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The ontological argument for the existence of God was first proposed by the medieval philosopher Saint Anselm in Chapter 2 of his Proslogion. Ontology as a philosophical sub-discipline is concerned with the nature of being. While Anselm himself did not propose an ontological system he was very much concerned with the nature of being. He argued that there are necessary beings -- things which cannot not exist -- and contingent beings -- things that can not exist. The ontological argument for the existence of God in all of its interpretations and forms ends with a statement like "God exists and is a necessary being". A very colloquial version of Anselm's conclusion is "God can't not exist." This is obviously a controversial position, and the ontological argument has a long history of detractors and defenders.
A quick summary of Anselm's original argument is as follows
Anselm presents the ontological argument as part of a prayer directed to God. He starts with a definition of God, or a necessary assumption about the nature of God, or perhaps both.
Then Anselm asks the big question - does God exist?
To answer this, first he tries to show that God exists 'in the understanding':
Anselm goes on to justify his assumption, using the analogy of a painter:
Now Anselm introduces another assumption:
Example: Most people would prefer a real £100 as opposed to an imaginary £100
Anselm's found a contradiction! From that contradiction, he draws his conclusion:
In order to understand the place this argument has in the history of philosophy, it is important to understand the essence of the argument as Anselm first conceived it.
A key to understand the ontological argument is understanding the idea of "perfections."
There are various kinds of so-called perfections. Size, intelligence, beauty, power, benevolence, and so forth -- all these qualities are called perfections. And there are various degrees of these perfections. What is more intelligent is more perfect as regards intelligence; what is more beautiful is more perfect as regards beauty; and so forth.
Here's a short, and very general description of the ontological argument.
One of the earliest recorded objections to Anselm's argument was raised by one of Anselm's contemporaries, Gaunilo. Gaunilo invited his readers to think of the greatest, or most perfect, conceivable island. As a matter of fact, it is likely that no such island actually exists. However, his argument would then say that we aren't thinking of the greatest conceivable island, because the greatest conceivable island would exist, as well as having all those other desirable properties. Since we can conceive of this greatest or most perfect conceivable island, then it must exist. While this argument seems absurd, Gaunilo claims that it is no more so than Anselm's.
Defenders of Anselm's argument answered that the idea of an island does not include the notion of perfection, the perfection is merely tacked on, while the concept of God cannot be separated from the notion of perfection. This explains their claim that there is an explanation for the failure of Gaunilo's argument -- namely the fact that the island's perfection is contingent -- which doesn't affect the Ontological Argument.
It can be argued that nonexistence is greater and more perfect than existence. The elements of existence are asymmetric and interact because of their imperfections. If they were perfect they would be static. Nonexistence is boundless, timeless, omnipresent, simple, etc. Existence is defined by its limitations. Furthermore, for any number of things that exists, one can imagine twice as many that do not exist, or the set of all sets of them, etc.
Another rationale is attributed to Melbourne philosopher fallacy of equivocation.
Obviously Anselm thought this argument was valid and persuasive, and it still has occasional defenders, but many, perhaps most, contemporary philosophers believe that the ontological argument, at least as Anselm articulated it, does not stand up to strict logical scrutiny.
Some of those who have argued that the ontological argument fails are content to leave it at that, either because they do not believe that God exists, or because they believe the existence of God is demonstrated on other grounds. Others, like Kurt Gödel, Charles Hartshorne, Gottfried Leibniz, and Alvin Plantinga have reformulated the argument in an attempt to revive it.