About teeth Argumentation presupposes evidence, but is not limited to it. Proof is the logical basis of argumentation.

At the same time, argumentation requires, along with evidence, persuasive influence. The compelling, necessary nature of evidence, its impersonality, constitute the main difference between evidence and argumentation. The argumentation is non-forceful in nature; its correctness cannot be established mechanically. When comparing the results of argumentation and evidence, they sometimes say: “Proved, but not convinced.” (But logicians say differently: “When they can’t prove it, then they argue.”)

In general, if we characterize the relationship between logic and the theory of argumentation, we can say that both of these disciplines study the techniques and forms of organizing thinking. But in accordance with their objectives and methodology, they do this in different ways. Symbolic (i.e. modern formal) logic studies the problem of the validity of our reasoning in the aspect of their evidence, using rigorous mathematical methods. Methods of symbolic logic are effective for solving a range of problems that can be formalized. The theory of argumentation introduces into scientific consideration a wider class of contexts and living speech situations called discourses, which can only partially be formalized. These are the arguments of philosophy, jurisprudence, sociology, history and other humanities. And in this sense, for example, legal argumentation carefully developed over many centuries, based on empirically established judgments and material evidence, is not considered logically sound argumentation. argumentation is a rational form of persuasion, since in it the conviction is based on the arguments of reason and logic, and not on emotions, feelings, and especially not on volitional and other influences or coercion. Typically, argumentation takes on a logical character, although the person using it may not know the laws of logic, just as a competent writer cannot accurately name the rules of grammar. In this case, laws and rules are applied unconsciously, automatically, as self-evident norms, since they lead to the right results. But when errors occur in oral reasoning or in writing, then the laws of logic or the rules of grammar make it possible not only to detect them, but also to explain the reasons for their occurrence. This is why logic and grammar play such an important role in the persuasion process.

Since the judgments of logic express the relationship of our thoughts to reality and they are characterized as true or false, logic has priority in rational argumentation. Of course, the most convincing arguments in argumentation are ultimately facts, but they must be properly ordered and systematized, and this can only be achieved with the help of logical judgments and inferences. Ultimately, rational belief is achieved through logically correct reasoning in which conclusions are deduced or supported by true premises. If the conclusion follows from the premises according to the rules of logical inference, the reasoning is called deductive. If the conclusion is only confirmed and justified by premises, then the reasoning will not be deductive, but, for example, a conclusion by induction or analogy, or a statistical inference.

Argumentation is the science and art of making your opinion justified and convincing another person of it.

Rationale And belief - These two fundamental principles of argumentation give it duality. On the one hand, the theory of argumentation is a logical discipline based on logical methodology, since proof is a prerequisite when advancing and defending one’s position both in scientific research and in public discussion. On the other hand, argumentation includes a rhetorical component due to the fundamentally communicative nature of proof: we always prove something to someone - a person, an audience.

The most important area of ​​application of argumentation is disputes and discussions. Argumentative debate in antiquity was called dialectics, which meant the art of verbal interaction, the intellectual game of questions and answers. This understanding of dialectics distinguishes it from simple dispute - eristics. A dispute arises on the basis of a confrontation of opinions; it can take place like a game without rules, where there are gaps in reasoning and there is no logical coherence of thoughts. Dialectics, on the contrary, presupposes as a necessary condition the presence of logical contacts, connections that give the flow of thought the character of consistent reasoning. The dialectical process is a process aimed at seeking knowledge or reaching agreements.

In addition, Aristotle, who can rightfully be called the founder of not only logic, but also the theory of argumentation, as well as rhetoric, gave dialectics another meaning - the art of plausible (probabilistic) reasoning, which deals not with exact knowledge, but with opinions. Actually, this is exactly what we encounter in discussions where certain points of view are discussed - opinions on certain socially significant or scientific issues.

As we have already noted, the theory of argumentation deals with evidence in a broad sense - as everything that convinces of the truth of any judgment. In this sense argumentation is always dialogical and broader than logical proof(which is predominantly impersonal and monological), since argumentation assimilates not only the “technique of thinking” (the art of logical organization of thought), but also the “technique of persuasion” (the art of coordinating the thoughts, feelings and wills of interlocutors). That is, we can say that in argumentation, emotional, volitional and other actions, which are usually attributed to psychological and pragmatic factors, play no less a role than methods of reasoning. In addition to them, a person’s moral attitudes, social orientations, individual habits, inclinations, etc. have a noticeable influence on conviction.

The following levels of argumentation are distinguished:

• 1) informational - the level of content of the message sent to the addressee; that information (primarily about facts, events, phenomena, conditions) that they strive to bring to his attention;
• 2) logical - level of organization of the message, its construction (consistency and mutual consistency of arguments, their organization into a logically acceptable conclusion, systemic coherence);
• 3) communicative-rhetorical- a set of methods of persuasion and techniques (in particular, forms and styles of speech and emotional influence);
• 4) axiological - systems of values ​​(general cultural, scientific, group) that the arguer and the recipient adhere to and which determine the selection of arguments and methods of argumentation;
• 5) ethical - the level of “practical philosophy”, the application of a person’s moral principles in practice, during a communicative dialogue, the moral acceptability or unacceptability of certain arguments and techniques of argument and discussion;
• 6) aesthetic - level of artistic taste, aesthetics of communication, construction of dialogue as an intellectual game.

The fundamental concept of argumentation theory is the concept justifications. Justification, or giving reasons for an argument or judgment, requires critical steps to reflect on the essence of the subject under discussion. Along with rational arguments in the modern theory of argumentation, types of justification include arguments from personal experience, since for an individual, his personal experience is the most natural criterion of truth and persuasiveness, appeals to faith, and a number of others.

Argumentation includes evidence (validity in the objective sense) and persuasiveness (validity in the subjective sense). Evidence in science, as a rule, coincides with persuasiveness (though within the framework of one paradigm or another). In real communication, the opposite is often the case - for a number of argumentative practices (dispute, business negotiations), the art of persuasion comes to the fore.

As a result of the above consideration of the phenomenon of argumentation, the following complete definition can be given.

Argumentation - This is a verbal, social and rational activity aimed at convincing a rational subject of the acceptability (unacceptability) of a point of view by putting forward a certain set of statements that are compiled to justify or refute this point of view.

This definition was developed by the Amsterdam school of pragma-dialectics. By shortening and simplifying this (and others similar to it) definition, we get a “working” version: argumentation is a communicative activity aimed at forming or changing the views (beliefs) of another person by presenting rationally based arguments.

As a result of mastering this topic, the student should: know

• – structural elements of argumentation, proof, refutation,
• – similarities and differences between argumentation and evidence; be able to
• – distinguish between direct and indirect evidence; own
• – skills in using various methods of refutation.

Argumentation and proof. Argument structure

Logical thinking is manifested in evidence and validity of the judgments put forward. Evidence is the most important property of correct thinking. The first manifestation of incorrect thinking is unfoundedness, groundlessness, disregard for strict conditions and rules of evidence.

Every judgment made about something or someone is either true or false. The truth of some judgments can be verified by directly comparing their content with reality using the senses in the process of practical activity. However, this method of verification cannot always be used. Thus, the truth of judgments about facts that took place in the past or that may appear in the future can be established and verified only indirectly, logically, since by the time such facts are known they either cease to exist or do not yet exist in reality and therefore cannot be perceived directly. It is impossible, for example, to directly verify the truth of the proposition: “At the time of the commission of the crime, the accused N was at the crime scene." The truth or falsity of such judgments is established or verified not directly, but indirectly. Because of this, at the stage of abstract thinking there is a need for a special procedure - justification (argumentation).

The modern theory of argumentation as a theory of persuasion goes far beyond the logical theory of evidence, since it covers not only logical aspects, but also largely rhetorical ones, so it is no coincidence that the theory of argumentation is called “new rhetoric.” It also includes social, linguistic, psychological aspects.

