: Any argument you bring I'm aware might be not necessarily your own view but a representation of a particular logician's view; regardless I reply to the argument itself- I do not constrain you to actually holding to whatever logic you try out here!)
Any argument you bring can be questioned in the following way: What is the question at issue? To what grounds does the argument appeal?
What backs the argument?
Is the conclusion warranted by that backing?
Does it address the question at issue?
Now, some of your arguments appeal to the idea of regarding something as "a formal system".
But appealing to that idea already involves APPLYING the Law Of Non-contradiction. You would not have either "formality", or "system", if everything "kept contradicting" - I use quotation marks because I strictly cannot describe such a concept- violation of LNC is so impossible it cannot be described or conceived.
Of course you might string some words together that LOOK like LNC is in trouble; but that is just an illusion, a fantasy that cannot be conceived.
I can write: moon/cheese/martian/drink/universe but this doesn't mean I'm saying anything. Just because you can make something LOOK like something doesn't mean LNC is actually in trouble.
"We can look only in our conceptual framework to our own perspective": this makes an assumption that "WE cannot obtain an unlimited conceptual framework". The assumption is not proven.
I hold that by its very nature, the conceptual framework of honesty (paying attention to Existence) is unlimited; the only limit is our very being (created by Existence). That is: we ARE the limit.
We can know the ultimate reality. To know thyself, is to know the Existence of one-self ; to detect Existence through one's own Existence. In other words, in Heaven you will meet "YOU"; and be aware of "God (Existence) creates You".
I agree that we may guess about whether the apparant "order" in the universe is based on more fundamental orders not yet observed (eg. role of early-expansion, dark-matter, etc. See April issue "Astronomy" magazine).
But the question at issue is being muddled between this question about current scientific models and "how basic are they?" and a different question "a logic that determines what really exists?".
What really exists is not determined by logic; it IS logic. What really exists is the logic of existence. This is not circular, there are no more powerful grounds to which you can appeal than this. Even "freedom" has to exist to be real freedom. And on existing, an item has freedom (If a cat didn't exist, it would not have the freedom of being "cat" and doing "cat stuff".)
Of course if I said the ultimate reality of a thing is the ultimate reality of a thing; I didn't say anything new. There is a false assumption that one can always say "this" is "that".
But ultimately you run out of "that"s. Things just ARE. That is why you cannot treat LNC as just another axiom system. You have to apply LNC to address the issue at all. It permeates everything.
English language doesn't constrain the nature of the Universe; but Existence does. Because Existence is freedom-constraint; to BE is to have a boundary that gives a freedom.
The constraint is the freedom of being. God gives you so much "elbow room" He creates the elbows in the first place!
Your example of "Do you have?" and "Do you not have?" questions are about two categories, two perspectives. There is no contradiction; just a distinction not being made. They say in philosophy: when you meet a contradiction, make a distinction.
When you change the question from "What prevents something existing and not existing?" to what prevents a correct answer of something existing being correctly answered with something not existing?" you have completely changed the issue.
Ontologically, LNC is absolute. But DESCRIPTIVELY, that is in terms of making "models", of course you can get pseudo-contradictory answers wherever the modelling confuses DIFFERENT aspects of the one entity. The "contradiction" appears in the error-margin, in the generalisation, in the multiple-category or multiple-perspective inherent in the generalisation modelling upon which the questions are based.
The LNC is not JUST one axiom among many. Without it, you can have no axioms. It is far more than an "axiom". You cannot raise "axiom" as a higher power than LNC. All the power comes from LNC.
If "those logicians" claim they can deny LNC, they are kidding themselves. As I said, they can never refute my refutation; because they cannot appeal that I am contradicting myself. And anything they say can be contradicted; so they are so easy to refute. Yet they can just contradict that, and so avoid refutation. So they are left alone chasing their tails in delusion!
I am not confusing LNC with the other axioms; the Peano axioms sounds like what you are referring to, they come from LNC.
Identity is a myth. There is no such thing as identity. Things ARE. They are not things. They just ARE. "Identity" talk is "double defining". Note the quote: "When you mean "yes", say "yes"; when you mean "no", say "no"; all else comes from the evil one". (Quote goes something like that).
Since "identity logic" is a myth, Liebniz's law collapses into the field of pattern-matching of partially differentiated patterns. Whole patterns do not have identity; they ARE. To say they are themselves is double defining.
I am not taking a formal system with axioms and saying it is unseparable from reality. I am saying that calling LNC a formal system with axioms is invalid; it is the very means by which such systems are constructed. It cannot be reduced to such a system, it always occupies territory outside any such system, permeating everything.
Of course there is no chance of being wrong, when you say: that which defines right and wrong, and being itself. It is outside the system of things that can be right and wrong. It is beyond those categories. (Quote: "For my yoke is easy, my burden light").
I am not using the formal system to prove the case (I know what is wrong with that). I'm saying that "proof" as Aurino noted, is created by LNC. The assumption that something "must be proved" ceases to apply with LNC; LNC IS proof itself. If you ask for proof of "proof" itself, "proof" is your answer.
That is not circular in the usual sense; it is the beginning and the ending, the alpha and omega. It is valid circularity, if you like.
It is "validity" itself.
Logic of LNC does not limit "what is out there"; it liberates what is out there; it IS what IS out there! Whatever is out there: MUST BE. Or it isn't out there!
The formal system you describe that has a "obtains" bucket standing in as an "is true" bucket, is clearly a logic of approximation. It still invokes LNC; but allows pseudo-contradictions because it bundles together different categories and perspectives as singular.
A formal system based on "choice" also contains numerous pseudo-contradictions without difficulty (e.g. you cannot make all possible moves at a given state of play on a Chess board within the rules; only one move is allowed. The virtual moves however do not contradict: they are in the category: virtual. Devising a logic that muddles the distinction between virtual moves and real moves would be a logic of multiplicity-of-perspectives; contradictions between "real/virtual" moves are solved by re-distinguishing their "real" and "virtual" components. Waves are like virtual moves, particles are like real moves.
When you conserve a pattern of any kind, you appeal to LNC even if not stated. To state "snow is white and grey" a second time is to conserve this pattern (and require it hasn't changed into some contradictory pattern already when you next viewed it).
In fact, to even deal with it once, requires that it doesn't keep switching into something else.
Of course, if it is a pattern of options; what is conserved is whatever has those options. Or in Dick's generalised system; what is conserved is Existence itself- that something (unknown) IS.
Saying that P and not-P are not necessarily false is saying that they are not necessarily singular. That they might be a muddling of categories. This does not deny LNC; it is a pseudo-evasion of LNC. P and not-P is, because of LNC, a generality and not specific. Specific P: IS.
People who misleadingly label LNC as an axiom like "law of bivalence" are being rather silly; anyone can pseudo-dodge LNC by talking in generalisations and muddling distinctions.
BEING is what makes formal systems possible; ultimately everything is reality; whether formal systems are allegedly concerned with reality or not; they EXIST as thoughts etc. and the LNC transcends them and cannot be overruled by them.
Your last comment sums up the whole deal: "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 (I) call it). Otherwise it is easy to confuse the two."
Formal systems are by nature descriptive. They are approximations. The formal words "Jones's car" describes Jones's car, but reality just IS; his car just IS.
Another thing: words are unduly limiting and approximate. There comes a point where one is faced with having to JUST BE; and reality is perfectly obvious. i.e., verbal conversation places difficulties on philosophical debate. Look at the stars, and no words needed- sort of thing.