Earlier quoted context omitted.
I pretty much told him the same: That we declared such statements to be true merely for convenience - it makes the proofs shorter. And that none of the theorems he would normally deal with would have a different "outcome" if we don't accept such statements - they'd still be valid theorems. But he won't be convinced until he formally studies logic. No idea when that will happen.
Well I have "formally" studied formal logic and he's in for a disappointment. Formal logic is a game, especially boolean logic, which has no grounding in truth, the world, or anything. Sometimes it's a useful game, but there's no inherent meaning behind it. Tell your friend that implication works that way because that's what we've assumed, nothing more, nothing less.
Thing is: He knows this. He's not a "beginner" who wants to learn a bit more. He just hasn't spent enough time pondering vacuously true statements, and is assuming there is more to it than there is, and hopes studying logic will shed some light. So he refuses to study analysis until he has time to study logic.
No doubt, if he ever gets to logic, he'll end up with more questions and branching off further and further. He'll never get around to analysis. But like the top level comment - he doesn't like it when people tell him not to bother.