The New Riddle of Induction
All Emeralds Are Green.
All Emeralds Are Grue.
Same evidence, opposite predictions. This paradox breaks inductive reasoning itself.
You have observed thousands of emeralds. Every single one is green. This seems like overwhelming evidence that all emeralds are green.
But consider the predicate “grue”:
An object is GRUE if and only if:
Before 2050
It is GREEN
After 2050
It is BLUE
Every emerald you have ever observed was observed before 2050 and was green. Therefore, every emerald you have observed is grue.
The same evidence that confirms “all emeralds are green”
equally confirms “all emeralds are grue”.
Yet these hypotheses make opposite predictions about the future.
Predictions Diverge at 2050
Before the pivot year, both hypotheses make identical predictions. After the pivot year, they completely disagree.
| Year | GREEN predicts | GRUE predicts | Agreement? |
|---|---|---|---|
| 1900-2049 | Green | Green (grue = green before 2050) | Yes |
| 2051 | Green | Blue (grue = blue after 2050) | NO |
| 2060 | Green | Blue (grue = blue after 2050) | NO |
| 2080 | Green | Blue (grue = blue after 2050) | NO |
| 2100 | Green | Blue (grue = blue after 2050) | NO |
This is the core of the paradox: no amount of past observation can distinguish between these hypotheses. They are empirically equivalent until the pivot year arrives.
Same Evidence, Both Confirmed
Every observation of a green emerald before 2050 is simultaneously:
- Evidence FOR “All emeralds are green”
- Evidence FOR “All emeralds are grue”
Try it yourself. Observe some emeralds and watch both confirmation counters rise together:
Click “Observe Emeralds” to start collecting evidence
Hypothesis: GREEN
All emeralds are green
Prediction for year 2060:
GREEN
Hypothesis: GRUE
Green before 2050, blue after
Prediction for year 2060:
BLUE
Philosopher Nelson Goodman introduced this paradox in 1955. He called it the “New Riddle of Induction” because it shows that the old problem of induction (how can we justify inductive reasoning?) has a deeper issue: which hypotheses should induction even consider?
The Perfect Symmetry
Your first instinct might be: “Grue is obviously artificial! It has a time reference built in.”
But here is the twist. Imagine a society that uses grue and bleen as their basic color terms. From their perspective:
Your native predicate:
GREEN
Reflects light at ~520nm wavelength
You call your rival's predicate:
GRUE
Green before 2050, blue after
Your objection:
“"Grue" is artificial - it changes definition based on time!”
From your perspective, GREEN is the simple, natural predicate. Your rival's GRUE is the weird, time-dependent one.
But this is perfectly symmetric. There is no objective way to pick a winner.
The grue-speaker would say: “Your predicate GREEN is the artificial one! It is grue-before-2050-and-bleen-after. Why would you use such a complicated concept?”
The symmetry is perfect.
Each speaker can define their rival's predicate in terms of their own plus a time reference. Neither has a privileged position.
This symmetry is what makes the paradox so troubling. You cannot dismiss grue as “unnatural” without begging the question.
Why “Natural Kinds” Does Not Solve It
The most common response is to invoke “natural kinds” - the idea that some categories are more real or fundamental than others. Green is a natural kind, grue is not.
But this response faces serious problems. Click each objection to see why it fails:
The objection fails because:
So does "green"! Green = grue before 2050, bleen after. The time reference is symmetric.
The deeper issue is this: what makes a kind “natural”?
If natural = physically fundamental
Then neither green nor grue is natural. Physics deals in wavelengths, not color categories. 520nm is not more fundamental than “520nm before 2050, 470nm after.”
If natural = useful for prediction
This is circular. We are trying to figure out which predicates are useful for prediction. We cannot use predictive success to justify the predicates we use to make predictions.
The hard truth: We have no non-circular way to distinguish “projectible” predicates (good for induction) from “non-projectible” ones.
What This Means
The grue paradox is not just a philosopher's puzzle. It has deep implications for science, machine learning, and how we think about evidence.
Induction Requires Background Assumptions
We cannot do induction from scratch. We need prior beliefs about which hypotheses are worth considering. Pure empiricism is impossible.
The Problem of Priors
Bayesian reasoning requires prior probabilities. But where do priors come from? The grue problem shows we need priors over hypothesis space itself - and those cannot be justified empirically.
Overfitting in ML
Machine learning faces the same problem. Any finite dataset is consistent with infinitely many hypotheses. Why prefer simpler ones? The answer cannot come from the data alone.
Scientific Realism vs Anti-Realism
If we cannot justify our choice of predicates, can we claim our scientific theories describe reality? The grue paradox supports skepticism about scientific realism.
The paradox reveals a foundational problem:
Evidence underdetermines theory. The world cannot tell us how to carve it into categories. We bring our conceptual scheme to the evidence, not the other way around.
Every inference rests on assumptions we cannot justify.
That is the new riddle of induction.
Explore More Paradoxes
The grue paradox is just one of many mind-bending results in philosophy and mathematics. Each one reveals something deep about the nature of knowledge and reasoning.
Reference: Goodman, N. (1955). Fact, Fiction, and Forecast