Basic logical forms. How to develop logical thinking

It is very important to learn to think logically - such skills help not only to calculate your actions in advance, but also quickly find a way out in unforeseen situations. A person who has developed logical thinking will undoubtedly be able to succeed even under not the most favorable circumstances. So, how to develop logical thinking?

Logic and logical thinking

Logic is

What is logical thinking and its types

Developed memory and logical thinking are simply necessary in Everyday life– when performing professional tasks, as well as in household chores. Developing memory and logic, you will learn to concentrate your attention, control the train of thought. As a result, it will be easier for you to focus on current tasks and organize own life. Performing exercises for the development of memory, after a certain period of time you will begin to notice that when solving problems, you use very non-standard and effective methods that you would not have thought of before.

How to develop memory and thinking in adults

Tasks for the development of logical thinking

Learn to think logically with games, puzzles and strategies

10 ways to improve your memory

For 5 minutes, study any object, or landscape. Closing your eyelids, try to reproduce the image of this picture in your memory. It is also important that it be in color. The effectiveness of the training can be increased by fixing objects on a piece of paper. If you perform this exercise regularly, you can achieve considerable progress in the development of visual memory.

How to develop logic and thinking in a teenager

Online logic test - develop the ability to think logically

Test:

If you did: No more than 2 errors Your logical thinking is on top! You must have put in a lot of work to achieve such results, but it was worth it! 3 to 5 mistakes You can safely say that you have high level intelligence, despite the fact that, at times, you are prone to mistakes. Be that as it may, you are able to analyze most events correctly. 6 to 7 mistakes Your level of intelligence can be called average. The potential for growth is certainly there, and if you wish, you will succeed in this business. 8 or more errors It is difficult to say that you are often guided by logic in your judgments. Most likely, you act mainly under the influence of emotions.

Books that develop logical thinking

This book can be considered a simple introduction to the world of modern mathematical logic. The textbook was recommended by the State Committee of the Russian Federation for higher education, and has become an excellent source of knowledge for many university students.

This is a book that lays out the basics of logic in an accessible way. The main emphasis is on syllogistic. The textbook contains tasks with the help of which the student is able to learn to apply certain rules in practice in a short time.

A tutorial that talks about how to learn to think logically - who finds it difficult, and who easily overcomes barriers. Contains useful tasks.

Ten-minute exercises for the development of logic for every day

Feb 24 2016

The ability to solve problems in the mind and draw conclusions about phenomena indirectly allows logical thinking. A person does not even think about how much logic is a valuable acquisition in the course of the development of the intellect. What is logic? Logics- this is the science of the correctness of judgments, which includes the rules for observing the sequence of real facts, evidence, checking for the presence or search for arguments.

Logic gives the ability to confirm and prove one's theories, to competently answer opponents in a dispute. At the initial stage of development in school, the ability to think logically is equated to the ability to solve problems in mathematics. It is from mathematical operations that the child learns to abstract from concrete material and to connect abstractions with each other. Logic, figuratively speaking, cleans the concrete meaning from the information and brings the thought to an elementary formula.

What is logic?

The action of logical transformations in the mind stands out as a type of thinking. In this case, logic is a process in time, the way the mind builds connections between real objects. Such connections are more stable and substantive than those connections that are formed within the framework of simple perception. Connections are made not only between individual phenomena of reality, but also between words and whole sentences, which represent a graphic image of thought.

In addition, logic is involved in the creation of abstract concepts.

concept- this is an abstract entity, it combines several objects (or objects of reality) at once. The content of the concept becomes a generalized feature, which manifests itself to varying degrees in all these objects.

For example, the concept of "living organism" may include plants and animals, which are united by the presence of organic nitrogenous compounds ( nucleic acids). Further, the concept of "plant" includes any plant (rose, fern, Christmas tree). Then this chain can be decomposed into specific representatives of the genus - "flowering", "algae", "mosses". Thus, low-level concepts are raised as a result of generalization to higher ones. For example, to the concept of “life” in general, based on the concepts of “recreating oneself” and “energy exchange”.

Such a multi-level hierarchy of concepts forms a system of ordered knowledge, any phenomenon in its place, like books in a library. Unlike a word, it does not have clear boundaries of meaning. It cannot be expressed in one word. But it helps to better master the information and what is at stake, eliminates ambiguity in the exchange of facts and is inherent only to a living subject. Concepts are formed within the framework of any specific system. For example, within the framework of the science of sociology, there are concepts: "family", "city", "society" and so on.

Obtaining abstract units and linking them to each other begins with two main logical operations - analysis and synthesis. Analysis- this is the decomposition of the phenomenon of reality, object or information into elementary units. In the process, it is determined what and how the object consists, what lies in its essence, how the parts of the whole relate to each other.

