Logic proof help? 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?
 A: 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?
A: 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}

