Logical errors. Training course. Chapter 1. Where do logical errors come from?

2024 Author: Seth Attwood | [email protected]. Last modified: 2023-12-16 15:55

Repetition

In the introduction, you learned about such important concepts as truth and validity. A true statement corresponds to the real state of affairs, which can be verified in one way or another (for example, the statement "there are 3 windows in a room" can often be checked on the spot: we can count the windows, convincing or rejecting what has been said). Correct reasoning is reasoning in which thoughts are consistent with each other. Strictly speaking, this is when from a true premise we can get only a true consequence (for example, from the statements “all metals expand when heated” and “gold is a metal”, with correct logic, only the true conclusion follows: “gold expands when heated”), but from false premises by correct inferences, you can get any kind of consequence, both true and false.

The term “ consistency (Soundness). A sound argument is an argument deduced from true premises by the correct form of reasoning. That is, in fact, a valid argument is necessarily true. In our popular science presentation, we do not need to distinguish consistency from truth, therefore, unless otherwise stated, we consider them synonyms.

Chapter 1. Where do logical errors come from?

The following material is based on chapters II-IV of the book by A. I. Uemov “Logical errors. How they interfere with right thinking”(1958), as well as the personal teaching experience of the author of the course for many years. Additional supporting materials are given along the way.

Intent

First of all, logical errors arise on purpose, that is, they are made with special intent. The intent can be different: from a simple joke to maliciously misleading the interlocutor in order to extract benefits. Here's one example of a joke:

a²- a²= a²- a²

a (a-a) = (a-a) (a + a)

a = a + a

a = 2a

1 = 2

On the other hand, such a joke can also be a real task for students on a test or even in a job interview. Thus, there is another option for intent: to deliberately mislead a person in order to test his attention and ability to find errors. Sometimes a stressful situation is arranged along the way in order to also test the stress tolerance of a candidate for a very nervous position.

And here is an example of malice. A person comes to service the car at a car service, and after a while the main foreman informs him: "The guys changed the brake fluid, but your fitting has become stuck, you will have to change the wheel bearings, otherwise they turned out right away, I don't know how you got there at all." Who does not know, I will explain: from a malfunction of the "fitting" (which really does happen) does not at all follow a malfunction of the bearings, and even more so they cannot be "turned out" (there is no such concept in auto mechanics, at least in relation to these components of the car). But the client may not know the terms, and in order not to seem stupid, he begins to obediently nod his head. This form of logical error, specially made by the master, is designed to "dilute" the client by several thousand rubles. Similar convincing forms of malicious "divorce", in which logical errors are hidden either behind a pile of terms or behind the subtleties of the process, can be found in various areas of our life. I think that the reader can cope with the search for such examples without me, simply delving into his own memories.

The article by E. A. Yashina "Intentional logical errors as a means of creating alogism in a literary text" (Bulletin of the Vyatka State Humanitarian University, No. 2-2, 2010) provides examples of intentional alogisms - violations or ignoring the laws of logic for a specific purpose, one of which - creating a certain mood in the reader. Here is an example given in the article and taken from the novel by I. S. Turgenev "Fathers and Sons":

At the first hut there were two peasants in hats and cursing. "You are a big pig," one said to another, "and you are worse than a small pig." “And your wife is a witch,” argued another.

Alogism consists in an attempt to combine concepts that are incompatible in meaning, giving meaninglessness to the overall picture of this dispute.

Another example of intent is sophistry, which more than two and a half thousand years ago were used in public debate, in preparation for a political career, in court cases, etc. (see Wikipedia). Here is an example of sophistry: “The Law of Moses forbade theft. The Mosaic Law has lost its power. Therefore, theft is not prohibited. However, we can even now find the use of sophisms in political debates.

You can talk about deliberate errors in logic for quite a long time, however, I think the reader has grasped the main idea. But I want to warn you again: intent is not always malicious, even if it appears to be. And the intent is not always present at all, even if everything points to the opposite. Never jump to conclusions, because that too can be a logical fallacy.

Emotions and psychological state

Many have noticed that the more intense the dispute between the interlocutors, the more logical mistakes both parties make. Here is an example of an anecdote on this topic.

The wife and husband are quarreling. The wife, in a rage, expresses everything that she has accumulated over the years of patience:

- You can’t do anything right, you have to ask for the same thing month after month, you can’t do anything at all, you’re just a moron! You are such a moron that if there was a competition for morons, you would take second place!

- Why the second?.. - the husband was offended.

- Yes, because you are a moron!

