Philosophy on kenji.blog

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)”.

Is a flying arrow at rest?: Zeno's Arrow Paradox

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

An arrow shot from a bow is flying through the sky. This arrow is certainly moving. However, Zeno, a Greek philosopher from the 5th century BC, developed the following terrifying logic.

“The flying arrow is actually at rest.”

This is neither a joke nor sophistry, but the “Arrow Paradox” that mathematicians and philosophers have seriously debated for 2500 years.

Zeno’s Argument

Zeno’s argument starts with the concept of a “moment in time.”

Time is a sequence of “moments.”
If you capture a single moment (a point where the length of time is zero) like a “photograph,” the arrow “is” at a specific point in space.
At that moment, the arrow is merely “occupying” that space and is not moving. (If it were moving, it would require a “duration of time” rather than a “moment.”)
This holds true no matter which moment you pick out.
If the arrow is at rest at every moment in time, when does it move?

graph TD A["Flying arrow"] --> B["Time is a continuous sequence of moments"] B --> C["Moment t1: Arrow is at rest at position A"] B --> D["Moment t2: Arrow is at rest at position B"] B --> E["Moment t3: Arrow is at rest at position C"] C --> F{"At every moment, the arrow is at rest"} D --> F E --> F F --> G["Conclusion: The arrow is not moving!"] style A fill:#2196F3,color:#fff style F fill:#FF9800,color:#fff,stroke-width:2px style G fill:#F44336,color:#fff,stroke-width:3px

Intuition vs. Logic

“Ridiculous. The arrow is actually flying, isn’t it?” This is probably most people’s first reaction. However, it is actually extremely difficult to point out logically where Zeno’s argument is flawed.

In fact, the ancient Greek philosopher Diogenes is said to have simply stood up and walked around the room in response to Zeno, showing him, “Look, it is moving.” However, this does not constitute a refutation of Zeno’s logic. What Zeno is questioning is not “whether it can move,” but rather, “can we logically explain what it means to move without contradiction?”

Resolution by Calculus (An Attempt)

Calculus, invented by Newton and Leibniz in the 17th century, provided a mathematical answer (at least partially) to this paradox.

In calculus, “the velocity at a certain moment (instantaneous velocity)” is defined as follows:

$$ v(t) = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} $$

In other words, velocity is defined as the “limit” where the change in position $\Delta x$ is divided by the change in time $\Delta t$, as $\Delta t$ approaches infinitely close to zero.

The point here is that “instantaneous velocity” is not the distance traveled within a zero-time duration. It is a quantity defined as the “tendency,” or “limit,” of minute changes before and after that time.

Therefore, the answer from the perspective of calculus is as follows:

“Indeed, if you capture a moment of zero length, the arrow has not moved ‘within’ that moment. However, even at that moment, the arrow has the property of ‘instantaneous velocity (a non-zero limit value).’ Being ‘at rest’ means having an ‘instantaneous velocity of zero,’ but since the instantaneous velocity of a flying arrow is not zero, the arrow cannot be said to be ‘at rest.’”

Remaining Philosophical Questions

While calculus provided a practical solution to Zeno’s paradox, it hasn’t completely settled the philosophical debate.

The concept of a “limit” is strictly a mathematical tool (calculation procedure), and it does not rigorously answer fundamental questions such as “what physically is the smallest unit of time (moment),” “what is continuity,” and “what is the essence of motion.”

In modern physics (quantum mechanics), the possibility that minimum units for time and space (Planck time, Planck length) exist is discussed. If time is not “continuous” but “discrete (digital),” Zeno’s paradox might need to be re-evaluated in a completely different context.

Even after 2500 years, Zeno’s arrow continues to ask us, “What does it mean to move?” and “What is time?”

The Master and the Student, A Contradictory Trial No Matter Who Wins: Paradox of the Court

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

In ancient Greece, a young man named Euathlus became a student of Protagoras, the greatest of the Sophists (teachers of rhetoric). The two entered into the following contract regarding the payment of tuition fees.

