Argumentation theory, or argumentation, is the interdisciplinary study of how conclusions can be reached through a series of logical reasoning; that is, claims based, firmly or not, on premises. It includes the arts and sciences of civil debate, dialogue, conversation, and persuasion. She studies the rules of inference, logic and procedural rules in both artificial and real world settings.

Argumentation includes debate and negotiation to reach mutually acceptable conclusions. It also covers eristics, a branch of public debate in which victory over a rival is the primary goal. This art and science is often the means by which people defend their beliefs or personal interests in rational dialogue, in vernacular, and in the process of argument.

Reasoning is used in law, for example, in research, in preparing arguments to be presented in court, and in testing the soundness of certain types of evidence. In addition, scholars study post hoc rationalizations, in which organizational actors try to justify decisions they made irrationally.

Key Components of Argumentation

• The ability to understand and identify the explicit or implied arguments and goals of participants in various types of communication
• Identification of the premises from which the conclusions are drawn
• Establishing the "burden of proof" - determining who made the original statement, and thus is responsible for providing evidence why his/her position deserves recognition.
• To convince the opponent that you are right, you must provide weighty arguments in favor of your position. The method by which this is achieved is by finding reasonable, sounding and convincing arguments that do not have flaws that are not easy to refute.
• In a discussion, the fulfillment of the "burden of proof" creates the "burden of objection". The opponent attempts to find inaccuracies in the arguments in order to refute them, to provide counterexamples if possible, to expose any fallacies, and to show why a valid conclusion cannot be drawn from the reasoning provided by the burden of proof's arguments.

The internal structure of the argument

Usually, the argumentation has an internal structure, including the following points:

1. set of assumptions or premises (thesis)
2. method of reasoning or logic (arguments)
3. conclusion or summary (demo)

An argument must have at least two premises and one conclusion.

In its most common form, argumentation involves the interlocutor and/or opponent participating in a dialogue where each disputant defends his position and tries to convince the other. Other types of dialogue, besides persuasion, are the art of controversy, information seeking, inquiry, negotiation, discussion, and the dialectical method (Douglas Walton). The dialectical method was made famous by Plato and his use of the Socratic method, the critical interrogation of various characters and historical figures.

Argumentation and the foundations of knowledge

The theory of argumentation has its origins in fundamentalism, in the theory of knowledge (epistemology) in the field of philosophy. She sought to find grounds for claims in the logic and actual laws of the universal system of knowledge. But the arguments of scientists gradually rejected the systematic philosophy of Aristotle and the idealism of Plato and Kant. They questioned, and eventually abandoned, the idea that the premises of an argument derive their validity from formal philosophical systems. And so the field expanded.

Types of argumentation

Conversational Argument

The study of the nature of conversation arose from the field of sociolinguistics. It is commonly referred to as conversion analysis. Inspired by ethnomethodology, it was developed in the late 1960s and early 1970s, mainly by the sociologist Harvey Sachs and, in particular, by his close associates Emanuel Scheglof and Gail Jefferson. Sachs died early in his career, but his work was carried on and conversational reasoning gained acceptance in sociology, anthropology, linguistics, and psychology. It is particularly influential in integrative sociolinguistics, discourse analysis and discursive psychology, and is a coherent discipline in its own right. Recently, the methods of sequential analysis of colloquial argumentation have been used to investigate the subtle details of phonetic speech.

The empirical research and theoretical formulations of Sally Jackson and Scott Jacobs, and several generations of their students, have described argumentation as a form of managing conversational disagreement within contexts and communication systems that agreement naturally favors.

Mathematical reasoning

The basis of mathematical truth has been the subject of much debate. Frege, in particular, sought to demonstrate (see Frege, Foundations of Arithmetic, 1884, and Logicism in the Philosophy of Mathematics) that arithmetical truths can be deduced from purely logical axioms and therefore, after all, logical truths. The project was developed by Russell and Whitehead in their Principia Mathematica. If an argument can be given as propositions in symbolic logic, then it can be verified by applying recognized proof procedures. This work was carried out for arithmetic using Peano's axioms. Be that as it may, an argument in mathematics, as in any other discipline, can be considered valid only if it can be shown that it cannot have true premises and a false conclusion.

Scientific reasoning

Perhaps the most radical statement of the social foundations of scientific knowledge appears in Alan G. Gross's The Rhetoric of Science (Cambridge: Harvard University Press, 1990). Gross believes that science is rhetorical "without a trace", which means that scientific knowledge in itself cannot be considered as an idealized basis of knowledge. scientific knowledge is rhetorically worked out, which means that it has special epistemic power only in so far as its communal methods of verification can be trusted. This thinking represents an almost complete rejection of the fundamentalism upon which the argument was first based.

Explanatory reasoning

Explanatory reasoning is a dialogical process in which participants explore and/or resolve interpretations often in the text of any medium containing significant differences in interpretation.

Explanatory reasoning pertains to humanities, hermeneutics, literary theory, linguistics, semantics, pragmatics, semiotics, analytical philosophy and aesthetics. Topics in conceptual interpretation include aesthetic, judicial, logical and religious interpretations. Topics in scientific interpretation include scientific modeling.

