Yesterday I mentioned that deciding what counts as evidence for justifying the statement “all crows are black” is a somewhat of a classical problem case. And it is a problem that I often have in mind when theorizing about what justification from evidence consists in. However, upon looking back at my work, I realize that I have never actually provided the solution I have developed to the problem, even though all the apparatus to solve the problem has been. So today I am going to rectify that, by saying what does and doesn’t justify the conclusion that all crows are black, and why.
But first I suppose I should say what exactly the problem is for those of you just tuning in. I think the best way to approach the matter it is to consider how we logically formalize that statement. The usual way is to write ∀x(Cx → Bx), which asserts that for all objects if that object is a crow then it is black. But that statement is true if and only if ∀x(~Bx → ~Cx), or, in natural language, that all non-black things are not crows. Now let us suppose that some evidence, X, confirms ∀x(~Bx → ~Cx), say observing a number of colored items, none of which happen to be a crow. Furthermore, if evidence confirms a statement it must also confirm all statements that follow logically from it (simply because that is how deduction works, and if deduction was impossible then it might seem that we would be unable to conclude anything). Thus this evidence must also confirm the claim that ∀x(Cx → Bx), that all crows are black. But that seems absurd, since we haven’t even observed any crows how can we say that it supports the claim that they are all black? And, even worse, that same evidence, by similar equivalences, supports the claim that all crows are green, that all crows are blue, and so on. Clearly we have done something wrong, although it is not clear at this point exactly where the problem lies.
Of course one way out of this problem is simply to deny that the reversed version, that all non-black things are not crows, is in fact confirmed by the evidence that we supposed it was. This is an intriguing possibility because it heads off the problem from the start, and validates our usual ways of thinking about formalizing such general statements and the intersection between evidence and deduction. Unfortunately, I just can’t see how to make the claim stand up to any scrutiny. Why can’t a number of colored objects confirm the claim that all non-black objects fail to be crows? Any plausible answer would have to involve some claim that the negative classes, those things that aren’t black or aren’t crows, are in some way illegitimate, that we can’t even meaningfully generalize about them. And that is not a path completely untrodden because such negative classes can cause difficulties in other situations. (For example, consider the union of the set of all crows and the set of all not-crows. The result is the set of all things. But the set of all things is not a set. Since the union of two sets always produces sets either the set of all crows or the set of all not-crows must not really be a set at all. And if one of them has to fail to be a set it seems natural to say that the set of not-crows is the illegitimate one. Furthermore, if we associate properties with sets, as some do, then there must be no such thing as the property of not being a crow, which validates this way of thinking to some extent, but now we have no idea what to do with such things (what do they even mean?).) However, that response opens up a can of worms involving how to deal with negative statements in general, if there is something wrong with them, and I would rather not have to go down that road unless we absolutely have to.
Another solution to the difficulty, specifically the fact that all crows being black and all crows being blue seem to be entailed by the same evidence, is based upon the realization that, in a sense, the claim that there all crows are black is compatible with the claim that all crows are blue (and that nothing is both blue and black) when there are no crows. Given that perhaps the statement “all crows are black” is better captured by a joint assertion that there are crows and that all of them are black. When the logical character of the statement is captured in this way then it is indeed the case that “all crows are black” necessarily excludes “all crows are blue”. Formally we would now write the assertion as ∃xCx ∧ ∀x(Cx → Bx). Such a formalization also prevents us from verifying the statement with evidence that doesn’t include at least some crows. And if those were the only problems we were concerned with then we might very well be happy with this solution. However, this new formalization does not really solve the problems, rather they have simply been hidden away. Consider, for example, trying to determine how some measure of the strength with which the evidence supports the statement. For convenience we can think of support as coming in a range from 0 to 1. Since our claim consists of a logical conjunction it is natural to assume that the support of the entire claim by some evidence is equal to the product of the support the evidence provides for both terms in the conjunction. Now if our evidence contains even one crow then the support that the first term has goes to 1, or at least very close to it. Given that all we need to worry about is how well the second term is supported by the evidence. But the second term is just ∀x(Cx → Bx), the statement that we found so problematic. And we can reason the same way we did about support as we did about whether evidence confirmed the statement, namely that if evidence lends support to ∀x(Cx → Bx) it must also support ∀x(~Bx → ~Cx) to the same degree, and vice versa. And now we have a new problem, quite similar to our original, namely that a single black crow plus a large number of other variously colored objects seems to strongly support the claim, and that this support only gets stronger as we add more colored objects. But that seems quite irrational, what it seems should be the case is that the statement “all crows are black” receive additional support only when we come across another black crow, and that determining how well supported the claim is should be indifferent to how many other colored objects we have observed.
Which is why I said that the solution we have just been considering was a solution in name only, and that it really just hid the problem, because the problem really was in determining which kinds of evidence support general statements in a principled way. I propose that what has really gone wrong here is turning the claim that all crows are black into the statement ∀x(Cx → Bx) or any logical construction involving that one. What we are really asserting when we make the original statement is that the frequency of crows being black (versus being some other color) is 100%. Thus this claim is simply one instance of a family of claims involving such members as “50% of crows are black” and “99.9% of crows are black”. There is no easy way, to the best of my knowledge, to transform these assertions into first order logic, so we are free to make up our own notation. I propose %(a)(b,c); where a is the frequency, ranging from 0 to 1, b identifies the objects that the frequency holds over, and c is the property under consideration (b and c then might be taken to be formulas with one free variable). And there is the additional restriction that whatever we put in for b must not be able to “double count” objects; not only must it pick our the objects to be counted it must be such that if a particular physical object is picked out by b then it is impossible for b to pick out as a separate object something that overlaps with the first. For example, if we allow Cx to be the property of being a crow it must not be the case that a complete bird can be C and that a subset of matter that composes that bird, say the entire bird minus a single feather, can also be C. We must forbid such double counting because it makes talk of frequencies meaningless, we have to generalize over a number of discrete and distinct objects or the frequency taken to be observed given some evidence is held hostage to an indeterminacy regarding how many objects there are and how many of them have the property under consideration. Of course this doesn’t have any bearing on the question at hand, but is a problem that has a tendency to crop up in other contexts when using this construction, and I like to be thorough.
In these terms the assertion that all crows are black is %(1)(Cx,Bx), assuming Cx is satisfies the other requirements just discussed. And the assertion that all non-black things are not crows is simply %(1)(~Bx,~Cx). Determining how well these kinds of statements are supported by evidence is something I have discussed previously, and so there is no need to delve into any substantial discussion on that topic here. It suffices to point out that, given how evidence lends support to these claims, that there is no necessary connection between how likely we deem %(1)(Cx,Bx) and %(1)(~Bx,~Cx), except when the evidence completely refutes %(1)(Cx,Bx). But that is not a problem, because the only way that can occur is if we find some object that is a crow and not black. And that evidence also refutes %(1)(~Bx,~Cx), and so this relationship follows immediately from the evidence, and we don’t have to worry that this it is an illegitimate instance of the very kind of logical connection that we were trying to avoid. This formalization also has a number of other advantages, but to conclude I’ll mention just one of them, namely that lacking any crows all statements of the form %(y)(Cx,Bx) are equally likely. Which means that, without any crows, we can’t say anything about how many of them are black, the evidence we collect doesn’t point to any conclusion. And this is exactly how it should be.