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.