A paradox is a statement or argument that seems logically sound yet is self-contradictory or leads to an absurd conclusion. The word "paradox" is derived from a Greek word meaning "contrary to opinion." Paradoxes are studied by mathematicians, logicians, philosophers, and linguists for the light they shed on commonly held opinions in each field.
Some of the best-known paradoxes were first formulated in ancient times. Included among these are the liar paradox, which has been attributed to Epimenides (6th century B.C.), and the paradox of Achilles and the tortoise, attributed to Zeno (5th century B.C.).
The Liar Paradox
Epimenides the Cretan is said to have uttered this statement: "All Cretans are liars." Now if this statement is true and all Cretans are liars, then he must be lying; hence his statement must be false. A modern paradox employing the same principle as Epimenides' puzzle is the statement: "This sentence is false." If it is true, then it is false; but furthermore, if it is false, then it is true.
Achilles and the Tortoise
Zeno proposed a race between Achilles and a tortoise. Aware of AchiDes' speed and the tortoise's slowness, he suggested that, in all fairness, the tortoise be given a slight head start. The result of the race, according to Zeno, is that no matter how long the track, Achilles never even catches up to the tortoise. The reasoning behind this absurd result is that no matter how fast Achilles runs, by the time he reaches the place where the tortoise started, the tortoise will have moved ahead some distance to a new position; by the time Achilles reaches the new position, the tortoise will have crawled ahead again. Thus, Achilles will come closer and closer, but he will never be able to catch up to the tortoise.
Analysis of These Paradoxes
There are obvious similarities between these two paradoxes. First, both of them involve absurd conclusions. A sentence certainly cannot be true only on the condition that it be false, and vice versa. Achilles certainly will catch up with the tortoise; indeed, if the speed of each is known, along with the length of the tortoise's head start, the number of minutes it will take him to catch up can be determined precisely. Secondly, it is not immediately clear that there has been any mistake in the reasoning that leads to these conclusions. "This sentence is false" seems to be a perfectly good sentence in English; Achilles will indeed have to reach the spots the tortoise has occupied before he can catch it.
It is this second feature of the paradoxes that differentiates them from a familiar type of argument known as reductio ad absurdum (Latin for "reduction to absurdity"). To prove, for instance, that a man is innocent of some crime, he is assumed guilty, and absurd consequences are then derived from this assumption. Since the assumption has led to absurd results, it must be false and is therefore discarded. But in Zeno's and Epimenides' paradoxes, the assumptions do not appear to be suspect.
Nevertheless, something must be generating tne absurdity. The difficulty is that in a typical logical paradox, men do not suspect their assumptions or reasonings because they are committed to them. To alter them would involve an uncomfortable change in their ways of talking or thinking. It is in this respect that a paradox is "contrary to opinion." Because the study of paradoxes forces the logician to alter common opinions it exercises an important function.
Despite the similarities between the two paradoxes, it might be argued that Zeno's paradox is actually different from Epimenides' in the second respect. That is, it might be said that Zeno's reasoning contains a hidden flaw, and that his paradox should therefore be more properly called a fallacy. Zeno's argument is so complex that there is to this day a good deal of disagreement as to the precise source of the absurdity. Nevertheless, many feel that the error lies in his assumption that an infinite succession of intervals of time must add up to eternity and that this error forces him to conclude that Achilles will never catch the tortoise. Zeno pictures the relative positions of the runners at ever shorter periods of time. Since there seems to be no end of intervals to choose, Zeno concludes that Achilles never catches up. But it is not necessarily true that an infinite succession of time intervals must add up to eternity. Some infinite series, called convergent series, by definition do not add upto infinity. (The series 1 + 1 + 1 + 1 + 1 . . . , for instance, is a convergent series. It has infinitely many terms, yet its sum, far from being infinite, never exceeds 1.) To avoid the paradox, one need merely see that Zeno is wrongly assuming that an infinite series must have an infinite sum.
But with the liar paradox there is no such well-established escape. The best solution offered to date, one not developed until the 20th century, is that the paradox results from confusing two languages: the object language, or language used to talk about objects, and the meta-language, or language used to talk about the object language. The trouble with "This sentence is false" is that it attempts to be in the object language and the meta-language simultaneously. In referring to itself, the term "false" becomes incurably ambiguous; the sentence is thus meaningless.
The difference between Zeno's and Epimenides' paradoxes, then, is that the first can be solved by appealing to a well-established mathematical concept, whereas solution of the second requires a fundamental change in one's understanding of the nature of language. It must be recognized, for instance, that English or any other tongue is not a single language, but a set of increasingly complex object languages and meta-languages, each containing the simpler languages as parts. Still, there seems to be good reason for calling Zeno's race a paradox. It would have required a tremendous alteration of common opinion for an ancient Greek to accept the notion of a convergent series. And even for people in modern times, the concept is puzzling.
Some famous paradoxes have not yet been satisfactorily dealt with. An example is the paradox stated by Bertrand Russell in 1901. Russell's remarkable paradox, which results from a common-sense assumption concerning the nature of membership in mathematical sets, has given rise to several divergent schools among contemporary mathematicians, each grounding its work in a different alternative to the common-sense assumption. But each of these alternatives has led to new difficulties.