Open Journal of Philosophy
2012. Vol.2, No.1, 61-63
Published Online February 2012 in SciRes (
Copyright © 2012 SciRes. 61
Goodman’s New Riddle of Induction
Dean Lubin
Leyton Sixth Form College, London, UK
Received September 29th, 2011; re vised December 18th, 2011; accepted December 25th, 2011
In this paper, I consider Goodman’s new riddle of induction and how we should best respond to it. Notic-
ing that all the emeralds so far observed are green, we infer (project) that all emeralds are green. However,
all emeralds so far observed are also grue, so we could also infer that they are grue. Only one of these in-
ductive inferences or projections could, however, be valid. For the hypothesis that all emeralds are green
predicts that the next observed emerald will be green; whereas the hypothesis that they are grue predicts
that it will blue. Goodman’s new riddle is the problem of saying why the inductive inference involving
“green” is the valid one. Goodman’s own solution appeals to the idea of entrenchment. His idea is that
“green” is a more entrenched predicate than “grue” in the sense that it has figured many more times in our
past projections than has “grue”. In his view, this explains why “green” is projectible (can be used in valid
inductive inferences) whereas “grue” isn’t. I argue that this response doesn’t go far enough and that we
additionally need an explanation of why “green” is more entrenched than “grue”—that we are otherwise
left with the unsatisfactory view that its superior entrenchment is a mere linguistic accident. I try to sup-
plement Goodman’s solution with an explanation of this kind. I argue that “grue” is not entrenched be-
cause past successful inductions involving “green” show that past projections that could have been made
using what I call “grue-like” predicates—predicates which are like “grue” except that the times featuring
in their definitions are past—would have been unsuccessful.
Keywords: Induction; Grue; Entrenchment
In Fact, Fiction and Forecast, Nelson Goodman famously
poses a problem for induction—which he calls the “new riddle
of induction”. In this paper, I want to consider Goodman’s new
riddle and how we should best respond to it.
To illustrate his new riddle, Goodman introduces the predi-
cate “grue” (Goodman, 1983: p. 74) defined as follows:
An object is grue if it is first examined before t (some future
time) and is green; or is not first examined before t and is blue.
Consider all the emeralds we have so far observed. Say there
are N of them. These emeralds have all been observed to be
green. But they are also grue (because they have been exam-
ined before t and have been observed to be green). Since the N
observed emeralds are both green and grue, there are two dif-
ferent inductive inferences that can be made on the basis of the
observations made:
1) Premise: N emeralds are green.
Conclusion: All eme r a l ds are green.
2) Premise: N emeralds are grue.
Conclusion: All emer a l d s a r e grue.
The problem is that 1) and 2) can’t both be acceptable (i.e.
valid or justified) inductive inferences, for 1) predicts that an
emerald examined after t will be green; whereas 2) predicts that
it will be blue.
Of course, we intuitively think that 1) is the valid inference.
Goodman’s new riddle of induction is the problem of how to
distinguish valid from invalid inductive inferences—it is the
problem of saying why 1) is valid whereas 2) is not. He calls
this the problem of projection: when, he asks, can we project
from the observed to the unobserved?
In responding to the new riddle, we clearly need a way of
distinguishing projectible predicates like “green” from non-
projectible predicates like “grue”. But can this be done?
“Green” Is Simple, “Grue” Isn’t
In order to make this distinction, some philosophers have
tried to appeal to the idea of simplicity (As Goodman notes on
p. 78, this is the kind of approach adopted by Rudolf Carnap).
On this view, the simple predicates are the ones which are
projectible; and the problem with “grue” is that it isn’t simple.
But what is it for a predicate to be simple? One suggestion—
considered by Goodman—is that in the definition of simple
predicates there is no reference to time. On this view, the prob-
lem with “grue” is that in its definition there is a reference to
However, Goodman notes that predicates like “green” could
also be defined in a way that makes reference to time. To see
this, he introduces the predicate “bleen” (p. 79) defined as fol-
An object is bleen if it is first examined before t (some futur e
time) and is blue; or is not first examined before t and is green.
A projectible predicate like “green” can then be defined in a
way that does refer to time:
An object is green if it is first examined before t (some fu-
ture time) and is grue; or is not first examined before t and is
bleen (p. 80).
Goodman concludes that the problem isn’t that there is a ref-
erence to ti me in the predicat e “grue”.
Goodman’s Response
How, then, does Goodman himself res pond to the new ridd le?
Goodman considers actual projections that have been made
in the past—these are the “raw material” of his discussion:
“The fact is that whenever we set about determining the va-
lidity of a given projection from a given base, we have and use
a good deal of other relevant knowledge. I am not speaking of
additional evidence statements, but rather of the record of past
predictions actually made and their outcome. Whether these
predictions—regardless of their success or failure-were valid or
not remains in question; but that some were made and how they
turned out is legitimately available information.” (p. 85)
Now, clearly “green” has figured many more times in our
past projections than has “grue” (In fact, pre Goodman, “grue”
had never appeared in our projections!). Goodman puts this
point by saying that “green” is a more entrenched predicate
than “grue” (p. 94).
