The Ontological Argument

Argument by reductio ad absurdum

A form of argument in which one attempts to prove something by deducing a contradiction from the assumption that the conclusion is false.

assume the denial of what you want to prove [(1)]
(optional) add some premisses [(2),(3),(7),(10)]
deduce a contradiction [(6) & (7)]
conclude that the assumption is false [(8)]

The Ontological Argument

Let 'God' abbreviate 'the being than which nothing greater can be conceived'

God exists in the understanding but not in reality. (assumption for reductio ad absurdum)
Existence in reality is greater than existence in the understanding alone. (premiss)
God's existence in reality is conceivable. (premiss)
Therefore,
If God did exist in reality, then he would be greater than he is. (1)(2)
Therefore,
It is conceivable that there be a being greater than God is. (3)(4)
Therefore,
It is conceivable that there be a being greater than the being than which nothing greater can be conceived. (5) and the definition of 'God'
It is not conceivable that there be a being greater than the being than which nothing greater can be conceived. (premiss)
Therefore,
It is false that God exists in the understanding but not in reality. (1)-(7)
Therefore,
Either God doesn't exist in the understanding or he does exist in reality. (8)
God exists in the understanding. (premiss)
Therefore,
God exists in reality. (9)(10)

Some Definitions:

x exists in the understanding =df someone has thought of x
x exists in reality =df x exists
a state of affairs, S, is conceivable =df S is possible

3') It is possible that God exists.
6') It is possible that there be a being greater than the being than which it is not possible that there be a greater.

Gaunilo's objection:

I.

Replacing the word 'God' with the words 'the greatest island' yields an argument

(i) of the same form as Anselm's
(ii) with true premisses, and
(iii) a false conclusion

This shows that Anselm's argument is invalid.

Anselm's Reply:

II. A second version of Gaunilo's objection:

Replacing the word 'God' with the words 'the greatest possible island' yields an argument

(i) of the same form as Anselm's
(ii) with true premisses, and
(iii) a false conclusion

This shows that Anselm's argument is invalid.

A reply on behalf of Anselm:

A Different Objection:

How do we get from

1) God exists in the understanding but not in reality. (assumption for reductio ad absurdum)

and

2) Existence in reality is greater than existence in the understanding alone. (premiss)

4) If God did exist in reality, then he would be greater than he is. (1)(2) ?

There is an intermediate step:

2') For all x, if x does not exist in reality, then if x did exist in reality x would be greater

Therefore,

2.5) If God does not exist in reality, then if God did exist in reality then God would be greater. (2'), universal instantiation

Therefore,

4) If God did exist in reality, then he would be greater than he is. (1)(2)(2.5)