Examples of paradox in literature. Logical paradoxes
A literary text, being a work of literature, is one of the means of human knowledge of the surrounding world and is aimed at the formation of human personality. One of the most striking means of updating the recipient’s attention to important
E. A. Yashina
Typology of paradoxes in literary texts
in the semantic aspect of the artistic context there is a paradoxical contradiction, the main functions of which will be discussed in this article, since it is the functional orientation of the paradox that determines its main features,
allowing us to highlight the main types of paradoxical contradiction within the artistic context.
The concept of “paradox” arose in ancient Greek philosophy to characterize a new original opinion1. Paradox as a stylistic device dates back to ancient rhetoric, and in modern science is interpreted ambiguously from the positions of logic, philosophy and linguistics. Many works of modern scientists are devoted to the development of the logical-philosophical concept of paradox, among whom the work of the famous French philosopher Gilles Deleuze’s “Logic of Sense”2, where an attempt is made to apply the logical and philosophical foundations of the theory of the search for meaning and the emergence of paradoxes to the study of literary text. “Common sense asserts that all things have a clearly defined meaning; but the essence of the paradox is the affirmation of two meanings at the same time,” the author rightly notes in his book3.
Some illogicality, inconsistency and paradox that arise during the interaction of elements of a literary text are due to “polysemy, inaccuracy of meaning, exceptions to the rules, inexplicable language customs, discrepancies in the pronunciation and spelling of words, the lack of a universal logic of construction”4 in natural language, the means of which are used by the author of a work of art. It is obvious that a paradoxical statement must conflict with the principles and foundations stored in the experience of the recipient. It sharpens the reader’s perception, makes him look at familiar things from a different angle, and at the same time stimulates comprehension of the text as a whole. In the ontological structure of the recipient, a contradiction arises between what he is accustomed to and this new approach to the analysis of the surrounding reality, which refutes the usual. The essence of the paradox lies in the realization of the truth of both.
Considering the paradox in a literary text from the point of view of pragmatics, on a functional basis, it seems possible to identify several types of paradoxical contradictions that reflect the leading functional properties of works of literary and artistic creativity. The most voluminous and frequently encountered phenomenon in literary texts is a paradox that touches on eternal problems that are significant both for an individual person and for the development of society as a whole, and is classified as philosophical. This type of paradox seeks to gain a deeper understanding of the laws of coexistence and interaction of people in society, which often brings it closer to the historical paradox inherent in works of art that describe significant events in the life of a particular state and the world. The problem of reproducing the versatility and inconsistency of human character, captured in an artistic image, helps to reveal the so-called characterological paradox, which often accompanies an unexpected plot twist, which is, in turn, fixed by a plot paradox. The plot and characterological types of paradox are often united by the common function of characterizing a particular image of a literary text. A special role - the function of realizing the author's irony in an artistic context - is played by the type of paradox defined as ironic.
From the point of view of the syntactic organization of a paradoxical judgment in a literary text, paradoxes are distinguished that are realized in a macro context, i.e. within the text of a work of art as a whole (an example of this kind of paradox can be paradoxical plot twists), in a micro context (within a paragraph or several sentences) , as well as at the level of sentences and phrases. In this case, a characteristic feature of the paradox is often parallelism.
On a semantic basis, it seems possible to classify such types of artistic paradox as:
Paradoxes based on comparison;
Paradoxes based on opposition;
Paradoxes-periphrases based on famous statements.
Let us try to analyze the features of the functioning of the designated types of paradox in a literary text. Let us note in advance the fragility of the boundaries drawn between different types of paradoxically contradictory statements. For example, a philosophical paradox undoubtedly has the ability to reflect paradoxical moments in a person’s life, those aspects that seem true when viewed from two opposing and contradictory positions. At the same time, the paradoxical comparison is realized and fully comprehended by the reader only in conjunction with an analysis of the plot outline of the work, which emphasizes the closeness of the plot and philosophical types of paradox. The latter often acquires the ability to exist outside the context of a work of art, often as a catchphrase or aphorism. The interweaving of characterological and philosophical types of paradox gives the reader the opportunity to evaluate the artistic image more broadly, from the philosophical position of analyzing existence.
Despite the fact that the nature of the emergence of a paradoxical judgment and the deep meanings implied by the author can only be reflected in the quoted context of a work of art, a philosophical paradox, while preserving the main set of noemas that make up its original semantic mosaic, is capable of acquiring new additional meanings within the framework of a new citing situation. The philosophical type of paradox, classified from the point of view of functional analysis, is often found in the works of B. Shaw. Let's analyze a few examples from his play “The Man of Destiny”:
“Blood costs nothing; wine costs money"5.
Compare: “Blood costs nothing, wine costs money.”
“What else but a love could stir up so much hate?”6.
Wed: “What else, if not love, could cause so much hatred?”
According to the method of syntactic organization, the cited paradoxes are realized within the framework of one syntactic construction; according to the semantic feature, they are paradoxes based on opposition, the characteristic feature of which is the presence of an antithesis, i.e. a statement containing two words or two groups of words interconnected by relations lexical or contextual antonymy. Thus, the lexical unit blood (blood) is contextually contrasted with the lexical unit wine (wine), the expression costs nothing (costs nothing) is contrasted with costs money (costs money), while the opposition is strengthened by the use of antithesis elements in parallel constructions. In the second example, the antithesis love - hate is clearly visible, while the paradoxical nature is emphasized by the fact that the cause-and-effect conditionality of one antonym by another is explicitly highlighted. The meaning implied in the cited paradoxes, classified as philosophical in their functional orientation, can be easily traced outside the context of the work, since it is universal and, from a non-standard position, re-reveals some of the moral foundations of society. A paradoxical judgment is aimed at updating two contradictory meanings at the same time. When perceiving a paradox, such meanings are located side by side in the ontological construction of the recipient. The more universal the paradox is in the sphere of its philosophical existence, the more capable it is of being realized regardless of the original context, the brighter and more obvious the inconsistency and at the same time the similarity of the concepts and meanings compared in it, the more noticeable it is for the attention of the recipient.
