Why if the antecedent P is false, and the consequence Q true, then the implication P $\Rightarrow$ Q is true? I know that that's the definition but I wonder why logicians choose that thefinition to be true. It sounds strange to me and I cant make sense of it if someone tell me 'if the sky is red, then I'm Marco Polo'.
 A: This is known as the principle of explosion.
We usually think of implications as promises. Suppose I always say the truth, and suppose I say: "if it's sunday, then I'll take my dog for a walk". If I take my dog for a walk in a saturday, did I break the promise?, no!, I only break my promise if it is sunday and I haven't taken my dog out.
Secondly, this very same example helps understanding the equivalence with the contrapositive ($p\Rightarrow q \iff ¬q \Rightarrow ¬p$): if I haven't taken my dog for a walk, that implies it's not sunday.
And, lastly, a phrase from von Neumann fits perfectly (and I paraphrase): in mathematics, one doesn't understand objects, one just gets used to them. In Enderton's A mathematical introduction to logic, he suggests that we shouldn't really care about the relation with the way we define implication (or equivalence, or conjugation) with the way we usually understand it in our everyday life; we should just study the objects per se, but perhaps I'm misunderstanding what he says.
