Question 1171807: 1. (J v F) v M
2. (J v M) ⊃ ~ P
3. ~F/~(F v P)
4. M Assumption for Indirect Proof
5.
12. ~(F v P)
I think there are 12 lines to get to the conclusion?
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
Here's one way to do the derivation
I'm using an arrow symbol in place of the horseshoe, or sideways "U", symbol.
| Number | Statement | Lines Used | Reason | | 1 | | (J v F) v M | | | | 2 | | (J v M) -> ~P | | | | 3 | | ~F | | | | :. | ~(F v P) | | | | 4 | ~~(F v P) | | Assumption for Indirect Proof | | 5 | F v P | 4 | Double Negation | | 6 | P v F | 5 | Commutation | | 7 | ~~P v F | 6 | Double Negation | | 8 | ~P -> F | 7 | Material Implication | | 9 | (J v M) -> F | 2,8 | Hypothetical Syllogism | | 10 | J v (F v M) | 1 | Association | | 11 | J v (M v F) | 10 | Commutation | | 12 | (J v M) v F | 11 | Association | | 13 | ~~(J v M) v F | 12 | Double Negation | | 14 | ~(J v M) -> F | 13 | Material Implication | | 15 | ~F -> ~~(J v M) | 14 | Transposition | | 16 | ~F -> (J v M) | 15 | Double Negation | | 17 | ~F -> F | 16,9 | Hypothetical Syllogism | | 18 | F | 17,3 | Modus Ponens | | 19 | ~F & F | 3,18 | Conjunction | | 20 | | ~(F v P) | 4-19 | Indirect Proof |
The idea is to assume the opposite of the conclusion. So we assume the opposite of ~(F v P), which is ~~(F v P) or simply F v P.
Then we use the rules of inference to show that a contradiction happens because of this. The contradiction occurs in line 19 when we say that ~F & F.
For example, we could say that F = "an object can fly", meaning that ~F = "an object cannot fly". The statement ~F & F means "the object cannot fly AND the object can fly". This is one example showing why we get a contradiction.
Since we get a contradiction, we then can conclude the opposite of the assumption is the case. So it's the opposite of ~~(F v P), which is ~(F v P) and that concludes the proof.
It's probably possible to have the proof done in 12 lines. I used more lines to be more thorough with the step by step process. As I finished up the table, I realized that I probably could have taken a more efficient route.
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