Entertaining and paradoxical logic. Syllogisms

Since ancient times, scientists and thinkers have loved to entertain themselves and their colleagues by posing insoluble problems and formulating various kinds of paradoxes. Some of these thought experiments remain relevant for thousands of years, which indicates the imperfections of many popular scientific models and “holes” in generally accepted theories that have long been considered fundamental. We invite you to reflect on the most interesting and surprising paradoxes, which, as they now say, “blew the minds” of more than one generation of logicians, philosophers and mathematicians.

1. Aporia "Achilles and the Tortoise"

The Achilles and the Tortoise Paradox is one of the aporias (logically correct but contradictory statements) formulated by the ancient Greek philosopher Zeno of Elea in the 5th century BC. Its essence is as follows: the legendary hero Achilles decided to compete in a race with a turtle. As you know, turtles are not known for their agility, so Achilles gave his opponent a head start of 500 m. When the turtle overcomes this distance, the hero sets off in pursuit at a speed 10 times greater, that is, while the turtle crawls 50 m, Achilles manages to run the 500 m handicap given to him . Then the runner overcomes the next 50 m, but at this time the turtle crawls away another 5 m, it seems that Achilles is about to catch up with her, but the rival is still ahead and while he runs 5 m, she manages to advance another half a meter and so on. The distance between them is endlessly shrinking, but in theory, the hero never manages to catch up with the slow turtle; it is not much, but is always ahead of him.

Of course, from the point of view of physics, the paradox makes no sense - if Achilles moves much faster, he will in any case get ahead, but Zeno, first of all, wanted to demonstrate with his reasoning that the idealized mathematical concepts of “point in space” and “moment of time” do not too suitable for correct application to real movement. Aporia exposes the discrepancy between the mathematically sound idea that non-zero intervals of space and time can be divided indefinitely (so the turtle must always stay ahead) and the reality in which the hero, of course, wins the race.

2. Time loop paradox

Paradoxes involving time travel have long been a source of inspiration for science fiction writers and creators of science fiction films and TV series. There are several options for time loop paradoxes; one of the simplest and most graphic examples of such a problem was given in his book “The New Time Travelers” by David Toomey, a professor at the University of Massachusetts.

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

3. The paradox of a girl and a boy

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

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

4. Jourdain's card paradox

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

Imagine holding a postcard in your hands that says, “The statement on the back of the postcard is true.” Turning the card over reveals the phrase “The statement on the other side is false.” As you understand, there is a contradiction: if the first statement is true, then the second is also true, but in this case the first must be false. If the first side of the postcard is false, then the phrase on the second cannot be considered true either, which means that the first statement again becomes true... An even more interesting version of the liar paradox is in the next paragraph.

5. Sophistry “Crocodile”

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

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

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

6. Aporia "Dichotomy"

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

7. Aporia "Flying Arrow"

The famous paradox of Zeno of Elea touches on the deepest contradictions in scientists’ ideas about the nature of motion and time. The aporia is formulated as follows: an arrow fired from a bow remains motionless, since at any moment in time it is at rest and does not move. If at every moment of time the arrow is at rest, then it is always in a state of rest and does not move at all, since there is no moment in time at which the arrow moves in space.

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

8. Galileo's paradox

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

Based on this contradiction, Galileo concluded that reasoning about the number of elements applied only to finite sets, although later mathematicians introduced the concept of power of a set - with its help, the validity of Galileo’s second judgment was proven for infinite sets.

9. The Potato Bag Paradox

Let's say a certain farmer has a bag of potatoes weighing exactly 100 kg. Having examined its contents, the farmer discovers that the bag was stored in damp conditions - 99% of its mass is water and 1% are other substances contained in potatoes. He decides to dry the potatoes a little so that their water content drops to 98% and moves the bag to a dry place. The next day it turns out that one liter (1 kg) of water has indeed evaporated, but the weight of the bag has decreased from 100 to 50 kg, how can this be? Let's calculate - 99% of 100 kg is 99 kg, which means the ratio of the mass of dry residue to the mass of water was initially equal to 1/99. After drying, water accounts for 98% of the total mass of the bag, which means the ratio of the mass of the dry residue to the mass of water is now 1/49. Since the mass of the residue has not changed, the remaining water weighs 49 kg.

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

10. The Raven Paradox

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

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

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This episode with the clever missionary is one of the paraphrases of the paradox of the ancient Greek philosophers Protagoras and Euathlus.

But every researcher who tried to strictly define all the concepts in his theory encountered a similar paradox of formal logic. No one has ever succeeded in this, since everything ultimately came down to a tautology like: “Motion is the movement of bodies in space, and movement is the movement of bodies in space.”

Another version of this paradox. Someone has committed a crime punishable by death. At the trial he has the last word. He must say one statement. If it turns out to be true, the criminal will be drowned. If it is false, the criminal will be hanged. What statement must he make to completely confuse the judge? Think for yourself.

Puzzled by this paradox, Protagoras devoted a special essay to this dispute with Euathlus, “The Litigation of Payment.” Unfortunately, it, like most of what Protagoras wrote, has not reached us. The philosopher Protagoras immediately felt that behind this paradox there was hidden something essential that deserved special study.

Aporia of Zeno of Elea. According to the laws of formal logic, a flying arrow cannot fly. A flying arrow at every moment of time occupies an equal position, that is, it is at rest; since it is at rest at every moment of time, it is at rest at all moments of time, that is, there is no moment in time at which the arrow moves and does not occupy an equal place.

This aporia is a consequence of the idea of the discreteness of movement, that a moving body in discrete units of time passes discrete intervals of distance, and the distance is the sum of an infinite number of indivisible segments that the body passes. This aporia raises a deep question about the nature of space and time - about discreteness and continuity. If our world is discrete, then movement in it is impossible, and if it is continuous, then it is impossible to measure it with discrete units of length and discrete units of time.

Formal logic is based on the concept of discreteness of the world, the beginning of which should be sought in the teachings of Democritus about atoms and emptiness, and perhaps in the earlier philosophical teachings of ancient Greece. We do not think about the paradoxical nature of formal logic when we say that speed is the number of meters or kilometers traveled by a body, which it travels per second or per minute (physics teaches us that distance divided by time is speed). We measure distance in discrete units (meters, kilometers, versts, arshins, etc.), time - also in discrete units (minutes, seconds, hours, etc.). We have a standard distance - a meter, or another segment with which we compare the path. We measure time with the standard of time (essentially, also a segment). But distance and time are continuous. And if they are discontinuous (discrete), then what is at the junctions of their discrete parts? Other world? Parallel world? Hypotheses about parallel worlds are incorrect, because... are based on reasoning according to the laws of formal logic, which assumes that the world is discrete. But if it were discrete, then movement would be impossible in it. This means that everything in such a world would be dead.

