Philosopher Stephen Reed on the liar paradox, semantic paradoxes and their direct connection with the foundations of mathematics.

It’s worth starting a conversation about logical paradoxes with a short story that Cervantes tells in his book “Don Quixote.” At one point in Don Quixote, he leaves Sancho Panza as governor on the island of Barataria, and while he is governor, his “subjects” fool him. One morning he was woken up and told: “Before breakfast you have one matter to decide.” And in Spain at that time there were a lot of tramps, so you had to be very careful with people. And so one landowner has a river flowing through his lands, across which a bridge is thrown, and in order to make sure that all passers-by are trustworthy, this landowner placed a gallows and a guard near the bridge, who demands that each passerby explain where and why he is going. If the passer-by tells the truth, he is allowed to cross the bridge, but if he lies, then the gallows awaits him. And everything was fine, it helped to distinguish who was a tramp and who was a merchant, until one day a man came who said: “My goal is to be hanged on this gallows, and nothing more.” And the guard was amazed, because he thought: “Okay, if we hang him, it will turn out that he told the truth, then we should have let him through, but if we let him through, then it will turn out that he lied, then we should have let him through.” hang." “So, Sancho Panza, how should we judge this matter?” And Sancho Panza takes some time to appreciate the paradox, but in the end he makes his decision: hang the half of the person who lied, and let the half who told the truth pass.

This all sounds like fun for the mind, but for people who want to understand questions of truth, argument, language and so on, it points to something very disturbing in the nature of language. It seems very easy to fall into a paradox: we simply do not know whether what that person said was true or not, whether he lied or not. And this takes us back to the original liar paradox, formulated by Eubulides in the 4th century BC. He elevated it to a work of art, he said, "Think of the statement 'I'm lying.'" If I say: “I’m lying,” I, of course, can mean some other statement of mine, but if I use extremely careful formulations, then I can say: “No, I’m lying in the very phrase that I’m saying now, This statement of mine is false.” And again, if you think about it, you will say: “If it were true, then since he says that his statement is false, it follows that it must be false, not true, that is, it cannot be true - it must be false. But if it is false, because it says that it is false, that he lied, it must be true.” So we have a paradox neatly contained in one sentence.

There are a lot of such paradoxes, and it is easy to understand why they are called logical paradoxes: the contradiction contained in them is revealed with the help of logic. Some have heard of Epimenides: he was a native of Crete, and he was so disappointed in the ability of his countrymen to tell the truth that he once said: “All Cretans are liars.” If he was right, if indeed all Cretans were liars or other Cretans always lied, then his own statement must be paradoxical. After all, if he says: “All Cretans are liars,” then he says that his own statement is false, but in this case, indeed, every single Cretan would be liars, which means he was telling the truth when he said that all Cretans are liars. The way out of the paradox, of course, is that if some Cretans were telling the truth, then his statement would simply be false, not paradoxical.

So we have a huge number of such paradoxes. Here's a paradox that I particularly like: Take a card that says on one side, “The statement on the back of this card is true.” You turn it over and it says, “The statement on the back of this card is false.” And if you think about it, it's just paradoxical, because if the statement on the first side is true, then that means the statement on the other side is also true, because that's what the first statement says; but on the second side it is written that the first statement is false, that is, if the first statement is true, it is at the same time false. But this is impossible, which means the second statement must be false; but it says that the first statement is false, then the first statement cannot be false - it must be true. But we have already seen that if the first statement is true, then it is false, so we get a pure paradox.

Some medieval thinkers preferred to describe this paradox through Socrates and Plato or sometimes Plato and Aristotle. So Plato was Aristotle's teacher and considered him his best student, so one day he said: “Everything that Aristotle says is the truth.” But Aristotle was not a very good student in the sense that he wanted to challenge Plato's teachings, so he said, “Everything Plato says is false,” and this is very similar to the card paradox.

These were all paradoxes in the field of truth, lies and language. But in the 20th century we encountered paradoxes in mathematics. A brief history of the issue is this: after the advent of calculus, and then after the work with infinite series in the 18th century, the foundations of mathematics were found to be unstable, people asked the question “How do infinite series work without leading us to contradictions in mathematics?” And in the 19th century, a large movement unfolded, the goal of which was to search for stable foundations of mathematics. Then set theory became such a basis. A set is a collection of objects defined through some property: for example, there may be a set of all natural numbers, a set of even numbers, or even a set of rice puddings - you can take different sets. In mathematics, of course, only number sets are used.

And all this looked great until the end of the 19th century. Frege, Dedekind and many other thinkers established mathematics or what seemed to be the solid foundation of set theory. But then Bertrand Russell, the famous British philosopher, reading the works of Frege, thought: “You can give a lot of numbers, you can give a lot of sets; one can define a set of sets that include themselves, or one can define a set of sets that do not include themselves.” And then he thought: “Wait a minute, if we have a set of sets that do not include themselves, will this set include itself or not?” If such a set included itself, then it should not include itself, because by condition we take only those sets that do not include themselves. So it would be better if the set did not include itself, but if it does not include itself, then it is a set not including itself, and it must be part of that set. And, as I said, all these paradoxes at first look like fun for the mind, but now, at the beginning of the 20th century, we have found a paradox, a contradiction at the very heart of what should be the foundations of mathematics. As is widely known, this was a big blow for Frege: he was about to publish the second volume of his Fundamental Laws of Arithmetic, and he had to add an appendix in which he wrote: “Bertrand Russell pointed out a weak spot in the very heart of my theory, but I think I can solve this problem,” and he proposed a solution, but, as it turned out, it was not correct.

I'll turn to paradoxes in set theory for a moment, because there is another rather interesting paradox that brings us back to the conversation about paradoxes related to truth, or so-called semantic paradoxes. So, about 40 years later, around 1940, the American mathematician and logician Haskell B. Curry was thinking about Russell's paradox and said: "Russell's paradox is based on negation - it talks about the many sets that do not include themselves." Is it possible to get the same paradox without using negation? Is there a way? And he said that there is a way. Let us take the set of all sets; if they include themselves, then zero is equal to one. According to set theory, this is a completely admissible set. But if we begin to consider such a set, if it includes itself, then it will satisfy the condition that if it includes itself, then zero is equal to one.

And we assumed that it includes itself, therefore, zero is really equal to one. But it is quite obvious that zero cannot be equal to one, so we work everything back and assume that a set cannot include itself. If it does not include itself, it immediately follows that either it does not include itself, or zero is equal to one. But this is the same as saying that if it includes itself, zero is indeed equal to one - it is the same as saying: either it does not include itself, or zero is equal to one. And this is the same as saying that if a set includes itself, then it is not non-self-inclusive, then zero is equal to one. But then it includes itself, that is, we have proven that it includes itself, but since we have proven this, therefore zero is equal to one. Save! We just proved that zero is equal to one! So once again we have a real nightmare paradox right at the heart of mathematics.

And a few years later this paradox was turned into one of the semantic paradoxes that I spoke about earlier, and it took the form of the statement: “If this statement is true, therefore zero is equal to one.” Or even: “If this statement is true, then God exists.” And then we can prove in just a few lines that God exists or anything else: zero is equal to one, God exists, it is raining in Moscow today - we can prove anything with such a statement. People think a lot about the truth, so it's very dangerous: is the truth really like that? Is truth really a contradictory concept?

And I'll end by briefly talking about another paradox to show that paradoxes don't stop there. Here is a statement: “You do not know this statement” - you do not know the very statement that I am now uttering. Now let's assume you know him. The concepts of knowledge and truth tell us that you can only know what is true, so that if you know it, it is true, in which case you do not know it because it says so. So if we assume that you know him, then it turns out that you don’t know him. It turns out that we proved that you don't know him, but it says that you don't know him, so we proved him. And of course, if we have proven something, that means it is true, that means we know it, because we have proof. And it turns out that we have proven both that you know this statement and that you do not know it, so we again have an epistemic paradox.

