Examples of induction. Method of mathematical induction: examples of solutions

The induction method requires a scrupulous attitude, since too much depends on the number of parts of the whole studied: the greater the number studied, the more reliable the result. Based on this feature, scientific laws obtained by induction are tested for a long time at the level of probabilistic assumptions to isolate and study all possible structural elements, connections and influences. In science, an inductive conclusion is based on significant features, with the exception of random provisions. This fact is important in connection with the specifics of scientific knowledge. This is clearly seen in the examples of induction in science.

There are two types of induction in the scientific world (in connection with the method of study):

induction-selection (or selection);
induction – exclusion (elimination).

The first type is distinguished by the methodical (scrupulous) selection of samples of a class (subclasses) from its different areas. An example of this type of induction is the following: silver (or silver salts) purifies water. The conclusion is based on many years of observations (a kind of selection of confirmations and refutations - selection). The second type of induction is based on conclusions that establish causal relationships and exclude circumstances that do not correspond to its properties, namely universality, adherence to temporal sequence, necessity and unambiguity.

Induction in logic

Induction is a process of logical inference based on the transition from a particular situation to a general one. Inductive inference connects particular premises to a conclusion not strictly through the laws of logic, but rather through some factual, psychological, or mathematical ideas.

The objective basis of inductive inference is the universal connection of phenomena in nature.

There is a distinction between complete induction - a method of proof in which a statement is proven for a finite number of special cases that exhaust all possibilities, and incomplete induction - observations of individual special cases lead to a hypothesis, which, of course, needs proof. Also for proofs, the method of mathematical induction is used, which allows complete induction for an infinite countable set of objects.

Scientific induction is a combination of induction and deduction, theory and empirical research. In scientific induction, the basis for a conclusion is not only a listing of examples and a statement of the absence of a counterexample, but also a justification for the impossibility of a counterexample due to its contradiction to the phenomenon under consideration. Thus, the conclusion is made not only on the basis of external signs, but also on the idea of the essence of the phenomenon. This means that you need to have a theory of this phenomenon. Thanks to this, the probability of obtaining a true conclusion in scientific induction increases significantly.

Example. In order to verify the reliability of the conclusion “Always before the rain, swallows fly low above the ground,” it is enough to understand that before the rain, swallows fly low above the ground because the midges they hunt for fly low. And midges fly low because before the rain their wings swell from moisture.

If in popular induction it is important to review as many cases as possible, then for scientific induction this is not of fundamental importance.

Example. Legend has it that in order to discover the fundamental law of universal gravitation, Newton only had to observe one incident - an apple falling.

Rules of induction

In order to avoid mistakes, inaccuracies and irregularities in your thinking, to avoid oddities, you need to comply with the requirements that determine the correctness and objective validity of inductive inference. These requirements are discussed in more detail below.

The first rule states that inductive generalization provides reliable information only if it is carried out on essential features, although in some cases we can talk about a certain generalization of non-essential features. The main reason that they cannot be the subject of generalization is that they do not have such an important property as repeatability. This is all the more important because inductive research consists of establishing essential, necessary, stable features of the phenomena being studied.
According to the second rule, an important task is to accurately determine whether the phenomena under study belong to a single class, recognizing their homogeneity or same type, since inductive generalization applies only to objectively similar objects. The validity of the generalization of features that are expressed in particular premises can depend on this.
Incorrect generalization can lead not only to misunderstandings or distortion of information, but also to the emergence of various kinds of prejudices and misconceptions. The main cause of errors is generalization based on random characteristics of individual objects or generalization based on general characteristics when there is no need for these specific characteristics.

The correct use of induction is one of the pillars of correct thinking in general. As stated above, inductive inference is an inference in which thought develops from knowledge of a lesser degree of generality to knowledge of a greater degree of generality. That is, a particular subject is considered and generalized. Generalization is possible to certain limits.

Any phenomenon of the surrounding world, any subject of research is best studied in comparison with another similar subject. So is induction. Its features are best demonstrated in comparison with deduction. These features manifest themselves mainly in the way the inference process takes place, as well as in the nature of the conclusion. Thus, in deduction one concludes from the characteristics of a genus to the characteristics of a species and individual objects of this genus (based on volumetric relations between terms); in inductive inference - from the characteristics of individual objects to the characteristics of the entire genus or class of objects (to the volume of this characteristic).

Therefore, there are a number of differences between deductive and inductive reasoning that make it possible to separate them from each other.

There are several features of inductive inferences:

inductive inference includes many premises;
all premises of inductive inference are single or particular judgments;
Inductive inference is possible with all negative premises.

Induction from the position of philosophy

Looking back historically, the term “induction” was first mentioned by Socrates. Aristotle described examples of induction in philosophy in a more approximate terminological dictionary, but the question of incomplete induction remains open. After the persecution of Aristotelian syllogism, the inductive method began to be recognized as fruitful and the only possible one in natural science. Bacon is considered the father of induction as an independent special method, but he failed to separate induction from the deductive method, as his contemporaries demanded.

Induction was further developed by J. Mill, who considered the inductive theory from the perspective of four main methods: agreement, difference, residues and corresponding changes. It is not surprising that today the listed methods, when examined in detail, are deductive. The realization of the inconsistency of the theories of Bacon and Mill led scientists to study the probabilistic basis of induction.

However, even here there were some extremes: attempts were made to reduce induction to the theory of probability with all the ensuing consequences. Induction receives a vote of confidence through practical application in certain subject areas and thanks to the metric accuracy of the inductive basis.

An example of induction and deduction in philosophy can be considered the Law of Universal Gravitation. On the date of discovery of the law, Newton was able to verify it with an accuracy of 4 percent. And when checked more than two hundred years later, the correctness was confirmed with an accuracy of 0.0001 percent, although the verification was carried out by the same inductive generalizations. Modern philosophy pays more attention to deduction, which is dictated by the logical desire to derive new knowledge (or truths) from what is already known, without resorting to experience or intuition, but using “pure” reasoning. When referring to true premises in the deductive method, in all cases the output is a true statement.

