# Ramsey=God?

In a very interesting recent note in Analysis, “Ramsey + Moore = God,” David Chalmers and Alan Hájek have argued that the Ramsey test for conditionals, together with the rational requirement to avoid Moore-paradoxical beliefs, entail that all conditionals of the form “If p, then I believe p” and “If I believe p, then p” are acceptable. But I think that (a) Chalmers and Hájek’s result doesn’t apply to every rational subject, and (b) for those rational subjects to whom it does apply, appeal to Moore’s paradox is not necessary.

The Ramsey test for conditionals is the following:

‘If p then q’ is acceptable to a subject S iff, were S to accept p and consider q, S would accept q.

Suppose now that you apply the Ramsey test to (1):

(1) If p, then I believe p.

The Ramsey test has as an immediate consequence that you should accept (1) if and only if you satisfy (1’):

(1’) If I were to accept p and consider whether I believe p, I would accept that I believe p.

Now: either you satisfy (1’) or you don’t. If you do, that by itself entails that (according to the Ramsey test) you should accept (1), independently of any appeal to Moore’s paradox. If you don’t, then that by itself entails that (according to the Ramsey test) you shouldn’t accept (1), and no appeal to Moore’s paradox is going to change that. (Even if you fail to satisfy (1’), though, it could still be irrational for you to assert (or believe) “p and I don’t believe p.” That would mean that, for you, belief is not closed under entailment. But perhaps that’s not surprising, given that you don’t satisfy (1’).)

Suppose next that you apply the Ramsey test to (2):

(2) If I believe p, then p.

The Ramsey test has as an immediate consequence that you should accept (2) if and only if you satisfy (2’):

(2’) If I were to accept that I believe p and consider whether p, I would accept that p.

Now: either you satisfy (2’) or you don’t. If you do, that by itself entails that (according to the Ramsey test) you should accept (2), independently of any appeal to Moore’s paradox. If you don’t, then that by itself entails that (according to the Ramsey test) you shouldn’t accept (2), and no appeal to Moore’s paradox is going to change that. (Even if you fail to satisfy (2’), though, it could still be irrational for you to assert (or believe) “Not-p and I believe p.” That would mean that, for you, belief is not closed under entailment. But perhaps that’s not surprising, given that you don’t satisfy (2’).)

So: not all rational subjects are compelled to accept (1) and (2) by applying Ramsey’s test—only those who satisfy (1’) and (2’). But, for these subjects, the Ramsey
test alone entails that they should accept that they have the epistemic powers of a God.

1. Hi Juan, I haven’t seen the C/H piece, but I think I can see how the argument is supposed to go. The appeal to Moore’s paradox is supposed to make it impossible to fail to satisfy your (1′) and (2′). I don’t quite get your point about belief not being closed under entailment, though. Here’s what I thought you were going to say:

To fail to satisfy the two counterfactuals above, we first assume that the antecedents are true. For (2′), that involves accepting that you believe p and considering whether p is true. Then for (2′) to be false, you’d have to refrain from accepting p. But to so refrain doesn’t commit one to the belief that you believe p while p is false, only to believing that you believe p while having nothing to say, doxastically speaking, about p itself.

Is that the idea? Or am I missing the argument? If this is the argument, I don’t see how closure for belief (under entailment?) comes into play in the argument.

2. Hi Jon,

Yes, I guess the idea is that Moore’s paradox prevents one from failing to satisfy (1′) and (2′), but my argument is that it doesn’t. The idea is the following: Moore’s paradox prohibits believing “p and I don’t believe p”; if you fail to satisfy (1′), then you believe p and you also believe that you don’t believe p; but that would mean that you believe “p and I don’t believe p” only if belief were closed under entailment for you. If you fail to satisfy (1′), though, we would expect that belief is not closed under entailment for you.

3. If I fail to satisfy 1′, I believe p (by the antecedent), but it is not the case that I believe that I believe p. I don’t need to believe that I don’t believe p for this last conjunct to be true; I might come to no opinion at all. I must be missing something. . .

4. Oh, I see what you mean now. Chalmers and Hájek claim that “p and I suspend judgment with respect to whether I believe p” is just as Moore-paradoxical as “p and I do not believe p.” But my argument still applies: that you believe p and also suspend judgment with respect to whether you believe p doesn’t entail that you believe “p and I suspend judgment with respect to whether I believe p”.

5. Hi Juan and Jon,

I think Juan is right: C/H make several problematic assumptions in their note. That belief is closed under entailment is just one of them. Here are three more points (apologies for any overlap with what has already been said):

1. Take (1) again (If p, then I believe that p). C/H state that to accept ‘p’ while rejecting ‘I believe that p’ is tantamount to accepting the Moore-paradoxical sentence ‘p and I do not believe that p”. Part of the idea seems to be that rejecting ‘I believe p’ is tantamount to accepting ‘I do not believe that p’. But is it true, even for rational subjects, that whenever they reject ‘p’, they accept ‘not-p’? That might not even be the case if the subject explicitly thinks about p and not-p. And: what about intuitionist logicians?

2. The case of a subject who accepts ‘p’ but suspends judgment about ‘I believe p’ is even more tricky. It is certainly not Moore-paradoxical to accept ‘p” and simply not make a judgment about ‘I believe that p’. What if the subject is aware, in addition, that she is suspending judgment about ‘I believe p’? We would have to assume something like

(A) If S suspends judgment about ‘I believe p’ then S is aware of that.

Even with the help of some closure principle for belief she would only get to something like ‘p and I do suspend judgment about p’. This, however, also presupposes that the subject has the notion of judgment, suspension, etc. Only if the subject has notions like these and something like (A) plus some closure principle for belief is true, will Chalmers’ and Hájek’s conclusion follow. This does not seem realistic for even highly rational human subjects.

3. Finally, Chalmers and Hájek reach their conclusion by making the following two assumptions: “if one accepts all instances of (1), one should accept that one is omniscient. And if one accepts all instances of (2), one should accept that one is infallible.” But couldn’t a rational subject accept all instances of (1) (or of (2) for that matter) without accepting that she does so? It seems Chalmers and Hájek are implicitly presupposing something like

(B) A rational subject who accepts all propositions of a kind P also accepts that she accepts all propositions of a kind P.

Again, this principle is much in need of argument. At first sight at least, it seems much too strong. It implies that rational subjects would have infinitely many beliefs and is certainly much too strong for even optimally rational human subjects.

6. I see now, Juan (and Peter), thanks! It is certainly false that belief is closed under entailment. For their argument to fail will be for there to be a counterexample to such a closure principle, but it would also have to be a counterexample to much more specific claims that aren’t closure principles as well. Peter, I like your remarks as well, since these other assumptions could sustain the argument even if closure for belief under entailment is false. It’s especially nice when intuitionists get to play a role in the argument!