Q. So I need to make peace with false antecedents, and perhaps you can help me by giving me a little perspective. Is a truth value assigned to F->F merely because it doesn't matter one way or another, and choosing the value T makes it easier to prove things? Have there been logicians or mathematicians who have gone the other way?

A. I am not aware of any mathematician or philosopher who defined F->F differently. Start with a two statements p(x),q(x) such that q(x) is a consequence of p(x). (For example, take p(x) = "x is divisible by 10" and q(x) = "x is even".) Then it should be the case that (forall x)[p(x) -> q(x)]. In order to make that work we need to define F->F = T.