Why a false statement can imply anything? According to the truth table,
If $P$ is false,then $P->Q$ is true.
if pigs fly, then $1+1=3$. Why is this implication true? How do you prove it?
 A: We know "pigs can't fly" is true, and by the law of the excluded middle, only one of the statements of { "pigs can't fly" , "pigs can fly"} is true.
But if we now suppose "pigs can fly" is true, then two of the statements of { "pigs can't fly" , "pigs can fly"} are true. But we've already shown only one is true, hence $1=2$. Adding one to both sides gives $1+1=2+1=3$.
QED.
A: In the style of Bertrand Russell and the Pope:
Assume we have a set of pigs $S$.  Two can't fly and one can.  How many pigs are in set $S$?   Well $|S| = 2 + 1 = 3$.
Let's $S_F$ be the number of pigs in $S$ that can fly.  Let $S_{\lnot F}$ be the number of pigs in $S$ that can't fly.  Since pigs can't fly, $|S_F| = 0$.  And we are given that $S_{\lnot F} = 2$.  So $|S| = |S_F| + |S_{\lnot F}| = 0 + 2 = 1 + 1$.
So $3 = 1 + 1$.
A: Suppose you say that "If it's raining, then the ground is wet."  
Then someone responds: "But the ground is dry."  
Your response would be:  "So what?  It's not raining, so my statement is still valid!"
A: It's a convention, so you can say eg. for all $x\in\mathbb R$ the following is true:
$$x\geq0\Longrightarrow x^2\geq0$$
This would be false if you didn't define it that way, because $-1<0$ even though $(-1)^2>0$.
