Logic: Logical paradoxes. Types of paradoxes

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

But every researcher who tried to strictly define all the concepts in his theory faced a similar paradox of formal logic. Nobody has succeeded in this yet, since everything ultimately came down to a tautology like: "Movement 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. In court, he appears the last word. He must make 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 should he make to completely confuse the judge? Think for yourself.

Perplexed by this paradox, Protagoras devoted a special essay to this dispute with Euathlus, "A Litigation for Payment." Unfortunately, it, like most of what was written by Protagoras, did not reach us. The philosopher Protagoras immediately felt that behind this paradox lies something essential that deserves special study.

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

This aporia is a consequence of the notion of the discreteness of motion that a moving body in discrete units of time passes through 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 the discreteness of the world, the beginning of which should be sought in the teachings of Democritus on 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 that it passes 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.), while we measure time in discrete units (minutes, seconds, hours, etc.). We have a distance standard - a meter, or another segment with which we compare the path. With the standard of time (in fact, also a segment), we measure time. But distance and time are continuous. And if they are discontinuous (discrete), then what is at the junctions of their discrete parts? Other world? A 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. And that means that everything in such a world would be dead.

Indeed, this paradox is unresolvable 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.

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

Paradox of Plato and Socrates
Plato: "Socrates' next statement 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 is lying 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 lies in the fact that, within the framework of formal logic, a proposition can be both true and false at the same time. This statement, which constitutes the liar's 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 trinity 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. The thesis and antithesis are not enough to characterize some element. If a statement is not true, then it does not follow that it is false. Conversely, just because a statement is not false does not mean that it is true. It is not easy for our mind 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: we are told: "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 inferiority of formal (binary) logic. Let's think about how the phrase should be constructed correctly: "History teaches a person, but he does not learn anything from history." In such a formulation, with such a 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 correctly constructing a phrase.

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 it contains do not contain themselves (are empty sets)? Russell popularized it in the form of the "barber paradox": "The barber only shaves people who don't shave themselves. Does he shave himself?

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

The concept of "empty set" is absurd by definition. How can a set be empty, containing nothing? The barber is not among the many people 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 take payment from himself for this.

A paradox from the category of antinomies - 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 memorize. Teaching as rote 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 owns this skill almost all his life. But what will happen to people if, during training, they are allowed to memorize only those 10% of the information that they remember for a long time? Unfortunately, no one has done such an experiment yet. Although...

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

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 are not reduced to the properties of these parts. The whole is greater than the sum of its parts! Therefore, even at the age of 50, a person remains himself, although 95% of the atoms of his body have already been replaced by others many times during this time, 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 stream of a certain depth and width, with a certain flow rate, in a word, a river is dynamic system, not the sum of its parts.

Bald 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 of baldness lies in a different plane than the one in which it was formulated. To answer such a question, it is necessary to leave one plane of reasoning and perception into a completely different one. For example, publications of one scientist are cited 100 times a year, and another 1 time per year. Question: which of them is a brilliant scientist? There can be four different answers to this question: 1 - none, 2 - both, 3 - first, 4 - second. And all four answers in this case are equally likely, 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 (the state) is to be democratic, then equal representation should be made precisely from the voters. Equal representation from the states, if the population in them is different, is not a principle of democracy, but something else. Equal representation from parties is something third, from religious associations is fourth, and so on.

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

But in practice, Monte Carlo's conclusion can be false for another reason as well. After all, the condition on the independence of elementary events when playing roulette may not be fulfilled. And if elementary events are not independent, but "linked" to each other in ways known to us, as well as unknown yet... then in this case it is better to bet on black, not on red.

It may turn out that there are other carriers of energy and information in the Universe, and not just electromagnetic field fluctuations and elementary particle flows. 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 of atoms there must be an infinite number... Stop! The brains are starting to boil again.

This paradox stems from our misunderstanding of what time is. If time is a stream of the world with many channels (as is often the case with a river), and the speed of the flow in the channels is different, then a sliver that has fallen into a fast channel will then again fall into a slow one when the fast channel merges with the slow one in which another chip floats with which they once sailed alongside. But now one sliver will be ahead of its "friend" and will not meet with her. In order to meet them, the lagging "girlfriend" must fall into another fast channel, and the one who is ahead of them should swim in the slow channel at this time. It turns out that the twin brother, who flew away on a sub-light ship, cannot, in principle, return to the past and meet his brother. The slow flow of time (sub-light ship) delayed him in the flow of time. During this time, his brother not only got older, but he went into the future, with him everything that surrounded him went into the future. So the brother, who is behind in time, will not be able to get into the future in principle.

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

Paradox of the dead 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 will not be able to kill your grandfather.

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

And thank God! Therefore, grandfather is not threatened that his grandson will come from the future and kill him. And there are many such grandchildren who have smoked marijuana today.

Recently, China's Central Bureau of Film, Radio and Television 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 shows and movies, but the meaning of such works, as well as their presentation, is very doubtful. “Most of them are completely fictional, do not correspond to logic and do not correspond to historical realities. Producers and writers are taking the story too lightly, distorting it and forcing that image on the audience, and this should not be encouraged,” he added. Such works do not rely on science, but use it as a pretext for commenting on current events.

I believe that the Chinese hit the nail on the head when they realized the harm of such films. It is dangerous to fool people with nonsense, passing them off as science fiction. The fact is that such films shake people's sense of reality, the boundaries of reality. And this is a sure way to schizophrenia.

