Paradox



A paradox (Gk: παράδοξος, "aside belief") is an apparently true statement or group of statements that leads to a contradiction or a situation which defies intuition. 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 recognition of ambiguities, equivocations, and unstated assumptions underlying known paradoxes has led to significant advances in science, philosophy and mathematics.

The word paradox is often used interchangeably and wrongly with contradiction; but where a contradiction by definition cannot be true, many paradoxes do allow for resolution, though many remain unresolved or only contentiously resolved, such as Curry's paradox. Still more casually, the term is sometimes used for situations that are merely surprising, albeit in a distinctly "logical" manner, such as the Birthday Paradox. This is also the usage in economics, where a paradox is an unintuitive outcome of economic theory.

Examples
Sometimes supernatural or science fiction themes are held to be impossible due to resultant paradoxial conditions. The theme of time travel has staged many popular paradoxes arising from the traveler interfering with the past. Suppose Jones, who was born in 1950, travels back in time to 1901 and kills his own grandfather. It follows that neither his father nor he himself will be born; but then he would not have existed to travel back in time and kill his own grandfather; but then his grandfather would not have died and Jones himself would have lived; etc. This is known as the Grandfather paradox.

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 false it must be true, and if it is true it must be false. Therefore, it can be concluded the sentence is both true and false. 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.

For more examples see List of paradoxes.

Etymology
The etymology of paradox can be traced back to at least Plato's Parmenides, where Zeno of Elea, who lived from 490-430 BCE, used the word "paradoxon" to describe some of his seminal philosophic ideas.

Zeno sought to illustrate that equal absurdities followed logically from the denial of Parmenides' views. There were apparently 40 'paradoxes of plurality' and other paradoxes that Zeno used to attack the Greek understanding of the physical world. In fact, Zeno's paradoxes of multiplicity and motion, which revealed problems in the Greek idea of space and time, were resolvable only using mathematics discovered in the 19th century.

It is unknown if incarnations of paradox were used before Zeno of Elea. Later and more frequent usage of the word has been traced to the early Renaissance. Early forms of the word appeared in the late Latin paradoxum and the related Greek παράδοξος paradoxos, which means "contrary to expectation", or "incredible". The word is a fusion of the preposition para, meaning "against" or "beyond", and the noun stem doxa, meaning "belief" or "opinion".

Common themes
Common themes in paradoxes include direct and indirect self-reference, infinity, circular definitions, and confusion of levels of reasoning. 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 paradox quality.

In moral philosophy, paradox plays a central 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 an armed neighbour actively trying to kill you: if he or she succeeds, you will not be able to love him or her. But to preemptively attack them or restrain them is not usually understood as loving. This might be termed an ethical dilemma. Another example is the conflict between an injunction not to steal and one to care for a family that you cannot afford to feed without stolen money.

Types of paradoxes
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 person may be more than nine years old on his ninth birthday.  Likewise, Arrow's impossibility theorem involves behaviour of voting systems that is surprising but all too true.
 * A falsidical paradox establishes a result that not only appears false but actually is false; there is a fallacy in the supposed demonstration. The various invalid proofs (e.g. that 1 = 2) are classic examples, generally relying on a hidden division by zero. Another example would be the Horse paradox.
 * 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.