1. The concept of inference
inference- this is a form of abstract thinking, through which new information is derived from previously available information. In this case, the sense organs are not involved, i.e., the entire process of inference takes place at the level of thinking and is independent of the information received in this moment outside information. Visually, the conclusion is reflected in the form of a column in which there are at least three elements. Two of them are premises, the third is called the conclusion. Parcels and conclusions are usually separated from each other by a horizontal line. The conclusion is always written below, the premises - above. Both the premises and the conclusion are judgments. Moreover, these judgments can be both true and false. For example:
All mammals are animals.
All cats are mammals.
All cats are animals.
This conclusion is true.
Inference has a number of advantages before the forms of sensory knowledge and experimental research. Since the process of inference takes place only in the realm of thinking, it does not affect real objects. This is a very important property, since often the researcher does not have the opportunity to get a real object for observation or experiments due to its high cost, size or remoteness. Some items at the moment can generally be considered inaccessible for direct research. For example, space objects can be attributed to such a group of objects. As is known, human exploration of even the closest planets to the Earth is problematic.
Another advantage of inferences is that they provide reliable information about the object under study. For example, it was through inference that D. I. Mendeleev created his periodic system chemical elements. In the field of astronomy, the position of the planets is often determined without any visible contact, based only on the information already available about the regularities in the position of celestial bodies.
Inference flaw one can say that conclusions are often characterized by abstractness and do not reflect many of the specific properties of the subject. This does not apply, for example, to the above-mentioned periodic table of chemical elements. It is proved that with its help, elements and their properties were discovered, which at that time were not yet known to scientists. However, this is not the case in all cases. For example, when determining the position of a planet by astronomers, its properties are reflected only approximately. Also, it is often impossible to speak about the correctness of the conclusion until it has passed the test in practice.
Inferences can be true and probabilistic. The former accurately reflect the real state of affairs, the latter are uncertain. The types of reasoning are: induction, deduction and conclusion by analogy.
inference- this is primarily the derivation of consequences, it is applied everywhere. Every person in his life, regardless of profession, made conclusions and received consequences from these conclusions. And here the question of the truth of such consequences arises. A person who is not familiar with logic uses it at a philistine level. That is, he judges things, draws conclusions, draws conclusions based on what he has accumulated in the process of life.
Despite the fact that almost every person is trained in the basics of logic at school, learns from their parents, the philistine level of knowledge cannot be considered sufficient. Of course, in most situations this level is enough, but there is a percentage of cases when logical preparation is simply not enough, although it is in such situations that it is most needed. As you know, there is such a type of crime as fraud. Most often, scammers use simple and proven schemes, but a certain percentage of them are engaged in highly skilled deception. Such criminals know logic almost perfectly and, in addition, have abilities in the field of psychology. Therefore, it often costs them nothing to deceive a person who is not prepared. All this speaks of the need to study logic as a science.
Inference is a very common logical operation. By general rule, in order to obtain a true judgment, it is necessary that the premises are also true. However, this rule does not apply to evidence to the contrary. In this case, knowingly false premises are deliberately taken, which are necessary in order to determine the necessary object through their negation. In other words, false premisses are discarded in the process of deriving a consequence.
This text is an introductory piece.Immediate inferences An inference built by transforming a judgment and containing one premise is called immediate. There are four types of transformations of judgments: transformation, inversion, opposition to a predicate, inference
Inductive reasoning Inductive reasoning is called inference, in the form of which empirical generalization proceeds, when, on the basis of the repetition of a feature in phenomena of a certain class, it is concluded that it belongs to all phenomena of this class. For example: in history
3.8. Inferences with the union “or” Both premises and the conclusion of a simple or categorical syllogism are simple judgments (A, I, E, O). If one of the premises of the syllogism or both of its premises are represented by complex judgments (conjunction, non-strict and strict disjunction,
§ 2. DIRECT CONCLUSIONS A proposition containing new knowledge can be obtained by transforming the proposition. Since the original (transformed) judgment is considered as a premise, and the judgment obtained as a result of the transformation is considered as a conclusion,
A. DEDUCTIVE CONCLUSIONS In the process of reasoning, inferences that are not deductive are sometimes mistaken for deductive ones. The latter are called incorrect deductive inferences, and (actually) deductive ones are called correct. Identification of methods of reasoning,
B. INDUCTIVE CONCLUSIONS In contrast to deductive reasoning, in which there is a relation of logical consequence between premises and conclusion, inductive reasoning is such a connection between premises and conclusion according to logical forms, with
§ 4. CONCLUSIONS BY ANALOGY The word “analogy” is of Greek origin. Its meaning can be interpreted as “the similarity of objects in some features.” Inference by analogy is a reasoning in which, from the similarity of two objects in some features
§ 1. The Paradox of Inference We will gain an even deeper understanding of the nature of formal logic if we consider some of the critical arguments against it. Our discussion of traditional logic, as well as modern logic and mathematics, has been aimed at clarifying
38. Deductive inferences The following types of inferences are deductive: inferences of logical connections and subject-predicate inferences. Also, deductive inferences are direct. They are made from one premise and are called transformation, conversion and
1. The concept of inference Inference is a form of abstract thinking, through which new information is derived from previously available information. In this case, the sense organs are not involved, i.e., the entire process of inference takes place at the level of thinking and is independent of the received
2. Deductive reasoning Like much in classical logic, the theory of deduction owes its appearance to the ancient Greek philosopher Aristotle. He developed most of the issues related to this kind of reasoning. According to the works of Aristotle, deduction is
1. The concept of inference by analogy A significant characteristic of inference as one of the forms of human thinking is the conclusion of new knowledge. At the same time, in the inference, the conclusion (consequence) is obtained in the course of the movement of thought from the known to the unknown. To such a movement
LOGICAL CONCLUSIONS The vast majority of reasoning that claims to be considered logical, in fact, is not. They are pseudo-logical, logical, or at best only partially logical. Reasoning is logical
2. The concept of a micro-object as a concept of a transsubjective reality or a transsubjective object called the “object of science”, which is applicable to aesthetics. This is not an object of my external feelings, existing outside of me and my consciousness: not something objectively real. This is not an object
CHAPTER I THE CONCEPT OF THE MODEL AND THE CONCEPT OF IMITATION One should choose one of the people of goodness and always have him before our eyes - to live as if he were looking at us, and to act as if he were seeing us. Seneca. Moral Letters to Lucilius, XI, 8 Take yourself, at last, for
In the process of knowing reality, we acquire new knowledge. Some of them - directly, as a result of the impact of objects of external reality on our senses. But most of the knowledge we get by deriving new knowledge from the knowledge we already have. This knowledge is called indirect or inferential.
The logical form of obtaining inferential knowledge is a conclusion.
Inference is a form of thinking by means of which a new judgment is derived from one or more propositions.
Any conclusion consists of premises, conclusion and conclusion. The premises of the inference are the initial judgments from which the new judgment is derived. A conclusion is a new judgment obtained logically from the premises. The logical transition from premises to conclusion is called a conclusion.
For example: “A judge cannot take part in the consideration of a case if he is a victim (1). Judge N. is the victim (2). This means that Judge N. cannot take part in the consideration of the case (3).” In this inference, (1) and (2) are the premises, and (3) is the conclusion.
When analyzing the conclusion, it is customary to write the premises and the conclusion separately, placing them under each other. The conclusion is written under the horizontal line separating it from the premises and denoting the logical consequence. The words "hence" and those close to it in meaning (hence, therefore, etc.) are usually not written under the line. Accordingly, our example looks like this:
A judge cannot take part in the consideration of a case if he is a victim.
Judge N. is the victim.
Judge N. cannot take part in the consideration of the case.
