An excerpt from a book I compiled based on A.D.’s textbook. Getmanova reference manual on logic.

Terminology and classification

Proof is a set of logical techniques to substantiate the truth of a thesis.

Proof structure:

Thesis – it is a proposition whose truth must be proven.
Arguments – These are the true judgments that are used to prove a thesis.
Form of evidence (demonstration) – a way of logical connection between the thesis and arguments.

Types of arguments:

Certified isolated facts.
Definitions as arguments of proof.
Axioms and postulates.
Early proven laws of science and theorems as proof arguments.

Refutation– a logical operation of establishing the falsity or unfoundedness of a previously put forward thesis.

Refutation thesis- a proposition that needs to be refuted.

Refutation Argument- judgments with the help of which the thesis is refuted.

Refutation of the thesis:

Refutation with facts
Establishing the falsity (or inconsistency) of the consequences arising from the thesis.
Refuting the thesis through proof of the antithesis.

Paralogism – an unintentional mistake made by a person in thinking.

Sophism- a deliberate mistake made with the aim of confusing the enemy and passing off a false judgment as true.

Paradox - this is reasoning that proves both the truth and the falsity of a certain judgment, in other words, proving both this judgment and its negation.

Rules of evidential reasoning

Thesis:

The thesis must be logically defined, clear and precise.
The thesis must remain identical, i.e. the same throughout the entire proof or refutation.

Arguments:

The arguments given to prove the thesis must be true.
Arguments must be a sufficient basis to prove the thesis.
Arguments must be judgments whose truth is independently proven, regardless of the thesis.

Evidence form:

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.

Logical errors found in proof and refutation

Thesis:

“Substitution of the thesis”, i.e. a thesis is intentionally or unintentionally replaced by another, and this new thesis begins to be proven or refuted.
“Argument to man”, i.e. substitution of evidence for the thesis with references to the personal qualities of the one who put forward the thesis.
"Transition to another gender." There are two cases that are easier to describe with examples. First case - when instead of one true thesis they try to prove another, stronger thesis (if B follows from A, but A does not follow from B, then thesis A is stronger than thesis B), and the second thesis may turn out to be false- 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 someone saw it. Second case - when instead of thesis A we prove a weaker thesis B- if, trying to prove that this animal is a zebra, we prove that it is striped, then we will not prove anything, since the tiger is also a striped animal.

Arguments:

Falsehood of grounds ("Basic fallacy"), i.e. when not true, but false judgments are taken as arguments, which they pass off or try to pass off as true.
“Anticipation of reasons”, i.e. when the thesis is based on unproven arguments, the latter do not prove the thesis, but only anticipate it.
“Vicious circle”, i.e. when the thesis is substantiated by arguments, and the arguments are substantiated by the same thesis.

Evidence form:

“Imaginary following”, i.e. when the thesis does not follow from the arguments given in support of it.
“From what is said with a condition to what is said unconditionally,” i.e. an argument that is true only taking into account a certain time, relationship, measure cannot be presented as unconditionally true in all cases.

PLAN

1. The concept of argumentation and its meaning.

2. Proof and its structure. Demonstration of proof. Types of arguments.

Argumentation is a set of logical operations that serve to search and present the grounds for a certain point of view with the aim of understanding it and/or accepting it. The purpose of argumentation is the acceptance of the proposed provisions by the audience or opponent. This means that the oppositions “truth – false”, “good – evil” are not central either in the argumentation or in its theory.

Every argument has both logical and communicative aspects. In logical terms, argumentation is a procedure for finding support, grounds for a certain statement and expressing this in a strict form. In communicative terms, argumentation is the process of transmitting, interpreting and suggesting information present in the original position. Final goal This process is the formation of some belief. The goal can be considered achieved if a person understands and accepts our starting position. The need for argumentation arises at that stage of considering a problem when possible ways to solve it are formulated, but it is not clear which of them has advantages.