Actualization of contradictions in the behavior of one or another image of artistic production
The work performs the function of characterizing the image and relates to characterological paradoxes. The peculiarities of the functioning of a characterological contradiction, in contrast to a paradox defined by its functional orientation as philosophical, are quite difficult to trace in isolation from the holistic context of the work. Let us only note that the following characterological paradox, quoted from O. Henry’s story “Conscience in Art,” has the goal of focusing the reader’s attention on the characteristic features of the described images through an unusual combination of lexical means and a special syntactic organization of the utterance:
“They are rough but uncivil in their manners, and though their ways are boisterous and unpolished, under it all they have a great deal of impoliteness and discourtesy”7.
Wed. “Their manners are rude but impolite, and, despite their noisy and untrained behavior, they are largely disrespectful and discourteous.”
The master of the short story O. Henry succinctly characterizes the characters within one sentence. Moreover, instead of using antonyms in the antithesis, which is clearly visible in the quotation thanks to the adversative conjunctions but and though, the author uses synonyms (highlighted in the original text), which violates the laws of the usual construction of the antithesis, thereby concentrating the reader’s attention on this segment of the text. Thus, the paradoxical nature of this fraction of the text is based on the comparison of lexical units with similar meanings within the framework of the stylistic “figure of opposition”8. Consequently, this example, characterological in functional terms, represents a paradox based on comparison - on a semantic basis, as well as a paradox realized in a microcontext - on the basis of syntactic organization.
The means of emotional assessment of historical events is historical
a paradox that figuratively and vividly conveys contradictory moments in the historical development of society. The artistic depiction of historical events contributes to their deeper emotional and sensory perception and understanding by the reader. Let us note that identifying the causes of a historical paradox is the task of historical research. The author of a work of fiction only subjectively and figuratively reproduces in the context of the novel a historical contradiction that once arose. The most striking point that focuses the recipient’s attention on paradoxical moments in the historical development of Soviet society seems to be the following section of the text of B. Pasternak’s novel “Doctor Zhivago”, in which the author sums up the development of the Soviet state in the first decades of its existence:
“I think collectivization was a false, failed measure, and a mistake could not be admitted. To hide the failure, it was necessary to use all means of intimidation to wean people from judging and thinking and to force them to see what does not exist and to prove the opposite with evidence. Hence the unprecedented cruelty of the Yezhovshchina, the promulgation of a constitution not designed for application, the introduction of elections not based on the elective principle.
And when the war broke out, its real horrors, real danger and threat real death were a blessing compared to the inhuman dominion of fiction, and brought relief because they limited the witchcraft power of the dead letter.”9
This paradox is of particular interest because it combines several methods of syntactic organization: the entire cited example as a whole is a paradox realized in a microcontext, on the one hand, also containing smaller syntactic constructs of expressing paradoxicality, at the level of a phrase and an individual sentence ( the first sentence of the analyzed example) - on the other. At the same time, each
the stage of realization of the contradiction is harmoniously woven into the general outline of the paradoxical microcontext, which is closely interconnected with the macrocontext and represents a certain conclusion to the entire work as a whole. The paradoxical nature of the text fraction in question forces us to draw a certain cause-and-effect relationship between the historical events described in the novel and their influence on the fate of the heroes. On a semantic basis, the paradox in the first syntactic construction of the analyzed example is based on the comparison of lexical units false, failed measure, error and expressions cannot be admitted, which reflects the illogicality of actions. Paradoxes at the level of phrases (seeing something that does not exist, proving the opposite of evidence, a constitution not designed for application, elections not based on an elective principle) from the point of view of semantic organization are based on opposition in which antonyms refute the lexical meaning of each other. The second paragraph of the analyzed quotation on a semantic basis represents a paradox based on a comparison of poorly comparable lexical units, usually in a relationship of complete or partial antonymy: war, horror, danger, threat of death are paradoxically presented as good and relief. At the same time, the repeatedly repeated lexical unit real is contrasted in its meaning with the word fiction and the expression witchcraft power of a dead letter. A characteristic feature of a historical paradox is the absence of the author’s irony in the process of comparing incomparable concepts and describing inexplicable turns in the historical development of society. Such a pile-up of means of creating paradox makes the fraction of the text in question rhetorically marked, and the paradox serves as a means of projecting the reader’s attention onto it.
The following paradoxical quote is from George Orwell's story "Animal Farm"
“All animals are equal. But some animals are more equal than others”10.
Compare: “All animals are equal. But some animals are more equal than others."
The paradox lies in the use of the adjective equal (equal) in a comparative degree that is unusual for it, due to the set of semes that make up its lexical meaning. Thus, the paradox is based on a comparison of the lexical meaning of a word and one of the theoretically possible forms of its use. Most of the paradoxes that reflect the author's irony can be perceived as such only in the context of a particular situation, including the functional orientation of the analyzed quote as ironic and is realized in the general context of the narrative, which describes an attempt to create a society of universal equality. Just as it is paradoxical that it is theoretically possible to use the lexical unit equal to a comparative degree in practice, universal equality in society is practically impossible. The author's ironic attitude towards the principle proclaimed in the first syntactic construction of the analyzed example is highlighted on the basis of the subsequent paradox, realized, from the point of view of its syntactic organization, within the framework of the sentence. Irony, as Gilles Deleuze puts it, is the art of “depth and height.” Paradox, as the author rightly notes, is “the liberation of depth, the bringing of an event to the surface and the deployment of language along this limit”11.
