Inductive and deductive reasoning. Deductive and inductive reasoning Reliability of deductive and inductive reasoning Cartesians

Book: LOGIC FOR LAWYERS: LECTURES. / Law College of LNU named after. Franco

§ 3. Indirect deductive reasoning

Reasoning according to the “reduction to absurdity” scheme

Reasoning according to the scheme *mixing to the absurd* - This is an argument in which the falsity of a certain statement is proven on the basis that contradictions are deduced from this statement with the help of other considerations.

Scheme 18.

♦ For example, let's try to interpret the legal concept "source of increased danger" using the “reduction to absurdity” method.

“Who is the subject of activities related to the use of sources of increased danger to others? Car owners? So! The assumption arises that perhaps all vehicles should be considered sources of increased danger, especially since under certain circumstances they are such. Let's continue the reasoning and bring it to its logical conclusion: what about the owner of a horse, bicycle, wheelbarrow, stretcher, etc.? We come to a clearly absurd conclusion that contradicts common sense. Therefore, this interpretation cannot be considered correct. So it’s not about the vehicles, it’s about the power of that vehicle.”

Reasoning according to the “proof by contradiction” scheme- This is reasoning in which the truth of a certain statement is proven on the basis that the negation of this statement with the help of other considerations leads to contradictions.

The scheme of this reasoning is as follows:

Scheme 19

♦ The investigator reasons: “Most likely, G. is not guilty. But let's try to assume the opposite. Let G. wine. Then on April 27, 2001 he should be at the crime scene in Kiev. However, witness G. testifies that G. was in London that evening. Given the difficulties of crossing the border, it is unlikely that he could get there from London in two hours. Consequently, he was not in Kyiv on April 27, 2001. It follows that my version of G.’s guilt is incorrect. Thus, G. is not guilty.”

§ 4. Deductive inferences in legal activity

Deductive reasoning plays a significant role in the theoretical and practical work of a lawyer. In this regard, a professional lawyer must be able to competently, in accordance with the rules of logic, build deductive conclusions of various types.

Let us give examples of the use of deduction in the activities of a lawyer.

Deductive reasoning can be used by lawyers V the process of arguing one’s own point of view and criticizing the enemy’s position (see section 8 of the textbook). It should be noted that it is with the help of deduction that a lawyer can substantiate the truth of a certain position or refute it, that is, prove its fallacy. Using non-deductive (plausible) inferences, this is almost impossible to do.

Deduction is also widely used in the process of putting forward investigative versions. Quite often, the version is the conclusion of deductive reasoning. Let's look at an example.

♦ During the investigation of the murder of A., the investigator measured it this way: “It can be assumed that the murder of A. was not carried out for the purpose of robbery. But this seems unlikely because A. was dressed poorly And He had no valuables with him. The murder could have been committed out of revenge, but people who knew A. described him as a modest, quiet person. For the past three years he has worked as a school custodian and has not had any quarrels.

These circumstances lead to the idea that the murder was committed for hooligan reasons.”

In this example of justification for the conclusion “The murder was not carried out with hooligan motives” The investigator conducts it according to the scheme of divisive-categorical inference, namely, its perceptual-affirmative mode. First, he puts forward all possible versions of the reasons for the murder, and then excludes those that seem unlikely to him. What is left becomes the main version.

Along with the use of deduction in the process of putting forward versions, it is also used in the process of checking versions, which, as a rule, begins with deductive deduction of consequences from the put forward version, and at the final stage, using logical proof or refutation, its truth or falsity is substantiated.

Basic terms

♦ deductive reasoning

♦ direct deductive inference

♦ mediated deductive inference

♦ purely conditional inference

♦ conditional disjunctive inference

♦ reasoning according to the “reduction to absurdity” scheme

♦ reasoning according to the “proof from a bedsore” scheme

Test questions and exercises

1.what is deductive reasoning?

2. how do the concepts of “deductive inference” and “correct reasoning” relate?

3.Is it possible for deductive reasoning to be incorrect?

4. Is it possible to obtain (substantiate) reliable knowledge using deductive reasoning?

5. guarantees the truth of the assumptions in deductive reasoning and the truth of the conclusion?

6. Can deductive reasoning have false premises?

7. Or is it possible to obtain a flawed conclusion using the deductive inference scheme?

8. What types of direct deductive inferences do you know? Can you provide their diagrams?

9. What types of indirect deductive inferences do you know? Give their diagrams.

10.What is the significance of deductive reasoning in legal activity?

11.Analyze the given texts. Exercise deductive reasoning on what they contain. Determine their type and correctness.

♦ In Conan Doyle’s story “The Beryl Tiara,” banker Alexander Holder, in whose house a jewel was stolen—a beryl tiara—turned to Sherlock Holmes for help. Hallder was sure that his son Arthur was guilty of the theft, because on the night when the theft was committed, he saw in his hands a diadem in which one horn with three beryls was missing. But Holder established that his niece was also involved in the theft of the tiara, who passed the tiara through the window to her lover.

In telling Holder about the results of the investigation, Holmes said, in part, the following: “My old principle of investigation is to exclude all obviously impossible assumptions. Then what remains is the truth, no matter how implausible it may seem.

I reasoned something like this: it’s a common thing, you didn’t give up the tiara. Therefore, only your niece and the maid remain. But if the maids are involved in the theft, then why did your son agree to take responsibility. There is no basis for such an assumption. They said that Arthur loves his cousin. And I understood the reason for his silence: I didn’t want to show off to Mary. Then I remembered that you stood at the window and that she lost consciousness when she saw the tiara in Arthur’s hands. My assumptions turned into certainties.” 1

♦ When Doctor Watson asked how Holmes knew that he had been at the post office in the morning and sent a telegram, the latter replied: “I know that you did not write any letters in the morning, because I sat opposite you all morning. And in the open drawer of your bureau, I noticed a thick stack of postcards and a whole sheet of stamps. Why then go to the post office, if not to send a telegram? Throw away everything that cannot be true, and there will remain one single fact, which is the truth."

♦ In the morning, the manager of the store, B., called the trade department and the district police department and reported that a window had been broken in the service area of the store and a large amount of goods had been stolen. After examining the scene of the incident, an investigator from the prosecutor's office put forward a version that the theft of Bya was staged. The main thing that testified in favor of this version was the dubious possibility of the theft of goods worth such a large amount through the opening in the bars (the metal bars that covered the window from the inside of the room were not damaged). At the same time, it was not excluded that the theft could have been committed by unauthorized persons. To clarify the question of whether a person could steal goods from the office premises through a broken window and undamaged play, an investigative experiment was conducted. The results of the experiment proved that a criminal could steal through a window a certain amount of goods only from a nearby rack, and it turned out to be impossible to get something from other racks through the window. The presence of dust and cobwebs on the grille also cast doubt on the credibility of B.'s statement about the theft of goods through a broken window.

As a result of checking the version of the theft that B. committed by staging the theft, he was completely exposed in the systematic theft of state property. B. admitted to staging the theft in order to conceal the shortage of goods.

♦ During the investigation of a case of mass death of livestock, it was established by autopsy of each animal that the cause of death was the exhaustion of the animals’ bodies. Further investigation showed that the cause of depletion was a lack of feed, and the reason for the lack of feed was its theft and squandering.

