A simple categorical syllogism must have terms. A simple categorical syllogism and examples of its use in judicial practice

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.

From the Greek syllogismos - counting.

New knowledge obtained with the help of a simple categorical syllogism is calculated from the existing judgment.

Composition of the PKS: Consists of two premises and a conclusion.

For example:

All people are mortal.

All logicians are people.

This means that all logicians are mortal.

Above the line there are 2 premises, and then the conclusion.

In turn, premises and conclusion consist of 3 terms. These terms are called “PKS terms”:

S - the minor term - is the subject of the conclusion of the syllogism. In our case, these are “logics”. A premise that contains a minor term is called a minor premise.

P - the larger term - is the predicate of the conclusion of the syllogism. In our case, these are “mortals”. A premise that contains a larger term is a major premise.

In the clear logical form of the PKS, the major premise is written at the top, the minor one below the major one, and the conclusion below the line.

M - middle term is a term that is contained in both premises, but is absent in the conclusion. In our case, these are “people”.

Axiom of syllogism:

Has two interpretations:

1) Attributive: A sign of a sign of a certain thing is a sign of that thing itself; that which contradicts the sign of a thing also contradicts the thing (the sign of a sign is a sign of the thing).

2) Volumetric: Everything that is affirmed (or denied) regarding all objects of a class is affirmed (or denied) regarding each object and any part of the objects of this class (said about everything and none).

The attributive interpretation of our example says that the attribute of people is “mortal.” And the sign “people” of the sign “are mortal” is a sign of the “logic” of things “mortal”.

General rules of the PKS:

There are 7 rules in total, which are divided into 2 groups.

Group I - rules of terms:

1) A syllogism must have only three terms. Error: "Quadruplement of terms." In another way it is called: “substitution of terms.” For example, “All the secretaries are busy with their work. Some birds are secretaries. This means that some birds are busy with their own business” - an example of incorrectness. The term secretary in the first and second premises has different meanings. In one there is a secretary - there is a job. And in the second - a species of birds. You can't do that.

2) The middle term must be distributed in at least one of the premises. Distribution table:

For example, “All liver flukes eat the liver. Some people in the restaurant also eat liver. So some people in the restaurant are liver flukes.” The middle term is “eating the liver.” The smaller term is "people in a restaurant". And the bigger term is “liver flukes.” That is, it turned out that the average term in both cases is minus. It is not right.

3) If the extreme term (greater or lesser) is not distributed in the premise, then it should not be distributed in the conclusion. Error: "illegal term extension." For example, “I am a person (A). You are not me (E). So you are not a person (E).” We find the terms of the syllogism: The middle term is “I”. The lesser term is "You". The larger term is "Man". This syllogism is incorrect.

Group II - parcel rules:

1) There must be at least one general premise (no conclusion can be drawn from two particular ones). That is, one of the premises must be a general proposition.

2) There must be at least one affirmative premise (no conclusion is drawn from two negative premises).

3) If one of the premises of a syllogism is private, then the conclusion is private.

4) If one of the premises is negative, then the conclusion in the syllogism is negative.

Solving PCS problems:

3 types of tasks:

1) Checking the PKS for correctness.

Task:

“Every passionary can change the course of history. Not a single janitor is a passionary. This means that no janitor can change the course of history.”

Define terms and define distribution.

Solution:

Define terms:

S - wiper.

P is the one who can change the course of history.

M - passionary.

We arrange the distribution:

A All M+ are P-

E No S+ is M+

E No S+ is P+

Check for correctness (according to the rules): First, it is not violated. The second is not violated. Third, it is violated. That is, the ACL is incorrect.

Task:

“All public sector students at IU are students of group 111. Some students of group 111 attend consultations. This means that some public sector students at IU attend consultations.”

1) We are looking for the conclusion of the syllogism and the terms: “So, some public sector students at IU attend consultations”

S - public sector student at IU.

P is a student who attends lectures.

M - student of group 111.

2) We draw up a diagram:

And All S+ is M-.

I Some M- are R-.

I Some S- are P-.

3) Check if the rules are violated:

1) Violated. The rest need not be checked.

Task:

“All geese are grey. Goose Grisha is not gray. So Grisha the goose is not a goose.”

1) We are looking for a conclusion and terms: “This means that the goose Grisha is not a goose.”

R - Goose Grisha

M - to be gray.

And All S+ is M-

E All P+ is not M+

E All P+ is not S+

The syllogism is incorrect because the axiom of the syllogism is violated.

2) Deriving a conclusion from the premises.

Task:

“All pineapples taste good. Potatoes are not pineapple. Means…"

Since there is no conclusion, we cannot define the lesser and greater terms. The mistake is that students try to define terms.

Therefore, we must begin solving this problem by searching for the middle term.

