Conclusions in logic. The concept of propositional logic

Part one. Deductive and plausible reasoning

CHAPTER 1. Subject and tasks of logic

1.1. Logic as a science

Logic is one of the most ancient sciences, the first teachings of which about the forms and methods of reasoning arose in the civilizations of the Ancient East (China, India). The principles and methods of logic entered Western culture mainly through the efforts of the ancient Greeks. Developed political life in the Greek city-states, the struggle of different parties for influence on the masses of free citizens, the desire to resolve property and other conflicts that arose through the courts - all this required the ability to convince people, to defend one’s position at various popular forums, in government institutions, and court hearings. and so on.

The art of persuasion, arguing, the skill of reasonably defending one’s opinion and objecting to an opponent during an argument and polemic was cultivated within the framework of ancient rhetoric, focused on improving oratory, and eristics, a special teaching about argument. The first teachers of rhetoric did a lot to disseminate and develop knowledge about the skill of persuasion, methods of argument and construction of public speech, paying special attention to its emotional, psychological, moral and oratorical aspects and features. However, later, when the schools of rhetoric began to be headed by the sophists, they sought to teach their students not to seek truth through argument, but rather to win, to win a verbal competition at any cost. For these purposes, deliberate logical errors were widely used, which later became known as sophistry, as well as various psychological tricks and techniques for distracting the opponent’s attention, suggestion, switching the dispute from the main topic to secondary issues, etc.

The great ancient philosophers Socrates, Plato and Aristotle resolutely opposed this tendency in rhetoric, who considered the main means of persuasion to be the validity of the judgments contained in the oratorical speech, their correct connection in the process of reasoning, i.e. inferring some judgments from others. It was for the analysis of reasoning that Aristotle (IV century BC) created the first system of logic, called syllogistics. It is the simplest, but at the same time the most frequently used form of deductive reasoning, in which the conclusion (conclusion) is obtained from the premises according to the rules of logical deduction. Note that the term deduction translated from Latin means conclusion.

To explain this, let us turn to the ancient syllogism:

All people are mortal.

Kai is human.____________

Therefore, Kai is mortal.

Here, as in other syllogisms, the inference is made from general knowledge about a certain class of objects and phenomena to particular and individual knowledge. Let us immediately emphasize that in other cases deduction can be carried out from particular to particular or from general to general.

The main thing that unites all deductive inferences is that the conclusion follows from the premises according to the logical rules of inference and has a reliable, objective character. In other words, the conclusion does not depend on the will, desires and preferences of the reasoning subject. If you accept the premises of such a conclusion, then you must accept its conclusion.

It is also often stated that the defining feature of deductive inferences is the logically necessary nature of the conclusion, its reliable truth. In other words, in such inferences the truth value of the premises is completely transferred to the conclusion. This is why deductive reasoning has the greatest persuasive power and is widely used not only to prove theorems in mathematics, but also wherever reliable conclusions are needed.

Very often in textbooks logics determined as a science about the laws of correct thinking or the principles and methods of correct conclusions. Since, however, it remains unclear what kind of thinking is considered correct, the first part of the definition contains a hidden tautology, since it is implicitly assumed that such correctness is achieved by observing the rules of logic. In the second part, the subject of logic is defined more precisely, because the main task of logic is reduced to the analysis of inferences, i.e. to identifying ways of obtaining some judgments from others. It is easy to notice that when they talk about correct inferences, they implicitly or even explicitly mean deductive logic. It is precisely in it that there are completely definite rules for the logical derivation of conclusions from premises, which we will get acquainted with in more detail later. Often deductive logic is also identified with formal logic on the grounds that the latter studies the forms of inferences in abstraction from the specific content of judgments. This view, however, does not take into account other methods and forms of reasoning that are widely used both in experimental sciences that study nature, and in socio-economic and human sciences, based on facts and results of social life. And in everyday practice, we often make generalizations and make assumptions based on observations of particular cases.

