1) Logic- In the book: 1) the universal boundary of the givenness of things in the world, which itself remains invisible; 2) a technique for indirectly identifying this boundary.

2) Logic- Activity can provide only one half of wisdom; the other half depends on perceptive inactivity. Ultimately, the debate between those who base logic on "truth" and those who base it on "research" stems from differences in values ​​and at a certain point becomes meaningless. In logic it is a waste of time to consider conclusions concerning particular cases; we always deal with completely general and purely formal implications, leaving for other sciences the study of in which cases the assumptions are confirmed and in which they are not. Although we can no longer be content with defining logical statements as following from the law of contradiction, we can and must still recognize that they form a class of statements entirely different from those of which we come to know empirically. All of them have a property that we agreed to call “tautology” just above. This, combined with the fact that they can be expressed solely in terms of variables and logical constants (where a logical constant is something that remains constant in a statement even when all its constituents change), will give the definition of logic or pure mathematics.

3) Logic - - the doctrine of the connections and sequences of human thinking, the forms of its development, the various relationships of mental forms and their transformations. L. considers questions about the means of existence of thinking, languages ​​of consolidation, reproduction, and translation of thought processes. In a broad sense, philosophy is an examination of the connections not only of thinking, but also of being, that is, literature that reveals the “logic of things,” the “logic of events,” and the “connection of times.” In this aspect, L. comes close to ontology. In its substantive aspects, philosophy is associated with the teachings of cognition, its development, functioning, and conservation and is directly included in epistemology. Thus, philosophy is one of the main subdivisions of philosophy and constantly plays a leading role in philosophizing, since the latter always deals with the issue of thinking in one way or another. In the 19th century Philosophy, as a special science, is separated from philosophy and, as such, deals with the formal analysis of thinking and its languages. Questions of the development of thinking, the evolution of its means, its cultural, historical and social conditionality remain within the competence of philosophy. Philosophy itself, in its specific socio-historical and cultural forms, becomes an important branch of philosophical research. Within the framework of this approach, several main stages in the evolution of light and its understanding can be identified. In the ancient world, the development of logical problems was associated with the processes of classification of artificial and natural things, tools of human activity, and acts of human interactions. L. develops generalizing concepts and techniques for operating with them. As part of philosophy, it is an important tool for creating a picture of the world and using it in the practice of society. In the Middle Ages, literature was focused on research into forms of thinking and their relationships; meaningful cognition is considered from the perspective its correspondence to logical forms. The doctrine of stable (or unshakable) structures of human thinking that ensure its correctness turns out to be an important prerequisite for the emerging standards of scientific rationality. When, following natural science, formal philosophy is separated from philosophy, the question of the rationality of human thinking finds itself at the center of philosophical polemics. On the one hand, the insufficiency of formal rationality for the needs of modern science, for the development of the human personality and the expansion of its spiritual horizons is revealed. On the other hand, the need to preserve rationality and philosophy in the broadest sense as conditions for the reproduction of culture is confirmed (Baden neo-Kantianism). In the 20th century, philosophical criticism of rationality (usually interpreted as a rigid connection of logical forms) intensifies and is conducted from different positions (existentialism, Marxism, deconstructionism). At the same time, in philosophy there is an increasing tendency to treat literature from a cultural and historical perspective, to study various laws inherent in different cultures and types of human activity. In the light of these approaches, the emphasis in understanding the content of L is changing. If previously this quality was associated mainly with clarifying the objective orientation of thinking, now the focus is on the connection of mental forms that arises in the interaction of human subjects, this interaction is consolidating and reproducing. V. E. Kemerov

4) Logic- - the science of the laws and operations of correct thinking. According to the basic principle of logic, the correctness of reasoning is determined only by its logical form or structure and does not depend on the specific content of the statements included in it. A distinctive feature of correct reasoning is that if the premises are true, logical thinking leads to a true conclusion (the answer to the question). Incorrect reasoning can lead from true and untrue premises to both true and untrue conclusions (the truth of the conclusion is a matter of chance). Thus, what logic is is clear - these are the rules for using certain mental techniques when processing information. There is formal logic, humanistic logic, women's logic, children's logic, schizophrenic logic, dialectical logic, philosophical logic, etc. But besides logic, there is also thinking itself, which can obey its laws (correct thinking) and not obey (incorrect thinking). illogical thinking). Associative block. From our point of view, logic is a section of the theory of knowledge that studies the relationship and existence of things in the full sense of the last word.

5) Logic- (from Greek – logos): in the broadest sense – the science of thinking, the doctrine of laws, forms and means of reasoning. Most often, this term is identified with the term “formal logic,” the founder of which was Aristotle. The main goal of logical research is to analyze the correctness of reasoning, the formulation of laws and principles, the observance of which is a necessary condition for obtaining true conclusions in the process of inference. Logical processes are studied by representing them in formalized languages. Each of them includes a set of appropriately interpreted expressions (formulas), as well as methods for transforming some expressions into others according to the rules of deduction. Modern logic is composed of a large number of logical systems that describe individual fragments (types) of reasoning. Depending on the basis (criteria) of classification, classical and non-classical logic are currently distinguished. In the modern sense, logic is the science of forms of discourse.

6) Logic- - etymologically goes back to the ancient Greek word “logos”, meaning “word”, “thought”, “concept”, “reasoning”, “law”. This is the science of the laws and forms of human thinking. She studies mental procedures. There is a distinction between traditional logic, which was started by Aristotle, which studies inferences, concepts and operations on them. The use of formalization methods and mathematical methods led to the creation of classical logic (symbolic or mathematical). Non-classical (modal or philosophical) logic, which uses formal methods to analyze meaningful realities. A simplified understanding of logic - the flow of reasoning, the rules of reasoning.

7) Logic- - the science of generally valid forms and means of thought necessary for rational knowledge of any area of ​​reality.

