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Deductive reasoning. Deductive and inductive reasoning

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Reasoning and inference. Structure of inference. Deductive reasoning. Correct and incorrect conclusions. Deductive inferences from categorical judgments. Direct and indirect inferences.

Reasoning is a procedure for justifying a certain statement by step-by-step derivation from other statements.

The simplest form of reasoning is inference.

Inference is a direct transition from one statement or several statements A 1 , A 2 , …, A n to the statement of V.

Statements A 1 , A 2 , …, A n , from which the conclusion is drawn are called parcels, and the statement IN, which is derived from the premises, is called conclusion.

As an example of an inference, we give the reasoning that, according to legend, Caliph Omar conducted to justify the need to burn the Library of Alexandria:

“If your books agree with the Koran, then they are unnecessary. If your books do not agree with the Koran, then they are harmful. But harmful or unnecessary books should be destroyed. Therefore, your books should be destroyed."

In the above inference, the first three statements are premises, and the fourth is the conclusion.

In logic, inference is usually formulated as follows:

A 1 , A 2 , …, A n ,

Where the premises are written above the line, the conclusion is written below the line, and the line itself expresses the act of deriving a conclusion from the premises.

Inference is the simplest type of reasoning because the thesis being substantiated (its role is played by conclusion B) is directly, as if in one step, derived from the premises A 1 , A 2 , …, A n, which can be considered as arguments in favor of the thesis.

However, many arguments have a much more complex structure. Thus, in the course of reasoning, several inferences can be made, and the conclusions of some can become premises in others. Let's look at an example.

A bank robbery was committed in an English city. Suspicion fell on known repeat offenders Smith, Jones and Brown. During the investigation, the following emerged. Jones never goes on a case without Brown. At least one of the repeat offenders - Smith or Jones - is involved in the crime. Brown has a solid alibi. The police inspector who conducted the investigation, based on this information, charged Smith.

At the same time, he could reason as follows. Data obtained during the investigation indicate that:

(1) If Jones is involved in a crime, then Brown is also involved (Jones does not go to work without Brown).

(2) Brown is not involved in the crime (he has an alibi)

Hence,

(3)Jones is not involved in the crime.

But, according to the investigation,

(4) Smith or Jones is involved in a crime.

Therefore, taking into account Jones’s non-involvement in the crime, we can conclude:

(5) Smith is involved in a crime.

In the above reasoning, two conclusions are drawn. In the first of them, the premises are statements (1) and (2), and the conclusion is Statements (3). In the second inference, the premises are (3) and (4), and the conclusion is the statement (5).

Sometimes in the course of reasoning, to justify a certain statement (let’s call it C), so-called indirect methods of argumentation. In this case, auxiliary reasoning is constructed and additional assumptions, from which they strive to obtain consequences of a certain kind (the nature of the assumptions made and the consequences sought usually depend on the type of statement C). If these problems are successfully solved, the auxiliary reasoning is considered completed, and S’s statement appears in the main part of the reasoning.

An example of an indirect method of argumentation is the widespread reasoning by contradiction. Their structure is as follows. To substantiate the statement B, the contradictory statement “It is not true that B” is accepted as an additional assumption, while from the assumption and a certain set of arguments D they strive to obtain a contradiction - the statement “D and it is not true that D.” If this auxiliary reasoning is successfully carried out, the assumption is considered to be false, and B itself is justified by means of G's arguments.

Let us show how the police inspector in the example considered could come to the conclusion that Smith is guilty by reasoning by contradiction.

Let us first accept the assumption that

(1) Smith is not involved in the crime.

From this assumption and established fact:

(2) Smith or Jones are involved in a crime - we get the statement:

(3)Jones is involved in a crime.

From it, as well as from another fact established during the investigation:

(4) If Jones is involved in a crime, then Brown is also involved in it - we get the statement:

(5) Brown is involved in a crime.

However, the investigation found that

(6) Brown is not involved in the crime.

Thus, a contradiction is obtained in the argument:

(7) Brown is and is not involved in the crime.

Therefore, assumption (1) is false, and the statement

(8) Smith is involved in a crime

Considered to be justified by arguments (2), (4) and (6).

Deductive reasoning and inference.

Deduction(lat. deductio- deduction) - a method of thinking in which a particular situation is logically deduced from the general, a conclusion according to the rules of logic; a chain of inferences (reasonings), the links of which (statements) are connected by a relation of logical implication.

Logic is often defined as the science of reasoning. Indeed, the study of reasoning, its types and methods of implementation is among the main tasks of logic. Nevertheless, the methods of logical analysis considered so far concerned the verification of the correctness or incorrectness of ready-made reasoning and did not address the question of how they are carried out. Let's describe the procedure deductive reasoning, which are also called plausible.

In general, reasoning is understood as a procedure for sequential step-by-step transition from some statements, accepted as initial ones, to other statements. Each step of this process is carried out on the basis of some rule called rule of inference. The last statement obtained in this process is called conclusion reasoning. At the same time, we will further classify as deductive only those reasonings in which the relation of logical consequence is preserved between the statements accepted as initial statements and the conclusion. To answer now specifically the question of how deductive type reasoning is built, it is necessary to develop some theory - deductive reasoning theory. But before that, let’s briefly describe the main types theories.

