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.
> Formal logic is a game, especially boolean logic, which has no grounding in truth, the world, or anything.
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.
Though not implication, there is some reasoning to some other vacuously true statements. Take for example:
All purple cows are smart.
This statement is true, because the set of purple cows is empty. It might seem, odd, but it's because we can rephrase it as:
There does not exist a purple cow that is not smart.
When you phrase it like that, I think most people would intuitively agree that it should evaluate to true. In other words, to preserve symmetry between the universal and existential qualifiers, you need the first statement to also be true. Without the ability to transform "exists" into "foralls" would make first order predicate logic pretty much useless. In addition, if the emptiness of a set actually changed how the quantifiers worked, it would be a big problem: breaking referential transparency, if you will.
I don't recall there being a similar justification for implication in boolean logic, but I think the reasoning is similar. Hope that helps!
But he won't be convinced until he formally studies logic. No idea when that will happen.