The relationship of logical consequence between the premises and the conclusion implies a connection between the premises in terms of content. If the judgments are not related in content, then the conclusion from them is impossible. For example, from the judgments: “The judge cannot take part in the consideration of the case if he is the victim” and “The accused has the right to defense” one cannot obtain conclusions, since these judgments do not have a common content and, therefore, are not logically connected with each other. .
If there is a meaningful connection between the premises, we can obtain new true knowledge in the process of reasoning, subject to two conditions: firstly, the initial judgments - the premises of the conclusion must be true; secondly, in the process of reasoning, one should follow the rules of inference, which determine the logical correctness of the conclusion.
Inferences are divided into the following types:
1) depending on the severity of the inference rules: demonstrative - the conclusion in them necessarily follows from the premises, i.e. logical consequence in such conclusions is a logical law; non-demonstrative - the inference rules provide only a probabilistic following of the conclusion from the premises.
2) according to the direction of the logical consequence, i.e. by the nature of the connection between knowledge of varying degrees of generality, expressed in premises and conclusions: deductive - from general knowledge to particular; inductive - from particular knowledge to the general; inferences by analogy - from particular knowledge to particular.
Deductive reasoning is a form of abstract thinking in which thought develops from knowledge more generality to knowledge of a lesser degree of generality, and the conclusion that follows from the premises, with logical necessity, is reliable. The objective basis of the control is the unity of the general and the individual in real processes, objects of the environment. peace.
The deduction procedure takes place when the information of the premises contains the information expressed in the conclusion.
It is customary to divide all conclusions into types on various grounds: by composition, by the number of premises, by the nature of the logical consequence and the degree of generality of knowledge in the premises and conclusion.
By composition, all the conclusions are divided into simple and complex. Inferences are called simple, the elements of which are not inferences. Compound statements are those that are made up of two or more simple statements.
According to the number of premises, inferences are divided into direct (from one premise) and indirect (from two or more premises).
According to the nature of the logical consequence, all conclusions are divided into necessary (demonstrative) and plausible (non-demonstrative, probable). Necessary inferences are those in which the true conclusion necessarily follows from the true premises (i.e., the logical consequence in such conclusions is a logical law). Necessary inferences include all types of deductive reasoning and some types of inductive ("full induction").
Plausible inferences are those in which the conclusion follows from the premises with a greater or lesser degree of probability. For example, from the premises: “Students of the first group of the first year passed the exam in logic”, “Students of the second group of the first year passed the exam in logic”, etc. follows “All first-year students passed the exam in logic” with a greater or lesser degree of probability (which depends on the completeness of our knowledge about all the troupes of first-year students). Plausible inferences include inductive and analogical inferences.
Deductive reasoning (from lat. deductio - derivation) is such a conclusion in which the transition from general knowledge to particular is logically necessary.
By deduction, reliable conclusions are obtained: if the premises are true, then the conclusions will be true.
Example:
If a person has committed a crime, then he should be punished.
Petrov committed a crime.
Petrov must be punished.
Inductive inference (from Latin inductio - guidance) is such a conclusion in which the transition from particular knowledge to general is carried out with a greater or lesser degree of plausibility (probability).
For example:
Theft is a criminal offence.
Robbery is a criminal offence.
Robbery is a criminal offence.
Fraud is a criminal offence.
Theft, robbery, robbery, fraud are crimes against property.
Therefore, all crimes against property are criminal offences.
Since this conclusion is based on the principle of considering not all, but only some objects of a given class, the conclusion is called incomplete induction. In full induction, generalization occurs on the basis of knowledge of all subjects of the class under study.
In inference by analogy (from the Greek. analogia - correspondence, similarity), on the basis of the similarity of two objects in some one parameters, a conclusion is made about their similarity in other parameters. For example, based on the similarity of the methods of committing crimes (burglary), it can be assumed that these crimes were committed by the same group of criminals.
All kinds of inferences can be well-formed and incorrectly constructed.
2. Immediate inferences
Immediate inferences are those in which the conclusion is derived from a single premise. For example, from the proposition "All lawyers are lawyers" you can get a new proposition "Some lawyers are lawyers". Immediate inferences give us the opportunity to reveal knowledge about such aspects of objects, which was already contained in the original judgment, but was not explicitly expressed and clearly realized. Under these conditions, we make the implicit - explicit, the unconscious - conscious.
Direct inferences include: transformation, conversion, opposition to a predicate, inference according to the “logical square”.
A transformation is a conclusion in which the original judgment is transformed into a new judgment, opposite in quality, and with a predicate that contradicts the predicate of the original judgment.
To transform a judgment, it is necessary to change its connective to the opposite, and the predicate to a contradictory concept. If the premise is not expressed explicitly, then it is necessary to transform it in accordance with the schemes of judgments A, E, I, O.
If the premise is written in the form of the proposition “Not all S are P”, then it must be converted into a partial negative: “Some S are not P”.
Examples and transformation schemes:
BUT:
All first-year students study logic.
No first-year student studies non-logic.
Scheme:
All S are R.
No S is a non-P.
Elena: No cat is a dog.
Every cat is a non-dog.
No S is R.
All S is non-P.
I: Some lawyers are athletes.
Some lawyers are not non-athletes.
Some S are R.
Some S's are not non-P's.
A: Some lawyers are not athletes.
Some lawyers are non-athletes.
Some S's are not R's.
Some S's are not-P's.
Inversion is such a direct inference in which the place of the subject and the predicate is changed while maintaining the quality of the judgment.
The address is subject to the rule of distribution of terms: if a term is not distributed in the premise, then it should not be undistributed in the conclusion.
If the conversion leads to a change in the original judgment in terms of quantity (a new particular judgment is obtained from the general original), then such a conversion is called a treatment with a restriction; if the conversion does not lead to a change in the original judgment in terms of quantity, then such a conversion is a conversion without restriction.
Examples and circulation schemes:
A: A general affirmative judgment turns into a particular affirmative one.
All lawyers are lawyers.
Some lawyers are lawyers.
All S are R.
Some P are S.
General affirmative singling out judgments circulate without restriction. Any offense (and only an offense) is an unlawful act.
Every wrongful act is a crime.
Scheme:
All S, and only S, are P.
All P are S.
E: A general negative judgment turns into a general negative one (without limitation).
No lawyer is a judge.
No judge is a lawyer.
No S is R.
No P is S.
I: Particular affirmative judgments turn into private affirmative ones.
Some lawyers are athletes.
Some athletes are lawyers.
Some S are R.
Some P are S.
Particularly affirmative highlighting judgments turn into general affirmative ones:
Some lawyers, and only lawyers, are lawyers.
All lawyers are lawyers.
Some S, and only S, are P.
All P are S.
A: Particularly negative judgments do not apply.
The logical operation of judgment reversal is of great practical importance. Ignorance of the rules of circulation leads to gross logical errors. So, quite often a universally affirmative judgment is drawn without restriction. For example, the proposition "All lawyers must know logic" becomes the proposition "All students of logic are lawyers." But this is not true. The proposition "Some students of logic are lawyers" is true.
Opposition to a predicate is the successive application of the operations of transformation and conversion - the transformation of a judgment into a new judgment, in which the concept that contradicts the predicate becomes the subject, and the subject of the original judgment becomes the predicate; the quality of judgment changes.
For example, from the proposition "All lawyers are lawyers" one can, by contrasting the predicate, get "No non-lawyer is a lawyer." Schematically:
All S are R.
No non-P is S.