Deduction is a theoretical way of understanding the world around us. Therefore, deduction procedures are used in the case when empirical cognitive techniques (observations, experiments, measurements) are not enough to obtain some new knowledge. In this capacity, deduction is widely used in everyday life: after all, we often try to defend our point of view through this or that reasoning, convince our interlocutor of its truth, refute the opponent’s point of view, etc., that is, we try to reason theoretically. However, deduction procedures as a theoretical research method are of greatest importance in the construction of scientific (theoretical) knowledge.

Depending on the degree of clarity (identification) of deductive connections between individual statements (statements) of theories, several types are distinguished. The first type includes substantive theories. In their composition, if deduction is used, it is only to connect some individual provisions of the theory. In this case, the initial statements in the reasoning represent some assumptions called parcels. The premises do not have to be (and do not always be) true, and therefore any sentence that is deduced using them is considered not true, but conditionally true: The final sentence (conclusion) is true provided that the premises are true. Reasoning in everyday life, for example, is of a similar nature. Examples of meaningful theories are school arithmetic, as well as various kinds of scientific concepts developed in those sciences in which there are no strictly defined theories. Examples of logical content theories are propositional and predicate logics.

Another type make up formalized theories. These include theories whose content is interconnected and deductively derived from some initially accepted initial statements. The latter are called axioms, and the theories themselves are called axiomatized theories. Examples of them are: Newton's celestial mechanics, Einstein's theory of relativity, quantum mechanics, Euclid's geometry. Unlike Euclid’s geometry, which was formalized more than 2 thousand years ago, arithmetic developed as a meaningful theory until the 20th century, and only at the turn of the 19th – 20th centuries was it formalized by the Italian mathematician Peano.

Deductive reasoning

The beginning (premises) of deduction are axioms or simply hypotheses that have the nature of general statements (“general”), and the end is the consequences of the premises, theorems (“particular”). If the premises of a deduction are true, then its consequences are true. Deduction is the main means of proof. The opposite of induction.

Example of deductive reasoning:

1) All people are mortal.

2) Socrates is a man.

3) Therefore, Socrates is mortal.

Inferences in which one of the premises is a disjunctive judgment, and the second coincides with one of the members of the disjunctive judgment or denies all but one. In the conclusion, accordingly, all members except those indicated in the second premise are denied, or the missing member is affirmed.

Forms of correct modes of dividing-categorical conclusions

Conditional inferences

Inferences whose premises and conclusions are conditional propositions.

A special type of inference from two conditional propositions and one dividing.

Types of correct dilemmas:

constructive:

(i.e.: first premise: if A, then C; second premise: if B, then C; third premise: A or B; conclusion: therefore C);

(complex)

(i.e.: first premise: if A, then B; second premise: if C, then D; third premise: A or C; conclusion: therefore B or D);

destructive:

(i.e.: first premise: if A then B; second premise: if A then C; third premise: neither B nor not C; conclusion: therefore not A);

(complex)

(i.e.: first premise: if A then B; second premise: if C then D; third premise: not B or not D; conclusion: therefore not A or not C).

Correct and incorrect conclusions

In order to show that a certain conclusion is incorrect, it is enough to find at least one conclusion of the same logical form, all of whose premises are true and whose conclusion is false. Thus, we have identified criterion for incorrect inference. It can be formulated as follows.

An inference is incorrect if and only if its logical form does not guarantee that, given true premises, we will necessarily obtain a true conclusion, that is, there is an inference of a given logical form with true premises and a false conclusion.

Now it’s not difficult to formulate criterion for the correctness of inferences.

An inference is correct if and only if its logical form does not guarantee that if the premises are true, we will necessarily obtain a true conclusion, that is, there is no inference of this form with true premises and a false conclusion.

When the specified condition is met, they also say that between the premises and the conclusion there is logical consequence relation, that the conclusion logically follows from parcels.

Correct ones include, for example, conclusion (1). Let us reveal its logical form. To this end, let us replace the simple statements that make up its premises and conclusion with parameters: the statement “Your books agree with the Koran” - the letter p, « Your books are superfluous" - letter q, “Your books are harmful” - letter r, “Your books should be destroyed” - letter s. The result is an expression.

If p, That q

If it is not true that p, That r

If q or r, That s

Mediated and non-mediated inferences

Inference- this is a form of thinking through which a new judgment is derived from one or more interconnected judgments with logical necessity. The logical essence of inference consists in the movement of thought from the analysis of existing knowledge to the synthesis of new knowledge. This movement has objective nature and is determined by the real connections of reality. The objective connection reflected in consciousness provides a logical connection of thoughts. On the contrary, the lack of objective connections between reality leads to logical errors.

The structure of any conclusion includes three elements:

1)original knowledge expressed in premises;

2)substantiating knowledge expressed in the rules of inference;

3)inferential knowledge expressed in a conclusion or conclusion.

