Conclusions in logic. Deductive reasoning

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26. Logical analysis

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Logical way

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 civilizations 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 their position in various popular forums, in government institutions, court hearings, etc.

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, turning Special attention on 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.

Distinctive feature of such non-deductive reasoning is that the conclusion in them 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." Speaking modern language, the creators of inductive logic considered 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 opening complex connections and deep theoretical laws required the use of all means and methods of empirical and theoretical research, maximum application 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 forms of deductive reasoning, the emergence of relational logic and, in particular, the application mathematical methods for the analysis of deduction, which culminated in the creation of symbolic (or mathematical) logic, which 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 in important ways 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 mathematical model deductive reasoning. 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 is focused primarily on mathematical proofs, therefore, abstracts from the nature of the 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 an argument, discussion, polemic, the analysis and assessment of premises acquires a special 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 reasoning. Concepts are included in the structure of any judgment in the form of a subject, that is, 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 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 general process rational reasoning aimed both at finding the truth and at proving it.

The properties of the basic concepts are revealed in axioms- proposals accepted without proof.

For example, in school geometry there are axioms: “through any two points you can draw a straight line and only one” or “a straight line divides a plane into two half-planes.”

The system of axioms of any mathematical theory, revealing the properties of basic concepts, gives their definitions. Such definitions are called axiomatic.

The properties of concepts to be proved are called theorems, consequences, signs, formulas, rules.

Prove the theorem AIN- this means to establish in a logical way that whenever a property is satisfied A, the property will be executed IN.

Proof in mathematics they call a finite sequence of propositions of a given theory, each of which is either an axiom or is deduced from one or more propositions of this sequence according to the rules of logical inference.

The basis of the proof is reasoning - logical operation, as a result of which, from one or more sentences interconnected in meaning, a sentence containing new knowledge is obtained.

As an example, consider the reasoning of a schoolchild who needs to establish the “less than” relation between the numbers 7 and 8. The student says: “7< 8, потому что при счете 7 называют раньше, чем 8».

Let us find out what facts the conclusion obtained in this argument is based on.

There are two such facts: First: if the number A when counting, the numbers are called before b, That a< b. Second: 7 is called earlier than 8 when counting.

The first sentence is general character, since it contains a general quantifier - it is called a general premise. The second sentence concerns the specific numbers 7 and 8 - it is called a private premise. Received from two parcels new fact: 7 < 8, его называют заключением.

There is a certain connection between the premises and the conclusion, thanks to which they constitute an argument.

An argument in which there is an implication relation between the premises and the conclusion is called deductive.

In logic, instead of the term “reasoning,” the word “inference” is more often used.

Inference- this is a way of obtaining new knowledge based on some existing knowledge.

An inference consists of premises and a conclusion.

Parcels- these contain initial knowledge.

Conclusion- this is a statement containing new knowledge obtained from the original one.

As a rule, the conclusion is separated from the premises using the words “therefore”, “means”. Inference with premises R 1, R 2, …, рn and conclusion R we will write it in the form: or (R 1, R 2, …, рn) R.

Examples inferences: a) Number a =b. Number b = c. Therefore, the number a = c.

b) If the numerator in a fraction is less than the denominator, then the fraction is proper. In a fraction numerator is less than denominator (5<6) . Therefore, the fraction - correct.

c) If it rains, then there are clouds in the sky. There are clouds in the sky, therefore it is raining.

Conclusions can be correct or incorrect.

The inference is called correct if the formula corresponding to its structure and representing a conjunction of premises, connected to the conclusion by an implication sign, is identically true.

For that to determine whether the conclusion is correct, proceed as follows:

1) formalize all premises and conclusion;

2) write down a formula representing a conjunction of premises connected by an implication sign with a conclusion;

3) draw up a truth table for this formula;

4) if the formula is identically true, then the conclusion is correct; if not, then the conclusion is incorrect.

In logic, it is believed that the correctness of a conclusion is determined by its form and does not depend on the specific content of the statements included in it. And in logic, rules are proposed, following which, one can build deductive conclusions. These rules are called rules of inference or patterns of deductive reasoning.

