MORE EXAMPLES Kozma Prutkov's "Historical Materials" tells of Duke de Rogan, who was prescribed by a doctor to take a special medicine twenty drops in water. When the next day the doctor came to the patient, he sat in a cold bath and calmly drank with a spoon.

Examples of information Let's now take all the modes of the second, third and fourth figures, and reduce them in turn to the first figure. Figure 2. Mode Cesare P1: No zombie is a vegetarian. (E) P2: All participants of ru_vegetarian (http://ru_vegetarian.livejournal.com/) are vegetarians. (A) Z: None

CHAPTER 6 EXAMPLES Historical examples make everything clear and, moreover, they are the best evidence in sciences based on experience. More than anywhere else, this is observed in military art. General Scharnhorst, who in his Sputnik wrote best of all about

3. Examples When the Allies occupied Bonaparte's capital in 1814, the aim of the war was achieved. Political stratifications, based on Paris, began to show, and a huge crack caused the collapse of the emperor's power. All this must be considered from the point of view of

EXAMPLES PRIMARY PEOPLES Where did my soul go? Come back, come back. It climbed far to the South, South of our southernmost tribes. Come back, come back. tribes.Come back

3. Facts as examples Empirical data can be used in the course of the argument as examples, illustrations and models. Acting as an example, a fact or a particular case makes generalization possible; by way of illustration, he reinforces the already established

1. Introduction to the course of logic In its development, mankind has come a long way - from the distant times, when the first representatives of our kind had to huddle in caves, to the cities in which we and our contemporaries live. Such a time gap did not affect the essence

It is known that formulating a problem is often more important and more difficult than solving it. “In science,” wrote the English chemist F. Soddy, “a problem properly posed is more than half solved. The mental preparation process required to find out what exists specific task, often takes more time than the solution of the problem itself.
The forms in which the problem situation is manifested and realized are very diverse. Far from always, it reveals itself in the form of a direct question that arose at the very beginning of the study. The world of problems is as complex as the process of cognition that generates them. Identifying problems is at the core of creative thinking. Paradoxes are the most interesting case of implicit, questionless ways of posing problems. Paradoxes are common in the early stages of the development of scientific theories, when the first steps are being taken in an as yet unexplored area and the most general principles of approach to it are being groped for.

Paradoxes and logic

In a broad sense, a paradox is a position that sharply diverges from generally accepted, established, orthodox opinions. “Generally accepted opinions and what is considered a matter of long-term decision, most often deserve research” (Glichtenberg). Paradox is the beginning of such research.
A paradox in a narrower and more specialized sense is two opposing, incompatible statements, for each of which there are seemingly convincing arguments.
The sharpest form of paradox is antinomy, a reasoning that proves the equivalence of two statements, one of which is a negation of the other.
Paradoxes are especially famous in the most rigorous and exact sciences - mathematics and logic. And this is no coincidence.

Logics- abstract science. There are no experiments in it, not even facts in the usual sense of the word. In building its systems, logic ultimately proceeds from the analysis of real thinking. But the results of this analysis are synthetic, undifferentiated. They are not statements of any separate processes or events that the theory should explain. Obviously, such an analysis cannot be called an observation: a concrete phenomenon is always observed.
Constructing a new theory, the scientist usually starts from the facts, from what can be observed in the experiment. However free his creative imagination may be, it must reckon with one indispensable circumstance: a theory makes sense only if it agrees with the facts pertaining to it. A theory that disagrees with facts and observations is far-fetched and has no value.
But if there are no experiments in logic, no facts, and no observation itself, then what holds back logical fantasy? What factors, if not facts, are taken into account when creating new logical theories?
The discrepancy between logical theory and the practice of real thinking is often revealed in the form of a more or less acute logical paradox, and sometimes even in the form of a logical antinomy, which speaks of the internal inconsistency of the theory. This just explains the importance that is attached to paradoxes in logic, and the great attention that they enjoy in it.

Variants of the “Liar” Paradox

The most famous and perhaps the most interesting of all logical paradoxes is the “Liar” paradox. It was he who glorified the name of Eubulides from Miletus who discovered it.
There are variants of this paradox, or antinomy, many of which are only apparently paradoxical.
In the simplest version of "Liar" a person says only one phrase: "I'm lying." Or he says, "The statement I am now making is false." Or: "This statement is false."

If the statement is false, then the speaker told the truth, and therefore what he said is not a lie. If the statement is not false, and the speaker claims that it is false, then this statement is false. It turns out, therefore, that if the speaker is lying, he is telling the truth, and vice versa.

In the Middle Ages, the following wording was common:

“What Plato said is false,” says Socrates.

“What Socrates said is true,” says Plato.

The question arises, which of them expresses the truth, and which is a lie?
And here is a modern paradox of this paradox. Let us assume that only the words are written on the front side of the card: "On the other side of this card is written a true statement." It is clear that these words represent a meaningful statement. Turning over the card, we must either find the promised statement, or it is not there. If it is written on the back, then it is either true or not. However, on the back are the words: “There is a false statement written on the other side of this card” - and nothing more. Assume that the statement on the front side is true. Then the statement on the back must be true, and therefore the statement on the front must be false. But if the statement on the front is false, then the statement on the back must also be false, and therefore the statement on the front must be true. The end result is a paradox.
The Liar paradox made a huge impression on the Greeks. And it's easy to see why. The question that it poses at first glance seems quite simple: is he lying who says only that he is lying? But the answer "yes" leads to the answer "no", and vice versa. And reflection does not clarify the situation at all. Behind the simplicity and even routine of the question, it reveals some obscure and immeasurable depth.
There is even a legend that a certain Filit Kossky, desperate to resolve this paradox, committed suicide. They also say that one of the famous ancient Greek logicians, Diodorus Kronos, already in his declining years, made a vow not to eat until he found the solution to the “Liar”, and soon died without achieving anything.
In the Middle Ages, this paradox was referred to the so-called unsolvable sentences and became the object of systematic analysis. In modern times, the “Liar” did not attract any attention for a long time. They did not see any, even minor, difficulties regarding the use of the language. And only in our so-called modern times the development of logic has finally reached a level where the problems that seem to be behind this paradox can be formulated already in strict terms.
Now "Liar" - this typical former sophism - is often referred to as the king of logical paradoxes. An extensive scientific literature is devoted to him. And yet, as in the case of many other paradoxes, it remains not entirely clear what problems lie behind it and how to get rid of it.

Language and metalanguage

Now "The Liar" is usually considered a characteristic example of the difficulties that the confusion of two languages ​​leads to: the language in which one speaks of a reality that lies outside of it, and the language in which one speaks of the very first language.

In everyday language there is no distinction between these levels: we speak the same language about reality and about language. For example, a person whose native language is Russian does not see any particular difference between the statements: "Glass is transparent" and "It is true that glass is transparent", although one of them speaks of glass, and the other about a statement about glass.
If someone had the idea of ​​the need to talk about the world in one language, and about the properties of this language in another, he could use two different existing languages Let's say Russian and English. Instead of just saying “Cow is a noun”, I would say “Cow is a noun”, and instead of “The statement “Glass is not transparent” is false”, I would say “The assertion “Glass is not transparent” is false”. With this use of two different languages, what is said about the world would be clearly different from what is said about the language with which one speaks about the world. Indeed, the first statements would refer to Russian, while the second to English.

If further our expert on languages ​​would like to speak out about some circumstances that already concern the English language, he could use another language. Let's say German. To talk about this latter, one could resort, let's say, to Spanish etc.
It turns out, therefore, a kind of ladder, or hierarchy, of languages, each of which is used for a very specific purpose: in the first one they talk about the objective world, in the second - about this first language, in the third - about the second language, etc. Such a distinction between languages ​​according to their area of ​​application is a rare occurrence in everyday life. But in the sciences, which, like logic, deal specifically with languages, it sometimes turns out to be very useful. The language used to talk about the world is usually called object language. The language used to describe the subject language is called a metalanguage.

It is clear that if language and metalanguage are demarcated in this way, the statement "I am lying" can no longer be formulated. It speaks of the falsity of what is said in Russian, and, therefore, refers to the metalanguage and must be expressed in English language. Specifically, it should sound like this: “Everything I speak in Russian is false” (“Everything I say in Russian is false”); this English statement says nothing about itself, and no paradox arises.
The distinction between language and metalanguage makes it possible to eliminate the “Liar” paradox. Thus, it becomes possible to correctly, without contradiction, define the classical concept of truth: a statement is true that corresponds to the reality it describes.
The concept of truth, like all other semantic concepts, has a relative character: it can always be attributed to a particular language.

As the Polish logician Atarsky showed, the classical definition of truth must be formulated in a language broader than the language for which it is intended. In other words, if we want to indicate what the phrase “a statement true in a given language” means, we must, in addition to the expressions of this language, also use expressions that are not in it.
Tarski introduced the concept of a semantically closed language. Such a language includes, in addition to its expressions, their names, and also, which is important to emphasize, statements about the truth of the sentences formulated in it.

There is no boundary between language and metalanguage in a semantically closed language. Its means are so rich that they allow not only to assert something about extralinguistic reality, but also to evaluate the truth of such statements. These means are sufficient, in particular, to reproduce the antinomy “Liar” in the language. A semantically closed language thus turns out to be self-contradictory. Every natural language is obviously semantically closed.
The only acceptable way to eliminate antinomy, and hence internal inconsistency, according to Tarski, is to abandon the use of a semantically closed language. This path is acceptable, of course, only in the case of artificial, formalized languages ​​that allow a clear division into language and metalanguage. In natural languages, with their obscure structure and the ability to talk about everything in the same language, this approach is not very realistic. It makes no sense to raise the question of the internal consistency of these languages. Their rich expressive possibilities also have their downside - paradoxes.

Other solutions to the paradox

So there are statements that speak of their own truth or falsity. The idea that these kinds of statements are not meaningful is very old. It was defended by the ancient Greek logician Chrysippus.
In the Middle Ages, the English philosopher and logician W. Ockham stated that the statement “Every statement is false” is meaningless, since it speaks, among other things, of its own falsity. A contradiction directly follows from this statement. If every proposition is false, then so is the proposition itself; but that it is false means that not every proposition is false.

The situation is similar with the statement “Every proposition is true”. It must also be classified as meaningless and also leads to a contradiction: if every statement is true, then the negation of this statement itself is also true, that is, the statement that not every statement is true.
Why, however, cannot a statement meaningfully speak of its own truth or falsity?
Already a contemporary of Ockham, the French philosopher of the XIV century. J. Buridan did not agree with his decision. From the point of view of ordinary ideas about meaninglessness, expressions like “I am lying”, “Every statement is true (false)”, etc. quite meaningful. What you can think about, what you can say - such is the general principle of Buridan. A person can think about the truth of the statement that he utters, which means that he can speak about it. Not all statements about themselves are meaningless. For example, the statement "This sentence is written in Russian" is true, but the statement "There are ten words in this sentence" is false. And both of them make perfect sense. If it is admitted that a statement can speak about itself, then why is it not capable of speaking meaningfully about such a property of itself as truth?
Buridan himself considered the statement “I am lying” not meaningless, but false. He justified it like this.

When a person affirms a proposition, he thereby asserts that it is true. If the sentence says of itself that it is itself false, then it is only an abbreviated formulation of a more complex expression that asserts both its truth and its falsity. This expression is contradictory and therefore false. But it is by no means meaningless.

Buridan's argument is still sometimes considered convincing.
There are other lines of criticism of the solution to the "Liar" paradox, which was developed in detail by Tarski. Is there really no antidote against paradoxes of this type in semantically closed languages—and all natural languages ​​are, after all?
If this were the case, then the concept of truth could only be defined in a rigorous way in formalized languages. Only in them is it possible to distinguish between the objective language in which people talk about the surrounding world and the metalanguage in which they speak about this language. This hierarchy of languages ​​is modeled after the assimilation foreign language with the help of native. The study of such a hierarchy led to many interesting conclusions, and in certain cases it is essential. But she's not in natural language. Does it discredit him? And if so, to what extent? After all, the concept of truth is still used in it, and usually without any complications. Is introducing a hierarchy the only way to eliminate paradoxes like "The Liar?"

In the 1930s, the answers to these questions seemed undoubtedly in the affirmative. However, now there is no former unanimity, although the tradition of eliminating paradoxes of this type by “stratifying” the language remains dominant.
Recently, egocentric expressions have attracted more and more attention. They contain words like “I”, “this”, “here”, “now”, and their truth depends on when, by whom, where they are used.

In the statement “This statement is false”, the word “this” occurs. What object does it refer to? "Liar" may indicate that the word "it" does not refer to the meaning of the given statement. But then what does it refer to, what does it mean? And why can't this meaning still be denoted by the word "this"?
Without going into details here, it is only worth noting that in the context of the analysis of egocentric expressions, “Liar” is filled with a completely different content than before. It turns out that he no longer warns against the confusion of language and metalanguage, but points out the dangers associated with the misuse of the word "this" and similar egocentric words.
The issues that have associated over the centuries with "The Liar" have changed radically depending on whether it was seen as an example of ambiguity, or as an expression outwardly presented as an example of a mixture of language and metalanguage, or, finally, as a typical example of the misuse of egocentric expressions. And there is no certainty that other problems will not be associated with this paradox in the future.