When analyzing a conclusion, it is customary to write the premises and conclusion separately, placing them on top of each other. The conclusion is written under a horizontal line separating it from the premises and indicating a logical follow-up. In accordance with this, let's consider next example conclusions:

All citizens of the Republic of Belarus have the right to education - premise

Novikov - citizen of the Republic of Belarus - sprinkles

Novikov has the right to education - conclusion

If there is a meaningful connection between the premises, one can obtain new true knowledge in the process of reasoning, subject to two conditions.

Firstly, the initial propositions - premises - must be true. However, it should be borne in mind that sometimes false judgments can give a true conclusion. Thus, as a result of a special selection of false premises in the following reasoning, we obtain a true conclusion:

All elephants have wings

All birds are elephants

All birds have wings

This indicates that focusing only on the form (structure) of premises while ignoring their objectively true connections can create the appearance of a correct conclusion.

Secondly, in the process of reasoning, it is necessary to observe the rules of inference, which determine the logical correctness of the conclusion. Without this, even from true premises you can get a false conclusion. For example:

All caterpillars eat cabbage

I eat cabbage

Therefore, I am a caterpillar

There are quite a lot of rules, a number of them are enshrined in the main types of inferences.

Depending on the sequence of thought development, as well as on the logical validity of the conclusion, inferences are divided into the following types: deductive, inductive and analogical reasoning.

In deductive reasoning(from Latin deductio - deduction) the connections between premises and conclusion are formal logical laws, due to which, with true premises, the conclusion always turns out to be true.

Deductive reasoning- this 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 arising from the premises is, with logical necessity, reliable in nature. The objective basis of deductive conclusions is the unity of the general and the individual in real processes and objects of the surrounding world.

The deduction procedure occurs when the information in the premises contains (often in implicit form) the information expressed in the conclusion. Deductive reasoning is a way of extracting this information and presenting it in explicit form.

The rules of deductive inference are determined by the nature of the premises, which can be simple or complex propositions, as well as their number. Depending on the number of premises used from which the conclusion is drawn, deductive conclusions can be direct or indirect.

Deductive inference is an inference that results in new knowledge about an object or group of objects based on some existing knowledge of the objects under study and the application of a general rule to them that operates within a given class of objects. In other words, in a correct deductive inference, there must be a relationship of subordination, or logical consequence, between the premises and the conclusion.

A simple categorical syllogism is a deductive inference in which a third categorical judgment is derived from two categorical (unconditional) judgments. More precisely, a syllogism can be defined as a conclusion about the relationship of two terms (the subject and the predicate of the conclusion) based on the relationship of each of them in the premises to some common (third) term.

Words and phrases expressing concepts included in a syllogism are called syllogism terms. The terms between which a relationship is established, i.e., the subject and predicate of the conclusion, are called extreme, while the subject of the conclusion is called a smaller term, denoted by the letter S, and the predicate of the conclusion is a larger term, denoted by the letter P. Each of them is contained in one from parcels. The premise containing the major term is called the major premise and the premise containing the minor term is called the minor premise. In the premises, in addition to the extreme terms, there is one term common to them - it is called the middle and is denoted by the letter M. It is with its help that the relationships between the extreme terms are established.

For example, the conclusion “All fish breathe with gills, a pike is a fish, which means a pike also breathes with gills” is a simple categorical syllogism, since a third is derived from two categorical judgments, and in the conclusion the relationship between the terms “pike” and “breathing with gills” is established ” based on the relationship in the premises of each of these terms to the term “fish”. The terms “pike” and “gill-breathing” are extreme terms; the first is smaller, the second is larger, the term “fish” is middle.

A fairly common logical error when constructing a syllogism is the error of “quadrupling terms.” The essence of this error is that the middle term in each of the premises is given different meaning and this means that there are actually four different terms in the syllogism. As a result, the middle term cannot serve to establish relations between extreme terms and the conclusion of such a syllogism may be false. You can sit in the following syllogism: “Matter is eternal. Cloth is matter. Therefore, cloth is eternal.” The conclusion in this conclusion is erroneous. Because the meaning of the term “matter” is not the same in the major and minor premises. In the larger premise, this term has a universal, philosophical meaning, and in the smaller premise, it has an everyday meaning. Consequently, although the word “matter” is one, the meanings that are invested in it are different and the ambiguously interpreted middle term cannot logically correctly connect the extreme terms. The conclusion of this conclusion is false.

Syllogisms can be distinguished by the position of the middle term in them. These varieties of syllogisms, differing in the position of the middle term, are called figures. The middle term could be:

– in the major premise the subject, in the minor predicate – this is the first figure. For example: “All people are mortal. Socrates is a man. This means that Socrates is mortal.”

– the predicate in both premises is the second figure. For example: “All sciences study the laws of objective reality. No religion studies the laws of objective reality. This means that no religion is a science.”

– the subject in both premises is the third figure. For example: “All whales are mammals. All whales live in water. This means that some animals living in water are mammals.”

– in the major premise as a predicate, in the minor as a subject – this is the fourth figure. For example: “All metals are material things. All material things have heaviness. This means that some bodies that have heaviness are metals.”

