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Conclusion
Plan:
1. The essence of conclusion.
2. Types of conclusions:
a) making a deductive conclusion;
b) making an inductive conclusion;
c) analogy.
In the process of knowing reality, a person acquires new knowledge. This knowledge is created with the help of abstract thinking, based on existing knowledge. Generating such knowledge is called inference in the science of logic.
Inferring is a form of thinking that consists of generating new knowledge from one or more true considerations using certain rules.
The process of deriving a conclusion consists of premises, conclusion and transition from premises to conclusion. In order to draw a correct conclusion, first of all, the grounds must be true considerations, logically connected.
For example, it is impossible to draw a conclusion from two true statements: "Aristotle is the founder of the science of logic" and "Plato is a Greek philosopher." Because there is no logical connection between these considerations.
The grounds of the conclusion and the conclusion must be logically connected. The necessity of such connection will be noted in the conclusion rules. If these rules are violated, the correct conclusion will not be reached. For example, from the statement "Student A is excellent" it is not possible to conclude that "Student A is polite".
Conclusions are divided into several types according to the level of truth of the conclusion, more precisely, according to the strictness of the rules of conclusion, as well as the number of grounds of the conclusion and the direction of the thought.
In this classification, the division of inference into types according to the direction of thought is more perfect, and it provides an opportunity to provide information about other types of inference. In particular, deductive inference can be considered as necessary inference, inductive inference (not including complete induction) and analogy as probable inference, and direct inference can be studied as a type of deductive inference.
a) DEDUCTIVE CONCLUSION
An important feature of making a deductive conclusion is that the transition from general knowledge to partial knowledge is logically necessary. One of its types is direct inference.
The creation of new knowledge based on only one reasoning is called direct inference. Direct inference is expressed in symbolic logic as follows: XSPYSP, where X and Y represent simple fixed propositions (A, E, I, O), and S and P represent the subject and predicate of propositions. XSP is the basis of conclusion or antecedent, YSP is called conclusion or consequent. In the process of drawing direct conclusions, new knowledge is obtained by changing the form of considerations. In this case, the structure of the basic reasoning, that is, the quantitative and qualitative characteristics of the subject and predicate relations, is of great importance. There are the following logical methods of direct inference:
1. Reversal (lat.-obversio) is a logical method in which a new opinion is created by changing its quality while maintaining the quantity of the given opinion. A double negation occurs when the conclusion is drawn in this way, that is, first the predicate of the premise is negated, then the conjunction. This can be written as:
In the process of negation, negative predicates (-ma; -siz; mas) or concepts that contradict the concept being negated are used. A conclusion is drawn from all of the simple fixed considerations by the conversion method. The reasoning that is the basis of the conclusion is expressed in the conclusion as follows:
Conclusion Basis Conclusion
1 A all SP E none SP mas
2 E is no SP All A is P not S
3 I Some SP O Some SP not you
4 O Some are not SP I Some are not S-P
In rotation, it changes to A-Ye, Ye-A, IO, OI.
For example:
1. A. all scientific laws are objective in nature.
Eat. no scientific law is subjective.
2. Eat. no generous miser.
A. All ungenerous people are misers.
3. I. Some concepts are concrete in content.
0. Some concepts are not abstract in content.
4. 0. Some considerations are not complicated.
I. Some considerations are simple.
Therefore, when a conclusion is made by the conversion method, it is based on the rule that "the double negation of something is equal to its confirmation".
II. Conversion (lat.-conversio) is a method of making a logical conclusion, in which the conclusion is generated by replacing the subject and predicate in the given reasoning.
It is necessary to take into account the size of the terms in the judgment given in the exchange. If the size of the terms in the given reasoning is not taken into account, the conclusion may be incorrect: For example,
All people are living beings
all living things are human beings
The conclusion is wrong, because R — (living things) is not taken to its full extent in the given reasoning, but it is taken to its full extent in the conclusion. From the above premise, the conclusion that "Some living beings are human beings" is correct. Accordingly, three types of substitution are distinguished: narrowed, extended and pure substitution.
Summary Basis Summary Substitution Type
1 A total SP A total PS Pure substitution
2 Yes No SP No PS Pure replacement
3 I Some SP I Not some PS Pure substitution
4 A all SP I Some PS Condensed replacement
5 I Some SP A all PS Advanced replacement
Let's consider the above scheme with examples.
1. All living things have the ability to feel.
A. All sentient beings are living beings.
2. Eat. no miser is generous.
Eat. no generous miser.
3. I Some philosophers are naturalists.
I. Some naturalists are philosophers.
4. A. all doctors are highly educated.
I. Some highly educated people are doctors.
5. I. Some people are poets.
all poets are human.
A partial negative proposition (O) cannot be deduced by substitution, since the predicate of this proposition is taken in its entirety. Therefore, it should be taken in its entirety in the conclusion, that is, the conclusion should be a general negative reasoning (Ye). In that case, the predicate of the conclusion will have to be taken in full size, which is impossible, because it is not taken in full size in the subject of the premise. For example:
O. Some philosophers are not logicians.
Eat. no logician is a philosopher.
or
O. Some logicians are not philosophers.
In both cases, the conclusion is wrong.
So, when the substitution method is used, the size of the subject and predicate in the reasoning is determined, and on this basis, the terms in the reasoning are replaced and a conclusion is drawn. This method is especially important in determining the correctness of the definitions given to the concept.
III. Contraposition to the predicate (lat. contrapositio) is one of the logical methods of direct inference, when this method is used, the given judgment is first turned, and then replaced. As a result, the subject of the resulting reasoning (conclusion) contradicts the predicate of the basic reasoning, and the predicate corresponds to its subject:
In this case, the negative form of S in the conclusion is the result of the negation of the conclusion connector. When contrasted with a predicate, it changes to A-Ye, Ye-I, 0-I
Drawing conclusions from various considerations using this method is shown in the following scheme:
Conclusion Basis Conclusion
1 A all SP none P none S
2 Ye is no SP Some R is S not
3 O Some are not SP Some are S not P
For example,
1. A. All the sentences are expressed by a prepositional phrase.
Eat. An opinion not expressed by a sentence is not a judgment.
2. Eat. No patriot betrays his country.
I. Some who do not betray their country are patriots.
3. O. Some students are not philosophers.
I. Some non-philosophers are students.
When a conclusion is drawn from a partial negative reasoning by the method of contrasting the predicate, it is necessary to take into account that it is not possible to draw a conclusion from this reasoning by the method of substitution. Therefore from reasoning O
Not in the form 'Some are not SP' but in the form 'Some are not S–P'
"Some are not RS", "Some are R not S"
a conclusion is made in the form
A partial statement (I) cannot be deduced from the predicate by contrast. Because, if we turn the reasoning "Some SP is not mas", i.e. a partial negative verdict will come. It cannot be deduced by substitution.
Making a conclusion through a logical square.
In this case, taking into account the mutual relations of simple strict judgments (see: logical square), a conclusion is made about the truth or falsity of one of the judgments. These conclusions are based on relations of contradiction, opposition, partial correspondence and subordination between judgments.