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

There is no time, there is movement, and movement is time. All the paradoxes connected with time come from the fact that 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 in this way, then in the evenings we could go for a walk in the courtyard of our childhood and meet childhood friends there, and childhood friends would be children, and we would be adults. But this is impossible. Time is not a characteristic of any movement, but a characteristic of an irreversible movement. Even if you start moving in a circle - loop, then each cycle will differ in something from the previous one. Photo from the site: http://kluchikov.net/node/76

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

In mathematics, an attempt to escape from 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. Bar charts depict the dependence of discrete phenomena and processes, and graphs (lines) - continuous processes and phenomena. However, the transition from diagram to graph is a kind of sacrament - something like sacrilege. After all, all experimental data (results of specific measurements) are discrete. And the researcher instead of a diagram takes and draws a graph. What is this? If we approach strictly, then the situation here is as follows: a graph is a transformation of a diagram into a graph that approximates this diagram. Building a graph in the form of a solid line, we make the transition from the world of discrete phenomena and objects to the continuous world. This is an attempt to break out of the limits of formal logic and thereby avoid its paradoxes.

In philosophy, already in the 19th century, scientists realized the inferiority of formal logic, some began to try to solve this problem. They started talking in unison about dialectics, about the triad (Hegel), about a different theory of knowledge. Philosophers realized before scientists that formal logic leads cognition into a dead end. The result of the introduction of dialectics into science was, for example, the doctrine of evolution (development). After all, if one is strictly on the positions of formal logic, then development is impossible in principle. Preformism is a pathetic attempt by formal logic to explain the evolution going on everywhere. Preformists argue that everything is predestined in some program in the bud, and the observed development is only the implementation (deployment) of this program. Formal genetics was born out of preformism, but it could only explain the development of an organism in ontogeny. But formal genetics could not explain the change in species and macroevolution. I had to add a new building to that initial formal genetics, which turned out to be several orders of magnitude larger than the building classical genetics up to the negation of discrete genes. But even in such a modified form, genetics could only explain microevolution, and macroevolution was too tough for it. And the attempts that geneticists make to explain macroevolution give rise to 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 finds its way with difficulty in 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 kind of stone. On the other hand, if he cannot lift the stone he created, then he is not omnipotent!

In reality, this does not happen, since in the world there are no identical things, phenomena, bundles of hay, equivalent types of execution. Even if the bundles of hay are the same palatability and size, then one of them may be a little further than the other, or one eye of a donkey 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 not always trusted.

People in life and in their activities (including economic ones) do not behave at all like "ideal" balls in theory. In addition to benefits, people strive for stability and comfort in the broadest sense of the word. The unknown risk can be less than or greater than the known risk. You can, of course, win more and become richer. But you can lose more and become bankrupt. And non-poor people give money in growth, they have something to value, and they do not want to be 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 owed her 70 rubles. And I owed a 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 error 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.

In space physics, the theory of black holes has become very fashionable today. According to this theory, huge stars in which thermonuclear fuel "burns out" shrink - collapse. At the same time, their density increases monstrously - so that the electrons fall on the nuclei and the intra-atomic voids collapse. Such a collapsed superdense 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, its gravity becomes so powerful that even light quanta cannot escape it. 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. Rather, they have structure and therefore energy and information. Moreover, black holes cannot absorb matter and energy indefinitely. In the end, having "overfed", they "burst" and throw out clots of superdense 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 constructions.

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 transformation of light energy into heat. Consequently, the Universe is waiting for "thermal death" - i.e. cooling to absolute zero -273 degrees Celsius. Therefore, an infinitely long existence of a "warm" Universe in time is impossible, it must cool down.

The theory of heat death of the Universe, in all likelihood, is a beautiful theory, but false. Something thermodynamics does not take into account, since its postulates lead to such a conclusion. However, the physicists love this theory too much and do not want to part with it or at least severely 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 is going to black holes? After all, if there is a mechanism of 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. The heat, constantly passing from a warmer to a colder body, and thereby striving to even out the existing differences in temperature, will gradually receive a more and more uniform distribution, and a certain equilibrium will also come between the radiant heat present in the ether and the heat located in the bodies. And finally, in respect of their molecular disposition, the bodies will approach a certain state in which, as regards the prevailing temperature, the total scattering will be the greatest possible. And further: “We must, therefore, deduce the conclusion that in all natural phenomena the total value of entropy can always only increase, not decrease, and we therefore obtain, as a short expression of the transformation process that is always and everywhere taking place, the following proposition: the entropy of the Universe tends to a certain maximum. (http://msd.com.ua/vechnyj-dvigatel/teplovaya-smert-vselennoj-i-rrt-2/)

But everything goes well 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. Capitals live only at the expense of turnover through production. As they say: "Airplanes live only in flight." And capital lives only in the processes of production and consumption. And without production and consumption, capitals disappear - they turn into nothing (yesterday it was, but today it is not), because of this, the balance of payments deficit in the USA is growing - the airbags of other countries in US banks have disappeared without a trace. The United States, having made the dollar an international currency, put itself on a dollar needle. The world economic crisis sharply aggravates the situation and the health of the dollar "addict". In an effort to acquire another "dose", the addict goes to any lengths, he becomes aggressive.

China is also developing well under socialism. Not at all because there is little private property, 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 possible only in a market economy.

The paradox of thrift. If everyone saves money during an economic downturn, then 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 austerity budgets in the countries of the United Europe, politicians have sharply reduced the demand of the population for goods and services. The reduction in demand led to a reduction in production, including in Germany and France themselves.