Of course, you can influence beliefs not only with the help of verbally formulated arguments, but also in many other ways: gestures, facial expressions, visual images, hypnosis, subconscious stimulation, medications, etc. Even silence can be a compelling argument. These methods of influence are studied by psychology and the theory of art, but are not affected by the theory of argumentation, even if its subject is interpreted extremely broadly. Argumentation is a speech act addressed to the mind of a person who is capable, after reasoning, of accepting or rejecting an opinion. Argumentation presupposes the intelligence of those who perceive it, their ability to rationally weigh arguments, consciously accept them or challenge them.

For the theory of argumentation, two properties of reasoning are important: evidence and persuasiveness. Their combinations give three different characteristics of reasoning. The first of them is unproven persuasiveness - a characteristic of reasoning that is not logically sound, but nevertheless recognized as sufficient within the framework of a certain attitude. These are various plausible reasoning based on induction, analogy, and probabilistic deduction. The persuasiveness of this kind of reasoning is sometimes achieved through oratory and skillful manipulation of expectations and prejudices. Most of the unproven but convincing statements are in the subject areas of non-deductive knowledge.

Inconclusive evidence characterizes reasoning that meets strict standards of validity (based on reliable conclusions), but is too complex for an inexperienced person to evaluate its correctness. Such reasoning includes, for example, some mathematical or complex logical proofs. Such argumentation has a narrowly professional purpose.

Convincing evidence is a property of strictly evidential reasoning that has a fairly transparent structure or a well-known, familiar method of construction. How to achieve this characteristic of reasoning? This question is difficult to answer briefly. You should learn to use the rules of evidential reasoning, and for this you should become familiar with the theory of argumentation in as much detail as possible.

The fundamental logical actions in the set of actions called argumentation are proof and refutation. A proof is a reasoning that establishes the truth of a statement by citing other, related and reliably true statements. In epistemology, evidence is considered one of the most common and accessible criteria of truth.

In any proof, there are three elements: thesis, argument (argument, basis) and demonstration. A thesis is a statement whose truth needs to be proven. Arguments are true judgments from which the truth of the thesis is deduced. Demonstration is a form of evidence, a way of logical connection between thesis and arguments. Example: “Thesis statement: platinum is electrically conductive. Arguments: platinum is a metal, and all metals are electrically conductive. Demonstration: modus Barbara of a simple categorical syllogism."

A demonstration is usually a form of inference or several inferences. The demonstration may take the form of one or another correct mode of a simple categorical syllogism; it can be a polysyllogism or an epicheireme, a demonstration can be an affirmative or a negating mode of a conditionally categorical syllogism; both modes of dividing-categorical syllogism can be used as demonstrations of evidence. In proofs, forms of inductive inferences are also possible (a ₁, a ₂, a ₃ → T). In the case of incomplete induction, as well as in reasoning by analogy, the thesis is substantiated only with a greater or lesser degree of probability; additional argumentation is needed for reliable proof. Various forms of substantiating a thesis can be used either independently or in combination.

The most important component of any proof is arguments. What kind of judgments can and should be arguments of evidence? In logical theory there are several types of arguments.

1. Certified isolated facts. This is primarily data from observations and experiments, statistical data, results of sociological research, some evidence (signatures on documents, witness statements), etc.

2. Definitions as arguments of proof. Without definitions, it is impossible to build clear and unambiguous proofs (to comply with the law of identity). Both the terms that make up the thesis and the terms that make up the arguments must have definitions.

3. Axioms. In the theory of argumentation they are accepted as truths without proof. Aristotle believed that axioms are reliably true because they are completely clear and simple. Euclid considered the geometric axioms he accepted as self-evident truths. Later, axioms were interpreted as eternal and immutable truths that exist before any experience and do not depend on it. In non-classical science, axiomatic justification has been rethought. Thus, K. Gödel substantiated that axioms are statements that are both unprovable and irrefutable. Axioms are not justified on their own, but as necessary components of a theory: confirmation of the latter is the simultaneous confirmation of the system of axioms. The criteria for choosing axioms vary from theory to theory and are largely pragmatic. Axioms are simply postulates, initial and accepted provisions of a theory, which can become the basis for proving its other provisions.