– The figures depict graphically, or more precisely, the path of movement of thought between the predicate and the subject of the conclusion is graphically depicted.

(To determine the figure of a syllogism, it is necessary that the major premise comes first and the minor premise comes second.)

It should be noted that all figures of a simple categorical syllogism can be reduced to the first figure. Reduction aims to verify the correctness of the syllogistic conclusion. Since the first figure of the syllogism most clearly shows the correspondence of the reasoning to the requirements of the axiom of a categorical syllogism. This axiom has the following formulation: “Everything that is affirmed (or denied) regarding each of the objects that make up a given set (class) is affirmed (or denied) regarding any object included in this set (class).”

In addition to figures, syllogisms have modes. Modes of a syllogism are varieties of its figures that differ from each other in the quality and quantity of those judgments that make up its premises and conclusion. The modes of a syllogism are usually written in three capital letters, which denote general affirmative (A), general negative (E), particular affirmative (I), and particular negative (O) propositions. For example, the first mode of the first figure is indicated by three letters: AAA. Of the 64 possible modes in formal logic, 19 are considered correct, that is, they can be used to obtain a correct conclusion. In mathematical logic, 15 modes are recognized as correct.

To determine the figure and mode of a syllogism, you need to find its terms, see how they are arranged, and establish the types of judgments included in it.

Take the syllogism: “Not a single plant can exist without moisture. (M–P), (E). All grains are plants (S–M) (A). No cereal can exist without moisture (S–P), (E)” In it, “cereal” is the lesser term, “exist without moisture” is the greater (subject and predicate of the conclusion), “plant” is the middle term. The big premise is “No plant can exist without moisture” (with a big term). To determine the figure and mode, it must be placed in first place. The middle term in it is the subject, and the larger one is the predicate (M - P). In the minor premise, the middle term is the predicate and the minor is the subject (S – M). This means that the syllogism refers to the first figure. And since its major premise is a generally negative proposition, its minor one is generally affirmative, its conclusion is generally negative, this is a syllogism of the EAE mode.

Having recognized the figure and mode of a syllogism, one can determine whether a given syllogism is a correct conclusion. The correct modes are:

in the 1st figure: AAA, EAE, AII, EIO;

in the 2nd figure: EAE, AEE, EIO, AOO;

in the 3rd figure: AAI, IAI, AII, EAO, OAO, EIO;

in the 4th figure: AAI, AEE, IAI, EAO, EIO.

The name of modes in the form of a special mnemonic poem was introduced in the Middle Ages by the famous logician Peter of Spain, later Pope John XXI (died 1277). Here is the poem:

Barbara, Celerent, Darii, Ferio – que prioris;

Cesare, Camestres, Festino, Baroko, secundae;

Tertia, Darapti, Disamis, Datisti, Felapton, Bokardo, Ferison habet;

Quatra insuper addit Bramantip, Camenes, Dimaris, Fesapo, Fresison.

There are general rules for obtaining a true conclusion in a simple categorical syllogism. They include two rules of terms and five rules of premises.

Rules of terms:

1. The middle term must be distributed in at least one of the premises.

2. A term that is not distributed in the premise cannot be distributed in the conclusion.

Parcel rules:

1. At least one of the premises must be affirmative.

2. If both premises are affirmative, then the conclusion is affirmative.

3. If one of the premises is negative, then the conclusion is negative. (Sometimes all these three rules are formulated in one: the number of negative premises is equal to the number of negative conclusions).

4. No conclusion can be obtained from two particular premises using a syllogism.

5. If one of the premises is private, then the conclusion, if possible, can only be private.

To check a syllogism according to general rules, you need to find its terms (and make sure that there are three of them), determine the types of judgments - premises and conclusions - and the distribution of terms in them. After this, they check whether each of the rules is satisfied. If even one rule is found to be violated, the syllogism is invalid.

Let us now check the same syllogism: “Not a single plant can exist without moisture. All grains are plants. Not a single grain can exist without moisture,” using general rules. We find the terms: the subject of the conclusion is the smaller term of the syllogism – “cereal”, the predicate of the conclusion – the larger term of the syllogism – “to exist without moisture”, the middle term is “plant”. There are three of them. The major premise – “No plant can exist without moisture” – is generally negative, which means that both terms are distributed in it. The minor premise - “All grains are plants” - is generally affirmative, its subject - “cereals” - is distributed, and the predicate - “plants” - is not distributed. The conclusion – “No cereal can exist without moisture” – is generally negative, both of its terms are distributed.

Now let's see if each of the rules is met. The first rule of terms is fulfilled - in this syllogism the middle term is distributed in the larger premise; the second rule of terms is fulfilled - the greater and lesser terms, being distributed in the conclusion, are also distributed in the premises; the first rule of premises is satisfied - there is one affirmative premise; the second and third rules of the premises are satisfied - with one negative premise the conclusion is negative. So, not a single rule has been broken, the syllogism is correct.

