Logic 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)”.

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.

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.

When Words Describe Themselves: The Grelling-Nelson Paradox

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

Words are tools for describing the world, but when we try to describe the words themselves, logic can fall into unexpected pitfalls.

Devised in 1908 by Kurt Grelling and Leonard Nelson, the “Grelling-Nelson Paradox” is a famous semantic paradox that confronts the limits of “words defining words.”

Classifying Words into Two Categories

Grelling and Nelson considered that all adjectives (words) could be classified into the following two groups:

Autological: A word that possesses the property it expresses.
Heterological: A word that does not possess the property it expresses.

Let’s Look at Some Examples

Examples of Autological Words:

“short”: The word itself is short.
“English”: The word itself is English.
“noun”: The word is a noun.
“pentasyllabic”: The word “pen-ta-syl-lab-ic” in English has five syllables.

Examples of Heterological Words:

“long”: The word itself is short, not long.
“German”: This word is English, not German.
“invisible”: This word is currently clearly visible on your screen or paper.

Up to this point, it looks like mere wordplay. Every word should theoretically fall into one of the two categories: either it embodies its meaning or it doesn’t.

The Fatal Question: The Emergence of the Paradox

Now, here begins the paradox. Consider the following single word:

Is the word “heterological” itself autological or heterological?

Whatever answer we choose for this question, we face a contradiction.

Case 1: Assume “heterological” is “autological”

If the word “heterological” is “autological,” by definition, it “possesses the property it expresses.” However, the meaning of this word is “heterological.” In other words, having the property of being “heterological” means that it is “heterological.” We assumed it was autological, but the result turned out to be heterological. (Contradiction)

Case 2: Assume “heterological” is “heterological”

If the word “heterological” is “heterological,” by definition, it “does not possess the property it expresses.” Since the meaning of this word is “heterological,” not having that property means that it is “autological.” We assumed it was heterological, but the result turned out to be autological. (Contradiction)

Whichever way it goes, logic collapses.

graph TD A["The word 'heterological'"] --> B{"How is it classified?"} B -->|Autological| C["Definition: Possesses the property it expresses"] C --> D["Its meaning is 'heterological'"] D --> E["Result: It is heterological!"] E -->|Contradiction| B B -->|Heterological| F["Definition: Does not possess the property it expresses"] F --> G["Its meaning is 'heterological'"] G --> H["Result: It is autological!"] H -->|Contradiction| B style A fill:#4CAF50,stroke:#333,stroke-width:2px,color:#fff style B fill:#FF9800,stroke:#333,stroke-width:2px,color:#fff style E fill:#F44336,stroke:#333,stroke-width:2px,color:#fff style H fill:#F44336,stroke:#333,stroke-width:2px,color:#fff

Connection to Math and Logic: A Relative of Russell’s Paradox

This paradox is not a simple miscalculation or illusion like the “Missing Dollar Riddle.” It shares essentially the same structure as Russell’s Paradox (“Does the set of all sets that do not contain themselves contain itself?”), which shook the foundations of mathematics.

The Grelling-Nelson Paradox can be considered the semantic (word meaning) version of Russell’s Paradox.

Russell’s Paradox in set theory: When defining a set

$$ R = \\{ x \mid x \notin x \\} $$

asking whether $R \in R$ or $R \notin R$ leads to a contradiction.

The Grelling-Nelson Paradox in semantics: When defining $Het(x)$ as “the word $x$ does not possess the property $x$ (is heterological)”,

$$ Het(\text{"Het"}) \iff \neg Het(\text{"Het"}) $$

leads to a logical contradiction.

Why is This Paradox Important?

When words refer to themselves (self-reference), there is always a latent danger of infinite loop-like errors occurring.

This is not just a problem in philosophy or linguistics. In the fields of computer science and artificial intelligence, similar logical walls are encountered when programs attempt to evaluate or modify their own code, or when natural language processing models interpret semantic contradictions.