This very important characteristic should not overshadow the value of the inductive method. Since induction, based on the achievements of experience, also becomes a means of processing it (including generalization and systematization).

Deduction and induction in psychology

Unfortunately, on psychology pages on the Internet there is practically no justification for the integrity of the deductive-inductive method. Although professional psychologists more often encounter manifestations of induction, or rather, erroneous conclusions. An example of induction in psychology, as an illustration of erroneous judgments, is the statement: my mother is deceiving, therefore, all women are deceivers.

You can glean even more “erroneous” examples of induction from life:

a student is incapable of anything if he gets a bad grade in math;
he is a fool;
he is smart;
I can do anything;
and many other value judgments based on completely random and, at times, insignificant premises.

It should be noted: when the fallacy of a person’s judgment reaches the point of absurdity, a frontier of work appears for the psychotherapist.

One example of induction at an appointment with a specialist: “The patient is absolutely sure that the color red is only dangerous for him in any form. As a result, the person excluded this color scheme from his life - as much as possible. There are many opportunities for a comfortable stay at home. You can refuse all red items or replace them with analogues made in a different color scheme. But in public places, at work, in a store - it’s impossible. When a patient finds himself in a stressful situation, each time he experiences a “tide” of completely different emotional states, which can pose a danger to others.”

This example of induction, and unconscious induction, is called “fixed ideas.” If this happens to a mentally healthy person, we can talk about a lack of organization of mental activity. A way to get rid of obsessive states can be the elementary development of deductive thinking. In other cases, psychiatrists work with such patients. The above examples of induction indicate that “ignorance of the law does not exempt you from the consequences (of erroneous judgments).”

Psychologists, working on the topic of deductive thinking, have compiled a list of recommendations designed to help people master this method. The first point is problem solving. As can be seen, the form of induction used in mathematics can be considered “classical”, and the use of this method contributes to the “discipline” of the mind.

The next condition for the development of deductive thinking is broadening one’s horizons (those who think clearly express themselves clearly). This recommendation directs the “suffering” to the treasuries of science and information (libraries, websites, educational initiatives, travel, etc.). Accuracy is the next recommendation. Indeed, from examples of the use of induction methods it is clearly seen that it is in many ways the guarantee of the truth of statements. The flexibility of the mind was also not spared, implying the possibility of using different ways and approaches in solving a given problem, as well as taking into account the variability of the development of events.

And, of course, observation, which is the main source of accumulation of empirical experience. Special mention should be made of the so-called “psychological induction”. This term, although not often, can be found on the Internet.

All sources do not provide at least a brief formulation of the definition of this term, but refer to “examples from life,” while passing off as a new type of induction either suggestion, or some forms of mental illness, or extreme states of the human psyche. From all of the above, it is clear that an attempt to derive a “new term” based on false (often untrue) premises dooms the experimenter to obtain an erroneous (or hasty) statement.

The concept of induction in physics

Electromagnetic induction

The phenomenon of electromagnetic induction is the phenomenon of the occurrence of electric current in a conductor under the influence of an alternating magnetic field.

It is important that in this case the conductor must be closed. At the beginning of the 19th century. After the experiments of the Danish scientist Oersted, it became clear that electric current creates a magnetic field around itself. Then the question arose about whether it was possible to obtain an electric current due to a magnetic field, i.e. perform the reverse action. If an electric current creates a magnetic field, then probably the magnetic field should also create an electric current. In the first half of the 19th century, scientists turned to just such experiments: they began to look for the possibility of creating an electric current due to a magnetic field.

Faraday's experiments

For the first time, the English physicist Michael Faraday managed to achieve success in this (that is, to obtain an electric current due to a magnetic field). So, let's turn to Faraday's experiments.

The first scheme was quite simple. Firstly, M. Faraday used a coil with a large number of turns in his experiments. The coil was short-circuited to a measuring device, a milliammeter (mA). It must be said that in those days there were not enough good instruments for measuring electric current, so they used an unusual technical solution: they took a magnetic needle, placed a conductor next to it through which current flowed, and by the deflection of the magnetic needle they judged the current flowing. So in this case, the currents could be very small, so a mA device was used, i.e. one that measures small currents.

M. Faraday moved a permanent magnet along the coil - the magnet moved up and down relative to the coil. Please note that in this experiment, for the first time, the presence of an electric current in a circuit was detected as a result of a change in the magnetic flux that passes through the coil.

Faraday also drew attention to the fact that the mA needle deviates from its zero value, i.e. shows that an electric current exists in a circuit only when the magnet is moving. As soon as the magnet stops, the arrow returns to its original position, to the zero position, i.e. in this case there is no electric current in the circuit.

Faraday's second achievement is the establishment of the dependence of the direction of the induction electric current on the polarity of the magnet and the direction of its movement. As soon as Faraday changed the polarity of the magnets and passed the magnet through a coil with a large number of turns, the direction of the induction current immediately changed, the one that arises in a closed electrical circuit.

So, some conclusion. A changing magnetic field creates an electric current. The direction of the electric current depends on which pole of the magnet is currently passing through the coil, in which direction the magnet is moving.

And one more thing: it turns out that the number of turns in the coil affects the value of the electric current. The more turns, the greater the current value will be.

Conclusions from experiments

What conclusions were drawn by M. Faraday as a result of these experiments? An induced electric current appears in a closed circuit only when there is an alternating magnetic field. Moreover, this magnetic field must change.

Electrostatic induction

Electrostatic induction is the phenomenon of inducing one’s own electrostatic field when an external electric field acts on a body. The phenomenon is caused by the redistribution of charges inside conducting bodies, as well as the polarization of internal microstructures of non-conducting bodies. The external electric field can be significantly distorted in the vicinity of a body with an induced electric field.

