The most sustained forms of logical paradoxes in which. Logic deadlocks (paradoxes)

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What is a paradox? The paradox is called two incompatible and opposite statements that have convincing arguments each in their direction. The most pronounced form of paradox is the antinomy - reasoning, which proves the equality of statements, one of which is a clear denial of the other. And the paradoxes in the most accurate and strict sciences, such as, for example, logic deserve special attention.

Logic, as you know, is abstract science. It has no place to experiments and any specific facts in their usual understanding; She always assumes an analysis of real thinking. But discrepancies in the theory of logic and practices of real thinking still have a place to be. And the most apparent confirmation of this is logical paradoxes, and sometimes even logic antinomy, personifying the inconsistency of the most logical theory. This is exactly what explains the value of logical paradoxes and the attention that is paid to these paradoxes in logical science. Below we will introduce you to you with the most vivid examples of logical paradoxes. This information will certainly be interesting to those who are in-depth studying the logic and those who just love to recognize new and interesting information.

Let's start with paradoxes compiled by the ancient Greek philosopher Zeno Elaisian who lived in the V century BC. His paradoxes were called "ARCRY OF SENON" and even have their interpretation.

Aprira Zenona

The Cornon Apriology is externally paradoxical reasoning about the movement and multiple. In total, Zenon's contemporaries were mentioned over 40 aquities (by the way, the word "aporia" from the ancient Greek language is translated as "difficulty") of his authorship, but only nine of them reached our time. If you wish, you can familiarize yourself with them in the works of Aristotle, Diogen Lanertsky, Plato, Femistry, Philopona, Elia and Sipmlikia. We will give an example three most famous.

Achilles and Turtle

Imagine that Achilles runs at a rate of ten times greater than the speed of the turtle, and is from her at a distance of a thousand steps behind. While Achilles will run a thousand steps, the turtle will make only a hundred. While Achilles overcomes more than a hundred, the turtle will have time to do ten, etc. And this process will continue indefinitely and Achilles will never catch up to the turtle.

Dichotomy

In order to overcome a certain way, it is necessary to initially overcome it half, and to overcome half, you need to overcome half of this half, etc. Based on this, the movement never starts.

Flying arrows

The flying boom always remains in place, because At any time it is at rest, and since it is at rest at any time, it is always at rest.

It will also be appropriate to bring another paradox.

Paradox Liaza

The authorship of this paradox is attributed to the ancient Greek priest and Providant Epimera. The paradox sounds like this: "The fact that I am currently saying is false, i.e. It turns out: either "I am LSU", or "My statement is false." This means that if the statement is truthful, then based on its content, it is a lie, but if this statement is initially false, then it is a statement - a lie. It turns out, falsely, that this statement is a lie. Consequently, the statement truthfully - this conclusion returns us to the beginning of our reasoning.

Nowadays, the paradox of the liar is considered as one of the wording of the Russell Paradox.

Paradox Russell

The Russell Paradox was opened in 1901 by the British philosopher Bertrand Russell, and later it was independently rebounded by the German mathematician Ernst Chermelo (sometimes this paradox is called "Paradox Russell-Cermer"). This paradox demonstrates the contradiction of the logical system of Freg, in which mathematics comes down to logic. The paradox of Russell has several wording:

The paradox of omnipotence - is able to create anything almighty creature, which can limit its omnipotence?
Suppose some kind of library has set the task to make one big bibliographic catalog in which all the bibliographic directories should include, which do not contain references to themselves. Question: Do I need to include a link to it in this directory?
For example, in some country a law was released that the mayors of all cities are forbidden to live in their city, and it is allowed to live only in the "City of Mayors". Where, in this case, will the mayor of this city?
The paradox of Brandobray - in the village only one brainy, and he ordered to shave everyone who does not shave himself, and not shave those who shave himself. Question: Who should shave brand?

The following paradoxes are equally interesting.

Paradox Bural-Forti

The assumption that the idea of \u200b\u200bthe possibility of a set of ordinal numbers may lead to contradictions, which means that the theory of sets in which the set of ordinal numbers can be constructed with contradictory.

Paradox Cantor

The assumption of the possibility of a multitude of all sets can lead to contradictions, which means that the theory is contradictory, according to which it is possible to build such a set.

Paradox Hilbert.

The idea that if all rooms in the hotel with an endless number of rooms are occupied, in it in any case, you can settle more people, and their number can be infinite. In this paradox, it is explained that the laws of logic are absolutely unacceptable to the properties of infinity.

False conclusion Monte Carlo

The conclusion that, playing in roulette, you can safely put on a red color if the black fell ten times in a row. This conclusion is considered false for the reason that, according to the theory of probability, the event has no effect on the occurrence of any subsequent event, it is preceding.

Paradox Einstein-Podolsky-Rosen

The question of whether developing far from each other processes and events have an impact on each other? For example, does the birth in the remote galaxy of the supernova in the weather in Moscow affect the remote galaxy? As an answer, the following can be given: based on the laws of quantum mechanics, such an effect is impossible due to the fact that both the speed of light and the speed of information transfer are the end values, and the universe is infinite.

Paradox twins

Question: Will a twin traveler who returned from the space wander on an exhumular star pilot younger than his brother left all this time on Earth? If we proceed from the theory of relativity, then on Earth (for earthly flow) passed more time than in a starship flying with superluminal speed, which means that the twin traveler will be younger.

Paradox of killed grandfather

Imagine that you were in the past and killed your grandfather before his acquaintance with your grandmother. It follows that you do not appear on the light and you cannot return to the past to kill my grandfather. The presented paradox clearly demonstrates the impossibility of traveling in the past.

Paradox predestination

For example, a person turns out to be in the past, has sex with his great-grandmother and conceives her son, i.e. His grandfather. It becomes the cause of a turn of descendants, including the parents of this person, as well as himself. It turns out that if this person did not take a trip to the past, he would never have emerged at all.

These are just a few logical paradoxes who occupy the minds of many people today. An inquisitive mind will not be difficult to find not yet one dozen of the like (for example,). The study, refutation or proof of each of them can be devoted to a considerable amount of time and strength. And, quite likely, about each paradox you can form your personal original conclusions. But this also tells us that, despite the predominance of laws of logic and causal relationships in our lives, not everything in our lives depends on them. Sometimes, similar to the logical paradoxes of contradictions arise in everyday life of each person. In any case, it is a wonderful food for mind and reason for reflection.

By the way, concern to reflections: there is a very interesting book called "Gödel, Eshort and Bach" on the topic of logical paradoxes. Its author is American physicist and informress Douglas Hofstadter.

Dear readers would be wonderful if in their comments you led several examples of logical paradox to you. And we will also be interested and your opinion on the importance of logic in our life is to vote for one of the statements below.

It is known to formulate the problem often more important and more difficult than solving it. "In science," wrote the English chemist F. Soddy, the task, properly supplied, more than half solved. The process of mental preparation required to find out that there is a certain task, often takes more time than the solution itself. "
The forms in which the problem situation is manifested is very diverse. It does not always detect himself in the form of a direct question that has stuck at the very beginning of the study. The world of problems is as complicated as the breeding process of knowledge. Identification of problems is associated with the very essence of creative, thinking. Paradoxes are the most interesting case of implicit, irrepressive ways to make problems. Paradoxes are common in the early stages of the development of scientific theories when the first steps are made in an even unexplored area and the most general principles of approach to it are fastened.

Paradoxes and logic

In the broad sense, the paradox is a position dramatically diverging with generally accepted, established, orthodox opinions. "Recognized opinions and what are considered to be a long-solved, most often deserve research" (Glychtenberg). Paradox - the beginning of such a study.
Paradox in a narrower and special meaning is two opposing, incompatible assertions, for each of which there are seemingly convincing arguments.
The most sharp shape of the paradox - antinomy, reasoning, proving the equivalence of two statements, one of which is the denial of the other.
Paradoxes in the most stringent and accurate sciences - mathematics and logic are especially fame. And it is not by chance.

Logics - Abstract science. There are no experiments in it, there are no facts in the usual sense of the word. Building its systems, logic is ultimately from the analysis of real thinking. But the results of this analysis are synthetic, unexplored. They are not statements of any individual processes or events that the theory should explain. Such analysis cannot be obviously called observation: there is always a concrete phenomenon.
Constructing a new theory, the scientist is usually sent from the facts, from what can be observed in the experience. No matter how free his creative fantasy is, it should be considered with one indispensable circumstance: the theory makes sense only when it is consistent with the facts relating to it. The theory, divergent with facts and observations, has no controversial and value.
But if there are no experiments in logic, there are no facts and there is no observation, which is contained by a logical fantasy? What if not facts, then factors are taken into account when creating new logical theories?
The discrepancy between the logical theory with the practice of valid thinking is often found in the form of a more or less acute logical paradox, and sometimes even in the form of a logical anti-rope speaking on the internal contradictions of the theory. This is just explained by the value that is attached to paradoxes in logic, and then much attention they enjoy in it.

Paradox options "Liaza"

The most famous and, perhaps, the most interesting of all logical paradoxes is a paradox "Liar". He mainly and glorified the name of the Ebbulid that opened him from the Mileta.
There are variants of this paradox, or antinomy, many of which are only visible paradoxical.
In the simplest version of the "liar", a person pronounces only one phrase: "I LDU". Or says: "The statement I am now pronounced is false." Or: "This statement is false."

If the statement is false, then the speaker said the truth, and it means that they did not lie. If the statement is not false, and the speaker claims that it is false, then this is his statement false. It turns out that in such a way that if the talking lies, he tells the truth, and vice versa.

In the Middle Ages, this wording was common:

"Plato said - false," Socrates says.

"What Socrates said is truth," says Plato.

The question arises, which of them expresses the truth, and who is a lie?
But the modern paradox is rephrased. Suppose that only words are written on the front side of the card: "True statement is written on the other side of this card." It is clear that these words are meaningful assertion. Turning the card, we must either detect the promised statement, or it is not. If it is written on the turn, it is either true or not. However, there are words on the turnover: "A false statement is written on the other side of this card" - and nothing more. Suppose that the affirmation on the front side is true. Then the assertion on the turn should be true and, it means that the approval on the front side should be false. But if the approval on the front side is false, then the assertion on the turnover must also be false, and, therefore, the approval on the front side should be true. As a result - paradox.
Paradox "Liar" made a huge impression on the Greeks. And it's easy to understand why. The question that is put in it, at first glance, it seems quite simple: is he who says only what he is lying? But the answer is "yes" leads to the answer "no", and vice versa. And thinking does not clarify the situation. For simplicity and even the usual issue, it opens some kind of unclear and immeasurable depth.
Even the legend walks that a certain Cranosie, desperate to resolve this paradox, committed suicide. It is also said that one of the well-known ancient Greek logists, a diodor of the Kronos, already on the slope of the years he made a vow not to take food until he finds the "liar" decision, and soon died, so having achieved anything.
In the Middle Ages, this paradox was attributed to the so-called unresolved proposals and became the object of systematic analysis. In the new time, "liar" did not attract any attention for a long time. It has not seen any, even insignificant difficulties relating to language consumption. And only in our, the so-called the latest time, the development of logic has finally reached the level when the problems that seem to be at this paradox, it became possible to formulate already in strict terms.
Now "liar" - this typical former sophis - is often referred to as the king of logical paradoxes. He is devoted to extensive scientific literature. And yet, as in the case of many other paradoxes, it remains not quite clear which problems are hiding behind it and how to get rid of it.

Language and meta-language

Now the "liar" is usually considered a characteristic example of those difficulties to which two languages \u200b\u200bare confused: a language on which the actual language is said, and the language that speaks about the very first language.