With Wikium, you can organize the process of developing memory and logical thinking according to an individual program

Synthesis is an association various elements. For example, the union of two objects into one concept, or the connection of parts of objects to obtain a new abstraction or model of reality. good example demonstration of the concept synthesis” may be the unification of all sensory signals of the body into one meaning, into a component of consciousness. However, logic, as the ability of the mind, is engaged in combining ready-made meanings into judgments, and judgments into inferences. Although, by its nature, the brain (mind) seeks to combine everything into a coherent picture of consciousness, and only logic helps to achieve the correctness of its perception.

Logic is engaged in the search for true knowledge, the identification of correct ideas about reality with the state of affairs in the world.

Language is the main sign system and a tool with which you can see and feel the reflection of logical connections.

Sign- this is a dual entity, which consists of a form (sound, graphic), perceived with the help of the senses and its meaning or content. These two sides of the sign have an associative, conditional correspondence between them, formed in the process of communication between people and their cultural development. A sign can be a single word, a phrase, a complete sentence, and even a whole text.

Each sign has its own designat, that is, what this sign stands for. Under designatum a real thing is understood - a specific person, essence, object, its interpretation and concept. The relationship between sign and designatum is called meaning- what property or attribute of an object is meant by its phonetic shell. The specific item gets practical value in one situation or another. For example, the word “fire” means both “heat”, “light”, and “fire”. The concept of "warmth" has in its content both "warmth" from fire, and "warmth" from the human body and the metaphorical meaning of "warmth" of the soul. Each meaning is included in the content of the concepts of each of them.

Two or more signs in one situation (context) form syntactic links that allow you to realize one of the meanings of the sign at a more specific level (lexical) and get a detailed idea of ​​the world. Another type of connection between a sign and a designatum in relation to the subject is pragmatic, which relates to a specific situation and how the speaker understands it.

With the help of language, you can build any sentences (in logic - judgments), even those that will not make sense in the real world. Language in this regard does not care about the correctness of ideas and thoughts.

For example, the sentence "green thoughts sleep furiously" may be meaningless from the point of view of logic, however, it complies with all the rules of the grammar of the language and is recognized on the basis of elementary meanings. Also, interrogative, exclamatory sentences are built in the language, which go beyond the limits of formal logic and mean various human emotions. They are neither true nor false, so they are of no value to logic.

Some language theories put forward versions that any, even the most absurd sentence, can get its meaning with the help of imagination. For example, there is a theory about parallel worlds: conceptually, it means that you should not discard a meaningless assumption, but try to imagine a world in which it will have a real meaning.

Logic, in contrast to the language system, deals with the consideration of affirmative sentences that correlates with real facts. Such proposals are called true judgments.

Stages of development of logical thinking in humans

Logical thinking is classified according to the stages of development, and is also divided into types depending on the predominance of one or another element of consciousness:

1. Formation of logic begins with visual-effective thinking. On the early stage young children lack stable logical connections. In this case, the thought process is based on a real situation - the construction of words from cubes, figures from a constructor.
2. The second stage in the development of logical thinking is visual-figurative develops during the preschool period. At this stage, there is a separation of specific images from the real object. The child operates not with real objects, but with the images of these objects recalled from memory. At this stage, there is still no analysis, the image of the object is not divided into components.
3. The next stage in the development of logic occurs in the primary school period. At this stage of development, all practical actions transformed into an internal thought process. Child school age successfully captures the elementary connections, similarities and differences of objects. Thinking reaches the abstract level, there is the ability to ignore the specific properties of objects and combine them into categories, classes.

How to develop logical thinking?

Mind games contribute to the development of logical thinking.

1. Chess, poker and the like are some of the best training methods for the mind.
2. Use of consonant words making up rhymes can be an exercise in the development of logical thinking. Popular English game limericks- inventing absurd rhymes. Also inventing a parody rhyme of a popular verse or song. Excellent parodies are poems from Carroll's book "Through the Looking-Glass".
3. Another exercise in the development of logic can be retelling or paraphrasing a sentence and text . Try to highlight the deep, abstract meaning and designate it in other words. Try to collapse into one word or expand into several of the same meaning.
4. Analogy game. Take any object - structure, try to see its essence (meaning). Try to imagine this object or meaning in another system. For example, take the characters of your friends and try to imagine them as chemical elements: "gold" - rich, "lead" - lazy, "arsenic" - malicious, harmful, and so on.
5. Suitable for the development of logic solving crosswords, puzzles and related computer games , including online simulators.
6. The development of intellectual abilities is influenced by training for combining any words into classes, or detailed object detailing . For example, take a few words: "fish", "square", "mug", "weather" and consider them in detail, what elements they consist of and what they can be associated with. “Square” is “straight”, “angle”, “parallel lines”, “plane”. "Weather" - "atmosphere". Use the matrix of connections (relationships of the word): cause-effect, part-whole, species-genus, sequence, opposite.
7. Engage in the study of explanatory dictionaries, invent own interpretations phenomena.
8. For improvement verbal-logical thinking psychologists recommend keep a diary entry . Concrete your thoughts with their help. When reading any information (articles, books), try to write down all the new knowledge.
9. Reading philosophical treatises and scientific books also improves logic, structures thoughts.

