Types of paradoxes

There are paradoxes that arise in a certain area scientific knowledge in the process of the historical development of science, when a contradiction is found between a certain established system of knowledge and new facts, between the directions of research fixed in certain paradigms and new discoveries that do not fit into these paradigms. Thus, scientific discoveries in cosmology, quantum physics, and biology made in the 20th century contradict classical theories in these branches of science and are interpreted as paradoxical from the point of view of classical theories.

In every industry scientific knowledge specific paradoxes appear - physical, chemical, biological, mathematical, etc.

The paradoxes that arise within the framework of a certain scientific theory reveal the inconsistency of the very movement of material objects that science studies, the "duality" of the nature of the object of study itself, predetermining the rethinking of the fundamental principles and paradigms of a particular science. For example, in the theory of quantum chemistry, it was found that an electron around a nucleus at any moment is located at every elementary point in space, although an electron is an elementary particle.

Types of paradoxes

Paradoxes according to the types of logic were classified into semantic and logical.

Semantic paradoxes arise in reasoning:

In the process of linking language expressions with their objective meaning, that is, denotation;

When two levels of symbolic representation of the objects of considerations are mixed, namely, the level of object language and metamovies;

When using abstract indefinite timeline, under which you can bring any object;

When there is a problem of determining the truth or falsity of statements in a particular context.

The semantic paradoxes include: the "Liar" paradox, the heterological paradox, the paradox of the theory of names, the paradox (antinomy) of the naming relation.

The "Liar" paradox of logic is classified as an antinomy. It was first formulated by the ancient Greek philosopher Eubulides of Miletus, and it has two variants of expression: 1. Someone says "I'm lying"; 2. The Cretan Epimenides said: "All Cretans are liars."

The meaning of the "Liar" paradox lies in the fact that it is impossible to unequivocally determine the truth or falsity of the statement "I'm lying." So, if Epimenides does not lie, then his statement is true and, therefore, Epimenides is a liar; if Epimenides lies, then his statements are false, therefore, Epimenides is not a liar. We get the antinomy - "Epimenides lies and does not lie," or "The statement "I lie" is true because it is false, and false because it is true."

Another modification of the "Liar" paradox was formulated by the English logician P. Jourdain: "The statement written on the first side of this card is true; and on the other side of the same card it is written: The statement written on the other side of this card is false." If the first proposition is true, then the second proposition is also true, since the first proposition states that the second proposition is true. But if the second statement is true, then "the first statement is false" is false. So, from the two possible assumptions of the truth of these two statements, a contradiction arises.

Scholars have proposed many ways to solve the Liar paradox. For example, the Polish logician A. Tarski proposed to clearly distinguish between the levels of language - object and metamovie. It is the statements "I'm lying" that are formulated object language, and the fact that it is paradoxical is determined at the level of its metalogical analysis by means of metamovies. To do this, it is necessary to create a formalized language that contains statements A, the truth predicate G. The formula P1 (A) g A (the statement A is true if and only if A). This means: the statement A is true if and only if the statement A is true, that is, it fixes (reflects) the existence of the object referred to in the statement.

The statement of the Cretan Epimenides "All Cretans are liars" is also expressed in object language. According to metalogical analysis, Epimenides is also a liar, since he, as a Cretan, belongs to the class of inhabitants of the island of Crete. If Epimenides were not a Cretan, then the statement "All Cretans are liars" would not be paradoxical.

The heterological paradox was formulated by K. Grelling (1886 - 1941). This is a paradox that arises as a result of highlighting such expressions of speech as adjectives, the meaning of which is properties, for example, "red", "new", "old", "Ukrainian". A word that has the property P, whose name it is, is called autological. A word that is not autological is called heterological. If a word (adjective) denotes a property inherent in itself, then it is called autological. This is, for example, the word "Ukrainian", and the words "white", "black" are not autological words, therefore, they are heterological. To what kind of words - autological or heterological does the word "heterological" belong? We get an antinomy: "If the word "heterological" is heterological, then it is not heterological, and if it is not heterological, then it is heterological."

The paradox of the theory of names is a semantic paradox that arose within the framework of the theory of logical semantics, which was developed by G. Frege, B. Russell, G. Carnap and other logicians, replacing given name description and vice versa, a proper name description (see 2.2.4). A proper name is a simple sign that denotes a single (individual) object. Description - complex sign, which defines the properties of an object or relationships between classes. If in a certain context one replaces one's own name with a description, then a semantic paradox arises. For example, for By. Russell, the proper name "Walter Scott" and the description "author of Waverley" point to one subject, respectively, the statement. "King Henry IV wants to know whether Walter Scott is the author of Waverley" does not contain a paradox, but if the proper name is replaced " Walter Scott" description of "the author of Waverley", we get the statement: "King Henry IV wants to know whether Walter Scott is Walter Scott", which is paradoxical.

Logical paradoxes are paradoxes that have arisen within a certain logical theory in the process of developing the science of logic. Logical paradoxes include paradoxes of material implication, paradoxes of strict implication, paradoxes of epistemic logic, paradoxes of the logic of existence, etc. (the content of these paradoxes will be determined in the context of the analysis of a particular logical theory where these paradoxes arose).

