Inferences are divided into the following types.

1. Depending on the strictness of the rules of inference, demonstrative (necessary) and non-demonstrative (plausible) inferences are distinguished. Demonstrative inferences are characterized by the fact that the conclusion necessarily follows from the premises, i.e. logical consequence in such conclusions is a logical law. In non-demonstrative inferences, the rules of inference provide only the probabilistic conclusion of the conclusion from the premises.

2. The classification of inferences according to the direction of logical consequence is important, i.e. by the nature of the connection between knowledge of varying degrees of generality, expressed in premises and conclusion. From this point of view, three types of inferences are distinguished: deductive (from general knowledge to particular), inductive (from particular knowledge to general), inferences by analogy (from particular knowledge to particular).

Deductive (from the Latin deductio - “inference”) is an inference in which the transition from general knowledge to specific knowledge is logically necessary.

The rules of deductive inference are determined by the nature of the premises, which can be simple (categorical) or complex propositions. Depending on the number of premises, deductive conclusions from categorical judgments are divided into direct, in which the conclusion is deduced from one premise, and mediated, in which the conclusion is deduced from two premises.

Types of deductive inferences: purely conditional or hypothetical syllogisms.

This is an inference with both premises and a conclusion, which are conditional propositions. If the ball is heated, it will increase in volume. If the ball increases in volume, it will not fit into the ring. If the ball is heated, it will not fit into the ring. In order for the inference to be reliable, one more thing is needed - the conditions must be sufficient.

Conditional categorical silogism is a conclusion in which one of the premises is a conditional proposition, and the other premise, as well as the conclusion, is a simple categorical proposition. It has 2 modes: affirmative and negating.

The assertor (M Ponens) in the affirmative mode finally makes an assertion of the truth of the consequent of the conditional premise, based on the assertion of the truth of the anti-incident in the 2nd categorical premise.

If water is heated to 100 it will boil. She was heated. She's seething.

Wrong affirmative mode.

The point is that the mode is only probabilistic. If you are smart, then you are rich. Rich. Smart

Denying (modus tolens).

If it's gold, it glitters. If it doesn't shine. It's not gold.

Wrong affirmative mode. The conclusion is only probabilistic.

If the firewood is birch, then it gives a lot of heat. They are not birch. They provide little heat.

Affirmative-denying. In the 2nd premise of this mode, one and only one member of the disjunction is affirmed, and in the conclusion, all the others are negated. You can pass the exam either well, excellent, or satisfactory. The student scored well in the exam. This means that I did not receive excellent and satisfactory results.

A valid conclusion follows from the premises if and only if the following rules are met: the disjunctive premise must be strictly a disjunction.

Denying-affirming. This means that in the second (negating premise) all members of the disjunction are negated, except one, and on this basis, at the end the truth of one and only one member of the disjunction is affirmed.

Rules: 1) The dividing premise does not have to be a disjunction, but it must contain all possible alternatives. Violation of this rule does not guarantee the accuracy of the conclusion. The constituent of a simple proposition can be either P or S. This constituent is S, therefore it is P

Purely divisive syllogism.

This is an inference and the premises and conclusion of which are divisive (disjunctive) judgments.

You can pass or fail the exam.

You can pass O, X, Y

You can pass O, X, Y, or not pass

Explained by the presence of a strict disjunction in the premises.

Conditionally dividing syllogism. Dilemma.

This is a conclusion of which one of the premises is a conditional proposition, the other premise and conclusion are divisive (strict disjunction). If you go to the left - the horse, to the right - the head. Either one or the other.

A dilemma is a type of conditionally divisive syllogism in the conclusion of which two alternatives are stated.

Knowledge in any field of science and practice begins with empirical knowledge. In the process of observing similar natural and social phenomena, attention is paid to the repeatability of certain signs in them. Stable repeatability suggests (induces) that each of these signs is not individual, but general, inherent in all phenomena of a certain class. The logical transition from knowledge about individual phenomena to general knowledge is made in this case in the form of inductive reasoning, or induction (from the Latin inductio - “guidance”).

Inductive inference is an inference in which, based on the attribute’s belonging to individual objects or parts of a certain class, a conclusion is drawn about its belonging to the class as a whole.

Complete induction is an inference in which, based on the membership of each element or each part of the class of a certain feature, a conclusion is drawn about its membership in the class as a whole.

Incomplete induction is an inference in which, based on the attribute’s belonging to some elements or parts of a class, a conclusion is made about its belonging to the class as a whole.

The incompleteness of inductive generalization is expressed in the fact that not all, but only some elements or parts of the class are studied - from Si to Sn. The logical transition in incomplete induction from some to all elements or parts of a class is not arbitrary. It is justified by empirical grounds - the objective dependence between the universal nature of signs and their stable repeatability in experience for a certain kind of phenomena. Hence the widespread use of incomplete induction in practice. For example, during harvesting, conclusions are drawn about the contamination, moisture and other characteristics of a large batch of grain based on individual samples. In production conditions, based on random samples, conclusions are drawn about the quality of one or another mass product, for example, detergents in the chemical industry; pipes, metal sheets, wire - in rolling production; milk, cereals, flour - in the food industry.

The inductive transition from some to all cannot claim logical necessity, since the repeatability of a feature may be the result of a simple coincidence.

Thus, incomplete induction is characterized by a weakened logical consequence - true premises provide not a reliable, but only a problematic conclusion. Moreover, the discovery of at least one case that contradicts the generalization makes the inductive conclusion untenable.

In science and practical affairs, the object of research is often isolated events, objects and phenomena that are unique in their individual characteristics. When explaining and evaluating them, it is difficult to use both deductive and inductive reasoning. In this case, they resort to the third method of reasoning - inference by analogy: they liken a new single phenomenon to another, known and similar single phenomenon and extend previously received information to the first.

For example, a historian or politician, analyzing revolutionary events in a particular country, compares them to a similar revolution previously accomplished in another country and, on this basis, predicts the development of political events. Thus, Russian politicians substantiated their idea of ​​​​the need to conclude a peace treaty with Germany in 1918 (the Treaty of Brest-Litovsk) by referring to a similar historical situation at the beginning of the 19th century, when the Germans themselves concluded an enslaving treaty with Napoleon in 1807 (the Treaty of Tilsit) , and then after 6-7 years, having gathered their strength, they came to their liberation. A similar solution was proposed for Russia.

The conclusion proceeded in the same form in the history of physics, when, when elucidating the mechanism of sound propagation, it was likened to the movement of a liquid. Based on this comparison, the wave theory of sound arose. The objects of comparison in this case were liquid and sound, and the transferred feature was the wave method of their propagation.

Inference by analogy is a conclusion about the belonging of a certain feature to the individual object under study (subject, event, relation or class) based on its similarity in essential features with another already known individual object.

Inference by analogy is always preceded by the operation of comparing two objects, which makes it possible to establish similarities and differences between them. At the same time, analogy does not require any coincidences, but similarities in essential features while the differences are insignificant. It is such similarities that serve as the basis for the comparison of two material or ideal objects.