Legal reasoning

Legal arguments are heard in the speech of a lawyer in court or in the speech of the court of appeal, or in the presentation of parties representing themselves and justifying legally why they should prevail. An oral argument at the appellate level is accompanied by a note, which is also argued in advance by each of the parties to the legal dispute. The final argument is a closing statement by each party's lawyer reiterating the important arguments in court case. The closing speech is given after the evidence has been presented.

Political reasoning

Political arguments are used by academics, media professionals, candidates for political office, and government officials. Political arguments are also used by citizens in ordinary relationships to comment on and understand political events. The rationality of the public is one of the main questions in this line of research. Political scientist Samuel L. Popkin coined the term "low-informed voter" to describe the majority of voters who know very little about politics or the world at large.

Psychological aspects

Contribution by Stephen E. Toulmin

By far the most influential theorist was Stephen Toulmin, who was educated as a philosopher at Cambridge as a student of Wittgenstein. What follows is a sketch of his ideas.

An alternative to absolutism and relativism

Toulmin argued that absolutism (in the face of theoretical or analytical arguments) was of limited practical value. Absolutism comes from Plato's idealized formal logic, which stands for universal truth. Thus, it is believed that the moral issues of absolutism can be resolved by adhering to standard sets of moral principles, regardless of context. On the contrary, Toulmin argues that many of these so-called standard principles are not relevant to the real situation that a person faces in Everyday life.

To describe his vision of everyday life, Toulmin introduced the concept of a field argument. In Ways to Use Argumentation (1958), Toulmin argues that some aspects of an argument differ from field to field and are hence called "field-dependent", while other aspects of an argument are the same for all fields and are called "field-invariant". ". According to Toulmin, the disadvantage of absolutism lies in its ignorance of the "invariant" aspect of the argument, absolutism admits that all aspects of the argument are "field-dependent".

Recognizing the omissions inherent in absolutism, Toulmin in his theory avoids the shortcomings of absolutism by not resorting to relativism, which, in his opinion, does not provide grounds for separating moral and immoral arguments. In Human Understanding (1972), Toulmin argues that anthropologists were swayed to the side of the relativists because they were the ones who drew attention to the impact of cultural change on rational reasoning, in other words, anthropologists and relativists give too much great importance the importance of the "field-dependent" aspect of the argument, and are unaware of the existence of the "invariant" aspect. In an attempt to solve the problems of absolutists and relativists, Toulmin in his work develops standards that are neither absolutist nor relativistic and will serve to assess the value of ideas.

Toulmin believes that a good argument can be successful in verification and will be resistant to criticism.

Argument Components

In Ways to Use Argumentation (1958), Toulmin proposed a layout containing six interrelated components for argument analysis:

1. Approval: Approval must be complete. For example, if a person tries to convince the listener that he is a British citizen, then his statement will be "I am a British citizen" (1)
2. Data: Facts that are cited as the basis for a statement. For example, a person in the first situation can support his statement with other data "I was born in Bermuda" (2)
3. Reasons: A statement that allows you to move from the evidence (2) to the statement (1). In order to move from evidence (2) "I was born in Bermuda" to the statement (1) "I am a British citizen" a person must use grounds to bridge the gap between statement (1) and evidence (2) by stating that "A person born in Bermuda can legally be a British citizen."
4. Support: Additions aimed at confirming the statement expressed in the grounds. Support should be used when reasons alone are not convincing enough for readers and listeners.
5. Rebuttal / Counterarguments: A statement showing the limitations that may apply. An example of a counterargument would be: "A person born in Bermuda can legally be a British citizen only if he has not betrayed Britain and is not a spy for another country."
6. Qualifier: Words and phrases expressing the author's degree of confidence in his statement. These are value judgments, such as, "probably", "possibly", "impossible", "certainly", "presumably" or "always". The statement "I am definitely a British citizen" carries a much greater degree of certainty than the statement "I am presumably a British citizen".

The first three elements, 'assertion', 'evidence' and 'reasons', are seen as the main components of practical argumentation, while the last three 'qualifier', 'support' and 'refutations' are not always necessary.

Toulmin did not expect this scheme to be applied in the field of rhetoric and communication, since originally this argumentation scheme was to be used to analyze the rationality of arguments, usually in a courtroom; in fact, Toulmin had no idea that this scheme would apply to the field of Rhetoric and Communication until his work was introduced by Wayne Brockriede and Douglas Ehninger. Only after posting Introduction to reasoning(1979) this scheme has gained acceptance.

evolutionary model

In Human Understanding (1972), Toulmin argues that the development of science is an evolutionary process. This book criticizes Thomas Kuhn's point of view regarding the conceptual change in the structure scientific revolutions. Kuhn believed that conceptual change is a revolutionary (as opposed to evolutionary) process in which mutually exclusive paradigms compete with each other. Toulmin was critical of Kuhn's relativistic ideas and was of the opinion that mutually exclusive paradigms do not provide a basis for comparison, in other words, Kuhn's statement is a relativist error, and it consists in excessive attention to the "field - dependent" aspects of the argument, while simultaneously ignoring, "field - invariant or the commonality shared by all arguments (scientific paradigms).