The paradox of the theory of classes (sets). In the logico-mathematical theory of classes (sets), the English logician and mathematician B. Russell discovered a logical inconsistency, which was called the paradox (antinomy) of classes (sets). All sets can be divided into the following types: 1. Sets that are not elements of themselves. Such sets are called proper. For example, the set of all states, all natural numbers, all books in scientific library University of the city of N., etc. 2. Sets that are elements of themselves. Such sets are called improper. The first type of sets is denoted by the symbol M., and the second - by the symbol M2. Further, we assume that it is possible to form a set M of those and only those sets that are proper, that is, all those sets that do not contain themselves as elements. This is plural- - is contradictory, because, by definition, it belongs to the number of its elements if and only if it does not belong to their number.

To solve the paradox of set theory By. Russell developed a theory of types, the essence of which is this. All sets can be divided into types, each of which separates elements that belong only to one type and do not belong to another. This is how a hierarchy of set types is created: the null type contains only elements that have the property P, the first type contains elements that have the properties G.; the second type - has properties P2 and under. Each type means a certain level of abstraction and generalization of sets: a) an ordinary set; b) an unusual set (the set of all sets), i.e. a set that contains itself as an element. To which set do the set of all ordinary sets belong? According to B. Russell, the theory of types also makes it possible to single out the hierarchy of sets and thereby overcome the paradox of set theory.

Popular versions of the paradox of set theory are the paradoxes "Municipality Mayor" and "Hairdresser".

The "Municipality Mayor" paradox was formulated by the American logician S. Kleene (1909-1994) as a popular variant of the set theory paradox. "Every municipality in Holland must have a mayor, and two different municipalities cannot have the same mayor. Sometimes it turns out that the mayor does not live in his municipality. We assume that a law has been passed according to which a certain area is allocated only for such mayors who do not live in their municipalities, and he obliges all mayors to settle in this territory. Let's also assume that there are so many of these mayors that this territory of N. forms a municipality. Where should the mayor of the municipality Y. live?"

The Barber Paradox is the second popular variant of the set theory paradox. "The barber shaves those and only those men of one village who do not shave themselves. Or does the barber shave himself?"

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Jolls K. Introduction to modern logic. - K., 1992.

Ivin A. The art of thinking correctly. - M., 1986.

Ivin A. Logic. - K., 1996.

Kayberg G. Probability and inductive logic. - M., 1978. Kant I. Works: In 6 volumes - M., 1964. Konversky A. Logic (traditional and modern). - K., 2004. Kondakov N. Logical dictionary-reference book. - M., 1975. Leibniz G. Works: In 4 volumes - M., 1984. Logical dictionary "Defort". - M., 1994. Minto V. Deductive and inductive logic. - S.-Pb., 1995.

Frege G. Logic and logical semantics. - M., 2000. Khomenko I. Logic for lawyers. - K., 2001. Shuman A. Modern logic: Theory and practice. - M., 2004.

Kotarbinski T. Kurs logiki. - Warzawa, 1955.

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 task often takes more time than the task 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 implicit, non-questioning 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 approach to her.

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 long since decided, most often deserve research” (G. Lichtenberg). Paradox is the beginning of such research.

A paradox in a narrower and more specialized sense is two opposite, 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 in the most rigorous and exact sciences- mathematics and logic. And this is no coincidence.

Logic is an 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 the truth,” 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: “On the other side of this card is written false statement" - 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 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. It is also said 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 of the “Liar”, and soon died without achieving anything.

In the Middle Ages, this paradox was referred to the so-called undecidable 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.

AT everyday language there is no difference 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 much 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 the Russian language, while the second would refer 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 phenomenon in ordinary 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 A. Tarsky showed, classic definition truth must be formulated in a language wider than the language for which it is intended. In other words, if we want to indicate what the phrase “a statement true in given language”, it is necessary, in addition to the expressions of this language, to use also 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 statement 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, French philosopher 14th century J. Buridan did not agree with his decision. From the point of view of ordinary ideas about meaninglessness, expressions like "I'm lying", "Every statement is true (false)", etc. quite meaningful. What you can think about, what you can say - this 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 about itself that it is itself false, then it is only an abbreviated formulation of a more complex expression asserting 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 to 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 subject language, in which they talk about the world around them, 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 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 that outwardly appears as an example of a mixture of language and metalanguage, or, finally, as typical example 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 on 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 Wright, “have puzzled us since their discovery and will probably always puzzle us. 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 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 students were asked as a homework assignment—as a warm-up, so to speak—to find the mistake made in this simple-looking statement: "I'm 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. Threatened over the most simple and important logical methods, the most common and useful concepts.

It immediately became obvious that neither in logic nor in mathematics for the whole long history their existence, absolutely nothing was worked out that could serve as a basis for eliminating the 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 research antinomy, the conviction in the need for a fundamentally new approach grew steadily. 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. Any element of the first set will be individual person, the element of the second is 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 says 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. 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. However, 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 well-known reasoning about the directory.

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 refer 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 they denote are called aliases, or heterological. 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 it does not exist in Russian compound name, 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

At the heart of one famous paradox lies what seems to be a small incident that happened more than two thousand years ago and has not been forgotten to this day.

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 Protagoras and Euathlus Paradox

Perplexed by this turn of the matter, Protagoras devoted a special essay to this dispute with Euathlus, "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 correct resolution based on 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 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 Evatl, i.e. by virtue of an agreement with Protagoras.

This appeal to life finally 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 networks. 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 was over, 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 so 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 of the crocodile and the 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. 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 was thrown across this river, and right there on the edge stands a gallows and there is something like a court, in which four people usually sit. judges, 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: 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 unhindered, then this will mean that he has violated 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, because they are still perplexed and hesitant ...