Contract Terms: After completing the entire course in rhetoric, Euathlus will pay the remaining balance of the tuition to Protagoras at the moment he wins his first court case.

Euathlus was an excellent student and successfully completed the entire rhetoric course. However, after completing it, he refused to take on any court cases for some reason. If he never went to court, the condition of “winning his first court case” would never be met, meaning he would not have to pay the tuition.

Exasperated, Protagoras sued Euathlus in court. “Pay the tuition,” he demanded.

And from here, a labyrinth of logic begins.

The Logic of the Master, Protagoras

Protagoras argued in court as follows:

“Oh judge, no matter what happens, I win.

If I win this case, by the court’s verdict, Euathlus must pay me the tuition.
If I lose this case, it means Euathlus has ‘won his first court case.’ In other words, the terms of the contract have been fulfilled, and he must pay the tuition according to the contract.

In either case, he has an obligation to pay the tuition.”

The Logic of the Student, Euathlus

In response, Euathlus also held his ground:

“Oh judge, no matter what happens, I win.

If I win this case, by the court’s verdict, I do not have to pay the tuition.
If I lose this case, I still have not ‘won my first court case.’ In other words, because the terms of the contract have not been fulfilled, contractually, I have no obligation to pay the tuition.

In either case, I do not need to pay the tuition.”

graph TD A["Result of the trial"] --> B["Protagoras wins"] A --> C["Euathlus wins"] B --> B1["Verdict: Euathlus must pay"] B --> B2["Contract: Euathlus has not won -> Does not have to pay"] C --> C1["Verdict: Euathlus does not have to pay"] C --> C2["Contract: Euathlus's first win -> Must pay"] B1 --> D{"Contradiction! Verdict vs Contract"} B2 --> D C1 --> E{"Contradiction! Verdict vs Contract"} C2 --> E style A fill:#ECEFF1,stroke:#333,stroke-width:2px style B fill:#4CAF50,color:#fff style C fill:#2196F3,color:#fff style D fill:#F44336,color:#fff,stroke-width:3px style E fill:#F44336,color:#fff,stroke-width:3px

Why Does It Contradict?

The root cause of this paradox is that two different rule systems (law and contract) make contradictory judgments against each other.

Rule of Law: Obey the court’s verdict.
Rule of Contract: Obey the condition of “pay if you win your first court case.”

Normally, law and contract function as independent domains, but because Protagoras made the “payment of tuition” the issue of the trial, the result of this trial itself affected the condition of the contract, causing the two systems to fall into a self-referential loop.

Answers From Legal Scholars

The ancient Roman jurist Aulus Gellius proposed the following solution to this problem:

“The court should rule in favor of Euathlus (no payment required) because it is a fact that the condition of the contract has not yet been met. However, after this verdict, Protagoras can sue Euathlus again. Because Euathlus won the first trial, the condition of the contract has been fulfilled. In the second trial, Protagoras will win.”

In other words, attempting to “solve the paradox simultaneously in a single trial” creates a contradiction, but handling it “in two stages” resolves it.

Connection to Self-Referential Paradoxes

The Paradox of the Court has the same self-referential structure as the “Liar Paradox (‘This sentence is false’)” and “Russell’s Paradox.” A proposition (the conclusion of the trial) affects the condition (the fulfillment of the contract) that determines its own truth or falsity.

This kind of paradox is deeply related to problems that demonstrate the fundamental limits of logic and computation, such as the “Halting Problem (it is impossible to create a program that determines whether a given program will halt or not)” in modern computer science, and Gödel’s Incompleteness Theorems.

The Paradox of the Court is a 2,400-year-old warning teaching us that systems of human-made rules (laws and contracts) can internally collapse through clever self-reference.

