Ben A. answered • 05/05/20

PhD student with 10+ years experience in Math, CS, and Logic

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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