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Lecture notes, cheat sheets
Logics. Lecture notes: briefly, the most important
Directory / Lecture notes, cheat sheets Table of contents
LECTURE No. 1. Introduction to the course of logic In its development, mankind has come a long way - from distant times, when the first representatives of our kind had to huddle in caves, to the cities in which we and our contemporaries live. Such a time gap did not affect the essence of man, his natural desire to know the world around him. However, the knowledge of something is impossible without the ability to separate the true from the false and the truth from the lies. It so happened that the truth has always been an ambiguous phenomenon. She generously endowed some, brought misfortune and sorrow to others. And here everything depends on the person himself, his upbringing, will and fortitude. But everyone should understand that only the truth contributes to the development of a person, both spiritually and scientifically. Science has not always followed the path of establishing the truth, and this path has shown its inconsistency. There were attempts to characterize a person's personality by the shape of his head, and many more no less absurd directions. But if such mistakes were not made in the development of science, it would be impossible to determine the value of correct approaches. The achievement of the desired result is also hindered by the fact that the path to true knowledge has always been thorny. Many scientists, fighting for their idea and the discoveries that they managed to make (sometimes centuries ahead of schedule), sacrificed their lives. Suffice it to recall the Italian scientist Giordano Bruno, who burned at the stake for not wanting to renounce his theory of the infinity of the Universe and the countlessness of its worlds. Or modern nuclear physicists, or microbiologists who were exposed to radioactive radiation and experimented on themselves for the benefit of others. However, despite this, not all useful discoveries now benefit people. Some projects are closed due to lack of funding, others serve the opposite purpose. For example, the atomic reaction from the very moment of discovery had a dual character. On the one hand, it effectively serves people, giving huge amounts of energy, and hence heat and light. On the other side of the scale lie the lives of those who died, exposed to deadly radiation. Therefore, I want to believe that in the future such knowledge will be used only for the benefit of man. Learning is light and ignorance is darkness. Knowledge is power. These are sayings known to everyone since childhood. Indeed, the greater the knowledge of a person, the greater his power. However, it is almost impossible to obtain true knowledge without the help of special techniques. There is an opinion that it is possible to think correctly without using the laws of logic and without even knowing about them, on the basis of worldly experience and common sense. However, it is not. For example, you can solve a mathematical problem by reaching, as they say, "with your own mind", but another such problem will no longer be obeyed, because it is based on rules not known to the solver. Or he can easily make a mistake that will result in a completely wrong answer. This is also the case with thinking. Only the study of logic and the constant training of logical abilities allow a person to think correctly, clearly and without errors. And a mistake, even the smallest one, can cost an individual and even humanity very dearly. For example, fascism, as a political phenomenon that led to the most devastating war in the modern world, was based on an ideology that was deliberately wrong. However, there was no person who could refute the ideas of fascism in time, expose them. This is just one example that makes it clear how necessary logic is in the life of a person, not only engaged in science or politics, but also an ordinary citizen, in order not to get into trouble, not to be deceived, not to be subjected to undesirable consequences of a carelessly spoken word. Thus, logic as the doctrine of the correctness of thinking, questions and answers, the construction of new hypotheses and evidence is necessary for every reasonable person. LECTURE No. 2. Logic. The main stages in the development of science The history of logic is long. As mentioned above, at all times man has strived for the truth, however, certain conditions were necessary for the emergence of the doctrine of the correctness of thinking. Here is the general mental development of a person, and the peculiarities of culture. And, of course, the presence of a spoken language is necessary. All the necessary factors were combined more than two thousand years ago in India, China, Greece. Initially, logic was born and developed as part of philosophy. Word "philosophy" comes from two Greek words "philo" and "sophos", "love" and "science" respectively. Thus, "philosophy" literally means "love of science". Philosophy is a science that combines all human knowledge about the world around us, the features of human consciousness and the laws of being. In general, the process of development of logic can be divided into several stages: the logic of the Ancient World, ancient logic, the logic of the Middle Ages, the logic of the Renaissance, the New Age, and, finally, modern logic. Let's pass to consideration of each stage which is passed by logic in the development. 1. The logic of the ancient world The logic of the ancient world owes its appearance to the philosophers of China, India and Greece. It is known that in the early stages of development, logical knowledge was of an ontological nature, i.e., the laws of thinking were equated with the laws of being. Much attention during this period was paid to inference, and the latter was practically identified with proof. Rhetoric gave impetus to the development of logic. Oratory used the rudiments of logical knowledge to achieve the main goal of the speaker - to convince the listeners, and not to establish the truth, as is the case in later periods. The logical element here is subordinate in nature, it is, as it were, an integral part of oratory. Philosophy as a body of scientific knowledge originated and developed simultaneously in ancient states that had different views on the world around them, with different approaches to its study and with a different body of accumulated knowledge. Therefore, the philosophical knowledge of the Ancient World can be divided into two depending on the state in which it originated. One of these movements arose in Ancient Greece, the other was fundamentally an Eastern approach to science, characteristic of philosophers in India and China. Modified under the influence of time, the Greek direction of philosophy is now represented in Russia, Western Europe and America, where it came through the Roman Empire and Byzantium along with the belief in one God. The Indo-Chinese direction of philosophy was adopted in Mongolia, Japan, Korea, Indonesia and other countries [1]. It is necessary to consider in more detail the logic of the ancient states. 2. Ancient India and Ancient China Ancient india. Ancient India is a very original country. It is known for great thinkers and numerous philosophical movements. Ancient Indian philosophy to this day is considered a meaningful and well-developed system that accurately reflects many features of the surrounding world. The logical knowledge accumulated by ancient Indian scientists also has a fairly clear structure and, what is especially important, contains logical concepts, approaches and methods that became known in the system of Western logic only several centuries later. Philosophical ideas in ancient India were developed by representatives of 16 schools, the main of which were the Charvaka, Lokayata (founded by Brihaspati and his student Charvaka), Vaisheshika (the founder of Canada), Nyaya (Gautama) and Jainism (Vardhamana Mahavira) schools. These schools belonged to the materialistic direction of philosophy, i.e., their representatives believed that the material world exists objectively, and matter is primary in relation to consciousness and exists forever. They were opposed by representatives of philosophical schools preaching an idealistic approach to the study of the world. They considered the spiritual principle, consciousness and thinking to be primary, and pushed the material world into the background. Yoga and Buddhism, as well as Mimamsa and Vedanta, adhered to such ideas. It is necessary to mention the school that adheres to an intermediate position, i.e., assigns equal positions to the material and spiritual (ideal) principles. In connection with such a variety of philosophical approaches, disputes between representatives of different philosophical schools were of considerable, or rather, even decisive importance in the development of the logic of ancient India. Today, the Vedas are considered the main and oldest literary monument of ancient Indian philosophy. It is a collection of philosophical ideas and thoughts. However, the Vedas are of a general nature, which led to the creation of the Upanishads by the Brahmins, which interpret and interpret the provisions contained in the Vedas. Logical knowledge, on the other hand, did not have a systematic consolidation for a long time, but was written down in the form of brief aphorisms and systematized only in the XNUMXth century. BC e., starting from Dinang. The development of the logic of ancient India has about two millennia, and partly because it has not yet been fully studied. This is also seen in the works devoted to the logic and philosophy of Ancient India. Despite the considerable number of such publications, they do not contain a unified approach to the issue under consideration. However, this does not prevent the recognition of the fact that ancient Indian logic has an original character and features that distinguish it from the logic of ancient Greece. So, the syllogism here is divided not into ten, but into five members (thesis, basis, example, application, conclusion); deduction and induction are considered inseparable; mental and verbal speech are distinguished; the basis of perception is the acquired experience, and the judgment is considered part of the inference. Despite a long period and a special approach to the development of logic, in ancient India there is only one complete system of it - navya-nyaya, translated as "new logic". Here, logic is seen as a new science, contributing to a more complete and objective knowledge of oneself and the world around, as well as obtaining truthful information. However, the traditional approach to categories makes the original Navya-nyaya logical teaching somewhat awkward. Also, as its disadvantage, one can point out the lack of differences between the abstract conclusion and a specific example. All approaches to the study of logic can be divided into two branches: classical and non-classical. The first is characterized by the presence of two truth values, that is, judgments can be either true or false. The second implies an infinite set of truth values, constructiveness of methods of proof and modality of judgments. Sometimes negations contained in classical logic can be excluded. It should be mentioned that modern, mathematical logic contains elements of both classical and non-classical logic. Late Navya-nyaya, according to some scholars, in many ways surpassed the achievements of Aristotle's logic. However, despite the high level of development and an enviable understanding of the laws of logic, the philosophers of ancient India did not use symbols. They were replaced by a complex system of cliches, using which many different expressions could be obtained. Ancient China. In Ancient China, much attention was paid to ethical, philosophical and political issues, which were enshrined in a large number of treatises. This is how the science of names (theory of names) developed, the laws of thinking and the specifics of reasoning and statements were revealed. The origin of the logic of Ancient China, according to modern historians, took place in the periods of Chuncu and Zhangguo, which are known for the emergence of a new concept of "philosophical discussion". Also, this period (722-221 BC) is characterized by the emergence and development of a process called "the rivalry of a hundred schools." Among the well-known representatives of philosophical teachings, who also develop the ideas of logic, are the names of Confucius and Mozi. The philosophical schools that existed in China at that time include mingjia (school of names), fajia (school of laws), zhujia (developing Confucian ideas) and mojia (school of Mohists). As a result of the activities of these schools, a more or less harmonious system of logic gradually began to take shape. However, since logical knowledge was fragmented, fixed not in one source, but in many treatises, they required systematization. A school was needed that would unite all knowledge about logic in a single act, which would greatly simplify the use of logical achievements. The Mojia school became such a school. Later Mohists, using the philosophy of Mozi, wrote the first treatise on logic in China called "Mobian". Logic in ancient China dealt with a number of problems specific to Chinese society of that period. Among them are theories of names, statements, reasoning and disputes. As can be seen, the logical science of ancient China was closely connected with writing and especially the spoken language, and was, as it were, hampered by it. Thus, the main efforts of philosophers were concentrated around the concepts of "min" and "tsy", i.e., the theory of names and statements, but no differences were made in the meaning of these concepts. China has always been a very distinctive country with a rich culture, a developed social system and a strong sense of submission. The younger in age must obey the elder, the latter obeyed the elder in position, etc. Wise men and elders always enjoyed certain privileges. This situation could not but be reflected in the logic of ancient China. Political and ethical doctrines had a strong influence on logical theories here, and logic itself was applied in nature and was used to achieve rhetorical goals. Therefore, there was practically no clear system of knowledge about inferences. Preference was given to the content of thinking over form. As a result, although logic in Ancient China arose in time earlier than ancient Greek, its structure was never built and remained in its infancy. 3. Ancient Greece It was here that the problems of logic were considered and developed most thoroughly. Logical questions are considered here by such philosophers as Parmenides and Zeno (representatives of the Elean philosophical school), Heraclid, the sophists Protagoras, Gorgias and others, Democritus and Aristotle. The activities of these philosophers directly or indirectly touched upon questions of logic. The ideas of the representatives of the Eleatic direction and the adherents of the logic of Heraclid came into conflict due to their opposite. The Eleatic school preached metaphysical theories, that is, a way of studying phenomena in which they are considered separately from each other and in an unchanged state. Heraclitus philosophy adhered to the ideas of dialectics (phenomena are studied in development and interaction). The main feature that characterizes the philosophical approach of the sophists is that they human beings were proposed as a research object, and not the surrounding world, as it was before. The Sophists viewed logic not as a science that allows one to establish the truth, but as a means of achieving victory in an argument. To do this, they deliberately violated the laws of logic. First opposed the sophists Democritus (460-370 BC), who belonged to the materialistic philosophical school. The philosophical system created by Democritus contains the doctrine of being, the theory of knowledge, ethics and aesthetics, cosmology, physics, biology, politics and logic. He also developed and consolidated in his Treatise "On Logic" ("Canons") the first system of logic. Democritus is considered one of the founders of inductive logic, since his treatise is based on empirical principles. Considering judgments, Democritus distinguishes subject and predicate in them. Problems of logic were also dealt with Socrates (469-399 BC) and Plato (428-347 BC). In the teachings of Socrates, the method was considered the main one, which made it possible to obtain the truth, and also contained the idea that knowledge of any subject becomes possible only if it is reduced to a general concept and on this basis this concept is judged. To achieve the truth, Socrates suggested that his students give a definition to any phenomenon, feature or characteristic feature inherent in the surrounding world or a person. Then, if such a definition turned out, in his opinion, to be insufficiently complete or correct, he, using examples from life, pointed out the mistakes made by the interlocutor, and then changed and supplemented it. Socrates considered the achievement of knowledge to be the discovery of patterns and the definition of a concept for a number of things. In the process of achieving knowledge, the common features of objects and the differences between them were taken into account. Ancient Greek philosopher Plato was a student of Socrates and developed theories of knowledge and logicbased on the teacher's ideas. Using his theories, Plato first received new concepts, and then tried to break them down into types and systematize them. To do this, he used his favorite technique called "dichotomy", that is, the division of the concept of A into B and not B (for example, crimes can be intentional and unintentional, and animals can be vertebrates or invertebrates). As in the school of Socrates, the students of Plato's Academy were busy getting new definitions. In modern philosophical science there is a mention of a curious case connected precisely with definitions. Plato, describing man, said that man "is a two-legged animal without feathers." Having learned about this definition, the famous philosopher Diogenes plucked a chicken and brought it to Plato's Academy during a lecture with the words: "Here is Plato's man." Plato was forced to admit the insufficiency of his definition and made changes according to which "man is a two-legged animal without feathers and with flat nails." Plato created a system of objective idealism, according to which the spiritual principle (as opposed to subjective idealism) exists independently of human consciousness. In this theory, Plato used the division of the world into material and ideal (spiritual) and made the first dependent on the second. In other words, the material world, according to Plato, is unstable and changeable, in contrast to the ideal world, which exists independently of matter and human consciousness. He considered ideas to be eternal and unchanging, and the material world, as it were, a projection of the ideal. In other words, a thing is only a reflection of an idea. Plato developed the theory of judgment, created two rules for the division of concepts, and also distinguished the relationship of difference from the relationship of opposites. Thus, many philosophers of ancient Greece worked on questions of logic, but its founder is considered to be Aristotle Stagirsky (Aristotle was born in the city of Stagir - this is where his nickname came from). He devoted himself to the study of many sciences, such as philosophy, logic, physics, astronomy, psychology, rhetoric, etc. Many of his works are devoted to these subjects. It was Aristotle who formalized the knowledge of logic into a clear system and discovered that knowledge, no matter where it comes from, always has a linguistic expression. From this he concluded that scientific knowledge is a sequence of statements united by logical connections and deduced from one another. Aristotle's logic is called formal or traditional. It includes such sections as concept, judgment, laws of correct thinking, inferences, argumentation and hypothesis. Aristotle's important achievement is that he first formulated laws of correct thinking: the law of identity, the law of non-contradiction and the law of excluded middle, and also began to study human thinking in order to derive its logical forms. These laws were formulated in the most important work of Aristotle "Metaphysics". Aristotle created syllogism theory, reviewed theory of definition and division of concepts and theory of proof. The main works in this area are treatises "First Analytics" и "Second Analytics", which subsequently, along with other works, were combined into "Organon" - a method, means or instrument of cognition of reality. This work contains the opinion that the laws of logic are inextricably linked with the surrounding world and with man and cannot exist in isolation from them. This conclusion also confirms that the logic corresponds to the culture of a particular society and reflects the features that characterize this culture. For example, in Indian logic there is no law of the excluded middle, which is characteristic of Aristotle's logic. According to scientists, this trend can be traced in the cultures of these countries as a whole. Thus, the population of countries in which the logic of Aristotle has become widespread tends more towards straight lines, which is clearly seen in judgments about good and evil, which are characterized by uncompromisingness, as well as in architecture (ancient columns) and weapons (a straight sword). Eastern countries are closer to the curve line (Muslim crescent, crooked swords, greater freedom of judgment). Aristotle considers a statement to be true if it corresponds to the situation of the surrounding world, i.e., reflects the real state of things. False, therefore, were considered judgments that are used not to reflect objective reality, but to consciously or accidentally change this reality, i.e., "fitting" the phenomena of the surrounding world to the required answer. In other words, what is false is that which breaks existing connections between things or creates new ones that exist only in words. Starting from this concept of truth, Aristotle creates his own logic. In conclusion, it is necessary to mention Stoic logic - a system of knowledge developed by adherents of the megaro-stoic school, the Stoics Zeno and Chrysippus and megarics Diodorus, Stilpo, Philo and Eubulides. As a result of the activities of this school, modern logic received an analysis of logical concepts negation, conjunction, disjunction and implication. They saw the task of logic as getting rid of errors and creating the opportunity to correctly judge things. Logic must study not only verbal signs, but also the thoughts expressed in them. Going beyond formal logic, representatives of the Megaro-Stoic school divided logic into dialectics and rhetoric. Unfortunately, the ideas of this philosophical school in the field of logic have only partially survived to our time. 4. Medieval logic Medieval logic is, for the most part, an interpretation and analysis of ancient philosophical theories. Mainly studied questions modal logic, the theory of logical implication, the theory of semantic paradoxes, and also an analysis of selecting and excluding judgments was carried out. The main directions considering questions of logic were the direction of realists and nominalists. The first believed that general concepts exist independently of individual things. Nominalists took opposite positions and believed that general concepts only name individual things that are real. It should be noted that both of these approaches are incorrect. The most famous scientists who worked on questions of logic in the Mediterranean are William of Ockham, Duns Scotus, Raymond Lull, Jean Buridan, Albert of Saxony. Special mention should be made of William Occam, who is famous for creating a logical tool called "Occam's blade". The science developed in Syria served as a conductor between ancient and Arabic logic. Questions of logic in the Arab world were dealt with by such scholars as al-Farabi, considered the founder of Syriac logic, Ibn Sina (Avicenna), Ibn Rushd (Averroes). Al-Farabi was an ideological follower of Aristotle. He commented Aristotle's main work "Organon". Al-Farabi's logic is aimed at studying scientific thinking and examines questions of truth, based on the concept of truth developed by Aristotle. The structure of his logic consists of two parts, one considers representations and concepts, and the other studies the theory of judgments, inferences and evidence. Al-Farabi paid special attention to issues of the theory of knowledge and grammar. The interpretation of the works of Aristotle was continued by Ibn-Sina. He used translations and commentaries of ancient works created by al-Farabi. Avicenna studied Aristotelian syllogistics, traced the dependencies and connections between categorical and conditional propositions, as well as the expression of implication through disjunction and negation. The scientist consolidated his ideas in the textbook "Logic". The most famous and used work on logic is treatise "Summulae logicales", containing a number of new ideas in the field of propositional logic. This work was written by Peter of Spain. LECTURE No. 3. The logic of the Renaissance and the New Age 1. Logic of the Renaissance A characteristic feature of the Renaissance is the ever-increasing importance of science. This is a time of scientific and geographical discoveries and an increase in the influence of mathematics. The logic of this time is characterized by the strengthening of empirical tendencies. One of the scientists working during the Renaissance was Francis Bacon (1561-1626), who is considered to be the founder of English materialism. He made a significant contribution to the development of the materialistic logical approach. F. Bacon believed that the only correct approach to the study of the subject is not only the collection of information, but also its intellectual processing and, thus, the creation of scientific theories. The main achievement of F. Bacon is his work "New Organon", which was intended to replace the “Organon” (means of knowledge), written by the ancient Greek philosopher Aristotle. F. Bacon's work discusses issues of induction, methods for determining the causal relationship between objects and phenomena (similarities and differences of accompanying changes, residues and the combined method of similarities and differences). It should be noted that F. Bacon studied the works of Aristotle in translations and revisions of medieval scholars, as a result of which he was unfair to his Organon. In the Renaissance, other scientists also dealt with questions of logic, among which the French philosopher is especially famous. Rene Descartes (1596-1650). He formulated four rules for the correct approach to scientific research. R. Descartes created a scientific work "Logic, or the art of thinking", the main idea of which was the liberation of Aristotle's logic from the changes introduced by medieval scientists. 2. The logic of modern times Immanuel Kant (1724-1804), a famous scientist of the modern period, proposed the division of logic into two types - formal and transcendental. Ordinary logic deals with the study of concepts, judgments and inferences. Transcendental logic examines the forms of thinking, and considers knowledge as prior to experience and independent of it. A priori (a priori - “from previous”) knowledge, thus, is a condition of experimental knowledge that gives it a formalized, universal and necessary character. A priori forms of logical knowledge, according to I. Kant, are designed to organize the chaos of sensations and provide complete and reliable information. I. Kant distinguished logical causes and effects from real causes and effects, which is an important contribution to the theory of science. I. Kant considered judgment as an expression of knowledge and divided the latter into two types: analytical and synthetic. Analytical judgments do not create new knowledge, but only define what already exists. Synthetic judgments can be a posteriori (a posteriori - “from what follows”), which are placed in direct dependence on experience, originating from it, and a priori, independent of experience and, moreover, even preceding it. From this it is clear that these two types are opposite to each other. It should be noted that even today among logicians and philosophers there is no unity of opinion regarding the a priori judgments of I. Kant. Georg Wilhelm Friedrich Hegel (1770-1831) is considered the most famous German philosopher of the classical school. He, relying on an objective-idealistic foundation, developed a systematic theory of dialectics. The main concept of this theory is development, which is understood as a characteristic of the activity of the world spirit (absolute). The Absolute is characterized by a supertemporal movement in the realm of pure thought in an ascending series of increasingly concrete categories (being, nothingness, quality, quantity, measure, etc.). G. Hegel identifies logic with dialectics. In this regard, formal logic is not only criticized by scientists, but also denied by them. This relation can be seen in the work of the scientist "Science of Logic". G. Hegel also criticizes the views of I. Kant. LECTURE No. 4. The subject of logic 1. Sensation, perception and representation as forms of knowledge of the surrounding world The subject of logic is understood differently by different scientists. Some indicate reasoning as the subject [2], others adhere to a broader interpretation and call thinking as the subject [3]. However, on the main points of this issue, the views of scientists coincide. Let's move on to a more specific consideration of this problem. The subject of logic is inextricably linked with such concepts as cognition, thinking, logical forms and logical laws. Logic is a science that studies the methods and principles of cognitive activity, its means. Such a study is impossible without defining two levels of knowledge: empirical and theoretical. Empirical level has the object of reality, directly reflected by the human senses. In relation to it, observation is possible, influence on its characteristic features through experiments, experiments. Thus, empirical knowledge provides information about the subject through observation, experience, experiment. Theoretical way of knowing often studies objects and phenomena that are inaccessible to direct sensory reflection. Human thinking arises only on the basis of knowledge and is impossible without it. Human knowledge does not exist without the mediation of sensations. Any information that a person receives comes from the outside world. Thus, the only source of information is the sense organs. It is through these organs that we become aware of the properties of the surrounding world. Each item has not one, but several properties (for example, weight, size, shape, texture, etc.). The sense organs, like the human brain, are amenable to training and, depending on training, provide more or less information for cognition. The training of the brain is characterized by its ability for a more fruitful process of thinking. Through sensations, the connection of consciousness with the outside world is carried out more fully, the more sense organs are involved at a given moment. There are cases when one or more of a person’s senses are damaged or do not function at all. Then the sensitivity of the others is heightened and even, to one degree or another, fills in the functions of those who are missing. Feeling - this is a reflection of the individual properties of the object at the time of its direct impact on the senses. Perception - this is a holistic image of the totality of the properties of an object that arises at the moment of the direct impact of the latter on the senses. Human perception is manifested in determining the specific properties of an object and their expression. In other words, a person pays attention to a specific property of an object (shape, color, smell, taste, etc.), as well as the degree of this property (round or oval, more or less sweet, heavy or light). From this we can conclude that perception is individual for each person. It depends on the characteristics of his senses and the experience acquired by a person; his education and attitude to the subject, mood. Thus, an electric discharge (artificial lightning) will be perceived differently by a person not involved in science, a physicist and, for example, an artist. An “ordinary” person will simply be impressed by the beauty of the spectacle; the artist will note the riot of colors and the polymorphism of the discharge. A physicist will be most interested in the readings of instruments. The connection between perception and human experience can be illustrated by the example of I. A. Krylov’s fable “The Monkey and the Glasses.” At the instigation of others, Monkey purchased several glasses in order to improve her vision. Then, not knowing the use of this item and based on her life experience, Monkey unsuccessfully tried to find a use for the glasses, using them as decoration. The following phrase highlights this situation very clearly: Unfortunately, this is what happens to people: // No matter how useful a thing is, without knowing its value, // The ignoramus turns his understanding of it for the worse... From sensations and perceptions, an idea is formed, an image of an object that is not perceived at the moment, but was previously perceived in one way or another. Representation is divided into reproducing and creative. reproducing - this, as the name implies, is an idea of \uXNUMXb\uXNUMXba object or phenomenon that was previously perceived by the human senses directly and remembered. creative performance based on stories, descriptions of an object or phenomenon. Such an idea can also arise in the imagination of a person. For example, the image of a non-existent person or animal that arises in the process of the artist's activity. Or a geographical place where a person has never been can be recreated by him from eyewitness accounts. Also, there may be an idea about the appearance of a person. An example would be a stereotype. For example, if a person is asked to imagine a top model, he will immediately remember a number of features characteristic of top models. We cognize with the help of sensory perception only the external characteristics of the object, but not its essence. For a deep knowledge of objects and phenomena, one sensory perception is not enough. A more complex form of cognition is needed - abstract thinking. It reflects the surrounding world and its processes much deeper. If sensory cognition reflects facts, then abstract thinking makes it possible to determine laws. 2. Abstract thinking: concept, judgment and conclusion Abstract thinking has several forms and these forms are concepts, judgments and inferences. Concept is a form of thinking that reflects an object or a group of objects in one or more essential features. In colloquial speech, a concept can be expressed in one or several words. For example, "horse", "tractor" or "worker of a research institute", "explosive bullet", etc. Judgment - this is a form of thinking containing an affirmation or denial about the world, its objects, patterns and relationships. Judgments are simple and complex. The difference between them is that a complex proposition consists of two simple propositions. Simple judgment: "The karateka strikes." Complicated proposition: "The train has departed, the platform is empty." As you can see, the form of judgment is a declarative sentence. Inference - this is a form of thinking that allows one or more interconnected judgments to draw a conclusion in the form of a new judgment. An inference is made up of several propositions that Example of a judgment. All trees are plants. Maple is a tree. Maple is a plant. Concept, judgment and inference - these are categories that are unthinkable without reference to everyday life and human activities. They are only tested in practice. Practice is a daily social, material, industrial and other human activity under certain conditions. It can be in the field of politics, law, industry, agriculture, etc. In other words, practice is a test of theoretical knowledge in terms of their applicability in the real world. Any product passes such check before the start of operation. Trains, cars, planes are being tested. Theories and concepts are tested. Definitions are also tested in practice (recall the case of "Plato's man"). All these difficulties are necessary to achieve real knowledge, truth. Truth - knowledge that adequately reflects in the human mind the phenomena and processes of the surrounding world. In addition to abstract thinking, sensations, perception, and representation can provide truth, but their level of knowledge is often not enough. Abstract thinking thus enables us to grasp the deeper layers of truth. Abstract thinking is the most important tool in the hands of a person, allowing to know the unknown, separate the truth from lies, create a work of art and make a discovery. This is a very significant phenomenon, and therefore it has characteristic features: 1) reflects the features of the surrounding world without the direct impact of any phenomena on the senses. In other words, a person does not always need direct contact with an object or phenomenon to obtain new information. He comes to this result, relying on his knowledge gained earlier (a student of a mathematical institute, solving an unfamiliar problem, applies the knowledge gained earlier when solving similar problems), on experience (an old hunter participating in a raid guesses which way he will go beast), on the imagination (a person who has never been to the Hawaiian Islands makes up an idea about them according to the description of the interlocutor); 2) it is always a generalization of the phenomena of reality in order to identify existing patterns. Any person instinctively strives to simplify the process of thinking, which increases its speed and efficiency. This is the result of the generalization. Information about an object or phenomenon is compressed, as it were, access to it is accelerated due to the connections formed in the brain. In other words, finding in the process of thinking something in common between different objects, a person, as it were, puts these objects in one row. Thus, he does not need to remember all the data about one object from a series, but only its characteristic features. The common thing for all these items needs to be remembered only once. To confirm, you can give an example with a car. If you ask a person to imagine a car, an object will appear in his imagination, just characterized by common features - four wheels, several doors, a hood, a trunk, etc. Further, it is only necessary to specify the brand, type, belonging of the car; 3) it is impossible without a direct connection with the linguistic expression of thought. The process of thinking can be conditionally divided into two types - thinking without the mediation of language and "internal conversation", that is, proceeding in the form of communication with oneself. Be that as it may, it should be noted that most of the information, especially complex information (created not on the basis of sensory reflection), a person receives through communication, through books, magazines, and the media. All this is carried out mainly through spoken (written) language. Thus, a situation is created when a person receives information from the outside world, processes it, creating something new, and reinforces it again. Therefore, language acts not only as a means of expression, but also as a means of fixing information. 