Electrostatic induction in conductors

Redistribution of charges in well-conducting metals under the action of an external electric field occurs until the charges inside the body almost completely compensate for the external electric field. In this case, opposite induced charges will appear on opposite sides of the conducting body.

Electrostatic induction in conductors is used when charging them. So, if a conductor is grounded and a negatively charged body is brought to it without touching the conductor, then a certain amount of negative charges will flow into the ground, being replaced by positive ones. If we now remove the ground and then the charged body, the conductor will remain positively charged. If you do the same without grounding the conductor, then after removing the charged body, the charges induced on the conductor will be redistributed, and all its parts will again become neutral.

True knowledge at all times has been based on establishing a pattern and proving its truthfulness in certain circumstances. Over such a long period of existence of logical reasoning, formulations of rules were given, and Aristotle even compiled a list of “correct reasoning.” Historically, it has been customary to divide all inferences into two types - from the concrete to the multiple (induction) and vice versa (deduction). It should be noted that the types of evidence from particular to general and from general to particular exist only in conjunction and cannot be interchanged.

Induction in mathematics

The term “induction” has Latin roots and is literally translated as “guidance.” Upon closer study, one can highlight the structure of the word, namely the Latin prefix - in- (denotes a directed action inward or being inside) and -duction - introduction. It is worth noting that there are two types - complete and incomplete induction. The full form is characterized by conclusions drawn from the study of all objects of a certain class.

Incomplete - conclusions that apply to all subjects of the class, but are made based on the study of only some units.

Complete mathematical induction is an inference based on a general conclusion about the entire class of any objects that are functionally connected by the relations of a natural series of numbers based on knowledge of this functional connection. In this case, the proof process takes place in three stages:

the first one proves the correctness of the position of mathematical induction. Example: f = 1, induction;
the next stage is based on the assumption that the position is valid for all natural numbers. That is, f=h is an inductive hypothesis;
at the third stage, the validity of the position for the number f=h+1 is proven, based on the correctness of the position of the previous point - this is an induction transition, or a step of mathematical induction. An example is the so-called if the first stone in a row falls (basis), then all the stones in the row fall (transition).

Both jokingly and seriously

For ease of understanding, examples of solutions using the method of mathematical induction are presented in the form of joke problems. This is the “Polite Queue” task:

The rules of conduct prohibit a man from taking a turn in front of a woman (in such a situation, she is allowed to go ahead). Based on this statement, if the last one in line is a man, then everyone else is a man.

A striking example of the method of mathematical induction is the “Dimensionless flight” problem:

It is required to prove that any number of people can fit on the minibus. It is true that one person can fit inside a vehicle without difficulty (basis). But no matter how full the minibus is, 1 passenger will always fit on it (induction step).

Familiar circles

Examples of solving problems and equations by mathematical induction are quite common. As an illustration of this approach, consider the following problem.

Condition: there are h circles on the plane. It is required to prove that, for any arrangement of figures, the map they form can be correctly colored with two colors.

Solution: when h=1 the truth of the statement is obvious, so the proof will be constructed for the number of circles h+1.

Let us accept the assumption that the statement is valid for any map, and there are h+1 circles on the plane. By removing one of the circles from the total, you can get a map correctly colored with two colors (black and white).

When restoring a deleted circle, the color of each area changes to the opposite (in this case, inside the circle). The result is a map correctly colored in two colors, which is what needed to be proven.

Examples with natural numbers

The application of the method of mathematical induction is clearly shown below.

Examples of solutions:

Prove that for any h the following equality is correct:

1 2 +2 2 +3 2 +…+h 2 =h(h+1)(2h+1)/6.

1. Let h=1, which means:

R 1 =1 2 =1(1+1)(2+1)/6=1

It follows from this that for h=1 the statement is correct.

2. Assuming that h=d, the equation is obtained:

R 1 =d 2 =d(d+1)(2d+1)/6=1

3. Assuming that h=d+1, it turns out:

R d+1 =(d+1) (d+2) (2d+3)/6

R d+1 = 1 2 +2 2 +3 2 +…+d 2 +(d+1) 2 = d(d+1)(2d+1)/6+ (d+1) 2 =(d( d+1)(2d+1)+6(d+1) 2)/6=(d+1)(d(2d+1)+6(k+1))/6=

(d+1)(2d 2 +7d+6)/6=(d+1)(2(d+3/2)(d+2))/6=(d+1)(d+2)( 2d+3)/6.

Thus, the validity of the equality for h=d+1 has been proven, therefore the statement is true for any natural number, as shown in the example solution by mathematical induction.

Task

Condition: proof is required that for any value of h the expression 7 h -1 is divisible by 6 without a remainder.

Solution:

1. Let's say h=1, in this case:

R 1 =7 1 -1=6 (i.e. divided by 6 without remainder)

Therefore, for h=1 the statement is true;

2. Let h=d and 7 d -1 be divided by 6 without a remainder;

3. The proof of the validity of the statement for h=d+1 is the formula:

R d +1 =7 d +1 -1=7∙7 d -7+6=7(7 d -1)+6

In this case, the first term is divisible by 6 according to the assumption of the first point, and the second term is equal to 6. The statement that 7 h -1 is divisible by 6 without a remainder for any natural h is true.

Errors in judgment

Often incorrect reasoning is used in proofs due to the inaccuracy of the logical constructions used. This mainly happens when the structure and logic of the proof is violated. An example of incorrect reasoning is the following illustration.

Task

Condition: proof is required that any pile of stones is not a pile.

Solution:

1. Let's say h=1, in this case there is 1 stone in the pile and the statement is true (basis);

2. Let it be true for h=d that a pile of stones is not a pile (assumption);

3. Let h=d+1, from which it follows that when adding one more stone, the set will not be a heap. The conclusion suggests itself that the assumption is valid for all natural h.