Making a conclusion based on contradiction (contradiction) relations. It is known that there is a contradiction between the considerations of AO and Ye I, and the third is subject to the law of exclusion. According to this relation, if one proposition is true, the other is false, and conversely, if one is false, the other is true. Conclusions are drawn up according to the following scheme:
For example,
A. All people have the right to life
0. Some people do not have the right to live.
I. Some philosophers are statesmen.
Eat. no philosopher is a statesman.
In this example, it follows from the truth of the premise that the conclusion is false (by the third law of exclusion).
draw a conclusion based on opposition (contrary) relations. the relation of opposition exists between considerations A and E and is subject to the law of contradiction. From the truth of one of the statements in this relationship, it is concluded that the other is false. But the error of one does not justify the truth of the other, because both opinions can be wrong. For example, from the truth of the general statement (A) that "all people want to live well", the error of the general negative statement (Ye) that "no one wants to live well" follows.
A. All concepts are concrete.
Eat. no concept is concrete.
In this example, the premise is a judgment and the conclusion is a fallacy. So, it is possible to draw a conclusion from the opposite relationship.
drawing a conclusion based on the partial correspondence (subcontrary) relationship. This relationship exists between partially affirmative (I) and partially negative (O) judgments. Both of these statements can be true at the same time, but not false at the same time. If one of them is clearly wrong, the other is true. it appears to draw a conclusion based on a partial correspondence relationship.
For example:
O. Some scientific laws are not objective in nature.
I. Some scientific laws are objective in nature.
In this case, the conclusion is true because the premise is false.
I. Some philosophers are statesmen.
O. Some philosophers are not statesmen.
In this example, the basis is both reasoning and conclusion. Sometimes it is impossible to determine the truth or falsity of the conclusion when the premise is true.
Making a conclusion based on the relationship of subordination. This relationship exists between general and partial judgments (A and I; Ye and O) that have the same qualities. If the general subordinating propositions are true, the partial subordinating propositions are also true. But it is not possible to draw a conclusion about the truth of the subordinate - partial judgments, or the truth of the subordinate - general judgments. Because in such a case general considerations can be true or false. Accordingly, the conclusion based on the attitude of submission will be as follows:
A I; Ye O.
For example:
A. All independent countries are members of the UN.
I. Some independent countries are members of the UN.
Since statement A is true, statement I is also true.
O. Some Uzbek women do not have higher education.
Eat. no Uzbek woman has a higher education.
In this example, O is true, but E is false.
Summarizing the above relations, the following situations can be indicated according to the level of truth of the basic reasoning and conclusion.
1. The basic reasoning and conclusion are true:
A — I, Ye — I.
2. The premise is true and the conclusion is false:
3. The premise is false and the conclusion is true.
When drawing a conclusion through a logical square, when one of the propositions in the relation of contradiction is false, when one of the propositions in the relation of partial compatibility is true, and when the partial propositions in the relation of subordination are true, the conclusion drawn from them is uncertain.
Direct inference methods provide an opportunity to identify an existing idea in knowledge, to correctly understand its essence, as well as to express the same idea in different ways, to create new knowledge.
A simple strict syllogism.
As you know, deductive reasoning is actually in the form of a syllogism. Syllogism means to calculate by addition. The term is used in logic to denote a simple strict syllogism, which is generally considered to be the more commonly used type of deductive reasoning. A syllogism is a form of inference in which a third, new, logical conclusion necessarily follows from two logically connected statements. In this case, one of the first judgments will necessarily be either a general affirmative or a general negative. the new reasoning produced will not be more general than the original reasoning. Accordingly, a syllogism can be called a conclusion based on generality. For example, given the following considerations:
no miser is generous.
Some rich people are greedy.
From these considerations, the third consideration necessarily follows - "Some rich people are not generous." Since the composition of the syllogism consists of simple strict propositions, it is called a simple strict syllogism.
The composition of the syllogism consists of premises (praemissae) and conclusion (conslusio). The bases of the conclusion and the concepts in the conclusion are called terms. The logical owner of the conclusion - S - is called a small term (terminus minor), its logical section - R - is called a large term (terminus major). The concept common to the bases of the conclusion, but not found in the conclusion – M — (terminus medius) is called the middle term. In bases, reasoning containing a large term is called a large basis, and reasoning containing a small term is called a minor basis.
Whether terms are called big or small depends on the size of the concepts they represent. The relationship between the terms can be expressed using circles as follows.
S is a small term.
M is the middle term.
R is a big term.
the middle term is considered the element that logically connects the large and small terms.
Axiom of syllogism.
Axioms are theoretical propositions accepted as true without proof, by means of which other ideas and propositions are justified. The axiom of the syllogism expresses the logical basis of the conclusion. The axiom of the syllogism can be defined according to the size or content of the terms, i.e. attributive.
The necessary derivation of the conclusion of the syllogism from the premises is based on the following rule: "if one item is located in the second item, and the second item is inside the third item, then the first item is also located inside the third item" or "one item is in the second item is located, and the second item is outside of the third item, then the first item is also located outside of the third item." This rule can be clearly expressed using the following schemes.
This rule explains the essence of the axiom of the syllogism based on the size relationship of the terms. The essence of the syllogism axiom is as follows: an opinion expressed affirmatively or negatively about a class of objects and events is considered an affirmative or negative opinion that applies to each or some part of all objects and events included in this class.
For example:
Thought forms are objective in nature.
A concept is a form of thought.
The concept has an objective character.
When expressing the syllogism axiom attributively, it is based on the relationship between the subject and its sign: an object is a sign of a sign of an event, this object is a sign of an event; Things that are contrary to the sign of an object or event are also contrary to the object or event itself.
In the axioms of the syllogism, the form and content of the thought represent some aspects of a continuous, objectively connected whole. On the one hand, this means that all generalities are characterized by specificity, partiality, and individuality, and that each individuality, partiality, and specificity have the character of generality, and on the other hand, that the object and the sign are inextricably linked, i.e. if a class of goods has a certain characteristic sign, this sign means that it will be a common sign for all objects of this class. These, in turn, are a unique manifestation of the dialectical relationship between individuality and generality, between quantity and quality in the process of thinking.
General rules of syllogism.
The fact that the premises of the conclusion are true is not sufficient for the conclusion to be true. In order for the conclusion to be true, it is necessary to follow certain rules. This is called the general rule of syllogism. They are the rules that apply to the terms and premises of a syllogism and include:
1. A syllogism should have three terms: major, minor and middle terms. It is known that the conclusion of a syllogism is based on the relationship of the major and minor terms to the middle term; for this reason, the number of terms is required to be no more or less than three. If the number of terms is less than three, the conclusion does not provide new knowledge.
For example: all orators have mastered the art of words.
Among those who master the art of words, there are also speakers.
No conclusion can be drawn from these two considerations because the number of terms is two. Exceeding the number of terms by three is associated with a violation of the requirements of the law of specifics and leads to the so-called quaternion of terms (quarternio termunorum):
It is a political expression of state-economic relations.
health is the greatest state for every person.
In these considerations, the use of the concept of "state" in two different meanings does not allow the logical connection of peripheral terms. The presence of more than three terms also causes a break in the logical connection between the bases:
All speakers are ambitious.