In everyday language there is no difference between these levels: and about reality, and we speak the same language about the language. For example, a person whose native language is Russian, does not see any special difference between the allegations: "Glass transparent" and "It's right that the glass is transparent," although one of them speaks about glass, and the other thing about the statement relative to the glass.
If someone had a thought about the need to talk about the world in the same language, and about the properties of this language - on the other, it could use two different existing languages, let us say Russian and English. Instead of just to say: "Cow is a noun," IS A Noun cow would say, and instead of: "Approval", the glass would not be transparently "falsely" the "The Assertion" glass is not transparently "IS FALSE". With this use of two different languages, the above-mentioned world would be clear from what has been said about the language with which they talk about the world. In fact, the first statements would relate to the Russian language, while the second - to English.

If further than our connoisseur of languages \u200b\u200bwanted to speak out about some circumstances concerning English already, he could take advantage of another language. Suppose German. To talk about this, the last one could be resorted, put, to the Spanish language, etc.
It turns out, therefore, a kind of ladder, or hierarchy, languages, each of which is used for a completely definite purpose: on the first they talk about the subject world, on the second - about this first language, on the third - about the second language, etc. Such a distinction of languages \u200b\u200bon their application is a rare phenomenon in ordinary life. But in the sciences, specially involved, like logic, languages, it sometimes turns out to be very useful. The language on which they argue about the world is usually called the subject. The language used to describe the subject language is called the metalanas.

It is clear that if the language and the meta language are delimited in the manner, the statement "LSU" cannot be formulated. It speaks of the falsity of what is said in Russian, and, it means, it belongs to the metalanak and should be expressed in English. Specifically, it should sound like this: "Everything I Speak in Russian IS False" ("All I said in Russian falsely"); In this English statement, nothing is said about himself itself, and no paradox arises.
The distinguishing of the language and the meta language allows you to eliminate the paradox "liar". Thus, it becomes possible correctly, without contradiction, it is determined to determine the classical concept of truth: the statement corresponding to the reality described by it is true.
The concept of truth, as well as all other semantic concepts, has a relative nature: it can always be attributed to a certain language.

As the Polish Logic of Artari showed, the classical definition of truth should be formulated in the language wider than the language for which it is intended. In other words, if we want to indicate what the turnover "saying, true in this language," is needed, in addition to the expressions of this language, we also use the expressions that are not in it.
Tarsky introduced the concept of a semantically closed language. Such a language includes, in addition to its expressions, their names, and also, it is important to emphasize, the statements about the truth of the proposals formulated in it.

The boundaries between the tongue and the metalanas in the semantically closed language do not exist. Its funds are so rich, which allows not only something to argue about non-speaking reality, but also to evaluate the truth of such statements. These funds are sufficient, in particular, in order to reproduce in the ninomial language "Liar". Semantically closed language turns out to be internally contradictory. Each natural language is obviously semantically closed.
The only acceptable way to eliminate the antinomy, and hence the internal contradictions, according to Tar, is the refusal to use a semantically closed language. This path is acceptable, of course, only in the case of artificial, formalized languages \u200b\u200bthat allow a clear division into tongue and metalas. In the natural languages \u200b\u200bwith their unclear structure and the ability to talk about everything in the same language, this approach is not very real. It makes sense to raise the question of the internal consistency of these languages. Their rich expressive opportunities have their own opposite direction - paradoxes.

Other decisions of Paradox

The situation is similar and with the statement of "Any statement truly". It should also be referred to meaningless and also leads to a contradiction: if each statement is truly true, then the denying of this statement itself is true, that is, the statement that not any statement is truly.
Why, however, the statement can not mean clearly about their own truth or falsity?
Already the contemporary of Okkama, French philosopher XIV century. J. Buridan, did not agree with his decision. From the point of view of ordinary ideas about meaninglessness, the expression of the type "I LSU", "Any statement is true (false)", etc. It is quite meaningful. What you can think about whether you can speak about whether the general principle of Buridan. A person may think about the truth of the allegation he says, it means that he can speak about it. Not all statements speaking about themselves belong to meaningless. For example, the approval "This proposal is written in Russian" is true, and the approval "In this sentence, ten words" falsely. And both are completely meaningful. If it is assumed that the statement can talk about itself, then why it is not capable of talking about this property, as a truth?
The Buridan himself considered the statement of "I LSU" is not meaningless, but false. He justified it.

When a person claims some proposal, he claims that it is true. If the proposal suggests itself that it is false itself, it represents only the reduced formulation of a more complex expression that approves simultaneously and its truth, and its falsity. This expression is contradictory and, therefore, false. But it is not meaningless.

The argument of the Buridan and is sometimes considered convincing.
There are other directions of criticism of the decision of the paradox of "Liar", which was developed in the details of Tarsky. Is it really in semantically closed languages \u200b\u200b- and these are all natural languages \u200b\u200b- there is no antidote against the paradoxes of this type?
If it were so, the concept of truth could be determined strictly only in formalized languages. Only they manage to distinguish between the subject language, on which they argue about the world around, and the meta language, which is talking about this language. This hierarchy of languages \u200b\u200bis based on a sample of a foreign language assimilation with the help of a native. The study of such a hierarchy led many interesting conclusions, and in certain cases it is essential. But it is not in the natural language. Does it discredit it? And if so, what exactly is? After all, in it, the concept of truth is still used, and usually without any complications. Is the Introduction of the hierarchy by the only way to exclude paradoxes like "liar?"

In the 30s, the answers to these issues were undoubtedly affirmative. However, now the former unanimity is no longer there, although the tradition to eliminate the paradoxes of this type by "separating the" language remains dominant.
Recently, egocentric expressions are attracted more and more attention. They meet words like "I", "this", "here", "now", and their truth depends on when, by whom they are used.

In the statement "This statement is false" meets the word "it". What exactly does it apply to? "Liar" may say that the word "it" does not apply to the meaning of this approval. But then what does it belong to what is it? And why does this meaning can not be labeled in the word "this"?
Without going into details here, it is worth noting that in the context of the analysis of the egocentric expressions of "Liar" is filled with completely different content than before. It turns out that it no longer warns against the mixing of the language and the metalanka, but indicates dangers associated with the wrong use of the word "this" and similar to it of egocentric words.
Problems that bind over the centuries with "liar" radically changed depending on whether it was considered as an example of ambiguity, or as an expression, externally appeared as a sample of a language and metalanak, or, finally, as a typical example of incorrect egocentric use expressions. And there is no confidence that with this paradox will not be associated in the future and other problems.

Famous modern Finnish logic and philosopher. Von Virigt wrote in his work dedicated to the "liar" that this paradox should not be understood as a local one, an isolated obstacle, eliminated by one inventive movement of thought. "Liar" affects many of the most important topics of logic and semantics. This is the definition of truth, and the interpretation of contradictions and evidence, and a whole series of important differences: between the proposal and the thought expressed to them, between the use of the expression and its mention, between the meaning of the name and the object denoted them.
It is similar to other logical paradoxes. "Antinomy of logic," Vrigg writes von Vygg, "they puzzled from the moment of their discovery and, likely, they will always pose. We must, I think, consider them not as much as problems waiting for solutions, how much as an inexhaustible raw material for reflection. They are important because the reflection on them affects the most fundamental questions of the whole logic, and therefore, and all thinking. "

In conclusion of this conversation about "Liaz", a curious episode can be remembered from the time when the formal logic was still taught at school. In the textbook of the logic, published in the late 40s, the schoolchildren of the eighth class were offered as a homework - in order, so to speak, warm-ups - find a mistake made in this simply on the type of approval: "I LDU". And, let it not seem strange, it was believed that the schoolchildren mostly successfully coped with such a task.

§ 2. Russell Paradox

The most famous from the open already in our age of paradoxes is the antinomy discovered by B. Russell and reported by him in a letter to the city of Ferge. The same antinomy was discussed simultaneously in Göttingen German mathematicians 3. Cermelo and D. Hilbert.
The idea was worn in the air, and her publication impressed the bomb. This paradox caused in mathematics, according to Hilbert, the effect of a complete catastrophe. The threat over the most simple and important logical methods, the most ordinary and useful concepts.
Immediately it became apparent that neither in logic, nor in mathematics for the entire long history of their existence it was decidedly nothing that could serve as the basis for. elimination of antinomy. It was clearly necessary to waste from the usual ways of thinking. But what place and in what direction? How radical was to be a refusal of established methods of theorization?
With a further study of the antinomy, the conviction of the need for a fundamentally new approach has grown steadily. After half a century after its opening, experts on the grounds of logic and mathematician L. Frenkel and I. Bar-Hillel, without any reservations, claimed: "We believe that any attempts to get out of the situation with the help of traditional (that is, those who have been bought before the 20th century) of thinking methods , I still have invariably failed, knowingly insufficient for this purpose. "
Modern American Logic X. Carry wrote a little later about this paradox: "In terms of logic, known in the XIX century, the situation simply did not give up an explanation, although, of course, in our educated age, people who will see (or think that they will see ) What is the error ".

Many ordinary sets

Regarding any arbitrarily taken set, it seems to be meaningful to ask, it is its own element or not. Sets that do not contain themselves as an element, we call ordinary. For example, the set of all people is not a person, as well as many atoms - this is not an atom. There will be unusual sets that are their own elements. For example, a set that combines all sets is a set and, it means it means itself as an element.
Consider now the set of all ordinary sets. Since it is a lot, it can also be asked about it, usual or unusual. The answer, however, turns out to be discouraged. If it is usual, then, according to its definition, must contain itself as an element, since it contains all conventional sets. But this means that it is an unusual set. The assumption that our set is the usual set, thus leads to contradiction. So it cannot be ordinary. On the other hand, it cannot also be unusual: an unusual set contains itself as an element, and only ordinary sets are elements of our set. As a result, we come to the conclusion that the set of all ordinary sets cannot be neither ordinary nor an unusual set.

So, the set of all sets that are not their own elements, there is an element in that and only when it is not such an element. This is an explicit contradiction. And it was obtained on the basis of the most believable assumptions and with the help of indisputable as if steps. The opponation says that such a set simply does not exist. But why can it not exist? After all, it consists of objects that satisfy a clearly defined condition, and the condition itself does not seem to be some exclusive or unclear. If such a simple and clearly specified set cannot exist, then what is actually the difference between possible and impossible sets? The conclusion about the non-existence of the set questioned sounds unexpectedly and inspires anxiety. He makes our overall concept of many amorphous and chaotic, and there is no guarantee that it is not able to generate some new paradoxes.

Paradox Russell is wonderful with its extreme community. For its construction, no complex technical concepts are needed, as in the case of some other paradoxes, there are enough concepts "set" and "element of the set". But this simplicity is just talking about his fundamentality: it affects the deepest foundations of our arguments about the sets, because it says not about any special cases, but about sets at all.

Other options Paradox

Russell paradox does not have a specifically mathematical nature. It uses the concept of multiple, but are not affected by some special, related to the mathematics of its properties.
It becomes obvious if we reformulate the paradox in purely logical terms.

Every property can, in all likelihood, ask, is applied to yourself or not.
The property of being hot, for example, is uncomfortable to yourself, because it is not hot; The property to be concrete also does not apply to yourself, because it is an abstract property. But here is an abstract property, being abstract, apparently to myself. Let's call these integrated properties to themselves uncomfortable. Is the property applies to be unpaved to itself? It turns out that uncompromising is unpaved only if it is not so. This is, of course, paradoxically.
Logical concerning properties The species of the anti-russell antinomy, as paradoxical, as well as mathematical, related to sets, its variety.
Russell also offered the next popular version of the paradox's open paradox.