Let's summarize. I have described several semantic paradoxes, mainly related to the concept of truth, and also shown that they are very similar to the paradoxes associated with set theory, which lie at the very heart of mathematics. In addition, we became acquainted with epistemic paradoxes that are associated not only with the concept of truth, but also with the concept of knowledge. So, we looked at a few semantic paradoxes, such as the liar paradox, the Epimenides paradox and the card paradox, which are based on the concept of truth (in them we talk about lies, untruths, truth, and so on), and then we looked at several paradoxes that arise in mathematics - they are related to set theory. And at the end we also talked about another type of paradox - epistemic paradoxes.

You can immediately see how important it is for us to find a solution to these paradoxes since mathematics is involved in them, because we were looking for solid foundations of mathematics to make sure that we did not make mistakes - and now we have discovered a contradiction in them. So we do need a solution when it comes to mathematical paradoxes related to set theory, but we also need one for semantic paradoxes. A lot of philosophers reflect on the concept of truth, and they want to understand the nature of truth, what a true statement is. It is natural to suppose that a statement is true if everything is as it says; and now look at the paradox of the liar: it is true, if I lie, this is paradoxical and leads to a contradiction. So we need to rethink the concept of truth, some want to rethink the logic behind it and the methods of evidence that led us to the contradiction. And it is very important that we do this if we are to gain a full understanding of the concepts of truth and knowledge.

gif: postnauka.ru/ Stephen Reed

According to the laws of logic, Ivin Alexander Arkhipovich

WHAT IS A LOGICAL PARADOX?

There is no exhaustive list of logical paradoxes, nor is it possible.

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

“Set-theoretic paradoxes are a very serious problem, not for mathematics, however, but rather for logic and the theory of knowledge,” writes the Austrian mathematician and logician K. Gödel. “The logic is consistent. There are no logical paradoxes, says Soviet mathematician D. Bochvar. - These kinds of discrepancies are sometimes significant, sometimes verbal. The point has a lot to do with what exactly is meant by “logical paradox.”

A logical dictionary is considered a necessary feature of logical paradoxes. Paradoxes classified as logical must be formulated in logical terms. However, in logic there are no clear criteria for dividing terms into logical and extra-logical. 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 be formulated in logical terms. Whether a particular paradox uses only purely logical premises is not always possible to determine unambiguously.

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

At the beginning of the study of logical paradoxes, it seemed that they could be identified by the violation of some, not yet studied, provision or rule of logic. The “vicious circle principle” introduced by B. Russell was especially active in claiming the role of such a rule. This principle states that a collection of objects cannot contain members definable only by that same collection.

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

In all paradoxes there is self-applicability, which means there is, as it were, a movement in a circle, ultimately leading to the starting point. In an effort to characterize an object of interest to us, we turn to the totality of objects that includes it. However, it turns out that for its definiteness it itself needs the object in question 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 appears in many completely non-paradoxical arguments. Circular is a huge variety of the most ordinary, harmless and at the same time convenient ways expressions. 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 important not only in ordinary language, but also in the language of science.

Mere reference to the use of self-applicable concepts is therefore not sufficient to discredit paradoxes. Some additional criterion is needed to separate self-applicability, leading to a paradox, from all its other cases.

There were many proposals on this matter, but a successful clarification of circularity was never 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 be a distinctive feature of all logical paradoxes, did not lead to anything definite.

Undoubtedly, some classification of paradoxes would be useful, dividing them into types and types, grouping some paradoxes and contrasting them with others. However, nothing lasting was achieved in this matter either.

The English logician F. Ramsay, who died in 1930, when he was not yet twenty-seven years old, proposed dividing all paradoxes into syntactic and semantic. The first includes, for example, Russell's paradox, the second - the paradoxes of the “liar”, Grelling, etc.

According to F. Ramsey, the paradoxes of the first group contain only concepts belonging to logic or mathematics. The latter include such concepts as “truth”, “definability”, “naming”, “language”, which are not strictly mathematical, but rather related to linguistics or even the theory of knowledge. Semantic paradoxes seem to owe their appearance not to some error in logic, but to the vagueness or ambiguity of some non-logical concepts, therefore the problems they pose concern language and must be solved by linguistics.

It seemed to F. Ramsey that mathematicians and logicians had no need to be interested in semantic paradoxes.

Later it turned out, however, that some of the most significant results of modern logic were obtained precisely in connection with a more in-depth study of precisely these “non-logical” paradoxes.

The division of paradoxes proposed by F. Ramsey was widely used at first and retains some significance even now. At the same time, it is becoming increasingly clear that this division is rather vague and relies primarily on examples rather than on an in-depth comparative analysis of the two groups of paradoxes. Semantic concepts are now precise definitions, and it is difficult not to admit that these concepts really relate to logic. With the development of semantics, which defines its basic concepts in terms of set theory, the distinction made by F. Ramsey is increasingly erased.

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2. Logical collapse - What can be demonstrated or what needs to be proven is the final knowledge of something special. Existence and transcendence, in the sense of this being, do not exist. If we think about them, then the thought takes on logical forms that

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3. The theological nature of the Kingdom of God In the tradition of the Old Testament and Judaism, the coming of the Kingdom of God means the coming of God. The center of eschatological hope was the divinely determined and realized “Day of Yahweh,” the day when God will be “all in all,” when

LOGICAL PARADOX

LOGICAL PARADOX

a proposition that is not yet obvious at first, but, contrary to expectations, expresses the truth. In ancient logic, a paradox was called a paradox, the ambiguity of which relates primarily to its correctness or incorrectness. In modern mathematics, paradoxes are actually mathematical ones. aporia.

Philosophical Encyclopedic Dictionary. 2010 .

LOGICAL PARADOX

The development of modern logical methods has led to new logical paradoxes. For example, Brouwer pointed out the following paradox of classical existence: in any sufficiently strong classical theory there is a provable formula of the form ExA(x), for which it is impossible to construct any specific t such that A(t) is provable.

In particular, it is impossible to construct a single non-standard model of real numbers in set theory, although such models can be proven. This paradox shows that the concepts of existence and constructability are irreversibly divergent in classical mathematics.

Further, non-standard models, which required an explicit distinction between language and metalanguage, led to the following paradox: “The set of all standard real numbers is part of a non-standard finite set. Thus, it can be part of the finite.”

This paradox sharply contradicts the ordinary understanding of the relationship between the finite and the infinite. It is based on the fact that “being standard” belongs to a metalanguage, but can be accurately interpreted in a non-standard model. Therefore, in the non-standard model, one can talk about the truth and falsity of any mathematical statements that include the concept of “being (non-standard", but for them the properties of the standard model are not required to be preserved, with the exception of logical tautologies. This paradox became the basis of the theory of semisets, in which there can be subclasses of sets .

And finally, the last class of logical paradoxes arises at the boundaries between formalized and informal concepts. Let's consider one of them (Simon); “Anything that can be expressed precisely can be expressed in the language of Turing machines. Therefore in humanities Only those models that can be expressed in the language of Turing machines can be considered. Moreover, according to the method of diagonalization, any precise objection to a given point of view is itself translated to and included in Turing machines.”

This paradox stimulated the emergence of the theory of non-formalizable concepts, but due to the fact that it was not immediately recognized as a paradox, at the same time it led to sad consequences, since this one, in which fundamental expressibility (requiring unrealistic resources) and real descriptions, was perceived as precise reasoning and, as noted in works on cognitive science, paralyzed Western psychology for almost 10 years. The rejection of Simon's argument after realizing its sophistic nature was structured in such a way that it led to a complete rejection of precise concepts and thereby essentially served as the motivation for movements such as postmodernism. IN in this case a logical mistake was made in replacing a contradictory judgment with the opposite one.

Ya. Ya. Nepeyvoda

New Philosophical Encyclopedia: In 4 vols. M.: Thought. Edited by V. S. Stepin. 2001 .