3. The value of thinking in reaching the truth. Logic forms Thinking - it is always an active process, as it is aimed at achieving a certain result, awareness, change, addition of information. Abstract thinking - this is a means of cognition, with the help of which logical science considers and studies the phenomena of the surrounding world, which are often impossible to know in any other way, and this shows the degree of necessity. To increase the efficiency of the thinking process, the concept of logical forms is used. These are the forms in which logical knowledge proceeds. They characterize the method of connection of the constituent parts of thought, its structure. Such a structure exists objectively, that is, it does not depend on a particular person, but characterizes the features of the surrounding world. Giving a definition to logical forms, it is necessary to say about such concepts as a quantified word, a connective, a subject and a predicate. Subject - this is a category that gives the concept of the subject of judgment, the logical form of which must be determined. Predicate - gives the concept of the sign of the subject. Bunch represented by the word "is" and may be absent. In this case, a dash is used instead. quantifier word is the word "everything". Thus judgments are expressed in forms like "All (quantifier) S (subject) is (copy) P (predicate)". As an example of a logical form "all S are P" the following judgments can be made: "All caterpillars are pests", "All people are mammals", etc. Perhaps the main thing in the process of thinking of each person, if he, of course, does not want to make logical errors, is the knowledge and correct application of logical laws. Compliance with these laws is the key to achieving the truth: 1) the law of identity; 2) the law of non-contradiction; 3) the law of the excluded middle; 4) the law of sufficient reason. It should also be mentioned that human thinking, in addition to formal logical laws, is subject to the general laws of dialectics: the laws of negation, the mutual transition of quality and quantity, the unity and struggle of opposites. These laws, like logical forms, have an objective character, that is, they do not depend on the will of man and exist independently of him. Therefore, even a person who has never studied logic and has not the slightest idea of the existence of its laws thinks on their basis, relying on common sense. This is typical not only for our time, but also for other historical eras. The significance of logical forms lies in the fact that they are used to achieve the truth of propositions, which can be either true or false. Truth and falsity - indicators of the specific content of a certain judgment. However, regardless of the truth of the judgments that act as premises, the conclusion, that is, the judgment derived from these premises, may be false. Reasoning as a process of obtaining a conclusion from the initial premises can only be right or wrong, but not false or true. It obeys the rules of logic and acts on their basis. It must be remembered that compliance with the rules of logic in reasoning is necessary, since if they are neglected, it is possible to obtain a false judgment even from true premises. There are also cases when, if one or more premises are false and the rules of logic are observed, the conclusion drawn can be true, as well as if the rules of logic are not observed if the premises are true. LECTURE No. 5. The concept as a form of thinking 1. General characteristics of concepts Concept - this is a form of thinking that reflects objects and phenomena in their essential features. As mentioned above, a person perceives this or that object, highlighting the characteristic properties (signs) of the latter (recall that sensation, perception and representation serve these purposes). It is due to these properties that we put objects either in one row, that is, we generalize them, or, conversely, we single out an object from a mass of homogeneous ones with different properties. For example, we all know that sugar is sweet and free-flowing, and salt is free-flowing, but salty. On the basis of flowability, we combine sugar with salt, but on the basis of taste we separate from each other. Features can be properties of an object that unite or separate objects from one another. In other words, evidence - These are the properties of objects in which they are similar to each other or differ. Any properties, features, state of an object that in one way or another characterize the object, distinguish it, help to recognize it among other objects, constitute its characteristics. Signs can be not only properties belonging to an object; an absent property (trait, state) is also considered as its sign [4]. Any object has a set, a whole complex of features that define it. Such signs can determine the properties of only this object and be single or reflect the characteristic features of a number of objects. Such signs are called by common. To confirm these words, the following example can be given: each person has a number of characteristics that characterize him, some of which characterize only him. These are facial features, physique, gait, facial expressions, as well as signs defined by law enforcement officials as “special features” and other striking signs. Other signs characterize an entire community of people and distinguish this community from the totality of other communities. Such characteristics include profession, nationality, social affiliation, etc. Here it is necessary to mention the characteristics that characterize all people and at the same time separate representatives of the human race from other living beings. They are inherent in every person. This is the ability for abstract thinking and articulate speech [5]. In addition to single (individual) and general features, logic distinguishes between essential and non-essential features. Signs that are characterized by mandatory belonging to an object (i.e., necessarily inherent in it) and express the essence of this object are usually called essential. They can be both general and individual. Thus, concepts that reflect a variety of objects include common essential features (the ability to express the thinking process in language and the thinking process itself). Concepts reflecting one subject include both general essential and individual characteristics. For example, the concept “Aniskin” includes general essential features (person, policeman) and individual features characteristic only of this person. Features that may or may not belong to the subject and which do not express its essence are called insignificant. The concept qualitatively differs from the forms of sensory knowledge, that is, sensations, perceptions and ideas. These forms exist in the human mind in the form of visual images that reflect individual objects or their properties. In other words, sensation It is a form of sensory knowledge. It, like representation, through perception forms a sensually visual image of an object or phenomenon. There is no visibility in the concept. In this way, notion - this is a form of thinking that reflects objects on an abstract basis, based on their essential features. This approach makes the concept a very convenient tool for scientific knowledge and therefore is widely used in various fields and branches of science, and also plays a huge role in building the educational process. This is true for both the natural sciences and the humanities. In the process of forming the concept, science reflects in the concept the objects and phenomena studied by it. It should be noted that concepts are characterized by a certain sensory poverty. Resorting to fixing only the essential features of objects and phenomena, generalizing them, concepts lose a significant number of individual features inherent in the object under consideration. From this point of view, the concept is much less saturated with sensory attributes. However, in return, concepts provide an opportunity for a deeper study of the surrounding world, its objects, processes, phenomena and allow you to reflect the information received with greater completeness compared to sensory cognition. Concepts have a linguistic expression and are inextricably linked with the main language unit - In short. Concepts are expressed both through the latter (words) and through phrases (word groups). It goes without saying that without words and phrases it is impossible to construct concepts or operate a name (words and phrases united by some meaning and denoting an object). It is necessary to mention special cases that sometimes cause confusion or misunderstanding. Words with ambiguous meanings can lead to such results. Homonyms (from the Greek homos - “same” and onyma - “name”) - different, but identical sounding and spelling units of language (words, morphemes, etc.) [6]. These are words that have the same sound, but different meanings (expressing different objects, processes or phenomena). For example, the word "onion", depending on the context, can mean an edible plant or small arms. Everyone knows the proposition "Peace to the world!". It contains two meanings of the word "world". There are many homonymous words in Russian, for example, the words "lynx", "bridge", "spit", "key" have several meanings at once. By devoting time to studying homonym words, you can sometimes get up to five or six meanings. However, it is unacceptable to take for homonyms concepts that include a separate word denoting similar phenomena, processes or objects. For example, the word "network" can be used in different expressions, such as "computer network", "electrical network"; "fishing net", "volleyball net", etc. In these examples, the word "net" is used in various combinations that change the context of its use, but not the semantic meaning. Recall that homonymous words have different meanings provided they sound the same. Synonyms (from the Greek synonymos - "of the same name") - these are words that differ in sound, but are identical or close in meaning, as well as syntactic and grammatical constructions that coincide in meaning. Synonyms are complete, for example, “linguistics” - “linguistics”, and partial, for example, “road” - “path” [7]. An example of the use of synonyms in context is the following sentences: “They had a long road ahead of them” - “There was a long road ahead”; “The severe frost chilled the travelers to the bones” - “It was January cold outside.” In connection with the foregoing, it should be noted that the ambiguity of words, the fuzziness of their semantic content can lead to errors in the definition of concepts, the construction of conclusions. Therefore, it is necessary to choose words with the clearest meaning, excluding duality and errors in reasoning. Terms are meant to be such words. Term (from Latin terminus - "border", "limit") - a word or phrase used with a touch of special scientific meaning. Thus, the term denotes a strictly defined concept and is characterized by unambiguity, at least within the framework of a particular science or group of sciences. 2. Types of concepts In modern logic, it is customary to divide concepts into: clear and blurry; single and general; collective and non-collective; concrete and abstract; positive and negative; non-relative and correlative. Let's move on to consider each type of concept separately. Clear and blurry. Depending on the content of concepts, they can reflect reality more or less accurately. It is this quality that forms the basis for the division of concepts into clear and vague. As you might guess, the clarity of reflection is much higher for clear concepts, while blurred ones often reflect the subject with insufficient completeness. For example, the clear concept of “inflation” contains in its characteristics a fairly clear indication of the degree of economic destabilization in the country. In various branches of science (mainly the humanities), concepts with a vague content (perestroika, glasnost) are used, which is often negative. This is especially true for law enforcement activities, in the course of which the lack of certainty of legal norms can lead to their free interpretation by subjects of law. Obviously this is unacceptable. Single and general concepts. This division is related to whether they involve one element or several. As you might guess, concepts in which only one element is implied are called singular (for example, “Venice”, “J. London”, “Paris”). Concepts in which several elements are thought of are called by common (for example, "country", "writer", "capital"). General concepts can be registering and non-registering. They differ in that in registering concepts many implied elements can be taken into account and can be recorded. Non-registering concepts are characterized by the fact that many of their elements cannot be counted; they have an infinite volume. Collective and non-collective concepts. Concepts containing signs of a certain set of elements included in one complex are usually called collective. As an example of collective concepts, we can cite the concepts of “team”, “pack”, “squad”. It should be noted that the content of a single concept cannot be attributed to a separate element included in its scope, since it applies to all elements at once. Collective concepts can be general (“team”, “flock”) and individual (“Team “Falcon””, “Team “Alpha””). Concepts containing signs not of the whole set, but of individual elements, are called non-collective. If the use of such a concept in speech refers to each of the elements that make up its volume, such an expression is called dividing. If all the elements are mentioned in a complex (totality) and without regard to each of the elements taken separately, such an expression is called collective. Concrete and abstract concepts. This division of concepts depends on the subject reflected in the content of the concept. This may be an object, or a certain set of objects, or a sign of this object (the relationship between objects). Accordingly, the concept, the content of which is information about the attribute of an object or the relationship between objects, is called abstract concept. On the contrary, the concept of an object or a set of objects is called specific. The main feature, the feature by which the division of concepts into concrete and abstract is carried out, is the ratio of the subject and its features. In other words, although the attributes of an object cannot exist without the latter, as a result of the logical method of "abstraction" they are distinguished into an independent object of thought and are considered without regard to their object. Accordingly, the concept is called abstract. We must not forget that specific and singular concepts are not synonymous, just as abstract ones must be separated from general ones. Thus, general concepts can be both concrete and abstract. For example, the concept of "merchant" is general and specific, while the concept of "mediation" is general and abstract. Positive and negative concepts. The classification of these concepts is based on the properties of an object, phenomenon or process. The type of concept here is made dependent on the presence or absence of characterizing properties of the object. In other words, a concept is called positive if it contains an indication of the presence of properties inherent in the object. In contrast to positive ones are negative concepts, which imply the absence of such properties. Thus, the positive concept will be “strong”, and the negative concept will be “weak”; positive - “calm”, negative - “restless”. Non-relative and correlative concepts. This classification is based on the presence or absence of a connection between the object that makes up the scope of the concept and other objects of the material world. Thus, concepts that exist separately from each other and do not have a significant impact on the existence of each of them will be irrelevant. Such concepts, for example, could be “nail” and “button”. Each of these objects exists separately and independently of the other. Based on the above, we can define correlative concepts as having a connection with each other, embedded in the features of the objects that make up their volume. Such concepts will be: "suzerain" - "vassal" or "brother" - "sister". The classification of concepts is inextricably linked with their logical characteristics. Determining the type of a particular concept, we thereby draw a conclusion regarding it, characterize it from the point of view of logic as a science. A logical characteristic helps to determine the content and scope of concepts and allows you to make as few mistakes as possible in the process of reasoning and use one or another concept with maximum efficiency in the process of proof. LECTURE No. 6. Formation of concepts, their content and scope 1. Logical methods of concept formation For a person engaged in scientific research, it is constantly necessary to receive new information. To do this, a scientist reads a lot of literature on a chosen subject, conducts observations, and makes experiments. However, all this activity would be useless if it did not lead to the formation of new concepts. In other words, the information received in such a case would remain only information, not clothed in a form suitable for consolidation and transmission. That is why it is necessary to know about the methods of concept formation. Such techniques are: abstraction, analysis, synthesis, comparison and generalization. Abstraction - this is a technique for the formation of concepts, in which it is necessary to abstract from a number of non-essential features of an object, discard them and leave only the essential ones. Comparison plays a significant role in the process of abstraction. Analysis - this is a mental fragmentation of an object, process or phenomenon into its constituent parts in order to establish the interaction of these parts and the relationships between them, as well as to identify the processes occurring inside the object under study. Analysis is necessary to obtain a reflection of an already existing concept. Synthesis - this is a mental assembly of the constituent parts of an object, phenomenon or process together. Synthesis is the reverse process of analysis and is usually used when the latter has already been carried out. Often, mental synthesis is preceded, if we are talking about an object, by the practical assembly of this object with strict observance of the sequence of setting the components. Synthesis is used to create new concepts on the basis of already existing ones subjected to synthesis, or to identify inaccuracies in a concept, as well as to make changes to these concepts. Comparison - this is a mental establishment of the similarity or difference of objects according to essential or non-essential features. Generalization - the mental association of a group of objects into a new row or the addition of one object to an existing one based on the characteristics inherent in these objects. Comparison and generalization make it possible to achieve greater accuracy in judgments, to separate one from the other, or, conversely, to combine several objects into one group (class). As an optional feature, they contribute to better assimilation of information by the human brain. All logical methods of concept formation are of great importance. They are interconnected, it is impossible to imagine one without the other. Often used together or precede one another. 2. Content and scope of concepts Any concept has content and scope. The content of the concept is a set of essential features characterizing its object, implied in this concept. The scope of the concept constitutes a set or set of objects that is conceived in a concept. Sufficient content for the formation of the concept of "isosceles right triangle" will be an indication of the presence in the composition of the geometric figure of two angles equal to 45 °. The scope of such a concept will be the entire set of possible isosceles triangles. Any concept can be fully characterized by defining its content (in other words, meaning) and establishing objects with which this concept has certain connections. Regardless of human consciousness, there are various objects in the world around us. These items are characterized by many. The set may be finite or infinite. If the number of items in a set is calculable, then the set is said to be finite. If such objects are incalculable, the set is called infinite. It is necessary to mention the relations of inclusion, belonging and identity. An inclusion relation is a relation of species and genus. Set A is a part or subset of set B if each element of A is an element of B. It is reflected in the form of a formula A with B (set A is included in set B). With regard to membership, the class a belongs to the class A and is written as a with A. The identity relation implies that the sets A and B are the same. This is fixed as A = B. The content of a concept is called its intensionality, and its relation to any objects is extensionality. Intensity of concepts. Most often, in the process of interpreting the term “content of a concept,” it is defined as a concept as such. In this case, it is implied that the content of a concept is a system of attributes through which the objects contained in the concept are generalized and distinguished from the mass of others. Sometimes content is understood as the meaning of a concept or all the essential features of an object contained in the concept taken together. In some studies, the content of a concept is identified with the entire complex of information that is known about a given subject. It can be seen from the above that the content of the concept is some information containing information about the objects, phenomena, processes included in this concept. This information is necessary for the formation of the concept, the definition of its form and rational consideration. Such information can be any information about an object that allows you to distinguish it from the mass of homogeneous (and heterogeneous) objects and clearly define its characteristics. In other words, this is information about the essential and other features of the subject. In the process of communication, from the point of view of the efficiency of information transfer, of particular interest is such an element of the content of the concept as connotation. It is more or less typical for the languages of different countries and to a very large extent - for the Russian language. These are all kinds of variations of pronunciation, intonation, stress on individual words, ethical, aesthetic, ethnic, professional, diminutive and other shades and colors of concepts used in speech. Such variations can lead to a change in the meaning of a concept without changing its verbal form, and a change in verbal form most often leads to a change in meaning. For example, the words "book" - "little book"; "grandmother" - "grandmother" - "grandmother" quite illustrate the connotation. It is necessary to say about the so-called magnitude of the content of concepts. It is inextricably linked with their volume. In this case, the ability of some concepts to be wider than others, and thus, as it were, "overlap" them, is implied. For example, the concept of "science" is much larger in content than the concept of "logic" and overlaps the latter. When characterizing the first concept, you can use, or you can not use the second, but replace it with another, or even get by with other means. However, when characterizing the concept of "logic", we will inevitably have to use the concept of "science". The concept of "science" in this case is subordinate, and "logic" is subordinate. Take for example two other concepts - "helicopter" and "aircraft". These concepts in relation to each other are not subordinate and subordinate. It is almost impossible to define one of them using the other. The only sign that connects these two concepts is that their objects are devices for making flights. The subordinating concept for both the first and the second will be "aircraft". Thus, only subordinate and subordinating concepts are subject to comparison in terms of the content of the volume. Extensionality of concepts. Any concept reflects an object and contains features that characterize and separate it from other objects. This object is always associated with other objects that are not included in the content of this concept, but have characteristics that partially repeat the characteristics of the object reflected in the concept. These items form a special group. Such a group can be defined as a set of objects characterized by the presence of common features, fixed by at least one concept. However, the mere reflection of the subject by one or another concept is not enough. An object that really exists and an object as an object of thought are not identical. This is connected with the representation of an abstract (imaginary, conceivable) and real (having a real embodiment) object. Abstract subject - this is a mental construction that can accurately reflect the signs, properties of an object, but may also contain an error or inaccuracy. In this context, one can define the scope of a concept as a set of abstract objects related to it. Thus, a real object is an object of the material world, which has characteristic features inherent only to it. An abstract object has no material embodiment and is characterized only by information about its belonging to a concept. There are two approaches to the question of belonging to the concept, according to which the scope of the concept can be the scope of diversity or quantitative. The first approach implies that the scope of a concept includes several other concepts. Accordingly, this last concept is common to all incoming. For example, the concept of "aircraft" includes "aircraft", "helicopter", "airship" and others, so it is general. This approach shows the presence of a sufficient number of elements included in the volume of the subject, respectively, such a volume is called the volume of diversity. Not only the objects themselves are related to the concept, but also the categories inherent in these objects. The scope of the same concept is the totality of objects associated with it. The concept, and accordingly, characterizing its content and volume, are mental formations. Therefore, the scope of a concept cannot consist of real objects, just as the thought of water cannot consist of water itself. It consists of mental reflections of these objects and their properties. The main condition is that such reflections, thoughts about objects, must fall under the signs implied in the concept. What makes a concept and the objects included in its scope real is the idea of the reality of these objects. Thus, the quantitative volume of a concept can be called a volume composed of mental reflections of real-life objects that correspond to a given concept. You should always remember the correct handling of any logical categories. Thus, a mistake related to the scope of concepts is possible. It is unacceptable to identify parts of the subject and parts of the scope of the concept of this subject. Otherwise, a part of a physical object (car wheel, aircraft wing, weapon striker) is identified with independent objects, mental reflections of which are included in the scope of the corresponding concept. It is also necessary to mention empty volumes. In some cases there may be so-called empty volumes. There are two options for the appearance of an empty volume: let us remember that the concept does not include the object itself, but only its mental reflection. Therefore, if an object reflected in a concept contradicts objective physical laws, the scope of such a concept is considered empty. This happens either with concepts containing fantastic objects, or with concepts about objects whose existence is impossible (for example, a perpetual motion machine). In another case, self-contradictory (false) concepts are implied. They have content when the volumes are empty. Different cases of the existence of volumes are studied by formal logic. She considers thinking from the point of view of its extensionality. Or, in other words, in an extensional context. Within the framework of formal logic, thinking is represented as a process of carrying out various operations with the volumes of concepts without considering the content of these concepts. The Purpose of Formal Logic - to determine the truth or falsity of concepts, relying only on their volumes. If there is a formal logic that studies only the scope of concepts, it would be reasonable to assume the existence of a logic of content that would study the content side of concepts and judgments. The object of consideration of the logic of content there must be an intensional part of thinking, the interaction of the content of various concepts and the degree of correctness of reflection in the concepts and judgments of the objective world. Logic studies concepts and judgments about objects in the real world. Concepts are only mental reflections of real-life objects. However, the concept implies the existence of its object. This is where the concept of modality comes into play. Modality is a way of existence of a certain object or process (ontological modality). There is also the notion of logical modality. This is a way of understanding, obtaining a conclusion about an object, phenomenon or process. Logical existence can be called absolute, since this concept defines existence in itself, existence as it is, without being tied to any particular object. Existence can be of the following types: 1) sensual. This is the existence of objects, processes and phenomena, perceived by man. Sense existence can be objective and subjective. The first implies the real existence of the object reflected in the perception of man. Such an object exists independently of the perceiver. The second (subjective) existence reflects not real objects, processes and phenomena, but only imaginary ones. It can be a person's fantasy, his thought about something, a dream, an image; 2) hidden existence. It is interesting that his objects are hidden from human perception for certain reasons. It can be objective and subjective. Objective. The reason for the impossibility of perceiving real-life objects is the inability of the human senses to perceive microscopic objects, various kinds of waves, electromagnetic fields and other similar phenomena. Subjective. This should include the existence of unconscious psychological characteristics that are part of and constitute the subconscious. These are various aspirations, instincts, drives, complexes, etc. The scope of a concept can exist either in a sensible or in a hidden form of existence, regardless of whether it is objective or not. However, such dependence occurs when a mistake is made. Being defined not in its kind of existence, the volume becomes empty. At the same time, we must not forget that the types of existence sometimes do not have clear boundaries. Depending on the circumstances, one of these types can flow into another - a hidden existence can become sensual, an objective - subjective. Therefore, often the scope of the concept may not be empty. It is necessary to consider the scope of the concept separately in each case. The relationship of categories within a concept is subject to logical laws and has its own specifics. Thus, the peculiarities of the effect of the content and scope of a concept on each other are reflected in the law of the inverse relationship between the content and scope of concepts. This law is based on the logical nature of concepts. Taking two concepts, we can notice that one of them is wider than the other in scope, while the other is included in the scope of the first. However, a concept that is included in the scope of another (having, accordingly, a smaller volume) in its content reflects more features and is more saturated with them. It is this phenomenon that forms the basis of the feedback law, which goes like this: the wider the scope of a concept, the narrower its content; the richer the content, the smaller the volume. The essence of this law is that the less information about an object is reflected in the content of the concept, the wider the class of objects and the more uncertain the composition. For example, the concept “airplane” is poor in content, but at the same time it includes aircraft of various types, brands and designs. Expanding the content, we add one more characterizing word and get the concept of “passenger aircraft”. Now the scope of the concept has narrowed significantly, but still contains a significant number of objects. The concept of “Boeing passenger aircraft” has almost the broadest possible content, but the class of objects included in the scope is now clearly defined and few in number. In this way, it is possible to narrow the scope of a concept by expanding its content down to one subject. LECTURE No. 7. Relations between concepts 1. General characteristics of the relationship between concepts The world around us by its nature is a very complex system. This nature is manifested in the fact that all objects that we can only imagine are always in relationship with some other objects. The existence of one is conditioned by the existence of the other. Considering the relationship between concepts, it is necessary to define the concepts comparable и incomparable. Incomparable concepts are far from each other in their content and do not have common features. Thus, “nail” and “vacuum” will be incomparable concepts. All concepts that cannot be called incomparable are comparable. They have some common features that allow us to determine the degree of proximity of one concept to another, the degree of their similarity and differences. Comparable concepts are divided into compatible и incompatible. This division is carried out based on the scope of these concepts. The scopes of compatible concepts coincide in whole or in part, and the content of these concepts has no features that exclude the coincidence of their scopes. Scope of incompatible concepts do not have common elements. For the sake of greater clarity and better assimilation of the relationship between concepts, it is customary to depict using circular diagrams, called Euler circles. Each circle denotes the volume of the concept, and each of its points - the object contained in its volume. Circular diagrams allow you to represent the relationship between different concepts. 2. Compatible concepts Compatibility relationships can be of three types. This includes equivalence, overlap и subordination. Equivalence. The relation of equivalence is otherwise called the identity of concepts. It arises between concepts containing the same object. The scope of these concepts coincides completely with different contents. In these concepts, one thinks of either one object or a class of objects containing more than one element. To put it more simply, the relation of equivalence refers to concepts in which one and the same object is conceived. As an example illustrating the relationship of equivalence, we can cite the concepts of "equilateral rectangle" and "square". These concepts contain a reflection of the same object - a square, which means that the volumes of these concepts completely coincide. However, their content is different, because each of them contains different features that characterize the square. The relationship between two similar concepts on the circular diagram is reflected in the form of two completely coinciding circles (Fig. 1).
