Complex inferences are those that consist of two or more simple inferences. Most often, this kind of complex reasoning, or, as they are also called in logic, chains of reasoning, are used in evidence. Consider such types of complex inferences as: a) polysyllogism; b) litters; c) epicheirema.
Polysyllogism is called chaining, a chain of syllogisms connected in such a way that the conclusion of the previous syllogism (prasyllogism) becomes one of the premises of the subsequent syllogism (episyllogism).
For instance:
No one capable of self-sacrifice is not an egoist.
All generous people are capable of self-sacrifice.
Not a single magnanimous person is an egoist.
All cowards are selfish.
No coward is generous.
Depending on which premise - greater or lesser - of the episyllogism becomes the conclusion of the prasyllogism, progressive and regressive chains of syllogisms are distinguished, respectively.
The example we have given is a progressive chain of syllogisms. In it, our thought goes from the more general to the less general.
Another example of a progressive chain of syllogisms.
All vertebrates have red blood.
All mammals are vertebrates.
All mammals have red blood.
All carnivores are mammals.
All carnivores have red blood.
Tigers are predatory animals.
Tigers have red blood.
In the regressive chain of syllogisms, the conclusion of the prasyllogism becomes the lesser premise of the episyllogism. In such a polysyllogism, thought moves from less general to ever more general knowledge.
For instance:
Vertebrates are animals.
Tigers are vertebrates.
Tigers are animals.
Animals are organisms.
Tigers are animals.
Tigers are organisms.
Organisms are destroyed.
Tigers are organisms.
Tigers are destroyed.
In order to check the logical consistency of a pollysyllogism, it is necessary to break it down into simple categorical syllogisms and check the consistency of each of them.
A sorite (translated from the Greek “heap”) is a complexly abbreviated syllogism in which only the last conclusion from a series of premises is given, and intermediate conclusions are not explicitly formulated, but only implied.
Sorit is built according to the following scheme;
All A is B.
All B is C.
All C is D.
Therefore, all A are D.
As you can see, the conclusion of the prasyllogism is missing here: “All A is C”, which should also act as a major premise of the second syllogism - episyllogism.
For instance:
Socially dangerous acts are immoral.
Crime is an essentially dangerous act.
Theft is a crime.
Theft is immoral.
Here the conclusion of the first syllogism (prasyllogism) is missing - “The crime is immoral”, which is the second, lesser premise of the second syllogism (episyllogism). This episyllogism in its entirety would look like this:
The crime is immoral.
Theft is a crime.
Theft is immoral.
There are two types of sorites - Aristotelian and Goclenian. They got their name from the authors who first described them.
Aristotle described a sorite that omits the conclusion of the prasyllogism, becoming the lesser premise of the episyllogism:
The horse is a quadruped.
Bucephalus is a horse.
A quadruped is an animal.
The animal is a substance.
Bucephalus is a substance.
In its full form, this polysyllogism will be as follows:
The horse is a quadruped.
Bucephalus is a horse.
Bucephalus is a quadruped.
A quadruped is an animal.
Bucephalus is a quadruped.
Bucephalus is an animal.
The animal is a substance.
Bucephalus is an animal.
Bucephalus is a substance.
Gokleniy (Professor of the University of Marburg, lived 1547-1628) describes the sorite, which omits the conclusion of the prasyllogism, which becomes the first, larger premise of the episyllogism. He cited this litter:
The animal is a substance.
A quadruped is an animal.
The horse is a quadruped.
Bucephalus horse.
Bucephalus is a substance.
In its full form, this polysyllogism looks like this:
1. An animal is a substance.
A quadruped is an animal.
The quadruped is a substance.
2. The quadruped is a substance.
The horse is a quadruped.
The horse is a substance.
3. Horse substance.
Bucephalus is a horse.
Bucephalus is a substance.
Epicheirema (translated from Greek "attack", "laying on of hands") is a syllogism in which each of the premises is an enthymeme.
