Grue-some confusion

Having concluded at the end of my previous post that the study of statistics has helped me to appreciate the value of the philosophy of science, it is only fitting to point to an instance of the reverse: Statistics can be a resource for solving philosophical problems.

Nelson Goodman’s new riddle of induction is supposed to show that no purely syntactical theory of confirmation is possible (i.e. one that does not depend on the meaning of the terms that appear in the argument, like “raven” or “black”). Here’s an outline of the argument. Goodman introduced the term “grue” and defined it thus: an object is grue if it is green before a certain date D or if it is blue after D. So an emerald is grue before D, while the sky is grue after D. (Note well that nothing changes color: This is all about the terminology we use to describe things.) Obviously, grue-like terms make it difficult to generalize from empirical observations. Even if we have examined a vast number of emeralds under all kinds of conditions and have found all of them to be grue, this fact does not generalize. After the future date D, we will encounter emeralds that are not grue. Thus, it is entirely hopeless to attempt to specify how many observations or how many variations of circumstances are needed before we can arrive at the general claim that “all emeralds are grue”.

It is a sound intuition to think that something must be fishy about grue-like terms. However, it has been difficult to show why precisely grue-like terms are inadmissible in science. Many attempts to solve the problem failed: Most famously, attempts to show that time-relative terms in general are inadmissible didn’t succeed, despite their intuitive plausibility. It was also proposed that the relevant distinction might be between terms that are “projectible” and those that are not, and this led to a search for criteria of projectibility. Others suggested that true confirmation is only possible where so-called “natural kinds” are concerned. In general, many philosophers concluded that the grue-problem may be intractable and may represent a deep problem for all theories of confirmation.

However, I think that a robust understanding of the problem (or much of the problem) was eventually found — an understanding based on statistical thinking. It is an excellent instance of progress in philosophy of science. Here’s a brief review of the proposed solution. Goodman was thinking about a sampling operation: You look at n members of a population in order to form an opinion about the properties of the entire population. To use his example, you conclude that all emeralds are green from sampling n emeralds. To use a more realistic example, you might want to predict how a country is going to vote based on a sample of 2000 likely voters. Now, it is well known that sampling fails if certain assumptions aren’t met. One of these assumptions is that the act of sampling must not alter the property being sampled. If my asking people “who will you vote for next Tuesday?” causes them to feel more established and therefore to vote for the incumbent party, my survey will overestimate the incumbents’ share of the vote. It turns out that something similar holds for the term grue: The fact that I sample an emerald before date D makes it “grue”, while otherwise it would be “not grue”. Thus, my sampling of an emerald can change its “grueness”. Clearly, then, this violates the rules of sampling. It is this statistical reasoning, and not some elaborate philosophical theory, that explains why “grue” is an inadmissible predicate.

The matter is, of course, more complex than I make it out to be; and there may indeed exist instances of grue-like problems that cannot be treated like this. For those who wish to go beyond my three paragraphs, I recommend this paper by Peter Godfrey-Smith.