8) Logic - (Greek logos - word, reasoning, concept, mind) - the science of forms, laws and methods of cognitive activity; the ability to think correctly (logically). Since ancient times, an important property of human cognitive thinking has been noticed: if at first some statements are made, then other statements can be recognized, but not any, but only strictly defined ones. Cognitive thinking, therefore, is subject to a certain compulsory force, its results are largely determined and predetermined by previous knowledge. This property was widely used by Socrates in his dialogues. By skillfully posing questions, he directed his interlocutor to the adoption of very specific conclusions. (Characterizing his method, Socrates explained that his manner of conversation is similar to what a midwife does, who does not give birth herself, but takes birth. So he only asks others, contributing to the birth of truth, but he himself has nothing to say.) Therefore, his method Socrates called maieutics - the art of the midwife.) Socrates' student Plato, then Aristotle made the determinism of thinking the subject of special study. Aristotle's results are particularly impressive. His success is due to the fact that he eliminated from the reasoning what can be called their content, retaining only the form. He achieved this by substituting letters (variables) in judgments instead of names with specific contents. For example, in an implicative argument: “If all Bs are Cs and all Aes are Bs, then all Aes are Bs.” Aristotle’s approach demonstrated the fact that the reliability of the results of reasoning with different contents depends not only on the truth of the initial positions (premises), but also on the relationships between them, the method of their connection, i.e. on the form of reasoning. Aristotle formulated the most important principles for the transition from true premises to true conclusions. Subsequently, these principles began to be called the laws of identity, contradiction and excluded middle. He proposed the first theoretical system of forms of reasoning - the so-called. assertoric syllogistic, which deals with propositions of the form “All A are B”, “Some A are B”, “No A is B”, “Some A are not B”. Thus, he laid the foundation for the science of generally valid means and forms of thinking, the laws of rational knowledge. Later this science began to be called L.L. did not limit itself to clarifying cases when the truth of the premises guarantees the truth of the conclusion. This type of reasoning became the subject of one of its branches - deductive L. But Democritus already discusses the problem of inductive inferences, through which the transition from particular statements to general provisions of a probabilistic nature is carried out. Particular interest in induction appears in the 17th and 18th centuries. when experimental sciences began to develop rapidly. The English philosopher F. Bacon made the first attempt at a theoretical understanding of induction, which, as he thought, could serve as the only method of understanding natural phenomena in order to use them for the benefit of people. Deductivism and inductivism were the main directions in the development of literature until the 19th century. Representatives of rationalistic philosophy (Descartes, Spinoza, Malebranche, Leibniz) preferred deduction, while representatives of empirical (sensualistic) philosophy (following F. Bacon - Hobbes, Locke, Condillac, Berkeley, Hume) were inductivists. Wolf, who proposed a comprehensive, in his opinion, system of philosophical knowledge as “the science of all possible objects, as far as they are possible,” tried to reconcile these directions. Being, in general, a rationalist, he nevertheless energetically emphasized the decisive importance of induction and experimental knowledge in certain scientific disciplines (for example, in physics). However, Wolffian ideas about the forms and laws of thinking and methods of cognition, which had developed in Leningrad by the 19th century, were unable to satisfy the needs of rapidly developing science and social practice. Kant and especially Hegel criticized the limitations of the rationalistic-metaphysical method. L. was faced with the task of developing means that would allow a conscious approach to the study of essential relationships. A serious attempt to solve this problem was made by Hegel. His outstanding merit is the introduction of the idea of ​​development and interconnection into literature. This allowed him to lay the foundations of dialectical literature as a theory of the movement of human thought from phenomenon to essence, from relative truth to absolute truth, from abstract knowledge to concrete knowledge. Based on the categories, principles, and laws of dialectical literature, methodological guidelines are developed for studying the content of objects in all their diversity and inconsistency. Currently, literature is a fairly extensive scientific discipline. Its most important and most mature section is formal literature. It received its name from the subject it has been dealing with since ancient times - forms of thoughts and reasoning that ensure the receipt of new truths on the basis of already established ones, and, first of all, the criteria for the correctness and validity of these forms. For a long time, formal literature was known primarily in the form that Aristotle and his commentators gave it. Hence the name corresponding to this stage is Aristotelian L. The tradition going back to Aristotle also gave rise to another equivalent term - traditional philosophy. The invariability of the problem and methods of resolving it within the framework of Aristotelian philosophy over many centuries gave the basis to Kant, who first used the term “formal philosophy,” to believe that over the two thousand years that have passed Since the time of Aristotle, this L. has not taken a single step forward and has an essentially complete character. Kant did not even imagine that just half a century after his death a “second wind” would begin in the development of formal logic. This qualitatively new stage was caused by the fact that the problems posed by the study of the logical foundations of mathematics could not be solved by means of Aristotelian logic. Almost simultaneously The processes of logicalization of mathematics and mathematization of L are underway. When solving logical problems, mathematical methods are actively used, logical calculus is created. Concrete steps are being taken to implement Leibniz's ideas on the use of computational methods in any science. J. Boole develops the first system of algebra L. Thanks to the work of O. de Morgan, W. Jevons, E. Schroeder, P.S. Poretsky, Peirce, Frege, J. Peano, and Russell created the main sections of mathematical mathematics, which became the most important branch of formal mathematics. In the 20th century, especially in the 20s and 30s, in the works of J. Lukasiewicz, E. Post, K Lewis, S. Yaskovsky, D. Webb, L. Brouwer, A. Heyting, A.A. Markova, A.N. Kolmogorov, G. Reichenbach, S.K. Kleene, P. Detouches-Fevrier, G. Birkhoff, and others lay the foundations of non-classical sections of formal linguistics: multi-valued linguistics, modal, probabilistic, intuitionistic, constructivist, and others. The transition to a number of truth values ​​greater than two (“true” ", and "false"), constitutes one of the characteristic features of non-classical, or, as they are often called, non-Chrysippian logic. In the 1930s, the development of formal logic was associated with the solution of many problems of metalogic (Greek meta - after, over), studying principles of construction and general properties of formal systems, for example, problems of consistency, completeness, independence of the system of axioms, solvability, the ability of these systems to express meaningful theories, etc. The foundations of the so-called. "machine thinking" The study of these problems was marked by outstanding discoveries that have important ideological and methodological significance and are associated with the names of Tarski, K. Gödel, A. Church. The most famous is K. Gödel’s theorem on the incompleteness of formalized systems, incl. arithmetic of natural numbers and axiomatic set theory. In accordance with this theorem, in each of these systems there are propositions that within their framework can neither be proven nor disproved. Thus, it was shown that not a single valid scientific theory can be squeezed into the framework of formalism. A. Church proved the theorem according to which there are no algorithms for solving many classes of problems, not to mention an algorithm that allows solving any problem (many outstanding logicians and mathematicians dreamed of inventing such an algorithm). Today, the development of formal logic is proceeding in two main directions: 1) the development of new systems of non-classical logic (the logic of imperatives, evaluations, questions, temporal, inductive logic, the theory of logical implication, etc.), the study of the properties of these systems and the relationships between them, the creation of their general theory; 2) expansion of the scope of application of formal L. The most important final result obtained in this direction is that formal L. has become not only an instrument of precise thought, but also the “thought” of the first precise instrument - the computer, directly in the role of a partner included by man in the sphere solving the problems facing him. L. (in the sum of all its sections) has become an integral part of human culture. Its achievements are used in a wide variety of areas of human activity. It is widely used in psychology and linguistics, management theory and pedagogy, law and ethics. Its formal sections are the original basis of cybernetics, computational mathematics and technology, and information theory. Without the principles and laws of literature, modern methodology of cognition and communication is unthinkable. The study of L. has always been given great importance. Parmenides already taught Socrates, who was still inexperienced in philosophy: “Your zeal for reasoning, rest assured, is wonderful and divine, but while you are still young, try to practice more in what most consider idle talk (i.e., operating with abstract concepts - V. B.) otherwise the truth will elude you." As we see, already in ancient times it was understood that the discipline, which was later given the name L., plays, first of all, a large methodological role - as a means of finding the truth. V.F. Berkov

9) Logic- - in a broad sense - this is the philosophical science of the laws of correct thinking; in a narrow sense - a sequence of necessities built in the search for truth.

10) Logic - (from the Greek logos - logos) 1) the ability to correctly, i.e. logically, think; 2) the doctrine of identity and its negation (G. Jacobi), the doctrine of consistency and methods of cognition (the science of logic). As "elementary formal logic" it deals with the most general properties inherent in all (existing) concepts. Basic properties of concepts are expressed in logical axioms (see Axiom). First, the doctrine of the concept is considered, then comes the doctrine of judgment and, finally, inference. The doctrines of logical axioms, concepts, judgments and inferences, taken together, form pure logic. Applied logic covers in traditional logic the doctrine of definition, proof, and method. It is often preceded not by scientific-logical, but by theoretical-cognitive, psychological teachings about experience, description and formulation (especially with the help of special language, terminology) and the formation of concepts. Sometimes the doctrine of the system is added to it. Logic (as a science) is only the doctrine of thinking in concepts, but not of knowledge through concepts; it serves to increase the formal accuracy of consciousness and the objectivity of the content of thinking and cognition. The founder of Western European logic (as a science) is Aristotle, the “father of logic.” The word "logic" first appeared among the Stoics; they and the Neoplatonists clarified certain aspects of it, and in the Middle Ages scholasticism developed it in the smallest detail, in subtleties. Humanism expelled scholasticism from logic, but could not renew it. The Reformation adopted the logic of Melanchthon, the Counter-Reformation - the logic of Suarez. Having risen in principle above scholasticism, Johannes Sturm from Strasbourg developed logic; Pierre Ramet became more famous. From the 17th century The influence on logic of the spheres of thought associated with mathematics became noticeable, and in Spinoza’s geometric method it was less than in Leibniz, who used improving natural science methods in logic. From Leibniz and mathematics, as well as from neo-scholasticism, came the logic of the Wolf school. Kant's "transcendental logic" is in reality a critical theory of knowledge, a logic of German. idealism (especially Hegel's logic) - speculative metaphysics. Schopenhauer, Nietzsche, Bergson and proponents of the philosophy of life rejected traditional logic. Currently, logic has split into many directions: 1) metaphysical logic (Hegelianism); 2) psychological logic (T. Lipps, partly W. Wundt); 3) epistemological, or transcendental, logic (neo-Kantianism); 4) semantic logic (Aristotle, Kulpe, modern nominalism); 5) subject logic (Remke, Meinong, Drish); 6) neo-scholastic logic; 7) phenomenological logic; 8) logic as methodology (neo-Kantianism) and logistics, which is at the center of debates about logic.

11) Logic- - see Dialectical logic. Mathematical logic, Formal logic.

Logics

In the book: 1) the universal boundary of the givenness of things in the world, which itself remains invisible; 2) a technique for indirectly identifying this boundary.

