(1) Earthenware is glazed. P a M
(2) This cup is not glazed. S e M
(3) This cup is not earthenware. S e P

Lines (1) and (2) represent premises, (3) – conclusion. The first premise notes the connection between the concept of “earthenware” and the concept of “glazed”, the second - a specific (single) cup with the same “glazed”. Thus, "glazed" acts as a middle term. From knowing the relationship of the other two terms to it, one can draw a conclusion about how they relate to each other: this cup is not earthenware. The subject of the conclusion (for us this is “this cup”) is usually denoted by the letter S. It is called the minor term and, in accordance with this, the premise in which it is contained is called the minor; it is always placed in second place (in the second line). The predicate of the conclusion (in our case it is “earthenware”) is denoted by the Latin letter P and is called a major term; hence the parcel where it is contained receives the name “large”; it is written down in the first line. The designation for the middle term is the Latin M. This term: as already said, is present in both premises. The premise (initial proposition) in which the subject of the inference (minor term) is found is called the minor premise, and the initial proposition in which the predicate of the inference (major term) is found is called the major premise. It is clear that the middle term in the premises acts as a link between the subject and the predicate of the conclusion, between these extreme terms of the conclusion.
Notice the abbreviation placed next to each proposition in the syllogism. The minor premise and conclusion are designated there as general negative judgments S e M and S e P. By S we mean “this cup” - a singular concept. And since individual concepts, let us recall, always involve the entire volume (for they simply do not have parts), then judgments with them in the place of the subject are always general and never private. In the theory of syllogism and the practice of its use, this is of fundamental importance.
The structure of a simple categorical syllogism is made up of three and only three terms: lesser, middle and greater. The premises in this syllogism can be the four types of simple categorical judgments known to us: general affirmative, general negative, particular affirmative and particular negative. Combinations of these judgments, which can be premises of inference, are subject to certain requirements of logic, acting as the laws of a given structured organization, the laws of a given form of thought, i.e. laws of simple categorical syllogism. These requirements form two groups of rules for a given inference: the rules of premises and the rules of terms.
Rules of premises: from two negative premises (i.e. from two initial simple categorical negative judgments), the conclusion does not necessarily follow; the conclusion also does not necessarily follow from two particular premises; if one of the premises is a negative judgment, then the conclusion will necessarily be negative; if one of the premises is a private judgment, then the conclusion will necessarily be private. It is clear that if among the premises one is partial and the other is negative, or if one of the premises is a partial negative judgment, then the conclusion will necessarily be partial negative; It is also clear that a negative conclusion does not follow from two positive premises (the first four rules of premises are decisive, the rest are derivative).
Rules of terms: in a simple categorical syllogism there must be three and only three terms: lesser, middle, greater; the middle term must be distributed (taken in its entirety, or in its entirety must be excluded from consideration), in at least one of the premises; a term not distributed in the premise cannot be distributed in the conclusion.
A syllogism is an inference about the relationship of two extreme terms based on their relationship to a third term called the middle. Depending on the position of the middle term in the premises (whether it is a subject or a predicate in the major and minor premises), four figures of the syllogism are distinguished. Graphically and using already accepted symbols, the figures are depicted in Fig. 1.
Each figure, in turn, contains several varieties of syllogism, called modes. A mode is a type (variety, modification) of a conclusion, determined by the premises included in this conclusion. In total, from the point of view of all possible combinations of premises and conclusions, there are 64 modes in each figure. In four figures 4? 64 = 256 modes. Syllogisms, like all deductive inferences, are divided into correct and incorrect.

The task of the logical theory of syllogism is to systematize correct syllogisms and indicate their distinctive features. Of all the possible modes of a syllogism, only 24 modes are correct, six in each figure. Of the 24 correct modes of a syllogism, 5 are weakened: the conclusions in them are particular affirmative or particular negative statements, although in the case of other modes these same premises give generally affirmative or generally negative conclusions. If we discard the weakened modes, there remain 19 correct modes of the syllogism. Their symbolic representation is shown in Table 1 of the modes of syllogism.

Modes of syllogism
Table 1.

The first figure of a syllogism is formed when the middle term in the major premise stands in the place of the subject, and in the lesser one - in the place of the predicate. In the list of modes they are collected in the first column on the left. The M symbol in all these modes is located, as it were, diagonally. Aristotle called this figure perfect. It is the most visual and easy to understand. This is explained by the fact that it expresses the simplest volumetric relationships between concepts#x2011;terms.
The small term is entirely contained in the middle term, the middle one is entirely included or not entirely included in the large term. In addition, only the first figure allows general affirmative conclusions; this means that it has the highest evidentiary power in deducing general laws by deduction. This figure has four modes in total, as can be seen from the table. We will present here only two of them by way of illustration.

All people (M) are mortal (P). M a P
Socrates (S) is a man (M). S a M
Socrates (S) is mortal (P). S a P

The criminal (M) is not law-abiding (P). M e P
Fraudster (S) – criminal (M). S a M
The swindler (S) is not law-abiding (P). S e P

The second figure of the syllogism is obtained when the middle term in both premises takes the place of the predicate. The example we first gave with earthenware represents exactly the second mode of this figure (second column, second line in the list of modes). This figure is characterized by the fact that one of the premises and the conclusion are always negative. It is therefore most often used in refutations or proofs by contradiction. The second figure gives four regular modes.
The third figure of the syllogism includes the middle term in place of the subject in both premises.

