After, for some reason or other, spending some 30 dense pages of Logic and Theism on the laughable ontological arguments of Descartes and Spinoza, Sobel moves on to the more interesting argument advanced by Anselm. (The next chapter deals with versions of the argument set in modern modal logic, such as those of Hartshorne and Plantinga.) In my view, the Descartes and Spinoza arguments don't even look good; the Anselm version at least produces puzzlement, insofar as the reasoning looks valid, yet it seems, intuitively, that no such strong conclusion could ever be derived from such weak premises.
Sobel (fairly uncontroversially, I think) takes it that the main premise of Anselm's argument is that a being than which none greater can be thought exists in the mind, i.e. is consistently conceivable. From this it is supposed to follow that such a being exists in reality. Now, Sobel appears to concede the validity of the argument. At any rate, he does not here attribute any logical fallacy to Anselm's argument, but seems to think that it is somehow dialectically inappropriate. He indicates that he will argue later that Hartshorne's argument "is strictly question-begging" (p. 65), but the problem with Anselm's argument seems to be slightly different, or at least less egregious. Now, the concepts of dialectical appropriateness and begging the question have recently come under scrutiny, and, indeed, even been looked upon skeptically by philosophers, so let me first explain how I understand these, and then I will explain the problem Sobel has with Anselm's argument. Incidentally, Sobel attributes this objection to Gaunilon and indicates that he thinks it is much more important than Gaunilon's more famous 'Lost Island' objection. I take it that Sobel's Gaunilon has discovered a limitation, but not a flaw, in Anselm's argument. (That is not to say that there are not other flaws.)
First, let's distinguish logical and dialectical fallacies. An argument is really a sort of abstract structure, consisting of premises and a conclusion. The premises are supposed to stand to the conclusion in some kind of support relation; in the case of deductively valid arguments, that relation is logical entailment. A logical fallacy occurs when the premises of an argument appear to support its conclusion, but actually don't.
Arguments, however, are not typically treated in this abstract way. More often they are used in some sort of inquiry or dispute. This is called a dialectical context. A dialectical fallacy occurs when an argument is put forward in a context where it can't be used. For instance, suppose I am inquiring into (or debating about) whether God exists, and I put forward an argument to the effect that he does. The argument should use premises which I already reasonably believe, even though I am still in doubt as to whether God exists. An argument commits the dialectical fallacy of begging the question if it contains premises that no one could reasonably believe without already believing the conclusion. Such an argument would always be dialectically inappropriate. Here is an example from Plantinga's Nature of Necessity (I don't have the page number handy):
Now, Gaunilon, as interpreted by Sobel, doesn't accuse Anselm of this sort of radical dialectical error. Anselm has set up the dialectical context as an argument with a stylized fictitious atheist known as 'the Fool' (after Psalms 14:1 and 53:1). According to Sobel's reading of Gaunilon, Anselm's argument appears stronger than it is because its main premise is ambiguous between a premise the Fool can safely reject and one that he cannot. The validity of the argument requires the former premise, so the Fool is not decisively refuted.
The premise Anselm needs is that a being than which none greater can be conceived exists in the Fool's mind. Anselm says that this must be the case, since the Fool understands what he says when he talks about a being than which none greater can be conceived. Here is how Sobel describes Gaunilon's objection:
'A being greater than all others that can be conceived,' I take Gaunilon to say, is undeniably in the doubter's mind when Anselm speaks those words to him, no more or less than would be 'an island greater than all others that can be conceived' when some dreamer speaks those words to someone who understands them is undeniably in the mind of the auditor. In neither case, I take Gaunilon to say, can the person who hears the words, but is not sure that they identify anything in reality, be sure that he has in mind an object ... He cannot deny that he has in mind the words, or that he has in mind what they mean, for he understands them. But he can wonder whether he has in mind a thing described by them, for he can understand that he does have a thing in mind, given how these words describe things, if and only if such things exist in reality. (p. 65)
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