Gusev D.A. Amazing logic. Sophisms and logical paradoxes Paradoxes of imprecise concepts

Something that is at odds with the usual expectations, common sense and life experience.
A logical paradox is such an unusual and amazing situation when two conflicting judgments are not only true at the same time (which is impossible due to the logical laws of contradiction and the excluded third, but also follow from each other, condition each other. either a trick, a deliberate logical error that can be detected, exposed and eliminated, then a paradox is an insoluble situation, a kind of mental impasse, a stumbling block in logic throughout its history, many different ways of overcoming and eliminating paradoxes have been proposed, but none of them is still not exhaustive, definitive generally accepted.
The most famous logical paradox is the liar's paradox. It is often called
“The king of logical paradoxes. It was discovered back in Ancient Greece. According to legend,
the philosopher Diodorus Kronos vowed not to eat until he resolved this paradox and died of hunger, without achieving anything, while another thinker, Philetus Kossky, fell into despair from the impossibility of finding a solution to the liar's paradox and committed suicide,
throwing herself off a cliff into the sea. There are several different formulations of this paradox. It is most briefly and simply formulated in a situation where a person utters a simple phrase I am a liar. An analysis of this elementary and seemingly unsophisticated statement leads to an overwhelming result. As you know, any statement
(including the above) can be true or false. Let us consider sequentially both cases, in the first of which this statement is true, and in the second it is false.
Let's say that the phrase I am a liar is true, those. the person who said it told the truth,
but in this case he really is a liar, therefore, having uttered this phrase, he lied.
Now suppose the phrase I am a liar is false, those. the person who uttered it lied, but in this case he is not a liar, a truth-lover, therefore, having uttered this phrase, he told the truth. It turns out something amazing and even impossible if a person told the truth, then he lied; if he lied, then he told the truth (two conflicting judgments are not only true at the same time, but also follow from each other).
Another well-known logical paradox, discovered at the beginning of the century by an English logician and philosopher
Bertrand Russell is the paradox of the country barber. Imagine
that in a certain village there is only one hairdresser who shaves those of its inhabitants who do not shave themselves. An analysis of this straightforward situation leads to an extraordinary conclusion.
Let us ask ourselves whether a village hairdresser can shave himself. Consider both options, in the first of which he shaves himself, and in the second he does not.
Let's say that the village hairdresser shaves himself, but then he belongs to those villagers who shave themselves and whom the hairdresser does not shave, therefore, in this case, he does not shave himself. Now suppose that the village hairdresser does not shave himself, but then he belongs to those villagers who do not shave themselves and whom the hairdresser shaves, therefore, in this case, he shaves himself. As you can see, it turns out incredible if the village hairdresser shaves himself, then he does not shave himself;
and if he does not shave himself, then he shaves himself (two conflicting judgments are simultaneously true and mutually condition each other

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The paradoxes of the liar and the village hairdresser, along with other similar paradoxes, are also called antinomies(from the Greek. antinomia - a contradiction in the law "),
that is, by reasoning in which it is proved that two statements that deny each other follow from one another. Antinomies are considered to represent the most extreme form of paradox. However, quite often the terms logical paradox and
"Antinomy" are considered synonymous.
Less surprising formulation, but not less famous than paradoxes
The "liar" and the village hairdresser, has the "Protagoras and Evatl" paradox, which appeared,
like the liar, back in ancient Greece. It is based on a seemingly unpretentious story, which lies in the fact that the sophist Protagoras had a student Evatl, who took lessons in logic and rhetoric from him.
(in this case, political and judicial eloquence. The teacher and student agreed that Evatl would pay Protagoras tuition fees only if he won his first trial. However, after completing his studies, Evatl did not participate in one process and money for the teacher, of course , did not pay. Protagoras threatened him that he would sue him, then Evatla would have to pay in any case. You will either be sentenced to pay a fee, or not, "Protagoras told him," if you are sentenced to pay, you will have to pay by and if you are not sentenced to payment, then you, as the one who won your first trial, will have to pay according to our agreement. To which Evatl answered him All right, I will either be sentenced to pay a fee, or not awarded if I am sentenced to payment, then I, as the loser of my first trial, will not pay according to our agreement, if I am not sentenced to payment, then I will not pay according to the court verdict. whether Evatl should pay Protagoras the fee or not is insoluble. The agreement between the teacher and the student, despite its completely innocent appearance, is internally, or logically, contradictory,
since he requires the execution of an impossible action, Evatl must pay the tuition fees and not pay at the same time. By virtue of this, the very treaty between Protagoras and Evatlus,
and the question of their litigation is something other than a logical paradox.
A separate group of paradoxes are aporias (from the Greek. aporia - difficulty, bewilderment of reasoning that show the contradictions between what we perceive with our senses (see, hear, touch, etc.) and what can be mentally analyzed (in other words, the contradictions between the visible, the thinkable. The most famous aporias were put forward by the ancient Greek philosopher Zeno Eleisky, who argued that the movement observed by us everywhere cannot be made an object of mental analysis, that movement can be seen, but cannot be thought. One of his aporias is called Dichotomy (Greek. dihotomia - halving. Let's say a body needs to go from point A to point B. There is no doubt that we can see
how the body, leaving one point, after some time reaches another. However, let's not trust our eyes, which tell us that the body is moving, and try to perceive the movement not with our eyes, but with a thought, try not to see it, but think. V
In this case, we get the following. Before going all the way from point A to point B, the body needs to go half of this way, because if it does not go half way,
then, of course, my wife will go all the way. But before the body has passed half the way, it needs to go 1/4 of the way. However, before it passes this 1/4 part of the path, it needs to go 1/8 of the way, and even earlier it needs to go 1/16 of the way, and before that -
1/32 part, and before that - 1/64 part, and before that - 1/128 part so to infinity. Means to pass from point A to point B, the body has to go through an infinite number of segments of this path. Is it possible to pass infinity Impossible Consequently, the body never

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will not be able to go its way. Thus, the eyes testify that the path will be traversed by thought, on the contrary, it denies it (the visible contradicts the thinkable).
Another famous aporia of Zeno of Elea - "Achilles and the tortoise - says that
that we can well see how the swift-footed Achilles catches up and overtakes the turtle slowly crawling in front of him, however, mental analysis leads us to the unusual conclusion that Achilles will never be able to catch up with the turtle, although they are moving at once faster than her. When he overcomes the distance to the turtle, then during the same time (after all, she is also moving) it will pass at once less (since it moves at once slower, namely part of the path that Achilles traveled, and this 1/10 part will be ahead of him.
When Achilles passes this 1/10 part of the path, the tortoise will travel at once less distance during the same time, those part of the path and this 1/100 part will be ahead of Achilles.
When he passes 1/100 part of the path separating him and the turtle, then during the same time it will cover 1/1000 part of the path, still remaining ahead of Achilles, so to infinity.
So, we are again convinced that the eyes tell us about one thing, and the thought about something completely different (the visible is denied by the thinkable).
Another aporia of Zeno - Arrow - invites us to mentally consider the flight of an arrow from one point in space to another. Our eyes, of course, indicate that the arrow is flying or moving. However, what will happen if we try, distracting from the visual impression, to think about its flight To do this, we ask ourselves a simple question where the flying arrow is now. now there, then all these answers will mean not the flight of the arrow, but just its immobility, because to be here, or here, or there - means exactly to rest, and not to move. How can wives answer the question - where is the flying arrow now - in such a way that the answer reflects her flight not immobility The only possible answer in this case should be like this She is everywhere now
and nowhere. But is it possible to be everywhere and nowhere at the same time? So, while trying to imagine the flight of an arrow, we came across a logical contradiction, an absurdity - the arrow is everywhere and nowhere. It turns out that the movement of the arrow can be clearly seen, but it cannot be conceived, as a result of which it is impossible, like any movement in general. In other words, moving, from the point of view of thought, and not sensory perceptions, means being in a certain place and not being in it at the same time, which, of course, is impossible.
In his aporias, Zeno confronted the data of the sense organs (talking about the multiplicity, divisibility and movement of everything that exists, assuring us that the swift-footed Achilles will catch up with the slow turtle, and the arrow will fly to the target) and speculation (which cannot conceive of the movement or multiplicity of objects in the world without falling into a contradiction).
Once, when Zeno was proving at a gathering of people the inconceivability and impossibility of movement, among his listeners was a philosopher no less famous in Ancient Greece
Diogenes of Sinop. Without saying anything, he got up and began to pace, believing that by doing this he proved the reality of the movement better than any words. However, Zeno was not taken aback and replied:
“Don't walk and don't wave your hands, but try to solve this difficult problem with your mind.”
There is even the following AC poem about this situation. Pushkin:
There is no movement, said the brown-haired sage,
Another was silent and began to walk before him.
He could not have argued more strongly;
Praised by all the answer is convoluted.
But, gentlemen, this funny case
Another example brings me to mind

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After all, the sun walks before us every day,
However, the stubborn Galileo is right.
Indeed, we see quite clearly that the sun moves across the sky every day from east to west, in fact, it is motionless (in relation to the Earth).
So why don't we assume that other objects that we see moving may in fact be motionless, and not rush to assert that the Eleatic thinker was wrong?
As already noted, many ways of resolving and overcoming paradoxes have been created in logic. However, none of them is free from objections and not generally accepted. Considering these methods is a long and tedious theoretical procedure, which in this case remains beyond our attention. An inquisitive reader will be able to get acquainted with various approaches to solving the problem of logical paradoxes in additional literature. Logical paradoxes are evidence in favor of the fact that logic, like any other science, is incomplete, but constantly evolving. Apparently, paradoxes indicate some deep problems of logical theory, slightly open the veil over something not yet fully known and understandable, outline new horizons in the development of logic

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I disagree with you (Terms and methods of discussion)
An important role in a dispute, or discussion (from lat. discussio - consideration, research, play argumentation, which is the practical application of species,
methods and logical rules of proof in their various combinations. Art criticism of the dispute, as well as the section of logic devoted to the study of its conditions, laws,
methods and techniques, called eristics from the Greek. eristikos -"disputant").
In order for the discussion to be fruitful, those. represented a valid search for truth, and a non-empty talk was a clash of ambition, certain conditions were required.
First, it is necessary to have a certain subject of dispute - a problem, question, topic, etc., otherwise the discussion will inevitably turn into meaningless verbal bickering.
Secondly, it is necessary that with respect to the subject of the dispute there should be a real opposition of the disputing parties, those. they must hold different beliefs about him. Otherwise, the discussion will turn into a discussion of words, opponents will talk about the same thing, but use different terms, thereby inadvertently creating the appearance of a divergence of views.
Third, it is important that there is some common basis for the dispute - some principles, beliefs, ideas, etc., which are recognized by both parties. If there is no such basis, those. the disputants do not agree on one position in general, then the discussion becomes impossible.
Fourth, some knowledge of the subject of the dispute is required. If the parties do not have the slightest idea about him, then the discussion will be devoid of any meaning.
Fifth, the dispute will not lead to any positive result if there are no certain psychological conditions, attentiveness of each discussing party to its opponent, the ability to listen and the desire to understand his reasoning, the willingness to admit his mistake and the correctness of the interlocutor. These are the basic conditions for an effective and fruitful discussion. The absence or violation of at least one of them leads to the fact that
that she does not reach her target. does not establish the truth or falsity of any thesis (statement, position, view, etc.).
The techniques used in the dispute are usually divided into loyal (correct,
acceptable) and disloyal (incorrect, unacceptable).
Loyal methods of controversy are few and simple.
It is possible from the very beginning to seize the initiative in the discussion to offer your own formulation of the subject of the dispute, a plan and timetable for the discussion, to direct the course of the polemic in the direction you need. To retain the initiative, one must not defend, but attack,
that is, to conduct the dispute in such a way that the opponent gets into the position of the defender, who will have to refute your arguments, respond to objections, etc. them.
In a dispute, it is permissible to place the burden of proof on the adversary to turn the discussion in such a way that it is not for you to confirm or deny something,
but to the opponent. Often this technique is enough to end the controversy in your favor, since a person who is poorly proficient in the methods of proof may get confused in his reasoning and will be forced to admit that he is defeated.