Sancho proposed, 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 stalemate 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 her own conditions was useless - it is 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, blurred. 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.

Inaccurate are the empirical characteristics of "big", "heavy", "narrow", etc. 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 after all, 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 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 the heading "Metal Hospital and Household Utensils" 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 the affections of the soul, cannot 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 a confirmation of the latter must also be a 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 "The cow is white", "The shoes are brown", 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 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 great multitude the most common, harmless and at the same time convenient ways expressions. 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-applied 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 important question: what, 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 habitual methods 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 somehow full consent in the question of the origin of paradoxes and ways to get rid of them, no.

“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 the same thing. To remove a paradox from a certain theory means to restructure 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 certain logical means, used in the derivation of the paradoxical statement, should be linked to our general considerations regarding the nature 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-conceived and careless rejection of too many or too strong assumptions can simply lead to what turns out, although not containing paradoxes, but much more weak theory with only private 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" refers to a certain object, the word "red" indicates a property inherent in this subject, "to be a color" - on the property of this property ("to be red") and "to be optical phenomenon” - points to the property of the “be color” property, which belongs 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 consensus that any of the proposed theories allows logical paradoxes rather than just discarding them without deep explanation. The problem of explaining paradoxes is still open and still important.

The future of paradoxes

G. Frege, the greatest logician last century, was, unfortunately, 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 to it special application, 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: long time unbearable. Think about it: 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, even if mathematical thinking misfire?"

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 found their definite, albeit troubled, place in a wide range logical research. 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 one, nor even, perhaps, main task. Although they are important, they are only an occasion for reflection on central themes logic. Continuing with the comparison of paradoxes with particularly pronounced symptoms of illness, one might say 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.

7. A few paradoxes, or what looks like them

And to conclude this brief discussion of logical paradoxes, here are a few problems that the reader will find useful to ponder. It is necessary to decide whether the statements and reasoning given are really logical paradoxes or only seem to be. To do this, obviously, one should somehow restructure the source material and try to derive a contradiction from it: both the affirmation and the denial of the same thing about the same thing. If a paradox is found, you can think about what causes its occurrence and how to eliminate it. You can even try to come up with your own paradox of the same type, i.e. built according to the same scheme, but on the basis of other concepts.

1. The one who says: "I know nothing" makes a seemingly paradoxical, self-contradictory statement. He states, in essence, "I know that I know nothing." But the knowledge that there is no knowledge is still knowledge. This means that the speaker, on the one hand, assures that he does not have any knowledge, and on the other hand, by the very assertion of this he says that he does have some knowledge. What's the matter here?

Reflecting on this difficulty, it may be recalled that Socrates expressed a similar idea more carefully. He said: "I only know that I know nothing." But another ancient greek, Metrodorus, asserted with complete conviction: “I don’t know anything and I don’t even know that I don’t know anything.” Is there a paradox in this statement?

2. Historical events are unique. History, if it repeats itself, is, according to a well-known expression, the first time like a tragedy, and the second time like a farce. Out of originality historical events sometimes the idea is derived that history teaches nothing. "May be, greatest lesson History, - writes O. Huxley, - really consists in the fact that no one has ever learned anything from history.

It is unlikely that this idea is correct. The past is precisely what is studied mainly in order to better understand the present and the future. Another thing is that the "lessons" of the past, as a rule, are ambiguous.

Isn't the belief that history teaches nothing self-contradictory? After all, it itself follows from history as one of its lessons. Wouldn't it be better for the proponents of this idea to formulate it in such a way that it does not apply to themselves: "History teaches the only thing - nothing can be learned from it," or "History teaches nothing but this lesson of hers"?

3. "Proved that there is no evidence." This seems to be an internally contradictory statement: it is a proof, or it presupposes a proof already done (“it has been proved that…”) and at the same time asserts that there is no proof.

The well-known ancient skeptic Sextus Empiricus proposed the following solution: instead of the above statement, accept the statement “It has been proven that there is no proof other than this” (or: “It has been proven that there is nothing proven other than this”). But isn't this way out illusory? After all, it is asserted, in essence, that there is only one and only proof - the proof of the non-existence of any evidence ("There is one and only proof: the proof that there are no other proofs"). What then is the operation of the proof itself, if, judging by this assertion, it was possible to carry it out only once? In any case, Sextus' own opinion of the value of evidence was not very high. He wrote, in particular: “Just as those who do without proof are right, so are those who, being inclined to doubt, unfoundedly put forward the opposite opinion.”

4. "No statement is negative", or more simply: "There are no negative statements." However, this expression itself is a statement and is precisely negative. It seems like a paradox. What reformulation of this statement could avoid the paradox?

The medieval philosopher and logician Zh. The donkey, like any other animal, strives to choose the best of two things. The two armfuls are completely indistinguishable from each other, and therefore he cannot prefer either of them. However, this "buridan donkey" is not in the writings of Buridan himself. In logic, Buridan is well known, and in particular for his book on sophisms. It contains the following conclusion, relevant to our topic: no statement is negative; therefore, there is a negative proposition. Is this conclusion justified?

5. N.V. Gogol's description of Chichikov's game of checkers with Nozdrev is well known. Their game never ended, Chichikov noticed that Nozdryov was cheating and refused to play for fear of losing. Recently, a draughts specialist reconstructed from the remarks of those who played the course of this game and showed that Chichikov's position was not yet hopeless.