Goodman claims that the hypothesis that all emeralds are
green is regarded as projectible and the hypothesis that all em-
eralds are grue as unprojectible, because “green” is a much
more entrenched predicate than “grue”. His basic response to
the new riddle is, then, effectively to say that valid inductive
inferences are those that accord with those past regularities that
we have picked out using our language; and that other inductive
inferences are invalid.
But should we accept Goodman’s response to the new rid-
A Problem with Goodman’s Response
What seems to be missing from his account is an explanation
of why the projectible predicate “green” is entrenched, whereas
the unprojectible predicate “grue” isn’t. Indeed, Goodman does
seem to regard this as something of a linguistic accident. He does
not, for example, rule out the possibility of new predicates—like
“grue”—becoming entrenched and therefore being projectible.
His view seems, then, to be one according to which it just so
happens that inductive inferences involving “green” are valid;
whereas those involving “grue” are not.
I think Goodman’s response needs to be supplemented. What
we need and haven’t yet got is an explanation of why “green” is
entrenched and “grue” isn’t. Goodman does seem to be aware
of this problem: “Are we not trusting too blindly to a capricious
Fate to see to it that just the right predicates get themselves
comfortably entrenched? Must we not explain why, in cases of
conflict like those illustrated, the really projectible predicate
happens to have been the earlier and more often projected?” (p.
98) But no such explanation is offered.
Why “Green” Is Entrenched, and “Grue” Isn’t
If we consider both projections actually made in the past and
those that could have been made (but weren’t), then “grue” and
“green” are on a par. After all, every time in the past where
“green” was projected, “grue” could also have been projected.
Goodman’s interest in distinguishing “grue” from “green”
therefore naturally leads him to focus on projections that have
actually been made in the past, rather than on those that could
have been made (pp. 94-95). As he puts it: “The significant dif-
ference appears only if we consider just those occasions when
each predicate was actually projected.” (p. 95).
But I think it is a mistake to completely disregar d all projec-
tions that weren’t, but could have been, made in the past. It
seems to me that we do need to also consider projections that
could have been made in the past but which would have been
unsuccessful. These projections contain valuable information
and so should also be part of the raw material of our discussion.
Past actual projections tell us which of the projections that
could have been made in the past would have been unsuccessful.
For example, consider a past successful projection actually
made involving “green” at or before t* (t* is any fixed past
Premise: N emeralds are green.
Conclusion: The next emerald observed (after t*) will be
Now consider the predicate “grue(t*)” defined as follows:
An object is grue(t*) if it is first examined before t* and is
green or is not first examined before t* and is blue.
Note that “grue(t*)” is a function of the past time t* (In a
similar way, the original predicate “grue” could be thought of
as a function of the future time t, and we could thus write
“grue(t)” instead of “grue”). So, “grue(t*)” is like grue, except
that the time t* is past. It is what we can call a “grue-like”
predicate. In fact, what is being defined here is a range of
grue-like predicates. If the past time (t*) is given in calendar
year, we are defining “grue(2011)”, “grue(2010)”, “grue (2009)”
Although the past successful projection made at or before t*
involved “green”, the following projection could have been
made at that time using “grue(t*)”:
Premise: N emeralds are grue(t*).
Conclusion: The next emerald observed (after t*) will be
Now, had this projection been made, it would have been un-
successful. After all, after t* the next emerald will have been
observed to be green, not blue; and so is not grue(t*). (This is
what the success of the past projection actually made involving
“green” tells us).
The success of past actual projections involving “green”
leads to the repeated use of “green” in our projections. In other
words, it leads to its entrenchment. But these projections have
taught us not to project using “grue(t*)”, because for each oc-
casion in the past when “green” was successfully projected,
projections using “grue(t*)” would have been unsuccessful.
This is valuable information that we should not (and do not)
ignore. It clearly explains why “grue(t*)” isn’t entrenched.
Moreover, and importantly, it also explains why “grue” isn’t
entrenched. We don’t seriously consider adopting “grue” in our
projections because we realise that projections involving “grue-
(t*)” would not have worked in the past.
After all, imagine someone who recognises that past project-
tions that could have been made using “grue(t*)” would all have
been unsuccessful, but who nevertheless thinks that projections
involving “grue” will be successful. For example, suppose that
he recognises that “grue(2009)”, “grue(2010)”, “grue(2011)”
etc are all unprojectible, but nevertheless thinks that “grue-
(2012)” will be projectible. We’d be inclined to think that he
was crazy! He would need a good reason for thinking that the
future will not be like the past—and what good reason does he
have to think this?
We have, then, an explanation (missing in Goodman’s ac-
Copyright © 2012 SciRes.
Copyright © 2012 SciRes. 63
count) of why “green” is, and “grue” isn’t, entrenched. Know-
ing that “grue-like” predicates would not have worked in the
past, we don’t adopt “grue” in our projections. But our not do-
ing so is explicable. Indeed, it is rational. That “green” is en-
trenched and “grue” is not, is certainly no lucky linguistic acci-
Carnap, R. (1947). On the application of inductive logic. Philosophy
and Phenomenological Research, 8, 133-147.
Goodman, N. (1983). Fact, fiction and forecast (4th ed.). Cambridge,
MA: Harvard University Press.