The mistake is that there is no definition of how many stones form a pile. Such an omission is called a hasty generalization in the method of mathematical induction. An example shows this clearly.

Induction and the laws of logic

Historically, they always “walk hand in hand.” Scientific disciplines such as logic and philosophy describe them in the form of opposites.

From the point of view of the law of logic, inductive definitions rely on facts, and the truthfulness of the premises does not determine the correctness of the resulting statement. Often conclusions are obtained with a certain degree of probability and plausibility, which, naturally, must be verified and confirmed by additional research. An example of induction in logic would be the following statement:

There is a drought in Estonia, a drought in Latvia, a drought in Lithuania.

Estonia, Latvia and Lithuania are Baltic states. There is drought in all the Baltic states.

From the example we can conclude that new information or truth cannot be obtained using the method of induction. All that can be counted on is some possible veracity of the conclusions. Moreover, the truth of the premises does not guarantee the same conclusions. However, this fact does not mean that induction languishes on the margins of deduction: a huge number of provisions and scientific laws are substantiated using the induction method. An example is the same mathematics, biology and other sciences. This is mostly due to the method of complete induction, but in some cases partial induction is also applicable.

The venerable age of induction has allowed it to penetrate almost all spheres of human activity - this is science, economics, and everyday conclusions.

Induction in the scientific community

In science, an inductive conclusion is based on significant features, with the exception of random provisions. This fact is important in connection with the specifics of scientific knowledge. This is clearly seen in the examples of induction in science.

There are two types of induction in the scientific world (in connection with the method of study):

induction-selection (or selection);
induction - exclusion (elimination).

The first type is distinguished by the methodical (scrupulous) selection of samples of a class (subclasses) from its different areas.

An example of this type of induction is the following: silver (or silver salts) purifies water. The conclusion is based on many years of observations (a kind of selection of confirmations and refutations - selection).

The second type of induction is based on conclusions that establish causal relationships and exclude circumstances that do not correspond to its properties, namely universality, adherence to temporal sequence, necessity and unambiguity.

Induction and deduction from the position of philosophy

Looking back historically, the term induction was first mentioned by Socrates. Aristotle described examples of induction in philosophy in a more approximate terminological dictionary, but the question of incomplete induction remains open. After the persecution of Aristotelian syllogism, the inductive method began to be recognized as fruitful and the only possible one in natural science. Bacon is considered the father of induction as an independent special method, but he failed to separate induction from the deductive method, as his contemporaries demanded.

The realization of the inconsistency of the theories of Bacon and Mill led scientists to study the probabilistic basis of induction. However, even here there were some extremes: attempts were made to reduce induction to the theory of probability with all the ensuing consequences.

Induction receives a vote of confidence through practical application in certain subject areas and thanks to the metric accuracy of the inductive basis. An example of induction and deduction in philosophy can be considered the Law of Universal Gravitation. On the date of discovery of the law, Newton was able to verify it with an accuracy of 4 percent. And when checked more than two hundred years later, the correctness was confirmed with an accuracy of 0.0001 percent, although the verification was carried out by the same inductive generalizations.

Modern philosophy pays more attention to deduction, which is dictated by the logical desire to derive new knowledge (or truths) from what is already known, without resorting to experience or intuition, but using “pure” reasoning. When referring to true premises in the deductive method, in all cases the output is a true statement.

Application of induction in economics

Induction and deduction have long been used as methods for studying the economy and forecasting its development.

The range of use of the induction method is quite wide: studying the fulfillment of forecast indicators (profits, depreciation, etc.) and a general assessment of the state of the enterprise; formation of an effective enterprise promotion policy based on facts and their relationships.

The same method of induction is used in “Shewhart maps”, where, under the assumption of the division of processes into controlled and uncontrollable, it is stated that the framework of the controlled process is inactive.

It should be noted that scientific laws are substantiated and confirmed using the induction method, and since economics is a science that often uses mathematical analysis, risk theory and statistics, it is not at all surprising that induction is on the list of main methods.

An example of induction and deduction in economics is the following situation. An increase in the price of food (from the consumer basket) and essential goods pushes the consumer to think about the emerging high cost in the state (induction). At the same time, from the fact of high prices, using mathematical methods, it is possible to derive indicators of price growth for individual goods or categories of goods (deduction).

Most often, management personnel, managers, and economists turn to the induction method. In order to be able to predict with sufficient truthfulness the development of an enterprise, market behavior, and the consequences of competition, an inductive-deductive approach to the analysis and processing of information is necessary.

A clear example of induction in economics related to erroneous judgments:

the company's profit decreased by 30%;
a competing company has expanded its product line;
nothing else has changed;
the production policy of a competing company caused a reduction in profits by 30%;
therefore, the same production policy needs to be implemented.

The example is a colorful illustration of how inept use of the induction method contributes to the ruin of an enterprise.

Deduction and induction in psychology

Since there is a method, then, logically, there is also properly organized thinking (to use the method). Psychology as a science that studies mental processes, their formation, development, relationships, interactions, pays attention to “deductive” thinking, as one of the forms of manifestation of deduction and induction. Unfortunately, on psychology pages on the Internet there is practically no justification for the integrity of the deductive-inductive method. Although professional psychologists more often encounter manifestations of induction, or rather, erroneous conclusions.

An example of induction in psychology, as an illustration of erroneous judgments, is the statement: my mother is deceiving, therefore, all women are deceivers. You can glean even more “erroneous” examples of induction from life:

a student is incapable of anything if he gets a bad grade in math;
he is a fool;
he is smart;
I can do anything;

And many other value judgments based on completely random and, at times, insignificant premises.