♦ During the investigation into the cause of the fire in the grocery store, it was determined that several months before the fire, major renovations had been carried out there. New electrical wiring was done, stoves and chimneys were rebuilt. At the same time, the night work was entrusted to a person who did not have sufficient qualifications in this matter; he laid the chimney with poor dressing of the seams, and used a low-quality solution. All this led to a weakening of the strength of the chimney, in which cracks appeared. On the day of the fire there was severe frost and the stove was heated with box containers, which create a long sparkling flame. Through the chimney cracks, hot fuel gases, and possibly sparks, acted on the ceiling structures. The combustion, which at first proceeded in the form of smoldering, could remain unnoticeable for a long time, because the smoke entered the attic, where it dissipated. The fire was noticed much later, when the combustion became open.

♦ During the investigation into the circumstances of K.’s death, it was established that the apartment doors were locked. There were no other keys that belonged to K. in the apartment. The lock design did not allow access to the door without a key. So, the doors were closed from the outside, while K. remained in the apartment. This could only be done by P., who was the last to leave the apartment.

♦ During the investigation into the murder of carriage conductor D., the investigator suggested that the murder was committed either by an acquaintance of the victim, or by the conductor of another carriage who was on the same train, or by one of the other railway workers. It was unlikely that unauthorized persons would enter the carriage at night. Moreover, before leaving, the leaders were specially instructed by Fr.

It was established that the murder was committed on the section of the road between stations G. and M. and that there were four other conductors on the train, Z., B., K. and S.

The assumption that D. was killed by conductor Z., which was at first very plausible, was not confirmed during the investigation. The participation of conductor S. in this crime was excluded, since the murder was accompanied by attempted rape. The version that conductor B. committed this crime, given his advanced age and relatively poor health, was unlikely. The version about the murder of D. by her acquaintances was also not confirmed.

♦ During the investigation of a murder case, the investigator came to the conclusion that murder for the purpose of robbery in this case was excluded. This is evidenced by the presence of the murdered person's clothes, jewelry and money. The assumption that the criminal was prevented from robbing the murdered man was also not confirmed: at the crime scene there were traces of the corpse being dragged away from the murder scene; a scarf, cap and handkerchief were hidden in one of the pipes, which happened nearby; There were also fingerprints of a person who had taken the snow, probably in order to wash her hands. All this allowed us to conclude that no one interfered with the criminal and he was in no hurry to leave the scene of the murder.

1.	LOGIC FOR LAWYERS: LECTURES. / Law College of LNU named after. Franco

2.

3.	3. Historical stages in the development of logical knowledge: logic of Ancient India, logic of Ancient Greece

4.	4. Features of general or traditional (Aristotelian) logic.

5.	5. Features of symbolic or mathematical logic.

6.	6. Theoretical and practical logic.

7.	Topic 2: THINKING AND SPEECH 1. Thinking (reasoning): definition and features.

8.	2. Activity and thinking

9.	3. Structure of thinking

10.	4. Correct and incorrect reasoning. Concept of logical fallacy

11.	5. Logical form of reasoning

12.	6. Types and types of thinking.

13.	7. Features of a lawyer’s thinking

14.	8. The importance of logic for lawyers

15.	Topic 3: Semiotics as the science of signs. Language as a sign system. 1. Semiotics as the science of signs

16.	2. The concept of a sign. Types of interchangeable signs

17.	3. Language as a sign system. Language signs.

18.	4. Structure of the sign process. Structure of the meaning of a sign. Typical logical errors

19.	5. Dimensions and levels of the sign process

20.	6. Language of law

21.	Section III. METHODOLOGICAL FUNCTION OF FORMAL LOGIC 1. Method and methodology.

22.	2. Logical methods of research (cognition)

23.	3. Method of formalization

24.	BASIC FORMS AND LAWS OF ABSTRACT LOGICAL THINKING 1. General characteristics of the concept as a form of thinking. Concept structure

25.	2. Types of concepts. Logical characteristics of concepts

26.	3. Types of relationships between concepts

27.

An even more complex form of thinking than judgment is inference. To understand the origin and essence of inference, 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 obtain with the help of our senses: sight, hearing, smell, etc. Such, for example, is knowledge expressed by judgments such as: “The tree is green,” “The snow is white,” “The bird is singing,” “The pine forest smells of resin.” They constitute a significant part of our knowledge and serve as its 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. In the Moscow region, large deposits of white stone were discovered, from which white-stone Moscow was built. 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, by derivation from other knowledge, is called indirect(or output). The logical form of their acquisition is inference. Thus, inference is a form of thinking through which new knowledge is derived from known knowledge.

6.2 General characteristics of deductive inferences

Deduction (translated from lat. deductio- inference) is often characterized as inference from the general to the specific. This not entirely correct characterization of deductive inferences is associated with their opposition to inductive inferences. The following definition is more correct:

Deductive inferences are those inferences that, given the truth of the premises, must guarantee the truth of the conclusion.

Parcels – these are those judgments from which the final judgment, called the conclusion, is derived; conclusion - This is a judgment that is deduced from previous judgments (premises).

The truth of the conclusion with the truth of the premises in deductive inferences is determined by the fact that in these inferences between the premises and the conclusion there is relation of logical consequence.

Due to the fact that in deductive reasoning the conclusion logically follows from the premises, they represent the most reliable method of proof. However, the reliability of deductive inferences exists at the expense of their information content, that is, they do not give new information about the world. The conclusions of these inferences contain the same information as the premises, and there is no new information. Therefore, conclusions of this type are reliable: if the information in the premises is true, then that part of it that is contained (derived) in the conclusion is true. Indeed, consider such deductive inferences as a simple categorical syllogism:

All people are mortal.

You are human.

Therefore you are mortal.

If it's raining outside, there are puddles outside.

Rain on the street.

Therefore, there are puddles on the street.

In neither one nor the other inference, judgments that are conclusions of deduction (located below the line) are of interest from the point of view of obtaining new information.

Nevertheless, deduction provides new knowledge, but in the sense that it changes the cognitive status of judgments, their place in the system of our knowledge about the world, that is, by substantiating opinions, guesses, proving hypotheses, assumptions, etc., it turns them into theorems, laws, beliefs, etc.

6.3 Direct inferences of propositional logic

Inferences of propositional logic are based on the structure of complex judgments (on the meaning of logical connectives that combine simple judgments into complex ones) and do not take into account the internal structure of simple judgments included in the premises.

The inferences of propositional logic can be direct or indirect. Direct are called inferences in which the conclusion is derived from a certain set of judgments. Indirect are inferences that are obtained by transforming other inferences.

Types of simple forms of direct inferences of the logic of judgments:

1. Conditionally categorical- these are inferences in which one premise is a conditional proposition, and the second premise and conclusion are categorical judgments. Conditional categorical inferences come in two varieties:

(In inference schemes above are written with a line parcels, under line - conclusion, trait means " hence»; A And IN– simple judgments).

Example 1. If a person has a cold ( A), then he is sick ( IN).

The man has a cold ( A).

He is sick ( IN).

Example 2. If a person has a cold ( A), then he is sick ( IN).

The person is not sick (ùIN ).

He doesn't have a cold (ù A).