Indeed, this paradox is unsolvable in binary logic. But it is precisely this logic that underlies most of our reasoning. From this paradox it follows that a true judgment about something cannot be built within the framework of this something. To do this you need to go beyond it. This means that the Cretan Epimenides cannot objectively judge the Cretans and give them characteristics, since he himself is a Cretan.

The liar paradox.“What I am saying now is false,” or “This statement is a lie.” This paradox was formulated by the philosopher of the Megarian school Eubulides. He said: “The Cretan Epimenides said that all Cretans are liars.” . If Epimenides is right that all Cretans are liars, then he is also a liar. If Epimenides is a liar, then he lies that all Cretans are liars. So are the Cretans liars or not? It is clear that this chain of reasoning is flawed, but in what way?

In science, this means that it is impossible to understand and explain a system based on the elements of only this system, the properties of these elements and the processes occurring within this system. To do this, we should consider the system as part of something larger - the external environment, a system of a larger order, of which the system we are studying is a part. In other words: in order to understand the particular, one must rise to the more general.

The paradox of Plato and Socrates
Plato: “The following statement of Socrates will be false.”
Socrates: “What Plato said is true.”
That is, if we assume that Plato is telling the truth, that Socrates is lying, then Socrates is lying, that Plato is telling the truth, then Plato is lying. If Plato lies that Socrates is lying, then Socrates is telling the truth that Plato is right. And the chain of reasoning returns to the beginning.

This paradox is that within the framework of formal logic, a judgment can be both true and false. This statement, which constitutes the liar paradox, is neither provable nor refutable in formal logic. It is believed that this statement is not a logical statement at all. An attempt to resolve this paradox leads to triple logic, complex logic.

This paradox shows the imperfection of formal logic, simply - its inferiority.

This paradox suggests that in order to characterize the elements of a system by the elements of this system, it is required that the number of elements in this system be more than two. Thesis and antithesis are not enough to characterize an element. If a statement is not true, then it does not follow that it is false. Conversely, if a statement is not false, this does not mean that it is true. It is not easy for our minds to agree with this statement, because we use formal alternative logic. And the case with the statements of Plato and Socrates suggests that this is possible. Judge for yourself: they tell us: “The ball in the box is not black.” If we think that it is white, then we may be mistaken, since the ball may turn out to be blue, red, or yellow.

In the last two examples we see that paradoxes are born from the defectiveness of formal (binary) logic. Let us think about how the phrase should be constructed correctly: “History teaches a person, but he learns nothing from history.” In such a formulation, with such clarification, there is no longer any paradox. The last two paradoxes are not antinomies; they can be eliminated within the framework of the laws of formal logic by constructing the phrase correctly.

The barber does not shave himself; Russell's paradox forbids him to do so. Photo from the site: http://positivcheg.ru/foto/837-solidnye-dyadenki.html

Russell's Paradox: Does the set of all sets contain itself if the sets included in it do not contain themselves (are empty sets)? Russell popularized it in the form of the “barber paradox”: “Barbers only shave people who don’t shave themselves. Does he shave himself?

There is a paradox of definition here: We began to build a logical construction without defining what a set is. If the barber is part of the multitude of people whom he shaves, then he must also charge himself for shaving. So what is the definition? But scientists often operate with concepts that they do not define in any way, which is why they cannot understand each other and argue pointlessly.

The concept of "empty set" is absurd by definition. How can a set be empty, containing nothing? The barber is not one of the many people whom he shaves as a barber. After all, any man shaves himself not like a barber, but like a shaving man. And a man who shaves is not a barber, since he does not charge himself for it.

A paradox from the category of antinomies is generated by an error in reasoning, in the construction of a phrase. The following paradox also applies to antinomies.

In this case, we must remember that a person must learn to think, and not just remember. Learning as mechanical memorization has no great value. Approximately 85-90% of what a person remembers while studying at school and university, he forgets during the first 3-5 years. But if he was taught to think, then he has mastered this skill almost all his life. But what will happen to people if, during training, they are given to memorize only those 10% of information that they remember for a long time? Unfortunately, no one has ever conducted such an experiment. Although...

There was one man in our village who completed only the 4th grade of school in the early 30s. But in the 60s, he worked as the chief accountant of a collective farm and did a better job than the accountant with a secondary technical education who later replaced him.

But if a ship is defined as a system, the essence of which is determined by its properties as a whole: weight, displacement, speed, efficiency and other characteristics, then even when all parts are replaced with similar parts, the ship remains the same. The properties of the whole differ from the properties of its parts and cannot be reduced to the properties of these parts. The whole is greater than the sum of its parts! Therefore, even at 50 years old, a person remains himself, although 95% of the atoms of his body have already been replaced many times during this time by others, and there are more atoms in his body than there were at the age of 10 years.

So the ancient philosopher was not entirely right when he said that you cannot enter the same river twice, since the water flows in it and all the time its molecules in the stream are replaced. In this case, it is implicitly postulated that the river is the sum of precisely these water molecules and no other water molecules. But this is not so, because we perceive a river not as a set of water molecules, but as a flow of a certain depth and width, with a certain flow speed, in a word, a river is a dynamic system, and not the sum of its parts.

Balding orangutan. Photo from the site: http://stayer.35photo.ru/photo_125775

Balding dandelion. Photo from the site: http://www.fotonostra.ru/4101.html

Often the answer to the question about baldness lies in a different plane than the one in which it was formulated. To answer such a question, one must move from one plane of reasoning and perception to a completely different one. For example, the publications of one scientist are cited 100 times a year, and another 1 time a year. Question: which of them is a brilliant scientist? There can be four different answers to this question: 1 - no one, 2 - both, 3 - the first, 4 - the second. And all four answers in this case are equally probable, since the number of citations, in principle, cannot be a sign of genius. The correct answer to this question can only be obtained in 100 years or a little less.

The absurdity in this case stems from the lack of a clear definition of the concept of “democracy”. If the social system (state) is to be democratic, then equal representation from voters should be achieved. Equal representation from states if their populations are different is not a principle of democracy, but something else. Equal representation from parties is something third, from religious denominations - fourth, etc.