Activity can provide only one half of wisdom; the other half depends on perceptive inactivity. Ultimately, the debate between those who base logic on "truth" and those who base it on "research" stems from differences in values ​​and at a certain point becomes meaningless. In logic it is a waste of time to consider conclusions concerning particular cases; we always deal with completely general and purely formal implications, leaving for other sciences the study of in which cases the assumptions are confirmed and in which they are not. Although we can no longer be content with defining logical statements as following from the law of contradiction, we can and must still recognize that they form a class of statements entirely different from those of which we come to know empirically. All of them have a property that we agreed to call “tautology” just above. This, combined with the fact that they can be expressed solely in terms of variables and logical constants (where a logical constant is something that remains constant in a statement even when all its constituents change), will give the definition of logic or pure mathematics.

The doctrine of the connections and sequences of human thinking, the forms of its development, the various relationships of mental forms and their transformations. L. considers questions about the means of existence of thinking, languages ​​of consolidation, reproduction, and translation of thought processes. In a broad sense, philosophy is an examination of the connections not only of thinking, but also of being, that is, literature that reveals the “logic of things,” the “logic of events,” and the “connection of times.” In this aspect, L. comes close to ontology. In its substantive aspects, philosophy is associated with the teachings of cognition, its development, functioning, and conservation and is directly included in epistemology. Thus, philosophy is one of the main subdivisions of philosophy and constantly plays a leading role in philosophizing, since the latter always deals with the issue of thinking in one way or another. In the 19th century Philosophy, as a special science, is separated from philosophy and, as such, deals with the formal analysis of thinking and its languages. Questions of the development of thinking, the evolution of its means, its cultural, historical and social conditionality remain within the competence of philosophy. Philosophy itself, in its specific socio-historical and cultural forms, becomes an important branch of philosophical research. Within the framework of this approach, several main stages in the evolution of light and its understanding can be identified. In the ancient world, the development of logical problems was associated with the processes of classification of artificial and natural things, tools of human activity, and acts of human interactions. L. develops generalizing concepts and techniques for operating with them. As part of philosophy, it is an important tool for creating a picture of the world and using it in the practice of society. In the Middle Ages, literature was focused on research into forms of thinking and their relationships; meaningful cognition is considered from the perspective its correspondence to logical forms. The doctrine of stable (or unshakable) structures of human thinking that ensure its correctness turns out to be an important prerequisite for the emerging standards of scientific rationality. When, following natural science, formal philosophy is separated from philosophy, the question of the rationality of human thinking finds itself at the center of philosophical polemics. On the one hand, the insufficiency of formal rationality for the needs of modern science, for the development of the human personality and the expansion of its spiritual horizons is revealed. On the other hand, the need to preserve rationality and philosophy in the broadest sense as conditions for the reproduction of culture is confirmed (Baden neo-Kantianism). In the 20th century, philosophical criticism of rationality (usually interpreted as a rigid connection of logical forms) intensifies and is conducted from different positions (existentialism, Marxism, deconstructionism). At the same time, in philosophy there is an increasing tendency to treat literature from a cultural and historical perspective, to study various laws inherent in different cultures and types of human activity. In the light of these approaches, the emphasis in understanding the content of L is changing. If previously this quality was associated mainly with clarifying the objective orientation of thinking, now the focus is on the connection of mental forms that arises in the interaction of human subjects, this interaction is consolidating and reproducing. V. E. Kemerov

The science of the laws and operations of correct thinking. According to the basic principle of logic, the correctness of reasoning is determined only by its logical form or structure and does not depend on the specific content of the statements included in it. A distinctive feature of correct reasoning is that if the premises are true, logical thinking leads to a true conclusion (the answer to the question). Incorrect reasoning can lead from true and untrue premises to both true and untrue conclusions (the truth of the conclusion is a matter of chance). Thus, what logic is is clear - these are the rules for using certain mental techniques when processing information. There is formal logic, humanistic logic, women's logic, children's logic, schizophrenic logic, dialectical logic, philosophical logic, etc. But besides logic, there is also thinking itself, which can obey its laws (correct thinking) and not obey (incorrect thinking). illogical thinking). Associative block. From our point of view, logic is a section of the theory of knowledge that studies the relationship and existence of things in the full sense of the last word.

(from Greek - logos): in the broadest sense - the science of thinking, the doctrine of laws, forms and means of reasoning. Most often, this term is identified with the term “formal logic,” the founder of which was Aristotle. The main goal of logical research is to analyze the correctness of reasoning, the formulation of laws and principles, the observance of which is a necessary condition for obtaining true conclusions in the process of inference. Logical processes are studied by representing them in formalized languages. Each of them includes a set of appropriately interpreted expressions (formulas), as well as methods for transforming some expressions into others according to the rules of deduction. Modern logic is composed of a large number of logical systems that describe individual fragments (types) of reasoning. Depending on the basis (criteria) of classification, classical and non-classical logic are currently distinguished. In the modern sense, logic is the science of forms of discourse.

Etymologically, it goes back to the ancient Greek word “logos”, meaning “word”, “thought”, “concept”, “reasoning”, “law”. This is the science of the laws and forms of human thinking. She studies mental procedures. There is a distinction between traditional logic, which was started by Aristotle, which studies inferences, concepts and operations on them. The use of formalization methods and mathematical methods led to the creation of classical logic (symbolic or mathematical). Non-classical (modal or philosophical) logic, which uses formal methods to analyze meaningful realities. A simplified understanding of logic - the flow of reasoning, the rules of reasoning.

The science of generally valid forms and means of thought necessary for rational knowledge of any area of ​​reality.