The well-known modern Finnish logician and philosopher G. von Wright wrote in his work devoted to the “Liar” that this paradox should by no means be understood as a local, isolated obstacle that can be removed by one inventive movement of thought. Liar touches on many of the most important topics in logic and semantics. This is the definition of truth, and the interpretation of contradiction and evidence, and a whole series of important differences: between a sentence and the thought expressed by it, between the use of an expression and its mention, between the meaning of a name and the object it denotes.
The situation is similar with other logical paradoxes. “The antinomies of logic,” writes von Wrigg, “have puzzled us since their discovery and will probably continue to puzzle us forever. We should, I think, regard them not so much as problems waiting to be solved, but as inexhaustible raw material for thought. They are important because thinking about them touches upon the most fundamental questions of all logic, and therefore of all thought.”

In conclusion of this conversation about the "Liar" we can recall a curious episode from the time when formal logic was still taught at school. In a logic textbook published in the late 1940s, eighth-grade schoolchildren were asked as a homework assignment—as a warm-up, so to speak—to find the mistake made in this simple-looking statement: “I am lying.” And, let it not seem strange, it was believed that the majority of schoolchildren successfully coped with such a task.

§ 2. Russell's paradox

The most famous of the paradoxes discovered already in our century is the antinomy discovered by B. Russell and communicated by him in a letter to G. Ferge. The same antinomy was discussed simultaneously in Göttingen by the German mathematicians Z. Zermelo and D. Hilbert.
The idea was in the air, and its publication produced the impression of an exploding bomb. This paradox caused in mathematics, according to Hilbert, the effect of complete catastrophe. The simplest and most important logical methods, the most common and useful concepts, are under threat.
It immediately became obvious that neither in logic nor in mathematics, in the entire long history of their existence, was anything decidedly worked out that could serve as a basis for. elimination of antinomy. Clearly a departure from habitual ways of thinking was necessary. But from where and in what direction? How radical was the rejection of established ways of theorizing supposed to be?
With further study of antinomy, the conviction in the need for a fundamentally new approach steadily grew. Half a century after its discovery, specialists in the foundations of logic and mathematics L. Frenkel and I. Bar-Hillel already stated without any reservations: , so far invariably failed, are obviously insufficient for this purpose.
The modern American logician H. Curry wrote a little later about this paradox: “In terms of the logic known in the 19th century, the situation simply defied explanation, although, of course, in our educated age there may be people who see (or think they see ), what is the error?

Russell's paradox in its original form is connected with the concept of a set, or a class.
We can talk about sets of different objects, for example, about the set of all people or about the set of natural numbers. An element of the first set will be any individual person, an element of the second - every natural number. It is also possible to consider sets themselves as some objects and speak of sets of sets. One can even introduce such concepts as the set of all sets or the set of all concepts.

Set of ordinary sets

With respect to any set arbitrarily taken, it seems reasonable to ask whether it is its own element or not. Sets that do not contain themselves as an element will be called ordinary. For example, the set of all people is not a person, just as the set of atoms is not an atom. Sets that are proper elements will be unusual. For example, a set that unites all sets is a set and therefore contains itself as an element.
Consider now the set of all ordinary sets. Since it is a set, one can also ask about it whether it is ordinary or unusual. The answer, however, is discouraging. If it is ordinary, then by definition it must contain itself as an element, since it contains all ordinary sets. But this means that it is an unusual set. The assumption that our set is an ordinary set thus leads to a contradiction. So it can't be normal. On the other hand, it cannot be unusual either: an unusual set contains itself as an element, and the elements of our set are only ordinary sets. As a result, we come to the conclusion that the set of all ordinary sets cannot be either ordinary or extraordinary.

Thus, the set of all sets that are not proper elements is a proper element if and only if it is not such an element. This is a clear contradiction. And it was obtained on the basis of the most plausible assumptions and with the help of seemingly indisputable steps. The contradiction suggests that such a set simply does not exist. But why can't it exist? After all, it consists of objects that satisfy a well-defined condition, and the condition itself does not seem to be somehow exceptional or obscure. If a set so simply and clearly defined cannot exist, then what, in fact, is the difference between possible and impossible sets? The conclusion about the non-existence of the considered set sounds unexpected and inspires anxiety. He makes our general concept set amorphous and chaotic, and there is no guarantee that it is not capable of generating some new paradoxes.

Russell's paradox is remarkable for its extreme generality. It does not require any complicated technical terms, as in the case of some other paradoxes, the concepts of “set” and “element of the set” suffice. But this simplicity just speaks of its fundamental nature: it touches on the deepest foundations of our reasoning about sets, since it speaks not about some special cases, but about sets in general.

Other variants of the paradox

Russell's paradox is not specifically mathematical. It uses the concept of a set, but does not touch on any special properties associated specifically with mathematics.
This becomes apparent when the paradox is reformulated in purely logical terms.

Of every property one can, in all probability, ask whether it is applicable to itself or not.
The property of being hot, for example, does not apply to itself, since it is not itself hot; the property of being concrete also does not refer to itself, for it is an abstract property. But the property of being abstract, being abstract, is applicable to oneself. Let us call these properties inapplicable to themselves inapplicable. Does the property of being inapplicable to oneself apply? It turns out that inapplicability is inapplicable only if it is not. This is, of course, paradoxical.
The logical, property-related variety of Russell's antinomy is just as paradoxical as the mathematical, set-related variety.
Russell also proposed the following popular version of the paradox he discovered.

Imagine that the council of one village defined the duties of a barber as follows: to shave all the men of the village who do not shave themselves, and only these men. Should he shave himself? If so, it will refer to those who shave themselves, and those who shave themselves, he should not shave. If not, he will belong to those who do not shave themselves, and therefore he will have to shave himself. We thus arrive at the conclusion that this barber shaves himself if and only if he does not shave himself. This, of course, is impossible. The argument about the barber is based on the assumption that such a barber exists. The resulting contradiction means that this assumption is false, and there is no such villager who would shave all those and only those villagers who do not shave themselves.
The duties of a hairdresser do not seem contradictory at first glance, so the conclusion that there cannot be one sounds somewhat unexpected. But this conclusion is not paradoxical. The condition that the village barber must satisfy is, in fact, self-contradictory and therefore impossible. There cannot be such a hairdresser in a village for the same reason that there is no person in it who would be older than himself or who would be born before his birth.
The argument about the hairdresser can be called a pseudo-paradox. In its course, it is strictly analogous to Russell's paradox, and this is what makes it interesting. But it is still not a true paradox.

Another example of the same pseudo-paradox is the well-known catalog argument.
A certain library decided to compile a bibliographic catalog that would include all those and only those bibliographic catalogs that do not contain references to themselves. Should such a directory include a link to itself?
It is easy to show that the idea of ​​creating such a catalog is not feasible; it simply cannot exist, because it must simultaneously include a reference to itself and not include.
It is interesting to note that cataloging all directories that do not contain references to themselves can be thought of as an endless, never ending process. Let's say that at some point a directory, say K1, was compiled, including all other directories that do not contain references to themselves. With the creation of K1, another directory appeared that does not contain a link to itself. Since the goal is to make a complete catalog of all directories that do not mention themselves, it is obvious that K1 is not the solution. He does not mention one of these directories - himself. Including this mention of himself in K1, we get the K2 catalog. It mentions K1, but not K2 itself. Adding such a mention to K2, we get KZ, which again is not complete due to the fact that it does not mention itself. And on without end.

§ 3. Paradoxes of Grelling and Berry

An interesting logical paradox was discovered by the German logicians K. Grelling and L. Nelson (Grelling's paradox). This paradox can be formulated very simply.

Autological and heterological words

Some words denoting properties have the very property they name. For example, the adjective “Russian” is itself Russian, “polysyllabic” is itself polysyllabic, and “five-syllable” itself has five syllables. Such words referring to themselves are called self-meaning or autological.
There are not so many such words, the vast majority of adjectives do not have the properties that they name. “New” is not, of course, new, “hot” is hot, “one-syllable” is one-syllable, and “English” is English. Words that do not have the property denoted by them are called heterologous or heterologous. Obviously, all adjectives denoting properties that are not applicable to words will be heterological.
This division of adjectives into two groups seems clear and unobjectionable. It can be extended to nouns: "word" is a word, "noun" is a noun, but "clock" is not a clock, and "verb" is not a verb.
A paradox arises as soon as the question is asked: to which of the two groups does the adjective “heterological” itself belong? If it is autological, it has the property it designates and must be heterological. If it is heterological, it does not have the property it calls, and must therefore be autological. There is a paradox.

By analogy with this paradox, it is easy to formulate other paradoxes of the same structure. For example, is or is not a suicidal person who kills every non-suicidal person and does not kill any suicidal person?

It turned out that Grellig's paradox was known in the Middle Ages as the antinomy of an expression that does not name itself. One can imagine the attitude towards sophisms and paradoxes in modern times, if the problem that required an answer and caused lively debate was suddenly forgotten and was rediscovered only five hundred years later!

Another, outwardly simple antinomy was indicated at the very beginning of our century by D. Berry.

The set of natural numbers is infinite. The set of those names of these numbers that are available, for example, in the Russian language and contain less than, say, one hundred words, is finite. This means that there are such natural numbers for which there are no names in Russian that consist of less than a hundred words. These numbers are obviously smallest number. It cannot be called by means of a Russian expression containing less than a hundred words. But the expression: “The smallest natural number for which there is no complex name in Russian, consisting of less than a hundred words” is just the name of this number! This name has just been formulated in Russian and contains only nineteen words. An obvious paradox: the named number turned out to be the one for which there is no name!

§ 4. Irresolvable dispute

The famous sophist Protagoras, who lived in the 5th century. BC, there was a student named Euathlus, who studied law. According to the agreement between them, Evatl had to pay for training only if he won his first trial. If he loses this process, he is not obliged to pay at all. However, after completing his studies, Evatl did not participate in the processes. It lasted quite a long time, the teacher's patience ran out, and he filed a lawsuit against his student. Thus, for Euathlus, this was the first trial. Protagoras substantiated his demand as follows:

“Whatever the decision of the court, Euathlus will have to pay me. He will either win his first trial or lose. If he wins, he will pay by virtue of our contract. If he loses, he will pay according to this decision.

Apparently Euathlus was a capable student, as he replied to Protagoras:

- Indeed, I either win the process or lose it. If I win, the court decision will release me from the obligation to pay. If the court decision is not in my favor, then I lost my first case and will not pay by virtue of our contract.

Solutions to the paradox "Protagoras and Euathlus"

Perplexed by this turn of the matter, Protagoras devoted a special essay to this dispute with Euathlus, “A Litigation for Payment.” Unfortunately, like most of written by Protagoras has not come down to us. Nevertheless, one must pay tribute to Protagoras, who immediately sensed a problem behind a simple judicial incident that deserves special study.

G. Leibniz, himself a lawyer by education, also took this dispute seriously. In his doctoral dissertation, "A Study of Intricate Cases in Law," he tried to prove that all cases, even the most intricate ones, like the litigation of Protagoras and Euathlus, must find a correct solution on the basis of common sense. According to Leibniz, the court should refuse Protagoras for the untimely filing of a claim, but leave, however, for him the right to demand payment of money by Evatl later, namely after the first process he won.

Many other solutions to this paradox have been proposed.

They referred, in particular, to the fact that the decision of the court must have great power than a private agreement between two people. It can be answered that without this agreement, no matter how insignificant it may seem, there would be neither a court nor its decision. After all, the court must make its decision precisely on its occasion and on its basis.

They also appealed to the general principle that every work, and therefore the work of Protagoras, must be paid. But it is known that this principle has always had exceptions, especially in a slave-owning society. Moreover, it is simply not applicable to the specific situation of the dispute: after all, Protagoras, guaranteeing high level training, he himself refused to accept payment in case of failure of his student in the first process.

Sometimes they talk like this. Both Protagoras and Euathlus are both right in part, and neither of them in general. Each of them takes into account only half of the possibilities that are beneficial to itself. Full or comprehensive consideration opens up four possibilities, of which only half is beneficial to one of the disputants. Which of these possibilities is realized, it will be decided not by logic, but by life. If the verdict of the judges will have more force than the contract, Euathl will have to pay only if he loses the process, i.e. by virtue of a court decision. If, however, a private agreement is placed higher than the decision of the judges, then Protagoras will receive payment only in the event of losing the process to Evatlus, i.e. by virtue of an agreement with Protagoras. This appeal to life completely confuses everything. What, if not logic, can judges be guided by in conditions when all the relevant circumstances are completely clear? And what kind of leadership will it be if Protagoras, who claims payment through the court, achieves it only by losing the process?

However, Leibniz's solution, which at first seems convincing, is a little better than the vague opposition of logic and life. In essence, Leibniz proposes retroactively to change the wording of the contract and stipulate that the first lawsuit involving Euathlus, the outcome of which will decide the question of payment, should not be the trial of Protagoras. This thought is deep, but not related to a particular court. Had there been such a clause in the original agreement, there would have been no need for litigation at all.

