A paradox, or logic riddle, by Nelson Goodman. The story is told of Martians who maintain that the color of emeralds is in fact "grue," which means "green until January 1, 2000 and blue thereafter." Note that this paradox predates the year 2000 by several decades, so you should really read this as any date conveniently off in the inconveniently distant future. Anyway, earthlings protest that in fact emeralds are green... but then, our reading of the evidence is no worse than theirs. "Why should they change color to blue?" we ask. They retort, "Change? The won't change color! They'll stay grue! It's you who are maintaining that they'll change color, from being grue until 1/1/2000 to being bleen thereafter!" ("bleen" being their word for the complementary color: we would call it blue until 1/1/2000 and green thereafter.) And until the Magic Date (whatever it is), there really is no way of knowing which interpretation is correct!

And yet, people never come up with a word or concept like "grue" naturally. What is it about the category "green" that makes it preferable to us? That's what this riddle is intended to help elucidate and make us think about. There are various discussions and answers.

Don't start in on the fact that some languages don't distinguish the colors "blue" and "green"; that doesn't enter into this discussion.

My take on this would be to seek refuge with Occam's razor. The categories "bleen" and "grue" are perfectly well-defined. But the categories "blue" and "green" would appear to be simpler.

For convenience, I shall be moving the critical date 100 years back, to allow us time to settle the argument.

Clearly, some care is warranted here. The fact that we humans have "blue" and "green" as simple concepts, but "bleen" and "grue" as concepts constructed out of simpler concepts (like "blue", "green", and "1 Jan 2100") does not necessarily imply that they really are simpler!

However, it would appear that "blue" is simpler than "bleen" in the following sense. It is easier to prove (empirically) that something is "blue" than that it is "bleen": look at its colour right now. To prove "bleenness", you'd need to stick around till 1 Jan 2100.

One important question is whether the Martians also believe in the colours "blue" and "green", or only in "bleen" and "grue". If it's the former, then we're home free: they have 4 concepts and empirical difficulties (which would appear unwarranted), we have 2 concepts and fewer problems. But if it's the latter, we're in trouble. How can we explain the one without the other? We can't, so it's hard to see how to explain one is simpler than the other...

Why the grue-bleen paradox is not solvable by Occam's Razor:

When I say an emerald is green it explicitly means that it caused a perception of greeness1 in my mind during the period I was observing it. The stone itself may in fact be green (see realism), may be some other colour (see hallucination), may not have a colour (see secondary qualities), or may not actually exist (see idealism).

So in the case of the grue-bleen paradox we must consider the phrase "that emerald is green" to implicitly expand to "that emerald appears green to me now and were I to observe it at any time in the future it would also appear green". This is not significantly less complex than "that emerald appears green to me now; were I to observe it at any time before 2000-01-01 it would also appear green; were I to observe it at any time after 2000-01-01 it would appear blue".

Now a programmer would consider this to be significantly less complex (measured by the number of lines of code necessary to predict the emerald's colour), however this is philosophy and therefore we must consider that there exists a bijection between the set of observations contained in both descriptions. IOW, both refer to the same amount of time, therefore there is no increase in plurality.


1: Materialists in the audience may wish to further expand on the concept of 'perception of greeness'. Be my guest but I'm only trying to complicate things by a moderate amount.

I've never understood the appeal of this so-called paradox; unlike other real paradoxes it doesn't appear to say anything logically interesting about concepts. It's obviously false and that's that. (At the moment I'm talking about the colour aspect of it, as discussed above, not the induction aspect.)

The Martians are saying something highly nontrivial about a physical property, which the Earthlings aren't. They're saying on such-and-such a date (it's not important -- say it's the Great Occultation of Canopus, some cosmic event we can both see and which is significant to both planets) -- on some future date, emeralds will change colour.

We say they won't change colour. They say they will change colour. But nothing ever changes colour on a specified date. There is absolutely zero physical evidence, or theoretical support, or inductive probability, for the Martians' strong claim that emeralds will change colour. And that's the end of the story. It's simple.

Saying they'll change from green to blue on 2100-01-01 is like saying all mongooses spontaneously turned magenta for one day only on the 17th of August, 278 BCE. No, we haven't got photographic evidence that proves it didn't happen, but we know perfectly well it didn't happen, and it doesn't count as any kind of interesting question or paradox.

Questions from the audience. You. The back. Blue sports coat. I'm sorry? Ah, yes. Colour names. Yes, if I may repeat the question for the rest of the audience. Why do they use the same colour name when the emerald changes colour? And do languages that have the same word for "green" and "blue" have anything to do with it? That's a good question, and it brings up one or two issues which, while not actually bearing on the pseudo-paradox, might confuse some people if they're not clear on the difference between a colour and the name of a colour.

Let's take three well-known languages, English, Russian, and Japanese, and three well-known gemstones, the emerald, the sapphire, and the aquamarine. I assume you all know a sapphire is a fairly deep blue, and an aquamarine is a delicate watery blue -- or rather, what we call in English, "blue".