Argumentation is a complete or partial justification of a judgment with the help of other judgments, where, along with logical methods, linguistic, emotional-psychological and other extra-logical techniques and methods of persuasive influence are also used.

Justify any judgment means to find other judgments that confirm it, which are logically related to the justified judgment.

There are two aspects to the study of argumentation: logical and communicative.

IN logical In terms of plan, the purpose of argumentation comes down to justifying a certain position, point of view, formulation with the help of other provisions called arguments. In the case of effective argumentation, it is also realized communicative aspect of argumentation, when the interlocutor agrees with the arguments and methods of proving or refuting the original position.

The core of argumentation, its deep essence, is evidence, which gives the argumentation the character of strict reasoning.

A proof is a logical technique (operation) that substantiates the truth of a judgment with the help of other logically related judgments, the truth of which has already been established.

Argumentation (like evidence) has a three-member structure, including thesis, arguments and demonstration, and has uniform rules for constructing the justification process, which are discussed below.

Thesis is a proposition whose truth needs to be proven.

Arguments (grounds, arguments) are true judgments with the help of which a thesis is justified.

In general, there are two types of arguments: correct and incorrect, correct or incorrect.

• 1. Arguments ad rem (concerning the case)) are correct. They are objective and relate to the essence of the thesis being proven. These are the following points of evidence:
  • A) axioms(Greek axioma– without proof) – unproven scientific provisions accepted as an argument in proving other provisions. The concept of “axiom” contains two logical meanings: 1) a true position that does not require proof, 2) the starting point of evidence;
  • b) theorems– proven scientific provisions. Their proof takes the form of a logical consequence of the axioms;
  • V) laws– special provisions of the sciences that establish essential, i.e. necessary, stable and repeating connections between phenomena. Each science has its own laws that sum up a certain type of research practice. Axioms and theorems also take the form of laws (axiom of syllogism, Pythagorean theorem);
  • G) judgments of fact– section of scientific knowledge of an experimental nature (observation results, instrument readings, sociological data, experimental data, etc.). As arguments, information about facts is taken whose truth is confirmed in practice;
  • d) definitions. This logical operation makes it possible to form in each scientific field a class of definitions that play a dual role: on the one hand, they allow you to specify a subject and distinguish it from other subjects in a given field, and on the other, to decipher the volume of scientific knowledge by introducing new definitions.
• 2. Ad hominem arguments (appealing to a person) are considered incorrect in logic, and the proof using them is incorrect. They are analyzed in more detail in the section “Unacceptable methods of defense and refutation.” Their goal is to convince at any cost - by citing authority, playing on feelings (pity, compassion, fidelity), promises, assurances, etc.

Proof pays "close attention" to the quality and composition of arguments. The form of transition from arguments to thesis can be different. It forms the third element in the structure of the proof - the form of proof (demonstration).

Form of evidence (demonstration ) called a method of logical connection between the thesis and arguments.

The course in formal logic is aimed at demonstrating the connection between natural language and thinking, the laws of the latter from the point of view of its structural organization, and the possibility of constructing logical calculus. It is built on the basis of traditional Aristotelian logic and propositional logic, and ends with the theory of argumentation.

About the course

The course is devoted to the structural, or formal, side of our thinking. It is basic, showing the relationship between thinking and language, the ideal content of the first and its material organization through the second.

This course is about why we, having accepted certain statements as initial and true, can and should come not to just any, but to a very definite – the only possible – conclusion.

Format

The course contains 19 topics. Each topic contains three sections - lecture, practical lesson, independent work. The lecture section includes a video, presentation, notes, glossary, test and list of recommended literature. The “Practical Lesson” section consists of methodological recommendations, examples of problem solving, the actual tasks and a list of references to which you can refer. Independent work involves one task aimed at solving non-standard problems or studying material on a topic that is not included in the lecture and practical lesson.

Requirements

Does not require special training

Course program

The course program includes 19 thematic lessons, each of which includes lecture material, practical assignments and assignments for the student’s independent work:

1. Subject and meaning of logic
2. Concept as a form of thinking
3. Logical operations with concepts
4. Judgment as a form of thinking
5. Logical analysis of questions
6. Complex judgment
7. Operations on complex judgments
8. Logical square
9. Logical law
10. Modal propositions
11. Inference as a form of thinking
12. Direct deductive reasoning
13. Simple categorical syllogism
14. Complex and abbreviated syllogisms
15. Deductive reasoning from complex premises
16. Non-deductive reasoning
17. Problem, hypothesis and theory and their place in scientific knowledge
18. Proof and refutation
19. Argumentation strategy and tactics

Learning outcomes

The ability to isolate the logical structure of natural language thinking and manipulate it according to the rules of logic. The skill of building argumentation in different ways. The skill of detecting logical errors in reasoning.

Formed competencies

OK-5: ability to communicate orally and in writing in Russian and foreign languages ​​to solve problems of interpersonal and intercultural interaction;

OK-7: ability for self-organization and self-education;

GPC-4: ability to carry out business communication and public speaking, conduct negotiations, meetings, carry out business correspondence and maintain electronic communications

Federal Agency for Education State educational institution of higher professional education "Omsk State Technical University"

A.S. Skachkov

Logic and theory of argumentation

Methodical instructions, course of lectures, homework

Tutorial

Reviewers:

N.I. Martinina, Doctor of Philosophy. sciences, prof. Department of History, Philosophy and Cultural Studies of Omsk State Transport University;

V.V. Nikolin, Doctor of Philosophy. sciences, prof. Department of Philosophy, Omsk State Pedagogical University

Skachkov A.S.

C42 Logic and theory of argumentation. Methodological instructions, course of lectures,

hometasks: Textbook. allowance. - Omsk: Omsk State Technical University Publishing House, 2005. - 184 p.

The textbook is intended for distance learning students of specialty 350400 - public relations, studying the discipline “Logic and Theory of Argumentation”, and can also be recommended for additional reading in the course “Philosophy” for students interested in the theory and methodology of knowledge.

Published by decision of the editorial and publishing council of Omsk State Technical University.

© A.S. Skachkov, 2005 © Omsk State

Technical University, 2005

PREFACE

The main task, the fulfillment of which this textbook should ensure, is the student’s complete mastery of the content of the discipline “Logic and Theory of Argumentation” without the involvement of traditional classroom teaching and control methods. To develop the skills and abilities of using logic in thinking - including in such an important form of intellectual activity for public relations specialists as conducting an argumentative process - without direct interaction with teachers, systematic and self-controlled training in the use of logical tools is provided. Thus, the fundamental goal that the student must see and progressively pursue is the individual and practical use of a diverse arsenal of logical means; The system of organizing educational material is called upon to indicate the ways to achieve this goal and to carry out the guided development of the logical abilities of a particular person.

In the textbook, you should pay attention to the methodological and content components. The first provides for the need for preliminary familiarization with the key semantic and didactic units of material in each new section, an indication of the logical diagram of their essential relationships and interdependence, as well as those particular goals and objectives, the solution and achievement of which will indicate a sufficient level of mastery of the material presented in these sections . In this connection, everyone who begins to study the discipline “Logic and Theory of Argumentation” using this textbook needs to carefully read the above programming partly, keep it in operational and then in long-term memory. Otherwise, unfortunately, a coherent, non-mosaic, systemically meaningful mastery of the material may not arise. The desired level of assimilation is supported by a variety of examples given in the text of the sections, their topics and specifications, which allow one to see and adopt techniques for consciously solving various logical problems, which are usually carried out outside the context of logic as a science at the level of common sense and logical intuition. Analysis of these examples allows you to master the transition from theoretical to practical content of the arsenal of logical tools for further use. In development of the aforesaid heuristic part, a set of options for homework is proposed, intersecting with examples as examples of solving standard logical problems, partially duplicating the latter, as well as expanding and deepening the content fixed in them. The main function of the latter becomes controlling, forcing self-assessment of the results achieved during the study of a particular section. Thus, each student is obliged to monitor the level of programmatic and heuristic skills he achieves. With the inevitable discovery of varying depths of discrepancy between the achieved level and its exemplary indicator - the ability to clearly and correctly comprehend and solve any logical problem that corresponds to the studied material,

chu – one should act in accordance with the most ancient principle of learning: “Repetition is the mother of learning.”