Example 3. From the premises “If a person has a cold ( A), then he is sick ( IN)" and "The man is sick ( IN)" does not necessarily follow "He has a cold ( A)". “A person is sick” can mean that his leg is broken, his blood pressure has risen, etc. And only with a certain degree of probability can it turn out that he is sick because he has a cold. The conclusion for the negating mode is similarly probable.

2. Separation-categorical- these are inferences in which one premise is a disjunctive judgment, and the other premise and conclusion are categorical judgments. Separation-categorical inferences also come in two varieties:

A) affirmative-denial scheme:	b) negative-affirmative scheme:
A KommersantIN, In sch A	A KommersantIN, And IN	A Kommersant(b) IN, sch A B	A Kommersant(b) IN, sch B A

Example. Negative-affirmative scheme:

Either we leave ( A), or we stay ( IN).

We are not leaving (ùA ).

We stay ( IN).

3. Dilemmas (conditional disjunctive syllogisms)- these are inferences in which two premises are conditional propositions, one is disjunctive, and the conclusion is either a simple judgment (in a simple dilemma) or a complex disjunctive (disjunctive) judgment (in a complex dilemma).

Types of dilemmas:

Example. "If you tell the truth ( A), people will curse you ( IN), and if you lie ( WITH), then the gods will curse you ( D). But you can only tell the truth ( A) or lie ( C). So the gods will curse you ( D) or people ( B)". If we write out from this reasoning only the letter designations of simple judgments, connecting them with appropriate logical connectives, we will obtain the form of a complex constructive dilemma.

There is another form of dilemma - constructive-destructive, or destructive-constructive. In these conclusions, some of the members of the dividing premise indicate the presence of the grounds of the conditional premises, and some deny the consequences (consequents) of other conditional premises. For example, a dilemma of the form is constructive-destructive:

A® IN, C® D

AÚù D

BÚù C

4. Purely conditional inferences- this is a conclusion from any number of premises, which represent conditional propositions and the conclusions of which are also conditional propositions. These inferences include, in particular, the transitivity of implication and the rule of contraposition.

A) transitivity of implication:

A® IN, IN® WITH

A® WITH

Example. “If the frontal cortex of the brain is damaged ( A), then the interaction of the individual with the external environment is disrupted ( B). In this case ( B) a person loses his real perception of reality ( C), which means ( C), turns into a slave to the situation ( D)". This inference has the form of transitivity of implication with three premises:

A® B, B® C, C® D

A® D

b) contraposition rule:

A® IN

sch IN®sch A

Example. "If a person knows geometry ( A), then he knows the Pythagorean theorem ( IN). Therefore, if he does not know the Pythagorean theorem (ù IN), then he does not know geometry (ù A).

All of the above forms of inference are correct, that is, their observance guarantees the correctness of the conclusion if the premises are true. Sometimes these forms are called rules corresponding conclusions.

To check the correctness of inferences that cannot be reduced to these types, the tabular method is used, first of all. It is based on the fact that between the premises and the conclusion of a deductive inference there must be a relation of logical consequence, meaning that the conclusion cannot be false if all the premises are true.

To check the correctness of a conclusion using a tabular method, you need to make formula this conclusion. To do this you should:

1) write down the premises and conclusion in the language of the logic of judgments;

2) connect the premises with each other using conjunction;

3) attach the conclusion to the premises using implication;

4) create a truth table for the resulting formula.

A conclusion will be correct (guaranteing the truth of the conclusion if the premises are true) only if its formula is identically true (in the last column of the table all values are “true”).

Example. “If a philosopher is a dualist, then he is not a materialist. If he is not a materialist, then he is a dialectician or a metaphysician. He is not a metaphysician. Therefore he is a dialectician or a dualist.”

This conclusion is quite difficult to bring to any traditional type, so let’s check its correctness using a tabular method.

Let us write down the premises and conclusion of our judgment in the language of the logic of judgments. Let's denote: R– philosopher – dualist; q– philosopher – materialist; r– philosopher – metaphysician; s– philosopher – dialectician.

Then the first premise is “If a philosopher is a dualist ( R), then he is not a materialist (ù q)” – in the language of judgment logic has the form:

RÉù q.

The second premise is “If he is not a materialist (ù q), then he is a dialectician ( s) or metaphysician ( r)" – will be written like this:

ù qÉ sÚ r.

The third premise is “He is not a metaphysician”:

Conclusion - “He is a dialectician ( s) or dualist ( R)»:

sÚ R.

Connecting the premises with a conjunction (Ù) and adding the conclusion to them with an implication (É), we obtain the formula:

[(R®ù q)Ù(ù q® sÚ r)Ùù r]®( sÚ R).

We create a truth table for this formula:

ù q

ù r

(R®ù q)

sÚ r

ù q® B

AÙ C

DÙù r

sÚ R

D® F

AND

The result is a feasible formula, since the last column of the truth table contains both true and false values. This suggests that the conclusion probable.

When checking the correctness of conclusions, you can not build a table in full, but, having received the truth values of the premises and conclusion, limit yourself to considering only those rows in which all premises are true. So, in this example, having received the values in columns 6 (third premise), 7 (first premise), 9 (second premise) and 12 (conclusion), we could examine only rows 6, 7, 8, 14.

The fact is that, on the one hand, it makes sense to talk about the truth of the conclusion only if truth of premises. With false premises, even a conclusion that is correct in form cannot guarantee the truth of the conclusion. And, on the other hand, by checking the correctness of the inference, we, in essence, check whether it observes logical consequence relation between premises and conclusion. It consists precisely in the fact that in all cases when the premises are true judgments, the conclusion is also a true judgment, and not a single row of the table shows a case where all the premises are true and the conclusion is false. If the premise is false, we cannot talk about the relation of logical implication at all.

6.4 Indirect inferences of propositional logic

Indirect inferences represent indirect reasoning. They have a rather complex structure, because they consist not of judgments, but of inferences. In them, one conclusion follows from another.

These forms of conclusions are often used in the process of argumentation, in particular as means of evidence and refutation. Indirect inferences include refutation “by reduction to the absurd,” proof “by contradiction,” and reasoning from cases.

Refutation "by reduction to absurdity" is an indirect inference in which the falsity of a certain judgment is proven on the basis that a contradiction can be deduced from this judgment using correct inferences.

The structure of this argument is as follows. First, some assumption is made. Then, using the correct inferences, a contradiction is obtained from it. Based on this, the proposed position is considered false. A simplified form of this output can be represented as follows:

A ├ IN sch sch IN

The basis for such reasoning is consistency as a property of our thinking. Contradiction is used as a sign of the incorrectness of any conclusion in our reasoning or the falsity of any judgment.

Example. Let's imagine that on some island only knights and knaves live. Moreover, liars always only lie, and knights always speak only the truth. A man arriving on the island meets two local residents and asks who they are. To which one of them replies: “At least one of us is a liar.” It is necessary to find out who the respondent is.

Let's assume he is a liar. We denote the proposition “The person who answered is a liar” A. But then he told a lie, therefore, neither of them is a liar, and both of them are knights. We got a contradiction: a knight who answered at the same time ( IN) and not a knight (ù IN). This means that our assumption is incorrect, and the one who answered is in fact not a liar, but a knight.

