Lawyer paradox (5TH CENTURY BC)

Ascribed to the sophist philosopher Protagoras(c.490-420 BC).

A lawyer teaches law to a student without fee on condition that the student will pay him when he qualifies and wins his first case.

However, when the student qualifies he takes up another profession. The lawyer sues him for his fees, on the grounds that if he wins, he is paid and if he loses, the student has won and so must pay by the agreement.

The student is unperturbed because if he wins he need not pay the fees, and if he loses he does not owe them. There is some confusion concerning the agreement here, but logical rules preventing the application of a condition to itself certainly resolve the paradox.

Logic

  • Barbershop paradox: The supposition that, ‘if one of two simultaneous assumptions leads to a contradiction, the other assumption is also disproved’ leads to paradoxical consequences. Not to be confused with the Barber paradox.
  • What the Tortoise Said to Achilles: If a presumption needs to be made that a specific result can be deduced from premises, then the result can never be deduced. Also known as Carroll’s paradox and is not to be confused with the “Achilles and the tortoise” paradox by Zeno of Elea.
  • Catch-22: A situation in which someone is in need of something that can only be had by not being in need of it. A soldier who wants to be declared insane to avoid combat is deemed not insane for that very reason and will therefore not be declared insane.
  • Drinker paradox: In any pub there is a customer such that if that customer is drinking, everybody in the pub is drinking.
  • Paradox of entailment: Inconsistent premises always make an argument valid.
  • Lottery paradox: If there is one winning ticket in a large lottery, it is reasonable to believe of any particular lottery ticket that it is not the winning ticket, but it is not reasonable to believe that no lottery ticket will win.
  • Raven paradox: (or Hempel’s Ravens): Observing a green apple increases the likelihood of all ravens being black.
  • Ross’ paradox: Disjunction introduction poses a problem for imperative inference by seemingly permitting arbitrary imperatives to be inferred.
  • Unexpected hanging paradox: The day of the hanging will be a surprise, so it cannot happen at all, so it will be a surprise. The surprise examination and Bottle Imp paradox use similar logic.

Self-reference

These paradoxes have in common a contradiction arising from either self-reference or circular reference, in which several statements refer to each other in a way that following some of the references leads back to the starting point.

  • Barber paradox: A male barber shaves all and only those men who do not shave themselves. Does he shave himself? (Russell’s popularization of his set theoretic paradox.)
  • Bhartrhari’s paradox: The thesis that there are some things which are unnameable conflicts with the notion that something is named by calling it unnameable.
  • Berry paradox: The phrase “the first number not nameable in under ten words” appears to name it in nine words.
  • Crocodile dilemma: If a crocodile steals a child and promises its return if the father can correctly guess exactly what the crocodile will do, how should the crocodile respond in the case that the father guesses that the child will not be returned?
  • Paradox of the Court: A law student agrees to pay his teacher after (and only after) winning his first case. The teacher then sues the student (who has not yet won a case) for payment.
  • Curry’s paradox: “If this sentence is true, then Santa Claus exists.”
  • Epimenides paradox: A Cretan says: “All Cretans are liars”. This paradox works in mainly the same way as the liar paradox.
  • Grelling–Nelson paradox: Is the word “heterological”, meaning “not applicable to itself”, a heterological word? (A close relative of Russell’s paradox.)
  • Hilbert-Bernays paradox: If there was a name for a natural number that is identical to a name of the successor of that number, there would be a natural number equal to its successor.
  • Kleene–Rosser paradox: By formulating an equivalent to Richard’s paradox, untyped lambda calculus is shown to be inconsistent.
  • Knower paradox: “This sentence is not known.”
  • Liar paradox: “This sentence is false.” This is the canonical self-referential paradox. Also “Is the answer to this question ‘no’?”, and “I’m lying.”
    • Card paradox: “The next statement is true. The previous statement is false.” A variant of the liar paradox in which neither of the sentences employs (direct) self-reference, instead this is a case of circular reference.
    • No-no paradox: Two sentences that each say the other is not true.
    • Pinocchio paradox: What would happen if Pinocchio said “My nose grows now”?[1]
    • Quine’s paradox: “‘Yields a falsehood when appended to its own quotation’ yields a falsehood when appended to its own quotation.” Shows that a sentence can be paradoxical even if it is not self-referring and does not use demonstratives or indexicals.
    • Yablo’s paradox: An ordered infinite sequence of sentences, each of which says that all following sentences are false. While constructed to avoid self-reference, there is no consensus whether it relies on self-reference or not.
  • Opposite Day: “It is opposite day today.” Therefore, it is not opposite day, but if you say it is a normal day it would be considered a normal day, which contradicts the fact that it has previously been stated that it is an opposite day.
  • Problem of Absolute Generality: It initially appears as if we can quantify over absolutely everything (including the expression itself), but this generates the liar paradox
  • Richard’s paradox: We appear to be able to use simple English to define a decimal expansion in a way that is self-contradictory.
  • Russell’s paradox: Does the set of all those sets that do not contain themselves contain itself?
  • I know that I know nothing: Purportedly said by Socrates

Vagueness

  • Ship of Theseus: It seems like one can replace any component of a ship, and it is still the same ship. So they can replace them all, one at a time, and it is still the same ship. However, they can then take all the original pieces, and assemble them into a ship. That, too, is the same ship they began with.
See also List of Ship of Theseus examples
  • Sorites paradox (also known as the paradox of the heap): If one removes a single grain of sand from a heap, they still have a heap. If they keep removing single grains, the heap will disappear. Can a single grain of sand make the difference between heap and non-heap?

