Complex inferences. Types of inferences The correctness of inferences depends primarily on

1. The concept of inference

inference- this is a form of abstract thinking, through which new information is derived from previously available information. In this case, the sense organs are not involved, 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 the outside. Visually, the conclusion is reflected in the form of a column in which there are at least three elements. Two of them are premises, the third is called the conclusion. Parcels and conclusions are usually separated from each other by a horizontal line. The conclusion is always written below, the premises - above. Both the premises and the conclusion are judgments. Moreover, these judgments can be both true and false. For instance:

All mammals are animals.

All cats are mammals.

All cats are animals.

This conclusion is true.

Inference has a number of advantages before the forms of sensory knowledge and experimental research. Since the process of inference takes place only in the realm of thinking, it does not affect real objects. This is a very important property, since often the researcher does not have the opportunity to obtain for observation or experiments the real thing due to its high cost, size or remoteness. Some items at the moment can generally be considered inaccessible for direct research. For example, space objects can be attributed to such a group of objects. As is known, human exploration of even the closest planets to the Earth is problematic.

Another advantage of inferences is that they provide reliable information about the object under study. For example, it was through inference that D. I. Mendeleev created his 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 information already available about the regularities in the position of celestial bodies.

Inference flaw one can say that conclusions are often characterized by abstractness and do not reflect many of the specific properties of the subject. This does not apply, for example, to the above-mentioned periodic table of chemical elements. It is proved that with its help, elements and their properties were discovered, which at that time were not yet known to scientists. However, this is not the case in all cases. For example, when determining the position of a planet by astronomers, its properties are reflected only approximately. Also, it is often impossible to speak about the correctness of the conclusion until it has passed the test in practice.

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

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

Despite the fact that almost every person is trained in the basics of logic at school, learns from their parents, the philistine level of knowledge cannot be considered sufficient. Of course, in most situations this level is enough, but there is a percentage of cases when logical preparation is simply not enough, although it is in such situations that it is most needed. As you know, there is such a type of crime as fraud. Most often, scammers use simple and proven schemes, but a certain percentage of them are engaged in highly skilled deception. Such criminals know logic almost perfectly and, in addition, have abilities in the field of psychology. Therefore, it often costs them nothing to deceive a person who is not prepared. All this speaks of the need to study logic as a science.

Inference is a very common logical operation. By general rule, in order to obtain a true judgment, it is necessary that the premises are also true. However, this rule does not apply to evidence to the contrary. In this case, knowingly false premises are deliberately taken, which are necessary in order to determine the necessary object through their negation. In other words, false premisses are discarded in the process of deriving a consequence.

This text is an introductory piece.

Immediate inferences An inference constructed by transforming a judgment and containing one premise is called immediate. There are four types of transformations of judgments: transformation, inversion, opposition to a predicate, inference

Inductive reasoning Inductive reasoning is called inference, in the form of which empirical generalization proceeds, when, on the basis of the repetition of a feature in phenomena of a certain class, it is concluded that it belongs to all phenomena of this class. For example: in history

3.8. Inferences with the union “or” Both premises and the conclusion of a simple or categorical syllogism are simple judgments (A, I, E, O). If one of the premises of the syllogism or both of its premises are represented by complex judgments (conjunction, non-strict and strict disjunction,

§ 2. DIRECT CONCLUSIONS A proposition containing new knowledge can be obtained by transforming the proposition. Since the original (transformed) judgment is considered as a premise, and the judgment obtained as a result of the transformation is considered as a conclusion,

A. DEDUCTIVE CONCLUSIONS In the process of reasoning, inferences that are not deductive are sometimes mistaken for deductive ones. The latter are called incorrect deductive inferences, and (actually) deductive ones are called correct. Identification of methods of reasoning,

B. INDUCTIVE CONCLUSIONS In contrast to deductive reasoning, in which there is a relation of logical consequence between premises and conclusion, inductive reasoning is such a connection between premises and conclusion according to logical forms, with

§ 4. CONCLUSIONS BY ANALOGY The word “analogy” is of Greek origin. Its meaning can be interpreted as “the similarity of objects in some features.” Inference by analogy is a reasoning in which, from the similarity of two objects in some features

§ 1. The paradox of inference We will gain an even deeper understanding of the nature of formal logic if we consider some of the critical arguments against it. Our discussion of traditional logic, as well as modern logic and mathematics, has been aimed at clarifying

38. Deductive inferences The following types of inferences are deductive: inferences of logical connections and subject-predicate inferences. Also, deductive inferences are direct. They are made from one premise and are called transformation, conversion and

1. The concept of inference Inference is a form of abstract thinking, through which new information is derived from previously available information. In this case, the sense organs are not involved, i.e., the entire process of inference takes place at the level of thinking and is independent of the received

2. Deductive reasoning Like much in classical logic, the theory of deduction owes its appearance to the ancient Greek philosopher Aristotle. He developed most of the issues related to this kind of reasoning. According to the works of Aristotle, deduction is

1. The concept of inference by analogy A significant characteristic of inference as one of the forms of human thinking is the conclusion of new knowledge. At the same time, in the inference, the conclusion (consequence) is obtained in the course of the movement of thought from the known to the unknown. To such a movement

LOGICAL CONCLUSIONS The vast majority of reasoning that claims to be considered logical, in fact, is not. They are pseudo-logical, logical, or at best only partially logical. Reasoning is logical

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

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

All mammals are animals.

All cats are mammals.

All cats are animals.

This conclusion is true.

Inference has a number of advantages before the forms of sensory knowledge and experimental research. Since the process of inference takes place only in the realm of thinking, it does not affect real objects. This is a very important property, since often the researcher does not have the opportunity to get a real object for observation or experiments due to its high cost, size or remoteness. Some items at the moment can generally be considered inaccessible for direct research. For example, space objects can be attributed to such a group of objects. As is known, human exploration of even the closest planets to the Earth is problematic.

Inference is a very common logical operation. As a general rule, in order to obtain a true judgment, the premises must also be true. However, this rule does not apply to evidence to the contrary. In this case, knowingly false premises are deliberately taken, which are necessary in order to determine the necessary object through their negation. In other words, false premisses are discarded in the process of deriving a consequence.

2. Deductive reasoning

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

According to the works of Aristotle deduction is the 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 reasoning, true knowledge must be obtained. This goal can only be achieved if necessary conditions, rules. There are two types of inference rules: direct inference rules and indirect inference rules. Direct inference means obtaining a conclusion from two premises that 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. Subject to these rules, one can speak of the correctness of thinking regarding the subject taken. This means that in order to obtain a true judgment, new knowledge, it is not necessary to have all the information. Part of the information can be recreated in a logical way and fixed. Consolidation is necessary, because without it the process of obtaining new information becomes meaningless. It is not possible to transfer 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 conjunction symbols, disjunctions, implications, literal expressions, brackets, etc.

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

Also deductive reasoning is direct.

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

Let's consider these conclusions. The transformation has a scheme:

S is not non-R.

This diagram shows that there is only one package. 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 a negation in a predicate by a negation in a connective. The first case is shown in the diagram above. In the second, the transformation is reflected in the scheme 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 non-P. No S is P - All S is non-P. Some S are P - Some S are not non-P. Some S's are not P - Some S's are not-P. Appeal- this is a conclusion in which the quality of the premise does not change when the places of the subject and the predicate are changed.

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

Appeal can be with or without limitation.(it is also called simple or pure). This division is based on a quantitative indicator of the judgment (meaning the equality or inequality of the volumes of S and P). This is expressed in whether the quantified word has changed or not and whether the subject and predicate are distributed. If such a change occurs, then the constraint has been handled. Otherwise, we can speak of pure conversion. Recall that a quantified word is a word - an indicator of quantity. Thus, the words "all", "some", "none" and others are quantified words.

