Complicated inferences. Types of inference The correctness of inference depends primarily on

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 senses are not involved, that is, the entire process of inference takes place at the level of thinking and is independent of the information received at the moment from outside. Visually, the inference 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 a conclusion. It is customary to separate premises and conclusions from each other with a horizontal line. The conclusion is always written at the bottom, the premise at the top. 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 several advantages before the forms of sensory knowledge and experimental research. Since the process of inference takes place only in the field of thinking, it does not affect real objects. This is a very important property, since often the researcher does not have the opportunity to obtain a real object for observation or experiments due to its high cost, size or distance. Some items at the moment can be generally considered inaccessible for direct research. For example, such a group of objects can include space objects. As you know, the study of even the planets closest to the Earth by man is problematic.

Another advantage of inference is that it allows you to obtain reliable information about the object under study. For example, it was through reasoning that D.I. Mendeleev created his own periodic system of chemical elements. In the field of astronomy, the position of the planets is often determined without any visible contact, based only on the already available information about the laws of the position of celestial bodies.

Lack of inference we can say that often the conclusions are characterized by abstractness and do not reflect many of the specific properties of the subject. This does not apply, for example, to the aforementioned periodic table of chemical elements. It was 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 astronomers determine the position of a planet, its properties are reflected only approximately. Also, it is often impossible to talk about the correctness of the conclusion until it has been tested in practice.

Inferences can be true and probabilistic. The former reliably reflect the real state of affairs, the latter are uncertain. The types of inference are: induction, deduction and conclusion by analogy.

Inference - this is primarily the derivation of consequences, it is used everywhere. Every person in his life, regardless of profession, built inferences and received consequences from these conclusions. And here the question arises of the truth of such consequences. A person who is not familiar with logic uses it at the philistine level. That is, he judges things, builds inferences, draws conclusions based on what he has accumulated in the process of life.

Despite the fact that almost everyone is taught 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 qualified 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.

Derivation of the investigation is a very common logical operation. As a general rule, in order to obtain a true judgment, it is necessary that the premises also be true. However, this rule does not apply to proof 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 premises are discarded in the process of inference.

This text is an introductory fragment.

Immediate inference Inference built by transforming a judgment and containing one premise is called direct. There are four types of transformations of judgments: transformation, inversion, opposition to a predicate, inference

Inductive inference Inductive inference is called inference, in the form of which empirical generalization takes place, when, based on the recurrence 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 conjunction "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. IMMEDIATE CONCLUSIONS A judgment containing new knowledge can be obtained by transforming the judgment. 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 INCLUSIONS In the process of reasoning, sometimes inferences that are not are taken as deductive. The latter are called incorrect deductive inferences, and (actually) deductive ones are called correct.

B. INDUCTIVE CONCLUSIONS In contrast to deductive reasoning, in which there is a relationship of logical consequence between premises and conclusions, inductive inferences are such connections between premises and conclusions in logical forms, when

§ 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 critical arguments against it. Our discussion of traditional logic, as well as modern logic and mathematics, aimed at clarifying

38. Deductive inferences The following types of inferences are deductive: conclusions of logical connections and subject-predicate conclusions. 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 senses are not involved, that is, 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 questions related to this kind of inference. According to the work 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 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 overwhelming majority of reasoning that claims to be logical, in fact, is not. They are pseudological, logical, or at best only partially logical. The reasoning is logical

2. The concept of a micro-object as a concept of a trans-subjective reality or a trans-subjective object, called the "object of science", which is applicable to aesthetics This is not an object of my external senses, existing outside me and my consciousness: not something objectively real. This is not an object

CHAPTER I THE CONCEPT OF A MODEL AND THE CONCEPT OF IMITATION One should choose one of the people of kindness and always have him before our eyes - in order to live as if he is looking at us and act as if he sees us. Seneca. Moral letters to Lucilius, XI, 8 Take yourself, at last, for

Inference is a form of abstract thinking, through which new information is derived from previously available information. In this case, the senses are not involved, that is, the entire process of inference takes place at the level of thinking and is independent of the information received at the moment from outside. Visually, the inference 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 a conclusion. It is customary to separate premises and conclusions from each other with a horizontal line. The conclusion is always written at the bottom, the premise at the top. 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 several advantages before the forms of sensory knowledge and experimental research. Since the process of inference takes place only in the field of thinking, it does not affect real objects. This is a very important property, since often the researcher does not have the opportunity to obtain a real object for observation or experiments due to its high cost, size or distance. Some items at the moment can be generally considered inaccessible for direct research. For example, such a group of objects can include space objects. As you know, the study of even the planets closest to the Earth by man is problematic.

Another advantage of inference is that it allows you to obtain reliable information about the object under study. For example, it was through reasoning that D.I. Mendeleev created his own periodic system of chemical elements. In the field of astronomy, the position of the planets is often determined without any visible contact, based only on the already available information about the laws of the position of celestial bodies.

Lack of inference we can say that often the conclusions are characterized by abstractness and do not reflect many of the specific properties of the subject. This does not apply, for example, to the aforementioned periodic table of chemical elements. It was 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 astronomers determine the position of a planet, its properties are reflected only approximately. Also, it is often impossible to talk about the correctness of the conclusion until it has been tested in practice.

Inferences can be true and probabilistic. The former reliably reflect the real state of affairs, the latter are uncertain. The types of inference are: induction, deduction and conclusion by analogy.

Inference - this is primarily the derivation of consequences, it is used everywhere. Every person in his life, regardless of profession, built inferences and received consequences from these conclusions. And here the question arises of the truth of such consequences. A person who is not familiar with logic uses it at the philistine level. That is, he judges things, builds inferences, draws conclusions based on what he has accumulated in the process of life.

Despite the fact that almost everyone is taught 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 qualified 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.

Derivation of the investigation is a very common logical operation. As a general rule, in order to obtain a true judgment, it is necessary that the premises also be true. However, this rule does not apply to proof 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 premises are discarded in the process of inference.

2. Deductive reasoning

Like much in classical logic, the theory of deduction owes its origin to the ancient Greek philosopher Aristotle. He developed most of the questions related to this kind of inference.

According to the works of Aristotle deduction - this is a transition in the process of inference from the general to the particular. In other words, deduction is the gradual concretization of a more abstract concept. It goes through several steps, each time deriving a consequence from several premises.

It must be said that in the process of deductive inference, true knowledge must be obtained. This goal can be achieved only if the necessary conditions and rules are met. Inference rules are of two kinds: direct and indirect inference rules. Direct inference means obtaining a conclusion from two premises, which will be true, provided that the rules of direct inference are observed.

Thus, the premises must be true and the rules for obtaining consequences must be observed. If these rules are observed, we can talk about the correctness of thinking regarding the taken object. This means that in order to obtain a true judgment, new knowledge, it is not necessary to have all the information. Some of the information can be logically recreated and consolidated. Reinforcement is necessary, since without it, the very process of obtaining new information becomes meaningless. It is not possible to transmit such information or use it in any other way. Naturally, such consolidation occurs through the language (spoken, written, programming language, etc.). Consolidation in logic occurs primarily with the help of symbols. For example, these can be symbols for conjunction, disjunction, implication, literal expressions, brackets, etc.

The following types of inferences are deductive: conclusions of logical connections and subject-predicate conclusions.

Also deductive inferences are direct.

They are made from one premise and are called transformation, inversion and opposition to the predicate; inferences by the logical square are considered separately. Such conclusions are derived from categorical judgments.

Consider these conclusions. The transformation has a scheme:

S is not not-P.

This diagram shows that there is only one premise. This is a categorical judgment. The transformation is characterized by the fact that when the quality of the premise changes in the process of inference, its quantity does not change, and the predicate of the consequence negates the predicate of the premise. There are two ways of transformation - double negation and replacement of the negation in the predicate by the negation in the bundle. The first case is reflected in the diagram above. In the second, the transformation is reflected in the diagram as S is not-P - S is not P.

Depending on the type of judgment, the transformation can be expressed as follows.

All S are P - No S is not-P. No S is P - All S are not-P. Some S are P - Some S are not non-P. Some S are not P - Some S are not-P. Appeal is an inference in which the quality of the premise does not change when the subject and predicate change places.

That is, in the process of inference, the subject takes the place of the predicate, and the predicate - in the place of the subject. Accordingly, the circulation scheme can be depicted as S is P - P is S.

Treatment can be limited and unlimited (also called simple or pure). This division is based on a quantitative measure of judgment (meaning equality or inequality of volumes S and P). This is expressed in whether the quantifier word has changed or not and whether the subject and the predicate are distributed. If such a change occurs, then the restriction is handled. Otherwise, we can talk about clean treatment. Let's remind that a quantifier word is a word - an indicator of quantity. So, the words "all", "some", "none" and others are quantifiable words.

Opposition to predicate characterized by the fact that the bundle in the consequence changes to the opposite, the subject contradicts the premise predicate, and the predicate is equivalent to the premise subject.

It must be said that direct inference with opposition to the predicate cannot be deduced from partially affirmative judgments.

Here are the opposition schemes depending on the types of judgments.

Some S are not P - Some not-P are S. None of S are P - Some not-P are S. All S are P - None of P are S.

Combining what has been said, we can consider the opposition to the predicate as a product of two immediate inferences at once. The first of these is the transformation. Its result is exposed.