Toulmin proposes an evolutionary model of conceptual development comparable to Darwin's model of biological evolution. Based on this reasoning, conceptual development includes innovation and choice. Innovation means the emergence of many variants of theories, and selection means the survival of the most stable of these theories. Innovation occurs when professionals in a particular field begin to perceive familiar things in a new way, not as they were perceived by their predecessors; selection exposes innovative theories to a process of discussion and exploration. The strongest theories that have been discussed and researched will take the place of traditional theories, or additions will be made to traditional theories.

From the point of view of absolutists, theories can be either reliable or unreliable, regardless of the context. From the point of view of relativists, one theory can neither be better nor worse than another theory from a different cultural context. Toulmin is of the opinion that evolution depends on a process of comparison that determines whether a theory will be able to improve standards better than another theory can.

Argumentation (from Latin argumentatio - bringing an argument) is a logical process during which the truth of a position is derived from the truth of an argument. Argument is an integral part of any proof.
The logical structure of the proof consists of three interrelated elements: 1) thesis; 2) arguments; 3) demonstrations.
When submitting a thesis, you must adhere to the following rules:
1) the thesis must be clearly formulated;
2) the thesis must remain unchanged throughout the proof;
3) the thesis should not contain a logical contradiction.

Types of evidence in oratory

Oratory uses direct and indirect evidence.
A proof is called direct, in which the truth of the thesis is justified by arguments without the help of additional constructions. For example, the dating of an old Russian manuscript of the 10th-12th centuries. can be proved by pointing out that it was written by an early charter (this type of writing was not used at other times); the presence of the suspect at the scene of the crime is proved by fingerprints.
Indirect evidence is usually realized in an apagogic version, which means “proof from the contrary” in this case, in addition to arguments in defense of the thesis put forward, an antithesis is also given: the proof “is carried out by establishing the falsity of a judgment that contradicts the thesis”. In public speech, circumstantial evidence has a greater effect than direct evidence in the forehead.
The polemic also uses this kind of apagogic proof, reductioabsurdum - reduction to absurdity. This is proof of the impossibility, the absurdity of the assumption of something. The famous philosopher I. Kant believed that the reproach of absurdity is always a personal censure that must be avoided, especially when refuting delusions.
In the process of proof, the speaker uses various arguments - arguments given in support of the thesis.
It is important to consider the logic requirements for arguments:
1) the arguments must be true (false arguments cannot prove the main idea);
2) the arguments must be sufficient for this thesis;
3) the truth of the arguments must be proven regardless of the thesis.
If the thesis is proved by false arguments, passing them off as true, a logical error arises - “false reason”. This mistake may be unintentional, but often it is deliberately used by unscrupulous politicians and economists. Then the juggling of numbers, distortion of statistical data, reference to non-existent documents is used. The error of "false reason" has a direct connection with another logical error - "anticipation of the reason." An unproven judgment is taken as an argument, it is used as a basis for a conclusion. This proposition cannot be considered deliberately false, but it itself needs proof in order to clarify the truth.
The error called "lack of reason" leads to a violation of the second rule of argumentation. The arguments given for the proof do not look convincing enough. Yes, to solve environmental issues it is not a set of thoughtful measures that is proposed, but another “lifeline” (for example, the replacement of project managers).
A logical fallacy, called the "vicious circle", occurs if the thesis is proved by arguments that are derived from the same thesis. Here is a convincing illustration of such a mistake.
AT Russian Academy Sciences Lomonosov asked Schumacher a question about why there are few Russian students in the academy. Schumacher explained this by the small number of professors who speak Russian. When Lomonosov was indignant at the fact that there were few Russian professors in the academy. Schumacher referred to the small number of Russian students.
Arguments are usually divided into two groups: logical (in ancient times they were called "arguments to the case") and psychological ("arguments to the person"). It is also traditional for rhetoric to distinguish between artificial and natural evidence. Artificial proofs are logical, they require the ability to reason. They resort to them in the absence of witnesses. Natural evidence is facts, documents.
Testimony that convinces in itself. Natural evidence is the most weighty, as it is based on eyewitness accounts, documents (forensic examination, doctor's certificate, protocol record of witnesses). This evidence is irrefutable when reconstructing events and evokes a heated emotional response from listeners.
Indirectly, documents can be presented as evidence. And then the facts reflected in the documents are used as part of the reasoning that is used in artificial evidence.
In the development of the theory of logical argumentation, Aristotelian syllogistics, the science of deriving inferences, played an important role.
Syllogism is a type of deductive reasoning. It consists of two categorical judgments. The first of them is the "large premise", the second is the "smaller premise". From these two judgments a new judgment is derived, which is called a conclusion. For example: Every experienced speaker knows how to convince his listeners, and Metropolitan Kirill has the gift of persuasion, therefore, he can be considered an experienced speaker. If we write down the syllogism, then the first two judgments are located one below the other, and the new one - the conclusion is separated from them by a line and written under them:
An experienced speaker knows how to convince listeners (big premise).
Metropolitan Kirill knows how to convince his listeners (minor premise).
Metropolitan Kirill is an experienced orator (conclusion).
The horizontal line between the second and third lines of the syllogism is the "signal" of the transition from premises to conclusion. Both premises serve as grounds on the basis of which it can be argued that the conclusion is true.
A simple categorical syllogism can often be found in everyday and business speech. We use syllogisms without noticing it and without thinking about this phenomenon. Here's a compelling example from a televised debate.
The facilitator states: “Every professional manager has the right to lead an enterprise. Sergey Petrov is a professional manager, therefore, he has the right to head a large enterprise.”
In this business speech, we see a clear construction of a simple categorical syllogism.
Simple reasoning Aristotle called enthymeme. An enthymeme is a categorical syllogism in which one of the premises or a conclusion is omitted. According to Aristotle, enthymemes form the basis of belief. They are more characteristic of rhetorical argument, since people usually do not speak in complete syllogisms. Speech saturated with them would be difficult to perceive, monotonous and inexpressive, and reasoning based on enthymeme makes speech lively and dynamic. So. instead of a complete syllogism about the orator Metropolitan Kirill, one can construct an enthymeme: Metropolitan Kirill is an experienced orator and therefore can convince listeners. Comparing the full syllogism with this enthymeme, we note that a large premise is missing: there is no line that an experienced (thinking) speaker can convince listeners. If we skip the smaller premise, we get the following scenario: A thinking orator knows how to convince his listeners, while Metropolitan Kirill is an experienced orator. The enthymeme can be read without a conclusion; let's omit it: An experienced orator knows how to convince his listeners, and Metropolitan Kirill also knows how to convince.
An enthymeme is an abbreviated syllogism. It is possible to omit any of its constituent parts provided that the content of the statement has not changed and is clear to the listeners.
Usually, an enthymeme is a well-known provision that does not require proof and comments. This explains the use of abbreviated syllogisms in everyday speech. For the sake of persuasiveness of what has been said, let us turn to the judgment from the TV debates. Let's represent it as a series of separate enthymemes.
1. Sergey Petrov - professional manager, has the right to manage the enterprise. (Large package missing.)
2. Every professional manager has the right to lead the enterprise, and Sergey Petrov has the right to lead. (Less premise omitted.)
3. Each professional manager has the right to manage the enterprise, and Sergey Petrov is a professional manager. (Conclusion omitted.)
We constantly hear such proposals - enthymemes in business, deliberative speech, in everyday life.
The logical order of argumentation is based on the fact that the proof is built in the form of a chain of inferences leading to a single conclusion.