4. Laws, previously proven theorems, solved problems. Previously proven judgments can serve as evidence arguments. When proving a thesis, as a rule, not one, but several of the listed types of arguments are used.

Ministry of Education and Science of the Russian Federation

Federal Agency for Education

TEST

DISCIPLINE: LOGIC

Logical Basics theories of argumentation

Test work of a student of group IYUB-11/3c:

Teacher:

1. /1.1 Concept of evidence……………………………………………………………3

1.2 Structure of evidence: thesis, arguments, demonstration…………5

1.3 Types of arguments………………………………………………………5

2. Direct and indirect (indirect) evidence……………………………6

3. The concept of refutation……………………………………………………….9

3.1 Refutation of the thesis…………………………………………………….8

3.2 Criticism of arguments……………………………………………………...10

3.3 Detection of demonstration failure………………………….11

4. Rules of evidential reasoning. Logical errors found in proofs and refutations…………………………………………...11

4.1 Rules in relation to the thesis……………………………………11

4.2 Errors regarding the thesis being proven…………………………….12

4.3 Rules regarding arguments………………………………….13

4.4 Errors in the grounds (arguments) of evidence…………………..14

4.5 Rule regarding the form of substantiation of the thesis (demonstration)...14

4.6 Errors in the form of evidence……………………………………..14

5. The concept of sophistry and logical paradoxes……………………………..16

5.1 The concept of logical paradoxes……………………………………17

5.2 Paradoxes of set theory…………………………………………….17

6. The art of discussion……………………………………………..18

Literature……………………………………………………………………..24

1. Concept of proof

1.1

Knowledge of individual objects and their properties begins with sensual 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 - 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, showing it important in a given life situation, etc. 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 has come
to us from antiquity (so, Ancient Greece- the birthplace of Plato’s dialogues, the technique of argument in the form of Socrates’ questions and answers, etc.). But dialogue is
external form of argumentation: the opponent can only be thought of (that
This is especially evident 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 a theory
and the practice of persuasion in the process of proving the truth of the thesis. “The Word is a great ruler, who, having very little and absolutely
invisible body, performs the most wonderful deeds. For it can and fear
expel, and destroy sadness, and 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 solar system, about the stars and galaxies of the Universe, theorems of mathematics, judgments about the directions of development of electronic technology, about the possibility of long-term weather forecasts, about the secrets of the World Ocean and space. All these judgments must be scientifically substantiated.

Proof - this 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 data from science and socio-historical practice, while beliefs can be based, for example, on religious faith, on prejudices, on people’s ignorance of issues of economics 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

Thesis- it is a proposition whose truth must be proven. Arguments - This - those true judgments that are used to prove a thesis. Form of evidence, or demonstration, called a 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, number of weapons - , 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. Axi-ohms - these 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 arguments for proof. 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 is divided into: straight And indirect (indirect). Direct the proof comes 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 daily life, but can lead to more serious consequences, therefore, along with jealousy, anger and hatred, it undoubtedly belongs to 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- this is a proof in which the truth of the thesis put forward is substantiated by proving the falsity of the antithesis. If the thesis is denoted by a letter A, then his denial (a) will be an antithesis, i.e. a judgment that contradicts the thesis.

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

Let A - a thesis or theorem that needs to be proven. We assume by contradiction that A false, i.e. true nope (or a ). From the assumption a we derive consequences that contradict reality or previously proven theorems. We have A V a , wherein a - false means its negation is true, i.e. a , which, according to the law of two-valued classical logic ( a > a ) gives A . So it's true A , Q.E.D.

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 AB and CD are straight 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 either A, or IN, or WITH.

It has been proven that they did not commit a crime A, nor IN.

Committed the crime WITH.