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It is advisable to concentrate attention and actions on the weakest link in
in the opponent's arguments, revealing the inconsistency of one or two of the opponent's arguments can lead to the destruction (destruction) of the entire system of his argumentation.
The correct method of discussion is to use the effect of surprise, the most important and strong arguments should be saved until the end of the dispute. Having said them at the end, when the opponent has already exhausted his arguments, you can confuse him and win.
It is quite permissible to take the last word in the discussion and, summing up the results, present its results in a light favorable to you (while, of course, not revising them and not substituting other results, those. false - for truth, etc.).
When the participants in the discussion set themselves the goal of establishing the truth or reaching an agreement, they use only loyal methods. If someone resorts to disloyal methods, it means that he is only interested in winning the dispute, and at any cost.
For such an opponent, discussion is the impossibility of researching something, understanding something, answering some questions, but a means of expressing and asserting one's own ambitions. You should not enter into an argument with such a person, because discussing with him is like speaking Russian with a foreigner who does not know a single Russian word, a lot of time and effort will be spent without any meaning and result.
However, it is advisable to know what constitutes disloyal dispute techniques. This helps to expose their use in a given discussion. Sometimes they are used involuntarily, unconsciously, often they are resorted to in passion. In such cases, an indication of the use of a disloyal technique is an additional argument indicating the weakness of the opponent's position.
Disloyal arguments are a variety of violations of the rules of proof. For example, false, hypothetical or contradictory judgments may be used as arguments, and the rules of inference may be violated.
Most often, the use of disloyal discussion methods is associated with the substitution of the thesis:
instead of proving one proposition, they prove another, which is only apparently similar to the first. For example, the thesis Any rhombus has equal angles is proved as follows: If a triangle has all sides equal, then it also has equal angles.
Therefore, if all sides of a quadrangle are equal, then all angles are equal for it.
A quadrilateral with equal sides is a rhombus, which means that any rhombus has equal
corners. In this case, the thesis is substantiated by replacing the reasoning about rhombuses with the reasoning about triangles from the fact that the equality of the sides of a triangle is equivalent to the equality of its angles, a conclusion is drawn according to which the equality of the sides of a quadrilateral also means the equality of its angles, however, what is true for some geometric objects can be unfair to others. Despite this, the evidence considered at first glance seems to be correct and convincing, those. substitution of the thesis,
on which it is based, it is not immediately noticeable.
Substitution of a thesis is expressed in various forms. Often, in the process of a dispute, a person seeks to formulate the opponent's thesis as broadly as possible, and to narrow his own as much as possible, since a more general position is more difficult to prove than an assertion of a lesser degree of generality. Sometimes one of the disputants begins to ask his opponent many questions, often even irrelevant, in order to divert his attention and drown the argument in lengthy reasoning.
Quite often, the substitution of a thesis manifests itself in the use of synonyms with different semantic connotations. For example, the words ask, beg, intercede, pray, plead

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nyat, being synonyms, denote one and the same action, however, depending on the use of each of these terms, the general meaning of what was said (the context in which they are used) changes somewhat. Synonyms can be positive or negative, laudatory or derogatory. So, the use of the word military instead of the term military or - boys instead of - young people is an implicit substitution of the thesis, it seems to be about the same thing, but the use of a certain synonym already means some kind of assessment, some kind of imperceptible at first glance statement. A variation of this technique is the labeling of the opposite, his position, statements.
Thesis substitution is at the heart of a very common mistake called gendering. It has two types of substitution of the private by the general; the substitution of the general by the private.
In the first case, instead of one proposition, they try to prove another - more general in relation to the first, and therefore stronger. Recall that the truth of a general judgment really determines the truth of a particular (If all crucians are fish, then
some of the crucians are also necessarily fish. However, it may well turn out
that a more general proposition will turn out to be false to substantiate a particular thesis with the help of it. For example, if instead of the statement the Diagonals of any rhombus are mutually perpendicular, they try to prove the more general statement of the Diagonals of any parallelogram
are mutually perpendicular on the basis that all rhombuses are parallelograms, then it turns out that this is impossible, since the second proposition is not true.
In the second case, on the contrary, instead of justifying the general position, they seek to prove the particular and from the truth of the particular statement to deduce the truth of the general, which is not true (If some mushrooms are edible, this does not mean that all mushrooms are edible).
For example, if instead of the statement Any rhombus has equal diagonals prove the particular position Any square has equal diagonals on the grounds that all squares are rhombuses, then the first judgment still remains unfounded, despite the truth of the second.
Very often, an unacceptable method of a dispute in the form of a substitution of the thesis of proof is associated with the use of arguments not on the merits of the case, those. not related to the subject of discussion. Arguments (arguments that are used in the discussion are usually divided into two types. Arguments ad rem (lat. To the case, on the merits) are directly related to the topic of discussion, are directly related to the issue under discussion and are aimed at valid confirmation or refutation of any thesis. Arguments ad

Contents

Introduction

1. Sophisms

1.2 Examples of sophisms

2. Logical paradoxes

2.2 Examples of logical paradoxes

Conclusion

Introduction

Objective, independent of our individual characteristics and desires, principles, or rules of thinking, the observance of which leads any reasoning to true conclusions, provided that the original statements are true, are called the laws of logic.

One of the most important and significant laws of logic is the law of identity. He argues that any thought (any reasoning) must necessarily be equal (identical) to itself, that is, it must be clear, precise, simple, definite. This law prohibits confusing and substituting concepts in reasoning (that is, using the same word in different meanings or putting the same meaning in different words), creating ambiguity, avoiding the topic, etc.

When the law of identity is violated involuntarily, out of ignorance, then there are simply logical errors; but when this law is violated deliberately, in order to confuse the interlocutor and prove to him some false thought, then not just errors appear, but sophisms.

Many sophisms look like a play with language devoid of meaning and purpose; a game based on the ambiguity of linguistic expressions, their incompleteness, understatement, the dependence of their meanings on the context, etc. These sophisms seem especially naive and frivolous.

Logical paradoxes are evidence in favor of the fact that logic, like any other science, is not complete, but constantly evolving.

Sophisms and paradoxes originated in antiquity. Using these logical techniques, our language becomes richer, brighter, more beautiful.

1. Sophisms

1.1 The concept of sophism and its historical origin

Sophism(from Greek - skill, skill, cunning invention, trick, wisdom) - a false conclusion, which, nevertheless, on a superficial examination seems correct. Sophism is based on a deliberate, deliberate violation of the rules of logic.

Being intellectual tricks or tricks, all sophisms are exposed, only in some of them the logical error in the form of violation of the law of identity lies on the surface and therefore, as a rule, is almost immediately noticeable. Such sophisms are not difficult to expose. However, there are sophisms in which the trick is hidden deep enough, well disguised, due to which you need to try to find it.

Example No. 1 simple sophism: 3 and 4 are two different numbers, 3 and 4 are 7, therefore 7 are two different numbers.In this seemingly correct and convincing reasoning, various, non-identical things are mixed or identified: a simple enumeration of numbers (the first part of the reasoning) and a mathematical addition operation (the second part of the reasoning); it is impossible to put an equal sign between the first and the second, violation of the law of identity.

Example No. 2 simple sophism: two times two (that is, two times two) will not be four, but three. Take a match and break it in half. This is one time two. Then take one of the halves and break it in half. This is the second time two. The result is three pieces of the original match. Thus, two times two will not be four, but three.In this reasoning, various things are mixed, the non-identical is identified: the operation of multiplying by two and the operation of dividing by two - one is implicitly substituted for the other, as a result of which the effect of external correctness and convincingness of the proposed "proof" is achieved.

Example No. 3 one of the ancient sophisms attributed to Eubulides: What you have not lost, you have. You haven't lost your horn. So you have horns.The ambiguity of the larger premise is masked here. If it is conceived as universal: "Everything that you did not lose ...", then the conclusion is logically flawless, but not interesting, since it is obvious that the big premise is false; if it is thought of as private, then the conclusion does not follow logically.

Using sophisms, you can also create some kind of comic effect, using the violation of the law of identity.

Example No. 4 : N.V. Gogol in his poem "Dead Souls", describing the landowner Nozdrev, says that he was a historical person, because wherever he appears, some story must have happened to him.

Example No. 5 : Do not stand anywhere, otherwise it will fall.

Example No. 6 : - I broke my arm in two places.

Don't go to these places again.

Examples No. 4,5,6 use the same technique: in the same words different meanings, situations, topics are mixed, one of which is not equal to the other, that is, the law of identity is violated.

2. Logical paradoxes

2.1 The concept of logical paradox and aporia

Paradox(from the Greek. unexpected, strange) - this is something unusual and surprising, something that is at odds with the usual expectations, common sense and life experience.

Logical paradox- this is such an unusual and surprising situation when two conflicting judgments are not only true at the same time (which is impossible due to the logical laws of contradiction and the excluded third), but also follow from each other, condition each other.

A paradox is an insoluble situation, a kind of mental impasse, a "stumbling block" in logic: throughout its history, many different ways have been proposed to overcome and eliminate paradoxes, but none of them is still exhaustive, final and generally recognized.

Some paradoxes (paradoxes of a "liar", "a village hairdresser", etc.) are also called antinomies(from the Greek. contradiction in the law), that is, arguments in which it is proved that two statements denying each other follow from one another. Antinomies are believed to represent the most dramatic form of paradox. However, quite often the terms "logical paradox" and "antinomy" are regarded as synonyms.

A separate group of paradoxes are aporia(from Greek - difficulty, bewilderment) - reasoning that shows the contradictions between what we perceive with the senses (see, hear, touch, etc.), and what can be mentally analyzed (contradictions between the visible and the thinkable) ...

sophism logical paradox language

The most famous aporias were put forward by the ancient Greek philosopher Zeno of Elea, who argued that the movement we observe everywhere cannot be made the subject of mental analysis. One of his famous aporias is called "Achilles and the Turtle". She says that we may well see how the swift-footed Achilles overtakes and overtakes a slowly crawling turtle; however, mental analysis leads us to the unusual conclusion that Achilles can never catch up with the turtle, although he moves 10 times faster than her. When he overcomes the distance to the turtle, then during the same time it will travel 10 times less, namely 1/10 of the path that Achilles traveled, and this 1/10 part will be ahead of him. When Achilles passes this 1/10 part of the path, the turtle will travel 10 times less distance in the same time, that is, 1/100 of the path, and this 1/100 part will be ahead of Achilles. When he passes 1/100 of the path separating him and the turtle, then during the same time it will cover 1/1000 of the path, still remaining ahead of Achilles, and so on ad infinitum. We are convinced that the eyes tell us one thing, and the thought is completely different (the visible is denied by the thinkable).

In logic, many ways have been created to resolve and overcome paradoxes. However, none of them is free from objections and not generally accepted.

