[Image Source Credit: TeX]
An argument is sound if and only if it is valid and the premises are true. If those conditions are met, the conclusion must be true.
Consider the following argument:
P1. If God does not exists, this argument is unsound.
P2. God does not exist.
C. Therefore, this argument is unsound.
The argument is valid (Modus Ponens), so it is sound if the premises are true. But, if both premises are true, the conclusion is would have to be true, and the argument would both be sound and unsound. So consistency demands that we deny the soundness of the argument. At lease one of the premises must be false.
Consider whether P1 is false. It is a material conditional, and so it is false when the antecedent is true (it is true that God does not exist) and when the consequent is false (it is false that this argument is unsound). So P1 is false only if the argument is sound, which means that the falsity of P1 leads to a contradiction, since the soundness of the argument entails P1 is true. So, P1 cannot be false.
P2 is the only premise that can be false. So given that the argument must be unsound, we must conclude that it is false that God does not exist.
So this unsound modus ponens proves the contradictory of the minor premise, whatever it might be!
I am probably not the first to note this, but it is new to me.
The truth-table for the Material Conditional is as follows:
p q | p → q
1. T T T
2. T F F*
3. F T T
4. F F T
*The material conditional is only false on line 2.