There are many rules, but the most commonly used are the following:

1. - rule of conclusion;

2. - rule of negation;

3. - the rule of syllogism.

Let's give example inferences made from rule conclusions:"If the recording of a number X ends with a number 5, that number X divided by 15. Writing a number 135 ends with a number 5 . Therefore, the number 135 divided by 5 ».

The general premise in this conclusion is the statement “if Oh), That B(x)", Where Oh)- this is a “record of number” X ends with a number 5 ", A B(x)- "number X divided by 5 " A particular premise is a statement that is obtained from the condition of the general premise when
x = 135(those. A(135)). A conclusion is a statement derived from B(x) at x = 135(those. V(135)).

Let's give example of a conclusion made according to the rule negatives:"If the recording of a number X ends with a number 5, that number X divided by 5 . Number 177 not divisible by 5 . Therefore it does not end with a number 5 ».

We see that in this conclusion the general premise is the same as in the previous one, and the particular one is the negation of the statement “number 177 divided by 5 "(i.e.). The conclusion is the negation of the sentence “Writing a number 177 ends with a number 5 "(i.e.).

Finally, let's consider example of an inference based on syllogism rule: "If the number X multiple 12, then it is a multiple 6. If the number X multiple 6 , then it is a multiple 3 . Therefore, if the number X multiple 12, then it is a multiple 3 ».

This conclusion has two premises: “if Oh), That B(x)" and if B(x), That C(x)", where A(x) is "the number X multiple 12 », B(x)- "number X multiple 6 " And C(x)- "number X multiple 3 " The conclusion is a statement “if Oh), That C(x)».

Let's check whether the following conclusions are correct:

1) If a quadrilateral is a rhombus, then its diagonals are mutually perpendicular. ABCD- rhombus Therefore, its diagonals are mutually perpendicular.

2) If the number is divisible by 4 , then it is divided by 2 . Number 22 divided by 2 . Therefore, it is divided into 4.

3) All trees are plants. Pine is a tree. This means that pine is a plant.

4) All students in this class went to the theater. Petya was not in the theater. Therefore, Petya is not a student in this class.

5) If the numerator of a fraction is less than the denominator, then the fraction is correct. If a fraction is proper, then it is less than 1. Therefore, if the numerator of a fraction is less than the denominator, then the fraction is less than 1.

Solution: 1) To resolve the question of the correctness of the inference, let us identify its logical form. Let us introduce the following notation: C(x)- "quadrangle" X- rhombus", B(x)- “in a quadrangle X the diagonals are mutually perpendicular." Then the first premise can be written as:
C(x) B(x), second - C(a), and the conclusion B(a).

Thus, the form of this inference is: . It is built according to the rule of conclusion. Therefore, this reasoning is correct.

2) Let us introduce the notation: Oh)- "number X divided by 4 », B(x)- "number X divided by 2 " Then we write the first premise: Oh)B(x), second B(a), and the conclusion is A(a). The conclusion will take the form: .

There is no such logical form among those known. It is easy to see that both premises are true and the conclusion is false.

This means that this reasoning is incorrect.

3) Let us introduce some notation. Let Oh)- "If X tree", B(x) - « X plant". Then the parcels will take the form: Oh)B(x), A(a), and the conclusion B(a). Our conclusion is built in the form: - rules of conclusion.

This means that our reasoning is structured correctly.

4) Let Oh) - « X- students of our class, B(x)- “students X went to the theater." Then the parcels will be as follows: Oh)B(x),, and the conclusion.

This conclusion is based on the rule of negation:

- it means it is correct.

5) Let’s identify the logical form of the inference. Let A(x) -"numerator of a fraction X less than the denominator." B(x) - “fraction X- correct." C(x)- "fraction" X less 1 " Then the parcels will take the form: Oh)B(x), B(x) C(x), and the conclusion Oh)C(x).

Our conclusion will have the following logical form: - the rule of syllogism.

This means that this conclusion is correct.

