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The classic example of scientific induction is the proposition "All ravens are black". But I know at least one text which instead uses

All swans are white.

This is cute because it elegantly demonstrates the limit of scientific induction (namely, you cannot draw conclusions about an infinite number of cases from some finite number of examples). Because when Australia was colonized, a species of black swans were discovered.

This leads into a nice discussion of what counts as inductive evidence. It's all very well to say "All swans are white": it sounds pretty clear-cut, and all you have to do is wait until a single counterexample comes along to disprove it.

What counts as a swan, and what counts as being non-white? What is the logic of our naming? The simple answer is that there is no one simple answer. "Swan" and "white" both vary somewhat depending on context.

The swan is a very distinctive bird. Although there are several species, they are all unmistakably like each other, and quite unlike geese or ducks. All swans are white, at least if we retreat to the England of the 1600s, where we are speaking English but have not yet been swamped by explorers such as Dirk Hartog (1616) or William Dampier (1688), who were the first to explore Western Australia, perhaps the first Europeans to see black swans. I don't know the true story*, but let's suppose we're in England in 1690, Captain Dampier is approaching Portsmouth after his navigation of the seven seas, and on board, pickled in rum, he has the corpses of several interesting New Holland animals, including a bird that looks almost exactly like a swan, except that it's black.

How do we assess "All swans are white" while we are still ignorant of these new-found creatures? How then do we re-assess it when Dampier docks and shows us his specimens?

First, the obvious dimension: pure white, snowy white, or an off-white or cream? This terrible word "really". Is an off-white swan really white? (Hint: If you use the word "really" as part of a philosophical argument you're probably on the wrong side.)

Does it make a difference if it's off-white because its feathers are that colour; - if it's greenish because it's been swimming in stagnant water? How about if it's pinkish because its diet consists of small crustacea that turn it pink? (I have an idea this is actually true of the flamingo, but even if it's not, it's enough that it be possible.) So this swan deprived of its habitat or diet, thoroughly washed and brushed, might be a somewhat different colour to its state out in the wild. (Which is its real condition?) Do we say "that's not a white swan, but it would be if you washed it"? Or do we say "that's a white swan but it's covered in green slime"? Does it matter?

Swans have two legs, two eyes, and a beak too. But if I chop off one of its legs, does it cease to be a swan? Of course not. Why not? Well instead of insisting "All swans have two legs" you could say "Most swans have two legs". Most? But they all do. Apart from (i) freak mutations, and (ii) accidental or deliberate mutilations. "All swans have two legs except those that have some other number"? No: there is a real truth about swans, a natural property of them, that they generically have two legs. Exceptions like these don't invalidate the rule.

Think about a case where it's clearly not true that "All X are Y". All cats are grey? No: you get grey, white, black, ginger, tortoiseshell, and tabby cats. You get yellow, green, and blue budgerigars. These are clear examples where species vary, in the way in which swans and ravens do not vary. A blue budgerigar is not a kind of queer exception to a general run of green budgerigars: you don't have to puzzle over whether it's "really" a green one with some odd dietary or environmental condition. But - of course you can get blue ones turned black with soot, etc.

The point I'm trying to make is that there's no simple one-size-fits-all answer to "What colour is X"? You can't just look at something and say what colour it is. The question needs context: do you mean, underneath, or at the moment, or during the day, or under this sodium light or in their deep-ocean habitat or...? A thing can be different colours in different circumstances (not just lighting but living conditions), and you just can't pick one of these out and say "that's its TRUE colour" (or REAL colour) or "its colour".

Dampier's ship arrives. He unloads these birds. They're remarkably like swans except they're black. Are they swans? Do we call them swans? We don't have to: Dampier might have learnt the native name (kookaburra, currawong, or whatever), and we might happily call it that. Then we say a currawong is very similar to a swan, but currawongs are black whereas swans are white.

Or we might choose to call this a "black swan" because it looks very similar to a swan, but is black, whereas swans are white. Why not? We did that with tigers. The Tasmanian tiger was so named because it looked (somewhat) and behaved like a tiger; but of course it's marsupial, not a close cousin to the Siberian tiger and Bengal tiger. It's just a name. The name "Tasmanian tiger" is fine, as long as you realize it's a descriptive, not a genetic name.

So this new West Australian bird, we can descriptively call it a name based on "swan": it looks like a black swan so let's call it "black swan": but that doesn't make it a swan. If this is why we named it that, the we have two animals, (i) the swan (= the traditional English white bird), and (ii) the black swan. All swans are still white, in this classification.

In fact, the black swan is of the same genus as the English swans, so it is reasonable to say they are swans. But a genus is not a natural kind: it doesn't automatically follow that you naturally lump anything in the same genus under the same common name "swan".

* sid's write-up on the black swan says the date of discovery was 1697; but there is an interesting earlier example: in 1660 there was a pub in Lincoln called the Black Swan. My source for this is some British postage stamps issued in August 2003 featuring pub signs. This does not imply prior zoological knowledge or even fortuitous guessing, since pub signs often feature heraldic beasts in unnatural colours.

