Lecture
A mistake in the proof is a rather ordinary thing. When conducting evidence, we rely on our logical intuition, on the spontaneously learned knowledge of the laws of logic. As a rule, it does not fail us. But in some, and especially in difficult cases, it may be unreliable.
Experiments conducted by psychologists show that almost every fourth of our conclusions do not rely on the laws of logic, and therefore is wrong. Logic is rarely studied specifically. Skills logical, i.e. consistent and evidential, thinking formed and improved in the practice of reasoning. But, as the English philosopher F. Bacon remarked, the exercises, not enlightened by theory, with equal success reinforce both correct and erroneous.
Our logical sense and our proof skills are not as perfect as it often seems. It is therefore useful not to miss a chance to improve them.
It is possible to draw a clear boundary only when it is known not only what is covered by it, but also what remains outside of it. A clear understanding of the proof implies, among other things, a certain idea of reasoning, having the form of proof, which in fact is not. Such “failed evidence” is the result of errors made — unintentionally or deliberately — in the course of the proof. Familiarity with the most typical of them contributes to the improvement of practical skills of proof and allows you to better understand what constitutes "infallible" evidence.
Proof is a logical, deductive connection between the accepted arguments and the thesis derived from them. Logical errors in the proof can be divided into related to the thesis, to the arguments and to their connection.
A formal error occurs when the conclusion is not based on a logical law and the conclusion does not follow from the accepted assumptions. Sometimes this error is abbreviated as “no leakage”.
Suppose someone reasons like this: “If I visit my uncle, he will give me a camera, when my uncle gives me a camera, I will sell it and buy a bicycle: it means that if I visit my uncle, I will sell it and buy a bicycle”.
Clearly, this is untenable reasoning. His conclusion about “selling uncle” is absurd. But the premises are harmless and may well be true, so the source of concern is not in them. The cause of the error is in the deduction itself, in deducing from the accepted assertions that they were not implied at all.
Deduction from the correct premise always gives the correct conclusion. In this case, the conclusion is false. Hence, inference does not rely on the law of logic and is wrong. The error is simple. The pronoun "him" may indicate different subjects. In the sentence “I will sell it and buy a bicycle,” it should point to the camera. But it turns out that in fact it refers to an uncle.
In order to refute this incorrect reasoning, it is necessary to show that between the received assumptions and the conclusion of a logical connection made on their basis.
The German physicist V. Nernst, who discovered the third law of thermodynamics (about the unattainability of absolute zero temperature), “proved” the completion of the development of the fundamental laws of this branch of physics: “The first principle was three authors: Meyer, Joule and Helmholtz; the second has two: Carnot and Clausius, and the third has only one: Nernst. Consequently, the number of authors of the fourth law of thermodynamics should be zero, i.e. there simply cannot be such a law. ”
This comic proof well illustrates the situation when there is clearly no logical connection between the arguments and the thesis. The illusion of a peculiar "logic" of reasoning is created by a listing that is purely external to the substance of the matter.
Wire was found in the tomb of the Egyptian pharaohs. On this basis, some "Egyptologist" suggested that the telegraph was known in ancient Egypt. Hearing about this, another “researcher” concluded that since no wire was found in the tombs of the Assyrian kings, the wireless telegraph was already known in Ancient Assyria.
The suggestion of an "Egyptologist" - if this is not a joke - is an obvious absurdity. Even more stupid - if this is again not a joke - the conclusion is “Assyriologist”. And, of course, there is no logical connection between these “assumptions” and the “conclusions” made on their basis.
Fortunately, quite rare, chaotic, amorphous reasoning occurs. Outwardly, they have the form of evidence and even claim to be considered as them. They contain the words “in this way”, “consequently”, “means” and similar to them, called to indicate the logical connection of the arguments and the position to be proved. But these arguments are not really evidence, since logical connections are replaced in them by psychological associations.