Reasoning of this kind, in which, on the basis of research and verification of any particular cases, one comes to a conclusion about unstudied cases or about all phenomena of the class as a whole, is called inductive. Term induction means guidance and well expresses the essence of such reasoning. They usually study the properties and relationships of a certain number of members of a certain class of objects and phenomena. The resulting general property or relationship is then transferred to unexplored members or to the entire class. Obviously, such a conclusion cannot be considered reliably true, because among the unexplored members of the class, and especially the class as a whole, there may be members that do not possess the supposed common property. Therefore, the conclusions of induction are not reliable, but only probabilistic. Often such conclusions are also called plausible, hypothetical or conjectural, since they do not guarantee the achievement of the truth, but only point to it. They have heuristic(search) rather than reliable in nature, helping to search for the truth rather than prove it. Along with inductive reasoning, this also includes conclusions by analogy and statistical generalizations.

A distinctive feature of such non-deductive reasoning is that the conclusion does not follow logically, i.e. according to the rules of deduction, from premises. Premises only to one degree or another confirm the conclusion, make it more or less probable or plausible, but do not guarantee its reliable truth. On this basis, probabilistic reasoning is sometimes clearly underestimated, considered secondary, auxiliary, and even excluded from logic.

This attitude towards non-deductive and, in particular, inductive logic is explained mainly by the following reasons:

Firstly, and this is the main thing, the problematic, probabilistic nature of inductive conclusions and the associated dependence of the results on the available data, inseparability from premises, and incompleteness of conclusions. After all, as new data becomes available, the likelihood of such conclusions also changes.

Secondly, the presence of subjective aspects in assessing the probabilistic logical relationship between the premises and the conclusion of the argument. These premises, such as facts and evidence, may seem convincing to one person, but not to another. One believes that they strongly support the conclusion, the other is of the opposite opinion. Such disagreements do not arise in deductive inference.

Thirdly, this attitude towards induction is also explained by historical circumstances. When inductive logic first arose, its creators, in particular F. Bacon, believed that with the help of its canons, or rules, it was possible to discover new truths in experimental sciences in an almost purely mechanical way. “Our path of discovery of sciences,” he wrote, “leaves little to the sharpness and power of talent, but almost equalizes them. Just as in drawing a straight line or describing a perfect circle, firmness, skill and testing of the hand mean a lot, if you act only with your hand, it means little or it means nothing at all if you use a compass and a ruler. This is the case with our method." In modern language, the creators of inductive logic viewed their canons as algorithms of discovery. With the development of science, it became more and more obvious that with the help of such rules (or algorithms) it is possible to discover only the simplest empirical connections between experimentally observed phenomena and the quantities that characterize them. The discovery of complex connections and deep theoretical laws required the use of all means and methods of empirical and theoretical research, the maximum use of the mental and intellectual abilities of scientists, their experience, intuition and talent. And this could not but give rise to a negative attitude towards the mechanical approach to discovery, which previously existed in inductive logic.

Fourthly, the expansion of the forms of deductive reasoning, the emergence of relational logic and, in particular, the use of mathematical methods for the analysis of deduction, which culminated in the creation of symbolic (or mathematical) logic, largely contributed to the advancement of deductive logic.

All this makes it clear why they often prefer to define logic as the science of the methods, rules and laws of deductive inferences or as the theory of logical inference. But we must not forget that induction, analogy and statistics are important methods of heuristic search for truth, and therefore they serve as rational methods of reasoning. After all, the search for truth can be carried out at random, through trial and error, but this method is extremely ineffective, although it is sometimes used. Science resorts to it very rarely, since it focuses on an organized, targeted and systematic search.

It must also be taken into account that general truths (empirical and theoretical laws, principles, hypotheses and generalizations), which are used as premises of deductive conclusions, cannot be established deductively. But it may be objected that they do not open inductively. However, since inductive reasoning is focused on the search for truth, it turns out to be a more useful heuristic means of research. Of course, in the course of testing assumptions and hypotheses, deduction is also used, in particular to draw consequences from them. Therefore, deduction cannot be opposed to induction, since in the real process of scientific knowledge they presuppose and complement each other.

Therefore, logic can be defined as the science of rational methods of reasoning, which cover both the analysis of the rules of deduction (deriving conclusions from premises) and the study of the degree of confirmation of probabilistic or plausible conclusions (hypotheses, generalizations, assumptions, etc.).

Traditional logic, which was formed on the basis of the logical teachings of Aristotle, was later supplemented by the methods of inductive logic formulated by F. Bacon and systematized by J.S. Millem. It is this logic that has been taught for a long time in schools and universities under the name formal logic.