Intersection (crossing). Concepts in relation to intersection are those whose volumes partially coincide. The volume of one, thus, is partially included in the volume of the other and vice versa. The content of such concepts will be different. The intersection relationship is schematically reflected in the form of two partially combined circles (Fig. 2). The intersection in the diagram is shaded for convenience. An example is the concepts of “villager” and “tractor driver”; "mathematician" and "tutor". That part of circle A that is not intersected with circle B contains the reflection of all villagers - not tractor drivers. That part of circle B that is not intersected with circle A contains the reflection of all tractor drivers who are not villagers. At the intersection of circles A and B, villagers-tractor drivers are imagined. Thus, it turns out that not all villagers are tractor drivers and not all tractor drivers are villagers.
Subordination (subordination). The relationship of subordination is characterized by the fact that the scope of one concept is completely included in the scope of the other, but does not exhaust it, but forms only a part. This relationship is genus -> species -> individual. In this relation are, for example, the concepts of "planet" and "Earth"; "athlete" and "boxer"; "scientist" and "physicist". As you can easily see, here the scope of some concepts is wider than others. After all, the Earth is a planet, but not every planet is the Earth. In addition to the Earth, there are also Mars, Venus, Mercury and many more planets, including those unknown to man. The same situation occurs in the other examples given. Not every athlete is a boxer, but a boxer is always an athlete; any physicist is a scientist, but speaking of a scientist, we do not always mean a physicist, etc. Here one of the concepts is subordinate, the other is subordinate. Obviously, it subordinates a concept that has a larger volume. The subordinate concept is denoted by the letter A, the subordinate - by the letter B. In the diagram, the relationship of subordination is displayed as two circles, one of which is inscribed in the other (Fig. 3). When two concepts enter into a subordination relation, each of which is general (but not singular), concept A (subordinate) becomes a genus, and B (subordinate) becomes a species. That is, the concept of "planet" will be a genus for the concept of "Earth", and the latter is a species. There are cases when a single concept can be both a genus and a species. This occurs if the concept of the genus, which contains the concept of the species, refers to the third concept, which is wider than the last in scope. It turns out a triple subordination, when a more general concept subordinates a less general one, but at the same time is in a relationship of subordination with another, which has a larger volume. The following concepts can be cited as an example: "biologist", "microbiologist" and "scientist". The concept of "biologist" is subordinate to the concept of "microbiologist", but is subordinate to the concept of "scientist".
A situation is possible when the general and singular concepts enter into the relationship of subordination. In this case, the general and concurrently subordinating concept is a species. The individual concept becomes an individual in relation to the general. This type of relationship illustrates the subordination of the concept of "Earth" to the concept of "planet". You can also give the following example: "Russian writer" - "N. G. Chernyshevsky". Thus, the relationship of subordination can be simplified in linear diagrams: "genus -> species -> species". Looking ahead, it can be noted that the relation -> view -> individual" is used in logical operations with concepts such as generalization, restriction, definition and division. 3. Incompatible concepts Incompatible are concepts whose volumes do not coincide either completely or partially. This happens as a result of the fact that the content of these concepts contains signs that completely exclude the coincidence of their volumes. Incompatibility relations are usually divided into three types, among which there are subordination, opposition and contradiction. Subordination. A relationship of subordination arises in the case when several concepts are considered that exclude each other, but at the same time have subordination to another, common to them, broader (generic) concept. Since such concepts exclude each other, it is quite natural that they do not intersect. For example, the concept of “firearm” includes in its scope “revolver”, “machine gun”, “rifle”, etc. Considering these concepts, it can be noted that not a single revolver can be a machine gun, just as not a single rifle is a revolver. Despite their mutual exclusion, these concepts are subordinated to the general. In a circular diagram, the relationship of subordination is depicted in the form of several circles (their number corresponds to non-overlapping concepts) inscribed in one, larger circle (Fig. 4). Concepts that are in a relationship of subordination to a more general concept for them, but do not intersect, are called subordinate. Subordinate concepts are types of a generic concept.
When defining the concepts included in the relationship of subordination, an error is sometimes possible. It lies in the fact that instead of mutually exclusive concepts, as an example, concepts are given that are subordinate to one another (for example, "writer" - "Russian writer" - "N.V. Gogol"). As a result, the relationship of subordination is replaced by a relationship of subordination, which is unacceptable. Opposite (contrast). Concepts that are in a relationship of opposition can be called such types of the same genus, the contents of each of which reflect certain characteristics that are not only mutually exclusive, but also replace each other. The volumes of two opposite concepts in their totality constitute only a part of the volume of the generic concept common to them, the types of which they are and to which they are subordinated. Each of these concepts in the content has features that, when superimposed on the opposite concept, overlap (replace) the features of the latter. It is characteristic that these concepts, by their linguistic nature, are antonymous words. These words reflect the contrast well, as a result of which they are widely used in the educational process. Antonym words expressing opposite concepts are: "top" - "bottom", "black" - "white", "heavy projectile" - "light projectile", etc. On the circular scheme, the relationship of opposites is depicted as a circle divided into several parts by opposite concepts. Opposite concepts, say "white" and "black", are on different sides of this circle and are separated from each other by other concepts, among which are, for example, "gray" and "green" (Fig. 5).
Contradiction (contradiction). A relation of contradiction arises between two concepts, one of which contains certain characteristics, and the other denies (excludes) these characteristics without replacing them with others. In this regard, two specific concepts that are in relation to contradiction occupy the entire scope of the concept that is generic for them. It should be especially noted that between two contradictory concepts there can be no other concept. Positive and negative concepts enter into the relation of contradiction. Words that make up contradictory concepts are also antonyms. Thus, on a linear diagram, the contradiction relation formula can be depicted as follows: a positive concept should be marked with the letter A, and a negative one (contradictory to the latter) should be designated as non-A. The concepts of "loud" and "quiet", "high" and "low", "pleasant" and "unpleasant" perfectly illustrate the relationship of contradiction. That is, the house can be large and small; chair comfortable and uncomfortable; bread fresh and stale, etc.
When using Euler circles for clarity, the contradiction relation is depicted as a circle divided into two parts, A and B (not-A) (Fig. 6). LECTURE No. 8. Generalization and limitation; definition of concepts 1. Generalization and restriction of concepts Generalization of the concept - this is the transition from a concept with a smaller volume, but more content to a concept with a larger volume and less content. When generalizing, a transition is made from a specific concept to a generic one. For example, generalizing the concept of "coniferous forest", we turn to the concept of "forest". The content of this new concept is narrower, but the scope is much broader. The content has decreased because we removed (removing the word "coniferous") a number of characteristic species features that reflect the characteristics of a coniferous forest. Forest is a genus in relation to the concept of "coniferous forest", which is a species. The initial concept can be both general and singular. For example, you can generalize the concept of "Paris" (single concept) by moving to the concept of "European capital", the next step will be the transition to the concept of "capital", then "city", "village". Thus, gradually excluding the characteristic features inherent in the subject, we are moving towards the greatest expansion of the scope of the concept, sacrificing content in favor of abstraction. Purpose of generalization - the maximum removal from the characteristic features. At the same time, it is desirable that such a removal should occur as gradually as possible, i.e., the transition from the genus should occur to the closest species (with the widest content). The generalization of concepts is not unlimited, and the limit of generalization is philosophical categories, for example, "being" and "consciousness", "matter" and "idea". Since the categories are devoid of a generic concept, their generalization is impossible. Concept constraint is a logical operation, the opposite of generalization. If the generalization follows the path of gradual removal from the attributes of the object, the restriction, on the contrary, enriches the totality of the attributes of the concept. Thus, there is a transition from the general to the particular, from species to genus, from single concepts to general ones. This logical operation is characterized by a decrease in volume due to the expansion of content. The operation of limitation cannot continue any further when a single concept is reached in its process. It is characterized by the most complete content and volume, in which only one object is conceived. In this way, restriction and generalization operations is a process of concretization and abstraction within the framework of a single concept to philosophical categories. These operations teach a person to think more correctly, contribute to the knowledge of objects, phenomena, processes of the surrounding world, their relationships. Through generalization and limitation, thinking becomes clearer, more precise, and more consistent. However, one should not confuse generalization and limitation with the selection of a part from the whole and consideration of this part separately. For example, a car engine consists of parts (carburetor, air filter, starter), parts consist of smaller ones, and those, in turn, of even smaller ones. In this example, the concept following the previous one is not its kind, but is only its component. 2. Definition The word "definition" comes from the Latin word definition. In the process of communication, work, just everyday life, a person often has problems understanding information and transferring this information to other people. This is due to the lack or ignorance of the definition of the subject given in the available information. Simply put, a person often does not understand the meaning of a particular concept. It is not necessary for the person who encountered the problem to explain a complex concept, to reveal its essence, but this can be done by a person whose profession the problem under consideration belongs to. To implement the interpretation, the logical operation of defining the concept is called upon. Definition of concept is a logical operation aimed at identifying the correct meaning of a term or the content of a concept. To define a concept means to fully reveal its content and to distinguish the scope of this concept from the scope of other concepts (that is, to determine the objects included in the concept and separate them from other objects). It is necessary to say about the relationship between definition and definition. Some scientists identify them, but some researchers separate the definition from the definition and call the latter a judgment that reveals the content of the concept. Thus, it turns out that definition is a logical operation, and definition - judgment. The concept, the content of which is required to be disclosed, is called the defined concept and denoted by Dfd (definendum). To reveal the content of this concept, a defining concept is used, denoted by Dfn (definence). The goal of a person who reveals the content of Dfd, using Dfn, is to achieve equivalence (equality) of both sides of the definition, i.e., the defined and the defining concept. The definition of a concept as a logical operation plays an important role in human activity, no matter what he does. At first glance, knowledge of the content of a particular concept is not necessary for people who are not involved in science. However, this is not so, because accurate knowledge of the signs of a concept not only increases the mass of a person’s knowledge, but also helps to avoid misunderstandings, incidents, and mistakes. The logical fallacy is all the more dangerous because at present the law plays a special role. Ignorance of the signs (content) of certain legal concepts makes a person vulnerable in legal relations. Needless to say, for science, the definition of concepts plays an even more significant role, because it is within the framework of science that new concepts appear and old ones are interpreted. And if we are talking about legal science, then we understand that the life of the state, society and the individual depends on how clear and correct the definitions are. The definition of a concept can be explicit and implicit. Explicit definitions contain the defined and the defining concept, with their equal volumes. In this form, the closest genus and species (species difference) containing the characteristic features of the concept being defined are used for definition. A variation of the definition through the genus and species difference is the genetic (from the Greek. genesis - "origin") definition. It indicates only the method of formation of this subject, its origin. The genetic definition plays a very important role for the sciences, where, due to their specificity, many concepts can be defined only through the method of formation or origin. Such sciences include mathematics, chemistry, physics. A genetic definition is a kind of definition through genus and specific difference, therefore it obeys the same rules and has a similar logical structure. As a separate type of definition through genus and species, nominal definitions can be called. They define a term denoting a concept, or introduce signs that replace it. Usually in such a definition there is the word "called". The definition through genus and specific difference is made in two steps. The first step of such a definition is the relation (submission) of the defined concept under a generic concept, characterized by a greater degree of generalization. In the second step, the concept being defined is separated from others belonging to the same genus, with the help of specific differences. The attributes of both the genus and the species, on the basis of which the concept is defined, are contained in the defining concept. For example: "A square is a rectangle with equal sides." The concept being defined here is "square"; generic - "rectangle"; specific difference - "with equal sides". For example: "The custom of business turnover is considered to be a rule of conduct that has developed and is widely used in any area of business activity, not provided for by law, regardless of whether it is recorded in any document." In this case, the concept of "usual business practice" is a defined concept. Generic for him will be the "rule of conduct" contained at the very beginning of the defining concept. Thus, we bring the defined concept under a more general one. Since the "rule of conduct" contains in its scope not only the custom of business turnover, but a whole set of rules, it becomes necessary to single out the latter from the general mass. To do this, we add signs of this phenomenon, thereby expanding the content and reducing the volume. The custom of business turnover is not enshrined in law, but may or may not be reflected in any document. Pointing to this characteristic feature, we reduce the number of objects contained in the volume to the desired ones. The signs by which we delimit the concept being defined from others corresponding to the generic concept are called species difference (kind). In the definition of species differences, there may be one or more. The definition through the genus and species difference can be reflected in the form of a formula A = Sun. Under А in this case, the concept being defined is implied, В is a genus, с - view. В и с taken together are the defining concept. Another way to reflect such a definition looks like this: Dfd = Dfn. Definition through the genus and specific difference is also called classical. It is the most common and widely used in various branches of scientific knowledge. Implicit Definitions. Definition through genus and species difference is a very convenient and effective tool for revealing the content of concepts. However, like any other tool, this type of definition has limitations. Thus, it is impossible to determine by referring to genus and species concepts that have no genus at all, such as general philosophical categories. Single concepts do not have a form, and, accordingly, also cannot be defined, because if we used only the genus to define a concept, we would get too many elements in its scope, which would also include this concept itself, which is impossible (for example, the concept “N. G. Chernyshevsky” cannot be defined only as “Russian writer”). When this situation arises, researchers use implicit definitions and techniques that replace definitions. In contrast to explicit definitions, where there are defined and defining concepts that are equal to each other, in implicit definitions, context, axioms, or a description of the way the defined object arises are substituted for the defining concept. There are several types of implicit definitions: contextual, inductive, ostensive, through axioms. contextual (from lat. contextus - "connection", "connection") definition characterized by the fact that it allows us to find out the essence, the meaning of a word, the meaning of which we do not know, through context, i.e. through a relatively complete piece of information that accompanies a given word, relates to it and contains its characteristics. Sometimes during a conversation we come across a situation where the interlocutor uses a word that is unfamiliar to us. Without asking again, we try to determine the meaning of this word, relying on the words accompanying it. This is definition through context. An example of such a definition is the following sentence: "...you will take a check there. It will be personal - in your name. You will receive money from it." Thus, even without knowing what a check is, you can understand from the context that this is a document by which funds are received. With some ingenuity, you can guess that there are also checks payable to bearer. Inductive definitions reveal the meaning of the term with the help of this term itself, through the concepts that contain its meaning. An example of this is the definition of natural numbers. So, if 1 is a natural number and n is a natural number, then 1 + n is also a natural number. Ostensive definition establishes the meaning of the term by resorting to the demonstration of the subject denoted by this term. Such definitions are used in revealing the essence of the objects of the sensory world, in other words, objects that are available for direct perception. Such a definition often focuses on the simplest properties of objects, such as taste, color, smell, texture, weight, etc. It is often used when learning a foreign language or explaining the meaning of an incomprehensible word. Sometimes, to characterize concepts, techniques are used that replace definitions. An axiom is a position that is accepted without logical proof due to direct persuasiveness. The definition through axioms is based on their quality. Characterization through axioms is widely used in mathematics. Comparison is a technique that allows you to quite clearly characterize an object by comparing its characteristic features and features with another, homogeneous object. Such a comparison leads to a fairly clear delimitation of the compared objects from each other by identifying not only similarities, but also differences in their features. When a comparison is used to define a concept, it will be defined the more fully, the more homogeneous objects the scope of this concept will be compared with. Comparison leads to the formation of an imaginary image of an object that has characteristic features. Description as a technique is simpler than comparison. The task of the researcher using the description is to consolidate as much information as possible about the subject, containing an indication of its characteristic features. In other words, when describing the image of an object directly perceived by the researcher, it is fixed in one form or another (drawing, diagram, text, etc.). When describing various kinds of characteristic features (weight, shape, size, etc.) should be reflected most fully and reliably. Characterization is the creation of an idea of an object by indicating some of its characteristic features. In this case, only one important sign is revealed. An example of a characteristic could be: “Gianfranco Pederzoli is the best Italian engraver of our time”; “According to K. Marx, Aristotle is “the greatest thinker of antiquity.” You can also find combinations of description and characteristics. Often used in both science and fiction. An example is used in cases where it is difficult to give a definition by genus and species difference, but you can resort to describing events, processes, phenomena, etc., illustrating this concept. An explanation with the help of an example is also the reflection of a complex concept through the enumeration of its elements. For example, the concept of "army" can be explained through the enumeration of its constituent units. Explanation by example is often used in the educational process of elementary grades. 3. Definition rules The truth of a definition depends not only on the correct presentation of its content, but also on how harmoniously and consistently its form will be built. If the truth of a definition depends on whether its content accurately reflects all the necessary features of the concept being defined, there is only one rational way to obtain such a definition - when formulating, strictly follow the requirements of the logical rules for the formation of definitions. Proportionality. The determination must be proportionate. This means that the defined concept must be equal to the defined one, that is, the defined and defining concepts must have equal volumes. If this rule is violated, a logical error occurs due to an incomplete definition or an overly broad interpretation of the subject. The definition in making such a mistake may be either too broad or too narrow; sometimes there are definitions that are both too narrow and too broad. Broader Definitions. They are characterized by the fact that the scope of the concept they define is greater than that of the one they define. In the form of a formula, this can be reflected as follows: Dfd ‹ Dfn. An example of a too broad definition would be the following: “a television is a means of satisfying the hunger for information” and “a chandelier is a source of light,” as well as “a wheel is a rubber circle.” In connection with this issue, we can recall the incident that occurred with the ancient Greek philosopher Plato, when he defined man as a “two-legged animal without feathers.” Subsequently, he had to admit the mistake and add the phrase “and with broad nails,” since Diogenes, another thinker of antiquity, brought a plucked chicken to a lecture at Plato’s school with the words: “Here is Plato’s man.” Too narrow a definition. This is a definition in which the scope of the defined concept is wider than the scope of the defining concept (Dfd › Dfn). This error is contained in the following definition: “an immovable thing is a house or other structure.” The mistake here is that a structure (including a house) does not exhaust the scope of the concept of “immovable thing”, since the latter also includes land plots, subsoil plots, separate water bodies, etc. The definition of “indivisible” is also too narrow a thing is a thing the division of which in kind is impossible.” One feature was not indicated here, namely, that the division of such a thing is impossible only if it changes its functional purpose. A definition that is too broad and at the same time narrow. They are characterized by a certain ambiguity. The same definition, depending on the direction in which its research is directed, becomes either too narrow or broader. For example, the concept of “a car is a device for transporting people” is broad, because a car is far from the only device for transporting people. However, on the other hand, the above concept is narrow, because a car can be used not only for transporting people (after all, you can also transport animals, building materials, for example, and other things). Absence in the definition of a circle. The circle in the definition arises in two cases. The first is called tautology and is characterized by the definition of a concept through the concept itself. In the second case, a circle is formed if the content of the defined concept is revealed through a concept that was previously (in a previous definition) defined through the concept being defined at the moment. Tautology - this is a simpler, in terms of structure and construction, erroneous definition. It is characterized by absolute uselessness, since it does not perform the main function of the definition - the disclosure of the content of the concept. In other words, after the definition-tautology, the concept remains as incomprehensible as it was before it. There are many examples of tautologies. You can often hear tautologies in colloquial speech, wherever you are - in line, at the market, at the circus and even the theater. People resort to tautology, often without noticing it. The following definitions are tautologies: "engine oil is an oily liquid with a pungent odor"; "an old person is one who has grown old in the process of life"; "funny is what causes laughter"; "an idealist is a person with idealistic convictions"; "a reminder is a reminder of something," etc. This shows that if we did not know the meaning of a concept and it was defined through itself, the meaning of this concept will not become clear, therefore, such a definition is useless. From a logical position, the expressions "given task" or, for example, "assigned task" are incorrect. It often happens that one person says to another: "Butter is oily, sugar is sugary." This is also a tautology, but in this context it is used to highlight the tautology in the speech of another person. Another case of a definition containing a circle is definition of the first concept by the second concept, which was previously defined first (the concept A is defined through the concept B, and then B is defined through A). A longer chain of definitions is possible, closing in a vicious circle. An example of such a circle is a definition derived from the proposition “the definition must be correct.” Here it is: “a correct definition is a definition that does not contain any signs of an incorrect definition.” This definition will be correct if we reveal the content of the concept “incorrect definition” (“this is a definition that contradicts the correct one”). The fact that there is a logical error here leads to the fact that this definition reveals something that does not reveal anything. Clarity of definition. The definition must discard ambiguity and use only true concepts that have been previously proven or do not need definition. If this rule is violated, that is, if the content of the defined concept is allowed to be revealed through a defining element, the meaning of which is also unknown, the logical error “defining the unknown through the unknown” arises. A definition that complies with the rule of clarity should not contain metaphors or comparisons. There are a number of aphorisms and metaphors that are true judgments, which, although they effectively convey information, serve instructive purposes and often play an important role in shaping a person’s worldview, are not definitions of the concepts they contain. For example, the following judgment does not define the concept: “The death of one person is a tragedy, the death of a thousand people is a statistic” (I.V. Stalin). Inadmissibility of negativity. This rule is due to the fact that a negative definition does not reveal the content of the concept being defined. An example of a negative definition would be the following proposition: “A car is not a carriage.” This judgment does not reveal the characteristics of a car, but only indicates that “car” and “carriage” are different concepts. Naturally, such an indication is not enough for a full definition. This rule does not apply to the definition of negative concepts, the content of which is revealed mainly through negative definitions: "an incomparable work is a work that has no equal." LECTURE No. 9. Division of concepts 1. General characteristics Definition - a very effective tool in the hands of the researcher. It allows you to get an idea of the content of the concept, i.e., reveals it. It is undeniable that the definition of concepts is one of the most important logical techniques. However, the application of the definition does not provide complete information about the concept under study, because, in addition to content, any concept also has volume. Division is a logical operation by which the volume of a concept, called a set, is divided into a number of subsets. With the help of this operation, the scope of the concept is revealed, while the definition reveals its content. The division operation contains a number of concepts: a shared concept, division members, division base. As the name implies, the divisible concept is the concept, the scope of which needs to be disclosed. Members of the division make up the scope of the divisible concept, but at the same time they are delimited from each other. These are the types into which the scope of the concept is divided. The base of the division is the basis by which the division is made. The presence of a division base is not required. Speaking about the performance of the operation of division, we mean the division of the volume of the concept subjected to division (generic concept) into the entire set of species contained in it. The shared concept is considered as a genus in relation to the elements of its scope related to this concept as species. Division allows you to understand the belonging of a certain species to a particular genus, to put several species in one row, based on various grounds, including generic affiliation. All this contributes both to a more effective knowledge of various kinds of information, and to its correct consolidation. 2. Rules for the division of concepts Dividing is an important and often difficult process. As a result, this process does not always lead to the correct result. It happens that the latter contains an element erroneously added not to its class. All this can lead to confusion, confusion, which deprives the division of the clarity inherent in any important tool of science. From what has been said, it is clear that it is necessary to establish rules that are mandatory for use in the process of the logical device "division". Such rules exist, there are four of them, and they effectively contribute to the elimination of logical errors in the process of division. Continuity of division. The main thing in the division process, from the point of view of this rule, is sequence. This means that when dividing the volume of a divisible (generic) concept into types, it is necessary to gradually move from one type, revealed last, to the next one, located closest to all the others. In this case, it is unacceptable to move from disclosing species of one order to species belonging to another order. This division leads to errors and omissions of certain types. It lacks consistency. In this case, a so-called division jump occurs. For example, you cannot divide sausage into smoked, raw smoked, “Doctor’s”, “Amateur”, etc. This is due to the fact that in the first level of division we had to indicate smoked, raw smoked and boiled. Only after this can you move on to dividing into types of a lower level and, among the types of boiled sausage, indicate “Doctoral” and “Amateur”. This error can be well illustrated by applying the Criminal Code, since it has a convenient generic structure. If we divide the concept of “crime” into crimes against the constitutional rights and freedoms of man and citizen, crimes against family and minors, against life and health, murder, beatings, leaving in danger, etc., it becomes obvious that the last three types are included in the scope of the generic The concepts of “crimes against life and health” are articles of the Criminal Code of the Russian Federation. They should be considered only after listing all the concepts of the same level, which are essentially chapters of the Criminal Code of the Russian Federation. Proportionality of division. It consists in fully revealing the scope of the concept under consideration, without omitting a single element, but without adding a single one. This is possible only in the case when the totality of the volumes of specific concepts is equal to the volume of the generic concept. This can be illustrated using the following example: all weapons are divided into bladed and firearms. The scope of the concept “weapon” is limited to these two types, each of which in turn is divided into types of the following series. The volume of the generic concept here is equal to the volume of the totality of species. If there are many species and their number is long or impractical to enumerate in their entirety, in order to avoid a logical error, the unfinished series is supplemented with the words "etc", "etc", "etc". Violation of the rule of proportionality of division leads to such errors as incomplete division and division with extra members. One base rule. A division base is a characteristic feature that is used in the division process to distinguish one division member from another. Having chosen a certain basis for division, the researcher must adhere to this basis until he fully reveals the terms delimited by this basis. Using several bases of division at the same time is unacceptable, as it leads to the crossing of the scope of concepts. An example of an incorrect division with crossing volumes is the following: “Bread can be wheat, rye, fresh and stale.” Two bases are used here - according to the grain from which the bread is made, and according to its condition. Mutual exclusion of division members. Division terms must always be mutually exclusive. Neither of them should be in an intersection relationship with the other (that is, it should not contain in its volume elements contained in the volume of another member). This result (partial intersection of the volumes of members (types) of division) is caused by a violation of the rule of division based on only one base, which determines the strong relationship between these two rules. An example of a correct division according to this rule is the following: “A substance can be in the following states: liquid, solid and gaseous.” Incorrect division with the same example: “A substance can be in the following states: liquid, solid, heated, gaseous, frozen.” Here the division members do not exclude each other precisely because the rule of one basis was violated. 