Again, we note that only regular exercises and constant training in this direction will give the expected result.

The development of logical thinking contributes to the improvement of a person's ability to reason and think consistently and consistently. Read more about the development of logical thinking.

Logical thinking and logic

Logic is the science of the forms, methods and rules of mental cognitive activity.

Logic is necessary for people in almost all life situations: starting with a simple conversation with an opponent, choosing goods in a store, and ending with solving complex technical or informational problems.

Thinking contributes to finding justification for certain phenomena. Logic helps to meaningfully evaluate the world around us and competently build speech and judgments.

5 features of logical thinking

The science of logic studies methods of achieving truth, excluding sensory experience, and is based on the process of studying and cognizing surrounding things on the knowledge that was obtained earlier.

There are interesting distinctive features and features of the development of logical thinking:

empirical knowledge

Empirical knowledge serves as the basis for logical laws. Special person formed the situation, became an eyewitness to the incident, saw their consequences and made his own conclusions and conclusions. The laws of logic are formed experimentally.

Acquired, not innate

Logic and logical thinking is an acquired, not an innate quality of people. A person studies and develops them throughout the life path.

The pursuit of comfort

People sometimes unconsciously do not want to develop thinking and make competent logical conclusions, trying to think in a way that is more comfortable and easier.

cynical calculation

Logical reasoning and thinking can become a tool for committing inhumane acts.

The world that surrounds people has two opposite sides: good and evil, positive and negative.

Therefore, logic, despite all the benefits it brings to a person, can bring a lot of harm.

Cynical calculation and logic put in the background such concepts as "self-sacrifice" and "love of neighbor."

Scientific

Science has certain axioms. Deviation from them is a sign of a mental disorder.

6 main axioms of logic

The development and improvement of logical thinking is impossible without the knowledge of logical axioms, which are the basis of a person's worldview:

The irreversibility of time

From childhood, people get acquainted with the concepts of "yesterday", "tomorrow", "today". That is, they begin to realize the difference between the past and the future.

Investigative connections, their sequence

The impossibility of the existence of the same facts in a certain period of time: with positive temperature regime water cannot freeze, and a woman who is expecting a child cannot get pregnant.

Deduction

The deductive method of thinking is based on logical laws and leads from the general to the particular: a heavy downpour has passed, the trees have become wet. The deduction method gives a 99.99% true answer.

Induction

This method of inference leads from the general to the particular and is based on similar properties of different objects and objects: trees, roads and cars are wet - it is raining. The inductive method has a 90% accuracy rate, as trees and other objects can get wet from more than just rain.

Sequencing

If a person performs several successive actions in stages, then he gets the expected and satisfactory result.

Man is an illogical being

Inferences very often run counter to morality and ethics, and in some cases to legislation.

After all, maniacs and people with a disturbed psyche believe that when they kill and carry out violent actions, they act logically.

Unnatural formation of logical thinking from childhood in the conditions of hostilities and extreme situations subsequently provoke people to commit terrible acts from the point of view of humanity.

Science is not perfect, so real life logic may be inferior to truth. A striking example is the situation when a woman makes a logical, in her opinion, conclusion: a man does not call, behaves aloof, which means he does not like me.

As practice shows, in 85% of cases, indifference from the opposite sex is a sign of interest in the formation and development of relationships. And in the conclusion of a woman, the errors of the inductive method are to blame.

Functions of logical thinking

The main task of science is to obtain true knowledge about the subject of reflection, based on reasoning and analysis of various aspects of the phenomenon and situation under consideration.

Logic is the main tool used in every science known today.

1. examine statements and derive other conclusions from them;
2. learn to think wisely, which will help in self-realization and achievement of goals.

How to develop logical thinking

People who aspire to inner harmony, success and well-being in the main areas of life, they ask a completely natural and relevant question: how to develop logical thinking?

Each person has it to some extent developed. But for the optimal and best understanding of reality and the ability to use it in certain situations, it is necessary to be able to think quickly and competently logically. How can you learn this?

brain training

It is necessary to regularly train the brain, not being lazy and not postponing for later.

Many mistakenly assume that people are born with a preliminary specific mental potential, therefore they cannot become smarter, wiser or dumber than genes and nature have.

This statement is not true, since any person, regularly training his thinking, develops until the end of his life path.

An effective method of self-improvement is constant exercise for the mind.

• Recommended in free time solve any logical tasks created for both children and adults. Puzzles need to be solved. Don't neglect simple riddles"spot the difference" type.

• You need to take IQ tests regularly. The result is not very important, the main thing is the process during which the development of mental and mental abilities takes place.

• You should play logic games with friends or acquaintances: chess, backgammon and other types.

• It is recommended to engage in self-education and study of sciences.

• It is necessary to learn to argue, based on facts and arguing your conclusions.

• You need to get into the habit of reading good detective stories.

