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I had the first completed but when doing its converse, since there is no CQ rule, I do not know where to proceed. Please help

  • $\begingroup$ What "CQ" means? $\endgroup$ – Mauro ALLEGRANZA Mar 1 at 7:30
  • $\begingroup$ @MauroALLEGRANZA Change of Quantifier. The rules of Quantifier Duality. $$\lnot\exists x~\varphi~\iff~ \forall x~\lnot\varphi\\\lnot\forall x~\varphi~\iff~ \exists x~\lnot\varphi$$ $\endgroup$ – Graham Kemp Mar 2 at 0:04
  1. $\lnot \exists x (P(x) \land Q(x))$ --- premise

  2. $P(a)$ --- assumed [a]

  3. $Q(a)$ --- assumed [b]

  4. $\exists x (P(x) \land Q(x))$

  5. $\bot$

  6. $\lnot Q(a)$

  7. $P(a) \to \lnot Q(a)$

  1. $\forall x (P(x) \to \lnot Q(x))$

As usual the key is to work backwards.

To introduce the universal, assume an arbitrary variable, $a$, then derive $P(a)\to\lnot Q(a)$.

To introduce that conditional, assume $P(a)$ then derive $\lnot Q(a)$.

To introduce that negation, assume $Q(a)$, then derive a contradiction.

To derive that contradiction under the assumptions of $\lnot\exists x~(P(x)\land Q(x))$, $P(a)$, and $Q(a)$, well...


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