Emergence mathematical logic radically changed the relationship between deductive and non-deductive logics that existed in traditional logic. This change was made in favor of deduction. Thanks to symbolization and the use of mathematical methods, deductive logic itself acquired a strictly formal character. In fact, it is quite legitimate to consider such logic as a mathematical model of deductive inferences. Therefore, it is often considered a modern stage in the development of formal logic, but they forget to add that we are talking about deductive logic.

It is also often said that mathematical logic reduces the process of reasoning to the construction of various systems of calculations and thereby replaces the natural process of thinking with calculations. However, the model is always associated with simplifications, so it cannot replace the original. Indeed, mathematical logic focuses primarily on mathematical proofs, therefore, it abstracts from the nature of premises (or arguments), their validity and acceptability. She considers such premises to be given or previously proven.

Meanwhile, in the real process of reasoning, in dispute, discussion, polemics, the analysis and evaluation of premises becomes especially important. In the course of argumentation, you have to put forward certain theses and statements, find convincing arguments in their defense, correct and supplement them, give counterarguments, etc. Here we have to turn to informal and non-deductive methods of reasoning, in particular to inductive generalization of facts, conclusions by analogy, statistical analysis, etc.

Considering logic as the science of rational methods of reasoning, we must not forget about other forms of thinking - concepts and judgments, with which any logic textbook begins. But judgments, and especially concepts, play an auxiliary role in logic. With their help, the structure of inferences and the connection of judgments in various types of reasoning become clearer. Concepts are included in the structure of any judgment in the form of a subject, i.e., an object of thought, and a predicate - as a sign characterizing the subject, namely, asserting the presence or absence of a certain property in the object of thought. In our presentation, we adhere to the generally accepted tradition and begin the discussion with an analysis of concepts and judgments, and then cover in more detail deductive and non-deductive methods of reasoning. The chapter where propositions are analyzed examines the elements of propositional calculus, which are usually the starting point for any course in mathematical logic.

Elements of predicate logic are covered in the next chapter, where the theory of categorical syllogism is considered as a special case. Modern forms of non-deductive reasoning cannot obviously be understood without a clear distinction between the logical and statistical interpretation of probability, since under probability what is most often implied is precisely its statistical interpretation, which has an auxiliary meaning in logic. In this regard, in the chapter on probabilistic reasoning, we specifically focus on clarifying the difference between the two interpretations of probability and explain in more detail the features of logical probability.

Thus, the entire nature of the presentation in the book orients the reader to the fact that deduction and induction, reliability and probability, the movement of thought from the general to the particular and from the particular to the general do not exclude, but rather complement each other in the general process of rational reasoning, aimed at both the search for truth and its proof.

Here F And G– formulas, and C– either a formula or ^.

The description of the inference system for propositional logic is now complete.

In each of the following problems, derive the given formula from the empty set of premises.

1) (pÚ q) É ( qÚ p).

2) (pÚ p) º p.

3) pÉ (( pÚ q) º q).

4) (p&(qÚ r)) º (( p&q) Ú ( p&r)).

5) pº p.

6) (pÚ q) º ( p&q).

I) Both rules for introducing a disjunction are correct.

J) The rule for removing a disjunction is correct.

Correctness theorem.If there is a conclusion F from G , Then G logically entails F.

Completeness theorem.For any formula F and any set of formulas G , If G implies F, then there is a derivation of F from the subset G.

The completeness of propositional logic (for another set of inference rules) was established by Emil Post in 1921.

Inference rule- this is a prescription or permission that allows from a judgment of the first logical structure, as premises, to derive judgments of a certain logical structure, as conclusions.

The peculiarities of the rules of conclusion are that the signs of the truth of the conclusion are made on the basis not of content, but of their structure. Inference rules are written in the form of a diagram, which consists of 2 parts (top and bottom), separated by a vertical line. Logical schemes of the premises are written above the line in the column, and logical schemes of the conclusion are written below the line.

All rules of inference of propositional logic are divided into 2 groups:

Basic and Derivatives.

- Basic– these are simple and obvious rules that do not need proof. The main ones are divided into direct and indirect.