The paradox of democracy(voting): “It is impossible to combine all the requirements for an electoral system in one system.” If you achieve equal representation in parliament from states or regions, then it is impossible to achieve equal representation in parliament from voters. But there are still religious denominations, etc.

But in politics, even formal logic is not held in high esteem, and often it is violated deliberately in order to fool the electorate. In the USA, “brain powdering” technologies are simply excellently developed. Their elections are not democratic, but majoritarian, but Americans firmly believe that they have a democratic state and are ready to tear apart anyone who thinks differently about their social system. They manage to pass off the aristocratic form of government as democratic. Are democratic elections possible in principle?

But in practice, the Monte Carlo conclusion may be false for another reason. After all, the condition about the independence of elementary events when playing roulette may not be satisfied. And if elementary events are not independent, but “linked” to each other in both ways known to us and still unknown... then in this case it is better to bet on black rather than red.

It may turn out that there are other carriers of energy and information in the Universe, and not just oscillations of the electromagnetic field and flows of elementary particles. If at its core the Universe is not discrete (vacuum), but continuous, then this paradox is inappropriate. Then every part of the Universe is influenced by the rest of it, then every atom of the universe is connected and interacts with all other atoms, no matter how far they are from it. But in the infinite Universe there must be an infinite number of atoms... Stop! The brains are starting to boil again.

This paradox stems from our misunderstanding of what time is. If time is the flow of the world with many channels (as is often the case with a river), and the speed of flow in the channels is different, then a sliver that falls into a fast channel will then fall into a slow channel again, when the fast channel merges with the slow one in which another sliver is floating , with which they once sailed next. But now one sliver will be ahead of its “friend” and will no longer meet with her. To meet them, the lagging “friend” must get into another fast channel, and the one ahead must swim in a slow channel at the same time. It turns out that the twin brother, who flew away on a sublight ship, in principle cannot return to the past and meet his brother. The slow flow of time (sublight ship) delayed him in the time flow. During this time, his brother not only grew older, but he went into the future, and with him everything that surrounded him went into the future. So, in principle, a brother who has fallen behind in time will no longer be able to get into the future.

And if the river of time does not have channels at different speeds, then there can be no paradox. Maybe the theory of relativity is incorrect, and time is not relative, but absolute?

The paradox of the murdered grandfather: you travel back in time and kill your grandfather before he met your grandmother. Because of this, you will not be able to be born and, therefore, you will not be able to kill your grandfather.

This paradox proves that traveling into the past is impossible. In order to get into the past, a person needs to turn into a different entity - move into the five-dimensional space of time, in which the past, present and future exist together - fused together, he will have to be born, die and live, and all this in the form of some kind of consubstantial phenomenon when "to be born, to live and to die" are not separate from each other. Becoming such a creature for a person means certain death - disintegration into subatomic particles. In general, we live in a four-dimensional world, and the way to the fifth-dimensional world is barred to us.

And thank God! Therefore, the grandfather is not in danger of his grandson coming from the future and killing him. And today there are many such grandchildren who have smoked marijuana.

China's Central Bureau of Film, Radio and Television recently banned time travel films because they "show disrespect for history." Film critic Raymond Zhou Liming explained the reasons for the ban by saying that now time travel is a popular topic in TV series and movies, but the meaning of such works, as well as their presentation, is very questionable. “Most of them are completely fictitious, do not follow logic and do not correspond to historical realities. Producers and writers are taking the story too lightly, distorting it and pushing this image on the audience, and this should not be encouraged,” he added. Such works are not based on science, but use it as an excuse to comment on current events.

I believe that the Chinese hit the nail on the head when they realized the harm of such films. Fooling people with nonsense, passing it off as science fiction, is dangerous. The fact is that such films undermine people’s sense of reality, the boundaries of reality. And this is the right path to schizophrenia.

Salvador Dali showed the absurdity of our ideas about time through painting. The current clock is not time yet. What is time? If there were no time, there would be no movement. Or maybe it would be more correct to say this: if there were no movement, then there would be no time? Or maybe time and movement are one and the same thing? No, rather, with the help of the categories time and space, we are trying to characterize and measure movement. In this case, time is something like an arshin malalan. To travel in time, you must stop being living (living) people and you must learn to move within the movement itself.

There is no time, there is movement, and movement is time. All paradoxes associated with time arise from the fact that the properties of space are attributed to time. But space is a scalar and time is a vector.

Past and present. If it were possible to connect the past with the present like this, then in the evenings we could go for a walk in the yard of our childhood and meet there with childhood friends, and our childhood friends would be children, and we would be adults. But this is impossible to do. Time is not a characteristic of any movement, but a characteristic of irreversible movement. Even if you start the movement in a circle - loop it, then each cycle will differ in some way from the previous one. Photo from the site: http://kluchikov.net/node/76

This is how we change over time. Traveling into the past is only possible by viewing old photographs and old movies. Also with the help of our memory. Maybe memory is precisely what makes us fifth-dimensional entities? Probably, memory is the only possible time machine that can rush us into the past. You just need to learn to remember everything. Photo from the site: http://loveopium.ru/page/94

Achilles and the tortoise: Swift-footed Achilles will never catch up with a leisurely tortoise if at the beginning of the movement the tortoise is in front of Achilles, since by the time he moves to the point where the tortoise was at the beginning of the competition, it will have time to move at least a little forward. By the time Achilles reaches the point where the turtle was, it will have time to move a certain distance forward. Now Achilles will have to run again some distance to the place where the turtle was, and during this time it will again move forward, and so on - the number of points of approach of Achilles to the turtle tends to infinity. It turns out that Achilles will never catch up with the turtle, but we understand that in reality he will easily catch up and overtake it.

Why does this happen, what caused this paradox? But the fact is that distance is not a collection of points. After all, a point has no size and on any geometric segment the number of points can be infinite. To visit an infinite number of points, Achilles will need an infinite amount of time. Therefore, it turns out that discrete mathematics and formal logic are not applicable to reality, and if they are applicable, then with great reservations.

This paradox is due to the fact that formal logic operates in a discrete world with discrete bodies consisting of points, and phenomena that also represent collections of points in four-dimensional space-time. This paradox is not so harmless. For 2.5 thousand years now, he has been showing scientists the absurdity of formal logic and the limitations of mathematics. But scientists stubbornly believe in formal logic and mathematics and do not want to change anything. Although... Timid attempts to change logic were made both in philosophy and mathematics.