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 - a logical operation of establishing the falsity or unfoundedness 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 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) revealing 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- the surest and most 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:

a = A > F

Where F - contradiction or 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 is put forward (i.e. nope), and judgment nope(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” (judgment A, general affirmative). For judgment A the judgment will be contradictory ABOUT - Partial negative: “Some dogs don’t bark.” To prove the latter, it is enough to give several examples, or at least one example: “Pygmies’ dogs never bark.” So, the proposition has been proven ABOUT. By virtue of the law of excluded middle, if ABOUT- true, then A- 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 a proposition I to the truth of a judgment A(analogically, from the truth of the judgment ABOUT to the truth of a 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, reports, lectures 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. Thesismust 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) “who
proves too little, he 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 from A should b, but from b do not do it
A, then the thesis A is stronger than a 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 a thesis A we will 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, not true, but false judgments are taken, which 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, they 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 those who are not familiar with the rules of logic and rely 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 defines the norms of human behavior, and in different conditions they can vary somewhat (for example, truthfulness - positive trait person, but if he betrays the secret to the enemy, it will be a crime).

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

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 century. The book by V.I. was very popular among students. Obreimov “Mathematical sophisms”, which contains many sophisms. And in a row modern books collected 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 equals positive";

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

5 = 1. Wanting to prove that 5 = 1, we will reason like this. From the numbers 5 and 1
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

Paradox - this is reasoning that proves both the truth and falsity of some judgment or (in other words) proving 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 the City" is as follows: every mayor of a city lives either in his own city or outside 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, dispute, 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: there is no point in entering 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 weakness of the opponent's position.

A.L. Nikiforov identifies the following loyal (acceptable) methods of argument, which are simple and few in number. Important from the very beginning seize the initiative: offer your formulation of the subject of the dispute, a discussion plan, and direct the course of the debate in the direction you need. It is important in a dispute 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 given to the one who manages to assign burden of proof or refutations on your opponent. And if he has poor command of evidence, he may get confused in his reasoning and will be forced to admit defeat. Recommended concentrate attention and actions on the most weak link in the opponent's argument, and not strive to refute all its elements. Loyal techniques also include the use of the effect 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 methods also include the desire have the last word in a discussion: by summing up the dispute, you can present its results in a light favorable to you.

Incorrect, disloyal methods 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 substitution of 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: arguments ad rem(to the point, to the point) and arguments ad hominem(to a 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 arguments ad hominem are incorrect and cannot be used in a discussion whose participants strive to clarify and substantiate the truth. The most common types of arguments ad hominem are the following:

1. Argument to the individual - 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 avmopumemy - reference to statements 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 to the public - 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. ArgumentTo vanity - lavishing excessive praise on an opponent in the hope of making him softer and more accommodating. 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. ArgumentTo strength(“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 - arousing 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. ArgumentTo ignorance - the use of facts and provisions about which the opponent knows nothing, 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.

Ministry of Education and Science of Ukraine

Department of Philosophy

test on logic No. 10

2nd year students, gr. MGKTS-3

Ponomareva Inna Alexandrovna.

House. address:

Kharkiv

2007-2008 academic year

Topic No. 10 Logical foundations of the theory of argumentation

2. The concept of refutation.

3. Rules of proof and refutation.

List of used literature.

1. The concept of proof and its structure.

Proof is a logical operation to substantiate the truth of judgments with the help of other true judgments.

Proof structure:

What is being proven?

What is the proof of the advanced position?

How is it proven?

The answers to these questions reveal: Thesis, Arguments, Demonstration.

A thesis is a judgment put forward by the proponent, which he substantiates in the process of argumentation. The thesis is the main structural element of the argument and answers the question: what is being justified.

Arguments are the initial theoretical or factual provisions with the help of which the thesis is substantiated. They serve as the basis, or logical foundation of the argument, and answer the question: what, with what help, is the thesis justified?

Demonstration is a logical form of constructing a proof, which, as a rule, takes the form of a deductive inference. The argument must always be true, while the conclusion is not always true.

There are two types of evidence:

Direct - the thesis logically follows from the arguments.

