Intuitionistic logic makes my head hurt

At the boundaries of language, Aidan has a post about (Duncan) "Pritchard's Dilemma," which was a new one on me. Aidan relates Scanlon's version thus:

Scanlon presents the original version as a dilemma facing any account of what makes an action wrong. We want to know why the fact (assuming for now that it is one) that an act is wrong gives us reason not to do it. The worry is that we'll either end up giving an uninformative answer like 'Because it's wrong!', or we'll end up citing some feature of the act which supposedly substantively explains why we have reason not to perform it which intuitively has no connection to its wrongness (that performing the act would lead to social ostracism, for example); as Scanlon notes, it is not such features of the act which we'd expect 'a moral person first and foremost to be moved by'.

But Aidan is concerned not with this version, but with the "theoretical analogue," where the analogous question is: "why does the fact that p is false give us reason not to believe that p?" Aidan is "inclined to think Prichard's dilemma bites in both cases," and airs some ideas for us about how to argue for this conclusion. I would have commented over there, but as I think I have mentioned before, I am reluctant to drop voluminous comments on other people's property (I've done it before and it looks funny). So let me do it here where I can stretch out. (Back to Wittgenstein some other day.)

Since I reject some of the premises in Aidan's discussion, I get more confused as the post continues, so I can't address it all; but there's plenty to talk about.

1. The two cases of the dilemma seem quite different. Let me address the moral case first.

Let's say you pose me this dilemma (i.e. ask me this question). If you really don't believe that the moral wrongness of an act gives us reason (conclusive or not) not to do it, you don't understand what I mean by these things. Your question ["why the fact that an act is wrong gives us reason not to do it"] concerns the conceptual relation between moral wrongness and rationality.

So if I reply "because it's wrong!", this is indeed "uninformative" in one sense; but that makes it not a failed argument, but instead a rejection of the need for same (and an annoyed refusal to supply the requested explanation). It has the sense of "what part of 'it's wrong' (or 'there's a reason') don't you understand?" Rather than an answer, it's a demand for a better question.

But let's say I do explain. My explanation will concern both concepts (relating them to other concepts still, like agency, belief, truth, normativity, and, I don't know, human being), and is not an argument taking us from one settled location (in conceptual space) to the other. After all, if the one doesn't entail the other, I don't know what you're talking about when you say "moral" (or, again, "reason"); so I'm hardly going to accept your challenge. It'd be like agreeing to build a bridge between the two peaks of Kilimanjaro.

Note that this is not just a semantic claim of analyticity; after all, we may speak as we like, and you might very well come back with: "so morality entails reasons for/against action; if so, why believe there is any such thing?" Instead, it's that if you don't show at least a minimal level of moral understanding, I'm not sure what sort of person I'm talking to, and I might start edging out of the room (or at least keeping an eye on the spoons when you're over for dinner). Theoretical analysis is supposed to explain this phenomenon, not defend it (yet it does, I believe, remain corrigible; but if so, it is you who must explain, not me). I think Anscombe has something about this. You may simply be, as she puts it, "wicked." If so, it's hardly something I can expect to be able to argue you out of, although I suppose it does happen.

So yes, an instrumental "explanation" of my reason not to do something would hardly capture the moral aspects of the act. But that doesn't mean I've got nothing to say except trivialities (although if you continue to play dumb I may run out of things to say to you). The dilemma seems to have no bite here. (Exercise for the reader: contrast Euthyphro and divine command conceptions of morality)

2. How about the "theoretical analogue" of this dilemma? Our question here, again, is "why does the fact that p is false give us reason not to believe that p?" and whether "because p is false!" is too trivial, in the way that "because it's wrong!" is claimed to be in the moral case. This strikes me as a quite different matter.

In the moral case, the reason I come out with "because it's wrong!" is that it's so obvious that an act's moral wrongness supplies a reason for refraining from performing it that I can't even accept the question (i.e., the demand for an inference from one to the other). If I do, I say something unsatisfactory (the second horn of the dilemma); so I deflect the question (not answer it) with my triviality. (Again, this means the "dilemma" is not a dilemma: a third option is available = starting over with a long explanation of the relevant conceptual relations, this time in a larger context, as above. Or simply breaking off the conversation as hopeless.)

But in this second case the answer to our question is simply: it doesn't. Perhaps surprisingly, this is true on either an internalist/evidentialist or externalist/reliabilist account of justification. Internalist first (more obvious). Something's being true (false) hardly constitutes evidence for its truth (falsity). The Andromeda galaxy either contains intelligent life or it does not. We have no evidence either way; yet one or the other is true. Once we see this, externalism gives the same result. No reliable process has produced a belief either that p or that not-p (in me anyway; I'm in doubt on the matter). So no external reason either. (I don't get (epistemological) externalism though; maybe Clayton can help us out here.)

