A Constrained Rant by The Famous Brett Watson, 11-Apr-2003.
Scientific statements or hypotheses tend to be extremely broad in their claims, if not altogether universal. For example, one scientific theory might claim that all massive objects are attracted to each other with a force that is proportional to their combined mass, and inversely proportional to the square of the distance between them. It's a neat and precise theory with some very practical and measurable consequences. Let's call it the theory of Universal Gravitational Attraction, since it makes a claim about all objects everywhere — a universal claim.
Such a theory is all very well and good, but science also requires evidence for a theory. The question of what constitutes good scientific evidence for a theory has been hotly contended by many scientists and philosophers of science over the years, and it seems unlikely that there will ever be a single universally-accepted formula for it. One relatively early idea was that of inductivism. This idea is based on the accumulation of favourable evidence. To prove my theory of Universal Gravitational Attraction, for example, I might devise an array of experiments in which I attempt to measure the attraction of massive bodies to each other, and the more results I obtain which match the claims of my theory, the more confirmation I accumulate.
I can never really prove that my theory is as universal as I say it is, but with increasing amounts of favourable evidence, I can feel increasingly confident that the theory is in fact a Law of Nature, and not just lucky coincidence, or something like that. This seems all very well and good, but certain people have thrown various spanners in the works of inductivism over the years. One such spanner-thrower was a logician by the name of Carl Hempel, who came up with a problem called the "raven paradox". Having become familiar with this problem myself, I thought I'd try it out on the Average Man In The Street, and the following conversation ensued.
I told Mr Average that I'd come up with a scientific theory, and I had some evidence for that theory which I'd like to share with him. My theory was that "all ravens are black" — a theory which didn't seem to impress him much, but no matter. I then showed him two ordinary computer floppy disks which I'd picked up off my desk beforehand: one was black, and the other was red. The black one, I explained, was not very interesting; but, triumphantly waving the red one, I claimed that here I had concrete evidence for my theory. Mr Average seemed pretty dubious about this claim, so I asked him if he understood why the red floppy disk was evidence, whereas the black one wasn't. Without much conviction, he suggested that perhaps I thought red was "the colour of evidence". But that would be base superstition; I was employing logic to the situation — Mister Spock would have been proud of me, I'm sure. So I explained the logic behind my claim.
The statement, "all ravens are black", is logically equivalent to a number of other statements. For example, instead of saying "all ravens are black", I could just as well say "no non-black thing is a raven", or "if something isn't black, then it's not a raven." These statements are all logically equivalent: there is no condition under which some can be true and some false; they all make the same claim, just in different ways. That being so, I can find confirming instances for any of these statements, and I'm still being a good inductivist. So far so good? Well, I didn't have any ravens handy, but I did have some floppy disks, and I noticed that one of them wasn't black, and it also wasn't a raven! Aha! My first piece of confirmatory evidence!
At this point in the conversation, Mr Average broke up with laughter and told me, "that's crazy!" But where had I gone wrong? I was being entirely logical in my approach to the problem. The red floppy disk was evidence for the theory because it was not black and not a raven; the black floppy disk was not evidence because it was black, but not a raven, and my theory makes no claims about black things which aren't ravens. In summary, I explained, there were two kinds of thing that I could present as evidence for my theory: black ravens, and non-black non-ravens. The red floppy disk fell into the latter category, but the black floppy disk fell into neither.
Alas, Mr Average still continued to laugh and insist that my claim was crazy. On the other hand, I'm sure that if Carl Hempel had been there, he would have been most impressed with my logical rigour. Mr Average thinks I have a problem, and that I'm being a nut case, not a scientist. Hempel would think I'm an admirably clear thinker, and that Mr Average's incredulity is a purely psychological issue with no real basis in fact. Although he never actually said it, I'm pretty sure Mr Average's objection to my approach was this: if I can present a red floppy disk as evidence for the theory that "all ravens are black", then I can present just about anything as evidence for just about any theory. To this, Hempel would probably shrug and say that the situation was a logical necessity, so get used to it.
Is inductivism in serious trouble at this point? Is it really the case that the theory "all ravens are black" is supported by the existence of a red floppy disk? If so, then why does the red floppy disk seem intuitively like it ought to be irrelevant? Is it just a matter of language and psychology, as Hempel suggests, or is there a deeper reason?
In my view, Hempel is absolutely right in his application of logic. Strange though it may seem, the existence of a red floppy disk (or any other thing which is not black and not a raven) is supporting evidence for the "all ravens are black" theory. It just so happens that it's not significant evidence, and the thing which made Mr Average laugh (even if he didn't realise it) was the fact that I was presenting a ludicrously weak and arbitrary piece of evidence as though it were a matter of vital significance.
To explain further, if we wanted to be absolutely sure that the "all ravens are black" theory were true, we would either have to examine every existing raven and determine that they were all black, or examine every existing non-black thing and determine that none of them were ravens. If you grant that the number of ravens in the world is very small compared to the number of things which aren't black, then you will also consider "one black raven" to represent a much larger portion of the available body of evidence than "one red floppy disk", and thus grant the raven much more weight as positive evidence. Assuming that ravens aren't too hard to find, my evidence-gathering is likely to be much more fruitful if I concentrate on ravens, even if non-black things also constitute technically valid evidence.
When viewed this way, the problem doesn't seem so bad, and can even be used to advantage. For example, if it's generally accepted that Sir Percival the Eccentric built one hundred pyramids in his lifetime, but neglected to document their whereabouts, and I theorise that "none of Percival's pyramids are in Perth", then I recognise that I can look for positive evidence in two ways: either by finding things in Perth which aren't Percival's pyramids, or by finding Percival's pyramids which aren't in Perth. If I manage to find all one hundred of Percival's pyramids, and determine that none are in Perth, or enumerate all the pyramids in Perth (if any), and determine that none are Percival's, then my case will be complete. If it's not possible to complete these tasks, then I can still accumulate positive evidence by either approach, and thus build a good inductive argument. One approach may be more fruitful than the other in practice, but both alternatives seem prima facie reasonable in this case.
It's also important to note that not all scientific theories are subject to this logical anomaly. The theory of Universal Gravitational Attraction which I formulated at the beginning of this essay is not one that can be manipulated logically, since it presents a mathematical relationship between things, not a categorical one. Mathematical relationships can be expressed in numerous different ways as well, but I can't offhand think of an example where this causes the kind of unexpected result we see with the paradox of the ravens.
Particularly alert readers may have noticed that the "all ravens are black" theory actually divides up the world into four categories, whereas I've only mentioned three. Two of the categories contain positive evidence for the theory: the category of black ravens, and the category of non-black non-ravens. One category contains irrelevant data: the black non-ravens, of which my black floppy disk was an example. The fourth category is the category of non-black ravens, and my theory makes a very important claim about this category: namely, that the category is empty; that no such thing exists. Each black raven or non-black non-raven supports the theory a little bit, in an inductive sense, but the discovery of just one non-black raven would conclusively disprove it.
If someone doesn't like my theory, then they have a pretty good target to aim at. All they have to do is produce one definitely non-black raven, and my theory is shot to pieces. This seems like a potentially fruitful approach, relative to the gathering of positive evidence, so long as you don't mind taking the approach of falsifying theories, as opposed to providing evidence for them. But falsificationism is a subject for another essay entirely!
Author: The Famous Brett Watson