Judgment (statement) is a form of thinking in which something is affirmed or denied. For example: "All pines are trees", "Some people are athletes", "No whale is a fish", "Some animals are not predators".

Consider several important properties of a judgment that at the same time distinguish it from a concept:

1. Any judgment consists of concepts related to each other.

For example, if we link the concepts " crucian carp" and " fish", then judgments can be obtained:" All crucian are fish”, “Some fish are crucian”.

2. Any judgment is expressed in the form of a sentence (remember, a concept is expressed by a word or phrase). However, not every sentence can express a judgment. As you know, sentences are declarative, interrogative and exclamatory. In interrogative and exclamatory sentences nothing is affirmed or denied, therefore they cannot express a judgment. A declarative sentence, on the contrary, always affirms or denies something, which is why the judgment is expressed in the form declarative sentence. Nevertheless, there are such interrogative and exclamatory sentences that are questions and exclamations only in form, but affirm or deny something in meaning. They're called rhetorical. For example, famous saying: « And what Russian does not like to drive fast?"- is a rhetorical interrogative sentence (rhetorical question), since it states in the form of a question that every Russian loves fast driving.

There is a judgment in such a question. The same can be said about rhetorical exclamations. For example, in the statement: Try to find black cat in a dark room if she's not there!"- in the form of an exclamatory sentence, the idea of ​​the impossibility of the proposed action is affirmed, due to which this exclamation expresses a judgment. It is clear that not rhetorical, but real question, for example: " What is your name?” - does not express a judgment, just as it does not express its present, and not a rhetorical exclamation, for example: “ Farewell, free element!

3. Any judgment is true or false. If the proposition is true, it is true, and if it is not true, it is false. For example, the statement: " All roses are flowers", is true, and the proposition: " All flies are birds", is false. It should be noted that concepts, unlike judgments, cannot be true or false. It is impossible, for example, to argue that the concept of " school" is true, and the concept of " institute" - false, the concept of " star" is true, and the concept of " planet"- false, etc. But are the concepts " Dragon», « Koschei the Deathless», « perpetual motion machine» not false? No, these concepts are null (empty), but neither true nor false. Recall that a concept is a form of thinking that denotes an object, and that is why it cannot be true or false. Truth or falsity is always a characteristic of some statement, statement or negation, therefore it is applicable only to judgments, but not to concepts. Since any proposition takes on one of two values ​​- true or false - Aristotelian logic is also often called two-valued logic.

4. Judgments are simple and complex. Compound propositions consist of simple propositions connected by some union.

As you can see, judgment is more complex shape thinking versus concept. It is not surprising, therefore, that the judgment has a certain structure, in which four parts can be distinguished:

1. Subject S) is what is being discussed in the judgment. For example, in the sentence: ", - we are talking about textbooks, so the subject of this judgment is the concept of" textbooks».

2. Predicate(denoted by the Latin letter R) is what is said about the subject. For example, in the same sentence: All textbooks are books”, - it is said about the subject (about textbooks) that they are books, therefore the predicate of this judgment is the concept of “ books».

3. Bundle is what connects the subject and the predicate. The role of the link can be the words “is”, “is”, “this”, etc.

4. Quantifier is a pointer to the volume of the subject. The role of the quantifier can be the words "all", "some", "none", etc.

Consider the statement: " Some people are athletes". In it, the subject is the concept of " people”, the predicate is the concept of “ athletes”, the role of the link is played by the word “ are", and the word" some" is a quantifier. If there is no connective or quantifier in some proposition, then they are still implied. For example, in the sentence: Tigers are predators", - the quantifier is missing, but it is implied - this is the word "all". By using symbols subject and predicate, one can discard the content of the judgment and leave only it logical form.

For example, if the judgment has: All rectangles are geometric shapes”, - discard the content and leave the form, then you get: “All S there is R". Logical form of judgment: " Some animals are not mammals", - "Some S do not eat R».

The subject and predicate of any judgment are always some concepts, which, as we already know, can be in various relationships with each other. There can be the following relations between the subject and the predicate of a judgment.

1. equivalence. In judgment: " All squares are equilateral rectangles", - subject " squares" and the predicate " equilateral rectangles"are in a relationship of equivalence, because they are equivalent concepts (a square is necessarily an equilateral rectangle, S = P and an equilateral rectangle is necessarily a square) (Fig. 18).

2. intersection. In judgment:

« Some writers are American", - subject " writers" and the predicate " Americans» are in relation to intersection, because they are intersecting concepts (a writer may or may not be an American, and an American may or may not be a writer) (Fig. 19).

3. Subordination. In judgment:

« All tigers are predators", - subject " tigers" and the predicate " predators» are in relation to subordination, because they represent species and generic concepts (a tiger is necessarily a predator, but a predator is not necessarily a tiger). Likewise in the sentence: Some predators are tigers", - subject " predators" and the predicate " tigers» are in a relationship of subordination, being generic and species concepts. So, in the case of subordination between the subject and the predicate of the judgment, two variants of relations are possible: the volume of the subject is completely included in the volume of the predicate (Fig. 20, a), or vice versa (Fig. 20, b).

4. Incompatibility. In judgment: " ", - subject " planets" and the predicate " stars» are in relation to incompatibility, because they are incompatible (subordinate) concepts (no planet can be a star, and no star can be a planet) (Fig. 21).

In order to establish the relationship between the subject and the predicate of this or that judgment, we must first establish which concept of the given judgment is the subject and which is the predicate. For example, it is necessary to define the relation between the subject and the predicate in a judgment: Some military personnel are Russians". First, we find the subject of judgment, - this is the concept of " military personnel»; then we establish its predicate, is the concept " Russians". Concepts " military personnel" and " Russians» are in relation to intersection (a serviceman may or may not be a Russian, and a Russian may or may not be a serviceman). Therefore, in the said proposition, the subject and the predicate intersect. Similarly, in the judgment: All planets are celestial bodies”, - the subject and the predicate are in the relation of subordination, and in the judgment: “ No whale is a fish

As a rule, all judgments are divided into three types:

1. Attribute judgments(from lat. attribute- attribute) - these are judgments in which the predicate is some essential, integral feature of the subject. For example, the statement: " All sparrows are birds", - attributive, because its predicate is an integral feature of the subject: to be a bird is main feature sparrow, its attribute, without which it will not be itself (if a certain object is not a bird, then it is necessarily not a sparrow). It should be noted that in an attributive judgment the predicate is not necessarily an attribute of the subject, and vice versa - the subject is an attribute of the predicate. For example, in the sentence: Some birds are sparrows”(as we see, in comparison with the above example, the subject and the predicate have changed places), the subject is an integral feature (attribute) of the predicate. However, these judgments can always be formally changed in such a way that the predicate becomes an attribute of the subject. Therefore, attributive judgments are usually called those judgments in which the predicate is an attribute of the subject.