Cicero was a statesman.
No conclusion can be drawn from these two considerations because they are not logically connected.
2. the middle term must be taken in full in at least one of the bases.
If the middle term is by no means taken to its full extent, the relation of the outer terms will be uncertain, and the truth or falsity of the conclusion cannot be ascertained.
Some philosophers are eloquent.
All members of our department are philosophers.
In this syllogism, the middle term is the subject of the partial sentence in the major premise and the predicate of the general affirmative sentence in the minor premise. Therefore, the relationship between the marginal terms is not defined. The conclusions drawn from these premises are ambiguous:
a) All members of our department are speakers.
b) Some members of our department are speakers.
3. The large and small terms should be in the same size in the conclusion as they are in the bases.
Violation of this rule results in inappropriate expansion of the small or large term size. For example:
all students take the exam.
no applicant is a student.
no applicant will take the exam.
In this example, an inappropriate expansion of the size of the small term caused the conclusion to be erroneous.
4. It is not possible to draw a conclusion from two negative judgments (premise). For example:
The unemployed are not entrepreneurs.
Students are not unemployed.
?
5. It is not possible to draw conclusions from two judgments. For example:
Some women are entrepreneurs.
Some statesmen are women.
?
6. If one of the grounds is a negative sentence, the conclusion is also a negative sentence. For example:
no crime goes unpunished.
Treason is a crime
Treason does not go unpunished.
7. If one of the grounds is a partial verdict, the conclusion is also a partial verdict. For example:
A good child respects his parents.
Some young people are good children.
Some young people respect their parents.
Figures and modes of syllogism.
Four figures of the syllogism are distinguished depending on the location of the middle term in the structure of a simple strict syllogism.
Figure I Figure II Figure III Figure IV
MP
SMRM
SMMP
MSPM
MS
SP SP SP SP
In figure I, the middle term is the subject of the large base and the predicate of the small base.
In Figure II, the middle term is the predicate of the major and minor bases.
In Figure III, the middle term is the subject of both bases.
In Figure IV, the middle term is the predicate of the large base, the subject of the small base.
The bases of syllogism consist of simple fixed sentences (A, Ye, I, 0). The occurrence of these judgments in a specific order (set) in two premises and conclusion is called modus. "Modus" means form. There are specific modes of syllogism figures. in determining the correct modes of each figure, in drawing a correct conclusion, along with the general rules of syllogism, the special rules of each figure are followed. The special rules of the figures are determined based on the specific connection of the terms of the syllogism.
The first figure of a simple strict syllogism has the following special rules:
1. There must be a general judgment of the great foundation.
2. The minor ground must be an affirmative judgment.
There are four correct modes of figure I:
AAA, Ye A Ye, AII, Ye I 0.
The first letter of the modus indicates the major basis, the second letter indicates the minor basis, and the third letter indicates the quality and quantity of the conclusion. In order to distinguish the modes of figures from each other, each of them is called by a different name.
AAA — Barbara's mode
A. all scientific laws are objective in nature.
A. The laws of thought are scientific laws.
A. The laws of thought are objective in nature.
YEAYE is Celarent mode.
Eat. no religious person is an atheist.
A. Imams are religious.
No imam is an atheist.
AII — Darii mode.
A All criminals deserve punishment.
I. Some people are criminals.
I. Some people deserve punishment.
YEIO — Ferio mode.
Eat. None of the moral people is without conscience.
I. Some young people are moral people.
O. Some young people are not without conscience.
The first figure of the syllogism gives conclusions for all types of simple definite sentences.
Figure II of a simple strict syllogism has the following special rules:
1. There must be a general judgment of the great foundation.
2. One of the grounds must be a negative sentence.
Figure II has four correct modes:
AYEE, YEAYE, AOO, YEIO.
AYEE is the mode of Camestres
A. All sentences are expressed by a prepositional phrase.
Eat. The question is not expressed through a sentence.
Eat. no question is a judgment.
YEAYE is Cesare's mode.
Eat. no atheist is religious.
A. Imams are religious.
Eat. no imam is an atheist.
AOO — Baroque mode.
A. all birds fly.
O. Some creatures do not fly.
O. Some creatures are not birds.
YEIO-Festino mode.
None of those who do not obey the laws are free.
I. Some citizens are free
O. Some citizens are not law abiding.
As can be seen from the above examples, the conclusions of figure II of the syllogism consist only of a negative sentence.
Figure III of the simple strict syllogism has one special rule: the minor premise must be an affirmative sentence.
The correct modes of figure III are six:
AAI, AII, IAI, YEAO, YEIO, OAO.
AAI — Darapti mode.
A. All logicians are philosophers.
A. All logicians are learned people.
I. Some learned people are philosophers.
AII — Datisi mode.
A. All simple definite sentences are grounds of conclusion.
I. Some simple fixed judgments are true opinions.
I. Some true ideas are grounds for conclusions.
IAI — Disamis mode.
I Some philosophers were logicians.
A. All philosophers are people of knowledge.
I. Some learned people were logicians.
YEAO — Phelapton mode.
Eat. no party can function without a program.
A. All parties are political organizations.
O. Some political organizations do not work without a program.
YEIO — Ferison mode.
Eat. no believer is without faith.
I. Some believers are young people.
O. Some young people are not without faith.
OAO — Vocardo mode.
O Some people don't tell the truth.
All people want to live well.
O Some people who want to live well are not telling the truth.
Conclusions of modes of figure III will consist of only a partial judgment.
Figure IV of a simple strict syllogism has the following special rules:
1. If one of the grounds is a negative sentence, the major ground is a general sentence.
2. If the major premise is an affirmative sentence, the minor premise is a general sentence.
There are five correct modes of figure IV:
AAI, AYEE, IAI, YEAO, YEIO.
AAI is Brahmalip mode.
A. All honest people are honest.
A. All conscientious people are fair people.
I. Some just people are honest people.
AYEE-Camenes mode.
A. Open-handed people are generous.
Eat. no generous miser.
Eat. no greedy person is open-handed.
IAI is Dimaris mode.
I. Some young people play sports.
A. All those who do sports are healthy people.
I. Some healthy people are young.
YEAO — Fesapo mode.
Eat. no sophist speaks the truth.
A. All those who do not tell the truth are liars.
O. Some liars are not sophists.
YEIO — Freesison mode.
Eat. no intelligent person is without knowledge.
I. Some scientists are young.
O. Some young people are not smart people.
Figure IV of the syllogism does not provide a conclusion in the form of a general affirmative sentence.
Transforming imperfect syllogisms into perfect syllogisms
Since Aristotle, all logicians have paid great attention to the I-figure of the syllogism and its modes. They considered the I-figure to be perfect, their conclusions were clear and obvious. They considered that other figures of syllogism are imperfect and that it is necessary to bring them to figure I in order to determine the truth of their conclusions. When this logic is executed, the name of the modes is taken care of:
1. If the name of the mode contains the letter "s", then the sentence expressed by the preceding vowel must be completely replaced (conversio simplex).
2. If the name of the modus contains the letter "r", the sentence expressed by the preceding vowel is partially replaced (per accidens).