Proof by contradiction close to a refutation “by reduction to the absurd.” However, in contrast to “reduction to absurdity”, which is aimed at refutation some judgment, proof “by contradiction” is aimed at proof any judgment, but at the same time it also uses a contradiction.

The structure of this conclusion is as follows. Let's say we need to prove the truth of some proposition. We temporarily assume as true a proposition that contradicts it, that is, its negation. Then, using correct inferences, we deduce a contradiction from the negation of the proposition being proven. And, if we manage to do this, we can consider it proven that we incorrectly assumed as true a proposition that contradicts what was being proven, and it is false. Consequently, the original proposition being proven itself is true, which is what was required to be proven.

In the form of a diagram, a proof “by contradiction” can be presented as follows:

sch A ├ IN sch sch IN

This inference uses the law of double negation: the negation of the negation of a certain proposition is equivalent to its affirmation ( schsch Aº A or schsch A® A).

Example. The same situation with knights and knaves can be used if the underlying assumptions are changed. Let's say we decide that the person who answered is a knight, and we want to prove it. Then we temporarily assume that he is a liar and derive a contradiction from this. Thus we prove the truth of the original statement.

Reasoning by Cases is used when it is necessary to draw a conclusion from a disjunctive judgment (disjunction). Since in practice it is quite difficult to draw conclusions directly from the disjunction, reasoning by cases seems to offer a workaround.

Its principle is as follows. First, we look to see if the judgment we are interested in follows from all the alternatives (cases) of the disjunction, and if it does, then it can be asserted as a consequence of the entire disjunction. The form of this inference:

A ├ WITH, IN ├ WITH

A Kommersant IN ├ WITH

This indirect inference differs from conditionally divisive inferences (dilemmas) in that its premises contain not judgments, but inferences (conclusions).

Example. “Condottieri have different mastery of their craft: some are excellent, others are mediocre. The first cannot be trusted, because they themselves will seek power... The second cannot be trusted, because they will lose the battle" ( Machiavelli).

The reasoning is based on the disjunctive premise “Condottieri have different mastery of their craft: some are excellent, others are mediocre.” In logical form, this complex judgment is formulated as follows: “Condottieres are excellent at their craft or condottieres are mediocre at their craft.” From this judgment, Machiavelli draws conclusions using indirect inference, namely reasoning by cases. He goes through the alternatives (cases) and shows that in both cases the condottieri cannot be trusted. Let us consider the reasoning scheme in more detail.

It contains the following simple propositions: s 1 – “Condottieri are excellent at their craft”; s 2 – “Condottieres are mediocre in their craft”; r– “Condottieres cannot be trusted”; R– “The condottieres themselves will seek power”; q- “The condottieri will lose the battle.”

s 1 and s 2 – these are the alternatives (cases) of the disjunctive premise underlying the conclusion. Let's see how conclusions are drawn from one and the other case.

First case: “Condottieri are excellent at their craft.” Machiavelli says: “If the condottieri master their craft excellently, then they themselves will seek power”:

R® r.

This means that they cannot be trusted. The output diagram will be like this:

s 1® R, s 1

Next step:

R® r, R

Second case: “Condottieri are mediocre at their craft.” Machiavelli argues that if condottieri are mediocre at their craft, they will lose the battle. If they lose the battle, then they cannot be trusted. From these premises it follows that they cannot be trusted. This results in the following output diagram:

s 2® q, s 2

Next step:

q® r, q

Thus we derived r from s 1 and s 2. This means that we can conclude r from s 1 b s 2, i.e.

s 1 b s 2 ├r.

The result is a case-by-case reasoning diagram:

s 1 ├ r, s 2 ├ r

s 1 b s 2 ├ r

6.5 Direct inferences

6.5.1 Concept and specifics of direct inferences

Direct inferences are inferences in which the conclusion is made from one premise, which is a categorical statement.

These include transformation, reversal, opposition to a predicate, opposition to a subject, and conclusions based on the “logical square.” Almost direct inferences (except for conclusions based on the “logical square”) are transformations of categorical judgments, as a result of which judgments of a different form are obtained, but expressing the same idea as the original judgments.

The need to apply direct inferences in human communication is based on the fact that different people express their thoughts in different ways. Therefore, it is difficult to recognize the same thought. This raises the problem of mutual understanding, which in logic boils down to finding out in what cases thoughts that are different in form have identical or similar content.

Resolving such issues in specific situations can sometimes be quite difficult. Indeed, let’s take two propositions:

a) Every transcendental synthesis is a priori.

b) No non-a priori synthesis is transcendental.

Not everyone will be able to immediately determine whether these judgments express the same thought or not. But if such judgments occur, for example, in a dispute, then you need to react quickly, and for this you need to have the skill of working with this kind of thoughts. You must be able to recognize the same thought expressed in different forms, and be able to prove that what is presented as different expressions of the same thought is not actually such.

Direct inferences allow one to develop the necessary skill of recognizing and identifying judgments of different forms with the same or similar meaning.

6.5.2 Transformation

Conversion is an inference consisting in the transformation of some categorical judgment into the opposite in quality with a predicate that contradicts the predicate of the original judgment.

In other words, when inferring using transformation, a negative judgment is transformed into an affirmative one and, conversely, an affirmative one into a negative one, and the predicate is taken with a negation (that is, P changes to non-P or non-P to P).

Forms of conclusions using transformation:

All S's are P's.

No S is a non-P.

No S is an R.

All Ss are non-Ps.

Some S's are P's.

Some Ss are not non-Ps.

4) for a partial negative judgment:

Some Ss are not Ps.

Some Ss are non-Ps.

Before transforming a judgment using the transformation operation (as well as using other direct inferences), it is advisable to write it down in logical form. This allows you not to make mistakes when defining those concepts that are the subject and predicate of categorical judgments, and thus avoid absurdities in the conclusion. Moreover, when writing a categorical judgment in a logical form, you need to remember that its subject and predicate must have a common gender.

Example. "All liquids are elastic." This is a generally affirmative proposition (A). Writing it in logical form (All S is P), we get the conclusion:

All substances that are liquids (S)

There are substances that are elastic (P).

No substance that is a liquid (S)

is not a substance that is not elastic (non-P).

The conclusions are also valid in the opposite direction - from the lower judgment to the upper one.

6.5.3 Contact

Conversion is a direct inference, consisting in the transformation of a categorical judgment into such a judgment, the subject of which is the predicate of the original, and the predicate is the subject of the original judgment.

In other words, when inferring by inversion, the subject and predicate change places. Moreover, in the case when the initial judgment (premise) is a general affirmative judgment, the quantity of the judgment also changes, that is, the conclusion becomes particular. This treatment is called "restricted treatment" or "pure treatment".

Forms of conclusions using appeal:

1) for a generally affirmative proposition:

All S's are P's.

Some P's are S's.

2) for a generally negative judgment:

No S is an R.

No P is an S.

3) for a private affirmative judgment:

Some S's are P's.

Some Ps are not Ss.

4) for a particular negative judgment, it is impossible to logically correctly derive any conclusion by means of inversion, since in this case the general rule of inferences from categorical judgments is violated: a term that is not distributed in the premises should not be distributed in the conclusion.

Example 1. “Every student is obliged to take exams.” This is a generally affirmative proposition, so we handle the constraint by writing the original proposition in logical form (All S are P):

All people who are students (S)

there are people who are required to take exams (R).