Contrasting with a predicate characterized by the fact that the link in the consequence is reversed, the subject contradicts the predicate of the premise, and the predicate is equivalent to the subject of the premise.

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

Let's give opposition schemes depending on the types of judgments.

Some S are not P - Some non-P are S. No S is P - Some non-P are S. All S are P - No P is S.

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

3. Conditional and disjunctive inferences

Speaking of deductive reasoning, one cannot but pay attention to conditional and disjunctive reasoning.

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

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

Above is a diagram of inferences, which are a kind of conditional. It is characteristic of such inferences that all of 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 propositions, one of them is a simple categorical proposition.

It is also necessary to mention modes - varieties of inferences. There are: affirming mode, denying mode and two probabilistic modes (first and second).

Approving mode has the widest distribution in thinking. This is due to the fact that it gives a reliable conclusion. Therefore, the rules of various academic disciplines are built mainly on the basis of the affirmative mode. You can display the affirmative mode as a diagram.

If a, then b.

Let us give an example of an assertive mode.

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

The ax fell into the water.

He will drown.

The two true propositions that are the premises of this proposition are transformed in the process of inference into a true proposition. Negative mode expressed in the following way. If a, then b. Non-b. Nope.

This judgment is based on the negation of the consequence and the negation of the foundation.

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

In this connection it is necessary to speak about probabilistic modes.

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

If a, then b.

Probably a.

As the name implies, the consequence deduced from the premises with the help of this mode is probable.

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

The yacht rolls to one side.

Probably a strong wind is blowing.

As we see from the assertion of the consequence to the assertion of the foundation, it is impossible to draw a true conclusion.

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

If a, then b. Nope.

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

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

This man does not lie under the sun.

It won't burn.

As can be seen from the above example, making a conclusion from the negation of the basis to the negation of the consequence, we will get not a true, but a probabilistic consequence.

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

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

In this way, inference is considered to be dividing, all or part of whose premises are disjunctive judgments. The structure of a simple disjunctive 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 can be with one transfer or with several transfers.

The path can be straight or with one transfer, or with several transfers.

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

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

Here it is necessary to mention conditional-separative inferences. They differ from the above inferences in their premises. One of them is a disjunctive proposition, which is not special, but the second premise of such propositions consists of two or more conditional propositions.

A conditional-separative judgment can be either a dilemma or a trilemma. in a dilemma the conditional premise consists of two members. In this case, the separation implies the presence of a choice. In other words, a dilemma is a choice between two options.

The dilemma can be simple constructive and complex constructive, as well as simple and complex destructive. The first 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 statement of the first premise (the conditional proposition).

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

You can press the pencil or bend the pencil.

The pencil will break.

A complex design dilemma involves a harder choice between alternatives.

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

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

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

The athlete will win.

There are cases when a conclusion or one of the premises is omitted in conditional, disjunctive or conditionally distributive inferences. Such conclusions are called abbreviated.

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

All living organisms feed on moisture.

All plants are living organisms.

=> All plants feed on moisture.

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

All birds are mammals.

All sparrows are birds.

=> All sparrows are mammals.

As you can see, in the above 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 us pay attention to the fact that inferences consist of judgments, and judgments consist of concepts, that is, one form of thinking enters into another as an integral part.

All inferences are divided into direct and indirect.

V immediate inferences, the conclusion is drawn from one premise. For instance:

All flowers are plants.

=> Some plants are flowers.

It is true that all flowers are plants.

=> It is not true that some flowers are not plants.

It is easy to guess that direct inferences are already known to us operations of transformation of simple judgments and conclusions about the truth of simple judgments in a logical square. The first example of a direct inference is a transformation of a simple judgment by inversion, and in the second example, by a logical square from the truth of a judgment of the form A a conclusion is made about the falsity of a judgment of the form O.

V mediated inferences, the conclusion is drawn from several premises. For instance:

All fish are living beings.

All carp are fish.

=> All carp are living beings.

Indirect inferences are divided into three types: deductive, inductive and inference by analogy.

Deductive inferences (deduction) (from lat. deductio-“inference”) are inferences in which a conclusion is drawn from a general rule for a particular case (a special case is derived from a general rule). For instance:

All stars radiate energy.

The sun is a star.

=> The sun radiates energy.

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

In deduction, reasoning proceeds from the general to the particular, from the greater to the lesser, knowledge narrows, due to which deductive conclusions are reliable, that is, accurate, obligatory, necessary. Let's look at the above example again. Could any other conclusion follow from these two premises than the one that follows from them? Could not. The following conclusion is the only possible one in this case. Let us depict the relationship between the concepts of which our conclusion consisted of Euler circles. The scope of the three concepts: stars(3); bodies that radiate energy(T) and The sun(C) schematically arranged as follows (Fig. 33).

If the scope of the concept stars included in the concept bodies that radiate energy and the scope of the concept The sun included in the concept stars, then the scope of the concept The sun automatically included in the scope of the concept bodies that radiate energy whereby the deductive conclusion is reliable.

The undoubted advantage of deduction lies in the reliability of its conclusions. 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 deduce the particular from the general. In one work, explaining to Dr. Watson the essence of his deductive method, he gives the following example. Near the murdered Colonel Ashby, Scotland Yard detectives found a smoked cigar and decided that the colonel had smoked it before his death. 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 set his mustache on fire. Therefore, the cigar was smoked by another person.

In this reasoning, the conclusion looks convincing precisely because it is deductive - from the general rule: Anyone with a big, bushy mustache can't finish a cigar, a special case appears: Colonel Ashby couldn't finish his cigar because he wore such a mustache. Let us bring the considered reasoning to the one adopted in logic standard form records of inferences in the form of premises and output:

Anyone with a big, bushy mustache can't finish a cigar.

Colonel Ashby wore a large, bushy mustache.

=> Colonel Ashby couldn't finish his cigar.

Inductive inference (induction) (from lat. inductio-“guidance”) are inferences in which a general rule is deduced from several special cases. For instance:

Jupiter is moving.

Mars is moving.

Venus is moving.

Jupiter, Mars, Venus are planets.

=> All planets move.

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, i.e., a certain general rule is formulated (following from three special cases).

It is easy to see that inductive reasoning is built on a principle opposite to that of deductive reasoning. In induction, reasoning goes from the particular to the general, from less to more, knowledge expands, due to which inductive conclusions (unlike 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 an error: it is quite possible that there are some exceptions in the group, and even if the set of objects from a certain group is characterized by some attribute, this does not mean that all objects of this group are characterized by this attribute. The probabilistic nature of the conclusions is, of course, a disadvantage of induction. However, its undoubted advantage and advantageous difference from deduction, which is a narrowing knowledge, is that induction is an expanding knowledge that can lead to a new one, 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 instance:

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

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

=> Probably, there is life on Mars.

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

When all judgments are simple (categorical syllogism)

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

Consider an example of a simple syllogism:

All flowers(M)are plants(R).

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

=> All roses(S)are plants(R).

Both premises and conclusion are simple judgments in this syllogism, and both premises and conclusion are judgments of the form A(general affirmative). Let us 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 the inference is present in the second premise of the syllogism, and the predicate of the inference is in the first. Also in both premises the term is repeated flowers, which, as it is easy to see, is a link: it is thanks to him that the unrelated, disjointed terms in the premises plants and roses can 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 the conclusion is located in the second premise of the syllogism and is called lesser syllogism term(the second premise is also called lesser).

The inference predicate is located in the first premise of the syllogism and is called the larger term of the syllogism(the first premise is also called greater). The inference predicate, as a rule, is a larger concept than the inference subject (in the given example of the concept roses and plants are in relation to generic subordination), which is why the inference predicate is called big term, and the subject of the output is smaller.