(Greek logos - word, reasoning, concept, mind) - the science of forms, laws and methods of cognitive activity; the ability to think correctly (logically). Since ancient times, an important property of human cognitive thinking has been noticed: if at first some statements are made, then other statements can be recognized, but not any, but only strictly defined ones. Cognitive thinking, therefore, is subject to a certain compulsory force, its results are largely determined and predetermined by previous knowledge. This property was widely used by Socrates in his dialogues. By skillfully posing questions, he directed his interlocutor to the adoption of very specific conclusions. (Characterizing his method, Socrates explained that his manner of conversation is similar to what a midwife does, who does not give birth herself, but takes birth. So he only asks others, contributing to the birth of truth, but he himself has nothing to say.) Therefore, his method Socrates called maieutics - the art of the midwife.) Socrates' student Plato, then Aristotle made the determinism of thinking the subject of special study. Aristotle's results are particularly impressive. His success is due to the fact that he eliminated from the reasoning what can be called their content, retaining only the form. He achieved this by substituting letters (variables) in judgments instead of names with specific contents. For example, in an implicative argument: “If all Bs are Cs and all Aes are Bs, then all Aes are Bs.” Aristotle’s approach demonstrated the fact that the reliability of the results of reasoning with different contents depends not only on the truth of the initial positions (premises), but also on the relationships between them, the method of their connection, i.e. on the form of reasoning. Aristotle formulated the most important principles for the transition from true premises to true conclusions. Subsequently, these principles began to be called the laws of identity, contradiction and excluded middle. He proposed the first theoretical system of forms of reasoning - the so-called. assertoric syllogistic, which deals with propositions of the form “All A are B”, “Some A are B”, “No A is B”, “Some A are not B”. Thus, he laid the foundation for the science of generally valid means and forms of thinking, the laws of rational knowledge. Later this science began to be called L.L. did not limit itself to clarifying cases when the truth of the premises guarantees the truth of the conclusion. This type of reasoning became the subject of one of its branches - deductive L. But Democritus already discusses the problem of inductive inferences, through which the transition from particular statements to general provisions of a probabilistic nature is carried out. Particular interest in induction appears in the 17th and 18th centuries. when experimental sciences began to develop rapidly. The English philosopher F. Bacon made the first attempt at a theoretical understanding of induction, which, as he thought, could serve as the only method of understanding natural phenomena in order to use them for the benefit of people. Deductivism and inductivism were the main directions in the development of literature until the 19th century. Representatives of rationalistic philosophy (Descartes, Spinoza, Malebranche, Leibniz) preferred deduction, while representatives of empirical (sensualistic) philosophy (following F. Bacon - Hobbes, Locke, Condillac, Berkeley, Hume) were inductivists. Wolf, who proposed a comprehensive, in his opinion, system of philosophical knowledge as “the science of all possible objects, as far as they are possible,” tried to reconcile these directions. Being, in general, a rationalist, he nevertheless energetically emphasized the decisive importance of induction and experimental knowledge in certain scientific disciplines (for example, in physics). However, Wolffian ideas about the forms and laws of thinking and methods of cognition, which had developed in Leningrad by the 19th century, were unable to satisfy the needs of rapidly developing science and social practice. Kant and especially Hegel criticized the limitations of the rationalistic-metaphysical method. L. was faced with the task of developing means that would allow a conscious approach to the study of essential relationships. A serious attempt to solve this problem was made by Hegel. His outstanding merit is the introduction of the idea of ​​development and interconnection into literature. This allowed him to lay the foundations of dialectical literature as a theory of the movement of human thought from phenomenon to essence, from relative truth to absolute truth, from abstract knowledge to concrete knowledge. Based on the categories, principles, and laws of dialectical literature, methodological guidelines are developed for studying the content of objects in all their diversity and inconsistency. Currently, literature is a fairly extensive scientific discipline. Its most important and most mature section is formal literature. It received its name from the subject it has been dealing with since ancient times - forms of thoughts and reasoning that ensure the receipt of new truths on the basis of already established ones, and, first of all, the criteria for the correctness and validity of these forms. For a long time, formal literature was known primarily in the form that Aristotle and his commentators gave it. Hence the name corresponding to this stage is Aristotelian L. The tradition going back to Aristotle also gave rise to another equivalent term - traditional philosophy. The invariability of the problem and methods of resolving it within the framework of Aristotelian philosophy over many centuries gave the basis to Kant, who first used the term “formal philosophy,” to believe that over the two thousand years that have passed Since the time of Aristotle, this L. has not taken a single step forward and has an essentially complete character. Kant did not even imagine that just half a century after his death a “second wind” would begin in the development of formal logic. This qualitatively new stage was caused by the fact that the problems posed by the study of the logical foundations of mathematics could not be solved by means of Aristotelian logic. Almost simultaneously The processes of logicalization of mathematics and mathematization of L are underway. When solving logical problems, mathematical methods are actively used, logical calculus is created. Concrete steps are being taken to implement Leibniz's ideas on the use of computational methods in any science. J. Boole develops the first system of algebra L. Thanks to the work of O. de Morgan, W. Jevons, E. Schroeder, P.S. Poretsky, Peirce, Frege, J. Peano, and Russell created the main sections of mathematical mathematics, which became the most important branch of formal mathematics. In the 20th century, especially in the 20s and 30s, in the works of J. Lukasiewicz, E. Post, K Lewis, S. Yaskovsky, D. Webb, L. Brouwer, A. Heyting, A.A. Markova, A.N. Kolmogorov, G. Reichenbach, S.K. Kleene, P. Detouches-Fevrier, G. Birkhoff, and others lay the foundations of non-classical sections of formal linguistics: multi-valued linguistics, modal, probabilistic, intuitionistic, constructivist, and others. The transition to a number of truth values ​​greater than two (“true” ", and "false"), constitutes one of the characteristic features of non-classical, or, as they are often called, non-Chrysippian logic. In the 1930s, the development of formal logic was associated with the solution of many problems of metalogic (Greek meta - after, over), studying principles of construction and general properties of formal systems, for example, problems of consistency, completeness, independence of the system of axioms, solvability, the ability of these systems to express meaningful theories, etc. The foundations of the so-called. "machine thinking" The study of these problems was marked by outstanding discoveries that have important ideological and methodological significance and are associated with the names of Tarski, K. Gödel, A. Church. The most famous is K. Gödel’s theorem on the incompleteness of formalized systems, incl. arithmetic of natural numbers and axiomatic set theory. In accordance with this theorem, in each of these systems there are propositions that within their framework can neither be proven nor disproved. Thus, it was shown that not a single valid scientific theory can be squeezed into the framework of formalism. A. Church proved the theorem according to which there are no algorithms for solving many classes of problems, not to mention an algorithm that allows solving any problem (many outstanding logicians and mathematicians dreamed of inventing such an algorithm). Today, the development of formal logic is proceeding in two main directions: 1) the development of new systems of non-classical logic (the logic of imperatives, evaluations, questions, temporal, inductive logic, the theory of logical implication, etc.), the study of the properties of these systems and the relationships between them, the creation of their general theory; 2) expansion of the scope of application of formal L. The most important final result obtained in this direction is that formal L. has become not only an instrument of precise thought, but also the “thought” of the first precise instrument - the computer, directly in the role of a partner included by man in the sphere solving the problems facing him. L. (in the sum of all its sections) has become an integral part of human culture. Its achievements are used in a wide variety of areas of human activity. It is widely used in psychology and linguistics, management theory and pedagogy, law and ethics. Its formal sections are the original basis of cybernetics, computational mathematics and technology, and information theory. Without the principles and laws of literature, modern methodology of cognition and communication is unthinkable. The study of L. has always been given great importance. Parmenides already taught Socrates, who was still inexperienced in philosophy: “Your zeal for reasoning, rest assured, is wonderful and divine, but while you are still young, try to practice more in what most consider idle talk (i.e., operating with abstract concepts - V. B.) otherwise the truth will elude you." As we see, already in ancient times it was understood that the discipline, which was later given the name L., plays, first of all, a large methodological role - as a means of finding the truth. V.F. Berkov

In a broad sense, it is a philosophical science about the laws of correct thinking; in a narrow sense - a sequence of necessities built in the search for truth.

(from Greek logos - logos) 1) the ability to correctly, i.e. logically, think; 2) the doctrine of identity and its negation (G. Jacobi), the doctrine of consistency and methods of cognition (the science of logic). As "elementary formal logic" it deals with the most general properties inherent in all (existing) concepts. Basic properties of concepts are expressed in logical axioms (see Axiom). First, the doctrine of the concept is considered, then comes the doctrine of judgment and, finally, inference. The doctrines of logical axioms, concepts, judgments and inferences, taken together, form pure logic. Applied logic covers in traditional logic the doctrine of definition, proof, and method. It is often preceded not by scientific-logical, but by theoretical-cognitive, psychological teachings about experience, description and formulation (especially with the help of special language, terminology) and the formation of concepts. Sometimes the doctrine of the system is added to it. Logic (as a science) is only the doctrine of thinking in concepts, but not of knowledge through concepts; it serves to increase the formal accuracy of consciousness and the objectivity of the content of thinking and cognition. The founder of Western European logic (as a science) is Aristotle, the “father of logic.” The word "logic" first appeared among the Stoics; they and the Neoplatonists clarified certain aspects of it, and in the Middle Ages scholasticism developed it in the smallest detail, in subtleties. Humanism expelled scholasticism from logic, but could not renew it. The Reformation adopted the logic of Melanchthon, the Counter-Reformation - the logic of Suarez. Having risen in principle above scholasticism, Johannes Sturm from Strasbourg developed logic; Pierre Ramet became more famous. From the 17th century The influence on logic of the spheres of thought associated with mathematics became noticeable, and in Spinoza’s geometric method it was less than in Leibniz, who used improving natural science methods in logic. From Leibniz and mathematics, as well as from neo-scholasticism, came the logic of the Wolf school. Kant's "transcendental logic" is in reality a critical theory of knowledge, a logic of German. idealism (especially Hegel's logic) - speculative metaphysics. Schopenhauer, Nietzsche, Bergson and proponents of the philosophy of life rejected traditional logic. Currently, logic has split into many directions: 1) metaphysical logic (Hegelianism); 2) psychological logic (T. Lipps, partly W. Wundt); 3) epistemological, or transcendental, logic (neo-Kantianism); 4) semantic logic (Aristotle, Kulpe, modern nominalism); 5) subject logic (Remke, Meinong, Drish); 6) neo-scholastic logic; 7) phenomenological logic; 8) logic as methodology (neo-Kantianism) and logistics, which is at the center of debates about logic.

The asymmetrical opposite of the absolute, characterized by negative extension, anti-substantiality, self-destructive...

Logic is a diverse concept that has become firmly entrenched in our life and culture of speech. In this article we will look at what logic is from a scientific point of view. Definition, types, laws of logic and historical background will help us with this.

general characteristics

So what is logic? The definition of logic is very multifaceted. Translated from Greek, it means “thought”, “mind”, “word” and “law”. In modern interpretation, this concept is used in three cases:

1. Designation of relationships and patterns that unite the actions of people or events in the objective world. In this sense, such concepts as “logical chain”, “logic of facts”, “logic of things” and so on are often used.
2. Designation of the strict sequence and regularity of the thinking process. In this case, expressions such as “logic of reasoning”, “logic of thinking”, “logic of speech” and so on are used.
3. Designation of a special science that studies logical forms and operations, as well as the laws of thinking associated with them.

