Thread: Rules of Inference - Stuck on discrete problem

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!

Originally Posted by Dfowj

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]

Another solution:







I used the following laws:

and