2. Existential judgments(from lat. existentia- existence) are judgments in which the predicate indicates the existence or non-existence of the subject. For example, the statement: " There are no perpetual motion machines", - is existential, since its predicate " can not be” testifies to the non-existence of the subject (or rather, the object that is designated by the subject).

3. Relative judgments(from lat. relativus- relative) - these are judgments in which the predicate expresses some kind of relation to the subject. For example, the statement: " Moscow was founded before St. Petersburg', is relative because its predicate ' founded before St. Petersburg» indicates the temporal (age) relationship of one city and the corresponding concept to another city and the corresponding concept, which is the subject of judgment.

Check yourself:

1. What is a judgment? What are its main properties and differences from the concept?

2. In what language forms is the judgment expressed? Why can't interrogative and exclamatory sentences express judgments? What are rhetorical questions and rhetorical exclamations? Can they be a form of expressing judgments?

3. Find the language forms of judgments in the expressions below:

1) Don't you know that the earth revolves around the sun?

2) Farewell, unwashed Russia!

3) Who wrote the philosophical treatise Critique of Pure Reason?

4) Logic appeared around the 5th century. BC e. in Ancient Greece.

5) America's first president.

6) Turn around on the march!

7) We all learned a little...

8) Try to move at the speed of light!

4. Why concepts, unlike judgments, cannot be true or false? What is two-valued logic?

5. What is the structure of the judgment? Think of five judgments and indicate in each of them the subject, predicate, connective and quantifier.

6. In what relationship can there be a subject and a predicate of a judgment? Give three examples for each case of the relationship between the subject and the predicate: equivalence, intersection, subordination, incompatibility.

7. Determine the relationship between the subject and the predicate and depict them using Euler's circular schemes for the following judgments:

1) All bacteria are living organisms.

2) Some Russian writers are world famous people.

3) Textbooks cannot be entertaining books.

4) Antarctica is an ice continent.

5) Some mushrooms are inedible.

8. What are attributive, existential and relative judgments? Give, on your own choosing, five examples for attributive, existential and relative judgments.

2.2. Simple Judgments

If a judgment contains one subject and one predicate, then it is simple. All simple judgments according to the volume of the subject and the quality of the bundle are divided into four types. The volume of the subject can be general (“all”) and particular (“some”), and the connective can be affirmative (“is”) and negative (“is not”):

The volume of the subject ……………… “all” “some”

Bond quality ……………… “Yes” “Not available”

As you can see, based on the volume of the subject and the quality of the link, only four combinations can be distinguished, which exhaust all types of simple judgments: “everything is”, “some is”, “everything is not”, “some is not”. Each of these species has its own name and symbol:

1. General affirmative judgments A) are judgments with the total volume of the subject and an affirmative link: “All S there is R". For example: " All students are students».

2. Private affirmative judgments(denoted by the Latin letter I) are judgments with a particular volume of the subject and an affirmative link: “Some S there is R". For example: " Some animals are carnivores».

3. General negative judgments(denoted by the Latin letter E) are judgments with the total volume of the subject and a negative link: “All S do not eat R(or "None S do not eat R"). For example: " All planets are not stars», « No planet is a star».

4. Private negative judgments(denoted by the Latin letter O) are judgments with a particular volume of the subject and a negative link: “Some S do not eat R". For example: " ».

Next, you should answer the question of which judgments - general or particular - should include judgments with a unit volume of the subject (that is, those judgments in which the subject is a single concept), for example: “ The sun is heavenly body”, “Moscow was founded in 1147”, “Antarctica is one of the continents of the Earth”. A judgment is general if it is about the entire volume of the subject, and particular if it is about a part of the volume of the subject. In judgments with a unit volume of the subject, we are talking about the entire volume of the subject (in the examples given, about the entire Sun, all of Moscow, all of Antarctica). Thus, judgments in which the subject is a single concept are considered general (general affirmative or generally negative). So, the three judgments cited above are generally affirmative, and the judgment: “ The famous Italian Renaissance scientist Galileo Galilei is not the author of the theory electromagnetic field ' is generally negative.

In the future, we will talk about the types of simple judgments, without using their long names, with the help of conventional notation - Latin letters A, I, E, O. These letters, taken from two Latin words: a ff i rmo- approve and n e g o - to deny, were proposed as a designation for the types of simple judgments back in the Middle Ages.

It is important to note that in each of the types of simple judgments, the subject and the predicate are in a certain relationship. Thus, the total volume of the subject and the affirmative link of judgments of the form A lead to the fact that in them the subject and the predicate can be in relations of equivalence or subordination (other relations between the subject and the predicate in judgments of the form A it can not be). For example, in the sentence: All squares (S) are equilateral rectangles (P)", - the subject and the predicate are in a relationship of equivalence, and in the judgment:" All whales (S) are mammals (P)', in relation to submission.

Partial scope of the subject and affirmative link of judgments of the form I determine that in them the subject and the predicate can be in a relationship of intersection or subordination (but not in others). For example, in the sentence: Some athletes (S) are blacks (P)”, - the subject and the predicate are in relation to the intersection, and in the judgment: “ Some trees (S) are pines (P)', in relation to submission.

The total volume of the subject and the negative link of judgments of the form E lead to the fact that in them the subject and the predicate are only in the relation of incompatibility. For example, in judgments: All whales (S) are not fish (P)”, “All planets (S) are not stars (P)”, “All triangles (S) are not squares (P)”, – subject and predicate are incompatible.

