Paradox

A paradox is a special kind of statement that seems to say two opposite things at once. It looks like it should not make sense, but it often comes from ideas that seem true. These statements can make us think very hard about what is really happening.

Paradoxes are important because they help us learn how to think carefully and deeply. Some famous paradoxes have even changed the way we understand math and logic. For example, Russell's paradox asked a simple question about lists that led to big changes in how we study groups of things.

Paradoxes are not just about words. They can also appear in pictures and other kinds of art. The famous artist M. C. Escher made drawings with stairs that seem to go up forever and walls that look like floors, showing us how tricky our eyes and minds can be.

Sometimes people use the word paradox to talk about results that seem surprising or hard to believe at first. These can make us see things in a new way.

Common elements

Self-reference, contradiction, and infinite regress are key parts of many puzzles called paradoxes. Other common parts include circular definitions and mixing up ideas at different levels of abstraction.

Self-reference

Self-reference happens when a sentence, idea, or formula talks about itself. While a sentence can talk about itself without causing trouble ("This sentence is written in English" is true and not puzzling), self-reference is often part of paradoxes. For example, the liar paradox is phrased as "This sentence is false." Another example is the barber paradox, which asks if a barber who shaves everyone who does not shave themselves will shave himself. In this puzzle, the barber is a concept that refers to itself.

Contradiction

Contradiction is another main feature of many paradoxes. The liar paradox, "This sentence is false," shows contradiction because the sentence cannot be both false and true at the same time. The barber paradox is contradictory because it suggests the barber shaves himself if and only if the barber does not shave himself.

Just like self-reference, a sentence can have a contradiction without being a paradox. "This sentence is written in French" is an example of a contradictory self-referential sentence that is not a paradox—it is simply false.

Vicious circularity, or infinite regress

Another key part of paradoxes is never-ending recursion, like circular reasoning or infinite regress. When this never-ending loop creates an impossible situation through contradiction, the loop or circle is vicious. The liar paradox is a good example: "This sentence is false"—if the sentence is true, then the sentence is false, which makes the sentence true, which makes the sentence false, and so on.

The barber paradox also shows this never-ending loop: The barber shaves those who do not shave themselves, so if the barber does not shave himself, then he shaves himself, then he does not shave himself, and so on.

Other elements

Other paradoxes involve tricky or incomplete statements and half-truths or depend on quick, wrong assumptions (For example, a father and his son are in a car crash; the father is hurt and the boy is taken to the hospital quickly. The doctor says, "I can't help this boy. He is my son." There is no puzzle here—the doctor is the boy's mother.)

Paradoxes that do not come from a hidden mistake often happen at the edges of what we mean or language, and need more explanation or a broader language to lose their puzzling nature. Paradoxes that come from normal uses of language are often studied by logicians and philosophers. "This sentence is false" is an example of the well-known liar paradox: it is a sentence that cannot be clearly seen as true or false, because if it is known to be false, it means it must be true, and if it is known to be true, it means it must be false. Russell's paradox, which shows that the idea of the set of all those sets that do not contain themselves leads to a problem, helped shape modern logic and set theory.

Thought experiments can also create interesting puzzles. The grandfather paradox, for instance, would happen if a time traveler were to stop his own grandfather before his mother or father were born, which would stop the time traveler from being born. This is one example of the butterfly effect—any change a time traveler makes in the past would change things so much that the world would be different, possibly stopping the time travel from ever happening.

Often, a puzzling result comes from an unclear or self-contradictory starting point. In the case of the time traveler stopping his own grandfather, the problem is the idea that the past he goes back to is different from the past that leads to his own future, but also that he must have come from that same future.

Quine's classification

W. V. O. Quine grouped puzzles into three types.

A veridical paradox looks strange at first but turns out to be true. For example, Condorcet's paradox shows that group decisions can sometimes surprise us. The Monty Hall paradox proves that a choice that seems like a 50-50 guess might actually have different chances than we expect.

A falsidical paradox seems true but is actually wrong because of a mistake in the reasoning used to prove it.

An antinomy is a puzzle that uses normal thinking rules to reach a result that contradicts itself. This helps us see where our understanding might be missing something important.

Ramsey's classification

Frank Ramsey separated paradoxes into two groups: logical paradoxes and semantic paradoxes. Russell's paradox is a logical paradox, while the liar paradox and Grelling's paradoxes are semantic paradoxes.

Logical paradoxes deal with math and logic terms, like "class" and "number." They show problems in our logic or math. Semantic paradoxes involve ideas about thinking, language, and symbols. These paradoxes happen because of wrong ideas about how we think or talk, and they belong to the study of knowledge.

In medicine

A paradoxical reaction to a drug is when something happens that is the opposite of what we expect. For example, a sedative might make someone feel more agitated instead of calm, or a stimulant might make someone feel sleepy. Some of these reactions are common and even helpful, like using Adderall and Ritalin to help people with attention deficit hyperactivity disorder, also called ADHD. But other reactions can be rare and dangerous, like feeling very agitated after taking a benzodiazepine.

Sometimes, the way antibodies work against antigens can surprise us. For instance, in a few cases, antibodies can actually make a disease stronger instead of fighting it off, in what is called antibody-dependent enhancement. There is also something called the hook effect, where the test results can be misleading. Even though these unusual cases happen, antibodies are usually very helpful in keeping us healthy.

There is also something called the smoker's paradox, where smoking, even though it causes many health problems, seems to be linked to lower rates of some diseases. Additionally, when artificial intelligence is used to help make medical decisions, it can sometimes lead to more problems, as juries might penalize doctors no matter whether they follow the AI's advice or not.

As an aspect of reality

Some thinkers believe that paradoxes show a deeper level of thinking. They suggest that to truly understand reality, we might need to think in ways that seem confusing or opposite. This is because the true nature of reality can be hard to describe with ordinary words, as it often contains contradictions.

Teaching tool

Paradoxes are sometimes used to help people learn, especially in management classes. They can also be found as questions in Zen Buddhism, where they are called koan.