Gospel afterword translation.  Sergievsky Church

All these theories are united by the desire to catalog such methods of reasoning that from true judgments-premises lead to true judgments-consequences; Cataloging is carried out, as a rule, within a logical framework. calculus. Special role in acceleration scientific and technological progress applications of logic play in computational mathematics, automata theory, linguistics, computer science, etc.

What is mathematical logic? MATHEMATICAL LOGIC, deductive logic, including mathematical methods research into methods of reasoning (inferences); mathematical theory deductive methods of reasoning. Mathematical logic is also called the logic used in mathematics.

Task. Three friends Belova, Krasnova and Chernova met. One of them had black dress, on the other - red, on the third - white. A girl in a white dress says to Chernova: “We need to change dresses, otherwise the color of the dresses for all three of them does not match their surnames.” Who was wearing what dress?

There is an opinion that a person can think correctly without knowing the exact rules and laws of logic, using them only on an intuitive level. After all, there are musicians who play some kind of musical instrument, without knowing musical (in particular, musical notation) reading.
But such musicians are limited in their creativity: they can neither perform a piece written down using notes, nor write down a melody they composed.

A person who has mastered logic thinks more clearly, his arguments are more convincing than those of someone who does not know logic. He makes mistakes much less often and is mistaken. But a misconception that led, for example, to a simple error in calculations during design spaceship, will then lead to an accident. Their delusions cost people dearly!

Logical thinking is not innate, so it can and should be developed different ways. The systematic study of the science of logic is one of the most effective ways development of logical abstract thinking.

A specific technique for developing logical thinking is solving logical problems.

Thus, the American mathematician R. Smullian, the author of many ingenious problems, offers the following: “One man was tried for participation in a robbery, the prosecutor and defense attorney stated the following during the trial.

Prosecutor: If the defendant is guilty, then he had an accomplice.
Defender: Wrong!
The defense attorney couldn't have said anything worse.

Why? Having answered the question posed, you can once again be convinced of the need for Everyday life correct, clear verbal formulation of thoughts.

Or this example.

An employee of one of the enterprises lost his wallet. Only one of the five employees of the department where the victim worked could steal the wallet: Alexey, Leonid, Dmitry, Tatyana or Olga.

When interviewing these employees, each of them gave the following answers:
Alexey: I didn’t take the wallet; I have never taken someone else's in my life; Tatyana did it.
Leonid: I didn’t take the wallet; my father is quite rich; Besides, I make good money myself.
Dmitry: I don't know anything about the theft; I did not know Olga before joining this company; Tatyana did it.
Tatyana: I'm innocent; Olga did it; Alexey lies, claiming that I did it.
Olga: I didn’t take the wallet; Leonidas is guilty of this; Dmitry can vouch for me, since he has known me since school..

Subsequently, each of the suspects admitted that only two of his three statements were true. This was enough for Inspector Borisov, who led the investigation into this theft, to immediately name the culprit. Who is to blame?

An interesting technique for developing thinking is thinking about paradoxes. This is without a doubt one of the best tests our logical abilities and one of the most effective means their training.

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 and its proposed solutions evaluated. It is difficult to claim to invent new solutions logical paradoxes, but already familiarization with their proposed solutions is a good school of practical logic.

The paradox of the rural barber

Let's imagine that the council of one village defined the duties of the barber of this village as follows: to shave all the men of the village who do not shave themselves, and only these men. Should this barber 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, of course; impossible. Could such a hairdresser exist?

Is there a taxi driver who transports all those and only those who do not drive a car themselves?

The Liar Paradox

In the simplest version of the “liar,” a person utters only one phrase: “I am lying.” Is it possible to determine whether this proposition is true or false?

The traditional laconic formulation of this paradox is that if a liar says that he is lying, then he is both lying and telling the truth.

Is the statement “The statement now expressed is false” true or false? If it is true and claims to be false, then it is false. If it is false and claims to be false, then it is true.

What a paradox!

Logic is not just a sequence of reasoning, but also a whole science that is studied and researched. If curiosity or necessity led you to this textbook, and there is little time and energy for activities that do not bring visible benefits, then a reasonable question arises: why is logic needed?

Getting to know logic will allow you, first of all, to learn to think accurately and express your thoughts. Even if you previously spoke incoherently and vaguely, logic will contribute to the formation of clear and coherent speech.