The Grelling-Nelson Paradox is a thought experiment that beautifully visualizes the bugs (limitations) inherently contained within the system of “language.”

Berry Paradox: The Contradiction That Occurs When Trying to Define a "Number" Using "Words"

Thu, 10 Sep 2026 11:00:00 +0900

1. Expressing Numbers with Words

We routinely express numbers not only using “Arabic numerals (1, 2, 3…)” but also using “words (Japanese, English, etc.)”.

For example, the number “$10$” can be expressed in various words as follows:

“じゅう” (jū: 3 characters)
“ごの2ばい” (twice five: 5 characters)
“ひゃくの10ぶんの1” (one tenth of a hundred: 9 characters)

In this way, let’s consider explaining a certain number using “Japanese characters”. We will set a limit on the number of characters we can use. Here, let’s consider numbers that can be expressed in Japanese using “19 characters or less”.

Naturally, there is a limit to the numbers that can be expressed in 19 characters or less. The number of Japanese character types (hiragana, katakana, kanji, etc.) is finite, and the number of combinations arranging them in 19 characters or less is also finite (it will be an astronomical number, but it is not infinite).

In other words, there absolutely must exist “huge integers that simply cannot be fully expressed in Japanese using 19 characters or less”.

2. Birth of the Paradox

Now, here is the main point. There are countless “integers that cannot be expressed in Japanese using 19 characters or less”. Suppose we find the “smallest one (the least integer)” among those countless unexpressible numbers.

Let’s call that number $X$. Since $X$ is by definition the smallest among the “numbers that cannot be expressed in Japanese using 19 characters or less”, we can refer to it as follows:

“じゅうきゅうもじいないであらわせないさいしょうのせいすう” (the smallest integer not expressible in nineteen characters or less)

Let’s count the number of characters. “じゅ・う・きゅ・う・も・じ・い・な・い・で・あ・ら・わ・せ・な・い・さ・い・しょ・う・の・せ・い・す・う” …Wait? Even without the punctuation, there are 25 characters. This exceeds “19 characters”.

So, let’s tweak the expression a bit and use kanji to make it shorter.

「十九文字以内で表せない最小の整数」

Now, please count the number of characters in this Japanese phrase.

Surprisingly, it is only “16 characters”.

Something strange has happened. We have just expressed the number $X$ using the “16 Japanese characters” in the phrase “十九文字以内で表せない最小の整数”!

graph TD Define["Definition:
X = The smallest integer not expressible in 19 characters or less"] --> CheckLength{"What is the character count of
『十九文字以内で表せない最小の整数』?"} CheckLength -->|It is 16 characters| Contradiction["Contradiction!
X was expressed in 『16 characters』!"] Contradiction --> Paradox["X is supposed to be 『not expressible in 19 characters or less』
but it is 『expressible in 19 characters or less (16 characters)』"] style Contradiction fill:#ff9999,stroke:#333 style Paradox fill:#ff4444,color:#fff,stroke:#333,stroke-width:2px

Even though $X$ is supposed to be a number that “cannot be expressed in 19 characters or less”, the very words defining it perfectly express $X$ in “16 characters (which is 19 characters or less)”. This is the “Berry Paradox”.

3. Who Created This Paradox?

This paradox was devised in 1904 by a person named G. G. Berry, a librarian at Oxford University. It spread worldwide after the genius mathematician and philosopher representing the 20th century, Bertrand Russell, introduced it in his own paper.

(*In the original English paper, the expression “The least integer not nameable in fewer than nineteen syllables” was used, and the paradox was constructed to work with the number of syllables in English.)

4. Why Did the Contradiction Occur?

The fundamental cause of this paradox lies in the ambiguity and self-reference of the “natural languages (such as Japanese or English)” that we usually use.