All goods (M) are exchanged for money (P). M a P
Some products (M) are products (S). M i S
Some items (S) are exchanged for money (P). S i P

This figure gives only partial conclusions. But one should not conclude from this that it is unsuitable for science. The fact is that the division into general and specific is to some extent relative. Let's say there is a general law of conservation and transformation of energy. It applies to all forms of movement. Consequently, it can be extended with the help of a third figure to some of their types. But in relation to these particular types of motion - thermal, electrical and others - the resulting laws are general, not particular. Therefore, this figure is used in scientific knowledge no less than others. It includes the most modes - six.
The fourth figure of a syllogism is formed when the middle term in the major premise is in the place of the predicate, and in the minor premise in the place of the subject.

No bird (P) – no mammal (M). P e M
All mammals (M) are vertebrates (S). M a S
Some vertebrates (S) are not birds (P). S o P

This figure of syllogism appeared after Aristotle. Its modes were studied by the disciples of the great thinker Theophrastus and Eudemus. And she was introduced into logic as an independent figure by the physician, scientist, and researcher of logic C. Galen (130–200). Sometimes this figure is considered dependent, artificial. There is a certain amount of truth in this. Let's say that for each of the other three figures, special rules can be formulated. We have already given them: volume ratios, the presence of a negative premise, etc. The fourth figure does not have such rules. Nevertheless, its five modes should not be overlooked, if only for the sake of completeness of the classification.
The basis of syllogistic inferences is one, fairly self-evident proposition about the relationship between parts and the whole. It is therefore called the axiom of the syllogism. It is formulated in two versions, each of which has its own strengths and weak sides. The most recognized formulation is:
Everything that is affirmed or denied regarding all objects of a given class is affirmed or denied regarding each object of a given class.
Another option: The sign of a sign is a sign of the thing itself.
Both formulations repeat each other in some ways, but there are also discrepancies between them. Most experts consider the first of them preferable, but there are also supporters of the second.
The most immediate applicability of the syllogism axiom is noticeable in the first figure with its simple three-dimensional relationships between concepts and terms. The remaining figures are reducible to the first. Basically, for this it is enough to subject the premises and conclusions of the second, third and fourth figures to the operations of transformation and inversion, as well as to rearrange the premises. Only in two cases is it necessary to resort to more complex reasoning. The proposition, called the axiom of syllogism, unites, in the theoretical sense of the word, the entire set of syllogistic conclusions into a single, harmonious system.
In the Middle Ages, all modes of simple categorical syllogism were given Latin names: Barbara, Cesare, Darii and others. For example, here are the traditionally accepted names of the correct modes of the first two figures:
1#x2011;I figure: Barbara, Celarent, Darii, Ferio, Barbari, Celaront;
2nd figure: Cesare, Camestres, Festino, Baroco, Cesaro, Camestros.

Each of these names contains three vowels. They indicate which categorical statements are used in the mode as its premises and conclusion. Thus, Barbara means a syllogism in which all three propositions are generally affirmative. This is the first figure, the first mode. The name Celarent means that in this mode of the first figure the major premise is a general negative statement (SeP), the minor is a general affirmative (SaP) and the conclusion is a general negative statement (SeP). Nowadays such names are rarely used.
When performing logical operations using syllogism schemes, you need to know its rules. We will present only the rules common to all figures (along with them, as already noted, there are also rules for each of the first three figures separately).
1. A categorical syllogism must have three and only three terms. Often, due to the ambiguity of words, actually four terms are mistakenly taken for three terms.
2. The middle term must be distributed in at least one of the premises.
3. A term cannot be distributed in the conclusion if it is not distributed in the premises.
4. A conclusion cannot be drawn from two negative premises.
5. If one premise is a negative judgment, then the conclusion must be negative.
6. A conclusion cannot be derived from two particular premises.
7. If one of the premises is a private judgment, then the conclusion must be private.
It is useful to know the most typical violations of the rules of syllogism. One of them is a violation of the first rule and is called the error of quadrupling terms, that is, instead of three terms, four are actually taken. The reason for this is the polysemy of words. When one word in one premise has one meaning, and in another or in the conclusion - another, then instead of three terms there are four. Here's what it might look like:

Black (M) is not bitter (P). M e P
Pepper (S) – black (M). S a M
Pepper (S) is not bitter (P). S e P

The word "black" in the first premise means blackness (which is really not a kind of taste sensation), and in the second - a black object. The conclusion was ridiculous. Although in the table of syllogisms such a mode is present in the first figure. There are errors associated with violation of the rules for the distribution of terms (rules 2 and 3).

The stolen (P) items were buried in the garden (M). P a M
The things seized from the criminal (S) were buried in the garden (M). S a M
The items seized from the criminal were stolen. S a P

Rule 2 is violated, since the middle term - the predicate of two general affirmative premises - is not distributed in any of them. This means that he is not known to us in full, either as having the property or as not having it. Therefore, in fact, the conclusion does not follow from these premises (there is no such mode in the table of syllogisms, just as there are no other modes constructed in violation of the rules of the syllogism).

