Lecture
The most famous and perhaps the most interesting of all logical paradoxes is the “Liar” paradox. It was he who mainly glorified the name of Eubulid who opened it from Miletus.
There are variants of this paradox, or antinomy, many of which are only paradoxical in appearance.
In the simplest version of “Liar”, a person utters just one phrase: “I lie”. Or he says: "The statement that I am saying now is false." Or: "This statement is false."
If the statement is false, then the speaker said the truth, and it means that what he said is not a lie. If the statement is not false, and the speaker claims that it is false, then his statement is false. It turns out, therefore, that if the speaker lies, he speaks the truth, and vice versa.
In the Middle Ages, the following wording was common:
“What Plato said is false,” says Socrates.
“What Socrates said is true,” says Plato. The question arises, which of them expresses the truth, and who lies?
But the modern rephrasing of this paradox. Suppose that on the front side of the card only the words are written: “Pa a true statement is written to the other side of this card.” It is clear that these words are a meaningful statement. Turning the card over, we must either find the promised statement, or it will not. If it is written on the back, then it is either true or not. However, on the back are the words: “A false statement is written on the other side of this card” - and nothing more. Suppose that the statement on the front side is true. Then the statement on the back must be true and, therefore, the statement on the front side must be false. But if the statement on the front side is false, then the statement on the back must also be false, and therefore, the statement on the front side must be true. In the end - a paradox.
The “Liar” paradox made a huge impression on the Greeks. And it's easy to understand why. The question that is put in it, at first glance, seems to be quite simple: Is the one a liar who says only that he lies? But the answer “yes” leads to the answer “no”, and vice versa. And thinking does not clarify the situation. Behind the simplicity and even the commonness of the issue, it opens up some unclear and immeasurable depth.
There is even a legend that a certain Filit Kossky, desperate to resolve this paradox, committed suicide. It is also said that one of the well-known ancient Greek logicians, Diodorus Kronos, already in his declining years made a vow not to take food until he found the solution of "Liar", and soon died, having achieved nothing.
In the Middle Ages, this paradox was attributed to the so-called intractable sentences and became the object of systematic analysis.
And the time “Liar” has long attracted no attention. They did not see any, even insignificant difficulties concerning the use of language. And only in our so-called modern times, the development of logic has finally reached a level where it seems possible to formulate the problems behind this paradox in strict terms.
Now "The Liar" - this typical former sophism - is often referred to as the king of logical paradoxes. He is devoted to extensive scientific literature. And, nevertheless, as in the case of many other paradoxes, it is not entirely clear what exactly the problems lie behind him and how to get rid of him.
Now "Liar" is usually considered a characteristic example of the difficulties to which the mixing of two languages leads: a language that speaks of reality outside of it, and a language that speaks of the very first language.
In everyday language, there is no difference between these levels: we speak the same language as about reality and about language. For example, a person whose native language is Russian does not see any particular difference between the statements: “Glass is transparent” and “True, that glass is transparent”, although one of them speaks of glass, and the other speaks of speaking about glass.
If someone had the idea of having to speak 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 just saying: “A cow is a noun,” I would say instead of: “The statement“ Glass is not transparent ”would falsely say
"The assertion" Glass is not transparent "is false." With this use of two different languages, what was said about the world would clearly differ from what was said about the language with which it is spoken about the world. In fact, the first statements would refer to the Russian language, while the second - to the English.
If further our language expert wanted to comment on some circumstances concerning the English language, he could use another language. Let's say German. To talk about this last one could resort to, say, the Spanish language, etc.
It turns out, thus, a kind of ladder, or hierarchy, of languages, each of which is used for a well-defined goal: the first one talks about the objective world, the second one talks about this first language, the third one about the second language, etc. Such a distinction between languages according to their area of application is a rare phenomenon in ordinary life. But in the sciences, which are specially engaged, like logic, in languages, it sometimes turns out to be very useful. A language that speaks of the world is usually called the objective language. The language used to describe a subject language is called a metalanguage.
It is clear that if a language and a metalanguage are delimited in this way, the statement “I lie” can no longer be formulated. It speaks of the falsity of what is said in Russian, and, therefore, refers to a metalanguage and should be expressed in English. Specifically, it should sound like this: “Everything I speak in Russian is false” (“Everything I have said in Russian is false”); this English statement says nothing about himself, and no paradox arises.
Distinguishing between language and metalanguage eliminates the “Liar” paradox. Thus, it becomes possible to correctly define the classical concept of truth without contradiction: a statement that corresponds to the reality described by it is true.
The concept of truth, like all other semantic concepts, has a relative character: it can always be attributed to a specific language.
As the Polish logician A. Tarski showed, the classical definition of truth should be formulated in a language broader than the language for which it is intended. In other words, if we want to indicate what the “statement that is true in a given language” means, we must, in addition to the expressions of this language, also use expressions that are not in it.
Tarsky introduced the concept of semantically closed language. Such a language includes, in addition to its expressions, their names, as well as, which is important to emphasize, statements about the truth of the sentences formulated in it.
