A chain of syllogisms from the word old-timer. Complex inferences. Compound and compound abbreviated syllogisms

Ways to check the correctness of a simple categorical syllogism can be demonstrated in the following example (second figure, mode AAA):

By general rules syllogism: the rules of the terms of the syllogism are violated: there is a quadrupling of terms, since in the larger premise the term M 1 -"materially support each other", and in a smaller premise M 2 - "support each other", the middle term is not distributed in any of the premises.

According to the special rules of the figures of the syllogism, the rule of the second figure of the syllogism is violated, namely: according to the rules of the second figure, one of the premises is a negative judgment, and in this example both premises are affirmative judgments.

With a counterexample: if instead of the concept "G and F"substitute the concept of" true friends ", then a false conclusion will be obtained from true premises.

According to the modes of figures: mode AAA- the wrong mode of the second figure of the syllogism.

With the help of diagrams: for this we write the structure of premises and conclusions as follows:

Based on this entry, we will depict the relationship between terms using circular diagrams (Fig. 8.8, 8.9).

Rice. 8.8

Rice. 8.9

As can be seen from the diagrams, the conclusion does not necessarily follow from the premises, i.e. necessary connection between S and R cannot be set, because in our example the middle term M is not distributed in any of the premises and there is a quadrupling of terms.

Violation of at least one of the rules means: the syllogism is incorrect (the conclusion does not necessarily follow from the premises).

Inference from Judgments with Relationships

An inference whose premises and conclusion are judgments with relations is called an inference with relations.

The most important logical properties of relations are reflexivity, symmetry, transitivity, functionality (uniqueness).

reflective This relationship between objects is called A and V in which the object is in the same relation to itself. If R has the property of reflexivity, then it is expressed by the formula

A R B → A R A∩B R B.

For example: "If A ≡ V, then A ≡ A and V ≡ V".

symmetrical is a relationship that takes place both between objects A and V, as well as between objects V and A. The logical property of symmetry can be written as a formula

A R B → B R A.

For example, the property of symmetry is possessed by the relation "to be a relative": if A relative V, then V- relative A.

transitive such a property of relations is called when, in the presence of this relation between objects A and V, V and WITH it is possible to establish this relationship between A and WITH, i.e. A R C. The logical property of transitivity can be expressed by the formula

(A R B) ∩ (B R C) → A R C.

For instance:

A > B 6 > 4

B > C 4 > 2

A > C 6 > 2

functional(unique) a relation is called if and only if each value of the relation at relationship x R y matches only one value X . For instance: " x father at ", because every person (at) there is only one father.

The logical property of functionality can be symbolically written as the following axiom:

(A R B ∩ C R B) → A ≡ WITH.

Abbreviated, complex compound abbreviated syllogisms

The varieties of simple categorical syllogism formed from simple judgments also include abbreviated syllogism (enthymeme), complex syllogism (polysyllogism) and compound abbreviation (epicheirema).

Enthymeme

Enthymeme is an abbreviated categorical syllogism. Translated from Greek, enthymeme means "in the mind, in thoughts." This name indicates that one or another part of the syllogism is implied, and not expressed. In the process of thinking, we often do not express all parts of the syllogism, but think in terms of enthymemes.

An enthymeme is a syllogism in which either one of the premises or the conclusion is omitted.

There are the following types of enthymemes:

a) with a missed larger parcel, for example:

b) with a missed smaller parcel, for example:

All chemical elements (M) have an atomic weight (P); (implied)

Hence, helium (5) has an atomic weight (P).

c) with a missing conclusion, for example:

All chemical elements (M) have an atomic weight (P)

Enthymeme structure:

The restoration of enthymemes to a complete syllogism is of great educational value. Sophistic tricks, false premises, as a rule, are veiled in the missing part of the enthymeme. This psychological feature is actively used by the enemy when deliberately misleading. For example, the following false conclusions can be found in enthymemes: "He is a pianist, because he has long flexible fingers", "All monkeys like bright things, and all women do too."

Restoring the missing part of the syllogism allows you to check both the truth and the correctness of enthymemes.

Like any conclusion, an enthymeme can be correct (correct) or incorrect (incorrect).

Enthymeme with missed parcel counts correct , if it is restored to a correct syllogism and the missing premise is not false.

Enthymeme with omitted conclusion counts correct if the conclusion is derived from the premises.

To restore the enthymeme to a full syllogism, one should be guided by the following rules.

1. Find a conclusion and formulate it in such a way that the larger and smaller terms are clearly expressed.
2. When finding premises and conclusions, one should proceed from the fact that the conclusion is usually placed after the words "means", "therefore", etc. or before the words "because", "because", "because". Another judgment, of course, will be one of the premises.
3. If one of the premises is omitted, but the conclusion is present, then it is necessary to establish which of them (larger or smaller) is present. This is done by checking which of the extreme terms is contained in the given premise. If the term is larger, then there is a larger premise; if there is a smaller term in the premise, then there is a smaller premise.
4. Knowing which of the premises is omitted, and also knowing the middle term, it is possible to determine both terms of the missing premise.

For example: "Jupiter, you are angry, so you are wrong." In this entisms, the big premise is implied, and therefore omitted: "Whoever gets angry is wrong." Let's restore the whole syllogism in full:

Inferences can also take the form of enthymemes, the premises of which are conditional and disjunctive judgments.

For example, let's check the enthymeme: "He must be an educated person, because he competently answers all the questions that he is asked."

Let us determine whether a premise or conclusion is missing in it and write down the conclusion, if it is, under the line, the premise (or both) above the line.

The presence of a conclusion in an enthymeme is usually indicated by the words: "since", "because", "because", etc. or "means", "therefore", "thus". The words of the first group show that the conclusion is in front of them, and after them comes the premise, the words of the second group show that they are followed by the conclusion. If there are no such words, then the conclusion is missing in the enthymeme. This etyme has a conclusion. The judgment, "He must be an educated man," is a conclusion, as it comes before the word "because." Let us define the structure of this judgment, i.e. find in it a subject and a predicate. The subject is "he", the predicate is "an educated person".

