philosophy assignment

The conclusion "S & I" results because you can derive a contradiction from the premises. You can get "H & ~H" on a line fairly early on in the proof. Since contradictions are false, you are allowed to derive anything you want from them; that is where your conclusion comes from. If you know reductio ad absurdum too, you can always show it that way.

Thank you so much.
Any thoughts on

P
Therefore, A ⇒(S ⇒P)

I'm a little lost on where to start

Ill give you a useful strategy for doing proofs: every proof in propositional logic and predicate logic can be proven via reductio ad absurdum. Simply assume the negation of the conclusion and you can go from there. So for your proof...

1. P / A=>(S=>P)
2. ~(A=>(S=>P)) ASSUMPTION FOR REDUCTIO
3. ~(~A v (S=>P)) 2, IMPLICATION
.
.
.
.
n. P & ~P CONJUNCTION INTRO
n+1. A=>(S=>P) 2-n REDUCTIO


Hope this helps.

Sorry to bother you again
I really don't understand this whole proof thing
I don't understand how many steps there are supposed to be in order to prove P.
I've read over my lecture a million times and it doesn't explain it very well. So far I have...

1. P/A⇒ (S⇒P) Premise
2. ¬ A⇒ (S⇒P) Assumption
2. A Assumption
3. S Assumption
4. P 1, 2 modus ponens
4. ¬ (A⇒ (S⇒P)) 2-4 Reductio
5. ¬ (¬A v (S⇒P)) 2, Assumption
6. P Constructive dilemma

This is probably very wrong cause I have no idea what is going on.

To be honest, I have no idea what you are doing, but that's ok. The assumptions that you make at the beginning have to be discharged with either reductio ad absurdum or conditional proof. Let me show you what your proof should look...
1. P / A=>(S=>P)
2. ~(A=>(S=>P)) ASSUMPTION FOR REDUCTIO
3. ~(~A v (S=>P)) 2, IMPLICATION
4. ~~A & ~(S=>P) 3, DEMORGAN
5. ~(S=>P) 4, CONJUNCTION ELIMINATION
6. ~(~S v P) 5, IMPLICATION
7. ~~S & ~P 6, DEMORGAN
8. ~P 7, CONJUNCTION ELIMINATION
9. P & ~P 1,8 CONJUNCTION INTRODUCTION
10. A=>(S=>P) REDUCTIO 2-9

I think that you were doing something along those lines but you have far too many assumptions; remember, when you make an assumption, you have to discharge it at some point. You made 4 assumptions and only discharged one. Thats does not work.

Also with reductio ad absurdum, you assume the negation of the entire statement. So when I assumed the negation of the conclusion, I put the negation symbol outside of the entire statement.

Does that make sense?