Can these propositions be equivalent?

Question

We all know that:

$P \rightarrow Q $

and

$(Not)Q \rightarrow (Not)P$

are equivalent.

Is it possible that in specific cases $P \rightarrow Q $ is equivalent with $Q \rightarrow P $ or $(Not)Q \rightarrow P $ or $P \rightarrow (Not)Q $

Can this happen or is that impossible? If it is, how should I attempt to prove it?

Answer 1

From a truth table $P\to Q$ is equivalent to $Q\to P$ iff $P=Q$, to $\neg Q\to P$ iff $Q$, and to $P\to\neg Q$ iff $P\land Q$. You can easily choose $P,\,Q$ to achieve these conditions.

Answer 2

There is no such thing as two statements being 'sometimes logically equivalent' or 'possibly logically equivalent'. Logical equivalence means always having the same truth-value. If it's not always, it's not equivalence.