3. Conditional and dividing inferences

Speaking about deductive inferences, one cannot but pay attention to conditional and dividing inferences.

Conditional inferences are called so because they use conditional propositions (if a, then b) as premises. Conditional inferences can be reflected in the following diagram.

If a, then b. If b, then c. If a, then c.

The above is a diagram of inferences that are a kind of conditional. It is characteristic of such inferences that all their premises are conditional.

Another type of conditional inference is conditionally categorical judgments. According to the name in this conclusion, not both premises are conditional judgments, one of them is a simple categorical judgment.

It is also necessary to mention the modes - varieties of reasoning. There are: an affirming mode, a negating mode, and two probabilistic modes (the first and the second).

Affirmative modus is most widespread in thinking. This is due to the fact that he gives a reliable conclusion. Therefore, the rules of various academic disciplines are built mainly on the basis of an assertive modus. You can display the assertive mode as a diagram.

If a, then b.

Let's give an example of an asserting mode.

If the ax falls into the water, it will sink.

The ax fell into the water.

He will drown.

The two true judgments that are the premises of this judgment are converted in the process of inference into a true judgment. Negative mode expressed as follows. If a, then b. Not-b. Nope.

This judgment is based on the denial of the effect and the denial of the reason.

Inferences can give not only true, but also indefinite judgments (it is not known whether they are true or false).

In this regard, it should be said about probabilistic modes.

The first probabilistic mode on the diagram is displayed as follows.

If a, then b.

Probably a.

As the name implies, the consequence deduced from the premises using this modus is probable.

If a strong wind blows, the yacht heels to one side.

The yacht heels to one side.

A strong wind is probably blowing.

As we can see from the statement of the consequence to the statement of the reason, it is impossible to deduce a true conclusion.

The second probabilistic mode in the form of a diagram can be represented as follows.

If a, then b. Nope.

Probably not-b. Let's give an example.

If a person lies under the sun, he will tan.

This person is not lying under the sun.

It won't light up.

As can be seen from the above example, making an inference from denial of reason to denial of consequence, we get not true, but probabilistic consequence.

The formulas of the affirming and denying modes are the laws of logic, while the formulas of the probabilistic ones are not.

Dividing inferences are divided into simple dividing and dividing categorical inferences. In the first case, all parcels are dividing. Accordingly, separating-categorical judgments have as one of the premises a simple categorical judgment.

Thus, inference is considered dividing, all or part of the premises of which are dividing judgments. The structure of a simple dividing inference is reflected as follows.

S is A or B or C.

And there is A1 or A2.

S is A1 or A2 or B or C.

An example of such a conclusion is the following.

The path can be straight or circular.

The roundabout route can be with one change or with several changes.

The route can be direct or with one change, or with several changes.

S is A or B. S is A (B). S is not B (A). For example:

The shot can be accurate and inaccurate. This shot is accurate. This shot is not inaccurate.

Here it is necessary to mention the conditional dividing inferences. They differ from the above inferences in premises. One of them is a dividing judgment, which is not special, but the second premise of such judgments consists of two or more conditional judgments.

A conditional dividing judgment can be either a dilemma or a trilemma. In a dilemma the conditional premise consists of two members. At the same time, dividing implies a choice. In other words, a dilemma is one of two choices.

The dilemma is simple constructive and complex constructive, as well as simple and complex destructive. The first one has two premises, one of which asserts the same outcome of the two proposed situations, the other says that one of these situations is possible. The corollary summarizes the assertion of the first premise (conditional proposition).

If you press on a pencil, it will break; if you bend the pencil, it breaks.

You can click on the pencil or bend the pencil.

The pencil will break.

A complex constructive dilemma involves a harder choice between alternatives.

Trilemma consists of two premises and a corollary and offers a choice of three options or states three facts.

If the athlete strikes in time, he will win; if the athlete distributes the forces correctly, he will win; if the athlete jumps cleanly, he will win.

The athlete will strike in time or correctly distribute forces over the distance, or perform a jump cleanly.

The athlete will win.

There are cases when a conclusion or one of the premises is missed in conditional, dividing or conditionally dividing inferences. Such inferences are called abbreviated.

- This is a form of thinking in which from two or more judgments, called premises, a new judgment follows, called a conclusion (conclusion). For example:

All living organisms feed on moisture.

All plants are living organisms.

\u003d\u003e All plants feed on moisture.

In this example, the first two judgments are premises, and the third is a conclusion. The premises must be true judgments and must be related. If at least one of the premises is false, then the conclusion is false:

All birds are mammals.

All sparrows are birds.

\u003d\u003e All sparrows are mammals.

As you can see, in the given example, the falsity of the first premise leads to a false conclusion, despite the fact that the second premise is true. If the premises are not connected with each other, then it is impossible to draw a conclusion from them. For example, no conclusion follows from the following two premises:

All pines are trees.

Let's pay attention to the fact that inferences consist of judgments, and judgments - of concepts, that is, one form of thinking enters into another as a component.

All inferences are divided into direct and indirect.

IN direct inferences, the conclusion is made from one premise. For example:

All flowers are plants.

\u003d\u003e Some plants are flowers.

It is true that all flowers are plants.

\u003d\u003e It is not true that some flowers are not plants.

It is not hard to guess that direct inferences are already known to us operations of transformation of simple judgments and conclusions about the truth of simple judgments along the logical square. The first given example of direct inference is the transformation of a simple judgment by means of inversion, and in the second example along a logical square from the truth of a judgment of the form ANDit is concluded that the judgment of the form ABOUT.

IN mediated inferences, the conclusion is made from several premises. For example:

All fish are living beings.

All crucians are fish.

\u003d\u003e All crucians are living creatures.

Indirect inferences are divided into three types: deductive, inductive and analogous inferences.

Deductive inferences (deduction) (from lat. deductio -"Derivation") - these are inferences, in which a conclusion is made from the general rule for a particular case (a particular case is derived from the general rule). For example:

All stars radiate energy.

The sun is a star.

\u003d\u003e The sun emits energy.

As you can see, the first premise is a general rule, from which (with the help of the second premise) a special case follows in the form of a conclusion: if all stars emit energy, then the Sun also radiates it, because it is a star.

In deduction, reasoning goes from general to particular, from more to less, knowledge narrows, due to which deductive conclusions are reliable, that is, they are accurate, obligatory, necessary. Let's take another look at the above example. Could a different conclusion flow from these two premises than the one that follows from them? Could not. The resulting conclusion is the only possible one in this case. Let us depict the relationship between the concepts of which our inference consisted of Euler's circles. The scope of the three concepts: stars(3); bodies emitting energy(T) and The sun(C) will be schematically arranged as follows (Fig. 33).

If the scope of the concept starsincluded in the scope of the concept bodies emitting energy,and the scope of the concept The sunincluded in the scope of the concept stars,then the scope of the concept The sunis automatically included in the scope of the concept bodies emitting energy,by virtue of which the deductive conclusion is reliable.

The undoubted merit of deduction lies in the reliability of its conclusions. Let us recall that the famous literary hero Sherlock Holmes used the deductive method in solving crimes. This means that he built his reasoning in such a way as to derive the particular from the general. In one work, explaining to Dr. Watson the essence of his deductive method, he gives the following example. Scotland Yard detectives discovered a smoked cigar near the murdered Colonel Ashby and assumed that the colonel had smoked it before he died. However, Sherlock Holmes irrefutably proves that the colonel could not smoke this cigar, because he wore a large, lush mustache, and the cigar was smoked to the end, that is, if Colonel Ashby smoked it, he would certainly burn his mustache. Hence, another person smoked the cigar.

In this reasoning, the conclusion looks convincing precisely because it is deductive - from the general rule: Anyone with a big, bushy mustache cannot smoke a cigar to the end,a special case is displayed: Colonel Ashby could not completely smoke a cigar because he wore such a mustache.Let us bring the considered reasoning to the standard form of writing inferences in the form of premises and conclusions accepted in logic:

Anyone with a big, bushy mustache cannot completely smoke a cigar.

Colonel Ashby wore a large, lush mustache.

\u003d\u003e Colonel Ashby could not finish his cigar.

Inductive inferences (induction) (from lat. inductio -"Guidance") - these are inferences in which a general rule is derived from several special cases. For example:

Jupiter is moving.

Mars is moving.

Venus is moving.

Jupiter, Mars, Venus are planets.

\u003d\u003e All planets are moving.

The first three premises are special cases, the fourth premise brings them under one class of objects, unites them, and the conclusion speaks about all objects of this class, that is, a certain general rule is formulated (following from three special cases).

It is easy to see that inductive reasoning is built on the opposite principle to deductive reasoning. In induction, reasoning goes from particular to general, from less to more, knowledge expands, due to which inductive conclusions (as opposed to deductive ones) are not reliable, but probabilistic. In the example of induction considered above, the feature found in some objects of a certain group is transferred to all objects of this group, a generalization is made, which is almost always fraught with error: it is quite possible that there are some exceptions in the group, and even if a set of objects from a certain group is characterized by some feature, this does not mean that all objects of this group are characterized by such a feature. The probabilistic nature of the conclusions is, of course, a disadvantage of induction. However, its undoubted merit and advantageous difference from deduction, which is shrinking knowledge, lies in the fact that induction is expanding knowledge that can lead to something new, while deduction is an analysis of the old and already known.