For instance:
All students of the institute international relations engage in logic, as they must think correctly.
We, students of the Institute of International Relations, study at this institute.
That's why we do logic.
It can be seen that each of the premises of this epicheireme is an abbreviated syllogism - an enthymeme. Thus, the first premise in its entirety will be the following syllogism:
All those who have to think correctly are engaged in logic.
All students of the Institute of International Relations should think correctly.
All students of the Institute of International Relations are engaged in logic.
The restoration of the second premise to a complete syllogism and the entire chain of syllogisms is left to the reader.
Epicheirema quite often used by us in the practice of thinking and in oratory. The Russian logician A. Svetilin noted that epicheirema is convenient in oratory in that it makes it possible to arrange a complex conclusion according to its constituent parts and makes them easily visible, and, consequently, the whole reasoning more conclusive.
The exercise
Determine the type of inference and check its consistency
A. 3 is an odd number.
All odd numbers are natural numbers.
All natural numbers are rational numbers.
All rational numbers are real numbers.
Therefore, 3 is a real number.
B. Everything that improves health is useful.
Sport improves health.
Athletics is a sport.
Running is a type of athletics.
Running is helpful.
B. All organisms are bodies.
All plants are organisms.
All bodies have weight.
All plants are bodies.
All plants have weight.
D. Noble work deserves respect, since noble work contributes to the progress of society.
The work of a lawyer is a noble work, as it consists in protecting the legal rights and freedoms of citizens.
Therefore, the work of a lawyer deserves respect.
D, What is good, that should be desired.
What is to be desired is to be approved.
And what is to be approved is commendable.
Therefore, what is good is commendable.
(Example of M.V. Lomonosov)
We obtain most of the knowledge about the reality around us through reasoning. The conclusions in them will be true if they are the results of correct reasoning. Right reasoning consider reasoning built according to the rules of logic. The teacher needs knowledge about the rules according to which correct reasoning. Reasoning is the basis of evidence, without which it is difficult to imagine mathematics.
In logic, along with the term "reasoning", the term "inference" is used.
Inference(reasoning) is a logical operation, as a result of which new knowledge is obtained on the basis of some existing knowledge or from some statements A 1, A 2, A 3, A 4 ... A n (n\u003e 1) get new in relation to the original, statement B .
The inference consists of parcels and conclusions.
Parcels inferences are initial statements, and conclusion is called a new statement, i.e. statement containing new knowledge.
In logic, it is customary to indicate the premises first, and then the conclusion, but in a specific conclusion, their order can be arbitrary: first the conclusion - then the premises; the conclusion may lie between the premises.
Example 1. From the two statements “All liquids are elastic” and “Water is a liquid”, a new statement can be obtained as follows: “All liquids are elastic. Water is a liquid, so water is elastic. Here, the original statements “All liquids are elastic” and “Water is a liquid” are premises, and the new statement “Water is elastic” is the conclusion of the conclusion.
Consider examples of inferences that younger students perform when studying mathematics.
Example 2. The student is asked to explain why the number 35 can be represented as the sum of 30 and 5. He argues: “The number 23 is two-digit. Any two-digit number can be represented as a sum of bit terms. Therefore, 35=30+5.”
In this conclusion, the first and second sentences are premises, and the first is private (characterizes only 35), and the second is general; conclusion - this is the part of the sentence that comes after the word "therefore", and the conclusion is private.
Example 3. One of the ways to get acquainted with the commutative property of multiplication is as follows. Using various means of visualization, students together with the teacher establish that, for example, 2∙5=5∙2, 6∙3=3∙6, 4∙7=7∙4. And then, on the basis of the obtained equalities, they conclude: for all natural numbers a and v true equality a∙b=в∙а.
In this conclusion, the premises are the first three equalities, which state that for specific natural numbers such a property is satisfied, i.e. parcels will be private. The conclusion is a statement of a general nature - the conclusion drawn.
As can be seen from the examples considered, the conclusions are different. In examples 1 and 2, the conclusion follows logically from the premises, and we do not doubt its truth.
