Froggy Problem Validation of Conclusion
We assume the negative of the conclusion viz E & ~a & ~b & d and show this leads to a contradiction:
Original Premisses:
01. ~s ⇒ r
02. ~m & h ⇒ ~d
03. E ⇒ a & ~k
04. l & n ⇒ v
05. z & m ⇒ ~c
06. t ⇒ A
07. ~s & ~n & c ⇒ ~r
08. z & v ⇒ m
09. E ⇒ w
10. v & k & m ⇒ t
11. E ⇒ s & c & ~n
12. A ⇒ ~B
13. k & ~h ⇒ n
14. r ⇒ c
15. ~v & ~a & ~h ⇒ l
16. w & ~t & s ⇒ n
17. d ⇒ e & k
18. s & n & ~b ⇒ B
19. w & e ⇒ z
20. ~t & ~A & ~r & ~c ⇒ ~n
|
| H | E & ~a & ~b & d | Hypothesis |
| 21 | E ⇒ w | By 9 |
| 22 | w ⇒ z ∨ ~e | By 19 |
| 23 | d ⇒ e | By 17 |
| 24 | e | By H, 23 |
| 25 | w ⇒ z | By 22, 24 |
| 26 | z ⇒ ~c ∨ ~m | By 5 |
| 27 | E ⇒ c | By 11 |
| 28 | c | By H, 27 |
| 29 | z ⇒ ~m | By 26, 28 |
| 30 | E ⇒ ~m | By 21, 25, 29 |
| 31 | ~m | By H, 30 |
| 32 | E ⇒ z | By 21, 25 |
| 33 | z | By H, 32 |
| 34 | ~m ⇒ ~z ∨ ~v | By 8 |
| 35 | ~v | By 31, 33, 34 |
| 36 | ~v ⇒ ~l ∨ ~n | By 4 |
| 37 | ~l ∨ ~n | By 35, 36 |
| 38 | ~l ⇒ v ∨ a ∨ h | By 15 |
| 39 | a | By H |
| 40 | ~l ⇒ h | So by 35,39 |
| 41 | h ⇒ ~d ∨ m | By 2 |
| 42 | d | By H |
| 43 | ~h | By 31, 42, 41 |
| 44 | l | By 40 |
| 45 | ~v ⇒ ~n | By 36,44 |
| 46 | ~n | By 35, 45 |
| 47 | ~n ⇒ ~k ∨ h | By 13 |
| 48 | k | By 42, 17 |
| 49 | ~n ⇒ h | By 47, 48 |
| 50 | ~n ⇒ ~d ∨ m | By 49, 41 |
| 51 | d&~m | By 31, 42 |
| 52 | n | By 50, 51 |
| 53 | Contradiction | 46, 52 |
| 54 | ~(E & ~a & ~b & d) | By H, 53 |