Natural Language Cannot Withstand the Rigor of Mathematics

In the world of mathematics, “defining a number” is a highly rigorous process (using equations and symbols). However, in the Berry Paradox, an attempt was made to define a mathematical object (an integer) using the everyday language of humans, such as “expressible” or “not expressible”.

Everyday language is incredibly powerful and flexible, but due to that flexibility, it allows for acrobatic feats like “referring to its own character count”. As a result, it caused a self-contradiction (a paradox of self-reference) where “the definition itself breaks the rules of the definition.”

What Does It Mean to Be “Nameable”?

Furthermore, the definition of the phrase “expressible in 16 characters” is also ambiguous. The phrase “the smallest integer not expressible in 19 characters or less” does not directly point to a specific, concrete number (like $987654321...$, for example). It merely describes indirectly that “there must be a number satisfying the condition.”

Mathematically, a clear distinction must be made between “expressing in a directly calculable form” and “stating indirect conditions in words.” The logical trick is hidden in the fact that these two are conflated to claim, “It could be expressed in 16 characters!”

5. Summary and Impact on the Modern Era

At first glance, the Berry Paradox seems like a mere “wordplay” or “riddle”. However, this problem served as a catalyst that made 20th-century mathematicians deeply recognize the “danger of building the foundations of mathematics using everyday language”.

“We must not define numbers with words. Mathematics must be constructed entirely and exclusively with independent, rigorous symbols.”

This paradox became an important milestone leading to cutting-edge studies that changed the history of mathematics, such as “Gödel’s incompleteness theorems” (there are truths in mathematics that can never be proven) and “Kolmogorov complexity” (the theory of how short information can be compressed) in computer science.

Just 16 characters of Japanese exposed the limits of mathematics. That is the beauty of the Berry Paradox.

The Unexpected Hanging Paradox: The Day a Logically "Absolutely Impossible" Test Takes Place

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

1. The Teacher’s “Absolute Declaration”

On the way home one Friday, a math teacher made a terrifying announcement to his students.

“Next week, on some day from Monday to Friday, I will give a ‘surprise test’ exactly once. However, if you can definitely predict ’the test is today’ on the morning of that day, it won’t be a surprise, so the test will not be given on that day.”

Hearing this declaration, the students trembled. It meant they had to spend every day in fear, wondering when the test would be held. However, Student A, the brightest in the class, suddenly smiled and stood up.

“Everyone, you can rest easy. It is absolutely impossible for a surprise test to be held next week. It’s logically impossible!”

With full confidence, Student A began to write the following “perfect logic” on the blackboard.

2. Student A’s Proof by “Perfect Logic”

Student A’s proof uses a mathematical technique of thinking backwards from “Friday” (backward reasoning).

Step 1: Eliminate the Possibility of Friday

Suppose the test was not given for the four days of Monday, Tuesday, Wednesday, and Thursday. Then, the only day left is “Friday.” On the morning of Friday, the students would be able to definitely predict, “Today is the only day left, so the test is definitely today!” According to the teacher’s declaration, “It will not be given on a day it can be predicted,” so it is logically impossible to give a surprise test on Friday. Therefore, there will absolutely be no test on Friday.

Step 2: Eliminate the Possibility of Thursday

It is confirmed that there is no test on Friday. This means that the last possible day the test can be given is “Thursday.” Suppose the test was not given for the three days of Monday, Tuesday, and Wednesday. Then, the only remaining possibility is Thursday (Friday has already been eliminated). On the morning of Thursday, the students would be able to definitely predict, “The test is today!” Therefore, there will absolutely be no test on Thursday either.