Since self-esteem - with its reduced value - is a subjective, if not crafty, thing, and the studied discipline “Logic and Theory of Argumentation” is an integral element of the knowledge system controlled by society and the state of anyone claiming to be a public relations specialist, then doing homework also serves a form of external control that develops into reporting. You should independently work through the entire array of options. homework, be ready at the first request to publish the results of their implementation using electronic or other means of remote communication. This means that for each section and from all exercises as part of the homework, any of the students will be offered two or three randomly chosen examples after the deadline for preparing this material determined by the curriculum. If at the same time it is proposed to complete the homework on a traditional paper medium, then using A4 sheets, having drawn up the title page in accordance with the requirements of abstract works, the student is obliged to: 1) record the full wording of each task proposed to him; 2) provide a detailed answer for each specific example. In the case of an electronic version of monitoring the quality of homework, the sequence of these operations is carried out within the framework of the software used. For each homework assignment, the testing authority determines (with grades “passed”/“failed” or grades on a multi-point scale) either the sufficiency or insufficiency of the level of mastery of the material to move on to the next section. After successfully mastering the material in all sections, i.e. acquiring all the necessary and controllable skills and abilities of a logical nature, the student is presented with one of the randomly selected options complex task, containing a more formalized repetition of the material already mastered for the course as a whole. Completion (failure) of such a task gives a final assessment of the level of training. This block of tasks can be used as a basis for monitoring the level of residual knowledge, both during intra-university monitoring of the quality of education, and during state certification events.

The content of the presentation of material in the discipline “Logic and Theory of Argumentation,” meeting the requirements of the State Educational Standard for the training of public relations specialists, represents a system of five sections: 1) Subject, basic concepts and types of logic; 2) Syllogistic theory of deductive reasoning; 3) Logic of statements and predicates; 4) Theory of plausible reasoning; 5) Basics of the argumentative process.

Each section is preceded by a small methodological programming part. At the end of each section, a compact list of those educational sources is offered, the use of which can provide any complementary, clarifying, interpretive clarifications. In general, the educational material contained in the sections is divided into thirteen lecture topics; in each of them, the key subtopics are highlighted in the form of subheadings, the content of which is related to

Both examples and exercises for homework and complex assignments are crocheted. The textbook “List of Basic Symbols of Classical Formal Logic” and the generalizing “Bibliographic List” complete the educational tools.

SUBJECT, BASIC CONCEPTS AND VARIETIES OF LOGIC

Introduction

Every person is capable of logical thinking - this thought is guiding and encouraging for those who, in their education, become familiar with the treasury of logical thought. But the potential possession of this property, its brilliance and power, should not be confused with the actual one. To paraphrase the famous aphorism of the ancient philosopher, we can put forward the following as the most general requirement for anyone who begins to study logic: “Be so wise that you do not avoid your ignorance.” You should be aware that what is to be learned in logic, although intuitively clear, is by no means known.

When critically starting to study logic, first of all it is necessary to learn to differentiate the meanings of this term (objective logic, subjective logic, logic as a science), since this allows you to see and evaluate the possibilities and prospects for the study of logical problems in general. As a result of such differentiation, one should clearly realize that logic as a science is abstract, but integral from the ontological (objective) and psychological (subjective) contexts of human thinking, therefore it is impossible to master this discipline without being at the level of abstractions and without finding a place for sensory forms (sensations , perceptions, ideas) in many forms of cognition and the process of cognition in general.

The most important logical abstractions are the concepts of the form of thought and its content. Since the structures of thought should be understood in their specific logical features, careful mastery of the entire set of basic concepts of logic is required, the development of which is the historical outline of this discipline, which developed by branching from traditional classical logic, founded by Aristotle, to classical formal logic, so called by Kant, and then to a wide range of classical and non-classical modern logics. In this regard, a systematic elaboration of the concepts is needed: truth, formal correctness, sign, linguistic sign, semantic category, logical consequence, formal correctness of thinking, logical law.

Having mastered the general semantic categories developed by scientific logic, having learned to understand and apply the language of classical logic, one should work with special care in practice on the most important logical fundamentals - the principles of formal logic: the law of identity, non-contradiction (contradiction), excluded middle, sufficient reason. This requires synthesis

to develop an intuitive understanding of the essence of logical consequence, which any person has, with his scientific understanding, and also to acquire the skills of using these principles for the daily guidance of one’s own reasoning and analysis of acts of cognition carried out by others.

In addition, special attention should be paid to the fact that the questions studied here are given in their preliminary form, i.e., they can be mastered in full only with subsequent study using the material from all other sections.

Chapter first

SUBJECT, CONDITIONS OF ARISE, TYPES AND FUNDAMENTALS OF LOGIC

1.1. Object and subject meaning of logic

The existence and life of any creature, including humans, takes place only in connection with the existence and life of the world as a whole, or the universe. In the context of the universe, objects of infinite variety manifest themselves, being in complex relationships and conditions and often being involved in objective-practical and cognitive life activity. Humanity of the species Homo sapiens, which arose about forty-five thousand years ago on planet Earth, is an organic part, witness and participant in this process.

Having a rational form of receiving, processing and storing information, people strive to use the interconnections and interdependence of the world as a whole more deeply and comprehensively. The light of knowledge that once flared up in the solar system increases as a shock wave of intelligence to the scale of a cosmic phenomenon, providing man, as his carrier-subject, with an increasing potential for resistance to the entropic, destructive aspects of reality.

As modern researchers note, humanity at this period of its development “among the general entropy of nature” knows “the only completely ordered entropy-free phenomenon - the logical activity of the human brain” (L.V. Leskov). The source of the latter, by and large, should be considered the universe itself, i.e., that which is indestructible by its very essence.

Thus, in a broad sense, logic combines everything that has ever been thought, is being thought, and can be thought of in the future by any intelligent being.

The fact that the meaning of rationality is associated with the attribute of “being involved in logical activity” in its broad sense is already clear from the very original, which served as the basis for the formation of the term “logic”, the ancient Greek word λόγος, meaning “regularity”, “reason”, “thought”, "word". Identifying the patterns of parts, sides, factors of reality, equipping a person interacting through sign systems with other people with the tools of correct thinking is the main task of formal logic. The fundamental distinction between truth and lies is its main imperative.

The universe appears in relation to a person in the form of an ordered, naturally arranged reality only if a person manifests himself as a conscious, cognizing rational being, which is why the main content of the concept of “logic” is concentrated in such senses of this relationship as “regularity” and “ thinking". For a person, the object of logical problematics is the universe in the aspect of its semantic (comprehensible by reason, conceivable) order. In turn, the object of logical problematics in the substantive disclosure of its content appears in three main aspects.

IN In the first, extremely broad sense, the word “logic” means all-

what is the necessary pattern in the relationship between objective phenomena (conditionally

– this is "object logic"). In this sense, the presence of logic (logicality) is stated in relation to any objective pattern.

IN In relation to natural phenomena, people state the “logicality” of the fact that after solar flares, “flares” appear in the night sky of northern latitudes. In relation to culture, after barbarism, progressively developing human communities enter the stage of civilizations; in relation to the functioning parameters of the human body - as a result of “sensory hunger” a person is deprived of the ability to think. This is the “logic” of things, facts, historical development, etc.

