May 4, 2009
The Ontological Argument for the Non-Existence of God
1. g=Df (the x such that Px)
- God is defined as a perfect being. (premise)
- We reformulate (1) by saying that God’s existence necessarily entails its perfection. All we did here was explain in terms of existence what (1) means. (from 1)
- Let us assume, as the Ontological arguments do, that the perfection of x necessarily implies the existence of x, for all x. (premise)
- Instantiating the principle in (3) for God. (from 3)
- We now see that (2) and (4) can be combined into one proposition. If Eg implies Pg, and Pg implies NEg, then Eg implies NEg – the existence of God implies the necessary existence of God. (from 2 and 4)
We have to take a break here. As Pollock explains in his development, this is the furthest that we can take (1) by logical means. Even assuming the truth of the premise of the ontological arguments in (3), it is impossible to arrive at Eg, the proposition that God exists.
Rather, the best we can do is the proposition that IF God exists, then NEg necessarily obtains. This is important for two reasons: one because it shows that we cannot arrive at Eg, and two because we will use this conclusion again at the end of our argument.
6. (g=Df the x such that Px) -> N(Eg->NEg)
- Here we simplify the first half of our argument in one proposition. (from 1 to 5)
7. ~ [(g=Df the x such that Px) -> Eg]
- We can explain this proposition in two ways. The first is to remember, as I discussed before, that a definition cannot entail actual existence. The other is to point out that we already showed that we cannot logically obtain Eg from (1). Either way, it is a fact that Eg is unattainable from the definition alone. (premise)
8. NEg iif [(g=Df the x such that Px) -> Eg]
- This is obtained from the definition of logical necessity. Something is logically necessary iif it follows logically from its definition. (premise)
- If something is only logically necessary iif it follows logically from its definition, and God’s existence does not follow logically from its definition, then God’s existence is not logically necessary. (from 7 and 8 )
- But we saw in (5) that it is necessary that if God exists, he exists necessarily: N(Eg->NEg). Since it is not the case that NEg, it is logically necessary that God does not exist. (from 5 and 9)
Our conclusion in (10) proves the strongest form of strong-atheism (“God cannot exist”), but also implies the weaker claim that ~Eg (“God does not exist”).
It seems to me that the only avenue open to the theist is to dispute (7), but perhaps some of the theists on this site can chip in with responses I haven’t seen.