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

The three terms of the syllogism can be arranged in different ways in it. The mutual arrangement of terms relative to each other is called figure of a 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 is the arrangement of its terms such that the first premise begins with the middle term and the second ends with the middle term. For instance:

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

Helium(S)is a gas(M).

=> Helium(S)is a chemical element(R).

Taking into account 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 conclusion the subject is associated with the predicate, let's draw up a diagram of the location and relationship of terms in the above example (Fig. 34).

The straight lines in the diagram (with the exception of the one that separates the premises from the conclusion) show the relationship of terms in the premises and in the conclusion. Since the role of the middle term is to link the major and minor terms of the syllogism, the diagram connects the middle term in the first premise with a line to 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 be obtained (Fig. 35).

The second figure of the syllogism is the arrangement of its terms such that both the first and second premises end in the middle term. For instance:

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

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

=> All whales(S)not fish(R).

The diagrams 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 is such an arrangement of its terms in which both the first and second premises begin with the middle term. For instance:

All tigers(M)are mammals(R).

All tigers(M)- they are predators(S).

=> Some Predators(S)are mammals(R).

The diagrams 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 is the arrangement of its terms such that the first premise ends with the middle term and the second begins with it. For instance:

All squares(R)are rectangles(M).

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

=> All Triangles(S)are not squares(R).

The diagrams of the mutual arrangement of terms and 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 can 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 species ( A, I, E, O). A set of simple propositions included in a syllogism is called mode of simple syllogism. For instance:

All celestial bodies move.

All planets are celestial bodies.

=> All planets move.

In this syllogism, the first premise is a simple proposition of the form A(generally affirmative), the second premise is also a simple proposition of the form A, and the conclusion in this case is a simple proposition of the form A. Therefore, the considered syllogism has the mode 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 three letters are present in it a, symbolizing the mode of the syllogism AAA. Latin "words" for the modes of simple syllogism were invented in the Middle Ages.

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

All magazines are periodicals.

All books are not periodicals.

=> All books are not magazines.

And one more example. This syllogism has a mode aai, or darapti.

All carbons are simple bodies.

All carbons are electrically conductive.

=> Some electrical conductors are simple bodies.

In total, there are 256 modes in all four figures (that is, possible combinations of simple judgments in a syllogism). There are 64 modes in each figure. 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 signs of deduction (and, therefore, of a syllogism) is the reliability of its conclusions, it becomes clear why these 19 modes are called correct, and the rest are incorrect.

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

All substances are made up of atoms.

All liquids are substances.

=> All liquids are made up of atoms.

First of all, it is necessary to find the subject and the predicate of the conclusion, i.e., the minor and major terms of the syllogism. Next, the location of the smaller term in the second premise and the larger one in the first should be established. After that, you can determine the middle term and schematically depict the location of all terms in the syllogism (Fig. 39).

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

All liquids(S)are substances(M).

=> All fluids(S)made up of atoms(R).

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

Going to school forever (General rules of the syllogism)

Syllogism rules are divided into general and particular.

The general rules apply to all simple syllogisms, no matter what figure 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 the syllogism.

A syllogism should have only three terms. Let us turn to the already mentioned syllogism in which this rule is violated.

Movement is eternal.

Going to school is movement.

=> Going to school forever.

Both premises of this syllogism are true judgments, but a false conclusion follows from them, because the rule in question is violated. Word motion used in two parcels in two different meanings: movement as a universal world change and movement 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 senses (since one of the terms is used in two different senses), that is, an extra sense, as it were, implies an extra term. In other words, in the given example of a syllogism there were not three, but four (by meaning) terms. The error that occurs when the above rule is violated is called quadrupling 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 it is easiest to establish the distribution of terms in simple judgments using circular diagrams: it is necessary to depict the relationship between the terms of the judgment with Euler circles, while the full circle in the diagram will denote a distributed term (+), and an incomplete one - undistributed (-). Consider an example of a syllogism.

All cats(TO)are living beings(J. s).

Socrates(WITH)is also a living being.

=> Socrates is a cat.

Two true premises lead to a false conclusion. Let us depict the relations between the terms in the premises of the syllogism with Euler circles 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 - undistributed middle term in each premise.

A term that was unallocated in the premise cannot be distributed in the output. Let's look at the following example:

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

All pears(G)- These are not apples.

=> All pears are inedible items.

The premises of a syllogism are true propositions, but the conclusion is false. As in the previous case, we depict the relationship between the terms in the premises and in the derivation of the syllogism with Euler circles 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 it is undistributed (-), and in the output it is distributed (+), which is prohibited by the considered rule. The error that occurs when it is violated is called expansion of a larger term. Recall that a term is distributed when it refers to all the objects included in it, and undistributed when it comes to a part of the objects included in it, which is why the error is called the expansion of the term.

A syllogism should not have two negative premises. 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 the conclusion from them either cannot be drawn at all, or, if it is possible to do so, it will be false or, at least, unreliable, probabilistic. For instance:

Snipers cannot have bad eyesight.

All my friends are not snipers.

=> All my friends have poor eyesight.

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

A syllogism should not have two partial premises.

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

Some students are first graders.

Some students are tenth graders.

No conclusion follows from these premises, because both of them are particular. 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 instance:

No metal is an insulator.

Copper is a metal.

=> Copper is not an insulator.

As we 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 be private. For instance:

All hydrocarbons are organic compounds.

Some substances are hydrocarbons.

=> Some substances are organic compounds.

In this syllogism, no general conclusion can follow from two premises. It can only be private, since the second premise is private.

Here are a few more examples of a simple syllogism - both correct and with violations of some general rules.
All herbivores eat plant foods.
All tigers do not eat plant foods.
=> All tigers are not herbivores.
(correct syllogism)

All excellent students do not receive deuces.
My friend is not an excellent student.
=> My friend gets twos.

All fish swim.
All whales swim too.
=> All whales are fish.
(Error - the middle term is not distributed in any of the premises)

The bow is an ancient shooting weapon.
One of the vegetable crops is onions.
=> One of the vegetable crops is an ancient shooting weapon.

Any metal is not an insulator.
Water is not a metal.
=> Water is an insulator.
(Error - two negative premises in the syllogism)

No insect is a bird.
All bees are insects.
=> No bee is a bird.
(correct syllogism)

All chairs are pieces of furniture.
All cabinets are not chairs.
=> All cabinets are not pieces of furniture.

Laws are made by people.
Universal gravitation is a law.
=> Universal gravitation was invented by people.
(Error - quadrupling terms in a simple syllogism)

All people are mortal.
All animals are not people.
=> Animals are immortal.
(Error - expansion of a larger term in a syllogism)

All Olympic champions are athletes.
Some Russians are Olympic champions.
=> Some Russians are athletes.
(correct syllogism)

Matter is uncreated and indestructible.
Silk is matter.
=> Silk is uncreatable and indestructible.
(Error - quadrupling terms in a simple syllogism)

All graduates of the school take exams.
All fifth-year students are not graduates of the school.
=> All fifth-year students do not take exams.
(Error - expansion of a larger term in a syllogism)

All stars are not planets.
All asteroids are minor planets.
=> All asteroids are not stars.
(correct syllogism)

All grandfathers are fathers.
All fathers are men.
=> Some men are grandfathers.
(correct syllogism)

No first grader is of legal age.
All adults are not first graders.
=> 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 follow its clear logical structure. For instance:

All fish are non-mammals.

All whales are mammals.

=> Therefore, all whales are not fish.

Instead, we are more likely to say: All whales are not fish because they are mammals. or: All whales are not fish because fish are not mammals. It is easy to see that these two conclusions 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 omitted. It is clear that three enthymemes can be deduced from any syllogism. For example, take the following syllogism:

All metals are electrically conductive.

Iron is a metal.

=> Iron is electrically conductive.

Three enthymemes follow from this syllogism: Iron is electrically conductive because it is a metal(missing a large package); Iron is electrically conductive because all metals are electrically conductive(minor premise omitted); All metals conduct electricity, and iron is a metal(output omitted).