3. Dichotomy Dichotomy (from Latin dichotomia - "dividing into two parts") - this is a very effective type of division. It is characterized by the fact that the members of the division do not intersect (i.e., exclude each other), such a division is made only on one basis, and the rule of proportionality is also observed. However, despite the indisputable convenience of dichotomous division, it has a serious drawback - the dichotomy is not always applicable. In cases where it is impossible to clearly set the criterion for division, this type of division does not fulfill its function. This happens when trying to divide concepts with a "fuzzy" volume. The division operation is used in cases where it is necessary to determine the types of generic concepts. The examples given in the previous questions are divisions based on species-forming characteristics. This name is associated with the process of division itself, which is carried out on the basis of a characteristic, from which new species concepts are derived. For example: “Crimes can be against life and health, against family and minors, against sexual integrity and sexual freedom of the individual, etc.” The basis of division here and, accordingly, the species-forming feature is the object at which the criminal act is directed. The dichotomy is significantly different from the specified type of division, which determines the scope of its application. Dichotomy is the division of the volume of a certain concept into two contradictory (not having intersection) concepts. With the literal designation of the process of dichotomous division, the following picture arises: the concept A (the concept over which the division is made) is divided into two - В и not = B. This is a simple type of dichotomous division that is limited to one stage. In more “complicated” cases, division is possible not = B on С и not = C etc. An example of a dichotomous division is the division of crimes into intentional and unintentional; citizens for adults and minors; animals on vertebrates and invertebrates, etc. As can be seen, dichotomous division has a number of advantages. Thus, for example, there is no need to enumerate all types of a divisible concept, but it is enough just to single out one type and a concept that contradicts it. The latter includes all other species. It follows that the two concepts formed by the dichotomy exhaust the entire volume of the divisible concept, therefore the subject under consideration is reflected only in one of them. At the same time, the scope of the negative concept is too wide, which implies the appearance of vagueness and uncertainty. As already mentioned, the dichotomy is characterized by a strict and consistent character. However, the second and subsequent stages of the dichotomous division, to a greater or lesser extent, lose their rigor and consistency. In this regard, researchers most often limit themselves to the first stage of division. It is necessary to mention the problem that arises when identifying the division of concepts and their mental division into parts. The main difference between division and dismemberment is that parts of the whole are not types of a divisible (generic) concept. It is impossible to recognize as a division the division of the concept "ship" into bow, stern, mast, bottom, etc., just as the latter cannot be called types of the specified generic concept. Here we are dealing only with parts of the whole. Also parts, but not types of the concept of "computer" are the monitor, system unit, keyboard and mouse. The above can be illustrated in the following way: imagine that the indicated parts of the whole are members of the division, and therefore, types of a generic concept. In this case, we can say that, for example, the monitor is a computer (kind of computer). It is obvious that this is not the case. Despite what has been said above, the operation of dismemberment of concepts cannot be neglected. It is widely used in the educational process of both senior and junior high schools. This operation is used in botany, biology, physics, chemistry, etc. Purpose of division - getting an idea about the constituent parts of an object. For example, you can divide the human skeleton into parts, and also divide these parts into smaller ones. You can also divide, say, an egg into shell, protein and yolk. The application of dismemberment, of course, is not limited to the educational process of secondary schools, but is used in universities, in science and in everyday life. For example, in medicine, the human body is divided into the thoracic and abdominal sections. 4. Classification One of the special divisions is classification. This is a systematic, consistent division of concepts with the distribution of types into an interdependent system, within which the latter are divided into subspecies, subspecies are also divided into division members, etc. Classification is of great importance and is used for the most part for the purposes of science, and it is precisely because of this that it has existed for a long time. Classifications, often used in science, are subject to changes, additions, but, despite this, are more permanent than a simple division. The purpose of classification is to systematize and preserve knowledge. Therefore, it has high precision, clarity and stability. Members of the division are usually reflected in various tables, charts and codes. There are classifications of plants, animals, legal classifications. Often classifications have a huge number of elements. These elements within the framework of the classification are combined into a single system, which makes it convenient and quick to access its individual parts and elements. The lack of classification would lead to chaos in a large array of unsystematized information. It is impossible not to note the relativity of any classification, which is associated with the ambiguity of many objects, phenomena, processes. Therefore, it is often not possible to attribute this or that phenomenon to one group. From the question of the ambiguity of phenomena follows the problem of choosing the basis of classification. One and the same concept can, depending on the chosen basis, express various objects, phenomena, or be interpreted from one side or another. Scientific classification is always an evolving system. It changes, as information accumulates, its structure improves. It happens that a new, more complete and developed classification replaces the previous one. Therefore, it is impossible to allow the limitation of operations on classifications by their formation alone. It is necessary to take into account the change in the body of knowledge about the subject, the dynamics of social relations, and many other factors, since any information, including that which is fixed within the framework of various classifications, is obtained by a person exclusively from the outside world. Accordingly, it is necessary to make the necessary changes in a timely manner. As an example of an ambiguous phenomenon, one can cite a family. Despite the fact that this institution is called social, it is impossible to limit it to only one or two areas of social life. Classification can be carried out according to a species-forming feature, or it can be dichotomous. The classification of animals, numbering more than one and a half million species, is obviously based on the use of a species-forming trait. The dichotomous classification is based on the features of the dichotomous division of concepts. The classification is also natural и auxiliary. The difference between them is that the first is carried out on essential grounds, while the second - on non-essential ones. Natural classification allows you to determine the properties of an individual classification element, knowing the general characteristics of this classification or another element. Auxiliary classification is needed so that you can quickly and correctly solve emerging problems. This requires prompt, quick access to one or another classification element. Convenient search and selection of the desired item often serves as the basis for effective activities. It is the achievement of the goals of efficiency, speed and convenience that determines the use of non-essential grounds. Such a classification does not give us any idea about the properties of the object. We are all familiar with such classifications. There are many of them and they are widely used in human life. How often do we take a notebook with telephone numbers, designated alphabetically by the names of acquaintances. This is an auxiliary classification. Having picked up a book dedicated to a particular subject of science, first of all we open the alphabetical subject index. This is also an auxiliary classification. When creating classifications, operations on classes are used. They allow you to achieve the desired result and get the classification that is needed at the moment. There are operations of addition, subtraction, multiplication and negation. Addition (combining classes). When using this operation, several groups (classes) are combined into one classification containing all the elements of those classes that are combined. Subtraction extracts separate classes from a larger class. The result is a class from which the elements of the selected class are removed. Multiplication (intersection of classes). There is a class of elements that are common to several classes. They are determined using the multiplication operation. Denial (education, addition). With the help of this operation, a new class of objects is derived from a more general class and considered separately as a new one. LECTURE No. 10. Judgment 1. General characteristics of judgments This is a form of thinking in which something is affirmed or denied about the surrounding world, objects, phenomena, as well as the relationships and connections between them. Judgments are expressed in the form of statements regarding a specific subject. For example, the following expressions are propositions: “Mars is called the red planet”; "Man is a mammal"; "Moscow is capital of Russia". All these statements assert something about their subject, but the judgment can also deny it. For example, “Plato did not live in China”; “The driving force of a trolleybus is not fuel,” etc. Judgments are both true and false, and the truth or falsity of judgments depends on the objectivity of the reflection of the surrounding world. If objects, processes, phenomena of our world are reflected in the judgment correctly, correctly, the judgment is called true. Based on the foregoing, it can be noted that all the above judgments are true, since they reflect the state of affairs that exists in reality. If the judgment reflects the surrounding world with distortions, incorrectly determines the place of objects in relation to each other and does not correspond to reality at all, it is called false. False judgments can arise due to a person's oversight or with his direct intent. The falsity of judgments is not always obvious, but in most cases it is obvious. For example, the proposition "From the Earth the far side of the Moon is visible" is false. Also, for example, the proposition "All vehicles are equipped with an engine" will be false. All of the above refers to traditional logic, which is characterized by ambiguity of judgments. In other words, every proposition can be either true or false. In this case, no other options are allowed. However, since the birth of logic, it has been known that some judgments are of an indefinite nature. At the moment they are neither true nor false. One of the most famous such judgments is the proposition “God exists.” Not supported by anything other than faith, this expression does not make it possible to reliably verify the truth or falsity of the information contained in it. Other such judgments include the following: “There is life on Mars” or “The Universe is infinite.” Today, it is not possible to reliably verify and approve or refute these judgments. Judgments about future phenomena for which it is not yet known whether they will occur or not can also be considered uncertain. For example, the judgment “It will snow tomorrow.” It cannot be true, because there may not be snow, in which case the true nature of this judgment will necessarily be refuted. However, this judgment is not false, because there is a possibility that snow will still fall. Since it is unknown whether there will be precipitation or not, we cannot determine in advance the nature of the judgment (whether it is true or false). Such an approach to determining the nature of judgments is inherent in one of the varieties of many-valued logic - three-valued logic. Judgments consist of a subject (denoted by the Latin letter S), a predicate (denoted by P) and a connective. It is also possible to have a quantified word. Subject of judgment is his subject. Namely, this is what the judgment says. The predicate gives the concept of the attributes of the subject. The link is expressed by the words "is", "is", "essence". Sometimes it is replaced by a dash. Any subject of judgment is reflected in some concept. As we remember, the concept is characterized by content and volume. It is to determine the part that the judgment occupies in the scope of the concept that reflects its subject (subject), and the quantified word is intended. In a language, such a quantifier could be the words "all", "some", "none", etc. 2. Language expression of judgments In language, judgments are expressed in the form of sentences. As is known, a sentence consists of linguistic units - words. This means that the meaning of a sentence depends on the words, their meaning, and the coloring with which we express our thoughts. According to the purpose of utterance, sentences can be narrative, motivating, or interrogative. Each type of offer has its own specifics. When examining each individual proposal for the presence or absence of judgment in it, it is necessary to be guided primarily by the information that it carries. Every sentence carries information, but not every sentence contains a judgment.. This means that a judgment is not just information, but has features characteristic only of judgments. Such features are the way information is presented in judgments: firstly, judgments confirm the presence or absence of an object, and secondly, judgments may contain a denial of the existence of a particular fact, phenomenon, process. From the point of view of the convenience of expressing judgments, the most suitable declarative sentence. As is known from the Russian language course studied in high school, a narrative sentence contains actively conveyed information. That is, the narrative contains a direct reflection of the subject in question. For example, “The sun is shining brightly today” is a true (if the sun really is shining) proposition expressed in a declarative sentence. As an example, we can cite a few more narrative sentences: “L.N. Tolstoy is a great Russian writer”; "The morning fog penetrates to the bones"; "Sugar is not the opposite of salt." All these sentences contain a judgment about a particular object and affirm its existence or deny this fact. Since declarative sentences are convenient for expressing judgments, they are most often used for this purpose. However, there is controversy among scientists about the ability to convey judgments of other types of sentences. One-part impersonal sentences, such as "Shives"; "Skid"; "It's baking"; "Hurts", may contain judgments. However, by considering such sentences, it is impossible to determine the truth or falsity of these judgments. This situation is associated with an extreme lack of information, because such sentences consist of one word and are intended more to reflect the mood than to accurately convey information. In this regard, it is necessary to recognize that a one-part impersonal sentence can be considered as a judgment only if it is clarified and supplemented with the necessary data. All of the above also applies to denomination sentences, such as "Summer"; "Sea". Nominal sentences, in addition to coinciding with one-component impersonal sentences, have their own specifics. It lies in the fact that such proposals cannot be considered at all in isolation from the context. Most often, noun sentences play the role of a response to a previously spoken phrase. For example: “A multi-colored arc after the rain, what is it?” - "Rainbow". It should be mentioned that some narrative sentences also need to be supplemented and clarified, since otherwise they cannot contain judgments. For example: "It's always cold in our area in summer" needs to be clarified as to which regions we are talking about. Otherwise, it is not clear whether the proposition is true or does not reflect reality. Just like the sentence "This team is the best in science" does not give us an idea of what kind of science we are talking about and what kind of team is named the best. Accordingly, additions and clarifications regarding these subjects are required. The declarative sentences discussed above arise most often due to the separation of a specific sentence from the main statement, without making changes to its composition. In other words, when a sentence is taken out of context. Currently, there is no unambiguous point of view on the problem of judgments in incentive sentences. Incentive sentences are intended to convey information about the desire, impulse, general direction of the activity of the person pronouncing them. Probably, every person knows examples of such sentences from childhood. For example, slogans, calls like "Protect nature - your mother!", "Motherland is calling!", "Peace to the world!" are incentives. Such sentences are not judgments, despite the fact that they contain an affirmation or negation of something. For example: "Don't smoke!", "Go in for sports!" - these are incentive proposals, the first of which is aimed at denying a bad habit, and the second affirms the correct way of life. However, a number of scientists argue that orders, commands, appeals, slogans contain modal judgments. They are considered within the framework of modal logic (this is non-classical logic). Modal propositions contain so-called modal operators. These are words such as “possible”, “proven”, “necessary”, etc. Modal judgments will be discussed in more detail in the corresponding topic. Thus, the calls "Be hardened!", "Don't make a fuss", "Full speed ahead!", according to a number of researchers, contain judgment. As mentioned above, a single point of view on the issue under consideration has not been reached, and some scientists do not deny the presence of judgments in incentive sentences at all. This position is argued by the fact that incentive sentences do not contain negation or affirmation, and it is impossible to say about them whether they are true or false. The question is the main way to learn something new from a person who knows more than you do. Questions are expressed in the form of interrogative sentences. Do these sentences contain judgments? There is no definite answer to this question. Most of the interrogative sentences do not deny anything, just as they do not affirm anything, and it is not possible to determine the truth of such a sentence, and, accordingly, its falsity. From this point of view, interrogative sentences clearly cannot be carriers of judgments. However, we must not forget about sentences that contain rhetorical questions. Such questions definitely fill the sentence with meaning and new information. Such a sentence, although not explicitly, but with sufficient obviousness, expresses some truths. For example, this information may indicate the desire of each person to be happy, people's attitudes towards war and peace, poverty and wealth. This makes the interrogative sentence capable of expressing a judgment. An example of such interrogative sentences can be: "Will the war end?", "Who does not want happiness?" etc. LECTURE No. 11. Simple judgments. Concept and types 1. The concept and types of simple judgments As you know, all judgments can be divided into simple и complex. Almost all of the judgments given above are simple. Simple Judgments can be identified in contrast to complex. The latter consist of several simple judgments, therefore they are expressed in language by longer and more complex constructions. If we assume a tautology, complex judgments are “more difficult” than simple ones in every sense. Often such judgments accurately and correctly reflect the phenomena of the surrounding reality, objects, their properties and relationships. A feature of complex judgments is that they contain information about several heterogeneous objects at once, this makes them more complete. However, this does not mean that simple judgments are “worse.” Thanks to their simplicity and clarity, they can still be found more often. Since in simple judgments there is no need to reflect several heterogeneous objects at once, there is less opportunity for error. We can also say that the construction of such judgments is “simpler”, because it consists of a sentence containing information about only one object (class of objects). Simple judgments are categorical and assertoric. At the same time, simple assertoric judgments, in turn, can be attributive (reflect the properties of the object) and existential (associated with the idea of whether an object exists in reality). The third kind of simple assertive judgments is judgment about relationships between objects. Categorical judgments are affirmative and negative, as well as general, particular and singular. 2. Categorical judgments Considering judgments from the point of view of traditional logic, it can be noted that they are basically categorical. This means that they either affirm or deny this or that subject, and at the same time the third option is not allowed. In this way, categorical judgments can be affirmative and negative. For example, the propositions “The Moon is a satellite of the Earth” and “Great Britain is an island state” are affirmative. The propositions “No capital is a village” or “Some wines are not French” are negative. This division of categorical judgments is carried out according to the quality of the connective. As we remember, the connective can be distinguished by the words “is” and “is not” or “is” and “is not.” Thus, depending on what type of connective is used in this particular case, we can talk about the presence or absence of certain features in the objects of judgment. Presence is indicated by the copula “is”; absence is expressed by the copula “is not”. From the above it is clear that categorical judgments can be affirmative and negative. However, in order to get a more complete understanding of the relationship between these two types of judgments, it is necessary to become more familiar with each of them. affirmative categorical judgment has the ability to determine the characteristics inherent in a particular subject. This makes such a judgment more convenient when reflecting one or another object, because in this way its properties are distinguished more fully. This means that it is enough for a person who forms an idea about an object on the basis of an affirmative judgment to simply distinguish it from the mass of other homogeneous (and, accordingly, heterogeneous) objects. Negative categorical judgment does not have affirmative properties. In terms of reflecting the properties of the object, these two types are opposite. So, a negative judgment does not say that an object has this or that property, but gives us an idea of what property this object does not have. Thus, a rather blurred picture is often obtained. Knowing only what property an object does not possess, it is very difficult to judge its nature. That is, it is much easier to distinguish an object from others, knowing what properties it has, than vice versa. Of course, a negative judgment can also serve the purpose of reflecting a certain subject, but more often it still serves to clarify. The division into types described above was carried out depending on the quality of the ligament. Another basis for division is quantity. This means that the classification is based on the question of how many objects of a certain class are included in a given concept and reflected in it. A concept may contain an indication that it refers to all objects of the class, part of these objects, or even only one of them. Depending on this basis, simple categorical concepts can be divided into general, private and individual. As you can see, all such judgments have a quantitative expression (they contain an indication of the objects contained in them). Therefore, for convenience, a typology (combined classification) of such judgments was derived. This classification consists of four points. First represented by general statements. As the name implies, such judgments are affirmative and general. Accordingly, the structure of such a judgment is "All S is P". For example, "All humans are mammals." The second type judgments is called private affirmative. It has the structure "Some S are P". For example, "Some athletes are snowboarders." The third type of simple categorical judgments is generally negative. A structure of this type is "No S is a P" and an example is "No dog is a reptile". The last and fourth type of simple categorical judgments is the particular negative type. It is reflected in the form of the formula "Some S are not P". An example would be the proposition "Some lakes are not freshwater". All of these types of judgments have a literal reflection. In the case of the general affirmative and the particular affirmative, these are the letters A and I, respectively. General negative judgments are designated as E, and particular negative ones as O. These letters are taken from the words affirmo ("I affirm") and nego ("I deny"). Considering the structure of judgments, one cannot leave aside such an important issue as the distribution of concepts. As is known, any judgment contains at least subject and predicate, denoted in the diagram by the letters S and P. Both the subject and the predicate are concepts, and, like all concepts, they are characterized by volume and content. If the content consists of features that characterize a concept, then the volume contains information about subordinate concepts. It is by the scope of the concepts S and P that an opinion is formed about their distribution or non-distribution. Thus, the scope of a concept is considered unallocated if it is partially included or partially excluded from the scope of another concept. In contrast to non-distribution, distributed is a term whose scope is completely included in or excluded from the scope of another. The distribution of a term may depend on the type of judgment. There are cases when the subject of the judgment is not distributed, in contrast to the predicate. For example, in the proposition "Some athletes are biathletes", the subject is the term "athletes", the predicate is "biathletes", and the quantifier is "some". The scope of the concept (term), which in this case is a predicate, is narrower than the scope of the subject of judgment. The relationship between these two concepts can be expressed using Euler circles. In this case, the circle representing the predicate will be completely inscribed in the larger circle of the subject. The subject here is not distributed, since only a part of the athletes (biathletes) is thought of in it, and the predicate is distributed, since the term "biathletes" is fully included in the scope of the concept "athletes". The above judgment is privately affirmative. The proposition “Some boxers are world champions” is characterized by the fact that both its subject and predicate are undistributed. Expressing these judgments in the form of Euler circles, we get two intersecting radii, neither of which is completely included in the volume of the other, because only some boxers are world champions, but not all champions are boxers. Judgment "All squares are rectangles" universal. Here the subject is the concept of “squares”, the predicate is “rectangles”. The quantifier word is “all”. The predicate in this case is wider than the subject and completely includes the latter in its scope. So, all squares are rectangles, but not all rectangles are squares. This means that the subject of a given judgment is distributed, while the predicate is not distributed. If you change this judgment, you can get the case of mutual distribution of the subject and predicate. Let's add the word "equilateral" to the judgment and get the following: "All squares are equilateral rectangles." In this case, the volumes of the two concepts are equal, they are completely included in each other. The distribution of concepts is reflected in diagrams where the plus sign (+) expresses the distribution of the concept, and non-distribution - the minus sign (-). Let's move from affirmative to negative concepts. Private negative judgments have the structure "Some S are not P". In the judgment "Some servicemen are not engineers" the subject is the concept "servicemen", the predicate is "engineers", the quantifier word is "some". The subject is undistributed, since in its scope we mean only a part of the military personnel, while the predicate reflects all engineers, none of which is part of the subject's scope. On Euler's circular scheme, this judgment is reflected as two intersecting circles. None of them is completely included in the scope of the other. This example shows that sometimes you can make a mistake. This is due to the external similarity of the circular schemes of particular negative and particular affirmative judgments. In this case, the error may be as follows: based on the fact that the subject and the predicate are characterized by mutual intersection, these terms can be incorrectly defined as undistributed. In simple terms, we note that in this judgment we are not considering the entire set of military personnel (S), but only the part that is not engineers (P). In the predicate, however, we think of all engineers, none of whom is included in the scope of the subject. Since the subject does not contain a single engineer, the entire set of people of this profession is conceived in the predicate. Thus, the predicate, unlike the subject, is distributed. All-negative judgments have the structure "No S is a P". The proposition "No man is a bird" is generally negative. Here both the subject and the predicate are completely distributed. This is due to the fact that the volumes of the concepts "man" and "bird" do not intersect, they are completely excluded from each other. On a circular diagram, the relationship between these concepts looks like two circles standing side by side, but not intersecting with each other. Having considered all these cases, we can conclude that there is a pattern. The distribution of subject and predicate depends on the type of judgment. The subject is distributed in general judgments, but not distributed in particular ones. Regarding the predicate, we can say that it is distributed in affirmative and negative judgments, but if in negative it is always distributed, then in affirmative ones only if it is equal in volume to the subject or if the volume of the subject is wider. The possibility of establishing the distribution of terms is very important, as it is one of the mechanisms for checking the correctness of judgments. This mechanism allows you to check the correctness of the construction of categorical syllogisms. Direct inferences are also checked. 