Inference by analogy (analogy) (from the Greek. analogia -"Correspondence") are inferences in which, on the basis of the similarity of objects (objects) in some features, a conclusion is made about their similarity in other features. For example:

Planet Earth is located in the solar system, it has atmosphere, water and life.

The planet Mars is located in the solar system, it has an atmosphere and water.

\u003d\u003e There is probably life on Mars.

As you can see, two objects are compared (planet Earth and planet Mars), which are similar to each other in some essential, important features (being in the solar system, having an atmosphere and water). Based on this similarity, it is concluded that, perhaps, these objects are similar to each other in other features: if there is life on Earth, and Mars is in many ways similar to Earth, then the existence of life on Mars is not excluded. The conclusions of analogy, like those of induction, are probabilistic.

When all judgments are simple (Categorical syllogism)

All deductive inferences are called syllogisms (from the Greek. sillogismos -"Counting, summing up, deriving the investigation"). There are several types of syllogisms. The first of them is called simple, or categorical, because all judgments included in it (two premises and a conclusion) are simple, or categorical. These are judgments of the species already known to us A, I, E, O.

Consider an example of a simple syllogism:

All flowers(M) Are plants(R).

All roses(S) - this is flowers(M).

\u003d\u003e All roses(S) Are plants(R).

Both premises and conclusion are simple judgments in this syllogism, and both premises and inference are judgments of the form AND(generally affirmative). Let's pay attention to the conclusion presented by the judgment All roses are plants.In this conclusion, the subject is the term roses,and the predicate is the term plants.The subject of inference is present in the second premise of the syllogism, and the inference predicate is in the first. Also in both premises the term is repeated flowers,which, as it is easy to see, is a connecting: it is thanks to him that the terms that are not connected, separated in premises plantsand rosescan be linked in the output. Thus, the structure of the syllogism includes two premises and one conclusion, which consist of three (differently arranged) terms.

The subject of inference is located in the second premise of the syllogism and is called lesser term syllogism (the second package is also called lesser).

The inference predicate is located in the first premise of the syllogism and is called big term syllogism (the first package is also called more). The inference predicate, as a rule, is a larger concept in scope than the inference subject (in the given example, the concept rosesand plantsare in relation to generic subordination), due to which the inference predicate is called big term, and the subject of the output is smaller.

A term that is repeated in two premises and connects the subject with a predicate (lesser and greater terms) is called middle term syllogism and is denoted by a Latin letter M(from lat. medium -"middle").

The three terms of the syllogism can be located in it in different ways. The relative position of terms relative to each other is called figure of simple syllogism... There are four such figures, that is, all possible variants of the mutual arrangement of terms in the syllogism are exhausted by four combinations. Let's consider them.

The first figure of the syllogism - this is such an arrangement of his terms, in which the first premise begins with the middle term, and the second ends with the middle term. For example:

All gases(M) Are chemical elements(R).

Helium(S) Is gas(M).

\u003d\u003e Helium(S) Is a chemical element(R).

Considering that in the first premise the middle term is associated with the predicate, in the second premise the subject is associated with the middle term, and in the inference the subject is associated with the predicate, we will draw up a scheme of the arrangement and connection of the terms in the given example (Fig. 34).

Straight lines on the diagram (with the exception of the one that separates the premise from the conclusion) show the connection of terms in the premise and in the conclusion. Since the role of the middle term is to connect the greater and lesser terms of the syllogism, in the diagram the middle term in the first premise is connected by a line with the middle term in the second premise. The diagram shows exactly how the middle term connects the other terms of the syllogism in its first figure. In addition, the relationship between the three terms can be depicted using Euler circles. In this case, the following scheme will turn out (Fig. 35).

Second figure of the syllogism - this is the arrangement of his terms, in which both the first and the second premise end with a middle term. For example:

All fish(R) breathe with gills(M).

All whales(S) do not breathe with gills(M).

\u003d\u003e All whales(S) not fish(R).

Schemes of the mutual arrangement of terms and the relations between them in the second figure of the syllogism look as shown in Fig. 36.

The third figure of the syllogism - this is such an arrangement of his terms, in which both the first and second premises begin with the middle term. For example:

All tigers(M) Are mammals(R).

All tigers(M) Are predators(S).

\u003d\u003e Some predators(S) Are mammals(R).

Schemes of the mutual arrangement of terms and relations between them in the third figure of the syllogism are shown in Fig. 37.

The fourth figure of the syllogism - this is such an arrangement of his terms, in which the first premise ends with the middle term, and the second begins with it. For example:

All squares(R) Are rectangles(M).

All rectangles(M) Are not triangles(S).

\u003d\u003e All triangles(S) Are not squares(R).

Schemes of the mutual arrangement of terms and the relations between them in the fourth figure of the syllogism are shown in Fig. 38.

Note that the relationship between the terms of the syllogism in all figures may be different.

Any simple syllogism consists of three judgments (two premises and a conclusion). Each of them is simple and belongs to one of four types ( A, I, E, O). The set of simple judgments included in the syllogism is called mode of simple syllogism... For example:

All celestial bodies are moving.

All planets are celestial bodies.

\u003d\u003e All planets are moving.

In this syllogism, the first premise is a simple proposition of the form AND(generally affirmative), the second premise is also a simple proposition of the form AND,and the conclusion in this case is a simple judgment of the form AND.Therefore, the considered syllogism has the modus AAA,or barbara.The last Latin word does not mean anything and is not translated in any way - it is just a combination of letters, selected in such a way that it contains three letters and,symbolizing the modus of the syllogism AAA.Latin "words" to denote modes of simple syllogism were invented in the Middle Ages.

The next example is a syllogism with modus EAE,or cesare:

All magazines are periodicals.

All books are not periodicals.

\u003d\u003e All books are not magazines.

And one more example. This syllogism has the modus AAI,or darapti.

All carbons are simple bodies.

All carbons are electrically conductive.

\u003d\u003e Some electrical conductors are simple bodies.

There are 256 modes in all four figures (ie, possible combinations of simple judgments in syllogism). Each figure has 64 modes. However, out of these 256 modes, only 19 give reliable conclusions, the rest lead to probabilistic conclusions. If we take into account that one of the main features of deduction (and hence of a syllogism) is the reliability of its conclusions, then it becomes clear why these 19 modes are called correct, and the rest are called incorrect.

Our task is to be able to determine the figure and mode of any simple syllogism. For example, you need to set the figure and mode of the syllogism:

All substances are made up of atoms.

All liquids are substances.

\u003d\u003e All liquids are made of atoms.

First of all, it is necessary to find the subject and the predicate of the inference, that is, the lesser and greater terms of the syllogism. The next step is to establish the location of the smaller term in the second premise and the larger in the first. After that, you can define the middle term and schematically depict the location of all terms in the syllogism (Fig. 39).

All substances(M) are made of atoms(R).

All liquids(S) Are substances(M).

\u003d\u003e All liquids(S) are made of atoms(R).

As you can see, the syllogism under consideration is built according to the first figure. Now we need to find its modus. To do this, it is necessary to find out what kind of simple judgments the first and second premises and conclusion belong to. In our example, both premises and conclusion are judgments of the form AND(generally affirmative), that is, the modus of a given syllogism is AAA, or b arb ar a. So, the proposed syllogism has the first figure and mode AAA.

Going to school forever (General rules of syllogism)

The rules of the syllogism are divided into general and particular.

General rules apply to all simple syllogisms, no matter what shape they are built on. Private the rules apply only to each figure of the syllogism and are therefore often called figure rules. Consider the general rules of syllogism.

There should be only three terms in a syllogism.Let us turn to the already mentioned syllogism, in which this rule is violated.

The movement is eternal.

Going to school is movement.

\u003d\u003e Going to school is forever.

Both premises of this syllogism are true judgments, but a false conclusion follows from them, because the rule in question is violated. Word motionused in two premises in two different meanings: motion as a universal world change and motion as a mechanical movement of a body from point to point. It turns out that there are three terms in the syllogism: movement, going to school, eternity,and there are four meanings (since one of the terms is used in two different senses), that is, an extra meaning, as it were, implies an extra term. In other words, in the given example of the syllogism, there were not three, but four (in terms of meaning) terms. An error that occurs when the above rule is violated is called quadrupling of terms.

The middle term must be distributed in at least one of the premises.The distribution of terms in simple judgments was discussed in the previous chapter. Recall that the easiest way to establish the distribution of terms in simple judgments is using circular schemes: it is necessary to depict the relations between the terms of the judgment with Euler circles, while a full circle on the diagram will denote a distributed term (+), and an incomplete one - unallocated (-). Consider an example of a syllogism.

All cats(TO) Are living beings(J. with).

Socrates(FROM) - this is also a living being.

\u003d\u003e Socrates is a cat.

A false conclusion follows from two true premises. Let us depict by Euler circles the relations between the terms in the premises of the syllogism and establish the distribution of these terms (Fig. 40).

As you can see, the middle term ( living beings) in this case is not distributed in any of the premises, but according to the rule it must be distributed in at least one. The error that occurs when the rule in question is violated is called - non-distribution of the middle term in each parcel.

A term that has not been allocated in the premise cannot be allocated in the output.Let's look at the following example:

All apples(I AM) - edible items(S. p.).

All pears(D) Are not apples.