• Experts say that intuition plays an important role in the development of logic. As paradoxical as it sounds, a person needs to learn to trust her. After all, intuition is the result of inferences made at a subconscious level, when people unconsciously draw conclusions from information that was once already received by the brain.

3 exercises for the development of logical thinking

Collective exercises for logical thinking are very effective:

Coding of famous phrases, verses of songs and proverbs

The group of people is divided into two companies. Each of them invites its rivals to solve a semantic riddle that betrays the content of the text.

Example: The minister of the church owned creature. Despite great feelings for him and affection, the man inflicted violent acts on the biological species, which led to the death of the latter. The reason for this behavior was that a living being ate a piece of animal product that was not intended for him. The algorithm of such actions is infinite.

Answer: "The priest had a dog ...".

Arguments and reasons

One person from the team begins to look for the reasons for a particular action of someone, then the reasons for the reasons, and so on until the arguments of the behavior are clarified.

Remove excess

It is very useful to perform exercises where it is necessary to remove excess from a set of words, numbers or pictures, based on logical thinking.

Example: chair, wardrobe, stool, pouffe.

Answer: closet.

You can train thinking with the help of this exercise on your own, resorting to thematic games in in social networks, or in a team where each team independently comes up with tasks for opponents.

Exercises for the development of logical thinking will help any person in personal growth, self-affirmation and solving controversial life issues.

The concept of logical thinking

Before considering the development of logical thinking in children of primary school age, let's define what thinking is as a psychophysiological process as a whole.

Objects and phenomena of reality have such properties and relations that can be known directly, with the help of sensations and perceptions (colors, sounds, shapes, placement and movement of bodies in visible space), and such properties and relations that can be known only indirectly and through generalization. , i.e. through thinking. Thinking is the mental processes of reflection objective reality constituting the highest level of human knowledge.

Thinking is the highest cognitive mental process. The essence of this process lies in the generation of new knowledge based on the creative reflection and transformation of reality by a person.

Thinking as a special mental process has a number of specific characteristics and features. The first such feature is a generalized reflection of reality.

The second, no less important, sign of thinking is indirect knowledge of objective reality.

The next most important characteristic feature thinking is that thinking is always associated with the solution of a particular problem that arose in the process of cognition or in practical activity. Thinking always begins with a question, the answer to which is the goal of thinking. Moreover, the answer to this question is not found immediately, but with the help of certain mental operations.

Exclusively important feature thinking is an inextricable link with speech. We always think in words, i.e. we cannot think without uttering the word. So, thinking is a generalized reflected and mediated cognition of reality.

In general, with regard to the concept of “thinking”, several views should be noted.

First, as indicated dictionary S.I. Ozhegov, thinking is “a person’s ability to reason, which is the process of reflecting objective reality in representations, judgments, concepts” . Let's dissect this concept.

A person would know very little about the surrounding world if his knowledge was limited only to the testimony of his analyzers. The possibility of a deep and broad knowledge of the world opens up human thinking. There is no need to prove that the figure has four corners, since we see it with the help of the analyzer (vision). But that the square of the hypotenuse is equal to the sum of the squares of the legs, we can neither see, nor hear, nor feel. This kind of concept is indirect.

According to E.G. Revina, thinking is the highest level of human cognition of reality. Sensual basis of thinking are sensations, perceptions and representations. Through the sense organs - these are the only channels of communication between the body and the outside world - information enters the brain. The content of information is processed by the brain. The most complex (logical) form of information processing is the activity of thinking. Solving the mental tasks that life puts before a person, he reflects, draws conclusions and thereby cognizes the essence of things and phenomena, discovers the laws of their connection, and then transforms the world on this basis. Thinking is not only closely connected with sensations and perceptions, but it is formed on the basis of them. The transition from sensation to thought is a complex process, which consists, first of all, in the selection and isolation of an object or its attribute, in abstraction from the concrete, individual and the establishment of the essential, common to many objects.

In the works of V.V. Levites thinking acts mainly as a solution to problems, questions, problems that are constantly put forward before people by life. Solving problems should always give a person something new, new knowledge. The search for solutions is sometimes very difficult, so mental activity, as a rule, is an active activity that requires focused attention and patience.

Rogov E.I. under thinking understands the process of cognitive activity of the individual, characterized by a generalized and mediated reflection of reality. Starting from sensations and perceptions, thinking, going beyond the limits of sensory data, expands the boundaries of our knowledge by virtue of its nature, which allows indirectly - by inference - to reveal what is not directly given - by perception.

A. V. Petrovsky defines thinking as a socially conditioned mental process, inextricably linked with speech, of searching for and discovering something essentially new, a process of mediated and generalized reflection of reality in the course of its analysis and synthesis. Thinking arises on the basis of practical activity from sensory cognition and goes far beyond its limits.

S.L. Rubinstein interprets thinking as a generalized and mediated knowledge of objective reality.

In the Russian Pedagogical Encyclopedia, thinking is understood as "the process of human cognitive activity, characterized by a generalized and indirect reflection of objects and phenomena of reality in their essential properties, connections and relationships."