The most famous logical paradox is liar paradox ... He is often called the "king of logical paradoxes". It was discovered back in Ancient Greece. According to legend, the philosopher Diodorus Kronos made a vow not to eat until he resolved this paradox and died of hunger, never having achieved anything. There are several different formulations of this paradox. It is most briefly and simply formulated in a situation where a person utters a simple phrase: "I am a liar." The analysis of this statement leads to a startling result. As you know, any statement can be true or false. Suppose that the phrase "I am a liar" is true, that is, the person who uttered it told the truth, but in this case he is really a liar, therefore, having uttered this phrase, he lied. Let us assume that the phrase "I am a liar" is false, that is, the person who uttered it lied, but in this case he is not a liar, but a lover of truth, therefore, having uttered this phrase, he told the truth. It turns out something amazing and even impossible: if a person told the truth, then he lied; and if he lied, then he told the truth (two conflicting judgments are not only truths at the same time, but also flow from each other).

Another well-known logical paradox, discovered in the XX century. English logician and philosopher Bertrand Russell, is the "village hairdresser" paradox. Imagine that in a certain village there is only one hairdresser who shaves those of its inhabitants who do not shave themselves. An analysis of this straightforward situation leads to an extraordinary conclusion. Let us ask ourselves a question: can a country hairdresser shave himself? Let's say that the village hairdresser shaves himself, but then he belongs to those villagers who shave themselves and whom the hairdresser does not shave, therefore, in this case, he does not shave himself. Let's say that the village hairdresser does not shave himself, but then he belongs to those villagers who do not shave themselves and whom the hairdresser shaves, therefore, in this case, he shaves himself. The result is incredible: if a village hairdresser shaves himself, he does not shave himself; and if he does not shave himself, then he shaves himself (two contradictory judgments are both true and mutually dependent on each other).

The Protagoras and Evatl paradox appeared in Ancient Greece. It is based on a seemingly unpretentious story, which lies in the fact that the sophist Protagoras had a student Evatl, who took lessons in logic and rhetoric from him. The teacher and student agreed that Evatl would pay Protagoras tuition fees only if he won his first lawsuit. However, upon completion of his studies, Evatl did not participate in any process and, of course, did not pay the teacher money. Protagoras threatened him that he would sue him and then Evatl would have to pay anyway. "You will either be sentenced to pay a fee, or you will not be awarded," Protagoras said to him. you will have to pay according to our agreement. " To this Evatl replied to him: "Everything is correct: I will either be sentenced to pay the fee or not; if I am sentenced to pay, then, as the loser of my first trial, I will not pay according to our agreement; if I am not sentenced to pay. , then I will not pay by the court verdict. " Thus, the question of whether Evatl should pay Protagoras or not is insoluble. The agreement between the teacher and the student, despite its completely innocent appearance, is internally, or logically, contradictory, since it requires the execution of an impossible action: Evatl must pay for tuition and not pay at the same time. By virtue of this, the very agreement between Protagoras and Evatlus, as well as the question of their litigation, is something other than a logical paradox.

Conclusion

With the help of sophisms, a comic effect can be achieved. Many anecdotes are based on them, and they are also contained in the basis of many tasks and puzzles known to us from childhood. All tricks are based on violation of the law of identity. The magician is doing one thing, and the audience thinks that he is doing something else.

Quite often, sophistry is used by the publishers of mainstream newspapers and magazines for commercial purposes. Walking past the kiosk and seeing the headline, we think one thing, but when, having become interested, we buy this newspaper, it turns out to be completely different. For example: "A first grader ate a crocodile" - it turns out that a first grader ate a large chocolate crocodile.

As you can see, sophisms are used and found in various areas of life.

Paradoxes point to some deep problems of logical theory, lift the veil over something not yet fully known and understandable, outline new horizons in the development of logic. An exhaustive explanation and final resolution of logical paradoxes remains a matter of the future.

List of used literature

1) Getmanova A.D. Logic tutorial. M .: Vlados, 2009.

2) Gusev D.A. A textbook on logic for universities. Moscow: Unity-Dana, 2010

) Ivin A.A. The art of thinking right. Moscow: Education, 2011.

) Koval S. From entertainment to knowledge / Per. O. Unguryan. Warsaw: Primary Technical Publishing House, 2012.

Plan:
Introduction
1 Sophisms
2 Logical paradoxes
2.1 The concept of a logical paradox
2.2 Examples of logical paradoxes
Conclusion
List of used literature
Introduction
Objective, independent of our individual characteristics and desires, principles, or rules of thinking, the observance of which leads any reasoning to true conclusions, provided that the original statements are true, are called the laws of logic.
One of the most important and significant laws of logic is the law of identity. He argues that any thought (any reasoning) must necessarily be equal (identical) to itself, that is, it must be clear, precise, simple, definite. This law prohibits confusing and substituting concepts in reasoning (that is, using the same word in different meanings or putting the same meaning in different words), creating ambiguity, avoiding the topic, etc.
When the law of identity is violated involuntarily, out of ignorance, then there are simply logical errors; but when this law is violated deliberately, in order to confuse the interlocutor and prove to him some false thought, then not just errors appear, but sophisms.
Many sophisms look like a play with language devoid of meaning and purpose; a game based on the ambiguity of linguistic expressions, their incompleteness, understatement, the dependence of their meanings on the context, etc. These sophisms seem especially naive and frivolous.
Logical paradoxes are evidence in favor of the fact that logic, like any other science, is not complete, but constantly evolving.
Sophisms and paradoxes originated in antiquity. Using these logical techniques, our language becomes richer, brighter, more beautiful.
1 Sophisms
Sophism is a false conclusion, which, nevertheless, on a superficial examination seems to be correct. Sophism is based on a deliberate, deliberate violation of the rules of logic.
Aristotle called sophism "imaginary evidence" in which the validity of the conclusion is apparent and is due to a purely subjective impression caused by the lack of logical analysis. At first glance, the persuasiveness of many sophisms, their "consistency" is usually associated with a well-disguised mistake - semiotic: due to the metaphorical nature of speech, homonymy or polysemy of words, amphibole, etc., violating the unambiguity of thought and leading to confusion of the meanings of terms, or logical: the main idea (thesis) of the proof, acceptance of false premises as true, non-observance of acceptable methods of reasoning (rules of logical inference), the use of "unresolved" or even "forbidden" rules or actions, for example division by zero in mathematical sophisms.
Sophisms still appeared in Ancient Greece. They are closely related to the philosophical activities of the sophists - paid teachers of wisdom, who taught everyone philosophy, logic, and especially rhetoric (science and the art of eloquence). One of the main tasks of the sophists was to teach a person to prove (confirm or deny) anything, to emerge victorious from any intellectual competition. For this, they developed a variety of logical, rhetorical and psychological techniques. Sophisms belong to the logical methods of dishonest but successful discussion. However, sophistry alone is not enough to win any dispute. After all, if objective truth is not on the side of the disputant, then in any case he will lose the polemic, despite all his sophistic art. The sophists themselves understood this well. Therefore, in addition to various logical, rhetorical and psychological tricks in their arsenal, there was an important philosophical idea (especially dear to them), which consisted in the fact that no objective truth exists: how many people, so many truths. The Sophists argued that everything in the world is subjective and relative. If we accept this idea as fair, then sophistic art will be quite enough to win any discussion: the winner is not the one who is on the side of the truth, but the one who is better at using the methods of polemics.
The Sophists were ideologically opposed by the famous Greek philosopher Socrates, who argued that there is objective truth, only it is not known exactly what it is, what it is: by virtue of which the task of every thinking person is to seek this truth, common to all.
The debate between the sophists and Socrates about the existence of objective truth originated around the 5th century. BC. Since then, it has continued to the present day. Among our contemporaries, you can find many people who assert that there is nothing objective and universally significant, that everything is equally confirmable and refutable, that everything is relative and subjective. "How many people, so many opinions" is an expression familiar to all of us, which is the undoubted point of view of the ancient sophists. However, in the current era there are those who, following Socrates, believe that although the world and man are complex and multifaceted, nevertheless, something objective and universally significant exists, just as the sun exists in the sky - one for all. They argue that if someone does not notice the objective truth, this does not mean at all that it does not exist, just as if someone closes his eyes and turns away from the sun, he thereby does not cancel his existence in the firmament.
The question of truth is too complicated and always open. It belongs to the category of eternal, or philosophical questions. Most likely, it is impossible to know about its existence or non-existence. However, each of us, in his thoughts, feelings, actions and in general - in life, proceeds from the fact that a single truth does exist, or, conversely, from the fact that it does not exist. The same thing happens with faith in God: it is impossible to prove or disprove his existence, but despite this, one person lives on earth as if God exists, that is, he proceeds in his thoughts and deeds from his existence, and the other on the contrary, he builds his life in such a way as if there is no God, that is, he proceeds in his behavior from his non-existence. It is clear that the life of the first is significantly different from the life of the second and, most likely, one will never understand the other. All of the above applies not only to truth or God, but also to many other very important things, including goodness, conscience, justice, freedom, love. You can proceed in your life from the fact that there really, really or objectively is good, conscience, justice, etc., but you can also proceed from the fact that all these are empty words and do not really exist and behave in an appropriate way.
It can be assumed that a person is an exceptional being in the universe, which is outside the laws of nature and therefore every day in his life must correspond to the name of a person. It is also possible, on the contrary, to proceed from the fact that a person is just one of the natural creatures that obeys the main law of nature - the law of mutual eating and therefore should not at all correspond to some exclusive fictitious name of a person, that is, he can live like an animal. The main thing is that each of us voluntarily and independently chooses what to proceed from in our thoughts and actions, and how to live.
From the point of view of the sophists, if there is no objective truth, then the main thing for victory in any dispute is skillful mastery of the methods of confirming and refuting anything, among which sophisms occupy an important place, in which the law of identity is violated in various ways. Each sophism is based on the fact that in reasoning concepts are substituted, different things are identified, or, on the contrary, identical objects are different.
Historically, the concept of "sophism" is invariably associated with the idea of ​​deliberate falsification, guided by the admission of Protagoras that the task of the sophist is to present the worst argument as the best through clever tricks in speech, in reasoning, caring not about the truth, but about success in an argument or practical benefit. The "criterion of foundation" formulated by Protagoras is usually associated with the same idea: a person's opinion is a measure of truth. Plato already noted that the basis should not lie in the subjective will of a person, otherwise it will be necessary to recognize the legitimacy of contradictions (which, by the way, was argued by the sophists), and therefore any judgments should be considered justified. This thought of Plato was developed in the Aristotelian "principle of consistency" and, already in modern logic, in interpretations and the requirement for proofs of "absolute" consistency. Transferred from the field of pure logic to the field of "factual truths", it gave rise to a special "style of thinking" that ignores the dialectics of "interval situations", that is, situations in which Protagoras' criterion, however, understood more broadly as the relativity of truth to conditions and the means of its cognition turns out to be very essential. That is why many arguments that lead to paradoxes and are otherwise irreproachable qualify as sophisms, although in essence they only demonstrate the interval nature of the epistemological situations associated with them.

2. Logical paradoxes
2.1 The concept of logical paradox and aporia
A paradox is something unusual and surprising, something that is at odds with the usual expectations, common sense and life experience.
A logical paradox is such an unusual and amazing situation when two conflicting judgments are not only true at the same time (which is impossible due to the logical laws of contradiction and the excluded third), but also follow from each other, condition each other.
A paradox is an insoluble situation, a kind of mental impasse, a "stumbling block" in logic: throughout its history, many different ways have been proposed to overcome and eliminate paradoxes, but none of them is still exhaustive, final and generally recognized.
Some
etc.................

One of the deliberate violations of the logical law of identity, as we already know, is sophisms(from the Greek. sophisma -"Fabrication, cunning"), which represent outwardly correct evidence of false thoughts. To be distinguished from sophisms paralogisms(from the Greek. paralogismus -"Incorrect reasoning") - logical mistakes made involuntarily, due to ignorance, inattention or other reasons.