If by the solution of this difficulty one understands the answer to the question whether Euathlus should pay Protagoras or not, then all these, like all other conceivable solutions, are, of course, untenable. They are nothing more than a departure from the essence of the dispute, they are, so to speak, sophistical tricks and cunning in a hopeless and insoluble situation. For neither common sense nor any general principles concerning social relations can settle the dispute.
It is impossible to carry out together the contract in its original form and the decision of the court, whatever the latter may be. To prove this, it is enough simple means logic. By the same means, it can also be shown that the treaty, despite its completely innocent appearance, is self-contradictory. It requires the realization of a logically impossible proposition: Euathlus must both pay for the education and at the same time not pay.

Rules that lead to a dead end

The human mind, accustomed not only to its strength, but also to its flexibility and even resourcefulness, finds it difficult, of course, to reconcile itself to this absolute hopelessness and admit that it has been driven into a dead end. This is especially difficult when the impasse is created by the mind itself: it, so to speak, stumbles out of the blue and falls into its own nets. Nevertheless, one has to admit that sometimes, and by the way, not so rarely, agreements and systems of rules, formed spontaneously or introduced consciously, lead to insoluble, hopeless situations.

An example from recent chess life will once again confirm this idea.

International rules for chess competitions oblige chess players to record the game move by move clearly and legibly. Until recently, the rules also stated that a chess player who missed the recording of several moves due to lack of time must, “as soon as his time trouble ends, immediately fill out his form, writing down the missed moves.” Based on this instruction, one judge at the 1980 Chess Olympiad (Malta) interrupted the game, which was going on in hard time trouble, and stopped the clock, declaring that the control moves had been made and, therefore, it was time to put the records of the games in order.

“But excuse me,” cried the participant, who was on the verge of losing and counted only on the intensity of passions at the end of the game, “after all, not a single flag has yet fallen and no one can ever (as it is also written in the rules) can tell how many moves have been made.
However, the referee was supported by the chief arbiter, who said that, indeed, since the time trouble had ended, it was necessary, following the letter of the rules, to start recording the missed moves.
It was pointless to argue in this situation: the rules themselves led to a dead end. It only remained to change their wording in such a way that similar cases could not arise in the future.
This was done at the congress of the International Chess Federation, which was taking place at the same time: instead of the words “as soon as the time trouble is over”, the rules now say: “as soon as the flag indicates the end of time”.
This example clearly shows how to deadlocks. It is useless to argue about which side is right: the dispute is insoluble, and there will be no winner in it. It remains only to come to terms with the present and take care of the future. To do this, you need to reformulate the original agreements or rules in such a way that they do not lead anyone else into the same hopeless situation.
Of course, such a course of action is not a solution to an insoluble dispute or a way out of a hopeless situation. It is rather a stop in front of an insurmountable obstacle and a road around it.

Paradox "crocodile and mother"

AT Ancient Greece the story about the crocodile and the mother was very popular, coinciding in its logical content with the paradox “Protagoras and Evatl”.
The crocodile snatched her child from an Egyptian woman standing on the river bank. To her plea to return the child, the crocodile, shedding, as always, a crocodile tear, answered:

“Your misfortune touched me, and I will give you a chance to get your child back. Guess if I'll give it to you or not. If you answer correctly, I will return the child. If you don't guess, I won't give it back.

Thinking, the mother replied:

You won't give me the baby.

"You won't get it," the crocodile concluded. You either told the truth or you didn't tell the truth. If it is true that I will not give up the child, then I will not give him up, because otherwise it will not be true. If what was said is not true, then you did not guess, and I will not give the child by agreement.

However, this reasoning did not seem convincing to the mother.

“But if I told the truth, then you will give me the child, as we agreed.” If I did not guess that you will not give the child, then you must give it to me, otherwise what I said will not be untrue.

Who is right: mother or crocodile? To what does the promise given to the crocodile oblige? In order to give the child, or, on the contrary, not to give it away? And to both at the same time. This promise is self-contradictory, and thus it cannot be fulfilled by virtue of the laws of logic.
The missionary found himself with the cannibals and arrived just in time for dinner. They let him choose how he will be eaten. To do this, he must utter some statement with the condition that if this statement turns out to be true, they will cook it, and if it turns out to be false, they will roast it.

What should the missionary say?

Of course, he should say: "You will fry me."

If he is really fried, it will turn out that he spoke the truth, and therefore he must be boiled. If he is boiled, his statement will be false, and he should just be fried. The cannibals will have no way out: from “fry” it follows “cook”, and vice versa.

This episode of the cunning missionary is, of course, another paraphrase of the dispute between Protagoras and Euathlus.

Paradox of Sancho Panza

One old paradox known in ancient Greece is played up in “Don Quixote” by M. Cervantes. Sancho Panza has become the governor of the island of Barataria and administers the court.
The first to come to him is some visitor and says: “Senior, a certain estate is divided into two halves by a deep river ... So, a bridge has been thrown across this river, and right there on the edge stands a gallows and there is something like a court, in which usually four judges sit, and they judge on the basis of a law issued by the owner of the river, the bridge and the whole estate, which law is drawn up in this way: let those through, and whoever lies, without any leniency, send them to the gallows located right there and execute them. From the time when this law was promulgated in all its severity, many managed to get across the bridge, and as soon as the judges were satisfied that the passers-by were telling the truth, they let them through. But then one day a man who was sworn in swore and said: he swears that he came in order to be hung up on this very gallows, and for nothing else. This oath perplexed the judges, and they said: “If this man is allowed to proceed without hindrance, then this will mean that he has broken the oath and, according to the law, is liable to death; if we hang him, then he swore that he came only to be hung up on this gallows, therefore, his oath, it turns out, is not false, and on the basis of the same law it is necessary to let him pass. And so I ask you, señor governor, what should the judges do with this man, for they are still perplexed and hesitating...
Sancho suggested, perhaps not without cunning, that the half of the person who told the truth should be let through, and the one that lied should be hanged, and in this way the rules for crossing the bridge would be observed in all forms. This passage is interesting in several respects.
First of all, it is a clear illustration of the fact that the hopeless situation described in the paradox may well be faced - and not in pure theory, but in practice - if not a real person, then at least a literary hero.

The way out proposed by Sancho Panza was not, of course, a solution to the paradox. But this was just the solution that only remained to be resorted to in his position.
Once upon a time, Alexander the Great, instead of untying the cunning Gordian knot, which no one has yet managed to do, simply cut it. Sancho did the same. Trying to solve the puzzle on its own terms was useless—it was simply unsolvable. It remained to discard these conditions and introduce your own.
And one moment. With this episode, Cervantes clearly condemns the exorbitantly formal scale of medieval justice, permeated with the spirit of scholastic logic. But how widespread in his time - and this was about four hundred years ago - were information from the field of logic! Not only Cervantes himself knows this paradox. The writer finds it possible to attribute to his hero, an illiterate peasant, the ability to understand that he faces an insoluble task!

§ 5. Other paradoxes

The above paradoxes are arguments, the result of which is a contradiction. But there are other types of paradoxes in logic. They also point out some difficulties and problems, but they do it in a less harsh and uncompromising way. Such, in particular, are the paradoxes discussed below.

Paradoxes of imprecise concepts

Most of the concepts of not only natural language, but also the language of science are inaccurate, or, as they are also called, blurry. Often this turns out to be the cause of misunderstanding, disputes, or even simply leads to deadlocks.
If the concept is inaccurate, the boundary of the area of ​​objects to which it is applicable is devoid of sharpness, blurred. Take, for example, the concept of "heap". One grain (a grain of sand, a stone, etc.) is not yet a pile. A thousand grains is already, obviously, a bunch. And three grains? And ten? What number of grains are added to form a heap? Not very clear. In the same way, it is not clear with the removal of which grain the heap disappears.
The empirical characteristics “large”, “heavy”, “narrow”, etc. are inaccurate. Such ordinary concepts as “wise man”, “horse”, “house”, etc. are inexact.
There is no grain of sand that, when removed, we can say that with its removal, what remains can no longer be called home. But this seems to mean that at no point in the gradual dismantling of the house - up to its complete disappearance - is there any 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 forming a heap 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. Therefore, no number of grains can form heaps.
The possibility of this and similar proofs leading to absurd conclusions means that the principle of mathematical induction has a limited scope. It should not be used in reasoning with inaccurate, vague concepts.

A good example of how these concepts can lead to irresolvable disputes is a curious trial that took place in 1927 in the United States. The sculptor C. Brancusi went to court demanding that his works be recognized as works of art. Among the works sent to New York for the exhibition was the sculpture "Bird", which is now considered a classic of the abstract style. It is a modulated column of polished bronze about one and a half meters high, which does not have any external resemblance to a bird. Customs officers categorically refused to recognize Brancusi's abstract creations as works of art. They put them under "Metal hospital utensils and household items" and gave them a large customs duty. Outraged, Brancusi sued.

Customs was supported by artists - members National Academy who defended traditional methods in art. They acted as witnesses for the defense at the trial and categorically insisted that the attempt to pass off the “Bird” as a work of art was simply a scam.
This conflict vividly emphasizes the difficulty of operating with the concept of “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 a sculptural image, increasingly moving away from the original, cease to be a work of art and become a “metal utensil”? This question is as difficult to answer as the question of where is the boundary between a house and its ruins, between a horse with a tail and a horse without a tail, and so on. 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.

The handling of imprecise concepts thus requires a certain amount of caution. Wouldn't it be better to avoid them altogether?

The German philosopher E. Husserl was inclined to demand such extreme rigor and precision from knowledge that is not found even in mathematics. In connection with this, Husserl's biographers recall with irony an incident that happened to him in childhood. He was presented with a penknife, and, deciding to make the blade as sharp as possible, he sharpened it until nothing was left of the blade.
More precise concepts are preferable to imprecise ones in many situations. The usual desire to clarify the concepts used is quite justified. But it must, of course, have its limits. Even in the language of science, a significant part of the concepts is inaccurate. And this is connected not with the subjective and random mistakes of individual scientists, but with the very nature of scientific knowledge. In natural language, imprecise concepts are overwhelming; this speaks, among other things, of his flexibility and latent strength. Anyone who demands the utmost precision from all concepts runs the risk of being left without a language altogether. “Deprive the words of any ambiguity, any uncertainty,” wrote the French aesthetician 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, will not be able to find its expression. But what am I saying: deprive ... I will say more. Deprive the word of any inaccuracy - and you will lose even axioms.
For a long time, both logicians and mathematicians did not pay attention to the difficulties associated with fuzzy concepts and their corresponding sets. The question was posed as follows: concepts must be precise, and anything vague is unworthy of serious interest. In recent decades, however, this overly strict attitude has lost its appeal. Logical theories are constructed that specifically take into account the uniqueness of reasoning with inaccurate concepts.
Actively developing mathematical theory so-called fuzzy sets, indistinctly defined collections of objects.
The analysis of problems of inaccuracy is a step towards bringing logic closer to 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 section of logic that does not have its own paradoxes.
Inductive logic has its own paradoxes, which have been actively, but so far without much success, being fought for almost half a century. Of particular interest is the confirmation paradox discovered by the American philosopher K. Hempel. It is natural to assume that general provisions, in particular scientific laws are confirmed by their positive examples. If, say, the proposition “All A is B” is considered, then its positive examples will be objects that have properties A and B. In particular, supporting examples for the proposition “All ravens are black” are objects that are both ravens and black. This statement is tantamount, however, to the statement "All things that are not black are not crows", and the confirmation of the latter must also be the confirmation of the former. But "Everything is not black is not a crow" is confirmed by every case of a non-black object that is not a crow. It turns out, therefore, that the observations “White cow”, “Brown shoes”, etc. confirm the statement "All crows are black."

An unexpected paradoxical result follows from seemingly innocent premises.

In the logic of norms, a number of its laws cause concern. When they are formulated in meaningful terms, their inconsistency with the usual notions of right and wrong becomes obvious. For example, one of the laws says that from the order “Send a letter!” the order “Send the letter or burn it!” follows.
Another law states that if a person has violated one of his duties, he gets the right to do whatever he wants. Our logical intuition does not want to put up with this kind of “laws of obligation”.
In the logic of knowledge, the paradox of logical omniscience is heavily discussed. He claims that a person knows all the logical consequences that follow from the positions he takes. For example, if a person knows the five postulates of Euclid's geometry, then, therefore, he knows all this geometry, since it follows from them. But it's not. A person can 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.

§ 6. What is a logical paradox

No exhaustive list of logical paradoxes exists, and it is impossible.
The considered paradoxes are only a part of all those discovered so far. It is likely that many other paradoxes will be discovered in the future, and even completely new types of them. The very concept of a paradox is not so definite that it would be possible to compile a list of at least already known paradoxes.
“Set-theoretic paradoxes are a very serious problem, not for mathematics, however, but rather for logic and the theory of knowledge,” writes the Austrian mathematician and logician K. Gödel. “The logic is consistent. There are no logical paradoxes,” says mathematician D. Bochvar. Such discrepancies are sometimes significant, sometimes verbal. The point is largely in what exactly is meant by a logical paradox.

