Category Archives: Paradoxes

The Rich Guest Paradox

There is a very poor quaint little town where everyone is in a huge debt with someone but with no money to pay for it. There is hotel which is hardly seeing any business anymore. They are to soon shut it down.

One day a very wealthy American guest shows up and he wants to spend a night there. However before he confirms he asks for a tour of the hotel. The receptionist asks for a security deposit which the american can take back in case he doesn’t like the rooms. The guest obliges.

It turns out by matter of luck this is the exact amount that the hotel owed to the chef as salary for three months which they hadn’t been able to pay. They gave the cash to the chef. The chef saw that this was the exact amount of cash he owed the grocer for months of groceries he hadn’t been able to pay for. He paid the grocer. The grocer realized it was the exact amount he owed the doctor for treating his wife’s arthritis. The doctor paid the money to the nurse for two months of service he couldn’t pay for. The nurse was new to the town so she had been staying in the hotel for a few days before she found a house to rent. She too was poor and couldn’t pay the hotel at that time. The money she received from the doctor was exactly what she owed the hotel so she paid.

Now the hotel had got back the exact amount it had paid the chef. Now the guest has finished his tour of the rooms. Turns out he doesn’t like it. He takes back his security deposit from the hotel and leaves, never to be seen again.

So everyone’s debt has been paid, but nothing is different from before, except everyone is now happy. 

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The paradox of tolerance

The paradox of tolerance arises when a tolerant person holds antagonistic views towards intolerance, and hence is intolerant of it. The tolerant individual would then be by definition intolerant of intolerance.

Philosopher Karl Popper defined the paradox in 1945 in The Open Society and Its Enemies Vol. 1. [1]

“Less well known is the paradox of tolerance: Unlimited tolerance must lead to the disappearance of tolerance. If we extend unlimited tolerance even to those who are intolerant, if we are not prepared to defend a tolerant society against the onslaught of the intolerant, then the tolerant will be destroyed, and tolerance with them.”

He concluded that we are warranted in refusing to tolerate intolerance:

“We should therefore claim, in the name of tolerance, the right not to tolerate the intolerant.”

In 1971, philosopher John Rawls concludes in A Theory of Justice [2] that a just society must tolerate the intolerant, for otherwise, the society would then itself be intolerant, and thus unjust. However, Rawls also insists, like Popper, that society has a reasonable right of self-preservation that supersedes the principle of tolerance:

“While an intolerant sect does not itself have title to complain of intolerance, its freedom should be restricted only when the tolerant sincerely and with reason believe that their own security and that of the institutions of liberty are in danger.”

In a 1997 work, Michael Walzer [3] asked “Should we tolerate the intolerant?” He notes that most minority religious groups who are the beneficiaries of tolerance are themselves intolerant, at least in some respects. In a tolerant regime, such people may learn to tolerate, or at least to behave “as if they possessed this virtue”.

References

  1. Popper, Karl, The Open Society and Its Enemies, volume 1, The Spell of Plato, 1945 (Routledge, United Kingdom); ISBN 0-415-29063-5 978-0-691-15813-6 (1 volume 2013 Princeton ed.)
  2.  Rawls, John, (1971). “A Theory of Justice”: 220
  3. Walzer, Michael, On Toleration, (New Haven: Yale University Press 1997) pp. 80-81 ISBN 0-300-07600-2

 

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Fitch’s paradox of knowability

Fitch’s paradox of knowability is one of the fundamental puzzles of epistemic logic. It provides a challenge to the knowability thesis, which states that every truth is, in principle, knowable. The paradox is that this assumption implies the omniscience principle, which asserts that every truth is known. Essentially, Fitch’s paradox asserts that the existence of an unknown truth is unknowable. So if all truths were knowable, it would follow that all truths are in fact known. The paradox is of concern for verificationist or anti-realist accounts of truth, for which the knowability thesis is very plausible, but the omniscience principle is very implausible.

A formal proof of the paradox is as follows.

Suppose p is a sentence which is an unknown truth; that is, the sentence p is true, but it is not known that p is true. In such a case, the sentence “the sentence p is an unknown truth” is true; and, if all truths are knowable, it should be possible to know that “p is an unknown truth”. But this isn’t possible, because as soon as we know “p is an unknown truth”, we know that p is true, rendering p no longer an unknown truth, so the statement “p is an unknown truth” becomes a falsity. Hence, the statement “p is an unknown truth” cannot be both known and true at the same time. Therefore, if all truths are knowable, the set of “all truths” must not include any of the form “something is an unknown truth”; thus there must be no unknown truths, and thus all truths must be known.

The proof has been used to argue against versions of anti-realism committed to the thesis that all truths are knowable. For clearly there are unknown truths; individually and collectively we are non-omniscient. So, by the main result, it is false that all truths are knowable. The result has also been used to draw more general lessons about the limits of human knowledge. Still others have taken the proof to be fallacious, since it collapses an apparently moderate brand of anti-realism into an obviously implausible and naive idealism.