It should be noted: when the fallacy of a person’s judgment reaches the point of absurdity, a frontier of work appears for the psychotherapist. One example of induction at a specialist appointment:

“The patient is absolutely sure that the color red is only dangerous for him in any form. As a result, the person excluded this color scheme from his life - as much as possible. There are many opportunities for a comfortable stay at home. You can refuse all red items or replace them with analogues made in a different color scheme. But in public places, at work, in a store - it is impossible. When a patient finds himself in a stressful situation, each time he experiences a “tide” of completely different emotional states, which can pose a danger to others.”

The above examples of induction indicate that “ignorance of the law does not exempt you from the consequences (of erroneous judgments).”

Psychologists, working on the topic of deductive thinking, have compiled a list of recommendations designed to help people master this method.

The first point is problem solving. As can be seen, the form of induction used in mathematics can be considered “classical”, and the use of this method contributes to the “discipline” of the mind.

Special mention should be made of the so-called “psychological induction”. This term, although not often, can be found on the Internet. All sources do not provide at least a brief formulation of the definition of this term, but refer to “examples from life,” while passing off as a new type of induction either suggestion, or some forms of mental illness, or extreme states of the human psyche. From all of the above, it is clear that an attempt to derive a “new term” based on false (often untrue) premises dooms the experimenter to obtain an erroneous (or hasty) statement.

It should be noted that the reference to the experiments of 1960 (without indicating the location, the names of the experimenters, the sample of subjects and, most importantly, the purpose of the experiment) looks, to put it mildly, unconvincing, and the statement that the brain perceives information bypassing all organs of perception (the phrase “is affected” would fit in more organically in this case), makes one think about the gullibility and uncriticality of the author of the statement.

Instead of a conclusion

It is not for nothing that the queen of sciences, mathematics, uses all possible reserves of the method of induction and deduction. The considered examples allow us to conclude that the superficial and inept (thoughtless, as they say) application of even the most accurate and reliable methods always leads to erroneous results.

In the mass consciousness, the method of deduction is associated with the famous Sherlock Holmes, who in his logical constructions more often uses examples of induction, using deduction in the right situations.

The article examined examples of the application of these methods in various sciences and spheres of human activity.

Deduction (Latin deductio - inference) is a method of thinking, the consequence of which is a logical conclusion, in which a particular conclusion is derived from the general. A chain of inferences (reasonings), where links (statements) are interconnected by logical conclusions.

The beginning (premises) of deduction are axioms or simply hypotheses that have the nature of general statements (“general”), and the end is the consequences of the premises, theorems (“particular”). If the premises of a deduction are true, then its consequences are true. Deduction is the main means of logical proof. The opposite of induction.

An example of the simplest deductive reasoning:

All people are mortal.
Socrates is a man.
Therefore, Socrates is mortal.

The method of deduction is opposed to the method of induction - when a conclusion is made on the basis of reasoning going from the particular to the general.

For example:

the Yenisei Irtysh and Lena rivers flow from south to north;
the Yenisei, Irtysh and Lena rivers are Siberian rivers;
therefore, all Siberian rivers flow from south to north.

Of course, these are simplified examples of deduction and induction. Conclusions must be based on experience, knowledge and specific facts. Otherwise, it would be impossible to avoid generalizations and draw erroneous conclusions. For example, “All men are deceivers, so you are also a deceiver.” Or “Vova is lazy, Tolik is lazy and Yura is lazy, which means all men are lazy.”

In everyday life, we use the simplest versions of deduction and induction without even realizing it. For example, when we see a disheveled man running headlong, we think that he is probably late for something. Or, looking out the window in the morning and noticing that the asphalt is strewn with wet leaves, we can assume that it rained and there was a strong wind at night. We tell the child not to sit late on a weekday, because we assume that then he will sleep through school, not have breakfast, etc.

History of the method

The term “deduction” itself was apparently first used by Boethius (“Introduction to Categorical Syllogism”, 1492), the first systematic analysis of one of the varieties of deductive inferences - syllogistic inferences- was implemented by Aristotle in the First Analytics and significantly developed by his ancient and medieval followers. Deductive reasoning based on the properties of propositional logical connectives, were studied in the Stoic school and especially in detail in medieval logic.

The following important types of inferences were identified:

conditionally categorical (modus ponens, modus tollens)
dividing-categorical (modus tollendo ponens, modus ponendo tollens)
conditional disjunctive (lemmatic)

In the philosophy and logic of modern times, there were significant differences in views on the role of deduction among other methods of cognition. Thus, R. Descartes contrasted deduction with intuition, through which, in his opinion, the human mind “directly perceives” the truth, while deduction provides the mind with only “indirect” (obtained through reasoning) knowledge.

F. Bacon, and later other English “inductivist logicians” (W. Whewell, J. St. Mill, A. Bain and others), especially noting that the conclusion obtained through deduction does not contain any “information” that would not be contained in the premises, they considered, on this basis, deduction a “secondary” method, while true knowledge, in their opinion, is provided only by induction. In this sense, deductively correct reasoning was considered from an information-theoretic point of view as reasoning whose premises contain all the information contained in its conclusion. Based on this, not a single deductively correct reasoning leads to the acquisition of new information - it just makes explicit the implicit content of its premises.

In turn, representatives of the direction coming primarily from German philosophy (Chr. Wolf, G. V. Leibniz), also, based on the fact that deduction does not provide new information, precisely on this basis came to the exact opposite conclusion: the obtained through deduction, knowledge is “true in all possible worlds,” which determines its “enduring” value, in contrast to “factual” truths obtained by inductive generalization of observational and experience data, which are true “only due to a coincidence of circumstances.” From a modern point of view, the question of such advantages of deduction or induction has largely lost its meaning. Along with this, the question of the source of confidence in the truth of a deductively correct conclusion based on the truth of its premises is of certain philosophical interest. Currently, it is generally accepted that this source is the meaning of the logical terms included in the reasoning; thus, deductively correct reasoning turns out to be “analytically correct.”