Step 3: All Days of the Week Disappear

We just need to repeat the same logic. If there’s no Thursday, the last day becomes Wednesday. Thus, if there is no test until Tuesday, it could be predicted on Wednesday morning, so Wednesday also disappears. If Wednesday disappears, Tuesday disappears, and Monday disappears too. Conclusion: As long as the teacher’s rules are followed, it is absolutely impossible to give a surprise test on any day from Monday to Friday!

graph TD Fri["Friday morning
(No test Mon-Thu)"] -->|"Predictable as 'Only Friday left'"| NoFri["Test impossible on Friday"] Thu["Thursday morning
(No test Mon-Wed)"] -->|"Predictable as 'Not Friday so it must be today'"| NoThu["Test impossible on Thursday"] Wed["Wednesday morning"] -->|"Predictable as 'Not Thu/Fri so it must be today'"| NoWed["Test impossible on Wednesday"] Tue["Tuesday morning"] -->|"Predictable similarly"| NoTue["Test impossible on Tuesday"] Mon["Monday morning"] -->|"Predictable similarly"| NoMon["Test impossible on Monday"] NoFri -.-> Thu NoThu -.-> Wed NoWed -.-> Tue NoTue -.-> Mon style NoFri fill:#ff9999,stroke:#333 style NoThu fill:#ff9999,stroke:#333 style NoWed fill:#ff9999,stroke:#333 style NoTue fill:#ff9999,stroke:#333 style NoMon fill:#ff9999,stroke:#333

The students in the class rejoiced. Student A’s logic seemed perfect, with no loopholes anywhere. They spent the weekend playing around and having fun, and welcomed Monday without studying for the test at all.

Monday… There was no test. “See!” Tuesday… There was no test. “Just like Student A said!”

And then, on Wednesday morning. Clatter! The classroom door opened, the teacher came in, and said:

“Alright, clear your desks. We’re going to start the surprise test now!”

The students fell into a panic. “W-Why!? We completely didn’t predict that there would be a test on Wednesday!”

The teacher smiled smugly. “See, you couldn’t predict it, could you? My ‘declaration’ was completely correct, and the surprise test was established according to the rules.”

3. Where Did the Logic Go Wrong?

Even though Student A’s proof looked perfect, why did a “perfect surprise test” actually take place in reality? This problem is originally called the “Unexpected Hanging Paradox,” and ever since it was devised by the Swedish mathematician Lennart Ekbom in the 1940s, it has continued to trouble philosophers and logicians.

Actually, there is still no unified view that “this is the one absolute correct answer” to this paradox. However, there are some leading approaches to resolving it.

Approach 1: “Paradox of Knowledge (Epistemology)”

The biggest pitfall in Student A’s reasoning was that he incorporated the premise that “the teacher’s declaration is 100% true” into his own prediction.

The teacher’s declaration consists of two conditions: “Give a test next week (P)” and “Do not give it on a day it can be predicted (Q).” If there has been no test up to Friday, the student thinks, “If the declaration is correct, it must be today,” but at the same time, room for doubt is born: “If I can predict it’s today, it violates Q of the declaration. In that case, wasn’t the declaration P (give a test) itself a lie in the first place?”

As a result of the collision between the belief that “the teacher’s words are absolutely correct” and “logical reasoning,” the students held the false conclusion (belief) that “the teacher will not give the test,” and as a result, no matter when the test was given, they ended up in an “unexpected (surprise)” state.

Approach 2: “Paradox of Self-Reference”

Let’s convert the teacher’s words into a logical formula. Let the teacher’s claim be $S$. $S =$ “I will give a test on a certain day $T$. And, you will not be able to predict that day $T$.”

This claim has a “self-referential structure” where its truth or falsehood changes depending on how the students receive it itself (the declaration). Just like the “Liar Paradox (‘This sentence is a lie’)”, it has the property of causing logical reasoning to loop infinitely.

4. “Surprise Tests” Lurking in Daily Life

This paradox is applied not only to mathematics but also to our everyday lives.

[The Dilemma of the Surprise Party]

Suppose a friend declares, “I’m going to throw a surprise party for your birthday this month!” Hearing this, you guess every day, “Is it today? Is it tomorrow?” If there is no party even by the last day of the month, you end up reasoning that in order to satisfy the condition of a “surprise (unpredictable)”, it absolutely cannot be done on the last day… However, in reality, if a cake suddenly appears around the middle of the month, you receive a perfect surprise, thinking, “I was really surprised!”