Depending on whether a relation of logical consequence exists between the premises and the conclusion, they distinguish two types of reasoning: deductive(lat. the word "deduction" means " breeding”), which in logic consider correct and non-deductive (incorrect).
deductive reasoning a conclusion is called, in which the premises and the conclusion are in relation to the logical consequence, i.e. whenever the premises are true, the conclusion is also true.
If the premises of the inference are denoted by the letters A 1, A 2, ... A n, and the conclusion - by the letter B, then the conclusion can be schematically represented as: A 1, A 2, ... A n ⇒ B.
Such a notation is also used in logic. The line in this entry replaces the word "therefore" ("means").
In deductive reasoning, true premises always lead to true conclusions. Deductive reasoning includes, for example:
Example 4. “If it rains, the ground becomes wet. It's raining. Therefore, the ground is wet."
Example 5. . Example 6. .
Example 7 If X∶2, then X- even number. Number 2002∶2.
The number 2002 is even.
Example 8 If X∶9, then X∶3. The number 122 is not divisible by 3.
The number 122 is not divisible by 9.
The correctness of an inference is determined by its form, and not by the truth of its constituent statements. When analyzing the correctness of an inference, it must be remembered that it is impossible to identify the correctness of an inference with the truth of the conclusion obtained. In logic, there are rules, observing which, you can build deductive reasoning. These rules are called inference rules or deductive reasoning schemes.
The most common deductive reasoning patterns are:
1. A( X)⇒B( X), A( a) - conclusion rule;
V( a)
2. A( X)⇒B( X), V( a) - negation rule;
A( a)
3. A( X)⇒B( X), V( X)⇒С( X) - syllogism rule.
A( X)⇒С( X)
In the examples 4 and 7 considered, the conclusion is built according to the rule of conclusion, in examples 5 and 6 - according to the rule of syllogism, in example 8 - according to the rule of negation, which means that they are all deductive conclusions.
Let us give examples of inferences (reasoning).
1) It is easy to verify the truth of the following statements:
3 + 2 < 3 · 2 (А 1),
4 + 3 < 4 · 3 (А 2),
7 + 5 < 7 · 5 (А 3).
Based on them, we can conclude (B): the sum of any two natural numbers is always less than their product.
2) If the number x is called before the number y when counting, then x is less than y (A 1). The number 7 is called when counting before the number 8 (A 2). Therefore, 7< 8 (В).
The rules of logic help to build deductive conclusions correctly and analyze them:
Statement A ( X) Þ В ( X) is called the common premise, A ( a) – private parcel, V ( a) is the conclusion. According to this rule, the conclusion is made in example 2.
Let us give an example of the use of this rule in working with preschoolers.
There are the same number of cups and saucers.
Assignment to the child: "Show that there are as many cups as there are saucers."
Reasoning of the child: "Let's put a cup on each saucer."
Here is an example of the inference according to this rule:
Consider an example of using the denial rule in working with preschoolers.
There are several cups and saucers.
Assignment to the child: "Set whether the cups and saucers are equal."
The reasoning of the child: "There is no cup on one saucer, so there are more saucers than cups."
Errors in reasoning, incorrect drawings, inability to use theorems and formulas lead to a false conclusion.
Mathematicians began to deliberately invent incorrect reasoning that had the appearance of being correct. Such reasoning is called sophistry. The analysis of sophisms forms the ability to reason correctly, helps to assimilate many mathematical facts.
Is equality true? 25 + 35 - 60 = 30 + 42 - 72
Let's take the common factor out of the bracket. 5 (5 + 7 - 12) = 6 (5 + 7 - 12)
Divide the right and left sides 5 = 6
equals to the expression in parentheses.
Where is the mistake? You can't divide by 0!
There are inferences other than deductive ones. An example of such inferences would be incomplete induction and analogy.
Incomplete induction- this is a conclusion in which, on the basis of the fact that some objects of the collection have a certain property, it is concluded that all objects of this collection have this property.