The private volume of the subject and the negative link of judgments of the form O cause the fact that they have a subject and a predicate, just as in judgments of the form I, can only exist in relations of intersection and subordination. The reader can easily pick up examples of judgments of the form O in which the subject and predicate are in this relationship.

Check yourself:

1. What is a simple proposition?

2. On what basis are simple judgments divided into types? Why are they divided into four types?

3. Describe all types of simple judgments: name, structure, symbol. Come up with an example for each of them. Which judgments - general or particular - are judgments with a unit volume of the subject?

4. Where did the letters come from to designate types of simple judgments?

5. In what relationship can there be a subject and a predicate in each of the types of simple judgments? Consider why in judgments of the form A subject and predicate cannot intersect or be incompatible? Why in judgments of the form I subject and predicate cannot be in a relationship of equivalence or incompatibility? Why in judgments of the form E subject and predicate cannot be equivalent, intersecting or subordinate? Why in judgments of the form O subject and predicate cannot be in relation of equivalence or incompatibility? Draw Euler Circles possible relationship between subject and predicate in all kinds of simple propositions.

2.3. Distributed and undistributed terms

terms of judgment its subject and predicate are called.

The term is considered distributed(expanded, exhausted, taken in in full), if the judgment refers to all the objects included in the scope of this term. The distributed term is denoted by the “+” sign, and on the Euler diagrams it is depicted as a full circle (a circle that does not contain another circle and does not intersect with another circle) (Fig. 22).

The term is considered undistributed(unexpanded, inexhaustible, not taken in full), if the judgment is not about all the objects included in the scope of this term. An undistributed term is indicated by the “–” sign, and on Euler’s diagrams it is depicted as an incomplete circle (a circle that contains another circle (Fig. 23, a) or intersects with another circle (Fig. 23, b).

For example, in the sentence: All sharks (S) are predators (P)”, - we are talking about all sharks, which means that the subject of this judgment is distributed.

However, in this judgment, we are not talking about all predators, but only about a part of predators (namely, those that are sharks), therefore, the predicate of this judgment is undistributed. Having depicted the relationship between the subject and the predicate (which are in relation to subordination) of the considered judgment by Euler schemes, we will see that the distributed term (subject " sharks”) corresponds to a full circle, and undistributed (to the predicate “ predators"") - incomplete (the circle of the subject falling into it, as it were, cuts out some part of it):

The distribution of terms in simple judgments can be different depending on the type of judgment and the nature of the relationship between its subject and predicate. In table. 4 shows all cases of distribution of terms in simple judgments:

Here we consider all four types of simple judgments and all possible cases of relations between the subject and the predicate in them (see section 2. 2). Pay attention to statements like O where the subject and predicate are in an intersection relationship. Despite the intersecting circles in Euler's scheme, the subject of this judgment is undistributed, and the predicate is distributed. Why is it so? Above, we said that the Euler circles intersecting in the diagram denote undistributed terms. Hatching shows that part of the subject, which is discussed in the judgment (in this case- about schoolchildren who are not athletes), due to which the circle denoting the predicate in the Euler scheme remained complete (the circle denoting the subject does not cut off some part from it, as happens in a judgment of the form I where subject and predicate are in an intersection relation).

Thus, we see that the subject is always distributed in judgments of the form A and E and is always not distributed in judgments of the form I and O, and the predicate is always distributed in judgments of the form E and O, but in judgments of the form A and I it can be both distributed and undistributed, depending on the nature of the relationship between it and the subject in these judgments.

The easiest way to establish the distribution of terms in simple judgments is with the help of Euler schemes (it is not necessary to memorize all cases of distribution from the table). It is enough to be able to determine the type of relationship between the subject and the predicate in the proposed judgment and depict them with circular diagrams. Further, it is even simpler - a full circle, as already mentioned, corresponds to a distributed term, and an incomplete one corresponds to an undistributed one. For example, it is required to establish the distribution of terms in the judgment: “ Some Russian writers are world famous people". Let us first find the subject and the predicate in this judgment: Russian writers"- subject, " world famous people' is a predicate. Now let's find out in what relation they are. A Russian writer may or may not be a world famous person, and a famous person may or may not be a Russian writer, therefore, the subject and predicate of the said judgment are in relation to the intersection. Let's depict this relationship on the Euler diagram, shading the part referred to in the judgment (Fig. 25):

Both the subject and the predicate are depicted as incomplete circles (some part of each of them is cut off), therefore, both terms of the proposed judgment are undistributed ( S –, P –).

Let's consider one more example. It is necessary to establish the distribution of terms in the judgment: ". Finding the subject and the predicate in this judgment: people"- subject, " athletes"- a predicate, and having established a relationship between them - subordination, we will depict it on the Euler diagram, shading the part referred to in the judgment (Fig. 26):

The circle denoting the predicate is complete, and the circle corresponding to the subject is incomplete (the circle of the predicate, as it were, cuts out some part of it). Thus, in this judgment, the subject is undistributed, and the predicate is distributed ( S –, P –).

Check yourself:

1. In what case is the term of judgment considered distributed, and in what case - undistributed? How can one establish the distribution of terms in a simple proposition using Euler's circular schemes?

2. What is the distribution of terms in all kinds of simple judgments and in all cases of relations between their subject and predicate?

3. Using Euler schemes, establish the distribution of terms in the following judgments:

1) All insects are living organisms.

2) Some books are textbooks.

3) Some students are not successful.

4) All cities are towns.

5) None of the fish are mammals.

6) Some ancient Greeks are famous scientists.

7) Some celestial bodies are stars.

8) All rhombuses with right angles are squares.

2.4. Transformation of a simple proposition

There are three ways of transforming, i.e., changing the form, of simple judgments: conversion, transformation, and opposition to a predicate.

