Pleae forgive the very basic question, but I know nothing really of formal logic and so would appreciate some feedback.
The truth table defining the implication operator
P Q P implies Q
T T T
T F F
F T T
F F T
together with the negation operator ~ defined in the obvious way enables one to construct the following table for ~Q $\implies$ ~P:
P Q ~P ~Q ~Q implies ~P
T T F F T
T F F T F
F T T F T
F F T T T
Evidently, truth values for ~Q $\implies$ ~P are the same as those for P $\implies$ Q. Is this enough to prove that $P \implies Q$ if and only if ~Q $\implies$ ~P ? My thinking is that yes, it is, because I belive that the logical operators involved are defined by their respective truth tables and this being the case the observations above should be sufficient to prove the equivalence.