The turtle felt sorry for Achilles and stopped. Only then was the exhausted and aged Achilles able to catch up with her and finally rest. Picture from the site: http://ecolours.pl/life.php?q=zeno-of-elea&page=2

Achilles runs after the turtle. In reality, he easily catches up with her, but in the logical design of this process, he cannot catch up with her. The turtle has a head start of 100 meters. Both runners start moving at the same time. While Achilles reaches point A, the turtle will move to point B, Achilles will again reduce the distance between himself and the turtle and move to point C. But at this time, the turtle will move forward and will be in front of Achilles at point D. Achilles will again reduce the distance between himself and the turtle and will end up at point E. But during this time the turtle will crawl forward again and end up at point J. And so on ad infinitum. The distance between Achilles and the turtle will decrease, but he will not be able to catch up with it. This conclusion follows from formal logic. Picture from the site: http://nebesa87.livejournal.com/

In mathematics, an attempt to break out of the captivity of formal logic was the creation of differential and integral calculus. Both presuppose a continuous change of some quantity depending on the continuous change of another quantity. Column diagrams depict the dependence of discrete phenomena and processes, and graphs (lines) depict continuous processes and phenomena. However, the transition from a diagram to a graph is a kind of sacrament - something like sacrilege. After all, all experimental data (results of specific measurements) are discrete. And the researcher takes and draws a graph instead of a diagram. What is this? If we approach strictly, then the situation here is like this: a graph is a transformation of a diagram into a graph that approximates this diagram. By constructing a graph in the form of a continuous line, we make a transition from the world of discrete phenomena and objects to the continuous world. This is an attempt to break out of the boundaries of formal logic and thereby avoid its paradoxes.

In philosophy, already in the 19th century, scientists realized the inferiority of formal logic, and some began to try to solve this problem. They started talking together about dialectics, about the triad (Hegel), about a different theory of knowledge. Philosophers understood earlier than scientists that formal logic leads knowledge to a dead end. The result of the introduction of dialectics into science was, for example, the doctrine of evolution (development). After all, if you strictly adhere to the positions of formal logic, then development is impossible in principle. Preformationism is a pathetic attempt by formal logic to explain the evolution occurring everywhere. Preformists argue that everything is predetermined in some program in embryo, and the observed development is only the implementation (deployment) of this program. Formal genetics was born from preformationism, but it could only explain the development of the organism in ontogenesis. But formal genetics could not explain the change in species and macroevolution. It was necessary to add a new building to that original formal genetics, which turned out to be several orders of magnitude larger than the building of classical genetics, even to the point of denying discrete genes. But even in this modified form, genetics could only explain microevolution, and macroevolution was too tough for it. And the attempts that geneticists make to explain macroevolution give paradoxes similar to those discussed above.

But even today the positions of formal logic are very strong in the minds of scientists: biologists, biophysicists, geneticists, biochemists. Dialectics has difficulty making its way into this science.

The paradox says that someone omnipotent can create any situation, including one in which he will be unable to do anything. In a simplified version, it sounds like this: can God create a stone that he himself cannot lift? On the one hand, he is omnipotent and can create any stone he wants. On the other hand, if he cannot lift a stone he created himself, then he is not omnipotent!

A pile of sand consists of 1,000,000 grains of sand. If you take one grain of sand out of it, it will still be a heap of sand. If you continue this action many times, it turns out that 2 grains of sand, and even one grain of sand is also a heap of sand. One can object to this that one grain of sand is just one grain of sand, but in this case the principle of interconnectedness of statements is violated, and we again come to a paradox. The only way to save this situation is to introduce an exception for one grain of sand that is not a heap. But two grains of sand can hardly be called a heap either. So how many grains of sand does a heap start with?

In reality, this does not happen, since in the world there are no identical things, phenomena, bundles of hay, or equivalent types of execution. Even if the bundles of hay are the same in taste and size, then one of them may be a little further than the other, or one of the donkey’s eyes may be more keen than the other, etc. Unfortunately, formal logic does not take this into account, so it should be used carefully and not in all judgments, and it should not always be trusted.

People in life and in their activities (including economic activity) do not behave at all like “ideal” balls in theory. In addition to profit, people strive for sustainability and comfort in the broad sense of the word. An unknown risk can be either less than or greater than the known one. You can, of course, win more and become richer. But you can lose more and become bankrupt. But non-poor people give money on loan; they have something to value, and they don’t want to end up homeless.

Let's say I took 100 rubles from a friend, went to the store and lost them. I met a friend and borrowed another 50 rubles from him. I bought a bottle of beer for 20 rubles, I had 30 rubles left, which I gave to my friend and I still owed her 70 rubles. And I owed my friend 50 rubles, a total of 120 rubles. Plus I have a bottle of beer for 20 rubles.
Total 140 rubles!
Where are the other 10 rubles?

Here is an example of a logical fallacy embedded in the reasoning. The error lies in the incorrect construction of the reasoning. If you “walk” in a given logical circle, then it is impossible to get out of it.

Let's try to reason. The logical error in this case is that the debt is counted along with what we have, what we did not lose - a bottle of beer. Indeed, I borrowed 100+50=150 rubles. But I reduced my debt by returning 30 rubles to my friend, after which I owed her 70 rubles and I owed my friend 50 rubles (70+50=120). In total, my debt now amounted to 120 rubles. But if I give a bottle of beer worth 20 rubles to a friend, then I will only owe him 30 rubles. Together with the debt to my friend (70 rubles), my debt will be 100 rubles. But this is exactly the amount I lost.

The theory of black holes has become very fashionable in cosmophysics today. According to this theory, huge stars in which thermonuclear fuel “burns” are compressed - collapse. At the same time, their density increases monstrously - so that electrons fall onto the nuclei and intra-atomic voids collapse. Such a collapsed super-dense extinct star has strong gravity and absorbs matter from outer space (like a vacuum cleaner). At the same time, such a neutron star becomes denser and heavier. Finally, her gravity becomes so powerful that not even light quanta can escape her. This is how a black hole is formed.

This paradox casts doubt on the physical theory of black holes. It may turn out that they are not so black after all. They most likely have structure and therefore energy and information. Moreover, black holes cannot absorb matter and energy indefinitely. In the end, having eaten too much, they “burst” and throw out clumps of super-dense matter, which become the cores of stars and planets. It is no coincidence that black holes are found in the centers of galaxies, and in these centers there is the highest concentration of stars escaping from these centers.

Any contradiction in the theoretical dogmas of science should encourage scientists to change (improve) the theory. Such a large number of paradoxes in logic, mathematics, and physics shows that not everything is going well in these sciences with theoretical constructs.

