I am pretty confused by this. We know that $\phi : = \exists x Px \land \exists x Qx $ does not imply $\psi : = \exists x (P x \land Q x)$, as for the model $M$ with domain $\{0,1\}$ with $P := \{0\}$ and $Q := \{1\}$, we have that $M \models \phi$, $M \not \models \psi$. But, seemingly:

\begin{eqnarray} \exists x Px \land \exists x Qx \implies & \lnot ( \lnot ( \exists x Px \land \exists x Qx))\\ \implies & \lnot ( \lnot \exists x Px \lor \lnot \exists x Qx)\\ \implies & \lnot (\forall x \lnot Px \lor \forall x \lnot Qx))\\ \implies & \lnot (\forall x (\lnot Px \lor \lnot Qx ))\\ \implies & \exists x \lnot ( \lnot P x \lor \lnot Q x) \\ \implies & \exists x (P x \land Q x) , \end{eqnarray}

where $(3) \implies (4)$ by the schema $\forall x A x \lor \forall x B x \implies \forall x (A x \lor B x)$; (2) $\implies (3), (4) \implies (5)$ by quantifier/negation relations; and $(1) \implies (2), (5) \implies (6)$ by De Morgan's laws. What went wrong here? Thanks!


$$ \lnot (\forall x \lnot Px \lor \forall x \lnot Qx)) \Rightarrow \lnot (\forall x (\lnot Px \lor \lnot Qx ))\\$$

is not correct.

It is equivalent to

$$ \varphi ~ =: ~\forall x \lnot Px \lor \forall x \lnot Qx \Leftarrow \forall x (\lnot Px \lor \lnot Qx ) ~ := \psi\\$$

Consider your examplary $\{P,Q\}$-structure $A$ over the universe $\{0,1\}$ with $P=\{0\}$ and $Q=\{1\}$.

Now $A \models \psi$ but $A \not \models \varphi$.

This the schema $\forall x A x \lor \forall x B x \Rightarrow \forall x (A x \lor B x)$ is correct but you have been using it the otherway around which is generally not a correct implication.

  • $\begingroup$ Great thank you! This answer also explains why moving from the last statement to the first is indeed valid. I made the error that I can substitute implications of logical expressions for those expressions e.g. P $\land$ Q implies P does not entail $\lnot(P \land Q)$ implies $\lnot(P)$ $\endgroup$
    – user40919
    Nov 21 '13 at 22:35
  • $\begingroup$ Results such as these are easy (and fun!) to check using the method of analytic tableaux. [I drew one up for you in an answer but my phone's not letting me share it for some reason.] I recommend learning how to use them :) $\endgroup$
    – Shaun
    Nov 21 '13 at 23:04

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.