Wolfgang Schwarz

RATs, PETs, Missed Clues, and Closure

Jonathan Schaffer argues (in Analysis 2001) that Relevant Alternatives Theories of knowledge (RATs) such as Lewis's fail because of Missed Clues cases:

Professor A is testing a student, S, on ornithology. Professor A shows S a goldfinch and asks, 'Goldfinch or canary?' Professor A thought this would be an easy first question: goldfinches have black wings while canaries have yellow wings. S sees that the wings are black (this is the clue) but S does not appreciate that black wings indicate a goldfinch (S misses the clue). So S answers, 'I don't know'.

We want to say that S doesn't know that the bird is a goldfinch. Yet it seems that S's evidence rules out all relevant alternatives. For situations with goldfinch-perceptions but no goldfinches are skeptical scenarios and usually regarded as irrelevant.

Anthony Brueckner (in Analysis 2003) argues that Lewis's theory falls prey to Missed Clues not because it is a RAT but because it is a PET, a Purely Evidentialist Theory on which knowledge is a matter of available evidence rather than justified belief. What the student lacks isn't evidence, but justified belief (based on her evidence).

However, Lewis's theory isn't a PET. On Lewis's theory, S knows that P iff S's evidence is incompatible with all relevant non-P possibilities. Justification and belief enter via the Rule of Belief, which says that if S assigns, or ought to assign, non-negligible credence to a possibility, then that possibility is relevant. (In stating the rule, Lewis speaks of belief rather than non-neglible credence, but his applications of the rule only make sense on the wider reading.)

Call an epistemic situation in which a subject has experiences E normal if the subject believes to have E and is justified in that belief.

Now at least on a certain, not too far-fetched understanding of justification, the Rule of Belief entails that in any normal situation, if the subject knows that P on Lewis's analysis, then the subject also has a justified, true belief in P.

Proof. Suppose S knows that P on Lewis's analysis but lacks justified true belief. P must be true by the Rule of Actuality, so the remaining possibilities are that either a) S fails to believe P or b) S fails to be justified in believing P.

Case a. S fails to believe P. Then S assigns non-negligible credence to some non-P possibilities. So by the Rule of Belief, these are relevant. Since S's evidence is incompatible with all relevant non-P possibilities (for S knows P), her evidence is therefore incompatible with some possibilities to which she assigns non-negligibly credence. Hence S assigns non-negligible credence to possibilities where she has other evidence, i.e. other experiences, than she actually has. The situation is not normal.

Case b. S isn't justified to believe P. Then S ought to assign non-negligible credence to some non-P possibilities. So by the Rule of Belief, these are relevant. Since S's evidence is incompatible with all relevant non-P possibilities (for S knows P), her evidence is therefore incompatible with some possibilities to which she ought to assign non-negligible credence. But if S ought to assign non-negligible credence to possibilities in which she doesn't have the evidence she actually has, she isn't justified in believing that she has that evidence. Again, the situation is not normal.

Now we have a puzzle: Missed Clues cases appear to be very common, so it is hard to believe that they always take place in non-normal situations. Indeed, can't we just stipulate that the student S in Schaffer's example is perfectly aware of her experiences, and justifiedly so? Certainly that wouldn't help her in identifying the bird! But if she doesn't believe that the bird is a goldfinch, it follows by what I've just proven that she also doesn't know that it is a goldfinch. So something must have gone wrong. But what?

Tim Black (in the Australasian Journal of Philosophy 2003) suggests that Schaffer has overlooked a non-skeptical alternative: S does assign non-negligible credence to possibilities where she has the same (goldfinchy) experiences without seeing a goldfinch. But these are not skeptical possibilities in which, say, somebody has painted a canary to look just like a goldfinch. Rather, they are possibilities where ordinary canaries look like actual goldfinches.

That sounds plausible to me. How lucky that it is possible for a canary to look just like a goldfinch! Unfortunately, we can't rely on such luck in all Missed Clues cases.

Consider this variation: S watches a DNA sample being taken from the bird and analysed. She sees the result of the analysis: the bird's entire DNA sequence. Obviously this won't help her to find out whether the bird is a goldfinch. What are the relevant alternatives she gives sufficiently high credence to? They can't be possibilities where ordinary canaries have the DNA that actual goldfinches have, for there are no such possibilities. (I assume that a bird's DNA decides whether it is a goldfinch or not. If you disagree, just replace the DNA evidence by evidence for whatever you think decides that the bird is a goldfinch.) This time, the only possibilities left open by S's evidence really are irrelevant skeptical possibilities where, say, the DNA sequencing went completely wrong.

So this time it seems we can't escape the conclusion that S knows that the bird is a goldfinch. If we still assume the situation is normal it follows that S believes that the bird is a goldfinch. More precisely, it follows that S assigns relatively high credence to that proposition. But didn't we assume the contrary?

No. Consider the worlds that might, for all S believes, be her world. In all of them, the bird she is looking at has such-and-such a DNA sequence. But all birds with such DNA in all possible worlds are goldfinches. So in all the worlds that might, for all S believes, be her world, the bird is a goldfinch.

Still, S doesn't believe that the bird is a goldfinch. The reason is that "S believes that P" is not true iff S assigns relatively high credence to the proposition ordinarily expressed by "P". On a coarse-grained conception of propositions this is obvious, as otherwise "S believes that P" would be true whenever "P" ordinarily expresses a necessary proposition, which clearly it is not. More generally, on a coarse-grained conception of propositions, belief is closed under strict implication, whereas belief ascriptions -- at least on their straight-forward interpretation -- are not.

The same is true for knowledge. On Lewis's account, the content of our knowledge is a set of (centered) worlds. It follows that knowledge is closed under strict implication. But this goes against (the straight-forward interpretation of) our knowledge ascriptions. The student knows that she sees a bird with such-and-such a DNA sequence. That strictly implies that she sees a goldfinch. But she doesn't know she sees a goldfinch. In the same manner, not everybody knows that Hesperus is Phosphorus, that all Ophtalmologists are eye doctors and that x^n + y^n = z^n has no solution for n > 2.

It turns out that what genuine Missed Clues cases -- those that can't be solved by Black's strategy -- really show is that knowledge ascriptions aren't closed under strict implication (on their straight-forward interpretation). Everyone who puts forward a coarse-grained analysis like a RAT is probably well aware of that. What is needed to answer Missed Clues cases is a plausible semantics for knowledge ascriptions.

How could that look like? The simplest idea (roughly Stalnaker's) is that when we attribute knowledge that P, we do not attribute knowledge of the proposition that is ordinarily expressed by "P". Rather, the relevant proposition is some other proposition somehow associated with "P" in the context at hand, namely something like the proposition that would ordinarily be expressed by "'P' is true" (in the given context). The student doesn't know that the bird is a goldfinch because she assigns non-negligible credence to possibilities in which "the bird is a goldfinch" is false. Not because in these possibilities, the bird isn't a goldfinch, but because the word "goldfinch" there denotes some other kind of bird. On this simple account (too simple I think), the evidence S lacks is linguistic evidence about the meaning of "goldfinch".