3. General, private, singular judgments General categorical judgments have the structure "All S is (is not) P". They can be selective and exclusive. First on the basis of certain features, one object is distinguished from a group of others and considered separately. Thus, the role of this subject, its connections, relations with other subjects are considered somewhat more thoroughly. The selection of an object from the class of others is carried out with the help of the word "only", which is used in all such judgments. An example would be the following sentences: "It was as if winter had come in all the rooms of the house, and only in the living room it was warm" or "Only Ivanov did not pass the exam on time." Exclusive judgments also separate one object from a group of others. They contain the words "except", "except", etc. For example: "All students passed the session on time, except for Ivanov"; "With the exception of the Moon, celestial bodies are not satellites of the Earth." Rules of the Russian language, mathematics, physics, logic, foreign languages and other sciences containing exceptions from the general should also be considered as excluding concepts. Private judgments can be reflected as "Some S are (are not) P". Scientists are considering a point of view regarding which such judgments can be uncertain and certain. According to researchers, uncertain judgments are those that do not contain a more or less precise indication of the range of subjects, the opinion of which is reflected in these judgments. So, for example, the proposition "Some cars are sports" is considered indefinite, since in it we do not say that all cars should be recognized as sports, but we do not give an indication that only a part of the cars can be considered sports. The word "some", which indicates that a given judgment belongs to particular ones, is considered by researchers who adhere to this point of view to be an insufficient limitation on the number of subjects in relation to which this judgment is derived. In order to change the meaning of this word and obtain certain judgments, it is proposed to clarify them with the word "only". For example, certain there will be a judgment "Only some cars are sports". Drawing the line of reasoning further, it must be said that the formula "Some S are (not are) P" is common to all particular judgments and they can be placed within the framework of this formula. This can be seen in the example of indefinite judgments. Certain propositions, which are also particular, obey the formula "Only some S are (are not) P". In certain private judgments one can meet the quantified words "a lot", "several", "majority", "minority", "many", etc. Singular categorical judgments have the structure "This S is (is not) P". Accordingly, their subject is a single concept, i.e., a concept, the scope of which is exhausted by only one element. Thus, single judgments are: "Moscow is the capital of Russia"; "J. London is not a Russian writer"; "The sun is not a planet." LECTURE No. 12. Complex judgments. Formation of complex judgments 1. The concept of complex judgments The concept of complex judgments is inextricably linked with conjunction, disjunction, implication, equivalence and negation. These are the so-called logical links. They are used as a unifying link, linking one simple proposition to another. This is how complex sentences are formed. That is complex judgments are judgments created from two simple ones. The ratio of the truth of judgments is displayed in the tables. These tables reflect all possible cases of truth and falsity of judgments, and each of the simple judgments, which is part of the complex one, is reflected in the "cap" of the table as a letter (for example, a, b). Truth or falsity is reflected in the form of the letters "I" or "L" (true and false, respectively). Before considering conjunction, disjunction, implication, equivalence and negation, it makes sense to give them a brief description. These logical connectives are called logical constants. In the literature, you can find their other name - logical constants, but this does not change their essence. In our language, these constants are expressed in certain words. So, the conjunction is expressed by the unions "yes", "but", "although", "but", "and" and others, and the disjunction is expressed by the unions "or", "or", etc. We can talk about the truth of the conjunction if both simple propositions included in it are true. A disjunction is true when only one simple proposition is true. This refers to a strict disjunction, while a non-strict disjunction is true provided that at least one of its constituent simple judgments is true. An implication is always true except in one case. Let's consider the above in more detail. conjunction (a^b) - this is a way of linking simple judgments into complex ones, in which the truth of the resulting judgment directly depends on the truth of the composite ones. The truth of such judgments is achieved only when both simple judgments (both a and b) are also true. If at least one of these judgments is false, then the new, complex judgment formed from them should also be recognized as false. For example, in the judgment "This car is of very high quality (a) and has only run ten thousand meters (b)", the truth depends on both its right side and its left side. If both simple propositions are true, then the complex one formed from them is also true. Otherwise (if at least one of the simple propositions is false), it is false. This judgment is a characteristic of a particular car. The falsity of one of the simple propositions, obviously, does not exclude the truth of the other, and this can lead to errors associated with determining the truth of complex propositions formed with the help of a conjunction. Of course, the truth of one simple proposition is not excluded by the falsity of another, but we should not forget that we are characterizing an object, and from this point of view, the falsity of one of the simple propositions is considered from the other side. This is due to the fact that with the falsity of the judgment on one of the points of this characteristic, the characteristic as a whole becomes false (in other words, it leads to the transmission of incorrect information about the machine as a whole). Disjunction (aVb) is strict and non-strict. The difference between these two types of disjunction is that in a non-strict form its members are not mutually exclusive. An example of a non-strict disjunction might be: "To obtain a workpiece, the part can be finished on the machine (a) or pre-processed with a file (b)". Obviously, here a does not exclude b and vice versa. The truth of such a complex judgment depends on the truth of its members in the following way: if both members are false, the disjunctive judgment formed through them is also considered false. However, if only one simple proposition is false, such a disjunction is recognized as true. Strict disjunction is characterized by the fact that its members exclude each other (in contrast to non-strict disjunction). The judgment "Today I will do my homework (a) or go for a walk outside (b)" is an example of a strong disjunction. Indeed, you can do only one thing at the moment - do your homework or go for a walk, leaving the lessons for later. Therefore, a strict disjunction is true only when only one of the simple propositions included in it is true. This is the only case in which a strict disjunction is true. Equivalent It is characterized by the fact that an educated complex proposition is true only in those cases when both simple propositions that make up its composition are true, and false if both of these propositions are false. In literal terms, equivalence looks like a = b. When negating the proposition, displayed as a, is true when the concept is falsely negated. This is due to the fact that negation and the negated simple proposition not only contradict, but also exclude (deny) each other. Thus, it turns out that when the concept a is true, the concept a is false. Conversely, if a is false, then a negating it is true. Implication (a - › b) true in all cases except one. In other words, if both simple propositions in the implication are true or false, or if proposition a is false, then the implication is true. However, when the proposition b is false, the implication itself becomes false. This can be seen with an example: "We will throw a working cartridge into the fire (a), it will explode (b)". Obviously, if the first judgment is true, then the second is also true, since the explosion of a cartridge thrown into a fire will inevitably occur. Therefore, considering the first case, we can conclude that if the second proposition is false, then the whole implication is false. All the above examples of conjunction, disjunction, implication consisted of two variables. However, this is not always the case. There may be three or more variables. Considering complex judgments for truth, we get literal formulas. The latter can be characterized as true or false. In this regard, a formula is called identically true if it is true for any combination of its variables. The name identically false has a formula that takes only a false value (the value "false"). The last kind of such formulas is the satisfiable formula. Depending on the combinations of variables included in it, it can take both the value "true" and the value "false". 2. Expression of statements Sentences are expressed using symbols. - variables and signs denoting logical terms. There are no other symbols for this purpose. Variable statements are expressed as letters of the Latin alphabet (a, b, c, d, etc.). Such letters are called variable statements, as well as propositional variables. In simple terms, this group of symbols refers to simple judgments that make up a statement. These judgments are expressed in the form of narrative sentences. Another group of characters, used to express statements in the form of formulas, these are signs. They represent logical terms such as conjunction and disjunction, which can be strict or non-strict, negation, equivalence and implication. A conjunction is displayed as an upward tick (^) and a disjunction as a downward tick (V). For a strict disjunction, a dot is placed above the checkbox. The implication has the sign "-›", negation (-), equivalence (=). The last type of symbols with which statements are expressed are parentheses. Symbols denoting logical terms and types of connectives are characterized by different strengths. Thus, the ligament ^ is considered the strongest, that is, it binds stronger than all the others. The V ligament is stronger than the -, which is only important in some cases. Thus, determining the strength of connectives becomes important when writing formulas without using parentheses. If we have a statement expressed by the formula (a^b)Vc, you don’t have to write parentheses, but directly indicate that a^bVc. The same rule applies when using the symbol - ›. However, this rule is not true in all cases. That is, in many cases it is unacceptable to omit parentheses. For example, when the conjunctive connective of the concept a is carried out with two other concepts connected by the relation of implication and separated by parentheses, it is unacceptable to omit the latter (a^(b - c)). This is obvious, since otherwise it would be necessary to first carry out the conjunction and only then the implication. From a school mathematics course we know that it is impossible to omit parentheses in such a case. The following example can illustrate such a situation: 2 X (2 + 3) = 10 и 2 X 2 + 3 = 7. The result is obvious. In connection with the above, it can be noted that not every symbolic expression of statements is a formula. This requires the presence of certain signs. For example, the formula must be constructed correctly. Examples of such a construction could be: (a^b), (aVb), (a - b), (a = b). This construction is noted as a PPF, i.e. a correctly constructed formula. Examples of incorrectly constructed formulas could be: a^b,aVb, Vb,a - b, (a^b) etc. In the first three cases, the incorrectness of the formula lies in the fact that the concepts united by sheaves must be enclosed in brackets. The last formula has an open bracket, while the third example is characterized by the fact that one simple concept is not combined with another, despite the fact that there is a disjunction symbol. In our daily life, we often, sometimes without noticing it, use not only simple, but also complex judgments. Such judgments, as already mentioned above, are formed from two or more simple judgments with the help of logical connectives, which are called disjunction, conjunction, implication and negation, as well as equivalence. These links are expressed using signs: ^ for the conjunction V for disjunction, -> for implication. familiar = display equivalence, and the sign a means negation. There are two options for displaying the disjunction. The first one is a simple downward tick for a simple disjunction. For complex, the same checkmark is used, but with a dot on top. The graphic representation of the formulas of complex judgments is very important, as it allows you to more clearly understand their structure, nature and meaning. Logical connectives unite simple propositions, which are essentially declarative sentences. And there are quite a lot of options here. Sentences can consist of nouns and adjectives, verbs, participles, etc. Some sentences are simple propositions, others are complex. Complex judgments or statements are characterized by the fact that they can be divided into two simple ones, united by a logical constant. However, this is not possible with all complex sentences. When, as a result of dismemberment, a statement changes its meaning, such an operation is unacceptable. For example, when we say “The area was old, and the houses in it had long since fallen into disrepair,” we mean a conjunction where one side, “the area was old,” is united by the conjunction “and” with the second part, “the houses in it have long since fallen into decay.” . The meaning of the statement has not changed, despite the fact that we examined simple propositions in isolation from each other. However, in the statement “There is a beautiful and fast car parked in the parking lot,” an attempt to separate will lead to distortion of the originally transmitted information. So, considering simple propositions separately, we get: “a beautiful (car) is parked in the parking lot” - this is the first proposition combined with the second conjunction “and”. The second proposition is: “(there is a) fast car parked in the parking lot.” As a result, you might think that there were two cars - one beautiful, the other fast. Logic - this is, of course, an independent science, which has its own conceptual apparatus, tools, information base. Any independent science is separated from others and often radically differs in its approach to a particular subject. This should be kept in mind when we consider the constructions of the Russian language from the point of view of logic. Logic studies such constructions more in isolation. Thus, the time factor is often not taken into account when considering various judgments. In Russian, the time factor, in appropriate cases, is always taken into account. Here it should be said about the commutativity of the conjunction, which is inextricably linked with the above features of the language and logic. Commutativity - this is the equivalence of judgments (statements), when (a^b) = (b^a). In language, the law of commutative conjunction does not apply, since the time factor is taken into account. Indeed, it is impossible to imagine the equivalence of certain judgments, one of which is earlier in time than the other, and vice versa. For example, the statement "It started to rain and we got wet" would not be equivalent. (a^b) and "We got wet and it started to rain" (b^a). The same situation can be seen in the statements “A shot rang out, and the beast fell” and “The beast fell, and a shot rang out.” Obviously, the time factor is taken into account here, according to which one event or action, reflected in a complex judgment, precedes another, which determines the meaning of the entire statement. Logic abstracts from time and evaluates the judgment only from the point of view of its correct construction, as well as truth or falsity. In this regard, the above statements are equivalent, since in each individual case, both parts of them are true. In this way, Conjunctive statements in logic are commutative, the use of the conjunction “and” in judgments from the point of view of language (in the case when the time factor is taken into account) is non-commutative. Despite the fact that the prepositions with which the conjunction is formed were indicated above, it cannot be said that in the absence of these prepositions in the judgment, the conjunction is impossible. This is not true. Often in sentences that are complex judgments, different punctuation marks are used as connectives. For example, it can be a comma or a dash, and sometimes a period. Punctuation marks used in statements are placed between simple judgments and connect them with each other. An example of the use of punctuation marks as logical connectives is the sentence "The clouds parted, the sun came out" or "It was frosty outside, all living creatures hid, icicles formed on the roofs." In general, many scientists have dealt with the issues of linguistic expression of the conjunction. Therefore, this issue is well worked out and covered. A disjunction (recall that its symbolic designation is V, as well as a similar tick, but with a dot at the top) can be strict or nonstrict. The differences between these two types, as already mentioned, lie in the fact that the terms of a non-strict disjunction exclude each other, while the members of a strict disjunction do not. The law of commutativity with disjunction is valid regardless of what kind of disjunction is meant. Let's remember that disjunction is expressed by conjunctions, the main ones are definitely “or” and “either”. Let us give examples of strict and non-strict disjunction and use them to illustrate the operation of the law of commutativity. The proposition “I will drink sparkling water or still water” is an example of a weak disjunction, while the proposition “I will go to university or stay at home” is a strict one. The difference between them is that in the first case the action will still be performed, regardless of the selected type of water. In the second case, the action (I will go to university) is excluded if you choose the second option and stay at home. In many cases, the conjunction "or" can simply be replaced with the conjunction "or". For example, in the sentence “Either I ski down the mountain or fall along the way,” you can use the conjunction “or” without any changes. However, there is a conjunction that is used independently and is also a disjunctive connective. This is an “either or either” conjunction. It is quite often used when constructing sentences “Today either an auditor or an auditor came”; “He lives either on Moskovskaya or Komsomolskaya street,” etc. As mentioned above, the law of commutativity in disjunctive statements operates regardless of the type of disjunction. Take for example the following proposition: “I will drink water with or without gas” and “I will drink water without or with gas.” Obviously, there is no difference between them, the meaning remains the same. You can also check other examples, say, “I will go to university or stay at home” and “I will stay at home or go to university.” The content and scope of a complex judgment formed using a disjunction do not change from rearranging its members. That is why we talk about universal commutativity. The expression of logical connectives in the language is very diverse, there are many schemes according to which statements are built. For each of these schemes, you can build a huge number of complex judgments. This is especially characteristic of the Russian language in all its ambiguity. For example, the implication is built according to such schemes as, for example, "A needs B"; "A is enough for B"; "if A, then B", "A, only if B", etc. For example: "In order to know a lot, you need to study a lot"; "To jump from a tower, it is enough to push off with your feet correctly"; "If the car gets stuck, it will have to be pushed"; "You can turn in your session on time only if you start preparing immediately." A number of formulas exist for equivalence: "A if B, and B if A"; "for A, B is necessary and sufficient"; "And if and only if B", etc. Let us give examples of judgments built on the basis of these schemes. For example: "If a person is engaged in weightlifting, he will become stronger" and "A person will become stronger if he is engaged in weightlifting"; "To enter a university it is necessary and sufficient to pass the entrance exams"; "You have reached the summit when and only when you have set foot on the highest point of the mountain." In this regard, it is also necessary to mention the ambiguity of conjunctions expressing logical constants (conjunction, disjunction, implication, etc.). For example, the union "if" can often express not an implication, but a conjunction. It depends on the existence of a meaningful connection between judgments. In this regard, it is necessary to consider natural language expressions from the standpoint of their diversity and heterogeneity. In addition to logical connectives, expressed in the Russian language through conjunctions that are used in the formation of general and particular judgments, there are quantifiers. These are the existential quantifier and the general quantifier. General quantifier expressed in Russian by the words "each", "any", "all", "none", etc. Usually a formula with a general quantifier is read as "all objects have a certain property". Existence quantifier expressed by the words "majority", "minority", "some", "many" and "few", "many" and "few", "almost all", etc. This quantifier is expressed as "there are some objects that have a certain property". There is a variant of the existential quantifier in which "there are some objects that are greater than a certain value". In this construction, objects are understood as numbers. Some judgments constructed using implication are expressed in the subjunctive mood. They have the same formula as other implications (a - › b), but they are usually called counterfactuals. The subjunctive mood makes us understand that the basis and consequence of such judgments are false. However, this falsity is not universal, i.e., under certain circumstances, the truth of such statements is possible. In other words, such judgments can reflect the subject matter correctly and objectively. Truth is possible if the relationship between reason and effect implies that the truth of the effect follows from the truth of the reason. Otherwise, we can state the falsity of such a judgment. A statement constructed in the subjunctive mood has the structure "if A, then it would be B". For example, "If you went to all classes in logic, you would successfully pass the exam"; "If the train had not been late, we would have missed the train" and "If the patient had not fallen, his leg would not have hurt." Counterfactual statements are of great importance for history, philosophy, to a certain extent mathematics and some other sciences. They are used in constructing hypotheses, considering historical and other issues, and determining possible directions for certain processes. For example, discussions on the topic of the Great Patriotic War are still ongoing. As part of this discussion, the question of the possibilities of its alternative course and the results that could have occurred under a different set of circumstances is considered. Also, within the framework of chemistry, physics, and astronomy, counterfactual judgments are often used. For example, practical physics sometimes comes to the conclusion that it is not possible to theoretically determine the exact course of a process. In this case, to achieve the desired result, you have to use the intellectual search method and confirm the results with practice. The following statement may be an example of a counterfactual statement in physics: "If we pass an electric current through a copper conductor, then the discharge will be stronger." Since the truth of a counterfactual judgment is ambiguous, and by default both its basis and its consequence (and, accordingly, the entire judgment as a whole) are recognized as false, this judgment has to be verified in practice. In this case, the proposition can be either true or false. It depends on which conductor we used earlier. For example, if we took an iron conductor before a copper one, our judgment will be true, since copper gives less resistance when moving along an electric current conductor. However, if we previously used gold as a conductor, the judgment will turn out to be false, again for a reason related to the conductivity of materials - gold has a conductivity much greater than copper. Astronomy calls into question some properties of the orbits of celestial bodies and the features of the movement of the latter, the relative position of planets, stars, systems and galaxies, etc. As a result, counterfactual statements are also used. Sometimes, in order to justify themselves or to smooth over an acute situation, people say: "If this had not happened, then everything would have gone differently." This is also an example of using the subjunctive mood. However, it should be remembered that counterfactual propositions consist of false reasons and consequences. Therefore, when using such constructions in science, a certain amount of caution must be observed. Counterfactual propositions can be expressed using formulas. Such formulas reflect the number of terms of the statement, the type of connective between them and the sign of the implication. The implication in a counterfactual judgment has a certain specificity: it corresponds, among other things, to the conjunction “if... then”. On the left in such a formula are reflected the members of the counterfactual statement corresponding to the conjunction “if”, on the right - the conjunction “then”. The left and right sides are separated by an implication sign, different from that used in classical propositional logic. The difference between these two symbols is that on the back side of the arrow indicating the implication (classic version (-›)), in the counterfactual implication there is a vertical bar (| - ›). Such a sign is not used in classical propositional logic. 3. Denial of complex judgments Negation of judgment in logic - this is the replacement of an existing bundle within a complex statement with another, opposite to the last one. If we are talking about a formula in which the negation of complex judgments can be expressed, then it should be noted that the negation is graphically expressed as a horizontal line above the negated judgment. Thus, we get two concepts, united by a logical link, over which a horizontal line is drawn. If such a feature already exists, then in order to implement negation it is necessary to remove such a feature. All of the above applies to operations performed using conjunction and disjunction. However, what has been said above does not mean that the denial of complex judgments is possible only if they contain exclusively conjunctions of conjunction and disjunction. If it is necessary to carry out the operation of negation in relation to a judgment containing an implication, it is necessary to replace this judgment in such a way that, in the absence of any of its changes, the implication is discarded. This means that it is necessary to choose a judgment equivalent to the given one, which would not contain an implication. When we speak of a judgment that is equivalent to one containing an implication, but not containing it, we mean replacing this connective with a conjunction or disjunction. Graphically, this looks like (a - b) = (a V b). Then the operation described above is performed, in which the sign of conjunction is changed to disjunction, and vice versa. Usually in speech the expression of negation comes down to adding the prefix “not”. Indeed, since the specified prefix is negative, its use to establish the opposite is completely justified. It is necessary to mention the laws of de Morgan. They are used in the process of negating complex judgments and have a formulaic expression. There are only four such laws and, accordingly, formulas: one) _________ a^b = aVb; 2) _____ a ^ b = a V b; one) _________ a Vb = a ^ b; 4) _____ a Vb = a ^ b. Having considered the above, it can be noted that the negation of a complex proposition, which contains a conjunction or disjunction, is a "simple" option, in which it is sufficient to carry out the negation operation. The formula formed using de Morgan's laws is as follows: (a ^ b) V (c ^ e) = (a V b) ^ (c V e). Let us give examples of the negation operation. Negation of a complex proposition, in which there is no implication: "I will finish work and go for a walk and go to the store" - "I will finish work, but I will not go for a walk and will not go to the store." The denial of a complex proposition, in which it is necessary to first change the implication to a conjunction or disjunction, can be illustrated by the following example: "If I buy a car, I will go out of town or turn to the dacha" - "I will buy a car, but I will not go out of town and will not turn into dacha". In this example, for convenience, we have omitted the implication elimination step. It must be said that judgments that negate each other cannot be both true and false at the same time. The situation of contradiction or negation is characterized by the fact that one of the contradictory concepts is always true, while the other is false. There can be no other position in this case. It is impossible to identify the operation of negation, as a result of which a new judgment is formed, from the negation, which is a part of negative judgments. The negation of judgments can be made both in relation to the entire judgment and its parts and is expressed by the words “is not”, “is not the essence”, “is not”, as well as “wrong”, etc. Based on the foregoing, we can conclude that there are two types of denial - internal and external. As you might guess, the external denies the entire judgment as a whole. For example, "Some soldiers are not paratroopers" is an internal negation, while "It is not true that the Moon is a planet" is an external negation. Thus, the external negation is the negation of the entire judgment as a whole, while the internal one shows the fact of a contradiction or inconsistency between the predicate and the subject. The following types of negative judgments can be displayed in the form of formulas: "all S are P" and "some S are not P" (these are general judgments); "no S is P" and "some S are P" (private judgments). The last kind of negative propositions is "this S is P" and "this S is not P" (propositions called singular). LECTURE No. 13. Truth and modality of judgments 1. Modality of judgments modal judgment - this is a separate type of judgments, which has its own characteristics and is characterized both by the presence of features common with assertoric judgments, and by difference from the latter. Modal judgments are studied within the framework of modal logic, which is heterogeneous in its content and is divided into several branches. Among them: logic of time, logic of action, logic of norms, deontic logic, logic of decision making and more From the point of view of classical logic, one or another judgment can be called assertoric or modal. It is obvious that these two types are different from each other. Modal judgments can be called clarifying. Judgments of this kind do not just characterize this or that object, describe, define it and its inherent properties, but also clarify and supplement such a characterization. In a simplified form, we can say that modal judgments express our attitude towards the object in question. Of course, this feature of modal judgments is reflected in natural language. So, unlike assertoric judgments (read - simple), modal ones contain a number of special words. For example, "proven", "necessarily", "possibly", "good", "bad", etc. These words are called modal operators. One can show the difference between assertoric and modal judgments by giving the following sentences: "Tomorrow it will be cold" - this judgment is assertoric; "Perhaps it will be cold tomorrow" - as is already clear, this is a modal judgment. From these positions, it can be argued that modal judgments are assertoric judgments supplemented by a specific relation. However, the role of modal statements is not limited to a simple transfer of the speaker's attitude to the subject. There is a more complex and noticeable pattern that is not at first glance: modal judgments reflect the nature of the connection between the subject and the predicate. In a sense, they create it themselves. Modal judgments are judgments that reflect the relationship and connection between the subject and the predicate and show the relationship to the subject with the help of modal operators. In order to better understand the nature of this type of judgment, let us consider a number of examples. We will first give an example of an assertoric judgment, and then a modal one formed from it. “There is not a cloud in the sky, and the sun is shining brightly,” “It’s good that there is not a cloud in the sky and the sun is shining brightly”; “Correct posture improves performance”, “Correct posture has been proven to improve performance” and “Pouring cold water improves health”, “Dousing cold water has been proven to improve health.” And also: “The runner in the second lane will come first,” “It is possible that the runner in the second lane will come first”; “Two multiplied by two makes four,” “Obviously, two multiplied by two makes four”; “An electric current, as it passes, heats the conductor” and “It is imperative that the current, as it passes, heats the conductor.” The difference between assertoric and modal judgments in the examples given is obvious. Let's say the first pair of judgments. “There’s not a cloud in the sky...” is just a statement of fact, a description of two components of clear weather, devoid of evaluation, and with it any feelings and emotions. With the addition of the word “good” to the judgment comes the speaker’s assessment of this weather. From this judgment we can clearly conclude that he likes this kind of weather. The first type of judgment, like the second (i.e., both assertoric and modal judgments) can be either true or false. There is no third option. However, one cannot but agree that modal judgments have more variations and shades. They can often be interpreted differently, which makes it possible for errors to occur in determining their truth or falsity. Here it is necessary to mention that logic in general and modal logic in particular approach the consideration of the meaning of the words “possible”, “necessary”, “proven”, “obligatory”, as well as “necessity”, “obligatory”, “obligatory” derived from them. chance", "impossibility" from a special point of view. If, from the point of view of natural language, the above words are only words and have different shades and meanings, then logic elevates them to the rank of categories. From this point of view, their interrelations and dependencies are considered. These categories are also considered within the framework of philosophy, which is most interested in their substantive side. In this way, assertive judgments - These are simple judgments in which certain information about a particular subject is affirmed or denied. They are also characterized by what they say about the relationship between the objects reflected in them. There may be two or more such items. To clarify the above, we give an example: "All professional skiers are athletes." In this judgment, the concepts of "professional skiers" and "athletes" are correlated, and the first is narrower than the second and is fully included in its scope, but richer in content, due to the fact that it has more features. A modal judgment, in contrast to an assertoric one, indicates the proof or lack of proof of what is reflected in the judgment, the need for a connection between objects or its accident, the relation to the subject of judgment from the point of view of morality, morality, etc. Modal judgments have a structure: M (S is (or not to eat) P). It must be said that assertoric judgments (as already described in other chapters) can be combined into complex ones using logical connectives (conjunctions, disjunctions, equivalences, implications, denials). Modal operators are great for complex judgments as well. In other words, even complex judgments can be modal. In this case, their structure will be: M (a ^ b) or M (a V b), etc. It is only necessary to remember that there are five logical connectives and, accordingly, complex judgments formed from them. Words in a natural language (including Russian) are characterized by a certain ambiguity. In other words, many words have different meanings with the same sound. Others, despite the fact that they differ in sound and spelling, mean the same thing. The latter also applies to modal operators. Thus, one of the modal operators can easily be replaced by another, and without losing the implied meaning of the judgment. For example, the judgment "Probably this athlete will run first" will not lose what he has and will not gain a new one if you replace "probably" with "maybe". Judge for yourself: "Perhaps this athlete will come running first." This can be done in other cases as well. Combining the above, we can call complex modal judgments such complex judgments that, with the help of modal operators, reflect the relationship and connection between the simple judgments that make up its composition. As described above, modal statements are formed using modal operators. The modality of judgments has a number of modal concepts. They are well studied and systematized. At the same time, the systematization is based on the strength of modality, as well as its positivity or negativity. There are three basic modal concepts, although some scholars insist on the view that there are four of them. The three main modal concepts are characterized by the fact that the first of them is strong and positive, the second is a weak characteristic, and the third, in contrast to the first, is a strong negative characteristic. The fourth modal concept is intended in some cases to replace a strong positive concept and a weak characteristic. Modalities can be logical and ontological, diontic, epistemic, axiological and temporal. Logical modalities together with ontological ones form alethic modalities. Speaking about the modality of judgments, it was mentioned more than once about modal operators. They show the necessity of a judgment or its chance, possibility or impossibility. However, the process did not define either truth or falsity or other terms from this series. Meanwhile, knowing the exact meaning of the above categories is important. So, the necessity of a judgment means that this judgment is based on a law discovered within the framework of any science, including logic. In this case, all justified consequences derived from these laws are also recognized as necessary. The determining factor in this case is the factor of objectivity. In other words, the law must be real, not virtual, that is, it must correctly reflect the real state of affairs. Random judgments are defined as statements, although not based directly on the laws known to science, but not contradicting them. The same applies to the consequences of these laws. In the case of impossible judgments, everything is obvious. Such judgments are those that contradict scientifically confirmed laws or their consequences. Possible judgments are based on common sense and do not contradict scientific laws and their consequences. The above categories study alethic modalities. 