Question about the FT, FF cases in the conditional: The conditional operator, $\phi \implies \psi$, is True for the values $TT, FT, FF$ and false for $TF$. I can easily understand why it's true for $TT$ and false for $TF$, but why is it for $FT$ and $FF$? As far as I can see, we can't come to any conclusion in those cases:
If $\phi$ is false but $\psi$ is true, to me that doesn't imply anything about the truth or falsity of $\phi \implies \psi$. Couldn't it be that when $\phi$ is true then $\psi$ is false? Why does this situation imply that $\psi$ follows from $\phi$? e.g.
$\phi : n < 4$, $\psi : n > 2$. If $\phi$ is false then clearly $\psi$ is true, but why does that imply that the truth of $\psi$ follows from the truth of $\phi$? If $\phi$ is true then either $\psi$ is true or $\psi$ is false, depending on the situation. 
Similarly, $\phi: n < 4$, $\psi : n < 2$. If $\phi$ is false then $\psi$ is false but if $\phi$ is true then $\psi$ is not necessarily true. 
Where did I go wrong with my reasoning?

I think that this explanation of S.C.Kleene, Mathematical Logic (1967) [pag.10 - footnote 12] is the best "short" elucidation of it : 

The ordinary usage certainly requires that "If $A$ , then $B$" to be true when $A$ and $B$ are both true, and to be false when $A$ is false but $B$ is false. So only our choice for $T$ in the third and fourth lines can be questioned. But if we changed $T$ to $F$ in both these lines, we would simply get a synonym for $\land$; in the third line only, for $\lnot$ . If we changed $T$ to $F$ in the fourth line only, we would loose the useful property of our implication that "If $A$ , then $B$" and "If $\lnot B$ , then $\lnot A$"  are true under the same circumstances [...].

