Goodman's New Riddle of Induction
You've examined thousands of emeralds. Every single one has been green. It seems perfectly reasonable to conclude: "All emeralds are green."
But philosopher Nelson Goodman (1955) invented a strange new color called grue. And here's the problem: every emerald you've ever seen is also grue.
So why predict future emeralds will be green rather than grue?
Before T, grue things look green. After T, unobserved grue things are blue!
The mirror of grue: before T looks blue, after T unobserved ones are green.
Move the time slider to see how "grue" emeralds behave differently before and after time T:
All observed emeralds are green → All emeralds are green → Future emeralds will be green
All observed emeralds are grue → All emeralds are grue → Future emeralds will be grue = BLUE!
Both inductions use the exact same evidence (observed green/grue emeralds) but reach opposite conclusions about unobserved emeralds!
What makes "green" a better predicate for induction than "grue"?
You might think grue is artificial because it's defined in terms of green, blue, and time. But here's the twist:
Grue/bleen defined via green/blue + time
Green/blue defined via grue/bleen + time!
The symmetry is perfect. If we started with grue and bleen as "basic," then green would be the weird, time-dependent predicate! There's no objective reason to prefer one set of color concepts over the other.
David Hume formulates the original problem of induction: we can't justify the assumption that the future will resemble the past.
Nelson Goodman publishes "Fact, Fiction, and Forecast," introducing grue/bleen and the "new riddle of induction" — a successor problem that's arguably harder to solve.
Philosophers attempt various solutions: natural kinds, simplicity, conventionalism. None fully succeed.
The problem connects to machine learning and Bayesian inference. Solomonoff induction and Kolmogorov complexity offer partial solutions based on description length.
Goodman's own answer was entrenchment: some predicates have been successfully used in past projections, while others haven't.
But critics note this is descriptive, not justificatory. It explains what we do, not why we're right to do it.
Green picks out a "natural kind" — a real category in nature. Grue is gerrymandered and doesn't carve nature at its joints. But what makes a kind "natural"?
Green is "simpler" than grue. But simplicity is relative to the language you start with. In grue-language, grue is simpler!
Prefer hypotheses with shorter descriptions in a universal programming language. Green has lower complexity than grue. This is the basis of Solomonoff induction.
Green corresponds to a physical property (wavelength). Grue doesn't map onto any single physical feature. Induction should track causal/physical structure.
We have strong prior probability for "all emeralds are green" and low prior for "all are grue." But this just pushes the question back: why those priors?
It doesn't matter philosophically — we just evolved to use predicates like green because they worked. Natural selection, not logic, chose our concepts.
The grue paradox isn't just a philosophical puzzle — it has real implications:
"The problem is not merely to explain why we in fact project 'green' rather than 'grue.' The problem is to provide a justification for this practice."
— Nelson Goodman
Hume showed we can't prove that induction works. Goodman showed something arguably worse: even if induction works, we can't say which inductions are correct.
The evidence doesn't uniquely determine the hypothesis. Infinitely many predicates are consistent with any finite set of observations. Choosing between them requires something beyond the data itself — prior assumptions, simplicity metrics, or brute convention.
In a sense, all learning is theory-laden. We never observe the world neutrally; we always bring concepts that shape what we see and what we expect.
"All emeralds are green. All emeralds are grue. Both statements are supported by identical evidence."
— The uncomfortable truth of induction