Is the Problem of Grue the Problem of Old Evidence?

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.]


Is the Problem of Grue the Problem of Old Evidence? — 19 Comments

  1. (2) If your total set of beliefs entails p (and you know this), then you are justified in believing p.


    (2) is pretty clearly false, no? Minimally, such a closure principle would have to stipulate that the conjunction of your beliefs B (where B that entails p) is itself justified. If your beliefs B are inconsistent, and you know this, then you won’t be justified in believing B. If B is inconsistent and you don’t know it, you won’t know that it entails any arbitrary p.

  2. Double-checking Branden’s slides, he actually says “It would be reasonable for S to believe p,” rather than “justified”; but that does seem to be a good argument for the falsity of (2) one way or another. To be fair Branden presents this as an oversimplified version of the relevantist critique of CL, since he’s just using it to set up his account of the grue problem — he’s trying to introduce the idea of the bridge priniciples and then apply it to RTE as a bridge principle in the grue problem. (I think; I wasn’t there.)

  3. Good questions, Jon!

    We have to be very careful about the dialectic here. I have a draft of the paper written now. I suggest we read that, and address comments toward the paper. For now, let me just say that all I am offering in this paper is a reconstruction of the argument so that it is (generally) unsound, and it has the same sort of structure as a (naive) relevantist argument against classical logic (i.e., it fails because it rests on a false bridge principle). This does not mean I have a POSITIVE account to offer — on the EPISTEMIC side — that would allow the Bayesian to deny the problematic epistemic conclusion about “grue” (that’s something I’m not tackling here, but remember, Goodman’s argument is an argument against inductive logic QUA LOGIC, not inductive epistemology — that’s why the dialectic goes the way it does). NOTE: (2) is SUPPOSED to be false. That’s the point I’m making (by analogy with the “grue” case). Anyhow, I recommend that you look at the paper, and then we can get a discussion going about it. Here are links to the paper, and to my paper on the ravens, which has some other salient background.


  4. Branden, that’s eminently fair. And in fact saying that you aren’t offering a positive account pretty much answers my original question, which was essentially whether there was a working positive account in there. I look forward to reading the papers.

  5. Thanks, Matt. As you’ll see, you’re right about the deductive case being a “straw man”, merely for analogical setting-up purposes. All of this is made clear in the paper. There are some positive (historical) accounts on the LOGIC SIDE in there. I don’t discuss MY OWN positive view about inductive logic (qua LOGIC) here, though, as that is not necesary for the historico-dialectial structure I’m trying to restore in this paper. And, of course, the FULL story (both logical and epistemic) is only groped at here in a very non-helpful way. In my book, I will have a large section on “applications” of pure inductive logic. And, that’s where the stuff you and Jon seem to want here will be discussed. What I think is crucial, however, is to get clear on the arguments about CONFIRMATION (which was LOGICAL for all the historical figures involved) first. As such, this paper is merely designed just to orient people’s dialectical thinking properly. Although, I think what I have is still pretty nice, nonetheless!

  6. Would anyone mind posting

    J. MacFarlane, In what sense (if any) is logic normative for thought?, manuscript, 2004.

    referenced in the Grue paper?

  7. John tells me he’s not willing to post that paper. But, if you email him, he would be willing to send it to you (subject to some caveats about the ways in which he’s unhappy about the project, etc.).

  8. Interesting proposal, Branden!

    Quickly: one might think that there is an important difference between the problem that ‘grue’ predicates raise and the problem of old evidence, since there is a general problem of basing inference on disjunctive distributions and using disjunctive classifiers, of which the grue problem can be seen as a special instance, and which can be resolved without addressing the problem of old evidence.

    We can have statistics about a class A suitable for use in drawing an inference, and statistics for B that is suitable, but this does not mean that A or B is a suitable basis; likewise, disjunctive classifications (unless they denote cells of a specified partition) create troubles, gruesome troubles.