Thanks to this science, you will gain the ability to logically justify your ideas and decisions, and you will be able to convince other people that you are right. This skill can be useful in any field of activity; it will be the key to your success on the career ladder.

Gradually, logic will form in you the habit of analyzing both your own and other people’s reasoning, arming you with the means to identify and eliminate errors in conclusions, and cope with sophistry and demagoguery. Let’s say someone turns to you with the following words: “I am a person, but you are not me, which means you are not a person.” Can you respond to this with dignity, even if you feel that something is wrong here? Many will be confused and answer “I’m a fool.” But, knowing the basics of logic, you will be able to find and identify a violation, respond with dignity and shame the ignoramus.

Logic will teach you the art of argument, which will be useful in professional activity, and in everyday life. The inability to argue often leads to shouting, bickering, and even a fight. A logical basis for your opinion will help you find a compromise, refute your opponent’s erroneous beliefs, and expose dishonest tricks and techniques.

In a difficult life situation, when emotions overwhelm the mind, you will be able to sort it out and put everything “on the shelves” and “restrain” emotions. Then, when all the root causes and sources of the problem are clear, just start getting out of difficult situation using logic and psychology.

Surely among your friends there are many manipulators who cleverly arrange their affairs at your expense. Familiarity with the basics of logic will allow you to recognize their intentions in time and easily deal with them.

There is a lot of unverified information around today, you get it from advertising, television, from the Internet, from strangers. You need to learn to recognize a lie in time and protect yourself from it; logic can also help you with this.

Everything interesting

Thinking is the object of study in various scientific disciplines. Sections of logic, philosophy, psychology, genetics, linguistics and other sciences try to understand human thought processes and answer questions about what thinking is, what...

The word logic is translated as “the art of reasoning” and implies the ability to analyze information and draw conclusions, and solve problems based on it. Sometimes you can hear that women have no logic; there is also the expression “female...

Formal logic is a science that deals with the construction and transformation of statements. The objects of a statement, as well as its content, are not taken into account by formal logic: it deals only with form, which is why it is called that.
In history…

It is believed that the father of the European school of logic was Aristotle. It was to him that the first steps were taken to systematize and substantiate the main logical laws, as well as the forms and rules of logical constructions. Logic as an academic discipline

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The issue of male and female logic remains quite “hot” for a long time. Men believe that women have a rather strange logic or none at all, while women are sure that they have secret knowledge, inaccessible to men...

In the era of industrial and technological progress, philosophy has faded into the background; not every person can clearly answer the question of what kind of science it is and what it does. People are busy with pressing problems, they are of little interest...

Rhetoric is the science of oratory, more in the narrow sense– the ability to logically and competently express one’s thoughts and convince the interlocutor. This science was included in the compulsory course of subjects studied in Russian gymnasiums, but after the Revolution this knowledge...

The ability to think logically helps a person to see the essence of things, problems and events that he encounters every day in various situations. Logical thinking can be developed to a certain extent. And if you want to help your child with...

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Logical thinking is a quality that must be constantly developed. It will allow you to easily solve a wide variety of problems, from everyday issues to the most complex ones, which are characterized by great responsibility and...

Logical thinking - the ability to reason and draw conclusions - can and should be developed in children from the very beginning. early age. The ability to think logically, analyze and draw conclusions will be useful both in childhood and in life. adult life. Development of logical...

Every day we are faced with many tasks, the solution of which requires our ability to think logically. Logic as the ability to think and reason consistently and consistently is required in many life situations, starting from solving complex technical and business problems, ending with persuading interlocutors and making purchases in a store.

But despite the high need for this skill, we often commit logical errors, without even knowing it. Indeed, among many people there is an opinion that it is possible to think correctly on the basis life experience and so-called common sense, without using laws and special techniques " formal logic" For performing simple logical operations, expressing elementary judgments and simple conclusions, it may also be suitable common sense, and if we need to know or explain something more complex, then common sense often leads us to errors.