3. If the name of the mode contains the letter "m", then it is necessary to replace the premises of the syllogism (metathesis or mutatio pramissarum).
4. The initial letters of the modes (B, C, D, F) indicate which mode of the I-figure they are brought to. The Cesare, Camestres, and Camenes modes of figures II and IV are brought into the Celarent mode of figure I. The Darapti and Disamis modes of figure II are brought to the Darii mode of figure 1, and Fresission to the Ferio mode of figure 1.
5. The letter "k" in the name of the mode indicates that this mode is proved by a separate method through one of the modes of figure I. This method is called Reductio ad absurdum.
Now let's look at some examples based on these rules:
The Cesare mode of figure II is brought into the Celarent mode of figure I (Rule 4). According to rule 1, the large base of figure II is completely replaced.
Figure II Cesare Figure I Celerent
E. no RM Ye. not any MR.
A. all SM A. all SM
E. not any SP E. not any SP
A comparison of the schemes shows that figure II is brought to figure I by a complete replacement of the large base.
For example,
no animal is a conscious being.
Man is a conscious being
no man is an animal.
no sentient being is an animal.
Man is a conscious being
no man is an animal.
Another example. We bring the Darapti mode of figure III to the Darii mode of figure I. The minor base in Darapti is partially replaced (Rule 2).
Figure III Darapti Figure I Darii
A. all MR A. all MR
A. All MS I. Some SM
I. Some SP E. Some SR
For example,
all logicians are philosophers. all logicians are philosophers.
all logicians are learned men Some learned men
is a psychologist
Some learned people are philosophers. Some learned people are philosophers.
The Bramanlip mode of figure IV is brought to the Barbara mode of figure I by transposing the bases (Rule 3)
Figure IV Bramanlip Figure I Barbara
A. all RM A. all MS
A. all MS A. all RM
I. Some SP A. all SH
For example,
A. All honest people are honest.
A. All conscientious people are fair people.
I. Some just people are honest people.
A. All conscientious people are fair people.
A. All honest people are honest.
A. All honest people are fair people.
It is explained by rule 2 that the partial conclusion in Figure IV takes the form of the general conclusion in Figure I.
Now we bring the Camestres mode of figure II to the Celarent mode of figure I. For this, we use the third and first rules, that is, we change the positions of the bases of figure II and completely replace the small base.
Figure II Camestres Figure I Celarent
A. all RM Ye. not any MS
Eat. none SM A. all RM
Eat. not any SP Ye. not any RS or
not any SP
For example,
all people are living beings. no living thing is a stone.
no stone is a living thing. Every human being is a living being.
no stone is human. no man is a stone.
The reductio ad absurdum method is related to rule 5, that is, it is used in cases where the name of the mode has the letter "k". Examples of such modes are Baroque mode of figure II and Bocardo mode of figure III. These modes are brought to the Barbara mode of figure I. The reductio ad absurdum method is used in this case. The essence of this method is as follows: we come to a certain conclusion from two bases. Someone will deny that the conclusion is correct. We must prove that this denial is absurd. To do this, we argue that the conclusion cannot be denied, while admitting the grounds of the conclusion. For example:
Figure II Baroque
A. all RM
O. Not some SM.
O. So not some SP.
The conclusion that "not some SR" is denied. Then the sentence that contradicts this conclusion should be accepted as true: "all SR" is the true sentence. A sentence contrary to the conclusion is taken as a small basis. As a result, a Barbara modus syllogism with the middle term "R" is formed:
A. all RM
A. everyone SR
A. all SM
Thus, if the initial conclusion is denied, the conclusion that "everything is SM" is reached. But this conclusion contradicts the premise of the original syllogism. As a result, those who admit the premises of the initial syllogism and deny its conclusion face a contradiction. Thus, we reasoned that their objections were "absurd", that is, ad absurdum.
The Bocardo mode of figure III is also brought to figure I by the same method.
Bocardo:
O. Not some MR.
A. all MS
O. Not some SR.
If the truth of the conclusion "Some are not SR" is denied, the conclusion "all SP" contradicting it is considered true. This sentence together with the premise "all MS" forms a syllogism with "S" as its middle term:
A. all SP
A. all MS
A. everyone MR.
Thus, the conclusion drawn contradicts the premise that "Some MRs are not". Since the premises of the first syllogism are true, the conclusion of the next syllogism is false.
We can take the following example.
Figure III Bocardo
O. Some philosophers are not naturalists.
A. All philosophers are human.
O. Some people are not naturalists.
If the truth of the conclusion of this syllogism is denied, then the argument that contradicts it, "all people are naturalists", must be true. Substituting this reasoning for the major premise and combining it with the minor premise, we get Barbara's syllogism:
A. All people are naturalists.
A. All philosophers are human.
A. All philosophers are naturalists.
The conclusion of this syllogism contradicts the larger premise of the original syllogism, which is nonsense because the premise of the original syllogism is accepted as true. So, the conclusion of the initial syllogism was proved to be incorrect and absurd.
Thus, the truth of the modes of this syllogism can be established by bringing the modes of figure II, III, and IV into figure I.
Common mistakes in syllogistic inference.
When the small basis of the I-figure is negative, the conclusion is ambiguous (often wrong).
For example:
all teachers are pedagogues.
This woman is not a teacher.
This woman is not a teacher.
In Figure II, when both of the bases of the conclusion are affirmative, the conclusion is ambiguous (often wrong).
For example:
all teachers are pedagogues.
This is a female teacher.
This is a female teacher
Only teachers are not pedagogues, so both conclusions are ambiguous
Enthymeme. (abbreviated strict syllogism).
An enthymeme is a syllogism in which one of the premises or the conclusion is omitted. Enthymeme means in the mind, in the mind. Enthymeme remembers the omitted part of the syllogism. There are three types of enthymemes:
1. The big base is dropped.
2. The small base is omitted.
3. The summary is omitted.
Let us be given the following syllogism:
All students of the Faculty of Philosophy study logic.
Sobirov is a student of the Faculty of Philosophy
Sobirov studies logic.
Now let's turn this syllogism into an enthymeme:
1. Since Sobirov is a student of the Faculty of Philosophy, he studies logic (a major reason was left out).
2. All students of the Faculty of Philosophy study logic, including Sobirov (minor base omitted).
3. All students of the Faculty of Philosophy study logic, and Sobirov is a student of this faculty (summary omitted).
Enthymemes are widely used in the process of discussion, in the art of oratory.
Complex and complex reduced syllogisms.
A conclusion made of two or more simple strict syllogisms connected with each other is called a polysyllogism, that is, a complex syllogism. In polysyllogism, the conclusion of the first syllogism is the major or minor premise of the next one. Accordingly, progressive and regressive types of polysyllogism are distinguished.
In progressive polysyllogism, the conclusion of the first syllogism takes the place of the main premise of the next one. For example:
Things that make a person perfect are useful.
Acquiring knowledge makes a person perfect.
Acquiring knowledge is beneficial.
Learning a craft means acquiring knowledge.
Hence, vocational training is beneficial.
In regressive polysyllogism, the conclusion of the first syllogism is the sub-premise of the next one. For example:
plants are living things.
Trees are plants.