Indirect (indirect) is such evidence in which the truth of the thesis put forward is substantiated by proving the falsity of the antithesis; they are divided into two types:

Proof by contradiction is carried out by establishing the falsity of a judgment that contradicts the thesis. The truth of the antithesis is assumed and a consequence is derived from it; if at least one of the obtained consequences contradicts either the premise or another consequence, the truth of which has already been established, then this consequence, and after it the antithesis, is assumed to be false.

Separation proofs, method of elimination. The falsity of all members of the disjunction is established, except for one, which is a valid thesis.

As mentioned above, any proof has three components: thesis, arguments and demonstration. In principle, the structure of the proof follows the structure of the inference. There, too, there is a thesis obtained in the form of a conclusion from premises-arguments, and the conclusion itself as a whole is an analogue of a demonstration. Only in proof can a demonstration be a long chain of inferences that make up a more or less extensive argument or, perhaps, a large theorem. In addition, and this is even more important, the proof, as V.F. once correctly pointed out. Asmus in his textbook on logic is, in fact, a conclusion about a conclusion, that it is constructed in accordance with the rules of logic, its premises are true and, therefore, the conclusions made in it must be recognized as true judgments. The fact is that the inference itself does not provide this. Let's say we have the following reasoning: stringed musical instruments are divided into plucked and bowed; the piano is not a bowed instrument; This means that the piano is a plucked instrument. Can the conclusion obtained with the help of this divisive-categorical syllogism be considered justified? Obviously not. Because for this you also need to know whether the premises are true and whether the rules of such syllogisms are observed, in particular, the requirement to indicate all possible alternatives; in this case, by the way, it was not fulfilled, since there are also percussion-keyboard string instruments, which include the piano.

The final evaluative conclusion may not be stated directly, but only implied, as is often the case with many other components of reasoning. But, in essence, it always represents a conditional categorical syllogism, the modus ponens already known to us. Its first, conditional premise: if the arguments are true judgments, and the conclusion is constructed correctly, then its conclusion is a true (proven) judgment; second, categorical: the arguments are true, the conclusion is correct. This leads to the conclusion about the immutable truth of the thesis. Thus, the entire process of proof, in accordance with its structure, falls into three stages: formulating a thesis, finding arguments that satisfy a number of special requirements, and then constructing a demonstration and testing it. One more thing can be distinguished, the fourth - the formation of an evaluative conditional-categorical syllogism. But its preparation in any case dissolves in the first three stages. The modus ponens itself is so simple that after the work in the previous stages is completed, its separate formulation becomes unnecessary. The test result, of course, may turn out to be negative. After all, it cannot be ruled out that the proof was carried out with errors. Then we will be dealing with some version of the refutation.

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, the territory of the state, the implementation of the plan, the number 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.

At the cost of tens of thousands of experiments conducted, collection scientific facts I.V. Michurin created a harmonious system for breeding new plant varieties. At first, he became interested in working on the acclimatization of pampered southern and Western European fruit crops in the conditions of central Russia. Through hybridization, he managed to create over 300 varieties of fruit and berry crops. This is a vivid example of how a genuine scientist collects and processes enormous scientific factual material.

2. Definitions as arguments of proof. Definitions of concepts are usually given in every science.

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

4. Previously proven laws of science and theorems as arguments of proof.

2. The concept of refutation.

It is quite acceptable to give the term “proof” an expanded meaning, so that refutation becomes a variation of it. To a certain extent this is justified and is often done. Because as a result of a refutation, some firmly established truths also appear, even if their content is not the external reality, not objects or phenomena, but someone’s statements that are given a new assessment. A refutation also has the usual three components of any proof: thesis, arguments and demonstration. At the same time, their differences cannot be ignored either. Indeed, while a proof is a conclusion about a conclusion, a refutation, in contrast, is a conclusion about the evidence. The object of attention in this case is the provisions that have already been proven or seem to be so. The refutation aims to eliminate them. From this point of view, proof and refutation are opposite directions.