See what “LOGICAL PARADOX” is in other dictionaries:

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The style of this article is non-encyclopedic or violates the norms of the Russian language. The article should be corrected according to Wikipedia's stylistic rules. The paradox of unexpected execution (eng. Unexpected hanging par ... Wikipedia

Plan:

I. Introduction

II. Aporias of Zeno

Achilles and the tortoise

Dichotomy

III . The Liar Paradox

IV . Russell's paradox

I . Introduction.

A paradox is two opposing, incompatible statements, for each of which there are seemingly convincing arguments. The most extreme form of paradox is antinomy, reasoning proving the equivalence of two statements, one of which is the negation of the other.

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

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

Designing new theory, a scientist usually starts from facts, from what can be observed in experience. No matter how free his creative imagination may be, it must take into account one indispensable circumstance: a theory makes sense only if it is consistent with the facts relating to it. A theory that diverges from facts and observations is far-fetched and has no value.

But if in logic there are no experiments, no facts and no observation itself, then what is holding 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 actual 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 precisely explains the importance attached to paradoxes in logic, and the great attention they enjoy in it.

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

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

What Plato said is a lie.
Socrates

Socrates speaks only the truth.
Plato

II. Aporias of Zeno.

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

The Eleatics proved their assumptions by negating the opposite of the assumption. Zeno went further than his teacher, which gave Aristotle grounds to see in Zeno the founder of “dialectics” - this term then called the art of achieving truth in a dispute by clarifying the contradictions in the opponent’s judgment and by destroying these contradictions.

Achilles and the tortoise. Let us begin our consideration of Zeno’s difficulties with the aporia about motion “ Achilles and the tortoise". Achilles is a hero and, as we would say now, an outstanding athlete. The turtle is known to be one of the slowest animals. However, Zeno argued that Achilles would lose a race to a tortoise. Let's accept following conditions. Let Achilles be separated from the finish by a distance of 1, and the tortoise by ½. Achilles and the turtle begin to move at the same time. For definiteness, let Achilles run 2 times faster than a turtle (i.e., walk very slowly). Then, having run a distance of ½, Achilles will discover that the tortoise has managed to cover a distance of ¼ in the same time and is still ahead of the hero. Then the picture repeats: having run a quarter of the way, Achilles will see a turtle one-eighth of the way ahead of him, etc. Consequently, whenever Achilles overcomes the distance separating him from the turtle, the latter manages to crawl away from him and still remains ahead. Thus, Achilles will never catch up with the tortoise. Once Achilles starts a movement, he will never be able to complete it.

Those who know mathematical analysis usually indicate that the series converges to 1. Therefore, they say, Achilles will cover the entire path in a finite period of time and, of course, will overtake the turtle. But here is what D. Gilbert and P. Bernays write about this:

“Usually they try to get around this paradox by arguing that the sum of an infinite number of these time intervals still converges and, thus, gives a finite period of time. However, this reasoning absolutely does not touch upon one essentially paradoxical point, namely the paradox that lies in the fact that a certain infinite sequence of successive events, a sequence whose completion we cannot even imagine (not only physically, but at least in principle) , in fact, should still be completed.”

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

“Let’s imagine a computer that would perform the first operation in ½ minute, the second in ¼ minute, the third in ⅛ minute, etc. Such a machine could, by the end of the first minute, “recalculate” the entire natural series (write, for example, countable number of units). It is clear that work on the design of such a machine is doomed to failure. So why does a body leaving point A reach the end of segment B, “counting out” a countable set of points A 1, A 2, ..., A n, ...?”

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

There is no movement, said the bearded sage.

The other fell silent and began to walk in front of him.

He could not have objected more strongly;

Everyone praised the intricate answer.

But, gentlemen, this is a funny case

Another example comes to mind:

After all, every day the sun walks before us,

However, stubborn Galileo is right.

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

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

Let, however, Achilles begin to catch up with the more distant Turtle-2, without paying any attention to the closer one. The same way of reasoning allows us to say that Achilles will be able to get close to Turtle-2, but this means that he will overtake Turtle-1. Now we come to a logical contradiction.”

It is difficult to object to anything here if you remain captive of figurative ideas. It is necessary to identify the formal essence of the matter, which will allow the discussion to move into the mainstream of strict reasoning. The first aporia can be reduced to the following three statements:

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

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

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

At the same time, it is not stated anywhere that the decreasing sequences of segments a i for and a i " for must be the same. On the contrary, if the segments and are unequal in length, their partitions into infinite sequences of decreasing segments will turn out to be different. In the above reasoning, Achilles is separated from the turtles 1 and 2 different distances. Therefore, we have two different segments with a common starting point A. Unequal segments generate different infinite sequences of points, and it is unacceptable to use one of them instead of the other. Meanwhile, it is this “illegal” operation that is used in the arguments about two turtles.

If we do not confuse illustrations and the essence of aporia, then it can be argued that aporia Achilles And Dichotomy symmetrical in relation to each other. Indeed, Dichotomy also leads to the following three statements:

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

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

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

Thus, the aporia Achilles is based on the thesis about the impossibility of completing a movement due to the need to visit sequentially each of the points of an infinite series ordered by type ω (i.e., by the type of order on natural numbers), which does not have a last element. In its turn Dichotomy asserts the impossibility of starting movement due to the presence of an infinite series of points ordered by type ω* (this is how negative integers are ordered), which does not have a first element.

If we analyze more carefully the two aporia given above, we will find that both of them are based on the assumption that continuity space and time in their sense infinite divisibility. This assumption of continuity is different from the modern one, but occurred in ancient times. Without the assumption that any spatial or temporal interval can be divided into smaller intervals, both aporias collapse. Zeno understood this perfectly. Therefore, he makes an argument based on the assumption that discreteness space and time, i.e., assumptions about the existence of elementary, further indivisible, lengths and times.

Stages . So, let's assume the existence of indivisible segments of space and time intervals. Consider the following diagram, in which each cell of the table represents an indivisible block of space. There are three rows of objects A, B and C, each occupying three blocks of space, with the first row remaining stationary, and rows B and C starting simultaneous movement in the direction indicated by the arrows:

End position

Row C, Zeno claims, in an indivisible moment of time passed one indivisible place of the fixed row A (place A1). However, during the same time, row C passed two places of row B (blocks B2 and B3). According to Zeno, this is contradictory, since the moment of passage of block B2, shown in the following diagram, should have been met:

AT 3	AT 2	IN 1
C1	C2	C3

Intermediate position

But where was row A in this intermediate position? There is simply no appropriate place left for it. It remains to either admit that there is no movement, or agree that we divide series A not into three, but into large quantity places But in the latter case, we again return to the assumption of the infinite divisibility of space and time, again falling into the dead end of aporia Dichotomy And Achilles. Whatever the outcome, movement is impossible.

The main idea of ​​the aporia of Zeno of Elea is that discreteness, multiplicity and movement characterize only the sensory picture of the world, but it is obviously unreliable. The true picture of the world is comprehended only by thinking and theoretical research.

If you do not delve into the depths of the aporias, you can look down on them and wonder how Zeno did not think of obvious things. But people continue to argue about Zeno, and the history of science shows that if they argue about something for a long time, it is usually not in vain. Undoubtedly, reflection on the aporias helped create mathematical analysis, played a certain role in the physical revolution of the twentieth century, and, quite possibly, their significance will be even more significant in the physics of the twenty-first century.

III . The liar paradox.

For almost two and a half thousand years, one of the logical riddles that torment people trying to harmonize the foundations of their thinking is the “liar paradox.” Despite the fact that dozens of semantic, logical and mathematical paradoxes and aporias are currently known, the “liar paradox” occupies a special place:

First, it is the most accessible of the many paradoxes and, as such, the most famous of them.

Secondly, it is primary in relation to many other paradoxes and, therefore, the latter cannot be eliminated until the “liar paradox” is resolved.

The simplest version of the liar paradox is the statement “I am lying.” If the statement is false, then the speaker told the truth, and that means what he said is not a lie. If the statement is not false, but the speaker claims that it is false, then his statement is false. It turns out, therefore, that if the speaker is lying, he is telling the truth, and vice versa.