\u003d\u003e All pears are inedible.

The premises of the syllogism are true judgments, and the conclusion is false. As in the previous case, let us depict by Euler's circles the relations between the terms in the premises and in the derivation of the syllogism and establish the distribution of these terms (Fig. 41).

In this case, the inference predicate, or the larger term of the syllogism ( edible items), in the first premise is unallocated (-), and in the conclusion - distributed (+), which is prohibited by the rule in question. The error that occurs when it is violated is called expansion of the larger term... Recall that the term is distributed when it comes to all the objects included in it, and unallocated when it comes to a part of the objects included in it, which is why the error is called the extension of the term.

There should be no two negative premises in a syllogism.At least one of the premises of the syllogism must be positive (both premises can be positive). If two premises in the syllogism are negative, then a conclusion from them either cannot be drawn at all, or, if it is possible to do it, it will be false or, at least, unreliable, probabilistic. For example:

Snipers cannot have poor eyesight.

All my friends are not snipers.

\u003d\u003e All my friends have poor eyesight.

Both premises in syllogism are negative propositions, and, despite their truth, a false conclusion follows from them. The error that occurs in this case is called that - two negative premises.

There should be no two particular premises in a syllogism.

At least one of the premises must be common (both premises can be common). If two premises in the syllogism are private judgments, then it is impossible to draw a conclusion from them. For example:

Some schoolchildren are first graders.

Some schoolchildren are tenth graders.

No conclusion follows from these premises, because they are both private. The error that occurs when this rule is violated is called - two private parcels.

If one of the premises is negative, then the conclusion must also be negative.For example:

No metal is an insulator.

Copper is a metal.

\u003d\u003e Copper is not an insulator.

As you can see, an affirmative conclusion cannot follow from the two premises of this syllogism. It can only be negative.

If one of the premises is private, then the conclusion must also be private.For example:

All hydrocarbons are organic compounds.

Some substances are hydrocarbons.

\u003d\u003e Some substances are organic compounds.

In this syllogism, a general conclusion cannot follow from the two premises. It can only be private, since the second premise is private.

Here are some more examples of simple syllogism - both correct and in violation of some general rules.

All herbivores feed on plant foods.

All tigers do not eat plant foods.

\u003d\u003e All tigers are not herbivores.

(Correct syllogism)

All excellent students do not receive twos.

My friend is not an excellent student.

\u003d\u003e My friend gets deuces.

All fish swim.

All whales swim too.

\u003d\u003e All whales are fish.

(Error - the middle term is not distributed in any of the premises)

The bow is an ancient archery weapon.

One of the vegetable crops is onion.

\u003d\u003e One of the vegetable crops is an ancient shooting weapon.

Any metal is not an insulator.

Water is not metal.

\u003d\u003e Water is an insulator.

(Error - two negative premises in the syllogism)

No insect is a bird.

All bees are insects.

\u003d\u003e No bee is a bird.

(Correct syllogism)

All chairs are pieces of furniture.

All cabinets are not chairs.

\u003d\u003e All wardrobes are not pieces of furniture.

Laws are invented by people.

Gravity is a law.

\u003d\u003e Gravity was invented by people.

(Error - quadrupling of terms in simple syllogism)

All people are mortal.

All animals are not people.

\u003d\u003e Animals are immortal.

(Error - extension of a larger term in syllogism)

All Olympic champions are athletes.

Some Russians are Olympic champions.

\u003d\u003e Some Russians are athletes.

(Correct syllogism)

Matter is uncreate and indestructible.

Silk is matter.

\u003d\u003e Silk is uncreate and indestructible.

(Error - quadrupling of terms in simple syllogism)

All school graduates take exams.

All fifth-year students are non-graduates.

\u003d\u003e All fifth-year students do not take exams.

(Error - extension of a larger term in syllogism)

All stars are not planets.

All asteroids are minor planets.

\u003d\u003e All asteroids are not stars.

(Correct syllogism)

All grandfathers are fathers.

All fathers are men.

\u003d\u003e Some men are grandfathers.

(Correct syllogism)

No first grader is an adult.

All adults are not first graders.

\u003d\u003e All adults are minors.

(Error - two negative premises in the syllogism)

Brevity is the sister of talent (Types of abbreviated syllogism)

A simple syllogism is one of the most widespread types of inference. Therefore, it is often used in everyday and scientific thinking. However, when using it, we, as a rule, do not observe its clear logical structure. For example:

All fish are not mammals.

All whales are mammals.

\u003d\u003e Therefore, all whales are not fish.

Instead, we will most likely say: All whales are not fish, as they are mammalsor: All whales are not fish, because fish are not mammals.It is easy to see that these two inferences are an abbreviated form of the above simple syllogism.

Thus, in thinking and speech, not a simple syllogism is usually used, but its various abbreviated varieties. Let's consider them.

Enthymeme Is a simple syllogism, in which one of the premises or conclusion is missing. It is clear that three entimemes can be derived from any syllogism. For example, take the following syllogism:

All metals are electrically conductive.

Iron is a metal.

\u003d\u003e Iron is electrically conductive.

Three enthymemes follow from this syllogism: Iron is electrically conductive as it is a metal(big package missing); Iron is electrically conductive because all metals are electrically conductive(a smaller package is missing); All metals are electrically conductive and iron is a metal(output skipped).

Epicheirem Is a simple syllogism in which both premises are entimemes. Let's take two syllogisms and deduce the entimemes from them.

Syllogism 1

Anything that leads society to disaster is evil.

Social injustice leads society to disaster.

\u003d\u003e Social injustice is evil.

Omitting the big premise in this syllogism, we get the following entimeme: Social injustice is evil because it leads society to disaster.

Syllogism 2

Anything that contributes to the enrichment of some at the expense of the impoverishment of others is social injustice.

Private property contributes to the enrichment of some at the expense of the impoverishment of others.

\u003d\u003e Private property is a social injustice.

Skipping the big premise in this syllogism, we get the following entimeme: If we arrange these two enthymemes one after the other, then they will become the premises of a new, third syllogism, which will be the epicheire:

Social injustice is evil because it leads society to disaster.

Private property is a social injustice, as it contributes to the enrichment of some at the expense of the impoverishment of others.

\u003d\u003e Private property is evil.

As you can see, three syllogisms can be distinguished in the composition of the epicheireme: two of them are parcels, and one is built from the conclusions of parcel syllogisms. This last syllogism provides the basis for the final conclusion.

Polysillogism (complex syllogism) - these are two or more simple syllogisms interconnected in such a way that the conclusion of one of them is the premise of the next. For example:

Let's pay attention to the fact that the conclusion of the previous syllogism became a larger premise of the next one. In this case, the resulting polysillogism is called progressive... If the conclusion of the previous syllogism becomes a lesser premise of the next one, then the polysyllogism is called regressive... For example:

The conclusion of the previous syllogism is a lesser premise of the next. It can be noted that in this case, two syllogisms cannot be graphically connected in a sequential chain, as in the case of progressive polysillogism.

It was said above that polysillogism can consist not only of two, but also of a larger number of simple syllogisms. Let's give an example of a polysillogism (progressive), which consists of three simple syllogisms:

Litters (compound syllogism) is a polysillogism, in which the premise of the subsequent syllogism, which is the conclusion of the previous one, is omitted. Let's return to the example of progressive polysyllogism considered above and skip in it the big premise of the second syllogism, which is the conclusion of the first syllogism. The result is a progressive litter:

Anything that develops thinking is useful.

All mind games develop thinking.

Chess is an intellectual game.

\u003d\u003e Chess is useful.

Now let's turn to the example of regressive polysillogism considered above and skip in it the lesser premise of the second syllogism, which is the conclusion of the first syllogism. The result is a regressive sorite:

All stars are celestial bodies.

The sun is a star.

All celestial bodies participate in gravitational interactions.

\u003d\u003e The sun participates in gravitational interactions.

Whether it is rain or snow (Inference with the union OR)

Inferences that contain separative (disjunctive) judgments are called dividing separating categorical syllogism, in which, as the name implies, the first premise is a dividing (disjunctive) judgment, and the second premise is a simple (categorical) judgment. For example:

An educational institution can be elementary, or secondary, or higher.

MSU is a higher educational institution.

\u003d\u003e MSU is not an elementary or secondary educational institution.

IN affirmative-negative mode the first premise is a strict disjunction of several options for something, the second asserts one of them, and the conclusion negates all the others (thus, the reasoning moves from assertion to negation). For example:

Forests are coniferous, deciduous, or mixed.

This forest is coniferous.

\u003d\u003e This forest is neither deciduous nor mixed.

IN denial-assertive modus, the first premise is a strict disjunction of several options for something, the second denies all these options except one, and the conclusion asserts one remaining option (thus, the reasoning moves from denial to affirmation). For example:

People are Caucasians, or Mongoloids, or Negroids.

This person is not a Mongoloid or Negroid.

\u003d\u003e This person is Caucasian.

The first premise of the separating-categorical syllogism is a strict disjunction, that is, it is the already familiar logical operation of dividing a concept. Therefore, it is not surprising that the rules of this syllogism repeat the rules for dividing a concept known to us. Let's consider them.

The division in the first premise should be done on one basis.For example:

Transport is land, or underground, or water, or air, or public.

Suburban electric trains are public transport.