Inference on the "logical square". The "logical square" is a scheme expressing truth relations between simple propositions that have the same subject and predicate. In this square, the vertices symbolize the simple categorical judgments known to us according to the combined classification: A, E, O, I. The sides and diagonals can be considered as logical relations between simple judgments (except for equivalent ones). Thus, the upper side of the square denotes the relation between A and E - the relation of the opposite; the lower side is the relationship between O and I - the relationship of partial compatibility. The left side of the square (the relationship between A and I) and the right side of the square (the relationship between E and O) is the relationship of subordination. The diagonals denote the relationship between A and O, E and I, which is called a contradiction.
The relationship of opposition takes place between judgments generally affirmative and generally negative (A-E). The essence of this relationship is that two opposing propositions cannot be both true at the same time, but they can be simultaneously false. Therefore, if one of the opposite judgments is true, then the other is necessarily false, but if one of them is false, then it is still impossible to unconditionally assert that it is true about the other judgment - it is indefinite, i.e., it can turn out to be both true and false. For example, if the proposition "Every lawyer is a lawyer" is true, then the opposite proposition "No lawyer is a lawyer" will be false.
But if the proposition “All students of our course have studied logic before” is false, then the opposite statement “No student of our course has studied logic before” will be indefinite, i.e., it can turn out to be either true or false.
The relation of partial compatibility takes place between the judgments of particular affirmative and particular negative (I - O). Such judgments cannot be both false (at least one of them is true), but they can be both true. For example, if the proposition "Sometimes you can be late for class" is false, then the proposition "Sometimes you cannot be late for class" will be true.
But if one of the judgments is true, then the other judgment, which is in relation to it in relation to partial compatibility, will be indefinite, i.e. it can be either true or false. For example, if the proposition "Some people study logic" is true, then the proposition "Some people do not study logic" will be true or false. But if the proposition "Some atoms are divisible" is true, then the proposition "Some atoms are not divisible" will be false.
The subordination relation consists in the fact that the truth of the subordinate judgment necessarily follows from the truth of the subordinate judgment, but the converse is not necessary: if the subordinate judgment is true, the subordinate will be indeterminate - it can turn out to be both true and false.
But if the subordinate judgment is false, then the subordinate will be all the more false. Again, the converse is not necessary: if the subordinate judgment is false, the subordinate may turn out to be both true and false.
For example, if the subordinate proposition "All lawyers are lawyers" is true, the subordinate proposition "Some lawyers are lawyers" will be all the more true. But if the subordinate judgment "Some lawyers are members of the Moscow Bar Association" is true, the subordinate judgment "All lawyers are members of the Moscow Bar Association" will be either false or true.
If the subordinate judgment “Some lawyers are not members of the Moscow Bar Association” (O) is false, the subordinate judgment “No lawyer is a member of the Moscow Bar Association” (E) will be false. But if the subordinating judgment “No lawyer is a member of the Moscow Bar” (E) is false, the subordinate judgment “Some lawyers are not members of the Moscow Bar” (O) will be true or false.
Relations of contradiction exist between general affirmative and particular negative judgments (A - O) and between general negative and particular affirmative judgments (E - I). The essence of this relation is that of two contradictory judgments, one is necessarily true, the other is false. Two contradictory propositions cannot be both true and false at the same time.
Inferences based on the relation of contradiction are called the negation of a simple categorical judgment. By negating a proposition, a new proposition is formed from the original proposition, which is true when the original proposition (premise) is false, and false when the original proposition (premise) is true. For example, denying the true proposition "All lawyers are lawyers" (A), we get a new, false, proposition "Some lawyers are not lawyers" (O). Rejecting the false proposition "No lawyer is a lawyer" (E), we get a new, true proposition "Some lawyers are lawyers" (I).
Knowing the dependence of the truth or falsity of some judgments on the truth or falsity of other judgments helps to draw correct conclusions in the process of reasoning.
3. Simple categorical syllogism
The most widespread type of deductive reasoning is categorical reasoning, which, because of its form, is called syllogism (from the Greek sillogismos - counting).
A syllogism is a deductive conclusion in which two categorical propositions-parcels connected by a common term yield a third proposition - a conclusion.
In the literature, there is the concept of a categorical syllogism, a simple categorical syllogism, in which the conclusion is obtained from two categorical judgments.
Structurally, the syllogism consists of three main elements - terms. Let's look at this with an example.
Every citizen Russian Federation has the right to education.
Novikov is a citizen of the Russian Federation.
Novikov - has the right to education.
The conclusion of this syllogism is a simple categorical proposition A, in which the scope of the predicate "has the right to be formed" is wider than the scope of the subject - "Novikov". Because of this, the predicate of the inference is called the major term, and the subject of the inference is called the minor term. Accordingly, the premise, which includes the inference predicate, i.e. the larger term is called the major premise, and the premise with the smaller term, the subject of the conclusion, is called the minor premise of the syllogism.
The third concept "citizen of the Russian Federation", through which a connection is established between the larger and smaller terms, is called the middle term of the syllogism and is denoted by the symbol M (Medium - mediator). The middle term is included in every premise, but not in the conclusion. The purpose of the middle term is to be a link between the extreme terms - the subject and the predicate of the conclusion. This connection is carried out in the premises: in the major premise, the middle term is associated with the predicate (M - P), in the minor premise - with the subject of the conclusion (S - M). The result is the following scheme of the syllogism.
M - R S - M
S - M or M - R R - M - S
S - R S - R
In doing so, keep in mind the following:
1) the name "greater" or "lesser" premise does not depend on the location in the syllogism scheme, but only on the presence of a larger or smaller term in it;
2) from a change in the place of any term in the premise, its designation does not change - the larger term (the predicate of the conclusion) is denoted by the symbol P, the smaller one (the subject of the conclusion) - by the symbol S, the middle one - by M;
3) from a change in the order of the premises in the syllogism, the conclusion, i.e. the logical connection between the extreme terms is independent.
Therefore, the logical analysis of the syllogism must begin with the conclusion, with the clarification of its subject and predicate, with the establishment from here - the major and minor term of the syllogism. One way to establish the correctness of syllogisms is to check whether the rules of syllogisms are followed. They can be divided into two groups: rules of terms and rules of premises.
A widespread type of mediated inference is a simple categorical syllogism, the conclusion of which is obtained from two categorical propositions.
In contrast to the terms of the judgment - the subject ( S) and predicate ( R) - the concepts that make up the syllogism are called
terms of the syllogism.
There are lesser, greater and middle terms.
Lesser syllogism term
the concept is called, which in the conclusion is the subject.
Big syllogism term
a concept is called, which in the conclusion is a predicate (“has the right to protection”). The smaller and larger terms are called
extreme
and are denoted respectively by Latin letters S(smaller term) and R(larger term).
Each of the extreme terms is included not only in the conclusion, but also in one of the premises. A premise that includes a smaller term is called
smaller package,
a premise that includes a larger term is called
larger shipment.
For the convenience of analyzing the syllogism, the premises are usually arranged in a certain sequence: the larger one is in the first place, the smaller one is in the second. However, such an order is not necessary in the argument. The smaller premise can be in the first place, the larger premise in the second. Sometimes the parcels are after the conclusion.
The premises differ not in their place in the syllogism, but in the terms included in them.
A conclusion in a syllogism would be impossible if it did not have a middle term.
The middle term of the syllogism
is called a concept that is included in both premises and is absent in detention (in our example - "accused"). The middle term is denoted by a Latin letter M.
The middle term connects the two extreme terms. The relation of the extreme terms (subject and predicate) is established by their relation to the middle term. Indeed, we know from the major premise that the relation of the major term to the middle term (in our example, the relation of the concept “has the right to defense” to the concept of “accused”) from the minor premise is the relation of the minor term to the middle term. Knowing the ratio of the extreme terms to the mean, we can establish the relationship between the extreme terms.
The conclusion from the premises is possible because the middle term acts as a link between the two extreme terms of the syllogism.
