The Problem Of Grue Isn’t; Or, A Gruesome Non-Paradox About Induction

This emerald does not appear to be green, nor grue. Maybe Goodman was right!
This emerald does not appear to be green, nor grue. Maybe Goodman was right!

Skepticism about induction happens only among academic philosophers, and only in print. Tell an induction skeptic to take a long walk off a short dock or hint that his health insurance will be cancelled and you will find an immediate and angry convert to Realism.

Some philosophers come to their skepticism about induction from puzzles which they are unable to solve and reason that, since they cannot solve the puzzles, it’s a good bet to side with skepticism. Well, in some ways this is natural.

A classic puzzle is Nelson Goodman’s “grue”. Goes like this. Grue is a predicate, like green or blue, but with a built-in ad hoc time component. Objects are grue if they are green and observed before 21 October 1978 or blue and observed after that date. A green grape observed 20 October 1978 and a blue bonnet observed 22 October 1978 are grue. But if you saw the green grape yesterday, or remember the blue bonnet from 1976, then neither are grue. The definition changes with the arbitrary date.

So imagine it’s before the Date and you’ve seen or heard of only green emeralds. Induction says future, or rather all unobserved, emeralds will also be green. But since it’s before the Date, these emeralds are also grue, thus induction also says all unobserved emeralds will be grue. Finally comes yesterday—and lo!—a green and not a blue emerald appears, thus not a grue emerald. Induction, which told us it should be grue, is broken!

There have been several exposures of the grue fallacy before, and up until the other day (another date!) I had thought David Stove’s in his Rationality of Induction was best. But I now cast my vote for Louis Groarke’s in his An Aristotelian Account of Induction. He calls belief in Goodman’s fallacy “an adamant will to doubt rather than an evidence-based example of a deep problem with induction” and likens it to the fallacy of the false question (e.g. “Have you stopped cheating on your taxes yet?”).

Groarke says (p. 65):

The proposition, “emeralds are grue,” [if true] can be unpacked into three separate claims: emeralds are green before time t (proposition1); emeralds are blue after time t (proposition2); and emeralds turn from green to blue at time t (proposition3). Goodman illegitimately translates support for proposition1 into support for proposition2 and proposition3. But the fact that we have evidence in support of proposition1 does not give us any evidence in support of all three propositions taken together.

What does the arbitrary time have to do with the essential composition of an emerald? Not much; or rather, nothing. The reason we expect (via induction) unobserved emeralds to be green is we expect that whatever is causing emeralds to be green will remain the same. That is, the essence of what it is to be an emerald is unchanging, and that is what induction is: the understanding of this essence, and awareness of cause.

Groarke emphasizes that the time we observe something is not a fact about the object, but a fact about us. And what is part of us is not part of the object. Plus, the only evidence anybody has, at this point in time, is that all observed emeralds have been green. We even have a chemical explanation for why this is so, which paradox enthusiasts must ignore. Thus “there is absolutely no evidence that any emeralds are blue if observed after time t.”

Two things Groake doesn’t mention. First is that, in real life, the arbitrary time t is ever receding into the future. I picked an obviously absurd date above; it’s absurd because we have all seen green emeralds but no blue ones up to today, which is well past 1978. The ad hoc date highlights the manufactured quality of the so-called paradox. When, exactly, should we use a grue-like predicate for anything?

Secondly, nobody not in search of reasons to be skeptical would have ever thought to apply a predicate like grue to anything. It is entirely artificial. If you doubt that, consider that you can substitute any other predicate after the arbitrary date. It doesn’t have to be blue. Try salty, hot, tall, or fast. An emerald that is green up until t then fast? That’s ridiculous! Yes, it is.

After showing the paradox isn’t, Groake goes on to explain the possible reasons why the paradox has been so eagerly embraced. Cartesian corrosion. That bottomless skepticism which dear old Descartes introduced in the hope of finding a bedrock of certainty. There isn’t space here to prove that, but anybody who has read deeply in epistemology will understand what that means.

Update A glimpse of how much angst the “problem” of grue has created, try this (or similar) searches. Also note the New & Improved title.


  1. Scotian

    “take a long walk off a short dock”

    It’s “take a long walk off a short pier”. I think that you say these things just to bug me.

  2. Briggs


    Pier? What is that, some kind of Canadian anachronism?

    Which reminds me. Time to re-watch Strange Brew.