\u003d\u003e Suburban electric trains are not ground, underground, water or air transport.

The syllogism is built according to the affirmative-negative mode: in the first premise, several options are presented, in the second premise, one of them is asserted, due to which all the others are denied in the conclusion. However, a false conclusion follows from two true premises.

Why does this happen? Because in the first premise, the division was carried out on two different grounds: in what natural environment the transport moves and to whom it belongs. Already familiar to us division base substitution in the first premise of the separating-categorical syllogism leads to a false conclusion.

The division in the first premise must be complete.For example:

Mathematical operations can be addition, or subtraction, or multiplication, or division.

Taking logarithm is not addition, subtraction, multiplication, or division.

\u003d\u003e Taking logarithms is not a mathematical operation.

Known to us incomplete division error in the first premise of the syllogism causes a false conclusion arising from the true premises.

The division results in the first premise must not overlap, or the disjunction must be strict.For example:

The countries of the world are northern, or southern, or western, or eastern.

Canada is a northern country.

\u003d\u003e Canada is not a southern, western or eastern country.

In syllogism, the conclusion is false, since Canada is as much a northern country as it is western. False conclusion with true premises is explained in this case intersection of division results in the first premise, or, which is the same, - loose disjunction... It should be noted that a non-strict disjunction in a separating-categorical syllogism is admissible in the case when it is constructed according to a negative-asserting mode. For example:

He is naturally strong or is constantly involved in sports.

He is not naturally strong.

\u003d\u003e He is constantly involved in sports.

There is no error in the syllogism, despite the fact that the disjunction in the first premise was not strict. Thus, the rule in question unconditionally applies only for the affirmative-negative mode of the separating-categorical syllogism.

The division in the first premise must be consistent.For example:

Sentences can be simple, or complex, or complex.

This sentence is complex.

\u003d\u003e This sentence is neither simple nor complicated.

In syllogism, a false conclusion follows from true premises for the reason that in the first premise a mistake already known to us was made, which is called jump in division.

Let us give a few more examples of separating-categorical syllogism - both correct and with violations of the considered rules.

Quadrangles are either squares or rhombuses or trapezoids.

This figure is not a diamond or a trapezoid.

\u003d\u003e This shape is a square.

(Error - incomplete division)

Selection in wildlife can be artificial or natural.

This selection is not artificial.

\u003d\u003e This selection is natural.

(Correct inference)

People can be talented, or mediocre, or stubborn.

He is a stubborn person.

\u003d\u003e He is not talented or mediocre.

(Error - substitution of the base in division)

Educational institutions are elementary, or secondary, or higher, or universities.

MSU is a university.

\u003d\u003e MSU is not a primary, secondary or higher educational institution.

(Error - jump in division)

You can study natural sciences or humanities.

I study natural sciences.

\u003d\u003e I don't study humanities.

(Error - intersection of division results, or lax disjunction)

Elementary particles have a negative electric charge, or positive, or neutral.

Electrons have a negative electrical charge.

\u003d\u003e Electrons have neither positive nor neutral electric charge.

(Correct inference)

Publications are periodical, or non-periodical, or foreign.

This edition is foreign.

\u003d\u003e This publication is not periodic and is not non-recurrent.

(Error - substitution of the basis)

The divisional-categorical syllogism in logic is often called simply the divisional-categorical inference. Besides it, there is also purely dividing syllogism (purely dividing inference), both premises and the conclusion of which are dividing (disjunctive) judgments. For example:

Mirrors can be flat or spherical.

Spherical mirrors are either concave or convex.

\u003d\u003e Mirrors can be flat or concave or convex.

If a person is flattering, then he is lying (Inferences with the union IF ... THEN)

Inferences that contain conditional (implicative) judgments are called conditional... In thinking and speaking, it is often used conditionally categorical a syllogism, the name of which indicates that the first premise in it is a conditional (implicative) judgment, and the second premise is simple (categorical). For example:

Today the runway is covered with ice.

\u003d\u003e Airplanes cannot take off today.

Affirmative modus - in which the first premise is an implication (consisting, as we already know, of two parts - the basis and the effect), the second premise is the assertion of the basis, and the conclusion asserts the effect. For example:

This substance is metal.

\u003d\u003e This substance is electrically conductive.

Negative mode - in which the first premise is the implication of reason and effect, the second premise is the negation of the corollary, and in the conclusion the reason is denied. For example:

If the substance is a metal, then it is electrically conductive.

This substance is not electrically conductive.

\u003d\u003e This substance is not metal.

It is necessary to pay attention to the feature of the implicative judgment already known to us, which consists in the fact that cause and effect cannot be reversed.For example, saying If the substance is metal, then it is electrically conductiveis true, since all metals are electrical conductors (from the fact that a substance is a metal, its electrical conductivity necessarily follows). However, the saying If the substance is electrically conductive, then it is metalis incorrect, since not all electrical conductors are metals (from the fact that a substance is electrically conductive, it does not follow that it is a metal). This feature of the implication determines two rules of conditionally categorical syllogism:

1. It is possible to assert only from the basis to the effect,that is, in the second premise of the affirming mode, the basis of the implication (the first premise) should be asserted, and in the conclusion - its consequence. Otherwise, a false conclusion may follow from two true premises. For example:

If a word is at the beginning of a sentence, then it is always written with a capital letter.

Word« Moscow» always capitalized.

\u003d\u003e Word« Moscow» always at the beginning of a sentence.

In the second premise, the consequence was affirmed, and in the conclusion, the basis. This statement from the investigation to the foundation is the reason for the false conclusion with the true premises.

2. It is possible to deny only from the investigation to the basis,that is, in the second premise of the negating mode, the consequence of the implication (the first premise) should be denied, and in the conclusion - its basis. Otherwise, a false conclusion may follow from two true premises. For example:

If the word is at the beginning of a sentence, then it must be written with a capital letter.

In this sentence, the word« Moscow» not at the beginning.

\u003d\u003e In this sentence, the word« Moscow» do not capitalize.

In the second premise, the reason is denied, and in the conclusion, the effect. This denial from reason to effect is the reason for the false conclusion when the premises are true.

Here are a few more examples of conditionally categorical syllogism - both correct and with violations of the considered rules.

If the animal is a mammal, then it is a vertebrate.

Reptiles are not mammals.

\u003d\u003e Reptiles are not vertebrates.

If a person is flattering, then he is lying.

This man is flattering.

\u003d\u003e This person is lying.

(Correct conclusion).

If a geometric figure is a square, then all its sides are equal.

An equilateral triangle is not a square.

\u003d\u003e An equilateral triangle has unequal sides.

(Error - denial from reason to effect).

If the metal is lead, then it is heavier than water.

This metal is heavier than water.

\u003d\u003e This metal is lead.

If a celestial body is a planet of the solar system, then it moves around the sun.

Halley's comet moves around the sun.

\u003d\u003e Halley's comet is a planet of the solar system.

(Error is a statement from the investigation to the basis).

If water turns into ice, then it increases in volume.

The water in this vessel has turned to ice.

\u003d\u003e The water in this vessel has increased in volume.

(Correct conclusion).

If a person is a judge, then he has a higher legal education.

Not every graduate of the law faculty of Moscow State University is a judge.

\u003d\u003e Not every graduate of the law faculty of Moscow State University has a higher legal education.

(Error - denial from reason to effect).

If the lines are parallel, then they have no common points.

The intersecting lines have no common points.

\u003d\u003e Crossed lines are parallel.

(Error is a statement from the investigation to the basis).

If a technical product is equipped with an electric motor, then it consumes electricity.

All electronic products consume electricity.

\u003d\u003e All electronic products are equipped with electric motors.

(Error is a statement from the investigation to the basis).

Recall that among complex judgments, in addition to the implication ( a \u003d\u003e b) there is also an equivalent ( and<=> b). If the implication always highlights the basis and the effect, then in the equivalent there is neither one nor the other, since it is a complex proposition, both parts of which are identical (equivalent) to each other. The syllogism is called equivalent-categoricalif the first premise of the syllogism is not implication, but equivalence. For example:

If the number is even, then it is divisible by 2 without a remainder.

Number 16 is even.

\u003d\u003e Number 16 is divisible by 2 without remainder.

Since in the first premise of an equivalent-categorical syllogism, neither grounds nor consequences can be distinguished, the rules of a conditionally-categorical syllogism considered above are inapplicable to it (in an equivalent-categorical syllogism, you can both assert and deny as you like).

So, if one of the premises of the syllogism is a conditional, or implicative, judgment, and the second is categorical, or simple, then we have conditional-categorical syllogism (also often called conditional-categorical inference). If both premises are conditional judgments, then this is a purely conditional syllogism, or a purely conditional inference. For example:

If the substance is a metal, then it is electrically conductive.

If the substance is electrically conductive, then it cannot be used as an insulator.

\u003d\u003e If the substance is a metal, then it cannot be used as an insulator.

In this case, not only both premises, but also the conclusion of the syllogism are conditional (implicative) judgments. Another kind of purely conditional syllogism:

If the triangle is rectangular, then its area is half the product of its base and height.

If a triangle is not right-angled, then its area is half the product of its base and height.

\u003d\u003e The area of \u200b\u200ba triangle is half the product of its base and height.

As you can see, in this variety of purely conventional syllogism, both premises are implicative judgments, but the conclusion (unlike the first considered variety) is a simple proposition.

