Lecture
The law of the excluded middle, as well as the law of contradiction, establishes a connection between statements contradicting each other. And again, the idea expressed by him seems simple and obvious at first: of the two contradictory statements, one thing is true.
In the semi-symbolic form already used: A or non-Ay, i.e. true statement A or true negation, non-A statement .
Specific applications of this law are, for example, statements: “Aristotle died in 322 BC. or he did not die this year, "" The larvae of flies have a head or do not have it. "
The truth of negation is equivalent to a false statement. By virtue of this, the law of the excluded middle can be transmitted in the following way: every statement is true or false.
The name of the law itself expresses its meaning: the situation is as described in the statement in question, or as its denial says, and there is no third possibility.
They tell the story about the owner of the dog, who was very proud of raising his pet. On his team: “Hey! Come or don't come! ”- the dog always came or not. So the team was fulfilled anyway.
A person speaks in prose or does not speak in prose, someone sobs or does not sob, the dog executes a command or does not perform, etc. - There are no other options. We may not know whether some particular theory is contradictory or not, but on the basis of the law of the excluded middle still before the beginning of the study, we are entitled to state: it is either inconsistent or contradictory.
This law with irony is played up in fiction. The reason for the irony is clear: to say "Something is or is, or it is not," then, absolutely nothing to say. And it's funny if someone doesn't know that.
In the comedy of Moliere “The Bourgeois in the Nobility” there is such a dialogue:
Mr. Jourdain. ... And now I have to tell you a secret. I am in love with one lady of the world, and I would like you to help write her a little note, which I am going to drop at her feet.
Philosophy teacher. Of course, you want to write her poems?
Mr. Jourdain. No, no, just not poetry. Philosophy teacher. Do you prefer prose? Mr. Jourdain. No, I do not want prose or poetry. Teaching a philosophy. It is impossible: either this or that.
Mr. Jourdain. Why?
Philosophy teacher. For the reason, sir, that we can express our thoughts just as in prose or verses.
Mr. Jourdain. None other than prose or verse?
Philosophy teacher. Not otherwise, sir. All that is not prose, then verses, and that which is not poetry, is prose.
In the famous fairy tale by L. Carroll “Alice through the looking-glass”, the White Knight intends to sing Alice “a very, very beautiful song”.
- When I sing it, everyone sobs ... or ...
- Or what? - Alice asked, not understanding why the Knight suddenly stopped.
- Or ... do not cry ...
In the fairy tale of A. N. Tolstoy “The Golden Key, or the Adventures of Pinocchio”, the folk healer Mantis concludes after the inspection of Pinocchio:
- One of two things: either the patient is alive or he is dead. If he is alive, he will live or he will not live. If he is dead, he can be revived or not revived.
Both laws — the law of contradiction and the law of the excluded middle — were known even before Aristotle. However, he was the first to give them clear formulations, stressed the importance of these laws for the understanding of thinking and being, and at the same time expressed certain doubts about the universal applicability of the second of them.
“It is impossible,” wrote Aristotle, “that the same thing at the same time be and is not inherent in the same in the same respect (and everything else that we could clarify, let it be clarified in avoiding verbal difficulties) is, of course, the most reliable of all beginnings. ” Such is the formulation of the law of contradiction and at the same time a warning about the need to preserve the same point of view in the statement and its denial “in order to avoid verbal difficulties”. Here Aristotle argues with those who doubt the validity of this law: "... no one can consider the same as existing and nonexistent, as Heraclitus argues, according to some,".
On the law of the excluded middle: "... there can be nothing intermediate between two members of the contradiction, but with respect to one thing, it is necessary that either one be either approved or denied."
From Aristotle, there is also a tradition that still lives today: giving the law of contradiction, the law of the excluded middle, and other logical laws, three different interpretations.
In one case, the law of contradiction is interpreted as the principle of logic, speaking about statements and their truth: out of two contradictory statements, only one can be true.
In another case, the same law is understood as a statement about the structure of the world itself: it cannot be so that something simultaneously exists and does not exist.
In the third case, this law already sounds like the truth of psychology concerning the originality of our thinking: it is not possible to think so about some thing so that it turns out to be so and at the same time not so.
It is often assumed that these three options differ only in formulations. In fact, this is absolutely not the case. The structure of the world and the peculiarity of human thinking are the themes of empirical, experimental research. The positions obtained with it are empirical truths. The principles of logic are completely different from experience and are not empirical, but logically necessary truths. Later, when it comes to the general nature of logical laws and logical necessity, the inadmissibility of such a mixture of logic, psychology and theory of being will become clearer.