Important Terms

Deductive reasoning- an inference that ensures, given the truth of the premises and compliance with the rules of logic, the truth of the conclusion. In such cases, deductive reasoning is treated as a simple case of proof or some step of proof.

Deductive proof– one of the forms of proof when a thesis, which is some kind of individual or particular judgment, is brought under a general rule. The essence of such proof is as follows: you must obtain the consent of your interlocutor that the general rule under which a given individual or particular fact fits is true. When this is achieved, then this rule applies to the thesis being proven.

Deductive logic- a branch of logic in which methods of reasoning are studied that guarantee the truth of the conclusion when the premises are true. Deductive logic is sometimes identified with formal logic. Outside the limits of deductive logic are the so-called. plausible reasoning and inductive methods. It explores ways of reasoning with standard, typical statements; These methods are formalized in the form of logical systems, or calculi. Historically, the first system of deductive logic was Aristotle's syllogistic.

How can deduction be applied in practice?

Judging by the way Sherlock Holmes unravels detective stories using the deductive method, it can be adopted by investigators, lawyers, and law enforcement officers. However, mastery of the deductive method will be useful in any field of activity: students will be able to quickly understand and remember the material better, managers or doctors will be able to make the only correct decision, etc.

There is probably no area of human life where the deductive method would not be useful. With its help, you can draw conclusions about the people around you, which is important when building relationships with them. It develops observation, logical thinking, memory and simply makes you think, preventing the brain from aging ahead of time. After all, our brain needs training no less than our muscles.

Attention to details

As you observe people and everyday situations, notice the smallest cues in conversations to become more responsive to events. These skills became the trademarks of Sherlock Holmes, as well as the heroes of the TV series True Detective and The Mentalist. New Yorker columnist and psychologist Maria Konnikova, author of Mastermind: How to Think Like Sherlock Holmes, says Holmes' thinking technique is based on two simple things - observation and deduction. Most of us do not pay attention to the details around us, but in the meantime, outstanding (fictional and real) detectives have a habit of noticing everything down to the smallest detail.

How to train yourself to be more attentive and focused?

First, stop multitasking and focus on one thing at a time. The more things you do at once, the more likely you are to make mistakes and are more likely to miss important information. It is also less likely that the information will be retained in your memory.
Secondly, it is necessary to achieve the right emotional state. Anxiety, sadness, anger and other negative emotions that are processed in the amygdala impair the brain's ability to solve problems or absorb information. Positive emotions, on the contrary, improve this brain function and even help you think more creatively and strategically.

Develop memory

Having tuned in to the right mood, you should strain your memory to begin to put everything you observe there. There are many methods for training it. Basically, it all comes down to learning to attach significance to individual details, for example, the brands of cars parked near the house and their license plate numbers. At first you will have to force yourself to remember them, but over time it will become a habit and you will memorize the cars automatically. The main thing when forming a new habit is to work on yourself every day.

Play more often Memory"and other board games that develop memory. Set yourself the task of remembering as many objects as possible in random photos. For example, try to remember as many objects from photographs as possible in 15 seconds.

Memory competition champion and author of Einstein Walks on the Moon, a book about how memory works, Joshua Foer explains that anyone with average memory ability can greatly improve their memory abilities. Like Sherlock Holmes, Foer is able to remember hundreds of phone numbers at a time, thanks to the encoding of knowledge in visual pictures.

His method is to use spatial memory to structure and store information that is relatively difficult to remember. So numbers can be turned into words and, accordingly, into images, which in turn will take a place in the memory palace. For example, 0 could be a wheel, a ring, or a sun; 1 – a pole, a pencil, an arrow or even a phallus (vulgar images are remembered especially well, writes Foer); 2 – a snake, a swan, etc. Then you imagine some space that is familiar to you, for example, your apartment (it will be your “memory palace”), in which there is a wheel at the entrance, a pencil on the bedside table nearby, and behind her is a porcelain swan. This way you can remember the sequence "012".

Maintaining"field notes"

As you begin your transformation into Sherlock, start keeping a diary with notes. As the Times columnist writes, scientists train their attention in this way - by writing down explanations and recording sketches of what they observe. Michael Canfield, a Harvard University entomologist and author of Field Notes on Science and Nature, says this habit "will force you to make better decisions about what's really important and what's not."

Taking field notes, whether during a regular work meeting or a walk in a city park, will develop the right approach to exploring the environment. Over time, you begin to pay attention to small details in any situation, and the more you do this on paper, the faster you will develop the habit of analyzing things as you go.

Focus attention through meditation

Many studies confirm that meditation improves concentration and attention. You should start practicing with a few minutes in the morning and a few minutes before bed. According to John Assaraf, lecturer and renowned business consultant, “Meditation is what gives you control over your brain waves. Meditation trains your brain so you can focus on your goals."

Meditation can make a person better equipped to obtain answers to questions of interest. All this is achieved by developing the ability to modulate and regulate different frequencies of brain waves, which Assaraf compares to the four speeds in a car transmission: “beta” is the first, “alpha” is the second, “theta” is the third and “ delta waves" - from the fourth. Most of us function in the beta range during the day, and that's not a terribly bad thing. However, what is first gear? The wheels spin slowly, and the engine wears quite a lot. People also burn out faster and experience more stress and illness. Therefore, it is worth learning how to switch to other gears in order to reduce wear and the amount of “fuel” consumed.

Find a quiet place where there will be no distractions. Be fully aware of what is happening and watch the thoughts that arise in your head, concentrate on your breathing. Take slow, deep breaths, feeling the air flow from your nostrils to your lungs.