Europe, in order to cope with the crisis, 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 through 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 - the savings that the population keeps in banks. They say that the Greeks lived beyond their means under the euro, the Greek budget was in big deficit. But after all, receiving this money in the form of salaries and benefits, the Greeks bought goods produced in Germany, France, and thereby stimulated production in these countries. Production began to collapse, the number of unemployed increased. The crisis worsened in countries that considered themselves donors to the European economy. But the economy is not only production and its lending. It is also consumption. Ignoring the laws of the system is the reason for this paradox.

Conclusion

Finishing this article, I want to draw 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 for granted as quite obvious things. But are these axioms of mathematics so obvious?

What is a point that has no length, width or thickness? And how is it that the totality of these "incorporeal" points, if they are lined up, is a line, and if one layer, then a plane? We take an infinite number of points that do not have volume, line them up in a row, and we get a line of infinite length. In my opinion, this is some kind of nonsense.

I used to ask this question to my math teacher at school. She was angry with me and said: "What a stupid you are! After all, this is obvious." Then I asked her: "And 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 we get not a line, but a point. To get a line or a plane, it is necessary to place the points in a row at a certain distance from each other. You can’t even call such a dashed line, because the points have no area and volume. They seem to exist, but as if they do not exist at all, they are intangible.

At school, I often thought: are we doing arithmetic operations, for example, addition, correctly? In addition arithmetic, 1+1 = 2. But that may not always be the case. If you add one more apple to one apple, you get 2 apples. But if we look at it differently and count not apples, but abstract sets, then by adding 2 sets, we will 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).

How much is 1+1+1? In ordinary arithmetic, it turns out 3. And if we take into account all combinations of 3 elements, first 2 each, and then 3 each? That's right, in this case 1+1+1=6 (three combinations of 1 element each, two combinations of 2 elements each and 1 combination of 3 elements each). Combinatorial arithmetic at first glance seems stupid, but it is so only 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 get 100 water molecules. And if you take 300 hydrogen atoms and 100 oxygen atoms? You still get 100 water molecules and 100 hydrogen atoms remain. So, we see that in chemistry a different arithmetic finds its application. Similar problems take place in ecology. For example, Liebig's rule is known that plants are affected by a chemical element in the soil, which is at a minimum. Even if all other elements are in large quantities, the plant will be able to assimilate them as much as the element that is at a minimum allows.

Mathematicians boast of their alleged independence from real world, their world is an abstract world. But if this is so, then why do we use the decimal system of counting? And some tribes had a twenty-fold system. Very simply, those southern tribes who did not wear shoes used the vigesimal system - according to the number of fingers and toes, but those who lived in the north and wore shoes used only fingers to count. If we had three fingers on our hand, we would use the six-digit system. But if we evolved from dinosaurs, then 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, do not consider yourself the queen of sciences, but be their servant, it would develop much faster. And so it turns out that the non-mathematician Pearson came up with the mathematical criterion chi-square, 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 it was physicists, chemists, biologists, geologists who introduced everything fundamentally new into it, and mathematicians, at best, developed this - they proved it from the standpoint of formal logic.

Researchers in non-mathematics constantly pulled mathematics out of the orthodoxy into which "pure" mathematicians tried to plunge it. For example, the theory of similarity-difference was created not by mathematicians, but by biologists, the theory of information by telegraphers, the theory of thermodynamics by physicists-heat engineers. Mathematicians have always tried to prove theorems using formal logic. But some theorems are probably impossible to prove with the help of formal logic.

Information sources 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:

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LOGICAL PARADOXES

1. What is a paradox

In a broad sense, a paradox is a position that sharply diverges from generally accepted, established, "orthodox" opinions.

A paradox in a narrower and more specialized sense is two opposing, incompatible statements, for each of which there are seemingly convincing arguments.

The sharpest form of paradox is antinomy, a reasoning that proves the equivalence of two statements, one of which is a negation of the other.

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

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

Constructing a new theory, the scientist usually starts from the facts, from what can be observed in the experiment. However free his creative imagination may be, it must reckon with one indispensable circumstance: a theory makes sense only if it agrees with the facts pertaining to it. A theory that disagrees with facts and observations is far-fetched and has no value.

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

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

The specialized literature on the topic of paradoxes is practically inexhaustible. Suffice it to say that more than a thousand papers have been written about only one of them - the liar paradox.

Externally, logical paradoxes are usually simple and even naive. But in their crafty naivety they are like an old well: it looks like a puddle, but you can’t get the bottom.

A large group of paradoxes speaks of the circle of things to which they themselves belong. They are particularly difficult to separate from statements that appear paradoxical but do not actually lead to a contradiction.

Take, for example, the saying “There are exceptions to every rule.” It is itself, obviously, the rule. This means that at least one exception can be found from it. But this means that there is a rule that does not have a single exception. The statement contains a reference to itself and denies itself. Is there a logical paradox here, disguised as both an affirmation and a denial of the same thing? However, the answer to this question is quite simple.

One might also wonder whether the opinion that any generalization is wrong is not internally inconsistent, since this very opinion is a generalization. Or advice - never advise anything? Or the maxim "Believe nothing!", which also applies to itself? The ancient Greek poet Agathon once remarked: “It is very plausible that many improbable things are happening.” Does not the poet's plausible observation here turn out to be an improbable event itself?

2. Liar paradox

Paradoxes are not always easy to separate from what only resembles them. It is even more difficult to say where the paradox came from, than the most natural, it would seem, assumptions and repeatedly proven methods of reasoning do not suit us.