1) Middle term: M - pineapple.

2) We conventionally designate the extreme terms from which we obtain the conclusion:

A - things taste good.

B - potatoes.

3) We write the structure of syllogisms:

And All M+ is A-

E All B+ is not M+

O Some S- are not P+

We establish the distribution of terms.

The order of deriving a conclusion from the premises:

1) Define the connective in the conclusion. The connective is determined by the rules and axioms of the premises. The conclusion in our judgment is also negative. If one of the premises is negative, then the conclusion is negative.

2) Determine the type of judgment in the conclusion. The type of judgment in the conclusion is determined by the distribution of extreme terms. Extreme terms A and B. They have distribution - and +. When we draw a conclusion, we must not violate the 3rd rule of premise. Therefore, we cannot take a generally negative judgment as a conclusion, because both terms are distributed there.

3) Demolish the extreme terms of the conclusion. We do it according to the distribution of terms. In O S-, and P+, therefore, we substitute: A-=S-, and B+=P+

We change the terms of judgment to our terms.

We write down the conclusion: “Some things that taste good are not potatoes.”

Task:

“All Zelyuks are Momzyuks. Every snark is a zeluke. Means…".

1) M - Zelyuki.

2) A - momzyuks.

B - snark.

3) Write the structure:

And All M+ is A-.

And All B+ is M-.

And All B+ is A-

4) Conclusion - with “is”.

Type of judgment - E (General Negative).

Conclusion: “Every snark is a momzyuk.”

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

For example:

Realism (M) this is a clear and sober understanding of reality (R).

"The main quality of a leader (5) is realism (M)"

(Marcus Aurelius).

The main quality of a leader (5) is a clear and sober understanding of reality (/").

PCS is an indirect inference that has its own structure. In it, the connection between two concepts (in the conclusion) is established through a third concept present in both premises.

The terms included in the conclusion are called extreme terms. Among the extreme terms there are minor term (he acts as the subject of the conclusion) (5) and greater term (this is a conclusion predicate) - (R). In our example, the smaller term is the concept of “the main quality of a leader,” and the larger term is “a clear and sober understanding of reality.”

A premise that contains a larger term is called larger parcel, and the premise containing the minor term is called smaller parcel. In our example, the large parcel comes first, and then the smaller one.

The order of premises in reasoning is not important, but in standard records In a simple categorical syllogism, the major premise is put as the first, and the minor as the second. Violation of this requirement makes the logical analysis of this type of reasoning difficult. The PKS formula looks like 5 - M - R, those. the subject of the conclusion is related to the predicate of the conclusion through the middle term. It is no coincidence that Aristotle (384-322 BC), who deeply and comprehensively developed the theory of syllogisms, emphasized that in a syllogism “research is carried out for the sake of the middle term.”

• (£) “A person who is not busy (M) will never enjoy complete happiness (P)” (H. Heine).
• (L) A slacker (5) is a person who is not busy (M).
• (E) A slacker (5) will never enjoy complete happiness (R).

The diagram shows: if all items of class 5 are included in the volume M, a class M has no common elements with R, then 5 has nothing in common with R, which is stated in the conclusion.

Let's look at another example:

• (L) “To be able to manage (M) means to be able to choose” (F. Pananti).
• (L) The main thing for a leader (5) is to be able to manage (M).
• (L) The main thing for a leader (5) is to be able to choose (R).

The diagram shows: if all elements of class 5 are included in the volume M, and the whole class M - into the volume of class P, then it is obvious that all elements of class 5 will be included in the volume P. This is stated in the conclusion.

Before us are graphic diagrams of the axiom of the syllogism:

“Everything that is affirmed or denied about a class of objects as a whole is affirmed or denied about a part or individual object of this class.”

The axiom of a syllogism is accepted without proof and is the starting point for substantiating the general rules of a simple categorical syllogism.

General rules of simple categorical syllogism are such that each of them separately is a necessary condition the correctness of the conclusion, and all together they are sufficient condition the correctness of the conclusion. A rule is considered necessary if, in the case where it is not satisfied, the inference is incorrect. Sufficiency is expressed in the fact that the fulfillment of each of the general rules of the syllogism indicates the correctness of the conclusion. In other words, a syllogism is correct if all its rules are satisfied, and incorrect if at least one of them is not fulfilled. The general rules of syllogism include rules of terms and rules of premises.

Let's look at the rules of terms.

A syllogism must have only three terms.

An error that occurs when this rule is violated is called quadrupling of terms. It is caused by the fact that the concept, which should be the connecting link between premises (and this is the role of the middle term), is ambiguous and is used in different meanings. In other words, the formula of a simple categorical syllogism is violated: 5 - M - R. In this example, an attempt is made to connect the subject and the predicate of the conclusion through two “middle” terms: 5 - Ml - M, - R.

