Paradox

For other uses, see Paradox (disambiguation).

A paradox is a statement or group of statements that leads to a contradiction or a situation which defies intuition. The term is also used for an apparent contradiction that actually expresses a non-dual truth (cf. kōan, Catuskoti). Typically, either the statements in question do not really imply the contradiction, the puzzling result is not really a contradiction, or the premises themselves are not all really true or cannot all be true together. The word paradox is often used interchangeably with contradiction. Often, mistakenly, it is used to describe situations that are ironic.

The recognition of ambiguities, equivocations, and unstated assumptions underlying known paradoxes has led to significant advances in science, philosophy and mathematics. But many paradoxes, such as Curry's paradox, do not yet have universally accepted resolutions.

Sometimes the term paradox is used for situations that are merely surprising. The birthday paradox, for instance, is unexpected but perfectly logical. The logician Willard V. O. Quine distinguishes falsidical paradoxes, which are seemingly valid, logical demonstrations of absurdities, from veridical paradoxes, such as the birthday paradox, which are seeming absurdities that are nevertheless true. Paradoxes in economics tend to be the veridical type, typically counterintuitive outcomes of economic theory, such as Simpson's paradox. In literature a paradox can be any contradictory or obviously untrue statement, which resolves itself upon later inspection.

Logical paradox

See also: List of paradoxes

Common themes in paradoxes include self-reference, the infinite, circular definitions, and confusion of levels of reasoning.

Patrick Hughes outlines three laws of the paradox:

• Self reference - all Cretans are liars, said the Cretan, is self referential, because the Cretan describes all Cretans;
• Contradiction - Cretans are liars, said the Cretan, is contradictory because the Cretan is saying that Cretans are liars;
• Vicious circularity or infinite regress - so if all Cretans are liars, and the Cretan told us so, then it cannot be true, but if it is not true that ALL Cretans are liars, then SOME Cretan must be a liar, and if there is only one Cretan, the statement stands, but then it is true that all Cretans are liars, so it must be a lie... and so on ad infinitum This is not necessarily a paradox: If there exists a Cretan other than the Cretan making the claim, and this Cretan is not a liar, then the Cretan making the claim is a liar: He is saying that ALL Cretans are liars when there is in fact a Cretan who is not. A better example of vicious circularity is the statement: "This statement is false".

Other paradoxes involve false statements or half-truths and the resulting biased assumptions.

For example, consider a situation in which a father and his son are driving down the road. The car collides with a tree and the father is killed. The boy is rushed to the nearest hospital where he is prepared for emergency surgery. On entering the surgery suite, the surgeon says, "I can't operate on this boy. He's my son."

The apparent paradox is caused by a hasty generalization; if the surgeon is the boy's father, the statement cannot be true. The paradox is resolved if it is revealed that the surgeon is a woman, the boy's mother.

Paradoxes which are not based on a hidden error generally happen at the fringes of context or language, and require extending the context or language to lose their paradoxical quality. Paradoxes that arise from apparently intelligible uses of language are often of interest to logicians and philosophers. This sentence is false is an example of the famous liar paradox: it is a sentence which cannot be consistently interpreted as true or false, because if it is known to be false then it is known that it must be true, and if it is known to be true then it is known that it must be false. Therefore, it can be concluded that it is unknowable. Russell's paradox, which shows that the notion of the set of all those sets that do not contain themselves leads to a contradiction, was instrumental in the development of modern logic and set theory.

Thought experiments can also yield interesting paradoxes. The grandfather paradox, for example, would arise if a time traveler were to kill his own grandfather before his father was conceived, thereby preventing his own birth. This paradox can be resolved by postulating that time travel leads to parallel or bifurcating universes, or that only contradiction-free timelines are stable.

W. V. Quine (1962) distinguished between three classes of paradoxes.

• A veridical paradox produces a result that appears absurd but is demonstrated to be true nevertheless. Thus, the paradox of Frederic's birthday in The Pirates of Penzance establishes the surprising fact that a twenty-one-year-old would have had only five birthdays, if he was born on a leap day. Likewise, Arrow's impossibility theorem demonstrates difficulties in mapping voting results to the will of the people.