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Poe’s Law

Poe’s law is a kind of paradox which states that, without a clear indicator of the author’s intent (even just a smiley), it is impossible to tell the difference between a sincere expression of extreme views and a parody of those views.  This is because the actual views and their parody both seem equally irrational or absurd.

Poe’s Law was formulated by the writer Nathan Poe in August 2005. The law emerged at the online Creation & Evolution forum.  Like most such places, it had seen a large number of creationist parody postings. These were usually followed by at least one user starting a flame war (a series of angry and offensive personal attacks) thinking it was a serious post and taking it at face value.

The law caught on and has since slowly become an Internet meme. Over time it has been extended to include not just creationist parody but any parody of extreme ideology, whether it be religious, secular, anti-science, conspiracy theorist or just totally bonkers.  We even find it a problem in the Skeptics in Australia Facebook group, where it is sometimes difficult to tell whether the poster is joking or not.  That’s why we ask people to include their own comment instead of just providing a link.

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A ‘Poe’ as a noun has become almost as ubiquitous as Poe’s Law itself. In this context, a Poe refers to either a person, post or news story that could cause Poe’s Law to be invoked. In most cases, this is specifically in the sense of posts and people who are taken as legitimate, but are probably a parody.  The use of the term is most common in online skeptical and science-based communities.  Many blogs, forums and wikis will often refer to the law when dealing with cranks of any stripe.

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The Barbershop Paradox

The Barbershop paradox was proposed by Lewis Carroll in a three-page essay titled “A Logical Paradox,” which appeared in the July 1894 issue of Mind. The name comes from the ‘ornamental’ short story that Carroll uses to illustrate the paradox (although it had appeared several times in more abstract terms in his writing and correspondence before the story was published). Carroll claimed that it illustrated “a very real difficulty in the Theory of Hypotheticals” in use at the time. Modern logicians would not regard it as a paradox but simply as a logical error on the part of Carroll.

Briefly, the story runs as follows: Uncle Joe and Uncle Jim are walking to the barber shop. There are three barbers who live and work in the shop—Allen, Brown, and Carr—but not all of them are always in the shop. Carr is a good barber, and Uncle Jim is keen to be shaved by him. He knows that the shop is open, so at least one of them must be in. He also knows that Allen is a very nervous man, so that he never leaves the shop without Brown going with him. Uncle Joe insists that Carr is certain to be in, and then claims that he can prove it logically. Uncle Jim demands the proof. Uncle Joe reasons as follows.

Suppose that Carr is out. If Carr is out, then if Allen is also out Brown would have to be in, since someone must be in the shop for it to be open. However, we know that whenever Allen goes out he takes Brown with him, and thus we know as a general rule that if Allen is out, Brown is out. So if Carr is out then the statements “if Allen is out then Brown is in” and “if Allen is out then Brown is out” would both be true at the same time.

Uncle Joe notes that this seems paradoxical; the two “hypotheticals” seem “incompatible” with each other. So, by contradiction, Carr must logically be in.

However, the correct conclusion to draw from the incompatibility of the two “hypotheticals” is that what is hypothesised in them (– that Allen is out) must be false under our assumption that Carr is out. Then our logic simply allows us to arrive at the conclusion “If Carr is out, then Allen must necessarily be in”.

In modern logic theory this scenario is not a paradox. The law of implication reconciles what Uncle Joe claims are incompatible hypotheticals. This law states that “if X then Y” is logically identical to “X is false or Y is true” (¬X ∨ Y). For example, given the statement “if you press the button then the light comes on”, it must be true at any given moment that either you have not pressed the button, or the light is on.

In short, what obtains is not that ¬C yields a contradiction, only that it necessitates A, because ¬A is what actually yields the contradiction.

In this scenario, that means Carr doesn’t have to be in, but that if he isn’t in, Allen has to be in.

A more detailed discussion of this apparent paradox may be found on Wikipedia.


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Catch-22

A Catch-22 is a paradoxical situation from which an individual cannot escape because of contradictory rules.

The term originates from satirical novel by the American author Joseph Heller.  In this novel, a World War II US Air Force bombardier called Yossarian was caught in the original Catch-22. Stationed in Italy and afraid of being shot down, he wanted to be declared insane and sent home.

But military rules said that fear of death was a rational response, so anyone who asked to be grounded could not possibly be truly crazy. And those who were insane would not be aware of the fact, and therefore would be unable to ask to be grounded. Therefore, Catch-22 ensures that no airman can ever be grounded for being insane even if he is.

catch22-finaledit-1

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The Paradox of Thrift

The paradox of thrift (or paradox of saving) is a paradox of economics generally attributed to John Maynard Keynes, although it had been stated as early as 1714 in The Fable of the Bees and similar sentiments dating to antiquity.

Keynes argued that consumer spending contributes to the collective good, because one person’s spending is another person’s income.  Thus, when individuals save too much instead of spending, they can cause collective harm because businesses do not earn as much and have to lay off employees who are then unable to save.  The paradox is that total savings may fall even when individual savings attempt to rise.  In this way, individual savings rather than spending can worsen a recession, and therefore be collectively harmful to the economy.