Living things are made up of cells.
Trees are living things.
So, trees are made of cells.
The first, initial syllogism in a polysyllogism is called a prosyllogism, and the rest are called episyllogisms.
An abbreviated form of polysyllogism is called sorit.
The structure of the sorite is as follows:
all AB
everyone BV
all VG
everyone is GD
all AD
Sorites are either progressive or regressive. In the progressive sorite, the conclusion of the prosyllogism, the main basis of episyllogisms is omitted.
In the regressive sorite, the conclusion of the prosyllogism and the minor basis of episyllogisms are omitted.
A sorite in which the minor premise of the syllogism is omitted is called Aristotelian sorite, and a sorite in which the major premise of the syllogism is omitted is called a Gauquelin sorite.
Epicheyrema
An epichyrema is a compound reduced syllogism, both of its bases consist of reduced simple syllogisms (enthymemes). The scheme of Epicheyrema is as follows:
M is R because M is N.
S is M because SO is.
It is S-P.
Example:
Scientific laws are proven ideas because they are true.
The laws of physics are the laws of science because they are the laws of nature
The laws of physics are proven ideas.
The complete appearance of an epichyrema is as follows:
1. Truth is a proven opinion. is N-P
Scientific laws are truth. It is M-N
Scientific laws are proven ideas. M-P is
2. The laws of nature are scientific laws. It is O-M
The laws of physics are the laws of nature. Happen
The laws of physics are scientific laws. It is S-M
3. Scientific laws are proven ideas. M-P is
The laws of physics are scientific laws. It is S-M
The laws of physics are proven ideas. It is S-P
Epicheyrema was used in discussions and debates, in the art of oratory. Despite the fact that epicherema is a type of complex syllogism, it is widely used in the process of thinking because it is easy to separate and distinguish between the large and small basis, conclusion.
Deductive reasoning based on complex sentences
In making a deductive conclusion based on complex sentences, the bases of the conclusion are considered as simple sentences connected by logical connectors. The grounds of the conclusion can be in the form of either a conditional or a deductive sentence, or both a conditional and a deductive sentence. According to the type of judgments in the grounds, there are the following forms of making such a conclusion:
1. Making a conditional conclusion.
2. Deductive reasoning.
3. Making a conditional-deductive conclusion.
A conditional conclusion is a syllogism in which both bases or one of the bases is a conditional statement. They are divided into purely conditional and conditional-strict types.
A syllogism in which both the premise and the conclusion are conditional statements are called purely conditional inferences. Its formula is as follows:
1) pq
qr
pr or [(pq) (qr)] pr
2) pq
q
q or [(p q) ( q)] q
For example:
If an idea is proven, then it is true.
If the thought is true, then it cannot be rejected
If an idea is proven, then it cannot be rejected.
If the weather is good, we will go to a concert.
Even if the weather is not good, we will go to the concert.
We will go to the concert.
Since the conclusion of this type of syllogism is conditional (conditional sentence), they are rarely used in the cognitive process.
A syllogism in which the major premise is a conditional sentence and the minor premise is a simple definite sentence is called a conditional conclusion. There are two correct (conclusive) modes of making such a conclusion:
1. Confirmatory mode
modus ponens
pq
_r__
q or [(p q) p] q
2. Denial mode
total modus
rq
_ __
or [(p q) )]
For example:
1. If citizens follow the laws of society, then they will be free.
Citizens obey the laws of society.
So they will be free.
2. If the norm is violated, then changes in quantity lead to changes in quality.
Changes in quantity did not lead to changes in quality.
So, the norm has not been violated.
In order for the conclusion of a conditional strict syllogism to be clear and true, it is necessary to follow the following rules:
1. The truth of the result follows from the truth of the premise in the conditional sentence, and the error of the premise logically follows from the error of the result.
2. The truth of the result in the conditional sentence does not prove the truth of the premise, and the error of the premise does not prove the error of the result.
When these rules are violated, the formula of the conditional-strict syllogism is as follows:
pqpq
qp
Extimol r
[(pq) q] p
Extimol q
[(pq) p] q
The reason why the conclusions of the conditional strict syllogism are uncertain (probability) is that the conditional sentence (pq) is considered true in all cases except for the case where r-true and q-false.
For example:
If the patient's blood pressure rises, then his head hurts.
The patient has a headache.
Extimol, his blood pressure is elevated.
In this case, it is not possible to logically deduce the truth of the premise from the truth of the result. Because another basis can produce such a result. In the above example, the conclusion of the syllogism was ambiguous because the basis of the conditional sentence was false, ambiguous, and the result was true.
Now let's change the above example a bit and see:
If the patient's blood pressure rises, then his head hurts.
The patient's blood pressure is elevated.
He probably doesn't have a headache.
We know that headaches are not only caused by high blood pressure, there can be other reasons as well. This makes the conclusion ambiguous. the following table summarizes the above points.
Modus Name Summary
1
2 [(pq) p]q
modus ponens
[(pq) ]
total modus
confirming the basis
negating the result Exact
(chinese)
3
4 [(pq) q]p
[(pq) p]q
result confirmer
which negates the premise Vague
(error)
Deductive reasoning refers to a syllogism in which both premises or one of the premises is a deductive proposition.
Pure deductive reasoning refers to a syllogism in which both the premise and the conclusion are deductive propositions.
For example:
Concepts are general, single, or free-sized depending on their size.
every general concept is either subtractive or additive.
Therefore, concepts are subtractive, or additive, or singular, or empty-sized depending on their size.
The formula for a pure subtractive syllogism is:
One of the bases of the conclusion in making a deductive-assertive conclusion is a deductive judgment, and the other is a simple decisive judgment. There are two modes of making such a conclusion:
1. Affirmative-denial.
modus putting tollens
rq
p
2. Confirmation by negation.
modus tolendo ponens
rq
q
For example:
1. Concepts are concrete or abstract according to their content.
This is a concrete concept.
So, this is not an abstract concept.
2. sentences are simple or complex according to their structure.
The sentence given is not a simple sentence.
Therefore, the given sentence is a complex sentence.
In order to draw a correct conclusion in a deductive syllogism, it is necessary to follow the following rules:
1. The simple sentences in the divisive sentence must negate each other and not intersect according to the size, otherwise the conclusion will be wrong.
For example: Books are interesting or fantastic.
This book is interesting.
This book is not fiction.
A book can be both interesting and fantastic. In this case, the simple sentences in the subtractive sentence do not negate each other and intersect according to the size. Therefore, the conclusion is wrong.
2. The mutually exclusive alternatives must be fully specified in the divisive sentence.
The corners are sharp or obtuse.
This angle is not acute.
This angle is an obtuse angle.
The reason for the error in the conclusion is that the alternatives in the subtractive sentence are not fully specified, that is, the existence of a right angle is overlooked.
Deductive syllogisms are more often used to determine problems with several solutions, that is, to correctly choose one of the alternatives.
Conditional - deductive - lemmatic (approximate) conclusion is called a syllogism, one of the bases of which consists of two or more conditional sentences, and the second one consists of a deductive sentence. According to the number of members in the deductive basis, such conclusions are dilemma (the deductive basis consists of two members), trilemma (the deductive basis consists of three members) and polylemma (the deductive basis consists of four or more a is called "consisting of".