Appeal (conversion) is a transformation of a simple proposition in which the subject and predicate are reversed. For example, the statement: " All sharks are fish", - is transformed by turning into a judgment:" ". Here the question may arise why the original judgment begins with the quantifier " all", and the new one - from the quantifier " some"? This question, at first glance, seems strange, because you can’t say: “ All fish are sharks', so the only thing left is: ' Some fish are sharks". However, in this case, we turned to the content of the judgment and changed the meaning of the quantifier " all» to quantifier « some»; and logic, as has already been said, is abstracted from the content of thinking and is concerned only with its form. Therefore, the reversal of the judgment: " All sharks are fish”, - can be performed formally, without referring to its content (meaning). To do this, we establish the distribution of terms in this judgment using a circular scheme. Terms of judgment, i.e. subject " sharks" and the predicate " fish", are in this case in relation to subordination (Fig. 27):

The circular diagram shows that the subject is distributed (full circle), and the predicate is undistributed (incomplete circle). Remembering that the term is distributed when it comes to all the objects included in it, and undistributed when it is not about all, we automatically mentally put before the term “ sharks» quantifier « all", and before the term " fish» quantifier « some". Making the reversal of the indicated judgment, i.e., interchanging its subject and predicate and starting a new judgment with the term " fish”, we again automatically supply it with the quantifier “ some”, without thinking about the content of the original and new judgments, and we get an unmistakable version: “ Some fish are sharks". Perhaps all this will seem like an excessive complication of an elementary operation, however, as we will see below, in other cases it is not easy to transform judgments without using the distribution of terms and circular schemes.

Let us pay attention to the fact that in the example considered above, the original judgment was of the form A, and the new one is of the form I, i.e., the inversion operation led to a change in the form of a simple judgment. At the same time, of course, its form has changed, but the content has not changed, because in judgments: All sharks are fish" and " Some fish are sharks", they are talking about the same thing. In table. 5 shows all cases of conversion, depending on the type of simple judgment and the nature of the relationship between its subject and predicate:

Judgment of the Kind A I. Judgment of the Kind I turns either into itself or into a judgment of the form A. Judgment of the Kind E always turns into itself, and a judgment of the form O not reversible.

The second way of transforming simple propositions, called transformation (obversion), lies in the fact that the judgment changes the link: positive to negative, or vice versa. In this case, the predicate of the judgment is replaced by a contradictory concept (i.e., the particle “not” is placed before the predicate). For example, the same judgment that we considered as an example for the appeal: " All sharks are fish", - is transformed by turning into a judgment:" ". This judgment may seem strange, because it is not usually said so, although in fact we have a shorter formulation of the idea that no shark can be such a creature that is not a fish, or that the set of all sharks is excluded from the set of all creatures, which are not fish. Subject " sharks" and the predicate " not fish The judgment resulting from the transformation is in relation to incompatibility.

The above example of transformation demonstrates an important logical pattern: any statement is equal to a double negation, and vice versa. As we can see, the original judgment of the form A as a result of the transformation became a judgment of the form E. Unlike conversion, transformation does not depend on the nature of the relationship between the subject and the predicate of a simple judgment. Therefore, a judgment of the form A E, and a judgment of the form E- in a judgment of the form A. Judgment of the Kind I always turns into a judgment of the form O, and a judgment of the form O- in a judgment of the form I(Fig. 28).

The third way to transform simple judgments is opposition to predicate- consists in the fact that first the judgment undergoes transformation, and then conversion. For example, in order to transform the proposition by opposing the predicate: “ All sharks are fish", - you must first subject it to transformation. Get: " All sharks are non-fish". Now we need to make an inversion with the resulting judgment, i.e., swap its subject “ sharks" and the predicate " not fish". In order not to be mistaken, we will again resort to establishing the distribution of terms using a circular scheme (the subject and the predicate in this judgment are in incompatibility) (Fig. 29):

The circular diagram shows that both the subject and the predicate are distributed (a full circle corresponds to both terms), therefore, we must accompany both the subject and the predicate with the quantifier " all". After that, we will make an inversion with a judgment: “ All sharks are non-fish". Get: " All non-fish are not sharks". The judgment sounds unusual, but it is a shorter formulation of the idea that if some creature is not a fish, then it cannot possibly be a shark, or that all creatures that are not fish cannot automatically be sharks, including . The appeal could be made easier by looking at Table. 5 for the appeal above. Seeing that a judgment of the form E always turns into itself, we could, without using a circular scheme and without establishing the distribution of terms, immediately put before the predicate " not fish» quantifier « all". In this case, another method was proposed to show that it is quite possible to do without the table. for circulation, and it is not necessary to memorize it at all. Here, approximately the same thing happens as in mathematics: you can memorize various formulas, but you can do without memorization, since any formula is easy to deduce on your own.

All three transformation operations of simple judgments are easiest to perform with the help of circular schemes. To do this, it is necessary to depict three terms: subject, predicate and a concept that contradicts the predicate (non-predicate). Then it is necessary to establish their distribution, and four judgments will follow from the resulting Euler scheme - one initial and three results of transformations. The main thing to remember is that the distributed term corresponds to the quantifier " all", and unallocated to the quantifier " some»; that the circles touching on the Euler diagram correspond to the connection " is", and non-contiguous - a bunch of" is not". For example, it is required to perform three transformation operations with a judgment: " All textbooks are books". Let's depict the subject " textbooks', predicate ' books' and non-predicate ' not books» a circular scheme and establish the distribution of these terms (Fig. 30):

1. All textbooks are books(original judgment).

2. Some books are textbooks(appeal).

3. All textbooks are not non-books(transformation).

4. All non-books are not textbooks

Let's consider one more example. It is necessary to transform the judgment in three ways: All planets are not stars". Let's depict the subject " planets', predicate ' stars' and non-predicate ' not stars". Please note that the concepts planets" and " not stars are in a relationship of subordination: a planet is not necessarily a star, but a celestial body that is not a star is not necessarily a planet. Let's establish the distribution of these terms (Fig. 31):

1. All planets are not stars(original judgment).

2. All stars are not planets(appeal).

3. All planets are not stars(transformation).

4. Some non-stars are planets(as opposed to a predicate).

Check yourself:

1. How is the circulation operation carried out? Take any three judgments and make an appeal to each of them. How does the conversion take place in all kinds of simple propositions and in all cases of relations between their subject and predicate? What judgments are not reversible?

2. What is a transformation? Take any three judgments and perform the transformation operation with each of them.

3. What is the operation of opposition to a predicate? Take any three judgments and transform each of them by opposing a predicate.

