Monday, December 12, 2016

The Raven Paradox

Since I didn’t succeed to take a picture of a raven, instead I have photographed an orange.

Last week I talked about the liar’s paradox and the Sorites’ paradox. Paradoxes are self-contradictory reasonings, for short. They have to be distinguished from fallacies, which are invalid or otherwise faulty reasonings that often seem correct (or to some they do) and therefore deceive the mind. I have talked about paradoxes and fallacies in my blogs before.
Paradoxes seem unsolvable but once they have been solved they appear to be fallacies. An example of a paradox that became a fallacy is Zeno’s paradox about Achilles and the tortoise. (Zeno of Elea lived from 490-430 BC) Achilles and the tortoise are in a footrace and the tortoise starts, say, 100 metres ahead of Achilles. Then, so Zeno, Achilles cannot outrun the tortoise, for when he has done 50 metres, the tortoise has done, say, 5 metres in the same time, and when Achilles has done half the distance to the tortoise that then remains, the tortoise has advanced again, and so each time that Achilles has done half the distance separating him from the tortoise, the latter has moved forward again, etc. Only in modern times this paradox could be solved, which made it a fallacy. The flaw in the reasoning is that Zeno does as if the time doesn’t move on.
When browsing on the Internet for more paradoxes, I came upon one that is interesting from a methodological point of view, namely Carl G. Hempel’s Raven Paradox, which he expounded originally in an essay published in 1945 (republished in 1965; see footnote). Hempel himself calls it the “confirmation paradox”, and just this name shows why it is interesting, for Hempel’s idea is that we must try to find confirming evidence for our hypotheses, which brought him into conflict with Karl R. Popper, who says that we must try to falsify hypotheses (and must formulate them that way, that this is possible). I’ll quote the paradox not from Hempel’s essay, but from a website that explains it without the logical notation used by Hempel:
“[T]he Raven Paradox begins with the apparently straightforward and entirely true statement that ‘all ravens are black.’ This is matched by a ‘logically contrapositive’ (i.e. negative and contradictory) statement that ‘everything that is not black is not a raven’—which, despite seeming like a fairly unnecessary point to make, is also true given that we know ‘all ravens are black.’ Hempel argues that whenever we see a black raven, this provides evidence to support the first statement. But by extension, whenever we see anything that is not black [and not a raven; HbdW], like an apple, this too must be taken as evidence supporting the second statement—after all, an apple is not black, and nor is it a raven.” (http://mentalfloss.com/article/59040/10-mind-boggling-paradoxes)
So, the evidence that apples are not black while ravens are so by hypothesis seems to confirm that ravens are black, indeed.
In 1967, the British mathematician Jack Good wiped the floor with the Raven Paradox. I haven’t read his article, but only a short version of his argumentation. However, already immediately after I had read Hempel’s reasoning it seemed counterintuitive to me and with right, for it’s not correct: There is simply no relation between the colour of apples and the colour of ravens and between apples and ravens, so what could apples tell about ravens? Whether apples are red, white, yellow or black, it’s quite well possible that there are white ravens, and the colour of apples cannot confirm or disconfirm the existence of this variety of ravens. The Raven Paradox is simply a fallacy.
Talking about fallacies, what the author of the website just quoted, Paul Anthony Jones, probably didn’t realize is that also his explanation of the Raven Paradox contains a fallacy (or to say it more friendly: it’s not correct). For immediately after the passage I quoted he goes on:
“The paradox here is that Hempel has apparently proved that seeing an apple provides us with evidence, no matter how unrelated it may seem, that ravens are black. It’s the equivalent of saying that you live in New York is evidence that you don’t live in L.A., or that saying you are 30 years old is evidence that you are not 29.”
The first sentence of the quotation is okay, and that’s what the Raven Paradox mistakenly says. However, Hempel’s reasoning is not the equivalent of the two examples that follow then, for there is a relation between living in New York and living in LA in this case, and there is a relation between being 29 or 30 years old, and this relation is you. You can live in only one place, which can be described by its geographical coordinates. When those coordinates are those of NY, they are simply different from those that belong to LA, although this doesn’t say something, of course, about what the coordinates of LA are and whether or how they relate to the coordinates of NY. In the same way, the fact that your age is 30 years old does say that you are not 29, for your age is a measurement on a time scale and just like that you cannot be at two places at the same time, you cannot have two ages (and, by the way, there is a relation between 29 and 30 years as such, namely that they indicate points on the same time scale).
The upshot is that one must not compare apples with oranges, not to speak of apples and ravens.

Note: Carl G. Hempel, Aspects of Scientific Explanation and other Essays in the Philosophy of Science. New York: The Free Press, 1965; pp. 14ff.

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