Definitions traditional in psychological science usually fix two of its essential features:

• generality and
• mediation.

Thus, thinking is the highest, most generalizing and mediated process of reflecting reality in human consciousness, establishing connections and relationships between the cognizable and objects, revealing their properties and essence.

In the process of thinking, with the interaction of external and internal stimuli in the cerebral cortex, temporary nerve connections begin to be excited and function, which are the physiological mechanisms of the thinking process. Main Feature human thinking is that it is able to identify not only random, isolated, but also essential, necessary connections based on real dependencies, separating them from random coincidences. Any human thinking takes place in generalizations, going from the individual to the general and from the general to the individual, notes L.M. Wecker.

Thinking as a process appears most fully when a person solves any problem. This solution path can be divided into 4 phases:

• the first is the emergence of a difficulty, contradiction, question, problem;
• the second is the development of a hypothesis, proposal or project for solving a problem;
• the third is the implementation of the decision;
• the fourth is the verification of the solution by practice and subsequent evaluation.

The success of the task depends on how correctly mental operations are carried out, how various forms and types of thinking are used.

Thinking is a special kind of activity that has its own structure and types.

Most often, thinking is divided into theoretical and practical. At the same time, in theoretical thinking there are conceptual and figurative thinking, but in practicevisual-figurative and visually effective.

Conceptual thinking is thinking in which certain concepts are used.

Figurative thinking is a type of thought process in which images are used. These images are retrieved directly from memory or recreated by the imagination.

Visual-figurative thinking is a type of thought process that is carried out directly with the perception of the surrounding reality and cannot be carried out without it.

Visual-effective thinking is a special kind of thinking, the essence of which lies in the practical transformational activity carried out with real objects.

So thinking:

This is the highest cognitive process;

This is the movement of ideas, revealing the essence of things. Its result is not an image, but some thought, an idea;

This is theoretical and Practical activities, suggesting a system of actions and operations included in it, tentatively - research; transformative and cognitive character;

This is the highest level of human knowledge. Allows you to gain knowledge about such objects, properties and relationships real world which cannot be directly perceived at the sensitive level of knowledge.

If the problem is solved with the help of logical reasoning, then a person uses logical thinking.

Artemov A. K. calls logical thinking, which proceeds in the form of reasoning, is consistent, consistent, justified.

Logic studies logical forms of thinking - concept, judgment, conclusion. Operating them reflects the essence of logical thinking.

The concept is a thought that reflects the general, essential and distinctive (specific) features of objects and phenomena of reality. It is customary to distinguish between general and singular concepts.

General concepts are those that cover a whole class of homogeneous objects and phenomena that bear the same name. For example, the concepts of "chair", "building", "disease", etc. The general concepts reflect the features that are characteristic of all objects that are united by the corresponding concept.

Singular are called concepts denoting any one object. Single concepts are a collection of knowledge about a single subject, but at the same time reflect properties that can be covered by another, more general concept. For example, the concept of "Yenisei" includes the fact that it is a river that flows through the territory of Russia.

A judgment is a reflection of the connections between objects and phenomena of reality or between their properties and features.

Judgments are:

General;

Private;

Single.

In general judgments, something is affirmed (or denied) with respect to all objects of a given group, this class, for example: "All fish breathe with gills." In private judgments, affirmation or negation no longer applies to all, but only to some subjects, for example: “Some students are excellent students”; in single judgments - only to one, for example: "This student did not learn the lesson well."

Thinking is the process of producing inferences with logical operations on them (Vekker M.L.).

inference - a form of thinking that allows a person to draw a new conclusion from a series of judgments. In other words, based on the analysis and comparison of existing judgments, a new judgment is made.

There are two main types of reasoning - induction and deduction.

Induction - this is a conclusion from particular cases to a general position.

Deduction is such a conclusion in which the conclusion is from a general judgment to a single judgment or from a general position to a particular case.

Analogy is a way of reasoning characterized by the fact that from the similarity of two objects in several features and if one of them has an additional feature, it is concluded that the other object has the same feature.

The mental activity of people is carried out with the help of mental operations: comparison, analysis and synthesis, abstraction, generalization, concretization. All these operations are different aspects of the main activity of thinking - mediation, i.e. disclosure of more and more essential objective connections and relations between objects, phenomena, facts.

Comparison is a comparison of objects and phenomena in order to find similarities and differences between them. KD Ushinsky considered the operation of comparison to be the basis of understanding. He wrote: “... comparison is the basis of all understanding and all thinking. We know everything in the world only through comparison ... ".

Analysis and synthesis are the most important mental operations, inextricably linked. In unity, they give a complete and comprehensive knowledge of reality.

Analysis is the mental division of an object or phenomenon into its constituent parts or the mental separation of individual properties, features, qualities in it.

Synthesis is a mental combination of individual parts of objects or a mental combination of their individual properties.

Abstraction is a mental abstraction from any parts or properties of an object in order to highlight its essential features.