The legitimacy of the conclusion, i.e. logical transition from premises to conclusion, in a categorical syllogism is based on the position
(the axiom of the syllogism): everything that is affirmed or denied with respect to all objects of a certain class is affirmed or denied with respect to each object and any part of the objects of this class.
Figures and modes of categorical syllogism
In the premises of a simple categorical syllogism, the middle term can take the place of a subject or a predicate. Depending on this, four types of syllogism are distinguished, which are called figures (Fig.).
In the first figure the middle term takes the place of the subject in the major and the place of the predicate in the minor premise.
In second figure- the place of the predicate in both premises. AT third figure- the place of the subject in both premises. AT fourth figure- the place of the predicate in the major and the place of the subject in the minor premise.
These figures exhaust all possible combinations of terms. The figures of a syllogism are its varieties, which differ in the position of the middle term in the premises.
The premises of a syllogism can be judgments that are different in quality and quantity: generally affirmative (A), generally negative (E), particular affirmative (I) and particular negative (O).
Varieties of syllogism, differing in quantitative and quality characteristics premises are called modes of a simple categorical syllogism.
It is not always possible to get a true conclusion from true premises. Its truth is determined by the rules of the syllogism. There are seven of these rules: three pertain to terms and four pertain to premises.
Terms rules.
1st rule: in A syllogism should have only three terms.
The conclusion in a syllogism is based on the ratio of two extreme terms to the middle one, so there can be neither less nor more sin of terms in it. Violation of this rule is associated with the identification of different concepts, which are taken as one and are considered as a middle term. This error is based on violation of the requirements of the law of identity and is called a quadruple of terms.
2nd rule: the middle term must be distributed in at least one of the premises.
If the middle term is not distributed in any of the premises, then the connection between the extreme terms remains indefinite. For example, in the parcels “Some teachers ( M-) - members of the Union of Teachers ( R)”, “All employees of our team ( S) - teachers ( M-)" middle term ( M) is not distributed in the major premise, since it is the subject of a particular judgment, and is not distributed in the minor premise as a predicate of an affirmative judgment. Therefore, the middle term is not distributed in any of the premises, so the necessary connection between the extreme terms ( S and R) cannot be installed.
3rd rule: a term that is not distributed in the premise cannot be distributed in the conclusion.
Error, associated with a violation of the rule of distributed extreme terms,
is called an illegal extension of the smaller (or larger) term.
Parcel rules.
1st rule: at least one of the premises must be an affirmative proposition. From two negative premises, the conclusion does not necessarily follow. For example, from the premises “Students of our institute (M) do not study biology (P)”, “Employees of the research institute (S) are not students of our institute (M)”, it is impossible to obtain the necessary conclusion, since both extreme terms (S and P) are excluded from the middle. Therefore, the middle term cannot establish a definite relationship between the extreme terms. In conclusion, the minor term (M) may be included in whole or in part in the scope of the larger term (P) or completely excluded from it. In accordance with this, three cases are possible: 1) “Not a single employee of the research institute studies biology (S 1); 2) “Some research institute employees study biology” (S 2); 3) “All research institute employees study biology” (S 3) (fig.).
2nd rule: if one of the premises is a negative proposition, then the conclusion must also be negative.
The 3rd and 4th rules are derived from those considered.
3rd rule: at least one of the premises must be a general proposition.
A conclusion does not necessarily follow from two particular premises.
If both premises are particular affirmative judgments (II), then the conclusion cannot be made according to the 2nd rule of terms: in particular affirmative. neither the subject nor the predicate is distributed in the judgment, and therefore the middle term is not distributed in any of the premises.
If both premises are private negative propositions (00),
then the conclusion cannot be made according to the 1st rule of premises.
If one premise is partial affirmative and the other is partial negative (I0 or 0i), then in such a syllogism only one term will be distributed - the predicate of a particular negative judgment. If this term is the middle one, then the conclusion cannot be made, so, according to the 2nd rule of premises, the conclusion must be negative. But in this case, the predicate of the conclusion must be distributed, which contradicts the 3rd rule of terms: 1) a larger term that is not distributed in the premise will be distributed in the conclusion; 2) if the larger term is distributed, then the conclusion does not follow according to the 2nd rule of terms.
1) Some M(-) are P(-) Some S(-) are not (M+)
2) Some M(-) are not P(+) Some S(-) are M(-)
None of these cases gives the necessary conclusions.
4th rule: if one of the premises is a particular judgment, then the conclusion must also be particular.
If one premise is generally affirmative, and the other is particular affirmative (AI, IA), then only one term is distributed in them - the subject of a generally affirmative judgment.
According to the 2nd rule of terms, it must be the middle term. But in this case, the two extreme terms, including the smaller one, will not be distributed. Therefore, in accordance with the 3rd rule of terms, the lesser term will not be distributed in the conclusion, which will be a private judgment.
4. Inference from judgment with relations
An inference whose premises and conclusion are judgments with relations is called an inference with relations.
For example:
Peter is Ivan's brother. Ivan is Sergey's brother.
Peter is Sergey's brother.
The premises and conclusion in the above example are judgments with relations that have a logical structure xRy, where x and y are the concepts of objects, R are the relations between them.
The logical basis of inferences from judgments with relations are the properties of relations, the most important of which are 1) symmetry, 2) reflexivity and 3) transitivity.
1. A relation is called symmetrical (from the Greek simmetria - “proportionality”) if it takes place both between objects x and y, and between objects y and x. In other words, rearranging the members of a relation does not lead to a change in the type of relation. Symmetric relations are equality (if a is equal to b, then b is equal to a), similarity (if c is similar to d, then d is similar to c), simultaneity (if the event x happened simultaneously with the event y, then the event y happened). simultaneously with the event x), differences, and some others.
The symmetry relation is symbolically written:
xRy - yRx.
2. A relation is called reflexive (from the Latin reflexio - “reflection”) if each member of the relation is in the same relation to itself. These are the relations of equality (if a = b, then a = a and b = b) and simultaneity (if the event x happened simultaneously with the event y, then each of them happened simultaneously with itself).
The reflexivity relation is written:
xRy -+ xRx R yRy.
3. A relation is called transitive (from the Latin transitivus - “transition”) if it takes place between x and z when it takes place between x and y and between y and z. In other words, a relation is transitive (transitional) if and only if the relation between x and y and between y and z implies the same relation between x and z.
The relations of equality are transitive (if a is equal to b and b is equal to c, then a is equal to c), simultaneity (if the event x happened simultaneously with the event y and the event y happened simultaneously with the event z, then the event x happened simultaneously with the event z), relations “more”, “less” (a less than b, b less than c, which means a less than c), “later”, “to be north (south, east, west)”, “to be lower, higher”, etc.
The transitivity relation is written:
(xRy L yRz) -* xRz.
To obtain reliable conclusions from judgments with relations, it is necessary to rely on the rules:
For the symmetry property (xRy -* yRx): if xRy is true, then yRx is also true. For example:
A is like B. B is like A.
For the property of reflexivity (xRy -+ xRx - yRy): if xRy is true, then xRx and yRy are true. For example:
a = b. a = a and b = b.
For the property of transitivity (xRy l yRz -* xRz): if the proposition xRy is true and the proposition yRz is true, then the proposition xRz is also true. For example:
K. was at the scene before L. L. was at the scene before M.
K. was at the scene before M.
Thus, the truth of a conclusion from judgments with relations depends on the properties of the relations and is governed by the rules that follow from these properties. Otherwise, the conclusion may be false. Thus, from the judgments “Sergeev is acquainted with Petrov” and “Petrov is acquainted with Fedorov”, the necessary conclusion “Sergeev is acquainted with Fedorov” does not follow, since “to be acquainted” is not a transitive relation
Tasks and exercises
1. Indicate which of the following expressions - Consequence, "consequence", ""consequence"" - can be substituted for X in the following expressions to get true sentences:
b) X is a word of the Russian language;
c) X is an expression denoting a word;
d) X - has reached a dead end.