Let us assume that Chichikov nevertheless continued the game and eventually won the game, despite his partner's trickery. According to the agreement, the loser Nozdryov had to give Chichikov fifty rubles and "some middle-class puppy or a gold signet for a watch." But Nozdryov would most likely refuse to pay, pointing out that he himself cheated the whole game, and playing not by the rules is, as it were, not a game. Chichikov might have objected that talking about fraud is out of place here: the loser himself cheated, which means that he must pay all the more.

Indeed, would Nozdryov have to pay in such a situation or not? On the one hand, yes, because he lost. But on the other hand, no, since a game not according to the rules is not a game at all; There can be no winner or loser in such a “game”. If Chichikov himself had cheated, Nozdryov, of course, would not have been obliged to pay. But, however, it was the loser Nozdryov who cheated ...

Something paradoxical is felt here: “on the one hand ...”, “on the other hand ...”, and, moreover, on both sides it is equally convincing, although these sides are incompatible.

Should Nozdryov still pay or not?

6. "Every rule has exceptions." But this statement is itself a rule. Like all other rules, it must have exceptions. Such an exception would obviously be the rule "There are rules that have no exceptions." Isn't there a paradox in everything? Which of the previous examples resemble these two rules? Is it permissible to reason like this: every rule has exceptions; Does that mean there are rules without exceptions?

7. "Every generalization is wrong." It is clear that this statement sums up the experience of the mental operation of generalization and is itself a generalization. Like all other generalizations, it must be wrong. So, there must be true generalizations. However, is it correct to argue like this: every generalization is wrong, therefore, there are true generalizations?

8. A certain writer has composed an "Epitaph to All Genres" designed to prove that literary genres, the distinction between which caused so much controversy, are dead and can not be remembered.

But the epitaph, meanwhile, is also a genre in a certain way, the genre of tombstone inscriptions, which developed back in ancient times and entered literature as a kind of epigram:

Here I rest: Jimmy Hogg.
May God forgive me my sins,
What would I do if I were God
And he is the late Jimmy Hogg.

So the epitaph to all genres, without exception, sins as if with inconsistency. What is the best way to reformulate it?

9. "Never say never." Forbidding the use of the word "never", you have to use this word twice!

The same seems to be the case with the advice: "It's time for those who say 'it's time' to say something other than 'it's time'."

Is there a peculiar inconsistency in such advice, and can it be avoided?

10. In the poem "Do not believe", published, of course, in the section "Ironic Poetry", its author recommends not to believe in anything:

... Do not believe in the magical power of fire:
It burns while firewood is placed in it.
Do not believe in the golden-maned horse
Not for any sweet gingerbread!
Do not believe that star herds
Rushing in an endless whirlwind.
But what will be left for you then?
Don't believe what I said.
Don't believe.

(V. Prudovsky)

But is this general disbelief real? Apparently, it is contradictory and, therefore, logically impossible.

11. Suppose that, contrary to common belief, there are still uninteresting people. Let's collect them mentally together and choose from among them the smallest in height, or the largest in weight, or some other "most ...". This person would be interesting to look at, so we needlessly included him in the list of uninteresting. Having excluded it, we will again find among the remaining ones “the very…” in the same sense, and so on. And all this until there is only one person left with no one to compare with. But it turns out that this is exactly what he is interested in! As a result, we come to the conclusion that uninteresting people no. And the argument began with the fact that such people exist.

One can, in particular, try to find among the uninteresting people the most uninteresting of all the uninteresting. In this he will no doubt be interesting, and he will have to be excluded from uninteresting people. Among the rest, again, there is the least interesting, and so on.

There is definitely a touch of paradox in these arguments. Is there a mistake here, and if so, what is it?

12. Let's say that you were given a blank sheet of paper and instructed to describe this sheet on it. You write: this is a leaf rectangular shape, white, of such and such dimensions, made from pressed wood fibers, etc.

The description seems to be complete. But it is clearly incomplete! In the process of description, the object changed: text appeared on it. Therefore, it is also necessary to add to the description: and besides, on this sheet of paper it is written: this is a sheet of rectangular shape, white ... etc. to infinity.

It seems like a paradox here, doesn't it?

A well-known nursery rhyme:

The priest had a dog
He loved her
She ate a piece of meat
He killed her.
Killed and buried
And on the board he wrote:
"The priest had a dog..."

Could this dog-loving pop ever finish his tombstone? Does not the composition of this inscription remind Full description sheet of paper on it?

13. One author gives this "subtle" advice: "If small tricks do not allow you to achieve what you want, resort to big tricks." This advice is offered under the heading "Tricks of the trade". But is he really one of those tricks? After all, "little tricks" do not help, and just for this reason you have to resort to this advice.

14. We call a game normal if it ends in a finite number of moves. Examples of normal games are chess, checkers, dominoes: these games always end either in the victory of one of the parties, or in a draw. The game, which is not normal, continues indefinitely without any result. Let us also introduce the notion of a supergame: the first move of such a game is to determine which game should be played. If, for example, you and I intend to play a super game and I own the first move, I can say, "Let's play chess." Then you in response make the first move of the chess game, say, e2 - e4, and we continue the game until it ends (in particular, due to the expiration of the time allotted by the tournament regulations). As my first move, I can suggest playing tic-tac-toe and the like. But the game I choose must be normal; you can not choose a game that is not normal.

