apparently self-contradictory statement, the underlying meaning of which is revealed only by careful scrutiny. The purpose of a paradox is to arrest attention and provoke fresh thought. The statement ¡°Less is more¡± is an example. Francis Bacon's saying, ¡°The most corrected copies are commonly the least correct,¡± is an earlier literary example. In George Orwell's anti-utopian satire Animal Farm (1945), the first commandment of the animals' commune is revised into a witty paradox: ¡°All animals are equal, but some animals are more equal than others.¡± Paradox has a function in poetry, however, that goes beyond mere wit or attention-getting. Modern critics view it as a device, integral to poetic language, encompassing the tensions of error and truth simultaneously, not necessarily by startling juxtapositions but by subtle and continuous qualifications of the ordinary meaning of words.

When a paradox is compressed into two words as in ¡°loud silence,¡± ¡°lonely crowd,¡± or ¡°living death,¡± it is called an oxymoron.


liar paradox

also called Epimenides' paradox the paradox that if ¡°This sentence is not true¡± is true, then it is not true, and if it is not true, then it is true. This example shows that certain formulations of words, though grammatically correct, are logically nonsensical. The English philosopher Bertrand Russell, in developing the theory of types, used the following illustration: the statement, ¡°I am lying¡± is true only if it is false, and false if it is true. Epimenides, a 6th-century-BC Cretan prophet, first recorded such a paradox.


Russell's paradox

statement in set theory, devised by the English mathematician-philosopher Bertrand Russell, that demonstrated a flaw in earlier efforts to axiomatize the subject.

Russell found the paradox in 1901 and communicated it in a letter to the German mathematician-logician Gottlob Frege in 1902. Russell's letter demonstrated an inconsistency in Frege's axiomatic system of set theory by deriving a paradox within it. (The German mathematician Ernst Zermelo had found the same paradox independently; since it could not be produced in his own axiomatic system of set theory, he did not publish the paradox.)

Frege had constructed a logical system employing an unrestricted comprehension principle. The comprehension principle is the statement that, given any condition expressible by a formula f(x), it is possible to form the set of all sets x meeting that condition, denoted {x | f(x)}. For example, the set of all sets¡ªthe universal set¡ªwould be {x | x = x}.

It was noticed in the early days of set theory, however, that a completely unrestricted comprehension principle led to serious difficulties. In particular, Russell observed that it allowed the formation of {x | x x}, the set of all non-self-membered sets, by taking f(x) to be the formula x x. Is this set¡ªcall it R¡ªa member of itself? If it is a member of itself, then it must meet the condition of its not being a member of itself. But if it is not a member of itself, then it precisely meets the condition of being a member of itself. This impossible situation is called Russell's paradox.

The significance of Russell's paradox is that it demonstrates in a simple and convincing way that one cannot both hold that there is meaningful totality of all sets and also allow an unfettered comprehension principle to construct sets that must then belong to that totality. (Russell spoke of this situation as a ¡°vicious circle.¡±)

Set theory avoids this paradox by imposing restrictions on the comprehension principle. The standard Zermelo-Fraenkel axiomatization (ZF; see the table) does not allow comprehension to form a set larger than previously constructed sets. (The role of constructing larger sets is given to the power-set operation.) This leads to a situation where there is no universal set¡ªan acceptable set must not be as large as the universe of all sets.

A very different way of avoiding Russell's paradox was proposed in 1937 by the American logician Willard Van Orman Quine. In his paper ¡°New Foundations for Mathematical Logic,¡± the comprehension principle allows formation of {x | f(x)} only for formulas f(x) that can be written in a certain form that excludes the ¡°vicious circle¡± leading to the paradox. In this approach, there is a universal set.


Achilles paradox

in logic, an argument attributed to the 5th-century BC Greek philosopher Zeno, and one of his four paradoxes described by Aristotle in the treatise Physics. The paradox concerns a race between the fleet-footed Achilles and a slow-moving tortoise. The two start moving at the same moment, but if the tortoise is initially given a head start and continues to move ahead, Achilles can run at any speed and will never catch up with it. Zeno's argument rests on the presumption that Achilles must first reach the point where the tortoise started, by which time the tortoise will have moved ahead, even if but a small distance, to another point; by the time Achilles traverses the distance to this latter point, the tortoise will have moved ahead to another, and so on.