Solution
a) "consequence" - philosophical category;
Instead of X, you can substitute the word "consequence", taken in quotation marks. We get: "Reason" - a philosophical category.
b) "consequence" - the word of the Russian language;
c) ""consequence"" - an expression denoting a word;
d) the investigation has reached a "dead end"
2. Which of the following expressions are true and which are false:
a) 5 × 7 = 35;
b) "5 × 7" = 35;
c) "5 × 7" ≠ "35";
d) "5 × 7 = 35".
Solution
a) 5 x 7 = 35 TRUE
b) "5 x 7" = 35 TRUE
c) "5 x 7" ¹ "35" FALSE
d) "5 x 7 = 35" cannot be evaluated because it is a quoted name
b) Lao-tzu's mother.
Solution
a) If no member of the Gavrilov family is an honest person, and Semyon is a member of the Gavrilov family, then Semyon is not an honest person.
In this sentence, “if ... then ...” is a logical term, “none” (“all”) is a logical term, “a member of the Gavrilov family” is a common name, “not” is a logical term, “is” (“there are ”) is a logical term, “honest person” is a common name, “and” is a logical term, “Semyon” is a singular name.
b) Lao-tzu's mother.
"Mother" is an object functor, "Lao-Tzu" is a singular name.
4. Summarize the following concepts:
a) Correctional labor without imprisonment;
b) Investigative experiment;
c) the constitution.
Solution
The requirement to generalize a concept means a transition from a concept with a smaller volume, but with more content, to a concept with a larger volume, but with less content.
a) Corrective labor work without detention - corrective labor work;
b) investigative experiment - experiment;
c) The Constitution is the law.
a) Minsk is the capital;
Solution
a) Minsk is the capital. * Belongs to the category of things. In this case, the term "capital" acts as a predicate of the judgment, as it reveals the signs of the judgment.
b) The capital of Azerbaijan is an ancient city.
In this case, the term "capital" has a semantic judgment.
In this case, the term "capital" is the subject of the judgment, since the said judgment reveals its features.
6. What methodological principles are discussed in the following text?
Article 344 of the Code of Criminal Procedure of the Russian Federation specifies the condition under which the sentence is recognized as inconsistent with the act: "if there is conflicting evidence ...".
Solution
This text refers to the principle of non-contradiction.
7. Translate the following proposition into the language of predicate logic: "Every lawyer knows some (some) journalist."
Solution
This judgment is affirmative in terms of quality, and public in terms of quantity.
¬(А˄ V)<=>¬(A¬B)
8. Translate the following expression into the language of predicate logic: "The population of Ryazan is greater than the population of Korenovsk."
Solution
The population of Ryazan is larger than the population of Korenovsk
Here one should speak of a judgment about the relation between objects.
This sentence can be written as follows:
xRy
The population of Ryazan (x) is greater than (R) the population of Korenovsk (x)
9. In places of deprivation of liberty, a selective survey of those who committed serious crimes was conducted (10% of such persons were interviewed). Nearly all of them responded that the severe penalties did not affect their decision to commit a crime. They concluded that strict penalties are not a deterrent in the commission of serious crimes. Is this conclusion justified? If not substantiated, then what methodological requirements for scientific induction are not met?
Solution
In this case, it is necessary to talk about some statistical generalization, which is a conclusion of incomplete induction, within the framework of which quantitative information about the frequency of a certain trait in the group (sample) under study is determined in the premises and is transferred in the conclusion to the entire set of phenomena.
The message contains the following information:
case sample – 10%
the number of cases in which the feature of interest is present is almost all;
the frequency of occurrence of the feature of interest is almost 1.
Hence, it can be noted that the frequency of occurrence of the feature is almost 1, which can be said to be an affirmative conclusion.
At the same time, it cannot be said that the resulting generalization - severe penalties are not a deterrent in the commission of serious crimes - is correct, since the statistical generalization, being the conclusion of incomplete induction, refers to non-demonstrative conclusions. The logical transition from premises to conclusion conveys only problematic knowledge. In turn, the degree of validity of statistical generalization depends on the specifics of the studied sample: its size in relation to the population and representativeness (representativeness).
10. Limit the following concepts:
a) the state;
b) court;
c) revolution.
Solution
a) state - the Russian state;
b) the court - the Supreme Court
c) revolution - October revolution - world revolution
11. Give a complete logical description of the concepts:
a) People's Court;
b) worker;
c) out of control.
Solution
a) The people's court is a single, non-collective, concrete concept;
b) worker - a general, non-collective, specific, irrelevant concept;
c) lack of control is a single, non-collective, abstract concept.
The concept of deductive reasoning. Simple categorical syllogism Form of law
The properties of the basic concepts are revealed in axioms- proposals accepted without proof.
For example, in school geometry there are axioms: “a straight line can be drawn through any two points and only one” or “a straight line divides a plane into two half-planes.”
The system of axioms of any mathematical theory, revealing the properties of the basic concepts, gives their definitions. Such definitions are called axiomatic.
Proved properties of concepts are called theorems, consequences signs, formulas, rules.
Prove the theorem BUTAT- it means to set in a logical way that whenever the property is executed BUT, the property will be executed AT.
Proof in mathematics, a finite sequence of sentences of a given theory is called, each of which is either an axiom or is derived from one or more sentences of this sequence according to the rules of inference.
The proof is based on reasoning - a logical operation, as a result of which one or more sentences related in meaning result in a sentence containing new knowledge.
As an example, consider the reasoning of a schoolboy who needs to establish the ratio "less than" between the numbers 7 and 8. The student says: "7< 8, потому что при счете 7 называют раньше, чем 8».
Let us find out on what facts the conclusion obtained in this reasoning is based.
There are two such facts: First: if the number a when counting, they call before the number b, then a< b. Second: 7 is called earlier than 8 when counting.
The first sentence is general in nature, since it contains a general quantifier - it is called a general premise. The second sentence concerns the specific numbers 7 and 8 - it is called a private premise. A new fact is obtained from two premises: 7< 8, его называют заключением.
There is a certain connection between the premises and the conclusion, thanks to which they constitute an argument.
Reasoning, between the premises and the conclusion of which there is a relation of consequence, is called deductive.
In logic, instead of the term "reasoning", the word "inference" is more often used.
inference It is a way of obtaining new knowledge on the basis of some existing one.
An inference consists of premises and a conclusion.
Parcels- is containing the original knowledge.
Conclusion- this is a statement containing new knowledge obtained from the original.
As a rule, the conclusion is separated from the premises with the help of the words "therefore", "means". Inference with parcels R 1, R 2, …, рn and conclusion R we will write in the form: or (R 1, R 2, …, рn) R.
Examples inferences: a) Number a =b. Number b = c. Therefore, the number a = s.
b) If the numerator is less than the denominator, then the fraction is proper. In fraction numerator less than denominator (5<6) . Therefore, the fraction - correct.
c) When it rains, there are clouds in the sky. There are clouds in the sky, so it's raining.
Inferences can be right or wrong.
The inference is called correct if the formula corresponding to its structure and representing the conjunction of the premises, connected with the conclusion by the sign of the implication, is identically true.
For to determine whether the conclusion is correct, proceed as follows:
1) formalize all premises and conclusion;
2) write down a formula representing the conjunction of premises connected by an implication sign with the conclusion;
3) make up a truth table for this formula;
4) if the formula is identically true, then the conclusion is correct, if not, then the conclusion is incorrect.