A problem arises: is the supergame itself normal or not? Let's assume that this is a normal game. Since it can choose any of the normal games as its first move, I can say, "Let's play the super game." After that, the super game has begun, and the next move in it is yours. You have the right to say: "Let's play a super game." I can repeat: "Let's play the super game" and thus the process can continue indefinitely. Therefore, the supergame does not apply to normal games. But due to the fact that the supergame is not normal, I cannot suggest a supergame with my first move in the supergame; I have to choose the normal game. But the choice of a normal game that has an end contradicts the proven fact that the supergame does not belong to the normal ones.

So, is the supergame a normal game or not?

In trying to answer this question, one should not, of course, follow the easy path of purely verbal distinctions. The easiest way to say that a normal game is a game, and a super game is just a draw.

What other paradoxes does this paradox of the supergame being both normal and abnormal at the same time remind of?

Literature

Bayif J.K. Logic tasks. - M., 1983.

Bourbaki N. Essays on the history of mathematics. - M., 1963.

Gardner M. Come on guess! – M.: 1984.

Ivin A.A. According to the laws of logic. - M., 1983.

Klini S.K. Mathematical logic. - M., 1973.

Smallian R.M. What is the name of this book? – M.: 1982.

Smallian R.M. Princess or tiger? – M.: 1985.

Frenkel A., Bar-Hillel I. Foundations of set theory. - M., 1966.

test questions

What is the significance of paradoxes for logic?

What solutions were proposed for the Liar paradox?

What are the features of a semantically closed language?

What is the essence of the paradox of many ordinary sets?

Is there a solution to the dispute between Protagoras and Euathlus? What solutions were proposed for this dispute?

What is the essence of the paradox of inexact names?

What could be the peculiarity of logical paradoxes?

What conclusions for logic follow from the existence of logical paradoxes?

What is the difference between eliminating and explaining a paradox? What is the future of logical paradoxes?

Topics of abstracts and reports

The concept of a logical paradox

The Liar Paradox

Russell's paradox

Paradox "Protagoras and Euathlus"

The role of paradoxes in the development of logic

Prospects for resolving paradoxes

Distinction between language and metalanguage

Elimination and resolution of paradoxes

October 1st, 2014

Scientists and thinkers have long been fond of entertaining themselves and their colleagues by setting unsolvable problems and formulating all sorts of paradoxes. Some of these thought experiments remain relevant for thousands of years, which indicates the imperfection of many popular scientific models and "holes" in generally accepted theories that have long been considered fundamental.

We invite you to reflect on the most interesting and amazing paradoxes, which, as they say now, "blew the brain" of more than one generation of logicians, philosophers and mathematicians.

1. Aporia "Achilles and the tortoise"

The paradox of Achilles and the tortoise is one of the aporias (logically correct, but contradictory statements) formulated by the ancient Greek philosopher Zeno of Elea in the 5th century BC. Its essence is as follows: the legendary hero Achilles decided to compete in running with a turtle. As you know, turtles do not differ in quickness, so Achilles gave the opponent a head start of 500 m. When the turtle overcomes this distance, the hero starts chasing at a speed 10 times greater, that is, while the turtle crawls 50 m, Achilles manages to run the given 500 m head start . Then the runner overcomes the next 50 m, but at this time the turtle crawls back another 5 m, it seems that Achilles is about to catch up with it, but the opponent is still ahead and while he is running 5 m, she manages to advance another half a meter and so on. The distance between them is infinitely reduced, but in theory, the hero never manages to catch up with the slow turtle, it is not much, but always ahead of him.

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Of course, from the point of view of physics, the paradox does not make sense - if Achilles moves much faster, he will break ahead anyway, however, Zeno, first of all, wanted to demonstrate with his reasoning that the idealized mathematical concepts of “point in space” and “moment of time” do not too suitable for correct application to real motion. The aporia reveals the discrepancy between the mathematically sound idea that non-zero intervals of space and time can be divided indefinitely (so the tortoise must always stay ahead) and the reality in which the hero, of course, wins the race.

2. Time loop paradox

The New Time Travelers by David Toomey

The paradoxes that describe time travel have long been a source of inspiration for science fiction writers and creators of science fiction films and TV shows. There are several variants of time loop paradoxes, one of the simplest and most illustrative examples of such a problem was given in his book The New Time Travelers by David Toomey, a professor at the University of Massachusetts.

Imagine that a time traveler has bought a copy of Shakespeare's Hamlet from a bookstore. Then he went to England during the time of the Virgin Queen Elizabeth I and, having found William Shakespeare, handed him a book. He rewrote it and published it as his own work. Hundreds of years pass, Hamlet is translated into dozens of languages, endlessly reprinted, and one of the copies ends up in the very bookstore where the time traveler buys it and gives it to Shakespeare, who makes a copy, and so on... Who should be counted in this case? the author of an immortal tragedy?

3. The paradox of a girl and a boy

Martin Gardner / © www.post-gazette.com

In probability theory, this paradox is also called "Mr. Smith's Children" or "Mrs. Smith's Problems." It was first formulated by the American mathematician Martin Gardner in one of the issues of Scientific American magazine. Scientists have been arguing over the paradox for decades, and there are several ways to resolve it. After thinking about the problem, you can offer your own version.

The family has two children and it is known for sure that one of them is a boy. What is the probability that the second child is also male? At first glance, the answer is quite obvious - 50 to 50, either he really is a boy or a girl, the chances should be equal. The problem is that for two-child families, there are four possible combinations of the sexes of children - two girls, two boys, an older boy and a younger girl, and vice versa - an older girl and a younger boy. The first can be excluded, since one of the children is definitely a boy, but in this case there are three possible options, not two, and the probability that the second child is also a boy is one chance in three.