Analogy is not an arbitrary logical construction; it is based on the objective properties and relationships of objects in reality. Each specific object, having many characteristics, represents not a random combination of them, but a certain unity. No matter how small this or that sign may be, its existence and change are always determined by the state of other aspects of the object or external conditions.

Based on the nature of the objects being compared, two types of analogy are distinguished: (1) analogy of objects and (2) analogy of relationships.

1) Analogy of objects is an inference in which the object of likening is two similar individual objects, and the transferable attribute is the properties of these objects.

An example of such an analogy is the explanation in the history of physics of the mechanism of light propagation. When physics faced the question of the nature of light motion, the Dutch physicist and mathematician of the 17th century. Huygens, based on the similarity of light and sound in such properties as their linear propagation, reflection, refraction and interference, likened the movement of light to sound and came to the conclusion that light also has a wave nature.

The logical basis for the transfer of features in analogies of this kind is the similarity of the objects being compared in a number of their properties.

2) Analogy of relations is an inference in which the object of likening is similar relations between two pairs of objects, and the transferred feature is the properties of these relations.

For example, two pairs of persons x and y, m and n are in the following relationships:

1) x is the father (relation Ri) of the minor son y;

2) m is the grandfather (relation R2) and the only relative of n’s minor grandson;

3) it is known that in the case of parental relations (Ri), the father is obliged to support his minor child. Considering a certain similarity between the relations Ri and Ri, we can conclude that Rz is also characterized by the noted property, namely the obligation of the grandfather in a certain situation to support his grandson.

In the process of understanding reality, we acquire new knowledge. Some of them are direct, as a result of the influence of objects of external reality on our senses. But we obtain most of our knowledge by deriving new knowledge from existing knowledge. This knowledge is called indirect or inferential.

The logical form of obtaining inferential knowledge is inference.

Inference is a form of thinking through which a new judgment is derived from one or more propositions.

Any conclusion consists of premises, conclusion and conclusion. The premises of an inference are the initial judgments from which a 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: “The judge cannot take part in the consideration of the case if he is the victim (1). Judge N. – victim (2). This means that judge N. cannot take part in the consideration of the case (3).” In this inference (1) and (2) the propositions are premises, and (3) is the conclusion.

When analyzing a conclusion, it is customary to write the premises and conclusion separately, placing them one below the other. The conclusion is written under a horizontal line separating it from the premises and indicating logical consequence. The words “therefore” and those close in meaning (meaning, therefore, etc.) are usually not written below the line. In accordance with this, the example we gave 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 presupposes a connection between the premises in content. If judgments are not related in content, then a 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”, it is impossible to obtain conclusions, since these judgments do not have a common content and, therefore, are not logically related to each other .

If there is a meaningful connection between the premises, we can obtain new true knowledge in the process of reasoning if two conditions are met: firstly, the initial judgments - the premises of the inference must be true; secondly, in the process of reasoning, one must observe the rules of inference, which determine the logical correctness of the conclusion.

Inferences are divided into the following types:

1) depending on the strictness of the rules of inference: demonstrative - the conclusion in them necessarily follows from the premises, i.e. logical consequence in this kind of conclusions is a logical law; non-demonstrative - the rules of inference provide only the probabilistic conclusion of the conclusion from the premises.

2) according to the direction of logical consequence, i.e. by the nature of the connection between knowledge of varying degrees of generality, expressed in premises and conclusion: deductive - from general knowledge to particular; inductive - from particular knowledge to general knowledge; inferences by analogy - from particular knowledge to particular.

Deductive inferences are a form of abstract thinking in which thought develops from knowledge of a greater degree of generality to knowledge of a lesser degree of generality, and the conclusion following from the premises is, with logical necessity, reliable in nature. The objective basis of remote control is the unity of the general and the individual in real processes and environmental objects. peace.

The deduction procedure occurs when the information in the premises contains the information expressed in the conclusion.

All inferences are usually divided into types on various grounds: by composition, by the number of premises, by the nature of logical consequence and the degree of generality of knowledge in the premises and conclusion.

Based on their composition, all inferences are divided into simple and complex. Inferences whose elements are not inferences are called simple. Complex inferences are those that consist of two or more simple inferences.

Based on the number of premises, inferences are divided into direct (from one premises) and indirect (from two or more premises).

According to the nature of logical consequence, all conclusions are divided into necessary (demonstrative) and plausible (non-demonstrative, probable). Necessary inferences are those in which a true conclusion necessarily follows from true premises (i.e., logical consequence in such conclusions is a logical law). Necessary inferences include all types of deductive inferences and some types of inductive ones (“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., it 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 troupes of first-year students). Plausible inferences include inductive and analogical inferences.

Deductive inference (from Latin deductio - inference) is an inference in which the transition from general knowledge to particular knowledge is logically necessary.

Through 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 must be punished.

Petrov committed a crime.

Petrov must be punished.

Inductive inference (from Latin inductio - guidance) is an inference in which the transition from particular to general knowledge is carried out with a greater or lesser degree of plausibility (probability).

For example:

Theft is a criminal offense.

Robbery is a criminal offense.

Robbery is a criminal offense.

Fraud is a criminal offence.

Theft, robbery, robbery, fraud are crimes against property.

Therefore, all crimes against property are criminal offenses.

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 complete 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), based on 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 in the methods of committing crimes (burglary), it can be assumed that these crimes were committed by the same group of criminals.

All types of inferences can be correctly constructed or incorrectly constructed.

2. Direct conclusions

Direct inferences are those in which the conclusion is derived from one premise. For example, from the proposition “All lawyers are lawyers” one can obtain a new proposition “Some lawyers are lawyers.” Direct inferences give us the opportunity to identify knowledge about such aspects of objects, which was already contained in the original judgment, but was not clearly expressed and clearly realized. Under these conditions, we make the implicit explicit, the unconscious conscious.

Direct inferences include: transformation, reversal, opposition to a predicate, inference based on a “logical square”.

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, you need to change its connective to the opposite one, 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 a proposition “Not all S are P,” then it must be transformed into a partial negative: “Some S are not P.”

Examples and transformation schemes:

A:

All first year students study logic.

Not a single first-year student studies logic.

Scheme:

All S are P.

No S is a non-P.

E: No cat is a dog.

Every cat is a non-dog.

No S is an R.

All Ss are non-Ps.

I: Some lawyers are athletes.

Some lawyers are not non-athletes.

Some Ss are Ps.

Some Ss are not non-Ps.

A: Some lawyers are not athletes.

Some lawyers are non-athletes.

Some Ss are not Ps.

Some Ss are non-Ps.

Conversion is a direct inference in which the places of the subject and the predicate change while maintaining the quality of the judgment.

