Anselm's infamous ontological argument effectively defines God into existence: We can conceive of the perfect being (that which none greater can be conceived) -- let's call it 'God'. Suppose (for reductio) that God does not exist. Then we can conceive of a greater being yet -- namely, one just like God but with the added virtue of existing. That would contradict our premise that we're conceiving of the greatest conceivable being (see our definition of 'God' above). So, on pain of contradiction, we must reject the supposition that God does not exist. Hence, God exists. So argues Anselm.
Now, this argument is obviously problematic. This is brought out by the fact that it would seem to commit us the existence of all sorts of perfect entities, as per the following schema:
1) We can conceive of the perfect X, for which no greater X can be conceived.
2) It is greater for an X to exist than not.
3) Suppose for reductio that the perfect X does not exist.
4) Then we can conceive of a greater X, namely, a twin of the perfect X that has the further virtue of existing.
5) This is a contradiction; thus (3) is false, i.e. the perfect X exists.
We can plug anything we want into the X placeholder. Anselm's argument goes through with X = 'being' (which then concludes with the existence of "the perfect being" = 'God'), but we might as well follow Gaunilo in putting X = 'island', and thus conclude, absurdly, that the perfect island must exist. So, we might think, the schema must fail, and Anselm's argument with it.
It isn't quite so simple, however. As was noted by Andrew in the comments at FQI, it isn't obvious that the (single, unique) perfect island is conceivable. Perhaps we can always imagine one slightly better, thus forming an infinite series with no upper bound. But that merely shows that 'island' is a poor choice for X -- it doesn't satisfy premise 1 -- but perhaps some other choice would do the trick. I'll return to this question in a moment.
First, I should explain what work this reductio is doing. Brandon has offered a couple of posts wherein he objects that Gaunilo's island objection (and my more general schema) fail to properly parallel Anselm's own argument. I think he's rather missing the point, and suggested as much in an unusually frustrating exchange in his comments section. Anyway, to clarify, here's my meta-argument:
(1') Absurd consequences follow from the conjunction of (1) and (2) in the schema above, for any X other than 'being'.
(2') Thus, for each such X, either (1) or (2) in the schema must be false.
(3') It seems implausibly strong to claim that 'the perfect X' is inconceivable for all such X.
(4') So it's most likely that (2) must be false instead, for some such X.
(5') If (4') is true, then it seems simplest and most plausible to take (2) as being universally false, i.e. false for all X.
(6') If (2) is false for all X, then Anselm's argument is unsound.
(C') Anselm's argument is probably unsound.
Put more loosely, the point of my reductio schema is to show that Anselm's argument can only survive if "the perfect X" is inconceivable for every X other than 'being'. (Okay, it's logically possible for (2) to be non-universally false, i.e. false for X = those conceivable perfect other-than-beings, and true for X = Anselm's perfect being. But this is implausibly ad hoc. Hence my premise (5').) That is a very strong claim, so Anselm's argument looks to be in trouble.
I should add that if you don't like (4') and (5'), we can replace them with the following:
(4") If (2) is true for X = 'being', then, given (3'), (2) is probably going to be true for some X other than 'being' for which 'the perfect X' is conceivable. By (1'), this will yield absurdities.
(5") Thus (2) is probably false for X = 'being'.
(6") Anselm's argument depends upon the truth of (2) when X = 'being'.
(C") Thus Anselm's argument is (probably) unsound.
Note that my meta-argument does not depend upon the sort of strict parallel that Brandon is criticising, so his objections are quite irrelevant. Indeed, the only point where I really depend upon the analogy is my premise (6'/"), but that one is surely uncontroversial, and unaffected by the sorts of nit-picky differences Brandon highlights. Anselm clearly relies on the claim that it's greater for a being to exist than not. Otherwise he wouldn't be able to reach the conclusion that the perfect being must exist.
Anyway, my main purpose here is to explain why the reductio has some significant rational force, contrary to Brandon's claim that it is merely "a clever bit of philosophical sleight-of-hand, useful for fooling those who don't take the trouble to analyze it, and nothing more."
It is illuminating because we can now see that the core issue is whether there is any other conceivable 'perfect X', for any X other than 'being'. We discussed this in the later comments over at FQI, and I presented some plausible candidates, e.g. X = "malevolent being". If the Anselmian responds by taking 'perfection' to be domain-general (so that the most perfect anything will always tend towards God's attributes: omnipotence, omniscience, and benevolence) then we can stipulatively define a domain-specific evaluative term to take its place in the reductio. Let "sperfect" =df "perfect according to the appropriate domain-specific criteria". Let us say that the evaluative criteria for malevolent beings are just the same as those for beings generally except that the criterion of goodness is replaced by that of evilness. It then follows, by the Anslemian logic of my reductio schema, that the perfectly evil being exists. This alteration won't affect my meta-argument, because Anselm still assumes that it's greater for a being to exist than not, and so it's sgreater for a being qua being to exist than not, and hence (2) must be true when X = 'being'. And the whole point of the reductio is to cast doubt on this latter claim.
As a final point, I'd note that you can have a lot of fun using Anselmian logic with domain-specific perfection. Over at FQI I discussed the conceivability of 'the perfect melody', evaluated against criteria which include its being actually accessible to me whenever I want to hear it. It follows from my reductio schema that the perfect melody really is actually accessible to me right now (given that I want to hear it)! If only...
Readers are welcome to leave a comment suggesting other interesting possibilities for defining perfection into existence. Extra credit if you use Anselm's own logic to prove that there is a perfect counterargument that will negate his own attempts! ;)