4. Jourdain's card paradox

The problem proposed by the British logician and mathematician Philippe Jourdain at the beginning of the 20th century can be considered one of the varieties of the famous liar paradox.

Philippe Jourdain

Imagine - you are holding a postcard in your hands, which says: "Approval for reverse side postcards true. Flipping the card over reveals the phrase "The statement on the other side is false." As you understand, there is a contradiction: if the first statement is true, then the second is also true, but in this case the first must be false. If the first side of the postcard is false, then the phrase on the second also cannot be considered true, which means that the first statement becomes true again ... An even more interesting version of the liar's paradox is in the next paragraph.

5. Sophism "Crocodile"

A mother and a child are standing on the river bank, suddenly a crocodile swims up to them and drags the child into the water. The inconsolable mother asks to return her child, to which the crocodile replies that he agrees to give him back safe and sound if the woman correctly answers his question: “Will he return her child?” It is clear that a woman has two answers - yes or no. If she claims that the crocodile will give her the child, then it all depends on the animal - considering the answer to be true, the kidnapper will let the child go, but if he says that the mother was mistaken, then she will not see the child, according to all the rules of the contract.

© Corax of Syracuse

The woman's negative answer complicates things considerably - if it turns out to be true, the kidnapper must fulfill the terms of the deal and release the child, but in this way the mother's answer will not correspond to reality. To ensure the falsity of such an answer, the crocodile needs to return the child to the mother, but this is contrary to the contract, because her mistake should leave the child with the crocodile.

It is worth noting that the deal offered by the crocodile contains a logical contradiction, so his promise is unfulfillable. The author of this classic sophism is considered to be an orator, thinker and political figure Corax of Syracuse, who lived in the 5th century BC.

6. Aporia "Dichotomy"

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Another paradox from Zeno of Elea, demonstrating the incorrectness of the idealized mathematical model movement. The problem can be put like this - let's say you set out to go through some street in your city from beginning to end. To do this, you need to overcome its first half, then half of the remaining half, then half of the next segment, and so on. In other words - you walk half of the entire distance, then a quarter, one eighth, one sixteenth - the number of decreasing segments of the path tends to infinity, since any remaining part can be divided in two, which means it is impossible to go the whole way. Formulating a somewhat far-fetched paradox at first glance, Zeno wanted to show that mathematical laws contradict reality, because in fact you can easily cover the entire distance without a trace.

7. Aporia "Flying Arrow"

The famous paradox of Zeno of Elea affects the deepest contradictions in the ideas of scientists about the nature of motion and time. Aporia is formulated as follows: an arrow fired from a bow remains motionless, since at any moment in time it rests without moving. If at each moment of time the arrow is at rest, then it is always at rest and does not move at all, since there is no moment in time at which the arrow moves in space.

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The outstanding minds of mankind have been trying for centuries to resolve the paradox of a flying arrow, but from a logical point of view, it is absolutely correct. To refute it, it is necessary to explain how a finite time interval can consist of an infinite number of moments of time - even Aristotle, who convincingly criticized Zeno's aporia, failed to prove this. Aristotle rightly pointed out that a period of time cannot be considered the sum of some indivisible isolated moments, but many scientists believe that his approach does not differ in depth and does not refute the existence of a paradox. It is worth noting that by posing the problem of a flying arrow, Zeno did not seek to refute the possibility of movement, as such, but to reveal contradictions in idealistic mathematical concepts.

8. Galileo's paradox

Galileo Galilei / © Wikimedia

In his Conversations and Mathematical Proofs Concerning Two New Branches of Science, Galileo Galilei proposed a paradox that demonstrates the curious properties of infinite sets. The scientist formulated two contradictory friend of judgment. First, there are numbers that are the squares of other integers, such as 1, 9, 16, 25, 36, and so on. There are other numbers that do not have this property - 2, 3, 5, 6, 7, 8, 10 and the like. Thus, total There must be more exact squares and regular numbers than just perfect squares. Second proposition: for every natural number there is its exact square, and for every square there is an integer Square root, that is, the number of squares is equal to the number of natural numbers.

Based on this contradiction, Galileo concluded that reasoning about the number of elements is applied only to finite sets, although later mathematicians introduced the concept of the cardinality of a set - with its help, the correctness of Galileo's second judgment was also proved for infinite sets.

9. Potato sack paradox

© nieidealne-danie.blogspot.com

Suppose a certain farmer has a bag of potatoes weighing exactly 100 kg. After examining its contents, the farmer discovers that the bag was stored in dampness - 99% of its mass is water and 1% of the remaining substances contained in potatoes. He decides to dry the potatoes a little so that the water content in them drops to 98% and transfers the bag to a dry place. The next day, it turns out that one liter (1 kg) of water has really evaporated, but the weight of the bag has decreased from 100 to 50 kg, how can this be? Let's calculate - 99% of 100 kg is 99 kg, which means that the ratio of the mass of dry residue and the mass of water was originally 1/99. After drying, the water contains 98% of total weight bag, then the ratio of the mass of dry residue to the mass of water is now 1/49. Since the mass of the residue has not changed, the remaining water weighs 49 kg.

Of course, the attentive reader will immediately discover the rudest mathematical error in calculations - the imaginary comic “paradox of a sack of potatoes” can be considered an excellent example of how, with the help of seemingly “logical” and “scientifically supported” reasoning, you can literally build a theory that contradicts common sense from scratch.