In 1850, the German physicist R. Clausius came to the conclusion that heat passes only from a warm body to a cold one, and never vice versa, which is why the state of the Universe must change more and more in a certain direction. Physicist William Thomson argued that all physical processes in the Universe are accompanied by the conversion of light energy into heat. Consequently, the Universe faces “thermal death” - i.e. cooling to absolute zero -273 degrees Celsius. Therefore, the infinitely long existence of a “warm” Universe in time is impossible; it must cool down.

The theory of the heat death of the Universe is, in all likelihood, a beautiful theory, but false. Thermodynamics does not take something into account, since its postulates lead to such a conclusion. However, gentlemen physicists love this theory too much and do not want to give up on it or at least greatly limit its applicability.

Another revolution in physics is brewing. Someone brilliant will create a new theory in which energy can not only be dissipated in the Universe, but also collected. Or maybe it gathers in black holes? After all, if there is a mechanism for the dispersion of matter and energy, then there must necessarily be an opposite process of concentration of matter. The world is based on the unity and struggle of opposites.

Photo from the site: http://grainsoft.dpspa.org/referat/referat-teplovoy-smerti-vselennoy.html

Clausius wrote about it this way: “The work that can be produced by the forces of nature and contained in the existing movements of celestial bodies will gradually turn more and more into heat. Heat, constantly moving from a warmer to a colder body and thereby trying to equalize existing differences in temperature, will gradually receive a more and more uniform distribution and a certain equilibrium will also occur between the radiant heat present in the ether and the heat located in bodies. And finally, with regard to their molecular arrangement, the bodies will approach a certain state in which, as regards the prevailing temperature, the total dispersion will be greatest possible.” And further: “We must, therefore, draw the conclusion that in all natural phenomena the total value of entropy can always only increase and not decrease, and therefore we obtain, as a brief expression of the transformation process that is always and everywhere taking place, the following proposition: the entropy of the Universe tends to a certain to the maximum. (http://msd.com.ua/vechnyj-dvigatel/teplovaya-smert-vselennoj-i-rrt-2/)

But everything goes fine until a production crisis occurs. And with a production crisis in the United States, the balance of payments deficit disappears. A lot of capital has accumulated in banks, but there is nowhere to invest it. Capital lives only through circulation through production. As they say: “Airplanes only live in flight.” And capital lives only in the processes of production and consumption. And without production and consumption, capital disappears - it turns into nothing (yesterday it was, but today it is not), this causes the balance of payments deficit to grow in the USA - the airbags of other countries in US banks have disappeared without a trace. The United States, having made the dollar an international currency, has put itself on the dollar needle. The global economic crisis is sharply aggravating the situation and the health of the dollar “addict”. In an effort to acquire the next “dose,” the addict goes to great lengths and becomes aggressive.

China is developing well under socialism. Not at all because there is little private property there, but more state property. It’s just that the Chinese began to determine the price of goods by the demand for them. And this is only possible in a market economy.

The paradox of thrift. If everyone saves money during an economic downturn, aggregate demand will fall and, as a result, the total savings of the population will decrease.

I would call this paradox the paradox of Angela Merkel and Sarkozy. By introducing budget austerity in the countries of United Europe, politicians sharply reduced the population's demand for goods and services. The reduction in demand led to a reduction in production, including in Germany and France themselves.

In order to cope with the crisis, Europe must stop saving and must come to terms with the inevitability of inflation. In this case, part of the capital will be lost, but production will be saved due to consumption.

Photo from the site: http://www.free-lance.ru/commune/?id=11&site=Topic&post=1031826

But inflation will inevitably lead to the loss of capital - savings that the population keeps in banks. They say that under the euro, the Greeks lived beyond their means; the Greek budget had a large deficit. But by receiving this money in the form of salaries and benefits, the Greeks bought goods produced in Germany and France and thereby stimulated production in these countries. Production began to collapse and the number of unemployed increased. The crisis also worsened in countries that considered themselves donors to the European economy. But the economy is not only about production and its lending. It's also about consumption. Ignoring the laws of the system is the cause of this paradox.

Conclusion

Concluding this article, I would like to draw your attention to the fact that formal logic and mathematics are not perfect sciences and, boasting of their proofs and the rigor of their theorems, are based on axioms taken on faith as completely obvious things. But are these axioms of mathematics so obvious?

What is a point that has no length, width or thickness? And how does it happen that the totality of these “incorporeal” points, if they are lined up in a row, is a line, and if in one layer, then a plane? We take an infinite number of points that have no volume, line them up in a row, and get a line of infinite length. In my opinion, this is some kind of nonsense.

I asked my math teacher this question back in school. She was angry with me and said: “How stupid you are! It’s obvious.” Then I asked her: “How many points can be squeezed into a line between two adjacent points, and is it possible to do this?” After all, if an infinite number of points are brought close to each other without distances between them, then the result is not a line, but a point. To get a line or plane, you need to place the points in a row at some distance from each other. Such a line cannot even be called dotted, because dots have no area or volume. It’s as if they exist, but it’s as if they don’t exist at all, they are immaterial.

At school, I often wondered: do we do arithmetic operations, such as addition, correctly? In arithmetic, when adding, 1+1 = 2. But this may not always be the case. If you add another apple to one apple, you get 2 apples. But if we look at this differently and count not apples, but abstract sets, then by adding 2 sets, we get a third one, consisting of two sets. That is, in this case 1 + 1 = 3, or maybe 1 + 1 = 1 (two sets merged into one).

What is 1+1+1? In ordinary arithmetic it turns out to be 3. But what if we take into account all combinations of 3 elements, first by 2, and then by 3? Correct, in this case 1+1+1=6 (three combinations of 1 element, two combinations of 2 elements and 1 combination of 3 elements). Combinatorial arithmetic at first glance seems stupid, but this is only true out of habit. In chemistry, you have to count how many water molecules you get if you take 200 hydrogen atoms and 100 oxygen atoms. You will get 100 water molecules. What if we take 300 hydrogen atoms and 100 oxygen atoms? You will still get 100 water molecules and 100 hydrogen atoms remaining. So, we see that a different arithmetic finds application in chemistry. Similar problems occur in ecology. For example, Liebig's rule is known that plants are influenced by a chemical element in the soil that is at its minimum. Even if all other elements are in large quantities, the plant will be able to absorb as much of them as the minimum element allows.