· Direct- these are rules that indicate the direct deducibility of some judgments from others.

· Indirect– only provide an opportunity to conclude about the legitimacy of deducing some judgments from others.

- Derivatives- a shortened withdrawal process, derived from the main ones.

Basic straight lines.

Introduction of conjunction: A, B

Removing a conjunction: A ⋀ B

Introduction of disjunction: A B

A ⋁ B A ⋁ B

Removing disjunction: A ⋁ B

Removing the implication: A ⊃ B

Introduction of negation/removal: A; Ǟ

Introduction of equivalence: A ⊃ B, B ⊃ A

Removing equivalence: A<-->IN

A ⊃ B, B ⊃ A

The main ones are indirect.

The peculiarity is that the conclusion does not follow obviously from the premises, and therefore additional conditions are resorted to.

Introduction of implication.

2.A – assumption

4.B – removal of implication 1,2

5.C – removal of implication 3.4

6.A ⊃ C introduction of implication 2.5.

Rule of reduction to absurdity - if from premises and assumptions, in the course of reasoning or proof, two contradictory statements B and not B are deduced, then in the conclusion it is possible to write not A. B (not B)

Derivatives.

Rule of conditional (hypothetical) syllogism:

Negation of disjunction:

Contraposition rule:

Complex contraposition:

Import rule.

Export Rule:

A simple constructive dilemma:

Difficult design dilemma:

A simple destructive dilemma:

Complex destructive dilemma:

Implication through conjunction

Questions for self-control:

1. What is the difference between judgments, questions and norms?

2. What is the composition and what are the types of attributive judgments?

3. What are the types of relationship judgments?

4. What are the types of complex judgments?

5. How is the negation of attributive judgments and judgments about relationships made?

6. How are complex judgments denied?

7. What are the main types of relations between judgments?

8. The relationships between which judgments are expressed through a logical square?

9. How are attributive judgments and judgments about relations expressed in the language of predicate logic?

10. Which questions are incorrect? Name the types of incorrect questions.

11. How do the concepts “obligatory”, “allowed” and “prohibited” relate?

Tasks for independent work:

I. Are the following sentences judgments?

1. The Urals are far from us.

2. On a clean, smooth path

I passed, didn't follow...

Who was sneaking around here?

Who fell and walked here?

(S. Yesenin)

3. Scientific and technological progress is impossible without experiments.

4. A modern physical or biological experiment often provides so much information that it is almost impossible to process it without a computer.

5. He didn’t show up for work today.

6. What student doesn’t dream of getting a good grade in an exam?

7. It is necessary to more actively introduce computer science and computer technology into the educational process.

8. Sleep! Turn off the light!

9. What does the coming day have in store for me?

10. Where should we go now? Will you ever get out of here? (K. Paustovsky).

11. Lilies of the valley and strawberries bloom in the shade under oak trees near a forest ravine.

12. Evgeny is waiting: Lensky is coming

On a trio of roan horses,

Let's have lunch quickly!

“Well, what about the neighbors?

What about Tatyana?

Why is Olga your frisky?”

(A.S. Pushkin)
II. Determine the type, terms of judgment and their distribution in the following reasoning:

1. Some subjects are expressed by pronouns in the nominative case.
2. Some schoolchildren do not study a second foreign language.

3. Granite is widely used in construction.

4. Not a single dolphin is a fish.

V. Knowing the distribution of terms in a simple attributive assertoric judgment, construct the thought correctly:

5.1. Highway (S+), paved road (P-);

5.2. Russian scientist (S-), Nobel Prize laureate (P-);

5.3. Panther (S+), herbivore (P+);

5.4. Head of Government (S+), head of the highest body of executive government (P+);

5.5. Writer (S-), playwright (P+).

IV. Determine the type and logical form of the following complex judgments
and write down their structure with a formula.

1. “A child’s soul is equally sensitive to its native word, to the beauty of nature, and to a musical melody. If in early childhood the beauty of a musical work is conveyed to the heart, if the child feels the multifaceted shades of human feelings in sounds, he rises to a level of culture that cannot be achieved by any other means” (V.A. Sukhomlinsky).

2. The more blood flows through the vascular system per unit of time, the more abundant the supply of oxygen and nutrients to the organs, the more waste products flow away from the tissues.