Generalization is a mental association of objects and phenomena according to their common and essential features.

Concretization is a mental representation of something single, which corresponds to a particular concept or general position.

The ability to think logically, according to A.V. Petrovsky, includes a number of components: the ability to focus on the essential features of objects and phenomena, the ability to obey the laws of logic, build one’s actions in accordance with them, the ability to perform logical operations, consciously arguing them, the ability to build hypotheses and derive consequences from these premises, etc. .d. Therefore, for him, logical thinking includes a number of components: the ability to determine the composition, structure and organization of elements and parts of the whole and focus on the essential features of objects and phenomena; the ability to determine the relationship of an object and objects, to see their change in time; the ability to obey the laws of logic, to discover patterns and development trends on this basis, to build hypotheses and draw consequences from these premises; the ability to perform logical operations, consciously arguing them.

Psychologist L.F. Tikhomirova, in her study on the psychological and pedagogical foundations of schooling, rightly notes that the logic of thinking is not given to a person from birth. He masters it in the process of life, in training. Emphasizing the importance of mathematics in the education of logical thinking, the scientist highlights general provisions organization of such education:

The duration of the process of educating a culture of thinking, its implementation on a daily basis;

Inadmissibility of error in the logic of presentation and justification;

Involving children in constant work to improve their thinking, which would be considered by them as a personally significant task;

The inclusion in the content of training of a system of certain theoretical knowledge, firstly, knowledge about the methods of orientation in the performance of mental actions.

The development of a child's logical thinking is a process of transition of thinking from the empirical level of cognition (visual-effective thinking) to the scientific and theoretical level (logical thinking), followed by the formation of the structure of interconnected components, where the components are the methods of logical thinking (logical skills) that provide a holistic functioning of logical thinking.

Thus, logical thinking is a type of thinking, the essence of which is to operate with concepts, judgments, inferences based on the laws of logic, their comparison and correlation with actions, or a set of mental logically reliable actions or operations of thinking, connected by cause-and-effect patterns that allow to harmonize available knowledge in order to describe and transform objective reality.

Later, under the Qin Dynasty, this line of research disappeared in China, since then the philosophy of legalism severely suppressed all other philosophical schools. Again, logic appeared in China only with the penetration of the Indian logic of the Buddhists there and further lagged far behind the development of European and Middle Eastern logic.

Indian logic

The origins of logic in India can be traced back to the grammatical texts of the 5th century BC. e .. Two of the six orthodox Hindu (Vedic) schools of Indian philosophy - Nyaya and Vaisheshika - dealt with the methodology of knowledge, and logic emerged from this problematic field.

The very name of the school "nyaya" means "logic". Its main achievement was the development of logic and methodology, which later became common property (cf. Aristotelian logic in Europe). The main text of the school was the Nyaya Sutras of Akshapada Gautama (2nd century AD). Since the Nyāyiks considered the attainment of reliable knowledge to be the only way to liberation from suffering, they developed subtle methods of distinguishing reliable sources of knowledge from false opinions. There are only four sources of knowledge (four pramanas): , inference, comparison and evidence. A strict five-term scheme of reasoning included: the initial premise, the basis, the example, the application and the conclusion.

Buddhist philosophy(not one of the six orthodox schools) was the main opponent of the Nyāyiks in logic. Nagarjuna, the founder of the Madhyamika ("middle way"), developed a reasoning known as "katuskoti" or tetralemma. This quadripartite argument systematically tested and rejected the statement's assertion, its negation, the conjunction of affirmation and negation, and finally the rejection of both its assertion and its negation.

With Dignaga and his follower Dharmakirti, Buddhist logic reached its peak. The central point of their analysis was the establishment (definition) of the necessary logical inherence (inclusion in the definition), "vyapti", also known as "unchanging following" or "belief". For this purpose, they developed the doctrine of "apoha" or distinction, the rules for including features in a definition or excluding them from it.

School navya-nyaya("new nyaya", "new logic") was founded in the 13th century by Ganesha Upadhyaya of Mityla, the author of Tattvachintamami (Treasure of Thought on Reality). However, he relied on the work of his predecessors of the 10th century.

European and Middle Eastern logic

In the history of European logic, stages can be distinguished: Aristotelian, or traditional - the period of dominance of formal logic - lasted hundreds of years, during which logic developed very slowly; the scholastic stage of development, which peaked in the 14th century; modern stage.

The logic of antiquity

The ancient Greek philosopher Aristotle is considered the founder of logic in ancient Greek philosophy, as it is believed that he deduced the first logical theory. The forerunners of Aristotle in the development of logical science in Ancient Greece were Parmenides, Zeno of Elea, and Plato. Aristotle, for the first time, systematized the available knowledge about logic, substantiated the forms and rules of logical thinking. His cycle of writings "Organon" consists of six works devoted to logic: "Categories", "On Interpretation", "Topeka", "First Analytics" and "Second Analytics", "Sophistic Refutations".