Anecdote is anecdote, but a reasonable person is still not so reasonable when emotions dominate him or he is in a state of passion. I think any of the readers was faced with an unfair accusation: someone addressed himself, and someone himself was such an accuser. Let's say you have lost some expensive thing, and there is only one person to whom you showed it. And now, suspicions begin to mature in my head that that person stole it, because it disappeared almost immediately after you showed it to him! In addition to this, various emotional distortions of reality are added: this person suddenly became somehow suspicious, somehow looks wrong, averts his eyes, for example, or avoids communication. All the circumstances connected with him suddenly begin, as it were, to suggest that it was he who stole the thing. And then you find it under the bed (the cat has driven it) - and then that person becomes innocent. And the fact that the thing disappeared right after its demonstration is easier to explain: you forgot to put it back right away, being distracted by something, left it on the table, and the cat climbed in and lent it to play.

The above example is a variant of cognitive distortion, when, under the influence of emotions, the correctness of thinking can be disrupted. Similar distortions can occur for other reasons, but we will look at this phenomenon later, in other chapters of the course. Another example of such distortion has recently spread widely on the Internet:

I even wrote an article on this subject "On the surface conclusions of the sofa thinker."

Another example:

A certain author seems to want to say that the people around you greatly influence you, and the better the environment, the better you will be. And this seemingly noble idea spread across the Internet, approved by so many people. But in fact, something else is written here: you are a rotten apple, and you are offered to go and spoil some collective with your presence, so that it begins to rot from your being in it. Emotions sometimes distort the meaning to the opposite, you start to see what you want, and not what is actually written, isn't it?

It is also pertinent to recall the saying “fear has large eyes”, which quite clearly shows the influence of emotions on the consistency of inferences.

Evidence and credibility (truth and plausibility)

Sometimes logical errors can be the result of a desire to be persuasive in the face of evidence. Moreover, colorful persuasiveness with the presence of errors is far from always worse than dry, but strict and error-free logic. What is persuasiveness in general? Let's go in order.

There are two important concepts: evidenceand persuasiveness … Evidence means the same as consistency or truth. That is, this is when we have a conclusion free from false data and errors. Persuasiveness is when the inference is plausible, i.e. seemswealthy, but not necessarily so. It is the speaker's job to be persuasive. Few will listen carefully to a person who says everything is absolutely correct, but difficult for most listeners. However, credibility does not require truth, only plausibility is enough. The task of the scientist is to ensure the validity of his conclusions, because science should strive to comply with the criterion of truth, even if at the expense of the expressiveness of thought.

Persuasiveness versus evidence can be applied in many areas. For example, in political debates. If a politician, entering the stage, strictly using graphs, diagrams, sophisticated analytics and other science begins to explain something to his audience, then he is unlikely to be heard, and it is unlikely that many people will vote for him. If a politician speaks brightly and colorfully from the rostrum, falls into the mood of the crowd, then they will vote for him with a greater probability, regardless of the reliability of his statements.

Plausibility contrary to truth is widely used in art. Remember the Russian cosmonaut Lev Andropov, who in the movie "Armageddon" (1998), which contains a lot of different nonsense, got out to meet the Americans in a hat with earflaps and a star on a T-shirt?

(Still from the film)

Can this be? Unlikely! But how else could one convincingly show the typical (for a Western man in the street) image of a Russian cosmonaut? If shown as it is, it will not be convincing. Then, with the blows of an adjustable wrench on the instrument cluster, accompanied by abuse, Lev corrected some severe error in the system.

(Still from the film)

Could this be in reality? No. But how plausible it is! If he just took the key and screwed the nut there, it would be more reliable, but it's boring!

We can wonder for a long time whether there was an intent to denigrate the Russians (who themselves laughed at themselves quite well, as I remember), but there are cases when plausibility is also used for more righteous purposes. Here is an example from A. Molchanov's book "How to write a script":

Once Stanislavsky invited a real village grandmother - a performer of folk songs for a small role in one of his productions. However, as soon as the grandmother appeared on the stage, she destroyed the whole world of the performance. She didn't play anything, didn't act, just did on stage what she did every day at home - some simple homework. Reality, like rust, corroded the director's drawing. The audience became uncomfortable. They immediately realized that they were in the theater, that they were being deceived. That a person on the stage is in circumstances unusual for him.

Artist Moskvin in rags on stage was believable. The artist Katchalov, who delivered tramp lines in a well-trained voice, was believable. The village grandmother on stage was implausible. She was not supposed to be here - this was the place for Moskvin and Kachalov. Stanislavsky deprived the grandmother of her words - the effect was the same. She silently appeared on the stage - and immediately the truth began. The grandmother was removed backstage, where she sang a short song - and the effect was the same. And the grandmother was removed altogether.