The peculiarity of logical paradoxes

A necessary feature of logical paradoxes is the logical dictionary.
Paradoxes that are logical must be formulated in logical terms. However, in logic there are no clear criteria for dividing terms into logical and non-logical. Logic, which deals with the correctness of reasoning, seeks to reduce the concepts on which the correctness of practically applied conclusions depends to a minimum. But this minimum is not predetermined unambiguously. In addition, non-logical statements can also be formulated in logical terms. Whether a particular paradox uses only purely logical premises is far from always possible to determine unambiguously.
Logical paradoxes are not rigidly separated from all other paradoxes, just as the latter are not clearly distinguished from everything non-paradoxical and consistent with the prevailing ideas. At the beginning of the study of logical paradoxes, it seemed that they could be distinguished by the violation of some as yet unexplored position or rule of logic. The vicious circle principle introduced by B. Russell was especially active in claiming the role of such a rule. This principle states that a collection of objects cannot contain members defined only by the same collection.
All paradoxes have one thing in common - self-applicability, or circularity. In each of them, the object in question is characterized by some set of objects to which it itself belongs. If we single out, for example, the very cunning person, we do this with the help of a set of people, to which belongs and this person. And if we say: "This statement is false," we characterize the statement of interest to us by referring to the totality of all false statements that includes it.

In all paradoxes, there is a self-applicability of concepts, which means that there is, as it were, movement in a circle, leading in the end to the starting point. In an effort to characterize the object of interest to us, we turn to the set of objects that includes it. However, it turns out that, for its definiteness, it itself needs the object under consideration and cannot be clearly understood without it. In this circle, perhaps, lies the source of paradoxes.
The situation is complicated, however, by the fact that such a circle exists in many completely non-paradoxical arguments. Circular is a huge variety of the most common, harmless and at the same time convenient ways of expression. Such examples as “the largest of all cities”, “the smallest of all natural numbers”, “one of the electrons of the iron atom”, etc., show that not every case of self-applicability leads to a contradiction and that it is important not only in ordinary language, but also in the language of science.
A mere reference to the use of self-applicable concepts is thus insufficient to discredit paradoxes. Some additional criterion is needed to separate self-applicability, leading to a paradox, from all other cases of it.
There have been many proposals to this effect, but no successful clarification of circularity has been found. It turned out to be impossible to characterize circularity in such a way that every circular reasoning leads to a paradox, and every paradox is the result of some circular reasoning.
An attempt to find some specific principle of logic, the violation of which would distinctive feature all logical paradoxes, did not lead to anything definite.
Some kind of classification of paradoxes would undoubtedly be useful, subdividing them into types and types, grouping some paradoxes and opposing them to others. However, nothing sustainable has been achieved in this case either.

The English logician F. Ramsey, who died in 1930, when he was not yet twenty-seven years old, proposed to divide all paradoxes into syntactic and semantic ones. The first includes, for example, Russell's paradox, the second - the paradoxes of the "Liar", Grelling, etc.
According to Ramsey, the paradoxes of the first group contain only concepts belonging to logic or mathematics. The latter include such concepts as “truth”, “definability”, “naming”, “language”, which are not strictly mathematical, but rather related to linguistics or even the theory of knowledge. Semantic paradoxes seem to owe their appearance not to some error in logic, but to the vagueness or ambiguity of some non-logical concepts, therefore the problems they pose concern language and must be solved by linguistics.

It seemed to Ramsey that mathematicians and logicians need not be interested in semantic paradoxes. Later it turned out, however, that some of the most significant results of modern logic were obtained precisely in connection with a deeper study of precisely these non-logical paradoxes.
The division of paradoxes proposed by Ramsey was widely used at first and retains some importance even now. At the same time, it is becoming increasingly clear that this division is rather vague and relies primarily on examples, and not on an in-depth comparative analysis of the two groups of paradoxes. Semantic concepts are now well defined, and it is hard not to recognize that these concepts are indeed logical. With the development of semantics, which defines its basic concepts in terms of set theory, the distinction made by Ramsey is increasingly blurred.

Paradoxes and Modern Logic

What conclusions for logic follow from the existence of paradoxes?
First of all, the presence of a large number of paradoxes speaks of the strength of logic as a science, and not of its weakness, as it might seem.

It was no coincidence that the discovery of paradoxes coincided with the period of the most intensive development of modern logic and its greatest successes.
The first paradoxes were discovered even before the emergence of logic as a special science. Many paradoxes were discovered in the Middle Ages. Later, however, they turned out to be forgotten and were rediscovered already in our century.
Medieval logicians were not aware of the concepts of "set" and "element of the set", introduced into science only in the second half of the 19th century. But the flair for paradoxes was honed in the Middle Ages to such an extent that already at that early time certain concerns were expressed about self-applicable concepts. The simplest example of this is the notion of "being one's own element" that appears in many of today's paradoxes.
However, such fears, like all warnings about paradoxes in general, were not systematic and definite until our century. They did not lead to any clear proposals for revisiting habitual ways of thinking and expressing.
Only modern logic has taken the very problem of paradoxes out of oblivion, discovered or rediscovered most of the specific logical paradoxes. She further showed that the ways of thinking traditionally explored by logic are completely insufficient for eliminating paradoxes, and indicated fundamentally new methods of dealing with them.
Paradoxes pose an important question: where, in fact, do some of the usual methods of concept formation and reasoning fail us? After all, they seemed completely natural and convincing, until it turned out that they were paradoxical.

Paradoxes undermine the belief that the habitual methods of theoretical thinking by themselves and without any special control over them provide a reliable progress towards the truth.
Requiring a radical change in an overly gullible approach to theorizing, paradoxes are a harsh critique of logic in its naive, intuitive form. They play the role of a factor that controls and puts restrictions on the way of constructing deductive systems of logic. And this role of them can be compared with the role of an experiment that tests the correctness of hypotheses in such sciences as physics and chemistry, and forces them to make changes to these hypotheses.
A paradox in a theory speaks of the incompatibility of the assumptions underlying it. It acts as a timely detected symptom of the disease, without which it could have been overlooked.
Of course, the disease manifests itself in many ways, and in the end it is possible to reveal it without such acute symptoms as paradoxes. For example, the foundations of set theory would be analyzed and refined even if no paradoxes in this area were discovered. But there would not have been that sharpness and urgency with which the paradoxes discovered in it raised the problem of revising set theory.

An extensive literature is devoted to paradoxes, it is proposed big number their explanations. But none of these explanations is universally accepted, and there is no complete agreement on the origin of paradoxes and how to get rid of them.
“Over the past sixty years, hundreds of books and articles have been devoted to the goal of resolving paradoxes, but the results are amazingly poor in comparison with the efforts expended,” writes A. Frenkel. “It looks like,” H. Curry concludes his analysis of the paradoxes, “that a complete reform of logic is required, and mathematical logic can become the main tool for carrying out this reform.”

No exhaustive list of logical paradoxes exists, and it is impossible.

The considered paradoxes are only a part of all those discovered so far. It is likely that many other paradoxes will be discovered in the future, and even completely new types of them. The very concept of a paradox is not so definite that it would be possible to compile a list of at least already known paradoxes.

“Set-theoretic paradoxes are a very serious problem, not for mathematics, however, but rather for logic and epistemology,” writes the Austrian mathematician and logician K. Gödel. “The logic is inconsistent. There are no logical paradoxes,” says mathematician D. Bochvar. Such discrepancies are sometimes significant, sometimes verbal. The point is largely in what exactly is meant by a logical paradox.

The peculiarity of logical paradoxes

A necessary feature of logical paradoxes is the logical dictionary.

Paradoxes that are logical must be formulated in logical terms. However, in logic there are no clear criteria for dividing terms into logical and non-logical. Logic, which deals with the correctness of reasoning, seeks to reduce the concepts on which the correctness of practically applied conclusions depends to a minimum. But this minimum is not predetermined unambiguously. In addition, non-logical statements can also be formulated in logical terms. Whether a particular paradox uses only purely logical premises is far from always possible to determine unambiguously.

Logical paradoxes are not rigidly separated from all other paradoxes, just as the latter are not clearly distinguished from everything non-paradoxical and consistent with the prevailing ideas.

At the beginning of the study of logical paradoxes, it seemed that they could be distinguished by the violation of some as yet unexplored position or rule of logic. The vicious circle principle introduced by B. Russell was especially active in claiming the role of such a rule. This principle states that a collection of objects cannot contain members defined only by the same collection.

All paradoxes have one thing in common - self-applicability, or circularity. In each of them, the object in question is characterized by some set of objects to which it itself belongs. If we select, for example, the most cunning person, we do this with the help of a population of people to which this person belongs. And if we say: "This statement is false," we characterize the statement of interest to us by referring to the totality of all false statements that includes it.

In all paradoxes, there is a self-applicability of concepts, which means that there is, as it were, movement in a circle, leading in the end to the starting point. In an effort to characterize the object of interest to us, we turn to the set of objects that includes it. However, it turns out that, for its definiteness, it itself needs the object under consideration and cannot be clearly understood without it. In this circle, perhaps, lies the source of paradoxes.

The situation is complicated, however, by the fact that such a circle exists in many completely non-paradoxical arguments. Circular is a huge variety of the most common, harmless and at the same time convenient ways of expression. Such examples as “the largest of all cities”, “the smallest of all natural numbers”, “one of the electrons of the iron atom”, etc., show that not every case of self-applicability leads to a contradiction and that it is important not only in ordinary language, but also in the language of science.

A mere reference to the use of self-applicable concepts is thus insufficient to discredit paradoxes. Some additional criterion is needed to separate self-applicability, leading to a paradox, from all other cases of it.

There have been many proposals to this effect, but no successful clarification of circularity has been found. It turned out to be impossible to characterize circularity in such a way that every circular reasoning leads to a paradox, and every paradox is the result of some circular reasoning.

An attempt to find some specific principle of logic, the violation of which would be a distinctive feature of all logical paradoxes, did not lead to anything definite.

Some kind of classification of paradoxes would undoubtedly be useful, subdividing them into types and types, grouping some paradoxes and opposing them to others. However, nothing sustainable has been achieved in this case either.

The English logician F. Ramsey, who died in 1930, when he was not yet twenty-seven years old, proposed to divide all paradoxes into syntactic and semantic ones. The first includes, for example, Russell's paradox, the second - the paradoxes of the "Liar", Grelling, etc.

According to Ramsey, the paradoxes of the first group contain only concepts belonging to logic or mathematics. The latter include such concepts as "truth", "definability", "naming", "language", which are not strictly mathematical, but rather related to linguistics or even the theory of knowledge. Semantic paradoxes seem to owe their appearance not to some error in logic, but to the vagueness or ambiguity of some non-logical concepts, therefore the problems they pose concern language and must be solved by linguistics.

It seemed to Ramsey that mathematicians and logicians need not be interested in semantic paradoxes. Later it turned out, however, that some of the most significant results of modern logic were obtained precisely in connection with a deeper study of precisely these non-logical paradoxes.

The division of paradoxes proposed by Ramsey was widely used at first and retains some importance even now. At the same time, it is becoming increasingly clear that this division is rather vague and relies primarily on examples, and not on an in-depth comparative analysis of the two groups of paradoxes. Semantic concepts are now well defined, and it is hard not to recognize that these concepts are indeed logical. With the development of semantics, which defines its basic concepts in terms of set theory, the distinction made by Ramsey is increasingly blurred.

Paradoxes and Modern Logic

What conclusions for logic follow from the existence of paradoxes?

First of all, the presence of a large number of paradoxes speaks of the strength of logic as a science, and not of its weakness, as it might seem.

It was no coincidence that the discovery of paradoxes coincided with the period of the most intensive development of modern logic and its greatest successes.

The first paradoxes were discovered even before the emergence of logic as a special science. Many paradoxes were discovered in the Middle Ages. Later, however, they turned out to be forgotten and were rediscovered already in our century.

Medieval logicians were not aware of the concepts of "set" and "element of the set", introduced into science only in the second half of the 19th century. But the flair for paradoxes was honed in the Middle Ages to such an extent that already at that early time certain concerns were expressed about self-applicable concepts. The simplest example of this is the notion of "being one's own element" that appears in many of today's paradoxes.

However, such fears, like all warnings about paradoxes in general, were not systematic and definite until our century. They did not lead to any clear proposals for revisiting habitual ways of thinking and expressing.

Only modern logic has taken the very problem of paradoxes out of oblivion, discovered or rediscovered most of the specific logical paradoxes. She further showed that the ways of thinking traditionally explored by logic are completely insufficient for eliminating paradoxes, and indicated fundamentally new methods of dealing with them.

Paradoxes pose an important question: where, in fact, do some of the usual methods of concept formation and reasoning fail us? After all, they seemed completely natural and convincing, until it turned out that they were paradoxical.

Paradoxes undermine the belief that the habitual methods of theoretical thinking by themselves and without any special control over them provide a reliable progress towards the truth.

Requiring a radical change in an overly gullible approach to theorizing, paradoxes are a harsh critique of logic in its naive, intuitive form. They play the role of a factor that controls and puts restrictions on the way of constructing deductive systems of logic. And this role of them can be compared with the role of an experiment that tests the correctness of hypotheses in such sciences as physics and chemistry, and forces them to make changes to these hypotheses.

A paradox in a theory speaks of the incompatibility of the assumptions underlying it. It acts as a timely detected symptom of the disease, without which it could have been overlooked.

