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Well-Known Paradoxes

Let us take a closer look at some of the most well-known and influential paradoxes:

1. The Liar Paradox​

The liar paradox is one of the oldest and best-known semantic paradoxes. In its simplest form it reads:

"This sentence is false."

If we assume that the sentence is true, then it must, according to its content, be false. If, on the other hand, we assume that it is false, then its statement does not hold, which means that it must be true. We end up in an endless circle.

Variants:

  • "I am lying right now."
  • "The next sentence is true. The previous sentence is false."
  • "This sentence is not true."

Significance: The liar paradox has profound implications for logic and the philosophy of language. It highlights the problems that can arise from self-reference and the mixing of object language and metalanguage. In response, Alfred Tarski developed his theory of truth, which distinguishes between different levels of language.

2. Russell's Paradox​

Russell's paradox, discovered by Bertrand Russell in 1901, concerns set theory and reads:

"Let R be the set of all sets that do not contain themselves as an element. Does R contain itself?"

If R contains itself, then by definition it does not belong to R. If R does not contain itself, then it meets the criterion for membership in R and would therefore have to contain itself.

Everyday version: The barber paradox: "In a village, the barber shaves all the men who do not shave themselves. Who shaves the barber?"

Significance: Russell's paradox shook the foundations of naive set theory and led to the development of axiomatic set theories such as Zermelo-Fraenkel set theory, which avoid such paradoxes.

3. The Ship of Theseus​

The Ship of Theseus is a classic philosophical paradox about identity and reads:

"Theseus owns a ship. Over time, all the planks and parts of the ship are gradually replaced with new ones. Is it still the same ship in the end? What if someone builds a second ship from the old parts – which is then the 'real' ship of Theseus?"

Significance: This paradox raises fundamental questions about the identity of objects over time. It is relevant to discussions about personal identity (if all the cells of our body are replaced, are we still the same person?) and to legal and ethical questions.

4. Sorites Paradox (Paradox of the Heap)​

The sorites paradox (from the Greek "soros" for "heap") concerns vague concepts and reads:

"A heap of sand remains a heap if you remove a single grain of sand. If you continue this process, even a single grain of sand would still have to be a 'heap', which is obviously false."

Variants:

  • "At how many hairs is one no longer bald?"
  • "At what income is one rich?"
  • "When does an embryo become a human being?"

Significance: The sorites paradox highlights the problems that arise from vague concepts. It has implications for the philosophy of language, logic (the development of many-valued logics) and for practical fields such as law and ethics, where clear boundaries often have to be drawn.

5. Zeno's Paradoxes of Motion​

Zeno's paradoxes are a series of paradoxes formulated by the Greek philosopher Zeno of Elea. The most famous is "Achilles and the Tortoise":

"Achilles, the fastest runner, gives a tortoise a head start. When Achilles reaches the tortoise's starting point, it has already moved a little further. When he reaches this new point, the tortoise has again moved a little further, and so on. It seems that Achilles can never catch up with the tortoise."

Other variants:

  • "The flying arrow": A flying arrow must, at every moment, be in a particular place. If it is in one place, it is not moving at that moment. If it is not moving at any moment, how can it move at all?
  • "The dichotomy": To cover a distance, you must first cover half of it, and before that half of the half, and so on. Since there are infinitely many such steps, it seems impossible even to set off.

Significance: Zeno's paradoxes have profoundly influenced mathematics and physics. They led to the development of concepts such as limits, infinitesimal calculus and the mathematical treatment of infinity.

6. The Prisoner's Dilemma​

The prisoner's dilemma is a paradox from game theory:

"Two prisoners are interrogated separately. Each has the choice to confess or to remain silent. If both remain silent, each receives one year in prison. If both confess, each receives three years. If one confesses and the other remains silent, the confessor goes free while the silent one receives five years. Although it would be best for both together to remain silent, rational self-interest leads both to confess."

Significance: The prisoner's dilemma shows how individually rational behaviour can lead to collectively suboptimal outcomes. It has far-reaching applications in economics, politics, ethics and evolutionary biology and helps to explain phenomena such as pollution, arms races and social dilemmas.

7. The Surprise Test Paradox​

The surprise test paradox reads:

"A teacher announces that in the coming week there will be a surprise test – on a day that the students cannot predict. The students argue: the test cannot take place on Friday, because if no test has taken place by Thursday, Friday would be predictable. For the same reason it cannot take place on Thursday, and so on back to Monday. The students conclude that no surprise test is possible. Nevertheless, the test takes place on Wednesday and really is a surprise."

Significance: This paradox concerns epistemic logic and the concept of knowledge about knowledge. It illustrates the complexity of announcements that incorporate their own unpredictability.

8. Newcomb's Paradox​

Newcomb's paradox is a decision paradox:

"A superintelligent being (the predictor) presents you with two boxes: Box A is transparent and visibly contains €1,000. Box B is opaque and contains either €1,000,000 or nothing. You have two options: take only Box B, or take both boxes. The predictor has already predicted what you will do. If it predicted that you would take only Box B, it put €1,000,000 in it. If it predicted that you would take both boxes, it left Box B empty. What should you do?"

Significance: This paradox pits two decision principles against each other: the dominance principle (taking both boxes is always better) and the expected-utility principle (taking only Box B leads to higher expected utility if the predictor is reliable). It has implications for decision theory, free will and causality.