Every factory (M) must pay taxes (P). M a P
This enterprise (S) is not a factory (M). S e M
This enterprise (S) does not have to pay taxes (P). S e P

The major term was not distributed in the premise, but turned out to be distributed in the conclusion (rule 3 was violated). Therefore, the conclusion does not follow from the premises at all.
An example of an error caused by a violation of Rule 4 is the following syllogism: No dishonest man (M) can be a judge (P). M e P Lawyer Petrov (S) is not a dishonest person (M). S e M Lawyer Petrov (S) can be a judge (P). S e P
In fact, such a conclusion does not follow from these premises, since they are both negative in quality.
Finally, an example of a violation of the rule regarding the quantitative characteristics of premises (rule 6) could be the following syllogism:

Some students (P) are students (M). P i M
Some students (M) are minors (S). M i S
Some minors (S) are students (P). S i P

Although the conclusion is obviously a true proposition, it cannot be justified by such premises. It does not flow from them.
Other rules may also be broken. A special role is played by the error called “imaginary generality of the major premise.” It arises when collective or predominant characteristics are taken as generally affirmative or generally negative judgments. For example, they may say: “All people are responsible for their actions, therefore, such a person must also be responsible for his actions.” In most cases, people are truly responsible for their own affairs. But still not always. Actions committed under duress do not entail liability in a number of cases. Therefore, accepting the corresponding statement as generally affirmative is not entirely correct.

Categorical syllogism(or simply: syllogism) is a deductive inference in which a new categorical statement is derived from two categorical statements.

The logical theory of this kind of inference is called syllogistics. It was created by Aristotle and for a long time served as a model of logical theory in general.

In syllogistics, the expressions “All ... are ...”, “Some ... are ...”, “All ... are not ...” and “Some ... are not ...” are considered as logical constants, i.e. taken as a whole. These are not statements, but certain logical forms, from which statements are obtained by substituting some names instead of dots. The substitute names are called in terms of a syllogism.

The following traditional restriction is essential: the terms of the syllogism must not be empty or negative.

An example of a syllogism would be:

All liquids are elastic.

Water is a liquid.

Water is elastic.

Every syllogism must have three terms: lesser, greater and middle.

Lesser term the subject of the conclusion is called (in the example, this term is the term “water”).

Big term is called the predicate of conclusion (“elastic”). A term that is present in the premises but not in the conclusion is called the middle (“liquid”). The minor term is usually denoted by the letter S, larger - letter R and middle - letter M. A premise containing a larger term is called bigger. The premise with the smaller term is called less. The larger message is written first, the smaller one - second. The logical form of the above syllogism is:

All M There is R.

All S There is M.

All S There is R.

Depending on the position of the middle term in the premises (whether it is a subject or a predicate in the major and minor premises), they differ four figures syllogism. Schematically, the figures are depicted as follows:

A syllogism is constructed according to the diagram of the first figure:

All birds (M) have wings (R).

All ostriches (S)- birds (M).

All ostriches have wings.

A syllogism is constructed according to the diagram of the second figure:

All fish (P) breathe through gills (M).

Whales (S) do not breathe with gills (M).

All whales are not fish.

A syllogism is constructed according to the diagram of the third figure:

All bamboos (M) bloom once in a lifetime (R).

All bamboos (M)- perennial plants (S).

Some perennial plants bloom once in their lifetime.

A syllogism is constructed according to the diagram of the fourth figure:

All fish (R) swim (M).

All floating (M) live in water (S).

Some living in water are fish.

The premises and conclusions of syllogisms can be categorical judgments of four types: SaP, SiP, SeP And SoP.

Modes of syllogism Varieties of figures are called, differing in the nature of the premises and conclusion.

In total, from the point of view of all possible combinations of premises and conclusions, there are 64 modes in each figure. There are 4 x 64 = 256 modes in four figures.

Syllogisms, like all deductive inferences, are divided into correct And incorrect. The task of the logical theory of syllogism is to systematize correct syllogisms and indicate their distinctive features.

Of all the possible modes of a syllogism, only 24 modes are correct, six in each figure. Here are the traditionally accepted names of the correct modes of the first two figures:

1st figure: Barbara, Celarent, Darii, Ferio, Barbari, Celaront;

2nd figure: Cesare, Camestres, Festino, Baroco, Cesaro, Camestros.

Each of these names contains three vowels. They indicate which categorical statements are used in the mode as its premises and conclusion. Yes, the name Celarent means that in this mode of the first figure the greater premise is a generally negative statement (SeP), less - universally affirmative (SaP) and in conclusion - a generally negative statement (SeP).

Of the 24 correct modes of a syllogism, 5 are weakened: the conclusions in them are particular affirmative or particular negative statements, although in the case of other modes these same premises give generally affirmative or generally negative conclusions (cf. modes Caesar And Cesaro second figure). If we discard the weakened modes, there remain 19 correct modes of the syllogism.

To assess the correctness of a syllogism, Euler circles can be used to illustrate the relationships between the volumes of names.

Let's take, for example, a syllogism:

All metals (M) forging (R).

Iron (S)- metal (M).

Iron (S) malleable (P).

The relationships between the three terms of this syllogism (mode Barbara) are represented by three concentric circles. This scheme is interpreted as follows: if all M(metals) are included in the volume R(malleable bodies), then with necessity S(iron) will enter the volume R(malleable bodies), which is stated in the conclusion “Iron forged”.

Another example of a syllogism:

All fish (R) have no feathers (M).

All birds (S) there are feathers (M).

Not a single bird (S) is not a fish (R).

The relationship between the terms of a given syllogism (mode Caesar) are presented in the figure. It is interpreted as follows: if everything S(birds) are included in the volume M(having feathers), and M has nothing to do with R(fish), then S(birds) have nothing to do with R(fish), which is stated in the conclusion.