Some people who are required to take exams (P)

there are people who are students (S).

Please note that

that the subject of the premise becomes and the predicate of the premise -

predicate of conclusion, subject of conclusion.

Example 2. If we try to make an inversion from the particular negative proposition “Some trees are not pine trees,” then the conclusion will be clearly incorrect:

Some plants that are trees (S -)

do not eat plants that are pine trees (P+).

Some plants that are pine trees (P -),

do not eat plants that are trees (S+).

But we know that all pines are trees. Having indicated the distribution of terms, we see that the rule of inference from categorical judgments is violated. In this case, the subject (S -) undistributed in the premise, having become a predicate in the conclusion, turned out to be distributed (S +), and the rule requires that a term not distributed in the premise should not be distributed in the conclusion.

When making inferences using transformation and reversal, it is necessary to take into account the existing rules of inference: premises containing empty subjects and predicates (for example, “a creature capable of living without food”), as well as universal terms, that is, terms expressing universal concepts (for example, "a creature in need of food").

Example. Reversing the proposition “No man (S) can live without food (P)” will result in the conclusion “No creature that can live without food (P) is a creature that is a man (S).” However, the conclusion turns out to be completely illegitimate, since there are no such creatures at all. The fact is that the conclusion uses a predicate in which the predicate (“a creature that can live without food”) is an empty concept. This is precisely what caused the illegality of the conclusion.

6.5.4 Contrasting a predicate

Contrasting a predicate is a transformation of a categorical judgment, as a result of which the concept that contradicts the predicate becomes the subject, and the subject of the original judgment becomes the predicate.

This conclusion can be made by sequentially applying the transformation of the original judgment and then the conversion of the resulting judgment, or by following the rules for contrasting the predicate:

1) for a generally affirmative proposition:

All S's are P's.

No non-P is an S.

2) for a generally negative judgment:

No S is an R.

Some non-Ps are Ss.

3) for a partial negative judgment:

Some Ss are not Ps.

Some non-Ps are Ss.

4) for partial affirmative judgments, it is impossible to draw a conclusion by contrasting the predicate, since after transforming the original judgment, a partial negative judgment is obtained, for which the inversion operation is not applied.

Example. Contrast with the predicate for the partial negative proposition “Some lakes have no drainage”:

Some bodies of water that are lakes (S)

there are no bodies of water that have flow (P).

Some reservoirs that do not have a drainage (non-P)

there are bodies of water that are lakes (S).

6.5.5 Opposition to the subject

Opposition to the subject - This is a transformation of a categorical judgment, as a result of which the predicate of the original judgment becomes the subject, and the concept that contradicts the subject of the original judgment becomes the predicate.

Such a conclusion can be achieved by sequentially applying the reversal of the original judgment, and then the transformation of the resulting result, or immediately following the rules for contrasting the subject:

1) for a generally affirmative proposition:

All S's are P's.

Some Ps are not non-Ss.

2) for a generally negative judgment:

No S is an R.

All P's are non-S's.

3) for a private affirmative judgment:

Some S's are P's.

Some Ps are not non-Ss.

4) for partial negative judgments, inferences using opposition to the subject are not used, since in the process of this conclusion we would have to make a reversal of a private negative judgment, for which inference through reversal is not used.

Example. “No evil man can be completely just.” This is a generally negative judgment (E). Bringing it to a logical form (“No S is P”), we draw a conclusion in accordance with the form of opposition to the subject for a generally negative judgment:

No man who is evil (S)

there is no person who can be completely just (R).

All people who can be completely fair (R),

there are people who are not evil (non-S).

6.5.6 Inferences based on the “logical square”

Inferences using the “logical square” are made from simple categorical judgments based on the relationships between them, fixed in a “logical square”.

Forms of conclusions according to the “logical square”:

1) relation of contrariety (opposite) between the universal affirmative ( A) and generally negative ( E) judgments is characterized by the fact that these judgments cannot be true together, therefore:

ù E ù A

2) relation of subcontrary (partial compatibility) between private affirmative ( I) and often negative ( ABOUT) judgments is characterized by the fact that these judgments cannot be false together, that is:

ù I ù O

3) attitude of subordination between the universal affirmative ( A) and private affirmative ( IE) and partial negative ( ABOUT) judgments : the truth of the subordinating proposition determines the truth of the subordinate, and the falsity of the subordinate determines the falsity of the subordinating one:

A E ù ABOUT ù I

I O ù E ù A

4) contradictory relation between the universal affirmative ( A) and partial negative ( ABOUT) judgments, as well as between generally negative ( E) and private affirmative ( I) judgments are characterized by the fact that judgments cannot be both true and false at the same time:

A E ù A ù E O I ù O ù I

ù O ù I O I ù A ù E A A

Example. Using the “logical square” we will draw conclusions from the generally affirmative proposition “Every person dreams of being happy.” Let's assume that it true. Then we can draw conclusions based on the relations of contrariness, subordination and contradictority.

1. Contrary relation:

AS),

R).

ù E: It is not true that no creature that is a human being ( S),

is not a creature who dreams of being happy ( R).

2. Subordination relationship:

A: All creatures that are human ( S),

there are creatures who dream of being happy ( R).

IS),

there are creatures who dream of being happy ( R).

3. Relation of contradictorship:

A: All creatures that are human ( S),

there are creatures who dream of being happy ( R).

ù ABOUT: It is not true that some beings who are human ( S),

R).

Let us assume that the judgment false. Then we can draw a conclusion based on the contradictory relation:

ù A: It is not true that all creatures that are human ( S),

there are creatures who dream of being happy ( R).

ABOUT: Some creatures that are human ( S),

There are no creatures who dream of being happy ( R).

6.6 Simple categorical syllogism

All people are mortal.

Socrates is a man.

Socrates is mortal.

A simple categorical syllogism always contains only three concepts, called terms, which are included in its premises and conclusion. Subject of the conclusion ( S) in a syllogism is considered by a smaller term, conclusion predicate ( P) - big term. The lesser and greater terms are extreme terms syllogism. Each of the extreme terms is contained in both the conclusion and one of the premises.

Traditionally, the major premise in a syllogism should come first.

Average(M) is a term that is included in both premises, but is not included in the conclusion. Through it, the connection between those terms-concepts that constitute the subject and predicate of the conclusion (between extreme terms) is revealed. Thus, a simple categorical syllogism is indirect inference, that is, a conclusion in which the connection between two concepts in the conclusion is established through a third one present in both premises.

The concepts found in a syllogism as terms are content syllogism. The connection that is given to the terms is form syllogism.

Example.

All people ( M) are mortal ( P). Major premise of the syllogism

Socrates (S ) - Human (M ). Minor premise of a syllogism

Socrates ( S) mortal ( P).

The terms that make up this syllogism are: "mortal" - the greater term (conclusion predicate ( R)); "Socrates" is a lesser term (subject of conclusion ( S)); "people" is the middle term ( M) (included in both premises, but not in the conclusion). Judgment "Socrates ( S) - Human ( M)» - smaller premise, since it contains a smaller term ( S). The proposition “All people ( M) are mortal ( R)» - big premise, since it contains a larger term ( R).