For example:

• (A) “Historical figures (M]) are people who have had a significant influence on the development of society (R).
• (A) “Nozdryov (5) was in some respects a historical person (M)” (N.V. Gogol).
• (L) Nozdryov (5) in some respects had a noticeable influence on the development of society (R).

To understand the mistake that led to the absurd conclusion, let us turn to the context of Gogol’s phrase: “Nozdryov was in some respects a historical person. Not a single meeting where he was present was complete without a story. Some story would certainly happen: either they would take him out of the way.” gendarmes in the hall, or are forced to push out their own friends.”

As we can see, the word “history” in the syllogism is ambiguous: in the first case it means “social reality in its development,” and in the second case it means “an incident, an adventure, most often unpleasant” (“getting into history” is said in such situations).

In other words, the law of identity is grossly violated here in the form of substitution of concepts. In fact, in a syllogism there are not three, but four terms - the middle term, which should be the connecting link between the premises, a kind of “bridge” for the transition from the premises to the conclusion, is ambiguous. Having discovered this, we see that there is no semantic connection between the premises. Judge for yourself:

“Historical figures are people who had a significant influence on the development of society. And Nozdryov always found himself in unpleasant situations.”

• - And what next? It’s the same as “There’s an elderberry in the garden, and there’s a guy in Kyiv.” As we see, in the absence of a meaningful connection between the premises, logical reasoning is impossible.
• The middle term must be distributed in at least one of the premises.

If M is not distributed in both premises, the conclusion is impossible. An error when violating this rule is undistribution of the middle term.

For example, let’s take two statements on the topic of identity. The famous Persian poet Saadi (1184-1291) remarked: “A donkey that has been to Mecca will still remain a donkey.” And our compatriot, the famous poet G.R. Derzhavin (1743-1816) expressed this thought in his own way: “A donkey will remain a donkey, even if you shower it with stars.” Using these statements as premises, we construct a syllogism:

• (L) “A donkey who has been to Mecca (P+) will still remain a donkey (M-).”
• (L) A donkey showered with stars (5*) will still remain a donkey
• (m-)._
• (L) Donkey showered with stars (5+), this is a donkey who visited Mecca (P~).

If you like, you can formulate the conclusion differently:

“A donkey that visited Mecca is showered with stars,” but this does not change the essence of the error. In the premises, the middle terms - the circle of those who will always remain an ass - are taken incompletely (partially). And this circumstance turns out to be decisive, since there is no reason (except for the play of chance) to believe that both statements are talking about the same subset. In essence, this is an implicit violation of the law of identity.

Formalizing the premises of the syllogism:

"All R there is M",

“All 5 are M”, let’s build circular diagrams:

As we see, based on the same premises, four mutually exclusive conclusions can be drawn.

The diagrams show that unambiguous relationships cannot be established between the terms of a syllogism. This is an indication that the syllogism is incorrect.

A term not distributed in the premise should not be distributed in the conclusion.

An error when violating this rule is illegal extension of an extreme term. In other words, having initial information about part of the objects of a particular set, in the process of reasoning they extend this information to the entire set, which contradicts the logical nature of deduction - both in its traditional sense (the movement of thought from the general to the particular), and in the modern sense the rigor of the conclusion.

As an example, let's use a story from ancient Greek mythology about a giant robber named Procrustes. As is known, he forcibly placed travelers on a bed and chopped off the legs of those who were larger than him, and stretched the short ones to the size of the bed. This is where the name “Procrustean bed” comes from, which in a figurative sense means an artificial measure that does not correspond to the essence of the phenomenon; restrictions imposed on something by force. In passing, we note that logic also imposes restrictions, but has neither direct nor indirect relation to the case of Procrustes. So, the syllogism:

• (A) “Governing a state (M+) is a cruel matter (R~)” (D. Halifax).
• (E) Procrustes (5+) did not rule the state (M+).
• (E) Procrustes (5*) did not engage in cruel deeds (P+).

It is clear from the signs of distribution that the predicate (“cruel deeds”) in the premise was taken in part of its volume, and in the conclusion - in full, which is unacceptable in deductive conclusions.

Having formalized the parcels:

"Everything is Revenge R",

"5 don't eat M", Let's build circular diagrams:

Obviously, the information from the premises is not enough to establish unambiguous relationships between the terms. Based on the larger premise, we are all a multitude M placed in the set P, and based on a minor premise, they mutually excluded the sets Mi 5. But the relationship between the extreme terms 5 and R, since 5 may or may not belong to the set P. Both possibilities are equivalent, and the preference for one of them has no relation to the laws of logic.

Let's look at the parcel rules.