2. Truth of judgments Turning to the question of the truth of judgments, it should immediately be said that often the definition of this factor becomes a difficult task. This may be due to the ambiguity of the words used in statements, or to the incorrect construction of the judgment from the point of view of logic. The reason may be the complexity of the structure of the judgment itself or the impossibility of determining the falsity or truth at the moment due to the unknown or unavailability of the necessary information. Determining the truth of judgments is directly related to comparability and incomparability. Comparable judgments are divided into compatible and incompatible. Incompatible judgments can be in a relationship of contradiction and opposition. The concepts included in the relation of contradiction are characterized by the fact that they cannot be simultaneously true or false. If one of the contradictory propositions is true, then the other is false, and vice versa. If one of the opposite propositions is true, the other is necessarily false, since they completely exclude each other. At the same time, the falsity of one of the opposite judgments does not mean the falsity or truth of the other. Indeed, the opposite of judgments does not yet mean that one of them is always true, and the other is false. For example: "There is no life on Mars" and "There is life on Mars". These concepts are indefinite, that is, it is not known whether they are true or false. Both of them may be false. But only one of them can be true. Compatible judgments enter into a logical relationship subordination, equivalence and overlap (intersection). Subordinate compatible judgments. They bear such a name due to the fact that one of these judgments is included in the scope of the other, is subordinate to it. Such judgments have a common predicate. The definition of the truth of judgments that are in relation to subordination is associated with a certain specificity, since one of the judgments is included in the scope of the second. In this regard, the truth of the general judgment entails the truth of the particular, while the truth of the particular does not determine with certainty the truth of the general. The falsity of the general leaves the particular judgment indefinite, and the falsity of the particular does not mean that the general is also false. Let's give an example: "Ferrari is a good car" and "All cars are good". The second proposition is false. It is subordinate. At the same time, the private judgment subordinated to it is true. Relatively speaking, compatible equivalent judgments reflect the same phenomenon or object of the surrounding world, but they do it differently. So, if we take for consideration two different judgments about one object or phenomenon, that is, two compatible judgments, then we will notice a pattern: in one case, both of these statements will have one subject, but differently expressed (although having the same meaning) predicates . In another case the opposite situation arises. However, in this case we are talking only about equivalent, but in no case about all compatible judgments. It goes without saying that when two judgments are equivalent, identical in meaning, if one of them is false, the second is false, and vice versa. Examples of equivalent compatible propositions are the following statements: "The moon is a natural satellite of the Earth" and "The moon is a satellite of the Earth that arose as a result of natural causes." When determining the truth of compatible judgments that are not equivalent, it is necessary each time to proceed from the real state of things: since compatible concepts often reflect the same subject only partially, each of them in this case can be both true and false. The intersection relation is characterized by the fact that if one such judgment is false, the other is necessarily true. This is due to the fact that such judgments have the same subject and predicate, which nevertheless differ in quality. Moreover, if one of these judgments is true, then with respect to the other it is not clear whether it is true or false. LECTURE No. 14. Logical laws 1. The concept of logical laws The laws of logic have been known since ancient times - law of identity, non-contradiction and excluded middle. All of them were discovered by Aristotle. The law of sufficient reason was discovered by Leibniz. They are of great importance for science, they are the pillars of logic, because without these laws logic is unthinkable. logical laws - these are objectively existing and necessary applied rules for the construction of logical thinking. Like any laws of the surrounding world, discovered within the framework of science (for example, natural), the laws of logic are objective. Logical laws differ from the laws of jurisprudence in that they cannot be repealed or changed. Thus, they are characterized by constancy. You can compare the laws of logic, for example, with the law of universal gravitation. It exists independently of anyone's will. Therefore, logical laws are the same for everyone. However, despite the presence of common features with the laws of nature, logical laws have their own specifics. The laws of logic are the laws of correct thinking, but not of the surrounding world. As mentioned above, the laws of logic represent a kind of foundation for the science of logic. Everything in it is based on these fundamental rules. Sometimes they are also called principles, and their application is widespread. Consciously or unconsciously, every person in everyday life - at work, on vacation, in a store or on the street - applies logical laws in practice. Sometimes statements, whether by accident or design, do not obey logical laws. More often than not, this is immediately noticeable and, as they say, “catchy.” That is why many people talk about the uselessness of logic as a science - after all, it is always clear when a person makes his judgment incorrectly. However, we should not forget that, in addition to everyday life, where philistine logic is sufficient, there is science, which is characterized by a higher level of knowledge. This is where precision and correct thinking are needed. What can be forgiven in a simple conversation is unacceptable in a scientific discussion. And there should be no doubt about this. Just imagine for a moment a nuclear power plant designer drawing diagrams by eye, and the importance of logical laws becomes obvious. 2. The law of identity. Law of non-contradiction Identity law (a = a). To characterize it, it is first necessary to understand what identity is in general. In the most general sense Identity means equivalence, sameness. At the same time, it is rarely possible to talk about absolute identity, since it is difficult to find two completely identical objects. In this sense, it is logical to talk about the identity of an object with itself. However, there are pitfalls here too - the same object, taken at different periods of time, most likely will not be characterized by identity. For example, you can take a person at 3 years old, 20 and 60 years old. Obviously, this is the same person, but at the same time they are three “different” people. Therefore, absolute identity in the real world is impossible. But since the world does not live according to absolute laws, we can talk about identity, moving away from complete abstraction. The law of identity follows from what has been said above. It means that in the process of constructing judgments and statements, it is unacceptable to replace one object with another. That is, you cannot arbitrarily replace the subject from which the logical construction began with another. One cannot call objects identical that are not identical, and one cannot deny the identity of identical objects. All this leads to a violation of the law of identity. Also, a violation of the law of identity occurs when a person incorrectly names things. In this case, he can convey correct information, which nevertheless does not concern the named subject. There are cases when the subject is changed in a dispute. That is, the arguing imperceptibly move from a discussion of a previously chosen subject to a new one or narrow the concept of the subject to its linguistic expression. That is, they are no longer discussing the subject itself, but the words, phrases, etc. expressing it. This change can occur for various reasons. Here is the intent of one of the participants, and a mistake, also intentional or unintentional. Often the law of identity is violated when using ambiguous words. These can be pronouns, homonyms. For example, homonymous words in a sentence taken out of context are often difficult to limit to one or another of their meanings. That is, it is not clear in what sense the word was used. In this case, instead of one value, another can be taken, and then the law of identity will be violated. Often arising from ambiguity, violation of the law of identity also creates ambiguity, and with it confusion. Speaking about the law of identity and its violations, these violations must be named. The first one is called "change of concept" and means that the subject of the concept was lost, i.e., the originally understood meaning has changed. Substitution of the thesis - the second type. It means changing the thesis initially understood in the process of discussion. The law of identity is widely used not only within the framework of logic, but also by other, including applied, sciences: computer science and mathematics, physics, chemistry, jurisprudence, forensic science, etc. Law of non-contradiction. Probably, everyone in his life has encountered a situation when the subject he undertook to talk about turned out to be so difficult that the thread of reasoning soon slipped away and confusion began in his thoughts. This happens because the subject is not well known to the narrator or he has not made the necessary preparation. As soon as a clear “path” of reasoning is lost, contradictions begin. The reasoner can, often without noticing it, express contradictory judgments one after another. The law of non-contradiction speaks precisely about the inadmissibility of a contradiction between what was said earlier and what was said again. It is also a contradiction to attribute previously rejected properties to one and the same object, and vice versa. Such a contradiction is called formal-logical. Not to mention the time factor. In this case, it is of immediate importance. We are talking about the inadmissibility of a contradiction between two or more statements, that is, if it was previously approved, say, that an object has one or another feature, the subsequent denial of this feature is unacceptable. However, do not forget about time and the fact that everything in our world tends to change. So, a judgment is not contradictory, which, although it contains mutually exclusive information about the subject, but implies the same subject at different intervals of time. 3. Law of the excluded middle Law of the excluded middle associated with conflicting opinions. It means that there can be only two contradictory judgments, there cannot be a third. Hence the name of this law. If two judgments deny each other, one affirms something, and the other contradicts the existence of what is being affirmed, we can say that these judgments are contradictory. Each of these judgments is independent and is considered separately due to the fact that it contains information that denies the opposite judgment. Consideration of them in this regard is carried out in order to determine which of them is true and which is false. Since such judgments are completely mutually exclusive, that is, if one is true, the other is always false, there is no third option. That is, it means that there is no intermediate state between true and false. This means that there can be no third judgment regarding one object, reflecting the same properties that are reflected (affirmed or denied) by two contradictory judgments. For a more complete understanding of the issue, examples should be given. To begin with, consider the schematic reflections of the contradictory judgments: "No S is P" and "Some S are P"; "All S are P" and "Some S are not P"; "This S is P" and "This S is not P". As you can see, all three given pairs of judgments are, respectively, general, particular and singular, as well as contradictory (i.e., type A and non-A). The judgments "Yuri Gagarin is the cosmonaut who first flew into space" and "Yuri Gagarin is not the cosmonaut who first flew into space" are contradictory judgments. When considering the law of the excluded middle, the question always arises of its differences from the law of non-contradiction. This is due to the fact that both these laws apply to the contradictory judgments now considered. However, there is a difference between them. It becomes clear if one considers counter-judgments (for example, "All men have limbs" and "No man has limbs") judgments. The law of the excluded middle does not apply to them. 4. Sufficient reason Any assertion must have a basis. It is obvious. When one of the parties to a dispute claims something, the other often demands: "Justify." sufficient reason wherein is reliable information. Any true thought must be sufficiently substantiated. Of course, the absence of a sufficient reason does not entail the falsity of a judgment; it can be true. However, this fact remains unknown until justification is received. It must be said that only a true judgment needs justification. What is false cannot have sufficient reason at all. Despite the fact that in some cases there have been attempts to substantiate false judgments with varying success, this approach cannot be called correct. The law of sufficient reason is not expressed in the form of a formula, since there is no such formula. When we say that true information is a sufficient basis for a judgment, we mean various kinds of data based on reliable sources. For mathematics, these are digital expressions derived without errors using axioms, theorems, various systems that allow reliable calculations (such a system, for example, is the multiplication table). Information obtained on the basis of scientific laws will also be considered reliable. To substantiate a new judgment, one can use previously derived judgments, in relation to which it has been proven that they are true. The law of sufficient reason, perhaps more than any other, operates in the realm of human daily life, and also applies within various professions. This is due to the fact that in the process of cognition, a person first of all thinks about what the new, received information is based on. For example, you can often hear in the media that information was obtained "from reliable sources", or sometimes the expression "according to unverified data" is used. Of course, the law of non-contradiction and the excluded middle, as well as the law of identity, play a huge role in correct thinking. However, they seem to follow the law of sufficient reason. The need for them arises only when there is a substantiation of one or another fact, concept, judgment. What has been said should be attributed, of course, not to the scientific significance of the laws of logic, but rather to the necessity of these laws for the life and activity of the average person. Within the framework of this question, it is necessary to say about one feature that is characteristic of the logical reason and consequence in their relationship with the real reason and consequence. If in real life the foundation always comes first, and the consequence is derived from it, then in logic the opposite situation can take place. This is due to the order of things - in the real world, the foundation process first passes, and only then the consequence is derived from it. A person, who did not have the opportunity to observe the reason, can rely only on the consequence. Thus, having received a consequence, a person mentally, virtually can recreate the foundation. LECTURE No. 15. Inference. General characteristics of deductive reasoning 1. The concept of inference Inference - this is a form of abstract thinking, through which new information is derived from previously available information. In this case, the sense organs are not involved, that is, the entire process of inference takes place at the level of thinking and is independent of the information received at the moment from the outside. Visually, the conclusion is reflected in the form of a column in which there are at least three elements. Two of them are premises, the third is called the conclusion. Parcels and conclusions are usually separated from each other by a horizontal line. The conclusion is always written below, the premises - above. Both the premises and the conclusion are judgments. Moreover, these judgments can be both true and false. For example: All mammals are animals. All cats are mammals. All cats are animals. This conclusion is true. Inference has a number of advantages before the forms of sensory knowledge and experimental research. Since the process of inference takes place only in the realm of thinking, it does not affect real objects. This is a very important property, since often the researcher does not have the opportunity to get a real object for observation or experiments due to its high cost, size or remoteness. Some items at the moment can generally be considered inaccessible for direct research. For example, space objects can be attributed to such a group of objects. As is known, human exploration of even the closest planets to the Earth is problematic. Another advantage of inferences is that they provide reliable information about the object under study. For example, it was through inference that D. I. Mendeleev created his own periodic system of chemical elements. In the field of astronomy, the position of the planets is often determined without any visible contact, based only on the information already available about the regularities in the position of celestial bodies. Inference flaw one can say that conclusions are often characterized by abstractness and do not reflect many of the specific properties of the subject. This does not apply, for example, to the above-mentioned periodic table of chemical elements. It is proved that with its help, elements and their properties were discovered, which at that time were not yet known to scientists. However, this is not the case in all cases. For example, when determining the position of a planet by astronomers, its properties are reflected only approximately. Also, it is often impossible to speak about the correctness of the conclusion until it has passed the test in practice. Inferences can be true and probabilistic. The former reliably reflect the real state of affairs, the latter are of an uncertain nature. The types of inference are: induction, deduction and conclusion by analogy. Inference - this is primarily the derivation of consequences, it is applied everywhere. Every person in his life, regardless of profession, made conclusions and received consequences from these conclusions. And here the question of the truth of such consequences arises. A person who is not familiar with logic uses it at a philistine level. That is, he judges things, draws conclusions, draws conclusions based on what he has accumulated in the process of life. Despite the fact that almost every person is trained in the basics of logic at school, learns from their parents, the philistine level of knowledge cannot be considered sufficient. Of course, in most situations this level is enough, but there is a percentage of cases when logical preparation is simply not enough, although it is in such situations that it is most needed. As you know, there is such a type of crime as fraud. Most often, scammers use simple and proven schemes, but a certain percentage of them are engaged in highly skilled deception. Such criminals know logic almost perfectly and, in addition, have abilities in the field of psychology. Therefore, it often costs them nothing to deceive a person who is not prepared. All this speaks of the need to study logic as a science. Inference is a very common logical operation. As a general rule, in order to obtain a true judgment, the premises must also be true. However, this rule does not apply to evidence to the contrary. In this case, knowingly false premises are deliberately taken, which are necessary in order to determine the necessary object through their negation. In other words, false premisses are discarded in the process of deriving a consequence. 2. Deductive reasoning Like much in classical logic, the theory of deduction owes its appearance to the ancient Greek philosopher Aristotle. He developed most of the issues related to this kind of reasoning. According to the works of Aristotle deduction is the transition in the process of inference from the general to the particular. In other words, deduction is the gradual concretization of a more abstract concept. It goes through several steps, each time deriving a consequence from several premises. It must be said that true knowledge must be obtained through the process of deductive reasoning. This goal can be achieved only if the necessary conditions and rules are met. There are two types of inference rules: direct inference rules and indirect inference rules. Direct inference means obtaining a conclusion from two premises that will be true if the rules of direct inference are followed. Thus, the premises must be true and the rules for obtaining consequences must be observed. Subject to these rules, one can speak of the correctness of thinking regarding the subject taken. This means that in order to obtain a true judgment, new knowledge, it is not necessary to have all the information. Part of the information can be recreated in a logical way and fixed. Consolidation is necessary, because without it the process of obtaining new information becomes meaningless. It is not possible to transfer such information or use it in any other way. Naturally, such consolidation occurs through the language (spoken, written, programming language, etc.). Consolidation in logic occurs primarily with the help of symbols. For example, these can be conjunction symbols, disjunctions, implications, literal expressions, brackets, etc. The following types of inferences are deductive: conclusions of logical connections and subject-predicate conclusions. Also deductive inferences are direct. They are made from one premise and are called transformation, inversion and opposition to the predicate, the conclusions on the logical square are considered separately. Such conclusions are derived from categorical judgments. Let's consider these conclusions. The transformation has a scheme: S is P S is not non-R. This diagram shows that there is only one package. This is a categorical judgment. The transformation is characterized by the fact that when the quality of the premise changes in the process of inference, its quantity does not change, and the predicate of the consequence negates the predicate of the premise. There are two ways of transformation - double negation and replacement of a negation in a predicate by a negation in a connective. The first case is shown in the diagram above. In the second, the transformation is reflected in the scheme as S is not-P - S is not P. Depending on the type of judgment, the transformation can be expressed as follows. All S are P - No S is non-P. No S is P - All S is non-P. Some S are P - Some S are not non-P. Some S's are not P - Some S's are not-P. Treatment - this is a conclusion in which the quality of the premise does not change when the places of the subject and the predicate are changed. That is, in the process of inference, the subject takes the place of the predicate, and the predicate takes the place of the subject. Accordingly, the circulation scheme can be depicted as S is P - P is S. Appeal can be with or without limitation. (it is also called simple or pure). This division is based on a quantitative indicator of the judgment (meaning the equality or inequality of the volumes of S and P). This is expressed in whether the quantified word has changed or not and whether the subject and predicate are distributed. If such a change occurs, then the constraint has been handled. Otherwise, we can speak of pure conversion. Recall that a quantified word is a word - an indicator of quantity. Thus, the words "all", "some", "none" and others are quantified words. Contrasting with a predicate characterized by the fact that the link in the consequence is reversed, the subject contradicts the predicate of the premise, and the predicate is equivalent to the subject of the premise. It must be said that a direct inference with opposition to a predicate cannot be deduced from particular affirmative judgments. Let's give opposition schemes depending on the types of judgments. Some S are not P - Some non-P are S. No S is P - Some non-P are S. All S are P - No P is S. Combining what has been said, we can consider the opposition to the predicate as the product of two immediate inferences at once. The first one is the transformation. Its result is inverted. 3. Conditional and disjunctive inferences Speaking of deductive reasoning, one cannot but pay attention to conditional and disjunctive reasoning. Conditional inferences are called so because they use conditional propositions as premises (if a, then b). Conditional inferences can be reflected in the form of the following diagram. If a, then b. If b, then c. If a, then c. Above is a diagram of inferences, which are a kind of conditional. It is characteristic of such inferences that all of their premises are conditional. Another type of conditional inference is conditional categorical judgments. According to the name in this conclusion, not both premises are conditional propositions, one of them is a simple categorical proposition. It is also necessary to mention modes - varieties of inferences. There are: affirming mode, denying mode and two probabilistic modes (first and second). Approving mode has the widest distribution in thinking. This is due to the fact that it gives a reliable conclusion. Therefore, the rules of various academic disciplines are built mainly on the basis of the affirmative mode. You can display the affirmative mode as a diagram. If a, then b. a. b. Let us give an example of an assertive mode. If the ax falls into the water, it will sink. The ax fell into the water. He will drown. The two true propositions that are the premises of this proposition are transformed in the process of inference into a true proposition. Negative mode expressed in the following way. If a, then b. Non-b. Nope. This judgment is based on the negation of the consequence and the negation of the foundation. Inferences can give not only true, but also indefinite judgments (it is not known whether they are true or false). In this connection it is necessary to speak about probabilistic modes. The first probabilistic mode in the diagram is displayed as follows. If a, then b. b. Probably a. As the name implies, the consequence deduced from the premises with the help of this mode is probable. If a strong wind blows, then the yacht heels to one side. The yacht rolls to one side. Probably a strong wind is blowing. As we see from the statement of the consequence to the statement of the reason it is impossible to draw a true conclusion. The second probabilistic mode in the form of a diagram can be depicted as follows. If a, then b. Nope. Probably not-b. Let's take an example. If a person lies under the sun, he will tan. This man does not lie under the sun. It won't burn. As can be seen from the above example, making a conclusion from the negation of the basis to the negation of the consequence, we will get not a true, but a probabilistic consequence. The formulas of the affirming and denying modes are the laws of logic, while the formulas of probabilistic ones are not. Divisive reasoning are divided into simple disjunctive and divisive-categorical inferences. In the first case, all premises are separating. Accordingly, dividing-categorical judgments have a simple categorical judgment as one of the premises. In this way, inference is considered divisive, all or part of whose premises are disjunctive judgments. The structure of a simple disjunctive inference is reflected as follows. S is A or B or C. And there is A1 or A2. S is A1 or A2 or B or C. An example of such a conclusion is the following. The path can be straight or circular. The roundabout can be with one transfer or with several transfers. The path can be straight or with one transfer, or with several transfers. Separative-categorical inferences can be represented in the form of a diagram. S is A or B. S is A (B). S is not B(A). For example: The shot is accurate and inaccurate. This shot is accurate. This shot is not inaccurate. Here it is necessary to mention conditional-separative inferences. They differ from the above inferences in their premises. One of them is a disjunctive proposition, which is not special, but the second premise of such propositions consists of two or more conditional propositions. A conditional-separative judgment can be either a dilemma or a trilemma. in a dilemma the conditional premise consists of two members. In this case, the separation implies the presence of a choice. In other words, a dilemma is a choice between two options. The dilemma can be simple constructive and complex constructive, as well as simple and complex destructive. The first has two premises, one of which asserts the same outcome of the two proposed situations, the other says that one of these situations is possible. The corollary summarizes the statement of the first premise (the conditional proposition). If you press on a pencil, it will break; if you bend a pencil, it will break. You can press the pencil or bend the pencil. The pencil will break. A complex design dilemma involves a harder choice between alternatives. Trilemma consists of two premises and a consequence and offers a choice of three options or states three facts. If the athlete strikes in time, he will win; if the athlete correctly distributes the forces, then he will win; if the athlete performs the jump cleanly, he will win. The athlete will strike in time or correctly distribute the forces over the distance, or perform the jump cleanly. The athlete will win. There are cases when a conclusion or one of the premises is omitted in conditional, disjunctive or conditionally distributive inferences. Such conclusions are called abbreviated. LECTURE No. 16. Syllogism 1. The concept of syllogism. Simple categorical syllogism The word "syllogism" comes from the Greek syllogysmos, which means "conclusion". It's obvious that syllogism - this is the derivation of a consequence, a conclusion from certain premises. A syllogism can be simple, compound, abbreviated, and compound abbreviated. A syllogism whose premises are categorical propositions is called, respectively, categorical. There are two premises in the syllogism. They contain three terms of the syllogism, denoted by the letters S, P and M. P is the greater term, S is the lesser, and M is the middle, connecting term. In other words, the term P is wider in scope (although narrower in content) than both M and S. The narrowest term in a syllogism is S. Moreover, the larger term contains the predicate of the judgment, the smaller one - its subject. S and P are related to each other by the middle concept (M). An example of a categorical syllogism. All boxers are athletes. This man is a boxer. This person is an athlete. The word "boxer" here is the middle term, the first premise is the major term, the second is the minor. To avoid mistakes, we note that this syllogism refers to a given, specific person, and not all people. Otherwise, of course, the second premise would be much broader in scope. A categorical syllogism has four forms, depending on the position of the middle term in its structure. In the first case, the major premise must be general, while the minor premise must be affirmative. The second form of the categorical syllogism gives a negative conclusion, and one of its premises is also negative. The larger concept, as in the first case, must be general. The conclusion of the third form must be private, the minor premise must be affirmative. The fourth form of categorical syllogisms is the most interesting. From such conclusions it is impossible to draw a generally affirmative conclusion, and there is a natural connection between the premises. So, if one of the premises is negative, the larger one should be general, while the smaller one should be general, if the larger one is affirmative. In order to avoid possible errors, when constructing categorical syllogisms, one should be guided by the rules of terms and premises. The rules of terms are as follows. Average term distribution (M). Means that the middle term, the connecting link, must be distributed in at least one of the other two terms - the greater or the lesser. If this rule is violated, the conclusion is false. Absence of unnecessary syllogism terms. Means that a categorical syllogism must contain only three terms - the terms S, M and P. Each term must be considered in only one meaning. Distribution in custody. In order to be distributed in the conclusion, the term must also be distributed in the premises of the syllogism. Parcel rules. 1. Impossibility of withdrawal from private parcels. That is, if both premises are private judgments, it is impossible to draw a conclusion from them. For example: Some cars are pickups. Some mechanisms are machines. No conclusion can be drawn from these premises. 2. Impossibility of inference from negative premises. Negative premises make it impossible to draw a conclusion. For example: People are not birds. Dogs are not people. Conclusion is not possible. 3. The next rule says that if one of the premises of the syllogism is particular, then its consequence will also be particular. For example: All boxers are athletes. Some people are boxers. Some people are athletes. 4. There is another rule that says that if only one of the premises of the syllogism is negative, the conclusion is possible, but it will also be negative. For example: All vacuum cleaners are household appliances. This technique is not household. This technique is not a vacuum cleaner. 2. Complex syllogism In thinking, we operate with concepts, judgments and conclusions, including syllogisms. Like judgments, a syllogism can be simple (discussed above) and complex. Of course, the word "difficult" should not be understood in the usual sense of the word, as "heavy" or "difficult". A complex syllogism consists of several simple syllogisms. They form polysyllogism, or complex syllogism; these are synonyms. A polysyllogism is a series of simple syllogisms connected to each other in a sequential manner. In this case, the conclusion, the consequence of one of the simple syllogisms becomes a premise for the subsequent one. Thus, a kind of “chain” of syllogisms is obtained. All polysyllogisms are divided into regressive и progressive. A progressive syllogism is characterized by the fact that its conclusion becomes the larger premise of the next syllogism. The conclusion of the regressive syllogism becomes the lesser premise in the following. 3. Abbreviated syllogism For ease of use and saving time, and especially in cases where the conclusion is obvious, abbreviated syllogisms are used. When talking about abbreviated syllogisms, it means that in such a conclusion one of the premises is missing, and in some cases the conclusion. All birds have wings. All seagulls are birds. All seagulls have wings. This is an example of a simple categorical syllogism. In order to get an abbreviated syllogism, you can omit the big premise, i.e. "all seagulls have wings." Thus, we get: "All seagulls are birds, which means that all seagulls have wings." Naturally, in this case the consequence of the syllogism will be true. In other words, the reduction of the syllogism does not affect its truth or falsity. You can give this example: "All gases are volatile, therefore, oxygen is volatile." This is an abbreviated syllogism, and the full one is expressed as follows. All gases are volatile. Oxygen is a gas. The oxygen is volatile. Unlike the previous example, the smaller premise is omitted here. The conclusion is skipped in the case when there is no need to express the result obtained due to its obviousness, obviousness for others, which stems from the nature of the premises themselves (i.e., if the premises and related objects, phenomena are well known). For example: "Everything that is lighter than water does not sink in it. Styrofoam is lighter than water." In this case, the omitted conclusion is fairly obvious. The syllogism looks like this. Anything lighter than water does not sink in it. Styrofoam is lighter than water. Styrofoam does not sink in water. In these cases, the restoration of the syllogism is quite simple, but sometimes there are problems with the definition of the premise and conclusion and their separation from each other. Therefore, it must be borne in mind that the words “because”, “because”, etc. are usually placed before the premise. Words such as “therefore” or “therefore” are usually put before the conclusion. Since the abbreviated syllogism is convenient and compact, it is used more often than full categorical syllogisms. The abbreviated categorical syllogism is also called enthymema. 