Seeming logic can lead to absurd or meaningless statements {paradox}. Paradox is about conflict of opposites, conflict with accepted ideas, or category conflict.
resolution
Paradoxes can resolve by alternating truth-values in time, changing logical laws, identifying language or fact conflict and working around it, or choosing correct category level.
logical
Paradoxes {logical paradox} can use faulty laws or misapply logical laws. Logical paradoxes include Epimenides, Russell, Burali-Forti, and relation between two relations that are not so related.
semantic
Paradoxes {semantic paradox} can have ambiguity or error in thought or language. Semantic paradoxes include liar, Berry, Konig {least definable ordinal}, Richard, and Grelling.
Using axiom of choice, fixed-radius Euclidean spheres can map to finite parts that can then make two spheres of same fixed radius {Banach-Tarski paradox}.
A barber says he shaves all those who do not shave themselves and does not shave those who shave themselves, so he can and cannot shave himself {barber paradox}.
Sets can have the least integer not nameable in fewer than nineteen syllables, but this statement has only 18 syllables {Berry's paradox} {Berry paradox}. However, naming is arbitrary and not the same as nameable.
If surgeons operate, they should use anesthesia. If surgeons do not operate, they should not use anesthesia. Surgeons should operate but do not {Chisholm paradox}.
A crocodile tells a parent that he will return a child if parent can guess whether crocodile will return it or not. Parent says that crocodile will not return child {crocodile paradox}.
The more two alternatives are similar, the harder it is to choose, and the less it matters {Fredkin paradox}.
For two linked games, at both of which player tends to lose, randomly switching between games can win {gambler's paradox}. One game must have constant event probabilities, and other game must have varying event probabilities. Switching favors keeping gains made, while losses stay constant.
If someone regrets crime, person committed the crime {Good Samaritan paradox}. However, people should regret, but people should not commit crime.
Many words do not describe themselves. Words are heterological if they are not themselves the word. What if the word is the word heterological {heterological paradox}?
Lotteries have high odds against winning, so no one can believe that they will win. Someone has to win, though nobody can expect to win {lottery paradox}. Therefore, belief probability cannot completely explain belief.
Choosers can select only box 2 or both box 1 and box 2. Box 1 has $1000. Box 2 has $1000000 if predictor predicts chooser will take box 2. Box 2 has $0 if predictor predicts chooser will take both boxes or choose randomly. Predictions have always been correct before. Using expected utility, take box 2, but, using dominance, take both boxes {Newcomb's paradox} {Newcomb paradox}.
Prefaces can state that a book has at least one mistake and that the author stands behind what he or she wrote {preface paradox}. Authors can believe all book statements but also believe that at least one statement is false.
Protagoras gave law lessons to a student who agreed to pay Protagoras after winning a case. The student never got a case, so Protagoras brought the first case against the student, asking specifically for the pay. If student wins case, he both does not and does have to pay {Protagoras paradox}. If Protagoras wins case, he will receive pay, and he will not receive pay.
Induction can lead to statements but can also lead to statement contrapositives. Contrapositive statements are general, while statements are specific. Evidence for contrapositive statements cannot support statements. For example, "All ravens are black" has support from each raven observation. The statement is logically the same as its contrapositive, "All not black things are not ravens", which also has support from each raven observation {paradox of the ravens} {ravens paradox}.
A class can be about all things not in the class {Russell paradox} {Russell's paradox}, such as set of all sets that are not members of themselves.
sets
The set of all infinite sets is an infinite set and is a member of itself. The set of all sets is a member of itself. The set of all ideas can be an idea.
However, the set of all men is not a man. Therefore, sets with elements that are not classes cannot be members of themselves.
proof
Assume set of all sets that do not belong to themselves exists and is not a member of itself. Then it must belong to itself by its set definition, causing contradiction. If this set really does belong to itself, then it must not belong to itself by its set definition, causing contradiction. Therefore, set either does or does not belong to itself.
universal set
These two set types are mutually exclusive. The set of all sets that do not belong to themselves cannot be in either of these two set types. Therefore, no universal set exists.
class
Classes {class, set} have sets as members. Classes cannot be class members.
resolution
To resolve the paradox, replace the word "class" with the word "function" in paradox and proof.
Probability in combined population can favor one solution, even if probability in separate populations favors another solution {Simpson's paradox} {Simpson paradox}. For example, for population A, solution 1 has probability 2/3 with N = 3, and solution 2 has probability 1/2 with N = 2. For population B, solution 1 has probability 3/4 with N = 4, and solution 2 has probability 5/7 with N = 7. For combined population, solution 1 has probability 5/7 with N = 7, and solution 2 has probability 6/9 with N = 9. Average of population averages is not necessarily combined population average, because some populations have more weight.
mediant fraction
Simpson's paradox follows from mediant fraction properties.
change
If probability of two outcomes varies in one population or set and does not vary in another set, expected outcome can reverse.
"This statement is false", liar paradox, and masked man paradox {Eubulides paradox} have direct contradiction.
People know a brother, but, if brother is a masked man, they do not know their brother {masked man paradox}. It depends on opaque reference.
One thing is tautologically identical to itself, but two different things cannot be identical {paradox of identity} {identity paradox}. Identity cannot be one relation. Identity requires conjunction of two propositions.
John Locke imagined that he had a sock with a hole. Is a mended sock the same sock {Locke's sock} {Locke sock}?
Theseus returned from Crete in a thirty-oared ship. Athenians preserved his ship and, as years passed, replaced planks. Is a ship that has replaced parts the same ship {Theseus' paradox} {Theseus paradox} {Ship of Theseus}? Do things that grow maintain their identity? Is a second ship, built from the old planks, the original ship?
People can say that they are stating false statements, but they can be lying, so statements can be both true and not true in all cases {liar paradox}.
This statement is false, or I am lying {Epimenides paradox}.
An event will happen some day of the next n days, but the day must be such that no one can predict the day {prediction paradox}. Event cannot be on last day, because it is last possible day and so predictable. If it cannot be last day, then it cannot be next-to-last day, because that day has in effect become the last day, and so on, until first day, so event cannot happen.
Examination will be some day next week, but no one can know the day {examination paradox}. Exam cannot be on last day of week, because it is last possible day. Because it cannot be last day, it cannot be next-to-last day, because last day is not possible. It cannot be on other days, because it cannot be on next day.
People that are to hang at noon one day of next week cannot predict the day {hangman's paradox}. If seventh day comes and no hanging has happened yet, prediction is possible, so it is clear that they cannot hang on seventh day. Then they cannot hang on sixth day either, and so on, until first day, so no hanging can happen.
Removing small amounts makes little difference, but inconsequential-change series can add to consequential change {sorites principle, logic}.
Heaps of sand are still heaps after removing some grains, but are not heaps after removing too many {heap paradox} {paradox of the heap}. Removing some grains makes little difference, but a series of such inconsequential changes adds to consequential change {sorites principle, heap}. The same idea applies to having hair and being bald {bald man paradox}.
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Date Modified: 2022.0225