An example of incomplete induction is the inference in example 1. The conclusions in such inferences can be either true or false. In example 1, the conclusion is false.
To see this, it suffices to give a counterexample:
the numbers 3 and 1 are natural numbers, 3 + 1 = 4, 3 1 = 3, 4 is not less than 3, i.e. There are two natural numbers whose sum is not less than their product.
Consider another example of the use of incomplete induction. It is known that 15 is divisible by 5, 25 is divisible by 5, 35 is divisible by 5. Therefore, it can be argued that any number whose entry ends with the number 5 is divisible by 5. In this case, the conclusion is true - we know the sign of divisibility by 5.
The conclusions obtained with incomplete induction are in the nature of assumptions, hypotheses. They need to be proven or disproven.
The role of incomplete induction is great as a way of obtaining general knowledge, as a way of discovering patterns and rules. The use of incomplete induction in teaching contributes to the development of skills to compare, generalize, and draw conclusions.
Here is an example of using incomplete induction in working with preschoolers:
Visual material: "Wonderful bag" with three-dimensional geometric shapes.
Assignment to the child: "Get one figure and name it."
Answer options: - ball,
Here, probably, all the balls.
Sometimes, when teaching preschoolers, they use inference from analogy, in which knowledge is transferred from the studied object to another, less studied object.
1) "A quadrangle has 4 corners and 4 sides, therefore a pentagon has 5 corners and 5 sides."
2) “If a triangle divides in half,
you get two triangles, so
if the square is divided in half
two squares” (Fig. 10). Rice. 9
Conclusions obtained by analogy can be true or false, they must be proved in a deductive way or refuted by a counterexample. Analogy is important in that it leads us to conjecture, contributes to the development of mathematical intuition.
Exercise 1. What are the essential properties of A B
The figure shown in Figure 2.
CONCLUSION - THE THIRD FORM OF THINKING
What is an inference?
inference- this is the third (after the concept and judgment) form of thinking, in which one, two, or several judgments, called premises, follow a new judgment, called the conclusion, or conclusion.
In logic, it is customary to place the premises and the output one under the other and to separate the premises from the output with a line:
All living organisms feed on moisture.
All plants are living organisms.
All plants feed on moisture.
In the above example, the first two judgments are the premises, and the third is the conclusion. It is clear that the premises must be true judgments and must be connected with each other.
If at least one of the premises is false, then the conclusion is false:
All birds are mammals.
All sparrows are birds.
All sparrows are mammals.
As you can see, in the above example, the falsity of the first premise leads to a false conclusion, despite the fact that the second premise is true. If the premises are not connected with each other, then it is impossible to draw a conclusion from them.
For example, no conclusion follows from the following two premises:
All planets are celestial bodies.
All pines are trees.
Let us pay attention to the fact that inferences consist of judgments, and judgments - of concepts, i.e. one form of thought enters into another as an integral part.
All inferences are divided into direct and indirect. V direct inferences, the conclusion is made from one premises.
for instance:
All flowers are plants.
Some plants are flowers.
Another example:
It is true that all flowers are plants.
It is not true that some flowers are not plants.
It is not difficult to guess that direct inferences are operations for transforming simple judgments and conclusions about the truth of simple judgments in a logical square. The first example of direct inference given above is a transformation of a simple proposition by inversion, and in the second example, by the logical square, from the truth of a proposition of type A, a conclusion is drawn about the falsity of a proposition of type O.
V mediated inferences, the conclusion is drawn from several premises.
for instance:
All fish are living beings.
All carp are fish.
All carp are living beings.
Since direct inferences are various logical operations with judgments, then under inferences are meant, first of all, indirect inferences. In the future, we will talk about them.
Indirect inferences are divided into three types. They are deductive, inductive and reasoning by analogy.
deductive reasoning, or deduction - these are inferences in which a conclusion is drawn from a general rule for a particular case (a special case is derived from a general rule).
for instance:
All stars radiate energy.
