Correct. And definitions are completely arbitrary. It is the premise part of the symmetry axiom -- that for any objects x and y,ifx=y....

The axiom is the conclusion --theny=x.

In any case, the implication is that certain self-referencing argument are not necessarily false. As I said, circular arguments are not formal fallacies, rather, a defect in argument. They are not necessarily false (as formal fallacies are). A circular argument is a defective argument for the simple reason that it does not say anything other than something is what it is -- which we all know to be true in the first place.

Then you are making a definist fallacy. What you should have said isthat belief is not necessarily proof.The reflexive and symmetry axioms are proof of themselves, are they not?

Clearly, your statement is false.

You are correct, of course that his argument is circular. You are incorrect to presume it is false simply because it is circular (reflexive and symmetry axioms).

If you intend to claim that his argument is false, then you need to show that his premise leads to a contradiction. Have you done that?

In fact, the opposite was already demonstrated to be false -- via the cosmological argument.