3. If a person loves flowers, he will always treat them with care: he will water them, tie up the stems, pick off dry leaves.

4. “If our children are our old age, then proper upbringing is our happy old age, bad upbringing is our grief, these are our tears, this is our guilt before other people” (A.S. Makarenko).

V. Determine the type of modality in the following judgments:

1. It has been proven that S= n R2 where S is the area of ​​the circle and R - its radius.

2. The introduction of computer technology is impossible without training the people who will use it.

3. It is necessary that space be peaceful.

4. Perhaps tomorrow the weather will be good and we will go on an excursion to the forest.

5. Children give us the opportunity to leave our mark on earth - in their memory, in their activities, in the tradition and knowledge that we pass on to them.

VI. Are the following formulas laws of logic:

6.1.((p → q) ^ q) → q.

6.2. (p V q V r) = p^q^r.

6.3. ((p → q) ^ (p → r) ^ (q V r)) → p

6.4. ((p → q) ^ (r → s) ^ (p V r)) → (q Vs).

VII. Using tabular propositional logic, determine whether the following reasoning is correct.

7.1. It has been established that the crime could have been committed by Smith, Jones or Brown. Jones is known to never commit a crime without Brown. Therefore, if Brown did not commit the crime, then Smith did.

7.2. If a person is satisfied with his job and happy in his family life, then he has no reason to complain about fate. This man has a reason to complain about fate. This means that he is either satisfied and happy in his family life, or happy in his family life, but not satisfied with his work.

7.3. If a person tells a lie, then he is mistaken or deliberately misleading others. This man is not telling the truth, but he is clearly not mistaken. Consequently, he deliberately misleads others.

VIII. Using tabular propositional logic, determine the relationships between the following statements:

8.1. The contracting parties have no claims against each other or they agree on settlement.

If they agree on settlement, then they have entered into a new contract or have claims against each other.

8.2. If a philosopher is a dualist, then he is not an idealist.

If a philosopher is not an idealist, then he is a dialectician or a metaphysician.

8.3. If a person has committed a crime, then he is subject to criminal liability.

If a person has committed a crime and it is proven, then he is subject to criminal liability.

A person has committed a crime, but he is not subject to criminal liability.

Chapter V. CONCLUSION as a form of thought.

Inference is a form of thinking through which from one or more judgments, called premises, according to certain rules of inference, we obtain a new judgment, called a conclusion.

Aristotle gave the following example of a conclusion: “All men are mortal” and “Socrates is a man” are premises. “Socrates is mortal” - conclusion. The transition from premises to conclusion occurs according to the RULES OF INFERENCE and the laws of logic.

RULE 1: If the premises of the inference are true, then it is also true

conclusion.
RULE 2: If the inference is valid in all cases, then it is valid in every particular case. (This rule is DEDUCTION- transition from general to specific.)
RULE 3: If an inference is valid in some particular cases, then it is valid in all cases. (This rule is INDUCTION- transition from particular to general.)
Chains of inferences are formed into REASONINGS and EVIDENCE, in which the conclusion of the previous inference becomes the premise of the next one. The condition for the correctness of a proof is not only the truth of the initial judgments, but also the truth of each inference included in it. Proofs must be constructed according to the laws of logic:

1. LAW OF IDENTITY. Every thought is identical to itself, i.e. the subject of reasoning must be strictly defined and unchanged until its completion. A violation of this law is the substitution of concepts (often used in legal practice).
2. LAW OF NON-CONTRADITION. Two opposing propositions cannot be true at the same time: at least one of them is false.
3. LAW OF THE EXCLUDED THIRD. Either a proposition is true or its negation (“there is no third option”).
4. LAW OF SUFFICIENT GROUNDS. For the truth of any thought there must be sufficient grounds, i.e. the conclusion must be justified based on judgments whose truth has already been proven.

Let's take a look at some interesting types of inferences:
PARALOGISM- an inference containing an unintentional error. This type of inference is often found in your tests.
SOPHISM- an inference containing a deliberate error with the aim of passing off a false judgment as true.
Let's try, for example, to prove that 2 x 2 = 5:

4/4 = 5/5
4(1/1) = 5(1/1)
4 = 5.