A dilemma can be simple or complex. The conditional judgments of a simple dilemma are similar either in their condition or in their outcome. The judgments based on the conditional basis of the complex dilemma differ from each other according to both the condition and the result. Dilemmas are divided into constructive or destructive types. So, there are four types of dilemmas: 1. Simple constructive dilemma. 2. Simple destructive dilemma. 3. Complex constructive dilemma. 4. Complex destructive dilemma.
The formula for a simple constructive dilemma is:
a s, b s
a b Formulation of simple destructive dilemma
a b, a c
s
For example:
If young people learn science, they will find their way in life.
If young people learn a trade, they will find their way in life.
Young people learn either science or craft.
So, they find their place in life.
If a student knows a foreign language well, he will participate in the competition.
If a student knows a foreign language well, he goes to study abroad.
The student did not participate in the competition or study abroad.
The student does not know a foreign language well.
Formulation of complex constructive dilemma.
a b, c d
a c The formulation of the complex destructive dilemma
a b, c d
b d
For example:
If a person does good deeds, he will be remembered with a good name.
If a person does bad deeds, he will be remembered with a bad name.
A person can do good or bad deeds.
So, he is remembered either with a good name or with a bad name.
If a person does good to others, others will do good to him.
If a person does evil to others, others will also do evil to him.
Neither good nor evil returned to man.
So, he did neither good nor evil to others.
It is necessary to determine all the solutions of the problem in question in order to formulate and solve the dilemmas correctly. A dilemma can sometimes be refuted by another dilemma with the opposite content. Here is an example from the history of logic: "An Athenian woman advises her son: Do not interfere in public affairs, because if you speak the truth, people will hate you, and if you lie, then the gods will hate you. Aristotle comes up with the following answer: I take part in public affairs, because if I speak the truth, the gods will love me, if I lie, people will love me."
In the trilemma, three different solutions to the given problem are considered. Trillemma is also divided into four types:
1. Simple constructive trilemma.
a d, b d, c d
a b c 2. Simple destructive trilemma.
a b, a c, a d
d
3. Complex constructive trilemma
a b, c d, m n
a c m 4. Complex destructive trilemma
a b, c d, m n
b d n
For example:
If the person under investigation is directly involved in the crime, he will be severely punished.
If the person under investigation is indirectly involved in the crime, he will be punished lightly.
If the person under investigation is not involved in the crime, he will be released.
The person under investigation is either directly or indirectly related to the crime, or completely unrelated.
Therefore, the person under investigation is either severely punished, or lightly punished, or released.
This is a conclusion in the form of a complex constructive trilemma. It is recommended to independently give examples of other types of trilemmas.
Deductive reasoning reveals that there are several ways to solve a problem, each of which leads to different consequences. In the words of Amir Temur, one of these consequences is chosen according to the interests of the state and the nation, that is, "more meritorious or less dangerous".
The logic of reasoning
Classical logic is a branch of symbolic logic in which, as in conventional logic, every proposition is assumed to have one of two logical values (true or false). Logic of propositions is the simplest branch of classical logic. The object of study of this logical system is operations on reasoning. A judgment consists of a statement that is evaluated as true or false.
Two types of reasoning are distinguished: simple and complex reasoning. A simple reasoning is an idea whose constituent parts cannot be a reasoning. It is generally considered a logical object that cannot be divided into parts (other considerations). For example, the statement "Pharaobi is a great thinker of the Middle Ages" is a simple statement. From simple reasoning, complex reasoning is built using logical connections (conjunction, strong and weak conjunctions, implication, equivalence, and negation). For example, the opinion that "Pharaobi is a thinker who deeply studied ancient Greek science and culture, made a great contribution to the development of the science of logic" is a complex opinion. The logical value (true or false) of complex propositions depends on the logical value of the simple propositions that make them up and the meaning of the logical connection.
The structure of complex reasoning is analyzed using a special formalized language called the language of reasoning logic. Formulas play an important role in it.
Determining formulas of reasoning logic by inductive way requires paying attention to the following cases: 1) any propositional variable is a formula; 2) if r-formula, then r (not r) is also a formula; 3) if there are r and q-formulas, rq, rq, rq (-signifies a strong disjunction), rq, rq are also formulas.
the stated rules are sufficient and necessary to determine whether or not an expression is a logical formula of reasoning (is it a well-constructed formula or not?).
The existing formulas in the logic of reasoning can be divided into three types. The first are called executable or neutral formulas, which can be true or false depending on the combination of values of the propositional variables that make them up. the following formulas are examples of it.
(pq) r; (pq) q
The latter are true formulas, which are always true regardless of the values of the propositional variables in them. For example, the following expressions are true formulas:
p (pr)$ p (hq)
They express the laws of true formulaic logic. Finding them is one of the main tasks of reasoning logic. Proving the exact truth of a formula can be a sufficient basis for considering the discussion to be correct, because that formula is a formalized expression of this discussion.
The third is precisely the erroneous formulas, which are only erroneous in any set of true values of the propositional variables contained in them. The following expressions are examples of incorrect formulas:
q q; ((pq) (qp))
They consist of the negation of true formulas and express the logical contradictions in the discussion.
In reasoning logic, it is possible to determine which of the existing types an arbitrary formula belongs to by finding its logical value (true or false). One way to determine the value of formulas is the tabular or matrix method. Its essence is defined by the value (true or false) of the formula, the value of the propositional variables contained in it, and the logical functors that connect them (conjunction, disjunction, implication, equivalence, negation) and their semantic meanings. is to find in the dependent case.
This shows that the logic of reasoning can be constructed in the form of a tabular method, as a natural inference system (or axiomatic system).
To build in the tabular way, first of all, it is necessary to determine the logical relationship between the formulas, in particular, the relationship of logical origin. It can be expressed as follows: If each of the premises A1,... and An is true, then the premises (conclusion) V is also true, then V follows logically from the premises A1,..., An. This connection in the form A1...,AnV can be considered as an implication, and the logical derivation symbol () can be replaced with the implication symbol (). For example, the above expression can be written in the form A1 A2 … An V.
Table construction can be shown using the formula of pure conditional syllogism, i.e. (p q)(q r) (p r). Based on the structure of the formula, we determine the amount of rows and columns in the table. the number of rows is determined by the formula 2n. It represents variables. We have 3 variables (p, q, r), so there will be 8 lines. And the number of columns consists of the sum of variables and logical connections. So, the number of columns is 8 (3+5). We divide the above formula into 8 sub-formulas. The first three columns show different logical values (true-false) of p, q, r, the next two - members of conjunctions (r q and q r), the sixth column - the basis of implication ((r q) ( q r)), the seventh column represents the conclusion (r r), the eighth represents the formula in full. The variants of the logical value sets of the three variables are in the following sequence; a) all true values - one row, b) two true, one false values - three rows, c) two false, one true values - three rows, g) all false values - one row. The overview of the table is as follows;
pqrp qq r (r q)
(q r) r r (r q) (q r) (r r)
ч ч ч ч ч ч ч
ch ch x ch xxx ch
x x x x x x x x
х ч ч ч ч ч ч
ch xxx ch xx ch
x ch x ch xx ch ch
xx ch ch ch ch ch ch ch
xxx xxx xxx xxx
The disadvantage of determining the truth value of the formula by the tabular method is that as the number of variables increases, it becomes very large. Difficulties caused by this score can be avoided by normalizing formulas. A formula is considered to have a normal form if equivalence, implication, strong disjunction, double negations are removed from it by means of equally strong substitutions, and the negation sign remains only in the variables.
