Question 871022: Please help me create this proof using the 18 rules of inference. I have been working on this for hours.
1. (P ∨ F) ⊃ (A ∨ D)
2. A ⊃ (M • ~P)
3. D ⊃ (C • ~P)
∴~P
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! I've done a lot of logic derivations, but this is probably (one of) the longest and hardest derivations I've worked with. So I can see why you got stuck. There might be an easier way, but I've yet to find it.
1. (P v F) -> (A v D)
2. A -> (M & ~P)
3. D -> (C & ~P)
Therefore, ~P
-------------------------------------------------------
4. ~(P v F) v (A v D) 1 MI
5. (~P & ~F) v (A v D) 4 DM
6. (A v D) v (~P & ~F) 5 Comm
7. [(A v D) v ~P] & [(A v D) v ~F] 6 Dist
8. (A v D) v ~P 7 Simp
9. A v (D v ~P) 8 Assoc
10. A v (~P v D) 9 Comm
11. A v (P -> D) 10 MI
12. ~~A v (P -> D) 11 DN
13. ~A -> (P -> D) 12 MI
14. ~A v (M & ~P) 2 MI
15. (~A v M) & (~A v ~P) 14 Dist
16. (~A v ~P) & (~A v M) 15 Comm
17. ~A v ~P 16 Simp
18. ~P v ~A 17 Comm
19. P -> ~A 18 MI
20. P -> (P -> D) 19,13 HS
21. (P & P) -> D 20 Exp
22. P -> D 21 Taut
23. ~D v (C & ~P) 3 MI
24. (~D v C) & (~D v ~P) 23 Dist
25. (~D v ~P) & (~D v C) 24 Comm
26. ~D v ~P 25 Simp
27. D -> ~P 26 MI
28. P -> ~P 22,27 HS
29. ~P v ~P 28 MI
30. ~P 29 Taut
Abbreviations/Acronyms Used:
Assoc = Association
Comm = Commutation
DM = De Morgan's Law
DN = Double Negation
Dist = Distribution
Exp = Exportation
HS = Hypothetical Syllogism
MI = Material Implication
Simp = Simplification
Taut = Tautology
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