The reasons for these misconceptions lie in the principles of development and formation of the foundations of logical thinking in people, which are laid in childhood. Teaching logical thinking is not carried out purposefully, but is identified with mathematics lessons (for children at school or for students at the university), as well as with solving and passing a variety of games, tests, tasks and puzzles. But such actions contribute to the development of only a small proportion of logical thinking processes. In addition, they explain to us the principles of finding solutions to tasks in a rather primitive manner. As for the development of verbal-logical thinking (or verbal-logical), the ability to correctly perform mental operations, consistently come to conclusions, for some reason we are not taught this. That is why the level of development of people's logical thinking is not high enough.

We believe that a person’s logical thinking and his ability to cognition should develop systematically and on the basis of a special terminological apparatus and logical tools. During the classes of this online training, you will learn about self-education methods for the development of logical thinking, get acquainted with the main categories, principles, features and laws of logic, and also find examples and exercises for applying the acquired knowledge and skills.

What is logical thinking?

To explain what “logical thinking” is, let’s divide this concept into two parts: thinking and logic. Now let's define each of these components.

Human thinking- This mental process processing information and establishing connections between objects, their properties or phenomena of the surrounding world. Thinking allows a person to find connections between the phenomena of reality, but so that the connections found actually reflect true position affairs, thinking must be objective, correct or, in other words, logical, that is, subject to the laws of logic.

Logics translated from Greek has several meanings: “the science of correct thinking”, “the art of reasoning”, “speech”, “reasoning” and even “thought”. In our case, we will proceed from the most popular definition of logic as a normative science about the forms, methods and laws of human intellectual mental activity. Logic studies ways to achieve truth in the process of cognition in an indirect way, not from sensory experience, but from knowledge acquired earlier, therefore it can also be defined as the science of ways to obtain inferential knowledge. One of the main tasks of logic is to determine how to come to a conclusion from existing premises and gain true knowledge about the subject of thought in order to better understand the nuances of the subject of thought being studied and its relationships with other aspects of the phenomenon under consideration.

Now we can define logical thinking itself.

This is a thought process in which a person uses logical concepts and constructions, which is characterized by evidence, prudence, and the goal of which is to obtain a reasonable conclusion from existing premises.

There are also several types of logical thinking; we list them, starting with the simplest:

Figurative-logical thinking

Figurative-logical thinking (visual-figurative thinking) - various thought processes of the so-called “imaginative” problem solving, which involves a visual representation of the situation and operating with images of its constituent objects. Visual-figurative thinking, in fact, is synonymous with the word “imagination”, which allows us to most vividly and clearly recreate the whole variety of different actual characteristics of an object or phenomenon. This type human mental activity is formed in childhood, starting from approximately 1.5 years.

To understand how developed this type of thinking is in you, we suggest you take the IQ Test “Raven’s Progressive Matrices”

The Raven's Test is a progressive matrix scale for assessing IQ, mental ability, and logical thinking, developed in 1936 by John Raven and Roger Penrose. This test can give the most objective assessment of the IQ of the people tested, regardless of their level of education, social class, type of activity, language and cultural characteristics. That is, it is possible with high probability assert that the data obtained as a result of this test in two people from different points the world will evaluate their IQ equally. The objectivity of the assessment is ensured by the fact that this test is based solely on images of figures, and since Raven's matrices are among non-verbal intelligence tests, its tasks do not contain text.

The test consists of 60 tables. You will be offered drawings with figures connected to each other by a certain relationship. One figure is missing; it is given at the bottom of the picture among 6-8 other figures. Your task is to establish a pattern connecting the figures in the picture and indicate the number the right figure by choosing from the options provided. Each series of tables contains tasks of increasing difficulty, while at the same time, the complication of the type of tasks is observed from series to series.

Abstract logical thinking

Abstract logical thinking- this is the completion of a thought process with the help of categories that do not exist in nature (abstractions). Abstract thinking helps a person to model relationships not only between real objects, but also between abstract and figurative representations which thinking itself created. Abstract logical thinking has several forms: concept, judgment and inference, which you can learn more about in the lessons of our training.

Verbal and logical thinking

Verbal and logical thinking (verbal-logical thinking) is one of the types of logical thinking, characterized by the use linguistic means and speech structures. This type of thinking requires not only the skillful use of thought processes, but also competent command of one’s speech. We need verbal and logical thinking for public speaking, writing texts, arguing and in other situations where we have to express our thoughts using language.

