St Anselm’s version of the ontological argument appears in his Proslogium, Chapter II, and is the definitive statement of the argument. The argument has the form of a reductio ad absurdum, which means that it takes a hypothesis, shows that it has absurd or otherwise unacceptable implications, and so concludes that the hypothesis is false.

In the case of Anselm’s ontological argument, the hypothesis treated in this way is the hypothesis that God does not exist. Anselm’s argument rests upon the conception of God as “that than which no greater can be conceived”. It is this conception of God with which the hypothesis that God does not exist is supposed to conflict.

If God is that than which no greater can be conceived, Anselm argues, then nothing can be imagined that is greater than God. If God does not exist, though, then something can be imagined that is greater than God, namely a God that does exist.

The hypothesis that God does not exist thus seems to give rise to a logical absurdity: that there both is and is not something that can be imagined that is greater than God. There is, because it’s possible to imagine a God that does exist. There isn‘t, because it’s impossible to imagine something greater than the greatest thing imaginable.

A hypothesis that gives rise to a logical absurdity, though, must be false. The hypothesis that God does not exist, therefore, is false; God exists.

A formal statement of this argument might be constructed as follows:

Anselm’s Ontological Argument

(1) God is that than which no greater can be conceived.
(2) If God is that than which no greater can be conceived then there is nothing greater than God that can be imagined.
Therefore:
(3) There is nothing greater than God that can be imagined.
(4) If God does not exist then there is something greater than God that can be imagined.
Therefore:
(5) God exists.

The first premise of this argument, (1), is Anselm’s conception of God. (2) is a simple logical truth; if God is the greatest conceivable being then there is no greater conceivable being. (3) follows simply from (1) and (2).

Anselm argues in support of (4) by comparing a non-existent God with an existent God. An existent God, says Anselm, is greater than a non-existent God. If God were non-existent, therefore, then we could imagine a God greater than he, namely an existent God.

(5) follows simply from (3) and (4).