At least one of the premises must be an affirmative proposition. This means that it is impossible to construct a correct syllogism from two negative judgments.

• (E) "The path of evil (P+) does not lead to good (M+)" (W. Shakespeare).
• (E) Playing with fire (5+) will not lead to good things (M+).
• (L) Playing with fire (5*) is the way of evil (P~).

Having formalized the premises: “None R is not M", "Not a single 5 is L/", let's build circular diagrams: ___

As we see, there is no unambiguous relationship between the extreme terms 5 and P. Based on the information contained in the premises, a number of mutually exclusive conclusions can be drawn, namely:

"All 5 are P"

"Some 5 are P"

"Some 5s are not P"

"No 5 is P."

• At least one of the premises must be a general proposition. This means that it is impossible to construct a correct syllogism from two particular judgments. For example:
• (/) “Positions often (M~) change their character (P”)>> (Cervantes). (I) Some positions (M-) are vacant (5_). (G) Some vacancies (5_) change the character (P).

Already from the distribution of terms it is clear that a correct conclusion from these premises is impossible, since the middle term is not distributed in any of them. But this is only a passing remark relating to a particular case. The essence of the problem is different: if the middle terms are taken as parts of the volumes, then there is no reason to believe that these are identical parts. And if so, then the conclusion falls apart. The situation here is in many ways similar to the quadrupling of terms, only in an implicit form.

Let's look at the situation in more detail. Let's say there is a set of students, from which some parts (subsets) are taken and certain thoughts are expressed in relation to them. It is possible that these subsets will turn out to be incompatible, and then thoughts will be expressed in relation to different subjects.

Eg:

• (G) Some students take exams in control theory.
• (£") Some students are first graders.

Possible output options: “Some first-graders are taking exams in control theory”; "Some of those taking the control theory exams are first-graders." In both cases - absurdities. Why? Yes, because the subsets of students are incompatible: in one case they are schoolchildren, in the other they are undergraduate or graduate students (at least not schoolchildren).

Let's return to the original example.

Having formalized the parcels:

"Some M are not P"

"Some M there is 5", let's build circular diagrams:

From the constructions it is clear that volume 5, intersecting with volume M, finds itself in an ambiguous relationship with volume R. Possible output options: “All 5 are P”, “No 5 are P”, “Some 5 are P”.

This indicates that the syllogism is incorrect.

With one negative premise, the conclusion must be negative.

An example of a violation of this rule:

• (£") For introverts (M) sociability is not characteristic (R). (A) I am (5) an introvert (M).
• (A) However, I (5) am a sociable person (R).

Having formalized the parcels:

"None M is not P",

“5 is M”, and having constructed a diagram, we obtain the ratio of extreme terms:

“5 is not P”, corresponding to the rules of inference. However, in violation of these rules, the conclusion states the opposite: “5 is P.”

With one particular premise, the conclusion must be private.

An example of a violation of this rule:

• (A) “Disorder (M~) makes us slaves (R~)” (A. Amiel). (G) Sometimes cleanliness becomes a mess (M).
• (A) Cleanliness makes us slaves (R-).

Already in the distribution of terms, a violation is noticeable: the subject, not distributed in the premise, turned out to be distributed in the conclusion.

Having formalized the parcels:

"All Revenge R"

"Some 5 are M", Let's build circular diagrams:

The relationship between the extreme terms l and R such that in one case it turns out: “All.9 are P,” and in the other: “Some 5 are P.” Obviously, taking into account the distribution of terms, it is the second option that is acceptable.

For a deeper understanding of the structure of a simple categorical syllogism, it is also necessary to take into account the diversity of its figures and modes.

There are four syllogism figures in total.

I figure

The middle term in the first figure plays the role of subject in the major premise and the role of predicate in the minor premise.

• (L) "Conceit (M) is a hindrance to success (P)"
• (Bion Borisfensky). (L) Exaggerated assessment of one’s personality (5) - conceit (M).
• (L) An exaggerated assessment of one’s personality (L”) is an obstacle to success (R).

The first figure of a simple categorical syllogism is used as a way of extending some general knowledge, expressed in a major premise, to special cases. Class 5 is subsumed under class P, regarding which there is general knowledge.

This can be clearly seen in the diagram:

If "All 5 are M",

and all M there is P",

then "All 5 are P."

II figure

The middle term in the second figure plays the role of a predicate of both premises.

• (L) "Any truly effective government (R) when tested turns out to be a dictatorship (M)" (G. Truman).
• (£") Democracy (5) is not dictatorship (M).
• (E) Democracy (5) is not good governance

The second figure of the PKS is used mainly as a means of refuting incorrect subsuming of something under a certain concept. This is also clearly visible in the diagram: If “All 5 are M”, and “None M is not P", then "No 5 is P".

