The traditional version of the ontological argument is usually criticized for treating existence as a predicate. If existence is a predicate, then it is a predicate that always applies, and as Russel quipped that is the sign of a mistake. A predicate must be fail to apply to some objects in order to count as expressing a genuine property. But Kripke has shown that it is easy to introduce an existence predicate into our formal language that avoides this and related difficulties. We do so as (1),
(1) E(y) (y=x)
Where ‘E(y)’ is the standard existential quantifier and identity is understood normally.(1) is an existence predicate because it is an open sentence that can be satisfied by the values of the free variable x. Intuitively (1) says that x has the property of being identical to some thing (y) and this captures what we mean when we say that existence is a property.
(1) can fail to apply, the model is very easy to give. Imagine a universe of two distinct objects A and B. Now, say that there are two distinct worlds in this universe one containing only A and the other containing only B. E(y) (y=B) will be false at the world where only A exists.
As Kripke points out the problem only arises when one mistakenly thinks that the claim that existence is a predicate is the claim that (2) makes
(2) (x) (E(y) (y=x))
(2) says that every thing exists and this cannot fail to apply and is a necessary truth, but it is also not a predicate (it is not an open sentence, it is closed by the universal quantifier ‘(x)’).
So if existence is a property then it makes sense to think it is the kind of property that God must have, since He is a being who would be without equal and if He lacked the property of existence then I or you would be His equal and better. So, He must exist. To concieve of God not existing is to concieve of an object that has everything and yet lacks something which is just as contradictory as concieving of an object that is triangular yet lacking three sides.