Mathematicians boast of their supposed independence from the real world; their world is an abstract world. But if this is so, then why do we use the decimal counting system? And some tribes had a 20 system. Very simply, those southern tribes that did not wear shoes used the decimal system - according to the number of fingers and toes, but those who lived in the north and wore shoes used only their fingers when counting. If we had three fingers on our hands, we would use the six-digit system. But if we descended from dinosaurs, we would have three fingers on each hand. So much for the independence of mathematics from the outside world.

Sometimes it seems to me that if mathematics were closer to nature (reality, experience), if it were less abstract, if it were not the queen of the sciences, but if it were their servant, it would develop much faster. And it turns out that the non-mathematician Pearson came up with the mathematical chi-square criterion, which is successfully used when comparing series of numbers (experimental data) in genetics, geology, and economics. If you take a closer look at mathematics, it turns out that everything fundamentally new was introduced into it by physicists, chemists, biologists, geologists, and mathematicians, at best, developed it - they proved it from the standpoint of formal logic.

Non-mathematical researchers constantly pulled mathematics out of the orthodoxy into which “pure” mathematicians tried to plunge it. For example, the theory of similarity and difference was created not by mathematicians, but by biologists, the theory of information by telegraph operators, and the theory of thermodynamics by thermal physicists. Mathematicians have always tried to prove theorems using formal logic. But some theorems are probably impossible to prove in principle using formal logic.

Sources of information used

Mathematical paradox. Access address: http://gadaika.ru/logic/matematicheskii-paradoks

Paradox. Access address: http://ru.wikipedia.org/wiki/%CF%E0%F0%E0%E4%EE%EA%F1

The paradox is logical. Access address: http://dic.academic.ru/dic.nsf/enc_philosophy/

Paradoxes of logic. Access address: http://free-math.ru/publ/zanimatelnaja_matematika/paradoksy_logiki/paradoksy_logiki/11-1-0-19

Khrapko R.I. Logical paradoxes in physics and mathematics. Access address:

Since ancient times, scientists and thinkers have loved to entertain themselves and their colleagues by posing unsolvable problems and formulating various kinds of paradoxes. Some of these thought experiments remain relevant for thousands of years, which indicates the imperfections of many popular scientific models and “holes” in generally accepted theories that have long been considered fundamental.

We invite you to reflect on the most interesting and surprising paradoxes, which, as they now say, “blew the minds” of more than one generation of logicians, philosophers and mathematicians.

1. Aporia "Achilles and the Tortoise"

2. Time loop paradox

The New Time Travelers by David Toomey

3. The paradox of a girl and a boy

Martin Gardner / © www.post-gazette.com

4. Jourdain's card paradox

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

Philippe Jourdain

5. Sophistry “Crocodile”

6. Aporia "Dichotomy"

7. Aporia "Flying Arrow"

8. Galileo's paradox

9. The Potato Bag Paradox

10. The Raven Paradox

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

You and I have already had a selection of paradoxes - , and also in particular, and The original article is on the website InfoGlaz.rf Link to the article from which this copy was made -

If you are not completely confused after reading this collection, then you are not thinking clearly enough.
Since ancient times, scientists and thinkers have loved to entertain themselves and their colleagues by posing unsolvable problems and formulating various kinds of paradoxes. Some of these thought experiments remain relevant for thousands of years, which indicates the imperfections of many popular scientific models and “holes” in generally accepted theories that have long been considered fundamental. We invite you to reflect on the most interesting and surprising paradoxes, which, as they now say, “blew the minds” of more than one generation of logicians, philosophers and mathematicians.
Aporia "Achilles and the Tortoise"
The Achilles and the Tortoise Paradox is one of the aporias (logically correct but contradictory statements) formulated by the ancient Greek philosopher Zeno of Elea in the 5th century BC. Its essence is as follows: the legendary hero Achilles decided to compete in a race with a turtle. As you know, turtles are not known for their agility, so Achilles gave his opponent a head start of 500 m. When the turtle overcomes this distance, the hero sets off in pursuit at a speed 10 times greater, that is, while the turtle crawls 50 m, Achilles manages to run the 500 m handicap given to him . Then the runner overcomes the next 50 m, but at this time the turtle crawls away another 5 m, it seems that Achilles is about to catch up with her, but the rival is still ahead and while he runs 5 m, she manages to advance another half a meter and so on. The distance between them is endlessly shrinking, but in theory, the hero never manages to catch up with the slow turtle; it is not much, but is always ahead of him.

Of course, from the point of view of physics, the paradox makes no sense - if Achilles moves much faster, he will in any case get ahead, but Zeno, first of all, wanted to demonstrate with his reasoning that the idealized mathematical concepts of “point in space” and “moment of time” do not too suitable for correct application to real movement. Aporia exposes the discrepancy between the mathematically sound idea that non-zero intervals of space and time can be divided indefinitely (so the turtle must always stay ahead) and the reality in which the hero, of course, wins the race.
Time loop paradox
Paradoxes involving time travel have long been a source of inspiration for science fiction writers and creators of science fiction films and TV series. There are several options for time loop paradoxes; one of the simplest and most graphic examples of such a problem was given in his book “The New Time Travelers” by David Toomey, a professor at the University of Massachusetts.
Imagine that a time traveler bought a copy of Shakespeare's Hamlet from a bookstore. He then went to England during the time of the Virgin Queen Elizabeth I and, finding William Shakespeare, handed him the book. He rewrote it and published it as his own work. Hundreds of years pass, Hamlet is translated into dozens of languages, endlessly republished, and one of the copies ends up in that same bookstore, where a time traveler buys it and gives it to Shakespeare, who makes a copy, and so on... Who should be considered in this case? the author of an immortal tragedy?
The paradox of a girl and a boy
In probability theory, this paradox is also called "Mr. Smith's Children" or "Mrs. Smith's Problems." It was first formulated by the American mathematician Martin Gardner in one of the issues of Scientific American magazine. Scientists have been arguing over the paradox for several decades, and there are several ways to resolve it. After thinking about the problem, you can come up with your own solution.
The family has two children and it is known for sure that one of them is a boy. What is the probability that the second child is also male? At first glance, the answer is quite obvious - 50/50, either he is really a boy or a girl, the chances should be equal. The problem is that for two-child families, there are four possible combinations of the genders of the children - two girls, two boys, an older boy and a younger girl, and vice versa - an older girl and a younger boy. The first can be excluded, since one of the children is definitely a boy, but in this case there are three possible options left, not two, and the probability that the second child is also a boy is one chance out of three.
Jourdain's paradox with a card
The problem, proposed by the British logician and mathematician Philip Jourdain at the beginning of the 20th century, can be considered one of the varieties of the famous liar paradox.
Imagine holding a postcard in your hands that says, “The statement on the back of the postcard is true.” Turning the card over reveals the phrase “The statement on the other side is false.” As you understand, there is a contradiction: if the first statement is true, then the second is also true, but in this case the first must be false. If the first side of the postcard is false, then the phrase on the second cannot be considered true either, which means that the first statement again becomes true... An even more interesting version of the liar paradox is in the next paragraph.
Sophistry "Crocodile"
A mother and child are standing on the river bank, suddenly a crocodile swims up to them and drags the child into the water. The inconsolable mother asks to return her child, to which the crocodile replies that he agrees to give him back unharmed if the woman correctly answers his question: “Will he return her child?” It is clear that a woman has two answer options - yes or no. If she claims that the crocodile will give her the child, then everything depends on the animal - considering the answer to be true, the kidnapper will release the child, but if he says that the mother was mistaken, then she will not see the child, according to all the rules of the contract.
The woman’s negative answer complicates everything significantly - if it turns out to be correct, the kidnapper must fulfill the terms of the deal and release the child, but thus the mother’s answer will not correspond to reality. To ensure the falsity of such an answer, the crocodile needs to return the child to the mother, but this is contrary to the contract, because her mistake should leave the child with the crocodile.
It is worth noting that the deal proposed by the crocodile contains a logical contradiction, so his promise is impossible to fulfill. The author of this classic sophism is considered to be the orator, thinker and politician Corax of Syracuse, who lived in the 5th century BC.
Aporia "Dichotomy"