The example is often used as a philosophical football in the philosophy of science. The assumption usually made by philosophers of science is that items of scientific knowledge (or scientific theories) have a special property that distinguishes them from other knowledge (or theories). The task is then to exactly delineate what this specialness consists in.

Karl Popper used the example to make clear his own differences with what's sometimes called (at least by Popper) naive inductivism: the view that scientific experiments serve to corroborate scientific hypotheses (after they are done in sufficient numbers). Popper describes himself as a deductivist, and holds that "all swans are white" counts as a scientific theory for the simple reason that one may find a non-white swan, and that the statement is therefore falsifiable. Popper holds that it's this characteristic of falsifiability that distinguishes scientific theories from non-scientific ones, and that a theory that isn't falsifiable should never be thought scientific.

Popper was very fond of saying that while any finite number of experimental observations cannot succeed in verifying a single scientific hypothesis, it only takes one observation to falsify it.

Popper's arch-antagonist in the field, Thomas Kuhn, prefers to define scientific theories as simply those thought up by scientists - a sort of social definition. By introducing his famous idea of a ruling paradigm - a sort of background knowledge that's assumed by most workers in a given scientific field, he distinguishes between normal science and extraordinary science: normal science is science which is done on the assumption of the ruling paradigm, and extraordinary science is science which aims (or succeeds) at creating a paradigm shift - a change in the basic methods or assumptions, the ruling paradigm, of the field in question. The classic examples of extraordinary science are the Copernican and Einsteinian revolutions.

In terms of the swan example, Kuhn supposes that some explorers discover a black creature that otherwise resembles a swan, and points out that contrary to Popper's view, where this straightforwardly falsifies the hypothesis that "all swans are white", we in fact have the choice of whether to count "whiteness" as a necessary property of swanhood, or whether to count the other resemblances as enough to lead us to count the creature in question as a swan. Kuhn points to the fact that we use genetics as a paradigm for informing us whether a given creature is to be counted as a swan or not, and that this paradigm may shift - who knows? It may shift in such a way that we no longer consider the newly discovered bird to be a swan.

Now, the point of this isn't that we should start thinking about reclassifying creatures according to what colour they are! Kuhn is a philosopher, not a naturalist. His concern is to point out that there's a flaw in Popper's methodology, because it takes no account of the paradigmatic features which act to inform (or at least so Kuhn claims) our interpretations of whether a given hypothesis is indeed falsified by a particular observation.

Also, as Imre Lakatos has mentioned, taking a different tack to Kuhn, what is often observed in the history of science is that when an observation is made that appears to contradict (i.e. to falsify) some long held scientific theory, the most common result is that auxilliary hypotheses are introduced in order to preserve both the theory and the observation. Suppose that we have a couple of reliable reports of a black, but otherwise swan-like, creature, which has unfortunately eluded the geneticists who chased after it (though they captured it on film). If the "all swans are white" theory is sufficiently entrenched, it may be hypothesised that some irresponsible vandal had in fact captured the swan and painted it black, and so the observation, after the introduction of that ad-hoc hypothesis, can be held to co-exist with the theory.

Another factor in the dispute is that scientific observations are often themselves wrapped up in theoretical considerations - as for example in particle physics, where heavy doses of physical theory are necessary in order to interpret the experimental results: obviously no-one has seen a subatomic particle with the naked eye, this is only done through much complex machinery and software - and therefore it may become hard to disentangle the observations from the theories which they are supposed to falsify.

Technically, the idea that we can only deduce that "something is wrong somewhere" - and not necessarily in the theory under question - from a mismatch between the predictions of a theory and an experimental result, is known as the Quine-Duhem thesis.

Under the onslaught of such anti-falsificationist arguments, Popper later (for example in his Postscript to the Logic of Scientific Discovery) weakened his position slightly, introducing the idea of a 'metaphysical research programme' - theories which "are usually not yet of the character of testable scientific theories. They may become scientific theories [...] much harder to criticize than are [scientific] theories--and much easier to retain uncritically." (Popper, Quantum Theory and the Schism in Physics, 1982). We may think of the ancient Greek theory of atoms (that matter has some fundamental smallest division) as such a programme. The idea is strikingly similar to Kuhn's idea of a paradigm.

It might seem that Popper is throwing out the baby with the bathwater here, because the strength of his view is that it attempts to describe the logical structure of scientific knowledge, not to give a practical recipe for achieving it, and stating that a hypothesis may "become testable" seriously undermines his idea of falsifiability - how do we know whether or not theories which are normally considered untestable (and hence not part of science) like Freudian Psychology, for example, may not one day become testable?

In a scientific hypothesis, in contrast to our ordinary use of language, we normally demand exact denotations from our terms - any fuzziness in the application of the term indicates that we have further work to do, either by experimentally determining whether the problem object is to be included in the denotation, or by coming up with a better (less fuzzy) term.

The terms 'science' and 'scientific' themselves, however, are perhaps not properly scientific ones, in that sense, and possibly it is simply a philosophical mistake to assume (as Popper seems to have done) that they are.

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