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?

The point behind the truth table is that you want it to behave in the following way: It is not the case that $\phi$ is true but $\psi$ is false. So you set it up to reflect that.
The reason it is DEFINED for other truth values as you mention above is this: Suppose you want to verify the truth of the statement If $x$ is even, then $x^{2}$ is even. Now suppose that $x=3$. Then $x^{2}=9$. If you didn't define the truth values in the way above, you have that the statement is false (since you would have defined $FF$ is $F$). But this doesn't reflect what we want (intuitively the statement should only be wrong if we can find $x$ even but $x^{2}$ false). Similar examples can be thought of for the other case: Think about "If $x>0$, then $x^{2}>0$". 
This does create some weird situations sometimes: If $x\neq{x}$, then dolphins are sharks is a true statement. However this doesn't affect things much when you do math: You use modus ponens to prove things (i.e. if $p\implies{q}$ and $p$, then $q$). Here while If $x\neq{x}$, then dolphins are sharks is a true statement, you don't have $x\neq{x}$ to conclude that dolphins are sharks, so it doesn't really hurt anything.  