Logic problems

As you can see, in each specific situation there may be at least one of several answers to the question: “What is logic?” The definition of logic problems is less extensive. The main task is to come to a conclusion based on premises and gain knowledge about the subject of reasoning in order to gain a deeper understanding of its relationships with other aspects of the phenomenon under consideration. In any science, one of the main tools is logic. It is not only an important subsection of philosophy, but also affects some mathematical teachings. "Algebra of logic" is a definition well known in mathematical circles. It is sometimes confused with which is the basis of computer science, but this is not entirely true.

Informal logic

Logic is mainly classified into:

1. Informal.
2. Formal.
3. Symbolic.
4. Dialectical.

Informal logic is the study of argumentation in the original language. This term is most common in English literature. Thus, the main task of informal logic is the study of logical errors in speech. A conclusion made in natural language may have a purely formal content if it can be demonstrated that it is nothing more than a particular application of a universal rule.

Formal and symbolic logic

The analysis of inference, which reveals that very formal content, is called formal logic. As for it, it explores symbolic abstractions that fix the formal composition of logical inference.

Dialectical logic

Dialectical logic is the science of thinking that provides knowledge about a way of reasoning that expands the possibilities of formal inference. In this case, the concept of logic can be used both in its own logical sense and in the form of a certain metaphor.

Dialectical reasoning is partially based on the formal laws of logic. At the same time, by analyzing the dynamics of the transition of concepts into their opposites, it allows for the coincidence of opposites, and therefore is guided by dialectical laws.

Logic object

The definition of logic as a science implies that its object is the human, a complex, multilateral process that involves a person’s generalized reflection of things and relationships in the surrounding world. This process is studied by various sciences: philosophy, psychology, genetics, linguistics, and cybernetics. Philosophy examines the origin and essence of thinking, as well as its identification with the material world and knowledge. Psychology controls the conditions for the normal functioning of thinking and its development, as well as the influence of the environment on it. Genetics strives to study the mechanism of inheritance of the ability to think. Linguistics seeks connections between thinking and speech. Well, cybernetics are trying to build technical models of the human brain and thinking. Logic itself looks at the process of thinking from the point of view of the structure of thoughts, as well as the correctness or incorrectness of reasoning, while abstracting from the content and development of thoughts.

Subject of logic

The subject of this field of knowledge is the logical form, the operations associated with it and the laws of thinking. It is best to consider the subject of studying logic through the process of human cognition of the surrounding world. Cognition is the process during which an individual gains knowledge about the world. There are two ways to gain knowledge:

1. Sensory cognition. It is carried out using sensory organs or instruments.
2. Rational cognition. It is carried out using abstract thinking.

Cognition is based on reflection theory. According to this theory, judgments, things and phenomena of the objective world can influence the human senses and activate the system of transmitting information to the brain, as well as activate the brain itself, as a result of which an image of these very things and phenomena is created in human thinking.

Sensory cognition

Sensory image refers to knowledge about the external properties of certain things and phenomena. Sensory cognition can occur in three forms:

1. Feeling. Reflects individual properties of an object.
2. Perception. Reflects the object as a whole, represents its holistic image.
3. Performance. This is an image of an object preserved in memory.

At the stage of sensory cognition, the essence of things and processes, their internal properties, is not always accessible to a person. The Little Prince from the story of the same name by Exupery said: “You cannot see the most important thing with your eyes.” Reason or abstract thinking comes to the aid of the senses in such cases.

Rational cognition

Abstract thinking reflects reality in terms of basic properties and relationships. Cognition of the world through abstract thinking occurs indirectly, and not explicitly. It does not involve resorting to observations and practice, but is built on the basis of deeper reasoning about the properties and relationships of objects and phenomena. For example, using the footsteps of a criminal, you can recreate the picture of the incident; using a thermometer, you can find out what the weather is like outside, and so on.

An important feature of abstract thinking is its close connection with language. Each thought is formalized using words and phrases, spoken through internal or external speech. Thinking not only helps a person describe the world around him, but also allows him to formulate new ideas, abstractions, forecasts and predictions, that is, it solves numerous logical problems. The definitions of “logic” and “thinking” in this regard are closely related to each other. Thinking, regardless of whether it is abstract or rational, can occur in three main forms: concept, judgment and inference. Let's consider them separately.

Concept

It is a form of thinking with which a person creates mental images about objects, their characteristics and relationships. A concept is impossible without a definition. But we will look at the rules of definitions in logic a little lower. In the process of forming concepts, an individual is engaged in analyzing the object of interest to him, comparing it with other objects, highlighting its main distinctive features, abstracting from unimportant features and generalizing different objects based on these features. As a result, mental images of objects, their properties and relationships are created.

Concepts play an important role in human cognitive activity. Thanks to them, it is possible to generalize what in reality exists separately. In the objective world there are no such concepts as student, apprentice, clerk, athlete, etc.; they are all generalized images that can only exist in an ideal world, that is, in a person’s head.

Opens up the possibility of obtaining knowledge about objects and phenomena based on the basic properties of a class of similar objects or phenomena. Jonathan Swift talks about what the world would be like if people did not use concepts when communicating with each other in his story about Gulliver's travels. According to the story, one day a sage advised people in conversation to use not concepts about objects, but the objects themselves. Many followed his recommendation, but in order to have a normal conversation with their interlocutor, they had to carry bags with different things on their shoulders. Of course, such a conversation with a demonstration of objects even among the owners of the largest bags was very scarce.

A concept cannot exist without a definition. In different sciences, the definition can be interpreted with some differences. Definition of concepts in logic is the process of assigning a specific meaning to a certain linguistic term. At its core, the concept is infinite, since it is developed by the universal mind. The definition is finite, since it represents the result of rational (logical) activity. According to Hegel, definition does not correspond to the Absolute and corresponds to representation. is to translate concepts into representations, getting rid of finite definitions.

The concept contains the meaning. And the definition of concepts in logic is an action aimed at identifying this meaning. Thus, a concept can be called a word that has received a definition through logical conclusions. Consequently, without a definition, a word is not a concept, even if it has a distribution. To define a concept means to describe its meaning, clarifying all the main nuances. Moreover, if you do this outside the framework of a certain knowledge system, then errors in definitions may occur. Everyone has their own logic, just like their understanding of a particular word. Therefore, when speaking on philosophical topics, it is important to define concepts.

The types of definitions in logic are presented very widely. The definition is: intensional, real, axiomatic, nominal, explicit, implicit, genetic, contextual, inductive and ostensive.

Judgment

Based on concepts about objects, a person can make judgments about them and draw conclusions. A judgment is a form of thinking in which something is affirmed or denied about the object of thought. From one judgment you can get another. For example, based on the fact that all people are mortal, we can conclude that the one who died is a person. During the construction of concepts, judgments and conclusions, everyone can make mistakes, both conscious and unconscious. To avoid them, you need to know the basics of correct thinking.

Correct thinking is one in which new true knowledge is obtained from true knowledge. Wrong thinking can also result in false knowledge. For example, there are two propositions: “If Ivan committed a robbery, he is a criminal” and “Ivan did not commit a robbery.” The judgment “Ivan is not a criminal”, obtained on the basis of this information, may be false, since the fact that he did not commit robbery does not indicate that he did not commit other crimes.

Inferences

When speaking about the correctness of inferences, scientists mean compliance with the rules of their construction and interrelation. This is the basis for the definition of the laws of logic as the science of thinking. Formal logic abstracts from the specific content and development of thoughts. At the same time, she emphasizes the truth and falsity of these thoughts. It is often called logical, with emphasis on the name of the science that studies a certain aspect of thinking.

The question of the truth or falsity of judgments and conclusions is a question of the correspondence or non-compliance of what they say with the objective world. A true judgment objectively reflects the state of things in objective reality. A false judgment, on the contrary, does not correspond to reality. The question of what truth is and how sensory knowledge relates to abstract thinking is no longer dealt with by logic, but by philosophy.

Conclusion

Today we learned what logic is. The definition of this concept is very capacious and multifaceted; it covers a wide area of ​​knowledge. Such a variety of manifestations of logic illustrates its relationship with other sciences, some of which are quite materialistic. The article also examined the main aspects of human thinking: inferences, judgments, concepts and definitions (in logic). Real-life examples helped us understand this material more easily.

Logics. Textbook Gusev Dmitry Alekseevich

Introduction, Or what is logic and why is it needed?

When starting to get acquainted with any science, we first of all answer the question of what it studies, what it is dedicated to, what it does. Logic is the science of thinking. But psychology, pedagogy, and many other sciences deal with thinking. This means that logic does not deal with all questions and problems related to thinking, not with all its areas or aspects, but only with some of them. What interests logic in thinking?

