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

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) (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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