We explain what a paradox is, the types of paradoxes that exist and what their characteristics are.
Paradoxes result in a circular argument that leads to two contradictory and, at the same time, plausible propositions.
What is a paradox?
A paradox, also called logical antinomy, is a logical reasoning in which, through correct reasoning, one arrives at a conclusion that is apparently contradictory.
This happens because, by affirming the truth of the premises, we arrive at their falsehood, and once this same falsehood is affirmed, we return to the initial truth. This type of reasoning results in a circular argument. which results in two contradictory and, at the same time, plausible propositions.
A paradox can also be defined as an argument in which, starting from the same premise, and following two methods of reasoning that are consequently logical (that is, that are deduced by a series of logical steps), one can reach contradictory conclusionsThis definition is the one adopted by the philosopher and mathematician Rudolf Carnap (1891-1970) in his book Meaning and Need.
In logic, paradoxes They serve the function of showing that not all rational arguments give a single conclusive proposition that determines the validity or invalidity of the argument.The American logician WVO Quine (1908-2000) argued that there are three types of paradoxes: false paradoxes, true paradoxes, and antinomies.
Logical paradoxes
Logic is the study of reasoning, which is statements that are composed of propositions that act as premises and conclusions. A proposition is a meaningful characterization stated by a descriptive sentence in natural language. This means that a sentence in natural language (language in its everyday use), when it describes something, also serves the function of being a propositionand thus characterizes, through meaning, what is described. Every proposition is either true or false. A reasoning, on the other hand, cannot be predicated of truth or falsehood, but rather validity or invalidity.
A logical paradox consists of an argument such that, starting from the same premise, a conclusion is reached whose validity or invalidity cannot be determined, since both things are plausible.
WVO Quine, an American logician, classified paradoxes into three types:
- Falsidious paradoxes. These are arguments that, in addition to seeming absurd, are false. That is, they are arguments that result in a fallacy. Zeno’s paradox is an example of a falsidical paradox. However, Quine clarifies that a falsidical paradox is not, categorically, a fallacy. A fallacy is an argument where the conclusion does not follow from the premises, so the argument is invalid. However, the conclusion of a fallacy can be true. However, in the case of falsidical paradoxes, the conclusion is always false.
- True paradoxes. These are arguments that support, in appearance, an absurdity in their conclusions, but which in the background hide something true that is not seen at first glance. They are arguments that are usually called pseudo-paradoxes or paradoxes of true expression. Frederic’s paradox (in which a man reaches the age of 21 passing through only five birthdays because he was born on February 29) and Bertrand Russell’s paradox of the Barber of Seville are examples of true paradoxes.
- Antinomies. These are arguments that lead to crises in thought. They are produced by a self-contradiction of the logical rules. Some famous antinomies are the Grelling-Nelson antinomies, which has to do with denotation and shows how natural language leads to paradoxes, and the Russell antinomies, which is a rediscovery of the Cantorian paradox of logic by Gottlob Frege (1848-1925).
Famous paradoxes
The paradoxes They have existed since logic existed as a science that studies the way of arguing.The ancient Greek world, in particular, is a field rich in paradoxes and paradoxical stories, the best known of which are Zeno’s paradox and Protagoras’ paradox.
Zeno’s paradox
The paradox of Zeno of Elea (490-430 BC) is not a paradox in the strict sense, but it is in the broad sense. Generally, It is known as the argument of Achilles and the tortoise..
Zeno relates how Achilles, the fastest of the Greeks, must compete with a tortoise that he can never catch. Aristotle, in Physical (IV, 3, 210 and 1, 209), tells the story as follows:
The second argument is called the Achilles argument, which consists in this: that the slowest will never be overtaken in a race by the swiftest, since it is necessary that the pursuer arrive first at the place from which the pursued has set out, so that the slowest will necessarily always precede him by some distance. This is the same argument as in the dichotomy, but it differs in not dividing the amount obtained in two. The conclusion of the argument is therefore that the slowest is not overtaken; and it follows the same path as in the dichotomy; so that it is necessary that there be also the same solution.
(Aristotle, Physicaliv, 3,210 and 1,209)
Zeno’s argument can be summed up as follows: To reach the tortoise, Achilles must always travel half the distance between him and the tortoise.and before that half of half, and so on to infinity, which will prevent it from reaching the turtle.
It is for this reason that Zeno is known as the creator of the reduction to absurdity. The reduction to absurdity is a logical method that consists in showing how the proposition contradictory or opposite to an affirmed proposition implies an absurd, impossible consequence.
Protagoras’ paradox
The paradox of Protagoras (480-410 BC) is one of the oldest known. It consists of a discussion between Protagoras, a Greek sophist, and one of his disciples, Eualzus.
The history narrates how Protagoras agreed to teach Eualzo without charging himIn return, and once he had won his first lawsuit, Eualzus would pay Protagoras a certain sum. However, after completing his studies, Eualzus did not take up any legal cases and, after a time, Protagoras sued his disciple to pay what he owed him.
Eualzo’s argument was that he could not pay Protagoras. since, if he were to enter into litigation, only two things could happen: win or lose. If the result were victory, the law would not oblige Eualzo to pay Protagoras. If it were adverse, it would not have been the case that he had won his first case and, therefore, he would not be able to pay Protagoras.
Protagoras’ response, which marks the end of the paradox, is decisive. What the master maintained was that, if it went to court, two things could happen: either Eualzo would win or he would win. If Protagoras wins, the law would force Eualzo to pay him and if Eualzo won, he would have won his first case and, according to the initial agreement, he would have to pay Protagoras.
Protagoras’ paradox comes closer than Zeno’s to the definition of paradox proposed. That is: starting from the same assumption, one can arrive at contradictory conclusions.
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References
- Aristotle, F. (1995). Physical. Madrid: Editorial Gredos SA.
- Carnap, R. (1947/56). Meaning and Necessity. A Study in Semantics and Logic Chicago: The University of Chicago Press.
- Mondolfo, R. (1945). Ancient thought; history of Greco-Roman philosophy (Vol. 2). Editorial Losada, sa.
- Gamut, LTF, & Durán, C. (2002). Introduction to logic. Buenos Aires, Argentina: Eudeba.