The “liar paradox” has a number of other formulations that are similar to each other. Below are just a few of them:

- “All Cretans are liars” (thesis expressed by the Cretan Epimenides);

- “I am now making a false proposal”;

- “Everything that X claims in the period of time P is a lie”;

- “This statement is false”;

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

While this list is far from complete, it gives some idea of ​​the problem. The logical problem is that the assumption that the given statements are false leads to their truth and vice versa.

The ancient Greeks were very interested in how a seemingly completely meaningful statement could be neither true nor false without a contradiction arising. The philosopher Chrysippus wrote six treatises on the liar paradox, none of which survive to this day. There is a legend that a certain Filit Kossky, despairing of resolving this paradox, committed suicide. They also say that one of the famous ancient Greek logicians, Diodorus Kronos, already in his declining years made a vow not to eat until he found the solution to the “Liar”, and soon died without achieving anything.

In the Middle Ages, this paradox was classified among the so-called undecidable sentences and became the object of systematic analysis. Now the “Liar” - this typical former sophism - is often called the king of logical paradoxes. An extensive scientific literature is devoted to it. And yet, as with many other paradoxes, it remains not entirely clear what problems are hidden behind it and how to get rid of it.

Consider the first formulation: the statement attributed to Epimenides is logically inconsistent if we assume that liars always lie and non-liars always tell the truth. Under this assumption, the statement “All Cretans are liars” cannot be true, for then Epimenides would be a liar and, therefore, what he claims would be a lie. But this statement cannot be false, for this would mean that the Cretans speak only the truth and, therefore, what Epimenides said is also true.

The history of logic knows many attempts and approaches to resolving this paradox. One of the first is an attempt to present the “liar paradox” as sophism. The essence of this idea is that real life no liar speaks only lies. Therefore, a paradox is a sophism based on a false premise.

But such an explanation is acceptable only for the first (early) formulation of the paradox, but does not “remove” the paradox in its more precise modern formulations. There are several solutions to the modern liar paradox. Which solution is correct? Everyone is correct. How can this be? Because a paradox is a reasoning leading to a contradiction. You can get rid of the contradiction different ways. They all boil down to replacing some dubious piece of reasoning with a more correct one. The result is an argument similar to the previous one, but without visible contradictions. In addition, various solutions are given through different types logician.

You can replace different pieces. In each case, different solutions will be obtained, and which one to prefer is a matter of taste. To one, one piece seems most dubious, to another, another. Sometimes the very first questionable piece is noticeable and obvious.

Perhaps the most common solution to the liar paradox is the separation of language and metalanguage:

"The Liar" is now generally considered to be a characteristic example of the difficulties that arise from the confusion of two languages: the language in which one speaks of a reality lying outside itself, and the language in which one speaks of the first language itself.

In everyday language there is no distinction between these levels: we speak about both reality and language in the same language. For example, a person whose native language is Russian does not see any particular difference between the statements: “Glass is transparent” and “It is true that glass is transparent,” although one of them is about glass, and the other is about a statement about glass.

If someone had the idea of ​​​​the need to talk about the world in one language, and about the properties of this language in another, he could use two different existing languages, let's say Russian and English. Instead of simply saying, “Cow is a noun,” one would say “Cow isanoun,” and instead of saying, “The statement “Glass is not transparent” is false,” one would say “Theassertion “Glass is not transparent” isfalse.” With two different languages ​​used in this way, what is said about the world would clearly differ from what is said about the language with which the world is spoken. In fact, the first statements would relate to the Russian language, while the second would refer to English.

If our language expert further wanted to speak out about some circumstances related to the English language, he could use another language. Let's say German. To talk about this last point, one could resort, for example, to the Spanish language, etc.

Thus, what emerges is a kind of ladder, or hierarchy, of languages, each of which is used for a very specific purpose: in the first they speak about the objective world, in the second about this first language, in the third about the second language, etc. Such a distinction between languages ​​according to their area of ​​application is a rare occurrence in everyday life. But in sciences that specifically deal with languages, like logic, it sometimes turns out to be very useful. The language in which one speaks about the world is usually called subject language. The language used to describe the subject language is called metalanguage.

It is clear that if language and metalanguage are distinguished in this way, the statement “I am lying” can no longer be formulated. It speaks of the falsity of what is said in Russian, and, therefore, belongs to the metalanguage and must be expressed in English language. Specifically, it should sound like this: “EverythingIspeakinRussianisfalse” (“Everything I said in Russian is false”); this English statement says nothing about himself, and no paradox arises.

The distinction between language and metalanguage allows us to eliminate the “Liar” paradox. Thus, it becomes possible to correctly, without contradiction, define the classical concept of truth: a statement is true if it corresponds to the reality it describes.

The concept of truth, like all other semantic concepts, is relative in nature: it can always be attributed to a specific language.

As the Polish logician ATarski showed, the classical definition of truth must be formulated in a language broader than the language for which it is intended. In other words, if we want to indicate what the phrase “a statement that is true in a given language” means, we must, in addition to expressions of this language, also use expressions that are not in it.

Tarski introduced the concept semantically closed language. Such a language includes, in addition to its expressions, their names, and also, what is important to emphasize, statements about the truth of the sentences formulated in it. There is no boundary between language and metalanguage in a semantically closed language. Its means are so rich that they allow not only to assert something about extra-linguistic reality, but also to evaluate the truth of such statements. These means are sufficient, in particular, to reproduce the antinomy “Liar” in the language. A semantically closed language thus turns out to be internally contradictory. Every natural language is obviously semantically closed.

The only acceptable way to eliminate antinomy, and therefore internal inconsistency, according to Tarski, is to refuse to use a semantically closed language. This path is acceptable, of course, only in the case of artificial, formalized languages ​​that allow a clear division into language and metalanguage. In natural languages, with their unclear structure and the ability to talk about everything in the same language, this approach is not very realistic. It makes no sense to raise the question of the internal consistency of these languages. Their rich expressive capabilities also have their own reverse side- paradoxes.

There are other solutions to the liar paradox, such as Occam's solution and Buridan's solution:

So, there are statements that speak about their own truth or falsity. The idea that these kinds of statements are not meaningful is a very old one. 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, about its own falsity. A contradiction directly follows from this statement. If every statement is false, then this applies to the given statement itself; but the fact that it is false means that not every statement is false. The situation is similar with the statement “Every statement is true.” It should also be classified as meaningless and also leads to a contradiction: if every statement is true, then the negation of this statement itself is 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 Occam, the French philosopher of the 14th century. J. Buridan did not agree with his decision. From the point of view of ordinary ideas about meaninglessness, expressions like “I am lying”, “Every statement is true (false)”, etc. quite meaningful. What you can think about, you can speak out about - 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 self-talk is nonsensical. 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 allowed that a statement can speak about itself, then why is it not capable of speaking meaningfully about such a property as truth?

Buridan himself considered the statement “I am lying” not meaningless, but false. He justified it like this. When a person asserts a proposition, he thereby asserts that it is true. If a sentence says about itself that it is itself false, then it is only a shortened formulation of a more complex expression that asserts both its truth and its falsity. This expression is contradictory and therefore false. But it is by no means meaningless.

Buridan's argument is still sometimes considered convincing.

There are other areas of criticism of the solution to the Liar paradox, which was developed in detail by Tarski. Is it really true that in semantically closed languages ​​- and all natural languages ​​are such - there is no antidote to paradoxes of this type?

If this were so, then the concept of truth could be strictly defined only in formalized languages. Only in them is it possible to distinguish between the subject language in which one talks about the world around us, and the metalanguage in which one speaks about this language. This hierarchy of languages ​​is built on the model of acquisition foreign language with the help of a native. The study of such a hierarchy has led to many interesting conclusions, and in certain cases it is significant. But it is not in natural language. Will this discredit him? And if so, to what extent? After all, the concept of truth is still used in it, and usually without any complications. Is introducing hierarchy the only way to eliminate paradoxes like Liar?