The plot paradox is entirely based on a logically structured text as a single whole. A classic example of a paradox in the development of a plot is considered to be the plot outline of O. Wilde’s novel “The Picture of Dorian Gray”, according to which it is the portrait of the main character that ages, and not himself, when
In this case, the hero’s base actions remain externally imprinted on his image. The essence of the plot paradox is succinctly characterized by the following paradoxical proposition, which semantically represents another type of paradox - a paraphrase paradox of a well-known statement:
“It is only shallow people who do not judge by appearances”12.
Compare: “Only superficial people do not judge by appearance.”
Obviously, the basis of the periphrasis was the well-known English proverb Never judge by appearances13, which is refuted in a paradoxical statement through the use of the lexical unit shallow, which contains in its meaning negative connotations in the characteristics of people acting in accordance with the stereotypical model of behavior reflected in the proverb. Being, in its functional orientation, a means of explicating the paradox in the development of the plot, the analyzed paradoxical judgment, implemented at the sentence level, is semantically a paraphrase paradox based on a well-known statement.
Thus, a practical analysis of factual material quoted from well-known works fiction in Russian and English, made it possible to classify the types of paradoxical contradictions in a literary text according to three main characteristics: according to the method of syntactic and semantic organization, as well as in accordance with the functions of artistic paradox in the text. As a result of the study of the functions of paradoxical judgments in a literary text, characterological, historical, ironic, plot and philosophical types paradox. According to the method of syntactic organization, contradictions in a literary text are divided into paradoxes, realized in
at the level of phrases, sentences, as well as in the microcontext and context of the work as a whole. The study of the semantic features of the interaction of lexical units as part of an artistic paradox made it possible to identify paradoxes based on comparison; paradoxes based on opposition or contrast; Paradoxes-paraphrases of famous sayings. The idea expressed in a literary text through paradox attracts attention by the unusual duality of the approach to the problem treated in the literary work, hardly goes unnoticed and stimulates the reader’s search for his own answers to the questions considered by the author.
1 Brief literary encyclopedia: in 8 volumes. M., 1968. T. 5. P. 592.
2 Deleuze J. Logic of meaning. M., 1995.
3 Ibid. P. 13.
4 Kasavin I. T. The language of everyday life between logic and phenomenology // Questions of Philosophy. 2003. No. 5. P. 25.
5 Shaw B. The Man of Destiny // Selected Works. Moscow, 1958. P. 101.
6 Ibid. P. 123.
7 O. Henry. The Room in the Attic and Other Stories: a book to read on English language. M., 1972. P. 14.
8 Brandes M. P. Stylistics of the text. Theoretical course. M., 2004. P. 376.
9 Pasternak B. L. Doctor Zhivago: A Novel. M., 1989. P. 179.
10 Orwell J. Animal Farm and a collection of essays: a book to read in English. St. Petersburg, 2004. P. 118.
11 Deleuze J. Logic of meaning. M., 1995. S. 22-23.
12 Wilde O. The Picture of Dorian Gray // Selected. production in 2 volumes (in English). M., 1979. T. 1. P. 114.
13 Rideout R. Whitting K. Dictionary English proverbs. St. Petersburg, 1997. P. 141.
What is a paradox? A paradox is two incompatible and opposing statements, each with convincing arguments in its own direction. The most pronounced form of paradox is antinomy - reasoning that proves the equivalence of statements, one of which is an explicit denial of the other. And it is the paradoxes in the most precise and rigorous sciences, such as, for example, logic, that deserve special attention.
Logic, as you know, is an abstract science. There is no place in it for experiments and any specific facts in the usual sense; it always involves an analysis of real thinking. But discrepancies in the theory of logic and the practice of real thinking still take place. And the most obvious confirmation of this is logical paradoxes, and sometimes even logical antinomy, which personifies the inconsistency of the logical theory. This is precisely what explains the meaning of logical paradoxes and the attention paid to these paradoxes in logical science. Below we will introduce you to the most striking examples of logical paradoxes. This information will certainly be of interest to both those who study logic in depth and those who simply like to learn new and interesting information.
Let's start with the paradoxes compiled by the ancient Greek philosopher Zeno of Elea, who lived in the 5th century BC. His paradoxes are called “Zeno’s Aporias” and even have their own interpretation.
Aporias of Zeno
Zeno's aporias are seemingly paradoxical arguments about motion and multitude. In total, Zeno’s contemporaries mentioned over 40 aporias (by the way, the word “aporia” from ancient Greek is translated as “difficulty”) of his authorship, but only nine of them have survived to this day. If you wish, you can familiarize yourself with them in the works of Aristotle, Diogenes Laertius, Plato, Themistius, Philoponus, Aelius and Symplykius. We will give examples of the three most famous ones.
Achilles and the tortoise
Let's imagine that Achilles is running at a speed ten times faster than the speed of the tortoise, and is a thousand steps behind it. While Achilles runs a thousand steps, the tortoise will only take a hundred. While Achilles will overcome another hundred, the tortoise will have time to do ten, etc. And this process will continue indefinitely and Achilles will never catch up with the turtle.
Dichotomy
In order to overcome a certain path, you must first overcome half of it, and in order to overcome half, you need to overcome half of this half, etc. Based on this, the movement will never begin.
Flying arrow
A flying arrow always remains in place, because... at any moment of time it is at rest, and since it is at rest at any moment of time, it is always at rest.
It would be appropriate to bring up another paradox here.
The Liar Paradox
The authorship of this paradox is attributed to the ancient Greek priest and seer Epimenides. The paradox sounds like this: “What I am saying at the moment is a lie,” i.e. comes out: either “I am lying” or “My statement is false.” This means that if a statement is true, then based on its content it is a lie, but if the statement is inherently false, then its statement is a lie. It turns out that this statement is false. Therefore, the statement is true - this conclusion takes us back to the beginning of our reasoning.