An example of an incorrect syllogism:

All tigers (M)- mammals (R).

All tigers (M)- predators (S).

All predators (S) are mammals (P).

The relationships between the terms of a given syllogism can be represented in two ways, as shown in the figure. In both the first and second cases, everything M(tigers) are included in the volume R(mammals) and all M also included in the scope S(predators). This corresponds to the information contained in the two premises of the syllogism. But the relationship between the volumes R And S can be twofold. Covering M, volume S can be completely included in the volume R or volume S can only intersect with volume R. In the first case, one could make the general conclusion “All predators are mammals,” but in the second case, only the particular conclusion “Some predators are mammals” is legitimate. The messages do not contain information allowing you to choose between these two options. This means that we have no right to make a general conclusion. The syllogism is not correct.

In a syllogism, as in any deductive conclusion, the conclusion cannot contain information that is not present in the premises. The conclusion only expands the information of the premises, but cannot introduce new information, missing from them.

In ordinary reasoning, there are often syllogisms in which one of the premises or the conclusion is not clearly expressed. Such syllogisms are called enthymemes. Examples of enthymemes: “Generosity deserves praise, like any virtue,” “He is a scientist, so curiosity is not alien to him,” “Kerosene is a liquid, so it transmits pressure in all directions evenly,” etc. In the first case, the minor premise “Generosity is a virtue” is omitted; in the second case, the major premise “Every scientist is curious is not alien”; in the third case, the major premise “Every liquid transmits pressure evenly in all directions” is omitted.

To assess the correctness of the reasoning in the enthymeme, it should be restored into a complete syllogism.

Simple categorical syllogism

Indirect inferences are those inferences in which the conclusion follows from two or more judgments that are logically related to each other. There are several types of indirect inferences: a) categorical syllogism; b) conditional inferences; c) divisive inferences.

Categorical syllogism (syllogism - from the Greek word "syllogismos" - counting) is a type of deductive inference in which from two true categorical judgments connected by one term, a third judgment is obtained - a conclusion.

For example:

All students study hard foreign language

Ivanov - student

Ivanov is diligently studying a foreign language

In contrast to the terms of judgment - S and P - the concepts included in the syllogism are called syllogism terms. There are lesser, greater and middle terms.

The minor term of a syllogism is the concept that is the subject in conclusion. The major term of a syllogism is a concept that in conclusion is a predicate. The lesser and greater terms are called extreme. They are designated respectively by the Latin letters S (minor term) and P (major term). Each of the extreme terms is included not only in the conclusion, but also in one of the premises. A premise that contains a minor term is called a minor premise; a premise that contains a larger term is called a major premise.

The middle term of a syllogism is a concept that is included in both premises and is absent in the conclusion. The middle term is denoted by the Latin letter M (from the Latin medius - middle).

Putting syllogism terms in place of the terms of judgment in our example, we get:

All students (M) diligently study a foreign language (R)

Ivanov(S) - student(M)

Ivanov (S) diligently studies a foreign language (R)

Varieties of syllogism forms, distinguished by the position of the middle term in the premises, are called syllogism figures, each of which has its own special rules. There are four figures.

The first figure is a type of syllogism in which the middle term takes the place of the subject in the major premise (M - P) and the place of the predicate in the minor (S - M), schematically expressed as follows:

All students (M) diligently study the history of the Fatherland (R)

Ivanov (S) - student (M)

Ivanov (S) diligently studies the history of the Fatherland (R)

Rules for the first figure: 1. The minor premise must be affirmative; 2. The large parcel must be general (A, E).

The second figure is a type of syllogism in which the middle term takes the place of a predicate in both premises (P - M; S - M), schematically expressed:

No book (P) is a periodical (M)

Magazine (S) - periodical(M)

A magazine (S) is not a book (P)

Rules of the second figure: 1. One of the premises must be negative (E, 0),2. The major premise must be general (A, E).

The third figure is a type of syllogism in which the middle term takes the place of the subject in both premises (M - P; M - S). His diagram:

Some wars (M) are just (R)

War (M) is violence (S)

Some violence(S) is fair(R)

Rules for the third figure: 1. The minor premise must be affirmative (A, I),2. The conclusion must be private (I, O).

The fourth figure is a type of syllogism in which the middle term takes the place of the predicate in the larger and the place of the subject in the minor premise (P - M, - M - S), schematically expressed:

All officers (P) are military personnel (M)

Not a single soldier (M) is a worker (S)

No worker(S) is an officer(R)

Rules of the fourth figure: 1. If the major premise is affirmative (A, I), then the minor premise must be general (A, E), 2. If one of the premises is negative (E, O), then the major premise must be common (A, E)

Rules of terms (RT)

PT - 1. Each syllogism should have only three terms. If this rule is violated, a “quadrupling of terms” error occurs, consisting in the fact that one of the terms is used in two meanings.

For example:

Life is a fight

Karate - wrestling

Life is karate

PT - 2. 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 in the conclusion remains uncertain.

For example:

Some plants(M)poisonous(P)

Porcini mushrooms (S) - plants (M)

Porcini mushrooms (S) - poisonous (P)

PT - Z. A term undistributed in the premises cannot be distributed in the conclusion. If this rule is violated, an "illegal term extension" error occurs.

For example:

All teachers (M) are well-mannered (R)

He (S) is not a teacher (M)

He (S) is not brought up (R)

Parcel Rules (PP):

PP - 1. If one premise is private, then the conclusion will be private.

