13.6 Other paradoxes in logic Paradoxes of inaccurate concepts Paradoxes of inductive logic

The above paradoxes are arguments, the outcome of which is a contradiction. But there are other types of paradoxes in logic. They also point to some difficulties and problems, but they do so in a less harsh and uncompromising manner. These, in particular, are the paradoxes discussed below.

Paradoxes of inaccurate concepts

Most of the concepts of ns only natural language, but the language of science are inaccurate, or, as they are called, blurred. Often this turns out to be the cause of misunderstanding, controversy, or simply leads to dead ends.

If the concept is inaccurate, the boundary of the area of ​​objects to which it is applied is devoid of sharpness, is blurred. Take, for example, the concept of "heap". One grain (a grain of sand, a stone, etc.) is not a pile yet. A thousand grains - this is obviously a bunch. And three grains? And ten? With the addition of what is the grain of the pile is formed? Nc is very clear. In the same way, it is not clear with which grain the heap disappears.

The empirical characteristics “big”, “heavy”, “narrow”, etc. are inaccurate. Such usual concepts as “sage”, “horse”, “house”, etc. are inaccurate.

There is no grain of sand, removing which we could say that with its elimination the rest can no longer be called home. But after all, it seems as if at no point in the gradual dismantling of the house - right up to its complete disappearance - there is no reason to declare that there is no house! The conclusion is clearly paradoxical and discouraging.

It is easy to see that the argument about the impossibility of heap formation is carried out using the well-known method of mathematical induction. One grain does not form a heap. If n grains do not form heaps, then n + 1 grains do not form heaps. Consequently, no number of grains can form heaps.

The possibility of this and similar evidence, leading to ridiculous conclusions, means that the principle of mathematical induction has a limited scope of application. It should not be used in reasoning with inaccurate, vague concepts.

A good example of the fact that these concepts can lead to intractable disputes can serve as a curious legal process, held in 1927 in the United States. The sculptor C. Brancusi went to court with a demand to recognize his works as works of art. Among the works sent to New York for the exhibition was the sculpture "The Bird", which is now considered a classic abstract style. It is a modulated column of polished bronze about one and a half meters in height, which has no external resemblance to a bird. Customs officers categorically refused to recognize Brancusi’s abstract creations as works of art. They conducted them under the column “Metallic hospital supplies and household goods” and imposed a large customs duty on them. Angered by Brancusi sued.

Customs was supported by artists - members of the National Academy, who defended the traditional techniques in art. They spoke at the trial as witnesses for the defense and categorically insisted that the attempt to extradite “The Bird” as a work of art was simply a scam.

This conflict in relief emphasizes the difficulty of operating with the term “work of art”. Sculpture is traditionally considered a form of fine art. But the degree of similarity of the sculptural image to the original can vary within very wide limits. And at what point does the sculptural image, increasingly removed from the original, cease to be a work of art and become "metal utensils"? This question is as difficult to answer as the question of where the border between the house and its ruins lies, between the horse with the tail and the horse without the tail, etc. By the way, modernists are generally convinced that sculpture is an object of expressive form, and it does not have to be an image at all.

Dealing with inaccurate concepts thus requires some caution. Is it not better then to abandon them at all?

The German philosopher E. Husserl was inclined to demand knowledge of such extreme rigor and accuracy, which is not found even in mathematics. Husserl's biographers ironically recall in connection with this the incident that happened to him in childhood. He was presented with a penknife, and, deciding to make the blade extremely sharp, he sharpened it until nothing remained of the blade.

More accurate concepts in many situations are preferable to inaccurate. The usual desire to clarify the concepts used is justified. But it must, of course, have its limits. Even in the language of spiders, a significant part of the concepts is inaccurate. And this is not connected with the subjective and accidental mistakes of individual scientists, but with the very nature of scientific knowledge. In natural language inaccurate concepts, the vast majority; it speaks, among other things, of its flexibility and hidden strength. The one who demands the utmost precision from all concepts risks in general to remain without language. “Deprive the words of any ambiguity, of any uncertainty,” wrote French aestheticist J. Joubert, “turn them ... into single digits — the game will leave speech, and with it, eloquence and poetry: everything that is mobile and changeable in attachments soul, can not find expression. But what I say: deprive ... I will say more. Depress the words of any inaccuracy - and you will lose even axioms. ”

For a long time, both logic and mathematics did not pay attention to the difficulties associated with vague concepts and their corresponding sets. The question was posed as follows: concepts should be accurate, and all vague unworthy of serious interest. In recent decades, this overly strict installation has lost, however, its attractiveness. Constructed logical theories, specifically taking into account the peculiarity of reasoning with inaccurate concepts.

The mathematical theory of so-called fuzzy sets, unclearly outlined sets of objects is actively developing.

The analysis of inaccuracy problems is a step on the path of convergence of logic with the practice of ordinary thinking. And we can assume that it will bring many more interesting results.

Paradoxes of inductive logic

There is, perhaps, no such section of logic in which there would not be its own paradoxes.

In inductive logic, there are paradoxes that have been actively, but so far without much success, almost half a century of struggle. The paradox of confirmation, discovered by the American philosopher K. Hempel, is especially interesting. It is natural to assume that the general provisions, in particular scientific laws, are confirmed by their positive examples. If we consider, say, the statement “All A is B”, then its positive examples will be objects with properties A and B. In particular, the supporting examples for the statement “All crows are black” are objects that are both crows and black. This statement is equivalent, however, to the statement “All objects that are not black, not crows,” and the confirmation of the latter must also be a confirmation of the first. By "Everything is not black is not a raven" is confirmed by each case of a non-black object that is not a raven. It turns out, therefore, that the observations “White cow”, “Brown shoes”, etc. confirm the statement "All ravens are black."

The innocent, seemingly parcels result in an unexpected paradoxical result.

In the logic of norms, anxiety is caused by a number of its laws. When they are formulated in meaningful terms, inconsistency with their usual ideas of what is due and forbidden becomes obvious. For example, one of the laws says that the order “Send a letter!” Implies the order “Send a letter or burn it!”.

Another law states that if a person has violated one of his duties, he gets the right to do anything. Pasha's logical intuition does not want to put up with this kind of “laws of responsibility”.

In the logic of knowledge, the paradox of logical omniscience is intensely discussed. He argues that a person knows all the logical consequences arising from the provisions he makes. For example, if a person knows five postulates of Euclidean geometry, then it means that he knows all this geometry, because it follows from them. But it is not. A person may agree with the postulates and at the same time not be able to prove the Pythagorean theorem and therefore doubt that it is generally true.