Each of us knows well that the content of human thinking is infinitely diverse, because you can think (think) about anything, for example, about the structure of the world and the origin of life on Earth, about the past of humanity and its future, about books read and films watched, about today's activities and tomorrow's rest, etc., etc.

But the most important thing is that our thoughts arise and are built according to the same laws, obey the same principles, fit into the same patterns or forms. Moreover, if the content of our thinking, as has already been said, is infinitely diverse, then the forms in which this diversity is expressed are very few.

To illustrate this idea, let's give a simple example. Let's look at three statements that are completely different in content:

1. All crucian carp are fish;

2. All triangles are geometric figures;

3. All chairs are pieces of furniture.

Despite the different content, these three statements have something in common, something unites them. What? They are united not by content, but by form. While differing in content, they are similar in form: after all, each of these three statements is constructed according to a pattern or form - "All A's are B's", where A and B are any objects. It is clear that the statement itself "All A's are B's" devoid of any content (What exactly does it talk about? Nothing!). This statement is a pure form, which, as you might guess, can be filled with any content, for example: All pines are trees; All cities are populated areas; All schools are educational institutions; All tigers are predators etc.

Let's give another example. Let’s take three statements with different contents:

1. If autumn comes, then the leaves fall;

2. If it rains tomorrow, there will be puddles on the street;

3. If a substance is metal, then it is electrically conductive.

Although different in content, these three statements are similar to each other in that they are constructed according to the same form: "If A, then B". It is clear that a huge number of different meaningful statements can be selected for this form, for example: If you don’t prepare for the test, you can get a bad mark; If the runway is covered with ice, planes cannot take off; If a word appears at the beginning of a sentence, it must be capitalized etc.

So, we noticed that our thinking is infinitely diverse in content, but all this diversity fits into just a few forms. So logic is not interested in the content of thinking (other sciences deal with this), it studies only the forms of thinking, it is not interested in what What we think, otherwise How we think, which is why it is also often called formal logic. So, for example, if the content of the statement All mosquitoes are insects is normal, understandable, meaningful, and the statement All Cheburashkas are aliens is meaningless, absurd, absurd, then for logic these two statements are equivalent: after all, it deals with forms of thinking, and the form of these two statements was the same - "All A's are B's".

Thus, form of thinking- this is the way we express our thoughts, or the scheme by which they are built. There are three forms of thinking.

1. Concept– is a form of thinking that denotes an object or a feature of an object (examples of concepts: pencil, plant, celestial body, chemical element, courage, stupidity, carelessness and so on.).

2. Judgment- this is a form of thinking that consists of concepts related to each other and affirms or denies something (examples of judgments: All planets are celestial bodies; Some schoolchildren are poor students; All triangles are not squares and so on.).

3. Inference is a form of thinking in which a new judgment or conclusion follows from two or more initial judgments. Examples of inferences:

All planets are moving.

Jupiter is a planet.

Jupiter is moving.

Iron is electrically conductive.

Copper is electrically conductive.

Mercury is electrically conductive.

Iron, copper, mercury are metals.

All metals are electrically conductive.

The entire endless world of our thoughts is expressed in concepts, judgments and conclusions. We will talk about these three forms of thinking in detail on other pages of the book.

In addition to forms of thinking, logic also deals with laws of thinking, that is, such rules, the observance of which always leads reasoning, regardless of its content, to true conclusions and protects against false ones (provided the initial judgments are true). There are four basic laws of thinking (or laws of logic). Here we will only list (name) them, and consider each of them in detail after we consider all forms of thinking.

1. Law of identity.

2. The law of contradiction.

3. The law of the excluded middle.

4. The law of sufficient reason.

Violation of these laws leads to various logical errors, as a rule, to false conclusions. Sometimes these laws are violated involuntarily, not on purpose, out of ignorance. The errors that occur in this case are called paralogisms. However, sometimes this is done deliberately, in order to confuse the interlocutor, confuse him and prove to him some false idea. Such deliberate violations of logical laws for the outwardly correct proof of false thoughts are called sophistry, which will be discussed below.

So, Logic is the science of the forms and laws of correct thinking.

Logic appeared around the 5th century. BC e. in Ancient Greece. Its creator is considered to be the famous ancient Greek philosopher and scientist Aristotle (384–322 BC). As you can see, logic is 2.5 thousand years old, but it still retains its practical significance. Many sciences and arts of the Ancient world are forever a thing of the past and represent for us only “museum” significance, interesting to us exclusively as monuments of antiquity. But some few creations of the ancients have survived the centuries, and today we continue to use them. These include Euclid's geometry (which is what we study at school) and Aristotle's logic, which is also often called traditional logic.

In the 19th century it appeared and began to develop rapidly symbolic either mathematical or modern logics, which is based on ideas put forward long before the 19th century. German mathematician and philosopher Gottfried Leibniz (1646–1716), about the implementation of a complete transition to an ideal (i.e., completely freed from content) logical form using a universal symbolic language, similar to the language of algebra. Leibniz talked about the possibility of representing a proof as a mathematical calculation. The Irish logician and mathematician George Boole (1815–1864) interpreted inference as the result of solving logical equalities, as a result of which the theory of inference took the form of a kind of algebra, differing from ordinary algebra only in the absence of numerical coefficients and powers. Thus, one of the main differences between symbolic logic and traditional logic is that the latter uses ordinary or natural language to describe correct thinking; and symbolic logic explores the same subject (correct thinking) through the construction of artificial, special, formalized languages, or, as they are also called, calculus.

Traditional and symbolic logic are not, as it might seem, different sciences, but represent two successive periods in the development of the same science: the main content of traditional logic entered symbolic logic, was refined and expanded in it, although much of it turned out to be rethought.

Now let’s answer the question why we need logic, what role it plays in our lives. Logic helps us construct our thoughts correctly and express them correctly, convince other people and understand them better, explain and defend our point of view, and avoid errors in reasoning. Of course, it is quite possible to do without logic: common sense and life experience alone are often enough to solve any problems. For example, anyone unfamiliar with logic can find a catch in the following reasoning:

Movement is eternal.

Going to school is movement.

Therefore, going to school is eternal.

Everyone will notice that a false conclusion is obtained due to the use of the word “movement” in different senses (in the first initial judgment it is used in a broad, philosophical sense, and in the second - in a narrow, mechanical sense). However, finding errors in reasoning is not always easy. Consider this example:

All my friends speak English.

The current president of America also speaks English.

Therefore, the current President of America is my friend.

Any person will see that there is some kind of catch in this reasoning, that something is wrong or wrong in it. But what? Anyone who is not familiar with logic will most likely not be able to accurately determine what error was made here. Anyone who is familiar with logic will immediately say that in this case a mistake was made - “the non-distribution of the middle term in a simple syllogism.” Or this example:

All cities in the Arctic Circle have white nights.

St. Petersburg is not located beyond the Arctic Circle.

Consequently, there are no white nights in St. Petersburg.

As we see, a false conclusion follows from two true judgments. It is clear that there is also something wrong in this reasoning, there is some error. But which one? It is unlikely that a person unfamiliar with logic will be able to immediately find it. And anyone who has a logical culture will immediately identify this error - “an extension of a larger term in a simple syllogism.”

After reading this book, you will learn not only how logical laws are violated in such reasoning, but also a lot of other interesting and useful information.

So, common sense and life experience are usually enough to navigate various difficult situations. But if we add logical culture to our common sense and life experience, then we will not lose at all from this, but, on the contrary, we will gain. Of course, logic will never solve all problems, but it can certainly help in life.

Common sense is often called practical, or intuitive logic. It is formed spontaneously in the process of life experience, by about 6–7 years, i.e. by school age or even earlier, and we all master it. So, for example, the word itself "logics", most likely, was familiar to you long before you started reading this book. In life we ​​often come across expressions such as “logical reasoning”, “illogical action”, “iron logic” etc. Even if we have never studied logic, we still fully understand what we are talking about when we talk about logic, logical or illogical.

Consider this example: anyone not familiar with logic will notice the logical incorrectness and even absurdity of the statement: I'm going in new trousers, and you're going to the gymnasium. And everyone will say that the following statement would be correct and meaningful: I'm walking in trousers, and you're walking in shorts or: I'm going to the gymnasium, and you're going to the lyceum. When we study logic, we learn that in the above example the logical law of identity is violated, since it mixes two different (unequal or non-identical to each other) situations: walking in some clothes and going somewhere. It turns out that even before becoming familiar with the law of identity, we already practically use it, we know about it, only implicitly, intuitively. In the same way, the law of identity is violated in the statement: Today we will dig a trench from this pillar until lunchtime. Even if a person knows nothing about the law of identity and about its various and numerous violations, he, nevertheless, will definitely pay attention to the fact that there is some kind of logical error in this statement (even if he could not determine which one). ).