4. How can knowledge about the distribution of terms in simple judgments and the ability to establish it with the help of circular schemes help in carrying out judgment transformation operations?

5. Take some kind of judgment A and perform all transformation operations with it using circular schemes and establishing the distribution of terms. Do the same with some kind of judgment E.

2.5. Logic square

Simple judgments are divided into comparable and incomparable.

Comparable (identical in material) propositions have the same subjects and predicates, but may differ in quantifiers and connectives. For example, judgments: », « Some students don't study math, are comparable: they have the same subjects and predicates, but the quantifiers and connectives are different. Incomparable judgments have different subjects and predicates. For example, judgments: All students study mathematics», « Some athletes are Olympic champions, are incomparable: their subjects and predicates do not coincide.

Comparable judgments, like concepts, are compatible and incompatible and can be in different relationships with each other.

Compatible are judgments that can be true at the same time. For example, judgments: Some people are athletes», « Some people are not athletes", are both true and compatible propositions.

Incompatible are called judgments that cannot be simultaneously true: the truth of one of them necessarily means the falsity of the other. For example, judgments: All students learn math", "Some students do not learn math", - cannot be both true and are incompatible (the truth of the first judgment inevitably leads to the falsity of the second).

Compatible judgments can be in the following relationships:

1. equivalence is a relation between two propositions whose subjects, predicates, connectives, and quantifiers are the same. For example, judgments: Moscow is an ancient city»,

« The capital of Russia is an ancient city", are in the relation of equivalence.

2. Subordination is a relationship between two propositions in which predicates and connectives are the same, and the subjects are in relation to species and genus. For example, judgments: All plants are living organisms», « All flowers (some plants) are living organisms", are in a relationship of subordination.

3. Partial match (subcontrarality) Some mushrooms are edible», « Some mushrooms are not edible, are in a partial match relationship. It should be noted that in this respect there are only private judgments - private affirmative ( I) and partial negatives ( O).

Incompatible judgments can be in the following relationships.

1. Opposite (contrarality) is a relation between two propositions in which the subjects and predicates are the same, but the connectives are different. For example, judgments: All people are truthful», « ", are in relation to the opposite. In this regard, there can only be general judgments - generally affirmative ( A) and generally negative ( E). An important feature of opposing propositions is that they cannot be both true, but they can be both false. Thus, the two opposite propositions given cannot be simultaneously true, but can be simultaneously false: it is not true that all people are true, but it is also not true that all people are not true.

Opposite judgments can be false at the same time, because between them, denoting some extreme options, there is always a third, middle, intermediate option. If this middle option is true, then the two extremes will be false. Between opposite (extreme) judgments: " All people are truthful», « All people are not truthful", - there is a third, middle option:" Some people are truthful and some people are not.”, - which, being a true judgment, causes the simultaneous falsity of two extreme, opposite judgments.

2. Contradiction (contradiction)- this is the relationship between two judgments, in which the predicates are the same, the ligaments are different, and the subjects differ in their volumes, that is, they are in a relationship of subordination (type and genus). For example, judgments: All people are truthful", "Some people are not truthful", are in contradiction. An important feature of contradictory judgments, in contrast to the opposite ones, is that there cannot be a third, middle, intermediate option between them. Because of this, two contradictory judgments cannot be simultaneously true and cannot be simultaneously false: the truth of one of them necessarily means the falsity of the other, and vice versa - the falsity of one determines the truth of the other. We will return to opposite and contradictory judgments when we talk about the logical laws of contradiction and the excluded middle.

The considered relations between simple comparable judgments are depicted schematically using a logical square (Fig. 32), which was developed by medieval logicians:

The vertices of the square represent four types of simple propositions, and its sides and diagonals represent the relationships between them. So, judgments of the form A and type I, as well as judgments of the form E and type O are in a relationship of subordination. Judgments of the Kind A and type E are in relation to opposites, and judgments of the form I and type O- partial match. Judgments of the Kind A and type O, as well as judgments of the form E and type I are in conflict. It is not surprising that the logical square does not depict the relation of equivalence, because in this relation there are judgments of the same kind, i.e., equivalence is the relationship between judgments A and A, I and I, E and E, O and O. In order to establish a relation between two propositions, it is sufficient to determine what kind each of them belongs to. For example, it is necessary to find out in what relation the judgments are: All people studied logic», « Some people didn't study logic". Seeing that the first judgment is universally affirmative ( A), and the second partly negative ( O), we can easily establish the relationship between them using a logical square - a contradiction. Judgments: " All people studied logic (A)», « Some people studied logic (I)", are in relation to subordination, and judgments:" All people studied logic (A)», « All people did not study logic (E)", are in relation to the opposite.

As already mentioned, important property judgments, unlike concepts, is that they can be true or false.

As far as comparable propositions are concerned, the truth-values ​​of each of them are connected in a certain way with the truth-values ​​of the others. Thus, if a judgment of the form A is true or false, then the other three ( I, E, O), judgments comparable to it (having subjects and predicates similar to it), depending on this (on the truth or falsity of a judgment of the form A) are also true or false. For example, if a judgment of the form A: « All tigers are predators, is true, then a judgment of the form I: « Some tigers are predators”, is also true (if all tigers are predators, then some of them, i.e. some tigers are also predators), the judgment of the species E: « All tigers are not predators, is false, and a judgment of the form O: « Some tigers are not predators", is also false. Thus, in this case, from the truth of a judgment of the form A the truth of a judgment of the form follows I and the falsity of judgments of the form E and type O(of course, we are talking about comparable judgments, i.e., having the same subjects and predicates).

Check yourself:

1. Which judgments are called comparable and which are incomparable?

2. What are compatible and incompatible judgments? Give three examples of compatible and incompatible judgments.

3. In what respects can there be compatible judgments? Give two examples each for equivalence, subordination, and overlapping relationships.

4. In what ways can there be incompatible judgments?

Give three examples each of opposite and contradictory relationships. Why can opposing judgments be false at the same time, but contradictory ones cannot?

5. What is a logical square? How does he depict the relationship between judgments? Why does the logical square not represent an equivalence relation? How to use the logical square to determine the relationship between two simple comparable propositions?