Another paradox from Zeno of Elea, demonstrating the incorrectness of the idealized mathematical model of movement. The problem can be posed like this: let's say you set out to walk some street in your city from beginning to end. To do this, you need to overcome the first half of it, then half of the remaining half, then half of the next segment and so on. In other words, you walk half the entire distance, then a quarter, one eighth, one sixteenth - the number of decreasing sections of the path tends to infinity, since any remaining part can be divided in two, which means it is impossible to walk the entire path. Formulating a somewhat far-fetched paradox at first glance, Zeno wanted to show that mathematical laws contradict reality, because in fact you can easily cover the entire distance without leaving a trace.
Aporia "Flying Arrow"
The famous paradox of Zeno of Elea touches on the deepest contradictions in scientists’ ideas about the nature of motion and time. The aporia is formulated as follows: an arrow fired from a bow remains motionless, since at any moment in time it is at rest and does not move. If at every moment of time the arrow is at rest, then it is always in a state of rest and does not move at all, since there is no moment in time at which the arrow moves in space.

Outstanding minds of mankind have been trying to resolve the paradox of the flying arrow for centuries, but from a logical point of view it is composed absolutely correctly. To refute it, it is necessary to explain how a finite time period can consist of an infinite number of moments of time - even Aristotle, who convincingly criticized Zeno’s aporia, was unable to prove this. Aristotle rightly pointed out that a period of time cannot be considered the sum of certain indivisible isolated moments, but many scientists believe that his approach is not deep and does not refute the existence of a paradox. It is worth noting that by posing the problem of a flying arrow, Zeno did not seek to refute the possibility of movement as such, but to identify contradictions in idealistic mathematical concepts.
Galileo's paradox
In his Discourses and Mathematical Proofs Concerning Two New Branches of Science, Galileo Galilei proposed a paradox that demonstrates the curious properties of infinite sets. The scientist formulated two contradictory judgments. First, there are numbers that are the squares of other integers, such as 1, 9, 16, 25, 36, and so on. There are other numbers that do not have this property - 2, 3, 5, 6, 7, 8, 10 and the like. Thus, the total number of perfect squares and ordinary numbers must be greater than the number of perfect squares alone. The second proposition: for each natural number there is its exact square, and for each square there is an integer square root, that is, the number of squares is equal to the number of natural numbers.
Based on this contradiction, Galileo concluded that reasoning about the number of elements was applied only to finite sets, although later mathematicians introduced the concept of power of a set - with its help, the validity of Galileo’s second judgment was proven for infinite sets.
The Potato Bag Paradox

Let's say a certain farmer has a bag of potatoes weighing exactly 100 kg. Having examined its contents, the farmer discovers that the bag was stored in damp conditions - 99% of its mass is water and 1% other substances contained in potatoes. He decides to dry the potatoes a little so that their water content drops to 98% and moves the bag to a dry place. The next day it turns out that one liter (1 kg) of water has indeed evaporated, but the weight of the bag has decreased from 100 to 50 kg, how can this be? Let's calculate - 99% of 100 kg is 99 kg, which means the ratio of the mass of dry residue to the mass of water was initially equal to 1/99. After drying, water accounts for 98% of the total mass of the bag, which means the ratio of the mass of the dry residue to the mass of water is now 1/49. Since the mass of the residue has not changed, the remaining water weighs 49 kg.
Of course, an attentive reader will immediately discover a gross mathematical error in the calculations - the imaginary comic “sack of potatoes paradox” can be considered an excellent example of how, with the help of seemingly “logical” and “scientifically supported” reasoning, one can literally build a theory from scratch that contradicts common sense. sense.
Raven paradox
The problem is also known as Hempel's paradox - it received its second name in honor of the German mathematician Carl Gustav Hempel, the author of its classic version. The problem is formulated quite simply: every raven is black. It follows from this that anything that is not black cannot be a raven. This law is called logical contraposition, that is, if a certain premise “A” has a consequence “B”, then the negation of “B” is equivalent to the negation of “A”. If a person sees a black raven, this strengthens his belief that all ravens are black, which is quite logical, but in accordance with contraposition and the principle of induction, it is logical to state that observing objects that are not black (say, red apples) also proves that all crows are painted black. In other words, the fact that a person lives in St. Petersburg proves that he does not live in Moscow.
From a logical point of view, the paradox looks impeccable, but it contradicts real life - red apples in no way can confirm the fact that all crows are black.

Contents

Introduction

1. Sophistry

1.2 Examples of sophistry

2. Logical paradoxes

Conclusion

Introduction

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

One of the most important and significant laws of logic is the law of identity. He claims 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 into different words), creating ambiguity, deviating from the topic, etc.

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

So many sophisms look like a meaningless and purposeless game with language; a game based on the polysemy of linguistic expressions, their incompleteness, understatement, dependence of their meanings on context, etc. These sophisms seem especially naive and frivolous.

Logical paradoxes provide evidence that logic, like any other science, is not complete, but constantly evolving.