But by no means always when using the term “logic” we are talking about objective phenomena. So, in the case of ascertaining the logicality of someone’s assumptions, we are dealing with the process of thinking as such, that is, with the necessary patterns in the interconnection of the thoughts of the cognizing subject. In this case, the object regularity acts as fundamental in relation to the subjective regularity, the latter being an indirect reflection of the object regularity. Thinking that does not distort the laws of objective phenomena, but follows these laws, at least intuitively, is recognized as objectively reliable and subjectively correct.

IN In general, when viewed in this way, the word “logic” means law-

difficulties in connections and development of thoughts (conditionally – “subjective logic”).

IN In relation to your reasoning or the reasoning of other people, you can notice such signs as coherence, consistency, completeness, etc., which allows you to talk about “logicality” (the presence of logic) in such reasoning: “Ivanov’s reasoning is logical (not devoid of logic).” Otherwise, we note the fact that the corresponding reasoning is illogical.

Using thinking involuntarily, as naturally as breathing or comprehending the world around us with the help of our senses, we can nevertheless comprehend the structure of these processes. In this case, in relation to the thinking process, we will be interested in those mental procedures and

operations, the implementation of which actually makes our thinking correct.

First of all, we note that our thoughts, based on information received through analyzers (sense organs) and organized in the forms of sensations, perceptions and ideas, i.e., information generated at the sensory level of cognition, themselves go beyond the scope of the spaces associated with feelings - temporal-veno-temporal characteristics, belong to the area of ​​ideal reality. This is the reality of the rational stage of cognition, where information is enclosed in organizational forms different from those characteristic of the sensory stage of cognition, namely: concepts, statements (judgments), conclusions.

The procedures, the constant execution of which is responsible for the existence of these forms, are analysis, synthesis, comparison, abstraction, generalization and etc.

The functioning of these forms in the holistic process of abstract thinking is normative in nature and takes place in the form of standard logical operations subject to general laws and particular rules, namely: generalizations and limitations, division and classification, definitions of concepts; evidence and refutation of statements. In this case, the subject of “logic” becomes a systemic set of forms of abstract thinking, taken in the aspect of their normative correct functioning, and this subject exists exclusively in the sphere of scientific comprehension of reality.

In relation to the identified subject, the word “logic” means the science of forms, laws, operations and methods of correct abstract thinking.

It is logical that if we know that each of our acquaintances has an education, then we know that all the women we know have an education; There is logic in the reasoning: since there is a planetary form of life in our solar system, and there are many star systems that are similar to ours, then, probably, the planetary form of life is not limited to that found in our solar system.

1.2. Varieties and historical aspect of logic as a science

Being a discipline abstracted from the real empirical content of the thinking process, studying precisely the forms of thinking, logic - in this aspect of its manifestation - is called formal. The definition of “formal” was first used by the German philosopher I. Kant (1724–1804), who deliberately emphasized the difference between logic with the dominance of this aspect from other possible logics (for example, from dialectical logic).

Formal logic arose in the 4th century. BC. and in its historical development demonstrates a steady tendency to increasingly formalize thinking procedures. According to the degree of this formalization, two stages of formal lo-

geeks: traditional and modern (symbolic, mathematical).

Traditional is called formal logic that studies correct thinking with the wide use of natural language capabilities, i.e. the language of such logic is not fully formalized.

Such logic does not eliminate the polysemy and uncertainty of the rules it studies for constructing expressions, assigning meanings, etc., which can only be achieved through the construction and use of artificial (symbolic) languages ​​designed to follow the logical form, reproducing it even at the expense of brevity and ease communication.

Sufficiently universal (not including words of ordinary spoken language) formalized languages ​​and corresponding theories of logical analysis began to be developed in the second half. XIX – first half. XX centuries, which marked the beginning of the modern stage in the historical development of formal logic.

The formal logic of the modern stage of its historical development is defined as “symbolic”, since it uses only formalized languages, and as “mathematical”, since the methods used in it are similar to the methods used in mathematics.

Mathematical logic explores the subject of logic by the method of constructing special formalized languages ​​- calculus, which allows one to avoid ambiguities and ambiguities of natural language. At the same time, the axioms of formal logic initially contained principle of two-valuedness (bivalence), according to which every meaningful statement is either true or false.

That part of formal logic (all traditional and some modern), which is based on the principle of two-valuedness, called classical

skoy (two-valued) logic.

The founder of formal logic, who laid down the principles of its classical version, is the ancient Greek philosopher Aristotle (384–322 BC). At the origins of modern classical formal logic are, along with many other researchers, J. Boole (1815–1864), A. de Morgan (1806–1871), C.S. Pierce (1839–1914), who gradually implemented what was proposed by G.V. Leibniz (1646–1716) the idea of ​​transferring mathematical methods into logic.

Doubts about the universality of the principle of two-valuedness were resolved within the framework of modern formal logic, which gave rise to the principle of plurality.

non-classical logic, including multi-valued logic.

The following order applies in the relationship between the varieties of formal logic: classical traditional formal logic serves as the basis for the apparatus of classical modern formal logic, the latter is considered the core of modern logic as a whole and retains its theoretical and practical significance for the latest non-classical logical theories. Many of these theories can be presented as extensions of classical logic, enriching its means of expression in comprehending an infinitely complex universe.

1.3. Basic provisions and concepts of classical formal logic

Classical formal logic initially appeared as a theoretical discipline designed to be a tool (as the ancients put it, an “organon”) of theoretical and worldview activity in all its manifestations.

The area of ​​use of logical knowledge is extremely wide: from the so-called sphere of common sense of everyday knowledge to the instrumental apparatus of specialized scientific disciplines. And, of course, logical knowledge, by and large, comes from philosophy as the sphere of its main functioning, which gives reason to consider logic primarily a philosophical science.

In fact, scientific logical problems emerged at the turn of the 5th–4th centuries. BC. in those ancient civilizations (Greece, India and China), where, within the framework of what appeared in the 7th–6th centuries. BC. philosophical worldview began a theoretical and conceptual clarification of the content of the connection between the truth of one or another human thought expressed in language (verbally) about reality with the structure of this thought itself.

Without being involved in clarifying the truth or falsity of a specific thought about reality, logic solves the problem of the correctness of the construction of the act of thinking itself, so that from true initial information it would be possible to obtain new true information, guaranteed or at least meaningfully. The aspects and semantic aspects of the above connection can be grouped into three circles of problems: linguistic ( logical-semiotic); building a theory of correct (deductive) reasoning; rules and ways of organizing knowledge systems ( logical-methodological).

In the course of solving these problems, fundamental logical concepts crystallized: truth (falsehood), attribute, sign with its meaning and meaning, logical form, logical law, formal correctness, logical consequence, logical theory etc.

Historically in logic they are used concepts of truth and the falsity of thought as relating only to the specific content of any judgment about reality (the principle of the concreteness of truth).

When we use a meaningful description of any object of the universe, then this description is in strictly defined spatial, temporal and other circumstances. Only in the context of such circumstances can various statements be recognized as corresponding (inappropriate) to the described state of affairs (reality): “You are a student”, “The student has now read this phrase”, “Today is a cloudy day”, “Fresh seas do not exist”, “Zinc harder than lead”, “Not a single meaningless statement can be accepted as a scientific axiom.”

Logical foundations of the theory of argumentation

Test on the discipline: logic

1. Concept of proof

Knowledge of individual objects and their properties begins with sensory forms (sensations and perceptions). We see that this house is not yet completed, we feel the taste of bitter medicine, etc. The truths revealed by these forms are not subject to special proof; they are obvious. However, in many cases, for example, at a lecture, in an essay, in a scientific work, in a report, during polemics, at court hearings, in defending a dissertation and in many others, we have to prove and justify the judgments we express.

