SOLUTION: 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 concl

Algebra ->  Proofs -> SOLUTION: 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 concl      Log On


   

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(3817) About Me  (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.
NumberStatementLines UsedReason
1(J v F) v M
2(J v M) -> ~P
3~F
:.~(F v P)
4~~(F v P)Assumption for Indirect Proof
5F v P4Double Negation
6P v F5Commutation
7~~P v F6Double Negation
8~P -> F7Material Implication
9(J v M) -> F2,8Hypothetical Syllogism
10J v (F v M)1Association
11J v (M v F)10Commutation
12(J v M) v F11Association
13~~(J v M) v F12Double Negation
14~(J v M) -> F13Material Implication
15~F -> ~~(J v M)14Transposition
16~F -> (J v M)15Double Negation
17~F -> F16,9Hypothetical Syllogism
18F17,3Modus Ponens
19~F & F3,18Conjunction
20~(F v P)4-19Indirect 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.