Why did we ever think otherwise? Well, let's take another look at the question. When you refer to "the fact that p is false," you pose a conversational dilemma. I can say "wait, how did we know that p is false? Maybe it's true." Given that you haven't said what p is, this would be a weird thing to say. What's going to convince me that "p" is true? I can hardly dispute it; nor can you give me evidence for it. Clearly your implied answer is "ex hypothesi, dummy."

So instead I let it go – I accept the implicit stipulation that p is false (i.e., that that p is false is a fact) – so that our conversation may continue and you may make your point. But now it looks weird to disclaim any reason to believe this "fact," or the relevance (to our belief) of its truth. I've just accepted it as true, without any argument, without even asking what it was. How can I then turn around and claim not to have any reason to believe it, since that it's a "fact" is all you told me? It is not making of the sense, to do this.

That's why it looks like "because p is false!" is trivially true, when in fact it's false (i.e, that p's falsity provides a reason for disbelieving p). It looks like the best reason there could possibly be, when it's no reason at all.

Here's where my head begins to throb (see title of post). For it seems to me that, given this, the difference between realism and (Dummetian) anti-realism, which Aidan goes on to discuss, doesn't seem to come up. The problem with getting from p's truth-value to my having a reason to believe it is that I don't have any idea what that truth value is until you tell me. As the Andromeda example shows, it could be true or false with no evidence either way. But now the (Dummetian) anti-realist objects: with no evidence either way, the statement has no truth-value (or: its truth-value is "indeterminate," neither true nor false). The Andromeda example is out.

But this doesn't help. Again, it's not that I have no evidence that p is false (or true); it's that I have no evidence about p's truth value at all, including what evidence there is for it. The anti-realist move simply pushes the problem back. The Andromeda example may not work, but a different example will – one, say, where I have no idea what would count as evidence one way or another. For any specific claim, of course, the Dummettian may object that this can't be a proposition that I understand. True enough; but that's hardly relevant, given that all I have been given is "p". Unless my interlocutor is messing with me, he understands it (i.e., he picked an intelligible example), and that means I have no reason to doubt that there is evidence for (the existence of evidence for or against or for the indeterminate nature of) "p". Again, to demand same here would be (as I believe they are putting it these days) teh weird.

Nor does is anti-realism relevant if we accept ex hypothesi, as before, that p is false. For the anti-realist, it follows that there is a proof (or conclusive evidence; Aidan uses a mathematical example, so let's talk about proof) from our axioms to not-p. Now that I have a proof, do I have a reason to believe that p is false?

Not in the relevant sense I don't, as the earlier reasoning still applies. I have no idea what this proof is (how could I – I don't even know what p is.) I'm in the same relation to (the existence of) this proof as I was earlier to p's truth-value: I can either demand to see it (weirdly) or I can accept your implicit claim that there is one (i.e., in declaring it false on an anti-realist construal) and (again weirdly) deny the relevance of that fact to the question of whether or not I should believe p, given the existence of said proof. This makes triviality (now of the form "because p is false, and so there is a proof that p is false!") the only apparent option.

Aidan actually says "If one asks why some set-theoretical statement s being false gives one reason not to believe that s, can we do better than to simply point out that s is false? Surely we can; usually we will be able to prove that ~s follows from some mutually shared set of axioms." So here we know what s is and have proved its negation. Now I do know what to believe. If so, then that's my reason for believing s is false: we proved its negation. But what about s's falsity? Did s's falsity provide (not a proximal, but maybe an ultimate) reason for my belief? Well, I didn't use it as a premise, obviously; so, no. If a, b, c entail not-s then not-s provides inductive evidence for a, b, c; but ex hypothesi again, I don't have independent access to not-s, so that doesn't help not-s provide ultimate reason for believing it.

Aidan asks: "Is it the case [...] that that fact that a statement's negation is provable is only a reason to believe it derivatively because of that close connection to truth?" My answer: if I know that the statement's negation is provable, then it's either a) that proof itself, or b) whatever other evidence I have for the existence of same (e.g., you told me), that provides my reason to believe its conclusion – not that conclusion itself. If I don't know this, I may have no reason to believe anything at all about it, whether it happens to be true or not. And again, it seems not to affect matters if we move from s's falsity to the provability of not-s.

And even in intuitionistic logic, if not-s is true, then s is false (right? right??), so I don't see how that's relevant. But by this point in the post I'm as confused as Aidan himself claims to be.