For example, the formula (( (rq) (rr)) (p q) is considered to be in normal form, and the formula (rq) is not in such form.
The logic of reasoning in the form of Natural Inference System (NXCHS) is built on the basis of inference rules close to natural reasoning. Inferring is understood as the consistency of formulas consisting of: 1) bases, 2) theorems - previously proven considerations, 3) conclusions - expressions derived using the rules of inference from preceding considerations. Rules of inference are accepted methods of logical transition from premises to conclusions, and at their basis lie the properties of logical connections. In NXCHS, the following are accepted as the main direct and indirect rules regarding the inclusion and exclusion of logical conjunctions (, , , , , ):
The main direct rules:
1. Conjunction insertion (KK) rule-
2. Rule of conjunction release (KCH)-
3. Disjunction insertion (DK) rule-
4. Disjunction Derivation (DCH) rule-
5. Rule of Implication Exclusion (ICH)-
6. Entry Equivalence (EC) Rule-
7. Equivalence derivation (ECH) rule-
8. double negative insertion (qIK) rule-
A
A
9. rule of double negation (qICH)-
A
A
Basic indirect rules:
1. Rule of Inclusion of Implication (IK)-
P (set of bases)
A (additional opinion)
V
AV
2. The rule of "bringing to conclusion" (BSC).
P (set of bases)
A (additional opinion)
V
V
A
Other (derivative) rules can be derived using the above basic rules. For example, the conditional syllogism rule is derived as follows:
1
2
(set of basics)
3
4
A 5
V
S (additional opinion)
(INSIDE: 1, 3)
(INSIDE: 2, 4)
6
Here, the expressions in parentheses indicate which rules and lines of inference the result to their left is based on. For example, "ICH: 1, 3" means that the "V" to its left is formed by applying the ICH rule to the expressions in lines 1 and 3. We write it as follows:
It should be said that the use of rules of inference ensures the correct construction of the discussion. and taken on their own, although they are a necessary condition for reaching true conclusions, they are not sufficient. In order to achieve true results when making a conclusion based on the natural inference system, it is necessary to comply with the requirements of conducting a reasonable (proving) discussion.
Proof in a formalized system means a certain consistency of formulas, in which, as a rule, the conclusion consists of the true formula (theorem) after removing redundant considerations. In proof, a true conclusion is drawn from true premises; when the conclusion is false, it implies that the premises cannot be true.
An example of direct proof in NXCHS is:
(pq) ((qr) (pr))
1)
2)
3)
considerations
4)
5)
(1, 3 Modus popens)
(2, 4 Modus popens)
The proof is considered complete, since r (the conclusion) is derived as a consequence of the initial expression.
Indirect proof is also used in NXCHS.
The main, logical properties of NXCHS are its non-contradiction and completeness. Non-contradiction of the system means that every formula in it is a true expression, that is, A and A cannot be proved in it.
The completeness of the system means that it has sufficient logical means to prove any formula (theorem) embodying the laws of logic.
The logic of judgments built in the style of an axiomatic system, along with the linguistic part, includes the exact formulas that perform the function of axioms in the system. All other formulas are accepted only if they are derived from the axioms of the system or introduced by definition.
Various axioms and basic logical symbols can be used to construct the logic of reasoning in the form of an axiomatic system. However different axiomatic systems may be, they are ultimately deductively equivalent. In other words, any theorem belonging to one system can be a theorem of another system.
Logic of predicates
Predicate logic is a logical system that analyzes reasoning processes based on the internal structure of reasoning. It contains the logic of reasoning. The language of predicate logic is derived from the language of propositional logic by adding additional symbols.
Using the semiotic categories related to the logic of predicates, various expressions can be generated. For example, the expression xR (x) (which reads: "The proposition that x has the property R applies to all x") is an optional reasoning scheme, which means "All objects belonging to a class have the property R." means bread. The expression x R (x) (which is read as follows: "There exists an object x with property R") is also an arbitrary reasoning scheme, which is "There is an object (at least one) such that it has property R has" meaning. And the expression xu R (x, u) is an arbitrary reasoning scheme like the above, "any subject x has some relationship R with u" (in short: "for any x there exists u: R x and it is read as "related to him". A predicate that cannot be separated into another predicate is called an elementary predicate. Adding a generality or availability quantifier to a predicate is called quantifier binding.
The act of connecting with a quantifier is considered one of the ways of deriving a conclusion from a predicate. Another method is to replace the variable with a name.
The result of correct replacement of the modifier is characterized by the derivation of only true expressions from true expressions. For example, if we put the names ``scientist'' instead of ``scientist'', ``one field of science'' instead of R, ``every scientist works in one field of science''. reasoning is formed.
The following rules are characteristic for the logic of first-order predicates:
1. the expressions that are put in place of the variable should touch the field of objects defined by the variable x;
2. Variable X can be replaced by a name (or individual variable) only if it is empty;
3. If we put a name on x in a certain expression, it is necessary to put it in place of all other x's in this expression;
4. no free variables should be bound as a result of substituting name.
Among the main rules of predicate logic are the rules of inference in the logic of propositions, as well as the rules of inclusion and exclusion of quantifiers. When these rules are followed, true conclusions will be drawn.
b) Making an inductive conclusion
In the previous topic, we got acquainted with making a necessary conclusion (on the basis of deductive conclusion). Probabilistic reasoning is also studied in logic.
Deductive reasoning can take many forms, including inductive reasoning. A characteristic feature of all of them is that the conclusion does not logically necessarily follow from the premises and is confirmed only to a certain extent. The degree to which the grounds confirm the conclusion is called logical probability.
Let's get acquainted with inductive reasoning in more detail.
Knowing, regardless of the field in which it takes place - whether it is at the level of common sense or scientific knowledge - always begins with the study of the perceptual properties and relationships of objects and phenomena. It is called the stage of empirical knowledge in philosophy and logic. At this stage, the subject observes the repetition of certain characteristics in different natural processes and social phenomena under similar conditions. This is the basis for coming to the opinion that this constant repeating property is not an individual property of some object, but a general property of objects belonging to a certain class. For example, in any country where the principles of democracy are well followed, it can be seen that the social standard of living of the population of that country is high. On this basis, it can be concluded that the standard of living of the population is high in any country where the principles and conditions of democracy are well practiced.
This logical transition from partial knowledge to general knowledge takes place in the form of induction (Latin inductio-to bring to a single basis).
Inductive inference takes place in the form of empirical generalization, in which, based on the observation of the repetition of a single sign in pedmets belonging to a certain class, a conclusion is made about the characteristic of this sign in all pedmets belonging to this class.