Sophisms appeared in ancient Greece. They are closely related to the philosophical activities of the sophists - paid teachers of wisdom, who taught everyone philosophy, logic and rhetoric (science and the art of eloquence). One of the main tasks of the sophists was to teach a person to prove (confirm or deny) anything, to emerge victorious from any intellectual competition. For this, they developed a variety of logical, rhetorical and psychological techniques. Sophisms belong to the logical methods of dishonest but successful discussion. However, sophistry alone is not enough to win any dispute. After all, if objective truth is not on the side of the disputant, then in any case he will lose the polemic, despite all his sophistic art. The sophists themselves understood this well. Therefore, in addition to various logical, rhetorical and psychological tricks in their arsenal there was an important philosophical idea (especially dear to them), which consisted in the fact that no objective truth exists: how many people, so many truths. The Sophists argued that everything in the world is subjective and relative. If we accept this idea as fair, then sophistic art will be quite enough to win any discussion: the winner is not the one who is on the side of the truth, but the one who is better at using the methods of polemics.

The Sophists were ideologically opposed by the famous Greek philosopher Socrates, who argued that there is objective truth, only it is not known exactly what it is, what it is; by virtue of which the task of every thinking person is to seek this truth, one for all.

The debate between the sophists and Socrates about the existence of objective truth arose around the 5th century BC. NS. Since then, it has continued to the present day. Among our contemporaries, you can find many people who assert that there is nothing objective and universally significant, that everything is equally confirmable and refutable, that everything is relative and subjective. “How many people, so many opinions,” they say. This is undoubtedly the point of view of the ancient sophists. However, in the current era there are those who, following Socrates, believe that although the world and man are complex and multifaceted, nevertheless, something objective and universally significant exists, just as the sun exists in the sky - one for all. They argue that if someone does not notice the objective truth, then this does not mean that it does not exist, just as if someone closes their eyes or turns away from the sun, he thereby will not cancel his existence in the firmament.

The question of truth is too complicated and always open. It belongs to the category of eternal, or philosophical, questions. It is most likely impossible to know about its existence or non-existence. However, each of us in his thoughts, feelings, actions and in general in life proceeds from the fact that the single truth still exists or, conversely, from the fact that it does not exist.

So, from the point of view of the sophists, if there is no objective truth, then the main thing for victory in any dispute is skillful mastery of the methods of confirming and refuting anything, among which sophisms occupy an important place, in which the law of identity is violated in various ways. Each sophism is based on the fact that in reasoning concepts are substituted, different things are identified, or, on the contrary, identical objects are different. Being intellectual tricks or tricks, all sophisms are exposed, only in some of them the logical error in the form of violation of the law of identity lies on the surface and therefore, as a rule, is almost immediately noticeable. Such sophisms are not difficult to expose. However, there are sophisms in which the trick is hidden deep enough, well disguised, due to which it is necessary to pretty much smash your head over them.

Let us recall the example of simple sophism that we have already considered: 3 and 4 are two different numbers, 3 and 4 are 7, therefore 7 are two different numbers. In this seemingly correct and convincing reasoning, various, non-identical things are mixed or identified: a simple enumeration of numbers (the first part of the reasoning) and a mathematical addition operation (the second part of the reasoning); it is impossible to put an equal sign between the first and the second, that is, there is a violation of the law of identity.

Consider another simple sophism: Two times two(i.e. twice two)there will be not four, but three. Take a match and break it in half. This is one time two. Then take one of the halves and break it in half. This is the second time two. The result is three pieces of the original match. Thus, two times two will not be four, but three. In this reasoning, as in the previous one, various things are mixed, the non-identical is identified: the operation of multiplying by two and the operation of dividing by two - one is implicitly substituted for the other, as a result of which the effect of external correctness and persuasiveness of the proposed "proof" is achieved.

Now let us consider a sophism, in which the conclusion, for all its absurdity, seems to be correct, that is, it follows from the original judgments, and the logical error is disguised quite skillfully. As you know, the Earth rotates on its axis from west to east, making a complete revolution in 24 hours. The length of the earth's equator is approximately 40,000 km. Knowing these values, it is easy to determine with what speed each point of the earth's equator is moving. To do this, you need to divide 40,000 km into 24 hours. This turns out to be approximately 1,600 kilometers per hour. At this speed, the Earth rotates at the equator.(Note that there is no catch yet: each point on the earth's equator really moves from west to east at a speed of about 1,600 km per hour). Now imagine that a rail track is laid on the equator, along which a train travels from east to west, that is, in the direction opposite to the rotation of the Earth(she moves east and the train west)... It turns out that this train must constantly overcome the speed of rotation of the Earth, that is, it must move at a speed exceeding 1,600 km per hour, otherwise it will be constantly blown back to the east, and it will not be able to move in the right direction at all. Therefore, such super-trains run at the equator, which develop speeds much higher than 1,600 km per hour. Another conclusion can be drawn from all that has been said: due to the impossibility for trains of such high speeds, they do not run at all at the equator, and there are no railways there. Both of these conclusions, obviously, are not only false, but also absurd, but they quite follow from the above reasoning, which, therefore, is a sophism containing a well-hidden error.

If you offer this sophism to your interlocutor, he will most likely immediately say that the conclusions about trains at the equator are false. However, the task of exposing sophisms is not to state the falsity of their conclusions (which the sophists not only do not hide, but also, on the contrary, emphasize), but to find out what exactly is the logical error of reasoning, what trick it contains how the law of identity is violated (that is, it is necessary to establish what is imperceptibly substituted with what, what is implicitly identified with what, being non-identical). It is unlikely that your interlocutor will be able to quickly cope with this task. Draw his attention to the formal correctness of the conclusions of the proposed reasoning, to the fact that they inevitably follow from the initial statements. To be more convincing, you can complete the sophism about a rotating earth and a moving train with the following comparison: Let's say that the escalator is moving down, and the person is running up it. If its speed is less than the speed of the escalator, it will constantly be driven down. If its speed is equal to the speed of the escalator, it will run in place. In order to get to the top of the escalator, a person must run at a speed greater than the speed of the escalator. In the same way, a train traveling along the equator to the west, against the rotation of the Earth, must move at a speed greater than the speed of rotation of the planet.(that is, it is necessary to overcome more than 1,600 km per hour).

Considering this sophism, one should pay attention to the fact that the point from which the train left and the point to which it should arrive move together with the Earth in the same direction and with the same speed, i.e., their relative position, and hence, the distance between them does not change. Thus, both of these points can be considered as motionless relative to each other. Consequently, no matter how fast a body moves, it will always leave one of the points and will certainly reach the other. Why, in our sophisticated reasoning, did it happen that a train coming from the east needs to develop a very high speed in order to reach its western destination? Because in sophism this western point is regarded as motionless, not taking part in the rotation of the Earth. Indeed, if we assume a certain point somewhere above the earth's surface, which is motionless, then the body moving towards it against the rotation of the Earth, of course, needs to develop a speed greater than the speed of the planet. However, this point (or point) moves with the Earth, and is not at all stationary. In reasoning, the fact of its movement is cunningly and imperceptibly replaced by an implicit statement about its immobility, as a result of which the effect required in sophism is achieved (the law of identity is violated by identifying non-identical phenomena: movement and immobility). Likewise, in the discussion about an escalator moving down, and a person running up it. In order to reach the upper, stationary part of the escalator, a person really needs to run faster than the escalator is moving. If he needs to get not to the upper, stationary part of the escalator, but to the passenger who, standing on the escalator, moves towards him, then in this case, no matter how fast the runner moves up, he will in any case reach the one who moves towards him. In sophism, the western point to which the train is heading is deliberately and incorrectly compared with the stationary part of the escalator, while it must be juxtaposed with some object that moves with the escalator (the fact of movement is imperceptibly replaced by the statement of immobility).

So, any sophism is fully disclosed, or exposed, only if we managed to clearly and definitely establish which non-identical things are deliberately and imperceptibly identified in this or that reasoning. Sophisms are found quite often in the most diverse areas of life. Russian writer V.V. Veresaev in his "Memoirs" says:

“... Pechernikov easily altered my words, slightly shifted my objections to another plane and victoriously refuted them, but I did not know how to keep track of where he moved my thoughts. It was sheer sophistry, and I was powerless against it ... ”. In order not to be powerless against sophistry, we must know well what sophisms are, how they are built, what logical errors they usually hide in ourselves, and always look for some non-identity, less or more disguised, in sophistic reasoning.

Here are a few more sophisms. Please note that in all examples, the conclusions are false, and somewhere their falsehood is obvious, and somewhere not at all.

1. Why does a person need ears? To see. Strange - these are the eyes in order to see, and the ears in order to hear. In fact, this is not the case. After all, the ears hold the cap, and if they were not there, the cap would slide over the eyes and nothing would be visible. Therefore, ears are needed to see.

2. One elderly person proves that his strength, despite his advanced years, has not diminished in the least:

- In my youth and youth, I could not lift a barbell weighing 200 kg and now I cannot, therefore, my strength remained the same.

3. A girl was born to a Chinese family. When she was one year old, a neighbor came to her parents and began to woo the girl for his two-year-old son. Father said:

- My girl is only one year old, and your boy is two, that is, he is twice her age, which means that when my daughter is 20 years old, your son will be 40. Why would I marry my daughter to an old groom ?!

The wife heard these words and objected:

- Now our daughter is one year old, and the boy is two, but in a year she will also be two and they will become the same age, so it is quite possible in the future to marry our girl off as a neighbor's boy.

4. Several people argued about which part of the human body is the most honorable. One said that it was eyes, the other that the heart, the third that the brain. One of the disputants said that the most honorable part of the body is the one on which we sit.

- How do you prove it? They asked him.

He replied:

- People say: whoever sits down first gets the most honor; and the part of the body that I have named always sits down first, therefore, it is the most honorable.

- Of course Africa, because you can see the moon from here, but not Africa!

6. Five excavators dig 5 meters of a ditch in 5 hours. Therefore, in order to dig 100 meters of a ditch in 100 hours, it will take one hundred excavators.

To be distinguished from sophisms logical paradoxes(from the Greek. paradoxes -"Unexpected, strange"). A paradox in the broadest sense of the word is something unusual and surprising, something that is at odds with the usual expectations, common sense and life experience. A logical paradox is such an unusual and amazing situation when two conflicting judgments are not only true at the same time (which is impossible due to the logical laws of contradiction and the excluded third), but also follow from each other, condition each other. If sophism is always some kind of trick, a deliberate logical error that can be detected, exposed and eliminated, then a paradox is an insoluble situation, a kind of mental impasse, a "stumbling block" in logic: throughout its history, many different ways have been proposed overcoming and eliminating paradoxes, however, none of them is still exhaustive, final and generally recognized.

The most famous logical paradox is the liar paradox. He is often called the "king of logical paradoxes." It was discovered back in Ancient Greece. According to legend, the philosopher Diodorus Kronos made a vow not to eat until he resolved this paradox and died of hunger, never having achieved anything; and another thinker, Philetus Kossky, fell into despair from the impossibility of finding a solution to the "liar" paradox and committed suicide by throwing himself off a cliff into the sea. There are several different formulations of this paradox. It is most briefly and simply formulated in a situation where a person utters a simple phrase: I am a liar. An analysis of this elementary and seemingly unsophisticated statement leads to an overwhelming result. As you know, any statement (including the above) can be true or false. Let us consider sequentially both cases, in the first of which this statement is true, and in the second it is false.

