1.) Bill Vallicella and Randall Rauser have both defended this argument:
A proof is a logically valid argument based on self-evident premises
We do not have a logically valid argument based on self-evident premises for the existence of God.
I think both premises are wrong, but in fairness to Vallicella and Rauser, I can’t prove them false by just showing them the sort of argument they had in mind. So why do I disagree with them?
2.) Every discourse has an appropriate standard of proof. Aristotle makes the point that we can’t expect politicians to give geometrical proofs, and he says this not because politicians are too dim to give them or because they should be allowed to shoot from the hip, but because demanding geometrical proof in political matters fails to recognize something very important about politics. Training politicians to seek mathematical certitude would make them indecisive, abstract, impersonal, etc. Again, politicians would not seek to cultivate the sorts of traits that one finds in Caesar, Lincoln, Robert E. Lee, etc. but the sorts of traits one finds in Archimedes and Cantor. The horror. Notice that the appropriate standard of proof applies not just to conclusions but to the premises one starts with, and so we not only have different standards for what is proven but also for what can count as given, axiomatic, or self-evident.
3.) One aspect of an appropriate level of proof is is tied up with the fact that some proofs are necessary because of the imperfection of knowledge. It would be irrational to say a prosecutor hadn’t proved his case because he failed to deliver a video of the defendant in the course of the act and a signed confession. This is not just because the level of proof demanded is inappropriate but, perhaps more importantly, because if all prosecutors has such evidence we wouldn’t need to have trials. As a consequence, we wouldn’t even need prosecutors. In this sense, we are giving argumentative evidence of guilt because we fall short of the highest sort of evidence we might have. We don’t just arbitrarily stipulate that prosecutors do not need to prove their case beyond a shadow of a doubt. If they were given such proof then we wouldn’t need prosecutors at all. To make a general rule: sometimes a doubt that cannot be eradicated by a proof is part of the reason the proof must be given.
4.) It’s given that there is an appropriate level of proof, but specifying what that level is with any exactitude requires not just rational considerations but also conventional ones set by authority. Why set the 5-sigma standard of proof and not, say, a somewhat larger or smaller one? Why demand that treason be proved by two witnesses and not three? Why ask prosecutors to prove beyond any reasonable doubt and not significantly beyond reasonable doubt? Why ask that a Saint be proven by two miracles and not more or fewer? All of these standards have both a rational and an arbitrary component set by some authority.
5.) So (a) some proofs require their own inability to eradicate some doubts (or to reach a highest level of evidence); and (b) in order to discuss whether something is proven requires some difficult antecedent work about what an appropriate level of proof and self-evidence would be, and (c) in order to specify this with any exactitude we need arbitrary conventions set by some governing authority.
6.) There’s more than one interpretation of these facts, but here’s mine: as opposed to other discourses philosophy has to recognize that its appropriate level of proof allows for a wider lack of consensus. All discourses recognize some reasonable level of dissent and absence of consensus, but philosophy needs to recognize its level as relatively much larger than other discourses. This seems to be the only way to preserve the facts that (a) we can’t deny that there is some appropriate level of proof in philosophy and (b) it’s against the very nature of philosophy to set up a universal authorities that might set conventional and arbitrary standard of proof.
7.) Under this description, the major premise of the original argument is true only in a sense that is not useful in deciding whether one has a philosophical proof (since “self-evidence” is relative to one’s appropriate level of proof). And the minor is false because our belief in its truth, as far as I can tell, rests on an idea of philosophical truth that has an inappropriate expectation of consensus.