With particular expressiveness, this is shown by one of the most ancient and, perhaps, the most famous of the logical paradoxes - the paradox of a liar. It refers to expressions that speak about themselves. It was opened by Eubulides from Miletus, who came up with many interesting problems that still cause controversy. But it was the paradox of the liar that brought true glory to Eubulides.

In the simplest version of this paradox, a person says only one phrase: "I'm lying." Or he says: "The statement I am now making is false." Or: "This statement is false."

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

In the Middle Ages, the following formulation was common: “What Plato said is false, says Socrates. "What Socrates said is the truth, says Plato."

The question arises, which of them expresses the truth, and which - a lie?

And here is a modern paradox of this paradox. Let us assume that only the words are written on the front side of the card: "On the other side of this card is written a true statement." It is clear that these words represent a meaningful statement. Turning the card over, we must either find what was promised or not. If the statement is written on the back, then it is either true or not. However, on the back are the words: "There is a false statement written on the other side of this card" - and nothing more. Assume that the statement on the front side is true. Then the statement on the back must be true, and so the statement on the front must be false. But if the statement on the front is false, then the statement on the back must also be false, and therefore the statement on the front must be true. The result is a paradox.

The liar paradox made a huge impression on the Greeks. And it's easy to see why. The question that it poses at first glance seems quite simple: is he lying who says only that he is lying? But the answer "yes" leads to the answer "no", and vice versa. And reflection does not clarify the situation at all. Behind the simplicity and even routine of the question, it reveals some obscure and immeasurable depth.

There is even a legend that a certain Filit Kossky, desperate to resolve this paradox, committed suicide. It is said that one of the famous ancient Greek logicians, Diodorus Cronus, already in his declining years, made a vow not to eat until he found the solution of the “liar”, and soon died without achieving anything.

In the Middle Ages, this paradox was referred to the so-called undecidable sentences and became the object of systematic analysis.

In modern times, the "liar" did not attract any attention for a long time. They did not see behind him any, even insignificant difficulties regarding the use of the language. And only in our so-called modern times did the development of logic finally reach a level where it became possible to formulate the problems behind this paradox in strict terms.

Now the "liar" is often referred to as the "king of logical paradoxes." An extensive scientific literature is devoted to him.

And yet, as in the case of many other paradoxes, it remains not entirely clear what problems lie behind it and how to get rid of it.

So there are statements that speak of their own truth or falsity. The idea that these kinds of statements are not meaningful is very old. It was defended by the ancient Greek logician Chrysippus.

In the Middle Ages, the English philosopher and logician W. Ockham stated that the statement "Every statement is false" is meaningless, since it speaks, among other things, of its own falsity. A contradiction directly follows from this statement. If every proposition is false, then so is the proposition itself, but the fact that it is false means that not every proposition is false. The situation is similar with the statement "Every statement is true." It must also be classified as meaningless and also leads to a contradiction: if every statement is true, then the negation of this statement itself is also true, that is, the statement that not every statement is true.

Why, however, cannot a statement meaningfully speak of its own truth or falsity?

Already a contemporary of Ockham, the French philosopher J. Buridan, did not agree with his decision. From the point of view of the usual ideas about the meaninglessness of expressions like "I'm lying", "Every statement is true (false)" are quite meaningful. What you can think about, what you can say - this is the general principle of Buridan. A person can think about the truth of the statement that he utters, which means that he can speak about it. Not all statements about themselves are meaningless. For example, the statement "This sentence is written in Russian" is true, but the statement "There are ten words in this sentence" is false. And both of them make perfect sense. If it is admitted that a statement can speak about itself, then why is it not capable of speaking meaningfully about such a property of itself as truth?

Buridan himself considered the statement "I am lying" not meaningless, but false. He justified it like this. When a person affirms a proposition, he thereby asserts that it is true. If the sentence says of itself that it is itself false, then it is only an abbreviated formulation of a more complex expression that asserts both its truth and its falsity. This expression is contradictory and therefore false. ^ it doesn't make sense at all.

Buridan's argument is still sometimes considered convincing.

According to the idea of ​​the Polish logician A. Tarsky, expressed in the 30s. of the last century, the reason for the liar's paradox is that the same language is used both about objects that exist in the world and about this “objective” language itself. Tarski called a language with this property "semantically closed". Natural language is obviously semantically closed. Hence the inevitability of the emergence of a paradox in it. To eliminate it, it is necessary to build a kind of ladder, or hierarchy of languages, each of which is used for a very specific purpose: the first one speaks of the world of objects, the second - about this first language, the third - about the second language, etc. Clearly, that in this case the statement, which speaks of its own falsity, can no longer be formulated and the paradox disappears.

This resolution of the paradox is, of course, not the only one possible. At one time it was generally accepted, but now the former unanimity is gone. The tradition of eliminating paradoxes of this type by “stratifying” the language has remained, but other approaches have also emerged.

As we can see, the issues that have been associated with the "liar" for centuries have changed radically depending on whether it was seen as an example of ambiguity, or as an expression that seems to make sense on the surface but is essentially meaningless, or as an example of a mixture of language and metalanguage. And there is no certainty that other problems will not be associated with this paradox in the future.

The Finnish logician and philosopher G. von Wright writes about his work on the “liar” that this paradox should by no means be understood as a local, isolated obstacle that can be removed by one inventive movement of thought. Liar touches on many of the most important topics in logic and semantics; this is the definition of truth, and the interpretation of contradiction and proof, and a whole series of important differences: between a sentence and the thought expressed by it, between the use of an expression and its mention, between the meaning of a name and the object it denotes.