4. Abbreviated compound syllogism Among compound abbreviated syllogisms, there are epicheirems и sorites. We should start with sorites, since their concept is used when considering the second type. Just like complex syllogisms, sorites can be progressive or regressive. Progressive sorites are obtained from progressive complex syllogisms, regressive ones - from regressive ones. As mentioned above, one of the premises of a complex syllogism is the conclusion of the previous one. When reducing a complex syllogism to the sorites form, this premise is omitted. The complex premise of the subsequent judgment in a polysyllogism may also be missed. The progressive sorite contains the predicate of the conclusion and its subject. It starts first and ends second. Unlike the progressive sorite, the regressive sorite begins not with the predicate of the conclusion, but with its subject. It ends with a predicate. Progressive sorites scheme. All A is B. All C is A. All D is C. All D is B. Regressive sorites diagram. All A is B. All B is C. All C is D. All A is D. LECTURE No. 17. Induction. Concept, rules and types 1. The concept of induction Concepts such as general and particular can only be considered in conjunction. None of them has independence, since when considering the processes, phenomena and objects of the surrounding world only through the prism of, say, a private picture, the picture will turn out to be incomplete, without many necessary elements. A too general look at the same objects and the picture will also give too general, the objects will be considered too superficially. In order to illustrate what has been said, a humorous story about a doctor can be given. One day the doctor had to treat a tailor who had a fever. He was very weak and the doctor thought that his chances of recovery were slim. However, the patient asked for ham and the doctor allowed it. After some time, the tailor recovered. In his diary, the doctor made a note that "ham is an effective remedy for fever." After a while, the same doctor treated the shoemaker, who also had a fever, and prescribed ham as a medicine. The patient died. The doctor wrote in his diary that "ham is a good remedy for fever in tailors, but not in shoemakers." Induction is the transition from the particular to the general. That is, this is a gradual generalization of a more particular, specific concept. In contrast to deduction, in which a true conclusion, reliable information, is derived from true premises, in inductive reasoning, even from true premises, a probabilistic conclusion is obtained. This is due to the fact that the truth of the particular does not uniquely determine the truth of the general. Since the inductive conclusion is probabilistic in nature, further construction of new conclusions on its basis can distort the reliable information received earlier. Despite this, induction is very important in the process of cognition, and one does not have to look far to confirm this. Any position of science, whether it be humanitarian or natural science, fundamental or applied, is the result of generalization. At the same time, generalized data can be obtained in only one way - by studying, considering the objects of reality, their nature and relationships. Such a study is a source of generalized information about the patterns of the world around us, nature and society. 2. Rules of induction In order to avoid mistakes, inaccuracies and inaccuracies in one's thinking, to avoid curiosities, one must comply with the requirements that determine the correctness and objective validity of an inductive conclusion. These requirements are discussed in more detail below. The first rule states that inductive generalization provides reliable information only if it is carried out according to essential features, although in some cases one can speak of a certain generalization of non-essential features. The main reason that they cannot be generalized is that they do not have such an important property as repeatability. This is all the more important because inductive research consists in establishing the essential, necessary, stable features of the phenomena being studied. According to second rule An important task is to accurately determine whether the phenomena under study belong to a single class, recognizing their homogeneity or same type, since inductive generalization applies only to objectively similar objects [8]. The validity of the generalization of features that are expressed in particular premises can depend on this. Incorrect generalization can lead not only to misunderstanding or distortion of information, but also to the emergence of various kinds of prejudices and misconceptions. The main reason for the occurrence of errors is generalization by random features of single objects or generalization by common features, when there is no need for these features. The correct application of induction is one of the pillars of correct thinking in general. As stated above, inductive reasoning - this is an inference in which thought develops from knowledge of a lesser degree of generality to knowledge of a greater degree of generality [9]. That is, a particular subject is considered and generalized. Generalization is possible to certain limits. Any phenomenon of the surrounding world, any subject of research is best studied in comparison with another similar subject. So is induction. Its features are best demonstrated in comparison with deduction. These features manifest themselves mainly in the way the inference process takes place, as well as in the nature of the conclusion. Thus, in deduction one concludes from the characteristics of a genus to the characteristics of a species and individual objects of this genus (based on volumetric relations between terms); in inductive inference - from the characteristics of individual objects to the characteristics of the entire kind or class of objects (to the volume of this characteristic) [10]. Therefore, there are a number of differences between deductive and inductive reasoning that allow us to separate them from each other. Can be identified several features of inductive reasoning: 1) inductive reasoning includes many premises; 2) all premises of inductive reasoning are single or private judgments; 3) inductive reasoning is possible for all negative premises. 3. Types of inductive reasoning First, let's talk about the fundamental division of inductive reasoning. They are complete and incomplete. Complete are called inferences, in which the conclusion is made on the basis of a comprehensive study of the entire set of objects of a certain class. Complete induction is used only in cases where it is possible to determine the entire range of objects included in the class under consideration, that is, when their number is limited. Thus, complete induction applies only to closed classes. In this sense, the use of complete induction is not very common. Moreover, such an inference gives a reliable value, since all the objects about which the conclusion is made are listed in the premises. The conclusion is made only concerning these subjects. In order to be able to talk about complete induction, it is necessary to verify compliance with its rules and conditions. Thus, the first rule says that the number of objects included in the class under consideration must be limited and determined; their number should not be large. Each element of the class taken, with respect to which an inference is created, must have a characteristic feature. And finally, the derivation of a complete conclusion must be justified, necessary, rational. The scheme of a complete inference can be reflected as: 51 - P 52 - P 53 - P Sn - R. An example of a complete inductive inference. All guilty verdicts are issued in a special procedural order. All acquittals are issued in a special procedural order. Guilty verdicts and acquittals are decisions of the court. All court decisions are issued in a special procedural order. This example reflects the class of objects - court decisions. All (both) of its elements were specified. The right side of each of the premises is valid in relation to the left. Therefore, the general conclusion, which is directly related to each case separately, is objective and true. Despite all the undeniable advantages and advantages of full induction, there are often situations in which its use is difficult. This is due to the fact that in most cases a person is faced with classes of objects, the elements of which are either unlimited or very numerous. In some cases, the elements of the class taken are generally inaccessible for study (due to remoteness, large dimensions, poor technical equipment or low level of available technology). Therefore, incomplete induction is often used. Despite a number of shortcomings, the scope of incomplete induction, the frequency of its use is much greater than the full one. Incomplete induction called an inference, which, on the basis of the presence of certain recurring features, ranks this or that object in the class of objects homogeneous to it, which also have such a feature. Incomplete induction is often used in human everyday life and scientific activity, as it allows one to draw a conclusion based on the analysis of a certain part of a given class of objects, saving time and effort. At the same time, we must not forget that as a result of incomplete induction, a probabilistic conclusion is obtained, which, depending on the type of incomplete induction, will fluctuate from less probable to more probable [11]. The scheme of incomplete induction can be represented as: 51 - P 52 - P 53 - P S1, S2, S3... constitute class K. Probably each element K - R. The above can be illustrated by the following example. The word "milk" changes by case. The word "library" changes by case. The word "doctor" changes by case. The word "ink" changes by case. The words "milk", "library", "doctor", "ink" are nouns. Probably all nouns change by case. Depending on how the conclusion of the conclusion is justified, it is customary to divide incomplete induction into two types - popular and scientific. Popular incomplete induction, or induction by simple enumeration, does not consider the objects and classes to which these objects belong in very depth. Thus, based on the repetition of the same characteristic in a certain part of homogeneous objects and in the absence of a contradictory case, a general conclusion is made that all objects of this kind possess this characteristic. As the name suggests, popular induction is very common, especially in non-scientific environments. The probability of such an induction is low. When forming a popular inductive reasoning, one should be aware of possible errors and prevent their occurrence. A hasty generalization means that the conclusion takes into account only that part of the facts that speaks in favor of the conclusion made. The rest are not considered at all. For example: Winter in Tyumen is cold. It is cold in Urengoy in winter. Tyumen and Urengoy cities. All cities are cold in winter. After, therefore, for a reason - means that any event, phenomenon, fact preceding the one under consideration is taken as its cause. The substitution of the conditional for the unconditional means that the relativity of any truth is not taken into account. That is, the facts in this case can be taken out of context, changed places, etc. At the same time, the truth of the results obtained continues to be affirmed. Scientific induction, or induction through the analysis of facts, is an inference, the premises of which, along with the repeatability of a characteristic in some phenomena of the class, also contain information about the dependence of this characteristic on certain properties of the phenomenon. That is, unlike popular induction, scientific induction is not limited to a simple statement. The subject under consideration is subjected to deep research. In scientific induction, it is very important to comply with a number of requirements: 1) research subjects should be selected systematically and rationally; 2) it is necessary to know as deeply as possible the nature of the objects under consideration; 3) understand the characteristic features of objects and their relationships; 4) compare the results with previously fixed scientific information. An important feature of scientific induction, which determines its role in science, is the ability to reveal not only generalized knowledge, but also causal relationships. It was through scientific induction that many scientific laws were discovered. LECTURE No. 18. Methods for establishing causal relationships 1. The concept of cause and effect relationships Before considering directly the methods of establishing cause-and-effect relationships, it is necessary to understand the concept of cause and effect. The reason called such a phenomenon, process or object, which, by virtue of its existence, causes certain changes in the surrounding world. The cause is characterized by the fact that it always precedes the result. It lies, as it were, at the basis of the consequences. Thus, no effect can be imagined without a cause, because the latter is a kind of starting point. Let's give an example: "Lightning struck - the forest caught fire." Obviously, lightning is the cause here, if it was she who provoked the fire. Without such a cause, there could be no effect. Of course, one can say that the fire could have started as a result of arson, but in this case, arson would have been the cause. Consequence is what the cause entails; it is always secondary and dependent, determined by it. It is on this relationship of cause and effect that the professional process of many people is built. Firefighters, rescuers, law enforcement officers, before starting work, first look for the cause. For example, firefighters start extinguishing a fire only when it is more or less clear what caused the fire and where. Otherwise, the risk to life would have increased several times. Of course, the final cause of the fire, whether it was set on fire, a malfunction of the electrical wiring or careless handling of fire, becomes clear only after the extinguishment is completed, but initially it must be determined at least approximately. A law enforcement officer, leaving the scene of an incident, first of all determines the causes of this incident. If a murder is reported, it is necessary to check whether the incident is actually a crime. That is, the cause of death is determined. At the same time, versions of suicide, accident, death from illness, etc. are eliminated. After that (if it is established that the murder took place), the reason for the crime is already determined - self-interest, revenge, etc. Rescuers, arriving at the place of the call, first determine the cause of the accident in order to develop the most effective rescue tactics. When it comes to a fall from a height, a car accident or other traumatic event, there is a need for a special transportation procedure. So, for example, the cervical, thoracic and lumbar spine should be fixed in case there is damage to the spinal column. The types of first aid provided also depend on what kind of event led to the emergence of dangerous situations, injuries. It is obvious that rescuers determine the causes of the events for the most effective organization of assistance to citizens. At first glance, it may seem that the definition of the cause is not important, does not matter much, but the above examples indicate the opposite. Establishing the cause is necessary, because otherwise the operational police officer would be looking for a non-existent criminal, investigating a confluence of circumstances similar to a crime (needless to say, establishing the cause is a large part of operational work), and firefighters and rescuers could not cope with the work. In this way, cause is called such an objective connection between two phenomena, when one of them causes the other - a consequence. The disclosure of a causal relationship between phenomena is a complex multifaceted process that includes a variety of logical means and methods of cognition. In logic, several methods have been developed to establish a causal relationship between phenomena. Of these methods, four are most commonly used: method of similarity, method of difference, method of concomitant changes and method of residuals. Often, combinations of these methods are used in scientific research, but to understand the essence of the issue, they should be considered separately [12]. 2. Methods for establishing causal relationships similarity method lies in the fact that if two or more cases of the phenomenon under study are similar in only one circumstance, there is a possibility that this particular circumstance is the cause or part of the cause of this phenomenon. For example: Under conditions ABC, the phenomenon a occurs. Under ADE conditions, phenomenon a occurs. Under AFG conditions, phenomenon a occurs. Probably circumstance A is the cause of a [13]. difference method consists in the following: two cases are defined. The first is the one in which the phenomenon under consideration occurs. The second case is the one in which the onset of this phenomenon does not occur. If these two cases differ from each other in only one circumstance, it is probably the cause of the occurrence of the phenomenon under consideration. For example: Under conditions ABC, the phenomenon a occurs. Under conditions of EHV, the phenomenon a. Probably circumstance A is the cause of a [14]. Accompanying change method is that if any particular phenomenon changes every time another phenomenon changes, with a certain degree of probability it can be assumed that the second phenomenon entails a change in the first and, therefore, they are in causal interdependence. For example: Under conditions A1BC, the phenomenon a1 occurs. Under conditions A2BC, the phenomenon a2 occurs. Under conditions A3BC, the phenomenon a3 occurs. Probably circumstance A is cause a [15]. Residual method means that, considering the causes of the complex phenomenon abc, which is caused by a number of circumstances ABC, one can move in stages. Having studied a certain part of the causal circumstances, we can subtract it from the phenomenon abc. As a result, we will get the remainder of this phenomenon, which will be a consequence of the circumstances remaining from the ABC complex. For example: The phenomenon abs is caused by the circumstances ABC. Part b of phenomenon abc is caused by circumstance B. The part c of the phenomenon abs is caused by the circumstance C. Probably, part a of the phenomenon abc is causally dependent on circumstance A [16]. Having considered the methods of establishing causal relationships, we can say that they, by their nature, relate to complex inferences. They combine induction with deduction, inductive generalizations are built using deductive consequences. Based on the properties of a causal connection, deduction acts as a logical means of excluding random circumstances, thereby it logically corrects and directs inductive generalization. The relationship of induction and deduction ensures the logical independence of reasoning when applying methods, and the accuracy of the knowledge expressed in the premises determines the degree of validity of the knowledge obtained. LECTURE No. 19. Analogy and hypothesis 1. The concept of inference by analogy A significant characteristic of inference as one of the forms of human thinking is the conclusion of new knowledge. At the same time, in the inference, the conclusion (consequence) is obtained in the course of the movement of thought from the known to the unknown. This movement of human thought includes deduction and induction. Along with them, there are other types of inferences, one of which is analogy. Analogy (Greek analogia - "similarity", "correspondence") is a similarity, similarity of objects (phenomena) in any properties, features, relationships. For example, the chemical composition of the Sun and the Earth is similar. Therefore, when the element helium, still unknown on Earth, was discovered on the Sun, by analogy they concluded: there is such an element on Earth. Inference by analogy is based on a number of undoubted data that science has at its disposal under specific historical conditions. It represents the movement of thought from the commonality of some properties and relations of compared objects (or processes) to the commonality of other properties and relations. Analogy plays an essential role in the natural and human sciences. Many scientific discoveries have been made by researchers through its use. For example, the nature of sound was established by analogy with a sea wave, and the nature of light - by analogy with sound. The analogy has its own specifics. So, it represents a certain likelihood of the object (or phenomenon) under study and expresses knowledge with an internally hidden probability. The process of formation and wide dissemination of the analogy began with everyday consciousness, and it is directly related to the daily life of people. The conclusions of the analogy are ambiguous, usually they do not have probative force. Therefore, one should move from a conclusion by analogy to a conclusion by necessity. Any apparent analogy needs to be verified through actual proof [17]. This requirement is due to the fact that it is possible to obtain a false conclusion, although it is constructed in accordance with the requirements of analogy. Diagram of inference by analogy. A has attributes a, b, c, d. B has features a, b, c. It is probable that B has feature d. 2. Types and rules of analogy Inferences by analogy can be divided into two groups. The first can be represented as an analogy of properties and qualities, or an analogy of relationships. In the first case, objects are considered - single or classes. The attributes of the analogy are the properties of these objects. Property analogy diagram. Object x has properties a, b, c, d, e, f. The object y has properties a, b, c, d. Probably the object y has the properties e, f. The basis of the analogy of properties is the relationship between the features of an object. Each object, having many properties, is an internal, interdependent unity in which it is impossible to modify some essential property without affecting its other features. The second kind is the analogy of relations. This is a conclusion in which not the objects themselves are considered, but their properties. Suppose there is a relation (aXb) and a relation (cX1b). The relations X and X1 are analogous, but not analogously with; b is not the same as d. second group analogies can be divided into two types - strict and non-strict analogy. A strict analogy contains a connection between common features and a transferred feature. The strict analogy is as follows. The object X has features a, b, c, d, e. Object Y has features a, b, c, d. From the set of signs a, e, c, d, an analogy necessarily follows. Strict analogy finds application in scientific research, as well as in mathematical proofs. The modeling method is based on the properties of inference by strict analogy. Modeling - this is a kind of analogy in which one of the similar objects is examined as an imitation of another. These objects are called the model and the original. The knowledge gained about the model is transferred to the original. At the same time, the model is both an object of study and a means of cognition. Non-strict analogy does not give a reliable, but only a probabilistic conclusion. This is due to the fact that the difference between the model and the original is not only quantitative, but also qualitative, and there are great differences between laboratory and natural conditions. In order to increase the degree of reliability of the hypothesis, it is necessary to observe a number of rules. first is a comprehensive study of objects and their properties. Second - identification of similar features between the objects under consideration. The third - identifying relationships between objects in order to find a transferable property between them. 3. Hypothesis Hypothesis called an assumption about any object or phenomenon, its causes, relationships, laws of nature, society and the state, based on scientific data. Proven hypotheses based on scientific knowledge can be called scientifically sound. Hypotheses not justified in this way should not be taken into account. Among such unfounded hypotheses, one can single out the hypotheses false. They can be created intentionally or out of ignorance. All hypotheses can be divided into general, particular and singular. General hypotheses are used to explain, to cover the whole class of phenomena. An example of a general hypothesis can be, for example, the hypothesis of the origin of life or the emergence of the world, Charles Darwin's hypothesis of the origin of man. Once proven, a hypothesis becomes a theory. Private hypotheses unlike the general ones, they do not cover the entire class of homogeneous objects, but only a part of it. At the same time, the object of interest is singled out from the entire class of homogeneous objects and is further considered separately from this class. Single hypotheses affect only one subject of a homogeneous class, the rest are excluded from consideration (it must be taken into account that the entire class can consist of only one subject). Such hypotheses arise when the object itself is single or it is necessary to consider its properties without taking into account the influence of objects of the same class. As an example of a single hypothesis, one can cite scientifically based assumptions about the phenomenon of the Tunguska meteorite and other similar phenomena. It is also necessary to mention such a type of hypotheses as working hypotheses. Their totality represents an intermediate stage between hypothesis and theory. That is, the construction of working hypotheses is used to prove the main hypothesis. Most often, working hypotheses arise at the beginning of the study. They do not have a very great depth of research, do not cover the entire range of issues, but they allow you to obtain the necessary information and establish some of the properties and connections of the subject. Working hypotheses are not final and in the process of work can be changed and replaced by others or simply discarded. It is also necessary to mention a special kind of hypotheses - false hypotheses. They can be created due to lack of information, unintentionally, or to achieve their goals, with intent. If a probabilistic conclusion is raised to the rank of a hypothesis, it can turn out to be either true or false, depending on whether the conclusion is true or false. Despite the fact that a false hypothesis conveys incorrect information about the subject under consideration, it cannot be said that it has a fairly large cognitive value. For example, a false hypothesis, if it contains a sound grain, can direct research in a new direction, add, so to speak, fresh blood to stagnant research and thereby lead to a scientific discovery. Also, a false hypothesis, when proven false, shows researchers (especially the next generation) a direction in which they definitely should not go. That is, new researchers are spared the need to test the guess underlying a false hypothesis. LECTURE No. 20. Argument in logic 1. Dispute. Dispute types In order to be able to reveal the essence of the dispute, it is necessary to say a little about the evidence. Without them, our world is unthinkable, every judgment requires proof. Otherwise, whatever the person said would be true. Exclusion of evidence in the absolute plan will lead the human world to chaos. Proof is necessary, because it is through it that we determine whether this or that proposition is true or not. The thought for which proof is constructed to substantiate the truth or falsity is called the thesis of the proof [18]. This is the ultimate goal of the discussion. Thesis in proof can be compared to the king in a chess game. A good chess player should always have the king in mind, no matter what move he is planning. Likewise, a good participant in a discussion or just a conversation: no matter what he talks about in the proof, he always ultimately has one main goal - the thesis, its statement, proof or refutation, etc. [19] Therefore, the main thing in the dispute can be called the clarification of the controversial thought, the identification of the thesis, that is, you need to penetrate into its essence and understand it so that it becomes completely clear in meaning. This saves a lot of time and guards against a lot of mistakes. There are three questions that need to be resolved when considering the thesis in order to be able to talk about a thorough study of the subject - whether all the words and expressions of the thesis are clear, whether their meaning is known. It is necessary to clarify each concept of the thesis until complete clarity is achieved. It is also necessary to accurately be aware of how many subjects are mentioned in the asserted judgment-thesis. Here, for clarity of thought, it is necessary to know whether we are talking about one object, about all objects of a given class, or about some (most, many, almost all, several, etc.). Often, when expressing his thoughts, an opponent in a dispute uses vague judgments - those in which it is impossible to understand, for example, how many objects are being discussed. The refutation of such theses is problematic, however, and simple at the same time. It is necessary to point out to the opponent his mistake. Then we need to find out what kind of judgment we consider the thesis to be true, reliable, false, or probable to a greater or lesser extent, or refutable. For example, a thesis seems to us only possible: there are no arguments for it, but there are no arguments against it either. Depending on all this, it is necessary to give various methods of proof, each of which plays its role only in certain cases, without touching the scope of others. It is these nuances that are most often overlooked when determining the asserted judgment. Since their value seems low, they are discarded as unnecessary. This cannot be done. In order to understand the meaning of seemingly unimportant information, one can turn to judicial practice, in which the outcome of a case often depends on one word. There are three types of dispute: scientific and business discussion and controversy. In the first case the purpose of the dispute is to solve some practical or theoretical problem that arises within the framework of a particular science. The second is aimed at reaching agreement on the main provisions put forward by the parties, finding a solution that corresponds to the real state of affairs. And the last kind of dispute, controversy, serves to achieve victory. In the most general form, we can say that this is an argument for the sake of an argument. However, a clear distinction between polemics and the two previous types of dispute cannot be drawn: every dispute, when conducted according to the rules of logic and without the use of unacceptable techniques, leads to the achievement of truth, no matter in what area it is started. The dispute can take place with the public, whose presence the parties to the dispute have to take into account, and without it. Disputes in public, especially as a demonstration of oratory skills, are more characteristic of Ancient Greece than of the present time. Then the sophist philosophers and adherents of the emerging logic deliberately and publicly staged disputes. This teaching method was used, for example, by Socrates in his school. Behind the scenes dispute, or an argument without spectators or listeners, has always been common. This is how, for example, deputies can argue before or after the adoption of a bill on its main points. Scientists can argue this way when discussing a new discovery or nuances of their work. The dispute may take place with or without an arbitrator. The role of arbitrator may be performed by the public when the dispute is public, but more often an individual is appointed to the role of judge. This is done because several people cannot always come to an unambiguous agreement themselves, and a dispute between two opponents can give rise to a dispute between the public, which does not have a very good effect on the efficiency of the dispute. The person who is elected as a judge, of course, must have a good knowledge of logic. dispute called a dispute between two people in which the public is present. In order for the dispute to proceed as calmly as possible, and the parties to be able to offer their arguments consistently, the order in which issues are discussed is often agreed in advance. The parties explain which theories they will appeal to. It must be said that such a "field of argumentation" is not always developed. Often the parties prefer to have an "ace in the hole" as a means to reach the truth. Many disputes also a priori begin not for the sake of truth, but to achieve certain goals. It goes without saying that the general course of such a dispute cannot be determined, since each of the parties can hide some particularly valuable material and use it at a decisive moment to turn the dispute in its favor. The dispute for the sake of achieving true knowledge is called dialectical. This name comes from ancient Greece, where dialectics was understood as the art of deducing the truth in a conversation with an opponent. Based on the foregoing, it can be summarized that discussion is always a dialectical dispute, while polemics and disputes are not. The dispute begins to achieve victory. The parties to the dispute are called differently, but most often - opponents. The term "proponent" is sometimes used. Proponent name the side that put forward the thesis for refutation by the other side. The latter is called the opponent. Also use the concept of "opponent". Basically, this is the name of the participants in the dispute aimed at achieving victory. Depending on the type of dispute, one or another strategy and tactics of argumentation and criticism are used. Strategy - this is a predetermined scheme, a plan for constructing an argument, proof or refutation. The strategy is to do the following:. 1. Logically flawless formulation of the thesis (the thesis must be consistent, clear, etc.). 2. Bringing arguments in defense of the thesis, criticism of competing concepts. 3. Logical assessment of the thesis in the light of the arguments found. This strategy is the simplest, although its use requires certain skills of the opponent and listeners. It happens that a thesis is formulated, arguments are given, but there is no conclusion about how much the arguments support the thesis. Sometimes discussions are held in the form of a round table. Basically, this is how the discussion of scientific and some other problems is organized. It is advisable to conduct such discussions in cases where it is necessary to discuss an “undeveloped” problem. A leader or presenter is appointed to conduct the round table, as well as a person who formulates the problem, if it is not known to everyone. Then solutions or solutions are proposed [20], the preferences of which are justified as theses of the argument. It is also worth mentioning such a type of dispute as business meeting. It is held as a round table, which was already mentioned above, and as a dispute between the parties - two or more people. In the second case, the existence of an already developed solution is assumed with the aim of improving or convincing those present of its truth. As the name implies, a business meeting is most often held to solve problems that arise in the course of the activities of any entity, whether it be an organization, body, government institution or their structural subdivisions. When conducting business meetings, in many cases it is important to comply with the regulations and maintain protocol, as well as to involve as participants persons who have the appropriate knowledge, are familiar with the problem statement in advance and are authorized to make appropriate decisions [21]. 