Use both the direct and indirect truth-table to show whether the above argument is valid.

Question

Symbolize the following argument using the symbols for simple statements and logical operators discussed in chapter 6.

 

If it is not the case that humans have free-will then God is to blame for the presence of evil. It is the case that humans have free-will. Thus God is not to blame for the presence of evil.

 

Use both the direct and indirect truth-table to show whether the above argument is valid.

(I figured out the formula, but I'm not quite sure how to do a truth table)

Answer 1

Hi Tasha,

It's a bit hard to represent a truth table here, but I'll give it a shot. Instead of the usual symbols, I'll use words.

Spoiler alert: This argument is not valid

Let's start with the formula:

Let W = "Humans have free will" and G = "God is to blame for the presence of evil."

Premise 1: not(W) implies G

Premise 2: W

Conclusion: not(G)

How do we use a truth table to show that an argument is invalid? We find an assignment of truth values to the variables so that the premises are true and the conclusion is false. To say this another way, make W and G true or false so that Premise 1 and Premise 2 are True but our Conclusion is False.

Premise 2: W is a premise, so we have to assume that's true.

Premise 1: We have not(W) implies G, but since W is true, not(W) is false.

You need to know that if you have {something False} implies G, then it doesn't matter whether G is true or false, the premise will be true no matter what.

We want our conclusion not(G) to be False, so set G to be true.

Our abbreviated truth table looks like this (I hope the formatting comes out well):

W | G | not(W) implies G | not(G)|

T | T | T | F |

We only need to write a single line because this line alone proves the argument is invalid. We found a way to make the premises all true but the conclusion false. So we know that the premises do not imply the conclusion.

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