For example:

All deputies are elected by the people

Some actors are MPs

From these premises no general conclusion is possible. It cannot be argued that all actors are chosen by the people, since we are talking only about part of the volume of a smaller term. Schematically it looks like this:

PP - 2. It is impossible to draw a conclusion from two particular premises. In this case, it is impossible to establish comprehensive relationships between the terms of the syllogism, so a definite conclusion will not be obtained. For example, from the premises “Some members of the Academy of Sciences are philosophers” and “Some sociologists are members of the Academy of Sciences”, no definite conclusion follows. The scope of the subject (“some sociologists”) may overlap to some extent with the scope of the predicate (“philosophers”), but may also be outside it.

PP - 3. A conclusion cannot be drawn from two negative premises. In this case, all terms exclude each other, eliminating any dimensional relationship between them. From the premises: “Not a single planet shines with its own light” and “An artificial satellite of the Earth is not a planet” - no conclusion follows.

PP - 4. If one of the premises is a negative judgment, then the conclusion must be negative. For example: “Every truly popular movement is progressive. Nationalism is not a progressive movement. Therefore, nationalism is not a truly popular movement.”

These are general rules, which must be taken into account when drawing up a categorical syllogism. Without observing them, it is impossible to draw a correct conclusion. By violating these rules, a person violates the axiom of the syllogism. The rules of inference are of great cognitive importance because they adequately reflect the relationships and properties of objective reality.

It is important to keep in mind that the premises of a syllogism can be judgments that differ in quality and quantity: general affirmative (A), general negative (E), particular affirmative (I) and particular negative (O). In this regard, modes of simple categorical syllogism are distinguished.

In four figures the number of combinations is 64. However, there are only 19 correct modes.

1st figure: AAA, EAE, AII, EIO, 2nd figure: EAE, AEE, EIO, AOO, 3rd figure: AAI, IAI, AII, EAO, OAO, EIO 4th figure: AAI, AEE , IAI, EAO, EIO.

In general, the analysis of simple categorical syllogisms in order to clarify the question of the nature of the conclusion involves a consistent determination of the following points:

• lesser, greater and middle terms;
• smaller and larger parcels;
• figures;
• mode;
• distribution of terms in premises and conclusion;
• the nature of the conclusion (necessary or probabilistic).

Let's take an example: "Laws are subject to observance. Instructions are not laws. Therefore, instructions are not subject to observance." The analysis of a syllogism should begin with the conclusion, since it contains extreme terms - greater and lesser. In our example, the concept of “instruction” is a smaller term as the subject of the conclusion. The concept of "compliance" or " legal act"to be observed" as a result of transforming the verbal form of the predicate into a nominal one - a larger term, since it is a predicate of the conclusion. The concept of "law", which is included in both premises, but is absent in the conclusion, is a middle term.

The premise “laws are to be observed” is major because it contains the larger term “legal act to be observed,” and the premise “Instructions are not laws,” which contains the lesser term “instructions,” is smaller. Since the middle term "law" is the subject of the major premise and the predicate of the minor, it is a syllogism of the first figure.

The major premise is a generally affirmative proposition (A), the minor one is a generally negative proposition (E), and the conclusion is also generally negative (E). Thus, here we have mode AEE. The middle term in the major premise is distributed as the subject of the general judgment ( symbol M+), and the larger term is not distributed as a predicate of an affirmative judgment (symbol P-). In the minor premise, the minor term is distributed as the subject of the general judgment (S +) and the middle term is distributed as the predicate of the negative judgment (M +). In conclusion, both extreme terms are distributed on the same basis as in the minor premise (S +) and (P +). Let us record the result of our analysis:

And Laws (M+) are subject to compliance (R-)

E Instruction (S +) is not a law (M +)

E Instructions (S +) are not subject to compliance (P +)

The nature of the conclusion is determined by the answer to the question whether the rules of the syllogism (rules of figure and general rules) are violated in this example: if violated, then the conclusion is probabilistic, if not, then reliable. Since our example is built on the first figure, it is easy to discover that one of its rules is not observed here - the minor premise must be affirmative, here it is negative. This means that the conclusion is probabilistic in nature. But since the rules of figures are consequences of general rules, it is also necessary to determine which general rules are violated. In this example, the PT-3 regarding the larger term is violated: the larger term in the premise is not distributed as a predicate of an affirmative judgment, but in the conclusion it is distributed as a predicate of a negative one. Therefore, the example contains the error “illegal extension of a larger term.”

Conditional and disjunctive inferences.

Inferences are built not only from simple, but also from complex judgments. Inferences are widely used, the premises of which are conditional and disjunctive judgments, appearing in various combinations with each other or with categorical judgments. The peculiarity of these inferences is that the derivation of a conclusion from the premises is determined not by the relations between terms, as in a categorical syllogism, but by the nature of the logical connection between judgments. Therefore, when analyzing premises, their subject-predicate structure is not taken into account. Let's consider the conclusions from complex judgments.

A conditional inference (conditional syllogism) is a type of mediated deductive inference in which at least one of the premises is a conditional proposition. There are purely conditional and conditionally categorical inferences.

A purely conditional inference is such an indirect inference in which both premises and the conclusion are conditional propositions. Its logical structure is:

If a, then b

If in, then with

If a, then c

For example

If a student does not have a developed sense of responsibility, then he does not develop the need to qualitatively master the legal profession.

