Paradox

For other uses, see Paradox (disambiguation).

Further information: List of paradoxes

A paradox is a statement or group of statements that lead to a contradiction or a situation which (if true) defies logic or reason, similar to circular reasoning. Typically, however, quoted paradoxical statements do not imply a real contradiction and the puzzling results can be rectified by demonstrating that one or more of the premises themselves are not really true, a play on words, faulty and/or cannot all be true together. But many paradoxes, such as Curry's paradox, do not yet have universally accepted resolutions. The word paradox is often used interchangeably with contradiction. The logician Willard V. O. Quine distinguishes between:

• Falsidical paradoxes, which are seemingly valid, logical demonstrations of absurdities
• Veridical paradoxes, such as the birthday paradox, which are seeming absurdities that are nevertheless true because they are perfectly logical.

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, infinite regress, circular definitions, and confusion between different levels of abstraction.

Patrick Hughes outlines three laws of the paradox:

Self reference

An example is "This statement is false", a form of the liar paradox. The statement is referring to itself. Another example of self reference is the question of whether the barber shaves himself in the barber paradox. One more example would be "Is the answer to this question no?" In this case, replying no would be stating that the answer is not "no". If the reply is yes, it would be stating that it is "no", as the reply was yes. But because the question was answered with a "yes", the answer is not "no". A negative response without saying the word "no", like "it isn't", would, however, render the question answered without bringing about a paradox. Another example is the term 'Nothing is Impossible', meaning that it is impossible for something to be impossible, thus contradicting itself.

The concept of "nothingness" is the simplest classical example of a self-referential paradox

The existence of the concept of "nothingness" invalidates it. It's the simplest paradox because it's a single word that means nothing.

Contradiction

"This statement is false"; the statement cannot be false and true at the same time.

Vicious circularity, or infinite regress

"This statement is false"; if the statement is true, then the statement is false, thereby making the statement true. Another example of vicious circularity is the following group of statements:

"The following sentence is true."

"The previous sentence is false."

"What happens when Pinocchio says, 'My nose will grow now'?"

Other paradoxes involve false statements or half-truths and the resulting biased assumptions. This form is common in howlers.

For example, consider a situation in which a father and his son are driving down the road. The car crashes into 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 mother or father was conceived, thereby preventing his own birth. 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 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 dialetheism. 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 assigned 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, and Borges, for instance, are all concerned with episodes and narratives designed around paradoxes. Two of literature's arguably most famous paradoxes is the Miltonic narrator's statement in Book One of 'Paradise Lost', that the fires of hell emit "no light, but darkness visible." Statements such as Wilde's "I can resist anything except temptation," Chesterton's "Spies do not look like spies" and Polonius' observation in Hamlet that "though this be madness, yet there is method in't" are examples of rhetorical paradox. Another example of a paradox is found in the dialogue between Touchstone and Corin in Act Three, Scene Two of As You Like It by William Shakespeare, when Touchstone says:

"Truly, shepherd, in respect of itself, it is a good

life, but in respect that it is a shepherd's life,

it is naught. In respect that it is solitary, I

like it very well; but in respect that it is

private, it is a very vile life. Now, in respect it

is in the fields, it pleaseth me well; but in

respect it is not in the court, it is tedious. As

is it a spare life, look you, it fits my humour well;

but as there is no more plenty in it, it goes much

against my stomach."

Paradox in philosophy

A taste for paradox is central to the philosophies of Laozi, Heraclitus, Meister Eckhart, Hegel, Kierkegaard, and Nietzsche, among many others. Søren Kierkegaard, for example, writes, in the Philosophical Fragments, that

one must not think ill of the paradox, for the paradox is the passion of thought, and the thinker without the paradox is like the lover without passion: a mediocre fellow. But the ultimate potentiation of every passion is always to will its own downfall, and so it is also the ultimate passion of the understanding to will the collision, although in one way or another the collision must become its downfall. This, then, is the ultimate paradox of thought: to want to discover something that thought itself cannot think.

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. However, for this to be a true example of a moral paradox, it must be assumed that "loving" and restraint cannot co-exist. In reality, this situation occurs often, notably when parents punish children out of love.

Paradox in media

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.

Paradoxology

Paradoxology, "the use of paradoxes." As a word it originates from Thomas Browne in his book Pseudodoxia Epidemica."

Alexander Bard and Jan Söderqvist developed a "paradoxology" in their book Det globala imperiet ("The Global Empire"). The authors emphasize paradoxes between the world as static and as ever-changing, while leaning on loose allegories from quantum mechanics. One may also include the philosopher Derrida in a list of users of paradoxes. Derrida's deconstructions attempt to give opposing interpretations of the same text by rhetoric arguments, similar to how lawyers in a court case may argue from the same text, the same set of laws that is, to reach opposite conclusions.

See also

• Animalia Paradoxa
• Dilemma
• Irony
• Oxymoron
• Ethical dilemma
• Formal fallacy
• Four valued logic
• Impossible object
• List of paradoxes
• Mu (negative)
• Paradoxes of material implication
• Self-refuting ideas
• Twin paradox
• Zeno's paradoxes

External links

• Stanford Encyclopedia of Philosophy:
  • "Paradoxes and Contemporary Logic" – by Andrea Cantini.
  • "Insolubles" – by Paul Vincent Spade.
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• "Zeno and the Paradox of Motion". MathPages.com.
• "The Legacy of Paradox: From Cognition to Cosmology"