III figure

The middle term in the third figure plays the role of subject in both premises.

• (A) “The word (M) is the shadow of the deed (P)” (Democritus).
• (A) "Word (M) there is an act (5)" (L.N. Tolstoy).
• (G) Some actions (5) are a shadow of the deed (P).

The third figure is often used as a way to refute unfounded generalizations. The diagram shows:

If "All 5 are M"

and "All P's are M's"

then "Some 5 are R".

In reasoning on the third figure, the fundamentally important point is the quantitative characteristic of the conclusion - it must always be private. By following this rule, we avoid unfounded generalizations.

IV figure

The middle term in the fourth figure acts as the predicate of the major and the subject of the minor premises.

• (£) “Strong words (P) cannot be strong evidence (A/)” (V. O. Klyuchevsky). (/) Strong evidence (M) usually convincing (5).
• (O) Usually, convincing arguments (6") do not need strong words (R).

The fourth figure is an artificial structure. Having no educational value, it is rarely used in practice. If we reverse both premises, then from the fourth figure we can obtain the first.

• (E) Strong evidence (L/) does not need strong words (R). (G) Usually convincing arguments (5) are supported by strong evidence (L/).
• (O) Usually, convincing arguments (5) do not need strong words (R).

For example, in the last example, all judgments in a syllogism are generally affirmative statements, therefore its mode AAA; and in the penultimate one: premises - generally affirmative statements (A), and the conclusion is privately affirmative (G), therefore its mode AAI. Actually, in all illustrative examples of PKS figures, to the left of the statements that are part of the syllogisms, there are letter designations, the sequence of which gives us modes.

Considering the presence of four types of categorical judgments (A, E, I, O), It can be calculated that there are 64 modes in each figure, and there are 256 in total! But not all of them are correct conclusions. There are only 24 correct modes (6 in each figure). Among them, there are 19 main (strong) correct modes and 5 weak ones (in which the conclusions are private judgments).

Syllogistics in traditional logic was developed in such detail that all strong regular modes received special names, which, facilitating memorization, contain all the information about the nature of the judgments that make up the mode. These names were invented by the Byzantine philosopher of the 11th century. named Michael Psellos (1018-c.1096). He wrote the Compendium to Aristotle's Logic, where he outlined his invention.

In order to make it easier to learn the strong correct modes of a simple categorical syllogism, medieval schoolchildren came up with a poem written in hexameter. Here it is.

Barbara, Celarent, Darii, Ferioquc prions; Cesare, Camestrcs, Festino, Baroko secundae;

Tcrtia, Darapti, Disamis, Datisi, Felapton, Bokardo, Ferison habet: Quarta insuper addit Bramantip, Camenes, Dimaris, Fcsapo, Fresison.

The vowels in the names of the modes indicate the types of judgments that play the roles of the major, minor premise and conclusion, respectively. For example, mode Felapton means that the major premise is a general negative judgment, the minor premise is a general affirmative, and the conclusion is a particular negative judgment.

Correct modes. For the first figure it is - AAA, EAE, AN, EY.

• Modus AAA (Barbara)
• (A) “Every name means something...” (A.F. Losev). (A) The word "Anna" is a name.
• (A) The word "Anna" means something.
• Modus EAE (Celarent).
• (E) “Not a single person can consider himself a smoky person” (N. A. Berdyaev). (A) I am human.
• (£) I cannot consider myself a complete person.
• Modus AN (Darii).
• (A) “A thought expressed is a lie” (F.I. Tyutchev). (G) Some of what I had in mind has been said.
• (G) Some of what I have in mind is false.
• Modus EY (Ferio).
• (E) “Nothing new is perfect” (Cicero). (/) Something is new in our lives.
• (G) Some things in our lives are not perfect.

The correct modes of the second figure are: EAE, AEE, EY, AOO.

The third figure has AAI, IAI, AN, EAO, JSC, EY.

The fourth has AAI, AEE, IAI, EAO, EY.

There is no need to specifically memorize modes, and especially their medieval names. The correct modes can be easily deduced logically, based on the general and special rules of a simple categorical syllogism (the so-called rules of figures).

“All lawyers are lawyers. - Large package

Petrov is a lawyer.- Smaller package

This means Petrov is a lawyer.” - Conclusion

The truth of this conclusion can be judged by analyzing the relationships illustrated above between the concepts “Petrov” - S, “lawyer” - M, “lawyer” - P. If the scope of the concept “Petrov” is included in the scope of the concept “lawyer”, and the scope of the concept “lawyer” ” - is included in the scope of the concept “lawyer”, then the scope of the concept “Petrov” is included in the scope of the concept “lawyer”.

