Goodman’s original definition of grue
The correct definition of Nelson Goodman’s famous predicate grue
has been the subject of some controversy.
Goodman’s original definition is somewhat difficult to find online
because most people do not quote Goodman verbatim,
but instead offer their own paraphrase.
Below are exact quotations from Goodman’s book
Fact, Fiction, and Forecast.
Differences in different editions are noted (however, I have not managed to
get a copy of the 3rd edition yet).
I give, in addition to the definition of grue,
the definition of other related words.
Chapter III, section 4:
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. And the statements that emerald a is grue, that
emerald b is grue, and so on, will each confirm the general hypothesis
that all emeralds are grue. Thus according to our definition, the
prediction is that all emeralds subsequently examined will be green and
the prediction that all will be grue are alike confirmed by evidence
statements describing the same observations. But if an emerald subsequently
examined is grue, it is blue and hence not green.
Consider...the predicate ‘bleen’
that applies to emeralds examined before time t
just in case they are blue and to other emeralds just in case they are green.
Let “emerose”
apply just to emeralds examined before time t,
and to roses examined later.
Chapter IV, section 4:
Suppose that the predicate “grund”
applies just to all things examined up
to a certain time t that are green and to all things not so examined
that are round.
[Note: The word “just” was omitted in the 2nd edition,
and “Suppose that” was changed to “Suppose, however,”
in the 4th edition.] Footnote [omitted in 4th edition]:
“All emeralds are grund” is not, it must be remembered,
equivalent to “All emeralds are green and round.”
All emeralds may be grund without
all being green, without all being round, and even without any emerald being
both green and round.
A thing is grare if either green and examined before t,
or not so examined and square. [4th edition only]
Let the predicate “emeruby”
apply to emeralds examined for color before t
and to rubies not examined before t.
I would also like to take this opportunity to
state my opinion about the oft-debated question about
whether, in a world in which all emeralds are grue,
emeralds change color.
It seems clear to me from the above quotations that
emeralds do not change color in such a world,
but if you are not yet convinced,
let me offer the following argument.
If I were Goodman, on the verge of inventing the concept of grue,
I would want to construct a maximally perverse example of
induction going wrong, to ram home my point.
Now what would be maximally perverse?
If all the emeralds that have ever been examined have been green,
then one would normally think that
induction would predict that all the emeralds that
have not yet been examined will also be green.
It would be maximally perverse for induction to predict the
exact opposite, namely that all the emeralds
that have not yet been examined
are in fact not green (and are blue, say).
This line of reasoning leads naturally to the definition of grue
as meaning that in a world where all emeralds are grue,
all the green emeralds get discovered first,
before some specific time t,
and all the blue emeralds get discovered after that.
In particular, no emerald changes its color at any time.
Now, it is true that after one understands Goodman’s basic idea,
one can devise more complicated versions of it.
In particular, given a single, specific emerald
that I examine occasionally,
I might note that every time I have examined that particular emerald,
it has been green. Induction, one would think, would predict
that on every subsequent examination,
the emerald will also be green.
The “maximally perverse” prediction
would be that on every subsequent examination,
the emerald will not be green (and will be blue, say).
That is, in conventional language,
the emerald would “change color.”
While this scenario does still exemplify the basic insight
behind the concept of grue,
I very much doubt that this was what Goodman was thinking
when he first came up with his new insight.
For one thing, if this were really what he had in mind,
it would have been more natural for him to
propose the statement, “This emerald is grue,”
rather than “All emeralds are grue.”
A second point to note is that if one examines the philosophical literature
on induction, one sees that it is customary to posit a scientist
collecting information about a species by examining specimens;
for example, Hempel’s raven paradox from the 1940s
imagines us examining ravens one at a time and noting that they are black,
or examining non-black things and noting that they are not ravens.
Hempel does not invite us to imagine examining the same non-black item
repeatedly in order to confirm that it never turns into a raven;
that much is taken for granted.
Similarly, I take Goodman to be implicitly assuming that
we are gathering information about emeralds by examining them one at a time,
rather than by examining the same emerald repeatedly.
Finally, if we take the color-changing version of the paradox seriously,
then we must be careful about what it means for
an emerald to be a certain color.
Implicit in our usual concept of an “emerald being green”
is that the green coloration of the emerald persists.
However, the persistence of the color is precisely what is being
called into question by a color-changing version of the paradox.
Therefore, to even state this version of the paradox properly,
we should really start with the concept of
being green at a particular time,
and then be careful to define what we mean
when we simply say that an emerald is green
without explicitly mentioning time.
That Goodman did not take pains to be precise about this point is,
I believe, evidence that he did not have
a color-changing version of the paradox in mind.
He was taking for granted that the color of an emerald persists,
so that one can speak of blue emeralds and green emeralds
without further explanation.
In short, the color-changing version of the grue paradox
is more complex and confusing
and I am convinced that Goodman originally had in mind the
simpler and crisper version in which individual emeralds do not change color.
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