# Can two propositions be both tautologies but not logically equivalent?

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?

Thank you

And note: $p \lor \top = q \lor \top = \top$
• Correct!! You can rewrite any tautology as $\top$. – Bram28 Feb 4 '17 at 23:55