10 Raven Paradox

Carl Gustav Hempel / © Wikimedia

The problem is also known as Hempel's paradox - it received its second name in honor of the German mathematician Carl Gustav Hempel, the author of its classical version. The problem is formulated quite simply: every raven is black. It follows from this that anything that is not black cannot be a raven. This law is called logical counterposition, that is, if a certain premise "A" has a consequence "B", then the negation of "B" is equivalent to the negation of "A". If a person sees a black raven, this reinforces his belief that all ravens are black, which is quite logical, however, in accordance with contraposition and the principle of induction, it is logical to argue that the observation of non-black objects (say, red apples) also proves that all crows are painted black. In other words, the fact that a person lives in St. Petersburg proves that he does not live in Moscow.

From the point of view of logic, the paradox looks impeccable, but it contradicts real life - red apples in no way can confirm the fact that all crows are black.

Here we already had a selection of paradoxes with you -, as well as in particular, and The original article is on the website InfoGlaz.rf Link to the article from which this copy is made -

Philosopher Stephen Reed on the liar paradox, semantic paradoxes and their direct connection to the foundations of mathematics.

It is worth starting a conversation about logical paradoxes with a little story that Cervantes tells in his book Don Quixote. At one point in Don Quixote, he leaves Sancho Panza as governor on the island of Barataria, and while he is governor, he is fooled by his "subjects". One morning he was woken up and told: "Before breakfast, you need to judge one thing." And in Spain at that time there were a lot of vagabonds, so you had to be very careful with people. And now, a river flows through the lands of one landowner, through which a bridge is thrown, and, in order to make sure that all passers-by are trustworthy, this landowner placed gallows and a guard near the bridge, who demands from each passerby to explain where and why he is going. If the passer-by tells the truth, he is allowed to cross the bridge, and if he lies, then the gallows awaits him. And everything was fine, it helped to distinguish who was a tramp and who was a merchant, until one day a man came who said: "My goal is to be hanged on this gallows, and nothing more." And the guard was amazed by this, because he thought: “Well, if we hang him, it will turn out that he told the truth, then we should have let him through, but if we let him through, it will turn out that he lied, then we should have him hang up." "So, Sancho Panza, how are we to judge this case?" And it takes Sancho Panza some time to appreciate the paradox, but in the end he makes his decision: hang the half of the person who lied and let the half that told the truth pass.

This all sounds like fun to the mind, but for people who want to get to the bottom of things like truth, reasoning, language, and so on, it points to something very unsettling about the nature of language. It seems very easy to fall into a paradox: we simply do not know whether the statement of that person was true or not, whether he lied or not. And this goes back to the original paradox of the liar, formulated by Eubulides in the 4th century BC. He elevated it to a work of art, he said, "Think of saying 'I'm lying'." If I say: “I am lying,” I, of course, can mean some other statement of mine, but if I use extremely careful wording, then I can say: “No, I am lying in the very phrase that I am saying now, this statement of mine is false. And again, if you think about it, you will say: “If this were true, then since he says that his statement is false, it follows that it must be false, and not true, that is, it cannot be true - it must be false. But if it's false, because it says it's false, that he lied, it must be true." So we end up with a paradox neatly wrapped up in one sentence.

There are a lot of such paradoxes, and it is easy to understand why they are called logical paradoxes: the contradiction contained in them is revealed with the help of logic. Some have heard of Epimenides: he was a native of Crete, and he was so disappointed in his countrymen's ability to tell the truth that he once said, "All Cretans are liars." If he was right, if indeed all Cretans were liars, or if other Cretans always lied, then his own statement must be paradoxical. After all, if he says: "All Cretans are liars," then he says that his own statement is false, but in this case, indeed, every single Cretan would be liars, which means that he was telling the truth when he said that all Cretans - liars. The way out of the paradox, of course, is that if some Cretans were telling the truth, then his statement would simply be false, not paradoxical.

So we have great amount such paradoxes. Here's one paradox that I particularly like: Take a card that says on one side, "The statement on the back of this card is true." You turn it over, and it says: "The statement on the back of this card is false." And if you think about it, it's just paradoxical, because if the statement on the first side is true, then the statement on the back side is also true, because the first statement says so; but on the second side it is written that the first statement is false, that is, if the first statement is true, it is also false. But this is impossible, so the second statement must be false; but it says that the first statement is false, then the first statement cannot be false - it must be true. But we have already seen that if the first statement is true, then it is false, so we get a pure paradox.

Some medieval thinkers preferred to describe this paradox in terms of Socrates and Plato, or sometimes Plato and Aristotle. So, Plato was Aristotle's teacher and considered him his best student, so one day he said, "Everything Aristotle says is true." But Aristotle was not the most exemplary student in the sense that he wanted to challenge Plato's teachings, so he said, "Everything Plato says is false," which is very similar to the card paradox.

All these were paradoxes in the field of truth, lies and language. But in the 20th century, we encountered paradoxes in mathematics. The brief history is this: after the advent of calculus, and then after working with infinite series in the 18th century, the foundations of mathematics became unstable, people wondered, “How do infinite series work without leading us to inconsistencies in mathematics?” And in the 19th century, a great movement unfolded, the purpose of which was to search for stable foundations of mathematics. Then the theory of sets became such a basis. A set is a collection of objects defined through some property: for example, there can be a set of all natural numbers, a set of even numbers, or even a set of rice puddings - you can take different sets. In mathematics, of course, only numerical sets are used.