The sun is a star.
The sun radiates energy.
As you can see, the first premise is general rule, from which (using the second premise) a special case follows in the form of a conclusion: if all stars radiate energy, then the Sun also radiates it, because it is a star. In deduction, reasoning goes from the general to the particular, from the greater to the lesser, knowledge is narrowed, due to which the deductive conclusions are reliable, i.e. accurate, obligatory, necessary, etc. Let's look again at the example above. Could any other conclusion follow from these two premises than the one that follows from them? Could not! The following conclusion is the only one possible in this case. Let us depict the relationship between the concepts of which our conclusion consisted, Euler circles. Volumes of three concepts: stars; body, radiating energy; The sun schematically arranged as follows.
If the scope of the concept stars included in the concept body, radiating energy, and the scope of the concept The sun included in the concept stars, then the scope of the concept The sun automatically included in the scope of the concept bodies that radiate energy, which makes the deductive inference valid.
The undoubted advantage of deduction, of course, lies in the reliability of its conclusions. Recall that the famous literary hero Sherlock Holmes used the deductive method in solving crimes. This means that he built his reasoning in such a way as to deduce the particular from the general. In one work, explaining to Dr. Watson the essence of his deductive method, he gives the following example. Near the murdered Colonel Morin, Scotland Yard detectives found a smoked cigar and decided that the colonel had smoked it before his death.
However, he (Sherlock Holmes) irrefutably proves that Colonel Morin could not smoke this cigar, because he wore a large, lush mustache, and the cigar was smoked to the end, i.e. if Morin had smoked it, he would certainly have set his mustache on fire. Therefore, the cigar was smoked by another person. In this reasoning, the conclusion looks convincing precisely because it is deductive: from the general rule ( Anyone with a big, bushy mustache can't finish a cigar.) a special case is displayed ( Colonel Morin could not finish his cigar because he wore such a mustache).
Inductive reasoning, or induction - these are inferences in which a general rule is deduced from several special cases (several special cases lead to a general rule).
for instance:
Jupiter is moving.
Mars is moving.
Venus is moving.
Jupiter, Mars, Venus are planets.
All planets are moving.
As you can see, the first three premises are special cases, the fourth premise brings them under one class of objects, combines them, and the output refers to all objects of this class, i.e. some general rule is formulated (following from three special cases). In induction, reasoning goes from the particular to the general, from less to more, knowledge expands, due to which inductive conclusions (unlike deductive ones) are not reliable, but probabilistic. The probabilistic nature of the conclusions is, of course, a disadvantage of induction. However, its undoubted advantage and advantageous difference from deduction, which is a narrowing knowledge, is that induction is an expanding knowledge that can lead to a new one, while deduction is an analysis of the old and already known.
Inference by analogy, or analogy- these are inferences in which, on the basis of the similarity of objects (objects) in some features, a conclusion is made about their similarity, and in other features, a conclusion is made about their similarity in other features.
for instance:
Planet Earth is located in the solar system, it has an atmosphere, water and life.
The planet Mars is located in solar system, it has an atmosphere and water.
There is probably life on Mars.
As you can see, two objects are compared (compared) (the planet Earth and the planet Mars), which are similar to each other in some essential, important features (being in the solar system, having an atmosphere and water). Based on this similarity, it is concluded that, perhaps, these objects are similar to each other in other ways: if there is life on Earth, and Mars is in many ways similar to Earth, then the presence of life on Mars is not excluded. The conclusions of analogy, like the conclusions of induction, are probabilistic.
What is an inference? This is a certain form of thinking and the only correct conclusion. The specifics are as follows: in the process of cognition, it becomes clear that the statements prompted by evidence are not all true, but only a certain part of them.
To establish the complete truth, a thorough investigation is usually carried out: to clearly identify questions, correlate already established truths with each other, collect the necessary facts, conduct experiments, check all conjectures that arise along the way and derive the final result. Here it will be - the conclusion.
