Counterexample for a statement

Question

Lets say I want to prove or disprove the statement $p \to q$. If I show that $p\to \lnot q$ for one particular case, isn't that a counterexample that shows that the statement is false?

Answer 1

No, that is not correct in general. For example, if $P$ is false, then you can prove that $P\implies \neg Q$, but you can also prove $P\implies Q$.

Maybe your question is abstracted from a problem in which this is not the case. You would need to give more details, in that case. As a general logical deduction, proving $P\implies \neg Q$, doesn't imply $\neg(P\implies Q)$.