And all this looked fine until the end of the 19th century. Frege, Dedekind, and many other thinkers established mathematics, or what seemed to be the solid foundation of set theory. But then Bertrand Russell, the famous British philosopher, reading the works of Frege, thought: “You can set a lot of numbers, you can give a lot of sets; you can specify a set of sets that include themselves, or you can specify a set of sets that do not include themselves. And then he thought, “Wait a minute, if we have a set of sets that do not include themselves, will this set include itself or not?” If such a set were to include itself, then it must not include itself, because by convention we take only those sets that do not include themselves. So it would be better if this set did not include itself, but if it does not include itself, then it is a set that does not include itself, and it must be part of this set. And, as I said, all these paradoxes at first look like entertainment for the mind, but now, at the beginning of the 20th century, we have found a paradox, a contradiction at the very heart of what should be the foundations of mathematics. As is widely known, it was big blow for Frege: he was about to publish the second volume of his Fundamental Laws of Arithmetic, and he had to add an appendix in which he wrote: "Bertrand Russell pointed out a weakness at the very heart of my theory, but I think I can solve this problem”, and he suggested a solution, but as it turned out, it was not correct.

I will turn briefly to paradoxes in set theory, because there is another rather interesting paradox that brings us back to the discussion of paradoxes related to truth, or the so-called semantic paradoxes. So, about 40 years later, around 1940, the American mathematician and logician Haskell B. Curry pondered Russell's paradox and said, "Russell's paradox is based on negation - it talks about a multitude of sets that do not include themselves." Is it possible to get the same paradox without using negation? Is there a way? And he said there is a way. Take the set of all sets; if they include themselves, then zero equals one. According to set theory, this is a completely admissible set. But if we start considering such a set, if it includes itself, then it will satisfy the condition that if it includes itself, then zero is equal to one.

And we assumed that it includes itself, therefore, zero is really equal to one. But it's quite obvious that zero can't be equal to one, so we play everything backwards and assume that a set can't include itself. If it does not include itself, it immediately follows that either it does not include itself or zero equals one. But that's the same as saying that if it includes itself, zero really equals one - that's the same as saying either it doesn't include itself, or zero equals one. And this is like saying that if a set includes itself, then it is not non-inclusive, then zero is equal to one. But then it includes itself, that is, we have proved that it includes itself, but since we have proved it, therefore, zero is equal to one. Save! We just proved that zero equals one! So right in the heart of mathematics, we have a real nightmarish paradox again.

And a few years later, this paradox was turned into one of the semantic paradoxes that I spoke about earlier, and it took the form of the statement: "If this statement is true, then zero is equal to one." Or even: "If this statement is true, then God exists." And then in just a few lines we can prove that God exists or anything else: zero is equal to one, God exists, it's raining in Moscow today - we can prove anything with such a statement. People think a lot about the truth, so it's very dangerous: is the truth really like that? Is truth really a controversial concept?

And I'll end by briefly talking about another paradox to show that paradoxes don't stop there. Here is the statement: "You do not know this statement" - you do not know the very statement that I am now pronouncing. Now suppose you know him. The concepts of knowledge and truth tell us that you can only know what is true, so that if you know it, it is true, in which case you do not know it because it says so. So if we assume that you know him, then it turns out that you do not know him. So we've proven that you don't know him, but it says you don't know him, so we've proven him. And of course, if we have proved something, then it is true, then we know it, because we have proof. And it turns out that we have proved both that you know this statement and that you do not know it, so we again have an epistemic paradox.

Let's summarize. I have described several semantic paradoxes, mainly related to the concept of truth, and also shown that they are very similar to the paradoxes associated with set theory, which lie at the very heart of mathematics. In addition, we got acquainted with the epistemic paradoxes that are associated not only with the concept of truth, but also with the concept of knowledge. So, we have analyzed several semantic paradoxes, such as the liar paradox, the Epimenides paradox and the card paradox, which are based on the concept of truth (in them we are talking about lies, untruth, truth, and so on), and then we have analyzed several paradoxes that arise in mathematics, they are associated with set theory. And in the end, we also talked about another type of paradox - epistemic paradoxes.

You can immediately see how important it is for us to find a solution to these paradoxes, since mathematics is involved in them, because we were looking for solid foundations of mathematics to make sure that we do not make mistakes - and now we have found a contradiction in them. So we really need a solution when it comes to mathematical paradoxes related to set theory, but we also need one for semantic paradoxes. A lot of philosophers are thinking about the concept of truth, and they want to understand the nature of truth, what is a true statement. It is natural to suppose that a proposition is true if everything is as it says; now look at the paradox of the liar: it's true if I'm lying - it's paradoxical and leads to a contradiction. So we need to rethink the concept of truth, some want to rethink the logic behind it and the methods of proof that led us to the contradiction. And it is very important that we do this if we are to gain a full understanding of the concepts of truth and knowledge.

gif: postnauka.ru/ Stephen Reid

According to the laws of logic Ivin Alexander Arkhipovich

WHAT IS THE 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 and even completely new types will be discovered in the future. 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 the Soviet mathematician D. Bochvar. - Such discrepancies are sometimes significant, sometimes verbal. The point is largely in what exactly is meant by "logical paradox".

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 extralogical ones. 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, a person as the most cunning in a class, we do this with the help of a set of people to which this person also belongs (with the help of "his class"). 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, self-applicability takes place, which means that there is, as it were, a 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 also 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 F. Ramsey, 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 F. 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 F. Ramsey was widely used at first and retains some significance 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 F. Ramsey is increasingly blurred.

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