In logic, it is believed that the correctness of an inference is determined by its form and does not depend on the specific content of the statements included in it. And in logic, such rules are proposed, observing which, one can build deductive conclusions. These rules are called inference rules or schemes of deductive reasoning.
There are many rules, but the following are the most commonly used:
1.
- conclusion rule;
2.
- the rule of negation;
3.
- the rule of syllogism.
Let's bring example inference made by rule conclusions:"If the entry of a number X ends with a number 5, that number X divided by 15. Writing a number 135 ends with a number 5 . Therefore, the number 135 divided by 5 ».
As a general premise in this conclusion, the statement “if Oh), then B(x)", where Oh) is a "record of a number X ends with a number 5
", a B(x)- "number X divided by 5
". A private premise is a statement that results from the condition of a general premise when
x = 135(those. A(135)). The conclusion is a statement derived from B(x) at x = 135(those. B(135)).
Let's bring an example of a conclusion made according to the rule negations:"If the entry of a number X ends with a number 5, that number X divided by 5 . Number 177 not divisible by 5 . Therefore, it does not end with a number 5 ».
We see that in this conclusion the general premise is the same as in the previous one, and the private one is a negation of the statement "the number 177 divided by 5 » (i.e.). The conclusion is the negation of the sentence "Recording the number 177 ends with a number 5 » (i.e.).
And finally, consider example of an inference based on syllogism rule: "If the number X multiple 12, then it is a multiple 6. If number X multiple 6 , then it is a multiple 3 . Therefore, if the number X multiple 12, then it is a multiple 3 ».
There are two premises in this conclusion: “if Oh), then B(x)" and if B(x), then C(x)”, where A (x) - “number X multiple 12 », B(x)- "number X multiple 6 " and C(x)- "number X multiple 3 ". The conclusion is the statement "if Oh), then C(x)».
Let's check if the following conclusions are correct:
1) If a quadrilateral is a rhombus, then its diagonals are mutually perpendicular. ABCD- rhombus. Therefore, its diagonals are mutually perpendicular.
2) If the number is divisible by 4 , then it is divisible by 2 . Number 22 divided by 2 . Therefore, it is divided into 4.
3) All trees are plants. Pine is a tree. So pine is a plant.
4) All students of this class went to the theatre. Petya was not in the theatre. Therefore, Petya is not a student of this class.
5) If the numerator of a fraction is less than the denominator, then the fraction is correct. If the fraction is correct, then it is less than 1. Therefore, if the numerator of the fraction is less than the denominator, then the fraction is less than 1.
Solution: 1) To resolve the issue of the correctness of the conclusion, we will identify its logical form. Let us introduce the notation: C(x)- quadrilateral X- rhombus, B(x)- in a quadrilateral X diagonals are mutually perpendicular. Then the first message can be written as:
C(x) B(x), second - C(a), and the conclusion B(a).
Thus, the form of this inference is as follows: 
. It is built according to the rule of conclusion. Therefore, this reasoning is correct.
2) Let's introduce the notation: Oh)- "number X divided by 4
», B(x)- "number X divided by 2
". Then we write the first message: Oh)B(x), second B(a), and the conclusion is A(a). The conclusion will take the form: 
.
There is no such logical form among the known ones. It is easy to see that both premises are true and the conclusion is false.
This means that this reasoning is wrong.
3) Let us introduce the notation. Let Oh)- "if X wood", B(x) - « X plant". Then the messages will look like: Oh)B(x), A(a), and the conclusion B(a). Our conclusion is built in the form: 
- Conclusion rules.
So our reasoning is correct.
4) Let Oh) - « X- Students in our class B(x)- “students X went to the theatre." Then the messages will be as follows: Oh)B(x),, and the conclusion.
This conclusion is built according to the rule of negation:
- means it is correct.
5) Let's reveal the logical form of the conclusion. Let A(x) -"numerator of a fraction X less than the denominator. B (x) - "fraction X- correct. C(x)- "fraction X less 1 ". Then the messages will look like: Oh)B(x), B(x) C(x), and the conclusion Oh)C(x).
Our conclusion will be of the following logical form: 
- the rule of syllogism.
So this conclusion is correct.
In logic, various methods of checking the correctness of inferences are considered, among which analysis of the correctness of inferences using Euler circles. It is carried out as follows: the conclusion is written in the set-theoretic language; depict the parcels on the circles of Euler, considering them to be true; they look to see if the conclusion is always true. If so, then the conclusion is said to be correct. If a drawing is possible from which it is clear that the conclusion is false, then the conclusion is said to be wrong.
Table 9
Verbal formulation of the sentence | Recording in set-theoretic language | Image on Euler circles | ||
Anything BUT there is AT |
| |||
Some BUT there is AT Some BUT do not eat AT | ||||
None BUT do not eat AT |
a there is BUT | ||
a do not eat BUT |
Let us show that the inference made according to the rule of conclusion is deductive. Let us first write this rule in set-theoretic language.
Package Oh)B(x) can be written in the form TATV, where TA and TV- truth sets of propositional forms Oh) and B(x).
private package A(a) means that aTA, and the conclusion B(a) shows that aTV.
The whole inference, built according to the rule of conclusion, will be written in the set-theoretic language as follows: 
.
| ||||
| ||||
Rice. 58.
Examples.
1. Is the conclusion correct “If the entry of a number ends with a number 5, then the number is divisible by 5. Number 125 divided by 5. Therefore, writing a number 125 ends with a number 5 »?
Solution: This conclusion is made according to the scheme 
, which corresponds 
. There is no such scheme among the known to us. Let's find out if it is a rule of deductive reasoning?
Let's use the Euler circles. In set-theoretic language
The resulting rule can be written as follows:
. Let us represent on the Euler circles the sets TA and TV and denote the element a from many TV.
It turns out that it can be contained in the set TA, or maybe not belong to him (Fig. 59). In logic, it is believed that such a scheme is not a rule of deductive reasoning, since it does not guarantee the truth of the conclusion.
This conclusion is not correct, since it is made according to a scheme that does not guarantee the truth of the reasoning.
| |||
Rice. 59.
b) All verbs answer the question "what to do?" or “what to do?”. The word "cornflower" does not answer any of these questions. Therefore, "cornflower" is not a verb.
Solution: a) Let us write this conclusion in the set-theoretic language. Denote by BUT- a lot of students of the pedagogical faculty, through AT- many students who are teachers, through FROM- many students over 20 years old.
Then the conclusion will take the form: 
.
If you depict these sets on circles, then 2 cases are possible:
1) sets A, B, C intersect;
2) set AT intersects with many FROM and BUT, and the set BUT intersects AT, but does not intersect with FROM.
b) Denote by BUT many verbs, and AT many words that answer the question "what to do?" or “what to do?”.
Then the conclusion can be written as follows:
Let's look at a few examples.
Example 1 The student is asked to explain why the number 23 can be represented as the sum 20 + 3. He argues: “The number 23 is two-digit. Any two-digit number can be represented as a sum of bit terms. Therefore, 23 = 20 + 3."
The first and second sentences in this inference of the premise, and one of a general nature is the statement “any two-digit number can be represented as a sum of bit terms”, and the other is private, it characterizes only the number 23 - it is two-digit. The conclusion - this sentence that comes after the word "therefore" - is also private, since it deals with the specific number 23.
Inferences that are commonly used in proving theorems are based on the concept of logical consequence. Moreover, from the definition of logical consequence it follows that for all values of propositional variables for which the original statements (premises) are true, the conclusion of the theorem is also true. Such inferences are deductive.
In the example discussed above, the above inference is deductive.
Example 2 One of the methods for introducing younger students to the commutative property of multiplication is as follows. Using various visual aids, students, together with the teacher, establish that, for example, 6 3 = 36, 52 = 25. Then, based on the obtained equalities, they conclude: for all natural numbers a and b true equality ab=ba.