Epicheirema is a simple syllogism in which both premises are enthymemes. Let us take two syllogisms and derive enthymemes from them.

Syllogism 1

Everything that brings society to disaster is evil.

Social injustice leads society to disasters.

=> Social injustice is evil.

Omitting the major premise in this syllogism, we get the following enthymeme: Social injustice is evil because it brings society to disaster.

Syllogism 2

Anything that makes some people rich at the expense of others is social injustice.

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

=> Private property is a social injustice.

Omitting a large premise in this syllogism, we get the following enthymeme: If you place these two enthymemes one after the other, then they will become the premises of a new, third syllogism, which will be the epicheireme:

Social injustice is evil because it brings 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.

=> Private property is evil.

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

polysyllogism (complex syllogism) are two or more simple syllogisms linked together in such a way that the conclusion of one of them is the premise of the next. For instance:

Let us pay attention to the fact that the conclusion of the previous syllogism has become a larger premise of the next one. In this case, the resulting polysyllogism is called progressive. If the conclusion of the previous syllogism becomes the minor premise of the next one, then the polysyllogism is called regressive. For instance:

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

It was said above that a polysyllogism can consist not only of two, but also of a larger number of simple syllogisms. Here is an example of a polysyllogism (progressive), which consists of three simple syllogisms:

sorite(complex abbreviated syllogism) is a polysyllogism in which the premise of the subsequent syllogism, which is the conclusion of the previous one, is omitted. Let us return to the example of a progressive polysyllogism discussed above and skip the big premise of the second syllogism, which is the conclusion of the first syllogism. You get a progressive sorite:

Everything that develops thinking is useful.

All intellectual games develop thinking.

Chess is an intellectual game.

=> Chess is useful.

Now let's turn to the example of a regressive polysyllogism discussed above and skip in it the minor premise of the second syllogism, which is the conclusion of the first syllogism. You get a regressive sorite:

All stars are celestial bodies.

The sun is a star.

All celestial bodies participate in gravitational interactions.

=> The sun participates in gravitational interactions.

Either rain or snow (Conclusions with the union OR)

Inferences that contain disjunctive (disjunctive) judgments are called separating divisive-categorical syllogism, in which, as the name implies, the first premise is a disjunctive (disjunctive) proposition, and the second premise is a simple (categorical) proposition. For instance:

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

Moscow State University is a higher educational institution.

=> Moscow State University is not a primary or secondary educational institution.

V affirmative-denying mode the first premise is a strict disjunction of several variants of something, the second affirms one of them, and the conclusion denies all the others (thus, the reasoning moves from affirmation to negation). For instance:

Forests are coniferous, or deciduous, or mixed.

This forest is coniferous.

=> This forest is neither deciduous nor mixed.

V denying-affirming mode, the first premise is a strict disjunction of several variants of something, the second denies all these variants except one, and the conclusion affirms one remaining variant (thus, the argument moves from denial to affirmation). For instance:

People are Caucasians, or Mongoloids, or Negroids.

This person is not a Mongoloid or a Negroid.

=> This person is Caucasian.

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

The division in the first premise must be carried out according to one base. For instance:

Transport can be ground, or underground, or water, or air, or public.

Suburban electric trains are public transport.

=> Suburban electric trains are not ground, underground, water or air transport.

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

Why is it so? 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 change in the first premise of the divisive-categorical syllogism leads to a false conclusion.

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

Mathematical operations are addition, or subtraction, or multiplication, or division.

Logarithm is not addition, subtraction, multiplication, and division.

=> Logarithm is not a mathematical operation.

known to us partial division error in the first premise of the syllogism, it causes a false conclusion that follows from the true premises.

The results of the division in the first premise must not intersect, or the disjunction must be strict. For instance:

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

Canada is a northern country.

=> Canada is not a southern, western or eastern country.

In the syllogism, the conclusion is false, since Canada is as much a northern country as it is a western one. False conclusion with true premises is explained in this case intersection of division results in the first premise, or, what is the same thing, - non-strict disjunction. It should be noted that a non-strict disjunction in a dividing-categorical syllogism is admissible in the case when it is built according to the negating-affirming mode. For instance:

He is strong by nature or constantly plays sports.

He is not strong by nature.

=> 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 under consideration is unconditionally valid only for the affirmative-negating mode of the separative-categorical syllogism.

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

Sentences are simple, or complex, or compound.

This sentence is complex.

=> This sentence is neither simple nor complex.

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

Let us give a few more examples of a divisive-categorical syllogism - both correct and with violations of the considered rules.
Quadrilaterals are squares, or rhombuses, or trapezoids.
This figure is not a rhombus or a trapezoid.
=> This figure is a square.
(Error - incomplete division)

Selection in living nature is either artificial or natural.
This selection is not artificial.
=> This selection is natural.
(correct inference)

People are talented, or untalented, or stubborn.
He is a stubborn person.
=> He is neither talented nor untalented.
(Error - substitution of the base in division)

Educational institutions are primary, or secondary, or higher, or universities.
MSU is a university.
=> Moscow State University is not an elementary, secondary or higher educational institution.
(Error - jump in division)

You can study the natural sciences or the humanities.
I study natural sciences.
=> I don't study the humanities.
(Error - intersection of division results, or non-strict disjunction)

Elementary particles have a negative electrical charge, or positive, or neutral.
Electrons have a negative electrical charge.
=> Electrons have neither positive nor neutral electric charge.
(correct inference)

Publications are periodic, or non-periodical, or foreign.
This edition is foreign.
=> This publication is not a periodical and is not a non-periodical.
(Mistake - base substitution)

A divisive-categorical syllogism in logic is often called simply a divisive-categorical inference. In addition to it, there is also purely divisive syllogism(purely disjunctive reasoning), both premises and the conclusion of which are disjunctive (disjunctive) judgments. For instance:

Mirrors are either flat or spherical.

Spherical mirrors are either concave or convex.

=> Mirrors are flat, or concave, or convex.

If a person flatters, then he lies (Conclusions with the union IF ... THEN)

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

Today the runway is covered with ice.

=> Planes cannot take off today.

Approving mode- in which the first premise is an implication (consisting, as we already know, of two parts - the foundation and the consequence), the second premise is the statement of the foundation, and the conclusion asserts the consequence. For instance:

This substance is a metal.

=> This substance is electrically conductive.

Negative mode- in which the first premise is an implication of the reason and the consequence, the second premise is the negation of the consequence, and the reason is denied in the conclusion. For instance:

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

This material is non-conductive.

=> This substance is not a metal.

It is necessary to pay attention to the peculiarity of the implicative judgment already known to us, which is that cause and effect cannot be interchanged. For example, the statement If the substance is a metal, then it is electrically conductive. is true, since all metals are electrical conductors (from the fact that a substance is a metal, its electrical conductivity necessarily follows). However, the statement If a substance is electrically conductive, then it is a metal. 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 the conditionally categorical syllogism:

1. It is possible to assert only from the basis to the consequence, i.e., in the second premise of the affirmative mode, the basis of the implication (the first premise) must be affirmed, and in the conclusion, its consequence. Otherwise, a false conclusion may follow from two true premises. For instance:

If a word is at the beginning of a sentence, it is always capitalized.

Word« Moscow» always capitalized.

=> Word« Moscow» always at the beginning of a sentence.

The second premise affirmed the consequence, and the conclusion, the foundation. This statement from the investigation to the basis is the cause of a false conclusion with true premises.

2. It is possible to deny only from the consequence to the basis, i.e., in the second premise of the negating mode, the consequence of the implication (the first premise) must be denied, and in the conclusion, its foundation. Otherwise, a false conclusion may follow from two true premises. For instance:

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.