If a student does not develop the need to master the legal profession in a quality manner, then he will be a poor specialist.

If the student does not have a developed sense of responsibility, then he will be a poor specialist.

In the example given, both premises are conditional propositions, and the basis of the second premise is the consequence of the first, from which, in turn, another consequence follows. The common part of the two premises allows us to connect the basis of the first and the consequence of the second. Therefore, the conclusion is also expressed in the form of a conditional proposition. The conclusion in a purely conditional inference is based on the rule: the consequence of the consequence is the basis of the reason.

If a, then b

This type inference has two modes - affirmative and negating. Each of them occurs in two forms: regular and irregular. In the correct forms, conclusions are reliable, in incorrect forms they are probabilistic.

The correct form of the affirmative mode is a type of conditional categorical inference, in which the course of inference is directed from the statement of the basis of the conditional premise to the statement of the consequence of the conditional premise.

For example:

The word "capital" comes at the beginning of the sentence (a)

The word "capital" in this sentence should be written with capital letters(b)

An incorrect form of the affirmative mode is a type of conditionally categorical inference, in which the course of inference is directed from the statement of the consequence to the statement of the reason.

For example:

If a word appears at the beginning of a sentence (a), then it must be written with a capital letter (b)

The word "Moscow" is written with a capital letter (b)

The word "Moscow" appears at the beginning of sentence (a)

The correct form of the negating mode is a type of conditionally categorical inference, in which the course of inference is directed from the negation of the consequence to the negation of the basis.

For example:

If a word appears at the beginning of a sentence (a), then it must be written with a capital letter (b)

The word "capital" in the sentence is not capitalized (- b)

The word "capital" does not appear at the beginning of the sentence(s)

The incorrect form of the negating mode is a type of conditionally categorical inference in which the course of inference is directed from the negation of the basis to the negation of the consequence.

For example:

If a word is at the beginning of a sentence (a), then it must be written in capital letters (b)

The word "Moscow" does not appear at the beginning of the sentence(s)

The word "Moscow" does not need to be capitalized (- b)

A disjunctive conclusion is a conclusion in which one or more premises are disjunctive judgments. There are divisive-categorical and conditionally divisive inferences

A disjunctive-categorical inference is a conclusion in which one of the premises is divisive, and the other premise and conclusion are categorical judgments. Separative-categorical inference has two modes: affirmative-negative and negating-affirmative.

The affirmative-negative mode is a type of separative-categorical inference, in which, by affirming one of the members of the separative judgment, all the others are negated. Its logical structure is:

For example:

The judgment can be either affirmative (a) or negative (b)

This proposition is affirmative (a)

This judgment is not negative (- b)

In a conclusion according to this mode, the following rule must be observed: the dividing premise must constitute a strict disjunction.

The negating-affirming mode is a type of dividing-categorical inference, in which by negating all members of the dividing judgment, except one, the remaining member is affirmed. Its logical structure is:

For example:

A judgment can be either affirmative (a) or negative (b)

This judgment is not affirmative

This judgment is negative (b)

In a conclusion according to this mode, the following rule must be observed: the major premise must list all possible alternatives, in other words, the major premise must be a complete (closed) disjunctive statement.

Conditionally disjunctive or lemmatic (from the Latin lemme - assumption) is a conclusion in which one premise consists of two or more conditional propositions, and the other is a disjunctive proposition. Based on the number of consequences of the conditional premise (alternatives), dilemmas, trilemmas and polylemmas are distinguished.

A dilemma is a conditional disjunctive conclusion with two alternatives. In the practice of reasoning, there are two types of dilemmas - constructive and destructive.

The conditional premise of a constructive dilemma establishes the possibility of two conditions and two consequences arising from them. The dividing premise limits the choice to only these two conditions, and the conclusion asserts the possibility of only one consequence.

For example:

If political theories are progressive (a), then they contribute to the development of society (b)

If political theories are reactionary (c), then they hinder the development of society (e)

But political theories can be either progressive (a) or reactionary (c)

Political theories either promote the development of society (b) or hinder it (c)

The conditional premise of a destructive dilemma states that two consequences can follow from two reasons, the dividing premise denies one of the possible consequences, and the conclusion denies one of the possible reasons.

For example:

If a philosopher recognizes the primacy of matter in relation to consciousness (a), then he is a materialist (b)

If a philosopher recognizes the primacy of consciousness in relation to matter (c), then he is an idealist (c) But the philosopher is either not a materialist (- b), or is not an idealist (- c)

The philosopher does not recognize either the primacy of matter in relation to consciousness
(- a), or the primacy of consciousness in relation to matter (- c).

Example of a syllogism:

Every man is mortal (major premise) Socrates is a man (minor premise) ------------ Socrates is mortal (conclusion)

Structure of a simple categorical syllogism

The syllogism includes exactly three term:

• S - minor term: subject of the conclusion (also included in the minor premise);
• P - major term: predicate of the conclusion (also included in the major premise);
• M is the middle term: included in both premises, but not included in the conclusion.

Subject S(subject) - that about which we express (divided into two types):

1. Definite: Singular, Particular, Plural
   • Single [judgments] - in which the subject is an individual concept. Note: “Newton discovered the law of gravity”
   • Particular judgment - in which the subject of judgment is a concept taken in part of its scope. Note: “Some S are P”
   • Multiple propositions are those in which there are several subject class concepts. Note: “insects, spiders, crayfish are arthropods”
2. Uncertain. Note: “it’s getting light”, “it hurts”, etc.