After Aristotle in ancient Greece, logic was also developed by representatives of the Stoic school. The orator Cicero and the ancient Roman theorist of oratory Quintilian made a great contribution to the development of this science.

Logic in the Middle Ages

As we approached the Middle Ages, logic became more widespread. It began to be developed by Arabic-speaking researchers, for example, Al-Farabi (c. 870-950). Medieval logic is called scholastic, and its heyday in the XIV century is associated with the names of scientists William of Ockham, Albert of Saxony and Walter Burley.

Logic in the Renaissance and Modern Times

This historical period in logic is marked by the appearance of many publications that are extremely significant for science.

Francis Bacon in 1620 publishes his "New Organon", containing the basics of inductive methods, improved later by John Stuart Mill and called the methods of establishing causal relationships between the phenomena of Bacon-Mill. The essence of Induction (Generalization) is that knowledge must be built into principles. You also need to look for the cause of your mistakes.

In 1662, the textbook "Logic of Port-Royal" was published in Paris, the authors of which are P. Nicole and A. Arno, who created a logical doctrine based on the methodological principles of Rene Descartes.

Modern logic

In the late XIX - early XX centuries, the foundations of the so-called. mathematical or symbolic logic. Its essence lies in the fact that mathematical methods can be used to detect the truth value of natural language expressions. It is the use of symbolic logic that distinguishes modern logical science from traditional.

A huge contribution to the development of symbolic logic was made by such scientists as J. Boole, O. de Morgan, G. Frege, C. Pierce, and others. In the 20th century, mathematical logic took shape as an independent discipline within the framework of logical science.

The beginning of the 20th century was marked by the formation of the ideas of non-classical logic, many of the important provisions of which were anticipated and/or laid down by N. A. Vasiliev and I. E. Orlov.

In the middle of the 20th century, the development of computer technology led to the emergence of logical elements, logical blocks and devices of computer technology, which was associated with the additional development of such areas of logic as the problems of logical synthesis, logical design and logical modeling of logical devices and computer technology.

In the 80s of the XX century, research began in the field of artificial logic programming based on languages ​​and systems. The creation of expert systems began with the use and development of automatic proof of theorems, as well as methods of evidence-based programming for the verification of algorithms and computer programs.

Changes in education also began in the 1980s. The appearance of personal computers in secondary schools led to the creation of computer science textbooks with the study of elements of mathematical logic to explain the logical principles of operation of logical circuits and computer equipment, as well as the principles of logical programming for fifth generation computers and the development of computer science textbooks with the study of the predicate calculus language for designing knowledge bases .

Basic concepts of the science of logic

Traditional logic

Deductive and inductive reasoning in traditional logic

• Induction
• Deduction

syllogistic

• Syllogism
• Syllogistic theories

classical mathematical logic

Apparatus of mathematical logic

mathematical logic(theoretical logic, symbolic logic) - a branch of mathematics that studies the proofs and questions of the foundations of mathematics. " The subject of modern mathematical logic is diverse.» According to the definition of P. S. Poretsky, « mathematical logic is logic by subject, mathematics by method". According to the definition of N. I. Kondakov, “ mathematical logic - the second, after traditional logic, stage in the development of formal logic, applying mathematical methods and a special apparatus of symbols and exploring thinking with the help of calculus (formalized languages)." This definition corresponds to the definition of S. K. Kleene: mathematical logic is “ logic developed with mathematical methods ". Also, A. A. Markov defines modern logic " exact science applying mathematical methods". All these definitions do not contradict, but complement each other.

The use of mathematical methods in logic becomes possible when judgments are formulated in some exact language. Such precise languages ​​have two sides: syntax and semantics. Syntax is a set of rules for constructing language objects (usually called formulas). Semantics is a set of conventions that describe our understanding of formulas (or some of them) and allow us to consider some formulas to be true and others not.

An important role in mathematical logic is played by the concepts of deductive theory and calculus. A calculus is a set of inference rules that make it possible to consider certain formulas as derivable. Inference rules are divided into two classes. Some of them directly qualify certain formulas as derivable. Such inference rules are called axioms. Others allow us to consider formulas derivable A, syntactically related in some predetermined way to finite sets of derivable formulas. A widely used rule of the second type is the modus ponens rule: if the derivable formulas A and , then we derive the formula B.

The relation of calculi to semantics is expressed in terms of semantic suitability and semantic completeness of the calculus. The AND calculus is said to be semantically suitable for the language I if any formula of the language I can be deduced in AND is true. Similarly, a calculus AND is said to be semantically complete in I if any valid formula in I is deducible in I.

Mathematical logic studies the logical connections and relationships underlying logical (deductive) inference using the language of mathematics.

Many of the languages ​​considered in mathematical logic have semantically complete and semantically useful calculi. In particular, K. Gödel's result is known that the so-called classical predicate calculus is semantically complete and semantically suitable for the language of classical first-order predicate logic. On the other hand, there are many languages ​​for which the construction of a semantically complete and semantically suitable calculus is impossible. In this area, the classic result is Gödel's incompleteness theorem, stating the impossibility of a semantically complete and semantically usable calculus for the language of formal arithmetic.