Of course, the disease manifests itself in many ways, and in the end it is possible to reveal it without such acute symptoms as paradoxes. For example, the foundations of set theory would be analyzed and refined even if no paradoxes in this area were discovered. But there would not have been that sharpness and urgency with which the paradoxes discovered in it raised the problem of revising set theory.

An extensive literature is devoted to paradoxes, a large number of their explanations have been proposed. But none of these explanations is universally accepted, and there is no complete agreement on the origin of paradoxes and how to get rid of them.

“Over the past sixty years, hundreds of books and articles have been devoted to the goal of resolving paradoxes, but the results are amazingly poor in comparison with the efforts expended,” writes A. Frenkel. “It looks like,” H. Curry concludes his analysis of the paradoxes, “that a complete reform of logic is required, and mathematical logic can become the main tool for carrying out this reform.”

Elimination and explanation of paradoxes

One important difference should be noted.

Eliminating paradoxes and resolving them are not at all the same thing. To eliminate a paradox from a certain theory means to reconstruct it in such a way that the paradoxical assertion turns out to be unprovable in it. Each paradox relies on a large number of definitions, assumptions and arguments. His conclusion in theory is a certain chain of reasoning. Formally speaking, one can question any of its links, discard it, and thereby break the chain and eliminate the paradox. In many works, this is done and is limited to this.

But this is not yet the resolution of the paradox. It is not enough to find a way to exclude it; one must convincingly justify the proposed solution. The very doubt of some step leading to a paradox must be well founded.

First of all, the decision to abandon some logical means used in the derivation of a paradoxical statement must be linked to our general considerations regarding the nature of logical proof and other logical intuitions. If this is not the case, the elimination of the paradox turns out to be devoid of solid and stable foundations and degenerates into a predominantly technical task.

Moreover, the rejection of some assumption, even if it does provide the elimination of some particular paradox, does not automatically guarantee the elimination of all paradoxes. This suggests that paradoxes should not be "hunted" one by one. The exclusion of one of them should always be so justified that there is a certain guarantee that other paradoxes will be eliminated by the same step.

Each time a paradox is discovered, A. Tarsky writes, “we must subject our ways of thinking to a thorough revision, reject some assumptions that we believed in, and improve the methods of argumentation that we used. We do this in an effort not only to get rid of antinomies, but also to prevent the emergence of new ones.

And finally, an ill-considered and careless rejection of too many or too strong assumptions can simply lead to the fact that although it does not contain paradoxes, it will turn out to be a much weaker theory that has only a particular interest.

What can be the minimum, least radical set of measures to avoid known paradoxes?

Logical grammar

One way is to single out, along with true and false sentences, also meaningless sentences. This path was adopted by B. Russell. Paradoxical reasoning was declared by him to be meaningless on the grounds that they violated the requirements of logical grammar. Not every sentence that does not violate the rules of ordinary grammar is meaningful - it must also satisfy the rules of a special, logical grammar.

Russell built a theory of logical types, a kind of logical grammar, whose task was to eliminate all known antinomies. Subsequently, this theory was substantially simplified and was called the simple theory of types.

The main idea of ​​the theory of types is the allocation of logically different types of objects, the introduction of a kind of hierarchy, or ladder, of the objects under consideration. The lowest, or null, type includes individual objects that are not sets. The first type includes sets of objects of zero type, i.e. individuals; to the second - sets of sets of individuals, etc. In other words, a distinction is made between objects, properties of objects, properties of properties of objects, etc. At the same time, certain restrictions are introduced on the construction of proposals. Properties can be attributed to objects, properties of properties to properties, and so on. But it is impossible to meaningfully assert that objects have properties of properties.

Let's take a series of suggestions:

This house is red.

Red is a color.

Color is an optical phenomenon.

In these sentences, the expression "this house" denotes a certain object, the word "red" indicates the property inherent in this object, "to be a color" - to the property of this property ("to be red") and "to be an optical phenomenon" - indicates the property of the property "be a color" belonging to the "be red" property. Here we are dealing not only with objects and their properties, but also with the properties of properties (“the property of being red has the property of being a color”), and even with the properties of properties of properties.

All three sentences from the above series are, of course, meaningful. They are built in accordance with the requirements of type theory. And let's say the sentence "This house is a color" violates these requirements. It ascribes to an object that characteristic which can belong only to properties, but not to objects. A similar violation is contained in the sentence "This house is an optical phenomenon." Both of these proposals must be classified as meaningless.

A simple theory of types eliminates Russell's paradox. However, to eliminate the paradoxes of the Liar and Berry, simply dividing the objects under consideration into types is no longer enough. It is necessary to introduce some additional ordering within the types themselves.

The elimination of paradoxes can also be achieved by avoiding the use of too large sets, similar to the set of all sets. This path was proposed by the German mathematician E. Zermelo, who connected the appearance of paradoxes with the unlimited construction of sets. The admissible sets were defined by him by some list of axioms formulated in such a way that known paradoxes would not be deduced from them. At the same time, these axioms were strong enough to deduce from them the usual arguments of classical mathematics, but without paradoxes.

Neither these two nor the other proposed ways of eliminating paradoxes are generally accepted. There is no common belief that any of the proposed theories resolves logical paradoxes, and not just discards them without deep explanation. The problem of explaining paradoxes is still open and still important.

The future of paradoxes

G. Frege, the greatest logician of the last century, unfortunately had a very bad character. In addition, he was unreserved and even cruel to his criticism of his contemporaries.

Perhaps that is why his contribution to the logic and foundation of mathematics did not receive recognition for a long time. And when fame began to come to him, the young English logician B. Russell wrote to him that a contradiction arises in the system published in the first volume of his book The Fundamental Laws of Arithmetic. The second volume of this book was already in print, and Frege could only add a special appendix to it, in which he outlined this contradiction (later called "Russell's paradox") and admitted that he was not able to eliminate it.

However, the consequences of this recognition were tragic for Frege. He experienced the greatest shock. And although he was then only 55 years old, he did not publish another significant work on logic, although he lived for more than twenty years. He did not even respond to the lively discussion caused by Russell's paradox, and did not react in any way to the many proposed solutions to this paradox.

The impression made on mathematicians and logicians by the newly discovered paradoxes was well expressed by D. Hilbert: “... The state in which we are now in relation to paradoxes is unbearable for a long time. Think: in mathematics - that model of certainty and truth - the formation of concepts and the course of inferences, as everyone studies, teaches and applies them, leads to absurdity. Where to look for reliability and truth, if even mathematical thinking itself misfires?

Frege was a typical representative of logic late XIX century, free from any paradoxes, logic, confident in its capabilities and claiming to be a criterion of rigor even for mathematics. The paradoxes showed that the absolute strictness achieved by supposedly logic was nothing more than an illusion. They undeniably showed that logic - in the intuitive form that it had at the turn of the century - needs a profound revision.

About a century has passed since the lively discussion of paradoxes began. The undertaken revision of the logic did not lead, however, to their unambiguous resolution.

And at the same time, such a state is hardly of concern to anyone today. Over time, attitudes towards paradoxes have become calmer and even more tolerant than at the time they were discovered. It's not just that paradoxes have become something familiar. And, of course, not that they put up with them. They still remain in the center of attention of logicians, the search for their solutions is actively continuing. The situation changed primarily because the paradoxes turned out to be, so to speak, localized. They have found their definite, albeit troubled, place in a wide range of logical studies. It became clear that absolute austerity, as it was portrayed at the end of the last century and even sometimes at the beginning of this century, is, in principle, an unattainable ideal.

It was also realized that there is no single problem of paradoxes that stands alone. The problems associated with them are of different types and affect, in fact, all the main sections of logic. The discovery of a paradox forces us to analyze our logical intuitions more deeply and engage in a systematic reworking of the foundations of the science of logic. At the same time, the desire to avoid paradoxes is neither the only, nor even, perhaps, the main task. Although they are important, they are only an occasion for reflection on the central themes of logic. Continuing the comparison of paradoxes with particularly pronounced symptoms of the disease, it can be said that the desire to immediately eliminate paradoxes would be like a desire to remove such symptoms without much concern for the disease itself. What is required is not just the resolution of paradoxes, but their explanation, which deepens our understanding of the logical patterns of thinking.

It is known that formulating a problem is often more important and more difficult than solving it. “In science,” wrote the English chemist F. Soddy, “a problem properly posed is more than half solved. The mental preparation process required to find out that there is a particular problem often takes more time than the solution of the problem itself.
The forms in which the problem situation is manifested and realized are very diverse. Far from always, it reveals itself in the form of a direct question that arose at the very beginning of the study. The world of problems is as complex as the process of cognition that generates them. Identifying problems is at the core of creative thinking. Paradoxes are the most interesting case of implicit, questionless ways of posing problems. Paradoxes are common in the early stages of the development of scientific theories, when the first steps are being taken in an as yet unexplored area and the most general principles of approach to it are being groped for.

Paradoxes and logic

In a broad sense, a paradox is a position that sharply diverges from generally accepted, established, orthodox opinions. “Generally accepted opinions and what is considered a matter of long-term decision, most often deserve research” (Glichtenberg). Paradox is the beginning of such research.
A paradox in a narrower and more specialized sense is two opposing, incompatible statements, for each of which there are seemingly convincing arguments.
The sharpest form of paradox is antinomy, a reasoning that proves the equivalence of two statements, one of which is a negation of the other.
Paradoxes are especially famous in the most rigorous and exact sciences - mathematics and logic. And this is no coincidence.

Logics- abstract science. There are no experiments in it, not even facts in the usual sense of the word. In building its systems, logic ultimately proceeds from the analysis of real thinking. But the results of this analysis are synthetic, undifferentiated. They are not statements of any separate processes or events that the theory should explain. Obviously, such an analysis cannot be called an observation: a concrete phenomenon is always observed.
Constructing a new theory, the scientist usually starts from the facts, from what can be observed in the experiment. However free his creative imagination may be, it must reckon with one indispensable circumstance: a theory makes sense only if it agrees with the facts pertaining to it. A theory that disagrees with facts and observations is far-fetched and has no value.
But if there are no experiments in logic, no facts, and no observation itself, then what holds back logical fantasy? What factors, if not facts, are taken into account when creating new logical theories?
The discrepancy between logical theory and the practice of real thinking is often revealed in the form of a more or less acute logical paradox, and sometimes even in the form of a logical antinomy, which speaks of the internal inconsistency of the theory. This just explains the importance that is attached to paradoxes in logic, and the great attention that they enjoy in it.

Variants of the “Liar” Paradox

The most famous and perhaps the most interesting of all logical paradoxes is the “Liar” paradox. It was he who glorified the name of Eubulides from Miletus who discovered it.
There are variants of this paradox, or antinomy, many of which are only apparently paradoxical.
In the simplest version of "Liar" a person says only one phrase: "I'm lying." Or he says, "The statement I am now making is false." Or: "This statement is false."

If the statement is false, then the speaker told the truth, and therefore what he said is not a lie. If the statement is not false, and the speaker claims that it is false, then this statement is false. It turns out, therefore, that if the speaker is lying, he is telling the truth, and vice versa.

In the Middle Ages, the following wording was common:

“What Plato said is false,” says Socrates.

“What Socrates said is true,” says Plato.

The question arises, which of them expresses the truth, and which is a lie?
And here is a modern paradox of this paradox. Let us assume that only the words are written on the front side of the card: "On the other side of this card is written a true statement." It is clear that these words represent a meaningful statement. Turning over the card, we must either find the promised statement, or it is not there. If it is written on the back, then it is either true or not. However, on the back are the words: “There is a false statement written on the other side of this card” - and nothing more. Assume that the statement on the front side is true. Then the statement on the back must be true, and therefore the statement on the front must be false. But if the statement on the front is false, then the statement on the back must also be false, and therefore the statement on the front must be true. The end result is a paradox.
The Liar paradox made a huge impression on the Greeks. And it's easy to see why. The question that it poses at first glance seems quite simple: is he lying who says only that he is lying? But the answer "yes" leads to the answer "no", and vice versa. And reflection does not clarify the situation at all. Behind the simplicity and even routine of the question, it reveals some obscure and immeasurable depth.
There is even a legend that a certain Filit Kossky, desperate to resolve this paradox, committed suicide. They also say that one of the famous ancient Greek logicians, Diodorus Kronos, already in his declining years, made a vow not to eat until he found the solution to the “Liar”, and soon died without achieving anything.
In the Middle Ages, this paradox was referred to the so-called unsolvable sentences and became the object of systematic analysis. In modern times, the “Liar” did not attract any attention for a long time. They did not see any, even minor, difficulties regarding the use of the language. And only in our, so-called modern times, the development of logic has finally reached a level when it has become possible to formulate the problems that seem to be behind this paradox in strict terms.
Now "Liar" - this typical former sophism - is often referred to as the king of logical paradoxes. An extensive scientific literature is devoted to him. And yet, as in the case of many other paradoxes, it remains not entirely clear what problems lie behind it and how to get rid of it.

Language and metalanguage

Now "The Liar" is usually considered a characteristic example of the difficulties that the confusion of two languages ​​leads to: the language in which one speaks of a reality that lies outside of it, and the language in which one speaks of the very first language.