3. Three irresolvable disputes

Another famous paradox is based on a small incident that happened more than two thousand years ago and has not been forgotten to this day.

The famous sophist Protagoras, who lived in the 5th century. BC, there was a student named Euathlus, who studied law. According to the agreement concluded between them, Euathlus had to pay for training only if he won his first lawsuit. If he loses this process, he is not obliged to pay at all. However, after completing his studies, Evatl did not participate in the processes. It lasted quite a long time, the teacher's patience ran out, and he filed a lawsuit against his student. Thus, for Euathlus, this was the first process; he would never have been able to get away from him. Protagoras substantiated his demand as follows: “Whatever the decision of the court, Evatl will have to pay me. He will either win his first trial or lose. If he wins, he will pay by virtue of our contract. If he loses, he will pay according to the court decision.”

Apparently Euathlus was a capable student, as he replied to Protagoras: “Indeed, I will either win the trial or lose it. If I win, the court decision will release me from the obligation to pay. If the court decision is not in my favor, then I lost my first trial and will not pay by virtue of our contract.

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

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

Many other solutions to this paradox have been proposed.

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

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

Sometimes they talk like this. Both Protagoras and Euathlus are both right in part, and neither of them in general. Each of them takes into account only half of the possibilities that are beneficial to itself. Full or comprehensive consideration opens up four possibilities, of which only half is beneficial to one of the disputants. Which of these possibilities is realized, it will be decided not by logic, but by life. If the verdict of the judges will have more force than the contract, Evatl will have to pay only if he loses the process, that is, by virtue of the decision of the court. If, however, a private agreement is placed higher than the decision of the judges, then Protagoras will receive payment only if the process is lost to Evatlus, that is, by virtue of an agreement with Protagoras.

This appeal to "life" completely confuses everything. What, if not logic, can judges be guided by in conditions when all the relevant circumstances are completely clear? And what kind of “leadership” will it be if Protagoras, who claims to be paid through the court, achieves it only by losing the process?

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

If by the solution of this difficulty we understand the answer to the question whether Euathlus should pay Protagoras or not, then all these, like all other conceivable solutions, are, of course, untenable. They are nothing more than a departure from the essence of the dispute, they are, so to speak, tricks and tricks in a hopeless and insoluble situation, since neither common sense, nor any general principles concerning social relations unable to resolve the dispute.

It is impossible to carry out together the contract in its original form and the decision of the court, whatever the latter may be. To prove this, simple means of logic are sufficient. By the same means, it can also be shown that the treaty, despite its completely innocent appearance, is self-contradictory. It requires the implementation of a logically impossible proposition: Euathlus must simultaneously pay for education and at the same time not pay.

IN Ancient Greece the story of the crocodile and the mother was very popular.

“The crocodile grabbed a child from a woman standing on the river bank. To her plea to return the child, the crocodile, shedding, as always, a crocodile tear, answered:

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

Thinking, the mother replied:

You won't give me the baby.

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

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

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

Who is right: mother or crocodile? To what does the promise given to the crocodile oblige? In order to give the child, or, on the contrary, not to give it away?

And to both at the same time. This promise is self-contradictory and thus unfulfillable by virtue of the laws of logic.

This paradox is played out in Don Quixote by M. Cervantes. Sancho Panza has become the governor of the island of Barataria and administers the court. The first to come to him is some visitor and says: “Senior, a certain estate is divided into two halves by a deep river ... A bridge is thrown across this river, and right there on the edge stands a gallows and there is something like a court, four judges usually sit in it , and they judge according to the law issued by the owner of the river, the bridge, and the whole estate. The law is drawn up in this way: “Everyone passing through the bridge over the river must declare under oath: where and why he is going. Whoever tells the truth, let those through, and whoever lies, send them to the gallows without any leniency and execute them. Since the time when this law was promulgated, many people managed to get across the bridge, and as soon as the judges were satisfied that the passers-by were telling the truth, they let them through. But one day a sworn man swore and said that he had come to be hung up on this very gallows, and for nothing else. This oath perplexed the judges, and they said: “If this man is allowed to proceed unhindered, it will mean that he has broken the oath and, according to the law, is liable to death; if they hang him, then he swore that he had come only to be hung up on the gallows, therefore, his oath is not false, and on the basis of the same law it is necessary to let him pass. I ask you, señor governor, what the judges should do with this man, for they are still perplexed and hesitant.

Sancho proposed, perhaps not without cunning, that the half of the person who told the truth should be let through, and the one that lied should be hanged, and in this way the rules for crossing the bridge would be observed in all forms.

This passage is interesting in several respects. First of all, it is a clear illustration of the fact that the stalemate described in the paradox may well be faced - and not in pure theory, but in practice - if not a real man, then at least a literary hero.

The solution proposed by Sancho Panza was, of course, not a solution to the paradox. But this was exactly the solution that only remained to be resorted to in his position.

Once upon a time, Alexander the Great, instead of untying the cunning Gordian knot, which no one has yet managed to do, simply cut it. Sancho did the same. Trying to solve the puzzle on its own terms was useless - it was simply unsolvable. It remained to discard these conditions and introduce your own.

With this episode, Cervantes clearly condemns the exorbitantly formalized scale of medieval justice, permeated with the spirit of scholastic logic. But how widespread in his time - and this was about four hundred years ago - were information from the field of logic! Not only Cervantes himself knows this paradox. The writer finds it possible to attribute to his hero, an illiterate peasant, the ability to understand that he faces an insoluble task!

