The Grue Paradox

 
The "grue paradox," originally conceived by Nelson Goodman in Fact, Fiction, and Forecast makes things even worse for the HD account of justification of universal statements by any amount of particular empirical evidence.

Typically the scientist will not be considering a single hypothesis but will be looking for which hypothesis is "best confirmed" by the available evidence. The grue paradox shows us, that if we eliminate induction  (as the HD-ist hoped to do) as a means of justification, for every hypothesis that is confirmed by some body of evidence, there are an infinite number of alternative hypotheses inconsistent with the first which are all equally well confirmed by that same evidence. Thus, based on the empirical evidence there is no justification for regarding the evidence as ever confirming one hypothesis more than another one!

This is a "paradox" because of course scientists frequently make judgments that the evidence confirms one hypothesis more favorably than another.

Here's how the paradox was presented by Goodman:

Take as our example this time the hypothesis that:

"All emeralds are green."

Clearly this hypothesis is confirmed by observations of green emeralds, i.e. its "positive instances." Now consider a rival hypothesis:

"All emeralds are grue."

Here "grue" is a new predicate which is defined as the property of being green before the year 2100 and blue afterwards. Thus this second hypothesis that all emeralds are grue will be confirmed by any observation of a green emerald before the year 2100, because "grue" means by definition being green before 2100. Of course, since all observational evidence available is before the year 2100, all the evidence we have confirms the grue hypothesis exactly as much as it confirms the green hypothesis!

Of course picking the year 2100 is absurdly arbitrary; we could have picked any date. So, in effect, there are an infinite number of alternative hypotheses (each of which would have the emeralds changing color at a different future date) which are all equally well confirmed by the observed positive instances of green emeralds. Since our justification for accepting a hypothesis as confirmed is the empirical evidence, it follows paradoxically that we have no rational justification for picking one hypothesis as better confirmed than an infinite host of alternatives!
 

NOTE: Obviously, this does not reflect what really is the case in
science. The hypothesis that all emeralds are green is in fact
considered highly confirmed and accepted by all mineralogists,
whereas no one believes "All emeralds are grue."
Why do we consider "green" as "reasonable" but "grue" as utterly "absurd"?
The reason seems obvious: no one has ever observed gems changing
color on an arbitrary date in the past, so no one has any grounds
for expecting any gem to change from green to blue in the year
2100.
But to say this is just to say that we expect the future
will resemble the past (the principle of the uniformity of nature),
which is of course the heart of Hume's problem of induction. No
doubt science proceeds on this assumption of the uniformity of
nature, but our task is to justify it. Yet the only way to justify
it is to reason from the evidence which we have accumulated from
the past, and that of course is to assume the very point which is
at issue, namely the reliability of inductive inference from the
past to the future. We must conclude that the confirmationist has
not escaped the problem of induction, and so Goodman calls this
"grue paradox" the "new riddle of induction."