Use rules of inference to show that if
and
then
Im doing test review stuff, and im just plain stuck here... I can get about halfway, then after that i can't make any connections. Any help would be great!
Here's an outline.
Assume you're in an NDS, with the standard prime rules.
1. You have a universally quantified conclusion. So think final step will be application of universal quantifier introduction.
2. Let u denote some arbitrary member of universe; call it a temporary constant if you like.
3. Drop universal quantifiers on the premisses (by universal quantifier elimination), and express them as Pu V Qu and (~Pu & Qu)->Ru
4. Assume ~Ru. Then assume ~Pu. (So essentially the derivation will be by CP (->I), with a subderivation by RAA (~I).)
5. Now hunt for a blatant contradiction, i.e., a self-contradiction. The one you might be looking for is: Ru & ~Ru.
6. Once you find it, you're almost home. The contradiction allows you to state, ~~Pu, by RAA. From this get Pu.
7. Now from your original assumption, ~Ru, you have, ~Ru->Pu, by CP.
8. Finally get the universally quantifed conclusion by an application of universal quantifier introduction mentioned above: (x)[~Rx->Px].
So then we have: (x)[Px V Qx], (x)[(~Px & Qx)->Rx] |- (x)[~Rx->Px]