Applying logic

Thinking using the tools of logic is necessary in almost any field human activity, including exact and humanities, in economics and business, rhetoric and oratory, in creative process and invention. In some cases, strict and formalized logic is used, for example, in mathematics, philosophy, and technology. In other cases, logic only supplies a person useful techniques to obtain a reasonable conclusion, for example, in economics, history, or simply in ordinary “life” situations.

As already mentioned, we often try to think logically on an intuitive level. Some people do it well, some do it worse. But when connecting the logical apparatus, it is better to know exactly what mental techniques we use, since in this case we can:

• It is more precise to choose the right method that will allow you to come to the right conclusion;
• Think faster and better - as a consequence of the previous point;
• It is better to express your thoughts;
• Avoid self-deception and logical fallacies,
• Identify and eliminate errors in other people’s conclusions, cope with sophistry and demagoguery;
• Use the necessary argumentation to convince your interlocutors.

The use of logical thinking is often associated with quickly solving logic tasks and passing tests to determine the level of intellectual development(IQ). But this direction is related to to a greater extent with bringing mental operations to automatism, which is a very insignificant part of how logic can be useful to a person.

The ability to think logically combines many skills in the use of various mental actions and includes:

1. Knowledge theoretical foundations logic.
2. The ability to correctly perform such mental operations as: classification, specification, generalization, comparison, analogy and others.
3. Confident use of key forms of thinking: concept, judgment, inference.
4. The ability to argue your thoughts in accordance with the laws of logic.
5. Ability to quickly and effectively solve complex problems logic problems(both educational and applied).

Of course, such operations of thinking using logic as definition, classification and categorization, proof, refutation, inference, conclusion and many others are used by every person in his mental activity. But we use them unconsciously and often with errors, without a clear idea of ​​the depth and complexity of those mental actions that make up even the most elementary act of thinking. And if you want your logical thinking to be truly correct and rigorous, you need to learn this specifically and purposefully.

How to learn this?

Logical thinking is not given to us from birth, it can only be learned. There are two main aspects of teaching logic: theoretical and practical.

Theoretical logic , which is taught at universities, introduces students to the basic categories, laws and rules of logic.

Practical training aimed at applying the acquired knowledge in life. However, in reality modern training practical logic is usually associated with the passage different tests and solving problems to test the level of intelligence (IQ) and for some reason does not affect the application of logic in real life situations.

To truly master logic, you need to combine theoretical and applied aspects. Lessons and exercises should be aimed at developing intuitive, automated logical tools and consolidating the acquired knowledge in order to apply it in real situations.

Based on this principle, the online training that you are reading now was compiled. Target this course- teach you to think logically and apply logical thinking methods. Classes are aimed at introducing the basics of logical thinking (thesaurus, theories, methods, models), mental operations and forms of thinking, rules of argumentation and laws of logic. In addition, each lesson contains tasks and exercises to train you to use the acquired knowledge in practice.

Logic lessons

Having collected a wide range of theoretical materials, as well as having studied and adapted the experience of teaching applied forms of logical thinking, we have prepared a series of lessons for the full mastery of this skill.

We will devote the first lesson of our course to a complex, but very important topic- logical analysis of language. It’s worth mentioning right away that this topic may seem abstract to many, loaded with terminology, and inapplicable in practice. Don't be scared! Logical analysis of language is the basis of any logical system and correct reasoning. The terms that we learn here will become our logical alphabet, without knowledge of which we simply cannot go further, but gradually we will learn to use it with ease.

A logical concept is a form of thinking that reflects objects and phenomena in their essential features. There are concepts different types: concrete and abstract, individual and general, collective and non-collective, irrespective and correlative, positive and negative, and others. Within the framework of logical thinking, it is important to be able to distinguish these types of concepts, as well as produce new concepts and definitions, find relationships between concepts and perform special actions on them: generalization, limitation and division. You will learn all this in this lesson.

In the first two lessons we said that the task of logic is to help us move from an intuitive use of language, accompanied by errors and disagreements, to a more orderly use of it, devoid of ambiguity. The ability to handle concepts correctly is one of the skills required for this. Another equally important skill is the ability to correctly define. In this lesson we will tell you how to learn this and how to avoid the most common mistakes.

