We're told that Mathematicians regard the propositions “P implies Q” and “if P then Q” as synonymous, so both have the same truth table. One example is:
“If Goldbach’s Conjecture is true, then $x^2$ ≥ 0 for every real number x.”
Since we have no idea whether Goldbach's conjecture is true, we are either looking at
TT or FT as far as PQ in the truth table. Either way, since Q is definitely true, the proposition as a whole is true.
But I think I'm missing something basic here. In what sense is this "if this then that"? In ordinary english, at least, "if this then that" means that the truth of "this" is what the truth of "that" depends on. But that "$x^2$ ≥ 0 for every real number x." is true regardless of the truth of Goldbach's Conjecture.
Like I said, I'm clearly missing something here. I just don't understand how either "P implies Q" or "if P then Q" corresponds in any way to what I see in the truth table. Could some explain this?