2. Tactics of the dispute The tactics of arguing, arguing, proving one's own theses and refuting the opponent's judgments have been studied quite well. Often it consists in the application of techniques developed over several thousand years. These techniques themselves originated much earlier than the science of logic. However, some of them were in their infancy, and some were subsequently recognized as incorrect and even unacceptable ways of conducting a dispute. All techniques can be conditionally divided into general techniques, which are also called general methodological, as well as logical and psychological (socio-psychological). This group also includes rhetorical tricks. The basis for the allocation of types of tactical techniques are aspects of argumentation, one of which is moral. Probably there is no absolute criterion according to which methods would be accepted from the point of view of morality or, on the contrary, rejected. General methodological tactics are: delay of expression, concealment of the thesis, prolongation of the dispute, as well as divide and conquer, placing the burden of proof on the opponent, cunctation, chaotic speech, Thomas' trick, ignoring intellectuals and simple speech. Each of these methods is discussed separately below. Pulling an expression occurs when a person who is arguing in a discussion suddenly finds himself in a difficult position in answering a question or selecting evidence arguments. However, he understands (or believes) that arguments exist and can be found, provided that he can buy time for reflection. Then you can ask your opponent to wait. Taking advantage of the respite, it is necessary to repeat the arguments that have already been given in the process of proof and refutation, to recall the main points that are worth paying attention to when considering this issue. Instead of asking the opponent to wait, sometimes they resort to a slight distraction, speaking not directly on the topic, but on the subject. This gives you more time to think. Relatively calm reflection after asking for a little time is still preferable. Concealment of the thesis is inextricably linked to the rule of clear definition. It says that a participant in a discussion, a lecturer speaking at a meeting, rally, conference, etc., must clearly formulate each thesis with its subsequent justification. This rule is intended to create comfortable conditions for those who are intended for the transmitted information (students, work colleagues, partners, etc.), as it contributes to the correct expression of thoughts, allows you to focus the attention of those present on the speaker and his thoughts. Argumentation can then proceed more easily, since its process is transparent. In some cases, it makes sense to reverse actions. First, the arguments are formulated clearly and correctly. Then you need to ask the opponent to express their attitude towards them. If he agrees, a thesis can be deduced from the stated judgments. And it is not necessary to do so. For example, if the thesis is obvious enough, you can provide its formulation to the opponent. In doing so, you can use additional means of persuasion - from the arguments expressed, one can conclude a false thesis, which clearly does not correspond to the general course of reasoning, and allow the opponent to independently find an error, having come to the correct conclusion. This will give him a sense of involvement in the proof and will involuntarily force him to treat the thesis as true, proven on his own. Due to its rather high efficiency, this technique is used when the opponent is not interested in proving the thesis. It is impossible to deny the opinion that emotions in a dispute on scientific topics, especially in the fundamental sciences, are excluded, since the theses that require proof or refutation are in this case strongly abstracted from the sensory side of human cognition. They belong more to the realm of the mind and do not affect the interests of people. Therefore, it is considered that the opponents remain impartial. However, it should be said that a subject that is important for a person, a subject to which he has devoted many years to the study, cannot but excite him, especially when an opposite point of view is expressed. This leads to the emergence of heated discussions and disputes regarding issues that, it would seem, cannot in any way affect such aspects of a person as his sensory sensations. In addition, many people simply have a disposition to get into arguments on any topic, whether that person is knowledgeable on a particular subject or not. It is necessary to mention the inertia of the mind of many people (probably, it is inherent, if not in all, then in most representatives of the human race). When a person has convinced himself of some fact, on which (if it concerns a scientist) he builds his concept, it is very difficult, and in some cases impossible, to make him believe that this fact is false. In such cases, the method of "hiding the thesis" can help to find the truth. The next method of discussion is prolongation of the dispute. This technique is used when the opponent cannot answer the objection, especially when he feels that he is wrong on the merits. Then he asks you to repeat your last thought, to formulate your thesis again. The only way to combat this type of dispute is to point out the incorrectness of the technique to the opponent, the arbitrator, and sometimes to the public. Cunctation (from lat. cunctator - "slow") lies in the fact that the opponent tries to take a wait-and-see position in the discussion in order to check his arguments, decide on the "aces in the hole" that should be held until the best moment, decide where to start the speech, and discard weak arguments. The goal is to speak in such a way as not to give the opponent the opportunity to object due to lack of time. Divide and Conquer is one of the hardest tricks. Its goal is to weaken the opponent in the event of a collective offensive, i.e., when the forces are unequal and one opponent has several opponents at once. To achieve this goal, differences in the opinions of the collective opponent are used, which are identified, put on public display (sometimes with exaggeration), and then one part of such an opinion is opposed to another. If the goal is achieved and a dispute arises within the group of opponents, you can proceed to the second part, namely, to invite the members of the group to digress from minor disagreements and defend the main idea, that is, their thesis. If there is no way to defend it even in this case, another statement can be proposed as the main idea, on which agreement has been reached among all members. Putting the burden of proof on the opponent due to the fact that in most cases it is easier to refute the argument of the opposite side than to substantiate your thesis. Therefore, the opponent using this technique tries to take as few steps as possible to substantiate the question put forward by himself, but to demand proof of the opponent's thesis. A lesser known and less commonly used name for this technique is "the truth is in silence". The trick called "Foma's Trick", has a number of disadvantages, but can sometimes have the necessary effect and contribute to the speedy achievement of results. The meaning of this technique comes down to denial. This technique is sometimes used out of conviction, and sometimes with the goal of remaining victorious in an argument. In the first case, the application of the technique is associated with ignorance or denial of the philosophical doctrine of the relationship between absolute and relative truths. This is due to the division of areas of science. They can be expressed as relative or absolute truth. The relativity of a doctrine means that it contains statements that are refuted in the process of developing its ideas. Absolute knowledge implies that the teaching contains statements that cannot be refuted in the future. When the denial is based on the fact that relative knowledge contains a number of contradictions, and the significance of these contradictions is clearly exaggerated, one can speak of agnosticism (from Greek - "inaccessible to knowledge"). The denial of absolute knowledge leads to dogmatism. Chaotic speech implies the use by an opponent who proposes a thesis to substantiate (many public people and authors of scientific works sin this), incoherent, ornate, complex speech. This is done when the thesis put forward cannot withstand the onslaught of the opponent, i.e. the arguing is not able to substantiate the defended opinion. Speech in this case abounds in place and out of place with the use of special terms, long and complex phrases, sometimes it is even characterized by the disappearance of the thread of thought. In other words, speech that seems normal at first glance, upon closer examination, turns out to be a set of words that do not express anything by and large. Ignoring intellectuals - this, as the name implies, is a way of expressing one's opinion, in which no attention is paid to inaccuracies in speech that can be revealed by the people present. This does not confuse the opponent, he can put forward inaccurate information about events, talk about the subject, incorrectly indicating dates, etc. simple speech at first glance, it is similar to ignoring intellectuals, but it is fundamentally different from the latter. The essence of this technique is the use of simple sentences, breaking the complex into parts, a detailed explanation, using examples to achieve the main goal - bringing to people who do not have, say, a special education, the intricacies of a particular issue. LECTURE No. 21. Argumentation and proof 1. Proof We cognize the world through the sense organs, and such cognition most often does not need proof, since it is quite obvious. For example, it does not require proof that fire is hot. Just reach out to him. However, not all phenomena, objects of the surrounding world are so clear that there is no need to prove them. In scientific activity and even in everyday life, one often has to face the need to prove, to defend one's point of view. Evidence - an important quality of correct thinking. Theories, proofs and refutations are the means in the hands of man to create new valid knowledge. Proof is necessary in the scientific world, it determines the truth of a phenomenon, judgment, conclusion. Without proof, any hypothesis will forever remain a hypothesis and will not acquire the value of a theory. It's good, because purpose of proof - obtaining true knowledge. Any new phenomenon, conjecture must be proved, whether it be secrets related to outer space or the depths of the ocean, mathematical research, etc. From these positions, one can define proof as a set of logical methods of substantiating the truth of a proposition with the help of other true and related propositions. In the ordinary sense, proof is often identified with the belief that it is unacceptable. These two concepts may coincide in part, but they are too different in many ways. So, the proof is based solely on scientifically substantiated facts, research, theories, etc. A belief, on the other hand, often does not depend on whether the asserted is scientifically proven or not. Persuasion is possible in relation to probabilistic or generally false theories. The structure of the proof is the thesis, arguments and demonstration. Thesis This is a statement that needs proof. Arguments are true propositions used in the process of proof. Demonstration is a way of logical connection between the thesis and arguments. There are rules for reasoning. Violation of these rules leads to errors related to the thesis being proved, the arguments, or the form of the proof itself. The proof is either direct or indirect. Direct proof proceeds from the consideration of arguments to the proof of the thesis, i.e., the truth of the proof is directly substantiated by the arguments. We can say that with direct proof, true propositions (k, m, l...) necessarily follow from the arguments (a, b, c...), and the thesis q to be proven follows from the latter. This type of evidence is used in judicial practice, in science, and in controversy. Direct evidence is widely used in statistical reports, in various kinds of documents, and in decrees. With indirect evidence the truth of the put forward judgment is substantiated by proving the falsity of the judgment that excludes it. The use of such a proof is justified when there are no arguments for direct proof. Depending on the form of the antithesis, two types of indirect evidence can be distinguished - from the opposite and divisive. proof by contradiction (apagogical) is carried out by establishing the falsity of a judgment that contradicts the thesis. This method is often used in mathematics. Partition proof produced on the basis of the negation of the antithesis. Provided that all antitheses are listed and their consistent negation (and rejection), one can speak of establishing the truth of the asserted judgment. 2. Argumentation As has been said, any proof needs arguments. The prover relies on them; they contain information that allows one to speak with certainty about a particular subject. In logic there are several arguments. These include certified individual facts, axioms and postulates, previously proven provisions and definitions. Certified Facts represent information fixed in any documents, works, databases and on various media. You can define this group of arguments as actual data. Such data include statistics, facts from life, testimonies, documents and documentary chronicles, etc. Such arguments play an important role in the proof process, as they are firm, irrefutable, and have already been proven. They can carry information about the past, which also makes authenticated facts important in terms of knowledge. Axioms. Many of us, when we hear the word “postulates,” remember school and mathematics lessons. Indeed, axioms are widely used in mathematical constructions, and mathematical logic is often based on them. Confirmed by experience, previously proven facts, and repeated repetition of evidence, these judgments do not require proof and are accepted as arguments. Statements of laws, theorems, which have been proven in the past, are accepted as arguments of proof, since their truth has already been determined and accepted. This group of arguments reminds us that all arguments underlying the evidence must be proven. The proof of the arguments of this group can be carried out either immediately before the proof of the axiom, or long before that. This group includes scientifically proven laws (for example, nature) and theorems. The last group of arguments is definitions. They are created within the framework of all sciences regarding the subjects under consideration and reveal the essence of the latter. The proof can be based on definitions accepted and applied in any science. However, we should not forget that many definitions are subject to debate and the proof based on them may not be accepted by the opponent. Here it is necessary to say about the inadmissibility of using unscientific definitions, since the main idea in them may be distorted, and the definitions themselves may be incomplete or even false. When proving a thesis, you can use several types of arguments - this will lead to greater persuasiveness. Do not forget also that the main factor in proving the theory is still practical application. If the theory has been confirmed in practice, it does not require other evidence or justification. LECTURE No. 22. Refutation 1. The concept of refutation A refutation is considered to be a logical operation in which the falsity or groundlessness of the thesis under consideration is shown (asserted). A thesis is a statement that needs to be refuted. It is refuted with rebuttal arguments - judgments, by means of which the thesis is refuted. Refutation can be direct and indirect. Wherein direct way there is only one refutation, while there are two indirect ones. Further, all methods are considered separately, starting with the first method of refutation - direct. direct way This is a refutation of the facts. From a scientific (and almost any) point of view, this method is the most convenient. Refutation by facts with the right approach fully shows the inconsistency of the thesis put forward. This is possible only with the correct selection of facts, their skillful use, depends on the person's abilities in the field of dialogue, as well as his knowledge in this area. The facts used to refute the thesis can be statistical data, axioms, proven positions, etc. As can be seen, due to the established truth of the indicated facts and their contradiction to the thesis under consideration, such a refutation has a correct, obvious character. Errors that can be easily refuted with facts are often found in Hollywood semi-historical films, where the chronological sequence of events is confused to achieve the desired effect. With such errors, it is sufficient to provide data on the real time of each event under consideration. The next two types of refutation are indirect. One of them is refutation through falsity of consequences. To do this, the consequences of the thesis are traced. During a refutation through the falsity of consequences, the thesis is accepted for discussion. This is done, firstly, so that the opponent temporarily feels superior (victory in this episode), and secondly, in order to reveal the falsity of the thesis. During the discussion, the consequences of the thesis are considered, which do not correspond to the real state of affairs. This makes obvious the inconsistency of the thesis itself. This approach is often called reduction to absurdity. It should be remembered that the contradiction of the consequences of the thesis to the truth must not only be quite clear and obvious, but also real. Another type of indirect refutation can be called refutation through antithesis. Obviously, the refutation here occurs on the basis of evidence from the opposite, i.e., antithesis. With this type of refutation, there is a concept, a judgment that contradicts the previously put forward statement. In order to prove the falsity of a thesis, the truth of its antithesis is proven, that is, a newly put forward judgment that contradicts the one being considered. The effectiveness of this method of refutation is based on the law of excluded middle (discussed in the corresponding chapter). In other words, after proving the truth of a proposition that contradicts the thesis under consideration, according to the law of excluded middle, the latter is inevitably recognized as false. Each of the two contradictory propositions can be either true or false, there is no third. It should be remembered that the truth of the antithesis must be fully proven. For an example of such a refutation, let's take the universally affirmative proposition "All athletes have well-developed muscles." Contradicting it will be a particular negative judgment "Some athletes do not have well-developed muscles." To prove this judgment, it is necessary to give examples proving that not all sports are aimed at developing muscles. For example, in chess, all attention is paid to the mental abilities of the athlete. Since the truth of a particular negative judgment has been established, it can be said that the refuted thesis is false. In this way, purpose of refutation is to identify the incorrect construction of evidence and the falsity or lack of evidence of the asserted judgment (thesis). 2. Refutation through arguments and form Other names for these methods of refutation are - criticism of arguments and failure of demonstration. As the name suggests, In the first case the refutation is directed not at the thesis itself, but at the arguments supporting it. Of course, the negation of the arguments in itself does not mean with certainty that the thesis itself is false, since false conclusions can be drawn from a true thesis. The essence of this method is, therefore, not to prove the falsity of the thesis, but to reveal, to show its lack of evidence. Any unproven thesis is not taken for granted, it needs proof. Therefore, criticism of arguments can be a fairly effective way of refutation. This is rather a way to achieve the truth, rather than effectively conducting a dispute, as it helps, first of all, to ensure that the opponent can prove his true judgment. False in this case will be rejected. The absence of true arguments in the proof may come from the falsity of the thesis being proved, the opponent's low awareness of the subject, and the lack of information about this subject in general. When using this method of refutation, one should not forget that it is impossible to conclude with certainty (as already mentioned above) from the denial of the foundation to the denial of the consequence. Another type of rebuttal is failure of demonstration. As in the first case, in the process of such a refutation the thesis is not affected, i.e. its falsity is not proven. Only errors made by the opponent during the proof process are revealed. Thus, just as when criticizing arguments, the fact that the thesis is unproven is shown. Mainly the arguments presented as evidence are considered. In this case, the task of refuting or confirming the thesis is not assigned to the refuter. It only reveals the shortcomings of the opponent’s evidence, forcing the latter to change arguments and correct mistakes that arise, as a rule, as a result of violation of one or another rule of deductive reasoning. In the process of proof, a hasty generalization can be made if, in the conclusion, only that part of the facts was taken into account that speaks in favor of the conclusion made. In this case, it is also necessary to point out to the opponent the mistake made. LECTURE No. 23. Sophisms. Logic paradoxes 1. Sophisms. Concept, examples Revealing this issue, it must be said that any sophism is a mistake. In logic, there is also paralogisms. The difference between these two types of errors is that the first (sophism) was made intentionally, while the second (paralogism) was made by accident. The speech of many people abounds in paralogisms. Conclusions, even seemingly correctly constructed ones, are distorted in the end, forming a consequence that does not correspond to reality. Paralogisms, despite the fact that they are allowed unintentionally, are still often used for their own purposes. You can call this tailoring to the result. Without realizing that he is making a mistake, a person in this case derives a consequence that corresponds to his opinion and discards all other versions without considering them. The accepted consequence is considered true and is not verified in any way. Subsequent arguments are also distorted in order to better suit the thesis put forward. At the same time, as mentioned above, the person himself does not realize that he is making a logical mistake, he considers himself to be right (moreover, he is more savvy in logic). Unlike a logical error that occurs involuntarily and is the result of a low logical culture, sophism is a deliberate violation of logical rules. It is usually carefully disguised as a true judgment. Deliberately allowed, sophisms aim to win the argument at any cost. Sophism is designed to knock the opponent off his line of thinking, to confuse, to draw into the analysis of errors that do not relate to the subject under consideration. From this point of view, sophism acts as an unethical way (and at the same time obviously wrong) of conducting a discussion. There are many sophisms created in antiquity and preserved to this day. The conclusion of most of them is curious. For example, the sophism "thief" looks like this: "The thief does not want to acquire anything bad; the acquisition of good is a good thing; therefore, the thief wants good." The following statement also sounds strange: "The medicine taken by the sick is good; the more good you do, the better; therefore, the medicine must be taken in large doses." There are other well-known sophisms, for example: "He who is sitting has risen; whoever has risen, he is standing; therefore, the one who is sitting is standing", "Socrates is a man; a man is not the same as Socrates; therefore, Socrates is something other than Socrates" , "These kittens are yours, the dog, their father is also yours, and their mother, the dog, is also yours. So, these kittens are your brothers and sisters, the dog and the bitch are your father and mother, and you yourself are a dog." Such sophisms were often used to mislead the opponent. Without such a weapon in their hands as logic, the rivals of the sophists in the dispute had nothing to oppose, although they often understood the falsity of sophistical conclusions. Disputes in the ancient world often ended in fights. With all the negative meaning of sophisms, they had a reverse and much more interesting side. So, it was sophisms that caused the emergence of the first rudiments of logic. Very often they pose the problem of proof in an implicit form. It was with sophisms that the comprehension and study of evidence and refutation began. Therefore, we can talk about the positive effect of sophisms, that is, that they directly contributed to the emergence of a special science of correct, demonstrative thinking. A number of mathematical sophisms are also known. To obtain them, numerical values are shuffled in such a way as to obtain one from two different numbers. For example, the statement that 2 x 2 = 5 is proven as follows: in turn, 4 is divided by 4, and 5 by 5. The result is (1:1) = (1:1). Therefore, four equals five. Thus, 2 x 2 = 5. This error is resolved quite easily - you just need to subtract one from the other, which will reveal the inequality of these two numerical values. A refutation is also possible by writing through a fraction. As before, so now sophisms are used to deceive. The above examples are quite simple, it is easy to notice their falsity and do not have a high logical culture. However, there are veiled sophisms, disguised in such a way that it can be very problematic to distinguish them from true judgments. This makes them a convenient means of deception in the hands of logically savvy scammers. Here are a few more examples of sophisms: “In order to see, there is no need to have eyes, since without the right eye we see, without the left we also see; apart from the right and left, we have no other eyes, therefore it is clear that the eyes are not necessary for sight" and "What you did not lose, you have; you did not lose your horns, so you have horns." The last sophism is one of the most famous and is often cited as an example. We can say that sophisms are caused by insufficient self-criticism of the mind, when a person wants to understand knowledge that is still inaccessible, not amenable at a given level of development. It also happens that sophism arises as a defensive reaction in the presence of a superior opponent, due to ignorance, ignorance, when the arguing does not show perseverance, not wanting to give up positions. It can be said that sophism interferes with the conduct of the dispute, but such a hindrance should not be classified as significant. With proper skill, sophism is easily refuted, although this leads to a departure from the topic of reasoning: one has to talk about the rules and principles of logic. 2. Paradox. Concept, examples Turning to the question of paradoxes, one cannot fail to say about their relationship with sophisms. The fact is that sometimes there is no clear line by which you can understand what you have to deal with. However, paradoxes are considered with a much more serious approach, while sophisms often play the role of a joke, nothing more. This is due to the nature of theory and science: if it contains paradoxes, then there is an imperfection in the underlying ideas. What has been said may mean that the modern approach to sophistry does not cover the entire scope of the problem. Many paradoxes are interpreted as sophisms, although they do not lose their original properties. paradox one can name an argument that proves not only the truth, but also the falsity of a certain judgment, i.e., proving both the judgment itself and its negation. In other words, paradox - these are two opposite, incompatible statements, for each of which there are seemingly convincing arguments. One of the first and certainly exemplary paradoxes was recorded Eubulides - Greek poet and philosopher, Cretan. The paradox is called "The Liar". This paradox has come down to us in this form: "Epimenides claims that all Cretans are liars. If he tells the truth, then he lies. Is he lying or is he telling the truth?" This paradox is called "the king of logical paradoxes". To date, no one has been able to solve it. The essence of this paradox is that when a person says: "I am lying", he does not lie and does not tell the truth, but, more precisely, he does both at the same time. In other words, if we assume that a person is telling the truth, it turns out that he is actually lying, and if he is lying, then he told the truth about it before. Both contradictory facts are asserted here. Of course, according to the law of the excluded middle, this is impossible, but that is why this paradox has received such a high "title". The inhabitants of the city of Elea, the Eleatics, made a great contribution to the development of the theory of space and time. They relied on the idea of the impossibility of non-existence, which belongs Parmenides. Every thought according to this idea is a thought about what exists. At the same time, any movement was denied: the world space was considered integral, the world was one, without parts. Ancient Greek philosopher Zeno of Elea known for compiling a series of paradoxes about infinity - the so-called paradoxes of Zeno. Zeno, a student of Parmenides, developed these ideas, for which he was named Aristotle "ancestor of dialectics". Dialectics was understood as the art of reaching the truth in a dispute, revealing contradictions in the opponent's judgment and destroying them. The following are the direct aporias of Zeno. "Achilles and the tortoise" represents an aporia about movement. As you know, Achilles is an ancient Greek hero. He had remarkable abilities in sports. The turtle is a very slow animal. However, in an aporia, Achilles loses the race to the tortoise. Suppose Achilles needs to run a distance of 1, and he runs twice as fast as a turtle, the last one needs to run 1/2. Their movement starts at the same time. It turns out that after running the distance 1/2, Achilles will find that the tortoise has managed to overcome the segment in the same time 1/4. No matter how much Achilles tries to overtake the tortoise, it will be ahead exactly by 1/2. Therefore, Achilles is not destined to catch up with the tortoise, this movement is eternal, it cannot be completed. The inability to complete this sequence is that it is missing the last element. Each time, having indicated the next member of the sequence, we can continue by indicating the next one. The paradox here lies in the fact that the endless sequence of successive events must actually come to an end, even if we could not imagine this end. Another aporia is called "dichotomy". The reasoning is based on the same principles as the previous one. In order to go all the way, you need to go halfway. In this case, half the path becomes a path, and in order to pass it, it is necessary to measure half (that is, already half of the half). This continues ad infinitum. Here the order of succession is reversed compared to the previous aporia, i.e. (1/2)n..., (1/2)3, (1/2)2, (1/2)one. The series here does not have the first point, while the aporia "Achilles and the tortoise" did not have the last. From this aporia, it is concluded that the movement cannot begin. Proceeding from the considered aporias, the movement cannot end and cannot begin. So it doesn't exist. Refutation of the aporia "Achilles and the Tortoise". As in the aporia, Achilles appears in its refutation, but not one, but two turtles. One of them is closer than the other. The movement also starts at the same time. Achilles runs last. During the time that Achilles runs the distance separating them at the beginning, the nearest tortoise will have time to crawl a little ahead, which will continue indefinitely. Achilles will get closer and closer to the tortoise, but he will never be able to overtake it. Despite the obvious falsity, there is no logical refutation of such an assertion. However, if Achilles begins to catch up with a distant tortoise, not paying attention to the near one, he, according to the same aporia, will be able to come close to it. And if so, then he will overtake the nearest turtle. This leads to a logical contradiction. In order to refute the refutation, i.e., to defend the aporia, which is strange in itself, it is proposed to throw away the burden of figurative representations. And to reveal the formal essence of the matter. Here it should be said that the aporia itself is based on figurative representations and to reject them means to refute it as well. And the rebuttal is quite formal. The fact that two turtles are taken instead of one in the refutation does not make it more figurative than an aporia. In general, it is difficult to talk about concepts that are not based on figurative representations. Even such philosophical concepts of the highest abstraction as being, consciousness, and others are understood only thanks to the images that correspond to them. Without the image behind the word, the latter would remain only a set of symbols and sounds. Stages implies the existence of indivisible segments in space and the movement of objects in it. This aporia builds on the previous ones. Take one immovable row of objects and two moving towards each other. Moreover, each moving row in relation to the immovable one passes only one segment per unit of time. However, in relation to the moving - two. which is considered contradictory. It is also said that in an intermediate position (when one row has already moved, as it were, the other has not) there is no room for a fixed row. The intermediate position comes from the fact that the segments are indivisible and the movement, even though it started simultaneously, must go through an intermediate stage when the first value of one moving series coincides with the second value of the second (movement, provided that the segments are indivisible, is devoid of smoothness). The state of rest is when the second values of all rows coincide. The fixed row, if we assume the simultaneity of the movement of the rows, must be in an intermediate position between the moving rows, and this is impossible, since the segments are indivisible. Notes 1. Makovelsky A. O. History of Logic. M., 1967. 2. V. S. Meskov, Essays on the Logic of Quantum Mechanics. M., 1986. 3. Demidov I. V. Logic: Textbook / Ed. B. I. Kaverina. 2nd ed. M.: Exam, 2006. 4. V. I. Kirillov and A. A. Starchenko, Logic. M., 2001. 5. Ibid. 6. Soviet Encyclopedic Dictionary / Ed. A. M. Prokhorova. 4th ed., rev. and additional M.: Sov. encycl., 1990. 7. Soviet Encyclopedic Dictionary / Ed. A. M. Prokhorova. 4th ed., rev. and additional M.: Sov. encycl., 1990. 8. Savchenko N. A. Course of lectures. Logics. M., 2002. 9. Savchenko N. A. Course of lectures. Logics. M., 2002. 10. Ibid. 11. Savchenko N. A. Course of lectures. Logics. Topic 4. M., 2002. 12. Savchenko N. A. Course of lectures. Logics. M., 2002. 13. Eryshev A. A. Logic. M., 2004. 14. Ibid. 15. Eryshev A. A. et al. Logic. M., 2004. 16. Savchenko N. A. Course of lectures. Logics. M., 2002. 17. Savchenko N. A. Course of lectures. Logics. M., 2002. 18. Povarnin S. I. Art of dispute: on the theory and practice of dispute. General information about the dispute. About proofs, questions of philosophy. N. 1990. 19. Ibid. 20. Ivin A. A. Logic: Textbook. M.: Gardariki, 2000. 21. Povarnin S. I. Art of dispute: on the theory and practice of dispute. General information about the dispute. About proofs, questions of philosophy. N. 1990. Author: Shadrin D.A.
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