Argumentation methods

Logical methods of presenting material can be classified as inductive and deductive. Deductive we call such a development of the message, when they go from the general to the particular. In philosophy and rhetoric, this method is considered as a method of searching for confirmations of the earlier generalization. The speaker has the opportunity to invite the audience to go through the path of knowing some particular through the general. Deduction as a method of presentation presupposes such a construction that leads from the effect to the cause (the principle of the composition of a detective novel). This is its attraction to the audience.
The inductive method of presentation involves the movement of thought from the particular to the general. The path to generalization runs through a series of single or particular factors. Induction as a form of inference also originated in antiquity. As Aristotle emphasized, "induction is convincing and simple, and from the point of view of sensory knowledge is more advantageous and accessible." In rhetoric and philosophy, induction is called the method of anticipating a foundation. This means that special cases separate facts lead to a certain pattern (logical basis). The inductive method of presentation from facts to generalization activates the attention of listeners, therefore it is indispensable in the conditions of oral propaganda, rally speeches. Effective induction for speeches in an insufficiently prepared audience.
An inference very close to induction is an analogy. Analogy is translated from Greek as "similarity, likeness". The analogy method involves a comparison of facts, phenomena, events. The new can be comprehended, understood only through the images of the old, known. Analogies play a huge role in cognition, as they lead to the formation of hypotheses, i.e. scientific assumptions and conjectures. An analogy with what is already known helps to understand what is unknown. An analogy sometimes allows you to tell more about an object than its longest description.
A classic example of the analogy of reasoning about life on Mars.
There is much in common between Mars and Earth: these are planets solar system, both have water and an atmosphere, almost the same temperature on their surface, etc. Since Mars is similar to Earth in terms of the conditions necessary for life, it means that life is possible on Mars.
However, reasoning by analogy still does not give reliable knowledge.
When using analogy in speech, the following rules must be observed:
1) analogy should be based not on superficial features, but on the essential properties of objects, phenomena;
2) general properties should characterize phenomena from different angles;
3) it is necessary to establish not only the similarity, but also the features of the difference of phenomena.