According to the subject and predicate of the conclusion, we establish the nature of the existing premise: "He competently answers all the questions that he is asked." It contains the subject of the conclusion: "he", therefore, is a minor premise. According to the predicate of the conclusion and the middle term, which is included in the minor premise, we restore the major premise missing in the enthymeme: "Everyone who correctly answers all the questions that are asked of him is an educated person."

The result is a complete syllogism:

Let's check the correctness of the resulting syllogism. It is built according I figure, both rules of this figure (see above) are observed. So this syllogism is correct. It can also be tested using a circular diagram (Fig. 8.10), which corresponds to the axiom of the syllogism.

Rice. 8.10

Polysyllogisms, sorites, epicheirema

In the process of thinking, syllogisms are interconnected, forming chains of syllogisms - complex syllogisms and polysyllogisms.

Polysyllogisms

A chain of syllogisms in which the conclusion of the preceding syllogism becomes the premise of the next is called a polysyllogism.

A syllogism that precedes another in a chain of syllogisms is called askedlogism .

A syllogism that follows another in a chain of syllogisms is called episyllogism .

There are progressive and regressive polysyllogisms.

progressive polysyllogism called polysyllogism, in which the conclusion of the previous polysyllogism (prosyllogism) becomes the larger premise of the episyllogism.

For instance:

Regressive polysyllogism is called a polysyllogism, in which the conclusion of the prologism becomes the lesser premise of the episyllogism.

All counterfeiters (E) - criminals (D)

All criminals(D) – offenders (C)

Hence,

All counterfeiters (E)– offenders (C)

Hence,

All counterfeiters (E) - people ( A)

All people ( A) are mortal ( V)

(E) - mortal (V)

Everything E there is D

EverythingD there is WITH

Everything E there is WITH

Everything WITH there isA

Everything E there is A

Everything A there is V

Everything E there is V

In each case, we fixed the conclusion by adding the word "therefore" to it. True, in the regressive polysyllogism we changed the usual arrangement of premises, placing the minor premise first.

sorite

A polysyllogism in which some premises (greater or smaller) are omitted is called a sorite (Greek. soros- heap, heap of parcels), or an abbreviated polysyllogism.

There are two types of sorites: progressive, or Goklenevsky, by the name of the author - the German logician R. Gauklen (1547–1628) and regressive, or Aristotelian.

Sorit, in which, starting from the second syllogism, a large premise is omitted in the chain of syllogisms, is called progressive (goklenevsky) .

Example.

All people (A) mortal (V)

All offenders (WITH) - people (A)

All criminals D) – offenders (WITH)

All counterfeiters E) - criminals(D)

Therefore, all counterfeiters (E) - mortal (V)

Everything A there is V

Everything WITH there is A

Everything D there is WITH

Everything E there isD

Everything E there is V

Sorit, in which, starting from the second syllogism, the minor premise is omitted in the chain of syllogisms, is called regressive (Aristotelian).

Example.

All counterfeiters E) - criminals (D)

All criminals (D)– offenders (C)

All offenders (C) are people ( A)

All people (A) mortal (V )

Therefore, all counterfeiters (E) mortal (V)

Everything E there is D

Everything D there is WITH

Everything WITH there is A

Everything A there is V

Everything E there is V

Epicheirema

Epicheirema (gr. epiheirema- conclusion) - this is such a complex abbreviated syllogism in which premises are enthymemes.

Example.

All diamonds ( A) are parallelograms ( WITH), since they (rhombuses) ( A) have pairwise parallel sides (V)

All squares ( D) – rhombuses ( A), since they are (squares) (O) have mutually perpendicular diagonals that bisect at the point of their intersection ( E)

Therefore, all squares (D)- parallelograms (C).

Everything A is C, because A there is V - enthymeme

EverythingD there isA, sinceD there is E - enthymeme

Everything D there is WITH

Conditional-separative inference

Simple judgments that make up a disjunctive (disjunctive) judgment are called members of the disjunction , or clauses. For example, the disjunctive judgment "Bonds can be bearer or registered" consists of two judgments - disjuncts: "Bonds can be bearer" and "Bonds can be registered", connected by the logical conjunction "or".

While affirming one term of the disjunction, we must necessarily deny the other, and, denying one of them, affirm the other. In accordance with this, two modes of divisive-categorical reasoning are distinguished: (1) affirmative-denying and (2) denying-asserting.

1. In affirmative-denying mode (modus ponendo tollens) the minor premise - a categorical judgment - affirms one member of the disjunction, the conclusion - also a categorical judgment - denies its other member. For instance:

Scheme of affirmative-denying mode:

Symbol of strict disjunction.

the major premise must be an exclusive disjunctive judgment, or a judgment of strict disjunction. If this rule is not observed, a reliable conclusion cannot be obtained. Indeed, from the premises “The theft was committed by K. or L.” and "Theft committed by K." conclusion L. did not commit theft” does not necessarily follow. It is possible that L. is also involved in the theft, is an accomplice of K.

2. In the negative-affirming mode(modus tollendo ponens) the minor premise denies one disjunct, the conclusion affirms another. For instance:

Scheme of the denying-affirming mode:

< >- the symbol of a closed disjunction.

An affirmative conclusion is obtained through negation: by denying one disjunct, we affirm another.

The conclusion on this modus is always reliable if the rule is observed: the major premise must list all possible propositions- disjuncts, in other words, the major premise must be a complete (closed) disjunctive statement. Using an incomplete (open) disjunctive statement, a reliable conclusion cannot be obtained. For instance:

However, this conclusion may turn out to be false, since not all possible types of transactions are taken into account in the larger premise: the premise is an incomplete, or open, disjunctive statement (the transaction can also be one-sided, for which it is enough to express the will of one person - issuance of a power of attorney, drawing up a will , renunciation of inheritance, etc.).

The separating premise may include not two, but three or more members of the disjunction. For example, in the process of investigating the causes of a fire in a warehouse, the investigator suggested that the fire could have occurred either as a result of careless handling of fire ( R), or as a result of self-ignition of stored materials ( q), or as a result of arson ( r). During the investigation, it was found that the fire was caused by careless handling of fire ( R). In this case, all other clauses are negated. The conclusion takes the form of an affirmative-denying mode and is built according to the scheme:

Another line of reasoning is also possible. Suppose the assumption that the fire was caused by careless handling of fire or as a result of spontaneous ignition of materials stored in the warehouse was not confirmed. In this case, the conclusion will take the form of a negative-affirming mode and will be built according to the scheme:

The conclusion will be true if all possible cases are taken into account in the conditional premise.