6. Take some true or false proposition of the form A and draw conclusions from it about the truth of judgments of the types comparable with it E, I, O. Take any true or false proposition of the form E and draw conclusions from it about the truth of judgments comparable to it A, I, O.

2.6. Complex judgment

Depending on the union with which simple judgments are combined into complex ones, five types of complex judgments are distinguished:

1. Conjunctive judgment (conjunction)- this is a complex proposition with a connecting union "and", which is indicated in logic by the conventional sign "?". With the help of this sign, a conjunctive judgment, consisting of two simple judgments, can be represented as a formula: a ? b(read " a and b"), where a and b- these are two simple judgments. For example, a complex proposition: Lightning flashed and thunder roared", - is a conjunction (connection) of two simple propositions: "Lightning flashed", "Thunder rumbled". A conjunction can consist not only of two, but also of more simple judgments. For example: " Lightning flashed and thunder rumbled and it started to rain (a ? b ? c)».

2. Disjunctive (disjunction)- this is a complex proposition with a divisive union "or". Let us remember that, speaking of logical operations addition and multiplication of concepts, we noted the ambiguity of this union - it can be used both in a non-strict (non-exclusive) sense, and in a strict (exclusive) one. It is not surprising, therefore, that disjunctive judgments are divided into two types:

1. Nonstrict disjunction- this is a complex proposition with a divisive union "or" in its non-strict (non-exclusive) meaning, which is indicated by the conventional sign "?". Using this sign, a non-strict disjunctive judgment, consisting of two simple judgments, can be represented as a formula: a ? b(read " a or b"), where a and b Is he studying English or is he studying German", - is a non-strict disjunction (separation) of two simple judgments: "He is learning English", "He is learning German". These judgments do not exclude each other, because it is possible to study both English and German at the same time, so this disjunction is not strict.

2. Strict disjunction- this is a complex proposition with a dividing union "or" in its strict (exclusive) meaning, which is indicated by the conventional sign "". Using this sign, a strict disjunctive judgment, consisting of two simple judgments, can be represented as a formula: a b(read "or a, or b"), where a and b Those are two simple sentences. For example, a complex proposition: Is he in 9th grade or is he in 11th grade”, is a strict disjunction (separation) of two simple propositions: "He is in 9th grade", "He is in 11th grade". Let us pay attention to the fact that these judgments exclude each other, because it is impossible to study both in the 9th and in the 11th grade at the same time (if he studies in the 9th grade, then he definitely does not study in the 11th grade, and vice versa), which is why this disjunction is strict.

Both non-strict and strict disjunctions can consist not only of two, but also of a larger number of simple judgments. For example: " He is learning English or he is learning German or he is learning French (a ? b ? c)», « He is in 9th grade or he is in 10th grade or he is in 11th grade (a b c)».

3. implicative judgment (implication)- this is a complex proposition with a conditional union "if ... then", which is indicated by the conditional sign ">". Using this sign, an implicative judgment, consisting of two simple judgments, can be represented as a formula: a > b(read "if a, then b"), where a and b Those are two simple sentences. For example, a complex proposition: If a substance is a metal, then it is electrically conductive.", - is an implicative judgment (causation) of two simple judgments: "The substance is a metal", "The substance is electrically conductive". In this case, these two judgments are connected in such a way that the second follows from the first (if the substance is a metal, then it is necessarily electrically conductive), but the first does not follow from the second (if the substance is electrically conductive, then this does not mean at all that it is a metal). The first part of the implication is called basis, and the second is consequence; the consequence follows from the reason, but the reason does not follow from the consequence. Implication formula: a > b, can be read like this: "if a, then necessarily b, but if b, then it is not necessary a».

4. Equivalent judgment (equivalent)- this is a complex proposition with the union "if ... then" not in its conditional meaning (as in the case of implication), but in the identical (equivalent) one. In this case, this union is denoted by the conventional sign "", with the help of which an equivalent proposition, consisting of two simple propositions, can be represented as a formula: a b(read "if a, then b, and if b, then a"), where a and b Those are two simple sentences. For example, a complex proposition: If the number is even, then it is evenly divisible by 2", - is an equivalent judgment (equality, identity) of two simple judgments: "The number is even", "The number is evenly divisible by 2". It is easy to see that in this case, two judgments are connected in such a way that the second follows from the first, and the first follows from the second: if the number is even, then it is necessarily divisible by 2 without a remainder, and if the number is divisible by 2 without a remainder, then it is necessarily even. . It is clear that in an equivalence, unlike an implication, there can be neither a foundation nor a consequence, since its two parts are equivalent judgments.

5. negative judgment (negation)- this is a complex proposition with the union "it is not true that ...", which is indicated by the conventional sign "¬". Using this sign, a negative judgment can be represented as a formula: ¬ a(read "it is not true that a"), where a is a simple judgment. Here the question may arise - where is the second part of the complex judgment, which we usually denoted by the symbol b? Recorded: ¬ a, there are already two simple propositions: a- this is some kind of statement, and the sign "¬" is its negation. Before us, as it were, two simple judgments - one affirmative, the other - negative. An example of a negative judgment: " It is not true that all flies are birds.».

So, we have considered five types of complex judgments: conjunction, disjunction (non-strict and strict), implication, equivalence and negation.

There are many conjunctions in natural language, but all of them in meaning are reduced to the considered five types, and any complex judgment refers to one of them. For example, a complex proposition: It's almost midnight, but Herman is still gone", - is a conjunction, because it contains the union" a" is used as a connecting union "and". A complex proposition in which there is no union at all: “ Sow the wind, reap the whirlwind”, - is an implication, since two simple judgments in it are connected in meaning by the conditional union “if ... then”.

Any complex proposition is true or false, depending on the truth or falsity of the simple propositions included in it. Table is given. 6 the truth of all types of complex judgments, depending on all possible sets of truth values ​​of the two simple judgments included in them (there are only four such sets): both simple judgments are true; the first judgment is true and the second is false; the first judgment is false and the second is true; both statements are false).