In the 1930s, the answers to these questions seemed undoubtedly affirmative. However, now the former unanimity is no longer there, although the tradition of eliminating paradoxes of this type by “stratifying” the language remains dominant.

Recently, more and more attention has been attracted egocentric expressions. They contain words like “I”, “this”, “here”, “now”, and their truth depends on when, by whom, and where they are used. In the statement “This statement is false” the word “this” appears. . Which object exactly does it refer to? "Liar" may be saying that the word "it" is not relevant to the meaning of the statement. But then what does it refer to, what does it mean? And why can’t this meaning still be designated by the word “this”?

Without going into details, it is only worth noting that in the context of the analysis of egocentric expressions, “Liar” is filled with a completely different content than before. It turns out that he no longer warns against confusing language and metalanguage, but points out the dangers associated with the incorrect use of the word “it” and similar egocentric words.

The problems associated with "The Liar" over the centuries have changed radically depending on whether it was seen as an example of ambiguity, or as an expression that appears externally as an example of a confusion of language and metalanguage, or, finally, as a typical example of the misuse of egocentric expressions. And there is no certainty that other problems will not be associated with this paradox in the future.

The famous modern Finnish logician and philosopher G. von Wright wrote in his work dedicated to “The Liar” that this paradox should in no case be understood as a local, isolated obstacle that can be eliminated with 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 evidence, and a whole series of important differences: between a sentence and the thought it expresses, between the use of an expression and its mention, between the meaning of a name and the object it denotes.

"The Liar Paradox" (surprisingly) is extremely close in its logical form and character logical error many other “paradoxes” that are considered to be completely independent. These include the famous "Russell's paradox".

III . Russell's paradox

The most famous of the paradoxes discovered already in the last century is the antinomy discovered by B. Russell and communicated by him in a letter to G. Ferge. Russell discovered his paradox, which relates to the fields of logic and mathematics, in 1902. The same antinomy was discussed simultaneously in Göttingen by the German mathematicians Z. Zermelo (1871-1953) and D. Hilbert. The idea was in the air, and its publication had the effect of a bomb exploding. This paradox caused, according to Hilbert, the effect of a complete catastrophe in mathematics. The most simple and important logical methods, the most common and useful concepts are under threat. It turned out that in Cantor's set theory, which was enthusiastically accepted by most mathematicians, there are strange contradictions that are impossible, or at least very difficult, to get rid of. The Russell paradox (more precisely, the Russell-Zermelo paradox) particularly clearly revealed these contradictions. The most outstanding mathematicians of those years worked on its resolution, as well as on the resolution of other found paradoxes of Cantor's set theory.

It immediately became obvious that neither in logic nor in mathematics for the entire long history their existence, absolutely nothing was developed that could serve as a basis for eliminating the antinomy. A departure from conventional ways of thinking was clearly necessary. But from what place and in what direction? How radical would it be to break away from established ways of theorizing? With further research into the antinomy, the conviction of the need for a fundamentally new approach grew steadily. Half a century after its discovery, specialists in the foundations of logic and mathematics L. Frenkel and I. Bar-Hillel already stated without any reservations: “We believe that any attempts to get out of the situation using traditional (that is, those in use before the 20th century) ways of thinking , which have so far consistently failed, are obviously insufficient for this purpose.” The modern American logician H. Curry wrote a little later about this paradox: “In terms of logic known in the 19th century, the situation simply could not be explained, although, of course, in our educated age there may be people who will see (or think that they will see ), what is the mistake?

Russell's paradox in its original form is associated with the concept of set, or class. We can talk about sets of different objects, for example, about the set of all people or about the set of natural numbers. An element of the first set will be every individual person, an element of the second set will be every natural number. It is also permissible to consider the sets themselves as some objects and talk about sets of sets. You can even introduce concepts such as the set of all sets or the set of all concepts. Regarding any arbitrary set, it seems reasonable to ask whether it is its own element or not. Sets that do not contain themselves as an element will be called ordinary. For example, the set of all people is not a person, just as the set of atoms is not an atom. Sets that are their own elements will be unusual. For example, a set that unites all sets is a set and therefore contains itself as an element.

Let us now consider the set of all ordinary sets. Since it is many, one can also ask about it, whether it is ordinary or unusual. The answer, however, turns out to be discouraging. If it is ordinary, then, according to its definition, it must contain itself as an element, since it contains all ordinary sets. But this means that it is an unusual set. The assumption that our set is an ordinary set thus leads to a contradiction. This means it cannot be ordinary. On the other hand, it cannot be unusual either: an unusual set contains itself as an element, and the elements of our set are only ordinary sets. As a result, we come to the conclusion that the set of all ordinary sets cannot be either an ordinary or an unusual set.

So, the set of all sets that are not proper elements is its own element if and only if it is not such an element. This is a clear contradiction. And it was obtained on the basis of the most plausible assumptions and with the help of seemingly indisputable steps. The contradiction suggests that such a set simply does not exist. But why can't it exist? After all, it consists of objects that satisfy a clearly defined condition, and the condition itself does not seem somehow exceptional or unclear. If such a simply and clearly defined set cannot exist, then what, exactly, is the difference between possible and impossible sets? The conclusion about the non-existence of the set in question sounds unexpected and causes concern. It makes our general concept of set amorphous and chaotic, and there is no guarantee that it cannot give rise to some new paradoxes.

Russell's paradox is remarkable for its extreme generality. To construct it, you do not need any complex technical concepts, as in the case of some other paradoxes; the concepts of “set” and “element of set” are sufficient. But this simplicity just speaks of its fundamental nature: it touches on the deepest foundations of our reasoning about sets, since it speaks not about some special cases, but about sets in general.

Other variants of the paradox Russell's paradox does not have a specifically mathematical character. It uses the concept of a set, but does not touch on any special properties related specifically to mathematics.

This becomes obvious if we reformulate the paradox in purely logical terms. For each property one can, in all likelihood, ask whether it applies to itself or not. The property of being hot, for example, does not apply to itself, since it is not itself hot; the property of being concrete also does not refer to itself, for it is an abstract property. But the property of being abstract, being abstract, is applicable to oneself. Let us call these self-inapplicable properties inapplicable. Does the property of being inapplicable to oneself apply? It turns out that an inapplicability is inapplicable only if it is not such. This is, of course, paradoxical. The logical, property-related version of Russell's antinomy is just as paradoxical as the mathematical, set-related version of it.

Russell also proposed the following popular version of the paradox he discovered. Let's imagine that the council of one village defined the duties of a barber as follows: to shave all the men in the village who do not shave themselves, and only these men. Should he shave himself? If so, then he will treat those who shave themselves, but those who shave themselves, he should not shave. If not, he will be one of 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 is, of course, impossible.

The argument about the barber is based on the assumption that such a barber exists. The resulting contradiction means that this assumption is false, and there is no resident of the village who would shave all those and only those villagers who do not shave themselves. The duties of a barber do not seem contradictory at first glance, so the conclusion that there cannot be one sounds somewhat unexpected. But this conclusion is not paradoxical. The condition that the village barber must satisfy is in fact internally contradictory and, therefore, impossible to fulfill. There cannot be such a barber in the village for the same reason that there is no person in it who is older than himself or who was born before his birth.

The argument about the barber can be called a pseudo-paradox. In its course, it is strictly similar to Russell’s paradox and this is why it is interesting. But it is still not a true paradox.

Another example of the same pseudo-paradox is the famous argument about the catalogue. A certain library decided to compile a bibliographic catalogue, which would include all those and only those bibliographic catalogs that do not contain links to themselves. Should such a directory include a link to itself? It is not difficult to show that the idea of ​​creating such a catalog is impracticable; it simply cannot exist, since it must simultaneously include a reference to itself and not include it.