Aristotle doubted the applicability of the law of the excluded middle to statements about future events. At the moment, the onset of some of them has not yet been predetermined. There is no reason for them to happen, or for them to happen. “It will rain in a hundred years on the same day,” this statement is now, most likely neither true nor false. The same is his denial. After all, now there is no reason for it to rain after a hundred years, or for a hundred years after a hundred years. According to the law of the excluded middle, he claims that either the statement itself or its negation is true. Hence, Aristotle concludes, although without much certainty, this law should be limited to statements about the past and the present and not to apply it to statements about the future.
Much later, in the twentieth century, Aristotle’s reasoning about the law of the excluded middle prompted the idea of the possibility of a fundamentally new direction in logic. But let's talk about this later.
In the XIX century. Hegel spoke very ironically of the law of contradiction and the law of the excluded middle.
He presented the latter, in particular, in the following form: “Spirit is green or not green”, and asked the “tricky” question: which of these two statements is true?
The answer to this question is not, however, labor. Neither of the two statements: “Spirit is green” and “Spirit is not green” is not true, since both of them are meaningless. The law of the excluded middle applies only to meaningful statements. Only they can be true or false. The senseless is neither true nor false.
Hegel's critique of logical laws relied, as is often the case, on giving them the meaning that they have in mind, and attributing to them those functions to which they have no relation. The case of criticism of the law of the excluded middle is one example of this approach.
The casual, scattered and insufficiently competent criticisms of Hegel addressed to formal logic were, unfortunately, widely circulated. In logic in the late XIX - early XX century. there was a scientific revolution that radically changed the face of this science. But even the tremendous successes achieved by logic could not completely eradicate the erroneous ideas about it, which Hegel stood at the source. It is not by chance that the German historian of logic X. Scholz wrote that the Hegelian critique of formal logic was so great an evil that it is difficult to overestimate it even now.
Dutch mathematician L. Brower subjected the law of the excluded third to harsh but well-founded criticism. At the beginning of the XX century. He published three articles in which he expressed doubt about the unlimited applicability of the laws of logic and, above all, the law of the excluded middle. The first of these articles did not exceed three pages, the second - four, and together they did not occupy seventeen pages. But the impression they made was extremely strong. Brower was convinced that logical laws are not absolute truths that are independent of what they are applied to. While opposing the law of the excluded middle, he insisted that there is still a third possibility between assertion and his denial, which cannot be ruled out. She finds herself on reasoning about infinite sets of objects.
Assume that the existence of an object with a certain property is asserted. If the set into which this object belongs, of course, then you can iterate through all the objects. This will make it possible to find out which of the following two statements is true: “In this set there is an object with the specified property” or “There is no such object in this set”. The law of the excluded middle is valid here.
But when the set is infinite, it is impossible to sort objects. If in the process of searching an object with the required property is found, the first of the stated assertions is confirmed. But if it is possible to find this NA object, it is impossible to say anything about the first or second of the statements, since the search was not carried out to the end. The law of the excluded middle does not work here: neither the statement about the existence of an object with a given property, nor the negation of this statement are true.
The restriction by Brouwer to the scope of this law significantly narrowed the range of methods of reasoning that are applicable in mathematics. This immediately caused a sharp opposition of many mathematicians, especially the older generation. “To remove the principle of the excluded middle from mathematics,” wrote the German mathematician D. Hilbert, “is the same as ... prohibiting a boxer from using his fists”.
The criticism by Brouwer of the law of the excluded middle led to the creation of a new trend in logic - intuitionistic logic. The latter does not accept this law and discards all those methods of reasoning associated with it. Among them - evidence by reduction to contradiction or absurdity.
It is interesting to take revenge that even before Brouwer doubts about the universal applicability of the law of the excluded third were expressed by the Russian philosopher and logician N. A. Vasiliev. He set as his task the construction of such a system of logic in which the scope of not only this law, but also the law of contradiction would be limited. According to Vasiliev, logic, limited in this way, is not capable of acting in the world of ordinary things, but it is necessary for a deeper understanding of the logical teachings of Aristotle.
Contemporaries could not adequately evaluate the seemingly paradoxical ideas of Vasiliev. In addition, he himself was inclined to substantiate his views with the help of arguments that have no direct relation to logic and the rules of logical technique, and sometimes simply confused. Looking back, however, it can be said that he turned out to be one of the forerunners of intuitionistic logic.
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Logics
Terms: Logics