    Old evidence seems quite different, and quite specifically a Bayesian problem. You could regiment your language to be careful with what can stand for either a classifier or a distribution, it seems to me, blocking the appearance of dubious disjunctions, and still face the problem of old evidence.

  9. Thanks, Greg. As I explain in the paper, I do not claim that “grue” and “old evidence” are THE SAME problem. I merely show that they can both be seen as being problematic FOR THE BAYESIAN for a common underlying reason (viz., adherence to a naive Bayesian precisification of the “requirement of total evidence”). As a result, certain Bayesian approaches to inductive LOGIC can manage to avoid BOTH problems at once (by going for a different rendition of the RTE). Note: this is a claim about inducitve LOGIC, NOT inductive epistemology. The key is understanding that, hustorically, “grue” was an argument against inductive LOGIC, not inductive EPISTEMOLOGY. What you are talking about here are EPISTEMIC pehenomena, and everything I say in the paper is consistent with there being strong disanalogies (on the purely epistemic side of things) betweem the two problems.

  10. I never actually replied to Jon’s question above. He says:

    “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.”

    That’s right. But, all I’m doing in this paper is trying to get clear on the “grue” ARGUMENT as an argument against inductive LOGIC. I’m not trying to tell you what you should say EPISTEMICALLY about grue contexts. As I mentioned to Matt, in my book, I’ll need to say more about positive proposals on both the logical and the epistemic sides of this dialectic. But, step #1 is to see that a Harmanian move can be made to defend inductive logic, just as it can be made to defend deductive logic (amazingly, nobody seems to have worked this out before, for “grue”). If all the relevance logicians had aimed to show was that classical logic isn’t very helpful epistemically in the case where one’s beliefs are inconsistent (i.e., that there is no “classical logic of inference”), that wouldn’t be very interesting. It’s obvious that there’s no classical, formal theory of INFERENCE. Harman’s point is that it doesn’t follow from this that there is no purely formal theory of ENTAILMENT (unless one assumes a dubious bridge principle). Similarly, it doesn’t follow from Goodman’s argument (which trades on the fact that there is no purely formal theory of evidential support — another epistemic concept) that there is no purely formal (Bayesian) theory of inductive LOGIC — UNLESS one assumes a dubious bridge principle like (RTE). That’s the main point of this particular paper. You are complaining that I don’t tell you what probabilities you should assign in grue contexts. That’s like complaining that Harman’s defensive maneuver on behalf of classical deductive logic doesn’t generate advise at to PRECISELY what you should believe, once you discover that your beliefs are inconsistent in a very particular way.

  11. Thanks, Branden. I was thinking about Evidential Probability, which is a species of inductive logic, although a non-Bayesian species. The construction of grue-like predicates is a problem for EP, but it is one that is resolved by restricting the construction of reference formulas and target formulas to, in effect, avoid arbitrary disjunctions of classifications (target formulas) and arbitrary distributions (reference formulas) from appearing in one’s total evidence.

    Old evidence is not a problem for EP, however.

    The answer to Jon’s question, from the point of view of EP, would be that grue-like predicates are instances of a general phenomena that occurs when allowing disjunctive classes to be formed from one’s evidence, in the EP sense of evidence, where we mean a set of statistical distributions, and set theoretical inclusion relations among the classes our reference formula denote, such as “all alligators are reptiles.” We don’t have “all alligators are creptiles” nor “the proportion of palligators which are creptiles in X is between l and u”, because “palligators” and “creptiles” are not terms we’ve constructed our language around, and our rules for regimenting evidence forbid them from appearing by logical manipulation alone. From an EP point of view, if you want ‘grue’ and ‘bleen’, ’emeroses’ and ‘palligators’, you’ve got to put them in the language yourself, by hand.

  12. The post is by me, not Jon! He is not to blame, except for giving me the keys to the blog.

    Branden, thanks for the additional clarification; though I thought your original post was a pretty good answer to the question. I think the key here is the difference between the inductive logic and the epistemic situation; I’m a little more concerned with grue as an epistemic problem—it’s an interesting point about grue being an argument against inductive logic rather than an argument for (a new form of?) skepticism about induction.