It is interesting to note that cataloging all directories that do not contain a reference to themselves can be thought of as an endless, never-ending process. Let's assume that at some point a directory, say K1, was compiled, including all directories different from it that do not contain links to themselves. With the creation of K1, another directory appeared that did not contain a link to itself. Since the problem is to create a complete catalog of all catalogs that do not mention themselves, it is obvious that K1 is not a solution. He doesn't mention one of those directories - himself. By including this mention of himself in K1, we get catalog K2. It mentions K1, but not K2 itself. By adding such a mention to K2, we get KZ, which is again incomplete due to the fact that it does not mention itself. And on and on without end.

One more logical paradox can be mentioned - the “paradox of the Dutch mayors”, similar to the barber’s paradox. Every municipality in Holland must have a mayor, and two different municipalities cannot have the same mayor. Sometimes it turns out that the mayor does not live in his municipality. Let us assume that a law has been passed according to which a certain territory S allocated exclusively to such mayors who do not live in their municipalities, and requiring all such mayors to settle in this territory. Let us further assume that there were so many of these mayors that the territory S itself forms a separate municipality. Where should the mayor of this Special Municipality S reside? Simple reasoning shows that if the mayor of a Special Municipality lives in the territory of S, then he should not live there, and vice versa, if he does not live in the territory, then he should live in this territory. That this paradox is similar to the barber's paradox is quite obvious.

Russell was one of the first to propose a solution to “his” paradox. The solution he proposed was called “type theory”: a set (class) and its elements belong to different logical types, the type of a set is higher than the type of its elements, which eliminates Russell’s paradox (type theory was also used by Russell to solve the famous “Liar” paradox) . Many mathematicians, however, did not accept Russell's solution, believing that it imposed too severe restrictions on mathematical statements.

The situation is similar with other logical paradoxes. “The antinomies of logic,” writes von Wrigg, “have puzzled us since their discovery and will probably always puzzle us. We must, I think, regard them not so much as problems awaiting solution, but as inexhaustible raw material for thought. They are important because thinking about them touches on the most fundamental questions of all logic, and therefore of all thinking.”

Bibliography:

1 Frenkel A.A., Bar-Hillel I. “Foundations of set theory”

2. B.Russell.“Introduction to mathematical philosophy.”

3. Russell B. “The principles of mathematics.”

4. Zadoya A.I. “Introduction to Logic”

5. Hilbert D. - Ackerman V., “Fundamentals of theoretical logic.”

6. Lakoff J. “Pragmatics in natural logic. New in linguistics.”

7. Jacobson R. “Boas's views on grammatical meaning.”

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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 meaning is two opposing, incompatible statements, for each of which there are seemingly convincing arguments.

The most extreme 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, there are not even facts in the usual sense of the word. When constructing its systems, logic ultimately proceeds from the analysis of real thinking. But the results of this analysis are synthetic and undifferentiated. They are not statements of any individual processes or events that the theory should explain. Such an analysis obviously cannot be called observation: a specific phenomenon is always observed.

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

But if in logic there are no experiments, no facts and no observation itself, then what is holding 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 actual 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 attached to paradoxes in logic and the great attention they enjoy in it.

Special literature on the topic of paradoxes is almost inexhaustible. Suffice it to say that more than a thousand works have been written about just one of them - the liar paradox.

On the surface, 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 reach the bottom.

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

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

One might also wonder whether the view that every generalization is false is not internally inconsistent, since the view itself is a generalization. Or advice - never advise anything? Or the maxim “Don’t believe anything!”, which also applies to yourself? The ancient Greek poet Agathon once remarked: “It is very plausible that many implausible things are happening.” Doesn't the poet's plausible observation turn out to be itself an implausible event?

2. The Liar Paradox

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

This is especially clearly demonstrated by one of the most ancient and, perhaps, the most famous of logical paradoxes - the liar paradox. It refers to expressions that speak about themselves. It was discovered by Eubulides of Miletus, who came up with many interesting problems that still cause controversy. But it was the liar paradox that brought Eubulides true fame.

In the simplest version of this paradox, a person utters only one phrase: “I am 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, but the speaker claims that it is false, then his 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 speaks the truth and which speaks a lie?

Here is a modern rephrasing of this paradox. Suppose the only words written on the front of the card are: “On the other side of this card is a true statement.” Clearly these words represent a meaningful statement. Turning the card over, we must either find what was promised or not. If a statement is written on the back, then it is either true or false. However, on the back are the words: “There is a false statement written on the other side of this card” - and nothing more. Let's assume that the statement on the front is true. Then the statement on the back must be true, and therefore the statement on the front must be false. But if the statement on the front side is false, then the statement on the back side must also be false, and therefore the statement on the front side 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 it poses seems quite simple at first glance: does he lie who only says that he lies? 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 routineness of the question, it reveals some obscure and immeasurable depth.

There is even a legend that a certain Filit Kossky, despairing of resolving this paradox, committed suicide. They say 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 to the “liar”, and soon died without achieving anything.

In the Middle Ages, this paradox was classified as one of 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 any, even minor, difficulties regarding the use of language. And only in our so-called Modern times did the development of logic finally reach a level where the problems behind this paradox became possible to formulate in strict terms.

Now the “liar” is often called the “king of logical paradoxes.” An extensive scientific literature is devoted to it.

And yet, as with many other paradoxes, it remains not entirely clear what problems are hidden behind it and how to get rid of it.

So, there are statements that speak about their own truth or falsity. The idea that these kinds of statements are not meaningful is a very old one. 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, about its own falsity. A contradiction directly follows from this statement. If every statement is false, then this applies to the given statement itself, but the fact that it is false means that not every statement is false. The situation is similar with the statement “Every statement is true.” It should also be classified as meaningless and also leads to a contradiction: if every statement is true, then the negation of this statement itself is 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 Occam's contemporary, the French philosopher J. Buridan, did not agree with his decision. From the point of view of ordinary ideas about meaninglessness, expressions like “I am lying”, “Every statement is true (false)” are quite meaningful. What you can think about, you can speak out about - 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 self-talk is nonsensical. 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 allowed that a statement can speak about itself, then why is it not capable of speaking meaningfully about such a property as truth?

Buridan himself considered the statement “I am lying” not meaningless, but false. He justified it like this. When a person asserts a proposition, he thereby asserts that it is true. If a sentence says about itself that it is itself false, then it is only a shortened formulation of a more complex expression that asserts both its truth and its falsity. This expression is contradictory and therefore false. ^ it is in no way meaningless.

Buridan's argument is still sometimes considered convincing.

According to the idea of ​​the Polish logician A. Tarski, expressed in the 30s. last century, the reason for the liar's paradox is that the same language speaks both about objects existing in the world and about this “object” language itself. Tarski called a language with this property “semantically closed.” Natural language is obviously semantically closed. Hence the inevitability of a paradox arising 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: in the first they speak about the world of objects, in the second - about this first language, in the third - about the second language, etc. Clearly, that in this case the statement that speaks of its own falsity can no longer be formulated and the paradox will disappear.

This resolution of the paradox is not, of course, the only possible one. At one time it was generally accepted, but now the former unanimity is no longer there. The tradition of eliminating paradoxes of this type by “layering” the language remains, but other approaches have emerged.

As we can see, the problems associated with the “liar” over the centuries have changed radically depending on whether it was seen as an example of ambiguity, or as an expression that appears meaningful on the surface but is essentially meaningless, or as an example of a confusion of language and language. 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 in no case be understood as a local, isolated obstacle that can be eliminated by one inventive movement of thought. "The 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 evidence, and a whole series of important differences: between a sentence and the thought it expresses, 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, Evatl had to pay for training only if he won his first trial. 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. This lasted quite a long time, the teacher’s patience ran out, and he sued his student. Thus, for Euathlus this was the first process; He would no longer be able to get away from him. Protagoras justified his demand as follows: “Whatever the court’s decision, Euathlus will have to pay me. He will either win this first trial or lose. If he wins, he will pay according to our agreement. If he loses, he will pay according to the court decision.”