Nowadays, the liar paradox is considered as one of the formulations of Russell's paradox.
Russell's paradox
Russell's paradox was discovered in 1901 by the British philosopher Bertrand Russell and was later independently rediscovered by the German mathematician Ernst Zermelo (sometimes called the "Russell-Zermelo paradox"). This paradox demonstrates the inconsistency of Frege's logical system, in which mathematics is reduced to logic. Russell's paradox has several formulations:
- The paradox of omnipotence - is an omnipotent being capable of creating anything that can limit his omnipotence?
- Let's say that some library has set the task of compiling one large bibliographic catalogue, which should include all and only those bibliographic catalogs that do not contain references to themselves. Question: Should I include a link to it in this directory?
- For example, in some country a law was passed stating that the mayors of all cities are prohibited from living in their city, and are only allowed to live in the “City of Mayors”. Where, then, will the mayor of this city live?
- The barber's paradox - there is only one barber in the village, and he is ordered to shave everyone who does not shave themselves, and not to shave those who shave themselves. Question: Who should shave the barber?
The following paradoxes are no less interesting and amusing.
The Burali-Forti paradox
The assumption that the idea of the possibility of a set of ordinal numbers can lead to contradictions, which means that set theory in which the construction of a set of ordinal numbers is possible will be contradictory.
Cantor's paradox
The assumption that a set of all sets is possible can lead to contradictions, which means that the theory according to which the construction of such a set is possible will also be contradictory.
Hilbert's paradox
The idea that if all the rooms in a hotel with an infinite number of rooms are occupied, more people can be accommodated in it anyway, and their number can be infinite. This paradox explains that the laws of logic are absolutely unacceptable to the properties of infinity.
Monte Carlo False Conclusion
The conclusion is that when playing roulette, you can safely bet on red if black comes up ten times in a row. This conclusion is considered false for the reason that, according to the theory of probability, the occurrence of any subsequent event is not influenced in any way by the event preceding it.
Einstein-Podolsky-Rosen paradox
The question is whether processes and events developing far from each other are capable of influencing each other? For example, does the birth of a supernova in a distant galaxy in any way affect the weather in Moscow? The answer can be given as follows: based on the laws of quantum mechanics, such an influence is impossible due to the fact that both the speed of light and the speed of information transfer are finite quantities, and the Universe is infinite.
Twin paradox
Question: will the twin traveler who returned from space travel on a superluminal spaceship be younger than his brother, who remained on Earth all this time? If we proceed from the theory of relativity, then more time has passed on Earth (according to the terrestrial flow of time) than in a spaceship flying at superluminal speed, which means that the traveler twin will be younger.
The paradox of the murdered grandfather
Imagine that you went back in time and killed your grandfather before he met your grandmother. The conclusion is that you will not be born and will not be able to go back in time to kill your grandfather. The presented paradox clearly demonstrates the impossibility of traveling into the past.
The paradox of predestination
For example, a person finds himself in the past, has sexual intercourse with his great-grandmother and conceives her son, i.e. his grandfather. This causes a succession of descendants, including the person's parents as well as the person himself. It turns out that if this person had not traveled into the past, he would never have been born at all.
These are just a few of the logical paradoxes that occupy the minds of many people today. It will not be difficult for an inquisitive mind to find dozens more similar ones (for example,). A considerable amount of time and effort can be devoted to studying, refuting or proving each of them. And, quite likely, you can form your own personal original conclusions about each paradox. But this tells us that, despite the predominance of the laws of logic and cause-and-effect relationships in our lives, not everything in our lives depends on them. Sometimes contradictions similar to logical paradoxes arise in Everyday life each person. In any case, this is excellent food for thought and food for thought.
By the way, regarding reflections: there is a very interesting book on the topic of logical paradoxes called “Gödel, Escher and Bach.” Its author is American physicist and computer scientist Douglas Hofstadter.
Dear readers, it would be great if in your comments you gave several examples of logical paradoxes that are familiar to you. We will also be interested in your opinion about the meaning of logic in our lives - Vote for one of the statements below.
Put down your Rubik's Cube quickly! Various puzzles and puzzles are often very attractive and very addictive. But there are also logical paradoxes - that is, logically correct reasoning leading to mutually exclusive conclusions - and they can be no less entertaining.Here's a classic funny example called the Omnipotence Paradox, which has puzzled many thinkers for centuries: Since God is omnipotent, can He make a stone so heavy that even He can't lift it? Is a subject capable of being so omnipotent as to create something that denies His own omnipotence?
There is another similar example on the same topic: “Could Jesus create a burrito so spicy that even He couldn’t eat it?” While you are thinking about these paradoxical questions, we will tell you about ten of the most unexpected logical puzzles that have interested people throughout time. (Don't worry, we chose the easiest ones that everyone will understand.)
10. The Heap Paradox
Let's go back a little and look into the fourth century BC. In those days there lived Eubulides of Miletus, a man who is considered the inventor of paradoxes. Eubulides came up with four fun puzzles that require a lot of thought to solve.
The heap paradox is the first of these classic paradoxes, and it deals with quantitative characteristics.
If a person has no hair on his head, then we say that he is bald. A person who has 10,000 hairs on his head is not considered bald. What happens if we add one hair to a bald man's head? He'll still be bald.
Now imagine that a person has only 1000 hairs. But the strands are evenly distributed and very thin. Will this person be bald?
Do you think that one grain of wheat is a “heap”? Definitely not. How about two grains? Probably not either. So, at what point does a few grains become a “heap” and a head with sparse hair begins to be considered bald? The problem is uncertainty. Where is the border between one and the other?
9. The Liar Paradox
What I am saying now is a lie. Stop for a second and think. Did I tell the truth or lie? This is called the liar paradox, and it was also formulated by Eubulides. This simple example can also be in another form: “This sentence is a lie” or “I am lying now.”