Mathematics

  • All horses are the same color: A fallacious argument by induction that appears to prove that all horses are the same color.
  • Ant on a rubber rope: An ant crawling on a rubber rope can reach the end even when the rope stretches much faster than the ant can crawl.
  • Cramer’s paradox: The number of points of intersection of two higher-order curves can be greater than the number of arbitrary points needed to define one such curve.
  • Elevator paradox: Elevators can seem to be mostly going in one direction, as if they were being manufactured in the middle of the building and being disassembled on the roof and basement.
  • Interesting number paradox: The first number that can be considered “dull” rather than “interesting” becomes interesting because of that fact.
  • Potato paradox: If potatoes consisting of 99% water dry so that they are 98% water, they lose 50% of their weight.
  • Russell’s paradox: Does the set of all those sets that do not contain themselves contain itself?

Statistics

  • Abelson’s paradox: Effect size may not be indicative of practical meaning.
  • Accuracy paradox: Predictive models with a given level of accuracy may have greater predictive power than models with higher accuracy.
  • Berkson’s paradox: A complicating factor arising in statistical tests of proportions.
  • Freedman’s paradox: Describes a problem in model selection where predictor variables with no explanatory power can appear artificially important.
  • Friendship paradox: For almost everyone, their friends have more friends than they do.
  • Inspection paradox: Why one will wait longer for a bus than one should?
  • Lindley’s paradox: Tiny errors in the null hypothesis are magnified when large data sets are analyzed, leading to false but highly statistically significant results.
  • Low birth weight paradox: Low birth weight and mothers who smoke contribute to a higher mortality rate. Babies of smokers have lower average birth weight, but low birth weight babies born to smokers have a lower mortality rate than other low birth weight babies. This is a special case of Simpson’s paradox.
  • Simpson’s paradox, or the Yule–Simpson effect: A trend that appears in different groups of data disappears when these groups are combined, and the reverse trend appears for the aggregate data.
  • Will Rogers phenomenon: The mathematical concept of an average, whether defined as the mean or median, leads to apparently paradoxical results—for example, it is possible that moving an entry from an encyclopedia to a dictionary would increase the average entry length on both books.

Probability

The Monty Hall problem: which door do you choose?

  • Bertrand’s box paradox: A paradox of conditional probability closely related to the Boy or Girl paradox.
  • Bertrand’s paradox: Different common-sense definitions of randomness give quite different results.
  • Birthday problem: What is the chance that two people in a room have the same birthday?
  • Borel’s paradox: Conditional probability density functions are not invariant under coordinate transformations.
  • Boy or Girl paradox: A two-child family has at least one boy. What is the probability that it has a girl?
  • Dartboard Puzzle: If a dart is guaranteed to hit a dartboard and the probability of hitting a specific point is positive, adding the infinitely many positive chances yields infinity, but the chance of hitting the dartboard is one. If the probability of hitting each point is zero, the probability of hitting anywhere on the dartboard is zero.[2]
  • False positive paradox: A test that is accurate the vast majority of the time could show you have a disease, but the probability that you actually have it could still be tiny.
  • Grice’s paradox: Shows that the exact meaning of statements involving conditionals and probabilities is more complicated than may be obvious on casual examination.
  • Monty Hall problem: An unintuitive consequence of conditional probability.
  • Necktie paradox: A wager between two people seems to favour them both. Very similar in essence to the Two-envelope paradox.
  • Nontransitive dice: One can have three dice, called A, B, and C, such that A is likely to win in a roll against B, B is likely to win in a roll against C, and C is likely to win in a roll against A.
  • Proebsting’s paradox: The Kelly criterion is an often optimal strategy for maximizing profit in the long run. Proebsting’s paradox apparently shows that the Kelly criterion can lead to ruin.
  • Sleeping Beauty problem: A probability problem that can be correctly answered as one half or one third depending on how the question is approached.
  • Three cards problem: When pulling a random card, how do you determine the color of the underside?
  • Three Prisoners problem: A variation of the Monty Hall problem.
  • Two-envelope paradox: You are given two indistinguishable envelopes, each of which contains a positive sum of money. One envelope contains twice as much as the other. You may pick one envelope and keep whatever amount it contains. You pick one envelope at random but before you open it you are given the chance to take the other envelope instead.

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