Let's say that the phrase I am a liar true, that is, the person who uttered it told the truth, but in this case he is really a liar, therefore, having uttered this phrase, he lied. Now suppose the phrase I am a liar false, that is, the person who uttered it lied, but in this case he is not a liar, but a lover of truth, therefore, having uttered this phrase, he told the truth. It turns out something amazing and even impossible: if a person told the truth, then he lied; and if he lied, then he told the truth (two conflicting judgments are not only true at the same time, but also follow from each other).

Another well-known logical paradox, discovered at the beginning of the 20th century by an English logician and philosopher

Bertrand Russell is the “country barber” paradox. Let's imagine that in a certain village there is only one hairdresser who shaves those of its inhabitants who do not shave themselves. An analysis of this straightforward situation leads to an extraordinary conclusion. Let us ask ourselves a question: can a country hairdresser shave himself? Consider both options, in the first of which he shaves himself, and in the second he does not.

Let's say that the village hairdresser shaves himself, but then he belongs to those villagers who shave themselves and whom the hairdresser does not shave, therefore, in this case, he does not shave himself. Now suppose that the village hairdresser does not shave himself, but then he belongs to those villagers who do not shave themselves and whom the hairdresser shaves, therefore, in this case, he shaves himself. As you can see, the result is incredible: if a village hairdresser shaves himself, he does not shave himself; and if he does not shave himself, then he shaves himself (two conflicting judgments are simultaneously true and mutually condition each other).

The "liar" and "country barber" paradoxes, along with other similar paradoxes, are also called antinomies(from the Greek. antinomia -"Contradiction in the law"), that is, by reasoning in which it is proved that two statements denying each other follow from one another. Antinomies are considered to represent the most extreme form of paradox. However, quite often the terms "logical paradox" and "antinomy" are regarded as synonyms.

A less surprising formulation, but no less famous than the paradoxes of the "liar" and "the village hairdresser", has the paradox "Protagoras and Evatl", which appeared, like the "liar", back in Ancient Greece. It is based on a seemingly unpretentious story, which lies in the fact that the sophist Protagoras had a student Evatl, who took lessons in logic and rhetoric from him.

(in this case, political and judicial eloquence). The teacher and student agreed that Evatl would pay Protagoras tuition fees only if he won his first lawsuit. However, upon completion of his studies, Evatl did not participate in any process and, of course, did not pay the teacher money. Protagoras threatened him that he would sue him and then Evatl would have to pay anyway. “You will either be sentenced to pay a fee, or you will not be awarded,” Protagoras told him. if you are not sentenced to payment, then you, as the winner of your first trial, will have to pay according to our agreement. " To this Evatl answered him: “Everything is correct: I will either be sentenced to pay a fee, or not; if I am sentenced to pay, then I, as the loser of my first trial, will not pay according to our agreement; if I am not sentenced to payment, then I will not pay by the court verdict. " Thus, the question of whether Evatl should pay Protagoras a fee or not is insoluble. The agreement between the teacher and the student, despite its completely innocent appearance, is internally, or logically, contradictory, since it requires the execution of an impossible action: Evatl must pay for tuition and not pay at the same time. By virtue of this, the very agreement between Protagoras and Evatlus, as well as the question of their litigation, is nothing more than a logical paradox.

A separate group of paradoxes are aporia(from the Greek. aporia -"Difficulty, bewilderment") - reasoning that shows the contradictions between what we perceive with the senses (see, hear, touch, etc.), and what can be mentally analyzed (in other words, the contradictions between the visible and the thinkable) ... The most famous aporias were put forward by the ancient Greek philosopher Zeno of Elea, who argued that the movement observed by us everywhere cannot be made the subject of mental analysis, that is, the movement can be seen, but not thought. One of his aporias is called "Dichotomy" (Greek. dihotomia -"bisection"). Let's say a body needs to pass from point A to point V. There is no doubt that we can see how the body, leaving one point, after some time reaches another. However, let's not trust our eyes, which tell us that the body is moving, and try to perceive the movement not with our eyes, but with a thought, try not to see it, but think. In this case, we get the following. Before you go all the way from the point A to point V, the body has to go half of this path, because if it does not go half the way, then, of course, it will not go all the way. But before the body has passed half the way, it needs to go 1/4 of the way. However, before it passes this 1/4 part of the path, it must pass 1/8 of the way; and even earlier, he needs to go through 1/16 part of the path, and before that - 1/32 part, and before that - 1/64 part, and before that - 1/128 part and so on ad infinitum. Means to pass from point A to point V, the body has to go through an infinite number of segments of this path. Is it possible to go through infinity? Impossible! Therefore, the body can never go its way. Thus, the eyes testify that the path will be traversed, but thought, on the contrary, denies it (the visible contradicts the thinkable).

Another famous aporia of Zeno of Elea - "Achilles and the Turtle" - says that we can well see how the swift-footed Achilles overtakes and overtakes the turtle slowly creeping in front of him; however, mental analysis leads us to the unusual conclusion that Achilles can never catch up with the tortoise, although he moves 10 times faster than her. When he overcomes the distance to the turtle, then in the same time (after all, she also moves) will pass 10 times less (since it moves 10 times slower), namely 1/10 of the path that Achilles traveled, and on this 1/10 part will be in front of him.

When Achilles passes this 1/10 part of the path, the tortoise will travel 10 times less distance in the same time, i.e. 1/100 of the path and this 1/100 part will be ahead of Achilles. When he passes 1/100 of the path separating him and the turtle, then during the same time it will cover 1/1000 of the path, still remaining ahead of Achilles, and so on ad infinitum. So, we are again convinced that the eyes tell us about one thing, and the thought about something completely different (the visible is denied by the thinkable).

Another aporia of Zeno - "Arrow" - invites us to mentally consider the flight of an arrow from one point in space to another. Our eyes, of course, indicate that the arrow is flying or moving. However, what will happen if we try, distracting from the visual impression, to contemplate its flight? To do this, let's ask ourselves a simple question: where is the flying arrow now? If, in answering this question, we say, for example, She's here now or She's here now or She's there now then all these answers will mean not the flight of an arrow, but just its immobility, because to be here, or here, or there - means exactly to rest, not to move. How can we answer the question - where is the flying arrow now - in such a way that the answer reflects its flight, and not immobility? The only possible answer in this case should be: She is now everywhere and nowhere. But is it possible to be everywhere and nowhere at the same time? So, when trying to think about the flight of an arrow, we came across a logical contradiction, an absurdity - the arrow is everywhere and nowhere. It turns out that the movement of the arrow can be clearly seen, but it cannot be conceived, as a result of which it is impossible, like any movement in general. In other words, moving, from the point of view of thought, and not sensory perceptions, means being in a certain place and not being in it at the same time, which, of course, is impossible.

In his aporias, Zeno confronted at the "confrontation" the data of the sense organs (which speak of the plurality, divisibility and movement of all that exists, assuring us that the swift-footed Achilles will catch up with the slow turtle, and the arrow will reach the target) and speculation (which cannot conceive of movement or plurality objects of the world, without falling into contradiction).

Once, when Zeno was proving the inconceivability and impossibility of movement at a gathering of people, among his listeners was the philosopher Diogenes of Sinop, no less famous in Ancient Greece. Without saying anything, he got up and began to pace, believing that by doing this he proved the reality of the movement better than any words. However, Zeno was not at a loss and replied: "Don't walk and wave your hands, but try to solve this difficult problem with your mind." About this situation there is even the following poem by A.S. Pushkin:

There is no movement, said the brown-haired sage,
The other was silent and began to walk before him.
He could not have argued more strongly;
Praised by all the answer is convoluted.
But, gentlemen, this funny case
Another example brings me to mind:
After all, the sun walks before us every day,
However, the stubborn Galileo is right.

Indeed, we see quite clearly that the Sun moves across the sky every day from east to west, but in fact it is motionless (in relation to the Earth). So why don't we assume that other objects that we see moving may in fact be motionless, and not rush to assert that the Eleatic thinker was wrong?

As already noted, many ways of resolving and overcoming paradoxes have been created in logic. However, none of them is free from objections and not generally accepted. Considering these methods is a long and tedious theoretical procedure, which in this case remains beyond our attention. An inquisitive reader will be able to get acquainted with various approaches to solving the problem of logical paradoxes in additional literature. Logical paradoxes are evidence in favor of the fact that logic, like any other science, is not complete, but constantly evolving. Apparently, paradoxes point to some deep problems of logical theory, lift the veil over something not yet fully known and understandable, outline new horizons in the development of logic.

An important role in a dispute, or discussion (from lat. discussio -"Consideration, research"), plays argumentation, which is the practical application of types, methods and logical rules of proof in their various combinations. The art of dispute, like the section of logic dedicated to the study of its conditions, patterns, methods and techniques, is called eristics(from the Greek. eristikos -"disputant").

In order for the discussion to be fruitful, that is, to represent a real search for truth, and not empty talk or a clash of ambitions, certain conditions must be met.

First, it is necessary to have a certain subject of dispute - a problem, a question, a topic, etc., otherwise the discussion will inevitably turn into meaningless verbal bickering.

Secondly, it is necessary that there is a real opposition of the disputing parties regarding the subject of the dispute, that is, they must adhere to different beliefs about it. Otherwise, the discussion will turn into a discussion of words: opponents will talk about the same thing, but use different terms, thereby inadvertently creating the appearance of a divergence of views.

Third, it is important that there is some common basis for the dispute - some principles, beliefs, ideas, etc., which are recognized by both parties. If there is no such basis, that is, the disputants do not agree on any position at all, then the discussion becomes impossible.

Fourth, some knowledge of the subject of the dispute is required. If the parties do not have the slightest idea about him, then the discussion will be devoid of any meaning.

Fifth, the dispute will not lead to any positive result if certain psychological conditions are absent: the attentiveness of each discussing party to its opponent, the ability to listen and the desire to understand his reasoning, the willingness to admit his mistake and the correctness of the interlocutor. These are the basic conditions for an effective and fruitful discussion. The absence or violation of at least one of them leads to the fact that it does not achieve its goal, that is, it does not establish the truth or falsity of any thesis (statement, position, view, etc.).

The techniques used in the dispute are usually divided into loyal (correct, acceptable) and disloyal (incorrect, unacceptable).

Loyal receptions disputes are few and simple.

Possibly from the start seize the initiative in the discussion: propose your own formulation of the subject of the dispute, a plan and timetable for the discussion, direct the course of the polemics in the direction you need. To retain the initiative, one must not defend, but attack, that is, conduct the dispute in such a way that the opponent finds himself in the position of the defender, who will have to refute your arguments, respond to objections, etc. Foreseeing possible arguments of the opponent, it is advisable to express them. before he does, and immediately respond to them.

In a dispute it is permissible place the burden of proof on the adversary: turn the discussion in such a way that it was not you, but your opponent, who had to confirm or deny something. Often this technique is enough to end the polemic in your favor, since a person who is poorly proficient in the methods of proof may get confused in his reasoning and will be forced to admit he is defeated.

Desirable focus attention and actions on the weakest link in the opponent's arguments: revealing the inconsistency of one or two of the opponent's arguments can lead to the destruction (destruction) of the entire system of his argumentation.

The correct method of discussion is using the surprise effect: the most important and strong arguments should be saved until the end of the dispute. Having said them at the end, when the opponent has already exhausted his arguments, you can confuse him and win.

It is quite acceptable take the last word in the discussion and, summing up, present its results in a favorable light for you (while, of course, not revising them and not replacing them with other results, that is, without presenting, for example, your defeat as a victory, doubtful - as reliable, false - as truth etc.).