5. Conclusion

“The Unexpected Hanging Paradox” brilliantly expresses the difficulty of including the human state of ‘knowing (predicting)’ itself into logical calculations.

What we think of as “perfect reasoning” might actually be nothing more than a castle built on sand, resting on the baseless belief that “the other party will absolutely follow the rules.” Next time a teacher says, “I’m giving a surprise test,” it seems the most rational thing to do is to stop twisting logic and just quietly study every day.

Russell's Paradox: Does the "set of all sets that do not contain themselves" contain itself?

Thu, 10 Sep 2026 04:00:00 +0900

1. The “Barber Paradox” That Struck a Peaceful Village

In a certain peaceful village, there was a single barber. At the entrance of the village stood a strange sign with the following rule:

“The barber of this village shaves all and only those villagers who do not shave themselves.”

The villagers were satisfied with this rule. Those who couldn’t shave themselves just had to go to the barber, and those who could shave themselves would just do it at home.

However, one day, the young barber looked in the mirror and was suddenly startled. He had stubble growing on his chin. “Now then, should I shave my own beard?”

He decided to think logically according to the rule on the sign.

What if he decides to “shave himself”? According to the rule, the barber must only shave those who “do not shave themselves.” Therefore, if he shaves himself, he is not eligible to be shaved by the barber (himself). In other words, “he must not shave himself.”
What if he decides “not to shave himself”? According to the rule, the barber must shave everyone who “does not shave themselves.” Therefore, if he doesn’t shave himself, he must be shaved by the barber (himself). In other words, “he must shave himself.”

“If he shaves himself, he must not shave himself.” “If he does not shave himself, he must shave himself.”

The barber completely panicked and became unable to take either action. This is the famous “Barber Paradox”.

graph TD Barber["Barber: Should he shave himself?"] Barber -->|YES: Shaves himself| Cond1["Rule violation!
(He must not shave the beard of someone who shaves himself)"] Barber -->|NO: Does not shave himself| Cond2["Rule violation!
(He must shave the beard of someone who does not shave himself)"] Cond1 --> Paradox["Contradiction (Paradox)"] Cond2 --> Paradox style Paradox fill:#ff4444,color:#fff,stroke:#333,stroke-width:2px

2. “Russell’s Paradox” That Shook the Mathematical World

This “Barber Paradox” is an allegory created by the British logician and philosopher Bertrand Russell to explain a mathematical paradox he discovered in a way that is easy for the general public to understand.

What he actually discovered was not about a barber, but a terrifying contradiction regarding “Sets”. It is called “Russell’s Paradox (1901)”.

The Concept of a “Set of Sets”

In mathematics, a “set” is a collection of things that satisfy a certain condition.

“The set of even numbers less than or equal to 10” = $\{2, 4, 6, 8, 10\}$
“The set of red apples”

And, as the contents (elements) of a set, you can also put in another “set”. For example, consider “the set of all books in the world”. Since this set itself is not a “book”, “the set of all books in the world” is not included within its own set.

On the other hand, consider “the set of things that are not books”. This set itself is also not a “book”. Therefore, “the set of things that are not books” is included within its own set.

In this way, sets in the world can be broadly divided into two types:

A: Sets that do not contain themselves (e.g., the set of books)
B: Sets that contain themselves (e.g., the set of things that are not books)

The Birth of the Demonic Set $R$

Here, Russell considered the following special set $R$:

Set $R$ = The set of all “sets that do not contain themselves (Type A)”

Written as a mathematical formula (intensional notation), it looks like this:

$$ R = \{ x \mid x \notin x \} $$

Now, here is the main point. Russell posed the following question regarding this set $R$:

“Does the set $R$ contain itself ($R$)?”

Let’s think about it.

