There can be an internal explanation: the existence of the first cause is explained inherently if its existence is part of its essence. — A Christian Philosophy

To me, "essence" suggests a set of necessary and sufficient properties that uniquely identify an existing, individual object. Existence isn't a property; that would imply there are objects in the world that lack it - which is absurd. All objects in the world exist.

Perhaps you mean something else. If so, explain what you mean, and why anyone should accept such a metaphysical framework. I'm on the lookout for contrivances that are devised to rationalize a God to the exclusion of a purely natural first cause.

Logic has a reason for existing, as provided in the OP under section "Argument in defence of the PSR". — A Christian Philosophy

Is a logical necessity a sufficient reason or a Brute Fact.

I agree that:
1) The Unrestricted PSR states that everything has a sufficient reason
2) "Logic is part of the fabric of reality"
3) Logically A = A

However, what is the reason that A = A?
What is the reason for the existence of logic?
If logic has no reason, then logic is a Brute Fact.
Brute Facts are unexplainable and uncaused.
That logic is part of the fabric of reality is not sufficient reason why logic is part of the fabric of reality.
If, for example, A = A is a brute fact, then the Unrestricted PSR is no longer valid.

As regards your argument that "logic has a reason for existing"

I agree that
1) Truth means conformity with Reality
2) We discover Truth using Reason
3) Reason uses Deduction and Induction
4) An example of Deduction = i) the sun rises in the east, ii) therefore tomorrow the sun will rise in the east
5) An example of Induction = i) for the past 100 days the sun has risen in the east, ii) therefore the sun rises in the east.
6) If reason can find truth, then reason must mirror reality.

However, I don't agree that
1) "deduction is equivalent to the principle (or laws) of logic, in that deduction is based on an axiom (the sun rises in the east) that may or may not be true.
2) "induction is equivalent to the principle of sufficient reason", in that induction is assuming the regularity of nature, which may or may not be true, as pointed out by Hume.
3) "Reason finds truth", in that there is no logical necessity that either deduction or induction find the truth.

IE, we have no reason to think that logic has a reason for existing.

If logic has no reason for existing, then logic is a brute fact.

If logic is a brute fact, then the Unrestricted PSR is not valid.

Good catch Tim. I would put it slightly differently - there are missing steps/facts in Corvus' "logic":

P -> Q
If John is in Tokyo, then John is in Japan.
R
John is in Paris (not in Tokyo). <=== A fact from real life situation.
S
Paris is not in Japan <=== Another fact from real life situation.
R & S ->~Q
Therefore John is not in Japan.

P -> Q
R
S
R & S->~Q
Therefore ~Q — Corvus

P is irrelevant to getting ~Q. Of course this is all loosey-goosey and not formal 1st order logic

If you don't understand the point from all the examples I have give out with the explanations, then I don't see any point carrying on. I shall not waste my time or yours. I suggest you do the same. Good luck.

What you're not seeing - I don't know why - is that you're making two different arguments.

If john is in Tokyo, then John is in Japan. John is not in Tokyo. Maybe he is in Osaka or Yokohama.

But you're argument really is, If John is in Tokyo then John is in Japan. John is in Paris, therefore he is not in Japan. In this argument is the extra premise.

You can conclude John is not in Japan not because he is not in Tokyo, but because he is in Paris. — tim wood

I thought you wanted to stop discussing on this topic from your last post. I am surprised to see you keep replying.

The point here is, that the proof is about whether John is in Japan or not. (Q or not Q). It is not about whether John is in Tokyo or Osaka or anywhere in Japan. If John is in Tokyo, then he is in Japan was an assumption for the proof (Q or not Q). But the assertion from the reality was John is in Paris, which proves John is not in Japan.

This is such a simple logic, but you are worrying about whether John is in any other part of Japan, which is irrelevant for the proof.

Anyhow, this was a sideline thought for proving the PSR is not valid. It is not related directly to the OP. Hence we better stop at here. If you feel that this is a worthy of a separate OP, please go and start one. I don't think it is worth for a new OP with this topic, because it is such a simple and basic stuff.
But if you feel so, do so. Thank you for your feedback. Good luck.

I am unclear on what you mean by "natural" vs "super-natural". How do you define those two terms? — A Christian Philosophy

Roughly, natural explanations do not introduce anything outside the natural world. It rejects the idea that the world is contingent and requires a necessary cause, that is, a super or supra-natural cause that it above or beyond the limits of the natural world and on which the world is dependent.

Ok. It's official. You do not know what you're talking about. Too bad, because sometimes that kind of ignorance can be a block to learning. Modus ponens, Modus tollens, as basic to basic logic as 2+2=4 to arithmetic. Learn them.