In the same way, any person, most likely, will not be able to help but notice some kind of logical violation in the following statements: He did not take verbal permission in writing; We'll leave tomorrow evening at dawn; She was a young girl of advanced age etc. Not everyone will be able to classify this error as a violation of the logical law of contradiction. However, even if we know nothing about this law, we sense or feel its violation.

Finally, in everyday life, each of us often hears and uses expressions such as: Why should I trust you? How will you prove this? On what basis? Justify! Motivate! etc. When we say this, we are using the logical law of sufficient reason. Anyone who has not studied logic is most likely unfamiliar with this law and has not heard anything about it. However, as we see, ignorance of this logical law does not prevent us from using it practically or intuitively.

These examples indicate that all people are proficient in logic, regardless of whether they have studied it or not. Thus, we practically use logic long before we begin to study it theoretically. The question arises: why do we need to study logic if we already know it?

Answering this question, it can be noted that the same thing happens with our native language: practically we begin to use it at 2.5–3 years of our life, and we begin to study it only from school age. Why do we study our native language at school, if long before school we already speak it well? At 2.5–3 years old, we use the language intuitively, or unconsciously: having practically mastered it, we know nothing not only about declensions and conjugations, but also about words and letters, and even about the very fact that in life we ​​constantly we use language. We learn about all this only when we begin to study it at school (or senior preschool) age, as a result of which our intuitive use of language gradually turns into conscious use - we begin to speak it much better.

It’s the same with logic: having mastered it intuitively and using it practically every day, we study it as a science in order to turn the spontaneous use of logic into a conscious one, master it even better and use it more effectively.

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LOGICS

Currently, logic is a branched and multifaceted science, which contains the following main sections: the theory of reasoning (in two versions: the theory of deductive reasoning and the theory of plausible reasoning), metalogic and logical methodology. Research in all these areas at the current stage of development of logic ch. O. and are primarily carried out within the framework of logical semiotics.

In the latter, linguistic expressions are considered as objects located in the so-called. sign situation, which includes three types of objects - the linguistic itself (the sign), the object designated by it (the meaning of the sign) and the interpreter of the signs. In accordance with this, language can be conducted from three relatively independent points of view: research into the logical syntax of language, that is, the relationship of sign to sign; studies of the logical semantics of language, i.e., the relationship of a sign to the object it denotes; and studies of logical pragmatics, that is, the relationship of the interpreter to the sign.

In logical syntax, language and the logical theories built on its basis are studied from their formal (structural) side. Here the alphabets of the languages ​​of logical theories are defined, the rules for constructing various complex language constructions from alphabetic signs are specified - terms, formulas, conclusions, theories, etc. The syntactic division of a set of language expressions into functors and arguments, constants and variables is carried out, the concept of the logical form of an expression is defined , the concepts of logical subject and logical predicate are defined, various logical theories are constructed and methods of operating in them are analyzed.

In logical semantics, language and logical theories are studied from their content side; Since LANGUAGE constructions not only denote, but also describe (have) something, in logical semantics a distinction is made between the theory of meaning and the theory of meaning. The first addresses the question of what objects signs denote and how exactly they do it. Similarly, the theory of meaning addresses the question of what is the semantic content of linguistic expressions and how they describe this content.

For logic as a science, logical terms are of particular importance, since the entire procedural side of our intellectual work with information is ultimately determined by the meaning (meaning) of these terms. Logical terms include connectives and operators. Among the first, predicative connectives “is” and “is not” and propositional connectives (logical connectives) stand out: conjunctions - “and” (“a”, “but”), “or” (“either”), “if, then”, phrases - “it is not true that”, “if and only if” (“then and only then”, “necessary and sufficient”) and others. Among the second, the formative statements are distinguished - “all” (“everyone”, “any”), “some” (“exists”, “any”), “necessary”, “possibly”, “randomly”, etc. and name-forming operators - “a set of objects such that”, “that object which”, etc.

The central concept of logical semantics is the concept of truth. In logic, it is subject to careful analysis, since without it it is impossible to clearly interpret a logical theory, and, consequently, to study and understand it in detail. It is now obvious that the powerful development of modern logic was largely determined by the detailed development of the concept of truth. Closely related to the concept of truth is another important semantic concept - the concept of interpretation, i.e., the procedure of attributing, through a special interpretive function, to linguistic expressions meanings associated with a certain class of objects, called the universe of reasoning. A possible implementation of a language is a strictly fixed pair , where Ü - reasoning, and I - interpretive, assigning names to elements of the universe, i-local predicators - sets of ordered i-ok elements of the universe, l-local subject functors - i-local functions mapping i-ki elements of the universe into elements universe. Expressions related to formulas are assigned two meanings - “true” or “false” - in accordance with the conditions of their truth.

The same class of sentences can be associated with different possible implementations. Those implementations in which each , included in the set of sentences G, takes the value “true” are called a model for G. The concept of a model is especially studied in a special semantic theory - model theory. At the same time, models of different types are distinguished - algebraic, set-theoretic, game-theoretic, probability-theoretic, etc.

The concept of interpretation is of the greatest importance for logic, since through it two central concepts of this science are defined - the concepts of logical law (see Logical Law) and logical implication (see Logical Consequence).

Logical semantics is a meaningful part of logic, and its conceptual apparatus is widely used for the theoretical justification of certain syntactic, purely formal constructions. The reason for this is that the total content of thought is divided into logical (expressed in logical terms) and (expressed in descriptive terms), and therefore, by highlighting the logical form of expressions, we are, generally speaking, not abstracting from any content. Such a distraction, i.e. consideration of the formal side of thoughts, is only a way of isolating in its pure form their logical content, which is studied in logic. This circumstance makes logic coming from Kant unacceptable as a purely formal discipline. On the contrary, logic is a deeply meaningful science in which each logical procedure receives its theoretical justification through substantive considerations. In this regard, “formal logic” as applied to modern logic is imprecise. In the true sense of the word, one can only talk about the formal aspect of research, but not about formal logic as such.

When considering certain logical problems, in many cases it is also necessary to take into account the intentions of the interpreter who uses linguistic expressions. For example, consideration of such a logical theory as the theory of argumentation, dispute, discussion is impossible without taking into account the goals and intentions of the participants in the debate. In many cases, the methods of polemics used here depend on the desire of one of the disputing parties to put its opponent in an uncomfortable position, confuse him, and impose on him a specific problem under discussion. Consideration of all these issues constitutes the content of a special approach to the analysis of language - “logical pragmatics”. The most fundamental branch of logic is the theory of deductive reasoning. Currently, this section in its hardware (syntactic, formal) part is presented in the form of various deductive theories - calculi. The construction of such an apparatus has a double meaning: firstly, theoretical, since it allows one to identify certain laws of logic and forms of correct reasoning, on the basis of which all other possible laws and forms of correct reasoning in a given logical theory can be substantiated; secondly, purely practical (pragmatic), since the developed apparatus can be and is used in the modern practice of scientific knowledge for the precise construction of specific theories, as well as for the analysis of philosophical and general scientific concepts, methods of cognition, etc.

Depending on the depth of analysis of statements, there are propositional calculi (see Propositional Logic) and quantifier theories - predicate calculi (see Predicate Logic). In the first, the analysis of reasoning is carried out with the precision of identifying simple sentences. In other words, in propositional calculi we are not interested in the internal structure of simple sentences. In predicate calculi, the analysis of reasoning is carried out taking into account the internal structure of simple sentences.

Depending on the types of quantified variables, predicate calculi of different orders are distinguished. Thus, in first-order predicate calculus, the only quantifiable variables are individual variables. In second-order predicate calculus, variables for properties, relations and objective functions of different localities are introduced and begin to be quantified. Predicate calculi of the third and higher order are constructed accordingly.

Another important division of logical theories is associated with the use of languages ​​with different categorical grids to represent logical knowledge. In this regard, we can talk about theories built in languages ​​of the Frege-Russell type (numerous variants of predicate calculus), syllogistic (various syllogistics, as well as Lesniewski, which is a modern form of singular syllogistics) or algebraic (various algebras of logic and class algebras - Boolean algebra, Zhegalkln algebra, de Morgan algebra, Hao Wang algebra, etc.). For many theories constructed in languages ​​with different categorical grids, their mutual translatability is shown. Recently, a category-theoretic language based on a new mathematical apparatus - category theory - has begun to be actively used in logical research.

