Hello this might sound like a play on word although I am kinda of confused. Can it be possible that two propositions are both tautologies but not logically equivalent?
Here is an example:
$(\neg p \wedge (p \vee q)) \rightarrow p$ and $(p \wedge(p\rightarrow q)) \rightarrow q$ now these two are both tautologies, and if I try to show that they are with identities they are both true at the end where:
$(p \vee T)$ and $(q \vee T)$ (Thus both true)
but when you try to show logical equivalence with identities, then you can't.
Is this right? and if yes why? How is this called?