In English we have two basic colour words to cover these. An emerald is called green and an aquamarine and a sapphire are both blue. In Japanese a single colour-name covers all three colours (note: one name, three different colours). An emerald, an aquamarine, and a sapphire are all aoi. In Russian there is no word meaning "blue": they have separate basic colour terms for light blue and dark blue. So an emerald is zelyonyy, an aquamarine is goluboy, and a sapphire is siniy.

Eye tests and psychological tests and comparison of cultures proves that Russian, Japanese, and English people (a) have the same visual acuity and perceptions, and (b) hold the same theories about the physical nature of objects. None of us believe that rubies spontaneously grow wings on Thursdays, or that sapphires suddenly changed from being amethyst-coloured to being sapphire-coloured in 4004 BCE.

But we are asked to believe that Martians do believe some such nonsense. That on 2100-01-01 Martians claim emeralds will change colour from colour X to colour Y.

A Japanese interpreter of this Martian nonsense would say the Martian claim is about a change of colour from one shade of aoi to a different one. An English interpreter agrees, but says in English one shade of green to one shade of blue. A Russian interpreter agrees, and says in Russian one shade of green to one shade of... either siniy or goluboy, it's not quite clear from our garbled translation of the Martian. A Martian interpreter agrees, and says the emeralds will change colour from one shade of grue to another.

The English, Russian, Martian, and Japanese interpreters all agree that the claimed physical property of Martian physics involves a shift in spectrograph readings from a peak of n1 nm to a peak of n2 nm. We are all agreed in the peculiar change that will take place, according to Martian metaphysics, on the special date.

The only area of disagreement is the trivial, unparadoxical one of which basic colour term different languages would use to name the two different colours we all agree are different. And why shouldn't the Martians have different languages? In North-Martian they say the stones will change from park grue to dale grue. In South Martian they say they'll change from groi to groluboi. But all Martians claim some bizarre physical change will happen at the Great Occultation.

In English we say aquamarines are blue and we say sapphires are blue. We do not claim aquamarines and sapphires are the same colour. We do not claim that aquamarines could change to being sapphire-coloured.

Problem of induction be blowed. We all know properties don't change like this. We assign fixed attributes (like "green", "aoi", "siniy") on the basis of observed consistencies. We do NOT name attributes according to their behaviour in the forty-third century. Our naming embodies our observations; it embodies the consistent continuity we have observed over the times the names have been in use. It does not contain inductive claims across time; but we do as a matter of empirical fact know that such physical attributes do not change across time. As do the Japanese. As do the Martians.


n6's discussion below adds a number of interesting points. However, I don't think the fact that a question leads to a lot of discussion necessarily makes it a good question. We all know arguments tend to gravitate to things we don't want to argue about. We can define peculiar predicates abundantly, but few of them will be interesting, even to the logician, even if they have logically strange properties.

Define something as grylva if either it's green or it was licked by the Queen of Roumania. All green things are unproblematically identifiable as grylva. Very few other things are. We can wander round the old royal palace in Bucharest staring intently at escritoires, gilded mirrors, and cake-mixture stirring spoons, wishing the walls could talk. All the available evidence for grylvaness is just the evidence for greenness, though we suspect they're not the same thing.

We could examine this more closely, but I don't think that's a very illuminating or paradoxical predicate, even though it had the honour of being invented by me.

Likewise, I don't think Goodman's idea adds any further problem to induction. Yes, evidence for being green is evidence for being grue, but I don't think we need to care.

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:

  1. All emeralds are green.
  2. 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 before.

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.

The colour green can (and most likely would, if talking to Martians, who no doubt see mostly in the deep infra-red) be defined as being electromagnetic radiation with wavelength in the range x to y. Yes, the boundaries are fuzzy, and the impression of green can be created in human eyes by mixing other wavelengths, but these wavelengths are stereotypically green to us.

So what is the Martian actually claiming?

Posibility 1: The wavelengths emited by the crystal will change, and this change will be measurable to any equipment. We martians have a colour naming scheme that accomodates this change.

Response: The physics of emision and absorbtion of photons from crystals is well-understood, and it won't happen. And if it did, you have already admitted that the emited light does change name, from "wavelength a" to "wavelength b". The fact that you have another name, translate grue as that-crystal's-light's-colour, doesn't change this.

Yes a mutable but as-yet-unchanged value might just be empirically indistinguishable from an eternal constant in some cases, but occam's razor and common sense would say that it's simpler to regard constants as constants until found otherwise, and when they change, say that they changed instead of inventing new gerrymander categories.

Posiblity 2: This is purely a name change. The emitted light stays the same to the measurements of a spectrometer. What was refered to as "blue" by us martians is now refered to as "green", so that-crystal's-light's-colour needs a new name, "grue", in order to keep it the same.

Response: This is self-consistent, but very counterproductive. Perhaps the martian language's complexity is what caused them to die out.

I really fail to see what's paradoxical about this flim-flam.

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