Of Grue Emerapphires, Bleen Sappheralds and Other Oddities

Ok, folks, here's your space to discuss Goodman's New Riddle of Induction.

Very briefly, the problem Goodman identifies is that any set of observations can justify any prediction about the future whatsoever, in which case, no set of observations really justifies any prediction at all.

The argument runs as follows. For any predicate, P, for which you have observed n positive instances--and hence expect P to be true of the n+1th instance--it is relatively straightforward to construct another predicate, P*, such that the n observed positive instances of P are also positive instances of P*. In which case, one can equally well expect P* to be true of the n+1th instance. However, P* is defined such that the n-1th instance cannot be both P and P*. Furthermore, we can repeat the reasoning, coming up with predicates P**, P***, etc., so that the n+1th instance will have any property at all. So, the n instances provide no more reason to expect the n+1th instance will be P than it does to expect P*, P**, P***, etc. In which case the n instances provide no reason to expect anything in particular.

That's the general formulation of the argument. It is easier to follow using the (now classic) example of the predicate "grue". An object is grue, recall, just in case it is either observed by some date t and green or unobserved by t and blue. Now, all emeralds observed by this date (whenever you're reading this) have been green, and using induction in the familiar way, we expect then next observed emerald to be green. But, having just defined "grue", it is clear that every observed emerald has also been grue. So, using induction in the familiar way, we should expect the next observed emerald to be grue. But, if we treat the time at which you--gentle reader--read this as t, then if the emerald is grue, it will be blue, not green. So the emeralds we have observed to this point provide no more reason to expect the next observed emerald to be green than grue--hence, blue. Furthermore, we can define predicates "gred", "grorange, "grellow", "grurple", etc. such that, by parity of reasoning, we have just as much reason to expect the next observed emerald to be red, orange, yellow, purple and so on. It follows that we really don't have any reason to expect the next observed emerald to be any particular color.

But of course this is absurd. Emeralds we have observed to this date do give us reason to expect the next to be green, and they give us much more reason to expect this than to expect the next emerald to be blue, red, orange, etc. (Nor do they provide reason to expect the next observed sapphire to be green, or blue, etc.--but this is a further development of Goodman's argument.) That is why Goodman's New Riddle is a paradox: an unacceptable conclusion reached from apparently acceptable premises via apparently acceptable reasoning.

The challenge is to come up with an explanation of why past observations of emeralds justify our expectation that the next will be green, rather than blue (etc.). That, it turns out, is surprisingly difficult to do.