Evidence is an important quality of correct thinking. Proof is related to argumentation, but they are not identical.

Argumentation is a method of reasoning, including proof and refutation, during which a belief in the truth of the thesis and the falsity of the antithesis is created both among the prover and among opponents; the expediency of accepting the thesis is substantiated in order to develop an active life position and implement certain action programs arising from the position being proven. The concept of “argumentation” is richer in content than the concept of “evidence”: the purpose of the proof is to establish the truth of the thesis, and the purpose of the argumentation is also to justify the expediency of accepting this thesis, to demonstrate its importance in a given life situation, etc. P. In the theory of argumentation, “argument” is also understood more broadly than in the theory of evidence, because the former refers not only to arguments confirming the truth of the thesis, but also arguments justifying the expediency of its adoption, demonstrating its advantages over others. with similar statements (sentences). Arguments in the process of argumentation are much more varied than in the process of proof.

The form of argumentation and the form of evidence also do not completely coincide. The first, like the last, includes various types of inferences (deductive, inductive, by analogy) or their chain, but, in addition, combining proof and refutation, provides justification. The form of argumentation most often has the character of a dialogue, because the arguer not only proves his thesis, but also refutes the opponent’s antithesis, convincing him and/or the audience witnessing the discussion of the correctness of his thesis, and strives to make them like-minded people.

Dialogue as the most reasoned form of conversation came to us from antiquity (for example, Ancient Greece is the birthplace of Plato’s dialogues, the technique of argument in the form of Socrates’ questions and answers, etc.). But dialogue is an external form of argumentation: the opponent can only be thought (which is especially clearly manifested in written argumentation). Internal

the form of argumentation is a chain of evidence and refutations of the person arguing in the process of proving the thesis and implementing the belief. In the process of argumentation, the development of beliefs in an interlocutor or audience is often associated with their persuasion. Therefore, in argumentation, the role of rhetoric in its traditional understanding as the art of eloquence is great. In this sense, Aristotle’s “Rhetoric” is still of interest, in which the science of eloquence is considered as the theory and practice of persuasion in the process of proving the truth of a thesis. “The Word is a great ruler who, possessing a very small and completely invisible body, performs the most wonderful things. For it can drive out fear, destroy sadness, instill joy, and awaken compassion,” wrote the ancient Greek scientist Gorgias about the art of argumentation. There has never been a period in history when people did not argue.

Without argumentation of statements, intellectual communication is impossible, for it is a necessary tool for knowing the truth.

The theory of proof and refutation is, in modern conditions, a means of forming scientifically based beliefs. In science, scientists have to prove a variety of propositions, for example, judgments about what existed before our era, to what period objects discovered during archaeological excavations belong, about the atmosphere of the planets of the solar system, about the stars and galaxies of the Universe, theorems of mathematics, judgments about directions the development of electronic technology, the possibility of long-term weather forecasts, the secrets of the World Ocean and space. All these judgments must be scientifically substantiated.

A proof is a set of logical techniques to substantiate the truth of a thesis. Proof is related to belief, but is not identical to it: evidence must be based on scientific data and socio-historical practice, while beliefs can be based, for example, on religious faith, on prejudices, on people’s ignorance of economic issues and politics, on the appearance of evidence based on various kinds of sophisms. Therefore, convincing does not mean proving.

1.2 Structure of evidence: thesis, arguments, demonstration

A thesis is a proposition whose truth must be proven. Arguments are those true judgments that are used to prove a thesis. The form of evidence, or demonstration, is the method of logical connection between the thesis and arguments.

Let's give an example of the proof. Paul S. Bragg expressed the following thesis: “You cannot buy health, you can only earn it through your own constant efforts.” He justifies this thesis as follows: “Only hard and persistent work on oneself will allow everyone to become an energetic long-liver enjoying endless health. I earned my health with my life. I am healthy 365 days a year, I do not have any pain, fatigue, or frailty of my body. And you can achieve the same results!”

1.3 Types of arguments

There are several types of arguments:

1. Certified isolated facts. This type of argument includes the so-called factual material, i.e. statistical data on the population, territory of the state, implementation of the plan, quantity of weapons, witness testimony, signatures on documents, scientific data, scientific facts. The role of facts in substantiating the propositions put forward, including scientific ones, is great.

Facts are the air of a scientist. Without them you will never be able to take off. Without them, your “theories” are empty attempts.

2. Definitions as arguments of proof. Definitions of concepts are usually given in every science. The rules for defining and types of definitions of concepts were discussed in the topic “Concept”, and numerous examples of definitions of concepts of various sciences were given there: mathematics, chemistry, biology, geography, etc.

3. Axioms. In mathematics, mechanics, theoretical physics, mathematical logic and other sciences, in addition to definitions, axioms are introduced. Axioms are judgments that are accepted as arguments without proof.

4. Previously proven laws of science and theorems as proof arguments. Previously proven laws of physics, chemistry, biology and other sciences, and theorems of mathematics (both classical and constructive) can be used as proof arguments. Legal laws are arguments in judicial evidence.

When proving a thesis, not one, but several of the listed types of arguments can be used.

2. Direct and indirect (indirect) evidence

Evidence by form is divided into direct and indirect (indirect). Direct proof goes from considering the arguments to proving the thesis, i.e. the truth of the thesis is directly justified by arguments. The scheme of this proof is as follows: from the given arguments (a, b, c, ...) the thesis q to be proved necessarily follows. This type of evidence is used in judicial practice, in science, in polemics, in the writings of schoolchildren, when a teacher presents material, etc.

Direct evidence is widely used in statistical reports, in various kinds of documents, in regulations, in fiction and other literature.

The teacher in the lesson, with direct proof of the thesis “The people are the creator of history,” shows, firstly, that the people are the creators of material wealth, and secondly, he substantiates the enormous role of the masses in politics, explains how in the modern era the people lead active struggle for peace and democracy, thirdly, reveals its great role in the creation of spiritual culture.

In the modern fashion magazine “Burda” the thesis “Envy is the root of all evil” is substantiated with the help of direct evidence with the following arguments: “Envy not only poisons people’s everyday lives, but can also lead to more serious consequences, therefore, along with jealousy, anger and hatred are undoubtedly among the worst character traits.

Creeping up unnoticed, envy hurts painfully and deeply. A person envies the well-being of others, and is tormented by the knowledge that someone is more fortunate.”

Indirect (indirect) evidence is evidence in which the truth of the thesis put forward is substantiated by proving the falsity of the antithesis. If a thesis is denoted by the letter a, then its negation (a) will be an antithesis, i.e. a judgment that contradicts the thesis.

Apagogical indirect evidence (or evidence “by contradiction”) is carried out by establishing the falsity of a judgment that contradicts the thesis. This method is often used in mathematics.

Let a be a thesis or theorem that needs to be proven. We assume by contradiction that a is false, i.e. true not-a (or a). From assumption a we derive consequences that contradict reality or previously proven theorems. We have a V a, while a is false, which means its negation is true, i.e. a, which, according to the law of two-valued classical logic (a > a) gives a. This means that a is true, which is what needed to be proven.

It should be noted that in constructive logic the formula a > a is not derivable, therefore in this logic and in constructive mathematics it cannot be used in proofs. The law of excluded middle is also “rejected” here (it is not a deducible formula), therefore indirect evidence does not apply here. There are a lot of examples of proof by contradiction in the school mathematics course. Thus, for example, the theorem is proven that from a point lying outside a line, only one perpendicular can be lowered onto this line. The following theorem is also proved using the “by contradiction” method: “If two straight lines are perpendicular to the same plane, then they are parallel.” The proof of this theorem directly begins with the words: “Let us assume the opposite, i.e. that straight lines AB and CD are not parallel.”