Depending on the method of constructing conclusions and proofs (see Logical inference) used in logical theories, the latter are divided into axiomatic calculi, calculus of natural deduction and sequential calculus (see Sequence calculus). In axiomatic systems, the principles of deduction are given by a list of axioms and rules of inference that allow one to move from some proven statements (theorems) to other proven statements. In systems of natural (natural) inference, the principles of deduction are given by a list of rules that allow one to move from some hypothetically accepted statements to other statements. Finally, in sequential calculi, the principles of deduction are specified by rules that allow one to move from some statements about deducibility (they are called sequents) to other statements about deducibility.

The construction of one or another calculus in logic constitutes a formal line of logical research, which it is always desirable to supplement with substantive considerations, i.e., the construction of a corresponding semantics (interpretation). For many logical calculi such semantics exist. They are represented by semantics of various types. These can be truth tables, so-called. analytical tables, Beta tables (see Semantic tables), various kinds of algebra, possible worlds of semantics, descriptions of states, etc. On the contrary, in the case when a logical system is initially constructed semantically, the question arises of formalizing the corresponding logic, for example, in the form of an axiomatic system.

Depending on the nature of statements, and ultimately on the types of relations of things that are studied in logic, logical theories are divided into classical and non-classical. The basis of such division is the adoption of certain abstractions and ideas when constructing the corresponding logic. In classical logic, for example, the following abstractions and idealizations are used: a) the principle of ambiguity, according to which every statement is either true or false, b) the principle of extensionality, i.e., permission for expressions that have the same meaning

understanding, their free replacement in any context, which suggests that in classical logic they are only interested in the meaning of expressions, and not their meaning, c) actual infinity, which allows one to reason about essentially non-constructive objects, d) the principle of existentiality, according to which the universe of reasoning must be a non-empty set, and each proper must have a referent in the universe.

These abstractions and idealizations form the point of view, the angle from which we see and evaluate the objective. However, no set of abstractions and idealizations can fully cover it. The latter always turns out to be richer, more flexible than our theoretical constructions, which makes the free variation of the original Principles justified. In this regard, a complete or partial rejection of any of these principles takes us into the realm of non-classical logics. Among the latter there are: many-valued logics, in particular probabilistic and fuzzy ones, in which the principle of double-valuedness is abandoned; intuitionistic logics and constructive logics, which explore reasoning within the abstraction of potential feasibility; modal logics (alethic, temporal, deontic, epistemic, axiological, etc.), relevant logics, paraconsistent logics, question logics, which consider statements with non-extensional (intensional) logical constants; logics free from existential assumptions, in which the principles of existentiality are abandoned, and many others.

The above shows that logic as a science that gives theoretical laws of thinking is not something once and for all. On the contrary, each time with the transition to the study of a new area of ​​objects that require the adoption of new abstractions and idealizations, taking into account new factors that influence the reasoning process, this theory itself changes. That. Logic is a developing science. But what has been said also demonstrates something more, namely, that the composition of the logic of a certain theory of the laws of thinking is directly related to the acceptance of certain ontological assumptions. From this point of view, logic is not only a theory of thinking, but also a theory of being (the theory of ontology).

An important section of modern logic is. The latter examines various problems related to logical theories. The main questions here are about the properties that logical theories possess: consistency, completeness, the presence of resolving procedures, independence of initial deductive principles, as well as various relationships between theories, etc. In this sense, metalogic is, as it were, a self-reflection of logic regarding its constructions. All metatheoretical research is carried out in a special metalanguage, which uses ordinary natural language, enriched with special terminology and metatheoretical deductive means.

Logical methodology is another branch of modern logic. Typically, methodology is divided into general scientific, within which cognitive techniques used in all areas of scientific knowledge are studied, as well as the methodology of individual sciences: the methodology of deductive sciences, the methodology of empirical sciences, as well as the methodology of social and humanitarian knowledge. In all these sections, logical methodology is involved as a specific aspect of the study. Thus, in general methodology, logical aspects include the study of such cognitive techniques as the development and formulation of concepts, the establishment of their types and various ways of operating with conceptual constructs (division, classification), definitions of terms, etc.

Particularly great success has been achieved in the field of methodology of deductive sciences. This was due both to the construction of logic itself in the form of a deductive apparatus, and to the use of this apparatus to substantiate such a deductive discipline as. All this required the development of significantly new cognitive methods and the introduction of new methodological concepts. In the course of the work carried out here, it was possible, for example, to generalize the concept of functions in such a way that it actually moved into the category of general methodological, epistemological concepts. We now have the opportunity to consider not only numerical functions, but also functions of any other nature, which has made it possible to make functional analysis of language the leading method for studying linguistic expressions. It was possible to work out such important methods of cognition as the method of axiomatization and formalization of knowledge with all care and rigor. For the first time, it was possible to define theoretical-evidential (deductive) methods of cognition in a clear and, most importantly, diverse form, to develop a theory of expressibility and definability of some terms through others as part of theories, and to define the concept of a computable function in various ways.

Currently, the logical problems of the methodology of empirical sciences are being actively developed. This area includes research on the construction and testing of hypotheses (in particular, the hypothetical-deductive method), the analysis of various types of plausible reasoning (induction and analogy), and measurement theory. Here, interesting results were obtained on the relationship between the empirical and theoretical levels of knowledge, procedures of explanation and prediction, and operational definitions. Various models of empirical theories are constructed to clarify their logical structure.

General methodological and logical principles include those laws and principles of knowledge that are studied within the framework of dialectical logic. In many cases, they act as some warning signs about what surprises we may encounter on the path of knowledge. In the field of methodology of empirical, as well as social and humanitarian knowledge, absolute and relative truth is of great importance; in the field of historical knowledge, the requirement for the coincidence of the historical and the logical becomes essential, which in fact means the usual requirement for the adequacy of knowledge, transferred to the sphere of historical disciplines. Recently, attempts have been made to construct deductive systems in which certain features of dialectical logic are formalized.

For thousands of years, logic was a compulsory discipline in school and university education, that is, it fulfilled its general cultural task - propaedeutics of thinking. Modern logic has fully retained this didactic and educational function. However, the recent development of the powerful apparatus of modern logic has made it an important applied discipline. In this regard, we point out the essential

Consolidated encyclopedia of aphorisms

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, from solving complex technical and business problems to persuading interlocutors and making purchases in a store.

But despite the high need for this skill, we often make logical mistakes without knowing it. Indeed, among many people there is an opinion that it is possible to think correctly on the basis of life experience and so-called common sense, without using the laws and special techniques of “formal logic”. To perform simple logical operations, express elementary judgments and simple conclusions, common sense can also be suitable, but if we need to understand 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 way. 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 is the mental process of 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 in order for the connections found to truly reflect the true state of 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 of 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 being tested, regardless of their level of education, social class, type of activity, linguistic and cultural characteristics. That is, it can be said with a high probability that the data obtained as a result of this test from two people from different parts of 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 that connects the figures in the picture and indicate the number of the correct figure by choosing from the proposed options. 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 model relationships not only between real objects, but also between abstract and figurative ideas that thinking itself has 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 of 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-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 area of ​​human activity, including the exact sciences and humanities, economics and business, rhetoric and public speaking, the 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 provides a person with useful techniques for obtaining 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 associated 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 of the theoretical foundations of 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. The ability to quickly and effectively solve complex logical 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 teaching of practical logic is usually associated with passing various tests and solving problems to test the level of intelligence development (IQ) and for some reason does not address 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. The purpose of this course is to 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 - the 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. Concepts come in 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 (what the judgment is about), a predicate (what is said about the subject), a copula (what connects the subject and the predicate) and a quantifier (the scope 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 correctly construct various types of judgments, understand their structural elements, features, 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 for constructing 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 examine the following laws of formal logic: the law of identity, the law of excluded middle, the law of contradiction, the law of sufficient reason, as well as De Morgan's laws, the laws of deductive inference, Clavius' law and the laws of division. By studying examples and completing special exercises, you will learn how to purposefully use each of these laws.

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 a general rule is derived from several particular cases. 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. This means that at this point it should 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 be the topic of the 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 distinctive feature is that the premises and conclusions in them are connected by a relation of strict logical consequence, 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 the theoretical material and practiced a little. 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 smartest people in 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" (educational and methodological complex). Dmitry Alekseevich Gusev;
• “Logic for Lawyers” (collection of problems). HELL. Getmanova;