In this conclusion, the premises are the first two equalities. They state that such a property holds for concrete natural numbers. The conclusion in this example is a general statement - the commutative property of multiplication of natural numbers.
In this conclusion, the premises of a particular nature show that some natural numbers have the property that the product does not change from a permutation of factors. And on this basis, it was concluded that all natural numbers have this property. Such reasoning is called incomplete induction.
those. for some natural numbers it can be argued that the sum is less than their product. So, based on the fact that some numbers have this property, we can conclude that all natural numbers have this property:
This example is an example of reasoning by analogy.
Under analogy understand a conclusion in which, based on the similarity of two objects in some features and in the presence of an additional feature, one of them concludes that the other object has the same feature.
The conclusion by analogy is in the nature of an assumption, a hypothesis and therefore needs either proof or refutation.
CONCLUSION
CONCLUSION
Lit.: Chelpanov G. I., Textbook of logic, M., 1946; Asmus V. F., Logic, M., 1947; his, Teaching of logic about proof and refutation, M., 1954; Tarsky A., Introduction to the logic and methodology of deductive sciences, trans. from English, M., 1948; Gorsky D.P., Logic, 2nd ed., M., 1963; Church A., Introduction to mathematical logic, trans. from English, vol. 1, M., 1960.
A. Subbotin. Moscow.
Philosophical Encyclopedia. In 5 volumes - M .: Soviet Encyclopedia. Edited by F. V. Konstantinov. 1960-1970 .
CONCLUSION
CONCLUSION - the procedure for the direct derivation of some statement from one or more statements. The statements from which the conclusion is drawn are called the premises of the inference, and the statement that is derived from the premises is called the conclusion. Inference is a cognitive technique by which the transformation of the information contained in the premises is carried out. It is the simplest kind of reasoning - the procedure for substantiating a statement by deriving it step by step from other statements; in a conclusion, the transition from arguments (their role is played by premises) to a substantiated thesis (conclusion) occurs in one step. In logic, the conclusion is usually formulated as follows: Αι,Α2,...,Αη
B --- where the premises are written above the line, the conclusion is written below the line, and the line itself expresses the derivation of the conclusion from the premises.
According to the degree of validity of deriving a conclusion from premises, inferences are usually divided into demonstrative and non-demonstrative. In demonstrative inferences, the simultaneous truth of the premises ensures the receipt of a true conclusion; the conclusions in them constitute part of the aggregate information of the premises. In non-demonstrative reasoning, on the contrary, when moving from premises to conclusion, it has an increment of information, however, the simultaneous truth of the premises does not guarantee the truth of the conclusion.
The most important and extensive variety of demonstrative reasoning is deductive reasoning. There is a logical consequence between their premises and the conclusion, i.e. the very logical form of these conclusions ensures the preservation of truth when deriving the conclusion from the premises.
In demonstrative reasoning of other types (these include, for example, complete induction, strict) the conclusion obtained from the true premises is determined not only by the logical form of the statements included in the conclusion, but also by the meanings of the descriptive terms contained in them, by the features of the universe of reasoning.
Among the non-demonstrative conclusions, the greatest are the so-called. plausible inferences, which include, for example, reverse, incomplete induction, non-strict analogy, statistical inference. Plausible inferences are characterized by the presence of a logical confirmation relationship between the premises and the conclusion. This relation has many different explications in modern logic. Thus, the interpretation of the confirmation relation in accordance with the criterion of positive relevance has become widespread: the premises confirm the conclusion if and only if the truth of the conclusion increases (but does not become equal to one) under the condition that the premises are simultaneously true. The main field of application of deductive reasoning is the exact sciences (primarily logic), in which special requirements are imposed on the rigor of evidence. Plausible Inferences, ch. O., are used in the empirical sciences to put forward and verify hypotheses, to obtain law-like statements related to the subject area under study.
V. I. Markin
New Philosophical Encyclopedia: In 4 vols. M.: Thought. Edited by V. S. Stepin. 2001 .
Synonyms:
Antonyms:
See what "INCLUSION" is in other dictionaries:
Inference... Spelling Dictionary
See the conclusion to build a conclusion ... Dictionary of Russian synonyms and expressions similar in meaning. under. ed. N. Abramova, M .: Russian dictionaries, 1999. inference conclusion, traduction, enthymeme, deduction, reasoning, syllogism, sophism, paralogy, ... ... Synonym dictionary
CONCLUSION, conclusions, cf. 1. The logical process of deriving a conclusion from two judgments, a syllogism (philosophical). Deductive reasoning. 2. Conclusion, conclusion (book). Make a conclusion. Correct inference. Explanatory Dictionary of Ushakov ... Explanatory Dictionary of Ushakov
inference- one of the logical forms of thinking (see also concept and judgment). U. is characterized by the conclusion based on the rules of logic of the conclusion or consequence of several judgments (parcels). In logic, U. classifications are being developed. Psychology ... Great Psychological Encyclopedia
Mental action based on the norms of conclusions inherent in individual consciousness, largely coinciding with the rules and laws of logic ... Big Encyclopedic Dictionary
Establishing a connection between any judgments. It is carried out in verbal form, due to which it is possible to get out of the influence of the perceptual field ... Psychological Dictionary
A form of thought by which a new judgment is made on the basis of one or more judgments already made. The initial judgments, on the basis of which a new judgment is obtained, are called the premises of U., and the new judgment resulting from ... ... The latest philosophical dictionary
CONCLUSION, I, cf. (book). Conclusion, conclusion (in 3 values). Make, withdraw from. Correct at. Explanatory dictionary of Ozhegov. S.I. Ozhegov, N.Yu. Shvedova. 1949 1992 ... Explanatory dictionary of Ozhegov
English conclusion/deduction; German Schlu?folgerung. Reasoning, in the course of which from one or several judgments, called U.'s premises, a new judgment is deduced, which logically follows from the premises. Antinazi. Encyclopedia of Sociology, 2009 ... Encyclopedia of Sociology
Ticket number 7
Reasoning and inference. The structure of the inference. deductive reasoning. Correct and incorrect inferences. Deductive reasoning from categorical judgments. Direct and indirect inferences.
Reasoning is the procedure for substantiating a certain statement by deriving it step by step from other statements.
The simplest kind of reasoning is inference.
Inference is a direct transition from one statement or several statements A 1 , BUT 2 , …, A n to the statement B.
sayings BUT 1 , BUT 2 , …, A n , from which the conclusion is drawn are called parcels, and the statement AT, which is derived from the premises, is called conclusion.
As an example of a conclusion, let us cite the reasoning that, according to legend, was carried out by Caliph Omar to justify the need to burn the Library of Alexandria:
“If your books agree with the Quran, then they are superfluous. If your books do not agree with the Qur'an, then they are harmful. But harmful or superfluous books should be destroyed. Therefore, your books should be destroyed."
In the above conclusion, the first three statements are premises, and the fourth is the conclusion.
In logic, the conclusion is usually formulated as follows:
BUT 1 , BUT 2 , …, A n ,
Where the premises are written above the line, the conclusion is written below the line, and the line itself expresses the act of deriving the conclusion from the premises.
Inference is the simplest type of reasoning because the thesis being substantiated (conclusion B plays its role) is directly, as if in one step, deduced from the premises BUT 1 , BUT 2 , …, A n, which can be considered as arguments in favor of the thesis.
However, many arguments have a much more complex structure. So, in the course of reasoning, several conclusions can be made, and the conclusions of some can become premises in others. Consider an example.
In one English city, a bank robbery was committed. Suspicion fell on the well-known recidivists Smith, Jones and Brown. During the investigation, the following was revealed. Jones never goes to work without Brown. At least one of the repeat offenders - Smith or Jones - is involved in the crime. Brown has a solid alibi. The inspector of police who conducted the investigation, on the basis of this information, charged Smith.