Separation proof (by elimination method). Antithesis is one of the members of a disjunctive judgment, in which all possible alternatives must be listed, for example:

The crime could have been committed by either A, B, or C.

It has been proven that neither A nor B committed the crime.

The crime was committed by S.

The truth of the thesis is established by sequentially proving the falsity of all members of the disjunctive judgment, except one.

The structure of the negative-affirmative mode of the divisive-categorical syllogism is used here. The conclusion will be true if the disjunctive judgment provides for all possible cases (alternatives), i.e. if it is a closed (complete) disjunctive proposition:

As noted earlier, in this mode the conjunction “or” can be used both as a strict disjunction () and as a non-strict disjunction (v), therefore the scheme also corresponds to it:

3. The concept of refutation

Refutation is a logical operation of establishing the falsity or lack of validity of a previously put forward thesis.

A refutation must show that: 1) the evidence itself (arguments or demonstration) is constructed incorrectly; 2) the thesis put forward is false or not proven.

A proposition that needs to be refuted is called a refutation thesis. Judgments with the help of which a thesis is refuted are called refutation arguments.

There are three ways of refutation: I) refutation of the thesis (direct and indirect); ii) criticism of arguments; Ш) identifying the failure of the demonstration.

3.1 Refutation of the thesis

Refutation of the thesis is carried out using the following three methods (the first is the direct method, the second and third are indirect methods).

1. Refutation with facts is the most accurate and successful way of refutation. Earlier we talked about the role of selecting facts, about the methodology for operating with them; all this must be taken into account in the process of refuting facts that contradict the thesis. Actual events, phenomena, statistical data that contradict the thesis must be presented, i.e. refutable judgment. For example, to refute the thesis “Organic life is possible on Venus,” it is enough to provide the following data: the temperature on the surface of Venus is 470-480 ° C, and the pressure is 95-97 atmospheres. These data indicate that life on Venus is impossible.

2. The falsity (or inconsistency) of the consequences arising from the thesis is established. It is proven that this thesis entails consequences that contradict the truth. This technique is called “reduction to absurdity” (reductio ad absurdum). They do this: the thesis being refuted is temporarily recognized as true, but then consequences are drawn from it that contradict the truth.

In classical two-valued logic (as already noted), the method of “reduction to absurdity” is expressed in the form of a formula:

where F is a contradiction or a lie.

In a more general form, the principle of “reduction to absurdity” is expressed by the following formula: (a > b) > ((a >) > a).

3. Refuting the thesis through proof of the antithesis. In relation to the thesis being refuted (judgment a), a judgment contradicting it (i.e., not-a) is put forward, and the judgment not-a (antithesis) is proven. If the antithesis is true, then the thesis is false, and a third is not given according to the law of the excluded middle.

For example, it is necessary to refute the widespread thesis “All dogs bark” (proposition A, generally affirmative). For judgment A, judgment O will be contradictory - the partial negative: “Some dogs do not bark.” To prove the latter, it is enough to give several examples, or at least one example: “Pygmies’ dogs never bark.” So, the proposition O is proven. By virtue of the law of excluded middle, if O is true, then A is false. Therefore, the thesis is refuted.

3.2 Criticism of the arguments

The arguments that were put forward by the opponent in support of his thesis are criticized. The falsity or inconsistency of these arguments is proven.

The falsity of the arguments does not mean the falsity of the thesis: the thesis can remain true.

It is impossible to reliably conclude from the denial of a reason to the denial of a consequence, but sometimes it is enough to show that the thesis has not been proven. Sometimes it happens that a thesis is true, but a person cannot find true arguments to prove it. It also happens that a person is not guilty, but does not have sufficient arguments to prove it. When refuting arguments, these cases should be kept in mind.

3.3 Detection of demonstration failure

This method of refutation consists of being shown. errors in the form of evidence. The most common mistake is that the truth of the thesis being refuted does not follow from the arguments given in support of the thesis. A proof may be incorrectly constructed if any rule of deductive reasoning is violated or a “hasty generalization” is made, i.e. incorrect inference from the truth of judgment I to the truth of judgment A (analogously, from the truth of judgment O to the truth of judgment E).

But having discovered errors during the demonstration, we refute its course, but do not refute the thesis itself. The task of proving the truth of the thesis lies with the one who put it forward.

Often, all of the listed methods of refuting the thesis, arguments, and the course of proof are not used in isolation, but in combination with each other.

4. Rules of evidential reasoning. Logical errors found in proofs and refutations

If at least one of the rules listed below is violated, then errors regarding the thesis being proven, errors in relation to the arguments and errors in the form of evidence may occur.

4.1 Rules regarding the thesis

1. The thesis must be logically defined, clear and precise. Sometimes people in their speech, written statement, scientific article, report, lecture cannot clearly, clearly, unambiguously formulate the thesis. Thus, a speaker at a meeting cannot clearly formulate the main provisions of his speech and therefore convincingly argue for them in front of the audience. And the listeners are perplexed why he spoke in the debate and what he wanted to prove to them.

2. The thesis must remain identical, i.e. the same throughout the entire proof or refutation. Violation of this rule leads to a logical error - “substitution of the thesis”.

4.2 Errors regarding the thesis being proven

l. “Substitution of thesis.” The thesis must be clearly formulated and remain the same throughout the entire proof or refutation - these are the rules in relation to the thesis. If they are violated, an error occurs called “substitution of the thesis.” Its essence is that one thesis is intentionally or unintentionally replaced by another and they begin to prove or refute this new thesis. This often happens during an argument or discussion, when the opponent’s thesis is first simplified or expanded in its content, and then they begin to criticize it. Then the one who is criticized declares that the opponent “distorts” his thoughts (or words) and attributes to him something that he did not say. This situation is very common; it occurs when defending dissertations, and when discussing published scientific works, and at various kinds of meetings and sessions, and when editing scientific and literary articles.

Here there is a violation of the law of identity, since they try to identify non-identical theses, which leads to a logical error.

2. “Argument to man.” The mistake consists in replacing the evidence of the thesis itself with references to the personal qualities of the one who put forward this thesis. For example, instead of proving the value and novelty of the dissertation work, they say that the dissertation author is an honored person, he worked a lot on the dissertation, etc. A conversation between a class teacher and a teacher, for example of the Russian language, about the grade assigned to a student sometimes comes down not to an argument that this student deserved this grade with his knowledge, but to references to the student’s personal qualities: conscientious in his studies, was sick a lot this term, He succeeds in all other subjects, etc.

In scientific works, sometimes, instead of a specific analysis of the material, the study of modern scientific data and the results of practice, quotes from the statements of major scientists and prominent figures are given in support of this, and they limit themselves to this, believing that one reference to authority is enough. Moreover, quotes can be taken out of context and sometimes interpreted arbitrarily. The "argument to man" is often simply a sophistical device, rather than an error made unintentionally.

A variation of the “argument to the public” is the fallacy called “argument to the public,” which consists of an attempt to influence the feelings of people so that they believe in the truth of the thesis put forward, although it cannot be proven.

3. “Transition to another gender.” There are two types of this error:

a) “he who proves too much proves nothing”; b) “he who proves too little proves nothing.”

In the first case, an error occurs when, instead of one true thesis, they try to prove another, stronger thesis, and in this case the second thesis may turn out to be false. If a implies b, but b does not entail a, then thesis a is stronger than thesis b. For example, if instead of proving that this person did not start the fight first, they begin to prove that he did not participate in the fight, then they will not be able to prove anything if this person really fought and witnesses saw it.

The error “he who proves too little proves nothing” arises when, instead of thesis a, we prove a weaker thesis b. For example, if, trying to prove that this animal is a zebra, we prove that it is striped, then we will not prove anything, because the tiger is also a striped animal.

