The crux of the problem I was worried about is that there are various constraints on rational credences that ensure some valid probabilistic inferences are logically invalid. For example there are surely *some* constraints regarding higher order credences. Similarly, you would have thought there are some connections between credences and knowledge (e.g. if you know p, you should have a high credence in p or if you have a high credence that you know p, you should have a high credence in p, and you should have a high credence that you have a high credence in p. etc etc…) If you accept some of these connections you can construct more problems along the same lines.

Concerning the lottery case: there is an even simpler example. Just think of a typical heads-tails situation. Your credence in heads is 1/2. But your credence in that you know that the coin comes up heads is 0, since you are certain that you cannot know whether heads or tails. Do your intuitions still differ?

Actually I’ve never been quite convinced about the failure of KK. But will the Williamson argument carry over to certainty? I’m not sure – if you’re certain that p_k, then you’re credence in p_n where n!=k is 0 by finite additivity. Doesn’t that just mean the margin for error principle fails? You might also think you should have fuzzy credences in propositions like in the Williamson example, in which case I don’t know how the argument would run.

Eitherway, you don’t actually need anything as strong as S4 for certainty for the result I mentioned. You can do it with: C¬Cp->¬Cp, which is strictly weaker. Another route would be that there can’t be any rational credence function that gives full credence to p, and full credence that p has been assigned zero credence. The counterexamples to KK rely on small differences between the credences and the credences in the credences, but here you have the widest difference possible.

yes, I had the latter conception of probabilistic validity (a la Adams) in mind.

I don’t know about the S4 axiom for certainty. But the second principle you mention seems to be false (if I understand it correctly). Consider a lottery case. My credence in `My ticket will lose’ is high, but strictly less than 1. However, I am certain that I do not know that my ticket will lose. This seems to be a rational epistemic state. But it would violate the principle that my credence in “p” equals my credence in “I know that p”.

If we think of validity as preservation of certainty, we get

*
*

Assuming the principles and respectively. (Here Cp means full credence in p.) This consequence relation violates classical inferences rules though, for example the deduction theorem:

* If then

To see how the consequent fails, it seems perfectly rational to have a credence of 1/2 in p and 0 in Cp. (I think most of the Williamson arguments against validity=preservation of supertruth carry over here.)

The other relevant consequence relation is where the sum of the uncertainties of the premises must be greater-than-or-equal to the uncertainty of the conclusion. This gets us back classical inference rules, like DT. I assume you had this one in mind, while I was assuming the other one in my post. The Moore like inference I stated in the first post remains valid on this consequence relation given the S4 axiom for certainty (and it was more than I needed for validity on the first consequence relation.) For the second inference you get it if you assume that your credence in p should be equal to your credence that you know p: . I find the principles appealed to quite appealing – but maybe you disagree.

Hope you had a good time in Krakow. I was planning on going, but was too disorganised :-s.

Thanks for your comments! I like the method of using probabilistic consequence to get a handle on inferences where standard methods are left wanting in the way you describe. However, I think that this consequence relation differs from the relation we think of as ‘logical consequence’ in some crucial respects, that make its application here slightly spurious.

In fact – I think the probabilistic notion is actually *extensionally* inadequate in that it classifies some inferences as valid which aren’t (in the strictly logical sense.) The example I have in mind trades on Moore like paradoxes: ‘p but I have a low credence in p’. In particular the following inference is valid with respect to all rational credence functions:

*

(The argument trades on the S4 principle for rational certainty – I briefly spelled it out here if you’re interested.)

Now the reason I think this is relevant here is because the counterexample above trades on doxastic and epistemic attitudes in the premises and conclusion, and the counterexample to modus ponens presented trades on epistemic modals.

Another potentially damaging principle relevant to this context, which governs the interaction between credences and knowledge, is the principle that if you believe something, then you believe that you know it. If you accept the corresponding principle for full credence instead of belief that makes the following valid:

*

Even if you found the Moore like paradox above an acceptable inferenc, this one is clearly an awful inference.

sorry that it took me so long to read this entry. After Krakow we had another workshop (the “phloxshop”) here in Berlin.

I am still a bit troubled by how best to set up the debate about modus ponens within the restrictor’s framework. I need to think more about this.

Perhaps there is a way to strengthen your second exammple. Once we enter the realm of epistemic modals etc, it is not clear whether asking what is assertable in which contexts is a very safe guide to validity. For instance, in a context in which you assert “p”, you may also be committed to assert “Must p”. But that shouldn’t force us to think that “Must p” follows from “p”. Rather, we should focus on the fact that assertion is governed by a certain norm, perhaps knowledge. Now, asserting “p” without acknowledging that the norm governing this assertion is in place is pragmatically inappropriate. So, asserting “p” and refraining from asserting “I know p” seems to be pragmatically somewhat strange. Nevertheless, no way that the latter follows from the former.

Now, I think that your example sounds not as good as you would like it to be for precisely this reason. But perhaps we can look at another feature of this inference in order to see that it is not valid. It seems that we can have high credence in the first premise, high credence or even certainty in the second premise, but low credence in the conclusion. Usually, such a complex of epistemic attitudes is not rationally permitted with respect to a valid inference. Doesn’t this show what you want?