Consider the following example:

thrift boxes

In the above example, one consumer increased his savings by $100, but this cause no net increase in total savings.  Increased savings reduced income for other economic participants, forcing them to cut their savings. In the end, no new savings was generated while $200 income was lost.

This paradox is related to the fallacy of composition, which falsely concludes what is true of the parts must be true of the whole.  It also represents a prisoner’s dilemma, because saving is beneficial to each individual but deleterious to the general population.

The paradox of thrift is a central component of Keynesian economics, and has formed part of mainstream economics since the late 1940s, though it is disputed on a number of grounds by non-Keynesian economists such as Friedrich Hayek.  One of the main arguments against the paradox of thrift is that when people increase savings in a bank, the bank has more money to lend, which will generally decrease interest rates and thus spur lending and spending.

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Determinism vs ethics

This is a paradox about the relationship between determinism and ethics. Simply stated, the paradox is as follows:

Determinism affords no reason for doing anything, and is therefore ethically irrelevant. Therefore it does not render ethics irrelevant, since if it did, it would be ethically relevant.

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Buridan’s Ass

Buridan’s Ass is the name give to an apparent paradox related to the free will paradox; although there is some debate amongst philosophers as to whether it actually is a paradox (see below).

The paradox is named after the French priest and philosopher Jean Buridan (c.1300-1358CE), who studied under William of Ockham.  It refers to a hypothetical situation where a donkey finds itself exactly halfway between two equally big and delicious bales of hay.  There is no way of distinguishing between these two bales – they appear to be identical.  Because the donkey lacks a reason or cause to choose one over the other, it cannot decide which one to eat, and so starves to death. This tale is usually taken as demonstrating that there is no free will.

The corollary to this argument is that if the donkey eats one of the bales of hay, then the donkey is making a choice.  If the donkey is making a choice, then it must have free will, because there is no causal mechanism to make it choose one bale over another. And if donkeys have free will, then so must humans.

Deliberations_of_Congress

Deliberations of Congress (Source: Wikimedia Commons)

The paradox actually predates Buridan – it dates to antiquity, being found in Aristotle’s On the Heavens.[2] Aristotle, in ridiculing the Sophist idea that the Earth is stationary simply because it is circular and any forces on it must be equal in all directions, says that is as ridiculous as saying that:

…a man, being just as hungry as thirsty, and placed in between food and drink, must necessarily remain where he is and starve to death.  — Aristotle, On the Heavens, (c.350 BCE)

The 12th century Persian Islamic scholar and philosopher Al-Ghazali discusses the application of this paradox to human decision-making, asking whether it is possible to make a choice between equally good courses without grounds for preference.  He takes the attitude that free will can break the stalemate.

Suppose two similar dates in front of a man, who has a strong desire for them but who is unable to take them both. Surely he will take one of them, through a quality in him, the nature of which is to differentiate between two similar things. — Abu Hamid al-Ghazali, The Incoherence of the Philosophers (c.1100CE)

Professor Hauskeller of Exeter University takes a scientifically sceptical view of this paradox, using the donkey scenario:

If we could find a donkey which was dumb enough to starve between two piles of hay, we would have evidence against free will, at least as far as donkeys are concerned (or at least that particular donkey). But that’s not very likely. No matter how artfully we arrange the situation, a donkey will not hesitate very long, if at all, and will soon choose one of the piles of hay. He doesn’t care which, and he certainly won’t starve. However, even if we conducted thousands of experiments like this, and no donkey ever starved, we would still not have proved the existence of free will, because the reason no donkey ever starves in front of two equally attractive piles of hay may simply be that those piles aren’t really equally attractive. Perhaps in real life there aren’t any situations where the weighted reasons for a choice are equal.[1]

So Hauskeller’s suggested solution to the paradox is that the piles of hay are not equal in practice – the donkey detects a slight difference which causes it to choose one pile over the other. This solution is not very convincing when one considers the hypothetical possibility of the two piles of hay being exactly equal in appearance. So it seems that we still have a problem here.

Some proponents of hard determinism have acknowledged the difficulty the scenario creates for determinism, but have denied that it illustrates a true paradox, since a deterministic donkey could recognize that both choices are equally good and arbitrarily (randomly) pick one instead of starving. For example, there are deterministic machines that can generate random numbers, although there is some dispute as to whether such numbers are truly random.

References:

[1] Hauskeller, M. (2010) Why Buridan’s Ass Doesn’t Starve Philosophy Now, London. http://philosophynow.org/issues/81/Why_Buridans_Ass_Doesnt_Starve

[2] Rescher, N. (2005). Cosmos and Logos: Studies in Greek Philosophy . Ontos Verlag. pp. 93–99.

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The Liar’s Paradox

In philosophy and logic, the liar’s paradox  is the statement “this sentence is false.” Trying to assign to this statement a classical binary truth value leads to a contradiction (see paradoxes).

If “this sentence is false” is true, then the sentence is false, which is a contradiction. Conversely, if “this sentence is false” is false, then the sentence is true, which is also a contradiction.

This paradox is obliquely related to the City of Lies or Truth puzzle.

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