Categorical syllogism
A deductive categorical inference is one where a conclusion follows from two true propositions. The concepts that are part of the syllogism are denoted by terms. has three terms:
- conclusion predicate (P) is a larger term;
- subject of conclusion (S) - lesser term;
- a bundle of premises P and S missing from the conclusion (M) is the middle term.
Forms of a syllogism that differ in the middle term (M) in the premises are called figures in a categorical syllogism. There are four such figures, each with its own rules.
- 1 figure: general large premise, affirmative smaller;
- 2 figure: common large premise, negative smaller;
- 3 figure: affirmative minor premise, private conclusion;
- Figure 4: the conclusion is not a universally affirmative judgment.
Each figure can have several modes (these are different syllogisms according to the qualitative and quantitative characteristics of premises and conclusions). As a result, the figures of the syllogism have nineteen correct modes, each of which is assigned its own Latin name.
Simple categorical syllogism: general rules
In order for the conclusion in a syllogism to turn out to be true, one must use true premises, honor the rules of figures and a simple categorical syllogism. Inference methods require the following rules:
- Do not quadruple terms, there should be only three. For example, movement (M) - forever (P); going to university (S) - movement (M); the conclusion is false: going to university is eternal. The middle term is used here in different senses: one - in the philosophical, the other - everyday.
- The middle term is necessarily distributed in at least one of the premises. For example, all fish (P) can swim (M); my sister (S) can swim (M); my sister is a fish. The conclusion is false.
- The conclusion term is distributed only after the distribution in the premise. For example, in all polar cities - white nights; St. Petersburg is not a polar city; there are no white nights in St. Petersburg. The term conclusion contains more than premises, the larger term has expanded.
There are rules for the use of premises, which the form of inference requires, they must also be observed.
- Two negative premises do not give a conclusion. For example, whales are not fish; pike are not whales. So what?
- With one negative premise, a negative conclusion is obligatory.
- No conclusion is possible from two private premises.
- With one private premise, a private conclusion is obligatory.
Conditional inference
When both premises are conditional propositions, a purely conditional syllogism is obtained. For example, if A, then B; if B, then C; if A, then B. Clearly: if you add two, then the sum will be even; if the sum is even, then you can divide by two without a remainder; therefore, if you add two odd numbers, then you can divide the sum without a remainder. There is a formula for such a relation of judgments: the consequence of the consequence is the consequence of the foundation.
Conditionally categorical syllogism
What is a conclusion judgment is in the first premise, and in the second premise and conclusion - categorical judgments. The mode here can be either affirmative or negative. In the affirmative mode, if the second premise affirms the consequence of the first, the conclusion will only be probable. In the negative mode, if the basis of the conditional premise is denied, the conclusion is also only probable. These are conditional inferences.
- If you don't know, shut up. You are silent - probably you do not know (if A, then B; if B, then probably A).
- If it snows, winter has come. Winter has come - probably it's snowing.
- If it's sunny, the trees provide shade. The trees don't give shade - it's not sunny.
Separative syllogism
An inference is called a divisive syllogism if it consists of purely divisive premises, and the conclusion is also obtained as a divisive judgment. Thus, the number of alternatives increases.
Even more important is the dividing-categorical conclusion, where one premise is a divisive judgment, and the second is a simple categorical one. There are two modes here: affirming-negative and denying-asserting.
Conditionally dividing
The concept of inference also includes conditionally dividing forms, in which one premise is two or more conditional propositions, and the second is a disjunctive proposition. Otherwise it is called a lemma. The task of the lemma is a choice from several solutions.
The number of alternatives divides conditional-separative inferences into dilemmas, trilemmas and polylemmas. The number of options (disjunction - the use of "or") affirmative judgments is a constructive lemma. If the disjunction of negations is a destructive lemma. If the conditional premise gives one consequence - the lemma is simple, if the consequences are different - the lemma is complex. This can be traced by drawing conclusions according to the scheme.