    There may be a similar thing to be said here about Greg’s comments on EP. My initial reaction to Greg’s #9 was, “The question is which predicates are disjunctive.” But that seems to be taken care of by #12; the non-disjunctive predicates are the ones you put into the language to be non-disjunctive. The epistemic question then is, which predicates should you put in in order to get justified beliefs? (And it seems like Goodman gives a similar answer—the ones that you have put in!)

  13. Matt, sorry!

    The EP approach treats grue-like predicates as a technical problem, and offers a technical solution. So, by EP’s lights, Goodman’s riddle isn’t a problem for inductive logic per se.

    EP handles the epistemic question in the way you suggest: by ignoring it. The language we use to represent our evidence is built from atomic predicates selected by convention. EP is compatible with some other story for why these atomic predicates and not other ones, (a theory of natural kinds, an evolutionary account of natural language, or something else) which may be appended to the view.

  14. Matt (and Greg and Branden), I thought of trying to be humorous by confusing everybody’s comments with everybody else’s, but it wasn’t coming out funny. . .

    So, I’ll just say this: Matt has really good questions and I’m happy to take credit for all his good thoughts!

  15. Matt and Jon,

    It’s not as if the paper is irrelevant to the epistemic questions. People have assumed (I think) that evidential symmetry was FORCED on Bayesians (or any other formal theory of confirmation). The paper shows this is not the case (since it’s not even a theorem of Bayesian confirmation theory – broadly construed – that E confirms H1 iff it confirms H2). Here’s one positive remark for you on the (which will appear in the sequel).

    Ask yourself about Pr(Oa | H1) vs Pr(Oa | H2), where these are inductive probabilities (of the kind Keynes, Nicod, Hempel, and Carnap would have thought relevant to the “paradox”). It seems to me not implausible that Pr(Oa | H1) > Pr(Oa | H2). This is because in “grue” worlds, some otherwise would-be-unexamined-before-t objects are (intuitively) “hidden away”. But, again intuitively, this is NOT so in green worlds. If you think this is plausible, then it’s also plausible that Pr(E | H1) > Pr(E | H2), for inductive probabilties, which would already suggest that H1 is better confirmed than H2 by E.

  16. Brandon, that is very nice, and exactly what I say when teaching about grue, though I offer it as a suggestion for further thought rather than something to be taken as settled. But I think it is a really interesting idea, and maybe even extendable to skeptical hypotheses more generally.

  17. This is because in “grue” worlds, some otherwise would-be-unexamined-before-t objects are (intuitively) “hidden away”.

    This is intriguing; could you elaborate a little? Does this mean that we are obliged, by the stipulation that the emeralds are grue, not to examine the blue emeralds before time t?

    If so, would this affect a version of the argument on which ‘grue’ means “green before t, blue afterwards”; so that if all emeralds are grue then at time t all emeralds change color? Such a view raises other complications to be sure!

  18. Matt — there are various ways a “grue” world could be. There are possible “grue” worlds in which emeralds change color at t (but nobody really takes that possibility very seriously, it seems to me). And, there are worlds in which it just so happens that all the blue ermeralds remain unobserved until after t. If (like me) you think it is improbable that this would happen by chance alone, then you suspect some “funny business” in grue worlds concerning emeralds (i.e., some strange process that conspires to hide the blue ones away from otherwise would-be-before-t observers). If it’s not implausible to suspect such a thing, then it’s not implausible to think the (inductive) likelihoods are related in a way similar to those in my “urn example” in the paper. While that example is there mainly just to illustrate the FORMAL possibility of such a model (for LOGICAL purposes), it can also be motivated by the sort of considerations I’m raising here. It’s neat that Jon also has thought about this.

    I have more positive stuff on the “logic” side of things as well (i.e., a story that’s different than Carnap’s or Williamson’s approach), but I’ll save that for the book, I think…

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