Question 1029291: Hi! I really could use some help solving these two proofs below. I also included instructions!
Instructions: You must construct a proof for two of the following. All proofs include at least 4 steps (besides the original premises).
I have left the section and problem numbers so that you can check your answers online before posting them. Here is the website for checking your answers:
http://www.poweroflogic.com/cgi/menu.cgi
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Questions from The Power of Logic 5th Edition/Chapter 8/Section 8.3/Part C
15. (~D v E) • (~D v ~F), (E • ~F) → ~G, ~D → ~G ∴ ~G
24. ~O ∴ ~Q → ~(O • P)
Please & Thank you :)
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! Problem #15
| Number | Statement | Lines Used | Reason |
|---|
| 1 | (~D v E) & (~D v ~F) | | | | 2 | (E & ~F) -> ~G | | | | 3 | ~D -> ~G | | | | :. | ~G | | | | 4 | ~D v E | 1 | Simp | | 5 | ~D v ~F | 1 | Simp | | 6 | D -> E | 4 | MI | | 7 | D -> ~F | 5 | MI | | 8 | E -> (~F -> ~G) | 2 | Exp | | 9 | D -> (~F -> ~G) | 6,8 | HS | | 10 | (D & ~F) -> ~G | 9 | Exp | | 11 | (~F & D) -> ~G | 10 | Comm | | 12 | ~F -> (D -> ~G) | 11 | Exp | | 13 | D - > (D -> ~G) | 7,12 | HS | | 14 | (D & D) -> ~G | 13 | Exp | | 15 | D -> ~G | 14 | Taut | | 16 | G -> D | 3 | Trans | | 17 | G -> ~G | 16,15 | HS | | 18 | ~G v ~G | 17 | MI | | 19 | ~G | 18 | Taut |
Abbreviations Used:
Comm: Commutation
Exp: Exportation
HS: Hypothetical Syllogism
MI: Material Implication
Simp: Simplification
Taut: Tautology
Trans: Transposition
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Problem #24
| Number | Statement | Lines Used | Reason |
|---|
| 1 | ~O | | | | :. | ~Q -> ~(O & P) | | | | 2 | ~O v Q | 1 | Add | | 3 | Q v ~O | 2 | Comm | | 4 | (Q v ~O) v ~P | 3 | Add | | 5 | ~(~Q & O) v ~P | 4 | DM | | 6 | (~Q & O) -> ~P | 5 | MI | | 7 | ~Q -> (O -> ~P) | 6 | Exp | | 8 | ~Q -> (~O v ~P) | 7 | MI | | 9 | ~Q -> ~(O & P) | 8 | DM |
Abbreviations Used:
Add: Addition
Comm: Commutation
DM: De Morgan's Law
Exp: Exportation
MI: Material Implication
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