=> In this sentence, the word« Moscow» no need to capitalize.

The second premise denies the ground, while the conclusion denies the consequence. This negation from reason to effect is the cause of a false conclusion with true premises.

Let us give a few more examples of a conditionally categorical syllogism - both correct and with violations of the considered rules.
If an animal is a mammal, then it is a vertebrate.
Reptiles are not mammals.
=> Reptiles are not vertebrates.

If a person flatters, then he lies.
This person is flattering.
=> This person is lying.
(Correct conclusion).

If a geometric figure is a square, then all sides are equal.
An equilateral triangle is not a square.
=> The sides of an equilateral triangle are not equal.
(Mistake - negation from the basis to the consequence).

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

If the celestial body is a planet in the solar system, then it moves around the sun.
Halley's comet moves around the sun.
=> Halley's Comet is a planet in the solar system.
(Mistake - statement from the investigation to the base).

If water turns into ice, it expands in volume.
The water in this vessel turned into ice.
=> 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 Faculty of Law of Moscow State University is a judge.
=> Not every graduate of the Faculty of Law of Moscow State University has a higher legal education.
(Mistake - negation from the basis to the consequence).

If the lines are parallel, then they have no common points.
Intersecting lines do not have common points.
=> Crossing lines are parallel.
(Mistake - statement from the investigation to the base).

If a technical product is equipped with an electric motor, then it consumes electricity.
All electronic products consume electricity.
=> All electronic products are equipped with electric motors.
(Mistake - statement from the investigation to the base).

Recall that among complex propositions, in addition to the implication ( a => b) there is also an equivalent ( a<=>b). If a reason and a consequence are always distinguished in the implication, 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-categorical, if the first premise of the syllogism is not an implication, but an equivalence. For instance:

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

The number 16 is even.

=> The number 16 is divisible by 2 without a remainder.

Since in the first premise of an equivalent-categorical syllogism neither grounds nor consequences can be singled out, the rules of conditionally-categorical syllogism considered above are not applicable to it (in an equivalent-categorical syllogism, one can both assert and deny as one likes).

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 conditionally categorical syllogism(also often called conditional-categorical inference). If both premises are conditional propositions, then this is a purely conditional syllogism, or a purely conditional inference. For instance:

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

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

=> 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 a triangle is a right triangle, then its area is half the product of its base times its height.

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

=> The area of a triangle is half the product of its base times its height.

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

We are faced with a choice (Conditional-separative inferences)

In addition to dividing-categorical and conditionally-categorical inferences, or syllogisms, there are also conditionally dividing inferences. V conditional divisional inference(syllogism) the first premise is a conditional or implicative proposition, and the second premise is a disjunctive or disjunctive proposition. It is important to note that in a conditional (implicative) judgment there may be not 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 enter Moscow State University, then you need to study a lot or you need to have a lot of money From one reason two consequences follow. In judgment If you enter Moscow State University, then you need to study a lot, and if you enter MGIMO, then you also need to study a lot From two bases one consequence follows. In judgment If the country is ruled a wise man, then it prospers, and if it is controlled by a rogue, then it is poor From two bases two consequences follow. In judgment If I speak out against the injustice surrounding me, then I will remain a man, although I will suffer severely; if I indifferently pass by her, then I will cease to respect myself, although I will be safe and sound; and if I help her in every possible way, I will turn into an animal, although I will achieve material and career well-being From three bases three consequences follow.

If the first premise of a conditionally dividing syllogism contains two bases 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 bases or consequences, then the syllogism is polylemma. Most often in thinking and speech there is a dilemma, on the example of which we will consider a conditionally divisive syllogism (also often called a conditional divisive inference).

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

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

If you enter Moscow State University, then you need to study a lot, and if you enter MGIMO, then you also need to study a lot.

You can enter Moscow State University or MGIMO.

=> You have to do a lot.

In the first post difficult design dilemma from two bases two consequences follow, the second premise is a disjunction of the bases, and the conclusion is a complex judgment in the form of a disjunction of the consequences. For instance:

If a country is ruled by a wise man, then it prospers, and if it is ruled by a rogue, then it is poor.

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

=> A country can prosper or be poor.

In the first post simple destructive dilemma two consequences follow from one foundation, the second premise is a disjunction of the negations of the consequences, and the foundation is denied in the conclusion (there is a denial of a simple judgment). For instance:

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

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

=> I will not enter Moscow State University.

In the first post complex destructive dilemma two consequences follow from two bases, the second premise is a disjunction of the negations of the consequences, and the conclusion is a complex judgment in the form of a disjunction of the negations of the bases. For instance:

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 neither a materialist nor an idealist.

=> 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 a conditionally disjunctive syllogism is an implication, and the second is a disjunction, its rules are the same as the rules of conditionally categorical and disjunctive categorical syllogisms considered above.

Here are some more examples of the dilemma.
If learning English, then everyday conversational practice is necessary, and if learning German, then everyday conversational practice is also necessary.
You can study English or German.
=> Everyday speaking practice is essential.
(A simple design dilemma).

If I confess to my wrongdoing, then I will suffer the deserved punishment, and if I try to hide it, I will feel remorse.
I will either confess to my wrongdoing, or I will try to hide it.
=> I will suffer a well-deserved punishment or I will feel remorse.
(A difficult design dilemma).

If he marries her, he will suffer a complete collapse or will drag out a miserable existence.
He does not want to suffer a complete collapse or drag out a miserable existence.
=> He will not marry her.
(A simple destructive dilemma).

If the speed of the Earth during its orbital motion would be 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» to the sun.
Earth does not leave the solar system and does not« falls» in the sun.
=> The speed of the Earth during its orbital movement is not more than 42 km / s and not less than 3 km / s.
(A complex destructive dilemma).

All 10B students are Losers (Inductive reasoning)

In induction, a general rule is deduced from several particular cases, reasoning proceeds from the particular to the general, from the smallest to the largest, knowledge expands, which is why inductive conclusions are, as a rule, probabilistic. Induction is either complete or incomplete. V full induction all objects from any group are listed and a conclusion is made about this entire group. For example, if all nine major planets of the solar system are listed in the premises of inductive reasoning, then such an 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 major planets solar system.

V incomplete induction some objects from any group are listed and a conclusion is made about this entire group. For example, if the premises of the inductive reasoning do not list all nine major planets of the solar system, but only three of them, then such an induction is incomplete:

Mercury is moving.

Venus is moving.

The earth is moving.

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

=> All major planets of the solar system are moving.

It is clear that the conclusions of complete induction are reliable, and that of incomplete induction are probabilistic, but complete induction is rare, and therefore, by inductive reasoning, incomplete induction is usually meant.

To increase the degree of probability of conclusions of incomplete induction, 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. It is required to check the level of progress of students in a certain school. Let's say there are 1000 students. According to the method of complete induction, it is necessary to test for the progress of each student out of this thousand. Since this is quite difficult to do, you can use the method of incomplete induction: test some part of the students and draw a general conclusion about the level of performance in a given school. Various sociological surveys are also based on the use of incomplete induction. It is obvious that the more students are tested, the more reliable will be the basis for inductive generalization and the more accurate the conclusion will be. However, simply a larger number of initial premises, as required by the rule under consideration, is not enough to increase the degree of probability of inductive generalization. Suppose a considerable number of students pass the test, but, by chance, among them there will be only unsuccessful ones. In this situation, we will come to the 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 pick up a variety of parcels.

Returning to our example, we note that the set of testees should not only be as large as possible, but also specially (according to some system) formed, and not randomly selected, i.e., care must be taken to include students ( approximately in 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 (feature) is insignificant for the conclusion about his progress. However, if testing shows that a 10th grade student has a particle NOT writes together with the verb, then this fact (feature) should be recognized as significant (important) for the conclusion about the level of his education and academic performance.