Rules for Effective Argumentation

To make an argument convincing, the speaker should operate with facts. "The most reliable kind of argument is facts."
Facts that are close and understandable to people are perceived by them as reliable information. An experienced speaker knows how to present factual material. What conditions for the effectiveness of a fact must be taken into account in public speaking? First of all, it is the reliability of the factual material. The rhetorician must cultivate the habit of carefully checking and rechecking the truth of the facts. The use of unverified, inaccurate or false information undermines the credibility of the speaker and his speech. The second important condition for the professional use of facts is their consistency. Facts only in their totality help to reveal this or that phenomenon, to show its organic connection with reality. Let us turn to the sad experience of propaganda in the USSR.
The media reported on the success of agriculture, and collective farmers fled from the villages to the city. The most necessary goods became scarce, the housing problem became more and more difficult to solve, and the people were told about the growth in the consumption of meat, milk, and “inflated” figures were given about the commissioning of square meters of housing. All this false information given in the media. couldn't convince people.
It is necessary to take into account the appropriateness of using factual material, i.e. its correspondence to the interests, educational level of students, connection with everyday practice. The audience always has a special interest in a speech that talks about a local business, familiar people, problems. The relevance of a fact is manifested in its logical connection with the problem under discussion.
The completeness and evidence of the facts is the main task of the orator, who makes sure that his speech is convincing. Usage good examples helps to activate the consciousness of the listeners and will give the speech persuasiveness, because often people remember what was discussed only by examples, forgetting other fragments of the speech.
Referring to the facts, the speaker often operates with figures: they are also used as strong evidence. The modern audience is accustomed to the language of numbers, so statistics should be recognized as one of the most important means of proof. But the speaker needs to skillfully introduce figures into speech, not to abuse them. Statistical tables should not be given in speeches, because any table is designed for visual perception. Experienced speakers suggest rounding numbers, then they are better perceived by ear. You should not just name or list the numbers, you need to present them brighter, more convincingly. To do this, the numbers need to be analyzed, compared, weighed.
The role of convincing arguments can be performed by diagrams, illustrations, photographs, posters, tables. A well-designed diagram or table will help the speaker explain in a few minutes what would take an hour and a large number of words. Visual aids will help the audience to “see” the phenomena and processes behind the numbers.

Argumentation is a speech procedure that serves to substantiate a certain statement using other statements. Argumentation has two aspects - logical and communicative.

Logically argumentation acts as a substantiation of a certain statement (thesis) with the help of other statements (reasons, arguments, arguments). This way of reasoning is characteristic of science. Outside of science, the thesis and arguments may be based on religious faith, the strength of tradition, the opinion of authority, etc.

In terms of communication argumentation is the process of interaction between the argumentator (the person who justifies something) and the recipient (the person to whom the justification is addressed). The ultimate goal of this process is the formation of some belief. Argumentation achieves this goal if the recipient has perceived, understood and accepted the thesis of the argumentator.

The main elements of the logical structure of argumentation are thesis, arguments and demonstration.

Thesis- this is a statement that is justified in the process of argumentation, that which is argued. It is the main element of the argument. Someone's opinion, a hypothetical answer to a question, etc. can be accepted as a thesis. In all cases, the thesis is something that goes beyond the generally accepted in this community, so there is a need for its argumentation.

Arguments(reasons, arguments) - these are statements that are used in the argumentation, what this thesis is argued with. Arguments serve as the foundation of argumentation.

In scientific argumentation, the following types of arguments are distinguished:

1. sayings about verified facts - knowledge about events or phenomena established with the help of direct perception or experimental study of the subject of science.

2. Definitions- statements that involve the expression of an unknown name through known ones, so they must be true.

3. Axioms- provisions that are not proven in science, but are accepted as true when substantiating its other provisions. Their truth is confirmed by centuries of practice. Axiomatic
some provisions of mathematics, mechanics, physics, logic, etc. have a character.

If the argument is built on the basis of axioms, certain logical requirements are imposed on them:

]). The chosen system of axioms must be consistent, i.e., relying on it, it is impossible to prove any statement and the negation of this statement at once.

2). The system of axioms must be complete, i.e., all the true propositions of a given science can be derived from it.

3). The axioms must be independent, i.e., none of the axioms can be derived from other axioms of the same science.

4. Previously proven positions of science(laws, theorems).

The logical connection between arguments and thesis is called demonstration(lat. demonstratio - show). At deductive demonstrations the thesis necessarily follows from the arguments, its truth is guaranteed. At inductive demonstration (when the thesis of general content is substantiated by particular cases, examples) demonstration in the form of analogy, comparison, etc. ensures the probabilistic nature of the conclusion.

Types of argumentation are distinguished according to various criteria:

1) by the nature of the argument expressing reliable or hypothetical knowledge (proof, refutation, explanation, confirmation);

2) according to the specifics of the demonstration (deductive and non-deductive arguments);

3) by goal (scientific - achieving the truth, business - finding a mutually acceptable solution, controversy - a dispute for the sake of victory);

4) according to the form of conduct (calm exchange of views - report, lecture, conversation; dispute - debate, discussion, quarrel, etc.).

Consider proof and refutation as the main types of argumentation.