In logic, various ways of checking the correctness of inferences are considered, including analysis of the correctness of inferences using Euler circles. It is carried out as follows: write down the conclusion in set-theoretic language; depict premises on Euler circles, considering them true; they look to see whether the conclusion is always true. If yes, then they say that the inference is constructed correctly. If a drawing is possible from which it is clear that the conclusion is false, then they say that the conclusion is incorrect.

Table 9

Let us show that inference made according to the rule of inference is deductive. First, let's write this rule in set-theoretic language.

Package Oh)B(x) can be written as TATV, Where TA And TV- truth sets of propositional forms Oh) And B(x).

Private parcel A(a) means that ATA, and the conclusion B(a) shows that ATV.

The entire inference constructed according to the inference rule will be written in set-theoretic language as follows: .

Rice. 58.

Examples.

1. Is the conclusion “If a number ends in a number” correct? 5, then the number is divisible by 5. Number 125 divided by 5. Therefore, writing the number 125 ends with a number 5 »?

Solution: This conclusion is made according to the scheme , which corresponds to . There is no such scheme known to us. Let's find out whether it is a rule of deductive inference?

Let's use Euler circles. In set-theoretic language

The resulting rule can be written as follows:

. Let us depict the sets on Euler circles TA And TV and denote the element A from many TV.

It turns out that it can be contained in a set TA, or may not belong to him (Fig. 59). In logic, it is believed that such a scheme is not a rule of deductive inference, since it does not guarantee the truth of the conclusion.

This conclusion is not correct, since it is made according to a scheme that does not guarantee the truth of the reasoning.

Rice. 59.

b) All verbs answer the question “what to do?” or “what should I do?” The word "cornflower" does not answer any of these questions. Therefore, "cornflower" is not a verb.

Solution: a) Let us write this conclusion in set-theoretic language. Let us denote by A- many students of the Faculty of Education, through IN- many students who are teachers, through WITH- many students over 20 years old.

Then the conclusion will take the form: .

If we depict these sets on circles, then 2 cases are possible:

1) sets A, B, C intersect;

2) set IN intersects with many WITH And A, and a lot A intersects IN, but does not intersect with WITH.

b) Let us denote by A many verbs, and through IN a lot of words that answer the question “what to do?” or “what should I do?”

Then the conclusion can be written as follows:

Let's look at a few examples.

Example 1. The student is asked to explain why the number 23 can be represented as the sum of 20 + 3. He reasons: “The number 23 is two-digit. Any two-digit number can be represented as a sum of digit terms. Therefore, 23 = 20 + 3."

The first and second sentences in this conclusion are premises, and one of a general nature is the statement “any two-digit number can be represented as a sum of digit terms,” and the other is particular, it characterizes only the number 23 - it is two-digit. The conclusion - this sentence that comes after the word “therefore” - is also private in nature, since it refers to the specific number 23.

Inferences, which are usually used in proving theorems, are based on the concept of logical implication. Moreover, from the definition of logical implication it follows that for all values ​​of the propositional variables for which the initial statements (premises) are true, the conclusion of the theorem is also true. Such conclusions are deductive.

In the example discussed above, the inference given is deductive.

Example 2. One of the techniques for introducing primary schoolchildren to the commutative property of multiplication is as follows. Using various visual aids, schoolchildren, together with the teacher, establish that, for example, 6 3 = 36, 52 = 25. Then, based on the obtained equalities, they conclude: for all natural numbers a And b equality is true ab = ba.

In this conclusion, the premises are the first two equalities. They claim that such a property holds for specific natural numbers. The conclusion in this example is a general statement - the commutative property of multiplication of natural numbers.

In this conclusion, premises of a particular nature show that some Natural numbers have the following property: rearranging the factors does not change the product. And on this basis it was concluded that all natural numbers have this property. Such inferences are called incomplete induction.

those. for some natural numbers it can be argued that the sum is less than their product. This means that based on the fact that some numbers have this property, we can conclude that all natural numbers have this property:

This example is an example of analogical reasoning.