An inference in which one premise is conditional and the other is separative judgment, is called conditional-separative, or lemmatic 1 .

A disjunctive judgment may contain two, three or more alternatives 2 , so lemmatic reasoning is divided into dilemmas (two alternatives), trilemmas (three alternatives), etc.

Consider the example of a dilemma the structure and types of conditional-separative reasoning. There are two types of dilemmas: constructive (creative) and destructive (destructive), each of which is divided into simple and complex.

In a simple design dilemma the conditional premise contains two grounds from which the same consequence follows. The dividing premise affirms both possible grounds, the conclusion affirms the consequence. The reasoning is directed from the assertion of the truth of the grounds to the assertion of the truth of the consequence.

Diagram of a simple constructive dilemma:

1 From the Latin lemma - "assumption".

2 From the Latin alternare - "to alternate"; each of two or more mutually exclusive possibilities

Example:

If the accused is guilty of knowingly unlawful detention ( R), then he is subject to criminal liability for a crime against justice ( G); if he is guilty of knowingly unlawful detention ( q), then he is also subject to criminal liability for a crime against justice ( G).

The accused is guilty or knowingly illegal detention R), or in knowingly unlawful detention (q )

The accused is subject to criminal liability for a crime against justice ( r)

In a complex design dilemma the conditional premise contains two bases and two consequences. The separating premise asserts both possible grounds. The reasoning is directed from the assertion of the truth of the grounds to the assertion of the truth of the consequences.

Diagram of a complex design dilemma:

If the savings certificate is bearer (p), then it is transferred to another person by delivery (q); if it is nominal (r), then it is transferred in the manner established for the assignment of claims (s). But a savings certificate can be bearer (r) or nominal (r)

The savings certificate is transferred to another person by delivery (q) or in the manner prescribed for the assignment of claims (s)

In a simple destructive dilemma the conditional premise contains one basis, from which two possible consequences follow. The dividing premise denies both consequences, the conclusion denies the reason. The reasoning is directed from the denial of the truth of the consequences to the denial of the truth of the foundation.

Outline of a simple destructive dilemma:

If N. committed an intentional crime (p), then he was direct in his actions (q) or indirect intent (d). But in action there was neither direct (q) nor indirect intent (d).

The crime committed by N. is not intentional (r)

In a complex destructive dilemma the conditional premise contains two bases and two consequences. The dividing premise denies both consequences, the conclusion denies both grounds. Reasoning is directed from the denial of the truth of the consequences to the denial of the truth of the grounds.

Outline of a complex destructive dilemma:

If the enterprise is a rental enterprise (p), then it carries out entrepreneurial activity based on the leased property complex (q); if it is collective (d), then it carries out such activities on the basis of the property it owns (s)

This enterprise does not operate on the basis of leased property complex (non-q), nor on the basis of his property (non-s)

This business is not for rent. (not-r) or not collective (non-g)

§ 4. Abbreviated syllogism (enthymeme)

A syllogism in which all its parts are expressed - both premises and a conclusion - is called complete. Such syllogisms have been discussed in previous sections. However, in practice, syllogisms are more often used, in which one of the premises or the conclusion is not explicitly expressed, but implied.

A syllogism with a missing premise or conclusion is called an abbreviated syllogism or enthymeme 1.

The enthymemes of a simple categorical syllogism are widely used, especially inferences from the first figure. For example: "N. committed a crime and is therefore subject to criminal liability. A big premise is missing here: "A person who has committed a crime is subject to criminal liability." It is a well-known provision, which is not necessary to formulate.

A complete syllogism is built on the 1st figure:

Missing can be not only a large, but also a smaller package, as well as the conclusion: "The person who committed the crime is subject to criminal liability, which means that N. is subject to criminal liability." Or: "The person who committed the crime is subject to criminal liability, and N. committed the crime." Missing parts of the syllogism are implied.

Depending on which part of the syllogism is missing, there are three types of enthymeme: with a missing major premise, with a missing minor premise, and with a missing conclusion.

An inference in the form of an enthymeme can also be constructed according to the 2nd figure; according to the 3rd figure, it is rarely built.

The form of an enthymeme is also taken by inferences, the premises of which are conditional and disjunctive judgments.

Consider the most common types of enthymemes.

A big premise is missing here - the conditional proposition "If the event of the crime did not take place, then a criminal case cannot be initiated." It contains a well-known provision of the Code of Criminal Procedure, which is implied.

The big premise - the disjunctive judgment "In this case, either an acquittal or a guilty verdict can be passed" is not formulated.

Separative-categorical syllogism with a missing conclusion:“The death occurred either as a result of murder, or as a result of suicide, or as a result of an accident, or due to natural causes. Death was the result of an accident."

A conclusion that denies all other alternatives is usually not formulated.

The use of abbreviated syllogisms is due to the fact that the missing premise or conclusion either contains a well-known provision that does not need oral or written expression, or it is easily implied in the context of the expressed parts of the conclusion. That is why reasoning proceeds, as a rule, in the form of enthymemes. But, since not all parts of the conclusion are expressed in the enthymeme, the error hiding in it is more difficult to detect than in the full conclusion. Therefore, to check the correctness of the reasoning, it is necessary to find the missing parts of the conclusion and restore the enthymeme to a complete syllogism.

In the process of reasoning, simple syllogisms appear in a logical connection with each other, forming a chain of syllogisms, in which the conclusion of the previous syllogism becomes the premise of the next one. The previous syllogism is called askedlogism, subsequent - episyllogism.

The combination of simple syllogisms, in which the conclusion of the previous syllogism (prosyllogism) becomes the premise of the subsequent syllogism (episyllogism), is called a complex syllogism, or polysyllogism.

There are progressive and regressive polysyllogisms.