The reader has already noticed that we have moved somewhat away from the topic of logical errors and moved on to discussing the difference between truth and plausibility. But remember I said that there will still be a little philosophy? I believe that this lyrical digression successfully complements the logical theme, although it is connected with it only indirectly.

Lack of culture of thinking

This is another reason for the appearance of logical errors. A person may not be well educated enough (I mean not only formal academic education, but also life experience), his consciousness may be clogged with templates and cliches, as well as dogmas and stereotypes, and his logic of thinking may be too superficial and straightforward. Just one of these shortcomings is enough to become a source of a stream of errors.

Say, dogmatism can make you contradict itself. There is a dogma, a person does not question it. A situation arises in which dogma clashes with reality. A person tries in a plausible way to convince himself or others that the dogma remains true, and reality does not conflict with it.

Dogma is often associated with religion, they say, it is in religion that dogmas exist, and often people make a logical mistake, believing that since there are dogmas in religion, then it is initially vicious. There are dogmas both in science and in our everyday life, just few people notice it.

For example, a dogma is the belief in the objectivity of the surrounding world and its laws. You can argue with me, but you cannot prove the opposite either, because the subjective factor comes into force, whatever one may say.

In my life, I met with different scientists, and from one quite respected mathematician I heard this opinion, they say that it is impossible to prove a theorem using a computer, only what can be written with a pen (pencil) on paper is considered proven. Unfortunately, I was not able to convince him that there are formulas whose size exceeds millions of characters (I just worked on such), and I have to create a program that checks their truth based on the rules of mathematics. I don’t know why, but man could not go beyond the dogma of the need for strict manual proof without accepting the machine proof. He motivated this by the fact that one can make a mistake in the program, but not on paper, because "everything is before your eyes and everything is strict." What happened next? This scientist later recognized the possibility of computer proof when my scientific research was approved by higher authorities than him. Then he agreed with me and also approved my work, even invited me to work in his laboratory.

I will not name names so as not to offend anyone, but I think the reader does not need any confirmation of my words, because he himself must have come across situations when a wise person, maybe even an elderly person, for some unknown reason, insists on obvious absurdity.

Stereotypes can also lead to errors. For example, consider the Jewish question. A person who is superficially familiar with the topic may, having seen a person with Jewish facial features, inadvertently endow him with a number of negative qualities attributed to Jews. From this, he can begin to draw incorrect conclusions, for example, a priori consider a person greedy for money, using the widespread myth about the universal greed of the Jews.

You can see another example related to the lack of a culture of thinking in political debates. For example, one presidential candidate pulls a "trump card out of his sleeve" - a certain act of his rival, done by that ten years ago and declares: "How can a person be president who did this and said this !?" Of course, this fact can cause disgust among the crowd and the authority of the exposed person will plummet. The candidate who has exposed the opponent will rub his hands victoriously. At the same time, few people will ask the question of whether anything has changed in 10 years, and whether that act is connected at all with the ability to govern the state, because you must admit that in childhood everyone went to the toilet in their pants. More modern examples can be sliced from the Clinton-Trump debate. Let's say right here. I did not find in this collection of any logical arguments of either side. However, on my part, it would be a logical mistake to consider both (at that time) presidential candidates as people with an undeveloped culture of thinking. It is quite possible that they were only playing a game in which it is customary to please the audience with various emotional attacks towards the opponent.

Straightforward or superficial thinking can also lead to logical errors. For example, because of a hasty judgment, taking the first impression on faith, you can draw the wrong conclusions. An example is shown in this video:

Deception of the senses and imperfect thinking

In mathematics, there is the concept of "geometric proof", which has a right to exist. The essence of the proof is that a certain geometric figure is being constructed that in an obvious way reflects the assertion being proved. Either immediately, or with the help of some additional calculations associated with this figure, the desired consistent conclusion is obtained. For example, here's a slide with a geometric proof of the complete square formula

(a + b)²= a²+ 2ab + b²

You don't have to study the picture in detail, everything is correct there: based on the picture, we calculate the areas of the internal figures and the total area of the entire square figure. Since the area of a square is the sum of the areas of its parts, the final formula is obtained.

However, our senses are imperfect, and such evidence in some cases may turn out to be incorrect. Here's a classic example:

Shown is a square with a side of 8. It was cut into 4 pieces and folded in a different order. We got a rectangle with sides 13 and 5. The area of the square was 8 × 8 = 64, and the area of the resulting rectangle was 13 × 5 = 65. Where did the extra unit of area come from?