PARADOX is an inference that proves both the truth and falsity of a certain proposition.
For example:
General and barber. Each soldier can shave himself or be shaved by another soldier. The general ordered the appointment of one special soldier-barber, who would shave only those soldiers who did not shave themselves. Who should shave the soldier barber?

In logic they study inferences, carried out on the basis or using the features of the logical forms of premises and conclusions. Inference contains judgments (and, consequently, concepts), but is not reduced to them, but also presupposes their certain connection. Thanks to this, a special form with its specific functions is formed. Formally - logical analysis of this form means answering the following basic questions: what is the essence conclusions and what is their role and structure; what are their main types; what relationships do they have with each other? finally, what logical operations are possible with them. The significance of such an analysis is determined by the fact that it is in conclusions(and the evidence based on them) the “secret” of the coercive power of speeches is hidden, which amazed people in ancient times and with the comprehension of which logic as a science began. Exactly inferences provide what we now call the power of logic. That is why logic is often called the science of inferential knowledge. And there is a significant amount of truth in this. After all, the analysis of concepts and judgments, although important in itself, fully reveals its full meaning only in connection with their logical functions in relation to conclusions(and therefore evidence). We'll consider inference in two relationships: 1) as a form of reflection of reality, and 2) as a form of thinking, one way or another embodied in language.

To understand the origin and essence inferences, it is necessary to compare two types of knowledge that we have and use in the process of our life - direct and indirect. Direct knowledge is that which we receive with the help of our senses: vision, hearing, smell, etc. Such, for example, is knowledge expressed by judgments such as “the grass is green,” “snow is white,” “the sky is blue,” “the flower smells.” ", "birds are singing." They constitute a significant part of all our knowledge in the process of reflecting the objective world by human consciousness and serve as their basis. However, we cannot judge everything in the world directly. For example, no one has ever observed that the sea was once raging in the Moscow area. And there is knowledge about this. It is derived from other knowledge. The fact is that large deposits of white stone have been discovered in the Moscow region. It was formed from the skeletons of countless small marine organisms, which could only accumulate on the bottom of the sea. Thus, it was concluded that approximately 250 - 300 million years ago the Russian Plain, on which the Moscow region is located, was flooded by the sea. Such knowledge, which is obtained not directly, directly, but indirectly, that is, by derivation from other knowledge, is called indirect (or inferential). The logical form of their acquisition is inference. In its most general form, it refers to a form of thinking through which new knowledge is derived from known knowledge. The existence of such a form in our thinking, like concepts and judgments, is determined by objective reality itself. If the basis of the concept is the objective nature of reality, and the basis of the judgment is the connection (relationship) of objects, then the objective basis inferences constitutes a more complex mutual connection of objects, their mutual relationships. So if one class of objects (A) is included entirely in another (B), but does not exhaust its volume, this means the necessary feedback: a wider class of objects (B) includes a less wide one (A) as its part, but is not reduced to him. This can be seen from the diagram: B A A B. For example: “All scientists are smart people,” this means: “Some smart people are scientists.” Or a more complex case of the relationship of objects of thought: if one class of objects (A) is included in another (B), and this, in turn, is included in the third (C), then it follows that the first (A) is included in the third (C ). In the diagram: B C B C A A Example: “M. Lomonosov is a scientist, and all scientists are smart people, then M. Lomonosov is a smart person.” This is an objective possibility conclusions: - this is a structural cast of reality itself, but in an ideal form, in the form of a structure of thought. And their objective necessity, like concepts and judgments, is also connected with the entire practice of mankind. The satisfaction of some human needs and the emergence of others on this basis requires the progress of social production, and this, in turn, is unthinkable without the progress of knowledge. The necessary link in the implementation of this progress is inferences as one of the forms of transition from known knowledge to new ones.