However, he could argue as follows. The data obtained during the investigation indicate that:
(1) If Jones is involved in a crime, then Brown is also involved in it (Jones does not go to work without Brown).
(2) Brown is not involved in the crime (he has an alibi)
Consequently,
(3) Jones is not involved in the crime.
But, according to the investigation,
(4) Smith or Jones is involved in the crime.
Therefore, given the innocence of Jones in the crime, we can conclude:
(5) Smith is involved in a crime.
In the above discussion, two conclusions are made. In the first of them, statements (1) and (2) are premises, and statements (3) are conclusions. In the second inference, the premises are (3) and (4), and the conclusion is the statement (5).
Sometimes in the course of reasoning, in order to substantiate a certain statement (let's call it C), the so-called indirect ways of reasoning. In this case, auxiliary arguments are constructed, and additional arguments are introduced into their composition. assumptions, from which they seek to obtain consequences of a certain kind (the nature of the assumptions accepted and the desired consequences usually depend on the type of statement C). If the indicated problems are successfully solved, the auxiliary reasoning is considered completed, and the statement by S appears in the main part of the reasoning.
An example of an indirect method of argumentation is the widespread reasoning by contradiction. Their structure is as follows. To substantiate statement B, the statement “It is not true that C” is taken as an additional assumption, while from the assumption and a certain set of arguments D, they seek to obtain a contradiction - the statement “D and it is not true that D”. With the successful implementation of this auxiliary reasoning, it is considered that the assumption was false, and B itself is justified by means of the arguments of G.
Let us show how the police inspector in the considered example could come to the conclusion that Smith was guilty, arguing from the contrary.
Let us first make the assumption that
(1) Smith is not involved in the crime.
From this assumption and established fact:
(2) Smith or Jones are involved in a crime - we get the statement:
(3) Jones is involved in the crime.
From it, as well as from another fact established during the investigation:
(4) If Jones is involved in the crime, then Brown is involved in it - we get the statement:
(5) Brown is involved in the crime.
However, the investigation found that
(6) Brown is not involved in the crime.
Thus, in the argument, a contradiction is obtained:
(7) Brown is involved and not involved in the crime.
Therefore, assumption (1) is false, and the statement
(8) Smith is involved in a crime
It is considered justified from the arguments (2), (4) and (6).
Deductive reasoning and reasoning.
Deduction(lat. deductio- derivation) - a method of thinking in which a particular position is logically derived from a general one, a conclusion according to the rules of logic; a chain of inferences (reasoning), the links of which (statements) are connected by a relation of logical consequence.
Logic is often defined as the science of reasoning. Indeed, the study of reasoning, their types and methods of implementation is one of the main tasks of logic. Nevertheless, the methods of logical analysis considered so far concerned the verification of the correctness or incorrectness of ready-made reasoning and did not touch upon the question of how they are carried out. Let's describe the procedure deductive reasoning, which are also called believable.
In the general case, reasoning is understood as the procedure of a successive step-by-step transition from some statements, accepted as initial ones, to other statements. Each step of this process is carried out on the basis of some rule called inference rule. The last statement obtained in this process is called conclusion reasoning. At the same time, we will further include among the deductive reasoning only those reasonings in which the relation of logical consequence is preserved between the statements taken as initial ones and the conclusion. To answer now specifically the question of how deductive-type reasoning is constructed, it is required to develop some theory - theory of deductive reasoning. But before that, let's briefly describe the main types theories.
Deduction is a theoretical way of knowing the world around us. Therefore, deduction procedures are used when empirical cognitive methods (observations, experiments, measurements) are not enough to obtain some new knowledge. In this capacity, deduction is widely used already in everyday life: after all, we often try to defend our point of view through this or that reasoning, to convince our interlocutor of its truth, to refute the opponent’s point of view, etc., that is, we try to reason theoretically. However, the deduction procedure, as a theoretical research method, is of the greatest importance in the construction of scientific (theoretical) knowledge.
Depending on the degree of clarity (revelation) of deductive connections between individual statements (statements) of theories, several types of them are distinguished. The first type includes content theories. In their composition, if deduction is used, then only to connect some individual provisions of the theory. In this case, the initial statements in the reasoning are some assumptions, called parcels. The premises do not have to be (and do not always have to be) true, and therefore any sentence that is deduced using them is considered not true, but conditionally true: the final sentence (conclusion) is true provided that the premises are true. Such a character is, for example, reasoning in everyday life. Examples of meaningful theories are school arithmetic, as well as various kinds of scientific concepts developed in those sciences in which there are no strictly defined theories. Examples of logical content theories are propositional and predicate logics.
Another type is formalized theories. These include theories, the content of which is interconnected and deductively derived from some of the original assumptions. The latter are called axioms, and the theories themselves are called axiomatized theories. Examples of them are: Newton's celestial mechanics, Einstein's theory of relativity, quantum mechanics, Euclid's geometry. Unlike the geometry of Euclid, formalized more than 2 thousand years ago, arithmetic developed as a meaningful theory until the 20th century, and only at the turn of the 19th and 20th centuries was it formalized by the Italian mathematician Peano.
deductive reasoning
The beginning (premises) of deduction are axioms or simply hypotheses that have the character of general statements (“general”), and the end is consequences from premises, theorems (“special”). If the premises of the deduction are true, then so are its consequences. Deduction is the main means of proof. The opposite of induction.
An example of deductive reasoning:
1) All people are mortal.
2) Socrates is a man.
3) Therefore, Socrates is mortal.
Inferences in which one of the premises is a disjunctive judgment, and the second coincides with one of the members of the disjunctive judgment or denies all but one. In the conclusion, respectively, all terms are denied, except for the one indicated in the second premise, or the omitted term is affirmed.
Forms of the correct modes of divisive-categorical conclusions
Conditional inference
Inferences, premises and conclusions of which - conditional propositions.
A special kind of inference from two conditional propositions and one separating.
Types of right dilemmas:
constructive:
(i.e.: first premise: if A, then C; second premise: if B, then C; third premise: A or B; conclusion: therefore C);
(complex)
(i.e.: first premise: if A, then B; second premise: if C, then D; third premise: A or C; conclusion: therefore B or D);
destructive:
(ie: first premise: if A, then B; second premise: if A, then C; third premise: not B or not C; conclusion: therefore not A);
(complex)
(ie: first premise: if A, then B; second premise: if C, then D; third premise: not B or not D; conclusion: therefore not A or not C).
Correct and incorrect inferences
In order to show that a certain conclusion is wrong, it is enough to find at least one conclusion of the same logical form, all premises of which are true, and the conclusion is false. Thus, we have identified criterion for wrong inference. It can be formulated as follows.
An inference is incorrect if and only if its logical form does not guarantee that with true premises we will necessarily get a true conclusion, that is, there is an inference of this logical form with true premises and a false conclusion.
Now it is easy to formulate criterion for the correctness of inferences.
An inference is correct if and only if its logical form does not guarantee that if the premises are true, we will necessarily get a true conclusion, that is, there is no inference of this form with true premises and a false conclusion.
When this condition is satisfied, it is also said that between the premises and the conclusion there is logical consequence relation, that the conclusion logically follows from parcels.
Among the correct ones is, for example, inference (1). Let's find out its logical form. To this end, we will replace the simple statements that are part of its premises and conclusions with parameters: the statement "Your books agree with the Koran" - the letter p, « Your books are superfluous" - by letter q, "Your books are harmful" - letter r, “Your books should be destroyed” - letter s. We get an expression as a result.
If a p, then q
If it is not true that p, then r
If a q or r, then s
Mediated and non-mediated inferences
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