Ontological Argument for the Non-Existence of God

by Francois Tremblay

Ontological arguments, which seek to deduce God’s existence logically from its perfection, are generally considered to be the weakest of the classical arguments for theism, and tend to be seen as nothing more then sleight of hand.

There is some merit to that assertion. The common flaw to all ontological arguments is that they assume that “X is defined as existing” is the same as “x exists”.

The answer to this assertion is that, on one hand, we do say that things defined cannot exist because they contain contradictions: we know (as much as we know anything) that contradictions cannot exist. We cannot, on the other hand, define things into existence, because there is no corresponding metaphysical necessity. The relation is necessarily asymmetrical.

A simpler answer would be to point out that we can define contradictions as necessarily existing, but that doesn’t mean they exist. For instance, I can declare:

zorglub=Df (the x such that Bx & ~Bx)

Where Bx represents “x is a ball”, or whatever other property you want. Either way, Bx & ~Bx is a contradiction, and no zorglub can exist. I can declare that zorglubs exist necessarily all I want, but all I’ve done is define them as necessarily existing. It can never be the case that zorglub exists.

In section 3 of the book “The Impossibility of God”, called “Proving the Non-Existence of God”, John L. Pollock succinctly presents his own version of the ontological arguments. When I say “succinctly”, I am not kidding: the whole discussion holds on three pages. But since his argument is a straightforward modal presentation, no great expanse of discussion is necessary. Either one accepts the logic, or one does not.

Like the modal arguments presented by theists, Pollock starts with the premise that God is defined as perfect, and that perfection implies necessary existence. He does not, however, assume that one can jump immediately to Pg, because, as we have seen, definition cannot directly imply instantiation. Rather, he uses the only logical conclusion that we can draw from the definition, that N(Eg->Pg).

Modal arguments tend to be more arduous for the casual reader, so I will try to explain each step as much as possible. Let me go through the first half of the argument and explain what it means:

1. g=Df (the x such that Px)
   1. God is defined as a perfect being. (premise)
2. N(Eg->Pg)
   1. 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)
3. N(x)(Px->NEx)
   1. Let us assume, as the Ontological arguments do, that the perfection of x necessarily implies the existence of x, for all x. (premise)
4. N(Pg->NEg)
   1. Instantiating the principle in (3) for God. (from 3)
5. N(Eg->NEg)
   1. 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)
   
   1. 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]
   
   1. 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]
   
   1. This is obtained from the definition of logical necessity. Something is logically necessary iif it follows logically from its definition. (premise)


9. ~NEg
   
   1. 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)


10. N(~Eg)
    
    1. 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”).

After developing his argument, Pollock does not address any substantial objection, except briefly addressing the point that some theists may not equal necessity with logical necessity, but some other form of necessity. It is hard to argue with his brevity, however, given that the argument is logically ironclad.

The only problem with the argument is the use of “x such as Px”. Perfection can only be defined relatively, and to posit such a definition without explaining what this “perfection” contains seems to lead to meaninglessness.

However, while this objection applies to the theistic ontological arguments, it does not apply here since Pollock is demonstrating a contradiction between the theological framework of perfection, its consequence N(Eg->NEg), and the criteria of logical necessity. From this perspective, we can classify the Ontological Argument for the Non-Existence of God as an incoherency argument, since it shows a contradiction between a theological premise and the “real world”.

Is it possible to reformulate the argument without using perfection at all? I can see only one way to do so: define g as the x such that NEx. In such a case, we can skip right to (5), since we can directly expand this definition to N(Eg->NEg). So is it possible to claim that God is defined as a being that necessarily exists?

One modal argument does propose that Pg->NEg, as Pollock points out in his introduction. Therefore, some theologians would agree with such a definition. Or if we transpose this in possible worlds language, most (if not all) theologians would agree that God, if he exists, exists in all possible worlds.

Finally, does the argument apply to the god-concept in general? Since any hypothetical god would be logically necessary, N(Eg->NEg) would hold true for any god also. Of course, it is hard to make sense of the claim that a god is logically necessary, but that is a semantic flaw of the god-concept, which is addressed by non-cognitivism. If we presuppose that the god-concept is coherent in total and in parts, then N(Eg->NEg) must hold true. Therefore I see no reason not to apply the Ontological Argument to the god-concept.

Last updated: January 2, 2005