PKS structure:

The PKS distinguishes three terms: smaller, larger and medium. Minor term– S – subject of the conclusion. Greater term– P – predicate of conclusion. Middle term– M is a term included in the premises and not included in the conclusion. Smaller package– a premise that includes the minor term S. Large package– a premise that includes the larger term P. In the standard PKS form, the larger premise is first written down, followed by the smaller one. They draw a line, and below the line is a conclusion.

In example 1, the subject of the conclusion is the concept “Petrov”, the predicate of the conclusion is the concept “lawyer”, therefore the smaller term S is “Petrov”, the larger term P is “lawyer”. The concept “lawyer” is included in both premises and is not included in the conclusion, therefore “lawyer” is the middle term of M. The minor premise is “Petrov is a lawyer,” the major premise is “All lawyers are lawyers.” Conclusion - “Petrov is a lawyer.”

General rules of simple categorical syllogism:

Parcel rules:

1. A definite conclusion cannot be drawn from two negative judgments.
/Example 2: “Not a single prosecutor is a lawyer. Muravyov is not a prosecutor. So(?), he is (not) a lawyer” /

2. If one premise is negative, then the conclusion will be negative.
/Example 3: “All lawyers are lawyers. Prosecutors are not lawyers. So he is not a lawyer” /

3. A definite conclusion cannot be drawn from two particular judgments.
/Example 4: “Some people are merciful. Some people are cruel. So (?), the cruel are merciful" /

4. If one premise is private, then the conclusion will be private.
/Example 5: “All mammals are vertebrates. Some aquatic animals are mammals. This means that some aquatic animals are vertebrates” /

Rules of terms:

1. A syllogism must have only three terms.

/Example 6: “All lawyers are lawyers, and Petrov is a pop star” - there is no general term, so there is no connection between these judgments and no conclusion can be drawn/.

/Example 7: “Matter is eternal. Silk is matter. Therefore, silk is eternal” - the word “matter” here means two different concepts, which means that no conclusion can be drawn.

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

3. An extreme term (S, P) is distributed in the conclusion if and only if it is distributed in the premises.

Example 8: P+ M-

"All criminals must be held accountable for their actions."

« Petrov must be held accountable for his actions.”

“Petrov is a criminal.”

In this example, the second and third rules of terms are violated, since the middle term M is not distributed in any of the premises, and the larger term P is not distributed in the conclusion, but is distributed in the premise.

Often, in order to make sure that the PKS is correct, it is enough to check the private rules - the rules of figures.

Shape rules:

Depending on the location of the middle term, four ACL figures are distinguished:

Having determined the PKS figure, you should check whether the rules of the corresponding figure are fulfilled.

First figure rule: The major premise must be a general proposition, and the minor premise must be an affirmative one.

In the above example 1, the PKS of the first figure is given. In it, the major premise is a general judgment, and the minor premise is an affirmative judgment. The figure rule is observed. Therefore, the conclusion is reliable.

Second figure rule: The major premise must be a general proposition, one of the premises must be negative.

In example 8 the PKS of the second figure is given. It has a major premise - a general judgment, but there is not a single negative premise. The second figure rule is not observed. The conclusion is unreliable.

Third figure rule: The minor premise is an affirmative proposition, and the conclusion is a particular one.

Example 9: M P

The whale is an aquatic animal.

A whale is a mammal.

Some mammals are aquatic animals.

Fourth figure rule: If the major premise is an affirmative proposition, then the minor premise is a general proposition. If one of the premises is negative, then the major premise is a general proposition.

Example 10: P M

All elephants are mammals

No mammal is an invertebrate.

No invertebrate is an elephant.

Theory for problem 29: Enthymeme is a shortened syllogism. There are enthymemes with a missing major premise, with a missing minor premise, and with a missing conclusion. The propositions that make up the enthymeme are connected by expressions: “since”, “because”, “for”, “since”, “therefore”, “means”, “therefore and”, “a”, “but”, “ yes”, etc.

Problem 29: Enthymema. Restore it to a complete simple categorical syllogism and check it.

Example: Stealing a car is punishable by law, since any theft is punishable by law.

Solution: Determine the type of enthymeme (with a missing major premise, with a missing minor premise, or with a missing conclusion). It is clear that this enthymeme contains a conclusion - “Car theft is punishable by law.” There is also a major premise containing the larger term “punishable by law.” This means that in this case the minor premise is missing. We restore. Let's put down the terms. Define the figure. Let's check the rules.

Any theft is punishable by law. General rules are followed.

S+ M- First figure.

Car theft is theft. Rule of first

S+ P- figures are observed.

Car theft is punishable by law. The conclusion is correct.

Topic 9. Inferences from complex judgments

Inferences from complex judgments are divided into conditional, disjunctive and conditionally disjunctive. Conditional ones are divided into purely conditional and conditionally categorical. Dividing ones are divided into purely dividing and dividing categorical. Conditional dividing (lemmatic) ones are divided into dilemmas, trilemmas and, in general, polylemmas.

