SOLUTION: Use Indirect proof to solve the following: (P v F) ⊃ (A v D) A ⊃ (M • ~P) D ⊃ (C • ~P) / ~P

Question 1198762: Use Indirect proof to solve the following:
(P v F) ⊃ (A v D)
A ⊃ (M • ~P)
D ⊃ (C • ~P) / ~P
Answer by math_tutor2020(3816) (Show Source): You can put this solution on YOUR website!

This is one way to do the derivation.

Number		Statement	Line(s) Used	Reason
1		(P v F) ⊃ (A v D)
2		A ⊃ (M • ~P)
3		D ⊃ (C • ~P)
:.		~P
	4	~(~P)		Assumption For Indirect Proof
	5	P	4	Double Negation
	6	P v F	5	Addition
	7	A v D	1,6	Modus Ponens
	8	(M • ~P) v (C • ~P)	2,3,7	Constructive Dilemma
	9	(M v C) • ~P	8	Distribution
	10	~P	9	Simplification
	11	~P • P	10,5	Conjunction
12		~P	4 - 11	Indirect Proof

The idea is to start with the conclusion ~P and negate it to get ~(~P).
The goal is to show a contradiction arises when we assume the opposite of the conclusion.
As shown above, the contradiction happens on line 11 when we have ~P and P together.
This contradiction then leads us to conclude the opposite of the assumption ~(~P) must be the case, i.e. the original conclusion we started with is the case.
This fully wraps up the proof.

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