Proof - a type of argumentation in which the truth of a thesis is logically deduced from arguments whose truth has already been established. The proof is widely used in science in the study of objects, their properties and relationships, the knowledge of which excludes empirical procedures. For example, the American astronomer Lovell calculated the orbit of an unknown planet, which was discovered 14 years later and named Pluto.

Evidence by way of implementation is direct or indirect.

Direct is called a proof in which the thesis necessarily follows from the arguments found. For example, the proof that 1992 was a leap year is based on the following arguments:

1) a leap year is a year in numerical terms whom
tens with ones are divisible by 4;

2) 92 is divisible by 4, so 1992 is a leap year
year.

The conclusion was made on the basis of the definition and one true statement taken as arguments of the proof.

Indirect called proof, in which the truth of the thesis follows from the established falsity of the statement (statements) that is in a certain connection with the thesis.

The most common types of indirect evidence are apagogic and divisive.

At apagogic evidence the truth of the thesis is established by establishing the falsity of the position that contradicts it, i.e. antithesis. AT mathematical sciences apagogical proof is called “proof by contradiction” (the name is inaccurate, since the truth of the thesis being proved is derived from the falsity of not the opposite, but the statement that contradicts it).

The general form of an apagogic proof is as follows. It is necessary to prove the thesis A. We assume that the antithesis is not - A; from it we obtain as a consequence some statement B. We establish that B contradicts the truth of the previously proven statement, therefore, is false; from the falsity of the consequence B we conclude about the falsity of its foundation, i.e., the antithesis not - A. On the basis of the law of the excluded middle from the falsity of not - A we conclude that the statement A is true, which was the purpose of the proof.

The logical scheme of the apagogic proof corresponds to the negative mode of the conditionally categorical syllogism:

If not A, then B.

Therefore, not non-A.

Not non-A is equivalent to A, therefore A is proved.

Let's turn to an example and consider the proof of the geometric theorem: "Two perpendiculars to the same line cannot intersect, no matter how much they continue." To prove it, we formulate a statement that contradicts the theorem: "Two perpendiculars to the same line intersect when continuing." A consequence of this assumption will be the statement that from a point lying outside a line, two perpendiculars can be lowered onto this line. But this corollary is false, since the theorem was previously proved that "from a point lying outside a straight line, only one perpendicular can be dropped onto this straight line." The falsity of the conclusion testifies to the falsity of the antithesis, and the falsity of the antithesis testifies to the truth of the thesis.

At parting proof the falsity of all members of the disjunctive (disjunctive) statement is established, except for one, which is the thesis being proved. If, for example, it is established that there was a crime that could only be committed by persons A, B, C, and if, in addition, it is established that neither B nor C committed it, then it is proved that the crime was committed by person A The separative proof is built according to the negative-affirming mode of the separative-categorical syllogism and is correct if the rules of this mode are observed:

A or AT or with.

Not B and not C.

Therefore, A.

Refutation establishes the falsity of the thesis of some statement. It is a special case of proof, since it is a process of substantiating the truth of the negation of the original statement.

There are three ways to refute:

1) refutation of the thesis (direct and indirect);

2) refutation of arguments;

3) refutation of the demonstration.

At direct rebuttal the thesis, first an assumption is made about the truth of the refuted thesis, and consequences are derived from it. If at least one of the consequences does not correspond to reality, i.e., is false, then the refuted thesis will also be false. Refutation by establishing the falsity of the consequences arising from the thesis is known as "reduction to the absurd".

At indirect refutation of the thesis, the truth of the antithesis is proved. According to the law of contradiction, the truth of the latter means the falsity of the thesis.

Refutationarguments is expressed in what indicates the falsity or inconsistency of the grounds. The falsity of the arguments does not mean the falsity of the thesis. The logical scheme for the refutation of arguments has the form

If A, then B.
Not A ________

Probably not AT.

Demonstration rebuttal lies in the fact that it indicates a violation of the rules of inference, according to which the proof of the thesis is built. But this does not mean that we refute the thesis itself. There are many examples where a true proposition was considered strictly proven, although over time there were errors in the proof.

The listed methods of refuting the thesis, arguments, demonstrations are often used not in isolation, but in combination with each other. With the help of refutation, science is freed from false claims and delusions.

Argumentation is genetic, indirect and direct.

In genetic argumentation, the conclusion about the truth or falsity of the thesis is obtained from arguments that show the origin of the thesis and the way in which the thesis has come down to us. Therefore, the arguments in these types of argumentation do not concern the essence of the content of the thesis, but speak only about the sources and ways of conveying the thesis.

In genetic proofs and confirmations, the conclusion about the truth of the thesis is obtained from arguments indicating that some judgment, received from a reliable source, was transmitted without distortion and has come down to us in its original form. In genetic refutation and criticism, the conclusion that the thesis is false is justified by the fact that the thesis does not agree with information received from a reliable source and has come down to us in a reliable way, or by the established fact that during the transmission the original judgment was distorted.