Imagine that the Council of one village so determined the duties of the hairdresser: shave all the villages of the villages who do not shave themselves, and only these men. Should he shave himself? If so, he will relate to those who shave himself, and those who shave himself should not shave. If not, it will belong to those who do not shake himself, and, it means he will have to shave himself. We come, so to the conclusion that this hairdresser shaves itself in that and only the case when he does not shave himself. This, of course, is impossible. The reasoning about the hairdresser relies on the assumption that such a hairdresser exists. The resulting contradiction means that this assumption is false, and there is no such resident of the village, which would vomit all those and only those inhabitants who do not shave themselves.
The duties of the hairdresser do not seem to be contradictory at first glance, so the conclusion that it cannot be, he sounds somewhat unexpected. But this conclusion is not yet paradoxical. The condition to which the village of Bradobremy should satisfy is actually internally contradictory and, therefore, impracticable. There may be no such hairdresser in the village for the same reason why there is no person in it, who would be older himself or who would be born before his birth.
The reasoning about the hairdresser can be called pseudoparads. In his go, it is strictly similar to the paradox of Russell and this is interesting. But it is still not a genuine paradox.

Another example of the same pseudoparadox is a well-known argument about the directory.
A certain library decided to compile a bibliographic catalog, which would include all those and only bibliographic directories that do not contain references to themselves. Should such a catalog include a link to yourself?
It is easy to show that the idea of \u200b\u200bcreating such a catalog is impracticable; It simply cannot exist, because it must simultaneously and include a link to itself and not include.
It is interesting to note that the compilation of the catalog of all directories that do not contain references to themselves can be represented as an endless, never ending process. Suppose that at some point a catalog was drawn up, say K1, including all directories from it, not containing links to ourselves. With the creation of K1, another directory appeared that does not contain references to itself. Since the task is to draw up a full catalog of all directories that do not mention yourself, it is obvious that K1 is not its solution. He does not mention one of these directories - himself. Including in K1 this mention of himself, we obtain the Catalog K2. It is mentioned K1, but not K2 himself. By adding such a mention to K2, we obtain the KZ, which is again not full due to the fact that it does not mention himself. And then without end.

§ 3. Grellling and Berry Paradoxes

An interesting logical paradox was opened by German logic K. Grelling and L. Nelson (Grellling paradox). This paradox can be formulated very simple.

Outlood and heterological words

Some words denoting properties have the same property that they call. For example, the adjective "Russian" itself is Russian, "multi-line" - the very complex, and the "five hundred" itself has five syllables. Such words relating to themselves are called identical or autologous.
There are not so many such words, in the overwhelming majority of adjectives do not possess the properties they call. "New" is not, of course, new, "hot" - hot, "single-threshold" - consisting of one syllable, and "English" is English. Words that do not have the properties denoted by them are called an intertwined, or heterologists. Obviously, all adjectives denoting properties, unpaid to words, will be heterological.
This separation of adjectives into two groups seems clear and does not cause objections. It can be distributed and nouns: "The Word" is a word, "noun" - nouns, but "clock" is not a clock and "verb" - not verb.
The paradox arises as soon as the question is asked: which of the two groups includes the adjective "heterological" itself? If it is autologous, it has a property designated by them and should be a ge-thermal. If it is heterologous, it does not have the properties called them and should therefore be autologous. There is a paradox.

By analogy with this paradox it is easy to formulate other paradoxes of the same structure. For example, is or not a suicide. The one who kills everyone does not kill and does not kill a single suicide?

It turned out that the Grellig Paradox was known in the Middle Ages as an antinomy of expressions that did not call himself. It is possible to imagine the attitude to sophimons and paradoxes in a new time, if the problem that demanded the answer and caused lively disputes, it was suddenly forgotten and was renounced only five hundred years later!

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

Many natural numbers are infinite. Many of the same names of these numbers that are available, for example, in Russian and contain less than, permissible, one hundred words is the final. This means that there are such natural numbers for which there are no names in Russian in Russian in less than a hundred words. Among these numbers is obviously the smallest number. It can not be called through the Russian expression containing less than a hundred words. But the expression: "The smallest natural number for which it does not exist in Russian its complex name, who has been staging 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. Obvious paradox: called the number for which there is no name!

§ 4. Unresolved dispute

At the heart of one famous paradox lies as if a small incident, which happened two more than a thousand years ago and not forgotten so far.

The famous Sofist Protagora, who lived in V c. BC, there was a student named Evatl, who studied right. According to the concluded between them, Evatl Agreement was supposed to pay for training only if he wins his first trial. If he loses this process, it is not obliged to pay at all. However, finished learning, Evatl did not participate in the processes. It lasted quite a long time, the patience of the teacher was dried, and he filed on his student to court. Thus, for Evatla, it was the first process. Prostagigar substantiated its requirement:

- Whatever the court decision, Evutl will have to pay me. He either won this first process, or will lose. If you win, I will pay due to our contract. If you lose, I will pay according to this solution.

Apparently, Evatl was a capable student as he replied to Protagor:

- Indeed, I will either win the process, or lose it. If you win, the court decision will free me from the obligation to pay. If the court decision is not in my favor, it means that I lost my first process and I will not pay for our contract.

Paradox solutions "Protagor and Evutl"

Puzzled by such a turnover of the case, Protagoras dedicated to this dispute with Evatl a special essay of "Trial about the board". Unfortunately, it, like the most part written by Protagora, has not reached us. Nevertheless, it is necessary to pay tribute to the protagora, who immediately felt a problem that deserves a special study at a simple judicial chart.

Labitz, a lawyer for education, also treated this sphere seriously. In his doctoral dissertation, "Study on confusing incidents in the right", he tried to prove that all cases, even the most confusing, like the gravity of Protagora and Evatla, should find the right permission based on common sense. According to Leibnitsa, the court must refuse the prostagor for the incommodation of the claim, but to leave, however, for him the right to pay the payment by Evatl later, namely after the first process won.

There were many other solutions of this paradox.

They refer in particular to the fact that the court decision should have a great power than a private agreement of two persons. This can be answered that if it's not that agreement, no matter how little it seems, there would be no court or his decision. After all, the court must make his decision precisely on its occasion and on its basis.

Also also apply to the general principle that every work, which means, the work of the protagon, must be paid. But it is known that this principle has always had exceptions, especially in a slave-owned society. In addition, he will simply be uncompressive to the specific situation of the dispute: after all, Protagor, guaranteeing a high level of training, refused to accept the fee in the event of the failure of his student in the first process.

Sometimes reason so. And Protagoras and Evutl - both right partly, and none of them in general. Each of them takes into account only half the possibilities beneficial for themselves. Full or comprehensive consideration opens four capabilities, of which only half is beneficial for one of the arguing. Which of these features is implemented, it will decide not logic, but life. If the sentence of judges will have a greater force than the contract, Evatl will have to pay only if the process plays, i.e. By virtue of the court decision. If the private agreement is raised higher than the decision of judges, then Protagor will receive a fee only if the Evatla process is losing, i.e. By virtue of the contract with Protagogue. This appeal to life is completely confused. What, if not logic, can judge be guided in conditions when all the circumstances relating to the case are completely clear? And what will it be for the guidance, if Protagora, applying for payment through the court, will achieve it, only losing the process?

However, the decision of the Leibnic, which seemed to be convincing initially, a little better than the unclear oppression of logic and life. In essence, Leibniz suggests the rear date to replace the statement of the contract and specify that the first with the participation of Evatla court process, the outcome of which will decide the issue of payment, should not be a court on the suit of Protagor. This thought is deep, but not related to a specific court. If there was such a reservation in the initial agreement, the needs in the trial would not have arisen at all.

If, under the solution of this difficulty, understand the answer to the question, should Evatle pay the protagora or not, then all these, like all other impending decisions, are, of course, untenable. They are no more than care from the creature of the dispute, so to speak, sophistic tricks and tricks in a hopeless and intractable situation. For no common sense, nor any general principles relating to social relations are not able to resolve the dispute.
It is impossible to fulfill the contract together in its original form and the court decision, no matter how the latter. To prove that enough simple means of logic. With the help of the same means, it can also be shown that the contract, despite its quite innocent appearance, internally contradictory. It requires the implementation of a logically impossible position: Evatl must simultaneously and pay for training, and at the same time not to pay.

Defense Rules

The human mind who is accustomed to not only for his strength, but also to his flexibility and even quirkness, is difficult, of course, to accept this absolute hopelessness and recognize himself to be drunk in a dead end. This is especially difficult when the dead-end situation is created by the mind itself: he, so to speak, crepts on an even place and pleases its own networks. And nevertheless, it is necessary to recognize that sometimes, and however, it is not so rare, agreements and system of rules that have developed spontaneously or entered consciously lead to an unresolved, hopeless positions.

An example of a recent chess life will again confirm this idea.

The international rules for conducting chess competitions oblige chess players to record a batch of passage in progress is clear and picky. Until recently, the rules also indicated that the chess player who missed the recording of several moves due to lack of time, should, "as soon as his stoutness end, to immediately fill its form, writing the missed moves." Based on this indication, one judge at the 1980 Chess Olympiad (Malta) interrupted the party held in a tough stiffness and stopped the clock, stating that the control moves were made and, therefore, it's time to put parties in order.

- But let me, - cried the participant who was on the verge of loss and calculated only for passions in the end of the party, - after all, no check box fell and no one ever (so also recorded in the rules) can not suggest how much moved.
The judge supported, however, the chief referee, who said that, indeed, since Zeietnotes ended, it was necessary, following the letter of the rules, proceed to the record of missed moves.
It was meaningless to argue in this situation: the rules themselves were headed in a dead end. It remained only to change their wording in such a way that such cases could not arise in the future.
This was done on the Congress of the International Chess Federation at the same time: instead of the words "as soon as Zeietnotes end," the rules are now recorded: "As soon as the checkbox indicates the end of time."
This example shows how to act in deadlog situations. It is useless to argue about which side of the right, it is useless: the dispute is intractable, and there will be no winner. It remains only to accept the present and take care of the future. To do this, it is necessary to reformulate the source agreements or the rules so that they do not make any more hopeless situation.
Of course, such a way of action is no decision of an unsolvable dispute and not exit from hopeless position. It is more likely to stop in front of an irresistible obstacle and the road bypassing it.

Paradox "Crocodile and Mother"

In ancient Greece, the story of the crocodile and mother, which coincides with its logical content with the paradox "Protagor and Evutl" was very popular.
The crocodile snatched the Egyptians standing on the banks of the river, her child. On her molib to return the child a crocodile, strait, as always, a crocodile tear, replied:

- Your misfortune was touched by me, and I will give you a chance to get a back baby. Guess, I will give it to you or not. If you answer correctly, I will return the child. If you do not guess, I will not give it.

Thinking, mother replied:

- You won't give me a child.

"You won't get it," the crocodile concluded. "You said either the truth or a lie." If the fact that I will not give a child is true, I will not give it, because otherwise there will be no appraul. If said - not true, it means you have not guess, and I will not give the child to a persuade.

However, this reasoning did not seem convincing.

"But if I told the truth, then you give me a child, as we agreed." If I did not guess that you would not give a child, then you have to give it to me, otherwise I will not be wrong.

Who is right: mother or crocodile? What does the crocodile make a promise given to them? To send a child or, on the contrary, not to give it to it? And to that and to another at the same time. This promise is internally contradictory, and thus it is not fulfilled by the laws of logic.
The missionary found himself from the cannibals and got just to dinner. They allow him to choose, in what form they will eat it. To do this, he must utter any statement with the condition that if this statement is true, they boil it, and if it is false, it is fried.