4.3 Rules regarding arguments

1) The arguments given to prove the thesis must be true and not contradict each other.

2) Arguments must be a sufficient basis for proving the thesis.

3) Arguments must be judgments, the truth of which can be proven independently, regardless of the thesis.

4.4 Errors in the grounds (arguments) of evidence

1. Falseness of the grounds (“fundamental fallacy”). As arguments, they take not true, but false judgments that they pass off or try to pass off as true. The error may be unintentional. For example, before Copernicus, scientists believed that the Sun revolves around the Earth and, based on this false argument, built their theories. An error can also be deliberate (sophism) with the aim of confusing, misleading other people (for example, giving false testimony by witnesses or accused during a judicial investigation, incorrect identification of things or people, etc., from which then false conclusions are drawn).

2. “Anticipation of foundations.” The arguments are not proven, but the thesis is based on them. Unproven arguments only anticipate, but do not prove the thesis.

3. "Vicious circle." The mistake is that the thesis is justified by arguments, and the arguments are justified by the same thesis. For example, K. Marx revealed this error in the reasoning of D. Weston, one of the leaders of the English labor movement. Marx writes: “We therefore begin with the statement that the value of commodities is determined by the value of labor, and we end with the statement that the value of labor is determined by the value of commodities. Thus, we are truly spinning in a vicious circle and coming to no conclusion.”

4.5 Rule regarding the form of justification of the thesis (demonstration)

The thesis must be a conclusion that follows logically from the arguments according to the general rules of inference or obtained in accordance with the rules of indirect evidence.

4.6 Errors in proof form

1. Imaginary following. If the thesis does not follow from the arguments given in support of it, then an error occurs, called “does not follow”, “does not follow”. People sometimes, instead of correct proof, connect arguments to the thesis using the words “therefore”, “therefore”, “thus”, “as a result we have”, etc., believing that they have established a logical connection between the arguments and the thesis. This logical error is often unknowingly made by someone who is not familiar with the rules of logic and relies only on their common sense and intuition. The result is a verbal appearance of evidence.

2. From what is said with a condition to what is said unconditionally. An argument that is true only taking into account a certain time, relationship, measure cannot be presented as unconditionally true in all cases. So, if coffee is beneficial in small doses (for raising blood pressure, for example), then in large doses it is harmful. Similarly, while arsenic is added in small doses to some medicines, in large doses it is a poison. Doctors must select medications for patients individually. Pedagogy requires an individual approach to students. Ethics determines the norms of people's behavior, and in different conditions they can vary somewhat (for example, truthfulness is a positive trait of a person, but if he reveals a secret to the enemy, it will be a crime).

3. Violation of the rules of inference (deductive, inductive, by analogy):

A). Errors in deductive reasoning. For example, in a conditionally categorical inference, it is impossible to draw a conclusion from the statement of the consequence to the statement of the reason. So, from the premises “If a number ends in 0, then it is divisible by 5” and “This is a number. divisible by 5" does not lead to the conclusion: "This number ends in 0." Errors in deductive reasoning have been covered in detail previously.

b). Errors in inductive reasoning. “Hasty generalization”, for example, the statement that “all witnesses give biased testimony.” Another error is “after this - it means because of this” (for example, the loss of an item was discovered after being in this person’s house, which means he took it away).

V). Errors in inferences by analogy. For example, African pygmies incorrectly draw conclusions by analogy between a stuffed elephant and a living elephant. Before hunting an elephant, they arrange ritual dances, depicting this hunt, pierce a stuffed elephant with spears, believing (by analogy) that the hunt for a living elephant will be successful, i.e. that they will be able to pierce him with a spear.

5. The concept of sophistry and logical paradoxes

An unintentional mistake made by a person in thinking is called laralogism. Many people make paralogisms. A deliberate mistake with the aim of confusing one’s opponent and passing off a false judgment as true is called sophistry. Sophists are people who try to pass off lies as truth through various tricks.

In mathematics there are mathematical sophisms. At the end of the 19th - beginning of the 20th centuries. The book by V.I. was very popular among students. Obreimov “Mathematical sophisms”, which contains many sophisms. And a number of modern books contain interesting mathematical sophisms. For example, F.F. Nagibin formulates the following mathematical sophisms:

4) “All numbers are equal to each other”;

5) “Any number is equal to half of it”;

6) “A negative number is equal to a positive number”;

7) “Any number is equal to zero”;

8) “Two perpendiculars can be dropped from a point to a straight line”;

9) “A right angle is equal to an obtuse angle”;

10) “Every circle has two centers”;

11) “The lengths of all circles are equal” and many others.

2 = 5. You need to find the error in the following reasoning. We have a numerical identity: 4: 4 = 5: 5. Let’s take the common factor out of brackets in each part of this identity. We get 4 (1: 1) = 5 (1: 1). The numbers in brackets are equal. Therefore 4=5, or 2

5 = 1. Wanting to prove that 5 = 1, we will reason like this. From the numbers 5 and 1, we separately subtract the same number 3. We get the numbers 2 and - 2. When these numbers are squared, we get equal numbers 4 and 4. This means that the original numbers 5 and 1 must also be equal. Where is the error?

5.1 The concept of logical paradoxes

A paradox is a reasoning that proves both the truth and falsity of a certain proposition or (in other words) proves both this judgment and its negation. Paradoxes were known in ancient times. Their examples are: “Heap”, “Bald”, “Catalogue of all normal directories”, “Mayor of the city”, “General and the barber”, etc. Let’s consider some of them.

The Heap Paradox. The difference between a heap and a non-heap is not one grain of sand. Let us have a heap (for example, sand). We start taking one grain of sand from it each time, and the heap remains a heap. Let's continue this process. If 100 grains of sand are a heap, then 99 are also a heap, etc. 10 grains of sand - a heap, 9 - a heap, ... 3 grains of sand - a heap, 2 grains of sand - a heap, 1 grain of sand - a heap. So, the essence of the paradox is that gradual quantitative changes (decreasing by 1 grain of sand) do not lead to qualitative changes.

5.2 Paradoxes of set theory

In a letter to Gottlob Frege dated June 16, 1902, Bertrand Russell reported that he had discovered the paradox of the set of all normal sets (a normal set is a set that does not contain itself as an element).

Examples of such paradoxes (contradictions) are “Catalogue of all normal catalogues”, “Mayor of the city”, “General and the barber”, etc.

The paradox, called the “Mayor of a City,” is as follows: every mayor of a city lives either in his own city or outside of it. An order was issued to allocate one special city, where only mayors who did not live in their city would live. Where should the mayor of this special city live? A). If he wants to live in his city, then he cannot do this, since only mayors live there who do not live in their city, b). If he does not want to live in his own city, then, like all mayors who do not live in their cities, he must live in the allotted city, i.e. in his own. So, he cannot live either in his city or outside it.

Thus, logic includes the category of time, the category of change: we have to consider the changing volumes of concepts. And the consideration of volume in the process of its change is already an aspect of dialectical logic. The interpretation of the paradoxes of mathematical logic and set theory associated with violation of the requirements of dialectical logic belongs to S.A. Yanovskaya. In the example with the directory, it is possible to avoid a contradiction because the scope of the concept “catalog of all normal directories” is taken for some specific, precisely fixed time, for example, June 20, 1998. There are other ways to avoid contradictions of this kind.

6. The art of discussion

The role of evidence in scientific knowledge and discussions comes down to the selection of sufficient grounds (arguments) and to showing that the thesis of the proof follows with logical necessity.

The rules for conducting a discussion can be shown using the example of a youth debate. Dispute allows you to consider, analyze problem situations, develop the ability to defend your knowledge and your beliefs with arguments.

Disputes can be planned in advance or occur impromptu (during a hike, after watching a movie, etc.). In the first case, you can read the literature in advance and prepare; in the second, there is an advantage in emotionality. It is very important to choose the topic of the debate; it should sound sharp and problematic.