In everyday language there is no distinction between these levels: we speak the same language about reality and about language. For example, a person whose native language is Russian does not see any particular difference between the statements: "Glass is transparent" and "It is true that glass is transparent", although one of them speaks of glass, and the other about a statement about glass.
If someone had the idea of ​​the need to speak about the world in one language, and about the properties of this language in another, he could use two different existing languages, let's say Russian and English. Instead of just saying “Cow is a noun”, I would say “Cow is a noun”, and instead of “The statement “Glass is not transparent” is false”, I would say “The assertion “Glass is not transparent” is false”. With this use of two different languages, what is said about the world would be clearly different from what is said about the language with which one speaks about the world. Indeed, the first statements would refer to Russian, while the second to English.

If further our expert on languages ​​would like to speak out about some circumstances that already concern the English language, he could use another language. Let's say German. To talk about this latter one could resort, let us say, to the Spanish language, and so on.
It turns out, therefore, a kind of ladder, or hierarchy, of languages, each of which is used for a very specific purpose: in the first one they talk about the objective world, in the second - about this first language, in the third - about the second language, etc. Such a distinction between languages ​​according to their area of ​​application is a rare occurrence in everyday life. But in the sciences, which, like logic, deal specifically with languages, it sometimes turns out to be very useful. The language used to talk about the world is usually called object language. The language used to describe the subject language is called a metalanguage.

It is clear that if language and metalanguage are demarcated in this way, the statement "I am lying" can no longer be formulated. It speaks of the falsity of what is said in Russian, and, therefore, belongs to the metalanguage and must be expressed in English. Specifically, it should sound like this: “Everything I speak in Russian is false” (“Everything I say in Russian is false”); this English statement says nothing about itself, and no paradox arises.
The distinction between language and metalanguage makes it possible to eliminate the “Liar” paradox. Thus, it becomes possible to correctly, without contradiction, define the classical concept of truth: a statement is true that corresponds to the reality it describes.
The concept of truth, like all other semantic concepts, has a relative character: it can always be attributed to a particular language.

As the Polish logician Atarsky showed, the classical definition of truth must be formulated in a language broader than the language for which it is intended. In other words, if we want to indicate what the phrase “a statement true in a given language” means, we must, in addition to the expressions of this language, also use expressions that are not in it.
Tarski introduced the concept of a semantically closed language. Such a language includes, in addition to its expressions, their names, and also, which is important to emphasize, statements about the truth of the sentences formulated in it.

There is no boundary between language and metalanguage in a semantically closed language. Its means are so rich that they allow not only to assert something about extralinguistic reality, but also to evaluate the truth of such statements. These means are sufficient, in particular, to reproduce the antinomy “Liar” in the language. A semantically closed language thus turns out to be self-contradictory. Every natural language is obviously semantically closed.
The only acceptable way to eliminate antinomy, and hence internal inconsistency, according to Tarski, is to abandon the use of a semantically closed language. This path is acceptable, of course, only in the case of artificial, formalized languages ​​that allow a clear division into language and metalanguage. In natural languages, with their obscure structure and the ability to talk about everything in the same language, this approach is not very realistic. It makes no sense to raise the question of the internal consistency of these languages. Their rich expressive possibilities also have their downside - paradoxes.

Other solutions to the paradox

So there are statements that speak of their own truth or falsity. The idea that these kinds of statements are not meaningful is very old. It was defended by the ancient Greek logician Chrysippus.
In the Middle Ages, the English philosopher and logician W. Ockham stated that the statement “Every statement is false” is meaningless, since it speaks, among other things, of its own falsity. A contradiction directly follows from this statement. If every proposition is false, then so is the proposition itself; but that it is false means that not every proposition is false.

The situation is similar with the statement “Every proposition is true”. It must also be classified as meaningless and also leads to a contradiction: if every statement is true, then the negation of this statement itself is also true, that is, the statement that not every statement is true.
Why, however, cannot a statement meaningfully speak of its own truth or falsity?
Already a contemporary of Ockham, the French philosopher of the XIV century. J. Buridan did not agree with his decision. From the point of view of ordinary ideas about meaninglessness, expressions like “I am lying”, “Every statement is true (false)”, etc. quite meaningful. What you can think about, what you can say - such is the general principle of Buridan. A person can think about the truth of the statement that he utters, which means that he can speak about it. Not all statements about themselves are meaningless. For example, the statement "This sentence is written in Russian" is true, but the statement "There are ten words in this sentence" is false. And both of them make perfect sense. If it is admitted that a statement can speak about itself, then why is it not capable of speaking meaningfully about such a property of itself as truth?
Buridan himself considered the statement “I am lying” not meaningless, but false. He justified it like this.

When a person affirms a proposition, he thereby asserts that it is true. If the sentence says of itself that it is itself false, then it is only an abbreviated formulation of a more complex expression that asserts both its truth and its falsity. This expression is contradictory and therefore false. But it is by no means meaningless.

Buridan's argument is still sometimes considered convincing.
There are other lines of criticism of the solution to the "Liar" paradox, which was developed in detail by Tarski. Is there really no antidote against paradoxes of this type in semantically closed languages—and all natural languages ​​are, after all?
If this were the case, then the concept of truth could only be defined in a rigorous way in formalized languages. Only in them is it possible to distinguish between the objective language in which people talk about the surrounding world and the metalanguage in which they speak about this language. This hierarchy of languages ​​is modeled on the acquisition of a foreign language with the help of a native language. The study of such a hierarchy led to many interesting conclusions, and in certain cases it is essential. But it does not exist in natural language. Does it discredit him? And if so, to what extent? After all, the concept of truth is still used in it, and usually without any complications. Is introducing a hierarchy the only way to eliminate paradoxes like "The Liar?"

In the 1930s, the answers to these questions seemed undoubtedly in the affirmative. However, now there is no former unanimity, although the tradition of eliminating paradoxes of this type by “stratifying” the language remains dominant.
Recently, egocentric expressions have attracted more and more attention. They contain words like “I”, “this”, “here”, “now”, and their truth depends on when, by whom, where they are used.

In the statement “This statement is false”, the word “this” occurs. What object does it refer to? "Liar" may indicate that the word "it" does not refer to the meaning of the given statement. But then what does it refer to, what does it mean? And why can't this meaning still be denoted by the word "this"?
Without going into details here, it is only worth noting that in the context of the analysis of egocentric expressions, “Liar” is filled with a completely different content than before. It turns out that he no longer warns against the confusion of language and metalanguage, but points out the dangers associated with the misuse of the word "this" and similar egocentric words.
The issues that have associated over the centuries with "The Liar" have changed radically depending on whether it was seen as an example of ambiguity, or as an expression outwardly presented as an example of a mixture of language and metalanguage, or, finally, as a typical example of the misuse of egocentric expressions. And there is no certainty that other problems will not be associated with this paradox in the future.

The well-known modern Finnish logician and philosopher G. von Wright wrote in his work devoted to the “Liar” that this paradox should by no means be understood as a local, isolated obstacle that can be removed by one inventive movement of thought. Liar touches on many of the most important topics in logic and semantics. This is the definition of truth, and the interpretation of contradiction and evidence, and a whole series of important differences: between a sentence and the thought expressed by it, between the use of an expression and its mention, between the meaning of a name and the object it denotes.
The situation is similar with other logical paradoxes. “The antinomies of logic,” writes von Wrigg, “have puzzled us since their discovery and will probably continue to puzzle us forever. We should, I think, regard them not so much as problems waiting to be solved, but as inexhaustible raw material for thought. They are important because thinking about them touches upon the most fundamental questions of all logic, and therefore of all thought.”

In conclusion of this conversation about the "Liar" we can recall a curious episode from the time when formal logic was still taught at school. In a logic textbook published in the late 1940s, eighth-grade schoolchildren were asked as a homework assignment—as a warm-up, so to speak—to find the mistake made in this simple-looking statement: “I am lying.” And, let it not seem strange, it was believed that the majority of schoolchildren successfully coped with such a task.

§ 2. Russell's paradox

The most famous of the paradoxes discovered already in our century is the antinomy discovered by B. Russell and communicated by him in a letter to G. Ferge. The same antinomy was discussed simultaneously in Göttingen by the German mathematicians Z. Zermelo and D. Hilbert.
The idea was in the air, and its publication produced the impression of an exploding bomb. This paradox caused in mathematics, according to Hilbert, the effect of complete catastrophe. The simplest and most important logical methods, the most common and useful concepts, are under threat.
It immediately became obvious that neither in logic nor in mathematics, in the entire long history of their existence, was anything decidedly worked out that could serve as a basis for. elimination of antinomy. Clearly a departure from habitual ways of thinking was necessary. But from where and in what direction? How radical was the rejection of established ways of theorizing supposed to be?
With further study of antinomy, the conviction in the need for a fundamentally new approach steadily grew. Half a century after its discovery, specialists in the foundations of logic and mathematics L. Frenkel and I. Bar-Hillel already stated without any reservations: , so far invariably failed, are obviously insufficient for this purpose.
The modern American logician H. Curry wrote a little later about this paradox: “In terms of the logic known in the 19th century, the situation simply defied explanation, although, of course, in our educated age there may be people who see (or think they see ), what is the error?

Russell's paradox in its original form is connected with the concept of a set, or a class.
We can talk about sets of different objects, for example, about the set of all people or about the set of natural numbers. An element of the first set will be any individual person, an element of the second - every natural number. It is also possible to consider sets themselves as some objects and speak of sets of sets. One can even introduce such concepts as the set of all sets or the set of all concepts.

Set of ordinary sets

With respect to any set arbitrarily taken, it seems reasonable to ask whether it is its own element or not. Sets that do not contain themselves as an element will be called ordinary. For example, the set of all people is not a person, just as the set of atoms is not an atom. Sets that are proper elements will be unusual. For example, a set that unites all sets is a set and therefore contains itself as an element.
Consider now the set of all ordinary sets. Since it is a set, one can also ask about it whether it is ordinary or unusual. The answer, however, is discouraging. If it is ordinary, then by definition it must contain itself as an element, since it contains all ordinary sets. But this means that it is an unusual set. The assumption that our set is an ordinary set thus leads to a contradiction. So it can't be normal. On the other hand, it cannot be unusual either: an unusual set contains itself as an element, and the elements of our set are only ordinary sets. As a result, we come to the conclusion that the set of all ordinary sets cannot be either ordinary or extraordinary.

Thus, the set of all sets that are not proper elements is a proper element if and only if it is not such an element. This is a clear contradiction. And it was obtained on the basis of the most plausible assumptions and with the help of seemingly indisputable steps. The contradiction suggests that such a set simply does not exist. But why can't it exist? After all, it consists of objects that satisfy a well-defined condition, and the condition itself does not seem to be somehow exceptional or obscure. If a set so simply and clearly defined cannot exist, then what, in fact, is the difference between possible and impossible sets? The conclusion about the non-existence of the considered set sounds unexpected and inspires anxiety. It makes our general notion of a set amorphous and chaotic, and there is no guarantee that it cannot give rise to some new paradoxes.

Russell's paradox is remarkable for its extreme generality. For its construction, no complex technical concepts are needed, as in the case of some other paradoxes, the concepts of “set” and “element of the set” are sufficient. But this simplicity just speaks of its fundamental nature: it touches on the deepest foundations of our reasoning about sets, since it speaks not about some special cases, but about sets in general.

Other variants of the paradox

Russell's paradox is not specifically mathematical. It uses the concept of a set, but does not touch on any special properties associated specifically with mathematics.
This becomes apparent when the paradox is reformulated in purely logical terms.

Of every property one can, in all probability, ask whether it is applicable to itself or not.
The property of being hot, for example, does not apply to itself, since it is not itself hot; the property of being concrete also does not refer to itself, for it is an abstract property. But the property of being abstract, being abstract, is applicable to oneself. Let us call these properties inapplicable to themselves inapplicable. Does the property of being inapplicable to oneself apply? It turns out that inapplicability is inapplicable only if it is not. This is, of course, paradoxical.
The logical, property-related variety of Russell's antinomy is just as paradoxical as the mathematical, set-related variety.
Russell also proposed the following popular version of the paradox he discovered.

Imagine that the council of one village defined the duties of a barber as follows: to shave all the men of the village who do not shave themselves, and only these men. Should he shave himself? If so, it will refer to those who shave themselves, and those who shave themselves, he should not shave. If not, he will belong to those who do not shave themselves, and therefore he will have to shave himself. We thus arrive at the conclusion that this barber shaves himself if and only if he does not shave himself. This, of course, is impossible. The argument about the barber is based on the assumption that such a barber exists. The resulting contradiction means that this assumption is false, and there is no such villager who would shave all those and only those villagers who do not shave themselves.
The duties of a hairdresser do not seem contradictory at first glance, so the conclusion that there cannot be one sounds somewhat unexpected. But this conclusion is not paradoxical. The condition that the village barber must satisfy is, in fact, self-contradictory and therefore impossible. There cannot be such a hairdresser in a village for the same reason that there is no person in it who would be older than himself or who would be born before his birth.
The argument about the hairdresser can be called a pseudo-paradox. In its course, it is strictly analogous to Russell's paradox, and this is what makes it interesting. But it is still not a true paradox.