5.1. Role conclusions and their structure.

Inferences a very common form used in scientific and everyday thinking. This determines their role in the knowledge and practice of people. Meaning conclusions people is that they not only connect our knowledge into more or less complex, relatively complete complexes - mental constructs, but also enrich and strengthen this knowledge. Together with concepts and judgments inferences overcome the limitations of sensory knowledge. They turn out to be indispensable where the senses are powerless in comprehending the causes and conditions of the emergence of any object or phenomenon, its essence and forms of existence, patterns of development, etc. They participate in the formation of concepts and judgments, which often act as a result conclusions to become a means of further knowledge. At every step inferences are produced in everyday life. So I look out the window in the morning and, noticing the wet roofs of houses, we conclude that it rained during the night. Watching the crimson-red sunset in the evening, we expect windy weather for tomorrow. Play a special role inferences in legal practice. In his famous notes about Sherlock Holmes, A. Canon Doyle gave the classic image of a detective who was fluent in the art of conclusions and on their basis he unraveled the most complex and incredible forensic stories. In modern legal literature and practice conclusions also plays a huge role. So a preliminary consequence from the point of view of logic is nothing more than the construction of all possible conclusions about the alleged criminal, about the mechanism of formation of traces of the crime, about the motives that prompted him to commit the crime, about the consequences of the crime for society. An indictment is only one form inferences at all. Inference- a holistic mental formation, it is similar to how, for example, water, being a holistic, qualitatively defined aggregate state of matter, decomposes into chemical elements - hydrogen and oxygen, which are in a certain ratio with each other, and so is any inference has its own structure. It is determined by the nature of this thinking and its role in cognition and communication. In structure inferences There are two main more or less complex elements: premises (one or more) and a conclusion, between which there is also a certain connection. Premises are the initial and, moreover, already known knowledge that serves as the basis inferences. The conclusion is a derivative, and a new one, obtained from the premises and serving as their consequence. Conclusion is a logical transition from premises to conclusion. This is the connection between the premises and by inference, there is a necessary relation between them that makes it possible to move from one to another - a relation of logical consequence. This is the basic law of every inferences, allowing one to reveal its deepest and most intimate “secret” - the compulsory conclusion. If we have recognized any premises, then whether we want it or not, we are forced to recognize the conclusion, precisely because of a certain connection between them. This law, which is based on the objective relationship of the objects of thought themselves, manifests itself in many special rules that are specific to different forms conclusions. We have already considered what role they play inferences in the formation of concepts and judgments, and now let’s consider what role concepts and judgments play in conclusions. Since concepts and judgments are part of the structure conclusions It is important for us to establish their logical functions here. Thus, it is not difficult to understand that judgments perform the functions of either premises or conclusions. Concepts, being terms of judgment, perform here the functions of terms inferences. If we consider concepts dialectically, as a process of transition from one level of knowledge to another, higher one, then it will not be difficult to understand the relativity of dividing judgments into premises and conclusions. The same judgment, being the result (conclusion) of one cognitive act, becomes the starting point (premise) of another. This process can be likened to building a house: one row of logs (or bricks) placed on an existing foundation thereby becomes the foundation for another, subsequent row. The situation is similar with concepts - terms. inferences: one and the same concept can act either as a subject, or as a predicate of a premise or conclusion, or as a mediating link between them. This is how the endless process of cognition is carried out. Like any judgment, a conclusion can be true or false. But both are determined here directly by their relation not to reality, but primarily to the premises and their connection. The conclusion will be true if two necessary conditions are present: firstly, the initial propositions - premises - must be true inferences; secondly, in the process of reasoning one must follow the rules of inference that determine logical correctness inferences.

For example: All artists have a keen sense of nature

I. Levitan - artist

I. Levitan - has a keen sense of nature

A - I. Levitan, B - artists C - sensitive people A B C A And vice versa, the conclusion can be false if: 1) at least one of the premises is false or 2) the structure inferences wrong.

Example: All witnesses are truthful

Sidorov - witness

Sidorov is truthful

Here one of the premises is false, which is why a definite conclusion cannot be drawn. And about how important the correct structure is inferences , is evidenced by a well-known humorous example in logic, when an absurd conclusion follows from both known premises.

All savages wear feathers

All women wear feathers

All women are savages

That a certain conclusion with such a design inferences impossible, as the circular diagram shows. A - women B - savages C - wearing feathers C A B A A A From false premises or with incorrect structure inferences the true conclusion may arise purely by chance.

For example: Glass does not conduct electricity.

Iron is not glass.

Iron conducts electricity.