Genetic argumentation is used more often in cases where it is not possible to justify on the merits. AT historical sciences it is the predominant type of argumentation. The testimony of eyewitnesses or participants in some events, recorded in the annals, memoirs, and other documents are considered as a source of the thesis. And the first argument in favor of its truth is a judgment about the credibility of this source. (Sometimes such a judgment itself needs justification, in which case its justification is achieved in a new process of proof, which can be either genetic or substantive.)

Then the entire path of information transfer is traced - the content of the thesis, and each step of the transfer is evaluated from the point of view of the possibility of distorting this information. Here arguments are interspersed with demonstration, and the process of proof ends with the conclusion that in the course of transmission the information could not have undergone significant changes, so the thesis can be considered true or at least quite probable.

Reliable substantiation of the thesis by genetic means is possible only in the simplest arguments, such as everything that is written there is true (false). It's written right there. So it is true (false). In other types of genetic argumentation, the truth value of the thesis is justified only partially.

Genetic refutation and criticism is the rationale for judgment. The thesis is not consistent with information received from a reliable source and has come down to us without distortion. If sources or transmitters were used in the genetic argument, questionable, then the refutation fails. Unreliable or unsuccessful genetic evidence cannot be regarded as a genetic refutation. The thesis in this case is only unfounded (given genetic evidence), but not refuted.

A much more common type of argumentation than the genetic type is substantive argumentation, where the arguments relate to the content of the thesis. In substantive argumentation, the content of the thesis itself is subjected to analysis, and not its origin. This type of argumentation has a more interesting logical aspect, so in modern logic only substantive arguments are considered, and only they are meant when talking about argumentative processes. Argumentation on the merits is also practically more important, because its persuasive effect is greater than the genetic one, because only argumentation on the merits can be quite demonstrative.

Argumentative processes can also be direct and indirect.

Direct argumentation is called, in which the arguments and demonstration are addressed directly to the thesis, i.e. demonstrates the connection of arguments directly with the thesis, and not with any other judgment.

In direct argumentation, the thesis is directly derived from the arguments as a conclusion (if we are talking on the justification of its truth) or its incompatibility with the arguments (when substantiating its falsity) is established.

Argumentation is called indirect if the arguments are not aimed at the thesis, but at the refutation or confirmation of another judgment or several judgments that are alternative to the thesis.

Mediating judgments can be assumptions that look like antitheses. Then indirect argumentation is called argumentation by contradiction. Antithesis is a judgment that contradicts the thesis, i.e. a judgment whose truth necessarily entails the falsity of the thesis, while its falsity reliably testifies to the truth of the thesis.

If we are talking about proving a thesis, then the falsity of the antithesis must be demonstrated. Argument and demonstration in this case aim to show that the antithesis is false. After justifying the falsity of the antithesis, the last step of the demonstration is taken: since the antithesis is false, then the thesis is true. Since arguments are true propositions, it means that the only way to detect the falsity of an antithesis is to show its incompatibility with the arguments. This incompatibility may manifest itself in the direct contradiction of one or more arguments to the antithesis or its consequences.

In other cases, the incompatibility of antithesis and arguments is manifested in the fact that arguments with antithesis lead to two statements that contradict each other (in this case, indirect evidence is called apagogic, or evidence by reduction to absurdity).

An example of an apagogic proof is a proof of the correctness of a conclusion in an abbreviated tabular way; the thesis of this proof is a judgment: this conclusion is correct, i.e. between its premises and conclusion there is a relation of logical consequence. Antithesis - the assumption that the conclusion is wrong i.e. that the premises can be true and the conclusion false. Then, from this assumption, consequences are deduced regarding the truth values ​​of the subformulas. Proof arguments are tabular definitions of logical connectives. Having received contradictory consequences that the same subformula must be both true and false, one concludes that the antithesis is false, and then the thesis is true.

Another type of indirect argumentation is disjunctive.

An example of disjunctive argumentation is the proof by the method of analytical tables of the assertion that a certain propositional logic formula is actually feasible. It is known that each formula is identically true, actually feasible, or identically false. Using the method of analytic tables, it is directly impossible to prove that the formula is actually feasible, but it is possible to establish the identical truth and identical falsity of the formula. If it is proved that it is not identically - true and not identically - false, then it is proved that it is actually feasible.

Proof by contradiction can also be considered as dividing with two members of a strict disjunction (the logical equivalent of the union "or"; an operation formalizing the basic logical properties of this union): the thesis and antithesis can neither be true nor false together. However, proofs to the contrary are specific. In them, before the beginning of the proof, one proposition is declared a thesis, the other is a conditional assumption, an antithesis, and all operations with antithesis are aimed at substantiating the truth of the thesis. In disjunctive evidence, it is often not initially known which of the alternative judgments will be confirmed, therefore, in many cases it is impossible to call one of them a thesis until the end of the proof process. All alternative judgments from the beginning act as equal, although, perhaps, the prover prefers one of them. Only successive refutations of all alternatives, except for one, reveal the true one, which is subsequently declared the thesis of this proof. Divisive proof often acts, first of all, as a way of finding the truth, and only then - as a way of demonstrating it.