The appeal 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 an appeal leads to a change in the original judgment in quantity (a new particular judgment is obtained from the general initial one), then such an appeal is called an appeal with a limitation; if the appeal does not lead to a change in the original judgment regarding quantity, then such an appeal is an appeal without limitation.

Examples and circulation schemes:

A: A generally affirmative judgment turns into a particular affirmative one.

All lawyers are lawyers.

Some lawyers are lawyers.

All S are P.

Some Ps are Ss.

General affirmative emphasizing judgments are addressed without restrictions. Every offense (and only an offense) is an unlawful act.

Any illegal act is a crime.

Scheme:

All S, and only S, are P.

All P's are S's.

E: A generally negative judgment turns into a generally negative one (without restrictions).

No lawyer is a judge.

No judge is a lawyer.

No S is an R.

No P is an S.

I: Particularly affirmative judgments turn into privately affirmative ones.

Some lawyers are athletes.

Some athletes are lawyers.

Some Ss are Ps.

Some Ps are Ss.

Particularly affirmative distinguishing judgments turn into generally affirmative ones:

Some lawyers, and only lawyers, are lawyers.

All lawyers are lawyers.

Some S, and only S, are P.

All P's are S's.

A: Partial negative judgments are not addressed.

The logical operation of reversing a judgment is of great practical importance. Ignorance of the rules of circulation leads to gross logical errors. Thus, quite often a generally affirmative proposition is addressed without limitation. For example, the proposition “All lawyers should know logic” becomes the proposition “All students of logic are lawyers.” But this is not true. The statement “Some students of logic are lawyers” is true.

Contrasting a predicate is the sequential application of the operations of transformation and inversion - 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, obtain “No non-lawyer is a lawyer.” Schematically:

All S are P.

No non-P is an S.

Inference based on the “logical square”. A “logical square” is a diagram that expresses 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 unified 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 relationship between A and E - the relationship of opposites; 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 represent the relationship between A and O, E and I, which is called contradiction.

The relation of opposition takes place between generally affirmative and generally negative judgments (A-E). The essence of this relationship is that two opposing propositions cannot be simultaneously true, but can be false at the same time. Therefore, if one of the opposing judgments is true, then the other is certainly false, but if one of them is false, then it is still impossible to unconditionally assert about the other judgment that it is true - it is indefinite, that is, 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 the students in our course have studied logic before” is false, then its opposite “Not a single student in our course has studied logic before” will be indefinite, i.e. it can be either true or false.

The relation of partial compatibility takes place between partial affirmative and partial negative judgments (I - O). Such propositions cannot be both false (at least one of them is true), but they can be true at the same time. For example, if the proposition “Sometimes you can be late for class” is false, then the proposition “Sometimes you can’t be late for class” will be true.

But if one of the judgments is true, then the other judgment, which is in relation to partial compatibility with it, will be indefinite, i.e. it can be either true or false. For example, if the proposition “Some people study logic” is true, the proposition “Some people do not study logic” will be true or false. But if the proposition “Some atoms are divisible” is true, the proposition “Some atoms are not divisible” will be false.

The relation of subordination is that the truth of the subordinating judgment necessarily implies the truth of the subordinate judgment, but the converse is not necessary: ​​if the subordinate judgment is true, the subordinating judgment will be indefinite - it can turn out to be either true or false.

But if the subordinate proposition is false, then the subordinating one will be even more false. The converse is again not necessary: ​​if the subordinating judgment is false, the subordinate one can 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 proposition “Some lawyers are members of the Moscow Bar Association” is true, the subordinate proposition “All lawyers are members of the Moscow Bar Association” will be false or true.

If the subordinate proposition “Some lawyers are not members of the Moscow Bar Association” (O) is false, the subordinate proposition “Not a single lawyer is a member of the Moscow Bar Association” (E) will be false. But if the subordinate proposition “No lawyer is a member of the Moscow Bar Association” (E) is false, the subordinate proposition “Some lawyers are not a member of the Moscow Bar Association” (O) will be true or false.

Relations of contradiction exist between generally affirmative and particular negative judgments (A - O) and between generally negative and particular affirmative judgments (E - I). The essence of this relationship 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 judgment, a new judgment is formed from the original judgment, which is true when the original judgment (premise) is false, and false when the original judgment (premise) is true. For example, denying the true proposition “All lawyers are lawyers” (A), we obtain a new, false proposition “Some lawyers are not lawyers” (O). By denying the false proposition “No lawyer is a lawyer” (E), we obtain the 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 inferences are categorical inferences, which because of their form are called syllogism (from the Greek sillogismos - counting).

A syllogism is a deductive conclusion in which, from two categorical premise judgments connected by a common term, a third judgment is obtained - the conclusion.

The concept of categorical syllogism, a simple categorical syllogism, in which the conclusion is obtained from two categorical judgments, is found in the literature.

Structurally, a syllogism consists of three main elements - terms. Let's look at this with an example.

Every citizen of the 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 education” is wider than the scope of the subject – “Novikov”. Because of this, the predicate of inference is called the major term, and the subject of inference is called the lesser term. Accordingly, the premise, which includes the predicate of the conclusion, 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 - intermediary). The middle term is included in each premise, but is not included 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 inference. This connection is carried out in 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 syllogism diagram.

M - P S - M

S - M or M - R R - M - S

S - P S - P

The following must be kept in mind:

1) the name “major” or “minor” premise does not depend on the location in the syllogism diagram, but only on the presence of a greater or lesser term in it;

2) changing the place of any term in the premise does not change its designation - 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 premises in a syllogism, the conclusion, i.e. the logical connection between extreme terms does not depend.

Consequently, the logical analysis of a syllogism must begin with the conclusion, with an understanding of its subject and predicate, with the establishment from here of the greater and lesser terms of the syllogism. One way to establish the validity 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 indirect inference is a simple categorical syllogism, the conclusion of which is obtained from two categorical judgments.

In contrast to the terms of judgment - subject ( S) and predicate ( R) - the concepts included in a syllogism are called
in terms of a syllogism. There are lesser, greater and middle terms.

Lesser term of a syllogism is called a concept, which in conclusion is a subject.
Large term of the syllogism is called a concept that in conclusion is a predicate (“has the right to protection”). The lesser and greater terms are called
extreme and are designated accordingly in Latin letters S(minor 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 containing a minor term is called
smaller parcel, a premise containing a larger term is called
larger parcel.

For the convenience of analyzing a syllogism, it is customary to place the premises in a certain sequence: the larger one in the first place, the smaller one in the second. However, in reasoning this order is not necessary. The smaller parcel may be in first place, the larger one in second. Sometimes parcels remain after the conclusion.

The premises differ not in their place in the syllogism, but in the terms included in them.

The conclusion in a syllogism would be impossible if it did not have a middle term.
The middle term of the syllogism is a concept that is included in both premises and is absent V conclusion (in our example - “accused”). The middle term is indicated by a Latin letter M.