When the participants in the discussion set themselves the goal of establishing the truth or reaching an agreement, they use only loyal methods. If someone resorts to disloyal methods, it means that he is only interested in winning the dispute, and at any cost. For such an opponent, the discussion is not an opportunity to investigate something, understand something, answer some questions, but a means of expressing and asserting one's own ambitions. You should not enter into an argument with such a person, because discussing with him is the same as speaking in Russian with a foreigner who does not know a single Russian word: a lot of time and effort will be spent without any meaning and result. However, it is advisable to know what constitutes disloyal dispute techniques. This helps to expose their use in a given discussion. Sometimes they are used involuntarily, unconsciously, often they are resorted to in passion. In such cases, an indication of the use of a disloyal technique is an additional argument indicating the weakness of the opponent's position.

Disloyal receptions disputes represent a variety of violations of the rules of proof. For example, arguments can be false, hypothetical, or conflicting judgments; violations of the rules of inference are possible.

Most often, the use of disloyal discussion techniques is associated with substitution of thesis: instead of proving one proposition, they prove another, which is only apparently similar to the first. For example, the thesis Any rhombus has equal angles is proved as follows: If all sides of a triangle are equal, then all angles are also equal. Therefore, if all sides of a quadrangle are equal, then all angles are equal for it. A rectangle with equal sides is a rhombus, which means that any rhombus has equal angles. In this case, the thesis is substantiated by replacing the reasoning about rhombuses with the reasoning about triangles: from the fact that the equality of the sides of a triangle is equivalent to the equality of its angles, a conclusion is drawn according to which the equality of the sides of a quadrangle also means the equality of its angles; however, what is true for some geometric objects may not be true for others. Despite this, the considered proof at first glance seems to be correct and convincing, that is, the substitution of the thesis on which it is based is not immediately noticeable.

Substitution of a thesis is expressed in various forms. Often, in the process of a dispute, a person seeks to formulate the opponent's thesis as broadly as possible, and to narrow his own as much as possible, since a more general position is more difficult to prove than an assertion of a lesser degree of generality. Sometimes one of the disputants begins to ask his opponent many questions, often even irrelevant, in order to divert his attention and drown the argument in lengthy reasoning.

Quite often, the substitution of a thesis manifests itself in the use of synonyms with different semantic connotations. For example, the words beg, beg, intercede, beg, beg, being synonyms, they mean the same action, however, depending on the use of each of these terms, the general meaning of what is said (that is, the context in which they are used) changes somewhat. Synonyms can be positive or negative, laudatory or derogatory. So, the use of the word military instead of the term military or - boys instead of - young people represent an implicit substitution of the thesis: we are talking about the same thing, but the use of a certain synonym already means some kind of assessment, some kind of imperceptible at first glance statement. A variation of this technique is "labeling" the enemy, his position, statements.

Thesis substitution is at the heart of a very common mistake called transition to another genus... It has two varieties: the substitution of the private with the general and the substitution of the general with the private.

In the first case, instead of one proposition, they try to prove another - more general in relation to the first, and therefore more "strong". Recall that the truth of a general judgment really determines the truth of a particular ( If all crucians are fish, then some of the crucians are also necessarily fish.). However, it may well turn out that a more general position turns out to be false, and it will not be possible to substantiate a particular thesis with its help. For example, if instead of the statement Diagonals of any rhombus are mutually perpendicular trying to prove a more general statement Diagonals of any parallelogram are mutually perpendicular(on the basis that all rhombuses are parallelograms), then it turns out that it is impossible to do this, since the second proposition is not true.

In the second case, on the contrary, instead of justifying the general position, they seek to prove the particular and from the truth of the particular statement to deduce the truth of the general, which is not true ( If some mushrooms are edible, this does not mean that all mushrooms are edible.). For example, if instead of the statement Any rhombus has equal diagonals prove a private position Any square has equal diagonals(on the grounds that all squares are rhombuses), then the first judgment still remains unfounded, despite the truth of the second.

Very often, an unacceptable method of a dispute in the form of a substitution of the thesis of proof is associated with the use of arguments not on the merits of the case, that is, not related to the subject of discussion. Arguments (arguments) that are used in the discussion are usually divided into two types. Ad rem arguments(lat. "to the case, on the merits of the case") are directly related to the topic of discussion, are directly related to the issue under discussion and are aimed at valid confirmation or refutation of any thesis. Ad hominem arguments(lat. "to a person"), on the contrary, are not connected with the subject of the dispute, have nothing to do with it and are aimed not at proving the thesis under consideration, but at achieving victory in the discussion at any cost.

Let's look at the most common ad hominem arguments.

Personality argument is a substitution of a debatable thesis by a discussion of the opponent's personal characteristics: his appearance, biography, tastes, habits, etc .; moreover, all this is presented, as a rule, in a negative light. For example, the falsity or groundlessness of any statement of the enemy, the weakness of his position is "proved" in something like this: Just look at him! Can this ragamuffin be right ?! He does not have a higher education, and he got a secondary education with difficulty: he barely finished school with a C grade. What smart person can say, especially when you consider that he grew up in the provinces, and his parents grazed cows all their life ... etc.

An example of an argument to personality we find in N.V. Gogol in "The Tale of how Ivan Ivanovich quarreled with Ivan Nikiforovich": scandalous: his sister was a whore known to the whole world and went off for the ranger's company, which stood five years ago in Mirgorod; and wrote down her husband as a peasant. His father and mother were also lawless people, and both were unimaginable drunkards. "

The argument for vanity- this is a kind of argument to the personality: instead of talking on the merits of the case, the personality of the opponent is characterized, but in this case, not in a negative, but in an exaggerated positive light. The adversary is lavished with immoderate praise in the hope that, touched by explicit or veiled compliments, he will become softer and more agreeable, more likely to make any concessions in polemics. For example: I am surprised how you, such a respected and famous scientist, a man of vast knowledge and a sharp mind, the author of many talented books, can adhere to such an obviously untenable point of view ?!

The fable of S. V. Mikhalkov "A Hare in Hop" is an excellent example of an argument for vanity:

... Leo woke up, hearing a drunken cry, -
At that moment our Hare was making his way through the thicket.
Leo - a pin on his collar!
“So that's who got into my clutches!
So it was you who was making noise, you fool?
Wait, you, I see, are drunk -
Some rubbish got drunk! "
All the hops from the head of the Hare went out!
He began to seek salvation from trouble:
“Yes I ... Yes you ... Yes we ... Let me explain!
Have mercy on me! I was visiting now.
There was enough too much. But everything is for you!
For your lion cubs! For your Lioness!
Well, how could you not get drunk here ?! "
And, picking up his claws, Leo let go of the Oblique ...

Argument for authority is an attempt to confirm or deny any position by referring to the opinions, statements, ideas of famous scientists, philosophers, writers, public figures, etc. Let's give an example:

According to legend, the famous Italian scientist of the Renaissance Galileo, having constructed a telescope, discovered spots on the Sun with it and invited a theologian to verify this. He looked through the telescope and said:

- There are no spots on the Sun.

“But you've just seen them yourself! - the scientist was amazed.

- What about what I saw? - the theologian answered calmly. - I read the whole of Aristotle twice. So in his writings nothing is mentioned about spots on the Sun, therefore, they are not.

The fact that a certain famous person adhered or did not adhere to any beliefs does not indicate their truth or falsity. No matter how recognized the authority of this or that figure, one should never forget that it is human nature to make mistakes. Also, just because someone is authoritative in one area does not mean that they are equally authoritative in all other areas. Also, the authority of a person in a certain era cannot be extended to all other eras. And finally, let us remember that authorities are often exaggerated: behind various titles, regalia, positions, and even widespread fame and public recognition, there may be nothing really smart and talented.

Argument for authority Is not necessarily a reference to the beliefs of some famous person. They often refer to the authority of public opinion, the authority of the audience, and even to their own authority. Sometimes fictitious authorities are invented, or assertions are attributed to real authorities that they never made.

Argument for pity- this is the desire to arouse sympathy in the other side and, thereby, to get from her any concessions. For example, a student who is completely unprepared for the exam asks the teacher to show him condescension and put a three just like that (or even a four), motivating this by the fact that he needs to work, support a family, raise children, etc., as a result which time for study is not enough, and therefore he deserves not censure and condemnation, but pity and sympathy. Even if everything that this would-be student says is true, his arguments have nothing to do with the essence of the matter, that is, with the thesis on which he needs to put a three, because the assessment of the level of his knowledge and the circumstances of his personal life are in no way connected together.

An example of an argument to pity is from the story of A.P. Chekhov "A Case from Judicial Practice":

When the assistant prosecutor managed to prove that the defendant was guilty and did not deserve leniency ... the defender stood up ...

“We are human beings, gentlemen of the jury, and we will also be judged humanly! .. Before appearing before you, this man suffered a six-month pre-trial detention. For six months, the wife was deprived of her beloved spouse, the eyes of the children did not dry out from tears at the thought that there was no dear father near them! Oh, if you only looked at these children! They are hungry because there is no one to feed them, they cry because they are deeply unhappy ... But look! They hold out their little hands to you, asking you to return their father to them!

The bailiff stopped looking threateningly and reached into his pocket for a handkerchief ... The prosecutor ... turned restlessly in his chair, blushed and began to look under the table ...

- Take a look at his eyes! - continued the defender ... - Can these gentle, gentle eyes look at the crime with indifference? Oh no! They, those eyes, are crying! Fine nerves are hidden under these Kalmyk cheekbones! Under this rough, ugly breast, a heart that is far from being a crime beats! And you people dare to say that he is to blame ?!

The defendant himself could not bear it ... He blinked his eyes, burst into tears and moved restlessly ...

- I'm sorry! - he spoke, interrupting the defender. - I'm sorry! I admit my guilt! Stole and built fraud! I am a cursed man! .. I confess! It's all to blame!

An argument to the public is designed to attract the audience (present or casual listeners) to their side and turn them against the statements of the opponent. Usually, this effect is achieved by demonstrating that the thesis being defended is somehow connected with the welfare of the listeners, and the refuted position somehow affects and violates their interests, is fraught with certain consequences for them. Suppose an official or a politician running for election tells voters that if they vote for his opponent, then no positive changes will take place in their lives: prices will rise, living standards will fall, social programs will be curtailed, etc .; and if they vote for him, then everything will be different: their aspirations and hopes will certainly come true.

An argument to strength consists in the threat of using any means of coercion in order to persuade his opponent to agree. A person endowed with power, physical strength, or armed, as a rule, has a great temptation to resort to threats in an argument with an intellectually superior opponent. For example, the leaders of the Inquisition, trying to restrain the rapid growth of scientific knowledge that began in the Renaissance, forced advanced scientists, on pain of death, to renounce their views on the structure of the world, which contradicted medieval religious ideas.

In this case, it should be remembered that consent, torn out under the threat of violence, is worthless and does not oblige the consenting person to anything. The famous phrase attributed to Galileo: "But it still turns!" - testifies just to this.

Argument to ignorance is based on using facts unknown to the opponent, attracting ideas unfamiliar to him, mentioning works that he obviously did not read. Many are afraid to admit that they do not know something, it seems to them that this belittles their dignity. In a dispute with such people, the argument to ignorance works flawlessly: trying to hide their ignorance, they are ready to agree with any statements of the opposite side. However, if you freely admit your ignorance of something and ask your opponent to tell you more about it, then it may well turn out that his link has nothing to do with the subject of discussion. Moreover, the adversary may have a very vague idea of ​​what he is referring to, and then he himself will fall into the trap that he was preparing for the other. Finally, relying on the opponent's ignorance, they sometimes use fictitious facts and mention non-existent compositions.