  3. Scotian

    You need to google these expressions before you use them Briggs. Remember the “so’s your brother” fiasco. But as one of the damned what do I know?

  4. Ken

    Juliet: “What’s in a name? That which we call a rose By any other name would smell as sweet.” Romeo and Juliet* (II, ii, 1-2)

    It’s a very simple principle that pretty much everybody comprehends without need to resort to philosophical treatises gussied up with a variety of fancily-labeled precepts.

    A fundamental problem with philosophy is that it is anchored in simple principles made arcane & abstract, and, completely unmoored from the world of reality — from extenuating factors that impose a variety of limits & conditions.** This allows for all manner of conclusions to be reached under seemingly (but often/usually not actually) objective & logical analysis. In actual practice fancy labels for arcanely-presented principles are applied as desired, usually, to reach the philosopher’s desired conclusion.

    Or, as Eric Hoffer put it:
    “One of the surprising privileges of intellectuals is that they are free to be scandalously asinine without harming their reputations.”

    * Romeo Montague and Juliet Capulet meet and fall in love in Shakespeare’s tale of “star-cross’d” lovers. They are doomed from the start as members of two warring families. Here Juliet tells Romeo that a name is an artificial and meaningless convention, and that she loves the person who is called “Montague”, not the Montague name and not the Montague family. Romeo, out of his passion for Juliet, rejects his family name and vows, as Juliet asks, to “deny (his) father” and instead be “new baptized” as Juliet’s lover. This one short line encapsulates the central struggle and tragedy of the play, and is one of Shakespeare’s most famous quotes.

    People have applied (as Shakespeare did) the philosophical principles developed in Brigg’s essay for centuries, and more people (such as Shakespeare’s audiences) have comprehended the concept with no need to resort to philosophical legerdemain.

    Where there’s no need to create philosophical pole vaults to clear simple conceptual mouse turds (to paraphrase another precept) such vaults ought not be created nor resorted to — how’s that for a fundamental philosophical principle?!

    ** For example consider: “…it’s absurd because we have all seen green emeralds but no blue ones up to today,…” In the philosophical realm this makes sense, in the real world it depicts a certain ignorance as a “blue emerald” (vs. a ‘blue-green’ emerald) is almost certainly a mislabeled “sapphire” — emeralds & sapphires are composed of different chemistries. Concocting logical gymnastics devoid of physical reality creates a misleading crutch for sustaining ignorance of relevant facts and will routinely lead one unknowingly to a wrong conclusion (if one actually applies the logic objectively, which seems the exception) if not the fanciful.

  5. John B

    I’m originally from Minnesota (Land of10,000 Lakes – or alternately “The Land of Sky Blue Water”). The last bit is for people who remember “The GRUE that grew with the Great Northwesr”. (Not really the same brews, but Grue’s and Brews seem to fit todays theme.)

    So even though we had uncountable docks (not the same as infinite), it was “Take a long walk on a short pier” where we were.

    Of course we in Minneesota also know the game “Duck, Duck, Grey Duck”. (I have several theories on why Minnesota doesn’t recognize “Duck, Duck, Goose” and who knows it may be a Canadian thing.

    Cheers, as my Kiwi friends used to say.

    Don’t get pissed on the brew; get even.

  6. Scotian

    Briggs, you managed to come up with a couple of other people who misquoted the expression. It’s like finding misspelled words on the web. Try the obvious search fragment instead: “take a long walk off”. You get dictionary definitions and even pictures but no docks. Maybe by perusing this I am still annoyed from yesterday.

  7. A blue emerald is called aquamarine, both being Be3Al2(SiO3)6.
    Obviously Goodman doesn’t know about chrystallography.

  8. Bob Lince

    “Obviously Goodman doesn’t know about chrystallography.”

    This is why Socrates would spend some time in the commonsense world of the Agora. That he might sprout fewer inanities then he could have.

  9. Why are people in the comments being hard on Goodman but ignoring Hume?

  10. Sander van der Wal

    There are lots of things that change color over time, but nobody thought it would be a good idea to come up with a new word for those changing colors.

    Except a philosopher, from the age of the microwave dinners.

  11. Sander, whatever the plausibility of his claim, it isn’t principally about color, but any predicate, for pity’s sake.

  12. How do you call the color of a tree leaf, grerange perhaps?

  13. Sander van der Wal


    But because the example is about color, he could have proved inductively that people do not give colors that change over time special names. The argument can also be generalized to other attributes.