In addition to the rules for obtaining a true conclusion for all figures, there are also special rules for each figure separately. For the first figure: 1) the major premise must be a general proposition, 2) the minor premise must be an affirmative proposition. For the second figure: 1) the major premise must be a general proposition, 2) one of the premises must be negative. For the third figure: 1) the minor premise must be affirmative, 2) the conclusion is always a private judgment. For the fourth figure: 1) when the major premise is affirmative, then the minor premise must be common, 2) if one of the premises is negative, then the major premise must be common.

2. Reduced syllogisms

Abbreviated syllogisms differ in form from full syllogisms. They are used quite often in everyday life. The simplest abbreviated syllogism is called an enthymeme, which translated from Greek means “in the mind,” i.e. we're talking about that this or that part of the conclusion is not stated, but is only implied. An enthymeme is a shortened syllogism, that is, a syllogism in which one of the premises or the conclusion is missing.

An enthymeme may omit either one of the premises or the conclusion. For example, the enthymeme: “N. committed a crime and is therefore subject to criminal liability.” A big premise is missing here: “A person who commits a crime is subject to criminal liability.” It is a well-known provision, the formulation of which is not necessary. Further, the enthymeme: “Every craft is useful, which means plumbing is useful.” The smaller premise, “Plumbing is a craft,” is omitted here. And, finally, the enthymeme: “Not a single second-year student passed this exam, but Ivanov did,” the conclusion is missing here: “Ivanov is not a sophomore.”

A more complex type of abbreviated syllogism is the epicheirema. It is used primarily in disputes. An epicheireme is a shortened syllogism in which each of the premises is an enthymeme. For example, epicheyrema: “A lie causes distrust, since there is a statement that does not correspond to the truth. Flattery is a lie, since it is a deliberate distortion of the truth. This means that flattery breeds distrust.” First parcel in full form is a typical syllogism: “Any statement that does not correspond to the truth causes mistrust. A lie is a statement that does not correspond to the truth. This means that lies cause distrust. “But the second premise in the example given is also an enthymeme. In its full form it presents the syllogism: “Every deliberate distortion of the truth is a lie. Flattery is a deliberate distortion of the truth. This means that flattery is a lie. ”

The epicheyrema scheme is as follows:

M is P because it is N

S is M because it is O

The first premise can be constructed as follows:

All N's are P's

All M's are N's

All M's are P's

The second premise can be expressed as follows:

All O's are M's

All S's are O's

All S are M

3. Complex syllogisms

In both scientific and practical thinking, syllogisms are rarely used alone. In reasoning, there is most often a chain of sequential conclusions.

A sequence of syllogisms connected into a logically connected argument or proof is called a polysyllogism or complex syllogism.

The diagram of a complex syllogism is as follows:

All B is A

All C are B

All C are A

All D are C

All D are A

Simple inferences, depending on the place they occupy in a complex syllogism, are either prosyllogisms or episyllogisms. A prosyllogism is a syllogism that precedes a complex syllogism and thereby provides a basis for the premise of the subsequent syllogism. Accordingly, an episyllogism is a syllogism in which the premise is the conclusion of the previous syllogism.

There are two ways to construct a complex syllogism: progressive and regressive.

A progressive syllogism is a combination of syllogisms where the conclusion of one syllogism is a premise for another syllogism, and the conclusion goes from more general to less general. For example:

All vertebrates have red blood.

All mammals are vertebrates.

All mammals have red blood.

All carnivores are mammals.

All carnivores have red blood.

Tigers are predatory animals.

All tigers have red blood.

A regressive syllogism is a combination of syllogisms when the conclusion of one syllogism is a premise for another syllogism, and the conclusion goes from the less general to the more general. For example:

Vertebrates are animals.

Tigers are vertebrates.

Tigers are animals.

Animals are organisms.

Tigers are animals.

Tigers are organisms

Organisms are destroyed.

Tigers are organisms.

Tigers are destroyed.

A type of complex syllogism is sorites. It is a combination of complex and abbreviated syllogisms. In sorites, each concept enters the premises twice; the first time as a predicate, the second - as the subject of the premises (except for the first and last premises). In the chain of syllogisms that make up the sorites, each syllogism is a prosyllogism and an episyllogism (except the first). Sorites is used if it is necessary to consistently designate long chain subordinate units.

If the order of subordination can be seen by moving from subordinate concepts to subordinate ones, then the minor premise is omitted, such a sorites is called Aristotelian. For example:

Bucephalus is a horse.

A horse is a four-legged animal.

A four-legged animal is an animal.

An animal is a substance.

Bucephalus is a substance.

A horse is a four-legged animal.

Bucephalus is a horse.

Bucephalus is a quadruped.

A four-legged animal is an animal.

Bucephalus is a quadruped.

Bucephalus is an animal.

An animal is a substance.

Bucephalus is an animal.

Bucephalus is a substance.

If the order of subordination can be seen by moving from subordinate concepts to subordinate ones, then the major premise is omitted, and such a sorite is called a Hocleniian one. For example:

An animal is a substance.

A four-legged animal is an animal.

A horse is a four-legged animal.

Bucephalus is a horse.

Bucephalus is a substance.

This sorites combines the following three syllogisms:

An animal is a substance.

A four-legged animal is an animal.

The quadruped is a substance.

A horse is a four-legged animal.