The middle term connects the two extreme terms. The relationship of extreme terms (subject and predicate) is established through their relationship to the middle term. In fact, from the major premise we know the relation of the larger term to the middle (in our example, the relation of the concept “has the right to defense” to the concept “accused”) from the minor premise - the relation of the smaller term to the middle. Knowing the ratio of extreme terms to the average, we can establish the relationship between extreme terms.

The conclusion from the premises is possible because the middle term acts as a connecting link between the two extreme terms of the syllogism.

The validity of the conclusion, i.e. logical transition from premises to conclusion, in a categorical syllogism is based on the position
(axiom of syllogism): everything that is affirmed or denied regarding all objects of a certain class is affirmed or denied regarding 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 subject or predicate. Depending on this, there are four types of syllogism, 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 premises.

In second figure- place of the predicate in both premises. IN third figure- the place of the subject in both premises. IN 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, differing in the position of the middle term in the premises.

The premises of a syllogism can be judgments of different quality and quantity: general affirmative (A), general negative (E), particular affirmative (I) and particular negative (O).

Varieties of syllogism that differ in the quantitative and qualitative characteristics of the premises are called modes of simple categorical syllogism.

It is not always possible to obtain a true conclusion from true premises. Its truth is determined by the rules of the syllogism. There are seven of these rules: three relate to terms and four to premises.

Rules of terms.

1st rule: in A syllogism must have only three terms. The conclusion in a syllogism is based on the ratio of the two extreme terms to the middle, so there can be no less or more sin of terms in it. Violation of this rule is associated with the identification of different concepts, which are taken as one and considered as a middle term. This error is based on a violation of the requirements of the law of identity and is called quadrupling 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 relationship between the extreme terms remains uncertain. 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. Consequently, 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 violation of the rule of distributed extreme terms,
is called an illegal extension of a lesser (or greater) term.

Parcel rules.

1st rule: at least one of the premises must be an affirmative proposition. From The conclusion does not necessarily follow from two negative premises. 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 average. Therefore, the middle term cannot establish a definite relationship between the extreme terms. Finally, the smaller term (M) may be fully or partially included 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 employees of the research institute study biology” (S 2); 3) “All employees of the research institute study biology” (S 3) (fig.).

2nd rule: if one of the premises is a negative proposition, then the conclusion must be negative.

The 3rd and 4th rules are derivatives arising from those considered.

3rd rule: at least one of the premises must be a general proposition. From two particular premises the conclusion does not necessarily follow.

If both premises are partial affirmative judgments (II), then the conclusion cannot be drawn according to the 2nd rule of terms: in the partial affirmative. in a judgment, neither the subject nor the predicate is distributed, therefore the middle term is not distributed in any of the premises.

If both premises are partial negative propositions (00), then the conclusion cannot be drawn according to the 1st rule of premises.

If one premise is a partial affirmative and the other is a 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 average, then a conclusion cannot be drawn, 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) the larger term, 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 provide the necessary conclusions.

4th rule: if one of the premises is a private judgment, then the conclusion must be private.

If one premise is generally affirmative, and the other is particularly affirmative (AI, IA), then only one term is distributed in them - the subject of the generally affirmative judgment.

According to the 2nd rule of terms, it must be a middle term. But in this case, the two extreme terms, including the smaller one, will not be distributed. Therefore, according to the 3rd rule of terms, the lesser term will not be distributed in the conclusion, which will be a private judgment.

4. Inferences from judgments with relations

An inference whose premises and conclusion are propositions with relations is called an inference with relations.

For example:

Peter is Ivan's brother. Ivan is Sergei's brother.

Peter is Sergei's brother.

The premises and conclusion in the above example are propositions with relations that have the logical structure xRy, where x and y are concepts about 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 relationship is called symmetrical (from the Greek simmetria - “proportionality”) if it occurs 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. Symmetrical 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 event x occurred simultaneously with event y, then event y also occurred simultaneously with event x), differences and some others.

The symmetry relation is symbolically written:

xRy - yRx.

2. A relationship is called reflexive (from the Latin reflexio - “reflection”) if each member of the relationship is in the same relationship to itself. These are relations of equality (if a = b, then a = a and b = b) and simultaneity (if event x happened simultaneously with event y, then each of them happened simultaneously with itself).

The reflexivity relation is written:

xRy -+ xRx L yRy.

3. A relation is called transitive (from the Latin transitivus - “transition”) if it occurs between x and z when it occurs between x and y and between y and z. In other words, a relation is transitive if and only if the relation between x and y and between y and z implies the same relation between x and z.

Transitive relations are equality (if a is equal to b and b is equal to c, then a is equal to c), simultaneity (if event x occurred simultaneously with event y and event y simultaneously with event z, then event x occurred simultaneously with event z), relations “more”, “less” (a is less than b, b is less than c, therefore, a is less than c), “later”, “to be further 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 relationships, it is necessary to rely on the following rules:

For the symmetry property (xRy -* yRx): if the proposition xRy is true, then the proposition yRx is also true. For example:

A is like B. B is like A.

For the property of reflexivity (xRy -+ xRx l yRy): if the judgment xRy is true, then the judgments xRx and yRy will be true. For example:

a = b. a = a and b = b.

For the transitivity property (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 propositions with relations depends on the properties of the relations and is governed by rules arising from these properties. Otherwise, the conclusion may be false. Thus, from the judgments “Sergeev is familiar with Petrov” and “Petrov is familiar with Fedorov” the necessary conclusion “Sergeev is familiar with Fedorov” does not follow, since “to be familiar” 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 expressions below to obtain true sentences:

b) X is a word in the Russian language;

c) X – 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” is a philosophical category.

b) “consequence” is a word in the Russian language;

c) “consequence” is 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 quote name

b) Mother of Lao Tzu.

Solution

a) If not a single 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, “member of the Gavrilov family” is a common name, “not” is a logical term,” “is” (“is” ) is a logical term, “honest man” is a general name, “and” is a logical term, “Semyon” is a singular name.

b) Mother of Lao Tzu.

“Mother” is an object functor, “Lao-Tzu” is a singular name.

4. Summarize the following concepts:

a) Corrective labor without detention;

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 without detention - corrective labor;

b) investigative experiment - experiment;

c) Constitution – Law.

a) Minsk is the capital;

Solution

a) Minsk is the capital. * Refers to the category of things. In this case, the term “capital” acts as a predicate of judgment, thus revealing the signs of judgment.

b) The capital of Azerbaijan is an ancient city.

In this case, the term “capital” has a semantic proposition.

In this case, the term “capital” acts as the subject of judgment, since the said judgment reveals its characteristics.

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: “in the presence of contradictory evidence...”.

Solution

This text talks about 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 general in terms of quantity.

¬(А˄ В)<=>¬(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 we should talk about judgments about the relationship between objects.