During the debate, 3-4 questions must be asked, but in such a way that no definite answers can be given to them.

There are different types of dialogue: argument, polemic, discussion, disputation, conversation, debate, quarrel, debate, etc. The art of arguing is called eristics (from the Greek - dispute), the branch of logic that studies the methods of argument is also called. In order for the discussion and dispute to be fruitful, i.e. could achieve their goal, certain conditions must be met. A.L. Nikiforov recommends remembering to comply with the following conditions when conducting a dispute. First of all, there must be a subject of dispute - some problem, a topic to which the statements of the participants in the discussion relate. If there is no such topic, the dispute turns out to be pointless and degenerates into a meaningless conversation. Regarding the subject of the dispute, there must be a real opposition between the disputing parties, i.e. the parties must hold opposing beliefs regarding the subject of the dispute. If there is no real divergence of positions, then the dispute degenerates into a conversation about words, i.e. opponents talk about the same thing, but using different words, which creates the appearance of discrepancy. There also needs to be some common basis for the dispute, i.e. some principles, provisions, beliefs that are recognized by both sides! If there is not a single provision that both parties would agree to, then the dispute turns out to be impossible. Some knowledge about the subject of the dispute is required: it is pointless to enter into an argument about something about which you have not the slightest idea. The conditions for a fruitful argument also include the ability to be attentive to your opponent, the ability to listen and the desire to understand his reasoning, the willingness to admit your mistake and the rightness of your interlocutor. A dispute is not only a clash of opposing opinions, but also a struggle of characters. The methods used in a dispute are divided into acceptable and unacceptable (i.e. loyal and disloyal). When opponents strive to establish the truth or achieve general agreement, they use only loyal techniques. If one of the opponents resorts to non-loyal methods, then this indicates that he is only interested in victory, achieved by any means necessary. You should not enter into an argument with such a person. However, knowledge of disloyal methods of argument is necessary: ​​it helps people expose their use in a particular dispute. Sometimes they are used unconsciously or in a temper; in such cases, an indication of the use of disloyal techniques serves as an additional argument indicating the weakness of the opponent’s position.

A.L. Nikiforov identifies the following loyal (acceptable) methods of argument, which are simple and few in number. It is important to seize the initiative from the very beginning: propose your own formulation of the subject of the dispute, a plan for discussion, and direct the course of the debate in the direction you need. In a dispute, it is important not to defend, but to attack. Anticipating the possible arguments of your opponent, you should express them yourself and immediately respond to them. An important advantage in a dispute is gained by the one who manages to place the burden of proof or refutation on the opponent. And if he has poor command of evidence, he may get confused in his reasoning and will be forced to admit defeat. It is recommended to concentrate attention and actions on the weakest link in the opponent’s argumentation, and not strive to refute all its elements. Loyal techniques also include the use of the effect of surprise: for example, the most important arguments can be saved until the end of the discussion. By expressing them at the end, when the opponent has already exhausted his arguments, you can confuse him and win. Loyal techniques also include the desire to take the last word in the discussion: by summing up the dispute, you can present its results in a light favorable to you.

Incorrect, disloyal techniques are used in cases where there is no confidence in the truth of the position being defended or even its falsity is realized, but nevertheless there is a desire to win the dispute. To do this, you have to pass off lies as truth, unreliable things as verified and trustworthy.

Most of the disloyal techniques are associated with a deliberate violation of the rules of evidence. This includes the substitution of the thesis: instead of proving or disproving one position, they prove or disprove another position, only apparently similar to the first. In the process of a dispute, they often try to formulate the opponent’s thesis as broadly as possible, and to narrow their own as much as possible. A more general proposition is more difficult to prove than a proposition of a lesser degree of generality.

A significant part of disloyal methods and tricks in a dispute is associated with the use of unacceptable arguments. Arguments used in a discussion, in a dispute, can be divided into two types: ad rem arguments (to the point, on the merits of the matter) and ad hominem arguments (to the person). Arguments of the first type are relevant to the issue under discussion and are aimed at substantiating the truth of the position being proven. Judgments about certified individual facts can be used as such arguments; definitions of concepts accepted in science; previously proven laws of science and theorems. If arguments of this type satisfy the requirements of logic, then the proof based on them will be correct.

Arguments of the second type do not relate to the essence of the matter, are not aimed at substantiating the truth of the position put forward, but are used only to win the dispute. They affect the personality of the opponent, his beliefs, appeal to the opinions of the audience, etc. From a logical point of view, all ad hominem arguments are incorrect and cannot be used in a discussion whose participants strive to clarify and substantiate the truth. The most common types of ad hominem arguments are:

1. Argument to personality - a reference to the personal characteristics of the opponent, his beliefs, tastes, appearance, advantages and disadvantages. The use of this argument leads to the fact that the subject of the dispute is left aside, and instead the personality of the opponent is discussed, and usually in a negative light. A variation of this technique is “labeling the opponent, his statements, his position. There is an argument to the individual with the opposite direction, i.e. referring not to shortcomings, but, on the contrary, to the merits of a person. This argument is often used in legal practice by defense attorneys for the accused.

2. Argument to autopumemy - a reference to the statement or opinions of great scientists, public figures, writers, etc. in support of his thesis. The argument to authority has many different forms: they refer to the authority of public opinion, the authority of the audience, the authority of the opponent, and even their own authority. Sometimes they invent fictitious authorities or attribute to real authorities such judgments that they never expressed.

3. Argument to the public - a reference to the opinions, moods, feelings of listeners. A person using such an argument no longer addresses his opponent, but those present or even random listeners, trying to attract them to his side and with their help exert psychological pressure on the enemy. One of the most effective types of argument to the public is a reference to the material interests of those present. If one of the opponents manages to show that the thesis defended by his opponent affects his financial situation, income, etc. those present, their sympathy will undoubtedly be on the side of the first.

4. An argument for vanity - lavishing excessive praise on an opponent in the hope of making him softer and more flexible. Expressions like: “I believe in the deep erudition of my opponent,” “The opponent is a person of outstanding merit, etc.,” can be considered veiled arguments for vanity.

5. Argument to force (“to the stick”) - a threat of unpleasant consequences, in particular the threat of using or direct use of any means of coercion. Any person endowed with power, physical strength or armed is always tempted to resort to threats in a dispute with an intellectually superior opponent. However, it should be remembered that consent extracted under the threat of violence is worth nothing and does not oblige the consenter to anything.

6. Argument for pity - arousal of pity and sympathy on the other side. This argument is unconsciously used by many people who have acquired the habit of constantly complaining about the hardships of life, difficulties, illnesses, failures, etc. in the hope of awakening in listeners sympathy and a desire to give in, to help in some way.

7. Argument to ignorance - the use of facts and provisions about which the opponent knows nothing, a reference to works that he, as is known, has not read. People are often afraid to admit that they don’t know something, believing that they are allegedly losing their dignity. In a dispute with such people, the argument of ignorance works flawlessly. However, if you are not afraid to admit that you don’t know something and ask your opponent to tell you more about what he is referring to, it may turn out that his reference has nothing to do with the subject of the dispute.

All of the above arguments are incorrect and should not be used in a strictly logical and ethically correct dispute. Having noticed an argument of this kind, you should point out to your opponent that he is resorting to incorrect methods of arguing, and therefore is not confident in the strength of his positions. A conscientious person will have to admit that he was mistaken. It is better not to enter into an argument with an unscrupulous person at all.

Bibliography

1. Getmanova A.D. Logics. M., 2002

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3. Ivlev Yu.V. Logics. M., 2002

4.Getmanova A.D. Logics. M., 2011

5.Kirillov V.I., Starchenko A.A. Logics. M., 2002.