As we see, a conjunction is true only when both simple propositions included in it are true. It should be noted that a conjunction consisting not of two, but of a larger number of simple judgments is also true only if all the judgments included in it are true. In all other cases, it is false. A non-strict disjunction, on the contrary, is true in all cases except when both simple propositions included in it are false. A non-strict disjunction, consisting not of two, but of a larger number of simple propositions, is also false only when all the simple propositions included in it are false. A strict disjunction is true only if one of the simple propositions included in it is true and the other is false. A strict disjunction, consisting not of two, but of a larger number of simple propositions, is true only if only one of the simple propositions included in it is true, and all the others are false. The implication is false only in one case - when its reason is true, and the consequence is false. In all other cases it is true. An equivalence is true when the two simple propositions that make it up are true, or when both are false. If one part of an equation is true and the other part is false, then the equation is false. The truth of a negation is most simply defined: when a statement is true, its negation is false; when a statement is false, its negation is true.

Check yourself:

1. On what basis are the types of complex judgments distinguished?

2. Describe all types of complex judgments: name, union, symbol, formula, example. What is the difference between a non-strict disjunction and a strict one? How to distinguish an implication from an equivalence?

3. How can you determine the type of a complex judgment if instead of the unions “and”, “or”, “if ... then” any other unions are used in it?

4. Give three examples for each type of complex judgments, without using the unions “and”, “or”, “if ... then”.

5. Determine what type the following complex judgments belong to:

1. Creature is human only when it has thinking.

2. Humanity can die either from the depletion of earth's resources, or from ecological disaster or as a result of World War III.

3. Yesterday he received a deuce not only in mathematics, but also in Russian.

4. A conductor heats up when an electric current passes through it.

5. The world around us is either cognizable or not.

6. Either he is completely mediocre, or he is a complete lazy person.

7. When a person flatters, he lies.

8. Water turns into ice only at a temperature of 0 ° C and below.

6. What determines the truth of complex judgments? What truth values ​​do conjunction, non-strict and strict disjunction, implication, equivalence and negation take depending on all sets of truth values ​​of simple propositions included in them?

2.7. Logic formulas

Any statement or whole reasoning can be formalized. This means discarding its content and leaving only its logical form, expressing it with the help of the already known conventional symbols of conjunction, non-strict and strict disjunction, implication, equivalence and negation.

For example, to formalize the following statement: He is engaged in painting, or music, or literature”, - you must first highlight the simple judgments included in it and establish the type of logical connection between them. The above statement includes three simple propositions: "He does painting", "He does music", "He does literature".

These judgments are united by a disjunctive connection, but they do not exclude each other (you can engage in painting, music, and literature), therefore, we have a non-strict disjunction, the form of which can be represented by the following conditional notation: a ? b ? c, where a, b, c- the above simple judgments. Shape: a ? b ? c, can be filled with any content, for example: " Cicero was a politician, or an orator, or a writer", "He studies English, or German, or French", "People move by land, or air, or water transport».

We formalize the reasoning: He is in 9th grade, or 10th grade, or 11th grade. However, it is known that he does not study in either 10th or 11th grade. So he is in 9th grade.". We single out the simple statements included in this reasoning and denote them by small letters of the Latin alphabet: “He is in 9th grade (a)”, “He is in 10th grade (b)”, “He is in 11th grade (c)”. The first part of the argument is a strict disjunction of these three statements: a ? b ? c. The second part of the argument is the negation of the second: ¬ b, and the third: ¬ c, statements, and these two negations are combined, i.e. connected conjunctively: ¬ b ? ¬ c. The conjunction of negations is attached to the strict disjunction mentioned above. three simple judgments :( a ? b ? c) ? (¬ b ? ¬ c), and already from this new conjunction, as a consequence, the assertion of the first simple proposition follows: “ He is in 9th grade". Logical consequence, as we already know, is an implication. Thus, the result of the formalization of our reasoning is expressed by the formula: (( a ? b ? c) ? (¬ b ?¬ c)) > a. This logical form can be filled with any content. For example: " For the first time a man flew into space in 1957, or in 1959, or in 1961. However, it is known that for the first time a man flew into space not in 1957 and not in 1959. Therefore, for the first time a man flew into space in 1961"Another option:" The philosophical treatise Critique of Pure Reason was written either by Immanuel Kant, or by Georg Hegel, or by Karl Marx. However, neither Hegel nor Marx are the authors of this treatise. Therefore, Kant wrote it».

The result of the formalization of any reasoning, as we have seen, is a formula consisting of small letters of the Latin alphabet, expressing simple statements included in the reasoning, and symbols of logical connections between them (conjunctions, disjunctions, etc.). All formulas are divided into three types in logic:

1. Identically true formulas are true for all sets of truth values ​​of the variables included in them (simple propositions). Any identically true formula is a logical law.

2. Identical false formulas are false for all sets of truth values ​​of their variables.

Identical false formulas are a negation of identically true formulas and are a violation of logical laws.

3. Doable (neutral) formulas for different sets of truth values ​​of the variables included in them are either true or false.

If as a result of the formalization of any reasoning, an identically true formula is obtained, then such reasoning is logically flawless. If the result of formalization is an identically false formula, then the reasoning should be recognized as logically incorrect (erroneous). A feasible (neutral) formula testifies to the logical correctness of the reasoning, of which it is a formalization.