It should be noted that in practice, many elementary logical operations are an obligatory part of the instruction set of all modern microprocessors and, accordingly, are included in programming languages. This is one of the most important practical applications of mathematical logic methods studied in modern computer science textbooks.

Propositional logic

• (Propositional logic)

Predicate Logic

• Logic of quantifiers
• First Order Logic
• Second Order Logic

Calculus and logical methods

• Resolvability,
• semantic tree
• Tables Beta
• axiomatics
• natural conclusion
• Sequence (logic)

Boolean semantics

• Algebraic semantics
• Set-theoretic semantics
• Relational semantics of possible worlds
• The problem of meaningfulness of the semantics of logical systems
• Categorical semantics
• Theory of semantic categories

Laws of logic

• Law of Identity
• Law of the excluded middle
• Law of contradiction
• Law of Sufficient Reason
• De Morgan's laws
• Laws of deductive reasoning
• Law of Clavius
• Division laws

Model theory

Branch of mathematical logic that deals with the study of the relationship between formal languages ​​and their interpretations, or models. Name model theory was first proposed by Tarski in 1954. The main development of the theory of models was in the works of Tarsky, Maltsev and Robinson.

proof theory

This is a section of mathematical logic that presents evidence in the form of formal mathematical objects, analyzing them using mathematical methods. Proofs are usually presented as inductively defined data structures such as lists and trees, constructed according to the axioms and inference rules of formal systems. So proof theory is syntactic, Unlike semantic model theory. Together with model theory, axiomatic set theory, and the theory of computation, proof theory is one of the so-called "four pillars" of the foundations of mathematics.

Theories of inference

• Theories of inference (inference theory)
• Theories of succession (theory of succession)
• Theories of implications (theory of implications)
• material implication

Non-classical logics

Logic with non-classical understanding of consequence

• Relevant logic
• Paraconsistent logic
• Nonmonotonic logics
  • Dynamic Logic

Logic that cancels the law of the excluded middle

• intuitionistic logic
• constructive logic
• Logics quantum mechanics(Quantum logic)

Logic that changes truth tables

• Multivalued logic
• Two Value Logic
• Three Value Logic

Logic that extends the composition of the statement

• Question Logic
• Grade Logic
• Logic of norms

modal logic

• Modality
• Alethic modalities (alethic modality, alethic modal logic, alethic modal logics)
• Deontic modalities (deontic modality, deontic modal logic, deontic modal logics)
• Epistemological modalities (epistemological modality, epistemological modal logic, epistemological modal logics)
• Temporal modalities (temporal modality, temporal modal logics, temporal modal logic)

• Strict implication
• material implication

Non-deductive logical theories

• inductive logic
• Probabilistic logic
• Decision logic
• Logic of fuzzy concepts (logic of fuzzy sets, fuzzy logic)
• Analogy (inference by analogy).

Other non-classical logics

• Category logic
• Combinatorial logic is logic that replaces variables with functions in order to clarify intuitive operations on variables such as substitution. Built on the basis of combinatorial logic, the arithmetic system contains all partially recursive functions and avoids Gödel's incompleteness.
• Conditional logic (conditional logic). Its subject is the truth of conditional sentences (in particular, the subjunctive mood). The logic of counterfactual claims.

Applications of logic

Applied problems of logic and logical semantics

• Applications of logic in philosophy
• Applications of logic in theology
• Applications of logic in legal sciences
• Applications of logic in other disciplines

Applications of logic in the analysis of cognitive procedures

Logical analysis of forms and methods of cognition

• Forms of thought
• Definition
• Classification
• Abstraction
• Idealization
• Axiomatization
• Formalization
• Logical problems of argumentation
• The Logic of Evidence

Applications of logic in the methodology of science

• Methodology of Science
• The logic of science
• Logic and empiricism

Applications of logic in philosophy

• Applications of logic in philosophy
• Applications of logic in ontology
• Applications of logic in epistemology
• Applications of logic in ethics
• Logical problems of argumentation (argumentation theory)
• Analytical philosophy

Applications of logic in psychology

• cognitive science
• cognitive psychology
• Logic of discovery

Since logic establishes laws and patterns of thought, there is a problem of correlating logic with , which relies on intuition. Creativity without limits is an idealization: it is limited by the psychological laws of perception or, for example, by the laws of composition in fine arts. Creativity involves not only the ability to put forward an interesting idea, but also the ability to convincingly substantiate it and put it into practice. certain rules, therefore, must follow some rules of thought.

Applications of logic in linguistics

• Logical language analysis
• Analytical philosophy

Applications of logic in computer science

• Dynamic logics (dynamic logic)
• Program logics (program logic)
• Expert system logic (expert system logic)
• Logic in computer science
• Evidence-based programming
• Automatic theorem proving
• Logic programming