  14. Nullius in Verba

    “What does the arbitrary time have to do with the essential composition of an emerald?”

    It’s nothing to do with the composition or color of emeralds. It’s about how our choice of vocabulary misleads us about what is going on in predictive induction.

    The definitions of ‘green’ and ‘blue’ from a purely logical point of view are arbitrary. The time-invariant choice is just one way from many semantically equivalent ways to define color terms. We can as an alternative use ‘grue’ meaning green before October 21st 1978 and blue after, while we use ‘bleen’ to mean blue before October 21st 1978 and green after.

    The new vocabulary is logically equivalent to the original one. We can still express the color of any object at any moment of time unambiguously. It’s like a change in coordinate system – it only changes how we describe things, it doesn’t change what they are. However, using our new vocabulary, it is the case that emeralds are grue up until October 21st 1978, and then they turn bleen on that date and thereafter.

    It’s supposed to be an objection to Hume’s claim that we can predict the future by assuming that the same predicates true in the past will be true in the future. The objection is pointing out that this isn’t a valid logical step based on the nature of predicates generally, it’s a side-effect of our conventional choice of time-invariant predicates for commonly time-invariant properties. That is to say, our choice of terminology is assuming precisely the conclusion the argument is trying to justify – that colors are time-invariant.

    The alternative predicates are just as valid considered as predicates, so if this was really a valid logical step, it ought to be just as valid using bleed and grue as it is using green and blue. But it isn’t. Under one convention, the color of emeralds changes arbitrarily on a certain date. In the other it’s fixed. Of course, because we know that the color of emeralds is time invariant, this isn’t a big surprise, but this is about how we come about that knowledge. To say “X has always been Y in the past, so we will induce that it will be so in the future” only works when Y is of a specific form – having precisely the time-invariance property that matches the real behavior of emeralds that we’re trying to deduce (or rather, ‘induce’). A valid logical argument would work just as well with ‘bleen’ and ‘grue’ – since it’s just a coordinate change.

    It’s like predicting future sea level rise by saying “as always, it’ll come about half way up a duck”. It produces a correct prediction, but only because we’ve chosen a coordinate system to describe it that happens to match the behavior we’re trying to predict. It’s circular logic, and so invalid.

    I will grant you that as is common in philosophy, it’s a complicated and roundabout way of illustrating a point that is so simple and obvious as to be trite, and one can’t help thinking there must be more to it than that.

  15. DAV

    Shouldn’t that be “purr” instead of “duck”? They can easily be confused with each other like other similar words as noted in an earlier exchange. In any case, an interesting observation from the world of Null-V.

  16. Gail Finke

    This is seriously supposed to be an unsolvable paradox? You MAKE UP a word that has a definition that CHANGES, and then say “My God, something described by this non-existent word before the definition changes and something else described by it after the definition changes are different, how can logic survive?” Philosophers have even less common sense than I thought.

  17. Gail Finke

    Nullius: But it doesn’t show that.

  18. Nullius in Verba

    “Nullius: But it doesn’t show that.”

    But it doesn’t show what?

  19. John B

    So this whole GRUE thing…

    Is the statement: “This changes everything!” sort of a corollary to it?

    So some “This Changes Everything” event occurred on Oct. 21,1978 and all emeralds turned blue.

    How many times have we heard or read some scientist or politician claim that: “This Changes Everything!”.

    Like watching the reworked “COSMOS” series and how through the Epochs they mention Climate Change or even Cataclysmic Climate Change where 95% of species was wiped out from the face of the earth. Rapid cooling and warming several times allowed Homo Sapiens to supplant the Neanderthal as the dominant species.

    Ultimately these Climate Changes had the attribute of being “good” or “natural” (NOT Man-caused).

    So Cosmos can blithely discuss “Climate Change” of previous epochs as good or natural AND recoverable (which is ultimately why it’s good).

    But the present “Climate Change” is evil, and unnatural; one from which we can NEVER RECOVER from.

    Is THAT GRUE or what?

  20. Joy

    There once was a man called Buck
    Who had the most terrible luck
    He got on a punt
    and fell off the front,
    and got bitten to death by a duck!

  21. YF

    Sorry, but I don’t think you have dissolved the problem.

    The essence of the ‘grue’ problem is the recognition that the laws of physics could change tomorrow. We cannot prove that they will not change, nor can we prove that they will. Alas, induction is based on animal faith or ‘habit’, not logic- just as Hume taught us long ago.

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