What should I say a missionary?

Of course, he must say: "You roam me."

If it is really fried, it turns out that he expressed the truth, and then it must be welded. If it is boiled, his statement will be false, and it should be fried just. The exit from the cannibals will not be: "Cook" follows from "fried", and vice versa.

This episode with a cunning missionary is, of course, another one of the rephrases of the dispute of Protagora and Evatla.

Paradox Sancho Pansy

One old, known in the ancient Greece, the paradox beats in the Don Quixote M.Servantes. Sancho Pansy became the governor of the island of Brataria and pecks the court.
The first to him is some consequences and says: - Senor, a certain estate is divided into two half a multi-water river ... So, through this river, the bridge is gone, and there is a gallows with the edge and there is something like a court, in whom He meets four judges, and they are judged on the basis of the law, published by the owner of the river, bridge and the whole estate, which the law is designed in this way: "Any passing on the bridge through this river must declare under the oath: Where and why he goes, and who will tell the truth, Those who are skipping, and who salting, those without any condescevement to send to the above gallows and execute. " Since that time when this law was published in all its rigor, many managed to go through the bridge, and how soon the judges were satisfied that passersby speak the truth, they missed them. But one day a certain man, cited by the oath, swore and said: He dreamed that he had come to pull him on this very gallows, and for nothing. Oath Siah led judges in bewilderment, and they said: "If you allow this person to be easily followed, it will mean that he broke the oath and according to the law of death; If we hang him, then he swore, that he only came to pull him on this gallows, therefore, the oath of him, it comes out, is not false, and on the basis of the same law it comes to miss it. " And here I ask you, Señor Governor, what to do to the judges with this person, for they still perplex and fluctuate ...
Sancho suggested, perhaps, not without truth: that half of the man who said the truth, let them miss, and the one that lied, let them hang, and thus the rules for the transition through the bridge will be observed in all form. This passage is interesting in several ways.
First of all, it is a clear illustration of the fact that with the hopeless position described in the paradox may well face - and not in a pure theory, but in practice - if not a real person, then at least a literary hero.

The output proposed by Sancho Pansa was not, of course, the solution of the paradox. But it was just the decision to which only remained to resort in his position.
Once, Alexander Macedonsky, instead of unleashing the cunning proud of the knot, which else could not do anyone, simply ruined him. Similarly, and Sancho came. Trying to solve the puzzle on her own conditions was useless - she simply is unresolute. It remained to discard these conditions and introduce your own.
And one moment. Cervantes This episode clearly condemns the exorbitant formal, permeated with the spirit of scholastic logic, the scale of medieval justice. But what prevalent in his time - and it was about four hundred years ago - there were information from the logic area! Not only the very servant is known this paradox. The writer finds it possible to attribute his hero, an illiterate peasant, the ability to understand that in front of him is an unresolved task!

§ 5. Other paradoxes

The above paradoxes are reasoning, the total of which is a contradiction. But in logic there are other types of paradoxes. They also indicate some difficulties and problems, but do it in a less sharp and uncompromising form. Such, in particular, paradoxes considered below.

Paradoxes of inaccurate concepts

Most concepts are not only a natural language, but also the science language is inaccurate, or, as they are also called, blurred. Often, this is the cause of misunderstanding, disputes, and even just leads to dead-end situations.
If the concept is inaccurate, the boundary of the object of the objects to which it is applied is devoid of sharpness, blurred. Take, for example, the concept of "bunch". One grain (grain, stone, etc.) is not a bunch. Thousand grains are already obviously a bunch. And three grains? And ten? With the addition of which grain is a bunch of? Not very clear. Just as it is not clear, with the withdrawal of what grain a bunch disappears.
Inaccurate are the empirical characteristics of "big", "heavy", "narrow", etc. Inclosure such ordinary concepts like "Sage", "Horse", "House", etc.
There are no sands, removing which we could say that with its elimination the remaining can not be called home. But this means it seems to be at what time of gradual disassembly house - right up to its full disappearance - there is no reason to declare that there is no house! The output is clearly paradoxical and discouraging.
It is easy to see that the argument on the impossibility of the formation of the heap is carried out with the help of a well-known method of mathematical induction. One grain does not forms heaps. If the grains do not form heaps, then n + 1 grain does not form heaps. Consequently, no number of grains can form heaps.
The possibility of this and similar evidence leading to ridiculous conclusions means that the principle of mathematical induction has a limited area of \u200b\u200bthe application. It should not be applied in reasoning with inaccurate, vague concepts.

A good example of the fact that these concepts are able to lead to unresolved disputes can serve as a curious trial that took place in 1927 in the United States. Sculptor K. Brankuzya appealed to the court demanding to recognize his works by works by art. Among the works sent to New York to the exhibition, there was a sculpture "Bird", which is now considered to be a classic abstract style. It is a modulated column of polished bronze about one and a half meters of height, having no external similarity with a bird. Customs officers categorically refused to recognize abstract creations by Brankuzy artwork. They conducted them according to the column "Metal Hospital Utensils and Household Objects" and imposed a large customs duty on them. Restricted Brankuzy sued.

Customs was supported by artists - members of the National Academy, defending traditional techniques in art. They performed on the process of witness protection and categorically insisted that an attempt to "bird" for the work of art is just a scam.
This conflict relief emphasizes the difficulty of operating the concept of "work of art." The sculpture by tradition is considered a type of visual art. But the degree of similarity of the sculptural image is the original may vary in very wide limits. And at what point the sculptural image, increasingly removing from the original, ceases to be a piece of art and becomes "metal utensil"? It is also difficult to answer this question as to the question of where the border between the house and its ruins goes, between the horse with the tail and the horse without a tail, etc. By the way, modernists are generally convinced that the sculpture is an expressive object and it is not at all obliged to be an image.

The treatment of inaccurate concepts requires thus known caution. Is it not better to give up them at all?

The German philosopher E. Gusserl was inclined to require knowledge of such extreme rigor and accuracy, which is not even found in mathematics. Gusserly's biographers with irony recall the case that happened to him in childhood. He was given a penny knife, and, having decided to make a blade extremely sharp, he sharpened him until he left from the blade.
More accurate concepts in many situations are preferable inaccurate. It is quite justified by the usual desire to clarify the concepts used. But it should, of course, have its limits. Even in science, a significant part of the concepts of inaccurate. And this is not connected with subjective and random errors of individual scientists, but with the very nature of scientific knowledge. In the natural language of inaccurate concepts, the vast majority; It says, among other things, about his flexibility and hidden strength. The one who requires from all the concepts of marginal accuracy risks at all staying without a language. "Lush the words of any ambiguity, every uncertainty," wrote the French aesthetic J. Juber, "turn them ... In unambiguous numbers - the game will go from speech, and with it - eloquence and poetry: everything that is movable and volatile in affection Souls, can not find his expression. But what I say: deprive ... I will say more. Lush the words of any inaccuracies - and you even lose the axiom. "
For a long time and logic, and mathematics did not pay attention to the difficulties associated with the blurred concepts and the corresponding sets. The question was as follows: the concepts must be accurate, and all vaguely unworthy of serious interest. In recent decades, this overly strict installation has lost, however, attractiveness. Logical theories were built, specifically taking into account the originality of reasoning with inaccurate concepts.
The mathematical theory of so-called blurred sets, fuzzy outlined sets of objects is actively developing.
Analysis of the problems of inaccuracies is a step towards convergence of logic with the practice of ordinary thinking. And it can be assumed that it will bring many more interesting results.

Paradoxes inductive logic

No, perhaps, such a section of logic, in which there would be no own paradoxes.
In inductive logic there are its own paradoxes with which actively, but so far, almost half a century are fighting without much success. The paradox of confirmation, the open American philosopher K.Gempel, is particularly interesting. It is natural to believe that general provisions, in particular, scientific laws are confirmed by their positive examples. If it is considered, let's say, the statement of "all A is in", then the objects with the properties of A and V. In particular, confirming the examples for the statement of "all black crows" are objects that are crowdes and black. This statement is equivalent, however, the statement "All items that are not black, not crows", and confirmation of the latter should also be a confirmation of the first. But "everything is not black not a crow" is confirmed by each case not a black subject that is not a raven. It turns out, so that the observations of the "white cow", "brown shoes", etc. Confirm the statement of "all black crows".

Of the innocent, it would seem, the parcels follows the unexpected paradoxical result.

In the logic of norms, anxiety causes a number of its laws. When they are formulated in meaningful terms, the inconsistency of their usual ideas about due and prohibited becomes obvious. For example, one of the laws says that from the order "send a letter!" The order follows the "Send a letter or burn it!".
Another law claims that if a person violated one of his duties, he gets the right to do whatever. From this kind of "laws of ownership", our logical intuition does not want to put up.
In the logic of knowledge, the paradox of logical omniscience is strongly discussed. He argues that a person knows all the logical investigations arising from the provisions made by him. For example, if a person knows five postulates of Euclidean geometry, then he also knows all this geometry, because it follows from them. But it is not. A person may agree with the postulates and at the same time not to be able to prove to the theorem of Pythagora and therefore doubt that it is generally true.

§ 6. What is a logical paradox

There is no exhaustive list of logical paradoxes, but it is impossible.
Considered paradoxes are only part of all detected to date. It is likely that many other paradoxes will be opened in the future, and even completely new types of them. The very concept of paradox is not so definite to be a list of at least already known paradoxes.
"The theoretical and multiple paradoxes are a very serious problem, not for mathematics, however, rather, for the logic and theory of knowledge," writes Austrian mathematician and the logic to the Monda. "The logic is consistent. There is no logical paradoxes, "says mathematician D. Bochar. This kind of discrepancy is sometimes significant, sometimes verbal. The case is in many respects that it is understood under a logical paradox.

Originality of logical paradoxes

The necessary sign of logical paradoxes is the logical dictionary.
Paradoxes attributed to logical must be formulated in logical terms. However, there are no clear criteria for dividing terms on logical and illogical. The logic that is engaged in the correctness of reasoning seeks to reduce the concepts on which the correctness of almost applied conclusions depends to a minimum. But this minimum is not predetermined unequivocally. In addition, in logical terms you can formulate and illogical allegations. It uses a specific paradox only purely logical packages, it is not always possible to determine uniquely.
Logical paradoxes are not separated hard from all other paradoxes, just as the latter are not rewarded clearly from all non-paradsal and consistent with the dominant ideas. At first, the study of logical paradoxes it seemed that they could be distinguished from a violation of some, not yet studied position or rule of logic. Especially actively applied for the role of such a rule introduced by B.Rashely the principle of a vicious circle. This principle argues that the set of objects cannot contain members defined only through the same totality.
All paradoxes have one general property - self-proof, or circularity. In each of them, the object in question is characterized by a certain set of objects to which he himself belongs. If we allocate, for example, the most cunning person, we do this with the help of a combination of people to which this person belongs. And if we say: "This statement is false," we characterize the statement of interest to us by reference to including its totality of all false statements.

In all paradoxes there is a self-proofness of concepts, and therefore, there is a movement in a circle, leading in the end to the initial item. In an effort to characterize the object of interest to us, we appeal to the combination of objects that includes it. However, it turns out that she herself needs to be subject to the object and cannot be clearly understood without it. In this circle, it is possible, and the source of paradoxes lies.
The situation is complicated, however, the fact that such a circle is available in many completely non-paradsal 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-proof leads to a contradiction and that it is important not only In the usual language, but also in the language of science.
A simple reference to the use of self-made concepts is insufficient, therefore, to discredit paradoxes. Another additional criterion is needed separating the self-proof leading to the paradox from all other cases.
There were many proposals on this account, but there was no successful clarification of circularity. It was impossible to describe the circularity in such a way that each circular reasoning led to the paradox, and each paradox was the result of some circular reasoning.
An attempt to find some specific principle of logic, whose violation would be a distinctive feature of all logical paradoxes, did not lead to anything defined.
There would be no doubt that some kind of classification of paradoxes, dividing them on types and types, grouping some paradoxes and opposing them to others. However, in this matter, nothing stable was reached.