A horse is a substance.

Bucephalus is a horse.

Bucephalus is a substance.

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

The logical form of obtaining inferential knowledge is inference.

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

Any conclusion consists of premises, conclusion and conclusion. Parcels inferences are the initial judgments from which a new judgment is derived.
Conclusion is a new proposition obtained logically from premises. The logical transition from premises to conclusion is called
conclusion.

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

The rules of deductive inference are determined by the nature of the premises, which can be simple (categorical) or complex propositions. Depending on the number of premises, deductive conclusions from categorical judgments are divided into
immediate, in which the conclusion is derived from one premise, and
mediated, V depending on the number of premises, inferences are divided into two groups: 1) inferences in the improper sense, or direct inferences; 2) inferences in the proper sense. This last group includes the following types of inferences: 1) induction, 2)deduction, 3)analogy, etc.

The premises of deductive inferences can be simple categorical judgments and judgments of all types of logical conjunctions - connecting, dividing, conditional, or a variety of their combinations that determine the nature of the conclusion. In accordance with this, the following types of deductive inferences are distinguished:

- dividing-categorical;

- conditionally dividing.

The most widespread type of deductive inferences are categorical inferences, which because of their form are called syllogism (from the Greek sillogismos - counting).

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

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

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

Every citizen Russian Federation has the right to education.

Novikov is a citizen of the Russian Federation.

Novikov has the right to education.

The conclusion of this syllogism is a simple categorical proposition A, in which the scope of the predicate “has the right to education” is wider than the scope of the subject – “Novikov”. Because of this, the predicate of inference is called the major term, and the subject of inference is called the lesser term. Accordingly, the premise, which includes the predicate of the conclusion, i.e. the larger term is called the major premise, and the premise with the smaller term, the subject of the conclusion, is called the minor premise of the syllogism.

The third concept “citizen of the Russian Federation”, through which a connection is established between the larger and smaller terms, is called the middle term of the syllogism and is denoted by the symbol M (Medium - intermediary). The middle term is included in each premise, but is not included in the conclusion. The purpose of the middle term is to be a link between the extreme terms - the subject and the predicate of the inference. This connection is carried out in premises: in the major premise, the middle term is associated with the predicate (M - P), in the minor premise - with the subject of the conclusion (S - M). The result is the following syllogism diagram.

M - P S - M

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

S - P S - P

The following must be kept in mind:

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

2) changing the place of any term in the premise does not change its designation - the larger term (the predicate of the conclusion) is denoted by the symbol P, the smaller one (the subject of the conclusion) by the symbol S, the middle one by M;

3) from a change in the order of premises in a syllogism, the conclusion, i.e. the logical connection between extreme terms does not depend.

Consequently, the logical analysis of a syllogism must begin with the conclusion, with an understanding of its subject and predicate, with the establishment from here of the greater and lesser terms of the syllogism. One way to establish the validity of syllogisms is to check whether the rules of syllogisms are followed. They can be divided into two groups: rules of terms and rules of premises.

2. GENERAL RULE OF CATEGORICAL SYLLOGISM, FIGURES AND MODES OF CATEGORICAL SYLLOGISM

A widespread type of indirect inference is a simple categorical syllogism, the conclusion of which is obtained from two categorical judgments.

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

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

Each of the extreme terms is included not only in the conclusion, but also in one of the premises. A premise containing a minor term is called
smaller parcel, a premise containing a larger term is called
larger parcel.

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

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

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

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

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

The validity of the conclusion, i.e. logical transition from premises to conclusion, in a categorical syllogism is based on the position
(axiom of syllogism): everything that is affirmed or denied regarding all objects of a certain class is affirmed or denied regarding each object and any part of the objects of this class.

Figures and modes of categorical syllogism

In the premises of a simple categorical syllogism, the middle term can take the place of subject or predicate. Depending on this, there are four types of syllogism, which are called figures (Fig.).

In the first figure the middle term takes the place of the subject in the major and the place of the predicate in the minor premises.

In second figure- place of the predicate in both premises. IN third figure- the place of the subject in both premises. IN fourth figure- the place of the predicate in the major and the place of the subject in the minor premise.

These figures exhaust all possible combinations of terms. The figures of a syllogism are its varieties, differing in the position of the middle term in the premises.

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

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

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

Rules of terms.

1st rule: in A syllogism must have only three terms. The conclusion in a syllogism is based on the ratio of the two extreme terms to the middle, so there can be no less or more sin of terms in it. Violation of this rule is associated with identification different concepts, which are taken as one and treated as the middle term. This error is based on a violation of the requirements of the law of identity and is called quadrupling of terms.

2nd rule: the middle term must be distributed in at least one of the premises. If the middle term is not distributed in any of the premises, then the relationship between the extreme terms remains uncertain. For example, in the parcels “Some teachers ( M-) - members of the Union of Teachers ( R)", "All employees of our team ( S) - teachers ( M-)" middle term ( M) is not distributed in the major premise, since it is the subject of a particular judgment, and is not distributed in the minor premise as a predicate of an affirmative judgment. Consequently, the middle term is not distributed in any of the premises, so the necessary connection between the extreme terms ( S And R) cannot be installed.

