Are Emeralds Green or 'Grue'?: Goodman's New Riddle of Induction

Thu, 10 Sep 2026 21:00:00 +0900

We predict the “future” from “past experiences”. “The sun rose from the east yesterday, so it will rise from the east tomorrow as well.” “All emeralds we have seen so far were green, so the next emerald unearthed will also be green.”

Such reasoning is called “induction”, and it is the foundation of all science. However, in 1955, philosopher Nelson Goodman devised a bizarre concept of color to show that this induction has a fundamental flaw. That is the “Grue” paradox.

Definition of the New Color “Grue”

Goodman defined a new property (color) called “Grue”, which is a synthesis of “Green” and “Blue”, as follows.

Definition of Grue: An object is “grue” if it is observed before a specific time $t$ (e.g., January 1, 2030) and is “green”, and if it is observed at or after time $t$ and is “blue”.

$$ \text{Grue} = \begin{cases} \text{Green} & (\text{Time} < t) \\ \text{Blue} & (\text{Time} \ge t) \end{cases} $$

According to this definition, the green emerald you hold in your hand right now (before time $t$) is simultaneously “green” and “grue”.

Why is it a Paradox?

The paradox occurs when we try to predict the future. All emeralds humanity has observed so far have been “green”. Therefore, using induction, we predict the following:

Hypothesis A: “All emeralds are ‘green’”

But wait a minute. Since all emeralds observed so far were from before time $t$, they must have also all been “grue”. Therefore, from exactly the same observational data, the following prediction also holds true.

Hypothesis B: “All emeralds are ‘grue’”

If we follow the rules of induction, past observations support Hypothesis B with “exactly the same strength” as they support Hypothesis A.

graph TD A["Past observation: All emeralds were green"] -->|Simultaneously| B["Past observation: All emeralds were 'grue'"] A --> C["Inductive prediction A: Future emeralds will also be 'green'"] B --> D["Inductive prediction B: Future emeralds will also be 'grue'"] C --> E["Remain green even after time t"] D --> F["Turn 'blue' after time t!"] style C fill:#4CAF50,stroke:#333,color:#fff style D fill:#2196F3,stroke:#333,color:#fff style F fill:#F44336,stroke:#333,color:#fff,stroke-width:2px

Will Emeralds Turn Blue?

If Hypothesis B is correct, the moment time $t$ arrives, all emeralds in the world must simultaneously turn “blue” (from the definition of grue).

Intuitively, we think, “That’s absurd. Hypothesis B is unnatural wordplay, and Hypothesis A (green) must be the correct one.”

However, Goodman’s question lies much deeper. Even though both “green” and “grue” hypotheses perfectly match past data, why do we consider only the “green” prediction as valid and eliminate the “grue” prediction? What is the “logical basis” for that?

Challenge to the “Uniformity of Nature”

To avoid this problem, an objection comes to mind: “We should use simple concepts like ‘green’ and not complex, time-dependent concepts like ‘grue’.”

However, Goodman showed the opposite: if we define a color “Bleen” (blue until time $t$, green thereafter), the very concept of “green” becomes a complex, time-dependent concept (“grue” until time $t$, “bleen” thereafter). In other words, which words we take as “fundamental” is merely a habit of our language.

Goodman’s “Grue” paradox (the new riddle of induction) proved that scientific theories are not determined merely by objective data alone, but depend heavily on “what conceptual framework (language) we use to carve up the world”.

Even in the context of AI and machine learning, this paradox continues to hold significant meaning today as the problem of “overfitting” and “bias”, where even with the same training data, predictions for the future can completely change depending on the “structure of the model (which features it focuses on)”.

Does seeing a blue apple prove 'ravens are black'? : Hempel's Ravens

Thu, 10 Sep 2026 21:00:00 +0900

How do scientists prove theories? Usually, they use “induction,” gathering data by observing the world. For example, if you wanted to prove the hypothesis that “all ravens are black,” you would observe ravens around the world and confirm one by one that they are black.