Another example of the same pseudo-paradox is the well-known catalog argument.
A certain library decided to compile a bibliographic catalog that would include all those and only those bibliographic catalogs that do not contain references to themselves. Should such a directory include a link to itself?
It is easy to show that the idea of ​​creating such a catalog is not feasible; it simply cannot exist, because it must simultaneously include a reference to itself and not include.
It is interesting to note that cataloging all directories that do not contain references to themselves can be thought of as an endless, never ending process. Let's say that at some point a directory, say K1, was compiled, including all other directories that do not contain references to themselves. With the creation of K1, another directory appeared that does not contain a link to itself. Since the goal is to make a complete catalog of all directories that do not mention themselves, it is obvious that K1 is not the solution. He does not mention one of these directories - himself. Including this mention of himself in K1, we get the K2 catalog. It mentions K1, but not K2 itself. Adding such a mention to K2, we get KZ, which again is not complete due to the fact that it does not mention itself. And on without end.

§ 3. Paradoxes of Grelling and Berry

An interesting logical paradox was discovered by the German logicians K. Grelling and L. Nelson (Grelling's paradox). This paradox can be formulated very simply.

Autological and heterological words

Some words denoting properties have the very property they name. For example, the adjective “Russian” is itself Russian, “polysyllabic” is itself polysyllabic, and “five-syllable” itself has five syllables. Such words referring to themselves are called self-meaning or autological.
There are not so many such words, the vast majority of adjectives do not have the properties that they name. “New” is not, of course, new, “hot” is hot, “one-syllable” is one-syllable, and “English” is English. Words that do not have the property denoted by them are called heterologous or heterologous. Obviously, all adjectives denoting properties that are not applicable to words will be heterological.
This division of adjectives into two groups seems clear and unobjectionable. It can be extended to nouns: "word" is a word, "noun" is a noun, but "clock" is not a clock, and "verb" is not a verb.
A paradox arises as soon as the question is asked: to which of the two groups does the adjective “heterological” itself belong? If it is autological, it has the property it designates and must be heterological. If it is heterological, it does not have the property it calls, and must therefore be autological. There is a paradox.

By analogy with this paradox, it is easy to formulate other paradoxes of the same structure. For example, is or is not a suicidal person who kills every non-suicidal person and does not kill any suicidal person?

It turned out that Grellig's paradox was known in the Middle Ages as the antinomy of an expression that does not name itself. One can imagine the attitude towards sophisms and paradoxes in modern times, if the problem that required an answer and caused lively debate was suddenly forgotten and was rediscovered only five hundred years later!

Another, outwardly simple antinomy was indicated at the very beginning of our century by D. Berry.

The set of natural numbers is infinite. The set of those names of these numbers that are available, for example, in the Russian language and contain less than, say, one hundred words, is finite. This means that there are such natural numbers for which there are no names in Russian that consist of less than a hundred words. Among these numbers there is obviously the smallest number. It cannot be called by means of a Russian expression containing less than a hundred words. But the expression: “The smallest natural number for which there is no complex name in Russian, consisting of less than a hundred words” is just the name of this number! This name has just been formulated in Russian and contains only nineteen words. An obvious paradox: the named number turned out to be the one for which there is no name!

§ 4. Irresolvable dispute

The famous sophist Protagoras, who lived in the 5th century. BC, there was a student named Euathlus, who studied law. According to the agreement concluded between them, Euathlus had to pay for training only if he won his first lawsuit. If he loses this process, he is not obliged to pay at all. However, after completing his studies, Evatl did not participate in the processes. It lasted quite a long time, the teacher's patience ran out, and he filed a lawsuit against his student. Thus, for Euathlus, this was the first trial. Protagoras substantiated his demand as follows:

“Whatever the decision of the court, Euathlus will have to pay me. He will either win his first trial or lose. If he wins, he will pay by virtue of our contract. If he loses, he will pay according to this decision.

Apparently Euathlus was a capable student, as he replied to Protagoras:

- Indeed, I either win the process or lose it. If I win, the court decision will release me from the obligation to pay. If the court decision is not in my favor, then I lost my first case and will not pay by virtue of our contract.

Solutions to the paradox "Protagoras and Euathlus"

Perplexed by this turn of the matter, Protagoras devoted a special essay to this dispute with Euathlus, “A Litigation for Payment.” Unfortunately, it, like most of what was written by Protagoras, did not reach us. Nevertheless, one must pay tribute to Protagoras, who immediately sensed a problem behind a simple judicial incident that deserves special study.

G. Leibniz, himself a lawyer by education, also took this dispute seriously. In his doctoral dissertation, "A Study of Intricate Cases in Law," he tried to prove that all cases, even the most intricate ones, like the litigation of Protagoras and Euathlus, must find a correct solution on the basis of common sense. According to Leibniz, the court should refuse Protagoras for the untimely filing of a claim, but leave, however, for him the right to demand payment of money by Evatl later, namely after the first process he won.

Many other solutions to this paradox have been proposed.

They referred, in particular, to the fact that a court decision should have greater force than a private agreement between two persons. It can be answered that without this agreement, no matter how insignificant it may seem, there would be neither a court nor its decision. After all, the court must make its decision precisely on its occasion and on its basis.

They also appealed to the general principle that every work, and therefore the work of Protagoras, must be paid. But it is known that this principle has always had exceptions, especially in a slave-owning society. In addition, it is simply not applicable to the specific situation of the dispute: after all, Protagoras, guaranteeing a high level of education, himself refused to accept payment in case of failure of his student in the first process.

Sometimes they talk like this. Both Protagoras and Euathlus are both right in part, and neither of them in general. Each of them takes into account only half of the possibilities that are beneficial to itself. Full or comprehensive consideration opens up four possibilities, of which only half is beneficial to one of the disputants. Which of these possibilities is realized, it will be decided not by logic, but by life. If the verdict of the judges will have more force than the contract, Euathl will have to pay only if he loses the process, i.e. by virtue of a court decision. If, however, a private agreement is placed higher than the decision of the judges, then Protagoras will receive payment only in the event of losing the process to Evatlus, i.e. by virtue of an agreement with Protagoras. This appeal to life completely confuses everything. What, if not logic, can judges be guided by in conditions when all the relevant circumstances are completely clear? And what kind of leadership will it be if Protagoras, who claims payment through the court, achieves it only by losing the process?

However, Leibniz's solution, which at first seems convincing, is a little better than the vague opposition of logic and life. In essence, Leibniz proposes retroactively to change the wording of the contract and stipulate that the first lawsuit involving Euathlus, the outcome of which will decide the question of payment, should not be the trial of Protagoras. This thought is deep, but not related to a particular court. Had there been such a clause in the original agreement, there would have been no need for litigation at all.

If by the solution of this difficulty one understands the answer to the question whether Euathlus should pay Protagoras or not, then all these, like all other conceivable solutions, are, of course, untenable. They are nothing more than a departure from the essence of the dispute, they are, so to speak, sophistical tricks and cunning in a hopeless and insoluble situation. For neither common sense nor any general principles concerning social relations can settle the dispute.
It is impossible to carry out together the contract in its original form and the decision of the court, whatever the latter may be. To prove this, simple means of logic are sufficient. By the same means, it can also be shown that the treaty, despite its completely innocent appearance, is self-contradictory. It requires the realization of a logically impossible proposition: Euathlus must both pay for the education and at the same time not pay.

Rules that lead to a dead end

The human mind, accustomed not only to its strength, but also to its flexibility and even resourcefulness, finds it difficult, of course, to reconcile itself to this absolute hopelessness and admit that it has been driven into a dead end. This is especially difficult when the impasse is created by the mind itself: it, so to speak, stumbles out of the blue and falls into its own nets. Nevertheless, one has to admit that sometimes, and by the way, not so rarely, agreements and systems of rules, formed spontaneously or introduced consciously, lead to insoluble, hopeless situations.

An example from recent chess life will once again confirm this idea.

International rules for chess competitions oblige chess players to record the game move by move clearly and legibly. Until recently, the rules also stated that a chess player who missed the recording of several moves due to lack of time must, “as soon as his time trouble ends, immediately fill out his form, writing down the missed moves.” Based on this instruction, one judge at the 1980 Chess Olympiad (Malta) interrupted the game, which was going on in hard time trouble, and stopped the clock, declaring that the control moves had been made and, therefore, it was time to put the records of the games in order.

“But excuse me,” cried the participant, who was on the verge of losing and counted only on the intensity of passions at the end of the game, “after all, not a single flag has yet fallen and no one can ever (as it is also written in the rules) can tell how many moves have been made.
However, the referee was supported by the chief arbiter, who said that, indeed, since the time trouble had ended, it was necessary, following the letter of the rules, to start recording the missed moves.
It was pointless to argue in this situation: the rules themselves led to a dead end. It only remained to change their wording in such a way that similar cases could not arise in the future.
This was done at the congress of the International Chess Federation, which was taking place at the same time: instead of the words “as soon as the time trouble is over”, the rules now say: “as soon as the flag indicates the end of time”.
This example clearly shows how to deal with deadlock situations. It is useless to argue about which side is right: the dispute is insoluble, and there will be no winner in it. It remains only to come to terms with the present and take care of the future. To do this, you need to reformulate the original agreements or rules in such a way that they do not lead anyone else into the same hopeless situation.
Of course, such a course of action is not a solution to an insoluble dispute or a way out of a hopeless situation. It is rather a stop in front of an insurmountable obstacle and a road around it.

Paradox "crocodile and mother"

In ancient Greece, the story of a crocodile and a mother was very popular, coinciding in its logical content with the paradox of Protagoras and Euathlus.
The crocodile snatched her child from an Egyptian woman standing on the river bank. To her plea to return the child, the crocodile, shedding, as always, a crocodile tear, answered:

“Your misfortune touched me, and I will give you a chance to get your child back. Guess if I'll give it to you or not. If you answer correctly, I will return the child. If you don't guess, I won't give it back.

Thinking, the mother replied:

You won't give me the baby.

"You won't get it," the crocodile concluded. You either told the truth or you didn't tell the truth. If it is true that I will not give up the child, then I will not give him up, because otherwise it will not be true. If what was said is not true, then you did not guess, and I will not give the child by agreement.

However, this reasoning did not seem convincing to the mother.

“But if I told the truth, then you will give me the child, as we agreed.” If I did not guess that you will not give the child, then you must give it to me, otherwise what I said will not be untrue.

Who is right: mother or crocodile? To what does the promise given to the crocodile oblige? In order to give the child, or, on the contrary, not to give it away? And to both at the same time. This promise is self-contradictory, and thus it cannot be fulfilled by virtue of the laws of logic.
The missionary found himself with the cannibals and arrived just in time for dinner. They let him choose how he will be eaten. To do this, he must utter some statement with the condition that if this statement turns out to be true, they will cook it, and if it turns out to be false, they will roast it.

What should the missionary say?

Of course, he should say: "You will fry me."

If he is really fried, it will turn out that he spoke the truth, and therefore he must be boiled. If he is boiled, his statement will be false, and he should just be fried. The cannibals will have no way out: from “fry” it follows “cook”, and vice versa.

This episode of the cunning missionary is, of course, another paraphrase of the dispute between Protagoras and Euathlus.

Paradox of Sancho Panza

One old paradox known in ancient Greece is played up in “Don Quixote” by M. Cervantes. Sancho Panza has become the governor of the island of Barataria and administers the court.
The first to come to him is some visitor and says: “Senior, a certain estate is divided into two halves by a deep river ... So, a bridge has been thrown across this river, and right there on the edge stands a gallows and there is something like a court, in which usually four judges sit, and they judge on the basis of a law issued by the owner of the river, the bridge and the whole estate, which law is drawn up in this way: let those through, and whoever lies, without any leniency, send them to the gallows located right there and execute them. From the time when this law was promulgated in all its severity, many managed to get across the bridge, and as soon as the judges were satisfied that the passers-by were telling the truth, they let them through. But then one day a man who was sworn in swore and said: he swears that he came in order to be hung up on this very gallows, and for nothing else. This oath perplexed the judges, and they said: “If this man is allowed to proceed without hindrance, then this will mean that he has broken the oath and, according to the law, is liable to death; if we hang him, then he swore that he came only to be hung up on this gallows, therefore, his oath, it turns out, is not false, and on the basis of the same law it is necessary to let him pass. And so I ask you, señor governor, what should the judges do with this man, for they are still perplexed and hesitating...
Sancho suggested, perhaps not without cunning, that the half of the person who told the truth should be let through, and the one that lied should be hanged, and in this way the rules for crossing the bridge would be observed in all forms. This passage is interesting in several respects.
First of all, it is a clear illustration of the fact that the hopeless situation described in the paradox may well be faced - and not in pure theory, but in practice - if not a real person, then at least a literary hero.

The way out proposed by Sancho Panza was not, of course, a solution to the paradox. But this was just the solution that only remained to be resorted to in his position.
Once upon a time, Alexander the Great, instead of untying the cunning Gordian knot, which no one has yet managed to do, simply cut it. Sancho did the same. Trying to solve the puzzle on its own terms was useless—it was simply unsolvable. It remained to discard these conditions and introduce your own.
And one moment. With this episode, Cervantes clearly condemns the exorbitantly formal scale of medieval justice, permeated with the spirit of scholastic logic. But how widespread in his time - and this was about four hundred years ago - were information from the field of logic! Not only Cervantes himself knows this paradox. The writer finds it possible to attribute to his hero, an illiterate peasant, the ability to understand that he faces an insoluble task!