All these statements contradict themselves: if I really lie, then I told the truth, but if I told the truth, then my statement is false.
So what do you think? Is this sentence a lie?
8. The paradox of the infinite and the finite
The following paradox was formulated by a philosopher named Zeno of Elea, who lived around 495-430 BC. He came up with quite a few puzzles that still remain unsolvable. Have you ever thought about the similarities between the micro and macro worlds? Have you ever thought that perhaps our entire Universe is just a small atom in the Universe of a larger being?
Zeno wanted to show that the idea of a plurality of worlds (which exist side by side in time and space) led to some serious logical inconsistencies. And this shows the Paradox of the infinite and the finite. If separate substances (things, worlds) coexist, then what separates one from the other? Where is the border between them?
This is often also called the Paradox of Multiplicity. It can be illustrated by the example of many objects, but let's focus on two. If there are two substances, what separates them? To separate two substances, there must be something third between them.
There are many substances that could be used in this example, but the main point you already got it. So, let's assume that there is a single huge object called the Universe, which consists of many individual objects. They are also divisible - but to what extent? Will this go on forever? Or is there some extremely small point at which division becomes impossible? The best scientific minds of humanity continue to think about this issue today.
7. The paradox of dichotomy
Another classic example of paradoxes attributed to Zeno is the Dichotomy Paradox. From his discussion of distance and motion, Zeno concluded that, in fact, motion is impossible at all. Just like the Multiplicity Paradox, this example is based on infinite division.
Let's say you decide to go to the store and buy soda. To get there, you will first have to cross halfway. No problem, this statement is completely understandable. But after that, you have to walk half of the remaining half way (that is, three-quarters of the distance from your house to the store). Then you will once again have to overcome half of the remaining, then again, and so on ad infinitum. Each time you cover less and less distance, which means you will never get to the store.
Just a minute. We all know very well that we can safely go to the store and buy soda. So how is this possible? At what point do we cross the last half of the last half of the journey? Zeno seems to have been obsessed with this question. Where is the line beyond which we end up in the store?
6. Achilles and the tortoise
Another famous puzzle from Zeno concerns Achilles and the tortoise, and is very similar to the Dichotomy Paradox. In this example, Achilles is competing with a tortoise. The well-prepared guy Achilles (who is also a demigod) gives the turtle a 100-meter head start. Achilles is an extremely fast runner, and the tortoise... well, he is a tortoise.
As soon as they take off, Achilles rushes after the turtle. In the blink of an eye, he crosses the 100 meters separating them - but during this time the turtle manages to crawl another 10 meters, that is, Achilles has not yet caught up with the turtle.
Achilles continues to run and covers another 10 meters. But during this time the turtle crawls another meter.
By this logic, Achilles will never be able to catch up with the tortoise, because every time he gets closer, the tortoise moves further away. Does this mean that achieving the goal is impossible in principle - even if we are daily convinced of the opposite?
We invite you to guess for yourself what Zeno wanted to show with this example.
5. The paradox of cognition
The paradox of knowledge (also known as Meno's paradox) was described in Plato's Dialogues. Meno enters into a discussion with Socrates about virtue, which leads to questions about the methodology of knowledge. If we don't know what we don't know, how will we know what we need to know?
It turns out that if we want to know something that we don’t know, then we can’t ask the appropriate question? Therefore, we can only learn something new by stumbling upon it by chance, and we will never learn anything by asking questions, which is clearly absurd. Questions are the foundation of any scientific research, and they are always the first step in knowledge.
As Meno said: “And how will you know about it if you are completely ignorant of what it is? Even if you come across it by chance, how will you know that it is something you didn’t know?”
Socrates paraphrased this paradox as follows: “Man cannot seek either what he knows or what he does not know. He cannot look for what he knows, because if he knows it, then he does not need to find out, and if he does not know it, then he does not know what he should look for.” If we know the answer to the question we are asking, then what can we learn by asking questions?
4. The double lie paradox
Let's move on to more modern toys and look at the entertaining sequel to The Liar Paradox called The Double Lie Paradox. Let's start with the riddle that was formulated by the mathematician Philippe Jourdain: take a card or a piece of paper. On one side write: “The sentence on the other side of this card is true.” Now turn it over and write on the other side: “The sentence on the other side of this card is false.”
If the second sentence is true, then the first sentence is false. (Turn over the card.) Here you end up again facing an endless contradiction. If the first sentence is true, then the second is false, but this contradicts the first sentence. Thus, both sentences are correct and incorrect at the same time. Check it out for yourself.
3. Monty Hall Paradox
You may have seen this on many game shows. Let's say there are three boxes. Two of them contain a brick, but the third contains one million dollars. You can choose a box and see if you win a million.
Let's say you choose box "A". And you hope for a million. Then the presenter opens any other box at random, say “B”, and shows that there was a brick there. There are two boxes left and your chances improve.
You just have to choose between the remaining two boxes. And you have the right to change your original choice. Since you don't know what's in your drawer, you're still choosing between the two, and your chances become 50x50, right? Since there are only two boxes left, it means your chances are one of two, couldn’t anything be simpler? Wrong.
It seems (unless you changed your original decision) that it would be counterintuitive to say that your chances are still one in three, but it is. Can you guess why?
2. The hairdresser's paradox
Another modern compiler of paradoxical puzzles is the philosopher Bertrand Russell, the author of Russell's Paradox, one of the variations of which is called the Barber's Paradox. The puzzle is simple: the barber says that he shaves all those people who do not shave themselves. Question: who then shaves the barber?
If he does this himself, then the statement that he shaves only those who do not shave themselves will cease to be true. And if he does not do this, then the statement that he shaves everyone who does not shave himself will be false.