In progressive polysyllogism the conclusion of the askedlogism becomes the larger premise of the episyllogism. For instance:

In regressive polysyllogism the conclusion of the askedlogism becomes the lesser premise of the episyllogism. For instance:

Both of the above examples are a combination of two simple categorical syllogisms built according to the AAA modus of the 1st figure. However, a polysyllogism can be a combination of a larger number of simple syllogisms built according to different modes of different figures. A chain of syllogisms can include both progressive and regressive links.

Purely conditional syllogisms that have a scheme can be complex:

It can be seen from the diagram that, as in a simple purely conditional inference, the conclusion is an implicative connection between the basis of the first premise and the consequence of the latter.

In the process of reasoning, a polysyllogism usually takes an abbreviated form; some of his parcels are omitted. A polysyllogism in which some premises are omitted is called a sorite. . There are two types of sorites: progressive polysyllogism with omitted major premises of episyllogisms and regressive polysyllogism with omitted minor premises. Here is an example of a progressive polysyllogism:

Epicheirema also belongs to compound abbreviated syllogisms. An epicheirema is a complex abbreviated syllogism, both premises of which are enthymemes. For instance:

1) Dissemination of deliberately false information that discredits the honor and dignity of another person is criminally punishable, as it is slander

2) The actions of the accused are the dissemination of deliberately false information that discredits the honor and dignity of another person, since they were expressed in a deliberate distortion of facts in application for citizen P.

3) The actions of the accused are criminally punishable

Let us expand the premises of the epicheireme into complete syllogisms. To do this, we restore the first enthymeme into a complete syllogism:

Defamation (M) is a criminal offense (P)

Spread knowingly false information discrediting the honor and dignity of another person (S) is slander (M)

Dissemination of knowingly false information that discredits the honor and dignity of another person (S) is a criminal offense (P)

As we can see, the first premise of the epicheirema is the conclusion and the minor premise of the syllogism.

Now let's restore the 2nd enthymeme.

Deliberate distortion of facts in a statement against citizen P. (M) is the dissemination of deliberately false information that discredits the honor and dignity of another person (P) The actions of the accused (S) were expressed in a deliberate distortion of facts in application for citizen P. (M)

The actions of the accused (S) represent the dissemination of knowingly false information that discredits the honor and dignity of another person (P)

The second premise of the epicheirema also consists of the conclusion and the minor premise of the syllogism.

The conclusion of the epicheirema is derived from the conclusions of the 1st and 2nd syllogisms:

Dissemination of knowingly false information that discredits the honor and dignity of another person (M) is criminally punishable (P) The actions of the accused (S) constitute the dissemination of knowingly false information discrediting the honor and dignity of another person (M)

The actions of the accused (S) are criminally punishable (P)

Expanding the epicheireme into a polysyllogism makes it possible to check the correctness of the reasoning, to avoid logical errors that may go unnoticed in the epicheireme.

The term "enthymeme" in Greek means "in the mind", "in thoughts".

Enthymemoi, or abbreviated categorical syllogism, A syllogism is called a syllogism in which one of the premises or conclusion is omitted.

An example of an enthymeme is the following conclusion: “All sperm whales are whales, therefore, all sperm whales are mammals.” Let's restore the enthymeme:

All whales are mammals.

All sperm whales are whales

All sperm whales are mammals.

There's a big package missing here.

The enthymeme "All hydrocarbons are organic compounds, so methane is an organic compound" misses a minor premise. Let's restore the categorical syllogism:

All hydrocarbons are organic compounds.

Methane is a hydrocarbon.

Methane is an organic compound.

The enthymeme “All fish breathe with gills, and the perch is a fish” misses the conclusion.

When restoring the enthymeme, it is necessary, firstly, to determine which judgment is the premise, and which is the conclusion. The premise usually comes after the unions “because”, “because”, “because”, etc., and the conclusion comes after the words “therefore”, “therefore”, “because”, etc.

Students are given an enthymeme: "This physical process is not evaporation, since there is no transition of matter from liquid to vapor." They restore this enthymeme, that is, they formulate a complete categorical syllogism. The judgment after the words "because" is a premise. The enthymeme misses a big premise that students formulate on the basis of knowledge about physical processes:

Evaporation is the process by which a substance changes from liquid to vapor.

This physical process is not a process of transition of a substance from a liquid to vapor .

This physical process is not evaporation.

This categorical syllogism is built according to figure II; its special rules are observed, since one of the premises and the conclusion are negative, the big premise is the general one, which is the definition of the concept of “evaporation”.

Enthymemes are used more often than full categorical syllogisms.

§ 6. Complex and complex abbreviated syllogisms:

(polysyllogisms, sorites, epicheirema)

In thinking, there are not only individual complete abbreviated syllogisms, but also complex syllogisms consisting of two, three or more simple syllogisms. Chains of syllogisms are called polysyllogisms.

polysyllogism(complex syllogism) are called D1 or several simple categorical syllogisms connected with each other in such a way that the conclusion of one of them becomes the premise of the other. There are progressive and regressive polysyllogisms.

In progressive polysyllogism the conclusion of the previous polysyllogism (prosyllogism) becomes the larger premise of the subsequent syllogism (episyllogism). Let us give an example of a progressive polysyllogism, which is a chain of two syllogisms and has the following scheme:

Scheme:

Sport (A) improves health (B) All A are B.

Gymnastics (C) - sport (A). All C are A.

So, gymnastics (C) improves health (B). So all C are B.

Aerobics (D) - gymnastics (C). All D are C.

Aerobics (D) improves health (B). All D are B.

V regressive polysyllogism the conclusion of the askedlogism becomes the lesser premise of the episyllogism. For instance:

All planets (A) - space bodies (V).

Saturn (C) - planet (A).

Saturn (C) - cosmic body (V).

All cosmic bodies (V) have mass (D)

Saturn (WITH) - cosmic body (V).

Saturn (C) has mass (D).

Putting them together and not repeating twice the judgment "All WITH essence V", we get a scheme of regressive polysyllogism for general affirmative premises:

Everything A essence V.

Everything C is the essence A.

Everything V essence D.

Everything C is the essence V.