3rd rule: a term that is not distributed in the premise cannot be distributed in the conclusion.

Error, associated with violation of the rule of distributed extreme terms,
is called an illegal extension of a lesser (or greater) term.

Parcel rules.

1st rule: at least one of the premises must be an affirmative proposition. From The conclusion does not necessarily follow from two negative premises. For example, from the premises “Students of our institute (M) do not study biology (P)”, “Employees of the research institute (S) are not students of our institute (M)” it is impossible to obtain the necessary conclusion, since both extreme terms (S and P) are excluded from average. Therefore, the middle term cannot establish a definite relationship between the extreme terms. Finally, the smaller term (M) may be fully or partially included in the scope of the larger term (P) or completely excluded from it. In accordance with this, three cases are possible: 1) “Not a single employee of the research institute studies biology (S 1); 2) “Some employees of the research institute study biology” (S 2); 3) “All employees of the research institute study biology” (S 3) (fig.).

Figure 2 - Explanation of the premises rule

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

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

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

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

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

If one premise is a partial affirmative and the other is a partial negative (I0 or 0I), then in such a syllogism only one term will be distributed - the predicate of a particular negative judgment. If this term is average, then a conclusion cannot be drawn, so, according to the 2nd rule of premises, the conclusion must be negative. But in this case, the predicate of the conclusion must be distributed, which contradicts the 3rd rule of terms: 1) the larger term, not distributed in the premise, will be distributed in the conclusion; 2) if the larger term is distributed, then the conclusion does not follow according to the 2nd rule of terms.

1) Some M(-) are P(-) Some S(-) are not (M+)

2) Some M(-) are not P(+) Some S(-) are M(-)

None of these cases provide the necessary conclusions.

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

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

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

3. CONDITIONAL AND SEPARATE SYLLOGISMS, CONDUCTED AND COMPLEX SYLLOGISMS

Conditional deductive inferences contain two related general concepts the initial premises are conditional propositions and have the structure:

Where the premises are written above the line and the conclusion below it.

The law of transitivity, which underlies all direct inferences, corresponds to them in its pure form. However, there are unstructured forms of purely conditional inferences. One of them, " If a, That b, if not- a, then too b " contains a predefined response equivalent to a statement b.

More significant are unstructured modes of hypothetical syllogism.

Approver: « If a, That b; a: b " And " If a, That b; a: b ».

Allows you to build reliable conclusions from the statement of the basis to the statement of the consequence.

Probabilistic-affirmative: « If a, That b; b: probably a " And " If a, That b; b: probably a " Does not give a reliable conclusion (since, when checked, the formula a → b ≡ b → a does not turn out to be identically true) and means that, the validity of the cause cannot be inferred from the statement of the effect(one cannot reliably conclude from the statement of the consequence to the statement of its basis).

The conclusion will be only probabilistic. Cause leads to effect: if there is no effect, there was no cause. However, an effect can also have other causes, so the statement of the reason affirms the effect, but the statement of the effect does not mean the statement of the reason.

Denying: « If a, That b; Not- b: Not- a " And " If a, That b; Not- b (b): Not- a " Allows you to build reliable conclusions from the negation of the consequence to the negation of the reason: there is no consequence - there was no reason.

Probabilistic-denier: « If a, That b; Not- a: probably not- b " And " If a, That b; Not- a: probably b(Not- b) " Means that from denying the reason it is impossible to deny the reliability of the consequence(one cannot reliably conclude from the negation of a reason to the negation of a consequence).

Denial of a consequence means negation of the basis: but since the consequence can have other reasons, then negation of the basis does not mean negation of the consequence.

The combination of these modes gives a single hypothetical syllogism: If the first entails the second, then the affirmation of the first affirms the second, and the negation of the second denies the first; Moreover, the affirmation of the second does not affirm the first, and the negation of the first does not deny the second.

An example of constructing a hypothetical syllogism (in in this case asserting and relying on consequences, that is, probabilistic):

“If the text contains the letter “i”, then it is probably Russian; the letter "i" is present. Conclusion: he may indeed turn out to be Russian-speaking, and he is Russian-speaking if other languages do not have the letter “i”. But even if he is Russian-speaking, this does not necessarily mean the presence of “and”, and if he does not have it, this does not mean that he is not Russian-speaking.”

Conditional syllogisms correspond to additional rules that make it easier to draw conclusions:

If the first clearly means the second(is its sufficient condition) , then negation of the second is impossible without negation of the first(“Water is liquid: not liquid is not water”) a → b ≡ a → b.

If it is known for certain that without the first there can be no second(if it is a necessary condition for the second) , then the second can only be due to the presence of the first(“The unsuitable is not used, the suitable is used”) a → b ≡ b → a.

If the first refutes the second, then the second refutes the first.(“If there is air, then it is not a vacuum; if there is a vacuum, then there is no air in it”) a → b ≡ b →a.

If refuting the first means the second, then refuting the second means affirming the first.(“Lifeless is dead; not dead is alive”) a → b ≡ b → a.

