As discussed thus far here, the predicate "grue" presents no logical difficulty and implies no paradox. In showing this, I will assume for the moment that grue is defined as above. That is, an object is said to be "grue(t)" if it is green before time t and blue thereafter. Let us attempt to concoct some sort of paradox by applying this predicate to the problem of determining the color of emeralds:
Once we allow this new predicate, the statement that "all observed emeralds have appeared green" seems to support both the conclusions 1 and 2 below:
All emeralds are green.
All emeralds are grue.
But intuitively, we all know that emeralds are in fact green, and not this silly new grue color. Some would say that this constitutes a paradox: how can the premise confirm both the correct conclusion 1 and the obviously false conclusion 2? There must be something very wrong with odd time-indexed predicates like "grue" which is not wrong with non-temporal predicates like "green". Right?
The answer is quite simple, in fact: There is nothing wrong with either conclusion, there is nothing wrong with time-indexed predicates, and there is no paradox. Certainly we know that emeralds to not go about changing color (or growing wings, or turning into black holes) at some particular time t, or ever for that matter. Conclusion 2 is clearly false. But how do we know this? We understand a bit of physics that leads us to believe that if we shine white light through your average crystal (say an emerald) and observe the resulting spectrograph readings at any two times we like during the life of the crystal, they will look pretty much the same, with no nasty shifts in a peak from, say, green to blue between the two measurements.
The crux of our confusion is that we have smuggled this additional information about the behavior of emeralds into the problem and still presumed it was not one of our premises in concluding that 1 is true while 2 is false. If we truly remove all other premises than "all observed emeralds have appeared green," then there is no reason to suppose that this premise does not support both conclusions.
For any who remain unconvinced of the validity of gruesome predicates (as defined above, of course), let me introduce the gruemerald, which I made up, just now. Bully for me. A gruemerald is indistinguishable from an emerald in observed color until midnight of January 1, 3001, Eastern time. Just as Dick Clark proudly announces the US-centric dawn of the 4th millennium C.E., the gruemerald undergoes some hitherto undiscovered radioactive transformation which causes its transmission spectrum to shift a few nanometers down. Hey guess what, it was grue all along. Perhaps we could have predicted the physical change which precipitated the gruemerald's radical shift in observed color (i.e. that it was, in fact, not an emerald but a gruemerald), but certainly not merely by observing that it looked green. And if someone had handed us a big sack full of seemingly identical green-seeming gemstones way back in 2001, we would have no way of knowing which of them might be gruemeralds (and thus grue) and which might be emeralds (and thus green) merely by noting their colors.
A much more convincing paradox arises from Nelson Goodman's actual definition of the offending term:
Now let me introduce another predicate less familiar than "green". It is the predicate "grue" and it applies to all things examined before t just in case they are green but to other things just in case they are blue. Then at time t we have, for each evidence statement asserting that a given emerald is green, a parallel evidence statement asserting that that emerald is grue.
This is a much more troubling formulation
of the problem. For the sake of convenience, let us say time t
is right now, this very second. For all emeralds to be grue, we no longer require that so much as one real emerald
do something as ridiculous as spontaneously changing color before our eyes, merely that all emeralds which we have not yet observed
be blue. Emeralds which were observed before now were green
before we observed them and go on being green
for as long as you're likely to be around to observe them. Other emeralds, on the other hand, will be blue
whenever we get around to taking a look at them and were always blue
The two conclusions still lead to incompatible predictions: if all emeralds are green, the next emerald observed will appear green, while if all emeralds are grue, the next emerald observed will appear blue. Notice, though, that this time it is not so easy to explain the problem away as requiring more information. Let's call G1 the conclusion that all emeralds are "grue" as defined earlier (i.e. they physicaly change color) and G2 the conclusion that they are "grue" as defined by Goodman.
While the previous argument does show that the unaided premise that all observed emeralds have appeared green supports G1 (and the conclusion that all emeralds are green, of course), we would certainly like to rule out the possibility that it supports G2, which concludes that all currently unobserved emeralds are in fact a different color from all the emeralds observed to date. If the inference to G2 were sensible, induction would fail miserably under the weight of a multitude of similar predicates. Every observation would seem to confirm every possible hypothesis about the unobserved.
It is at this point that we cart out the notion that temporal notions of green and blue can be defined in terms of non-temporal notions of grue and bleen to defend against claims that all this time-indexing business is a load of rubbish. We are finally left with Goodman's real grue-bleen paradox.
References taken from Nelson Goodman's The New Riddle of Induction
Other authors have attempted to unravel the paradox as stated by Goodman, and you're free to go read about it yourself. Frank Jackson's Grue is one of the more easily understandable treatises on the subject.
StrawberryFrog: There is absolutely nothing paradoxical about the grue that refers to objects which physically change color at some time. Of course you're correct: the standard green-blue naming convention covers this case quite nicely. The word "grue" is used merely as a shorthand for "green before t and blue thereafter", which is precisely why no paradox arises. The problem comes not from this definition of "grue", but the one I've outlined in the 2nd half of the writeup, which in no way implies that a given crystal will ever change color. It is not a mutable predicate by any stretch of imagination. Under this definition, a green object observed before t is both green and grue for all time, and a blue object not observed before t is both blue and grue for all time.