Predicate P(predicate) - what we express (2 types of judgments):

• Narrative is a judgment regarding events, states, processes or activities that are passing quickly. Note: “A rose is blooming in the garden.”
• Descriptive - when some property is attributed to one or many objects. The subject is always a certain thing. Note: “Fire is hot,” “snow is white.”

Relationship between subject and predicate:

1. Identity judgments - the concepts of subject and predicate have the same scope. Note: “every equilateral triangle is an equiangular triangle”
2. Judgments of subordination - a concept with a less wide scope is subordinate to a concept with a wider scope. Note: “A dog is a pet”
3. Judgments of relation - namely space, time, relationship. Note: “The house is on the street”

When determining the relationship between the subject and the predicate, a clear formalization of the terms is important, since a stray dog, although not a domestic dog from the point of view of living in a house, still belongs to the class of domestic animals from the point of view of belonging on a socio-biological basis. That is, it should be understood that a “domestic animal” according to the socio-biological classification in some cases may be a “non-domestic animal” from the point of view of its habitat, that is, from a social and everyday point of view.

Classification of simple attributive statements by quality and quantity

Based on quality and quantity, four types of simple attributive statements are distinguished:

Note. For conventional lettering of statements, vowels from Latin words are used affirmo(I affirm, I say yes) and nego(I deny, I say no).

Single statements (those in which the subject is a single term) are equated to general ones.

Distribution of terms in simple attributive statements

The subject is always distributed in a general statement and never distributed in a particular statement.

The predicate is always distributed in negative judgments; in affirmative judgments it is distributed when, in terms of volume P<=S.

In some cases, the subject can act as a predicate.

Rules for a simple categorical syllogism

• The middle term must be distributed in at least one of the premises.
• A term not distributed in the premise should not be distributed in the conclusion.
• The number of negative premises must be equal to the number of negative conclusions.
• Each syllogism must have only three terms.

Figures and modes

Figures of a syllogism are forms of a syllogism that differ in the location of the middle term in the premises:

Each figure corresponds to modes - forms of syllogism that differ in the quantity and quality of premises and conclusion. Modes were studied by medieval schools, and mnemonic names were invented for the correct modes of each figure:

Examples of each type of syllogism.

All animals are mortal. All people are animals. All people are mortal.

Celarent

No reptile has fur. All snakes are reptiles. No snake has fur.

All kittens are playful. Some pets are kittens. Some pets are playful.

No homework is fun. Some reading is homework. Some reading is not fun.

No healthy food makes you fat. All cakes are full. No cake is a healthy food.

Camestres

All horses have bloat. No person has bloating. No man is a horse.

No lazy person passes exams. Some students are taking exams. Some students are not lazy.

All informative things are useful. Some sites are not useful. Some sites are not informative.

All fruits are nutritious. All fruits are delicious. Some delicious foods are nutritious

Some mugs are beautiful. All mugs are useful. Some useful things are beautiful.

All the good boys in this school are red-haired. Some of the studious boys in this school are boarders. All the diligent boarding boys at this school are red-haired.

Felapton

Not a single jug in this cabinet is new. All the jugs in this cabinet are cracked. Some of the cracked items in this closet are not new.

Some cats are tailless. All cats are mammals. Some mammals are tailless.

Not a single tree is edible. Some trees are green. Some green things are not edible.

Bramantip

All the apples in my garden are healthy. All healthy fruits are ripe. Some ripe fruits are apples in my garden.

All bright flowers are fragrant. Not a single fragrant flower is grown indoors. No flower grown indoors is bright.

Some small birds feed on honey. All birds that feed on honey are colored. Some colored birds are small.

No person is perfect. All perfect creatures are mythical. Some mythical creatures are not human.

Fresison

No competent person makes mistakes. Some fallible people work here. Some people working here are incompetent.

According to the rules, shapes can be transformed into other shapes, and all shapes can be transformed into one of the shapes of the first shape.

Story

The doctrine of syllogism was first expounded by Aristotle in his First Analytics. He speaks of only three figures of the categorical syllogism, without mentioning a possible fourth. He examines in particular detail the role of the modality of judgments in the process of inference. Aristotle's successor, the founder of botany, Theophrastus, according to Alexander of Aphrodisius (in his commentary on Aristotle's first Analytics), added five more modes (modi) to the first figure of the syllogism; these five modes were subsequently distinguished by Claudius Galen (who lived in the 2nd century AD) into a special fourth figure. In addition, Theophrastus and his student Eudemus began analyzing conditional and disjunctive syllogisms. They allowed five types of inferences: two of them correspond to the conditional syllogism, and three to the disjunctive one, which they considered as a modification of the conditional syllogism. This ends the development of the doctrine of syllogism in ancient times, except for the addition that the Stoics made in the doctrine of conditional syllogism. According to Sextus Empiricus, the Stoics recognized certain types of conditional and disjunctive syllogism αναπόδεικτοι , that is, not requiring proof, and considered them as prototypes of a syllogism (as, for example, Sigwart looks at a syllogism). The Stoics recognized five types of such syllogisms, coinciding with Theophrastus. Sextus Empiricus gives the following examples for these five species:

1. If it is day, then there is light; but now it is day, therefore there is light.
2. If it is day, then there is light, but there is no light, therefore there is no day.
3. There cannot be day and night (at the same time), but day has come, therefore there is no night.
4. It may be day or night, but now it is day, therefore there is no night.
5. It may be day or night, but there is no night, therefore it is now day.