Examples would be something like this:
- A simple constructive lemma: ab+cb+db= b; a+c+d=b. If the son goes to visit (a), he will do his homework later (b); if the son goes to the cinema (c), then before that he will do his homework (b); if the son stays at home (d), he will do his homework (b). The son will go to visit or to the cinema, or stay at home. He will still take lessons.
- Complex constructive: a + b; c+d. If the power is hereditary (a), then the state is monarchical (b); if the government is elected (c), the state is a republic (d). Power is inherited or elected. The state is a monarchy or a republic.
Why do we need a conclusion, a judgment, a concept
Inferences do not live by themselves. Experiments are not blind. They only make sense when combined. Plus, synthesis with theoretical analysis, where by means of comparisons, comparisons and generalizations, conclusions can be drawn. Moreover, it is possible to draw a conclusion by analogy not only about what is directly perceived, but also about what is impossible to “feel”. How can one directly perceive such processes as the formation of stars or the development of life on the planet? Here such a game of the mind as abstract thinking is needed.
concept
It has three main forms: concepts, judgments and inferences. The concept reflects the most general, essential, necessary and decisive properties. It contains all the signs of reality, although sometimes reality is devoid of visibility.
When a concept is formed, the mind does not take most of the individual or unimportant accidents in signs, it generalizes all perceptions and representations of as many similar objects as possible in terms of homogeneity and collects from this what is inherent in all and specific.
Concepts are the results of generalizing the data of one or another experience. V scientific research they play one of the main roles. The path of studying any subject is long: from simple and superficial to complex and deep. With the accumulation of knowledge about the individual properties and features of the subject, judgments about it also appear.
Judgment
With the deepening of knowledge, the concepts are improved, and judgments about the objects of the objective world appear. This is one of the main forms of thinking. Judgments reflect the objective connections of objects and phenomena, their inner content and all patterns of development. Any law and any position in the objective world can be expressed by a certain judgment. Inference plays a special role in the logic of this process.
The Phenomenon of Inference
A special mental act, where a new judgment about events and objects can be derived from the prerequisites - the ability to reason that is characteristic of mankind. Without this ability it would be impossible to know the world. For a long time it was impossible to see the globe from the side, but even then people were able to come to the conclusion that our Earth is round. The correct connection of true judgments helped: spherical objects cast a shadow in the form of a circle; The Earth casts a round shadow on the Moon during eclipses; The earth is spherical. Inference by analogy!
The correctness of inferences depends on two conditions: the premises from which the conclusion is built must correspond to reality; the connections of the premises must be consistent with logic, which studies all the laws and forms of building judgments in the conclusion.
Thus, the concept, judgment and inference as the main form of abstract thinking allow a person to cognize the objective world, to reveal the most important, most essential aspects, patterns and connections of the surrounding reality.
CONCLUSION
CONCLUSION
Lit.: Chelpanov G. I., Textbook of logic, M., 1946; Asmus V. F., Logic, M., 1947; his, Teaching of logic about proof and refutation, M., 1954; Tarsky A., Introduction to the logic and methodology of deductive sciences, trans. from English, M., 1948; Gorsky D.P., Logic, 2nd ed., M., 1963; Church A., Introduction to mathematical logic, trans. from English, vol. 1, M., 1960.
A. Subbotin. Moscow.
Philosophical Encyclopedia. In 5 volumes - M .: Soviet Encyclopedia. Edited by F. V. Konstantinov. 1960-1970 .
CONCLUSION
CONCLUSION - the procedure for the direct derivation of some statement from one or more statements. The statements from which the conclusion is drawn are called the premises of the inference, and the statement that is derived from the premises is called the conclusion. Inference is a cognitive technique by which the transformation of the information contained in the premises is carried out. It is the simplest kind of reasoning - the procedure for substantiating a statement by deriving it step by step from other statements; in a conclusion, the transition from arguments (their role is played by premises) to a substantiated thesis (conclusion) occurs in one step. In logic, the conclusion is usually formulated as follows: Αι,Α2,...,Αη
B --- where the premises are written above the line, the conclusion is written below the line, and the line itself expresses the derivation of the conclusion from the premises.