Think critically and ask questions

Once you learn to pay close attention to detail, begin to transform your observations into theories or ideas. If you have two or three puzzle pieces, try to understand how they fit together. The more puzzle pieces you have, the easier it will be to draw conclusions and see the whole picture. Try to derive specific provisions from general ones in a logical way. This is called deduction. Remember to apply critical thinking to everything you see. Use critical thinking to analyze what you observe closely, and use deduction to build a big picture from those facts. It is not easy to describe in a few sentences how to develop your critical thinking abilities. The first step to this skill is to return to childhood curiosity and the desire to ask as many questions as possible.

Konnikova says the following about this: “It is important to learn to think critically. So, when acquiring new information or knowledge about something new, you will not just memorize and remember something, but learn to analyze it. Ask yourself: “Why is this so important?”; “How can I combine this with the things I already know?” or “Why do I want to remember this?” Questions like these train your brain and organize information into a network of knowledge.”

Let your imagination run wild

Of course, fictional detectives like Holmes have the superpower of seeing connections that ordinary people simply ignore. But one of the key foundations of this exemplary deduction is nonlinear thinking. Sometimes it’s worth giving free rein to your imagination to replay the most fantastic scenarios in your head and go through all possible connections.

Sherlock Holmes often sought solitude to think and freely explore a problem from all sides. Like Albert Einstein, Holmes played the violin to help him relax. While his hands were busy playing, his mind was immersed in a meticulous search for new ideas and problem solving. Holmes even mentions at one point that imagination is the mother of truth. By detaching himself from reality, he could look at his ideas in a completely new way.

Expand your horizons

It is obvious that an important advantage of Sherlock Holmes is his broad outlook and erudition. If you can also easily understand the works of Renaissance artists, the latest trends in the cryptocurrency market, and discoveries in the most advanced theories of quantum physics, your deductive methods of thinking have a much greater chance of success. You should not place yourself within the framework of any narrow specialization. Strive for knowledge and cultivate a sense of curiosity about a wide variety of things and areas.

Conclusions: exercises for developing deduction

Deduction cannot be acquired without systematic training. Below is a list of effective and simple methods for developing deductive thinking.

Solving problems in the fields of mathematics, chemistry and physics. The process of solving such problems increases intellectual abilities and contributes to the development of such thinking.
Expanding your horizons. Deepen your knowledge in various scientific, cultural and historical fields. This will not only allow you to develop your personality from different angles, but will also help you gain experience, rather than relying on superficial knowledge and guesswork. In this case, various encyclopedias, trips to museums, documentaries and, of course, travel will help.
Pedantry. The ability to thoroughly study an object of interest to you allows you to comprehensively and thoroughly gain a complete understanding. It is important that this object evokes a response in the emotional spectrum, then the result will be effective.
Flexibility of mind. When solving a task or problem, it is necessary to use different approaches. To choose the best option, it is recommended to listen to the opinions of others, thoroughly considering their versions. Personal experience and knowledge, combined with outside information, as well as the availability of several options for resolving the issue, will help you choose the most optimal conclusion.
Observation. When communicating with people, it is recommended not only to hear what they say, but also to observe their facial expressions, gestures, voice and intonation. Thus, one can recognize whether a person is sincere or not, what his intentions are, etc.

Objective-logical thinking presupposes a general line; an example is the transition of society from one formation to another.

The objective-historical method is a concrete manifestation of a certain pattern in the infinite variety of its individual manifestations and features. In society, as an example, we can use the connection of individual destinies with the real history of the country.

Methods

These types of knowledge are analyzed by two methods: logical and historical. Any phenomenon can be understood and explained only in its historical development. In order to understand an object, it is necessary to reflect the history of its appearance. Without an idea of the development path, it is difficult to understand the final result. History proceeds in zigzags and leaps; in order to ensure that the sequence is not interrupted during its analysis, a variant of logical research is necessary. To study history you need:

analysis;
synthesis;
induction;
deduction;
analogy.

Logical thinking presupposes a generalized reflection of historical development and explains its importance. This method often means a certain state of the object being studied at a specific time interval. This depends on many factors, but the objectives of the study, as well as the nature of the object, are decisive. Thus, to discover his law, I. Kempler did not study the history of the planets.

Research methodology

Induction and deduction are distinguished as separate research methods. Let's analyze the features of each of them and try to identify their characteristic features. What is the difference between induction and deduction? Induction is the process of identifying particular (individual) facts on the basis of general provisions. There is a division of it into two parts: incomplete and complete. The second is characterized by conclusions or judgments about objects based on information about the entire set. In practice, both induction and deduction are used; the choice depends on the specific situation. The use of incomplete induction is considered a common occurrence. In this case, conclusions about the object being studied are made on the basis of partial information about the subject. Reliable information can be obtained from experimental studies conducted repeatedly.

Application in modern times

Induction and deduction are still widely used today. Deduction involves reasoning from the general to the individual (particular). All conclusions that are obtained in the course of such reasoning are reliable only if the correct methods have been chosen for analysis. In human thinking, induction and deduction are closely interrelated. Examples of such unity allow a person to analyze current events and look for the right ways to resolve a problem situation. Induction directs human thought to the conclusion of empirically verifiable consequences from general hypotheses, their experimental confirmation or refutation. An experiment is characterized by a scientifically staged experiment conducted to study the phenomenon caused by it. The researcher works under certain conditions, monitors the results obtained, using a variety of instruments and materials, and directs him in the right direction.

Examples

What is the difference between induction and deduction? Examples of the use of these methods can be found in any field of activity of modern man. When considering the deductive method of thinking as an example, the image of the legendary detective Sherlock Holmes immediately appears. This technique is associated with logic, analysis of many details, and decision-making based on the information received.

