SOLUTION: prove the argument is valid using the method of natural deduction: 1. (~N wedge R) horseshoe B 2. A wedge ~(M hor

Question 1208943: prove the argument is valid using the method of natural deduction:
1. (~N wedge R) horseshoe B
2. A wedge ~(M horseshoe N) / therefore A horsehoe B
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

The argument is invalid.

An invalid argument happens when all true premises lead to a false conclusion.

Let's make the conclusion A --> B to be false.
I'm using an arrow instead of a horseshoe.

Here is the truth table for A --> B

A	B	A --> B
T	T	T
T	F	F
F	T	T
F	F	T

Refer to this lesson for more info.

We see that A --> B is false when A is true leads to B being false.

A = true = T
B = false = F
A --> B is false

Since A is true, it means the 2nd premise A v ~(M --> N) is also true. The truth value of the portion ~(M --> N) won't affect the truth value of the entire premise.

B is false, so the 1st premise is of the template (~N v R) --> F
To make sure this premise is true, we just need the ~N v R portion to be false. If it was true then the entire premise would be false.

~N v R is false only when both ~N is false and R is false.
~N being false means N is true.

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Recap:
If we have these truth values,
A = T
B = F
M = either T or F. Doesn't matter.
N = T
R = F
then we'll have
premise 1: (~N v R) --> B becomes (~T v F) --> F simplifies to T
premise 2: A v ~(M --> N) becomes T v ~(T --> T) simplifies to T
conclusion: A --> B becomes T --> F then turns into F

In short,
premise 1 = true
premise 2 = true
conclusion = false

We have shown that the premises are all true, but they lead to a false conclusion.
Therefore the argument is invalid. Your teacher made a typo somewhere.

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