All considered ad hominem arguments, as a rule, are used not in isolation, but in one combination or another. Together with other ways of substituting the thesis and other errors in the proof, they constitute disloyal methods of discussion. Having noticed them in a dispute, one should point out to the enemy that he is resorting to unacceptable methods of conducting polemics and, therefore, is not sure of the strength of his positions. A conscientious person in this case will have to admit that he was mistaken. With an unscrupulous opponent, as already mentioned, it makes no sense to enter into an argument.

In conclusion, we present an excerpt from the story of V. M. Shukshin "Cut". The original character of this story - Gleb Kapustin became famous in his village for the fact that in discussions with visiting "noble people" (scientists, writers, etc.) he always came out the winner, "cut" them. Pay attention to what disloyal arguments he uses in a dispute with Ph.D. Konstantin Zhuravlev.

“- In what area do you reveal yourself? - he asked.

- Where do I work, or what?

- At the philology department.

- Philosophy?

- Not really…

- A necessary thing. - Gleb needed philosophy. He perked up. - Well, what about primacy?

- What is the priority? - the candidate did not understand. And he looked closely at Gleb.

- The primacy of spirit and matter ...

- As always ... Matter is primary.

- And the spirit is secondary. And what?

- Is this the minimum? Excuse me, we are here ... far from the public centers, I want to talk, but you don't really run away - there is no one with whom. How does philosophy now define the concept of weightlessness?

- As always determined. Why now?

- But the phenomenon has been discovered recently, that's why I ask. Natural philosophy, for example, will define it this way, strategic philosophy - in a completely different way ...

- Yes, there is no such philosophy - strategic! - the candidate chuckled.

“Let's say, but there is a dialectic of nature,” Gleb continued with general attention. - And nature is defined by philosophy. Weightlessness has recently been discovered as one of the elements of nature. That is why I ask: is there no confusion among philosophers? The candidate burst out laughing. But he laughed alone ... and felt awkward ...

- Let's establish, - the candidate began seriously, - what are we talking about? What is the subject of our conversation?

- Good. The second question is, how do you personally feel about the problem of shamanism in certain regions of the North? ..

- No such problem! - the candidate slashed from the shoulder.

Now Gleb laughed. And he summed up:

- Well, no, and no trial! A woman with a cart is easier for a horse, - added Gleb. - There are no problems, but these ... - Gleb showed something intricate with his hands, - they are dancing, ringing bells ... Yes? But if you wish ... they seem to be absent. Because if ... Okay! Another question: how do you feel about the fact that the moon is also the work of reason? Scientists have suggested that the moon lies in an artificial orbit, it is assumed that intelligent beings live inside ...

The candidate looked intently at Gleb.

- Where are your calculations of natural areas? Where in general can all space science be applied? The peasants listened attentively to Gleb.

- Assuming the idea that mankind will increasingly visit our, so to speak, neighbor in space, we can also assume that at one fine moment intelligent beings will not stand it and will crawl out to meet us. Are we ready to understand each other?

- Who are you asking?

- You thinkers ...

- Are you ready?

- So, so ... - the candidate looked pointedly at his wife ...

- Invite your wife to laugh? - Gleb asked ... - It's a good thing ... But maybe we will first learn at least to read newspapers? A? What do you think? They say that this does not bother the candidates either.

- Listen!

We've already listened! They had, so to speak, pleasure. So let me tell you, comrade candidate, that the candidacy is not a suit that you bought once and for all. But even the suit must be cleaned sometimes. And the candidacy, if we have already agreed that this is not a suit, all the more it is necessary ... to support. - Gleb spoke quietly, edifying ... It was embarrassing to look at the candidate: he was clearly taken aback, looked now at his wife, then at Gleb, then at the men ... - Of course, you can surprise us here: drive up to the house in a taxi, pull five suitcases out of the trunk ... One can hope that the candidates have not been seen here, but they have been seen here - both candidates, professors, and colonels ... So my advice to you, comrade candidate: get down to earth more often. Honestly, there is a reasonable beginning in this. And it’s not so risky: falling will not hurt so much. ”

An assumption of a scientific nature put forward in order to explain any objects, phenomena, events, etc., is called hypothesis... A hypothesis differs from a simple assumption, for example, a guess, in greater complexity and validity. She plays an important role in the scientific knowledge of the world.

The last two or three centuries are characterized by the fact that such a form of spiritual culture as science has come to the fore in the intellectual life of mankind, having crowded out its other forms - religion, philosophy, art. The present time can rightfully be called the scientist era (from lat. scientia -"Science"), because the face of the modern world is determined primarily by science.

As you know, sciences are divided into natural (or natural science) and humanitarian (also often called social and humanitarian). The subject of natural sciences is nature, studied by astronomy, physics, chemistry, biology, and other disciplines; and the subject of the humanities is man and society, comprehended by psychology, sociology, culturology, history, etc. Let us pay attention to the fact that the natural sciences, in contrast to the humanities, are often called exact. Indeed, the humanities lack the degree of precision and rigor that is characteristic of the natural. Therefore, science in the full sense of the word is usually considered natural science. Even on an intuitive level, science means first of all it. When the word “science” sounds, then first of all thoughts about physics, chemistry and biology come to mind, and not about sociology, cultural studies and history. In the same way, when the word "scientist" sounds, the image of a physicist, chemist or biologist first appears before the mind's eye, and not a sociologist, culturologist or historian. In addition, the natural sciences are far superior to the humanities in their achievements. For 2.5 thousand years (science originated around the 5th century BC in Ancient Greece), natural science and the technology based on it have achieved truly fantastic results: from primitive tools to space flights and the creation of artificial intelligence.

The successes of the humanities are much more modest. Questions related to the comprehension of man and society, by and large, to this day remain unanswered. We know a thousand times more about nature than about ourselves. If a person knew as much about himself as he knows about nature, general happiness would reign on Earth. However, this is not the case. A long time ago, a person fully realized that one cannot kill, steal, lie, etc., that one must live according to the law of mutual assistance, not mutual eating. Nevertheless, the entire history of mankind, starting with the Egyptian pharaohs and ending with the current presidents, is a history of disasters and crimes, which suggests that for some reason a person cannot live as he sees fit and right, cannot do himself and society as they should be according to its ideas. All this is evidence in favor of the fact that, having advanced far in the development of the surrounding world, or nature, a person almost failed to know himself, society and history ... That is why the concepts science, scientific knowledge, scientific achievements etc., as a rule, everything connected with natural science is meant. Therefore, speaking further about science and scientific knowledge, we will mean the natural sciences.

The structure of scientific knowledge includes two levels, or stages: empirical and theoretical. Empirical level(from the Greek. empeiria -"Experience") is the accumulation of various facts observed in nature. Theoretical level(from the Greek. theoria -"Mental contemplation, speculation") is an explanation of the accumulated facts.

You can often hear the erroneous statement that theory follows from facts, or, in other words, that from the first "floor" of scientific knowledge (empirical) to the second (theoretical) there is a smooth transition in the form of some convenient "ladder". In reality, the situation is different and more complicated. The theory does not follow from the facts, for the reason that by themselves they do not testify to anything. Facts are silent, and nothing follows from them, except ... the facts themselves. For example, there is a fact that we constantly observe the slow daytime movement of the Sun across the sky from east to west. What is he talking about? That the Sun revolves around the motionless Earth? Or, perhaps, that, on the contrary, the Earth revolves around the stationary Sun? Or about the fact that both the Sun and the Earth rotate relative to each other? Or maybe not about that, and not about the other, and not about the third, but about something else? As you can see, there are several different and even mutually exclusive explanations for one fact. However, if the explanation of the facts (or theory) followed directly from them, then there would be no disagreement: only one definite explanation would strictly correspond to one fact.

If theory does not follow from facts, then where does it come from? A theory is put forward by the human mind and applied to facts for the purpose of explaining them. Moreover, initially the mind creates not a theory, but a hypothesis, a theoretical assumption, a kind of pre-theory, which is mentally superimposed on the facts. If the hypothesis coordinates (joins) them with each other, connects them into a single picture and even anticipates the discovery of new, still unknown facts, then it will turn into a theory and for a long time will occupy dominant positions in one or another section of scientific knowledge. If, on the contrary, the hypothesis fails to reconcile all the facts available in any area of ​​reality and connect them into a single picture, then it will be discarded and replaced by a new hypothesis. It is impossible to answer the exact question of why a certain scientist puts forward just such a hypothesis and not another to explain some facts, because its creation is in many ways an intuitive act, which is the secret of scientific creativity. Only after correlating a hypothesis with facts is its greater or lesser consistency found out, and its confirmation or refutation occurs. As already mentioned, a hypothesis can be superimposed on facts more or less successfully, and its further fate will depend on this.

The interaction of the empirical and theoretical levels of scientific knowledge can be conditionally compared with the well-known game of children's cubes, which depict fragments of various pictures. Let's say there are nine cubes in a set. Each face of any cube is a fragment of a picture, thus consisting of nine parts. Since the cube has six faces, six different pictures can be made from the set. To make it easier for the child to put the cubes in a certain sequence, six stencil pictures, or drawings, are attached to the set, looking at which he finds the necessary fragments. So, randomly scattered cubes in our analogy are facts, and stencil pictures are mental constructions (hypotheses and theories), on the basis of which they try to organize and connect facts into a certain system. If the desired picture from the cubes is not obtained using the selected stencil pattern, then the wrong pattern is selected and it should be replaced with another one corresponding to the picture that you intend to build. Likewise, if with the help of a hypothesis an ordered picture is not formed from the available facts, then this hypothesis must be replaced by some other. A correctly chosen stencil when drawing cubes is the very hypothesis that is successfully superimposed on the facts, finds its confirmation and turns into a theory.

So, scientific knowledge consists of two "levels": lower - empirical and upper - theoretical. Moreover, the second "floor", being built on top of the first, must crumble without it: the theory is created for this, in order to explain the facts (if they do not exist, then there is nothing to explain). The theoretical level of knowledge is impossible without the empirical, but this does not mean, as already mentioned, that theory follows from facts. For all the interconnection of these two levels, they are, nevertheless, quite autonomous: there is no direct and convenient “ladder” between the lower and upper “floors” of scientific knowledge; is nothing more than the advancement of a hypothesis with its subsequent confirmation and transformation into a theory, or - a refutation and replacement with a new hypothesis.

Most of modern scientific knowledge is built with the help of hypothetical-deductive method , assuming the execution of an algorithm that consists of four links. First, certain facts related to a certain area of ​​reality are discovered. Then an initial hypothesis is put forward, usually called a working hypothesis, which, on the basis of a certain recurrence of the found facts, constructs their simplest explanation. Further, facts are established that do not fit (do not fit) into it. And finally, already taking into account these facts that fall out of the original explanation, a new, more developed, or scientific, hypothesis is created, which not only reconciles all available empirical data, but also allows predicting the receipt of new ones, or, in other words, from which one can derive ( deduce) all known facts, as well as point out unknowns (i.e., not yet discovered). For example, when crossing plants with red and white flowers, the resulting hybrids often have pink flowers. These are discovered facts, on the basis of which it can be assumed (to create a working hypothesis) that the transmission of hereditary traits occurs according to the principle of mixing, that is, parental traits are transferred to the offspring in some intermediate version (such ideas about heredity were widespread in the first half of the 19th century) ... However, other facts do not fit into this explanation. When crossing plants with red and white flowers, albeit infrequently, hybrids still appear not with pink, but with purely red or white flowers, which cannot be the case with averaging inheritance of traits: by mixing, for example, coffee with milk, you cannot get black or white liquid. In order to fit these facts into the general picture, some other explanation of the mechanism of heredity is required, the invention of another, more perfect (scientific) hypothesis is necessary. As you know, it was created in the 60s of the XIX century by the Austrian scientist Gregor Mendel, who suggested that the inheritance of traits does not occur by mixing them, but, on the contrary, through separation. Inherited parental traits are passed on to the next generation using small particles called genes. Moreover, the gene of one of the parents (dominant) is responsible for any trait, and the gene of the other parent (recessive), also passed on to the descendant, does not manifest itself in any way. That is why when crossing plants with red and white flowers in the new generation there can be either only red or only white flowers (one parental trait is manifested, and the other is suppressed). But why do plants with pink flowers also appear? Because often none of the parental traits is suppressed by the other, and both of them are manifested in the offspring. This hypothesis, which so successfully explained and reconciled various facts, later turned into a coherent theory, which laid the foundation for the development of one of the important areas of biology - genetics.

