5.1 was easy,
5.2 was a bit more challenging but doable (I got stuck for a while at the end since I had not played that much with contradictions and quantifiers, but if you had spent some time doing the questions in the text book, this should have been fairly easy â€“ most students would have gotten more than half the marks though)
First sub proof starts with b
then you can infer
Funny(b) v Clever(b) v Pretty(b)
as well as
(Funny(b) v Clever(b)) -> ~Angry(b)
both are sited by Universal Elimination
Now, clearly we have to prove by case (3 in fact)
Sub proof one ( Assume Funny(b)) should not be a problem (you wish to get ~Angry(b) as end result)
Sub proof two (Assume Clever(b)) should also be easy (you wish to get ~Angry(b) as end result)
Sub proof three is the tricky one..
Now make a new sub proof under this one and assume Angry(b).
Now, you can state Angry(b) & Pretty(b) (Conditional Intro from Pretty(b) and Angry(b))
Since we now have Angry(b) & Pretty(b), we can state Ex(Angry(x) & Pretty(x)), but guess what, this contradicts Premise 1, so indicate contradiction and state your two lines, end the sub proof and now state ~Angry(b) with rule ~Intro and state the lines of the sub proof you just finished. Now, you should see that you have three cases and all of them ends with ~Angry(b), so end this last case and write ~Angry(b). State as rule Disjunction elimination and state the lines for your disjunction sentence and your three cases. End the sub proof and state Ax ~Angry(x) and state as rule, Universal Intro specifying the lines for your b sub proof
And that's that..
Ooooh I see. So you don't actually assume anything next to the little block (b in this case) in your first subproof? Man, I couldn't find a single example like this anywhere! (At least not one we have the answer to).
That's pretty tricky imo, considering ALL the examples in the assigmnet have you assuming something next to the little block thing in your subproof first.