In the main part of both informing and argumentative speech, arguments (argument, proof) are used, so these two types of speeches are very close to each other.

The arguments are divided into two groups:

1) rational arguments, or “arguments on the case”;

2) irrational (psychological) arguments, or “arguments to the person”, “arguments to the audience”.

Rational arguments include:

a) Facts. Cf.: Facts are stubborn things. However, it should be borne in mind that the speaker does not always have all the data. Most often, the speaker (or arguing) has only individual facts at his disposal, they can be both typical and private, and against their background a general conclusion is made. Therefore, the argument-fact should be treated critically, analytically. This also applies to the statistical data of the results. opinion polls, since errors in the method of collecting these data can lead to a distortion of facts, reality.

b) Appeal to authorities is one of the common types of arguments. At the same time, the speaker should know that in this audience the mentioned authorities are really recognized and respected. Currently in general philosophical questions an authoritative source is, for example, the Bible, as well as folk wisdom, for example, proverbs, sayings. In scientific matters, the authorities are the founders of this branch of knowledge, prominent scientists.

c) Laws, theories, axioms traditionally accepted in a given society.

To irrational arguments include appeal to the feelings, desires, interests of the addressee. These arguments most often affect the self-esteem of the audience (those present are evaluated as reasonable, noble, sensible people, i.e. a positive characterization of the audience is given), material, social interests of the public, well-being, freedom, habits of listeners.

It is thanks to this type of arguments in discussions that they often move from the case “to the faces”, when it is no longer the subject of the dispute that is being evaluated, but the opponent.

Both types of arguments in rhetoric differ in their strength and are distinguished exhaustive, main and controversial arguments.

Exhaustive arguments, most often one, are those arguments that fully prove the correctness of some opinion, position. Such arguments are rare.

The main arguments are various facts convincing of the reality of something. Judicial speech theorists point out that the strongest arguments should be given at the end of the forensic speech.

Controversial arguments can serve as "for" and "against" the position being proved.

When choosing arguments to prove the proposed position (thesis), the speaker must remember the requirements for arguments. Arguments must be true, consistent, proven regardless of the thesis, sufficient.

If the arguments are not true, this is either a special trick to deceive listeners (often a propaganda trick), or their use leads to a logical error, which is called "false reason" or "false fallacy."

The insufficiency of arguments leads to the fact that the position to be proved does not follow from the arguments given. The truth of the argument must be proven regardless of the thesis. Violation of this rule leads to a logical error “vicious circle”, when the thesis is proved by arguments, and the arguments are the thesis (The team succeeded because it worked successfully).

For the speaker, it is also important how definitely, clearly, precisely, and consistently the thesis put forward and defended by him is formulated.

If the thesis is formulated not quite definitely, it can easily be replaced by another in a dispute, it can be interpreted ambiguously, as a result, “substitution of the thesis” is very often observed in discussions when they proceed to discuss another problem. If a discussion is underway, then it is necessary to make sure not only of the accuracy and certainty of one's own thesis, but also of the thesis put forward by the opponent, to make sure that the opponent's thesis is understood exactly.

Uncertainty, the generality of the formulation of the thesis can also lead to the second mistake, which is often made by inexperienced speakers - “losing the thesis”, when the speaker easily loses the main thread of reasoning and begins to speak “in general”. A kind of “substitution of the thesis” is the “default figure”, i.e. suppression of unfavorable facts and events. This conscious “mistake” is very often encountered in the interpretation of entire historical periods in the development of society.

So, any proof consists of three elements: the thesis, arguments, logical connective (form of logical connection) between the thesis and arguments. Arguments must not only be selected, but also correctly used to prove the put forward position (thesis).

Distinguish direct and indirect proof.

A direct proof is constructed as follows:

Arguments given;

True judgments are derived from them;

A true judgment proves the thesis put forward by the speaker.

Such proof is called inductive proof. It is especially productive when the speaker has irrefutable vivid facts as arguments. This proof is productive because the audience, especially in a dispute, is most convincingly affected by the concrete, figurative.

The deductive method in the proof most often relies on general provisions known to the audience, the truth of which is not in doubt. Such a proof consists, therefore, of the well-known general position(major premise), the judgment associated with it, leading to its application, and the conclusion.

for example:

No dishonest person will be elected mayor.

X is dishonest.

Therefore, X will not be elected mayor.

Indirect evidence lies in the fact that the speaker substantiates the falsity of the opposite thesis. Firstly, this is done either by the method of proof from the contrary, or by the method of exclusion (the alibi method). The method of proof by contradiction is often used in science (see geometry). The “method of exclusion” is also called the “alibi method”, as it is often used in judicial practice. In this case, the truth of the thesis is proved by revealing the falsity of all possible alternatives (cf., for example, discussing candidates for a position).

Based on the foregoing, we can conclude about the methods of refuting the opposite thesis. The simplest and most reliable way is to refute a false thesis with facts. Secondly, the arguments of the opponent are criticized, as a result of which the entire system of proof collapses; thirdly, the illogical conclusion of the opponent from a false thesis is substantiated.