If several causes together lead to an effect, then the presence of some of them means that the effect will appear with the appearance of others (“Having enough money means that, with the appearance of a desire, you can buy a car”).

When using logical rules, one should not forget that logic is a formal science. She doesn't take into account dialectical laws and, relying only on it, one cannot be sure of the physical truth of conclusions that are true from its point of view. For example: From the premises “If the court comes to the conclusion that a document is forged, it removes it from the evidence” and “This document is not removed from the evidence”, one can obtain a reliable conclusion by means of the negating mode of the conditional syllogism:

However, the denying mode presupposes the presence of “fatum” - the inevitability of the consequence. There is no inevitability in these statements. They are not formulated, for example, as “If the court comes to the conclusion that a document is falsified, it removes it from evidence under any circumstances of any case" Therefore, the denying mode is applicable here only with the proviso that “any circumstances of any case” are implied. In fact, the document may not be removed from evidence exactly due to the fact that the court came to the conclusion that it was forged (for example, if the fact of forgery itself is considered in court).

This means that, contrary to formal laws, it is impossible to obtain a reliable conclusion based only on the information contained in these judgments.

Disjunctive deductive reasoning (disjunctive syllogisms) consist of premises that are disjunctive (disjunctive) judgments.

Every such S is A, S is B... is an alternative. Example:

Separation-categorical A syllogism, along with one dividing premise, contains one categorical one. His affirmative-negative mode

Equivalent denying-affirming.

A prerequisite for a divisive-categorical syllogism is to take into account all possible alternatives, otherwise the conclusion will not be reliable. In the case of two mutually exclusive alternatives, the divisive-categorical syllogism, based on law of exclusion of third, allows you to choose one of them by refuting the other and, conversely, refuting it by choosing another, which is clearly represented by the diagram:

If one is accepted, the other is refuted

One accepted

Other refuted

If one is refuted, another is accepted

One thing is refuted

Other accepted

Which is easy to use by inserting true variables into it. For example:

Conditionally disjunctive inferences (conditionally separative syllogisms) have a more complex form.

A simple design dilemma in the first (conditional) premise states that, of the two various reasons the same consequence follows, and in the second (separation) it asserts that one (or both) of the grounds is true, thereby predetermining the conclusion.

A simple destructive dilemma in the first premise, it states that two different consequences follow from one basis, and in the second, it indicates the unacceptability of one or both consequences, thereby denying the acceptability of the basis (in the scheme, the dividing premise, for convenience, is changed according to the equivalence rule ab ≡ ).

An example of a simple destructive dilemma can be conveniently demonstrated using the reasoning of K. F. Ushinsky:

A simpler example:

A difficult design dilemma has two different consequences of two different reasons, one of which is approved.

The outcome is not clear - a difficult dilemma only indicates the presence of a tough alternative. The choice depends on the person. In this case, one reason or consequence may turn out to be the antipode of another, as in the following example:

Complex destructive dilemma has two grounds with different consequences, one of which is denied without affirming the other.

No answer

Polylemmas, with complete analogy to dilemmas, have larger number alternatives.

A simple constructive polylemma: If everyone possible reasons the same consequence, it is predetermined(in the example, the first premise is simplified according to the equivalence rule):

If a person is sick or suspects that he is sick, or believes that he may be sick, then he should consult a doctor.

After studying Chapter 6, the bachelor should:

know

Basics of logical analysis of deductive reasoning;

be able to

find deductive reasoning contained in the text, determine its types;
draw rational conclusions from available information in accordance with the rules of deductive reasoning;
logically correctly build deductive reasoning and find errors in the reasoning of other people;

own

Deductive reasoning skills.

The concept of deductive reasoning. Types of Deductive Reasoning

Deductive reasoning - This is an argument in which there is a relation of logical consequence between the premises and the conclusion.

An example of deductive reasoning could be the following text: “A crime can be committed intentionally or through negligence. This crime was committed intentionally. Therefore, it was not committed but by negligence.”

As a rule, the premises of deductive reasoning contain general knowledge, and the conclusion contains particular knowledge.

In all cases when we need to consider a phenomenon on the basis of already known knowledge, a general rule, and draw the necessary conclusion regarding this phenomenon, we reason on the basis of deduction. Thus, deductive reasoning makes it possible to obtain new truths from the knowledge that we already have on the basis of pure reasoning, without resorting to experimental data. Deduction provides a complete guarantee of success in substantiating the truth of a conclusion if the starting points and premises are true statements. It is no coincidence that deductive reasoning is also called necessary or forced reasoning.

Highlight different kinds deductive reasoning. Among them:

- straight - reasoning in which the conclusion follows directly from the premises;
- indirect - reasoning in which the conclusion from the premises follows indirectly with the help of additional reasoning.

Distinguish a large number of patterns of direct and indirect deductive reasoning. For example, there are deductive reasoning schemes based on the structure of complex statements, as well as reasoning schemes based on the structure of simple statements.

However, among the whole variety of such schemes, the most typical ones can be identified, which people prefer in practice. These are the ones that will be the focus of this section.

We will analyze four types direct deductive reasoning: