Goodman Grue Paradox

Formulation of the Paradox

The Goodman grue paradox is a central problem in the philosophy of induction, introduced by Nelson Goodman as part of what he called the “new riddle of induction.” It aims to show that familiar examples of inductive reasoning are not as straightforward as they appear, because equally well-supported but intuitively absurd hypotheses can be generated by redefining our predicates.

Goodman introduces the artificial predicate “grue.” An object is defined as grue if and only if:

• it is observed before some fixed future time t and is green, or
• it is not observed before time t and is blue.

Up to time t, anything that is actually green and has been observed (e.g., all the emeralds we have so far examined) is both green and grue. Suppose that:

1. All emeralds observed before time t have been green.
2. Therefore, all emeralds observed before time t have been grue (since they meet the definition).

If we now apply standard inductive reasoning—inferring general laws from repeated observation—we seem equally entitled to adopt the hypotheses:

• H₁: All emeralds are green.
• H₂: All emeralds are grue.

Yet these hypotheses diverge in their future predictions. H₁ says that emeralds observed after time t will be green; H₂ says that emeralds observed after time t will be blue (since unobserved-before-t emeralds count as grue only if blue). The same body of observational evidence appears to support incompatible generalizations about the future.

Goodman’s paradox is thus not about emeralds or colors per se, but about the structure of inductive inference and the role of the predicates we use in formulating hypotheses.

Projectibility and the New Riddle of Induction

Goodman’s central claim is that the traditional problem of induction—made famous by Hume—asks why induction is justified at all, whereas his new riddle asks a different question: Which inductive generalizations are legitimate, or “projectible”?

A predicate is said to be projectible if past instances in which it applies provide genuine inductive support for hypotheses about unobserved instances. For example, “green” is ordinarily taken to be projectible: observing many green emeralds supports the hypothesis that unobserved emeralds are also green. By contrast, “grue” is intuitively non-projectible: observing grue emeralds before t does not, we feel, support the claim that emeralds after t will be blue.

The grue paradox shows that this intuition is not captured by the logical form of the generalization or by the raw observational data alone. Both “All emeralds are green” and “All emeralds are grue”:

• are universal generalizations of the same logical type,
• are equally compatible with all past observations of green emeralds before t,
• receive exactly the same confirmatory evidence up to time t.

Thus, Goodman argues, a theory of induction must explain why “green” is projectible while “grue” is not, even though the data confirm both equally up to the present. This leads him to focus on the entrenchment or past usage of predicates: predicates that have figured successfully in past inductive inferences (like “green”) are entrenched, whereas artificial constructions like “grue” are not.

On this view, a predicate’s history of use in successful generalizations partly determines its projectibility. The new riddle of induction is therefore about:

• the choice of language and predicates in science,
• the constraints on which hypotheses count as lawlike,
• and why some generalizations (e.g., “All emeralds are green”) are inductively respectable while others (e.g., “All emeralds are grue”) are not.

Responses and Criticisms

Philosophers and philosophers of science have offered a variety of responses to Goodman’s paradox, often aimed at explaining the intuitive difference between “green” and “grue” without appealing merely to entrenched usage.

1. Naturalness and similarity

Some theorists, influenced by David Lewis and others, appeal to natural properties or objective similarity. On this account:

• Predicates like “green” correspond to relatively simple, natural properties in the world.
• Predicates like “grue” carve across these natural joints, combining color with an arbitrary time condition.

Because “green” tracks a more natural, simpler property, generalizations using “green” are privileged. “Grue” is rejected not because of its history of use, but because it is metaphysically gerrymandered. Critics, however, question whether such appeals to naturalness are themselves independent of our inductive practices.

2. Simplicity and theory choice

Others appeal to simplicity in scientific theory choice. The hypothesis that all emeralds are green is embedded in a relatively simple theory about color, light, and material composition. The hypothesis that all emeralds are grue requires more complex temporal conditions and additional stipulations. On this view:

• Theories using “green” are simpler and more unified, and hence better supported.
• “Grue” hypotheses are penalized for their complexity under standard criteria of theory choice.

Debate here focuses on whether simplicity is an objective guide to truth, or merely a pragmatic preference.

3. Time-dependence and lawlikeness

Some responses argue that predicates whose application depends explicitly on arbitrary times (like the fixed time t in “grue”) cannot figure in genuine laws of nature. Laws are typically taken to be time-invariant in form. “Green” fits into such lawlike generalizations; “grue” does not.

Goodman, however, counters that this distinction itself relies on prior judgments about which generalizations are lawlike, and thus may not solve the problem independently of our inductive practice.

4. Goodman’s own entrenchment account

Goodman’s own solution emphasizes entrenchment: predicates become projectible through repeated successful use in accepted inductive inferences. On this view:

• “Green” is entrenched; past practices have confirmed its inductive reliability.
• “Grue” is not entrenched and therefore lacks projectibility.

Critics contend that this threatens to make projectibility historically and socially relative, and may not provide a non-circular justification for induction. Supporters reply that induction may not admit a deeper, non-pragmatic justification.

Overall, the Goodman grue paradox remains a focal point in discussions of induction, confirmation theory, and the language of science. It highlights that answering Hume’s question—why induction at all?—does not dissolve the further puzzle of which inductive patterns count as rational or lawlike, and why some bizarre but well-confirmed hypotheses are excluded from serious scientific consideration.