Logical judgment is a form of thinking in which something is affirmed or denied about the surrounding world, objects, phenomena, as well as relationships and connections between them. Judgments in logic consist of a subject (about which we're talking about in a judgment), predicate (what is said about the subject), connective (what connects the subject and predicate) and quantifier (the volume of the subject). Judgments can be of various types: simple and complex, categorical, general, particular, individual. The forms of connectives between the subject and the predicate also differ: equivalence, intersection, subordination and compatibility. In addition, within the framework of composite (complex) judgments there can be their own connectives, which define six more types of complex judgments. The ability to think logically presupposes the ability to construct correctly different kinds judgments, understand them structural elements, signs, relationships between judgments, and also check whether a judgment is true or false.

Before moving on to the last third form of thinking (inference), it is important to understand what logical laws exist, or, in other words, objectively existing rules building logical thinking. Their purpose, on the one hand, is to help build inferences and argumentation, and on the other hand, to prevent errors and violations of logic associated with reasoning. this lesson will cover the following laws formal logic: the law of identity, the law of the excluded middle, the law of contradiction, the law of sufficient reason, as well as De Morgan's laws, the laws of deductive reasoning, Clavius' law and the laws of division. After studying the examples and doing special exercises, you will learn to use each of these laws purposefully.

Inference is the third form of thinking in which from one, two or more propositions, called premises, a new proposition, called a conclusion or conclusion, follows. Inferences are divided into three types: deductive, inductive and analogical inferences. In deductive inference (deduction), a conclusion is drawn from a general rule for a particular case. Induction is inference in which, from several particular cases, one can deduce general rule. In inferences by analogy, based on the similarity of objects in some characteristics, a conclusion is drawn about their similarity in other characteristics. In this lesson you will become familiar with all types and subtypes of inferences and learn how to build various cause-and-effect relationships.

This lesson will focus on multi-premise inferences. Just as in the case of single-premise conclusions, all the necessary information in a hidden form will already be present in the premises. However, since there will now be many premises, the methods for extracting them become more complex, and therefore the information obtained in conclusion will not seem trivial. In addition, it should be noted that there are many different types of multi-premise inferences. We will focus only on syllogisms. They differ in that both in the premises and in the conclusion they have categorical attributive statements and, based on the presence or absence of some properties in objects, they allow one to draw a conclusion about the presence or absence of other properties in them.

In previous lessons we talked about various logical operations that form an important part of any reasoning. Among them were operations on concepts, definitions, judgments and inferences. So, on this moment It must be clear what components the reasoning consists of. However, we have not yet touched upon the questions of how reasoning as a whole can be organized and what types of reasoning there are in principle. This will become the topic last lesson. Let's start with the fact that reasoning is divided into deductive and plausible. All types of inferences discussed in previous lessons: inferences using a logical square, appeals, syllogisms, enthymemes, sorites, are precisely deductive reasoning. Their hallmark consists in the fact that the premises and conclusions in them are connected by a relation of strict logical implication, while in the case of plausible reasoning there is no such connection. First, let's talk more about deductive reasoning.

How to take classes?

The lessons themselves with all the exercises can be completed in 1-3 weeks, having mastered theoretical material and with a little practice. But to develop logical thinking, it is important to study systematically, read a lot and constantly train.

For maximum effect, we recommend that you first simply read all the material, spending 1-2 evenings on it. Then take 1 lesson daily, doing the necessary exercises and following the suggested recommendations. After you have mastered all the lessons, engage in effective repetition in order to remember the material for a long time. Next, try to apply logical thinking techniques more often in life, when writing articles, letters, when communicating, in disputes, in business, and even in your leisure time. Reinforce your knowledge by reading books and textbooks, as well as using additional material, which will be discussed below.

Additional material

In addition to the lessons in this section, we tried to select a lot of useful material on the topic under consideration:

• Logic problems;
• Tests for logical thinking;
• Logic games;
• The most smart people Russia and the world;
• Video lessons and master classes.

As well as books and textbooks, articles, quotes, auxiliary trainings.

Books and textbooks on logic

On this page we have selected useful books and textbooks that will help you deepen your knowledge of logic and logical thinking:

• "Applied Logic". Nikolai Nikolaevich Nepeyvoda;
• "Textbook of Logic". Georgy Ivanovich Chelpanov;
• "Logic: lecture notes." Dmitry Shadrin;
• "Logics. Training course" (training and metodology complex). Dmitry Alekseevich Gusev;
• “Logic for Lawyers” (collection of problems). HELL. Getmanova;