In fact, if you do this operation carefully, you will notice that a very long, but narrow "hole" is formed in the center of the figure, the area of which will be that extra unit. It is very difficult to cut everything out of paper so evenly and fold it in order to notice such a "hole". But she is:

Our imperfect consciousness is not always able to notice such trifles in what previously seemed obvious. Deception of the senses, such as vision, can occur especially frequently. The brain tries to interpret the spots of color seen in a familiar way, but sometimes it turns out not what it wants. Here's another classic example:

This is actually dolphins jumping into the sea, not a hugging couple. They say that children just see these dolphins right away, but adults do not.

And here I have a question related to the upbringing of children. Have parents thought about how their parenting logic will affect the child? For example, a mother says to her son: “if you don’t wash your face, Moidodyr will come and eat all your sweets!”. Obviously, the logic is broken, but the child does not understand this, this logic seems to him quite real. Later, he begins to notice that Moydodyr still does not eat candy, if he does not wash his face … and no other arguments in favor of washing were offered. So you don't have to wash your face anymore! And my mother, it turns out, can lie! And let someone think that in adulthood a person will still understand everything, my personal practice shows that this does not always happen. Here is an example of superstition: “If I don’t surpass my left shoulder now, then…” Does it not resemble Moidodyr’s logic? However, behind some superstitions there may be a certain correct meaning, unconscious of a person, but analysis of this topic will lead us into the jungle of primitive culture, and this is not included in my plans now.

Language reasons

These are the reasons associated with the peculiarity of expressing thoughts in natural language. For instance, ambiguity … Remember the famous statement of Alexander Grigorievich Lukashenko:

You will live badly, but not for long

A situation may arise when a sentence is intended to play on emotions, while its real meaning is not defined at all. Here is an example from the monologue of a court speaker (there are other similar examples here):

The rise in crime depends on how persistently and effectively the fight against offenders is carried out

That is, the higher the efficiency, the stronger the growth? Here, the premise and the consequence are generally inconsistent, but for a "catchphrase" and greater persuasiveness it is suitable.

Also this includes play on words … Once on an exam, I saw this picture. The teacher says to the student who answered the question:

- I rate it as "good".

- And why "good", because I told everything correctly! You didn't even ask questions.

- Well, you told everything well, didn't you? - the teacher clarified.

- Yes! - answered the student, convinced of the righteousness.

- Well, since they told it well, then the assessment should be “good”! - concluded the teacher.

It was "iron logic" in the arsenal of a teacher of mathematical analysis. Of course, the student failed to convince him.

Language is ambiguous and is not a perfect means of transmitting thought, and therefore logical errors can arise not only due to the illiteracy of the speaker (writer), but also due to the illiteracy of his listener (reader). The inability to read correctly is a separate topic of conversation related to culture in general.

Outcome

Today you learned about where logical errors can come from. Let me briefly recall the list of reasons: intent (both malicious and not, for example, the desire to be convincing), emotions and psychological state (including cognitive distortions), lack of a culture of thinking (straightforward thinking, hasty conclusions), deception of the senses, imperfection of thinking, as well as linguistic reasons.

Homework

REGULATIONS: You do your homework exclusively for yourself. You can do it, or you can not do it, but in any case, I ask you NOT to discuss these tasks in the comments, unless you found a glaring mistake on my part in their wording (and if you are sure that I did not do it on purpose). The reference (but not necessarily correct) solution of all problems will be described in the next chapter of the course.

In addition to the correct answer to the question of the problem, I ask you to additionally think about the philosophical component of each problem and your answer to it. I always give life-related tasks, but this is not always obvious.

Problem 1

Two arguments are given: “all the coins in my pocket are gold” and “I put a coin in my pocket”. Does it follow from this that "the coin put in the pocket will become gold"?

Task 2

Consider a typical example of an unsuccessful student returning home from school, parents start scolding their son.

Act I

- You got a deuce again?

- But there was a difficult job, everyone did a bad job!

- We are not interested in what everyone has, we are interested in what you have! Take responsibility for yourself!

Act II

- Well, what is the control?

- "Three".

- Why “three”, everyone got “four” and “five”, and you - “three” ?!

Both acts took place in the same family with the same child. Find the logical error of the parents and try to explain the reason for its occurrence, which is the most probable, in your opinion.

Problem 3

A moderate alcohol drinker's argument might be:

"Wine is made from grapes, and grapes are good for the heart, so drinking wine is good." What is the error and what is its cause? Do you think the moderate drinker himself knows about this mistake?

Problem 4

One person on a forum on the Internet proves his point of view to another, there is a long exchange of views, but at some point the interlocutor stopped responding. “I won,” the first thinks, “I wrote everything to him so clearly that he cannot object, so I’m right!” The question is the same: what is the error and what is its cause?

Problem 5

The person blames the other for something that he is not really to blame. However, the second cannot prove his innocence and blushes. "Yeah, an honest man will not blush when he is scolded, then you are to blame!" The question is still the same …