Theory for problem 31Purely conditional inferences- this is an inference in which all premises and conclusion are conditional propositions.

Problem 31: Construct the given text in the form of a purely conditional inference, draw a conclusion, construct an inference diagram.

Example: “A student will learn to construct correct reasoning if he masters logic well. Then his speech will become more convincing.”

Solution: In order to construct this reasoning in purely conditional form, we can introduce the following notation: A - “The student will master logic well.” B - “He will learn to build correct reasoning.” C - “His speech will become more convincing.” Then this thought will take the form of a purely conditional conclusion: “If a student masters logic well, then he will learn to build correct reasoning. If a student learns to build correct reasoning, his speech will become more convincing.” Conclusion: “So, if a student masters logic well, his speech will become more convincing.”

Inference diagram: (A®B)Ù(B®C)

So, A®C.

Theory for problems 32-33 Conditional categorical syllogism (CCS)- this is an inference in which one premise is a conditional proposition, and the other premise and conclusion are categorical judgments. It has two correct modes (giving a reliable conclusion) and two incorrect modes (not giving a reliable conclusion). It should be noted that the logical expressions of the correct modes are logical laws, and the logical expressions of the incorrect modes are not logical laws.

Problem 32 Conditional categorical syllogism. Draw a conclusion, write down the formula, determine the mode and nature of the conclusion.

Example: “If a person has a high temperature, then he is sick. This man is sick."

Conclusion: “The person may have a high fever.”

Formula: ((A®B)ÙB)®A.

Mode: Incorrect approver.

Nature of the conclusion: Unreliable.

Problem 33 Based on this premise, construct a conditional categorical syllogism based on correct and incorrect modes.

Example: “If it rains, the asphalt is wet.”

A) correct affirmative mode:((A®B)ÙA)®B.

It's raining at the moment.

The asphalt is wet now.” The conclusion is reliable.

B) Correct negating mode: ((A®B)ÙØB)®ØA.

“If it rains, the asphalt is wet.

The asphalt is not wet at the moment.

It is not raining now". The conclusion is reliable.

IN) incorrect affirmative mode: ((A®B)ÙB)®A.

“If it rains, the asphalt is wet.

At the moment the asphalt is wet.

It might be raining." The conclusion is unreliable.

G) incorrect negating mode:((А®В)ÙØА)®ØВ.

“If it rains, the asphalt is wet

There is no rain at the moment.

The asphalt is not wet." The conclusion is unreliable.

Theory for problems 34, 35 Separation-categorical called a syllogism (RSS), in which one premise is a disjunctive judgment, and the other premise or conclusion is a categorical judgment. RKS has two forms: ((A Ú B)ÙA)®ØB – affirmative-negative mode; ((АÚВ)ÙØА)®В – negative-affirmative mode.

The rule of the affirmative-negative mode: the disjunction must be strict, i.e. the alternatives in the dividing premise must be mutually exclusive.

If the disjunction is not strict in the affirmative-negative mode, then the conclusion will be probable / “He suffers from illness or poverty. He is sick. He is probably not poor” - the conclusion is unreliable, since the alternatives are not mutually exclusive.

The rule of the negative-affirmative mode: the disjunction must be complete, i.e. the separating premise must list all alternatives.

Problem 34 Dividing-categorical syllogism . Draw a conclusion. Write down the formula, determine the mode and nature of the conclusion

Example: “Animals are either vertebrates or invertebrates.

This animal is not a vertebrate."

Solution: “So it’s an invertebrate.”

AÚV negative-affirmative

ØB mode.
A The conclusion is reliable (the rule of mode is observed).

Problem 35: Using a dividing premise, build a dividing-categorical conclusion: a) according to the affirmative-denying mode; b) in a negative-affirming mode. Determine the nature of the conclusion (credible or probable).

Example: “Simple propositions are either affirmative or negative.”

Solution: To this dividing premise we add a simple categorical premise:

a) Affirming: b) Denying:

“This judgment is negative.” « This judgment is not negative.”

“So it’s not affirmative.” “So it’s affirmative.”

If the premise is affirmative, then the conclusion must be negative, and vice versa.

The conclusion in both cases is reliable, since all the rules are followed.