The Achilles paradox cuts to the root of the problem of the continuum. Aristotle's solution to it involved treating the segments of Achilles' motion as only potential and not actual, since he never actualizes them by stopping. In an anticipation of modern measure theory, Aristotle argued that an infinity of subdivisions of a distance that is finite does not preclude the possibility of traversing that distance, since the subdivisions do not have actual existence unless something is done to them, in this case stopping at them. See also paradoxes of Zeno.


paradoxes of Zeno

statements made by the Greek philosopher Zeno of Elea, a 5th-century-BC disciple of Parmenides, a fellow Eleatic, designed to show that any assertion opposite to the monistic teaching of Parmenides leads to contradiction and absurdity. Parmenides had argued from reason alone that the assertion that only Being is leads to the conclusions that Being (or all that there is) is (1) one and (2) motionless. The opposite assertions, then, would be that instead of only the One Being, many real entities in fact are, and that they are in motion (or could be). Zeno thus wished to reduce to absurdity the two claims, (1) that the many are and (2) that motion is.

Plato's dialogue, the Parmenides, is the best source for Zeno's general intention, and Plato's account is confirmed by other ancient authors. Plato referred only to the problem of the many, and he did not provide details. Aristotle, on the other hand, gave capsule statements of Zeno's arguments on motion; and these, the famous and controversial paradoxes, generally go by names extracted from Aristotle's account: the Achilles (or Achilles and the tortoise), the dichotomy, the arrow, and the stadium.

The Achilles paradox (q.v.) is designed to prove that the slower mover will never be passed by the swifter in a race. The dichotomy paradox is designed to prove that an object never reaches the end. Any moving object must reach halfway on a course before it reaches the end; and because there are an infinite number of halfway points, a moving object never reaches the end in a finite time. The arrow paradox endeavours to prove that a moving object is actually at rest. The stadium paradox tries to prove that, of two sets of objects traveling at the same velocity, one will travel twice as far as the other in the same time.

If, in each case, the conclusion seems necessary but absurd, it serves to bring the premise (that motion exists or is real) into disrepute, and it suggests that the contradictory premise, that motion does not exist, is true; and indeed, the reality of motion is precisely what Parmenides denied.


Logical paradoxes

Highly amusing and often tantalizing, logical paradoxes generally lead to searching discussions of the foundations of mathematics. As early as the 6th century Bc, the Cretan prophet Epimenides allegedly observed that ¡°All Cretans are liars,¡± which, in effect, means that ¡°All statements made by Cretans are false.¡± Since Epimenides was a Cretan, the statement made by him is false. Thus the initial statement is self-contradictory. A similar dilemma was given by an English mathematician, P.E.B. Jourdain, in 1913, when he proposed the card paradox. This was a card on one side of which was printed:

¡°The sentence on the other side of this card is TRUE.¡±

On the other side of the card the sentence read:

¡°The sentence on the other side of this card is FALSE.¡±

The barber paradox, offered by Bertrand Russell, was of the same sort: The only barber in the village declared that he shaved everyone in the village who did not shave himself. On the face of it, this is a perfectly innocent remark until it is asked ¡°Who shaves the barber?¡± If he does not shave himself, then he is one of those in the village who does not shave himself and so is shaved by the barber, namely, himself. If he shaves himself, he is, of course, one of the people in the village who is not shaved by the barber. The self-contradiction lies in the fact that a statement is made about ¡°all¡± the members of a certain class, when the statement or the object to which the statement refers is itself a member of the class. In short, the Russell paradox hinges on the distinction between those classes that are members of themselves and those that are not members of themselves. Russell attempted to resolve the paradox of the class of all classes by introducing the concept of a hierarchy of logical types but without much success. Indeed, the entire problem lies close to the philosophical foundations of mathematics.