This judgment can be written as follows:

xRy

The population of Ryazan (x) is larger (R) than the population of Korenovsk (x)

9. A sample survey of those who committed serious crimes was conducted in places of deprivation of liberty (10% of such persons were surveyed). Almost all of them responded that strict penalties did not influence their decision to commit a crime. They concluded that strict penalties are not a deterrent to the commission of serious crimes. Is this conclusion justified? If not justified, 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 feature in the studied group (sample) is defined in the premises and is transferred in the conclusion to the entire set of phenomena.

This message contains the following information:

sample cases – 10%


the number of cases in which the characteristic of interest is present is almost all;


the frequency of occurrence of the characteristic of interest is almost 1.


From this we can note 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 - strict penalties are not a deterrent when committing serious crimes - is correct, since statistical generalization, being a conclusion of incomplete induction, refers to non-demonstrative inferences. 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 sample studied: its size in relation to the population and representativeness (representativeness).


10. Limit the following concepts:


a) state;


b) court;


c) revolution.


Solution


a) state – Russian state;


b) court – Supreme Court


c) revolution - October Revolution - world revolution


11. Give a complete logical description of the concepts:


a) People's Court;


b) worker;


c) lack of control.


Solution


a) People's Court is a single, non-collective, specific concept;


b) worker – a general, non-collective, specific, non-relative concept;


c) lack of control is a single, non-collective, abstract concept.
The concept of deductive reasoning. Simple categorical syllogism Form of law

Complex inferences are those that consist of two or more simple inferences. Most often, complex inferences of this kind, or, as they are also called in logic, chains of inferences, are used in proofs. Let us consider such types of complex inferences as: a) polysyllogism; b) litters; c) epicheyrema.

A polysyllogism is a chain, a chain of syllogisms connected in such a way that the conclusion of the previous syllogism (prasyllogism) becomes one of the premises of the subsequent syllogism (episyllogism).

For example:

No one capable of self-sacrifice is an egoist.

All generous people are capable of self-sacrifice.

No one is generous, not an egoist.

All cowards are selfish.

No coward is generous.

Depending on which premise - greater or lesser - of the episyllogism becomes the conclusion of the prasyllogism, progressive and regressive chains of syllogisms are distinguished, respectively.

The example we have given is a progressive chain of syllogisms. In it our thought moves from the more general to the less general.

Another example of a progressive chain of syllogisms.

All vertebrates have red blood.

All mammals are vertebrates.

All mammals have red blood.

All carnivores are mammals.

All carnivores have red blood.

Tigers are predatory animals.

Tigers have red blood.

In a regressive chain of syllogisms, the conclusion of the prasyllogism becomes the lesser premise of the episyllogism. In such polysyllogism, thought moves from less general to increasingly general knowledge.

For example:

Vertebrates are animals.

Tigers are vertebrates.

Tigers are animals.

Animals are organisms.

Tigers are animals.

Tigers are organisms.

Organisms are destroyed.

Tigers are organisms.

Tigers are destroyed.

In order to check the logical consistency of a pollysyllogism, it is necessary to break it down into simple categorical syllogisms and check the consistency of each of them.

Sorites (from the Greek “heap”) is a complex abbreviated syllogism in which only the last conclusion from a series of premises is given, and intermediate conclusions are not explicitly formulated, but only implied.

Sorites is built according to the following scheme;

All A's are B's.

All B's are C's.

All C's are D's.

Therefore, all A's are D's.

As you can see, the conclusion of the prasyllogism is missing here: “All A is C,” which should also act as a greater premise of the second syllogism - episyllogism.

For example:

Socially dangerous acts are immoral.

A crime is a significantly dangerous act.

Theft is a crime.

Stealing is immoral.

Here the conclusion of the first syllogism (prasyllogism) is missing - “Crime is immoral”, which is the second, lesser premise of the second syllogism (episyllogism). This episyllogism in its entirety would look like this:

Crime is immoral.

Theft is a crime.

Stealing is immoral.

There are two types of sorites - Aristotelian and Hoklenian. They received their name from the authors who first described them.

Aristotle described a sorites in which the conclusion of a prasyllogism is omitted, becoming the lesser premise of an episyllogism:

A horse is a quadruped.

Bucephalus is a horse.

A four-legged animal.

An animal is a substance.

Bucephalus is a substance.

In its full form, this polysyllogism will look like this:

A horse is a four-legged animal.

Bucephalus is a horse.

Bucephalus is a quadruped.

A four-legged animal.

Bucephalus is a quadruped.

Bucephalus is an animal.

An animal is a substance.

Bucephalus is an animal.

Bucephalus is a substance.

Goklenius (prof.. University of Marburg, lived 1547-1628) wrote a description of sorites, in which the conclusion of the prasyllogism is omitted, becoming the first, greater premise of the episyllogism. He cited the following litter:

An animal is a substance.

A four-legged animal.

A horse is a four-legged animal.

Bucephalus horse.

Bucephalus is a substance.

In its full form, this polysyllogism looks like this:

1. An animal is a substance.

A four-legged animal.

Quadruped is a substance.

2. Quadruped - substance.

A horse is a four-legged animal.

A horse is a substance.

3. Horse substance.

Bucephalus is a horse.

Bucephalus is a substance.

Epicheyrema (translated from Greek “attack”, “laying on of hands”) is a syllogism in which each of the premises is an enthymeme.

For example:

All students at the Institute of International Relations study logic because they must think correctly.

We, students of the Institute of International Relations, because we study at this institute.

That's why we deal with logic.

It can be seen that each of the premises of this epicheireme is an abbreviated syllogism - an enthymeme. Thus, the first premise in its entirety will be the following syllogism:

Everyone who should think correctly is engaged in logic.

Everyone, students of the Institute of International Relations must think correctly.

All students at the Institute of International Relations study logic.

We leave the restoration of the second premise to a complete syllogism and the entire chain of syllogisms to the reader.

Epicheyrema We use it quite often in the practice of thinking and in oratory. The Russian logician A. Svetilin noted that epicheyrema is convenient in oratory because it makes it possible to more conveniently arrange a complex conclusion according to its component parts and makes them easily visible, and, consequently, the whole reasoning more conclusive.

Exercise

Determine the type of inference and check its consistency

A. 3 is an odd number.

All odd numbers are natural numbers.

All natural numbers are rational numbers.

All rational numbers are real numbers.

Therefore, 3 is a real number.

B. Everything that improves health is useful.

Sport improves health.

Athletics is a sport.

Running is a type of athletics.

Running is good for you.

B. All organisms are bodies.

All plants are organisms.

All bodies have weight.

All plants are bodies.

All plants have weight.

D. Noble work deserves respect because noble work contributes to the progress of society.

The work of a lawyer is a noble work, as it consists of protecting the legal rights and freedoms of citizens.

Therefore, the work of a lawyer deserves respect.

D, What is good must be desired.

What must be desired must be approved.

And what must be approved is commendable.

Therefore, whatever is good is praiseworthy.