40. Complex and complex abbreviated syllogisms.

Compound and compound abbreviated syllogisms

In the process of reasoning, simple syllogisms appear in a logical connection with each other, forming a chain of syllogisms in which the conclusion of the previous syllogism becomes the premise of the next one. The previous syllogism is called askedlogism, subsequent - episyllogism

Distinguish between progressive and regressive polysyllogisms

In progressive polysyllogism the conclusion of the previous syllogism (prosyllogism) becomes the larger premise of the subsequent one (episyllogism). For instance:

A socially dangerous act (A) is punishable (B)

Crime (C) - socially dangerous act (A)

Crime (C) punishable (B) -conclusion of syllogism 1 (large premise in syllogism 2)

Giving a bribeD) - crime (C)

Giving a bribe (D) is punishable (B) - conclusion 2 syllogism

In regressive polysyllogism the conclusion of the previous syllogism (prosyllogism) becomes the lesser premise of the subsequent one (episyllogism). for instance

Economic crimes (A) - socially dangerous acts (B)

Illegal business (C) - economic crime (A)

Illegal entrepreneurship (C) - socially dangerous act (C) -

Socially dangerous acts (B) are punishable (D)

Illegal entrepreneurship (C) - socially dangerous act (C) - conclusion of syllogism 1 (minor premise in syllogism 2)

Illegal business (C) punishable (D)

Both of these examples are a combination of two simple categorical syllogisms built according to the AAA modus of the 1st figure. However, a polysyllogism can be a combination of a larger number of simple syllogisms built according to different modes of different figures. A chain of syllogisms can include both progressive and regressive connections.

Varieties of polysyllogism - sorit and epicheirema.

A sorite is an abbreviated polysyllogism that omits the conclusions of the previous syllogisms and one of the premises of the subsequent syllogism. There are two types of sorites: progressive polysyllogism with omitted major premises of episyllogisms and regressive polysyllogism with omitted minor premises.

Progressive sorite scheme:

All A is B

All C is A

EverythingDhave C

All D is B

Scheme of regressive sorite:

All A is B

All B is C

All C isD

All A are D

Here is an example of a progressive polysyllogism:

A socially dangerous act (A) is punishable (B).

Crime (C) - socially dangerous act (A)

Giving a bribeD) - crime (C)

Giving a bribe (D) is punishable (B)

Epicheirema also belongs to compound abbreviated syllogisms.

An epicheirema is a complex abbreviated syllogism, both premises of which are enthymemes.

For instance:

1) Dissemination of deliberately false information that discredits the honor and dignity of another person is criminally punishable, as it is slander

2) The actions of the accused represent the dissemination of deliberately false information that discredits the honor and dignity of another person, as they were expressed in a deliberate distortion of facts in a statement against citizen P.

3) The actions of the accused are criminally punishable.

Let us expand the premises of the epicheireme into complete syllogisms. To do this, we restore the first enthymeme into a complete syllogism:

Defamation (M) is a criminal offense (P)

Dissemination of deliberately false information discrediting the honor and dignity of another person (S), is slander (M)

Dissemination of knowingly false information discrediting the honor and dignity of another person (S) is a criminal offense (P)

As we can see, the first premise of the epicheirema is the conclusion and the minor premise of the syllogism.

Now let's restore the 2nd enthymeme.

Deliberate distortion of facts in an application against citizen P. (M) is the dissemination of deliberately false information that discredits the honor and dignity of another person (R).

The actions of the accused (S) were expressed in a deliberate distortion of facts in a statement against citizen P. (M)

The actions of the accused (S) represent the dissemination of knowingly false information that discredits the honor and dignity of another person (P)

The second premise of the epicheirema also consists of the conclusion and the minor premise of the syllogism.

The conclusion of the epicheirema is derived from the conclusions of the 1st and 2nd syllogisms:

Dissemination of knowingly false information discrediting the honor and dignity of another person (M) is a criminal offense (P)

The actions of the accused (S) represent the dissemination of knowingly false information that discredits the honor and dignity of another person (M)

The actions of the accused (S) are criminally punishable (P)

This lesson will focus on multi-premise inferences. Just as in the case of one-parcel inferences, all the necessary information in a hidden form will already be present in the premises. However, since there will now be a lot of parcels, the methods for extracting them become more complex, and therefore the information obtained in the conclusion will not seem trivial. In addition, it should be noted that there are many different types multi-message inferences. We will focus only on syllogisms. They differ in that both in the premises and in the conclusion they have categorical attributive statements and, based on the presence or absence of some properties of objects, allow us to conclude that they have or do not have other properties.

Simple categorical syllogism

A simple categorical syllogism is one of the simplest and most common inferences. It consists of two parcels. The first premise talks about the relationship between terms A and B, the second about the relationship between terms B and C. Based on this, a conclusion is made about the relationship between terms A and C. Such a conclusion is possible because both premises contain the general term B, which mediates the relationship between terms A and C.

Let's take an example:

All fish cannot live without water.
All sharks are fish.
Therefore, all sharks cannot live without water.

In this case, the term "fish" is a common term for two premises, and it helps to connect the terms "sharks" and "creatures that can live without water." The common term for two premises is usually called the middle term. The subject of confinement (in our example it is "sharks") is called a lesser term. The predicate of conclusion ("creatures capable of living without water") is called a larger term. Accordingly, the premise containing the smaller term is called the minor premise ("All sharks are fish"), and the premise containing the larger term is called the major premise ("All fish cannot live without water").

Naturally, in the argument, the premises can be in any order. However, for the convenience of checking the correctness of syllogisms, the major premise is always placed first, and the smaller one is placed second. Then, depending on the location of the terms, all simple categorical syllogisms can be divided into four types. These types are called figures.

A figure is a form of a simple categorical syllogism that is determined by the location of the middle term.

At the top is the major premise, followed by the minor premise, below the line is the conclusion. The letter S denotes the smaller term, the letter P denotes the larger term, and the letter M denotes the middle term.