English logic F. Macey, who died in 1930, when he was not yet and twenty-seven years old, proposed to divide all paradoxes on syntactic and semantic. The first refers, for example, the paradox of Russell, to the second - Paradoxes "Liaza", Grellling, and others.
According to Ramsey, the paradoxes of the first group contain only the concepts belonging to logic or mathematics. The second includes such concepts as "truth", "definability", "naming", "language", not strictly mathematical, and relating to linguistics or even the theory of knowledge. Semantic paradoxes are obliged, as it seems, its occurrence is not some kind of error in logic, but the confusion or ambiguity of some illogical concepts, so the problems they have concerned and should be solved by linguistics.

Ramsey seemed that mathematicians and logics had no reason to be interested in semantic paradoxes. In the future, it turned out, however, that some of the most significant results of modern logic were obtained just in connection with a deeper study of precisely these illogical paradoxes.
The division of paradoxes proposed by Ramsey was widely used at first and retains some meaning and now. At the same time, it becomes clearer that this division is quite vague and relies on the advantage of examples, and not on an in-depth comparable analysis of two groups of paradoxes. Semantic concepts have now received accurate definitions, and it is difficult not to recognize that these concepts really relate to logic. With the development of semantics, which defines its basic concepts in terms of the theory of sets, the difference conducted by Ramseym is increasingly erased.

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 is talking about the power of logic as science, and not about its weakness, as it may seem.

The detection of paradoxes did not accidentally coincide with the period of the most intensive development of modern logic and its greatest success.
The first paradoxes were discovered before the occurrence of logic as special science. Many paradoxes were discovered in the Middle Ages. Later, they were, however, were forgotten and were again open in our age.
The medieval logics were not known the concepts of "set" and "element of the set", entered into science only by the second half of the XIX century. But the flair on the paradoxes was calculated in the Middle Ages so much that there were certain concerns about self-proof concepts. The simplest example is the concept of "to be their own element", which appears in many current paradoxes.
However, such fears, as in general, all warnings concerning paradoxes were not to our century in due measure systematic and defined. They did not lead to any clear proposals on the revision of the usual ways of thinking and expression.
Only modern logic learned from oblivion to the problem of paradoxes, opened or overcluded most of the specific logical paradoxes. She further showed that ways of thinking, traditionally studied by logic, are completely insufficient to eliminate paradoxes, and indicated fundamentally new techniques of treatment with them.
Paradoxes put an important question: what actually give us some ordinary methods of the formation of concepts and methods of reasoning? After all, they seemed completely natural and convincing, until they revealed that they were paradoxical.

Paradoxes undermine the faith in the fact that the usual techniques of theoretical thinking themselves and without any special control over them ensure reliable promotion to truth.
Requires radical changes in an excessively trusty approach to theorization, paradoxes are a sharp criticism of logic in its naive, intuitive form. They play the role of a factor controlling and putting restrictions on the way of designing deductive logic systems. And this role can be compared with the role of the experiment that verifies the correctness of the hypotheses in the sciences such as physics and chemistry, and the changes to make changes to these hypotheses.
Paradox in theory speaks of incompatibility of assumptions underlying it. He acts as a timely discovered symptom of the disease, without which it could be looked.
Of course, the disease is manifested diverse, and in the end it is possible to reveal and without such sharp symptoms as paradoxes. For example, the bases of the theory of sets would be analyzed and clarified if even no paradoxes in this area were discovered. But there would be no sharpness and urgency, with which the problem of the revision of the theory of sets found in it paradoxes.

Paradoxes are devoted to extensive literature, a large number of their explanations have been proposed. But none of these explanations are generally accepted, and there is no complete agreement on the origin of paradoxes and methods of deliverance from them.
"Over the past sixty years, hundreds of books and articles were devoted to the goals of the permission of paradoxes, but the results are strikingly poor in comparison with the efforts spent," writes A.frenkel. "It looks like that," C. Paradoxes is an analysis, which requires a complete logic reform, and mathematical logic can become a major tool for carrying out this reform. "

Plan:

I. Introduction

II. Aprira Zenona

Achilles and Turtle

Dichotomy

III . Paradox Liaza

IV. . Paradox Russell

I. . Introduction

Paradox - these are two opposing, incompatible statements, for each of which there are seemingly convincing arguments. The most sharp shape of the paradox - antinomy, The argument proving the equivalence of two statements, one of which is the denial of the other.

Paradoxes in the most stringent and accurate sciences - mathematics and logic are especially fame. And it is not by chance.

Logic - abstract science. There are no experiments in it, there are no facts in the usual sense of the word. Building its systems, logic is ultimately from the analysis of real thinking. But the results of this analysis are synthetic. They are not statements of any individual processes or events that the theory should explain. Such analysis cannot be obviously called observation: there is always a concrete phenomenon.

Constructing a new theory, the scientist is usually sent from the facts, from what can be observed in the experience. No matter how free his creative fantasy is, it should be considered with one indispensable circumstance: the theory makes sense only when it is consistent with the facts relating to it. The theory, divergent with facts and observations, has no controversial and value.

But if there are no experiments in logic, there are no facts and there is no observation, which is contained by a logical fantasy? What if not facts, then factors are taken into account when creating new logical theories?

The discrepancy between the logical theory with the practice of valid thinking is often found in the form of a more or less acute logical paradox, and sometimes even in the form of a logical anti-rope speaking on the internal contradictions of the theory. This is just explained by the value that is attached to paradoxes in logic, and then much attention they enjoy in it.

One of the first and, perhaps, the best paradoxes was recorded by Evbulid, the Greek poet and the philosopher who lived in Crete in the VI century BC. e. In this paradox, Christine Epimyda argues that all the critical liars. If he tells the truth, he lies. If he is lying, he tells the truth. So who is epimeque - a liar or not?

Another Greek philosopher Zenon Elayky was a series of paradoxes about infinity - the so-called "Aritiani" of Zenon.

What Plato said is a lie.
Socrates

Socrates speaks only the truth.
Plato

II. Aporish of Zenona.

A great contribution to the development of the theory of space and time, eleauts (residents of the city of Elea in Southern Italy) were introduced into the study of the problems of movement. The philosophy of Eleaitov relied on a nominated Parmenide (Teacher of Zenon) an idea of \u200b\u200bthe impossibility of non-existence. Any thought, claimed Parmenid, there is always the idea of \u200b\u200bexisting. Therefore, there is no non-existent. There is no movement, since the world space is filled with entirely, and therefore the world is one, there are no parts in it. Everything is a lot of feelings. From this implies the conclusion about the impossibility of occurrence, destruction. According to Parmeno, nothing arises and is not destroyed. This philosopher was the first one who began to prove the positions put forward by thinkers

Eleata proved their assumptions with the denial of approval, reverse assumption. Zenon went on his teacher, which gave the foundation of Aristotle to see the Dialectics Xenona, "Dialectics" - this term was then called the art of achieving the truth in the dispute by clarifying the contradictions in the judgment of the enemy and by destroying these contradictions.

Achilles and turtle. Let's start considering the zenonic difficulties with the worker about the movement " Achilles and Turtle " . Achilles - hero and, no matter how we say, an outstanding athlete. Turtle, as you know, one of the most slow animals. Nevertheless, Zeno argued that Achill would lose the turtle competition in Run. We will take the following conditions. Let Achille separate the distance of 1 from the finish line, and the turtle - ½. Move Ahill and Turtle start at the same time. Let for definiteness ahill runs 2 times faster than the turtle (i.e. very slowly goes). Then, running the distance ½, Achille will discover that the turtle managed to overcome the segment ¼ and is still ahead of the hero. Next, the picture is repeated: running the fourth part of the way, Achille will see the turtle on the same eighth part of the path in front of himself, etc. Consequently, whenever Achill overcomes the distance separating it from the turtle, the latter has time to complete him and still remains ahead. Thus, the Achill will never catch up the turtle. Starting the move, Achill never be able to complete it.

Knowing mathematical analysis usually indicate that the series converges to 1. Therefore, they say, Achill overcomes all the way over the final period of time and, of course, will overtake the turtle. But here's what they write on this occasion D. Hilbert and P. Bernays:

"Usually, this paradox is trying to circumvent the argument that the sum of the infinite number of these time intervals is still converged and thus gives a finite time interval. However, this reasoning absolutely does not affect one substantially paradoxual moment, namely the paradox, which consists in the fact that a certain endless sequence of the events following each other, the sequence, the completion of which we cannot even imagine (not only physically, but at least in principle) , in fact, still have to complete. "

The principal incovering of this sequence lies in the fact that it does not have the last element. Whenever by specifying another member of the sequence, we can specify the following for it. Interesting remark, also indicating the paradoxicality of the situation, we meet with the city of Vaila:

"Let us imagine a computing machine that would perform the first operation for ½ minutes, the second - for ¼ minutes, the third - for ⅛ minutes, etc. Such a car could be the end of the first minute" to recalculate "the entire natural row (write, for example, Accounting number of units). It is clear that the work on the design of such a car is doomed to failure. So why the body, released from point A, reaches the end of the segment in, "counting" the counting set of points A 1, and 2, ..., and n, ...? "

Dichotomy . Reasoning is very simple. In order to pass the whole way, the moving body must first pass half the way, but to overcome this half, it is necessary to pass half of the half, etc. to infinity. In other words, under the same conditions as in the previous case, we will deal with an inverted near the points: (½) n, ..., (½) 3, (½) 2, (½) 1. If in case of an aporish Achilles and Turtle The corresponding row did not have the last point, then in Dichotomy This series has no first point. Therefore, concludes Zeno, the movement cannot begin. And since the movement not only can not end, but can not begin, there is no movement. There is a legend that A. S. Pushkin recalls in the motion poem:

No movement, said the sage bradyt.

The other has grown and began to walk before it.

It would not be more strongly to argue;

Praised all the answer intricate.

But, gentlemen, funny case

Another example of memory leads to me:

After all, every day, before us, the sun walks,

However, the rights are stubborn Galilee.

Indeed, according to legend, one of the philosophers and "objected" Zenon. Zenon ordered him to beat him with sticks: after all, he was not going to deny the sensual perception of movement. He spoke about his untion The fact that strict thinking about movement leads to unresolved contradictions. Therefore, if we want to get rid of the aporish in the hope that it is generally possible (and Zeno just thought that it was impossible), then we must resort to theoretical arguments, and not refer to sensual evidence. Consider one curious theoretical objection that was put forward against the aporish Achilles and Turtle .

"Imagine that on the road in one direction a quick-legged Achille and two turtles are moving, of which Turtle-1 is somewhat closer to Achillu than the Turtle-2. To show that Achilles will not be able to overtake the turtle-1, we argue as follows. During the time Ahill runs the distance dividing them at the beginning, the turtle-1 will be able to wage a little forward, while Achills will run this new segment, it will move further, and this position will be infinitely repeated. Ahill will be closer and closer to the turtle-1, but will never be able to overtake it. This conclusion, of course, contradicts our experience, but there are no logical contradiction yet.

