# Distribution of double negation

Can we use double negation and distribute it like this

$$(\neg \neg p \lor -q ) \equiv \neg(\neg p \lor q)$$

But De Morgan's law isn't like what I did above!

• $\neg$ \neg – Kenny Lau Sep 5 '17 at 16:19

Can we use double negation and distribute it like this

$$(\neg \neg p \lor -q ) \equiv \neg(\neg p \lor q)$$

No, you can't. Here is why:

When $p$ and $q$ are true, then $\neg \neg p \lor q$ is true, but $\neg(\neg p \lor q)$ is false. So these are not equivalent.

• You're right; I've deleted my answer. – Theoretical Economist Sep 5 '17 at 16:49
• @TheoreticalEconomist I almost added that comment to my Post, as I figured several people would do just what you did :) – Bram28 Sep 5 '17 at 16:51
• I was wondering why you hadn't pointed that out. It turns out that you just read the OP more carefully than I did. :) – Theoretical Economist Sep 5 '17 at 16:52
• I wasn't aware that we should evaluate () before the negation.Thanks! – Sun292 Sep 5 '17 at 16:54
• @Sun292 Yes, it's just like other mathematical expressions. Glad I could help! – Bram28 Sep 5 '17 at 16:56