However, in the 1940s, logician Carl Hempel pointed out a strange logical loophole hidden in this commonplace scientific method. This is the paradox of Hempel’s Ravens, which states that “simply seeing a blue apple or a red shoe serves as evidence that ‘ravens are black’.”

Logical Substitution: The Magic of the Contrapositive

To understand Hempel’s argument, we must recall the concept of the “contrapositive” learned in high school mathematics.

In logic, if a proposition “If A, then B” is true, its contrapositive “If not B, then not A” must also be true (this is called logical equivalence).

Hypothesis $H_1$: “All ravens are black (If it is a raven, then it is black)”

Let’s take the contrapositive of this hypothesis $H_1$. It becomes “If it is not black, then it is not a raven.”

Hypothesis $H_2$: “Everything that is not black is not a raven”

According to the rules of logic, $H_1$ and $H_2$ have exactly the same meaning (equivalence). If one is proven, the other is automatically proven as well.

Proving Ravens Without Seeing Ravens

Now, to confirm hypothesis $H_1$ (ravens are black), every time you find a black raven, the certainty (evidence) of the hypothesis gets a little stronger. This is something everyone can agree on.

However, since $H_1$ and $H_2$ mean the same thing, finding evidence for hypothesis $H_2$ (things that are not black are not ravens) should directly serve as evidence for hypothesis $H_1$.

So, what constitutes evidence for $H_2$? You just need to find something that is “not black and not a raven.”

Suppose there is a “blue apple” on the table. It is not black, and it is not a raven. Therefore, it is evidence supporting $H_2$.
There are “red shoes” in the closet. These are also not black and not ravens. They are evidence for $H_2$.
A “white cloud” is floating in the sky. This is also evidence for $H_2$.

Since evidence for $H_2$ holds the same value as evidence for $H_1$, the following bizarre conclusion is logically derived:

“The more you observe blue apples and red shoes in a room, the more the hypothesis ‘all ravens are black’ is proven to be true.”

graph TD A["Proposition H1: All ravens are black"] <-->|Logical equivalence (Contrapositive)| B["Proposition H2: What is not black is not a raven"] C["Observation: Black raven"] -->|Serves as evidence for| A D["Observation: Blue apple"] -->|Serves as evidence for| B D -.->|Therefore, this should also be evidence for?| A style A fill:#4CAF50,stroke:#333,stroke-width:2px,color:#fff style B fill:#4CAF50,stroke:#333,stroke-width:2px,color:#fff style C fill:#2196F3,stroke:#333,color:#fff style D fill:#FF9800,stroke:#333,color:#fff

Why Is It Counterintuitive?

No ornithologist anywhere in the world grows more convinced that “ravens are black” by looking at a blue apple. Although it should be perfectly correct logically, why does our common sense reject this?

Several approaches have been proposed in the fields of philosophy and statistics to address this paradox.

1. The Bayesian Solution (Difference in Information Content)

The most compelling counterargument from the perspective of modern statistics (Bayesian probability) focuses on the difference in the “strength of evidence (information content).”

In the world, there are overwhelmingly more “things that are not black” than “black things,” and astronomically more “things that are not ravens” than “ravens.”

When you see a blue apple, it certainly serves as evidence that “all ravens are black,” but its value as evidence (the increase in probability) is infinitely close to zero. Confirming just one of the countless “non-black things” in the vast universe raises the probability that “ravens are black” by an amount comparable to the effect of removing a single grain of sand from a desert. On the other hand, directly finding one black raven carries an overwhelmingly greater evidentiary value.

In other words, the Bayesian solution is that logically “a blue apple is evidence,” but practically “its evidentiary value is equal to zero and can be ignored.”

2. The Limits of “Indoor Ornithology”

This paradox highlights how the foundation of science known as “induction (deriving general laws from observation)” rests on a fragile premise. Relying solely on logical equivalence would enable “indoor ornithology,” where one could verify any universal law (“all swans are white,” “no aliens are green,” etc.) simply by observing the junk in a room without ever going outside.

Hempel’s Ravens is a fascinating paradox that shows that the words “evidence” and “proof” we unconsciously use cannot be fully captured by the rules of pure symbolic logic alone.

Induction on kenji.blog