Sophisms and paradoxes originated in ancient times. By using these logical techniques and phrases, our language becomes richer, brighter, more beautiful.

1. Sophistry

1.1 The concept of sophism and its historical origin

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

Aristotle called sophistry “imaginary evidence”, in which the validity of the conclusion is apparent and is due to a purely subjective impression caused by the insufficiency of logical analysis. The persuasiveness of many sophisms at first glance, their “logicality” is usually associated with a well-disguised error - a semiotic one<#"center">1.2 Examples of sophistry

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

Example #1 simple sophistry: 3 and 4 are two different numbers, 3 and 4 are 7, therefore 7 are two different numbers.In this outwardly correct and convincing reasoning, various, non-identical things are mixed or identified: a simple enumeration of numbers (the first part of the reasoning) and the mathematical operation of addition (the second part of the reasoning); It is impossible to put an equal sign between the first and second, a violation of the law of identity.

Example No. 2 simple sophistry: two times two (that is, twice two) is not four, but three. Let's take a match and break it in half. It's one time two. Then take one of the halves and break it in half. This is the second time two. The result was three parts of the original match. Thus, two times two is not four, but three.In this reasoning, various things are mixed up, the non-identical is identified: the operation of multiplication by two and the operation of division by two - one is implicitly replaced by 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 haven't lost, you have. You didn't lose your horns. So you have horns.This masks the ambiguity of the larger premise. If it is thought of as universal: “Everything that you haven’t lost...”, then the conclusion is logically flawless, but uninteresting, since it is obvious that the major 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 a violation of the law of identity.

Example No. 4 : N.V. Gogol, in his poem “Dead Souls,” describing the landowner Nozdryov, says that he was a historical person, because wherever he appeared, some story was sure to happen to him.

Example No. 5 : Don't stand just anywhere, otherwise you'll get hit.

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

Don't go to these places again.

In examples No. 4,5,6, the same technique is used: different meanings, situations, themes are mixed in identical words, 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) is something unusual and surprising, something that diverges from usual expectations, common sense and life experience.

Logical paradox- this is such an unusual and surprising situation when two contradictory judgments are not only simultaneously true (which is impossible due to the logical laws of contradiction and the excluded middle), 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 to overcome and eliminate paradoxes have been proposed, but none of them is still exhaustive, final and generally accepted.

Some paradoxes (paradoxes of the “liar”, “village barber”, etc.) are also called antinomies(from the Greek contradiction in law), that is, by reasoning in which it is proven that two statements that deny each other follow from one another. It is believed that antinomies represent the most extreme form of paradoxes. However, quite often the terms “logical paradox” and “antinomy” are considered synonymous.

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

sophistry logical paradox language

The most famous aporia was 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 Tortoise." She says that we may well see how the fleet-footed Achilles catches up and overtakes the slowly crawling turtle; 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 it. When he covers the distance to the turtle, during the same time it will cover 10 times less, namely 1/10 of the path that Achilles traveled, and this 1/10 will be ahead of him. When Achilles travels this 1/10th of the way, the tortoise will cover 10 times less distance in the same time, that is, 1/100th of the way, and will be ahead of Achilles by this 1/100th. When he passes 1/100th of the path separating him and the turtle, then in the same time it will cover 1/1000th of the path, still remaining ahead of Achilles, and so on ad infinitum. We become convinced that the eyes tell us one thing, but the thought tells us something completely different (the visible is denied by the conceivable).

Logic has created many ways to resolve and overcome paradoxes. However, none of them is without objections and is not generally accepted.

2.2 Examples of logical paradoxes

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 vowed not to eat until he resolved this paradox and died of hunger, having achieved nothing. There are several different formulations of this paradox. It is most briefly and simply formulated in a situation when a person utters a simple phrase: “I am a liar.” Analysis of this statement leads to a stunning result. As you know, any statement can be true or false. Let us assume 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, by uttering 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 truth-teller, therefore, by uttering 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 contradictory judgments are not only simultaneously true, but also follow from each other).

Another famous logical paradox discovered in the 20th century. English logician and philosopher Bertrand Russell, is paradox of the "village barber". Let's imagine that in a certain village there is only one barber who shaves those residents who do not shave themselves. Analysis of this simple situation leads to an extraordinary conclusion. Let's ask ourselves: can a village barber shave himself? Let us assume that the village barber shaves himself, but then he is one of those village residents who shave themselves and whom the barber does not shave, therefore, in this case he does not shave himself. Let us assume that the village barber does not shave himself, but then he is one of those village residents who do not shave themselves and whom the barber shaves, therefore, in this case he shaves himself. It turns out incredible: if a village barber shaves himself, then he does not shave himself; and if he does not shave himself, then he shaves himself (two contradictory judgments are simultaneously true and mutually condition each other).

Paradox "Protagoras and Euathlus" appeared in Ancient Greece. It is based on a seemingly simple story, which is that the sophist Protagoras had a student Euathlus, who took lessons in logic and rhetoric from him. The teacher and student agreed in such a way that Euathlus would pay Protagoras a tuition fee only if he won his first trial. However, upon completion of the training, Evatl did not participate in any process and, of course, did not pay the teacher any money. Protagoras threatened him that he would sue him and then Euathlus would have to pay in any case. “You will either be sentenced to pay a fee, or you will not be sentenced,” Protagoras told him, “if you are sentenced to pay, you will have to pay according to the verdict of the court; if you are not sentenced to pay, then you, as having won your first lawsuit, You 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 I will not be sentenced; 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 pay , then I will not pay the court verdict." Thus, the question of whether Euathlus should pay Protagoras or not is unanswerable. The contract between teacher and student, despite its completely innocent appearance, is internally, or logically, contradictory, since it requires the implementation of an impossible action: Evatl must both pay for training and not pay at the same time. Because of this, the agreement itself between Protagoras and Euathlus, as well as the question of their litigation, represents something other than a logical paradox.

Conclusion

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

Quite often, sophisms are used by publishers of mass newspapers and magazines for commercial purposes. Passing by a 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: “The first grader ate a crocodile” - it turns out that the first grader ate a large chocolate crocodile.

As we 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 entirely known and understood, and outline new horizons in the development of logic. A comprehensive explanation and final resolution of logical paradoxes remains a matter of the future.

List of used literature

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

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

) Ivin A.A. The art of thinking correctly. M.: Education, 2011.

) Koval S. From entertainment to knowledge / Transl. O. Unguryan. Warsaw: Scientific and Technical Publishing House, 2012.

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