These are the basic rules of incomplete induction. Now let's turn to her most common mistakes. Speaking of 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 seen as the result of the simultaneous violation of all the rules, and at the same time, the violation of each rule can be considered as the cause leading to any of the errors.

The first error often encountered in incomplete induction is called hasty generalization. Most likely, each of us is well acquainted with it. Everyone has heard statements like All men are callous, all women are frivolous, etc. 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 at all that the whole group without exception is characterized by this feature. From the true premises of inductive reasoning, a false conclusion can follow if a hasty generalization is allowed. For instance:

K. studies poorly.

N. studies poorly.

S. studies poorly.

K., N., S. are students 10« A».

=> All students 10« A» study poorly.

It is not surprising that hasty generalization underlies many allegations, rumors and gossip.

The second mistake is long and at first glance strange name:after that, therefore, 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 by just a temporal 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, we make a logical error. For example, in the following inductive reasoning, the generalizing conclusion is false, despite the truth of the premises:

The day before yesterday, a black cat ran across the road to a bad student N., and he received a deuce.

Yesterday a black cat ran across the road to N.'s loser, and his parents were called to school.

Today, a black cat ran across the road to a loser N., and he was expelled from school.

=> A black cat is to blame for all the misfortunes of the loser N..

It is not surprising that this common mistake has given rise to many tall tales, superstitions and hoaxes.

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

At home, water boils at 100°C.

Outdoor water boils at 100°C.

In the laboratory, water boils at 100°C.

=> Water everywhere boils at 100 °C.

We know that high in the mountains water boils at a lower temperature. On Mars, the temperature of boiling water would be about 45°C. So the question Is boiling water always and everywhere hot? is not ridiculous, as it may seem at first glance. And the answer to this question will be: Not always and not everywhere. What appears in one setting may not appear in another. In the premises of the considered example, there is a conditional (occurring in certain conditions), which is replaced by an unconditional (occurring in all conditions in the same way, independent of them) in the conclusion.

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

Here are a few 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, and the grandmother did not pull it out either. The granddaughter, the bug and the cat also did not pull out the turnip. She managed to pull 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 = a. However, later it turned out that equations of the fifth degree cannot be solved in radicals.
(Mistake is a hasty generalization).

3. In classical, or Newtonian, natural science, it was believed that space and time are unchanging. This belief was based on the fact that, wherever different material objects are located and whatever happens to them, time flows the same for each of them 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 by no means immutable. So, for example, when material objects move at speeds close to the speed of light (300,000 km/s), time slows down significantly for them, and space becomes 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. V 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 enumeration of facts, but also on the basis of knowledge of their cause. Therefore, scientific induction (unlike the popular one) is characterized by much more accurate, almost reliable conclusions.

For example, primitive people see how the sun rises in the east every day, moves slowly through the sky during the day and sets in the west, but they do not know why this happens, 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 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 conclusion that we make is a 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 the Earth has been rotating around its axis for several billion years 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 an earthly observer, the Sun has always risen and will rise in the east.

The main difference between scientific induction and popular induction lies in the knowledge of the causes of events. 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 the cause (Methods for establishing causal relationships)

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

Single Similarity Method built according to the following scheme:

Under conditions ABC, the phenomenon x occurs.

Under the conditions of ADE, the phenomenon x occurs.

Under conditions AFG, the phenomenon x occurs.

We have three situations in which the conditions apply A, B, C, D, E, F, G, and one of them ( A) is repeated in each. This recurring condition is the only thing in which these situations are similar. Further, it is necessary to pay attention to the fact that in all situations there is a phenomenon X. From this, one can plausibly conclude that the condition A is the cause of the phenomenon. X(one of the conditions is repeated all the time, and the phenomenon constantly arises, which gives reason to combine the first and second by a causal relationship). For example, it is required to establish which food causes an allergy in a person. Let's say that within three days an allergic reaction invariably occurred. At the same time, on the first day, a person ate food A, B, C, on the second day - products A, D, E, on the third day - products A, E, G i.e. for three days only the product was re-ingested A, which is most likely the cause of the allergy.

We will demonstrate the method of unique similarity 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, then the conductor heats up;

If the word is at the beginning of a sentence, then it must be capitalized;

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

2. Analyzing the examples, he drew the attention of students to the same union IF ... THEN, connecting simple judgments into a complex one, and concluded that this circumstance gives grounds to write down all three complex judgments with the same formula.

3. Once E. F. Burinsky poured red ink on an old unwanted 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 ink-filled text showed through. Subsequent experiments with inks of different colors led to the same result - the text was revealed. Therefore, the reason for the manifestation of the text is its photographing through the red glass. Burinsky was the first to apply his method of photographing in forensics.

Single difference method is built like this:

Under conditions A BCD, the x phenomenon occurs.

Under BCD conditions, the x phenomenon does not occur.

=> Probably condition A is the cause of phenomenon x.

As we can see, the two situations differ only in one respect: in the first condition A is present and the other is absent. Moreover, in the first situation, the phenomenon X occurs, and in the second - does not occur. Based on this, it can be assumed that the condition A and there is a cause 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 the feather. However, if this experiment is carried out 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, i.e. with the same acceleration. Seeing that different accelerations of falling bodies take place in an air medium, but not in an airless medium, we can conclude that, in all likelihood, air resistance is the cause of the fall of different bodies with different accelerations.

Examples of applying the single difference method are given below.
1. The leaves of a plant grown in the basement do not have a green color. 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 almost at the same latitudes, not far from 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 lies in the influence of sea currents.

Accompanying change method built like this:

Under the 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.

=> Probably condition A is the cause of phenomenon x.

A change in one of the conditions (with the other conditions unchanged) is accompanied by a change in the occurring phenomenon, due to which it can be argued that this condition and the specified phenomenon are united by a cause-and-effect relationship. For example, if the speed of movement is doubled, the distance traveled is also doubled; If the speed increases three times, then the distance traveled becomes three times greater. Therefore, an increase in speed is the cause of an increase in the distance traveled (of course, in the same period of time).

Let us demonstrate the method of concomitant changes with examples.
1. Even in antiquity, it was noticed that the periodicity of the sea tides and the change in their height correspond to changes in the position of the moon. The highest tides occur on the days of new moons and full moons, the smallest - on the so-called quadrature days (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 due to the action of the moon.

2. Anyone who has held a ball in his hands knows that if you increase the external pressure on him, the ball will decrease. If you stop this pressure, then the ball returns to its original size. The 17th-century French scientist Blaise Pascal was apparently the first to discover this phenomenon, and he did it in a very peculiar and quite convincing way. Going with his assistants to the mountain, 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 ascended, and began to decrease on the way back. When the researchers 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 built as follows:

Under ABC conditions, the xyz phenomenon occurs.

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

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

=> It is likely that condition A is the cause of phenomenon X.

In this case, the occurring phenomenon is divided into 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. It is known that the editor V edited the manuscript in blue ink ( at), and editor C in red ( z). However, the manuscript contains notes made in green ink ( X). It can be concluded that, most likely, they were left by the editor A.

Examples of applying the residual method are given below.
1. Observing the movement of the planet Uranus, 19th-century astronomers noticed that it deviated somewhat from its orbit. It was found that Uranus is deflected by magnitudes 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 in size d. From this, a hypothetical conclusion was made about the presence of a still 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 the telescope he designed, found it on the celestial sphere. So in the 19th century, the planet Neptune was discovered.

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

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

In conclusions by analogy, on the basis of 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:

Object A has attributes a, b, c, d.

Object B has signs a, b, c.

=> Probably item B has a feature d.

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

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

« Thought» in series« Philosophical heritage»

=> 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. Similar features of these two books are that they are published by the same publishing house, in the same series, provided with introductory articles and commentaries. Based on this, it can be argued with a high degree of probability that if the works of Sextus Empiricus are equipped with a subject-nominal index, then the works of Francis Bacon will also be equipped with them. Thus, the presence of a subject index is a portable feature in the considered example.