With such a structure inferences It is enough to replace “hardware” with “rubber” to understand the randomness of the correct conclusion. The connection between the premises and the conclusion must not be accidental, but necessary, unambiguous, justified; one must really follow and follow from the other. If the connection is random or ambiguous in relation to the conclusion, as they say when exchanging apartments, “options are possible,” then such a conclusion cannot be drawn, otherwise an error is inevitable.

5.2.Inference and connection of sentences.

Like any other form of thinking, inference one way or another embodied in language. If a concept is expressed by a separate word (or phrase), and a judgment is expressed by a separate sentence (or combination of sentences), then inference there is always a connection between several (two or more) sentences, although not every connection between two or more sentences is necessarily inference(for example, complex judgments). In Russian, this connection is expressed by the words “therefore”, “means”, “thus”, “because”, “since”, etc. Inference may end with a conclusion (conclusion), but may also begin with it; finally the output can be in the middle inferences, between parcels. General rule of linguistic expression inferences is as follows: if the conclusion comes after the premises, then the words “therefore”, “means”, “therefore”, so “,” hence follows “, etc. are placed before it. If the conclusion comes before the premises, then the words "are placed after it" because "," since "," for "," because "and others. If, finally, it is located between the premises, then the corresponding words are used simultaneously before and after it. In the example given, the following logical ones are possible, and therefore, language constructions: 1) All scientists are smart people, and M. Lomonosov is a scientist, therefore, he is an intelligent person (conclusion at the end); 2) M. Lomonosov is an intelligent person, because he is a scientist, and everyone scientists are smart people, (conclusion at the beginning); 3) All scientists are smart people, therefore, M. Lomonosov is a smart person, because he is a scientist, (conclusion in the middle). It is not at all difficult to guess that we have not exhausted all possible options for logical constructions conclusions, but it is important to know them in order to be able to identify more or less stable mental structures in the stream of living speech - written or oral - in order to subject them to strict logical analysis in order to avoid possible or already made mistakes and misunderstandings.

5.3. Kinds conclusions.

Acting as a more complex form of thinking than concept and judgment, inference At the same time, it represents a form richer in its manifestations. Reviewing the practice of thinking, one can discover a great variety of very diverse types and varieties conclusions, but three main fundamental types can be distinguished inferences, classified according to the direction of logical consequence, i.e., according to the nature of the connection between knowledge of varying degrees of generality, expressed in premises and conclusions. This inferences: deduction, induction and traduction.

Deduction (from the Latin deductio - "deduction") is inference, in which the transition from general knowledge to particular knowledge is logically necessary. The rules of deductive inference are determined by the nature of the premises, which can be simple or complex propositions. Depending on the number of premises, deductive conclusions are divided into direct, in which the conclusion is deduced from one premise, and indirect, in which the conclusion is deduced from several (two or more) premises.

Example: All metals conduct electricity.

Copper is a metal.

Copper conducts electricity.

Inductive inferences (from the Latin inductio - “guidance”) are inferences, in which, based on the attribute’s belonging to individual objects or parts of a certain class, a conclusion is made about its belonging to the class as a whole. The main function of inductive inferences in the process of cognition is generalization, i.e., obtaining general judgments. In terms of their content and cognitive significance, these generalizations can be of a different nature - from the simplest generalizations of everyday practice to empirical generalizations in science or universal judgments expressing universal laws. Depending on the completeness and regularity of empirical research, two types of inductive research are distinguished: conclusions: complete induction and incomplete induction. Example: Having determined that every metal conducts electricity, we can conclude: “All metals conduct electricity.”

Traductive inferences (from the Latin traductio - “translation”, “movement”, “transfer”) are inferences in which both the premises and the conclusion are of the same degree of generality, i.e. these are inferences from judgments of attitude and inference by analogy, which represent a conclusion about the belonging of a certain feature to the individual object under study (subject, event, relation or class) based on its similarity in essential features with another already known individual object. Inference by analogy, it is always preceded by the operation of comparing two objects, which allows us to establish similarities and differences between them. At the same time, analogy does not require any coincidences, but similarities in essential features while the differences are insignificant. It is these similarities that serve to compare two material or ideal objects. An example can be given in the history of physics about the mechanisms of propagation of sound and light, when they were likened to the movement of a liquid. Based on this, wave theories of sound and light arose. The objects of comparison in this case were liquid, sound and light, and the transferred sign was the wave method of their propagation.