Every M is a P
Every S is an M
Every S is a P

No M is P
Some M's are S's
Some S's are not P's

These different combinations of statements in the figures form the so-called modes. Each figure has 64 modes, so there are 256 modes in all four figures. If you think about all the variety of inferences that have the form of syllogisms, then 256 modes is not so much. In addition, not all modes form correct inferences, that is, there are such modes that, if the premises are true, do not guarantee the truth of the conclusion. Such modes are called incorrect. The right modes are called those modes, with the help of which we always get a true conclusion from true premises. There are 24 correct modes in total - six for each figure. This means that throughout the classical syllogistic, which exhausts the lion's share reasoning produced by people, there are only 24 types of correct conclusions. This is a very small number, so the correct modes are not that hard to remember.

Each of these modes received a special mnemonic name back in the Middle Ages. Each type of categorical attributive statement was designated with just one letter. Statements like "All S are P" were denoted by the letter " a”, the first letter in the Latin word “affirmo” (“I affirm”), and their record turned into “S a P". Statements like "Some S are P" were written with the letter " i", the second vowel in the word "affirm", so they looked like "S i P". Statements of the form "No S is P" were denoted by the letter " e”, the first vowel in the Latin word “nego” (“I deny”), they began to be written in the form “S e P". As you probably already guessed, statements like "Some S are not P" were marked with the letter " O", the second vowel in the word "nego", their formal notation looked like "S o P". Therefore, the modes of regular syllogisms are traditionally denoted precisely with the help of these four letters, which are presented as words for ease of remembering. The table of all correct modes looks like this:

		Figure III

For example, the modus of the second figure Cesare (eae) in expanded form will look like this:

No P is M
All S are M
No S is P

Although 24 modes is not much at all and some regularities can be seen in the table (for example, the modes eao and eio are correct for all figures), it is still difficult to remember it. Fortunately, this is completely optional. You can also use model diagrams to check syllogisms. Only, unlike those schemes that we built earlier, they should already contain not two, but three terms: S, P, M.

Let's take the mode of the fourth figure Bramantip (aai) and check it with the help of model diagrams.

Every P is M
Every M is an S
Some S's are P's

First, you need to find such model schemes for which both premises are simultaneously true. There are only four such schemes:

Now, on each of these diagrams, we have to check whether the statement "Some S are P", representing the conclusion, is true. As a result of the check, we find that on each diagram this statement will be true. Thus, the conclusion according to the Bramantip (aai) modus of the fourth figure is correct. If there were at least one diagram in which this statement would be false, then the conclusion would be wrong.

The method of checking syllogisms with the help of model diagrams is good, as it allows you to visualize the relationship between terms. However, for some premises, many schemes may turn out to be true at once. As a result, their construction and verification will be a laborious and time-consuming task. Thus, the method of model schemes is not always convenient.

Therefore, logicians have developed another method for determining whether a syllogism is correct or not. This method is called syntactic and consists of two lists of rules (rules of terms and rules of premises), under which the syllogism will be true.

Terms rules

A simple categorical syllogism should include only three terms.
The middle term must be distributed in at least one of the premises.
If a major or minor term is not distributed in the premise, then it must also be undistributed in the conclusion.

Parcel rules:

At least one of the premises must be affirmative.
If both premises are affirmative, then the conclusion must be affirmative.
If one of the premises is negative, then the conclusion must also be negative.

The rules of premises are clear, but the rules of terms require some explanation. Let's start with the rule of three terms. Although it seems obvious, it is quite often violated due to the so-called substitution of terms. Look at the following syllogism:

Gold is an element of the 11th group, the sixth period of the periodic system of chemical elements of D. I. Mendeleev, with atomic number 79.
Silence is gold.
Silence is an element of the 11th group, the sixth period of the periodic system of chemical elements of D. I. Mendeleev, with atomic number 79.

First of all, if you remember the figures and the correct modes, you can immediately tell that this syllogism is incorrect, since it refers to the second figure and has a mode aaa, which does not belong to the list of correct modes for this figure. But if you don't remember them, you can still expose its falsity, because there are clearly four terms here instead of three. The term "gold" is used in two completely different senses: as a chemical element and as something of value. Let's look at a more complex example:

All the books from the collection of the Russian State Library cannot be read in a lifetime.
"Fathers and Sons" by Ivan Turgenev - a book from the collection of the Russian State Library.
"Fathers and Sons" by Ivan Turgenev cannot be read in a lifetime.

This syllogism seems to fit the Barbara mode of the first figure. However, the premises are true and the conclusion is false. The problem is that in this example there is again a quadrupling of terms. This syllogism seems to contain three terms. A smaller term is "Fathers and Sons" by Ivan Turgenev. The larger term is "books that one cannot read in a lifetime". The middle term is "books from the collection of the Russian State Library". If you look closely, it becomes clear that the subject of the first premise is not the term "books from the collection of the Russian State Library", but the term " all books from the collection of the Russian State Library. In this case, “all” is not a general quantifier, but a part of the subject, since this word is used not in a separative sense (each separately), but in a collective sense (all together). If we were to replace the word “all” with the words “each one individually”, then the first premise would simply become false: “Each individual book from the collection of the Russian State Library cannot be read in a lifetime.” Thus, we get four terms instead of three, and therefore this conclusion is false.

Now let's move on to the rules about the distribution of terms. First, let's explain what this feature is. A term is called distributed if the statement refers to all objects included in its scope. Accordingly, the term is not distributed if the statement does not refer to all the objects that make up its volume. Roughly speaking, a term is distributed if we are talking about all objects, and not distributed if we are talking about only some objects, about a part of the scope of the term.

Let's take the types of statements and see which terms are distributed in them and which are not. A distributed term is marked with a “+” sign, an unallocated term is marked with a “-” sign.

All S + are P - .

No S + is P + .

Some S - are P - .

Some S - are not P + .

and + is P - .

a + is not P + .

As you can see, the subject is always distributed in general and singular statements, but not distributed in particular ones. The predicate is always distributed in negative statements, but not distributed in affirmative ones. If we now transfer this to our rules for terms, then it turns out that the middle term in at least one of the premises must be taken in its entirety.

Penguins are birds.
Some birds cannot fly.
Penguins can't fly.

Although both the statements above the line and the statement below the line are true, there is no inference as such. There is no logical transition from premises to conclusion. And this can be easily identified, since the middle term "birds" is never taken in its entirety.