What if $R$ “contains itself ($R \in R$)”? The condition to be included in $R$ is “not containing oneself”. Therefore, $R$ does not satisfy the condition, and cannot be included in $R$. (Leading to $R \notin R$, a contradiction)
What if $R$ “does not contain itself ($R \notin R$)”? The condition to be included in $R$ is “not containing oneself”. Therefore, $R$ perfectly satisfies the condition, and must be included in $R$. (Leading to $R \in R$, a contradiction)

Written as a formula, it is a logical collapse in just one line:

$$ R \in R \iff R \notin R $$

“If it contains itself, it is not contained.” “If it is not contained, it contains itself.” This has exactly the same structure as the Barber Paradox. However, while the village barber story can be laughed off with “the mayor who put up a sign with such rules is just stupid”, in the world of mathematics, it’s not that simple.

This was because the mathematical community at the time was right in the middle of trying to rebuild all of mathematics based on the naive rule (Naive Set Theory) that “as long as you clearly define the condition, you can freely create a ‘set’ out of anything”.

3. Frege’s Tragedy

The person to whom Russell sent a letter detailing this was the great German logician Gottlob Frege. Frege had just sent the second volume of his magnum opus, The Basic Laws of Arithmetic, to the printing press, having dedicated his entire life to it. This book was a culmination of his attempt to prove the completeness of mathematics based on the rule that “a set can be created from any condition”.

Upon reading Russell’s letter, Frege despaired. It proved that using the “most fundamental of foundations” rules in his book, an “absolutely contradictory set” like Russell’s Paradox could be created. If the foundation collapses, hundreds of pages of mathematical formulas built upon it all become invalid.

At the very end of his book, right before publication, Frege left the following agonizing postscript:

“Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This was the position I was placed in by a letter of Mr. Bertrand Russell, just when the printing of this volume was nearing its completion.”

4. Overcoming the Crisis: The Birth of Axiomatic Set Theory

Russell’s Paradox triggered a massive panic in the mathematical community, known as the “foundational crisis of mathematics”. The freewheeling rule that “as long as you decide the condition, you can freely create a set” had given birth to a monster called contradiction.

To resolve this crisis, mathematicians set out to strictly enforce the rules. Mathematicians such as Zermelo and Fraenkel established a rulebook (axiomatic system) that strictly distinguishes between “sets that are allowed to be created” and “sets that must not be created (too large)”. This is called the “ZFC Axioms (Axiomatic Set Theory)”.

Under the ZFC axioms, a “set $R$ that collects all ‘sets that do not contain themselves’” as conceived by Russell was banned from the world of mathematics, deemed “too huge and dangerous, so it is no longer recognized as a ‘set’ (it is merely a ‘class’)”.

graph LR subgraph "Naive Set Theory (Before Russell)" Free["You can freely create a set
with any condition!"] --> Monster["The Monster of Contradiction R
(Russell's Paradox)"] end subgraph "Axiomatic Set Theory (Modern Mathematics)" Strict["Only those that follow strict
rules (axioms) are 'Sets'"] --> Safe["Contradiction R is not recognized
as a 'Set', so it's safe!"] end Monster -.->|Crisis in the Mathematical World| Strict

5. Conclusion: Paradoxes are a “Drastic Medicine” for Fixing “Logic Bugs”

Russell’s Paradox is the ultimate logic bug caused by self-reference (referring to oneself), similar to “a snake eating its own tail (Ouroboros)” or “the liar who says ‘I am a liar’”.

At first glance, paradoxes may seem like mere sophistry or wordplay, but they destroyed the very foundation of mathematics, the most rigorous of disciplines, and consequently forced it to evolve into something stronger and more rigorous.

If the genius Russell had not noticed this “barber’s bug”, modern mathematics, and computer science which lies on the extension of that logic, might have developed while harboring a fatal contradiction somewhere. A paradox is the most stimulating drastic medicine that teaches us the limits of human logic.