Theory for problem 36 Dilemma is a conditional disjunctive inference in which one premise consists of two conditional propositions, and the other is a disjunctive proposition. Dilemmas can be constructive or destructive. Constructive dilemmas are characterized by the fact that thought in them moves from the approval of variants of the bases of conditional propositions to the approval of consequences. Design dilemmas can be simple or complex. IN simple design dilemma the first premise consists of two conditional propositions, the bases of which are different, but the consequences coincide; the second premise contains the disjunction of both bases:

(A®B)Ù(C®B)

IN difficult design dilemma the first premise consists of two conditional propositions, the grounds and consequences of which are different; the second premise contains the disjunction of both bases:

(A®B)Ù(C®D)

Destructive dilemmas are characterized by the fact that thought in them moves from the denial of variants of consequences to the denial of grounds. Destructive dilemmas can also be simple or complex.

In a simple destructive dilemma the first premise consists of two conditional propositions, the bases of which are the same, but the consequences are different. The second premise contains a disjunction of negations of both consequences:

(A®B)Ù(A®C)

ØVÚØS

IN complex destructive dilemma the first premise consists of two conditional propositions, the grounds and consequences of which are different. The second premise also contains a disjunction of negations of both consequences:

(A®B)Ù(C® D)

ØВÚØ D

Problem 36. Determine the type of dilemma. Draw a conclusion, build a diagram. Determine the nature of the output.

Example: “If the statement about a crime is oral, then it is entered into the protocol, which is signed by the investigator, prosecutor or judge who accepted the statement; if the application is written, then it must be signed by the person from whom it comes. But a statement about a crime can be oral or written.”

Conclusion: “This means that it is entered into the protocol, which is signed by the investigator, prosecutor or judge who accepted the statement, or signed by the person from whom it comes.”

Diagram: (A ® B) Ù (C ® D)

A Ú C

A difficult design dilemma. The conclusion is reliable.

Topic 10. Inductive inferences

Theory for problem 37: All the inferences discussed above were deductive. Deduction- inference from more general knowledge to less general knowledge. Deductive inferences are usually constructed in the form of a simple categorical syllogism, conditional categorical syllogism, or other standard forms of reasoning described above. /“All lawyers are lawyers. Petrov is a lawyer. So he is a lawyer.” In this case, the thought goes from knowledge about all lawyers to knowledge about specific Petrov.

Inductive called inferences from less general knowledge to more general knowledge. In induction, the data of experience “points” to the general. (From Latin inductio - guidance).

A distinction is made between complete and incomplete induction.

Full induction is obtained if, firstly, all elements of the class of objects are examined, and, secondly, it is established that each of them has the same property. /“Monday is a sunny day; Tuesday, Wednesday, Thursday, ..., Sunday were sunny days. So the whole week was sunny.” Complete induction produces a valid conclusion.

Incomplete induction- inference from knowledge of only some objects of the class to knowledge of all elements of the class. The conclusion is plausible.

Types of incomplete induction: A) popular– this is ordinary induction without the use of special (scientific or statistical) methodology; b) statistical– this is induction based on the use of special techniques for selecting and analyzing class objects that establish the probability of an event occurring; V) scientific– induction, based on identifying the causal relationship between phenomena.

When making inductive inferences, the following errors are possible:

a) “hasty generalization” - when studying a clearly insufficient number of subjects in a class / for example, when a teacher, having interviewed three students of one large group, and, without receiving proper answers, concludes that the entire group is unprepared /;

b) “after this, therefore, because of this” - if the causal relationship between the phenomena is not established / “eating cucumbers is life-threatening, since 99.9% of people involved in car and plane accidents ate cucumbers; 99.9% of people who died from various diseases if cucumbers…”/ .

Methods for determining the causal relationship between phenomena (scientific induction) were discovered by Francis Bacon and refined by John Stuart Mill. There are five main methods for determining the causal relationship between phenomena: 1) the method of single similarity; 2) single difference method; 3) combined method of similarity and difference. 4) method of accompanying changes; 5) method of residuals.

The English physicist D. Brewster discovered the reason for the iridescent colors on the surface of mother-of-pearl shells in the following way. By chance, he received an imprint of a mother-of-pearl shell on wax and discovered on the surface of the wax the same play of rainbow colors as on the shell. He made prints on plaster, resin, rubber and other substances and became convinced that it was not the special chemical composition of the substance of the mother-of-pearl shell, but the certain chemical structure of its inner surface that caused this beautiful play of colors.

Until the 80s of the 19th century, there was a simplified idea of ​​the nutritional needs of the animal body. Scientists argued that the body only needs protein and small amounts of various salts. In 1880, Russian doctor N.I. Lunin. took several dozen mice and divided them into experimental and control ones. He began to feed the first with artificial milk, made from purified substances that make up natural milk - protein, fat, casein, sugar and corresponding salts; other control mice - natural milk. The experimental mice fell ill and died, while the control mice remained healthy. Based on this, Lunin concluded that natural food contains still unknown substances that are necessary for the body. With his experiments carried out using the method of distinction, N.I. Lunin laid the foundation for the study of vitamins.

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