What's with conditionals in mathematical logic?

Question

Having a bit of difficulty understanding the conditional ($\rightarrow$) in mathematical logic. I read up on the already-existing questions and it did help me understand it better (the 'promise' analogy really helps!).

But then I continued reading my text-book, which contains something that muddled me up a bit, again. Here it is:

Consider the following conditionals.
(i) If $x$ is an odd integer, then 4 divides $x^2 - 1$.
(ii) If $x$ is an odd integer, then 4 does not divide $x^2 - 1$.
(iii) If $x$ is not an odd integer, then 4 divides $x^2 - 1$.
(iv) If $x$ is not an odd integer, then 4 does not divide $x^2 - 1$.

You certainly know that 4 divides $x^2 - 1$, if $x$ is an odd integer. You will easily see that (ii) is false, while the rest are true, because in all the three statements conclusion is a fact.

Okay, so I understand why (i) is true, and (ii) is false.

Then I cross-checked (iii).
At first I thought that if $x$ isn't an odd integer, then 4 doesn't divide $x^2 -1$, so it couldn't be true. After which I realised that $x$ could be $\sqrt{13}$ and satisfy both conditions.

But then how is (iv) true? If x isn't an odd integer, it could be $\sqrt{13}$. In that case 4 would divide $x^2 - 1$. That's what's confusing me.

$P.S:$ There's also the possibility that the example isn't a good one. Is it a good idea to have both statements related to each other? (As they are in this case)

Answer 1

For the correct explanation of this passage, see Peter Smith's answer to this post.

The discussion dates back to :

• A.N.Whitehead & B.Russell, Principia Mathematica (1st ed - 1910) : pages 20-21 of the Introduction.

The context is the elucidation of formal implication [in modern symbols : $\forall x (\phi (x) \rightarrow \psi (x))$ ]:

A formal implication states that, for all possible values of $x$, if the hypothesis $\phi(x)$ is true, the conclusion $\psi(x)$ is true. Since " $\phi (x) \rightarrow \psi (x)$ " will always be true when $\phi (x)$ is false, it is only the values of $x$ that make $\phi (x)$ true that are important in a formal implication; what is effectively stated is that, for all these values, $\psi (x)$ is true.