Translation, Validity and
Proof in PL
Solution, Question 5
5. Oxygen isn't a sufficient condition for fire. Fire not being present is a sufficient condition for chaos not being present. Therefore oxygen is a necessary condition for chaos.
Dictionary:
O: Oxygen is present.
F: Fire is present.
C: Chaos is present.
Translation:
~(O
⊃
F)
~F
⊃ ~C
C
⊃ O
This argument is valid. (Verify with a truth-tree.)
Proof:
| 1. | ~(O ⊃ F) | prem | |
| 2. | ~F ⊃ ~C | prem | |
| 3. | ~(~O v F) | CE | 1 |
| 4. | ~~O & ~F | DEM | 3 |
| 5. | ~F | simp | 4 |
| 6. | ~C | mp | 2,5 |
| 7. | ~C v O | disj | 6 |
| 8. | C ⊃ O | CE | 7 |