We are faced with a choice (conditionally dividing inferences)

In addition to dividing categorical and conventionally categorical inferences, or syllogisms, there are also conventionally dividing inferences. IN conditional dividing inference (syllogism), the first premise is a conditional or implicative proposition, and the second premise is a dividing, or disjunctive, proposition. It is important to note that in a conditional (implicative) judgment there may be more than one reason and one consequence (as in the examples that we have considered so far), but more reasons or consequences. For example, in the judgment If you enroll in Moscow State University, you need to study a lot or you need to have a lot of moneytwo consequences follow from one foundation. In judgment If you enroll at Moscow State University, you need to do a lot, and if you enter MGIMO, you also need to do a lot.one consequence follows from two reasons. In judgment If a country is ruled by a wise man, then it flourishes, and if it is ruled by a rogue, then it is in poverty.two consequences follow from two reasons. In judgment If I oppose the injustice around me, then I will remain human, although I will suffer severely; if I pass by her indifferently, I will cease to respect myself, although I will be whole and unharmed; and if I help her in every possible way, then I will turn into an animal, although I will achieve material and career well-beingfrom three reasons, three consequences follow.

If the first premise of the conditionally dividing syllogism contains two reasons or consequences, then such a syllogism is called dilemma, if there are three reasons or consequences, then it is called trilemma, and if the first premise includes more than three reasons or consequences, then the syllogism is polyilemma... Most often, in thinking and speech, there is a dilemma, by the example of which we will consider the conditionally dividing syllogism (also often called the conditionally dividing inference).

The dilemma can be constructive (affirming) and destructive (denying). Each of these types of dilemma, in turn, is divided into two types: both constructive and destructive dilemmas can be simple or complex.

IN simple constructive dilemma one consequence follows from two grounds, the second premise is a disjunction of grounds, and the conclusion asserts this one consequence in the form of a simple proposition. For example:

If you enroll at Moscow State University, you have to do a lot, and if you enter MGIMO, you also need to do a lot.

You can enroll in Moscow State University or MGIMO.

\u003d\u003e You have to do a lot.

In the first package complex constructive dilemma two reasons follow from two reasons, the second premise is a disjunction of reasons, and a conclusion is a complex judgment in the form of a disjunction of consequences. For example:

If a country is ruled by a wise man, then it flourishes, and if it is ruled by a rogue, then it is in poverty.

A country can be ruled by a wise man or a rogue.

\u003d\u003e A country can prosper or be poor.

In the first package simple destructive dilemma from one reason two consequences follow, the second premise is a disjunction of negations of consequences, and in the conclusion the reason is denied (there is a denial of a simple proposition). For example:

If you enter Moscow State University, you have to study a lot or you need a lot of money.

I don't want to study a lot or spend a lot of money.

\u003d\u003e I will not go to Moscow State University.

In the first package complex destructive dilemma Two consequences follow from two grounds, the second premise is a disjunction of negations of consequences, and a conclusion is a complex judgment in the form of a disjunction of negations of grounds. For example:

If a philosopher considers matter to be the origin of the world, then he is a materialist, and if he considers consciousness to be the origin of the world, then he is an idealist.

This philosopher is not a materialist or an idealist.

\u003d\u003e This philosopher does not consider matter to be the origin of the world, or he does not consider consciousness to be the origin of the world.

Since the first premise of the conditionally dividing syllogism is an implication, and the second is a disjunction, its rules are the same as the rules of the conditionally categorical and dividing categorical syllogisms considered above.

Here are some more examples of the dilemma.

If you learn English, you need daily speaking practice, and if you learn German, you also need everyday speaking practice.

You can study English or German.

\u003d\u003e Daily speaking practice is essential.

(Simple constructive dilemma).

If I confess to my wrongdoing, I will bear the punishment I deserve, and if I try to hide it, I will feel remorse.

I will either confess to my wrongdoing, or try to hide it.

\u003d\u003e I will suffer the punishment I deserve or feel remorse.

(Difficult constructive dilemma).

If he marries her, he will fail completely, or will drag out a miserable existence.

He doesn't want to crash completely or drag out a miserable existence.

\u003d\u003e He won't marry her.

(Simple destructive dilemma).

If the speed of the Earth during its orbital motion were more than 42 km / s, then it would leave the solar system; and if its speed were less than 3 km / s, then it« fell» would be in the sun.

Earth does not leave the solar system and does not« falls» in the sun.

\u003d\u003e The speed of the Earth when it moves in orbit is not more than 42 km / s and not less than 3 km / s.

(Complex destructive dilemma).

All students 10B are Losers (Inductive Inference)

In induction, a general rule is derived from several particular cases, the reasoning goes from particular to general, from less to more, knowledge expands, due to which inductive conclusions are, as a rule, probabilistic. Induction is complete and incomplete. IN full induction all objects from any group are listed and a conclusion is made about this whole group. For example, if all nine major planets of the solar system are listed in the premises of inductive inference, then such induction is complete:

Mercury is moving.

Venus is moving.

The earth is moving.

Mars is moving.

Pluto is moving.

Mercury, Venus, Earth, Mars, Pluto are the large planets of the solar system.

=>

IN incomplete induction some objects from any group are listed and a conclusion is made about this whole group. For example, if in the premises of inductive inference, not all nine large planets of the solar system are listed, but only three of them, then such induction is incomplete:

Mercury is moving.

Venus is moving.

The earth is moving.

Mercury, Venus, Earth are the large planets of the solar system.

\u003d\u003e All major planets of the solar system are moving.

It is clear that the conclusions of full induction are reliable, and incomplete ones are probabilistic, however, full induction is rare, and therefore, by inductive inference, incomplete induction is usually meant.

To increase the likelihood of incomplete induction conclusions, the following important rules should be observed.

1. It is necessary to select as many initial premises as possible.For example, consider the following situation. You want to check the level of student performance in a certain school. Let's say there are 1000 students in it. Using the full induction method, each student in this thousand should be tested for academic performance. Since it is rather difficult to do this, you can use the method of incomplete induction: test some part of the students and draw a general conclusion about the level of academic performance in a given school. Various opinion polls are also based on the use of incomplete induction. Obviously, the more students are tested, the more reliable the basis for inductive generalization will be and the more accurate the conclusion will be. However, just a larger number of initial premises, as required by the rule under consideration, is not enough to increase the probability of inductive generalization. Let's say that a considerable number of students will pass the test, but, by chance, there will be only those who are not successful among them. In this situation, we will come to a false inductive conclusion that the level of achievement in this school is very low. Therefore, the first rule is supplemented by the second.

2. It is necessary to select a variety of premises.

Returning to our example, we note that the set of test-takers should not only be as large as possible, but also specially (according to some system) formed, and not randomly selected, that is, it is necessary to make sure that students are included in it ( in approximately the same quantitative ratio) from different classes, parallels, etc.

3. It is necessary to draw a conclusion only on the basis of essential features.If, for example, during testing it turns out that a 10th grade student does not know by heart the entire Periodic Table of Chemical Elements, then this fact (sign) is insignificant for the conclusion about his progress. However, if testing shows that a 10th grade student has a particle NOTwrites together with the verb, then this fact (feature) should be recognized as essential (important) for the conclusion about the level of his education and academic performance.

These are the basic rules for incomplete induction. Now let's turn to its most common mistakes. Speaking about deductive reasoning, we considered this or that error together with the rule, the violation of which gives rise to it. In this case, the rules of incomplete induction are first presented, and then, separately, its errors. This is because each of them is not directly related to any of the above rules. Any inductive error can be viewed as the result of the simultaneous violation of all the rules, and at the same time, violation of each rule can be presented as a cause leading to any of the errors.

The first mistake often encountered in incomplete induction is called hasty generalization... Most likely, each of us is familiar with her. Everyone has heard such statements as All men are callous, All women are frivolousand so on. These common stereotypical phrases are nothing more than a hasty generalization in incomplete induction: if some objects from a group have a certain feature, this does not mean that the whole group is characterized by this feature, without exception. True premises of inductive inference can lead to a false conclusion if a hasty generalization is allowed. For example:

K. studies poorly.

N. studies poorly.

S. studies poorly.

K., N., S. are students 10« AND».

\u003d\u003e All students 10« AND» study badly.

Unsurprisingly, a hasty generalization is at the heart of many allegations, rumors and gossip.

The second mistake has a long and seemingly strange name: after that, then, because of this (from lat. post hoc, ergo propter hoc). In this case, we are talking about the fact that if one event occurs after another, then this does not necessarily mean their causal relationship. Two events can be connected only by a time sequence (one earlier, the other later). When we say that one event is necessarily the cause of another, because one of them happened before the other, then we are committing a logical error. For example, in the following inductive inference, the generalizing conclusion is false, despite the truth of the premises:

The day before yesterday a black cat crossed the road to a poor student, and he received a deuce.

Yesterday a black cat ran across the road to a poor student, and his parents were summoned to school.

Today a black cat ran across the road to N.'s poor student, and he was expelled from school.

\u003d\u003e A black cat is to blame for all the misfortunes of N.'s poor student.

Unsurprisingly, this common mistake has spawned many tales, superstitions and hoaxes.

The third mistake, widespread in incomplete induction, is called substitution of conditional by unconditional... Consider inductive inference in which a false conclusion follows from true premises:

At home, water boils at 100 ° C.

