I have been trying to solve the question as mentioned above and I have failed to so do. Please provide me a solution.

  • 1
    $\begingroup$ What have you tried? What logical rules are you allowed to use? $\endgroup$ – ZeroXLR Apr 8 at 9:28
  • $\begingroup$ See the answer to similar post. $\endgroup$ – Mauro ALLEGRANZA Apr 8 at 9:33
  • $\begingroup$ @ZeroXLR This is my 1st post on logic. I am new in logic. I would like to request you to use very basic logical rules and please solve. $\endgroup$ – user1942348 Apr 8 at 9:34
  • 1
    $\begingroup$ Please, provide details about the proof system you are asked to use : Natural Deduction ? $\endgroup$ – Mauro ALLEGRANZA Apr 8 at 9:49
  • $\begingroup$ Natural Deduction $\endgroup$ – user1942348 Apr 8 at 9:55

First of all for a formula $\varphi(x)$ we have

$\exists x\varphi(x) \iff \bigvee_a{\varphi(a)}$


$\forall x\varphi(x)\iff\bigwedge_a{\varphi(a)}$

In your case

$\neg\exists x\varphi(x) \iff \neg\bigvee_a{\varphi(a)} \iff \bigwedge_a{\neg\varphi(a)} \iff \forall x\neg\varphi(x)$

The above follows from the de morgan rules

Finally negating again gives

$\exists x\varphi(x) \iff \neg\neg\exists x\varphi(x) \iff \neg\forall x\neg\varphi(x)$

You need to make it a bit more formal but thats the idea


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.