Euathlus appears to have been a capable student, for he replied to Protagoras: “Indeed, I will either win the trial or lose it. If I win, the court's decision will release me from the obligation to pay. If the court’s decision is not in my favor, it means I lost my first case and will not pay by virtue of our agreement.”

Puzzled by this turn of events, Protagoras devoted a special essay to this dispute with Euathlus, “The Litigation for Payment.” Unfortunately, it, like most of what Protagoras wrote, has not reached us. Nevertheless, we must pay tribute to Protagoras, who immediately sensed a problem behind a simple judicial incident that deserved special study.

The German philosopher G. W. Leibniz, a lawyer by training, also took this dispute seriously. In his doctoral dissertation, “A Study on Confused Cases in Law,” he tried to show that all cases, even the most complicated ones, like the litigation of Protagoras and Euathlus, must find the correct resolution based on common sense. According to Leibniz, the court should refuse Protagoras for untimely filing of the claim, but should, however, retain the right to demand payment of money from Euathlus later, namely after the first case 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. To this we can answer that if it were not for 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 about it and on its basis.

They also turned 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. Moreover, it is simply not applicable to the specific situation of the dispute: after all, Protagoras, while guaranteeing a high level of training, himself refused to accept payment if his student failed in the first trial.

Sometimes they argue like this. Both Protagoras and Euathlus are both partially right, and neither of them is right in general. Each of them takes into account only half of the possibilities that are beneficial to themselves. Full or comprehensive consideration opens up four possibilities, of which only half are beneficial to one of the disputants. Which of these possibilities is realized will be decided not by logic, but by life. If the judges' verdict has greater force than the contract, Evatl will have to pay only if he loses the case, that is, by virtue of a court decision. If a private agreement is placed higher than the decision of the judges, then Protagoras will receive payment only if Euathlus loses the case, 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 relevant circumstances are completely clear? And what kind of “leadership” will it be if Protagoras, who claims payment through the court, achieves it only by losing the process?

However, Leibniz’s solution, which seems convincing at first, is only to a few best advice court than the unclear opposition between “logic” and “life”. In essence, Leibniz proposes to retroactively change the wording of the treaty and stipulate that the first with the participation of Euathlus trial, the outcome of which will decide the issue of payment, there should not be a trial based on the claim of Protagoras. The thought is deep, but not related to a specific court. If there had been such a clause in the original agreement, the need for trial would not have arisen at all.

If by the solution to this difficulty we mean 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 represent 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, are unable to resolve the dispute.

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

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

“The crocodile snatched 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 has touched me, and I will give you a chance to get your child back. Guess whether 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 away.

After thinking, the mother replied:

You won't give me the child.

You won’t get it,” concluded the crocodile. - You told either the truth or a lie. If it is true that I will not give the child away, I will not give him away, since otherwise what is said will not be true. If what was said is not true, then you did not guess correctly, and I will not give up the child by agreement.

However, the mother did not find this reasoning convincing.

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

Who is right: the mother or the crocodile? What does the promise he makes oblige the crocodile to? To give the child away, or, on the contrary, not to give him away?

And to both at the same time. This promise is internally contradictory and, therefore, impossible to fulfill due to the laws of logic.

This paradox is played out in “Don Quixote” by M. Cervantes. Sancho Panza became governor of the island of Barataria and administers court. The first to come to him is a visitor and says: “Sir, a certain estate is divided into two halves by a high-water river... A bridge is thrown across this river, and right there on the edge there is a gallows and there is something like a court, in which four judges usually sit. , and they judge according to the law made by the owner of the river, the bridge and the entire estate. The law is drawn up as follows: “Everyone passing over a bridge over a river must declare under oath where and why he is going. Those who tell the truth will be let through, and those who lie will be sent to the gallows and executed without any mercy.” From the time this law was promulgated, many people managed to cross the bridge, and as soon as the judges were sure that the passers-by were telling the truth, they let them through. But one day a certain man, sworn in, swore and said that he had come to be hanged on this very gallows, and for nothing else. This oath perplexed the judges, and they said: “If this man is allowed to continue unhindered, it will mean that he has broken his oath and, according to the law, is guilty of death; if he is hanged, then he swore that he came only to be hanged on the gallows, therefore, his oath is not false, and on the basis of the same law he should be let through.” I ask you, Señor Governor, what should the judges do with this man, because they are still perplexed and hesitant.

Sancho suggested, perhaps, not without cunning: let the half of the person who told the truth be let through, and the half who lied should be hanged, and thus the rules for crossing the bridge will be respected in their entirety.”

This passage is interesting in several ways. First of all, it is a clear illustration of the fact that the hopeless situation described in the paradox may well be encountered - 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 precisely the solution that was the only thing left to resort to in his situation.

Once upon a time, Alexander the Great, instead of untying the tricky Gordian knot, which no one had ever managed to do, simply cut it. Sancho did the same. Trying to solve the puzzle on its own terms was futile - it was simply unsolvable. All that remained was to discard these conditions and introduce our own.

With this episode, Cervantes clearly condemns the excessively 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 - was information from the field of logic! Not only Cervantes himself is aware of this paradox. The writer finds it possible to attribute to his hero, an illiterate peasant, the ability to understand that he is faced with an insoluble task!

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

The missionary ended up with the cannibals and arrived just in time for lunch. They allow him to choose in what form he will be eaten. To do this, he must utter some statement with a condition: if this statement turns out to be true, they will boil him, and if it turns out to be false, they will fry him. What should you tell the missionary?

Of course he must say, “You will roast 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 choice: from “fry” comes “cook,” and vice versa.

4. Some modern paradoxes

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

Russell came up with a popular version of his paradox - the “barber paradox”. Let us assume that the council of a village has 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, he will refer to those who shave themselves; but those who shave themselves, he should not shave. If not, he will be one of 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 is, of course, impossible.

In its original version, Russell's paradox concerns sets, that is, collections of objects that are somewhat similar to each other. Regarding an arbitrary set, one can ask the question: is it its own element or not? Thus, the multitude of horses is not a horse, and therefore it is not a proper element. But a 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. Having divided all sets into those that are proper elements and those that are not, we 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 own element only in the case when it is not such an element.

This reasoning is based on the assumption that there is a set of all sets that are not their own elements. The contradiction resulting 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 an incentive to exclude from its consideration “too large sets” similar to the set of all sets, to limit the rules for operating with sets, etc. Despite the large number of methods proposed to date for eliminating paradoxes from set theory, complete There is no agreement yet on the reasons for their occurrence. Accordingly, there is no single, unobjectionable way to prevent their occurrence.

The above discussion about the barber is based on the assumption that such a barber exists. The resulting contradiction means that this assumption is false, and there is no resident of the village 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 it cannot exist sounds somewhat unexpected. But this conclusion is not paradoxical. The condition that the “village barber” must satisfy is in fact internally contradictory and, therefore, impossible to fulfill. There cannot be such a barber in the village for the same reason that there is no person in it who is older than himself or who was born before his birth.

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

Another example of the same pseudo-paradox is the famous argument about the catalogue.

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

It is not difficult 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 a reference to themselves can be thought of as an endless, never-ending process.

Let us assume that at some point a directory, say K1, was compiled, including all directories different from it that do not contain links to themselves. With the creation of K1, another directory appeared that did not contain a link to itself. Since the problem is to create a complete catalog of all catalogs that do not mention themselves, it is obvious that K1 is not a solution. He doesn't mention one of those directories - himself. By including this mention of himself in K1, we get catalog K2. It mentions K1, but not K2 itself. By 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 endlessly.

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

Some property words 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 many similar words; the vast majority of adjectives do not have the property they call. “New” is not, of course, new, “hot” is hot, “monosyllabic” is consisting of one syllable, “English” is English. Words that do not have the property they denote are called “foreign” or “heterological”. Obviously, all adjectives denoting properties that cannot be applied 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 autologous, it has the property it denotes and must be heterologous. 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 known back in the Middle Ages as the antinomy of an expression that does not name itself.

