Definition: Let $p$ and $q$ be two compound statements. I read that, If $p\implies q$ is a tautology, then $q$ is said to be a logical consequence of $p$.
Furthermore, it notes that the statement $p\implies q$ is automatically true when $p$ is false, and saying that $p\implies q$ is a tautology actually means that $q$ is true, when $p$ is true.
I do not really understand the last sentence (the bold ones). Can $p\implies q$ not be a tautology, when $p$ is false, or when $p$ and $q$ are both false?
For example, let $p$ and $q$ be simple statements, and we know that the compound statement $[(p\implies q)\wedge p]\implies q$ is a tautology, which can be shown by making a truth table, see for example URL (generate "((p>q)&p)>q" without the "-signs).
Does the note say that we only need to check when $(p\implies q)\wedge p$ is true, which only happens when $p$ and $q$ are true (as shown in the truth table), and we forget the rest of the combinations?