True, one could take into account the fact that when a refutation is correct, when as a result of its implementation the falsity of those truths that were considered proven is revealed, then in this case it is simultaneously revealed that the previous proof itself was not such in fact. This means that the refutation should then be recognized not as an inference about the evidence, but as an inference about the inference mistakenly accepted as evidence. Refutation as a logical action, taking into account such circumstances, fully falls within the definition of evidence and could be considered some kind of verification. And in addition, it can be divided into the same types as evidence.

Refutation is a type of evidentiary process aimed at already existing evidence in order to show their inconsistency.

It is not necessary that a new substantive truth be born as a result of a refutation (although sometimes it appears as a by-product). But a new, well-founded assessment of existing views is required. In this sense, refutation is not only destructive, but also creative; it frees knowledge from inaccurate, superficial, hasty conclusions and statements, clarifies ideas about things, although it never speaks directly about them. Refutation is just as necessary component knowledge, as well as evidence.

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; III) identifying the failure of the demonstration.

I. Refutation of the thesis (direct and indirect).

Logic and theory of argumentation

Introduction

It is difficult to overestimate the importance of logic and the theory of argumentation not only in the development of scientific knowledge, but also in everyday life. For science, the essential points are effective ways of processing information and research methods, forms of thought and operations with them, the basics of evidence, rules for constructing hypotheses and theories. In general, everything that forms the basis of logic and the theory of argumentation. In everyday life, it is very important to be able to defend your point of view and find a way out of a difficult life situation. This is greatly facilitated by the study of logic and the theory of argumentation.

This discipline was formed at the intersection of several sciences - logic, rhetoric, psychology, etc. Moreover, the theory of argumentation and logic can be studied as separate disciplines, each of which has its own area of study: logic - forms of thinking, their features and interaction, laws of thinking; theory of argumentation - methods of persuasion. The combination of logic and argumentation theory pursues the goal of forming a student’s logical culture, based on theoretical knowledge of the fundamentals of logic and the practical application of these fundamentals in the process of argumentation.

Developed logical thinking is one of the signs of a modern educated person. The ability to think clearly and quickly make the right decision based on an analysis of the current situation ensures that a person is in demand and successful in his professional activities. For example, the ability to use the entire arsenal of logical knowledge and methods of persuasion will be useful in professional activities that involve interaction with people, the opportunity to influence their opinions, tastes, and choice of a particular product. Therefore, people who have chosen a field of activity such as public relations, personnel management, etc. it is necessary to study logic and the theory of argumentation.

Topic 1. Subject of logic

define logic;

characterize the stages of development of formal logic;

indicate the features of non-classical logic;

understand the meaning of building logical formalized systems;

name the main aspects of the language;

understand the uniqueness of the logical approach to the study of thinking in comparison with other sciences.

Logics is the science of the forms, methods and means of correct thinking. Generally valid forms of thought include concepts, judgments, inferences, and generally valid means of thought include definitions, rules for the formation of concepts, judgments and inferences, rules for the transition from one judgment or inference to another as consequences of the first (rules of reasoning).

Formal logic went through two main stages in its development. The beginning of the first stage is associated with the works of the ancient Greek philosopher Aristotle, in which a systematic presentation of logic was first given. Aristotelian logic and all pre-mathematical logic are usually called "traditional" logic. Traditional logic identifies and describes some of the simplest forms of reasoning fixed in language. The second stage is the emergence of mathematical or symbolic logic. For the first time in history, ideas about constructing logic on a mathematical basis were expressed by the German mathematician G. Leibniz at the end of the 17th century. The first implementation of Leibniz's idea belongs to the English scientist D. Boole (mid-19th century). He created an algebra in which letters represent statements. Thanks to the introduction of symbols into logic, the basis was obtained for the creation of a new science - mathematical logic. The application of mathematics to logic made it possible to present logical theories in a new, convenient form and to apply the computing apparatus to solving problems that are inaccessible to human thinking due to their complexity.