Outside, water boils at a temperature of 100 ° C.

In the laboratory, water boils at 100 ° C.

\u003d\u003e Water boils everywhere at 100 ° C.

We know that high in the mountains, water boils at a lower temperature. On Mars, boiling water would have a temperature of about 45 ° C. So the question is Is boiling water always and everywhere hot?is not as absurd as it might seem at first glance. And the answer to this question would be: Not always and not everywhere.What manifests itself in some conditions may not manifest in others. In the premises of the considered example, there is a conditional (occurring under certain conditions), which is replaced by an unconditional (occurring in all conditions in the same way, not depending on them) in the output.

A good example of the substitution of the conditional by the unconditional is contained in the fairy tale about tops and roots, known to us from childhood, in which we are talking about how a man and a bear planted a turnip, agreeing to divide the crop as follows: to a peasant - roots, to a bear - tops. Having received the tops of the turnip, the bear realized that the peasant had deceived him, and made the logical mistake of substituting the conditional with the unconditional - he decided that only the roots should always be taken. Therefore, the next year, when it was time to divide the wheat harvest, the bear gave the peasant tops, and again took the tops for himself - and again was left with nothing.

Here are some more examples of errors in inductive reasoning.

1. As you know, grandfather, grandmother, granddaughter, bug, cat and mouse pulled out a turnip. However, the grandfather did not pull out the turnip, the grandmother did not pull it out either. The granddaughter, the Beetle and the cat also did not pull out the turnip. It was possible to pull it out only after the mouse came to the rescue. Therefore, the mouse pulled out the turnip.

(Error - "after this" means "because of this").

2. For a long time in mathematics it was believed that all equations can be solved in radicals. This conclusion was made on the basis that the studied equations of the first, second, third and fourth degrees can be reduced to the form x n \u003d a.However, later it turned out that equations of the fifth degree cannot be solved in radicals.

(Error is a hasty generalization).

3. In classical, or Newtonian, natural science, it was believed that space and time are unchanged. This belief was based on the fact that no matter where different material objects are and whatever happens to them, time for each of them flows the same way and space remains the same. However, the theory of relativity, which appeared at the beginning of the 20th century, showed that space and time are not at all constant. So, for example, when material objects move with speeds close to the speed of light (300,000 km / s), time for them slows down significantly, and space is curved and ceases to be Euclidean.

(The error of the classical concept of space and time is the substitution of the conditional by the unconditional).

Incomplete induction is popular and scientific. IN popular induction the conclusion is made on the basis of observation and a simple enumeration of facts, without knowing their cause, and in scientific induction the conclusion is made not only on the basis of observation and listing of facts, but also on the basis of knowledge of their cause. Therefore, scientific induction (as opposed to popular) is characterized by much more accurate, almost reliable conclusions.

For example, primitive people see how the sun rises every day in the east, slowly moves during the day across the sky and sets in the west, but they do not know why this is happening, they do not know the reason for this constantly observed phenomenon. It is clear that they can draw a conclusion using only popular induction and reasoning something like this: The day before yesterday the sun rose in the east, yesterday the sun rose in the east, today the sun rose in the east, therefore, the sun always rises in the east.We, like primitive people, observe the daily sunrise in the east, but unlike them we know the reason for this phenomenon: the Earth rotates around its axis in the same direction at a constant speed, due to which the Sun appears every morning in the eastern side of the sky ... Therefore, the inference that we make is scientific induction and looks something like this: The day before yesterday the sun rose in the east, yesterday the sun rose in the east, today the sun rose in the east; moreover, this happens because for several billion years the Earth has been rotating on its axis and will continue to rotate in the same way for many billions of years, being at the same distance from the Sun, which was born before the Earth and will exist longer than it; therefore, for the terrestrial observer, the Sun has always risen and will rise in the east.

The main difference between scientific induction and popular induction is the knowledge of the causes of the events taking place. Therefore, one of the important tasks of not only scientific, but also everyday thinking is the discovery of causal relationships and dependencies in the world around us.

Search for a cause (Methods for establishing causal relationships)

In logic, four methods of establishing causal relationships are considered. They were first put forward by the 17th century English philosopher Francis Bacon, and they were comprehensively developed in the 19th century by the English logician and philosopher John Stuart Mill.

Single Similarity Method is built according to the following scheme:

Under ABC conditions, the phenomenon x appears.

Under ADE conditions, the x phenomenon occurs.

Under AFG conditions, the x phenomenon occurs.

=>

Before us are three situations in which the conditions apply A, B, C, D, E, F, G,and one of them ( A) is repeated in each. This repeating condition is the only thing in which these situations are similar. Next, you need to pay attention to the fact that in all situations the phenomenon occurs x.From this, we can conclude that the condition ANDis the cause of the phenomenon x(one of the conditions repeats all the time, and the phenomenon constantly arises, which gives reason to combine the first and the second with a cause-and-effect relationship). For example, it is required to establish which food product causes an allergy in a person. Let's say an allergic reaction invariably occurs within three days. Moreover, on the first day, the person ate food A, B, C,on the second day - food A, D, E,on the third day - food A, E, G,i.e., for three days, only the product was re-eaten AND,which is most likely the cause of the allergy.

Let's demonstrate the single similarity method with examples.

1. Explaining the structure of a conditional (implicative) judgment, the teacher gave three examples of different content:

If an electric current passes through the conductor, the conductor heats up;

If the word is at the beginning of a sentence, then it must be written with a capital letter;

If the runway is covered with ice, then the planes cannot take off.

2. Analyzing the examples, he drew the students' attention to the same union IF ... THEN, which unites simple judgments into a complex one, and concluded that this circumstance gives the basis for all three complex judgments to be written in the same formula.

3. Once EF Burinsky poured red ink on an old unnecessary letter and photographed it through red glass. While developing the photographic plate, he did not suspect that he was making an amazing discovery. On the negative, the stain disappeared, but the text, filled with ink, appeared. Subsequent experiments with ink of different colors led to the same result - the text came to light. Consequently, the reason for the appearance of the text is its photographing through red glass. Burinsky was the first to apply his method of photography in forensic science.

Single difference method is built like this:

Under conditions A BCD, the x phenomenon occurs.

Under BCD conditions, no x phenomenon occurs.

\u003d\u003e Probably condition A is the cause of the phenomenon x.

As you can see, the two situations differ from each other only in one thing: in the first, the condition ANDis present, and in the second it is absent. Moreover, in the first situation, the phenomenon xarises, but in the second it does not arise. Based on this, we can assume that the condition ANDand there is a reason for the phenomenon x.For example, in air, a metal ball falls to the ground earlier than a feather thrown simultaneously with it from the same height, i.e., the ball moves to the ground with greater acceleration than a feather. However, if this experiment is performed in an airless environment (all conditions are the same, except for the presence of air), then both the ball and the feather will fall to the ground at the same time, that is, with the same acceleration. Seeing that different accelerations of falling bodies take place in an air environment, but not in an airless environment, we can conclude that, in all probability, air resistance is the cause of the fall of different bodies with different accelerations.

Examples of the application of the single difference method are given below.

1. The leaves of the plant growing in the basement are not green. The leaves of the same plant grown under normal conditions are green. There is no light in the basement. Under normal conditions, the plant grows in sunlight. Therefore, it is the cause of the green color of plants.

2. The climate of Japan is subtropical. In Primorye, which lies at almost the same latitudes near Japan, the climate is much more severe. A warm current passes off the coast of Japan. There is no warm current off the coast of Primorye. Consequently, the reason for the difference in the climate of Primorye and Japan is the influence of sea currents.

Method of accompanying changes built like this:

Under conditions A 1 BCD, the x 1 phenomenon occurs.

Under conditions A 2 BCD, the x 2 phenomenon occurs.

Under conditions A 3 BCD, the x 3 phenomenon occurs.

\u003d\u003e Probably condition A is the cause of the phenomenon x.

A change in one of the conditions (provided that other conditions remain unchanged) is accompanied by a change in the occurring phenomenon, by virtue of which it can be argued that this condition and the indicated phenomenon are united by a causal relationship. For example, when the speed of movement is doubled, the distance traveled also doubles; if the speed increases three times, then the distance traveled becomes three times greater. Therefore, an increase in speed is the reason for an increase in the distance traveled (of course, for the same period of time).

Let's demonstrate the method of accompanying changes with examples.

1. Even in ancient times it was noticed that the frequency of sea tides and changes in their height correspond to changes in the position of the moon. The greatest tides occur on the days of new and full moons, the smallest on the so-called days of quadratures (when the directions from the Earth to the Moon and the Sun form a right angle). Based on these observations, it was concluded that sea tides are caused by the action of the moon.

2. Anyone who has held the ball in their hands knows that if the external pressure on it is increased, the ball will decrease. If you stop this pressure, the ball returns to its previous size. The French scientist of the 17th century Blaise Pascal, apparently, was the first to discover this phenomenon, and he did it in a very peculiar and rather convincing way. Going uphill with his assistants, he took with him not only a barometer, but also a bubble, partially inflated with air. Pascal noticed that the volume of the bubble increased as it rose, and began to decrease on the way back. When the explorers reached the foot of the mountain, the bubble returned to its original size. From this it was concluded that the height of the mountain rise is directly proportional to the change in external pressure, that is, it is in a causal relationship with it.