• A falsidical paradox establishes a result that not only appears false but actually is false due to a fallacy in the supposed demonstration. The various invalid mathematical proofs (e.g. that 1 = 2) are classic examples, generally relying on a hidden division by zero. Another example is the inductive form of the Horse paradox, falsely generalizes from true specific statements.
• A paradox which is in neither class may be an antinomy, which reaches a self-contradictory result by properly applying accepted ways of reasoning. For example, the Grelling-Nelson paradox points out genuine problems in our understanding of the ideas of truth and description.

A fourth kind has sometimes been described since Quine's work.

• A paradox which is both true and false at the same time in the same sense is called a dialetheia. In Western logics it is often assumed, following Aristotle, that no dialetheia exist, but they are sometimes accepted in Eastern traditions and in paraconsistent logics. An example might be to affirm or deny the statement "John is in the room" when John is standing precisely halfway through the doorway. It is reasonable (by human thinking) to both affirm and deny it ("well, he is, but he isn't"), and it is also reasonable to say that he is neither ("he's halfway in the room, which is neither in nor out"), despite the fact that the statement is to be exclusively proven or disproven.

Paradox in literature

Main article: Paradox (literature)

The paradox as a literary device has been defined as an anomalous juxtaposition of incongruous ideas for the sake of striking exposition or unorthodox insight. It functions as a method of literary analysis which involves examining apparently contradictory statements and drawing conclusions either to reconcile them or to explain their presence.

Literary or rhetorical paradoxes abound in the works of Oscar Wilde and G. K. Chesterton; other literature deals with paradox of situation. Rabelais, Cervantes, Sterne, Borges, and Chesterton are all concerned with episodes and narratives designed around paradoxes. Statements such as Wilde’s “I can resist anything except temptation” and Chesterton’s “spies do not look like spies” are examples of rhetorical paradox. Further back, Polonius’ observation in Hamlet that “though this be madness, yet there is method in’t” is a memorable third.

A taste for paradox is central to the philosophies of Laozi, Heraclitus, Meister Eckhart, Kierkegaard, Nietzsche, and Tom Robbins, among many others.

Moral paradox

In moral philosophy, paradox in a loose sense plays a role in ethics debates. For instance, it may be considered that an ethical admonition to "love thy neighbour" is not just in contrast with, but in contradiction to armed neighbors actively intending murder. If the hostile neighbors succeed, it is impossible to follow the dictum. On the other hand, to attack, fight back, or restrain them is also not usually considered loving. This might be better termed an ethical dilemma rather than a paradox in the strict sense.

Another example is the conflict between a moral injunction and a duty that cannot be fulfilled without violating that injunction. For example, take the situation of a parent with children who must be fed (the duty), but cannot afford to do so without stealing, which would be wrong (the injunction). Such a conflict between two maxims is normally resolved through weakening one or the other of them: the need for survival is greater than the need to abide by the law. However, as maxims are added for consideration, the questions of which to weaken in the general case and by how much pose issues related to Arrow's impossibility theorem; it may not be possible to formulate a consistent system of ethics rules with a definite order of preference in the general case, a so-called "ethical calculus."

Paradoxes in a more strict sense have been relatively neglected in philosophical discussion within ethics, as compared to their role in other philosophical fields such as logic, epistemology, metaphysics, or even the philosophy of science. Important book-length discussions appear in Derek Parfit's Reasons and Persons and in Saul Smilansky's 10 Moral Paradoxes.

See also

• Dilemma
• Formal fallacy
• Impossible object
• List of paradoxes
• Mu (negative)
• Paradoxes of set theory
• Self refuting ideas
• Temporal paradox
• Four valued logic

External links

• Stanford Encyclopedia of Philosophy:
  • "Paradoxes and Contemporary Logic" -- by Andrea Cantini.
  • "Insolubles" -- by Paul Vincent Spade.

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• "MathPages - Zeno and the Paradox of Motion"