Despite its complexity, this paradox can be compared to an endless list in which we add items about completed tasks. Have you added an item to this list that you have made an item to add an item to your list?
1. Schrödinger's cat
Does the Moon exist in those moments when you are not looking at it? And how can you really know this?
Let's move on to a deeper logical statement, which may not be a paradox. Let's talk about Schrödinger's cat. The idea is that we take a cat and put it in a soundproof box. Now, if we don't open the lid, how can we know whether the cat is alive or dead?
Physicist Erwin Schrödinger came up with this logical example in 1935. It illustrates the Copenhagen interpretation of quantum mechanics: when we are not observing a particle (or matter), it can exist in all possible states. We can draw conclusions about her condition only at the moment of observation.
In a more complex version of the experiment, a cat is placed in a box with a jar of poison, a hammer that breaks the glass when the Geiger counter is triggered, and a radiation source so powerful that there is a 50 percent chance of the Geiger counter going off within an hour.
Science can tell us a lot about the cat and the likelihood that radiation could trigger the counter - but only about each of them individually. But science cannot tell us anything about the state of the cat at the moment if we do not see it with our own eyes.
Thus, after an hour, we can theoretically say equally that the animal is alive and that it is dead, which, as we understand, is absurd and impossible. This was a serious blow to the dominant theories of the time. Even the hardest physicists have begun to rethink their ideas about quantum mechanics.
In a nutshell, every time you look at something (like a chair), you get a certain answer regarding its condition. (He is.) When you turn your head, you can only guess what the likelihood is that he is still there. Yes, we can say with confidence that the chair has not gone anywhere. But if you don't see this, then you don't know what's really going on. So, can we be confident about something that we don't personally observe?
Here's a simpler version of the same paradox: "If there is a fallen tree in the forest and no one saw it fall, can we say that it actually fell?" Niels Bohr, another physicist of the time, would say no. First of all, because if we don’t see it, it doesn’t exist. This is what our famous scientists say. Is it funny?
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Secondary school No. 2.
Our life is pure mathematics
(paradox)
Gulevich Prokhor Andreevich,
student of class 10B
MOBU secondary school No. 2.
Supervisor research work:
Shirshova Elena Viktorovna,
mathematics teacher MOBU secondary school No. 2
Svobodny
2013
Content
Introduction………………………………………………………………………………3
Sophistry………………………………………………………………………………3
Paradoxes………………………………………………………………………………… 4
Paradoxes of our life…………………………………………………………… 6
Paradoxes in statements………………………………………………………8
Conclusion………………………………………………………………………………… 8
Literature………………………………………………………………………………… 9
Introduction
One of the branches of mathematics is logic. This is the science of methods of proof and refutation. Aristotle is considered the founder of logic.
Getting acquainted with mathematical logic, I became interested in the concepts of “sophism” and “paradox”, especially since they, especially paradox, are used in everyday life.
Yes, life is paradoxical. What seems right at first glance turns out to be fundamentally wrong after careful consideration. At first glance, the Sun revolves around the Earth. But in fact, everything is exactly the opposite.
Reading newspapers and watching TV, I began to think about the paradoxes occurring in our lives.
The purpose of my work : find out whether our life really consists of paradoxes.
Based on the above, I will put forwardhypothesis : our whole life is a complete paradox.
Based on the goal and hypothesis, we determine the followingwork tasks:
Study the concepts of “sophism” and “paradox”
Consider examples of sophistry and paradoxes
Find paradoxes in our lives
Sophistry
Sophism - a deliberate mistake made with the aim of confusing the enemy and passing off a false judgment as true. Sophistry (from the Greek “skill”, skill, cunning invention, trick, wisdom) is a false statement that, nevertheless, seems correct. Unlike an involuntary logical error, sophistry is a deliberate, disguised violation of the requirements of logic.
Historically, the concept of “sophism” was inextricably linked with the task of the sophist - to present, through cunning tricks in speech, in reasoning, the worst option as the best, caring not about the truth, but about success in an argument or practical benefit.
I have reviewed mathematical sophisms - amazing statements, the proof of which conceals imperceptible and sometimes quite subtle errors. This is how they differ from a paradox.
Examples:
A) 2 = 3
Proof: 10-10=0 and 15-15=0
10-10=15-15, put 2 and 3 out of brackets
2(5-5)=3(5-5)
2=3, which is what needed to be proven.
The mistake is that you cannot divide by zero (5-5).
B) 4 = 5
Proof: 16 – 36 = 25 – 45
16 – 36 + = 25 – 45+
4= 5 Error
C) A half-empty barrel is equal to a half-full one, which means that an empty barrel is equal to a full one.
It turns out that empty equals full .
G) What you haven't lost, you have. You didn't lose your horns. So you have horns. (sophism
Eubulida)
D) the weight of an elephant is equal to the weight of a mosquito
Let X- the weight of the elephant, and at – mosquito weight. Let us denote the sum of these weights2 n :
x+y=2 n
from this equality we can obtain two more:
x -2 n =- y And x =- y +2 n
multiply these two equalitiesx 2 -2 nx = y 2 -2 ny
add to both sides of the last equality byn 2 :
x 2 -2 nx + n 2 = y 2 -2 ny + n 2 or ( x - n ) 2 =( y - n ) 2
taking the square root of both sides of the last equality, we get:x – n = y - n , x = y ,
those. The weight of an elephant is equal to the weight of a mosquito!
Answer: this absurdity resulted from negligence in extracting the square root.
The persuasiveness of many sophisms at first glance, their logic is usually associated with a well-disguised error. Aristotle called sophistry
"sham evidence" in which the validity of the conclusion appears to be true and is due to a purely subjective impression caused by a lack of logical analysis.
Based on the above examples of sophisms, as well as many others, they can be divided into several groups.