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

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

Let's take a few more examples.
1. Gills are to fish what lungs are to mammals.

2. A story by A. Conan Doyle "The Sign of the Four" about the adventures of the noble detective Sherlock Holmes, which is distinguished by a dynamic plot, I really liked. I have not read A. Conan Doyle's The Hound of the Baskervilles, 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 M. M. Bongard and A. L. Challenge demonstrated an installation that simulated human color vision. When the lamps were turned on quickly, she unmistakably recognized the color and its intensity. Interestingly, this installation had a number of the same shortcomings as human vision.
For example, orange light after intense red in the first moment was perceived by her as blue or green.

V relationship analogies two groups of objects are compared, and the transferred attribute is some kind of relationship between objects within these groups. Relationship analogy example:

In a mathematical fraction, the numerator and denominator are inversely related: 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 about himself, how he evaluates himself.

=> 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 inverse relationship between objects is transferred from the first group to the second.

Let's take two more examples.
1. The essence of E. Rutherford's planetary model of the atom 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; in the same way, two point charges that are motionless relative to each other (according to Coulomb's law) interact with an electrostatic force that is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

Because 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 respects resemble the rules of incomplete induction already known to us.

Firstly, it is necessary to draw a conclusion on the basis of the largest possible number of similar features of like objects.

Secondly, these signs should be varied.

Thirdly, similar features should be essential for the compared items.

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

The first three rules of analogy actually repeat the rules of incomplete induction. Perhaps the most important is the fourth rule, about the relationship of similar features and the transferred feature. Let us return to the analogy example discussed at the beginning of this section. A portable feature - the presence of a subject-name index in a book - is closely related to similar features - publisher, series, introductory article, comments (books of this genre are necessarily supplied with a subject-name index). If a transferred feature (for example, the volume of a book) is not naturally associated with similar features, then the conclusion of the conclusion by analogy may turn out to be false:

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

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

=> Most likely, the published works of Francis Bacon, like those of Sextus Empiricus, have a volume of 590 pages.

Despite the probabilistic nature of the conclusions, reasoning by analogy has many advantages. Analogy is a good means of illustrating and explaining some complex material, it is a way of giving it artistic imagery, often leading to scientific and technical discoveries. So, on the basis of the analogy of relations, many conclusions have been drawn in bionics - a science that studies objects and processes of wildlife to create various technical devices. For example, snowmobiles have been built, the principle of movement of which is borrowed from penguins. Using the peculiarity of the jellyfish's perception of infrasound with a frequency of 8-13 oscillations per second (which allows it to recognize in advance the approach of a storm by storm infrasounds), scientists have created an electronic device capable of predicting the onset of a storm in 15 hours. Learning to fly bat, which emits ultrasonic vibrations and then picks up their reflection from objects, thereby accurately navigating in the dark, a person has designed radars that detect various objects and accurately determine their location, regardless of weather conditions.

As we can see, reasoning by analogy is 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 consequence in such conclusions is a logical law; non-demonstrative - the rules of inference provide only a probabilistic following of the conclusion from the premises.
2) according to the direction of the logical consequence, i.e. by the nature of the connection between knowledge of varying degrees of generality, expressed in premises and conclusions: deductive - from general knowledge to particular; inductive - from private knowledge to general; reasoning by analogy - from particular knowledge to particular.

Deductive reasoning is a form of abstract thinking in which thought develops from knowledge of a greater degree of generality to knowledge of a lesser degree of generality, and the conclusion that follows from the premises is logically reliable. The objective basis of remote control 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.

All conclusions are usually divided into types according to various grounds: by composition, by the number of premises, by the nature of the logical consequence and the degree of general knowledge in the premises and conclusion.

By composition, all the conclusions are divided into simple and complex. Inferences are called simple, the elements of which are not inferences. Compound statements are those that are made up of two or more simple statements.

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

According to the nature of the logical consequence, all conclusions are divided into necessary (demonstrative) and plausible (non-demonstrative, probable). Necessary inferences are those in which the true conclusion necessarily follows from the true premises (i.e., the logical consequence in such conclusions is a logical law). Necessary inferences include all types of deductive reasoning and some types of inductive ("full induction").

Plausible inferences are those in which the conclusion follows from the premises with a greater or lesser degree of probability. For example, from the premises: “Students of the first group of the first year passed the exam in logic”, “Students of the second group of the first year passed the exam in logic”, etc. follows “All first-year students passed the exam in logic” with a greater or lesser degree of probability (which depends on the completeness of our knowledge about all the troupes of first-year students). Plausible inferences include inductive and analogical inferences.

Deductive reasoning (from lat. deductio - inference) is such a conclusion in which the transition from general knowledge to particular is logically necessary.

By deduction, reliable conclusions are obtained: if the premises are true, then the conclusions will be true.

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

Since this conclusion is based on the principle of considering not all, but only some objects of a given class, the conclusion is called incomplete induction. In full induction, generalization occurs on the basis of knowledge of all subjects of the class under study.

In inference by analogy (from the Greek. analogia - correspondence, similarity), on the basis of the similarity of two objects in some one parameters, a conclusion is made about their similarity in other parameters. For example, based on the similarity of the methods of committing crimes (burglary), it can be assumed that these crimes were committed by the same group of criminals.

All kinds of inferences can be well-formed and incorrectly constructed.

Immediate inferences are those in which the conclusion is derived from a single premise. For example, from the proposition "All lawyers are lawyers" you can get a new proposition "Some lawyers are lawyers". Immediate inferences give us the opportunity to reveal knowledge about such aspects of objects, which was already contained in the original judgment, but was not explicitly expressed and clearly realized. Under these conditions, we make the implicit - explicit, the unconscious - conscious.

Direct inferences include: transformation, conversion, opposition to a predicate, inference according to the “logical square”.

Transformation is such a conclusion in which the original judgment is transformed into a new judgment, opposite in quality, and with a predicate that contradicts the predicate of the original judgment.

To transform a proposition, it is necessary to change its connective to the opposite, and the predicate to a contradictory concept.

Conversion is such a direct inference in which the place of the subject and the predicate is reversed while maintaining the quality of the judgment.

The address is subject to the rule of distribution of terms: if a term is not distributed in the premise, then it should not be undistributed in the conclusion.

If the conversion leads to a change in the original judgment in terms of quantity (a new particular judgment is obtained from the general original), then such a conversion is called a treatment with a restriction; if the conversion does not lead to a change in the original judgment in terms of quantity, then such a conversion is a conversion without restriction.

General affirmative singling out judgments circulate without restriction. Any offense (and only an offense) is an unlawful act.

Every wrongful act is a crime.

The logical operation of judgment reversal is of great practical importance. Ignorance of the rules of circulation leads to gross logical errors. So, quite often a universally affirmative judgment is drawn without restriction. For example, the proposition "All lawyers must know logic" becomes the proposition "All students of logic are lawyers." But this is not true. The proposition "Some students of logic are lawyers" is true.

Opposition to a predicate is the successive application of the operations of transformation and conversion - the transformation of a judgment into a new judgment, in which the concept that contradicts the predicate becomes the subject, and the subject of the original judgment becomes the predicate; the quality of judgment changes.

Inference on the "logical square". The "logical square" is a scheme that expresses truth relations between simple propositions that have the same subject and predicate. In this square, the vertices symbolize the simple categorical judgments known to us according to the combined classification: A, E, O, I. The sides and diagonals can be considered as logical relationships between simple judgments (except for equivalent ones). Thus, the upper side of the square denotes the relation between A and E, the relation of opposites; the downside is the relationship between O and I -- the relationship of partial compatibility. The left side of the square (the relationship between A and I) and the right side of the square (the relationship between E and O) is the relationship of subordination. The diagonals denote the relationship between A and O, E and I, which is called a contradiction.