(Example of M.V. Lomonosov)

In this lesson, we finally move on to the topic that forms the core of any reasoning and any logical system - inference. In the fourth lesson, we said that reasoning is a set of judgments or statements. Obviously, such a definition is not complete, because it does not say anything about why some different statements suddenly appeared nearby. To give a more precise definition, reasoning is the process of justifying a statement using its consistent conclusion from other statements. This conclusion is most often carried out in the form of inferences.

Inference- this is a direct transition from one or more statements A 1, A 2, ..., A n to the statement B. A 1, A 2, ..., A n are called premises. There can be one parcel, there can be two, three, four, in principle - as many as you like. The parcels contain information known to us. B is the conclusion. In the conclusion there is new information that we extracted from the parcels using special procedures. This new information was already contained in the parcels, but in a hidden form. So the task of inference is to make this hidden explicit. In addition, sometimes premises are called arguments, and the conclusion is called a thesis, and the conclusion itself in this case is called justification. The difference between inference and justification is that in the first case, we do not know what conclusion we will come to, and in the second, we already know the thesis, we just want to establish its connection with the premises-arguments.

To illustrate the conclusion, we can take the reasoning of Hercule Poirot from “Murder on the Orient Express” by Agatha Christie:

But I felt that he rebuilt as he went. Suppose he wanted to say: “Wasn’t she burned?” Therefore, McQueen knew both about the note and that it was burned, or, in other words, he was a murderer or an accomplice of a murderer.

The premises are located above the line, the conclusion is below the line, and the line itself denotes the relation of logical consequence.

Criteria for the truth of inferences

Just like for judgments, for inferences there are certain conditions for their truth. When determining whether a conclusion is true or false, you need to pay attention to two aspects. First aspect- this is the truth of the premises. If at least one of the premises is false, then the conclusion drawn will also be false. Since the conclusion is the information that was hidden in the premises and which we simply brought to light, it is impossible to accidentally obtain the correct conclusion from incorrect premises. This can be compared to trying to make a steak out of carrots. I guess you can give carrots the color and shape of a steak, but the inside will still be carrots and not meat. No cooking operation converts one into the other.

Second aspect- this is the correctness of the conclusion itself from the point of view of its logical form. The point is that the truth of the premises is an important but not sufficient condition for the conclusion to be correct. There are often situations where the premises are true, but the conclusion is false. An example of an incorrect inference when the premises are true is the inference of the dove from Carroll's Alice in Wonderland. Dove accuses Alice of not being a snake. Here's how she comes to this conclusion:

Snakes eat eggs.
Girls eat eggs.
So girls are snakes.

Although the premises are correct, the conclusion is absurd. The conclusion as a whole is made incorrectly. To avoid such errors, logicians have identified such conclusions, the logical forms of which, if the premises are true, guarantee the truth of the conclusion. They are usually called correct conclusions. Thus, in order for the conclusion to be drawn correctly, it is necessary to monitor the truth of the premises and the correctness of the form of the conclusion itself.

We will consider various forms of correct inferences using the example of syllogistics. In this lesson we will look at the simplest one-premise conclusions. The next lesson contains more complex conclusions: syllogisms, enthymemes, multi-premise conclusions.

To make it easier to remember exactly what types of inferences are possible between categorical attributive statements, logicians came up with a special logical square depicting the relationships between them. Therefore, some one-premise inferences are also called logical square inferences. Let's look at this square:

Let's start with relationships of subordination. We have already encountered them in the fourth lesson, when we considered the truth conditions for partial affirmative and partial negative statements. We said that from the statement “All S are P” it would be logical to deduce the statement “Some S are P”, and from the statement “No S is P” - “Some S are not P”. Thus, the following types of inferences are possible:

• All S's are P's
• Some S's are P's
• All birds have a beak. Therefore, some birds have beaks.
• No S is a P
• Some S's are not P's
• No goose wants to be caught and roasted. Consequently, some geese do not want to be caught and roasted.

In addition, according to the rule of contraposition, two more correct conclusions can be drawn from relations of subordination. The rule of contraposition is a logical law that states: if statement A implies statement B, then the statement “it is not true that B” will follow the statement “it is not true that A.” You can try to test this law using a truth table. So, the following conclusions regarding contraposition will also be true:

• It is not true that all S are P
• It is not true that some cars do not have wheels. Therefore, it is not true that all cars do not have wheels.
• It is not true that all S are not P
• It is not true that some wines are not spirits. Therefore, it is not true that all wines are not spirits.

Contrary relationship(opposites) means that statements like “All S are P” and “No S is P” cannot be both true, but they can be false at the same time. This is clearly seen from the truth table for categorical attributive statements, which we built in the last lesson. From this we can derive the so-called law of counter-contradiction: It is not true that all S are P and at the same time no S is P.

According to the law of counter contradiction, the following types of inferences will be true:

• All S's are P's
• All apples are fruits. Therefore, it is not true that no apple is a fruit.
• No S is a P
• It is not true that all S are P
• No whale can fly. Therefore, it is not true that all whales can fly.

Subcontrary relations(subopposites) mean that statements like “Some S are P” and “Some S are not P” cannot be both false, although they can be true at the same time. On this basis, the law of the subcontrary excluded middle can be formulated: Some S are not P or Some S are P.

• According to this law, the following conclusions will be correct:
• It is not true that some S are P
• Some S's are not P's
• It is not true that some foods are healthy. Therefore, some foods are not healthy.
• It is not true that some S are not P
• Some S's are P's
• It is not true that some students in our class are not poor students. Thus, some students from our class are poor students.

Relationships of contradiction(contradictory) say that the statements contained in them cannot be both true and false. Based on these relations, two laws of contradiction and two laws of excluded middle can be formulated. First law of contradiction: It is not true that all S are P and some S are not P. Second law of contradiction: It is not true that no S is P and some S are P. First law of excluded middle: All S are P or some S are not is P. Second Law of Excluded Middle: No S is P or some S is P.

The following types of inferences are based on these laws:

• All S's are P's
• It is not true that some S are not P
• All children need care. Therefore, it is not true that some children do not need care.
• Some S's are not P's
• It is not true that all S are P
• Some books are not boring. Therefore, it is not true that all books are boring.
• It is not true that all S are P
• Some S's are not P's
• It is not true that all employees of our company work hard. Therefore, some employees of our company do not work hard.
• It is not true that some S are not P
• All S's are P's
• It is not true that some zebras do not have stripes on their skin. Therefore, all zebras have stripes on their skin.
• No S is a P
• It is not true that some S are P
• Not a single painting in this room dates back to the 20th century. It is therefore not true that some of the paintings in this room date back to the 20th century.
• Some S's are P's
• It is not true that no S is P
• Some students play sports. Therefore, it is not true that no student plays sports.
• It is not true that no S is P
• Some S's are P's
• It is not true that no scientist is interested in art. Consequently, some scientists are interested in art.
• It is not true that some S are P
• No S is a P
• It is not true that some cats smoke cigars. So no cat smokes cigars.