1.Logical - based on violation of the rules of reasoning
2. Terminological - consist of inaccurate or incorrect word usage and phrase construction:"All the angles of a triangle less than 180 0 "in the sense that "every corner less than 180 0 "; "all angles of a triangleequal 180 0 " in the sense of "sum of the angles of a triangleequal to 180 0 »
The understanding and study of proof and refutation began with sophisms. Therefore, sophisms contributed to the emergence of a special science of correct, demonstrative thinking. In other words, sophistry is the reason for the emergence of the science of logic.
Scientists have this property: they will put all of humanity in a dead end, and then a whole generation or even several generations will struggle to get out of it, showing miracles of ingenuity.
Paradoxes
Paradox (from Greek - unexpected, strange) - a situation that can exist in reality, but has no logical explanation. This can be not only a situation, but also a statement, statement, judgment or conclusion. In other words, a paradox is a reasoning that both proves and denies a particular proposition.
Paradox - a statement that diverges from generally accepted opinion, strange. A phenomenon that seems incredible and unexpected (Ozhegov’s dictionary).
Paradox is also considered to be an antinomy (from the Greek literally, contradiction in the law). Paradox is a situation when two mutually exclusive propositions are proven in a theory, and each of these propositions is derived by means that are convincing from the point of view of this theory).
Now let’s look at examples of paradoxes: “People are cruel, but man is kind,” “All people are equal, however, there are great ones.”
Paradox is very close to sophistry. What distinguishes them is that a paradox is an unintentionally obtained contradictory result.
Let's consider a simple example of the paradox associated with the disappearance of lines.
Let's draw ten vertical lines of the same length on a rectangular sheet of paper and draw a diagonal with a dotted line, as shown in Figure 1. There are exactly ten of these lines. Then, cut the rectangle diagonally and move the lower part down to the left, as shown in Figure 2. By counting the number of vertical lines, we will find only nine lines. Which line disappeared and where? If you move the left part to its previous position, the disappeared line will appear again. What's going on?
Figure 1 Figure 2
What happens is this: eight of the ten lines are cut by a dotted line into two segments, and the resulting sixteen segments are “rebuilt”, forming nine lines, each of which is slightly longer than the original ones.
The paradox stimulates new research, a deeper understanding of the theory, its “obvious” postulates, and often leads to its complete revision.
Paradoxes of our life
I would like to draw a parallel between paradoxes and our lives. After all, we encounter paradoxes in everyday life.
Paradox 1 - "The Paradox of Adulthood":
At school:
Teachers scold us: “Put away your phones, you can’t use phones in class,” but they themselves regularly use phones to check the time, answer an urgent call or send an SMS. Or here’s another example, when the teacher has completely run out of patience: “Vasya Pupkin (for example), stop fooling around, otherwise I’ll rip your head off,” but this is impossible. Teachers don’t like it when we are late for lessons, but they themselves are always late at least for a minute.
At Olympiads, we encounter tasks on topics that have not yet been covered.
The history teacher told us how one day we accidentally got some cards ( test papers) sixth graders to eleventh graders – the high school students failed to complete the assignments. Paradox
As a child, many people often hear from their parents: “When will you grow up and become independent?” - well, this is understandable: combining work and caring for a small child is a very difficult matter, but the whole paradox is that when a growing child begins to defend his opinion and claim equality of position, then this begins to infuriate the parents: “How are you and Are you talking to your parents?! You’re too young to argue with us!”... So in the end, to grow up or not to grow up?
It’s even cooler in educational institutions:
Teachers oblige us to complete assignments and attend lessons/classes, but when we use an unconventional approach to completing assignments and try to defend our own point of view, different from the teacher’s, we immediately encounter irritation from the teacher and hear speeches from him about our lack of rights...
Paradox 2 "The Paradox of Tolerance":
It may not be strange, but most often people who seem simple are not simple at all. And those that seem difficult are much simpler than you think... Paradox...
Modern democratic society proclaims itself tolerant. It is interested in people with disabilities and fights racial discrimination, but has no tolerance for people with weak nerves. Judge for yourself: everywhere there is a principle of competition, rating systems, rivalry, a fast pace of life, high requirements for stress resistance - only strong nerves can withstand this! Any work and study is stressful. The one who copes is successful, the one whose nerves can’t stand it is a weakling and a loser. And, meanwhile, each of us has the right to happiness! What kind of new discrimination against people based on stress resistance? Why are there conditions in our century in which only those with strong nerves can survive? Where is tolerance and humanism?
People with weak nerves are also people, and they also deserve a normal life! If they are unsuccessful in constant bustle, this does not mean that they are mediocrity. In a calm environment, they could express themselves, reveal their talents and do a lot. So what kind of world is this in which there are all conditions for some, but not for others?
Paradox 3 - "The paradox of cynicism and kindness":
We are afraid of attention from strangers, forgetting that we are all people first and foremost. Everyone is a person, no matter how close they are to us. We can help, we can take care of any living creature, and at the same time we cannot cope with the health of a friend or colleague.
We come up with holidays, but we cannot enjoy the world around us, nature, birds and flowers. We live in the future, making plans for the month, for the year, but we lose that one moment that is priceless - the moment of the present, here and now.
We always find excuses for our cynicism and indifference, but we always get offended by others when they show this towards us. And even if we try to understand why people cannot and do not want to be kind, then all the same, somewhere deep down we expect that they will make an exception for us.
People want world peace, but at the same time they continue to improve weapons.
There have always been plenty of paradoxes in Russia. We often ask ourselves questions: why was this or that law adopted, who calculates the cost of the consumer basket, who insists on reducing literature lessons at school? And so on and so forth.
Here is another one of the paradoxes. This year, Don State Agrarian University became a laureate of the competition for the best higher education institutions. educational institutions Russia. And almost immediately he was included in the list of educational institutions with signs of inefficiency, along with three other universities in the region.