And finally, one of the modern rephrasings of the dispute between Protagoras and Euathlus.

The missionary found himself with the cannibals and arrived just in time for dinner. They let him choose how he will be eaten. To do this, he must utter some statement with the condition: if this statement turns out to be true, they will cook it, and if it turns out to be false, they will roast it. What should the missionary say?

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

4. Some Modern Paradoxes

The paradox discovered by the English logician and philosopher of the last century B. Russell had the most serious impact not only on logic, but also on mathematics.

Russell came up with such a popular version of his paradox - the "barber's paradox". Let us suppose that the council of some village defined the duties of the village barber as follows: to shave all men who do not shave themselves, and only these men. Should he shave himself?

If so, it will refer to those who shave themselves; but those who shave themselves, he must not shave. If not, he will belong to those who do not shave themselves, and therefore he will have to shave himself. We thus come to the conclusion that this barber shaves himself if, and only if, he does not shave himself. This, of course, is impossible.

In the original version, Russell's paradox concerns sets, that is, collections of objects that are somewhat similar to each other. Concerning an arbitrary set, one can ask the question: is it its own element or not? Thus, the set of horses is not a horse, and therefore it is not its own element. But the multitude of ideas is an idea and contains itself; a directory of directories is again a directory. The set of all sets is also its own element, since it is a set. Dividing all sets into those that are proper elements and those that are not, one can ask: does the set of all sets that are not proper elements contain itself as an element or not? The answer, however, turns out to be discouraging: this set is its element only if it is not such an element.

This reasoning is based on the assumption that there is a set of all sets that are not proper elements. The contradiction derived from this assumption means that such a set cannot exist. But why is such a simple and clear set impossible? What is the difference between possible and impossible sets?

Researchers answer these questions in different ways. The discovery of Russell's paradox and other paradoxes of mathematical set theory led to a decisive revision of its foundations. It served, in particular, as a stimulus for excluding from its consideration "too large sets", similar to the set of all sets, for restricting the rules for operating with sets, etc. Despite the large number of methods proposed to date for eliminating paradoxes from set theory there is still no agreement on the causes of their occurrence. Accordingly, there is no single, unobjectionable way to prevent their occurrence.

The above argument about the hairdresser is based on the assumption that such a hairdresser exists. The resulting contradiction means that this assumption is false, and there is no such villager who would shave all those and only those villagers who do not shave themselves.

The duties of a hairdresser do not seem contradictory at first glance, so the conclusion that there cannot be one sounds somewhat unexpected. However, this conclusion is not paradoxical. The condition that the "village barber" must satisfy is in fact self-contradictory and therefore impossible. There cannot be such a hairdresser in a village for the same reason that there is no person in it who would be older than himself or who would be born before his birth.

The argument about the hairdresser can be called a pseudo-paradox. In its course, it is strictly analogous to Russell's paradox, and this is what makes it interesting. But it is still not a true paradox.

Another example of the same pseudo-paradox is the well-known catalog argument.

A certain library decided to compile a bibliographic catalog that would include all those and only those bibliographic catalogs that do not contain references to themselves. Should such a directory include a link to itself?

It is easy to show that the idea of ​​creating such a directory is unfeasible: it simply cannot exist, since it must simultaneously include a link to itself and not include it.

It is interesting to note that cataloging all directories that do not contain references to themselves can be thought of as an endless, never ending process.

Let's say that at some point a directory, say K1, was compiled, including all other directories that do not contain references to themselves. With the creation of K1, another directory appeared that does not contain a link to itself. Since the goal is to make a complete catalog of all directories that do not mention themselves, it is obvious that K1 is not the solution. He doesn't mention one of those directories - himself. Including this mention of himself in K1, we get the K2 catalog. It mentions K1, but not K2 itself. Adding such a mention to K2, we get KZ, which is again incomplete due to the fact that it does not mention itself. And so on without end.

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

Some words denoting properties have the very property they name. For example, the adjective "Russian" is itself Russian, "polysyllabic" is itself polysyllabic, and "five-syllable" itself has five syllables. Such words referring to themselves are called "self-meaning" or "autological". There are not so many similar words, the vast majority of adjectives do not have the property they call. "New" is not, of course, new, "hot" - hot, "one-syllable" - consisting of one syllable, "English" - English. Words that do not have the property they signify are called "non-meaningful" or "heterological". Obviously, all adjectives denoting properties that are not applicable to words will be heterological.

This division of adjectives into two groups seems clear and unobjectionable. It can be extended to nouns: "word" is a word, "noun" is a noun, but "clock" is not a clock, and "verb" is not a verb.

A paradox arises as soon as the question is asked: to which of the two groups does the adjective "heterological" itself belong? If it is autological, it has the property it designates and must be heterological. If it is heterological, it does not have the property it calls, and must therefore be autological. There is a paradox.

It turned out that Grelling's paradox was already known in the Middle Ages as the antinomy of an expression that does not name itself.

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

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

5. What are the paradoxes about?

paradox liar logic argument

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

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

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

At the beginning of the study of logical paradoxes, it seemed that they could be distinguished by the violation of some yet unexplored rule of logic. Especially actively claimed the role of such a rule introduced by Russell "principle of a vicious circle". This principle states that a collection of objects cannot contain members defined only by the same collection.

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

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

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

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

There have been many proposals on this subject, but a successful clarification of circular™ has not been found. It turned out to be impossible to characterize circularity in such a way that every circular reasoning leads to a paradox, and every paradox is the result of some circular reasoning.

