SOLUTION: Premise: 1. (K ∨ L) ⊃ (M • N) 2. (N ∨ O) ⊃ (P • ~K) Conclusion: ~K Use either indirect proof or conditional proof (or both) and the eighteen rules of inference t

Question 1206424: Premise:
1.
(K ∨ L) ⊃ (M • N)
2.
(N ∨ O) ⊃ (P • ~K)
Conclusion:
~K
Use either indirect proof or conditional proof (or both) and the eighteen rules of inference to derive the conclusion of the following symbolized argument.
Answer by math_tutor2020(3816) (Show Source): You can put this solution on YOUR website!

Here are the Rules of Inference that you should have on a reference sheet or memorized.

I'll do an indirect proof. This is also known as a proof by contradiction.
The idea is to assume the opposite of the conclusion. Then show that assumption leads to a contradiction; which therefore must mean the original conclusion is indeed the case.

I'll use the ampersand & in place of the dot.
I'll use the arrow -> in place of the horseshoe.

Number		Statement	Line(s) Used	Reason
1		(K v L) -> (M & N)
2		(N v O) -> (P & ~K)
:.		~K
	3	~(~K)		Assumption for Indirect Proof
	4	K	3	Double Negation
	5	K v L	4	Addition
	6	M & N	1,5	Modus Ponens
	7	N	6	Simplification
	8	N v O	7	Addition
	9	P & ~K	2,8	Modus Ponens
	10	~K	9	Simplification
	11	K & ~K	4,10	Conjunction
12		~K	3 - 11	Indirect Proof

The original conclusion is ~K

Line 3 is where we assume the opposite of that conclusion.
Following the logic of lines 3 to 11, we arrive at K & ~K which is a contradiction. So that allows us to conclude ~K at the end.

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