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I've spent over 3 days trying to learn logical equivalences/proof but no matter how much I try I can't seem to do the exercises I've been given. At all.

For example: "Given that P $\land$ ¬P is an inconsistency, show that ¬(¬P $\lor$ Q) $\land$ ¬(¬Q $\lor$ P) is an inconsistency without using a truth table."

My proof is as follows:

  • Given: ¬(¬P $\lor$ Q) $\land$ ¬(¬Q $\lor$ P)
  • Double negation: ¬(¬P $\lor$ Q) $\land$ (Q $\lor$ P)
  • Commutativity: ¬(Q $\lor$ ¬P) $\land$ (Q $\lor$ P)
  • Distributivity: Q $\lor$ (¬P $\land$ P)
  • Commutativity: Q $\lor$ (P $\land$ ¬P)

I'm not sure what to do after this point, although I suspect it's because it's utterly wrong.

Please could someone share some guidance?

  • 0
    What are the axioms and the inference rules of your deductive system? – 2017-01-04

2 Answers 2

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Some remarks:

  • From the first to the second line, you used "double negation" to turn $\neg(\neg Q \vee P)$ into $Q \vee P$. This is false - the outer $\neg$ does not stand in front of $\neg Q$ alone!
  • From your third to your fourth line, you used distributivity but the $\neg$ in front of $(Q \vee \neg P)$ just vanished?

A correct solution:

\begin{align} \neg(\neg P \vee Q) \wedge \neg (\neg Q \vee P) &\iff (\neg (\neg P) \wedge \neg Q) \wedge (\neg(\neg Q) \wedge \neg P) & \text{(bring $\neg$ inside brackets)}\\ &\iff (P \wedge \neg Q) \wedge (Q \wedge \neg P)& \text{(double negation)}\\ &\iff (P \wedge \neg P) \wedge (Q \wedge \neg Q)& \text{(commutativity)}\\ \end{align}

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Your first step (the one you call "double negation") is incorrect. The symbols $\neg Q\vee P$ can be read as "not $Q$ or $P$". That is, either $Q$ is not true, or $P$ is true. We can negate this using one of DeMorgan's laws: \begin{equation*} \neg(\neg Q\vee P) \Leftrightarrow \neg(\neg Q) \wedge \neg P \Leftrightarrow Q\wedge \neg P. \end{equation*} Similarly we have \begin{equation*} \neg(\neg P\vee Q) \Leftrightarrow \neg(\neg P) \wedge \neg Q \Leftrightarrow P\wedge \neg Q. \end{equation*} Can you now see why the original statement is an inconsistency?