Under analogy understand an inference in which, based on the similarity of two objects in some characteristics and the presence of an additional characteristic in one of them, a conclusion is made about the presence of the same characteristic in the other object.

A conclusion by analogy is in the nature of an assumption, a hypothesis, and therefore needs either proof or refutation.

When drawing a conclusion, it is convenient to present the rules for introducing and removing logical connectives in the same way as the rules for inference:

Rule 1. If the premises $F_1$ and $F_2$ have the meaning “and”, then their conjunction is true, i.e.

$$\frac(F_1 ; F_2)((F_1\&F_2))$$

This entry, if the premises $F_1$ and $F_2$ are true, provides for the possibility of introducing a logical conjunction of a conjunction into the conclusion; this rule is identical to axiom A5 (see);

Rule 2. If $(F_1\&F_2)$ has the value “and”, then the subformulas $F_1$ and $F_2$ are true, i.e.

$$\frac((F_1\&F_2))(F_1) \: and \: \frac((F_1\&F_2))(F_2)$$

This notation, if $(F_1\&F_2)$ is true, provides for the possibility of removing the logical connective of the conjunction in the conclusion and considering the true values ​​of the subformulas $F_1$ and $F_2$; this rule is identical to axioms A3 and A4;

Rule 3. If $F_1$ has the value “and”, and $(F_1\&F_2)$ has the value “l”, then the subformula $F_2$ is false, i.e.

$$\frac(F_1;\left\rceil\right. \!\!(F_1\&F_2))( \left\rceil\right. \!\!F_2)$$

This entry, if $(F_1\&F_2)$ is false and one of the subformulas is true, provides for the possibility of removing the logical conjunction of the conjunction in the conclusion and considering the value of the second subformula to be false;

Rule 4. If at least one premise $F_1$ or $F_2$ is true, then their disjunction is true, i.e.

$$\frac(F_1)( (F_1\vee F_2)) \: or \: \frac(F_2)( (F_1\vee F_2))$$

This notation, if at least one subformula $F_1$ or $F_2$ is true, provides for the possibility of introducing a logical connective of disjunction in the conclusion; this rule is identical to axioms A6 and A7;

Rule 5. If $(F_1\vee F_2)$ has the value “and” and one of the subformulas $F_1$ or $F_2$ has the value “l”, then the second subformula $F_2$ or $F_1$ is true, i.e.

$$\frac((F_1\vee F_2); \left\rceil\right. \!\!F_1 )( (F_2) \: or \: \frac((F_1\vee F_2); \left\rceil\right . \!\!F_2 )( (F_1)$$

This notation, if $(F_1\vee F_2)$ is true, provides for the possibility of removing the logical connective of the disjunction in the conclusion and considering the true values ​​of the subformulas $F_1$ or $F_2$;

Rule 6. If the subformula $F_2$ has the value “and”, then the formula $(F_1\rightarrow F_2)$ is true for any value of the subformula $F_1$, i.e.

$$\frac(F_2)( (F_1\rightarrow F_2))$$

This notation, with a true value of $F_2$, provides for the possibility of introducing an implication into the conclusion of a logical connective for any value of the subformula $F_1$ (“truth from anything”); this rule is identical to axiom 1;

Rule 7. If the subformula $F_1$ has the value “l”, then the formula $(F_1\rightarrow F_2)$ is true for any value of the subformula $F_2$, i.e.

$$\frac(\left\rceil\right. \!\!F_1 )( (F_1\rightarrow F_2))$$

This notation, if the value of $F_1$ is false, provides for the possibility of introducing a logical connective of implication into the conclusion for any value of the subformula $F_2$ (“anything from false”);

Rule 8. If the formula $(F_1\rightarrow F_2)$ has the value “and”, then the formula $(\left\rceil\right. \!\!F_2\rightarrow \left\rceil\right. \!\!F_1)$ is true , i.e.

$$\frac((F_1\rightarrow F_2) )( (\left\rceil\right. \!\!F_2\rightarrow \left\rceil\right. \!\!F_1))$$

