Can someone please verify whether my proof is okay? It looks very tedious and possibly stating unnecessary statements. I might also be missing some important facts.

Prove that $p$ and $q$ are logically equivalent if and only if $p\leftrightarrow q$ is a tautology.

Assume $p$ is logically equivalent to $q$. Then every truth value gives $p$ and $q$ the same truth values. Then $p\leftrightarrow q$ is a tautology.

Assume $p\leftrightarrow q$ is a tautology. Then $p$ and $q$ have the same truth values. Then it must be that $p$ is logically equivalent to $q$.

  • 1
    $\begingroup$ Can you explain what do you mean by logically equivalent? The statement you are trying to prove seems true to me by definition. $\endgroup$ – Hugocito Jun 10 '18 at 1:59
  • $\begingroup$ @Hugocito Logically equivalent: $p \Leftrightarrow q$. Yes, the definition of logically equivalent proves it is a tautology already! So I am just wondering whether my proof is too tedious? $\endgroup$ – numericalorange Jun 10 '18 at 4:03
  • 1
    $\begingroup$ Correct; see Logical equivalence. $\endgroup$ – Mauro ALLEGRANZA Jun 10 '18 at 9:28

"Assume p is logically equivalent to q. Then every truth value gives p and q the same truth values."

That sounds fine. 'p' and 'q' have the same truth value by definition.

"Then p↔q is a tautology."

You might do better to refer to the values of ($\bot$↔$\bot$) and ($\top$↔$\top$) specifically. And you might do well to note that no other truth values than $\top$ and $\bot$ exist for how ↔ gets understood, because there exist three-valued logical systems where there exist connectives for a similar concept, but the same truth values for both 'p' and 'q' do not guarantee that (p↔q) is a tautology.

The same goes for the second part.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.