There is no border between a language and a 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 funds are sufficient, in particular, in order to reproduce the antinomy "Liar" in 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 the antinomy, and hence the internal inconsistency, according to Tarski, is to abandon the use of a semantically closed language. This path is acceptable, of course, only in the case of artificial, formalized languages that allow for a clear division into the language and the metalanguage. In natural languages with their obscure structure and the ability to speak about everything in the same language, this approach is not very real. Raising the question of the internal consistency of these languages does not make sense. Their rich expressive possibilities have their opposite side - paradoxes.
So, there are statements that speak of their own truth or falsity. The idea that such statements are not meaningful is very old. She was defended by the ancient Greek logician Chrysippus.
In the Middle Ages, the English philosopher and logician U. Ockham stated that the statement “Every utterance is false” is meaningless, since it also speaks of its own falsity. This statement directly implies a contradiction. If every utterance is .yuzhno, then this also applies to the statement itself; but the fact that it is false means that not every statement is false. The situation is similar with the statement "Every utterance is true." It should also be attributed to meaningless and also leads to a contradiction: if every utterance is true, then the negation of the utterance itself is true, i.e. saying that not every statement is true.
Why, however, cannot a statement speak intelligently about its own truth or falsity?
Already 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 lie”, “Every utterance is true (false)”, etc. quite meaningful. What you can think about, you can speak about - such is the general principle of Buridan. A person can think about the truth of the statement that he makes, which means that he can speak about it. Not all statements that speak about themselves are meaningless. For example, the statement “This sentence is written in Russian” is true, and the statement “In this sentence is ten words” is false. And both of them are completely meaningful. If it is assumed that an assertion can speak about itself, then why is it not able to speak with a meaning about such a property as truth?
Buridan himself considered the statement “I lie” not to be meaningless, but false. He justified it so. When a person approves a sentence, he claims that it is true. If the sentence says about itself that it is itself false, then it is only an abbreviated formulation of a more complex expression that asserts both its truth and its falsity. This expression is contradictory and therefore false. By no means senseless.
Argument Buridan and now sometimes considered convincing.
There are other areas of criticism of the decision of the “Liar” paradox, which was developed in detail by Tarski. Is there really no antidote against paradoxes of this type in semantically closed languages - and such are all natural languages?
If this were so, then the concept of truth could be defined in a strict way only in formalized languages. Only in them will it be possible to distinguish between the objective language, in which they talk about the world around them, and the metalanguage, in which they speak about this language. This language hierarchy is built on the model of mastering a foreign language with the help of the native. The study of such a hierarchy has led to many interesting conclusions, and in certain cases it is essential. But it is not in natural language. Does it discredit him? And if so, to what extent? After all, in him the concept of truth is still used, and usually without any complications. Is the introduction of a hierarchy the only way to eliminate paradoxes like the Liar?
In the 30s. XX century. The answers to these questions were undoubtedly affirmative. However, now the former unanimity is already pet, although the tradition to eliminate the paradoxes of this type by “stratifying” the language remains dominant.
Recently, egocentric expressions have attracted more and more attention . They contain words like “I,” “this,” “here,” “now,” and their truth depends on when, by whom, where they are used.
In the statement “This statement is false,” the word “this” is found. What particular object does it belong to? “Liar” can say that the word “this” does not refer to the meaning of this statement. But then what does it mean, what does it mean? And why this sense can not still be denoted by the word "it"?
Without going into details here, it is worth noting only 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 the confusion of language and the metalanguage, but points to the dangers associated with the misuse of the word "this" and similar egocentric words.
The problems that have been associated with “Liar” over the centuries have radically changed depending on whether it was considered as an example of ambiguity, or as an expression that looks like a pattern of language mixing and a 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 well-known modern Finnish logician and philosopher G. von Wrigt wrote in his work devoted to “The Liar” that this paradox and in no case should be understood as a local, isolated obstacle eliminated by a single inventive movement of thought. "Liar" covers many of the most important topics of logic and semantics. This is the definition of truth, and the interpretation of contradiction and evidence, and a whole series of important differences: between the sentence and the thought it expresses, between the use of the expression and its mention, between the meaning of the name and the object it denotes.
The situation is similar with other logical paradoxes. “The antinomies of logic,” writes von Wrigt, “have perplexed from the moment of their discovery and are likely to perplex us always. We must, I think, consider them not so much as problems to be solved, but as inexhaustible raw material for thought. They are important because thinking about them involves the most fundamental questions of all logic, and therefore of all thinking. ”
In conclusion of this talk about “The Liar”, we can recall a curious episode from the time when formal logic was still taught in school. In the textbook of logic, published in the late 40s. XX century., Eighth grade students were offered as homework - in order, so to speak, warm-ups - to find the mistake made in this simple-looking statement: “I lie”. And even if it does not seem strange, it was believed that the majority of schoolchildren successfully coped with this task.
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Logics
Terms: Logics