Another, apparently 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, for example, in the Russian language and contain less than, say, a hundred words, is finite. This means that there are natural numbers for which there are no names in Russian that consist of less than a hundred words. Among these numbers there are obviously smallest number. It cannot be named using a Russian expression containing less than a hundred words. But the expression: “The smallest natural number for which there is no compound name in the Russian language consisting of less than a hundred words” is precisely the name of this number! This name is just 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 do paradoxes say?

paradox liar logic argument

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

A logical dictionary is considered a necessary feature of logical paradoxes. Paradoxes classified as logical must be formulated in logical terms. However, in logic there are no clear criteria for dividing terms into logical and extra-logical. 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 be formulated in logical terms. Whether a particular paradox uses only purely logical premises is not always possible to determine unambiguously.

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

At the beginning of the study of logical paradoxes, it seemed that they could be identified by the violation of some, not yet studied, rule of logic. The “principle of a vicious circle” introduced by Russell was particularly active in claiming the role of such a rule. This principle states that a collection of objects cannot contain members definable only by that same collection.

All paradoxes have one common property - self-applicability, or circularity. In each of them, the object in question is characterized by a certain 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 the totality of people to which this person belongs (using “his class”). And if we say: “This statement is false,” we characterize the statement in question by reference to the set of all false statements that includes it.

In all paradoxes there is self-applicability, which means there is, as it were, a movement in a circle, ultimately leading to the starting point. In an effort to characterize an object of interest to us, we turn to the totality of objects that includes it. However, it turns out that for its definiteness it itself needs the object in question 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 appears 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.

Mere reference to the use of self-applicable concepts is therefore not sufficient to discredit paradoxes. Some additional criterion is needed to separate self-applicability, leading to a paradox, from all its other cases.

There were many proposals in this regard, but no successful clarification of circular™ was ever 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 be a distinctive feature of all logical paradoxes, did not lead to anything definite.

Undoubtedly, some classification of paradoxes would be useful, dividing them into types and types, grouping some paradoxes and contrasting them with others. However, nothing lasting was achieved in this matter either.

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

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

Gorgias' argument about the non-existence of nature unfolds like this. First it is proven 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 follows that what exists is incomprehensible to man. Once again a step back is taken and it is assumed, contrary to what seems to have already been proven, that what exists is still comprehensible. From the last assumption it follows that what is comprehensible is inexpressible and inexplicable to another.

What exactly were the problems Gorgias wanted to pose? It is impossible to answer this question unambiguously. It is obvious that Gorgias' reasoning confronts us with contradictions and encourages us to look for a way out to get rid of them. But what exactly are the problems that the contradictions point to, 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 carts. He, in particular, argued: “What does not have thickness cannot be accumulated, and yet its bulk can extend for a thousand miles. - Heaven and earth are equally low; mountains and swamps are equally level. - The sun, having just reached its zenith, is already at sunset; a thing that has just been born is already dying. - The southern side of the world has no limit and at the same time has a limit. “I just went to Yue today, but 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 confusing and stated that "his biased words never hit the mark." The ancient philosophical treatise “Zhuang Tzu”, in particular, 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 escape from an echo by making sounds, or trying to rush away from your own shadow. Isn't it sad?

Well said, but hardly fair.

The impression of confusion and inconsistency in Hui Shi's sayings is due to the external side of the matter, to the fact that he poses his problems in a paradoxical form. What one could reproach him for is that for some reason he considers posing 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 difficulty is hinted at by his statement that a person who has just set off somewhere has long since arrived there? This can be interpreted 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 heading, like Hui Shi, to Yue, constantly keeps this point in mind and during the entire time of moving towards it, he seems to remain in it. But if a person who just went to Yue has already been there for a long time, then why should he go there at all? It is not entirely clear what difficulty lies behind this simple statement.

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 is 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 were forgotten and were rediscovered in the last century.

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

Paradoxes pose an important question: where, in fact, do some conventional methods of concept formation and methods of 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 usual methods of theoretical thinking by themselves and without any special control over them provide reliable progress towards the truth.

Demanding a radical change in an overly credulous approach to theorizing, paradoxes represent a sharp critique of logic in its naive, intuitive form. They play the role of a factor that controls and sets restrictions on the way of constructing deductive systems of logic. And this role of theirs can be compared with the role of an experiment that tests the correctness of hypotheses in sciences such as physics and chemistry, and forces changes to be made 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 a variety of ways, and in the end it can be revealed without such acute symptoms as paradoxes. Let's say, the foundations of set theory would have been analyzed and clarified even if no paradoxes had been discovered in this area. But there would not have been the sharpness and urgency with which the paradoxes discovered in it posed the problem of revising set theory.

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

There is one important difference to note. Eliminating paradoxes and resolving them are not the same thing. To eliminate a paradox from a theory means to reconstruct it so that the paradoxical statement turns out to be unprovable in it. Each paradox relies on a large number of definitions and assumptions. His conclusion in theory represents a certain chain of reasoning. Formally speaking, you can question any of its links, eliminate them 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 a solution to the paradox. It is not enough to find a way to exclude it; one must convincingly justify the proposed solution. The doubt itself about any step leading to a paradox must be well founded.

First of all, the decision to abandon any logical means used in deriving 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, eliminating the paradox turns out to be devoid of solid and stable foundations and degenerates into a primarily technical task.

Moreover, the rejection of an assumption, even if it ensures the elimination of a particular paradox, does not automatically guarantee the elimination of all paradoxes. This suggests that paradoxes should not be “hunted” individually. 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 the result, although not containing paradoxes, is a significantly weaker theory that has only private interest.

G. Frege, who is one of the founders of modern logic, had a very bad character. In addition, he unconditionally and even cruelly criticized his contemporaries. Perhaps this is why his contribution to logic and the foundation of mathematics did not receive recognition for a long time. And when it began to come, the young English logician Russell wrote to him that a contradiction arose 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 numerous 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 we are in now with regard to paradoxes is unbearable for a long time. Think: in mathematics - this example of reliability 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 logic of the late 19th century, free from any 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 supposed logic was nothing more than an illusion. They showed indisputably that logic - in the intuitive form that it then had - needed a deep revision.

A whole century has passed since a lively discussion of paradoxes began. The attempted revision of logic did not, however, lead to an unambiguous resolution of them.

And at the same time, such a state hardly seems unbearable to anyone now. Over time, the attitude towards paradoxes became calmer and even more tolerant than at the time of their discovery.

The point is not only that paradoxes have become something, although unpleasant, but nevertheless familiar. And, of course, not that they have come to terms with them. They still remain the focus of attention of logicians, and the search for their solutions continues actively.

The situation has changed primarily in the sense that the paradoxes have become, so to speak, localized. They have found their definite, albeit troubled, place in the wide spectrum of logical research.

It became clear that absolute severity, as it was pictured at the end of the last century and even sometimes at the beginning of this one, 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 belong to different types and affect, in essence, all the main sections of logic. The discovery of a paradox forces us to deeply analyze our logical intuitions and engage in systematic reworking of the foundations of the science of logic. At the same time, the desire to avoid paradoxes is neither the only nor, perhaps, the main task. Although they are important, they are only a reason for thinking about the central themes of logic. Continuing the comparison of paradoxes with particularly distinct symptoms of a disease, we can say that the desire to immediately eliminate paradoxes would be similar to the desire to remove such symptoms without particularly caring about the disease itself. It is not just the resolution of paradoxes that is required, it is necessary to explain them, deepening our understanding of the logical laws of thinking.

Pondering paradoxes is, without a doubt, one of the best tests of our logical abilities and one of the most effective means of training them.

Getting to know paradoxes and getting to the heart of the problems behind them is not an easy task. It requires maximum concentration and intense thought into several seemingly simple statements. Only under this condition can the paradox be understood. It is difficult to claim to invent new solutions to logical paradoxes, but already familiarizing yourself with the proposed solutions is a good school of practical logic.

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