As for the third rule of terms, if the premises deal with only a part of the objects from the scope of terms, then in the conclusion we cannot say anything about all the objects of the scope of terms. We cannot move from the part to the whole. By the way, the reverse transition is possible: if we are talking about all the elements of the scope of terms, then we can draw a conclusion about some of them.

Enthymemes

During real discussions and disputes, we quite often omit certain parts of the argument. This leads to the emergence of enthymemes. An enthymeme is an abbreviated form of inference that omits premises or a conclusion. It is important not to confuse enthymemes with single-terminal inferences. An enthymeme is precisely a multi-message inference; its parts are simply omitted for one reason or another. Sometimes such omissions are justified, since both interlocutors are well versed in the problem, and they do not need to pronounce all the steps. Meanwhile, unscrupulous interlocutors may deliberately use enthymemes to obscure and confuse their reasoning and hide their true arguments or conclusions. Therefore, it is necessary to be able to distinguish correct enthymemes from incorrect ones. An enthymeme is called correct if it can be restored as a correct mode of a categorical syllogism, and if all the missing premises turn out to be true.

Let's talk about how to restore the enthymeme to a complete syllogism. First of all, you need to understand what exactly is missing. To do this, you need to pay attention to the marker words denoting causal relationships: “thus”, “hence”, “because”, “because”, “as a result”, etc. For example, let's take the argument: "Gold is a precious metal, because it practically does not oxidize in air." Here the conclusion is the statement "Gold is a precious metal". One of the premises: "Gold practically does not oxidize in air." Another shipment missed. I must say that most often they miss exactly one of the parcels. It is rather strange if the most important thing is missing in the reasoning - the conclusion.

So, we have established what exactly is missing. In our example, this is the package. Is it a big parcel or a smaller one? As you remember, the minor premise contains the subject of the conclusion ("gold"), and the major one contains the predicate of the conclusion ("precious metal"). We already know the premise containing the subject of the conclusion: "Gold practically does not oxidize in air." This means that we know the smaller premise, and we do not know the larger one. In addition, thanks to a well-known premise, we can also establish the middle term: "metals that practically do not oxidize in air," a term that is not contained in the conclusion.

Now we have the information known to us in the form of a syllogism:

3. Gold is a precious metal.

Or in diagram form:

2.S a M
3. S a P

The major premise must contain the predicate of the conclusion and the middle term: "precious metals" (P) and "metals that oxidize in air" (M). There are two options here:

1. P M
2.S a M
3. S a P

1. M P
2.S a M
3. S a P

This means that either the second figure or the first figure is a syllogism. Now we look at our tablet with the correct modes of syllogisms. In the second figure, there are no regular modes at all, where in the conclusion there would be a statement like a. There is only one such mode in the first figure - Barbara. We complete our syllogism:

1M a P
2.S a M
3. S a P

1. All metals that practically do not oxidize in air are precious.
2. Gold practically does not oxidize in air.
3. Gold is a precious metal.

Now we check if our restored premise is true. In our case, it is true, so the enthymeme was correct.

Sorites

The term sorites was used by Lewis Carroll to refer to complex syllogisms that have more than two premises. By and large, sorite is a hybrid of syllogism and enthymeme. It is structured as follows: a set of premises is given, intermediate conclusions are drawn from each pair of premises, which are usually omitted, new premises are added to the intermediate conclusions, new intermediate conclusions are made from them, to which new premises are again attached, and so on, until we sort through all available parcels and will not reach the final conclusion. In principle, people argue in this way in Everyday life. Therefore, it is very important to be able to solve sorites and evaluate whether they are correct or not.

We will give an example of a sorite from Lewis Carroll's book "The Knot Story":

2. A person with long hair cannot but be a poet.
3. Amos Judd never went to jail.

5. There are no other poets in this district, except for policemen.
6. No one dine with our cook except her cousins.

8. Amos Judd loves cold lamb.

Above the line are the premises, below the line is the conclusion.

How do you decide and check sorites? Let's give step by step instructions. First, it is necessary to bring all the parcels into a more or less standard form:

1. All the policemen from our district dine with our cook.
2. All people with long hair are poets.
3. Amos Judd was not in jail.
4. All cousins of our cook love cold lamb.
5. All poets from our district are policemen.
6. All the people who dine with our cook are her cousins.
7. All people with short hair were in prison.

Now we need to take two initial premises. By and large, it doesn’t matter what kind of packages you start with. The main thing is that your initial premises together contain only three terms. This means that we cannot accept the packages "Amos Judd hasn't been in jail" and "All our cook's cousins like cold lamb". They include four different terms, and therefore we cannot draw any conclusion from them. I will take premises 7 and 3 as initial ones and draw a conclusion from them according to the rules for simple categorical syllogisms.

1. All people with short hair were in prison.
2. Amos Judd was not in jail.
3. Amos Judd is not a man with short hair.

This syllogism corresponds to the mode Camestres (aee) of the second figure. Now, for convenience, I will reformulate our intermediate conclusion as follows: "Amos Judd is a man with long hair." I connect this intermediate conclusion with premise number 2:

1. All people with long hair are poets.
2. Amos Judd is a man with long hair.
3. Amos Judd is a poet.

This syllogism corresponds to the Barbara mode (aaa) of the first figure. Now I'm attaching this intermediate output to parcel number 5:

1. All the poets in our district are policemen.
2. Amos Judd is a poet.
3. Amos Judd is a policeman.

This syllogism again corresponds to the Barbara mode (aaa) of the first figure. We attach an intermediate conclusion to the parcel number 1:

1. All the policemen from our district dine with our cook.
2. Amos Judd is a policeman.
3. Amos Judd is having dinner with our cook.

This syllogism, as you probably already noticed, also represents the Barbara (aaa) mode of the first figure. We attach this conclusion to premise number 6:

1. All the people who dine with our cook are her cousins.
2. Amos Judd is having dinner with our cook.
3. Amos Judd is the cousin of our cook.