As you most likely noticed in all of these inferences, the statements above the line and below the line convey the same information, just presented in a different form. The important detail is that the meaning of some of these statements is perceived easily and intuitively, while the meaning of others is dark, and sometimes you have to rack your brain over them. For example, the meaning of affirmative statements is more easily perceived than the meaning of negative statements; the meaning of statements with one negation is more understandable than the meaning of statements with two negations. Thus, the main purpose of inferences using a logical square is to bring difficult-to-understand, incomprehensible statements to the simplest and clearest form.

Another type of single-premise inference is reversal. This is a type of inference in which the subject of the premises coincides with the predicate of the conclusion, and the subject of the conclusion coincides with the predicate of the premises. Roughly speaking, in conclusion, S and P are simply swapped.

Before moving on to inferences through inversion, let us construct a truth table for statements in which P takes the place of the subject, and S takes the place of the predicate.

Compare it with the table that we built in the last lesson. An inversion, like other inferences, can be correct only when the premise and conclusion are both true. When comparing the two tables, you will see that there are not so many such combinations.

So, there are two types of circulation: pure and limited. Pure circulation occurs when the quantitative characteristic does not change, that is, if the premise contained the word “all”, then the conclusion will also contain the words “all”/“none”; if the premise contains the word “some”, then the conclusion will also contain the word “some”. "some. Accordingly, when dealing with a restriction, the quantitative characteristic changes: there were “all”, but now there are “some”. For statements like “No S is P” and “Some S are P,” the correct pure inversion is:

• No S is a P
• No P is an S
• No person can survive without air. Therefore, no living creature that can survive without air is a human being.
• Some S's are P's
• Some P's are S's
• Some snakes are poisonous. Therefore, some poisonous creatures are snakes.
• For statements like “All S are P” and “No S is P,” the constraint treatment is true:
• All S's are P's
• Some P's are S's
• All penguins are birds. Thus, some birds are penguins.
• No S is a P
• Some P is not S
• No crocodile eats marshmallows. Therefore, some marshmallow-eating creatures are not crocodiles.
• Statements like “Some S are not P” are not addressed at all.

Although appeals, like inferences based on a logical square, are single-premise inferences, and we also extract all new information from the existing premise, the premise and conclusion in them can no longer be called simply different formulations of the same information. The information received relates to another subject, and therefore it no longer seems so trivial.

So in this lesson we started looking at the correct types of inferences. We talked about the simplest single-premise inferences: inferences using a logical square and inferences through inversion. Although these conclusions are quite simple and even trivial in some places, people everywhere make mistakes in them. It is clear that it is difficult to retain all types of correct inferences in memory, so when you do exercises or are faced with the need to test or make a single-premise inference in real life, do not be afraid to resort to the help of model diagrams and truth tables. They will help you check whether, when the premises are true, the conclusion is also true, and this is the main thing for correct inference.

Exercise “Pick up the key”

In this game you need to create a key of the correct shape. To do this, set the serifs to the desired length (from 1 to 3, 0 cannot be), and then click the “Try” button. You will be given 2 judgments, how many serifs of the selected length are present in the key (for simplicity, the value is “presence”), and how many of the selected ones are in place (for simplicity, the value is “in place”). Adjust your decision and try until you find the key.

Exercises

Make all possible conclusions from the following statements using a logical square:

• All bears hibernate for the winter.
• It is not true that all people are envious.
• Not a single gnome reaches two meters in height.
• It is not true that no man has ever been to the North Pole.
• Some people have never seen snow.
• Some buses run on schedule.
• It is not true that some elephants have flown to the moon.
• It is not true that some birds do not have wings.

Make appeals with those statements with which this is possible:

• No one has built a time machine yet.
• Some waiters are very annoying.
• All professionals are experienced in their field.
• Some books do not have a hard cover.

Check if the following conclusions are correct:

• Some rabbits do not wear white gloves. Consequently, some rabbits wear white gloves.
• It is not true that no one has been to the moon. So some people have been to the moon.
• All people are mortal. Therefore, all mortals are people.
• Some birds cannot fly. Therefore, some creatures that cannot fly are birds.
• No lamb has a taste for whiskey. Therefore, no creature who has a taste for whiskey is a lamb.
• Some sea animals are mammals. Thus, it is not true that no marine animal is a mammal.

Test your knowledge

If you want to test your knowledge on the topic of this lesson, you can take a short test consisting of several questions. For each question, only 1 option can be correct. After you select one of the options, the system automatically moves on to the next question. The points you receive are affected by the correctness of your answers and the time spent on completion. Please note that the questions are different each time and the options are mixed.

The properties of the basic concepts are revealed in axioms- proposals accepted without proof.

For example, in school geometry there are axioms: “through any two points you can draw a straight line 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 basic concepts, gives their definitions. Such definitions are called axiomatic.

The properties of concepts to be proved are called theorems, consequences, signs, formulas, rules.

Prove the theorem AIN- this means to establish in a logical way that whenever a property is satisfied A, the property will be executed IN.

Proof in mathematics they call a finite sequence of propositions of a given theory, each of which is either an axiom or is deduced from one or more propositions of this sequence according to the rules of logical inference.

The basis of the proof is reasoning - a logical operation, as a result of which from one or more sentences interconnected in meaning, a sentence containing new knowledge is obtained.

As an example, consider the reasoning of a schoolchild who needs to establish the “less than” relation between the numbers 7 and 8. The student says: “7< 8, потому что при счете 7 называют раньше, чем 8».

Let us find out what facts the conclusion obtained in this argument is based on.

There are two such facts: First: if the number A when counting, the numbers are called before b, That 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. From two premises a new fact is obtained: 7< 8, его называют заключением.

There is a certain connection between the premises and the conclusion, thanks to which they constitute an argument.

An argument in which there is an implication relation between the premises and the conclusion is called deductive.

In logic, instead of the term “reasoning,” the word “inference” is more often used.

Inference- this is a way of obtaining new knowledge based on some existing knowledge.

An inference consists of premises and a conclusion.

Parcels- these contain initial knowledge.

Conclusion- this is a statement containing new knowledge obtained from the original one.

As a rule, the conclusion is separated from the premises using the words “therefore”, “means”. Inference with premises R 1, R 2, …, рn and conclusion R we will write it in the form: or (R 1, R 2, …, рn) R.

Examples inferences: a) Number a =b. Number b = c. Therefore, the number a = c.

b) If the numerator in a fraction is less than the denominator, then the fraction is proper. In a fraction numerator is less than denominator (5<6) . Therefore, the fraction - correct.

c) If it rains, then there are clouds in the sky. There are clouds in the sky, therefore it is raining.

Conclusions can be correct or incorrect.

The inference is called correct if the formula corresponding to its structure and representing a conjunction of premises, connected to the conclusion by an implication sign, is identically true.