Every syllogism has a figure and a mode .

Syllogism figure shows the arrangement of terms ( P, S, M) in parcels. Depending on the location of the middle term, four figures of the syllogism are distinguished (Fig. 18).

Rice. 18. Figures of a simple categorical syllogism

Upper the edge of the figure always shows the location of the terms in greater parcel, lower- V less parcel.

IN first figure V greater MR). IN less SM).

In second figure V greater R), predicate – middle term ( M). IN less in the premise the subject is the lesser term ( S), predicate – middle term ( M).

IN third figure V greater in the premise the subject is the middle term ( M), predicate – larger term ( R). IN less in the premise the subject is the middle term ( MS).

IN fourth figure V greater in the premise the subject is the larger term ( R), predicate – middle term ( M). IN less in the premise the subject is the middle term ( M), predicate – smaller term ( S).

Example. To determine the figure of the above syllogism (about Socrates), you need to write out from its premises the letter designations of the terms in the order in which they are located there, connect the middle terms ( M) and draw lines from them to the extreme ones ( S And R). We get the first figure:

Modus a simple categorical syllogism shows the type of categorical judgments that make up the syllogism. Moreover first the letter in the mode always shows the form greater parcels, second - less parcels, third- view conclusions.

Example. In the syllogism about Socrates, both premises and conclusion are generally affirmative propositions ( A), which means its mode is AAA.

Simple categorical syllogisms can be correct or incorrect. The correctness of a syllogism does not depend on its content, but depends only on its form (figure and mode). Moreover, only a syllogism with the correct form ensures the truth of the conclusion when the premises are true. Otherwise, even with true premises, the truth of the conclusion is not guaranteed.

To determine whether a syllogism is correct, one can check whether it follows the general rules of syllogisms and the rules of figures.

General rules of syllogisms:

1. At least one of the premises must be a general proposition.

2. At least one of the premises must be an affirmative proposition.

3. In case of a private premise, the conclusion must be private.

4. If the premise is negative, the conclusion must be negative.

5. With two affirmative premises, the conclusion must be affirmative.

6. The middle term must be distributed in at least one of the premises.

7. A term not distributed in the premise should not be distributed in the conclusion.

Shape rules:

First figure: the minor premise must be affirmative, and the major premise must be general.

Second figure: one of the premises must be negative, and the larger one must be general.

Third figure: the minor premise must be affirmative, and the conclusion must be partial.

For fourth figure, no special rules are formulated, since practically they come down to listing the correct modes of this figure.

Example. Let's check whether the general rules and the rules of figures are observed in the following syllogism:

All lawyers ( R M -).

Everyone present (S +) there are people who know the signs of a crime ( M -).

Everyone present ( S+) there are lawyers ( R -).

It is easy to notice that in this case the sixth of the general rules of the syllogism is not observed, since the middle term ( M) turned out to be undistributed in both parcels.

The rule of the second figure is also not observed (and this syllogism has precisely the second figure), since both premises are affirmative judgments, and the rule of the second figure requires that one of the premises be negative. Therefore, the above syllogism is not correct.

You can verify the correctness of a syllogism in another way - by looking at whether its mode is one of correct modes of his figure.

There are a total of 256 modes of simple categorical syllogisms (64 modes in each figure). However, not all of them represent correct conclusions. There are only 24 correct modes (six modes in each figure). Among them there are 19 main, so-called strong modes. The rest - weak modes– can be presented as complex conclusions: combinations of conclusions in the form of a categorical syllogism with conclusions according to the rules of the “logical square” (Table 3).

Table 3

Correct modes of a simple categorical syllogism

Example. The above syllogism (about those present) has a second figure and mode AAA. However, among the regular modes of the second figure there is no mode AAA. This mode exists only in the first figure. This also shows that the syllogism is incorrect.

6.7 Enthymeme

In the process of reasoning, we do not always use syllogisms in their full, expanded form. Sometimes only the major premise and conclusion of a syllogism are stated, and the minor premise is only implied. In other cases, the major premise is not explicitly stated and only the minor premise and conclusion are stated. It often happens that only premises are given, the conclusion from which is left to the interlocutor or the reader himself. This implies that the conclusion is possible according to the rules of a syllogism.

A syllogism in which any of its parts (premise or conclusion) is released (not expressed explicitly) is called a shortened syllogism or enthymeme.

The conclusions of the logic of judgments can also be abbreviated (enthymematic). There may also be premises or conclusion missing. Therefore, a more general definition of enthymeme is possible:

An enthymeme is a conclusion in which one of the premises or conclusion is missing.

The meaning of this name (from Greek. ẻν θυμφ - in the mind) lies in the fact that some part of the syllogism is not expressed explicitly, but is pronounced as if in the mind.

The possibility of abbreviated expression of inferences is due to the fact that if two parts of a syllogism are given, then it is always possible to accurately establish the missing part in a logical way.

In discussions and disputes, when the interlocutor expresses his thought in the form of an abbreviated syllogism, it is always necessary to understand exactly what kind of judgment is not expressed, but only implied in this reasoning. Otherwise, it is impossible to fully understand this reasoning and refute it if it is incorrect. Often people base their reasoning on false or dubious propositions, but do not express them explicitly, using abbreviated forms of inference. To find an error in such reasoning and refute it, it is necessary to establish what is assumed in it, but is not expressed explicitly.

In simple cases, the premises or conclusion implied in the reasoning can be established without resorting to special techniques - according to the general meaning of the reasoning. But in many cases it is not so easy to restore the missing part of the syllogism according to the general meaning. However, this can be done by performing the operation of restoring the syllogism to its full form, which consists of several stages:

1) determination of the missing element of the syllogism (premise or conclusion). If the enthymeme contains expressions denoting a logical connection (“therefore”, “because”, “since”, etc.), this means that the enthymeme has a conclusion. If these words are missing, then most likely the conclusion is missing;

2) definition of syllogism terms (lesser, greater and average);

3) determination of the type of missed parcel (if it is the parcel that is missed) - large or small;

4) determination of the figure and mode of the syllogism;

5) formulation of the syllogism in full form.

Difficulties in restoring syllogisms using an enthymeme may be due to the fact that in order to correctly define the concepts (terms) from which the missing element (premise or conclusion) will be formulated, it is necessary to know the logical forms of the existing elements (two premises or a premise and conclusion). However, in real reasoning, standard logical forms of categorical judgments (of which syllogisms are made) are not always used. Before you can reduce judgments to a standard form, you need to understand their meaning, which can be difficult.

Example. Let us restore the syllogism from the enthymeme “This syllogism has three terms, and therefore it is correct.”

This enthymeme contains a word denoting a logical connection (“therefore”), which means it has a conclusion. The conclusion is the proposition following the word “therefore”: “It is correct.” The remaining proposition – “This syllogism has three terms” – is one of the premises. We need to restore the second, missing parcel.

We define the subject and predicate of the conclusion, formulating it in a logical form and taking into account that it refers to “this syllogism” and the pronoun “he” means “this syllogism”:

This syllogism ( S) is a correct syllogism ( R).

The premise in the enthymeme contains the subject of the conclusion or a lesser term (“a given syllogism”), i.e. is a minor premise. And since any premise always contains one of the extreme terms and a middle term, therefore, the second term of the premise (“a syllogism having three terms”) is the middle term of the syllogism ( M):

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

This syllogism ( SM).