Again Barbara, which is one of the most common mods. We attach the last premise number 4 to our last intermediate conclusion:

1. All cousins of our cook love cold lamb.
2. Amos Judd is the cousin of our cook.
3. Amos Judd loves cold lamb.

So, with the help of the same Barbara mode, we got our conclusion: "Amos Judd likes cold lamb." Thus, sorites are solved and tested by stepwise division into simple categorical syllogisms. In our example, the sorite turned out to be correct, but reverse situations are also possible. There are two conditions for correctness of sorites. First, each sorite must be broken down into a sequence of regular modes of syllogisms. Secondly, the conclusion you get when all the premises have been exhausted must be the same as the conclusion of the sorite. This condition is valid in those cases when you are dealing with someone else's reasoning, in which there is already some kind of conclusion.

So, we have considered various multi-premise inferences on the example of simple categorical syllogisms, enthymemes and sorites. By and large, if you know how to deal with them, then you are armed for any discussion with any opponents. The only thing that can cause some dissatisfaction at the moment is the need to spend a lot of time checking the correctness of the conclusions. Do not be upset about this: it is better to look slow-witted, who argues correctly, than a brilliant demagogue who does not notice his own and other people's mistakes. Moreover, with the accumulation of experience of an attentive attitude to conclusions, you will have a flair, an automatic skill that allows you to quickly separate correct arguments from incorrect ones. Therefore, there will be a lot of exercises for this lesson so that you have the opportunity to fill your hand.

Einstein's problems

This game is our version of the world-famous "Einstein's riddle" in which 5 foreigners live in 5 streets, eat 5 types of food, and so on. Read more about this task here. In such tasks, you need to make the correct conclusion based on the premises that, at first glance, are not enough for this.

Exercises

Exercises 1, 2 and 3 are taken from Lewis Carroll's book "History with knots", M .: Mir, 1973.

Exercise 1

Make conclusions from the following premises according to the rules for a simple categorical syllogism. Remember that a simple categorical syllogism should contain only three terms. Do not forget to bring statements to the standard form.

An umbrella is a very necessary thing when traveling.
When you go on a trip, leave everything you don't need at home.

The music that can be heard causes vibrations in the air.
Music that cannot be heard is not worth paying money for.

No Frenchman likes pudding.
All English people love pudding.

No old curmudgeon is cheerful.
Some old curmudgeons are skinny.

All non-gluttonous rabbits are black.
No old rabbit is inclined to abstinence in food.

Nothing sensible has ever puzzled me.
The logic baffles me.

None of the countries explored so far are inhabited by dragons.
Unexplored countries captivate the imagination.

Some dreams are terrible.
Not a single lamb inspires horror.

No bald creature needs a comb.
None of the lizards have hair.

All eggs can be broken.
Some eggs are hard boiled.

Exercise 2

Check if the following reasoning is correct. Try different verification methods. Don't forget to put the big premise on the first line.

Dictionaries are helpful.
Useful books are highly valued.
Dictionaries are highly valued.

Heavy gold.
Nothing but gold can silence him.
Nothing light can silence him.

Some ties are tasteless.
Anything done with taste delights me.
I don't like some ties.

No fossil animal can be unhappy in love.
An oyster can be unhappy in love.
Oysters are not fossil animals.

None of the hot muffins are helpful.
All raisin buns are useless.
Buns with raisins - not a muffin.

Some of the pillows are soft.
None of the pokers are soft.
Some pokers are not pillows.

Boring people are unbearable.
No boring person is begged to stay when he is about to leave the guests.
No intolerable person is begged to stay when he is about to leave the guests.

No frog has a poetic appearance.
Some ducks look prosaic.
Some ducks are not frogs.

All intelligent people walk with their feet.
All foolish people walk on their heads.
No man walks on his head and legs.

Exercise 3

Find the conclusions of the following sorites.

Little children are unintelligent.
Anyone who can tame crocodiles deserves respect.
Unreasonable people do not deserve respect.

Not a single duck waltzes.
Not a single officer will refuse to dance a waltz.
I have no other bird but ducks.

Anyone who is of sound mind can do logic.
No sleepwalker can be a juror.
None of your sons can do logic.

This box does not contain my pencils.
None of my lollipops are cigars.
All my property not in this box consists of cigars.

No terrier wanders among the signs of the Zodiac.
That which does not wander among the signs of the Zodiac cannot be a comet.
Only the terrier has a ring tail.

No one subscribes to The Times unless he has received a good education.
Not a single porcupine can read.
Those who cannot read have not received a good education.

No one who really appreciates Beethoven will make noise during the performance of the Moonlight Sonata.
Guinea pigs are hopelessly ignorant of music.
Those who are hopelessly ignorant of music will not observe silence during the Moonlight Sonata.

Things sold on the street are of little value.
Only rubbish can be bought for a penny.
The eggs of the great auk are of great value.
Only what is sold on the street is real rubbish.

Those who break their promises are not trustworthy.
Drinkers are very sociable.
A person who keeps his promises is honest.
No teetotaler is a usurer.
Someone who is very sociable can always be trusted.

Any thought that cannot be expressed as a syllogism is truly ridiculous.
My dream of buns is not worth writing down on paper.
None of my impossible dreams can be expressed as a syllogism.
I didn't have a single really funny thought that I wouldn't tell my friend about.
All I dream about is sweet buns.
I never expressed a single thought to my friend if it was not worth putting down on paper.

Exercise 4

Check the correctness of the following enthymemes.

Barsik is not a law-abiding cat, because he stole a sausage from me.
Mercury is liquid, therefore, it cannot be a metal.
No obedient child throws tantrums over trifles. Therefore, Tolya is a naughty child.
Some women are stupid, so some men can take advantage of it.
All girls want to get married, as each of them dreams of a fluffy white dress.
No student wants to get an A on an exam, which is why all students are nerds.
Someone stole my wallet, so I didn't have any money left.
Peacocks are narcissistic birds because they have a big beautiful tail.

Test your knowledge

If you want to test your knowledge on the topic of this lesson, you can take a short test consisting of several questions. Only 1 option can be correct for each question. After you select one of the options, the system automatically moves on to the next question. The points you receive are affected by the correctness of your answers and the time spent on passing. Please note that the questions are different each time, and the options are shuffled.

It might be useful to read: