double negation rule - properties of semantic equivalence

Question

I know the rule for double negation is

¬(¬P) ≡ P

However if I have:

¬(¬P V Q) 

does that give me

(P V Q)

or would the negation still apply to the Q:

(P V ¬Q)  

Thank you in advance.

Answer 1

NO.

In :

$¬(¬P \lor Q)$

we have not a case of double negation, because the outer $\lnot$ applies to all the sub-formula inside the parentheses.

We have to apply De Morgan's laws to transform it into the equivalent form :

$\lnot \lnot P \land \lnot Q$

and then apply *double negation to get :

$P \land \lnot Q$.