This entry, with a true value of $(F_1\rightarrow F_2)$, determines the possibility of swapping the poles of the implication while simultaneously changing their values; this is the law of contraposition;

Rule 9. If the formula $(F_1\rightarrow F_2)$ has the value “and”, then the formula $((F_1\vee F_3)\rightarrow (F_2\vee F_3)$ is true for any value of $F_3$, i.e.

$$\frac((F_1\rightarrow F_2) )(((F_1\vee F_3)\rightarrow (F_2\vee F_3)) $$

This entry, with a true value of $(F_1\rightarrow F_2)$, determines the ability to perform the disjunction operation for any value of the formula $F_3$ over each pole of the implication; this rule is identical to axiom A11.

Rule 10. If the formula $(F_1\rightarrow F_2)$ has the value “and”, then the formula $((F_1\&F_3)\rightarrow (F_2\&F_3)$ is true for any value of $F_3$, i.e.

$$\frac((F_1\rightarrow F_2) )(((F_1\&F_3)\rightarrow (F_2\&F_3))$$

This entry, with a true value of $(F_1\rightarrow F_2)$, determines the ability to perform the conjunction operation for any value of the formula $F_3$ over each pole of the implication; this rule is identical to axiom A10.

Rule 11. If the formulas $(F_1\rightarrow F_2)$ and $(F_2\rightarrow F_3)$ have the value “and”, then the formula $(F_1\rightarrow F_3)$ is true, i.e.

$$\frac((F_1\rightarrow F_2); (F_2\rightarrow F_3) )((F_1\rightarrow F_3))$$

This entry, with the true value of $(F_1\rightarrow F_2)$ and $(F_2\rightarrow F_3)$, provides for the possibility of forming the implication $(F_1\rightarrow F_3)$ (the law of syllogism); this rule is identical to axiom A2;

Rule 12. If the formulas $F_1$ and $(F_1\rightarrow F_2)$ have the value “and”, then the formula $F_2$ is true, i.e.

$$\frac(F_1; (F_1\rightarrow F_2) )( F_2)$$

This entry, given the true value of the premise $F_1$ and the implication $(F_1\rightarrow F_2)$, allows you to remove the logical connective of the implication and determine the true value of the conclusion $F_2$;

Rule 13. If the formulas are $\left\rceil\right. \!\!F_2 and (F_1\rightarrow F_2)$ have the meaning “and”, then the formula $\left\rceil\right is true. \!\!F_1$, i.e.

$$\frac(\left\rceil\right. \!\!F_2; (F_1\rightarrow F_2) )( \left\rceil\right. \!\!F_1)$$

This entry is given the true value of the premise $\left\rceil\right. \!\!F_2$ and implications $(F_1\rightarrow F_2)$ allows you to remove the logical connective of the implication and determine the true value of the conclusion $\left\rceil\right. \!\!F_1$;

Rule 14. If the formulas $(F_1\rightarrow F_2)$ and $(F_2\rightarrow F_1)$ have the value “and”, then the formula $(F_1\leftrightarrow F_2)$ is true, i.e.

$$\frac((F_1\rightarrow F_2); (F_2\rightarrow F_1) )( (F_1\leftrightarrow F_2))$$

This entry, with the true value of $(F_1\rightarrow F_2)$ and $(F_2\rightarrow F_1)$, allows you to introduce a logical equivalence connective and determine the value of the formula $(F_1\leftrightarrow F_2)$;

Rule 15. If the formula $(F_1\leftrightarrow F_2)$ has the value “and”, then the formulas $(F_1\rightarrow F_2)$ and $(F_2\rightarrow F_1)$ are true, i.e.

$$\frac((F_1\leftrightarrow F_2) )( (F_1\rightarrow F_2) ) \: and \: \frac((F_1\leftrightarrow F_2) )( (F_2\rightarrow F_1) )$$

This entry, with the true value of $(F_1\leftrightarrow F_2)$, allows you to remove the logical connective of equivalence and determine the true value of the formulas $(F_1\rightarrow F_2)$ and $(F_2\rightarrow F_1)$.