This syllogism ( S) is a correct syllogism ( R).

We are restoring a large parcel. A major premise always contains a larger term ( R) and middle term ( M). However, they can be arranged in different sequences: R-M or M-R. To determine the sequence of terms, as well as the type of premise (general affirmative, general negative, particular affirmative or particular negative), we determine the figure and mode of the syllogism. At the same time, we take into account that the restored syllogism must be correct.

In the minor premise the terms are arranged in order S-M. This arrangement of terms in the minor premise is possible either in the first or in the second figure (in the third and fourth terms the terms are arranged in the reverse order - M-S). This means that the syllogism will have either a first or a second figure.

Now we find the mode of the syllogism. Since the minor premise and conclusion are generally affirmative propositions ( A), the mode will end in... AA. Let's see which of the pre-selected figures (first or second) has the correct mode ending in... AA. There is such a mode in the first figure, and this is the mode AAA.

The required major premise is a generally affirmative proposition ( A), and the terms in it should follow in order M-R, since this is exactly how they are located in the larger parcel in the first figure. We get the following syllogism:

All syllogisms having three terms ( M), there are correct syllogisms ( R).

This syllogism ( S) is a syllogism having three terms ( M).

This syllogism ( S) is a correct syllogism ( R).

The resulting premise is not a true proposition, because the number of terms, as we already know, is not the only condition for the correctness of a syllogism. Consequently, the conclusion of the enthymeme about the correctness of “this syllogism” turns out to be unfounded.

Review questions and exercises

1. What conclusions are called deductive?

2. Why is deduction the most reliable method of proof?

3. What is the basis for the need to use direct inferences in human communication?

5. What conclusions are called enthymemes? What determines the possibility of expressing thoughts in the form of enthymemes?

6. If possible, carry out circulation and transformation operations:

a) All liquids are elastic.

b) Not everything new is progressive.

c) Some lakes have drainage.

d) Some philosophers are not rationalists.

e) No crime is moral.

7. Conduct a logical analysis of the syllogism (indicate its terms, figure and mode, determine the truth):

a) Some artists deserve admiration.

Some modernists are artists.

Some modernists deserve admiration.

b) No person can be completely impartial.

Every lawyer is a person.

No lawyer can be completely impartial.

c) No prudent person is superstitious.

Some well-educated people are superstitious.

Some well-educated people are unreasonable.

d) All philosophers have read the Critique of Pure Reason.

Some writers have read the Critique of Pure Reason.

Some writers are philosophers.

8. Based on the true premises, come up with one syllogism each of the first, second, third and fourth figures that have the correct modes.

9. Restore the enthymeme into a complete syllogism:

a) All jokes are intended to make people laugh; All jokes are jokes.

b) Some of the disputed provisions are worthy of attention, since some such provisions may turn out to be true.

c) A sign of combustion is the presence of a flame, so oxidation is not combustion.

d) Since all liquids are elastic, it means that metals are not elastic.

e) If even pleasure does not make you more humane, then you are by nature as cruel as a beast.

10. Determine the types of inferences and establish their correctness:

a) If the frontal cortex of the brain is damaged, then the interaction of the individual with the external environment is disrupted. In this case, a person loses his real perception of reality, which means he turns into a slave to the situation.

b) An exchange of residential premises may be declared invalid by a court if it was carried out in violation of the requirements stipulated by the Housing Code. If the exchange is declared invalid, the parties are subject to relocation to previously occupied premises.

c) If he were smart, he would have seen his mistake. And if he had been sincere, he would have confessed to her. However, his past behavior shows that he is either unintelligent or insincere, or maybe both. Thus, it is to be expected that he will either not see the mistake or not admit to it.

d) A victim is a person who has been inflicted moral, physical and property harm by a criminal. Neither moral nor physical harm was caused to the victim. Consequently, he suffered property damage.

e) If a straight line touches a circle, then the radius drawn to the point of contact is perpendicular to it. Thus, the radius of the circle is not perpendicular to this line because it is not tangent to the circle.

2. Deductive reasoning

Like much in classical logic, the theory of deduction owes its appearance to the ancient Greek philosopher Aristotle. He developed most of the questions related to this type of inference.

According to the works of Aristotle deduction- this is a transition in the process of inference from the general to the specific. In other words, deduction is the gradual specification of a more abstract concept. It goes through several stages, each time deducing a consequence from several premises.

It must be said that true knowledge must be obtained through the process of deductive reasoning. This goal can be achieved only if the necessary conditions and rules are met. There are two types of inference rules: direct inference rules and indirect inference rules. Direct inference means obtaining a conclusion from two premises that will be true if the rules of direct inference are followed.

Thus, the premises must be true and the rules for obtaining consequences must be observed. If these rules are observed, we can talk about the correctness of thinking regarding the taken subject. This means that to obtain a true judgment, new knowledge, it is not necessary to have all the information. Some information can be reconstructed logically and consolidated. Consolidation is necessary, because without it the process of obtaining new information becomes meaningless. It is not possible to transmit such information or use it in any other way. Naturally, such consolidation occurs through language (spoken, written, programming language, etc.). Consolidation in logic occurs primarily with the help of symbols. For example, these can be symbols of conjunction, disjunction, implication, literal expressions, parentheses, etc.

The following types of inferences are deductive: conclusions of logical connections and subject-predicate conclusions.

Also deductive inferences are direct.

They are made from one premise and are called transformation, reversal and opposition to the predicate; conclusions based on the logical square are considered separately. Such conclusions are derived from categorical judgments.

Let's consider these conclusions. The transformation has the following scheme:

S is not non-P.

This diagram shows that there is only one parcel. This is a categorical judgment. The transformation is characterized by the fact that when the quality of the premise changes in the process of inference, its quantity does not change, and the predicate of the consequence negates the predicate of the premise. There are two ways of transformation - double negation and replacing a negation in a predicate with a negation in a connective. The first case is reflected in the diagram above. In the second, the transformation is reflected in the diagram as S is not-P - S is not P.

Depending on the type of judgment, the transformation can be expressed as follows.

All S are P - No S is not-P. No S is P - All S is not-P. Some S are P - Some S are not non-P. Some S are not P - Some S are not-P. Appeal- this is an inference in which, when the places of the subject and predicate are changed, the quality of the premise does not change.

That is, in the process of inference, the subject takes the place of the predicate, and the predicate takes the place of the subject. Accordingly, the circulation scheme can be depicted as S is P - P is S.

Treatment can be with or without restrictions(it is also called simple or pure). This division is based on a quantitative indicator of judgment (meaning equality or inequality of volumes S and P). This is expressed by whether the quantifier word has changed or not and whether the subject and predicate are distributed. If such a change occurs, then the constraint is addressed. Otherwise, we can talk about pure circulation. Let us recall that a quantifier word is a word that is an indicator of quantity. Thus, the words “all”, “some”, “none” and others are quantifier words.

Contrast with predicate characterized by the fact that the connective in the consequence changes to the opposite, the subject contradicts the predicate of the premise, and the predicate is equivalent to the subject of the premise.

It must be said that a direct inference with a contrast to the predicate cannot be derived from particular affirmative judgments.

