Rationality

I came across a pretty convincing solution to the raven paradox given by Goodman and Scheffler*.  They grant that evidence that confirms a theory T also confirms any equivalent formulation of T; however, they propose that evidence only “selectively confirms” a theory if it both confirms T and falsifies the contrary of T. 

In order to understand the solution, one must know the difference between the contradictory of a proposition and the contrary of a proposition.  Given any universal conditional statement (All F’s are G’s), the contradictory of the statement is its negation: It is not the case that all F’s are G’s, and equivalently, Some F is not G.  The contrary to a universal conditional statement negates the consequent: All F’s are not G’s, and equivalently, No F’s are G’s.  In terms of ravens:

Theory (T) - All ravens are black. 
Contradictory (TCD) - There is some raven that is not black. 
Contrary (TCR) - No ravens are black.

Back to Goodman and Scheffler’s idea, the observation of a black raven selectively confirms our theory because it confirms T and it falsifies TCR.  In other words, an observation of a black raven supports our theory that all ravens are black, and it falsifies the contrary theory that no ravens are black. 

The ravens paradox is counterintuitive because it allows non-black non-ravens (such as a white shoe) to confirm T because they confirm the equivalent theory “all non-black things are non-ravens.”  However, the white shoe, on Goodman/Scheffler’s account, does not falsify TCR.  We can still beleive that no ravens are black (TCR), even in the face of our observation of a white shoe.  So, the white shoe confirms (or satisifes) T, but it does not selectively confirm T.  The reason that the raven paradox seems like a paradox is because we intuitively want our evidence to selectively confirm our hypotheses, not just minimally or trivially confirm them.  In other words, selective confirmation is what interests scientists.

The reason that this works as a solution is that the contraries of two logically equivalent statements are not themselves logically equivalent.  Using the ravens again: (T) “All ravens are black,” is logically equivalent to (N) “All non-black things are non-ravens,” but the contrary of T (TCR) “No ravens are black,” is NOT logically equivalent to the contrary of N (NCR) “All non-black things are ravens.”

The idea of selective confirmation still permits us to call the white shoe confirming evidence for T, and so does not solve the paradox in this sense.  However, it does tell us that the white shoe is not selectively confirming evidence for T, thereby justifying our intuitions that such evidence is irrelevant to sceintific inquiry.  QED?

*Scheffler, Israel and Nelson Goodman. “Selective Confirmation and the Ravens: A Reply to Foster.”  The Journal of Philosophy Vol. 69, No. 3. (Feb. 10, 1972), pp. 78-83.

Ok, I’m sorry, but a few more things need to be said about the ravens. I’ll be quick about it. The idea, again, is that the observation of a non-black non-raven should constitute evidence that confirms our theory that all ravens are black.

(Case 1) The Picture of a White Raven [Richard's example] – Suppose we observe a photograph of a white raven. The photograph itself is not black and it is not a raven (it’s a picture!), so it should be confirming evidence for “All ravens are black,” but obviously it would seem to do more to disconfirm the theory than it would to confirm it.

(Case 2) The White Crow – If the picture of a white raven seems like a cheap move, consider a more plausibly scientific example. Suppose an ornithologist (the outdoor variety) observes a white crow. The white crow is not black and it is not raven, so it confirms our notorious theory. However, like ravens, crows are usually black; they are also probably closely related to ravens (perhaps in the same family), so if albinism occurs in crows, we may have good reason to believe that it could happen in ravens. This is, perhaps, a way to tie explanation and evidence together. The explanation for why the crow is white seems as if it could serve as a plausible explanation for why ravens might be white. In any case, even if the observation of a white crow is (minimally) confirming our theory, it seems to be doing more to disconfirm it.

(Case 3) The 10-Foot Tall Man – This is just a related example to the two above. Suppose our theory is that no man on Earth is 10 feet tall. Imagine, then, that we come across a fellow who is 9′10”. He confirms our theory (because he is a man and he isn’t 10 feet tall), yet casts extreme doubt upon it. These kinds of cases show that there must be more to confirmation than the kinds of concepts that Hempel is dealing with in the paradox of the ravens.

This semester I have the biggest head start that I’ve ever had on a seminar paper. I’ve already had my topic approved for philosophy of science – and I already have the guts of the arguments I plan to make. The subject is the classic paradox of the ravens (which somehow always manages to rear its head whenever more than two philosophers congregate at a bar); hence the title of this post: Return of the Ravens.

I’ve been battling this paradox since my first semester as an undergrad philosophy major (thanks to Dr. Johnsen’s epistemology class). The paradox makes perfect logical sense, yet it is massively counter-intuitive. I once tried and failed at arguing against the equivalence condition. This condition, which is generally accepted, states that any evidence that confirms a theory (T) also confirms any theory to which it (T) is logically equivalent. Hence any evidence that confirms “All ravens are black” also confirms the logically equivalent theory “All non-black things are non-ravens.” Of course, most things are non-ravens. I am a non-raven who is writing a non-raven for my non-raven on a non-raven with my non-raven Amy sitting next to me on the non-raven. You get the idea.

