I have a homework assignment which declares that following:
If $P \implies Q$ is valid, and $P$ is valid, then $Q$ must also be valid (prove true or false).
My interpretation of validity in logic is that it basically means that there is a tautology. Its hard for me to wrap my head around the fact that we can declare $P \implies Q$ valid for any propositional formulas $P$ and $Q$ since logically $P \implies Q$ is not a tautology.
Should I be interpreting $P \implies Q$ as being true and that if $P$ is true then $Q$ must be true? That seems to go against the textbook interpretation of validity, but I know how to prove that statement.
Is there something I'm missing about the concept of validity? If $P \implies Q$ is declared as valid does that mean I should only be looking at the "true" sections of the truth table and then proceed with my proof? And that if we declare $P \implies Q$ as satisfiable it means to look at all the entries of the truth table? Just getting confused on this new terminology I am being introduced to.