Residual method is constructed as follows:

Under ABC conditions, the xyz phenomenon occurs.

It is known that part y from the phenomenon xyz is caused by condition B.

It is known that the z part of the xyz phenomenon is caused by condition C.

\u003d\u003e Condition A is probably the cause of the X phenomenon.

In this case, the occurring phenomenon is divided into its component parts and the causal relationship of each of them, except for one, with some condition is known. If only one part of the emerging phenomenon remains and only one condition from the set of conditions that give rise to this phenomenon, then it can be argued that the remaining condition is the cause of the remaining part of the phenomenon considered. For example, the author's manuscript was read by editors A, B,C, making notes in it with ballpoint pens. Moreover, it is known that the editor INruled the manuscript in blue ink ( at), and editor C - in red ( z). However, the manuscript contains notes in green ink ( x). We can conclude that, most likely, they were left by the editor. AND.

Examples of the application of the residual method are given below.

1. Observing the motion of the planet Uranus, astronomers of the 19th century noticed that it deviates somewhat from its orbit. It was found that Uranus is deflected by values a, b, c,moreover, these deviations are caused by the influence of neighboring planets A, B, C.However, it was also noticed that Uranus in its motion deviates not only by magnitudes a, b, c,but also by the amount d.From this, a presumptive conclusion was made about the presence of an unknown planet beyond the orbit of Uranus, which causes this deviation. The French scientist Le Verrier calculated the position of this planet, and the German scientist Halle, using a telescope designed by him, found it in the celestial sphere. This is how the planet Neptune was discovered in the 19th century.

2. It is known that dolphins can move at high speed in water. Calculations have shown that their muscular strength, even with a completely streamlined body shape, is not able to provide such a high speed. It was suggested that part of the reason lies in the special structure of the dolphin's skin, tearing off the turbulence of the water. Later, this assumption was confirmed experimentally.

Similarity in one - similarity in another (Analogy as a kind of inference)

In inferences by analogy, based on the similarity of objects in some features, a conclusion is made about their similarity in other features. The structure of the analogy can be represented by the following diagram:

Subject A has signs a, b, c, d.

Item B has signs a, b, c.

\u003d\u003e Probably item B has the attribute d.

In this scheme ANDand IN -these are objects (objects) compared or similar to each other; a, b, c -similar signs; d -it is a portable feature. Consider an example of inference by analogy:

« Think» in series« Philosophical heritage» , provided with an introductory article, comments and a subject index.

« Think» in series« Philosophical heritage»

\u003d\u003e Most likely, the published works of Francis Bacon, as well as the works of Sextus Empiricus, are provided with a subject index.

In this case, two objects are compared (juxtaposed): the previously published works of Sextus Empiricus and the published works of Francis Bacon. The similarities between these two books are that they are published by the same publisher, in the same series, with introductory articles and commentaries. On the basis of this, it can be argued with a high degree of probability that if the works of Sextus Empiricus are provided with a subject index, then the works of Francis Bacon will also be provided with it. Thus, the presence of a domain name index is a portable feature in the considered example.

Inferences by analogy are divided into two types: analogy of properties and analogy of relations.

IN property analogies two objects are compared, and the transferred attribute is some property of these objects. The above example is a property analogy.

Here are some more examples.

1. Gills are to fish what lungs are to mammals.

2. The story of A. Conan Doyle "The Sign of Four" about the adventures of the noble detective Sherlock Holmes, featuring a dynamic plot, I really liked. I have not read the novel by A. Conan Doyle "The Hound of the Baskerville", but I know that it is dedicated to the adventures of the noble detective Sherlock Holmes and has a dynamic plot. Most likely, I will also like this story very much.

3. At the All-Union Congress of Physiologists in Yerevan (1964), Moscow scientists MM Bongard and AL Vyzov demonstrated an installation that simulated human color vision. When the lamps were turned on quickly, she accurately recognized the color and its intensity. Interestingly, this attitude had a number of the same shortcomings as human vision.

For example, orange light after intense red in the first instant she perceived as blue or green.

IN relationship analogies two groups of objects are compared, and a transferable feature is any relation between objects within these groups. An example of a relationship analogy:

In a mathematical fraction, the numerator and denominator are in the opposite relationship: the larger the denominator, the smaller the numerator.

A person can be compared to a mathematical fraction: its numerator is what he really is, and the denominator is what he thinks of himself, how he evaluates himself.

\u003d\u003e It is likely that the higher a person evaluates himself, the worse he actually becomes.

As you can see, two groups of objects are compared. One is the numerator and denominator in a mathematical fraction, and the other is a real person and his self-esteem. Moreover, the relation of inverse dependence between objects is transferred from the first group to the second.

Let's give two more examples.

1. The essence of the planetary model of the atom by E. Rutherford is that negatively charged electrons move around a positively charged nucleus in different orbits; just like in the solar system, the planets move in different orbits around a single center - the sun.

2. Two physical bodies (according to Newton's law of universal gravitation) are attracted to each other with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them; similarly, two point charges that are stationary relative to each other (according to Coulomb's law) interact with an electrostatic force, which is directly proportional to the product of charges and inversely proportional to the square of the distance between them.

By virtue of the probabilistic nature of its conclusions, the analogy is, of course, closer to induction than to deduction. Therefore, it is not surprising that the basic rules of analogy, the observance of which makes it possible to increase the degree of probability of its conclusions, in many ways resemble the rules of incomplete induction already known to us.

Firstly,it is necessary to draw a conclusion on the basis of the greatest possible number of similar features of the objects being compared.

Secondly,these signs should be varied.

Thirdly,similar characteristics must be significant for the compared items.

Fourth,there must be a necessary (natural) connection between similar signs and the transferred sign.

The first three rules of analogy actually repeat the rules of incomplete induction. Perhaps the most important is the fourth rule, about the relationship between similar traits and a portable trait. Let's go back to the analogy example discussed at the beginning of this section. A portable feature - the presence of a subject-nominal index in a book - is closely related to similar features - a publishing house, a series, an introductory article, comments (books of this genre must be supplied with a subject-nominal index). If the transferred feature (for example, the volume of a book) is not naturally associated with similar features, then the conclusion of an inference by analogy may turn out to be false:

Works of the philosopher Sextus Empiricus, published by the publishing house« Think» in series« Philosophical heritage» , are provided with an introductory article, comments and have a volume of 590 pages.

The annotation to the book novelty - the works of the philosopher Francis Bacon - says that they were published by the publishing house« Think» in series« Philosophical heritage» and provided with an introductory article and comments.

\u003d\u003e Most likely, the published works of Francis Bacon, like those of Sextus Empiricus, are 590 pages long.

Despite the probabilistic nature of the conclusions, inferences by analogy have many advantages. Analogy is a good means of illustrating and explaining any complex material, is a way of giving it artistic imagery, and often leads to scientific and technical discoveries. So, on the basis of the analogy of relations, many conclusions are built in bionics - a science that studies objects and processes of living nature for the creation of various technical devices. For example, snowmobile machines have been built, the principle of movement of which is borrowed from the penguins. Using the peculiarity of the jellyfish's perception of infrasound with a frequency of 8-13 vibrations per second (which allows it to recognize in advance the approach of a storm by storm infrasound), scientists have created an electronic apparatus capable of predicting the onset of a storm in 15 hours. Studying the flight of a bat, which emits ultrasonic vibrations and then catches their reflection from objects, thereby accurately navigating in the dark, man has constructed radars that detect various objects and accurately determine their location regardless of weather conditions.

As you can see, inferences by analogy are widely used both in everyday and in scientific thinking.

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 following in this kind of conclusions is a logical law; non-demonstrative - the inference rules provide only the probabilistic following of the conclusion from the premises.
• 2) by the direction of logical following, i.e. by the nature of the connection between knowledge of various degrees of community, expressed in premises and conclusions: deductive - from general knowledge to particular; inductive - from private knowledge to general; inferences by analogy - from private knowledge to private.

Deductive inference is a form of abstract thinking in which thought develops from knowledge of a greater degree of community to knowledge of a lesser degree of community, and the conclusion arising from the premises is, with logical necessity, reliable. The objective basis of DM is the unity of the general and the individual in real processes, objects of the surrounding world.

The deduction procedure takes place when the information of the premises contains the information expressed in the conclusion.

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Inference{!LANG-5e5dcacb54f551ed7bfce9a6694333ea!}

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All pines are trees.

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All flowers are plants.

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It is true that all flowers are plants.

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IN mediated{!LANG-200bcc18d2e2a60d4fcab005202c837d!}

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If the scope of the concept stars{!LANG-ebda131749e089eb15c009fae9022c8e!} {!LANG-746ec04c5e68b64ce0efe53d73a941bf!}, {!LANG-c4003205a390fef655cfaff56864ec04!}{!LANG-408aaeb307b0ee66cc07673c8c379bef!} The sun{!LANG-ebda131749e089eb15c009fae9022c8e!} stars{!LANG-f026a1b8cf390cf642a641d20645b003!} The sun{!LANG-9d47bed209188de29d368a2d8ae8db58!} bodies emitting energy{!LANG-93c20ad3ca6a211bed0815ac57b48f11!}

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The planet Mars is located in the solar system, it has an atmosphere and water.

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