Let, however, Achilles will begin to catch up with a more far tortoise-2, not paying any attention to the near. The same way of reasoning suggests that Achille will be easy to get closer to the Turtle-2, but this means that it will distort the Turtle-1. Now we come to the logical contradiction. "

It is difficult to argue anything here if you remain in captivity of figurative views. It is necessary to identify the formal essence of the case, which will allow to translate the discussion in the row of strict arguments. The first apior can be reduced to the following three statements:

2. Any segment can be represented as an infinite sequence of decreasing segments in length ....

3. Since the infinite sequence A I (1 ≤ i< ω) не имеет последней точки, невозможно завершить движение, побывав в каждой точке этой последовательности.

You can illustrate the resulting output in different ways. The most famous illustration - "The fastest one will never be able to catch up with the most slow" - was considered above. But it is possible to offer a more radical picture in which the later Achill (released from point A) is unsuccessfully trying to fill the turtle, quietly heating in the sun (in paragraph B) and even not thinking to run away. The essence of the aporish from this does not change. The illustration will then become a much more acute statement - "the fastest can never catch up with a fixed." If the first illustration of the paradoxical, then the second is the more.

At the same time, it is not argued that the decreasing sequences of AI segments for and Ai "for must be the same. On the contrary, if the segments are unequal in length between themselves, their partitions to infinite sequences of decreasing segments will be different. In the above reasoning, the Achilla separates from the turtles 1 and 2 Different distances. Therefore, we have two different segments and with a common starting point A. Unequal segments and generate various endless sequences of points, and it is unacceptable to use one of them instead. Meanwhile, this "illegal" operation is applied in arguments about two turtles.

If you do not mix illustrations and the creature of an aporish, then it can be argued that apior Achilles and Dichotomy symmetrical in relation to each other. Indeed, Dichotomy Also lasting to the following three statements:

1. Whatever the segment moving from and to the body should be in all points of the segment.

2. Any segment can be represented as an infinite sequence of decreasing segments in length ... ....

3. Since the infinite sequence B i does not have the first point, it is impossible to visit each of the points of this sequence.

Thus, the aporia Achilles It is based on the thesis on the impossibility of completing the movement due to the need to visit consistently each of the points of the infinite series, ordered by type ω (i.e., by type of order on natural numbers), which does not have the last element. In turn Dichotomy Approves the impossibility of the beginning of the movement due to the presence of an infinite series of points, ordered by type ω * (soorly negative numbers are so ordered), which does not have the first element.

After analyzing more thoroughly two presented aporias, we will find that both of them rely on assumption continuity space and time in the sense of them infinite divisibility . Such an assumption of continuity differs from the modern, but took place in antiquity. Without the admission of the thesis that any spatial or time interval can be divided into smaller intervals in length, both aporities are crumbling. Zenon understood it perfectly. Therefore, it leads an argument emanating from making assumptions about discreteness space and time, i.e. assumptions about the existence of elementary, further indivisible, length and time.

Stadium . So, let's say the existence of indivisible segments of space and time intervals. Consider the following scheme, on which each cell of the table represents an indivisible block of space. There are three rows of objects A, B and C, which occupy three blocks of space, and the first row remains fixed, and the rows in and c begin simultaneous Movement in the direction indicated by arrows:

End position

A number of C, says Zeno, in the indelibious moment of time passed one indivisible place of the stationary series A (place A1). However, for the same time, a number of two places in (B2 and B3 blocks) were held. According to Zenon, it is controversial, since the time of the B2 block was depicted in the following scheme:

IN 3	AT 2	IN 1
C1.	C2.	C3.

Intermediate position

But where in this intermediate position was a series of? It simply does not have an appropriate place for him. It remains either to recognize that there is no movement, or agree that a row is not divided by three, but for more places. But in the latter case, we again return to the assumption of the infinite divisibility of space and time, again falling in a dead end Dichotomy and Achilles . With any outcome, the movement is impossible.

The main idea of \u200b\u200bthe Arsenius of Zenon Elaisky is that discreteness, multipleness and movement characterize only a sensual picture of the world, but it is obviously unreliable. The true picture of the world is comprehended only by thinking and theoretical research.

If you do not delve into the depths of the aquaries, you can relate to them down and wonder how this Zenon did not think to obvious things. But Zenon does not cease to argue, and the history of science shows that if something is arguing for a long time, then this is usually not in vain. Undoubtedly, reflections above the aporites helped create mathematical analysis, played a certain role in the physical revolution of the twentieth century and it is possible that in the physics of the XXI century their value will be even more significant.

III . Paradox liar.

Already almost two and a half thousand years one of the logical mysteries, tormenting people trying to harmonize the foundations of their thinking, is a "liar paradox". Despite the fact that there are currently dozens of semantic, logical and mathematical paradoxes and aporis, "Liaz Paradox" occupies a special place:

First, it is most affordable from many paradoxes and, by virtue of this, the most famous of them.

Secondly, it is primarily in relation to many other paradoxes and, therefore, the last non-resistant is not yet allowed a "liar paradox".

The simplest option of the paradox of the liar is the saying "I LDU". If the statement is false, then the speaker said the truth, and it means that they did not lie. If the statement is not false, and the speaker claims that it is false, then this is his statement false. It turns out that in such a way that if the talking lies, he tells the truth, and vice versa.

The "Liaz Paradox" has a number of other formulations similar to each other. Below are only some of them:

- "All Creaters are liars" (the thesis expressed by Christian Epimema);

- "I express a false offer now";

- "All that X approves in the period of time p - false";

- "This statement is false";

- "This statement does not belong to the class of true statements."

Although the listed list is not full, it gives some idea of \u200b\u200bthe essence of the problem. A logical problem is that the assumption of the listed statements leads to their truth and vice versa.

The ancient Greeks very much occupied how, it would seem that a completely meaningful statement could not be a true nor false without the contradictions. Philosopher Kharmpype wrote six treatises about the paradox of the liar, none of which was preserved to our time. There is a legend that a certain Cranosie, desperate to resolve this paradox, committed suicide. It is also said that one of the famous ancient Greek logists, a diodor of the Kronos, already on the slope of the years, did not make food until he finds the decision of the "liar", and soon died, so having achieved anything.

In the Middle Ages, this paradox was attributed to the so-called unresolved proposals and became the object of systematic analysis. Now "liar" - this typical former software is often referred to as the king of logical paradoxes. He is devoted to extensive scientific literature. And yet, as in the case of many other paradoxes, it remains not quite clear which problems are hiding behind it and how to get rid of it.

Consider the first wording: the approval attributed epimera is logically contradictory, assuming that liars are always lying, and the Negrezers always speak the truth. With this assumption, the approval of "all the critical liar" can not be true, because then Epimine would be a liar and, therefore, what he claims would be a lie. But this statement cannot be false, because it would mean that Christians speak only the truth and, therefore, what Epimyda said is also true.

The history of logic knows many attempts and approaches to the resolution of this paradox. One of the first is an attempt at the view of the "liar paradox" as a sofism. The essence of such a presentation is that in real life, no liar speaks only a lie. Consequently, the paradox is a sophis based on a false premise.

But such an explanation is acceptable only for the first (early) wording of the paradox, but not "removes" the paradox in its more accurate modern wording. There are several liar paradox solutions in its modern formulation. Which solutions is correct? All are correct. How can it be? Because the paradox is a reasoning leading to a contradiction. You can get rid of contradiction in different ways. All of them are reduced to replacing some dubious piece of reasoning to more correct. As a result, there is a reasoning similar to that, but without visible contradictions. In addition, various solutions are given through different types of logic.

You can replace different pieces. In each case, there will be different solutions, and which one preferred is a matter of taste. One very doubt seems to be one piece, the other - the other. Sometimes the very first dubious piece is noticeable and obvious.

Perhaps the most common option for solving a paradox of a liar is the separation of language and metalanas:

Now the "liar" is usually considered a characteristic example of those difficulties to which two languages \u200b\u200bare confused: the language on which the actual language is referred to, and the language that speaks about the very first language.

If someone had a thought about the need to talk about the world in the same language, and about the properties of this language - on the other, it could use two different existing languages, let us say Russian and English. Instead of simply to say: "Cow is a noun," ISANOUN Cow would say, and instead: "The" glass is not transparent "approval," the "Theassertion" of the glass would not transparently "ISFALSE." With this use of two different languages, the above-mentioned world would be clear from what has been said about the language with which they talk about the world. In fact, the first statements would relate to the Russian language, while the second - to English.

It turns out, therefore, a kind of ladder, or hierarchy, languages, each of which is used for a completely definite purpose: on the first they talk about the subject world, on the second - about this first language, on the third - about the second language, etc. Such a distinction of languages \u200b\u200bon their application is a rare phenomenon in ordinary life. But in the sciences, specially involved, like logic, languages, it sometimes turns out to be very useful. Language on the world is usually called subject. The language used to describe the subject language is called meta language.

It is clear that, if the language and metalanak are delimited in the manner, the statement "I LSU" can no longer be formulated. It speaks of the falsity of what is said in Russian, and, it means, it belongs to the metalanak and should be expressed in English. Specifically, it should sound like this: "Everythingispeakinrussianisfalse" ("Everything that I said in Russian is false"); In this English statement, nothing is said about himself itself, and no paradox arises.

The distinguishing of the language and the metabasics allows you to eliminate the paradox "Liaza". Thus, it becomes possible correctly, without contradiction, it is determined to determine the classical concept of truth: the statement corresponding to the reality described by it is true.

The concept of truth, as well as all other semantic concepts, has a relative nature: it can always be attributed to a certain language.

Tarsky introduced the concept semantically closed tongue. Such a language includes, in addition to its expressions, their names, and also, it is important to emphasize, the statements about the truth of the proposals formulated in it. The boundaries between the tongue and the metalanas in the semantically closed language do not exist. Its funds are so rich, which allows not only something to argue about non-speaking reality, but also to evaluate the truth of such statements. These funds are sufficient, in particular, in order to reproduce in the language of the Antinomy of "Liar". Semantically closed language turns out to be internally contradictory. Each natural language is obviously semantically closed.

The only acceptable way to eliminate the antinomy, and hence the internal contradictions, according to Tar, is the refusal to use a semantically closed language. This path is acceptable, of course, only in the case of artificial, formalized languages \u200b\u200bthat allow a clear division into tongue and metalas. In the natural languages \u200b\u200bwith their unclear structure and the ability to talk about everything in the same language, this approach is not very real. It makes sense to raise the question of the internal consistency of these languages. Their rich expressive opportunities have their own opposite direction - paradoxes.

There are other solutions of the paradox of a liar, for example, the decision of the Okkam and the dried decision:

So, there are statements that spell about their own truth or falsity. The idea that this kind of statement is not meaningful, very old. She defended her still ancient Greek logic of Chrysipp. In the Middle Ages, the English philosopher and the logic of U. Kokkova said that the statement "Any statement is falsely" is meaningless, since it speaks among other things and about his own lodge. From this statement directly follows the contradiction. If any statement is false, then this also applies to this statement itself; But the fact that it is false means that not any statement is false. The situation is similar and with the statement of "Any Statement True". It should also be referred to meaningless and also leads to a contradiction: if each statement is truly true, then the denying of this statement itself is true, that is, the statement that not any statement is truly.

Why, however, the statement can not mean clearly about their own truth or falsity? Already the contemporary of Okkama, French philosopher XIV century. J. Buridan, did not agree with his decision. From the point of view of ordinary ideas about meaninglessness, the expression of the type "I LSU", "Any statement is true (false)", etc. It is quite meaningful. What you can think about whether you can speak about whether the general principle of Buridan. A person may think about the truth of the allegation he says, it means that he can speak about it. Not all statements speaking about themselves belong to meaningless. For example, the statement "This proposal is written in Russian" is true, and the approval "In this proposal of ten words" is false. And both are completely meaningful. If it is assumed that the statement can talk about itself, then why it is not capable of talking about this property, as a truth?