Modern symbolic logic is a very ramified field of knowledge. Symbolic logic is divided into classical and non-classical. Non-classical logic is also divided into intuitionistic logic, modal logic, question logic, relevant logic, etc. Non-classical logic is based on the idea that the law of excluded middle is inapplicable in some cases, in particular when it comes to infinite sets. In addition, in a number of areas of non-classical logic, the initially two-valued logic of Aristotle is transformed into three-valued, four-valued, and then multi-valued.

Traditional logic was empirical in nature. She identified and described some of the simplest forms of reasoning from the so-called categorical judgments recorded in the language of everyday use. Modern logic has expanded the range of forms under consideration, introducing into it reasoning specific to scientific knowledge, in particular, mathematical knowledge. Moreover, modern logic determined the principles of theoretical substantiation of the conditions for the correctness of conclusions and evidence, using the concepts: logical law and logical consequence.

Unlike other sciences that study thinking, logic studies the features and properties of forms of thought, while abstracting from the specific content that these forms of thought can carry; she studies them from the point of view of structure, structure, i.e. internal natural connection of the elements that make up the form of thought.

It should be borne in mind that logical forms and laws are universal and objective, that is, they are not associated with any psychophysiological characteristics of people or with certain cultural and historical factors.

Thinking is closely related to language, however, these are not identical concepts. Language is a material formation, which is a certain sign system that allows you to express thoughts, store them and transmit them. Thinking is an ideal system. If the main elements of language are letters, words, phrases and sentences, then the elements of thinking are individual forms of thought (concepts, judgments, conclusions) and their combinations.

Natural language is a system of signs. When considering language as a system of signs, it is important to take into account three main aspects of language: syntax, semantics and pragmatics.

Syntactic aspect includes the variety of relationships of signs to other signs, the rules in the language for the formation of some signs from others, and the rules for changing signs.

Semantic aspect constitutes a set of relations of signs to objects of extra-linguistic reality, that is, to what they designate.

Pragmatic aspect includes all such features of the language that depend on who and in what situations it is used.

Based on the principle of objectivity of knowledge, in science they strive to exclude, when determining the semantic contents of linguistic expressions and when describing cognitive procedures, any possible influence of the subjective characteristics of cognizing people. There should be no, for example, uncertainties or ambiguities in the expression of thoughts in language. These requirements are met by specially constructed logical formalized languages.

The main goal of logic is to clarify the conditions for the truth of knowledge and develop effective cognitive procedures. Knowledge of logic improves the culture of thinking, promotes clarity, consistency and evidence of reasoning, enhances the effectiveness and persuasiveness of speech. Logical culture is not an innate quality. Logical culture is formed as a result of careful study of logic and accumulation of experience in the practical application of logical knowledge.

Logic is of great importance in the development and organization of the information process. Failure to comply with logical form and logical sequence in information processes is fraught with negative consequences in various spheres of human life and society.

Control questions:

Define logic as a science.

What is the difference between traditional logic and symbolic logic?

Who is the founder of logic?

What basic aspects of language do you know?

What principles form the basis of non-classical logic?

What practical significance does the study of logic have?

Name the main forms of thought.

Topic 2. Concept

After studying the topic materials, you will be able to:

understand the logical methods of forming concepts;

give a logical characterization of any concept, based on the classification of concepts;

determine the relationships between concepts by volume;

understand the essence of such logical operations on concepts as generalization, limitation, division and definition;

name the logical errors that arise when the rules of division and definition are violated.

understand the meaning of operations with classes.

A concept is a form of thought that reflects general, essential and specific characteristics of objects, phenomena, and processes.

The formation of a concept is possible through the use of logical techniques such as analysis, synthesis, abstraction, and generalization. Analysis– mental division of objects into their component parts, mental identification of features in them (i.e. properties and relationships). Synthesis– mental connection into a single whole of parts of an object or its features obtained in the process of analysis, which is carried out both in practical activity and in the process of cognition. Abstraction- mental selection, isolation of individual features, properties, connections and relationships of interest to us of a particular object or phenomenon and mental abstraction of them from many other signs, properties, connections and relationships of this object. Generalization– mental selection of some properties belonging to a certain class of objects; transition from the individual to the general, from the less general to the more general.