The relationship of opposition takes place between judgments generally affirmative and generally negative (A-E). The essence of this relationship is that two opposing propositions cannot be both true at the same time, but they can be simultaneously false. Therefore, if one of the opposite judgments is true, then the other is necessarily false, but if one of them is false, then it is still impossible to unconditionally assert that it is true about the other judgment - it is indefinite, i.e., it can turn out to be both true and false . For example, if the proposition "Every lawyer is a lawyer" is true, then the opposite proposition "No lawyer is a lawyer" will be false.

But if the proposition “All students of our course have studied logic before” is false, then the opposite statement “No student of our course has studied logic before” will be indefinite, i.e., it can turn out to be either true or false.

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

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

The relationship of subordination exists between general affirmative and particular affirmative judgments (A-I), as well as between general negative and particular negative judgments (E-O). In this case, A and E are subordinate, and I and O are subordinate judgments.

The relation of subordination consists in the fact that the truth of the subordinate judgment necessarily follows from the truth of the subordinate judgment, but the converse is not necessary: if the subordinate judgment is true, the subordinate will be indeterminate - it can turn out to be both true and false.

But if the subordinate judgment is false, then the subordinate will be all the more false. The converse, again, is not necessary: if the subordinate judgment is false, the subordinate may turn out to be both true and false.

For example, if the subordinate proposition "All lawyers are lawyers" is true, the subordinate proposition "Some lawyers are lawyers" will be all the more true. But if the subordinate judgment “Some lawyers are members of the Moscow Bar Association” is true, the subordinate judgment “All lawyers are members of the Moscow Bar Association” will be either false or true.

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

The relationship of contradiction exists between general affirmative and particular negative judgments (A - O) and between general negative and particular affirmative judgments (E - I). The essence of this relationship is that of two contradictory judgments, one is necessarily true, the other is false. Two contradictory propositions cannot be both true and false at the same time.

Inferences based on the relation of contradiction are called the negation of a simple categorical judgment. By negating a proposition, a new proposition is formed from the original proposition, which is true when the original proposition (premise) is false, and false when the original proposition (premise) is true. For example, denying the true proposition "All lawyers are lawyers" (A), we get a new, false, proposition "Some lawyers are not lawyers" (O). Rejecting the false proposition "No lawyer is a lawyer" (E), we get a new, true proposition "Some lawyers are lawyers" (I).

Knowing the dependence of the truth or falsity of some judgments on the truth or falsity of other judgments helps to draw correct conclusions in the process of reasoning.

The most widespread type of deductive reasoning is categorical reasoning, which, because of its form, is called syllogism (from the Greek sillogismos - counting).

A syllogism is a deductive conclusion in which two categorical propositions-parcels connected by a common term produce a third proposition - a conclusion.

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

CONCLUSION - THE THIRD FORM OF THINKING

What is an inference?

inference- this is the third (after the concept and judgment) form of thinking, in which one, two, or several judgments, called premises, follow a new judgment, called the conclusion, or conclusion.

In logic, it is customary to place the premises and the output one under the other and to separate the premises from the output with a line:

All living organisms feed on moisture.

All plants are living organisms.

All plants feed on moisture.

In the above example, the first two judgments are the premises, and the third is the conclusion. It is clear that the premises must be true judgments and must be connected with each other.

If at least one of the premises is false, then the conclusion is false:

All birds are mammals.

All sparrows are birds.

All sparrows are mammals.

For example, no conclusion follows from the following two premises:

All planets are celestial bodies.

All pines are trees.

Let us pay attention to the fact that inferences consist of judgments, and judgments - of concepts, i.e. one form of thought enters into another as an integral part.

All inferences are divided into direct and indirect. V immediate inferences, the conclusion is made from one premises.

for instance:

All flowers are plants.

Some plants are flowers.

Another example:

It is true that all flowers are plants.

It is not true that some flowers are not plants.

It is not difficult to guess that direct inferences are operations for transforming simple judgments and conclusions about the truth of simple judgments in a logical square. The first example of direct inference given above is a transformation of a simple proposition by inversion, and in the second example, by the logical square, from the truth of a proposition of type A, a conclusion is drawn about the falsity of a proposition of type O.

V mediated inferences, the conclusion is drawn from several premises.

for instance:

All fish are living beings.

All carp are fish.

All carp are living beings.

Since direct inferences are various logical operations with judgments, then under inferences are meant, first of all, indirect inferences. In the future, we will talk about them.

Indirect inferences are divided into three types. They are deductive, inductive and reasoning by analogy.

deductive reasoning, or deduction - these are inferences in which a conclusion is drawn from a general rule for a particular case (a special case is derived from a general rule).

for instance:

All stars radiate energy.

The sun is a star.

The sun radiates energy.

As you can see, the first premise is a general rule, from which (using the second premise) a special case follows in the form of a conclusion: if all stars radiate energy, then the Sun also radiates it, because it is a star. In deduction, reasoning goes from the general to the particular, from the greater to the lesser, knowledge narrows, due to which the deductive conclusions are reliable, i.e. accurate, obligatory, necessary, etc. Let's look again at the example above. Could any other conclusion follow from these two premises than the one that follows from them? Could not! The following conclusion is the only one possible in this case. Let us depict the relationship between the concepts of which our conclusion consisted, Euler circles. Volumes of three concepts: stars; body, radiating energy; The sun schematically arranged as follows.

If the scope of the concept stars included in the concept body, radiating energy, and the scope of the concept The sun included in the concept stars, then the scope of the concept The sun automatically included in the scope of the concept bodies that radiate energy, which makes the deductive inference valid.

The undoubted advantage of deduction, of course, lies in the reliability of its conclusions. 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 deduce the particular from the general. In one work, explaining to Dr. Watson the essence of his deductive method, he gives the following example. Near the murdered Colonel Morin, Scotland Yard detectives found a smoked cigar and decided that the colonel had smoked it before his death.

However, he (Sherlock Holmes) irrefutably proves that Colonel Morin could not smoke this cigar, because he wore a large, lush mustache, and the cigar was smoked to the end, i.e. if Morin had smoked it, he would certainly have set his mustache on fire. Therefore, the cigar was smoked by another person. In this reasoning, the conclusion looks convincing precisely because it is deductive: from the general rule ( Anyone with a big, bushy mustache can't finish a cigar.) a special case is displayed ( Colonel Morin could not finish his cigar because he wore such a mustache).

Inductive reasoning, or induction - these are inferences in which a general rule is deduced from several special cases (several special cases lead to a general rule).

for instance:

Jupiter is moving.

Mars is moving.

Venus is moving.

Jupiter, Mars, Venus are planets.

All planets are moving.

As you can see, the first three premises are special cases, the fourth premise brings them under one class of objects, combines them, and the output refers to all objects of this class, i.e. some general rule is formulated (following from three special cases). In induction, reasoning goes from the particular to the general, from less to more, knowledge expands, due to which inductive conclusions (unlike deductive ones) are not reliable, but probabilistic. The probabilistic nature of the conclusions is, of course, a disadvantage of induction. However, its undoubted advantage and advantageous difference from deduction, which is a narrowing knowledge, is that induction is an expanding knowledge that can lead to a new one, while deduction is an analysis of the old and already known.

Inference by analogy, or analogy- these are inferences in which, on the basis of the similarity of objects (objects) in some features, a conclusion is made about their similarity, and in other features, a conclusion is made about their similarity in other features.

for instance:

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

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

There is probably life on Mars.

As you can see, two objects are compared (compared) (the planet Earth and the 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 ways: if there is life on Earth, and Mars is in many ways similar to Earth, then the presence of life on Mars is not excluded. The conclusions of analogy, like the conclusions of induction, are probabilistic.

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