§ 5. Other paradoxes

The above paradoxes are arguments, the result of which is a contradiction. But there are other types of paradoxes in logic. They also point out some difficulties and problems, but they do it in a less harsh and uncompromising way. Such, in particular, are the paradoxes discussed below.

Paradoxes of imprecise concepts

Most of the concepts of not only natural language, but also the language of science are inaccurate, or, as they are also called, blurry. Often this turns out to be the cause of misunderstanding, disputes, or even simply leads to deadlocks.
If the concept is inaccurate, the boundary of the area of ​​objects to which it is applicable is devoid of sharpness, blurred. Take, for example, the concept of "heap". One grain (a grain of sand, a stone, etc.) is not yet a pile. A thousand grains is already, obviously, a bunch. And three grains? And ten? What number of grains are added to form a heap? Not very clear. In the same way, it is not clear with the removal of which grain the heap disappears.
The empirical characteristics “large”, “heavy”, “narrow”, etc. are inaccurate. Such ordinary concepts as “wise man”, “horse”, “house”, etc. are inexact.
There is no grain of sand that, when removed, we can say that with its removal, what remains can no longer be called home. But this seems to mean that at no point in the gradual dismantling of the house - up to its complete disappearance - is there any 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 forming a heap 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. Therefore, no number of grains can form heaps.
The possibility of this and similar proofs leading to absurd conclusions means that the principle of mathematical induction has a limited scope. It should not be used in reasoning with inaccurate, vague concepts.

A good example of how these concepts can lead to irresolvable disputes is a curious trial that took place in 1927 in the United States. The sculptor C. Brancusi went to court demanding that his works be recognized as works of art. Among the works sent to New York for the exhibition was the sculpture "Bird", which is now considered a classic of the abstract style. It is a modulated column of polished bronze about one and a half meters high, which does not have any external resemblance to a bird. Customs officers categorically refused to recognize Brancusi's abstract creations as works of art. They put them under "Metal hospital utensils and household items" and imposed a heavy customs duty on them. Outraged, Brancusi sued.

Customs was supported by artists - members of the National Academy, who defended traditional methods in art. They acted as witnesses for the defense at the trial and categorically insisted that the attempt to pass off the “Bird” as a work of art was simply a scam.
This conflict vividly emphasizes the difficulty of operating with the concept of “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 a sculptural image, increasingly moving away from the original, cease to be a work of art and become a “metal utensil”? This question is as difficult to answer as the question of where is the boundary between a house and its ruins, between a horse with a tail and a horse without a tail, and so on. 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.

The handling of imprecise concepts thus requires a certain amount of caution. Wouldn't it be better to avoid them altogether?

The German philosopher E. Husserl was inclined to demand such extreme rigor and precision from knowledge that is not found even in mathematics. In connection with this, Husserl's biographers recall with irony an incident that happened to him in childhood. He was presented with a penknife, and, deciding to make the blade as sharp as possible, he sharpened it until nothing was left of the blade.
More precise concepts are preferable to imprecise ones in many situations. The usual desire to clarify the concepts used is quite justified. But it must, of course, have its limits. Even in the language of science, a significant part of the concepts is inaccurate. And this is connected not with the subjective and random mistakes of individual scientists, but with the very nature of scientific knowledge. In natural language, imprecise concepts are overwhelming; this speaks, among other things, of his flexibility and latent strength. Anyone who demands the utmost precision from all concepts runs the risk of being left without a language altogether. “Deprive the words of any ambiguity, any uncertainty,” wrote the French aesthetician 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, will not be able to find its expression. But what am I saying: deprive ... I will say more. Deprive the word of any inaccuracy - and you will lose even axioms.
For a long time, both logicians and mathematicians did not pay attention to the difficulties associated with fuzzy concepts and their corresponding sets. The question was posed as follows: concepts must be precise, and anything vague is unworthy of serious interest. In recent decades, however, this overly strict attitude has lost its appeal. Logical theories are constructed that specifically take into account the uniqueness of reasoning with inaccurate concepts.
The mathematical theory of the so-called fuzzy sets, indistinctly defined collections of objects, is actively developing.
The analysis of problems of inaccuracy is a step towards bringing logic closer to 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 section of logic that does not have its own paradoxes.
Inductive logic has its own paradoxes, which have been actively, but so far without much success, being fought for almost half a century. Of particular interest is the confirmation paradox discovered by the American philosopher K. Hempel. It is natural to consider that general propositions, in particular scientific laws, are confirmed by their positive examples. If, say, the proposition “All A is B” is considered, then its positive examples will be objects that have properties A and B. In particular, supporting examples for the proposition “All ravens are black” are objects that are both ravens and black. This statement is tantamount, however, to the statement "All things that are not black are not crows", and the confirmation of the latter must also be the confirmation of the former. But "Everything is not black is not a crow" is confirmed by every case of a non-black object that is not a crow. It turns out, therefore, that the observations “White cow”, “Brown shoes”, etc. confirm the statement "All crows are black."

An unexpected paradoxical result follows from seemingly innocent premises.

In the logic of norms, a number of its laws cause concern. When they are formulated in meaningful terms, their inconsistency with the usual notions of right and wrong becomes obvious. For example, one of the laws says that from the order “Send a letter!” the order “Send the letter or burn it!” follows.
Another law states that if a person has violated one of his duties, he gets the right to do whatever he wants. Our logical intuition does not want to put up with this kind of “laws of obligation”.
In the logic of knowledge, the paradox of logical omniscience is heavily discussed. He claims that a person knows all the logical consequences that follow from the positions he takes. For example, if a person knows the five postulates of Euclid's geometry, then, therefore, he knows all this geometry, since it follows from them. But it's not. A person can 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.

§ 6. What is a logical paradox

No exhaustive list of logical paradoxes exists, and it is impossible.
The considered paradoxes are only a part of all those discovered so far. It is likely that many other paradoxes will be discovered in the future, and even completely new types of them. The very concept of a paradox is not so definite that it would be possible to compile a list of at least already known paradoxes.
“Set-theoretic paradoxes are a very serious problem, not for mathematics, however, but rather for logic and the theory of knowledge,” writes the Austrian mathematician and logician K. Gödel. “The logic is consistent. There are no logical paradoxes,” says mathematician D. Bochvar. Such discrepancies are sometimes significant, sometimes verbal. The point is largely in what exactly is meant by a logical paradox.

The peculiarity of logical paradoxes

A necessary feature of logical paradoxes is the logical dictionary.
Paradoxes that are logical must be formulated in logical terms. However, in logic there are no clear criteria for dividing terms into logical and non-logical. Logic, which deals with the correctness of reasoning, seeks to reduce the concepts on which the correctness of practically applied conclusions depends to a minimum. But this minimum is not predetermined unambiguously. In addition, non-logical statements can also be formulated in logical terms. Whether a particular paradox uses only purely logical premises is far from always possible to determine unambiguously.
Logical paradoxes are not rigidly separated from all other paradoxes, just as the latter are not clearly distinguished from everything non-paradoxical and consistent with the prevailing ideas. At the beginning of the study of logical paradoxes, it seemed that they could be distinguished by the violation of some as yet unexplored position or rule of logic. The vicious circle principle introduced by B. Russell was especially active in claiming the role of such a rule. This principle states that a collection of objects cannot contain members defined only by the same collection.
All paradoxes have one thing in common - self-applicability, or circularity. In each of them, the object in question is characterized by some set of objects to which it itself belongs. If we select, for example, the most cunning person, we do this with the help of a population of people to which this person belongs. And if we say: "This statement is false," we characterize the statement of interest to us by referring to the totality of all false statements that includes it.

In all paradoxes, there is a self-applicability of concepts, which means that there is, as it were, movement in a circle, leading in the end to the starting point. In an effort to characterize the object of interest to us, we turn to the set of objects that includes it. However, it turns out that, for its definiteness, it itself needs the object under consideration and cannot be clearly understood without it. In this circle, perhaps, lies the source of paradoxes.
The situation is complicated, however, by the fact that such a circle exists in many completely non-paradoxical arguments. Circular is a huge variety of the most common, harmless and at the same time convenient ways of expression. Such examples as “the largest of all cities”, “the smallest of all natural numbers”, “one of the electrons of the iron atom”, etc., show that not every case of self-applicability leads to a contradiction and that it is important not only in ordinary language, but also in the language of science.
A mere reference to the use of self-applicable concepts is thus insufficient to discredit paradoxes. Some additional criterion is needed to separate self-applicability, leading to a paradox, from all other cases of it.
There have been many proposals to this effect, but no successful clarification of circularity has been found. It turned out to be impossible to characterize circularity in such a way that every circular reasoning leads to a paradox, and every paradox is the result of some circular reasoning.
An attempt to find some specific principle of logic, the violation of which would be a distinctive feature of all logical paradoxes, did not lead to anything definite.
Some kind of classification of paradoxes would undoubtedly be useful, subdividing them into types and types, grouping some paradoxes and opposing them to others. However, nothing sustainable has been achieved in this case either.

The English logician F. Ramsey, who died in 1930, when he was not yet twenty-seven years old, proposed to divide all paradoxes into syntactic and semantic ones. The first includes, for example, Russell's paradox, the second - the paradoxes of the "Liar", Grelling, etc.
According to Ramsey, the paradoxes of the first group contain only concepts belonging to logic or mathematics. The latter include such concepts as “truth”, “definability”, “naming”, “language”, which are not strictly mathematical, but rather related to linguistics or even the theory of knowledge. Semantic paradoxes seem to owe their appearance not to some error in logic, but to the vagueness or ambiguity of some non-logical concepts, therefore the problems they pose concern language and must be solved by linguistics.

It seemed to Ramsey that mathematicians and logicians need not be interested in semantic paradoxes. Later it turned out, however, that some of the most significant results of modern logic were obtained precisely in connection with a deeper study of precisely these non-logical paradoxes.
The division of paradoxes proposed by Ramsey was widely used at first and retains some importance even now. At the same time, it is becoming increasingly clear that this division is rather vague and relies primarily on examples, and not on an in-depth comparative analysis of the two groups of paradoxes. Semantic concepts are now well defined, and it is hard not to recognize that these concepts are indeed logical. With the development of semantics, which defines its basic concepts in terms of set theory, the distinction made by Ramsey is increasingly blurred.

Paradoxes and Modern Logic

What conclusions for logic follow from the existence of paradoxes?
First of all, the presence of a large number of paradoxes speaks of the strength of logic as a science, and not of its weakness, as it might seem.

It was no coincidence that the discovery of paradoxes coincided with the period of the most intensive development of modern logic and its greatest successes.
The first paradoxes were discovered even before the emergence of logic as a special science. Many paradoxes were discovered in the Middle Ages. Later, however, they turned out to be forgotten and were rediscovered already in our century.
Medieval logicians were not aware of the concepts of "set" and "element of the set", introduced into science only in the second half of the 19th century. But the flair for paradoxes was honed in the Middle Ages to such an extent that already at that early time certain concerns were expressed about self-applicable concepts. The simplest example of this is the notion of "being one's own element" that appears in many of today's paradoxes.
However, such fears, like all warnings about paradoxes in general, were not systematic and definite until our century. They did not lead to any clear proposals for revisiting habitual ways of thinking and expressing.
Only modern logic has taken the very problem of paradoxes out of oblivion, discovered or rediscovered most of the specific logical paradoxes. She further showed that the ways of thinking traditionally explored by logic are completely insufficient for eliminating paradoxes, and indicated fundamentally new methods of dealing with them.
Paradoxes pose an important question: where, in fact, do some of the usual methods of concept formation and reasoning fail us? After all, they seemed completely natural and convincing, until it turned out that they were paradoxical.

Paradoxes undermine the belief that the habitual methods of theoretical thinking by themselves and without any special control over them provide a reliable progress towards the truth.
Requiring a radical change in an overly gullible approach to theorizing, paradoxes are a harsh critique of logic in its naive, intuitive form. They play the role of a factor that controls and puts restrictions on the way of constructing deductive systems of logic. And this role of them can be compared with the role of an experiment that tests the correctness of hypotheses in such sciences as physics and chemistry, and forces them to make changes to these hypotheses.
A paradox in a theory speaks of the incompatibility of the assumptions underlying it. It acts as a timely detected symptom of the disease, without which it could have been overlooked.
Of course, the disease manifests itself in many ways, and in the end it is possible to reveal it without such acute symptoms as paradoxes. For example, the foundations of set theory would be analyzed and refined even if no paradoxes in this area were discovered. But there would not have been that sharpness and urgency with which the paradoxes discovered in it raised the problem of revising set theory.

An extensive literature is devoted to paradoxes, a large number of their explanations have been proposed. But none of these explanations is universally accepted, and there is no complete agreement on the origin of paradoxes and how to get rid of them.
“Over the past sixty years, hundreds of books and articles have been devoted to the goal of resolving paradoxes, but the results are amazingly poor in comparison with the efforts expended,” writes A. Frenkel. “It looks like,” H. Curry concludes his analysis of the paradoxes, “that a complete reform of logic is required, and mathematical logic can become the main tool for carrying out this reform.”