via Jon’s post below and Gillian Russell at TAR, Greg Restall summarizes and links to Branden Fitelson’s very interesting talk (pdf of Branden’s slides) at the Banff Mathematical Methods in Philosophy workshop. I’ve only seen the slides, but I have some questions about Branden’s argument; I hope Branden or some Banffer will be able to say more about how it works.
Branden starts by discussing the new developments in formal epistemology that Jon mentioned below. But his main argument is an analogy between the problem of ‘grue’ and, on the one hand, the relevantist critique of classical logic, and on the other, the problem of old evidence.
Branden quickly summarizes the relevantist critique thus:
(1) On classical logic, if a set X of beliefs is inconsistent, then any p is a consequence of X.
(2) If your total set of beliefs entails p (and you know this), then you are justified in believing p.
(3) But even if you know that your beliefs are inconsistent, there are some things you aren’t justified in believing.
Conclusion: We should reject classical logic, on which an inconsistent set entails anything.
Branden points out that Harman acknowledges the inconsistency but rejects the bridge principle (2) linking entailment and inference. We can preserve the classical entailment relation by denying that we are entitled to infer whatever our beliefs entailed. When our beliefs are inconsistent the proper response is often to reject one of our beliefs rather than to infer all their consequences.
Branden argues that the grue argument against a Carnapian conception of confirmation uses the Requirement of Total Evidence as a similar bridge principle. Carnap sees confirmation as an a priori logical relation between sentences. But this relation alone doesn’t tell us how to apply itself in forming new beliefs. That requires the bridge principle of the Requirement of Total Evidence:
(RTE) E evidentially supports H for S in C iff E confirms H relative to K, where K is S’s total evidence in C.
(E confirms H relative to K means Pr(H|E&K) > Pr(H|K), for a suitable probability function Pr; Carnap thought ‘suitable’ meant ‘logical’, but Branden points out that the grue argument he canvasses doesn’t depend on which probability function is used.) The grue argument—if I’m reading Branden aright—works because we admit ‘the emerald is observed before time t’ into the total background evidence, and once we do this our new evidence confirms “This emerald is green” and thus “All emeralds are green” just in case it confirms “This emerald is grue” and thus “All emeralds are grue.” [See step (iii) in the lower right hand slide on Branden’s p. 3.]
But, as Branden points out, (RTE) has to be rejected anyway because of the Problem of Old Evidence, that hypotheses can be evidentially supported by evidence that is already in K. Bayesians and Carnapians are much better off defining evidential support in terms of confirmation relative to an empty set of background knowledge.
Branden concludes, “So, many Bayesians already reject (RTE). They shouldn’t be too worried about “Grue”. It’s a new twist on “Old Evidence”.
Here’s my main question: If “grue” is just a new twist on old evidence, then presumably the solution to old evidence is meant to work for grue. That solution is to replace (RTE) with the idea that evidential support depends on confirmation relative to an empty background. But I don’t see how that helps with the grue problem.
Consider the choice between two hypotheses, that all emeralds are green and that all emeralds are grue with respect to Jan. 1, 2150. Against an empty background, does the observation of a green emerald now support one hypothesis more than the other? I don’t see how; and answering the question why it should just is solving the problem of grue, it seems to me. Similarly, we’d need to solve the problem of grue to explain why, on an empty background, we might assign a higher probability to “All emeralds are green” than to “All emeralds are grue2150.” If we can’t do either of those things, the move from (RTE) to confirmation against an empty background just doesn’t seem to give us any progress in explaining why the observation of a green emerald now supports “All emeralds are green” rather than “All emeralds are grue.” Any thoughts?
[I have another question about a more minor point of Branden’s presentation. In the lower left slide on p. 3, he presents a counterexample to the hypothesis that the examination of a green emerald before t confirms “All emeralds are green” iff it confirms “All emeralds are grue.” This is based on I.J. Gold’s counterexample to the idea that a black raven always confirms “All ravens are black”; in Branden’s example are background knowledge is that either there are 1000 green emeralds 900 of which have been examined before t, no non-green emeralds, and 1 million other things, or there are 100 green emeralds all of which have been examined before time t, 900 non-green emeralds that have not been examined before time t, and 1 million other things. Then, if you randomly select an object and it turns out to be an examined green emerald, against this background knowledge this confirms that all emeralds are green but not that all emeralds are grue. My question is: Can we set up the background knowledge this way? Shouldn’t the observations that have already been carried out be part of the background knowledge? This problem, however, could be easily taken care of by changing the second hypothesis so that there are nine times as many of each kind of emerald, and in both cases we’ve examined 900 green emeralds; we might also have to add an unexamined green emerald to the second case, because on my view we should see the next examined object as drawn from the ones that haven’t been examined already.]