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

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

The paradox does not always appear in such a transparent form, as in the case of, say, the liar's paradox or Russell's paradox. Sometimes a paradox turns out to be a peculiar form of posing a problem, with respect to which it is even difficult to decide what exactly the latter consists of. Thinking about such problems usually does not lead to any definite result. But it is undoubtedly useful as a logical training.

The ancient Greek philosopher Gorgias wrote an essay with the intriguing title "On the Non-Existent, or On Nature."

Gorgias' reasoning about the non-existence of nature unfolds as follows. First, we prove that nothing exists. As soon as the proof is completed, a step back is taken, as it were, and it is assumed that something still exists. From this assumption it is deduced that the existing is incomprehensible to man. Once again a step is taken back and it is assumed, contrary to what has already seemed to be proven, that what exists is nevertheless comprehensible. From the last assumption it is deduced that the comprehensible is inexpressible and inexplicable for another.

What exactly did Gorgias want to pose? It is impossible to answer this question unambiguously. Obviously, Gorgias' reasoning confronts us with contradictions and encourages us to look for a way out in order to get rid of them. But what exactly are the problems indicated by the contradictions, and in what direction to look for their solution, is completely unclear.

It is known about the ancient Chinese philosopher Hui Shi that he was very versatile, and his writings could fill five wagons. He, in particular, argued: “That which does not have thickness cannot be accumulated, and yet its bulk can stretch for a thousand miles. - Heaven and earth are equally low; mountains and swamps are equally flat. - The sun, which has just reached its zenith, is already at sunset; a thing that has just been born is already dying. - The south side of the world has no limit and at the same time has a limit. "Only today, having gone to Yue, I arrived there a long time ago."

Hui Shi himself considered his sayings great and revealing the most hidden meaning of the world. Critics found his teaching contradictory and confused, and stated that "his partisan words never hit the mark." In the ancient philosophical treatise Zhuang Tzu, in particular, it says: “What a pity that Hui Shi thoughtlessly wasted his talent on unnecessary things and did not reach the sources of truth! He pursued the outer side of the darkness of things and could not return to their innermost beginning. It's like trying to run away from the echo by making sounds, or trying to run away from your own shadow. Isn't that sad?"

Well said, but hardly fair.

The impression of confusion and inconsistency in Hui Shi's sayings is connected with the external side of the matter, with the fact that he poses his problems in a paradoxical form. What could be reproached him for is that for some reason he considers the presentation of a problem to be its solution.

As with many other paradoxes, it is difficult to say with certainty what specific questions lie behind Hui Shi's aphorisms.

What intellectual embarrassment is hinted at by his statement that a person who has just gone somewhere has long since arrived there? One can interpret this in such a way that, before leaving for a certain place, one must imagine this place and thereby, as it were, visit there. A person who, like Hui Shi, is heading to Yue, constantly keeps this point in mind and during the entire time of moving towards it, as it were, stays in it. But if a person who just went to Yue has already been there for a long time, then why would he even go there? It is not entirely clear what difficulty lies behind this simple saying.

What conclusions for logic follow from the existence of paradoxes?

First of all, the presence of a large number of paradoxes speaks of the strength of logic as a science, and not of its weakness, as it might seem. It was no coincidence that the discovery of paradoxes coincided with the period of the most intensive development of modern logic and its greatest successes.

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

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

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

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

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

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

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

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

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

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

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

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

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

G. Frege, who is one of the founders of modern logic, had a very bad character. In addition, he unconditionally and even severely criticized his contemporaries. Perhaps that is why his contribution to the logic and foundation of mathematics did not receive recognition for a long time. And just as it began to come, the young English logician Russell wrote to him that there was a contradiction in the system published in the first volume of his most important book, The Fundamental Laws of Arithmetic. The second volume of this book was already in print, but Frege added a special appendix to it, in which he outlined this contradiction (Russell's paradox) and admitted that he was not able to eliminate it.

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

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

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

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

And at the same time, such a state hardly seems unbearable to anyone now. Over time, attitudes towards paradoxes have become calmer and even more tolerant than at the time they were discovered.

The point is not only that paradoxes have become something, although unpleasant, but nonetheless familiar. And, of course, not that they put up with them. They still remain in the center of attention of logicians, the search for their solutions is actively continuing.

The situation has changed, first of all, in the sense that the paradoxes have turned out to be, so to speak, localized. They found their definite, albeit troubled, place in a wide range logical research.

It became clear that absolute austerity, as it was portrayed at the end of the last century and even sometimes at the beginning of this century, is, in principle, an unattainable ideal.

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

Meditation on paradoxes is without a doubt one of the best tests of our logical powers and one of the most effective means of training them.

Acquaintance with paradoxes, penetration into the essence of the problems behind them is not an easy task. It requires maximum concentration and intense thought into a few seemingly simple statements. Only under this condition can the paradox be understood. It is difficult to pretend to invent new solutions to logical paradoxes, but already familiarization with the proposed solutions is a good school of practical logic.

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2. Paradox. Concept, examples

Turning to the question of paradoxes, it is impossible not to say about their relationship with sophisms. The fact is that sometimes there is no clear line by which you can understand what you have to deal with.

However, paradoxes are considered with a much more serious approach, while sophisms often play the role of a joke, nothing more. This is due to the nature of theory and science: if it contains paradoxes, then there is an imperfection in the underlying ideas.