Silliness aside, the paradox is usually downplayed by acknowledging that evidence comes with degrees of confirmation. The black raven is relatively strong evidence for the theory that all ravens are black; likewise, the blue pen is relatively weak evidence. None the less, the blue pen is, in some small degree, confirming evidence. If the blue object was a raven rather than a pen, it would, after all, falsify the theory. The paradox infamously opens the door to “indoor ornithology,” but given that non-black non-ravens only constitute a minutia of confirmation for our theory about ravens, indoor ornithology is hardly feasible even if we accept the consequences of the paradox.

Still, there is something fishy about the whole thing. On that note, suppose our evidence is a red herring; which theories does our new evidence confirm? It surely confirms the theory that “all non-black things are not ravens” and by that token, it confirms that “all ravens are black.” It also surely confirms the theory that “all non-white things are not ravens” and by that token, it confirms “all ravens are white.” Our red herring confirms contradictory theories! As it turns out, the fishy evidence does not pose much of a threat. By scientist’s lights, if a bit of evidence equally confirms theory A and theory not-A, then the evidence is simply ignored.

Ok, so scientific pragmatism aside, do we still want to accept the dust bunnies in the corner of the room as confirming evidence for our theory about ravens? I do not. I would rather not admit that the dust bunny is any kind of evidence whatsoever. I think the basis for rejecting the paradox lies in breaking the parallel between the two equivalent theories. One way to do so would be to show that the theories differ in meaning. By analogy, consider the terms “renate” (has a kidney) and “cordate” (has a heart): they have the same extension, which means that all and only things that have a kidney have a heart; but the two words have different intensions, they have different meanings. In order to throw a wrench in the ravens paradox, one must show that the two theories, “all ravens are black” and “all non-black things are non-ravens,” have the same extension (pick out the same objects), but different intensions (they differ in meaning).

This is where I step in. Well, really this is where one of my heroes, Fred Dretske, steps in. Back in 1972 he wrote an excellent article called “Contrastive Statements” in which he points out a bit of disparity in the raven theories. He briefly explains (in one paragraph and a footnote) that the reason or explanation for one theory might not be the same as that of the other. As is always the case, this idea is easier said than proven. Dretske gives the example that the presence of gene X might be a reason for believing “all ravens are black,” but the presence of gene X does not explain why we should believe “all non-black things are non-ravens.” More importantly, but harder to show, is the fact that the lack of gene X also does not constitute a reason for believing the “non-raven” theory. Sure, pick any non-black thing and you will find both that it lacks gene X and it is not a raven. But does the missing gene give us a reason to form such a belief about non-black things?

I have a few arguments in mind that can be used to show that logical equivalence does not entail semantic equivalence in this case. First, notice that the presence of gene X is not a reason to believe that anything other than a raven will be black. But the lack of gene X is a reason to believe that things other than non-black things will not be ravens. To put it another way, the presence of gene X is only relevant when the subjects are ravens. However, the absence of gene X could be relevant when the subjects are black things. The absence of gene X might explain, in some small way, why the black table is not a raven; but the presence of gene X will never explain why the black table is black (trivially, since no table will have gene X). These examples show that the explanatory power of gene X is different in the two raven theories.

The second argument may be a little easier to follow: (1) Take anything that is a raven; you cannot explain black pigmentation without the fact that it has gene X. [This rests on the assumption that if you do not know about gene X, then you don't know why the raven is black. Seems plausible to me.] (2) Take anything that’s not black; you can explain why it is not a raven without the fact that it lacks gene X. [At least, in so far as one can explain why something is not a raven, (2) seems true. How about: The blue iPhone is not a raven because it is inorganic?] The logical form of (1) is: For all R, X is necessary to explain B. And the form of (2) is: For all ~B, X is NOT necessary to explain ~R. This argument shows that even though the two raven theories pick out the same sets of objects (namely, things that are and are not ravens or black), they can be described by different sets of true propositions. If we accept (1) and (2), then the two logically equivalent theories are not semantically equivalent and thus have different intensional content.

You might have noticed that my argument does not prove that the two theories have different sets of confirming evidence (if you noticed this, a kudo to you!). You may have also noticed that because I did not produce such a proof, the paradox still remains a paradox. “Wait a second,” you interject, “the red herring is still confirming evidence for both theories.” You begin to feel a bit disgruntled and misled by my long-winded proof that didn’t solve anything. “What, exactly, did you accomplish here? And why have you started writing in prose?,” you ask, now quite befuddled. First, for your astute and well-timed inquisitiveness, I reward you with yet another kudo (that makes two kudos!). Second, the last of your questions will simply have to remain a mystery.

Finally, let me explain what my argument has accomplished. Even if the paradox cannot be solved, if the best we can do is accept random bits of seemingly irrelevant observations as minuscule, but confirming evidence, then at least we have more (and better) reasons to disregard that evidence. If nothing else, my argument shows that our intuitions against the raven paradox, and specifically against the equivalence condition (and non-ravens), are justified since the two theories have different meanings in a relevant scientific sense. A further project might be to tie together the concepts of explanation and evidence. Once that is done, it would be an easy step to show that since (i) two logically equivalent theories may have different sets of explanations (via my arguments), and (ii) evidence for a theory T is essentially connected to the explanation for T (via a new argument), we can deduce that two logically equivalent theories may have different sets of evidence (QED!).

If I can pull this one off, maybe the ravens will finally be put to rest. When it’s all said and done, though, I suspect the ravens will always have the last word.