I am studying entailment in classical first-order logic.

The Truth Table we have been presented with for the statement $(p \Rightarrow q)\;$ (a.k.a. '$p$ implies $q$') is:
$$\begin{array}{|c|c|c|}
\hline
p&q&p\Rightarrow q\\ \hline
T&T&T\\
T&F&F\\
F&T&T\\
F&F&T\\\hline
\end{array}$$

I 'get' lines 1, 2, and 3, but I do not understand line 4.    

Why is the statement $(p \Rightarrow q)$ True if both p and q are False?

We have also been told that $(p \Rightarrow q)$ is logically equivalent to $(~p || q)$ (that is $\lnot p \lor q$).

Stemming from my lack of understanding of line 4 of the Truth Table, I do not understand why this equivalence is accurate.




  Administrative note. You may experience being directed here even though your question was actually about line 3 of the truth table instead. In that case, see the companion question In classical logic, why is $(p\Rightarrow q)$ True if $p$ is False and $q$ is True? And even if your original worry was about line 4, it might be useful to skim the other question anyway; many of the answers to either question attempt to explain both lines.


Here are two explanations from the books on my shelf followed by my attempt. The first one is probably the easiest justification to agree with. The second one provides a different way to think about it.
From Robert Stoll's “Set Theory and Logic” page 165:

To understand the 4th line, consider the statement $(P \land Q) \to P$. We expect this to be true regardless of the choice of $P$ and $Q$. But if $P$ and $Q$ are both false, then $P \land Q$ is false, and we are led to the conclusion that if both antecedent and consequent are false, a conditional is true.

From Herbert Enderton's “A Mathematical Introduction to Logic” page 21:

For example, we might translate the English sentence, ”If you're telling the truth then I'm a monkey's uncle,” by the formula $(V \to M)$. We assign this formula the value $T$ whenever you are fibbing. In assigning the value $T$, we are certainly not assigning any causal connection between your veracity and any simian features of my nephews or nieces. The sentence in question is a conditional statement. It makes an assertion about my relatives provided a certain condition — that you are telling the truth — is met. Whenever that condition fails, the statement is vacuously true.
Very roughly, we can think of a conditional formula $(p \to q)$ as expressing a promise that if a certain condition is met (viz., that $p$ is true), then $q$ is true. If the condition $p$ turns out not to be met, then the promise stands unbroken, regardless of $q$.

That's why it's said to be “vacuously true”. That $(p \to q)$ is True when both $p$, $q$ are False is different from saying the conclusion $q$ is True (which would be a contradiction). Rather, this is more like saying “we cannot show $p \to q)$ to be false here” and Not False is True.