The Buridan himself considered the statement "I LSU" is not meaningless, but false. He justified it. When a person claims some proposal, he claims that it is true. If the proposal suggests itself that it is false itself, it represents only the reduced formulation of a more complex expression that approves simultaneously and its truth, and its falsity. This expression is contradictory and, therefore, false. But it is not meaningless.

The argument of the Buridan and is sometimes considered convincing.

There are other directions of criticism of the paradox of "liar", which was developed in the details of Tarsky. Is it really in semantically closed languages \u200b\u200b- and these are all natural languages \u200b\u200b- there is no antidote against the paradoxes of this type?

If it were so, the concept of truth could be determined strictly only in formalized languages. Only they manage to distinguish between the subject language, on which they argue about the world around, and the meta language, which is talking about this language. This hierarchy of languages \u200b\u200bis based on a sample of a foreign language assimilation with the help of a native. The study of such a hierarchy led many interesting conclusions, and in certain cases it is essential. But it is not in the natural language. Does it discredit it? And if so, what exactly is? After all, in it, the concept of truth is still used, and usually without any complications. Is the Introduction of the hierarchy by the only way to exclude paradoxes like "liar?"

In the 30s, the answers to these issues were undoubtedly affirmative. However, now the former unity is no longer there, although the tradition of eliminating the paradoxes of this type by "separating" the language remains dominant.

Recently, more and more attention is attracted egocentric expressions. They meet words like "I", "this", "here", "now", and their truth depends on when, by whom they are used. In the statement "This statement is false" the word "this" meets . What exactly does it apply to? "Liar" may say that the word "this" does not apply to the meaning of this approval. But then what does it belong to what is it? And why does this meaning can not be marked with the word "it"?

Without going into details, it is worth noting that in the context of the analysis of the egocentric expressions "liar" is filled with completely different content than before. It turns out that he no longer warns against the mixing of the language and the metalanka, but indicates dangers associated with the wrong use of the word "this" and similar to it of egocentric words.

The problems that bind over the centuries with the "liar" radically changed depending on whether it was considered as an example of ambiguity, or as an expression, externally appeared as a sample of a language and metalanak, or, finally, as a typical example of incorrect egocentric use expressions. And there is no confidence that with this paradox will not be associated in the future and other problems.

Famous modern Finnish Logic and Philosopher G. Vista Vrigt wrote in his work dedicated to the "Loze" that this paradox should not be understood as a local one, an isolated obstacle, eliminated by one inventive movement of thought. "Liar" affects many of the most important topics of logic and semantics. This is the definition of truth, and the interpretation of contradictions and evidence, and a whole series of important differences: between the proposal and the thought expressed to them, between the use of the expression and its mention, between the meaning of the name and the object denoted them.

Liaz Paradox (as surprising), extremely close in its logical form and character of a logical error to many other "paradoxes", which are considered quite independent. The famous "Russell Paradox" belongs to their number.

III . Paradox Russell

The most famous of the open centions already in the last century of paradoxes is the antinomy discovered by B. Russell and reported by him in a letter to the city of Ferge. Russell opened his paradox related to the field of logic and mathematics, in 1902. The same antinomy was discussed simultaneously in Göttingen German mathematicians 3. Cerm (1871-1953) and D. Hilbert. The idea was worn in the air, and her publication impressed the bomb. This paradox caused in mathematics, according to Hilbert, the effect of a complete catastrophe. The threat over the most simple and important logical methods, the most ordinary and useful concepts. It turned out that in the theory of Cantor's sets, which with delight was adopted by a majority mathematicians, there are strange contradictions, from which it is impossible, or at least very difficult to get rid of. Russell paradox (more precisely, Russell - Cermer) especially revealed these contradictions. Over its resolution, as well as the resolution of other found paradoxes of the Cantor's theory of sets, worked the most outstanding mathematicians of those years.

Immediately it became apparent that neither in logic, nor in mathematics for the entire long history of their existence was not developed decisively anything that could serve as the basis for eliminating the antinomy. It was clearly necessary to waste from the usual ways of thinking. But what place and in what direction? How radical was to be a refusal of established methods of theorization? With a further study of the antinomy, the conviction of the need for a fundamentally new approach has grown steadily. After half a century after its opening, specialists in the foundations of logic and mathematician L. Frenkel and I. Bar-Hillel, without any reservations, were claimed: "We believe that any attempts to get out of the situation with the help of traditional (that is, having to go before the 20th century) of thinking methods , I still have invariably failed, knowingly insufficient for this purpose. " Modern American Logic X. Carry wrote a little later about this paradox: "In terms of logic, known in the XIX century, the situation simply did not give in to the explanation, although, of course, in our educated century people who will see (or think that they will see ) What is the error ".

Paradox of Russell in its initial form is associated with the concept of set, or class. You can talk about the sets of various objects, for example, about the set of all people or about the multiple natural numbers. An element of the first set will be every separate person, the element of the second - every natural number. It is also permissible to consider themselves as some objects and talk about sets of sets. You can even enter such concepts as a set of all sets or many of all concepts. Regarding any arbitrarily taken set, it seems to be meaningful to ask, it is its own element or not. Sets that do not contain themselves as an element, we call ordinary. For example, the set of all people is not a person, as well as many atoms - this is not an atom. There will be unusual sets that are their own elements. For example, a set that combines all sets is a set and, it means it means itself as an element.

Consider now the set of all ordinary sets. Since it is a lot, it can also be asked about it, usual or unusual. The answer, however, turns out to be discouraged. If it is usual, then, according to its definition, must contain itself as an element, since it contains all conventional sets. But this means that it is an unusual set. The assumption that our set is the usual set, thus leads to contradiction. So it cannot be ordinary. On the other hand, it cannot also be unusual: an unusual set contains itself as an element, and only ordinary sets are elements of our set. As a result, we come to the conclusion that the set of all ordinary sets cannot be neither ordinary nor an unusual set.

So, the set of all sets that are not their own elements, there is an element in that and only when it is not such an element. This is an explicit contradiction. And it was obtained on the basis of the most believable assumptions and with the help of undisputed as if steps. The contradiction says that such a variety simply does not exist. But why can it not exist? After all, it consists of objects that satisfy a clearly defined condition, and the condition itself does not seem to be some exclusive or unclear. If such a simple and clearly specified set cannot exist, then what is actually the difference between possible and impossible sets? The conclusion about the existence of the set under consideration sounds unexpectedly and inspires anxiety. He makes our overall concept of many amorphous and chaotic, and there is no guarantee that it is not able to generate some new paradoxes.

Paradox Russell is wonderful with its extreme community. It does not need any complex technical concepts for its construction, as in the case of some other paradoxes, the concepts of "set" and "element of the set" are sufficient. But this simplicity is just talking about his fundamentality: it affects the deepest foundations of our arguments about the sets, because it says not about any special cases, but about sets at all.

Other paradox options Paradox Russell is not specifically mathematical. It uses the concept of multiple, but are not affected by some special, related to the mathematics of its properties.

It becomes obvious if we reformulate the paradox in purely logical terms. Every property can, in all likelihood, ask, is applied to yourself or not. The property of being hot, for example, is uncomfortable to yourself, because it is not hot; The property to be concrete also does not apply to yourself, because it is an abstract property. But here is an abstract property, being abstract, apparently to myself. Let's call these integrated properties to themselves uncomfortable. Is the property applies to be unpaved to itself? It turns out that no application is unpaid only if it is not so. This is, of course, paradoxically. Logical concerning properties The species of the anti-russell antinomy, as paradoxical, as well as mathematical, related to sets, its variety.

Russell also offered the next popular version of the paradox's open paradox. Imagine that the Council of one village so determined the responsibilities of Brand-having: shave all the villages of the village who do not shave themselves, and only these men. Should he shave himself? If so, he will relate to those who shave himself, and those who shave himself should not shave. If not, it will belong to those who do not shake himself, and, it means he will have to shave himself. We come in this way, to the conclusion that this brainy shaves himself in that and only the case when he does not shave himself. This, of course, is impossible.

The argument about the Brandobre is based on the assumption that there is such a browver. The resulting contradiction means that this assumption is false, and there is no such resident of the village, which would vomit all those and only those inhabitants who do not shave themselves. Responsibilities Brand-roll do not seem to be contradictory, so the conclusion that it cannot be, he sounds somewhat unexpected. But this conclusion is not still paradoxical. The condition to which the village of Bradobremy should satisfy is actually internally contradictory and, therefore, impracticable. There may be no such hairdresser in the village for the same reason why there is no person in it, who would be older himself or who would be born before his birth.

The argument about the brandobre may be called pseudoparads. In his go, it is strictly similar to the paradox of Russell and this is interesting. But it is still not a genuine paradox.

It is interesting to note that the compilation of the catalog of all directories that do not contain references to themselves can be represented as an endless, never ending process. Suppose that at some point a catalog was drawn up, say K1, including all directories from it, not containing links to ourselves. With the creation of K1, another directory appeared that does not contain references to itself. Since the task is to draw up a full catalog of all directories that do not mention yourself, it is obvious that K1 is not its solution. He does not mention one of these directories - himself. Including in K1 this mention of himself, we obtain the Catalog K2. It is mentioned K1, but not K2 himself. By adding such a mention to K2, we obtain the KZ, which is again not full due to the fact that it does not mention himself. And then without end.

You can mention another logical paradox - the "paradox of Dutch mayors", similar to the paradox of Brand-roll. Every municipality in Holland must have mayor, and two different municipalities cannot have the same mayor. It is sometimes it turns out that the mayor does not live in his municipality. Suppose that a law is published, according to which some territory S. Eliminated exclusively for such mayors who do not live in their municipalities, and prescribing all these mayors to settle in this territory. Suppose further that these mayors turned out so much that the territory S. Itself forms a separate municipality. Where should the mayor of this special municipality s? Simple reasoning shows that if the mayor of a special municipality lives on the territory S, he should not live there, and vice versa, if he does not live on the territory, then he should also live in this territory. The fact that this paradox is similar to the paradox of brand-roll, is completely obvious.

Russell one of the first suggested the solution to the "his" paradox. The solution suggested by him, got the name of the "Type theory": The set (class) and its elements relate to different logical types, the type of set above its elements, which eliminates the Russell Paradox (the theory of types was used by Russell and to solve the famous paradox "Liar") . Many mathematics, however, did not accept the Russell decision, believing that it imposes too hard restrictions on mathematical statements.

It is similar to other logical paradoxes. "Antinomy of logic," Vrigg writes von Vygg, "they puzzled from the moment of their discovery and, probably, they always will be puzzled. We must, I think, consider them not as much as problems waiting for solutions, how much as an inexhaustible raw material for reflection. They are important because reflection about them affects the most fundamental questions of the whole logic, and therefore, and all thinking. "

Bibliography:

1 Frankel A.A., Bar Hillel I. "The foundations of the theory of sets"

2. B.russell. "Introduction to Mathematic Philosophy".

3. Russell B. "The Principles of Mathematics".

4. Zadoya A.I. "Introduction to logic"

5. Hilbert D. - Akkerman V., "Fundamentals of theoretical Logic".

6. Lakoff J. "Pragmatics in natural logic. New in linguistics. "

7. Jacobson R. "Boas views on grammatical meaning."

Plan:

I.Introduction

II.Aprira Zenona

Achilles and Turtle

Dichotomy

III. Paradox Liaza

IV.. Paradox Russell

I.. Introduction

Paradox - these are two opposing, incompatible statements, for each of which there are seemingly convincing arguments. The most sharp shape of the paradox - antinomy,the argument proving the equivalence of two statements, one of which is the denial of the other.