When Does a Heap of Sand Stop Being a Heap? The Sorites Paradox

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

Imagine a fine heap of sand made up of 10,000 grains right in front of you. Anyone would agree that this is a “heap.” Now, let’s remove just one grain of sand. 9,999 grains. Still a heap, right? Remove one more grain. 9,998 grains. Still a heap.

Let’s keep repeating this process. Removing a single grain shouldn’t turn a “heap” into “not a heap.” However, if you keep applying this logic over and over, you’ll eventually be left with just one grain of sand.

Is a single grain of sand a “heap”?

Of course, no one would call a single grain of sand a “heap.” And yet, we never once rejected the premise that “removing one grain still leaves a heap.” The logic must break down somewhere, but at exactly which grain did the heap stop being a heap?

This is the Sorites Paradox (also known as the Paradox of the Heap), attributed to the ancient Greek philosopher Eubulides in the 4th century BCE.

Logical Structure

This paradox can be expressed in the form of a syllogism:

Premise 1: A collection of 10,000 grains of sand is a “heap.” Premise 2: A heap with one grain of sand removed is still a “heap.” Conclusion: Therefore, even a single grain of sand is a “heap.”

Premise 1 and Premise 2 each sound perfectly reasonable on their own. However, repeatedly applying Premise 2 leads to an obviously false conclusion.

Why This Paradox Is Unsolvable

The crux of the Sorites Paradox is that the word “heap” is inherently vague. There is no clear definition (threshold) for how many grains constitute a “heap.” Such concepts are called “vague predicates.”

Our everyday language is full of such vague terms:

“Tall” — how many centimeters qualifies as “tall”? A person who is 180 cm is “tall.” What if you shave off 1 mm? Another 1 mm?
“Rich” — how much wealth makes someone “rich”? 10 billion yen is “rich.” What if you lose 1 yen?
“Bald” — how few hairs qualifies as “bald”? Zero hairs is “bald.” What if one hair grows back?

All of these share exactly the same structure as the Sorites Paradox.

Philosophical Approaches

1. Epistemicism (The Boundary Exists)

This approach claims that “a precise boundary between heap and non-heap actually exists, but humans simply lack the ability to perceive it.” For example, there may be an exact boundary where “5,837 grains is a heap but 5,836 grains is not” — we just cannot know it.

This is logically tidy, but most people feel intuitively uneasy with this position.

2. Fuzzy Logic (Graduated Truth Values)

Classical logic deals in a binary of “true or false,” but fuzzy logic allows values anywhere between 0 and 1.

For example:

10,000 grains of sand → “Heap-ness = 1.0 (completely a heap)”
5,000 grains → “Heap-ness = 0.7”
100 grains → “Heap-ness = 0.1”
1 grain → “Heap-ness = 0.0 (completely not a heap)”

This method is practical, but it does not fully resolve the paradox. It introduces a new kind of vagueness: “What is the difference between a heap-ness of 0.7 and 0.699?”

3. Supervaluationism

This approach considers all conceivable reasonable boundaries for the word “heap” simultaneously. If something is judged a “heap” under every possible boundary, it is “definitely a heap.” If it is judged “not a heap” under every boundary, it is “definitely not a heap.” The area where opinions diverge is deemed “indeterminate.”

Impact on Modern Society

The Sorites Paradox is not merely a word game — it raises serious problems in the real world of law and policy.

Age of majority: At 17 years and 364 days, you are a “child”; at exactly 18 years and 0 days, you are an “adult.” What fundamentally changes in a single day?
Poverty line: If your income falls 1 yen below the threshold, you are “in poverty”; 1 yen above, and you are “not in poverty.”
Environmental regulations: If pollutant emissions exceed the standard by 0.001 mg, it’s illegal. Right at the standard, it’s legal.

Human language and thought are inherently imbued with vagueness, and attempting to carve the world into clear-cut binary categories may be fundamentally flawed. The Sorites Paradox is a paradox that has perplexed philosophers for over 2,400 years, revealing the fundamental limits of human intellect.