For that to determine whether the conclusion is correct, proceed as follows:

1) formalize all premises and conclusion;

2) write down a formula representing a conjunction of premises connected by an implication sign with a conclusion;

3) draw 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 a conclusion is determined by its form and does not depend on the specific content of the statements included in it. And in logic, rules are proposed, following which, one can build deductive conclusions. These rules are called rules of inference or patterns of deductive reasoning.

There are many rules, but the most commonly used are the following:

1. - rule of conclusion;

2. - rule of negation;

3. - the rule of syllogism.

Let's give example inferences made from rule conclusions:"If the recording 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 ».

The general premise in this conclusion is the statement “if Oh), That B(x)", Where Oh)- this is a “record of number” X ends with a number 5 ", A B(x)- "number X divided by 5 " A particular premise is a statement that is obtained from the condition of the general premise when
x = 135(those. A(135)). A conclusion is a statement derived from B(x) at x = 135(those. V(135)).

Let's give example of a conclusion made according to the rule negatives:"If the recording 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 particular one is the negation of the statement “number 177 divided by 5 "(i.e.). The conclusion is the negation of the sentence “Writing a number 177 ends with a number 5 "(i.e.).

Finally, let's consider example of an inference based on syllogism rule: "If the number X multiple 12, then it is a multiple 6. If the number X multiple 6 , then it is a multiple 3 . Therefore, if the number X multiple 12, then it is a multiple 3 ».

This conclusion has two premises: “if Oh), That B(x)" and if B(x), That C(x)", where A(x) is "the number X multiple 12 », B(x)- "number X multiple 6 " And C(x)- "number X multiple 3 " The conclusion is a statement “if Oh), That C(x)».

Let's check whether 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 divided by 2 . Number 22 divided by 2 . Therefore, it is divided into 4.

3) All trees are plants. Pine is a tree. This means that pine is a plant.

4) All students in this class went to the theater. Petya was not in the theater. Therefore, Petya is not a student in this class.

5) If the numerator of a fraction is less than the denominator, then the fraction is correct. If a fraction is proper, then it is less than 1. Therefore, if the numerator of a fraction is less than the denominator, then the fraction is less than 1.

Solution: 1) To resolve the question of the correctness of the inference, let us identify its logical form. Let us introduce the following notation: C(x)- "quadrangle" X- rhombus", B(x)- “in a quadrangle X the diagonals are mutually perpendicular." Then the first premise can be written as:
C(x) B(x), second - C(a), and the conclusion B(a).

Thus, the form of this inference is: . It is built according to the rule of conclusion. Therefore, this reasoning is correct.

2) Let us introduce the notation: Oh)- "number X divided by 4 », B(x)- "number X divided by 2 " Then we will write the first premise: 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 those known. It is easy to see that both premises are true and the conclusion is false.

This means that this reasoning is incorrect.

3) Let us introduce some notation. Let Oh)- "If X tree", B(x) - « X plant". Then the parcels will take the form: Oh)B(x), A(a), and the conclusion B(a). Our conclusion is built in the form: - rules of conclusion.

This means that our reasoning is structured correctly.

4) Let Oh) - « X- students of our class, B(x)- “students X went to the theater." Then the parcels will be as follows: Oh)B(x),, and the conclusion.

This conclusion is based on the rule of negation:

- that means it is correct.

5) Let us identify the logical form of the inference. 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 parcels will take the form: Oh)B(x), B(x) C(x), and the conclusion Oh)C(x).

Our conclusion will have the following logical form: - the rule of syllogism.

This means that this conclusion is correct.

In logic, various ways of checking the correctness of inferences are considered, including analysis of the correctness of inferences using Euler circles. It is carried out as follows: write down the conclusion in set-theoretic language; depict premises on Euler circles, considering them true; they look to see whether the conclusion is always true. If yes, then they say that the inference is constructed correctly. If a drawing is possible from which it is clear that the conclusion is false, then they say that the conclusion is incorrect.

Table 9

Let us show that inference made according to the rule of inference is deductive. First, let's write this rule in set-theoretic language.

Package Oh)B(x) can be written as TATV, Where TA And TV- truth sets of propositional forms Oh) And B(x).

Private parcel A(a) means that ATA, and the conclusion B(a) shows that ATV.

The entire inference constructed according to the inference rule will be written in set-theoretic language as follows: .

Rice. 58.

Examples.

1. Is the conclusion “If a number ends in a number” correct? 5, then the number is divisible by 5. Number 125 divided by 5. Therefore, writing the number 125 ends with a number 5 »?

Solution: This conclusion is made according to the scheme , which corresponds to . There is no such scheme known to us. Let's find out whether it is a rule of deductive inference?

Let's use Euler circles. In set-theoretic language

The resulting rule can be written as follows:

. Let us depict the sets on Euler circles TA And TV and denote the element A from many TV.

It turns out that it can be contained in a set TA, or may not belong to him (Fig. 59). In logic, it is believed that such a scheme is not a rule of deductive inference, 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 should I 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 set-theoretic language. Let us denote by A- many students of the Faculty of Education, through IN- many students who are teachers through WITH- many students over 20 years old.

Then the conclusion will take the form: .

If we depict these sets on circles, then 2 cases are possible:

1) sets A, B, C intersect;

2) set IN intersects with many WITH And A, and a lot A intersects IN, but does not intersect with WITH.

b) Let us denote by A many verbs, and through IN a lot of words that answer the question “what to do?” or “what should I 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 of 20 + 3. He reasons: “The number 23 is two-digit. Any two-digit number can be represented as a sum of digit terms. Therefore, 23 = 20 + 3."

The first and second sentences in this conclusion are premises, and one of a general nature is the statement “any two-digit number can be represented as a sum of digit terms,” and the other is particular, it characterizes only the number 23 - it is two-digit. The conclusion - this sentence that comes after the word “therefore” - is also private in nature, since it refers to the specific number 23.

Inferences, which are usually used in proving theorems, are based on the concept of logical implication. Moreover, from the definition of logical implication it follows that for all values ​​of the propositional variables for which the initial statements (premises) are true, the conclusion of the theorem is also true. Such conclusions are deductive.

In the example discussed above, the inference given is deductive.

Example 2. One of the techniques for introducing primary schoolchildren to the commutative property of multiplication is as follows. Using various visual aids, schoolchildren, 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 equality is true ab = ba.

In this conclusion, the premises are the first two equalities. They claim that such a property holds for specific natural numbers. The conclusion in this example is a general statement - the commutative property of multiplication of natural numbers.

In this conclusion, premises of a private nature show that some Natural numbers have the following property: rearranging the factors does not change the product. And on this basis it was concluded that all natural numbers have this property. Such inferences are called incomplete induction.

those. for some natural numbers it can be argued that the sum is less than their product. This means that 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 analogical reasoning.

Under analogy understand an inference in which, based on the similarity of two objects in some characteristics and the presence of an additional characteristic in one of them, a conclusion is made about the presence of the same characteristic in the other object.

A conclusion by analogy is in the nature of an assumption, a hypothesis, and therefore needs either proof or refutation.