Paradoxes in statements
For some reason The most long-awaited events happen exactly when you least expect them.
We spend more, but have less; we buy more, but enjoy less.
We have better education, but less intelligence, better knowledge, but assess the situation worse, we have more experts, but also more problems,better medicine, but worse health.
We know how to survive, but we don’t know how to live.
The greater the number of billionaires in a country, the lower the standard of living of the majority of the population.
The more promises candidates make, the less these promises are fulfilled.
The smaller the amount of social guarantees in a country, the higher the level of corruption in it.
In Russia, in order not to violate traffic rules, a double continuous line must also be two meters high!
Having bitten into an apple, it is always more pleasant to see a whole worm in it than half of it...
It's amazing how important your job is when you need to take time off from it, and how unimportant it is when you ask for a raise.
Paradoxes of our life: The more cheese, the more holes. But, on the other hand, the more holes, the less cheese...
Conclusion
By researching and analyzing sophisms and paradoxes, I tried to find out whether our life really consists of paradoxes.
People think about the end of the world for all humanity, without noticing the end of the world in their souls; worry about global warming, forgetting to warm their hearts.It’s strange, but sometimes there is a feeling that everything is turned upside down. People pass by a man lying on the street with indifferent faces, but they organize charity concerts, the preparation of which takes an incredible amount of time and money; they dream of a family and a career, but remain blind to those around them. People strive to get the best, but cannot do a nice little thing for a stranger. Paradox…
The contradictions on which most of the paradoxes of our life are found are:
Fathers and children (how to be independent and not ruin your relationship with your parents)
To live or not to live (how to be happy without wasting your talents on survival alone)
How cordially he treats everyone when he constantly “shits on your soul.”
So, I think that the hypothesis I put forward that our whole life is a complete paradox has been confirmed.
We live in a paradoxical world.
Life consists of paradoxes, paradoxes are the reason for the emergence of the science of logic, logic is a branch of mathematics. Means,our life is pure mathematics.
Literature
M. Gardner Mathematical miracles and mysteries - M., Nauka, 1978 - 127 p.
E. I. Ignatiev In the kingdom of ingenuity - M., Nauka, 1984 - 189 p.
The article talks about what a paradox is, provides examples of them and discusses their most common varieties.
Paradox
With the development of science, such directions as, for example, logic and philosophy appeared in it. They belong to the humanities, and at first glance it may seem that, unlike the disciplines that study the world around us (biology, physics, chemistry), they are not so significant. However, it is not. True, people most often associate these disciplines with paradoxes of various kinds, which is partly true. But in fairness, it is worth mentioning that paradoxes as such are also found in other areas of science. So what is a paradox and what could it be? We'll figure this out.
Definition
The word “paradox” itself comes from the ancient Greek language. Which is quite logical, because it was the times of the Roman Empire and Ancient Greece are considered the dawn of such sciences as logic and philosophy, which deal with paradoxes most often. So what is a paradox?
The concept has several similar definitions. For example, in everyday understanding, a paradox is a situation that can exist in reality, but at the same time have no logical explanation, or its essence is very difficult to perceive and blurred.
If we consider the meaning of this word in logic, then this is a formal-logical contradiction, which becomes such due to some special or unusual conditions. Now we know what logical paradoxes are.
The essence
If we consider this concept in in a broad sense, then it is usually understood as judgments, statements and other situations that strongly diverge from the usual opinion and seem objectively or subjectively very illogical. True, logic gradually appears if you begin to analyze the subject of discussion in more detail. But at the same time, it is important to remember - unlike an aphorism, a paradox strikes precisely with its unexpectedness and clear logical component.
But let's look at paradoxes in logic in more detail.
Logics
In short, a logical paradox is a kind of contradiction that has the form of a specific, clear and logically correct conclusion, but at the same time it is a reasoning that leads to the formation of two or more conclusions that exclude each other. So now we know what a paradox is.
There are also several types of logical paradoxes - aporia and antinomy.
The latter is characterized by the presence of two judgments that contradict each other, but both of them are equally provable.
Aporia is expressed by the presence of an argument or several arguments that strongly contradict common sense, conventional public opinion, or something else obvious. And these arguments are clear and provable.
The science
In sciences that use logic as one of the tools of cognition, situations sometimes occur when researchers come across contradictions of a theoretical kind or contradictions that emerged from the consequences of a theory with the verbal, practical result of a particular experience. True, this is not always a paradox in pure form, sometimes this occurs as a result of common errors, imperfections of current knowledge, methods of obtaining it, or inaccuracy of instruments.
Nevertheless, the presence of a paradox has always been an additional incentive to understand in more detail the seemingly obvious theory and some of its supposedly obvious evidence. Sometimes this led to the fact that even well-established and clear theories were subject to complete revision. Now we know the essence of such a thing as a paradox. Let's look at some examples below.
Photometric paradox
It belongs to the category of cosmological ones. Its meaning lies in the question of why it is dark at night if all the endless outer space is filled with stars emitting light? If this is so, then at every point in the night sky there will definitely be some distant luminary, and it will definitely not be black.
True, this paradox was resolved over time. To do this, you need to take into account the finite and finite speed of light, which means that the part of the Universe that is available for viewing will necessarily be limited by the so-called particle horizon.
In logic and philosophy
Many people have encountered similar paradoxes of life, both in everyday thoughts and in various books and textbooks. For example, one of the most popular is the paradox of God. After all, if we assume that he is omnipotent, then is he capable of creating a stone that he himself cannot move?
The second, also very common, is based on philosophy. Its meaning is that people almost never value what they have, and begin to value it only after a loss.
As we see, paradoxes are very multifaceted phenomena that exist in various fields of science and life.