Paradoxes in the most stringent and accurate sciences - mathematics and logic are especially fame. And it is not by chance.

Another Greek philosopher Zenon Elayky was a series of paradoxes about infinity - the so-called "Aritiani" of Zenon.

What Plato said is a lie.
Socrates

Socrates speaks only the truth.
Plato

II.Aporish of Zenona.

Achilles and turtle.Let's start considering the zenonic difficulties with the worker about the movement " Achilles and Turtle ". Achilles - hero and, no matter how we say, an outstanding athlete. Turtle, as you know, one of the most slow animals. Nevertheless, Zeno argued that Achill would lose the turtle competition in Run. We will take the following conditions. Let Achille separate the distance of 1 from the finish line, and the turtle - ½. Move Ahill and Turtle start at the same time. Let for definiteness ahill runs 2 times faster than the turtle (i.e. very slowly goes). Then, running the distance ½, Achille will discover that the turtle managed to overcome the segment ¼ and is still ahead of the hero. Next, the picture is repeated: running the fourth part of the way, Achille will see the turtle on the same eighth part of the path in front of himself, etc. Consequently, whenever Achill overcomes the distance separating it from the turtle, the latter has time to complete him and still remains ahead. Thus, the Achill will never catch up the turtle. Starting the move, Achill never be able to complete it.

Dichotomy. Reasoning is very simple. In order to pass the whole way, the moving body must first pass half the way, but to overcome this half, it is necessary to pass half of the half, etc. to infinity. In other words, under the same conditions as in the previous case, we will deal with an inverted near the points: (½) n, ..., (½) 3, (½) 2, (½) 1. If in case of an aporish Achilles and Turtle The corresponding row did not have the last point, then in Dichotomy This series has no first point. Therefore, concludes Zeno, the movement cannot begin. And since the movement not only can not end, but can not begin, there is no movement. There is a legend that A. S. Pushkin recalls in the motion poem:

No movement, said the sage bradyt.

The other has grown and began to walk before it.

It would not be more strongly to argue;

Praised all the answer intricate.

But, gentlemen, funny case

Another example of memory leads to me:

After all, every day, before us, the sun walks,

However, the rights are stubborn Galilee.

Indeed, according to legend, one of the philosophers and "objected" Zenon. Zenon ordered him to beat him with sticks: after all, he was not going to deny the sensual perception of movement. He spoke about his untionThe fact that strict thinking about movement leads to unresolved contradictions. Therefore, if we want to get rid of the aporish in the hope that it is generally possible (and Zeno just thought that it was impossible), then we must resort to theoretical arguments, and not refer to sensual evidence. Consider one curious theoretical objection that was put forward against the aporish Achilles and Turtle.

1. Whatever the segment moving from and to the body should be in all points of the segment.

2. Any segment can be represented as an infinite sequence of decreasing segments in length ....

Scientists and thinkers for a long time love to entertain themselves and colleagues by formulation of unsolvable tasks and formulating a different kind of paradoxes. Some of these mental experiments retain relevance for thousands of years, which indicates the imperfection of many popular scientific models and "holes" in generally accepted theories that have long considered fundamental. We invite you to reflect on the most interesting and amazing paradoxes, which are now expressed, "blew up" not one generation of logic, philosophers and mathematicians.

1. Apri region "Achilles and Turtle"

The paradox of Achilles and the Turtles are one of the aporis (logically loyal, but contradictory statements), formulated by the ancient Greek philosopher Zeno Elaisky in the V-M century BC. The essence of it is as follows: the legendary hero Achilles decided to compete in running with the turtle. As you know, the turtles are not distinguished, so Achilles gave an opponent to a 500 meter. When the turtle overcomes this distance, the hero is started in the chase at a speed of 10 times more, that is, while the turtle crawls 50 m, Achilles has time to run data 500 m for . The runner overcomes the next 50 m, but the turtle will deploys at this time another 5 m, it seems that Achilles is about it to catch it up, however, the rival is still ahead and while he runs 5 m, she is able to advance half meter and so on. The distance between them is infinitely reduced, but in theory, the hero is not possible to catch up with the slow turtle, it is not aimless, but always ahead of him.

Of course, from the point of view of physics, the paradox does not make sense - if Achilles moves much faster, in any case will break forward, but Zenon, first of all, wanted to demonstrate his arguments that idealized mathematical concepts "point of space" and "moment of time" Too suitable for correct use to real movement. Aporia reveals the discrepancy between the mathematically reasonable idea that the nonzero space intervals and time can be divided infinitely (so the turtle should always stay ahead) and the reality in which the hero, of course, wins the race.

2. Paradox of the time loop

Paradoxes describing travel travel for a long time serve as a source of inspiration for science fiction writers and creators of scientific fiction films and serials. There are several options for the paradox of a temporary loop, one of the most simple and visual examples of such a problem brought in his book "The New Time Travelers" ("New Travelers in Time") David Tumi, a professor at the University of Massachusetts.

Imagine that the traveler in time bought a copy of the Shakespearean "Hamlet" in the bookstore. Then he went to England the times of the Queen-Virgin Elizabeth I and finding William Shakespeare, handed him a book. He rewritten it and published as his own essay. Hundreds of years, the "Hamlet" is transferred to dozens of languages, infinitely reprint, and one of the copies turns out to be in the most bookstore, where the traveler buys it in time and gives Shakespeare, and he removes a copy and so on ... who in this case should be considered The author of the immortal tragedy?

3. Girl and boy paradox

In the theory of probability, this paradox is also called "Children Mr. Smith" or "Problems of Mrs. Smith." For the first time, it was formulated by the American mathematician Martin Gardner in one of the numbers of SCIENTFIC AMERICAN magazine. Scientists argue over the paradox for several decades and there are several ways to resolve it. After thinking over the problem, you can offer your own option.

In the family there are two children and just know that one of them is a boy. What is the likelihood that the second child also has a male floor? At first glance, the answer is quite obvious - 50 to 50, or he is really a boy or a girl, chances should be equal. The problem is that for two-order families there are four possible combinations of children's floors - two girls, two boys, a senior boy and a younger girl and the opposite - a senior girl and a junior boy. The first one can exclude, since one of the children is exactly exactly the boy, but in this case there are three possible options, and not two and the likelihood that the second child is also a boy - one chance of three.

4. Paradox Jourdin with a card

The problem proposed by British logic and mathematician Philip Zubdenov at the beginning of the 20th century can be considered one of the varieties of the famous paradox of a liar.

Imagine - you keep the postcard in your hands, on which it is written: "Approval on the reverse side of the postcard is true." Turning the postcard, you find the phrase "Approval on the other side false." As you understand, there is no contradiction: if the first statement is true, then the second is also true, but in this case the first should be false. If the first side of the idle card, the phrase on the second one can also be considered true, which means that the first statement is again becoming true ... An even more interesting version of the liar is in the next paradox.

5. Sophism "Crocodile"

On the banks of the river, a mother stands with a child, suddenly swims the crocodile and skews the child into the water. A luckless mother asks to return her child, to which the crocodile answers that he agrees to give it whole and unharmed if the woman responds correctly to his question: "Does he return her child?" It is clear that a woman has two options for answering - yes or no. If she claims that the crocodile will give her a child, it all depends on the animal - considering the answer to the truth, the kidnapper will release the child, if he says that the mother was wrong, then she does not see her child, according to all the rules of the contract.

The negative answer of the woman complicates significantly - if it turns out true, the kidnapper must fulfill the terms of the transaction and release the child, but in this way the mother's response will not correspond to reality. To ensure the deceit of such an answer, the crocodile needs to return the child of the mother, but this contradicts the contract, because its mistake should leave the Chado in the crocodile.

It is worth noting that the transaction proposed by a crocodile contains a logical contradiction, so his promise is impracticable. The author of this classic Sophism is the speaker, a thinker and politician Corax Syracuse, who lived in the V-M century to our era.

6. Aritia "Dichotomy"

Another paradox from Zeno Eleas, demonstrating the incorrectness of an idealized mathematical model of motion. The problem can be delivered so - say, you set out the goal to pass any street of your city from the beginning to the end. To do this, you need to overcome the first half of it, then half the remaining half, then half the next segment and so on. In other words, you pass half 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 half, it means that the entire path is impossible. Formulating a somewhat contrived at first view of the paradox, Zeno wanted to show that mathematical laws contradict reality, because in fact you can easily go all the distance without a balance.

7. Aproria "Flying Arrow"

The famous paradox of Zeno Elaysky affects the deepest contradictions in the ideas of scientists about the nature of movement and time. Aporia is formulated as follows: the arrow released from Luke remains stationary, as at any time it rests, without moving. If at each moment of time the arrow rests, it means it is always at rest and does not move at all, since there is no time in which the arrow moves in space.

The outstanding minds of humanity in centuries are trying to resolve the paradox of a flying boom, but from a logical point of view, it is absolutely true. To refute it, it is required to explain how the final time period may consist of an infinite number of time moments - to prove it did not even succeed in Aristotle, convincingly criticized the Aritation of Zenon. Aristotle rightly pointed out that the length of time cannot be considered the sum of some indivisible isolated moments, but many scientists believe that its approach does not differ in depth and does not refute the presence of a paradox. It is worth noting that the formulation of the problem of the flying arrow Zenon was striving not to disprove the possibility of movement, as such, and reveal contradictions in idealistic mathematical concepts.

8. Galilee paradox

In its work, "conversations and mathematical evidence relating to two new industries" Galileo Galilee proposed a paradox showing the curious properties of infinite sets. Scientist formulated two judgments contradictive to each other. The first: There are numbers that are squares of other integers, for example, 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, the total number of accurate squares and conventional numbers should be greater than the number of only accurate squares. The second judgment: for each natural number there is its exact square, and for each square there is a whole square root, that is, the number of squares is equal to the number of natural numbers.

Based on this contradiction, Galile concludes that reasoning about the number of elements is applied only to the final sets, although later mathematics introduced the concept, the power of the set - with its help, the loyalty of the second judgment of the Galilee and for infinite sets was proved.

9. Potato bag paradox

Suppose some farmer has a potato bag 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% the remaining substances contained in potatoes. It solves slightly dry potatoes so that the water content in it decreases to 98% and transfers the bag into 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 it be? Let's calculate - 99% of 100 kg is 99 kg, which means the ratio of the mass of dry residue and the mass of water was originally 1/99. After drying, the water has 98% of the total mass of the bag, which means the ratio of the mass of the 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 find a roughest mathematical error in calculations - an imaginary comic "Potato bag paradox" can be considered an excellent example of how with the help of "logical" and "scientifically supported" reasoning can be literally in an empty place to build a theory contrary to common meaning.

10. Paradox of Voronov

The problem is also known as the paradox of hempel - she received the second name in honor of the German mathematician Karl Gustav Gempel, the author of its classical option. The problem is formulated quite simple: each raven has a black color. It follows from this that everything that is not black, can not be raven. This law is called a logical contaposition, that is, if a certain package "A" has a consequence of "b", the denial of "b" is equivalent to denial "A". If a person sees a black crow, it strengthens his confidence that all crows have a black color, which is quite logical, however, in accordance with the contraposition and the principle of induction, it is natural to say that the observation of objects is not black (let's say, red apples) also proves that All crows are painted in black. In other words, the fact that a person lives in St. Petersburg proves that he lives not in Moscow.

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

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