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!

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- Sep 22nd 2009, 08:28 PMDfowjRules 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! - Sep 23rd 2009, 04:28 AMPiperAlpha167
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] - Sep 23rd 2009, 04:57 AMpaweld
Another solution:

I used the following laws:

and