Question 694575: I'm having a ton of trouble figuring out this proof for our take-home quiz. We're only supposed to use the first 8 rules of solving propositional logic, which include modus ponens, modus tollens, constructive dilemma, hypothetical syllogism, disjunctive syllogism, simplification, conjunction, and addition. Here's what I have so far:
1. R->S given
2. P->Q given
3.[(PvK) & N]->(K->R) given
4. PvK given
5.N given
6.(PvK) & N 4,5 conj.
7.K->R 6,6 M.P.
8.K->S 1,7 H.S.
...and that's where I get stuck. Any help would be great. I'm trying to reach a conclusion of QvS, which makes me think the last few steps are going to be constructive dilemma but it's really difficult to get there. Thanks!
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! You're doing great until you hit line 8, here's one way how you do it
1. R->S given
2. P->Q given
3.[(PvK) & N]->(K->R) given
4. PvK given
5. N given
--------------------------------------
6. (P v K) & N 4,5 Conj
7. K -> R 3,6 MP
8. K v P 4 Comm
9. ~~K v P 6 DN
10. ~K -> P 7 MI
11. ~P -> ~~K 10 Trans
12. ~P -> K 11 DN
13. ~P -> R 12,7 HS
14. ~~P v R 13 MI
15. P v R 14 DN
16. (P -> Q) & (R -> S) 2,1 Conj
17. Q v S 16,15 CD
Edit: I realized that you can only use the first 8 rules. Here is the fixed version. In this version you are so close that you just need two more lines.
Much easier version...
1. R->S given
2. P->Q given
3.[(PvK) & N]->(K->R) given
4. PvK given
5. N given
--------------------------------------
6. (P v K) & N 4,5 Conj
7. K -> R 3,6 MP
8. K -> S 7,1 HS
9. (P -> Q) & (K -> S) 2,8 Conj
10. Q v S 9,4 CD
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