According to the degree of validity of deriving a conclusion from premises, inferences are usually divided into demonstrative and non-demonstrative. In demonstrative inferences, the simultaneous truth of the premises ensures the receipt of a true conclusion, the conclusions in them form part of the aggregate information of the premises. In non-demonstrative reasoning, on the contrary, when moving from premises to conclusion, it has an increment of information, however, the simultaneous truth of the premises does not guarantee the truth of the conclusion.
The most important and extensive variety of demonstrative reasoning is deductive reasoning. There is a logical consequence between their premises and the conclusion, i.e. the very logical form of these inferences ensures the preservation of truth when deriving the conclusion from the premises.
In demonstrative reasoning of other types (these include, for example, complete induction, strict) the conclusion obtained from the true premises is determined not only by the logical form of the statements included in the conclusion, but also by the meanings of the descriptive terms contained in them, by the features of the universe of reasoning.
Among the non-demonstrative conclusions, the greatest are the so-called. plausible inferences, which include, for example, reverse, incomplete induction, non-strict analogy, statistical inference. Plausible inferences are characterized by the presence of a logical confirmation relationship between the premises and the conclusion. This relation has many different explications in modern logic. Thus, the interpretation of the confirmation relation in accordance with the criterion of positive relevance has become widespread: the premises confirm the conclusion if and only if the truth of the conclusion increases (but does not become equal to one) under the condition that the premises are simultaneously true. The main field of application of deductive reasoning is the exact sciences (primarily logic), in which special requirements are imposed on the rigor of evidence. Plausible Inferences, ch. O., are used in the empirical sciences to put forward and verify hypotheses, to obtain law-like statements related to the subject area under study.
V. I. Markin
New Philosophical Encyclopedia: In 4 vols. M.: Thought. Edited by V. S. Stepin. 2001 .
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Inference... Spelling Dictionary
See the conclusion to build a conclusion ... Dictionary of Russian synonyms and expressions similar in meaning. under. ed. N. Abramova, M .: Russian dictionaries, 1999. inference conclusion, traduction, enthymeme, deduction, reasoning, syllogism, sophism, paralogy, ... ... Synonym dictionary
CONCLUSION, conclusions, cf. 1. The logical process of deriving a conclusion from two judgments, a syllogism (philosophical). Deductive reasoning. 2. Conclusion, conclusion (book). Make a conclusion. Correct inference. Dictionary Ushakov ... Explanatory Dictionary of Ushakov
inference- one of the logical forms of thinking (see also concept and judgment). U. is characterized by the conclusion based on the rules of logic of the conclusion or consequence of several judgments (parcels). In logic, U. classifications are being developed. Psychology ... Great Psychological Encyclopedia
Mental action based on the norms of conclusions inherent in individual consciousness, largely coinciding with the rules and laws of logic ... Big Encyclopedic Dictionary
Establishing a connection between any judgments. It is carried out in verbal form, due to which it is possible to get out of the influence of the perceptual field ... Psychological Dictionary
A form of thought by which a new judgment is made on the basis of one or more judgments already made. The initial judgments, on the basis of which a new judgment is obtained, are called the premises of U., and the new judgment resulting from ... ... The latest philosophical dictionary
CONCLUSION, I, cf. (book). Conclusion, conclusion (in 3 values). Make, withdraw from. Correct at. Explanatory dictionary of Ozhegov. S.I. Ozhegov, N.Yu. Shvedova. 1949 1992 ... Explanatory dictionary of Ozhegov
English conclusion/deduction; German Schlu?folgerung. Reasoning, in the course of which from one or several judgments, called U.'s premises, a new judgment is deduced, which logically follows from the premises. Antinazi. Encyclopedia of Sociology, 2009 ... Encyclopedia of Sociology
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