Paradox

We explain what a paradox is, the types of paradoxes that exist and what their characteristics are.

What is a paradox?

A paradox, also called logical antinomy, is a logical reasoning in which, through correct reasoning, an apparently contradictory conclusion is reached.

This happens because, by affirming the truth of the premises, we arrive at their falsity, and once this same falsity is affirmed, we arrive again at 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 consequently logical methods of reasoning (that is, they are deduced by a series of logical steps), contradictory conclusions can be reached. This definition is the one adopted by the philosopher and mathematician Rudolf Carnap (1891-1970) in his book Meaning and Necessity.

In logic, the 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: falsidic paradoxes, veridical 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 of natural language (of language in its everyday use), when it describes something, also fulfills the function of being a propositionand in this way, characterizes, through meaning, what is described. Every proposition is either true or false. A reasoning, for its part, 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, American logician, classified paradoxes into three types:

• Falsidic 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 false paradox. However, Quine clarifies that a falsidic 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. On the other hand, in the case of falsidic paradoxes, the conclusion is always false.
• true paradoxes. They are those that support, in appearance, an absurdity in their conclusions, but that deep down hide something true not seen in a first approach. These are arguments that are often called pseudo-paradoxes or paradoxes of true expression. Frederic's paradox (in which a man reaches the age of 21 having only five birthdays because he was born on February 29) and that of the barber of Seville, by Bertrand Russell, are examples of true paradoxes.
• Antinomies. They are arguments that lead to crises in thought. They are produced by a self-contradiction of logical rules. Some famous antinomies are Grelling-Nelson's, which has to do with denotation and shows how natural language leads to paradoxes, and Russell's, 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 being the paradox of Zeno and the paradox of Protagoras.

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 tells how Achilles, the fastest of the Greeks, must compete with a tortoise whom he can never catch up with. Aristotle, in Physics (IV, 3, 210 and 1, 209), collects the story as follows:

The second (reasoning) is the so-called Achilles: it consists of this: that the slowest will never be overtaken, in a race by the fastest, since it is necessary for the pursuer to reach first the place from which the pursued started, so so that the slower one will necessarily always precede it, by some distance. This is the same argument as the dichotomy, but it differs by not dividing the amount obtained in two. The conclusion of the argument, then, is that the slowest is not reached; and it follows the same path as in the dichotomy; so it is necessary that there also be the same solution.

(Aristotle, Physicsiv, 3,210 and 1,209)

Zeno's argument is summarized in that, To reach the tortoise, Achilles must always travel half the way that separates him from it.and before that half of half, and so on to infinity, which will prevent him from reaching the turtle.

For this argument, Zeno is known as the creator of the reduction to the absurd. Reduction to absurdity is a logical method that consists of showing how the contradictory or opposite proposition to an affirmed proposition implies an absurd, impossible consequence.

The 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, Eualzo.

The story tells how Protagoras agreed to teach Eualzo without charging him. In exchange, and once he had won his first lawsuit, Eualzo would pay Protagoras a certain sum. However, upon completing his studies, Eualzo did not take up any legal cases and, after a time, Protagoras demanded that his disciple pay what he owed him.

Eualzo's argument was that he could not pay Protagoras since, if you participate in litigation, only two things could happen: win or lose. If the result were victory, the law would not force Eualzo to pay Protagoras. If it were adverse, it would not have been the case that he had won his first case and therefore would not be able to pay Protagoras.

Protagoras' response, which marks the end of the paradox, is decisive. What the teacher maintained was that, if he 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, would have to pay Protagoras.

Protagoras' paradox is closer to the proposed definition of paradox than Zeno's. That is: starting from the same assumption, contradictory conclusions can be reached.

References

• Aristotle, F. (1995). Physics. 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). The ancient thought; history of Greco-Roman philosophy (Vol. 2). Editorial Losada, sa.
• Gamut, LTF, & Durán, C. (2002). Introduction to logic. Buenos Aires, Argentina: Eudeba.