Let us present contrast schemes depending on the types of judgments.

Some S are not P - Some non-P are S. No S is P - Some non-P are S. All S are P - No P is S.

Combining the above, we can consider opposition to a predicate as a product of two immediate conclusions at once. The first of these is the transformation. Its result is subject to reversal.

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From the book Logic for Lawyers: Textbook. author Ivlev Yuri Vasilievich

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From the book Logic and Argumentation: Textbook. manual for universities. author Ruzavin Georgy Ivanovich

§ 3. Indirect deductive inferences In mediated inferences, the conclusion follows from two or more judgments that are logically related to each other. There are several types of indirect inferences: a) syllogisms; b) conditional inferences; V)

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1. Induction as a type of inference Express the structure of the following inductive inferences in schematic form and determine the nature of the conclusion: “Take, for example, Roger Bacon's study of the origin of the colors of the rainbow. At first, it seems, he had the idea of tying

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A. DEDUCTIVE CONCLUSIONS In the process of reasoning, conclusions that are not such are sometimes taken as deductive. The latter are called incorrect deductive inferences, and (actually) deductive inferences are called correct. Identification of methods of reasoning,

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B. INDUCTIVE INFLUENCES Unlike deductive inferences, in which there is a relation of logical consequence between the premises and the conclusion, inductive inferences represent such connections between the premises and the conclusion according to logical forms, with

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Chapter VII DEDUCTIVE INFERENCES. CONCLUSIONS FROM SIMPLE JUDGMENTS § 1. CONCLUSION AS A FORM OF THINKING. TYPES OF CONCLUSIONS In the process of cognition, we acquire new knowledge. Some of them - directly, as a result of the influence of objects of the external world on the organs

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§ 2. DIRECT CONCLUSIONS A judgment containing new knowledge can be obtained by transforming the judgment. Since the original (transformed) judgment is considered as a premise, and the judgment obtained as a result of the transformation is considered as a conclusion,

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Chapter VIII DEDUCTIVE INFERENCES. CONCLUSIONS FROM COMPLEX JUDGMENTS. CONDUCT AND COMPLEX SYLLOGISMS Inferences are built not only from simple but also from complex judgments. Inferences are widely used, the premises of which are conditional and disjunctive judgments,

Inference- this is a form of thinking through which a new judgment is derived from one or more interconnected judgments with logical necessity. The logical essence of inference consists in the movement of thought from the analysis of existing knowledge to the synthesis of new knowledge. This movement is objective in nature and is determined by the real connections of reality. The objective connection reflected in consciousness provides a logical connection of thoughts. On the contrary, the lack of objective connections between reality leads to logical errors.

The structure of any conclusion includes three elements:

1)original knowledge expressed in premises;

2)substantiating knowledge expressed in the rules of inference;

3)inferential knowledge expressed in a conclusion or conclusion.

When analyzing a conclusion, it is customary to write the premises and conclusion separately, placing them on top of each other. The conclusion is written under a horizontal line separating it from the premises and indicating a logical follow-up. Accordingly, consider the following example of an inference:

All citizens of the Republic of Belarus have the right to education - premise

Novikov - citizen of the Republic of Belarus - sprinkles

Novikov has the right to education - conclusion

If there is a meaningful connection between the premises, one can obtain new true knowledge in the process of reasoning, subject to two conditions.

Firstly, the initial propositions - premises - must be true. However, it should be borne in mind that sometimes false judgments can give a true conclusion. Thus, as a result of a special selection of false premises in the following reasoning, we obtain a true conclusion:

All elephants have wings

All birds are elephants

All birds have wings

This indicates that focusing only on the form (structure) of premises while ignoring their objectively true connections can create the appearance of a correct conclusion.

Secondly, in the process of reasoning, it is necessary to observe the rules of inference, which determine the logical correctness of the conclusion. Without this, even from true premises you can get a false conclusion. For example:

All caterpillars eat cabbage

I eat cabbage

Therefore, I am a caterpillar

There are quite a lot of rules, a number of them are enshrined in the main types of inferences.

Depending on the sequence of thought development, as well as on the logical validity of the conclusion, inferences are divided into the following types: deductive, inductive and analogical reasoning.

In deductive reasoning(from Latin deductio - deduction) the connections between premises and conclusion are formal logical laws, due to which, with true premises, the conclusion always turns out to be true.

Deductive reasoning- this is a form of abstract thinking in which thought develops from knowledge of a greater degree of generality to knowledge of a lesser degree of generality, and the conclusion arising from the premises is, with logical necessity, reliable in nature. The objective basis of deductive conclusions is the unity of the general and the individual in real processes and objects of the surrounding world.

The deduction procedure occurs when the information in the premises contains (often in implicit form) the information expressed in the conclusion. Deductive reasoning is a way of extracting this information and presenting it in explicit form.

The rules of deductive inference are determined by the nature of the premises, which can be simple or complex propositions, as well as their number. Depending on the number of premises used from which the conclusion is drawn, deductive conclusions can be direct or indirect.

The logical form of which guarantees the receipt of a true conclusion, subject to the simultaneous truth of the premises. In deductive reasoning, there is a relationship between premises and conclusion following logical ; the logical content of the conclusion (i.e., its information without taking into account the meanings of non-logical terms) forms part of the total logical content of the premises.

For the first time, a systematic analysis of one of the varieties of deductive inferences - syllogistic inferences, the premises and conclusions of which are attributive statements - was carried out 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. Such important types of inferences were identified as conditionally categorical (modus ponens, modus tollens), dividing-categorical (modus tollendo ponens, modus ponendo tollens), conditionally dividing (lemmatic), etc.

However, within the framework of traditional logic, only a small part of deductive reasoning was described and there were no precise criteria for the logical correctness of reasoning. In modern symbolic logic, thanks to the use of formalization methods, the construction of logical calculi and formal semantics, and the axiomatic method, the study of deductive inferences has been raised to a qualitatively different, theoretical level.

By means of modern logical theory, it is possible to define the entire set of forms of correct deductive inferences within the framework of a certain formalized language. If the theory is constructed semantically, then the transition from formulas A 1 A 2 , ..., A n to formula B declared to be a form of correct deductive reasoning in the presence of logical consequence B from A 1 A 2 , ..., A n ; this relation is usually defined as follows: for any interpretation of non-logical symbols admissible in a given theory, in which A 1 A 2 , ..., A n take the highlighted value (truth value), formula B also takes the highlighted value. In syntactically constructed logical systems (calculi), the criterion for the logical correctness of the transition from A 1 A 2 , ..., A n to B indicates the existence of a formal derivation of the formula B from formulas A 1 A 2 , ..., A n, carried out in accordance with the rules of this system (see. Logical output ).

The choice of a logical theory adequate for testing deductive inferences is determined by the type of statements included in its composition and the expressive capabilities of the language of the theory. Thus, inferences containing complex statements can be analyzed by means propositional logic , while the internal structure of simple statements within complex ones is ignored. Syllogistics explores inferences from simple attributive statements based on voluminous relations in the sphere of general terms. By means predicate logic correct deductive inferences are identified based on taking into account the internal structure of simple statements of a wide variety of types. Inferences containing modal statements are considered within the framework of systems modal logic , those that contain tense utterances - within temporal logic etc.