What has been said may mean that the modern approach to sophistry does not cover the entire scope of the problem. Many paradoxes are interpreted as sophisms, although they do not lose their original properties.

paradox one can name a reasoning that proves not only the truth, but also the falsity of a certain judgment, i.e., proving both the judgment itself and its negation. In other words, paradox- these are two opposite, incompatible statements, for each of which there are seemingly convincing arguments.

One of the first and certainly exemplary paradoxes was recorded Eubulides- Greek poet and philosopher, Cretan. The paradox is called "The Liar". This paradox has reached us in this form: “Epimenides claims that all Cretans are liars. If he is telling the truth, then he is lying. Is he lying or is he telling the truth? This paradox is called "the king of logical paradoxes". To date, no one has been able to solve it. The essence of this paradox is that when a person says: “I am lying”, he does not lie and does not tell the truth, but, more precisely, he does both at the same time. In other words, if we assume that a person is telling the truth, it turns out that he is actually lying, and if he is lying, then he told the truth about it before. Both contradictory facts are asserted here. Of course, according to the law of the excluded middle, this is impossible, but that is why this paradox received such a high “title”.

The inhabitants of the city of Elea, the Eleatics, made a great contribution to the development of the theory of space and time. They relied on the idea of ​​the impossibility of non-existence, which belongs Parmenides. Every thought according to this idea is a thought about the existent. At the same time, any movement was denied: the world space was considered integral, the world is one, without parts.

ancient greek philosopher Zeno of Elea known for compiling a series of paradoxes about infinity - the so-called paradoxes of Zeno.

Zeno, a student of Parmenides, developed these ideas, for which he was named Aristotle"ancestor of dialectics". Dialectics was understood as the art of reaching the truth in a dispute, revealing contradictions in the opponent's judgment and destroying them.

"Achilles and the tortoise" represents an aporia about movement. As you know, Achilles is an ancient Greek hero. He had remarkable abilities in sports. The turtle is a very slow animal. However, in an aporia, Achilles loses the race to the tortoise. Suppose Achilles needs to run a distance of 1, and he runs twice as fast as a tortoise, the latter needs to run 1/2. Their movement starts at the same time. It turns out that, having run the distance 1/2, Achilles will find that the tortoise managed to overcome the segment 1/4 in the same time. No matter how much Achilles tries to overtake the tortoise, it will be exactly 1/2 ahead. Therefore, Achilles is not destined to catch up with the tortoise, this movement is eternal, it cannot be completed.

The inability to complete this sequence is that it is missing the last element. Each time, having indicated the next member of the sequence, we can continue by indicating the next one.

The paradox here is that the endless sequence of successive events must actually come to an end, even if we could not imagine this end.

Another aporia is called "dichotomy". The reasoning is built on the same principles as the previous one. In order to go all the way, you need to go half way. In this case, half of the path becomes a path, and in order to pass it, it is necessary to measure out half (that is, already half of the half). This continues ad infinitum.

Here the order is reversed compared to the previous aporia, i.e. (1/2)n…, (1/2)3, (1/2)2, (1/2)1. The series here does not have the first point, while the aporia "Achilles and the tortoise" did not have the last.

From this aporia, it is concluded that the movement cannot begin. Proceeding from the considered aporias, the movement cannot end and cannot begin. So it doesn't exist.

Refutation of the aporia "Achilles and the tortoise".

As in the aporia, Achilles appears in its refutation, but not one, but two turtles. One of them is closer than the other. The movement also starts at the same time. Achilles runs last. During the time that Achilles runs the distance separating them at the beginning, the nearest tortoise will have time to crawl a little ahead, which will continue indefinitely. Achilles will get closer and closer to the tortoise, but he will never be able to catch up with it. Despite the obvious falsity, there is no logical refutation of such an assertion. However, if Achilles begins to catch up with a distant tortoise, not paying attention to the near one, he, according to the same aporia, will be able to come close to it. And if so, then he will overtake the nearest turtle.

This leads to a logical contradiction.

In order to refute the refutation, i.e., to defend the aporia, which is strange in itself, it is proposed to throw away the burden of figurative representations. And to reveal the formal essence of the matter. Here it should be said that the aporia itself is based on figurative representations and to reject them means to refute it as well. And the rebuttal is quite formal. The fact that two turtles are taken instead of one in the refutation does not make it more figurative than an aporia. In general, it is difficult to talk about concepts that are not based on figurative representations. Even such philosophical concepts of the highest abstraction as being, consciousness, and others are understood only thanks to the images that correspond to them. Without the image behind the word, the latter would remain only a set of symbols and sounds.

Stages implies the existence of indivisible segments in space and the movement of objects in it. This aporia builds on the previous ones. Take one immovable row of objects and two moving towards each other. Moreover, each moving row in relation to the immovable one passes only one segment per unit of time. However, in relation to the moving - two. which is considered contradictory. It is also said that in an intermediate position (when one row has already moved, as it were, the other has not) there is no room for a fixed row. The intermediate position comes from the fact that the segments are indivisible and the movement, even though it started simultaneously, must go through an intermediate stage when the first value of one moving series coincides with the second value of the second (movement, provided that the segments are indivisible, is devoid of smoothness). The state of rest is when the second values ​​of all rows coincide. The fixed row, if we assume the simultaneity of the movement of the rows, must be in an intermediate position between the moving rows, and this is impossible, since the segments are indivisible.

1. Sophisms. Concept, examples Expanding this issue, it must be said that any sophism is a mistake. There are also paralogisms in logic. The difference between these two types of errors lies in the fact that the first (sophism) is committed intentionally, while the second (paralogism) is accidental.