I previously posted a question to use the rules of inference to prove that Superman does not exist. To start solving this proof, it’d be convenient to label each of the hypothesis. So:

1. If Superman were able and willing to prevent evil, he would do so.
2. If Superman were unable to to prevent evil, he would be impotent.
3. If he were unwilling to prevent evil, he would be malevolent.
4. Superman does not prevent evil.
5. If Superman exists, then he is neither impotent or malevolent.

Let:

a = “Superman exists”
b = “Superman is able to prevent evil”
c = “Superman is willing to prevent evil”
d = “Superman would do so”
e = “Superman is impotent”
f = “Superman is malevolent”

Therefore, we can rewrite the hypothesis as follows:

Now we can come up with the conclusion that Superman doesn’t exist (not a)  using the Rules of Inference!

Note that Step #9 states that “If Superman exists, then he is neither impotent (not e) OR malevolent (not f)”. The above simplifies this statement to “If Superman exists, then he is not malevolent (not f)” in Step #10.

Step #10 can also be simplified by saying “If Superman exists, then he is not impotent (not e)”, and still reach the same conclusion — that he doesn’t exist (not a).