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

Algebra ->  Proofs -> 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      Log On


   

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) About Me  (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.
NumberStatementLine(s) UsedReason
1(K v L) -> (M & N)
2(N v O) -> (P & ~K)
:.~K
3~(~K)Assumption for Indirect Proof
4K3Double Negation
5K v L4Addition
6M & N1,5Modus Ponens
7N6Simplification
8N v O7Addition
9P & ~K2,8Modus Ponens
10~K9Simplification
11K & ~K4,10Conjunction
12~K3 - 11Indirect 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.