Here, for example, is a reasoning that looks like a proof: “The perpetual motion machine was considered impossible, because it contradicts the law of conservation of energy, or the first law of thermodynamics. When the second law of thermodynamics was discovered, they began to talk about the impossibility of a perpetual motion machine of the second kind. The same can be said about the eternal engine of the third kind, which is prohibited by the third law of thermodynamics. But the fourth start of thermodynamics pet! Consequently, nothing prevents the creation of a perpetual motion machine of the fourth kind. And the more perpetual motion of the fifth and so on kind! "
A characteristic mistake in relation to a thesis is the substitution of a thesis, its unconscious or deliberate replacement in the course of the proof by some other statement. The substitution of the thesis leads to the fact that not what is required to be proved is proved.
The thesis may be narrowed, and in this case it remains unproved. For example, to prove that the sum of the angles of a triangle is equal to two straight lines, it is not enough to prove that this sum is not greater than 180 ° C. To substantiate that a person should be honest, it is not enough to prove that a reasonable person should not lie.
Thesis may also expand. In this case, additional grounds are needed. And it may turn out that not only the initial thesis follows from them, but also some other, already unacceptable statement. “He who proves too much does not prove anything,” this old Latin proverb means such a danger.
Sometimes there is a complete substitution of the thesis, moreover, it is not as rare as it may seem. Usually it is obscured by any circumstances related to a particular situation, and due to this escapes attention. The case of the ancient Greek philosopher Diogenes, who was once beaten for substitution of the dispute thesis, was widely known. One philosopher argued that in the world, as it appears to our thinking, there is no movement, there are not many things, but there is only one single thing, moreover, motionless and round. In order of objection, Diogenes got up and began to walk slowly in front of the disputants. For this his, if you believe some old sources, and beat with a stick.
It was about the fact that for our mind the world is still. Diogenes, with his walk, tried to confirm another thought: there is movement in the sensually perceived world. But this was not contested. The author of the opinion that there is no movement, believed that feelings that speak of the multiplicity of things and their movement simply deceive us.
Of course, the opinion that there is no movement is mistaken how wrong the idea is that feelings do not give us a correct idea of the world. But once such an opinion was discussed, it was necessary to talk about him, and not about something else, even if it was true.
This is how A. S. Pushkin describes this dispute:
There is no movement, said the wise man, the Other said nothing and began to walk in front of him. Stronger he could not argue; Praised all the answer is convoluted. But, gentlemen, this is a funny case. Another example leads me to memory: After all, every day the sun goes before us, However, stubborn Galileo is right.
The most common mistake is an attempt to substantiate a thesis with the help of false arguments.
Tigers, as you know, do not fly. But the reasoning is “Only birds fly; tigers are not birds; therefore, tigers do not fly ”is not, of course, proof of this fact. In the reasoning, an incorrect premise is used that only birds are capable of flying: many insects and mammals (for example, bats) fly, planes, etc. With the help of the parcel “Only birds fly”, it is possible to derive not only the true, but also a false conclusion, say that the May beetles, since they are not birds, do not fly.
A rather common mistake is a circle in evidence: the validity of the position being proved is substantiated by the same position, expressed, perhaps in a slightly different form. If the basis of the proof is that which still needs to be proved, the substantiated thought is derived from itself, and it turns out not proof, but empty circulation in a circle.
Why do we see through glass? The usual answer is: it is transparent. But to call a substance transparent means to say what can be seen through it.
In the article “So what do we do?” Leo Tolstoy sharply accuses political economy in a clear vicious circle. “The question of economics,” writes Tolstoy, “is this: what is the reason that only people with land and capital can enslave those who do not have land and capital?”
The answer, which is common sense, is that it comes from money that has the ability to enslave people. But science denies this and says: this is not due to the quality of money, but because some have land and capital, while others do not have it. We ask: Why do people who have land and capital enslave the poor? We are answered: because they have land and capital.
Why, we just ask the same thing. Deprivation of land and instruments of labor is enslavement. After all, this is the answer: lulls, because it has hypnotic power. "
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Logics
Terms: Logics