When getting acquainted with the doctrine of the concept, it is important to clearly understand that the concept as a thought is not identical to either the word that expresses it or the object that it reflects.

A concept has only two elements of its structure - content and volume. Volume is a set of objects of thought united in a concept. Content– a set of attributes of objects united in a concept. There is the following relationship between the volume and content of a concept: the greater the volume, the less the content; The smaller the volume, the greater the content.

Isolating the elements of the structure of a concept and becoming familiar with their features and properties makes it possible to consider the types of concepts, the relationships between them and, finally, operations on concepts.

In count concepts are divided into general, isolated and “empty”. General are concepts whose scope contains two or more elements. For example, the concept of “book”. Single are concepts whose scope contains only one element. For example, the concept of “Russian Museum”. In fact, all proper names are singular concepts. Empty Concepts are concepts whose scope does not contain a single element. For example, the concept of “Koschei the immortal” or the concept of “square circle”.

By quality concepts are divided into positive, negative, concrete, abstract, correlative and non-relative, comparable, incomparable, collective and separative, registering, non-registering.

Positive concepts are concepts that indicate the presence of a particular quality or relationship in an object. For example, the concept of “decency”. Negative concepts are concepts that indicate the absence of some quality or relation in an object. For example, the concept of “futility”.

Specific concepts are concepts that reflect objects. For example, the concept of “house”. Abstract concepts are concepts that reflect the properties and relationships between objects. For example, the concept of “height”.

Collective concepts are concepts whose attributes relate not to each element of the set, but to the entire set as a whole. For example, the concept of “platoon”. Dividing concepts are concepts whose attributes relate to each element of a set of objects. For example, the concept of “soldier”.

Correlative concept is a concept whose content represents the presence or absence of a relationship between an object conceived in it and some other object. In the correlative concept, an object is conceived that determines the existence of another object. For example, the concept of “boss” determines the existence of the concept of “subordinate”. Irrelevant concept is a concept whose content is not connected by any relationship, where conceivable objects(features) exist completely independently, independently of other objects (properties). For example, the concept of “pencil”.

Comparable Concepts are concepts whose contents are closely related. For example, the concept of “man” and the concept of “living being”. Incomparable Concepts are concepts that have a distant connection in content. For example, the concepts “picture” and “mole” are incomparable concepts.

Registrants are called concepts in which the multitude of elements conceivable in it can be taken into account and registered (at least in principle). For example, “heroes of the Soviet Union”, “month”. Registering concepts have a finite scope. Non-registering concepts that relate to an indefinite number of elements are called. Thus, in the concepts of “machine” and “paper” the multitude of elements conceivable in them cannot be counted: all people, all cats are conceivable in them. Non-registering concepts have an infinite scope.

Relations between concepts are relations between types of concepts. Relationships between concepts are compatible And incompatible.

Compatible concepts are concepts whose scopes partially or completely coincide. Compatibility relationships: identity, subordination, intersection. Identical concepts are concepts whose scopes completely coincide. Subordinates concepts are concepts whose volumes have such a relationship that the volume of one of the concepts is completely included in the volume of the other, but does not coincide with it. Subordinate concepts reflect generic relations. Crossed(being in a relation of intersection) concepts are concepts whose scopes partially coincide.

Incompatible concepts are concepts whose volumes do not have common elements. Relationships of incompatibility: contradiction, opposition, subordination. Subordinates concepts are concepts whose scopes exclude each other, but at the same time are included in the scope of some broader (generic) concept. Contradictory concepts are concepts that are species of a certain kind, the characteristics of which are mutually exclusive, and the sum of their volumes exhausts the volume of the generic concept. Opposite concepts are concepts that are included in the scope of some generic concept and the scopes of which are mutually exclusive. The volumes of opposite concepts in their totality do not exhaust the volume of the generic concept.

For better memorization and orientation in these relationships, it is customary to depict all types of relationships using Euler circles.

Nature