Conclusions drawn on the basis of induction are reflected in the form of various empirical laws established in scientific knowledge, created generalizations, and lead to the expansion of our knowledge about subjects and phenomena.
Making an inductive conclusion is considered an indirect conclusion, that is, its basis consists of two or more considerations. They usually represent a single object or part of a class of objects. In the conclusion, a general opinion is formed regarding all subjects belonging to the same logical class.
So, in making inductive conclusions, we observe the dialectical connection of singularity, partiality and generality. Partial knowledge representing certain facts serves as a logical basis for generating general knowledge. Since recurrent stable relations usually consist of important necessary relations of objects, these common knowledges represent regularities. And the knowledge about single and partial facts in the foundations records the manifestation of these laws. Since inductive inference is related to the generalization of observation and experimental results, we will dwell on them briefly.
Observation is the simplest and most widely used method of studying objects and phenomena. In it, the subject (for example, a researcher) studies the observed phenomenon in its natural state, in its connections, without directly affecting it. In this, the subject works with his senses, research equipment (for example, a microscope, a night vision device, etc.).
Naturally, observation is not carried out sporadically, but on a consistent basis, often based on a previously prepared plan (for example, a research plan). For example, the head of the enterprise systematically and regularly monitors the work of the responsible employees and workers working in its various links, departments, and comes to certain conclusions based on induction. These conclusions serve as a basis for making certain changes to the management structure and personnel issues. Another example. Before arresting a person suspected of committing a crime, the police or prosecutor's office observes his actions on the basis of a pre-arranged plan, in different conditions, in a natural state, without disturbing him. This can help him find the facts he needs to make a firm decision.
Experiment (experiment) is a more complex method of studying phenomena, which requires influencing the object of knowledge in a certain way. The experiment, of course, is carried out on the basis of a previously prepared plan, in specially created conditions, using the necessary equipment, using the necessary logical methods.
Experience allows you to create the following facilities in the process of learning.
1. the scope of studied (experimented) subjects can be expanded or narrowed by the researcher at will;
2. The object of knowledge can be studied in a "pure" state, that is, when it is "separated" from the influence of other objects or in its interaction with them;
3. The conditions affecting the object of knowledge can be changed at will;
4. the experiment can be accelerated or slowed down;
5. The experiment can be repeated as many times as necessary to be sure of the truth of the result.
As we noted above, the basis of inductive inference is the expression of observation and experiment results, which record the information about steady repetition of the sign r in events S1, S2,...Sn belonging to any class S. In conclusion, an opinion is formed about the characteristic of this sign for the whole class of objects. The outline of the discussion will be as follows:
The phenomenon S1 has the symbol R
The phenomenon S2 has the symbol R
... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
The phenomenon Sn has the symbol R
S1, S2,..., Sn belong to class S
Every phenomenon of class C has the symbol R.
The symbolic expression is as follows:
R(x1)
R(x2)
... ... ...
R(xn)
x1, x2,…, xnS
x ((xS)R (x)
The constant connections of the subject, which are repeated many times in the experiment, consist of a generality that expresses causality, necessity, and it serves as a logical basis for moving from the foundations to the conclusion in making inductive conclusions.
The main task of making an inductive conclusion in knowledge is to generalize the partial score, that is, to create general knowledge based on raising (generalizing) the characteristic characteristic of some facts to the characteristic characteristic of all subjects belonging to the given class. according to its content and importance in cognition, this knowledge can be from the simplest generalizations based on the generalization of everyday experience to the level of empirical and theoretical laws, hypotheses, and scientific theories.
The history of scientific knowledge confirms that the discoveries made in various fields of science, such as electricity, magnetism, optics, have many causal relationships and laws that were established precisely by inductive means.
The logical value of the conclusions drawn on the basis of induction depends on the nature of the experiment conducted.
Observation and experience will be incomplete. New important features and relationships of objects and events can be determined in the next experiments and observations. This changes the perception of existing objects and events. In particular, the knowledge that was previously considered to be true is questioned and turns into probabilistic thoughts.
In logic, the concept of probability means the uncertainty of the drawn conclusion, the need for additional checks.
Two types of inductive reasoning are distinguished: complete and incomplete inductions.
Complete induction is a type of inductive inference, in which, based on the determination of the characteristics of any object belonging to a certain class, a conclusion is made that this symbol is a common symbol for the objects of the given class.
Complete induction is used to draw conclusions in relation to a small class of objects, closed systems whose elements are clearly visible and limited in quantity. For example, conclusions about the planets in the solar system, NATO member states, enterprises located in one city, and so on can be obtained by full induction. In particular, the conclusion that the direction of the movement of the planets entering the solar system is opposite to the direction of the clockwise movement is made using this method. In the same way, inferential knowledge expressed through general sentences such as "All metals conduct electricity", "NATO member states adhere to the charter of this organization", "All enterprises in the city of Tashkent are fully supplied with electricity" is formed based on the use of full induction.
In full induction, the construction scheme of the argument looks like this:
Subject S1 has the symbol R
Subject S2 has the symbol R
... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
The object Sn has the symbol R
Only S1, S2,…, Sn and only S
constitutes a class
Each subject of class C is R
has a sign
The symbolic expression is as follows:
R(x1)
R(x2)
... ... ...
R(xn)
{2}R (α)>0.
Certain conditions must be met in order to increase the level of accuracy of the conclusion made by analogy, that is, to increase the likelihood of the conclusion being true. These include:
1. As much as possible similar features of the objects being compared should be identified. Then the level of truthfulness of the conclusion, the possibility of making a true conclusion increases.
2. Similar signs of the objects being compared should be important signs for the objects. Then the conclusion is closer to the true idea.
3. Other signs of the objects being compared must be in a necessary relationship with the sign being copied. Then the conditions for the conclusion to be convincing and clear will be fulfilled.
4. Similar symbols of the objects being compared must be of the same type as the symbol being copied.
5. The amount of distinguishing features of the objects being compared must be small and these features must not be necessary or important. If the objects differ from each other in important, necessary features, the conclusion of the analogy will be a mistake.
Violation of the above rules leads to a false analogy, that is, a false conclusion. In a false analogy, the probability of the conclusion being true is equal to 0: R (a)=0. In the process of knowing, knowingly or unknowingly, a false analogy is made. Believing in various myths (for example, if salt is spilled, there will be a fight, etc.) is a clear example of a false analogy.
Many examples of all types of analogies can be given from fiction and folklore. For example: "A young man does not go back on his word, a tiger does not go back on his tracks."
Conclusions of analogy are as important as other types of inference as a method of knowledge.
The process of knowing begins with comparing the external and internal properties of objects and phenomena in objective reality, and determining their organic connection. In analogy, on the basis of comparison, similar, common characteristics are determined, knowledge about objects and phenomena is deepened and concreted. In natural and social sciences, analogy serves as a method of generating and expressing various assumptions, that is, hypotheses about various phenomena.
It is known that many laws were initially stated in the form of a hypothesis, in which the conclusion was made in the form of analogy. Comparison of two objects and phenomena, identification of their similarities allows to gain new knowledge. Analogy is widely used as a means of expanding human knowledge.
Its conclusions are used in the proof process when they are clear and free from extremism.
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