In Sextus Empiricus and skeptics in general we also encounter criticism of syllogism, but the purpose of criticism is to prove the impossibility of proof in general, including syllogistic proof. Scholastic logic did not add anything significant to the doctrine of syllogisms; it only broke the connection with the theory of knowledge that existed in Aristotle and thereby turned logic into a purely formal teaching. The exemplary manual of logic in the Middle Ages was the work of Marcian Capella, the exemplary commentary was the work of Boethius. Some of Boethius' commentaries deal specifically with the doctrine of syllogisms, for example "Introductio ad categoricos syllogismos", "De syllogismo categorico" and "De syllogismo hypothetico". Boethius's writings have some historical significance; they also contributed to the establishment of logical terminology. But at the same time, it was Boethius who gave logical teachings a purely formal character.

"logical square"

From the era of scholastic philosophy, Thomas Aquinas († 1274) deserves attention in relation to the doctrine of syllogism, especially his detailed analysis of false conclusions (“De fallaciis”). A work on logic, which had some historical significance, belongs to the Byzantine Michael Psellus. He proposed the so-called “logical square”, which clearly expresses the relationship of various types of judgments. He owns the names of various modi (Greek. τρόποι ) figures. These names, Latinized, passed into Western logical literature.

Michael Psellus, following Theophrastus, attributed the five modi of the fourth figure to the first. The naming of species had mnemonic purposes in mind. He also owns the commonly used designation by letters of the quantity and quality of judgments (a, e, i, o). Psellos's logical teachings are formal in nature. The work of Psellus was translated by William of Sherwood and gained currency through the adaptation of Peter of Spain (Pope John XXI). In Peter of Spain, the same desire for mnemotechnical rules is noticeable in his textbook. The Latin names of the types of figures given in formal logics are taken from Peter of Spain. Peter of Spain and Michael Psellus represent the flowering of formal logic in medieval philosophy. Since the Renaissance, criticism of formal logic and syllogistic formalism begins

The first serious critic of Aristotelian logic was Pierre Ramet, who died during the Night of Bartholomew. The second part of his Dialectics talks about syllogism; His teaching on syllogism, however, does not represent significant deviations from Aristotle. Beginning with Bacon and Descartes, philosophy follows new paths and defends methods of research: the unsuitability of the syllogistic method in the sense of a method of research, finding truth, becomes more and more obvious.

Syllogism in modern logic

The syllogism dominated logic until the 19th century and had limited application due in part to its association with the categorical syllogism. A replacement for the syllogism is a simpler and more powerful

The word "syllogism" comes from the Greek syllogysmos, which means "inference." It's obvious that syllogism- this is the derivation of a consequence, a conclusion from certain premises. A syllogism can be simple, complex, abbreviated and complex abbreviated.

A syllogism whose premises are categorical judgments is called, respectively, categorical. There are two premises in the syllogism. They contain three terms of the syllogism, denoted by the letters S, P and M. P is the greater term, S is the lesser, and M is the middle, connecting term. In other words, the term P is wider in scope (although narrower in content) than both M and S. The narrowest term in a syllogism is S. Moreover, the larger term contains the predicate of the judgment, the smaller one – its subject. S and P are related to each other by the middle concept (M).

All boxers are athletes.

This man is a boxer.

This man is an athlete.

The word "boxer" here is the middle term, the first premise is the greater term, the second the lesser. To avoid mistakes, we note that this syllogism refers to a given, specific person, and not all people. Otherwise, of course, the second parcel would be much wider in scope.

In the first case, the major premise must be general, and the minor must be affirmative. The second form of a categorical syllogism gives a negative conclusion, and one of its premises is also negative. The larger concept, as in the first case, must be general. The conclusion of the third form must be partial, the minor premise must be affirmative. The fourth form of categorical syllogisms is the most interesting. It is impossible to derive a generally affirmative conclusion from such conclusions, but there is a natural connection between the premises. So, if one of the premises is negative, the larger one must be general, while the smaller one must be general, if the larger one must be affirmative.

In order to avoid possible mistakes, when constructing categorical syllogisms, one should be guided by the rules of terms and premises. The rules of terms are as follows.

Distribution of the middle term (M). Means that the middle term, the connecting link, must be distributed in at least one of the other two terms - the greater or the lesser. If this rule is violated, the conclusion is false.

Absence of unnecessary syllogism terms. Means that a categorical syllogism must contain only three terms - the terms S, M and P. Each term must be considered in only one meaning.

Distribution in custody. In order to be distributed in the conclusion, the term must also be distributed in the premises of the syllogism.

Parcel rules.

1. Impossibility of withdrawal from private parcels. That is, if both premises are partial propositions, it is impossible to draw a conclusion from them. For example:

Some cars are pickups.

Some mechanisms are machines.

No conclusion can be drawn from these premises.

2. Impossibility of conclusion from negative premises. Negative premises make it impossible to draw a conclusion. For example:

People are not birds.

Dogs are not people.

No withdrawal possible.

3. The next rule states that if one of the premises of a syllogism is private, then its consequence will also be private. For example:

All boxers are athletes.

Some people are boxers.

Some people are athletes.

4. There is another rule that says that if only one of the premises of a syllogism is negative, the conclusion is possible, but it will also be negative. For example:

All vacuum cleaners are household appliances.

This appliance is not household appliances.

This technique is not a vacuum cleaner.