Research in Economics

Induction and deduction in economics are commonplace. Thanks to these methods, all analytical and statistical studies are carried out and specific decisions are made. For example, through deduction, economists study consumer demand for mortgage lending. The results obtained during the research are analyzed, the overall result is derived, and on its basis a decision is made to modernize the offer for this type of lending for the population. Economic research is carried out according to a certain algorithm. First, a research object is selected, which will become the basis for the work of statisticians. Next, a hypothesis is put forward; the final result of the study largely depends on the correctness of its formulation. In order to obtain reliable information, methods are selected and an algorithm of actions is created. The results are considered reliable only if the experiments are carried out not 1-2 times, but in several series of 2-3 studies.

Conclusion

We have analyzed important terms such as induction and deduction. Examples from different areas of human activity confirm the advisability of using two methods at once. For example, modern pedagogy is based on deductive methods. Before offering certain banking products to borrowers, they are carefully analyzed by specialists, all possible consequences of their appearance on the market are assumed. What exactly to choose: deduction or induction, professionals decide taking into account the specific situation. Deduction allows you to draw conclusions in which errors are practically eliminated. It is this technique that psychologists recommend that people study in order to protect themselves from constant stress and to seek strength to deal with complex problems.

Induction and deduction are interrelated, complementary methods of inference. A whole occurs in which a new statement is born from judgments based on several conclusions. The purpose of these methods is to derive new truth from pre-existing ones. Let's find out what it is and give examples of deduction and induction. The article will answer these questions in detail.

Deduction

Translated from Latin (deductio) it means “deduction”. Deduction is the logical conclusion of the particular from the general. This line of reasoning always leads to a true conclusion. The method is used in cases where it is necessary to derive the necessary conclusion about a phenomenon from a generally known truth. For example, metals are heat-conducting substances, gold is a metal, we conclude: gold is a heat-conducting element.

Descartes is considered the founder of this idea. He argued that the starting point of deduction begins with intellectual intuition. His method includes the following:

Recognizing as true only what is known with maximum obviousness. There should not be any doubts in the mind, that is, one must judge only on irrefutable facts.
Divide the phenomenon under study into as many simple parts as possible so that they can be easily overcome.
Move from simple gradually to more complex.
Compile the overall picture in detail, without any omissions.

Descartes believed that with the help of such an algorithm, the researcher would be able to find the true answer.

It is impossible to comprehend any knowledge except through intuition, reason and deduction. Descartes

Induction

Translated from Latin (inductio) it means “guidance”. Induction is the logical conclusion of the general from particular judgments. Unlike deduction, reasoning leads to a probable conclusion, all because several bases are generalized, and hasty conclusions are often drawn. For example, gold, like copper, silver, and lead, is a solid substance. This means that all metals are solids. The conclusion is not correct, since the conclusion was hasty, because there is a metal such as mercury, and it is a liquid. An example of deduction and induction: in the first case, the conclusion turned out to be true. And in the second - probable.

Economic sphere

Deduction and induction in economics are research methods on a par with such as observation, experiment, modeling, method of scientific abstractions, analysis and synthesis, systems approach, historical and geographical method. When using the inductive method, research begins with observation of economic phenomena, facts are accumulated, and then a generalization is made on their basis. When applying the deductive method, an economic theory is formulated, then hypotheses are tested based on it. That is, from theory to facts, research goes from general to specific.

Let us give examples of deduction and induction in economics. The increase in the cost of bread, meat, cereals and other goods forces us to conclude that prices are rising in our country. This is induction. The notification about the increase in the cost of living makes it seem that prices for gas, electricity, other utilities and consumer goods will increase. This is deduction.

Field of psychology

For the first time, the phenomena in psychology we are considering were mentioned in his works by an English thinker. His merit was the unification of rational and empirical knowledge. Hobbes insisted that there can only be one truth, achieved through experience and reason. In his opinion, knowledge begins with sensibility as the first step towards generalization. The general properties of phenomena are established using induction. Knowing the actions, you can find out the cause. After clarifying all the reasons, we need the opposite path, deduction, which makes it possible to understand new and different actions and phenomena. and deductions in psychology according to Hobbes show that these are interchangeable stages of one cognitive process, passing from each other.

Sphere of Logic

We are familiar with two types thanks to such a character as Sherlock Holmes. Arthur Conan Doyle introduced the deductive method to the whole world. Sherlock began the observation with the general picture of the crime and led to the specific, that is, he studied each suspect, every detail, motives and physical capabilities, and, using logical conclusions, figured out the criminal, arguing with iron-clad evidence.

Deduction and induction in logic are simple; without noticing, we use it every day in everyday life. We often react quickly, instantly jumping to the wrong conclusion. Deduction is longer thinking. To develop it, you need to constantly challenge your brain. To do this, you can solve problems from any field, mathematics, physics, geometry, even puzzles and crosswords will help develop thinking. Books, reference books, films, travel - everything that broadens one's horizons in various fields of activity will provide invaluable assistance. Observation will help you come to the correct logical conclusion. Every, even the most insignificant, detail can become part of one big picture.

Let's give an example of deduction and induction in logic. You see a woman about 40 years old, in her hand is a handbag with an unfastened zipper due to the large number of notebooks in it. She is dressed modestly, without frills or frilly details, on her hand is a thin watch and a white chalk mark. You will conclude that most likely she works as a teacher.

Sphere of pedagogy

The method of induction and deduction is often used in school education. Methodological literature for teachers is organized inductively. This type of thinking is widely applicable to studying technical devices and solving practical problems. And with the help of the deductive method it is easier to describe a large number of facts, explaining their general principles or properties. Examples of deduction and induction in pedagogy can be observed in any lesson. Often in physics or mathematics, the teacher gives a formula, and then during the lesson the students solve problems that fit this case.

In any field of activity, the methods of induction and deduction are always useful. And you don’t have to be a super detective or a genius in scientific fields to do this. Give your thinking a workout, develop your brain, train your memory, and in the future complex tasks will be solved on an instinctive level.