By the way, due to the notions of heredity widespread in the first half of the 19th century, according to which, when traits are transmitted from one generation to another, they are mixed, the evolutionary theory of Charles Darwin, which is based on the principle of natural selection, was under the threat of collapse for a long time. After all, if there is a mixing of inherited traits, it means that they are averaged. Consequently, any trait, even the most beneficial for the organism, which appeared as a result of mutation (sudden change), should eventually disappear, dissolve in the population, which implies the impossibility of natural selection. British engineer and scientist Francis Jenkin proved this rigorously mathematically. "Jenkin's nightmare" for many years poisoned the life of Charles Darwin, but he never found a convincing answer to the question, otherwise the glory of the creator of genetics would have been added to his fame as the author of the evolutionary theory ...

From the point of view of logic, hypotheses are statements, the truth or falsity of which has not yet been established. Therefore, their simplest classification is based on the form of judgments in which they are expressed. Thus, hypotheses are divided into general, particular and individual. General ones are assumptions about the entire set of objects under study, particular ones - about some elements of any set, single ones - about specific, separate objects or phenomena. For example, a hypothesis: The capabilities of any human organism in ordinary life conditions are involved to a very insignificant extent. is common. Another example of a general hypothesis:

In the 40s of the XIX century, G. T. Fechner expressed the idea that an electric current is the movement of positive and negative electrical particles along a conductor in opposite directions. He substantiated the probability of this position, proceeding from the phenomenon of electromagnetic induction, discovered by Faraday in 1831, and the law of interaction of two current elements, formulated by Ampere in 1820.

Hypothesis Some stars of our Galaxy have satellites-planets on which there are favorable conditions for the origin and further evolution of various forms of life. refers to private.

A particular hypothesis is also the following:

Electric charge is ... the most important characteristic of elementary particles. All known particles have a positive, negative or zero charge. Each particle, except for a photon and two mesons, corresponds to antiparticles with the opposite charge. Around 1963-1964, a hypothesis was put forward about the existence of quarks - particles with a fractional electric charge. This hypothesis has not yet found experimental confirmation.

Hypothesis Life on Mars may exist in the early stages of its development - in the form of viruses and bacteria is single. The hypothesis below, which explains the mystery of the Tunguska meteorite, is also a single one.

According to N. Dombkovsky, in the area of ​​the epicenter, where quite recently geologists have found a rich deposit of gas condensate, a huge cloud of explosive gases flowed out of the faults. Early in the morning, when calm reigned and the rays of the rising sun had not yet touched the gas, a red-hot fireball flew into this cloud. He played the role of a kind of trigger, a burning match brought to a barrel of gasoline. A powerful explosion turned the meteorite itself into steam, destroyed all living things around ...

Plan:

I. Introduction

II. Zeno's aporias

Achilles and the turtle

Dichotomy

III . The Liar's Paradox

IV . Russell's paradox

I ... Introduction.

The paradox is two opposite, incompatible statements, for each of which there are seemingly convincing arguments. The sharpest form of the paradox is antinomy, an argument proving the equivalence of two statements, one of which is the negation of the other.

Paradoxes are especially famous in the most rigorous and exact sciences - mathematics and logic. And this is no coincidence.

Logic is an abstract science. There are no experiments in it, not even facts in the usual sense of the word. In building its systems, logic ultimately proceeds from the analysis of real thinking. But the results of this analysis are synthetic. They are not statements of any separate processes or events that the theory should explain. Obviously, such an analysis cannot be called observation: there is always a concrete phenomenon.

When constructing a new theory, a scientist usually starts from facts, from what can be observed in experience. No matter how free his creative imagination is, it must reckon with one indispensable circumstance: a theory makes sense only if it is consistent with the facts related to it. A theory that is at variance with facts and observations is far-fetched and has no value.

But if in logic there are no experiments, no facts and no observation itself, then what is the restraining of logical fantasy? What factors, if not facts, are taken into account when creating new logical theories?

The discrepancy between logical theory and the practice of real thinking is often revealed in the form of a more or less acute logical paradox, and sometimes even in the form of a logical antinomy, which speaks of the internal inconsistency of the theory. This explains the importance that is attached to paradoxes in logic, and the great attention that they use in it.

One of the first and perhaps the best paradoxes was recorded by Euboulides, a Greek poet and philosopher who lived in Crete in the 6th century BC. NS. In this paradox, the Cretan Epimenides claims that all Cretans are liars. If he is telling the truth, then he is lying. If he is lying, then he is telling the truth. So who is Epimenides - a liar or not?

Another Greek philosopher Zeno of Elea compiled a series of paradoxes about infinity - the so-called “aporias” of Zeno.

What Plato said is a lie.
Socrates

Socrates speaks only the truth.
Plato

II. Aporias of Zeno.

The Eleatics (residents of the city of Elea in southern Italy) made a great contribution to the development of the theory of space and time, to the study of problems of movement. The philosophy of the Eleatics was based on the idea put forward by Parmenides (Zeno's teacher) about the impossibility of non-being. Every thought, Parmenides argued, is always a thought about existence. Therefore, there is no non-existent. There is also no movement, since the world space is filled with everything entirely, which means that the world is one, there are no parts in it. Any multitude is a deception of the senses. From this follows the conclusion about the impossibility of occurrence, destruction. According to Parmenides, nothing arises or is destroyed. This philosopher was the first who began to prove the positions put forward by thinkers.

The Eleatics proved their assumptions by denying the opposite of the assumption. Zeno went further than his teacher, which gave Aristotle the basis for seeing Zeno as the founder of "dialectics" - this term was then called the art of reaching truth in a dispute by clarifying the contradictions in the opponent's judgment and by eliminating these contradictions.

Achilles and the turtle. Let us begin our consideration of Zeno's difficulties with the aporias about motion “ Achilles and the tortoise "... Achilles is a hero and, as we would say now, an outstanding athlete. The turtle is known to be one of the slowest animals. However, Zeno claimed that Achilles would lose the race to the tortoise. Let us accept the following conditions. Let Achilles be separated from the finish line by distance 1, and the tortoise - by ½. Achilles and the turtle begin to move at the same time. For the sake of definiteness, let Achilles run 2 times faster than a turtle (that is, he walks very slowly). Then, having run a distance of ½, Achilles will find that the turtle has managed to overcome the segment ¼ in the same time and is still in front of the hero. Then the picture repeats itself: after running the fourth part of the path, Achilles will see the turtle one-eighth of the way ahead of him, etc. Consequently, whenever Achilles overcomes the distance separating him from the turtle, the latter manages to crawl away from him and still remains ahead. Thus, Achilles will never catch up with the turtle. Once Achilles starts moving, he can never complete it.

Those who know mathematical analysis usually indicate that the series converges to 1. Therefore, they say, Achilles will travel all the way in a finite period of time and will certainly overtake the turtle. But here is what D. Hilbert and P. Bernays write on this matter:

“Usually they try to get around this paradox by arguing that the sum of an infinite number of these time intervals nevertheless converges and, thus, gives a finite period of time. However, this reasoning does not at all touch on one essentially paradoxical moment, namely the paradox, which consists in the fact that some infinite sequence of successive events, a sequence, the completion of which we cannot even imagine (not only physically, but at least in principle) , in fact, it should be completed after all ”.

The fundamental incompleteness of this sequence lies in the fact that it lacks the last element. Each time, having indicated the next member of the sequence, we can indicate the next one after it. An interesting remark, also indicating the paradoxical nature of the situation, is found in G. Weil:

“Imagine a computer that would perform the first operation in ½ minute, the second in ¼ minutes, the third in ⅛ minutes, and so on. Such a machine could, by the end of the first minute,“ recalculate ”the entire natural series (write, for example, countable number of units). It is clear that work on the design of such a machine is doomed to failure. So why does the body, emerging from point A, reach the end of the segment B, having “counted off” the countable set of points A 1, A 2, ..., A n, ...? "

Dichotomy . The reasoning is very simple. In order to travel the entire path, a moving body must first travel half the way, but to overcome this half, it must travel half a half, and so on, ad infinitum. In other words, under the same conditions as in the previous case, we will deal with an inverted series of points: (½) n, ..., (½) 3, (½) 2, (½) 1. If in the case of aporia Achilles and the turtle the corresponding series did not have the last point, then in Dichotomies this row does not have the first point. Hence, Zeno concludes, the movement cannot begin. And since the movement not only cannot end, but also cannot begin, there is no movement. There is a legend about which A. Pushkin recalls in the poem "Movement":

There is no movement, said the sage, the brown-haired man.

The other was silent and began to walk before him.

He could not have argued more strongly;

Praised by all the answer is convoluted.

But, gentlemen, this funny case

Another example brings me to mind:

After all, every day the sun goes down before us,

However, the stubborn Galileo is right.

Indeed, according to legend, one of the philosophers "objected" to Zeno. Zeno ordered to beat him with sticks: after all, he was not going to deny the sensory perception of the movement. He talked about him unthinkable, that strict thinking about movement leads to insoluble contradictions. Therefore, if we want to get rid of aporias in the hope that this is generally possible (and Zeno just believed that it was impossible), then we must resort to theoretical arguments, and not refer to sensory evidence. Consider one curious theoretical objection that has been raised against the aporia Achilles and the turtle .

“Let us imagine that fast-footed Achilles and two turtles are moving along the road in the same direction, of which Turtle-1 is somewhat closer to Achilles than Turtle-2. To show that Achilles will not be able to overtake Turtle-1, we reason as follows. During the time Achilles runs the distance separating them at first, Turtle-1 will have time to crawl forward a little, while Achilles runs this new segment, she will again move further, and this position will repeat itself endlessly. Achilles will get closer and closer to Turtle 1, but he will never be able to overtake her. Such a conclusion, of course, contradicts our experience, but we do not yet have a logical contradiction.

Let, however, Achilles begin to catch up with the more distant Turtle-2, not paying any attention to the close one. The same line of reasoning allows us to assert that Achilles will be able to get close to Turtle-2, but this means that he will overtake Turtle-1. Now we come to a logical contradiction ”.

It is difficult to argue with anything here if you remain in captivity of figurative representations. It is necessary to identify the formal essence of the matter, which will translate the discussion into a channel of rigorous reasoning. The first aporia can be reduced to the following three statements:

1. Whatever the segment, the body moving from A to B must visit all points of the segment.

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

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

This conclusion can be illustrated in different ways. The most famous illustration - “the fastest can never catch up with the slowest” - was discussed above. But we can offer a more radical picture, in which Achilles sweating (who left point A) unsuccessfully tries to overtake a turtle, calmly basking in the Sun (at point B) and does not even think to run away. This does not change the essence of the aporia. An illustration will then be a much more poignant statement - "the fastest can never catch up with the motionless." If the first illustration is paradoxical, then the second is even more so.