In order to determine what kind this or that formula belongs to, and, accordingly, to evaluate the logical correctness of some reasoning, they usually make up a special truth table for this formula. Consider the following reasoning: Vladimir Vladimirovich Mayakovsky was born in 1891 or 1893. However, it is known that he was not born in 1891. Therefore, he was born in 1893.”. Formalizing this reasoning, we single out the simple statements included in it: "Vladimir Vladimirovich Mayakovsky was born in 1891." "Vladimir Vladimirovich Mayakovsky was born in 1893". The first part of our discussion is undoubtedly a strict disjunction of these two simple statements: a ? b. Further, the negation of the first simple statement is added to the disjunction, and the conjunction is obtained: ( a ? b) ? ¬ a. And, finally, the statement of the second simple proposition follows from this conjunction, and the implication is obtained: (( a ? b) ? ¬ a) > b, which is the result of the formalization of this reasoning. Now we need to make a table. 7 truths for the resulting formula:

The number of rows in the table is determined by the rule: 2 n , where n is the number of variables (simple statements) in the formula. Since there are only two variables in our formula, there should be four rows in the table. The number of columns in the table is equal to the sum of the number of variables and the number of logical unions included in the formula. In the formula under consideration, there are two variables and four logical unions (?, ?, ¬, >), which means that the table should have six columns. The first two columns represent all possible sets of truth values ​​for the variables (there are four such sets: both variables are true; the first variable is true and the second is false; the first variable is false and the second is true; both variables are false). The third column is the truth values ​​of the strict disjunction that it takes depending on all (four) sets of truth values ​​of the variables. The fourth column is the truth values ​​of the negation of the first simple statement: ¬ a. The fifth column is the truth values ​​of the conjunction consisting of the above strict disjunction and negation, and finally the sixth column is the truth values ​​of the entire formula or implication. We have broken the whole formula into its component parts, each of which is a two-term compound proposition, i.e., consisting of two elements (in the previous paragraph it was said that negation is also a two-term compound proposition):

The last four columns of the table present the truth values ​​of each of these binary complex propositions that form the formula. First, fill in the third column of the table. To do this, we need to return to the previous paragraph, where the truth table of complex judgments was presented ( see table. 6), which in this case will be basic for us (like a multiplication table in mathematics). In this table we see that a strict disjunction is false when both parts of it are true or both parts are false; when one part of it is true and the other is false, then the strict disjunction is true. Therefore, the values ​​of strict disjunction in the table being filled (from top to bottom) are as follows: "false", "true", "true", "false". Next, fill in the fourth column of the table: ¬ a: when the statement is true twice and twice false, then the negation ¬ a, on the contrary, is twice false and twice true. The fifth column is the conjunction. Knowing the truth values ​​of strict disjunction and negation, we can establish the truth values ​​of a conjunction that is true only when all of its constituent elements are true. Strict disjunction and negation, which form this conjunction, are simultaneously true only in one case, therefore, the conjunction takes the value “true” once, and “false” in other cases. Finally, you need to fill in the last column: for the implication, which will represent the truth values ​​of the entire formula. Returning to the basic truth table of complex propositions, remember that the implication is false only in one case: when its base is true and the consequence is false. The basis of our implication is the conjunction presented in the fifth column of the table, and the consequence is a simple proposition ( b) presented in the second column. Some inconvenience in this case lies in the fact that from left to right the consequence goes before the foundation, but we can always mentally swap them. In the first case (the first line of the table, not counting the "cap") the basis of the implication is false, and the consequence is true, which means that the implication is true. In the second case, both the reason and the consequence are false, so the implication is true. In the third case, both the reason and the consequence are true, so the implication is true. In the fourth case, as in the second, both the reason and the consequence are false, which means that the implication is true.

The considered formula takes the value "true" for all sets of truth values ​​of the variables included in it, therefore, it is identically true, and the reasoning, the formalization of which it acts, is logically flawless.

Let's consider one more example. It is required to formalize the following reasoning and establish what form the formula expressing it belongs to: “ If any building is old, then it needs overhaul. This building is in need of a major overhaul. Therefore, this building is old.". Let us single out the simple statements included in this argument: "Any building is old", "Any building needs a major overhaul". The first part of the argument is an implication: a > b, these simple statements (the first is its foundation, and the second is its consequence). Further, the statement of the second simple statement is added to the implication, and the conjunction is obtained: ( a > b) ? b. And finally, the assertion of the first simple statement follows from this conjunction, and a new implication is obtained: (( a > b) ? b) > a, which is the result of the formalization of the reasoning under consideration. To determine the type of the resulting formula, we will compile a table. 8 its truth.

There are two variables in the formula, which means there will be four lines in the table; there are also three unions (>, ?, >) in the formula, which means that the table will have five columns. The first two columns are the truth values ​​of the variables. The third column is the truth values ​​of the implication.

The fourth column is the truth values ​​of the conjunction. The fifth, last column is the truth values ​​of the entire formula - the final implication. Thus, we have broken the formula into three components, which are binary complex judgments:

Let's fill in the last three columns of the table sequentially according to the same principle as in the previous example, that is, based on the basic truth table of complex judgments (see Table 6).

The formula under consideration takes both the value “true” and the value “false” for different sets of truth values ​​of the variables included in it; the content of the reasoning, such a form of its construction could lead to an error, for example: “ If the word is at the beginning of a sentence, then it is capitalized. The word "Moscow" is always capitalized. Therefore, the word "Moscow" is always at the beginning of a sentence.».

Check yourself:

1. What is the formalization of a statement or reasoning? Come up with some reasoning and formalize it.

2. Formalize the following reasoning:

1) If a substance is a metal, then it is electrically conductive. Copper is a metal. Therefore, copper is electrically conductive.

2) The famous English philosopher Francis Bacon lived in the 17th century, or in the 15th century, or in the 13th century. Francis Bacon lived in the 17th century. Therefore, he did not live either in the 15th century or in the 13th century.

3) If you're not stubborn, then you can change your mind. If you can change your mind, then you are able to recognize this judgment as false. Therefore, if you are not stubborn, then you are able to recognize this judgment as false.

4) If the sum of the interior angles of a geometric figure is 180°, then the figure is a triangle. The sum of the interior angles of a given geometric figure is not equal to 180°. Therefore, this geometric figure is not a triangle.

5) Forests are coniferous, or deciduous, or mixed. This forest is neither deciduous nor coniferous. Therefore, this forest is mixed.

3. What are identically true identically false and satisfiable formulas? What can be said about reasoning if the result of its formalization is an identically true formula? What will be the reasoning if its formalization is expressed by an identically false formula? What, from the point of view of logical fidelity, are the arguments that, when formalized, lead to feasible formulas?

4. How can one determine the type of this or that formula, which expresses the result of the formalization of a certain reasoning?

What algorithm is used to build and fill in truth tables for logical formulas? Come up with some reasoning, formalize it and use the truth table to determine the form of the resulting formula.

2.8. Types and rules of the question

The question is very close to the judgment. This is manifested in the fact that any judgment can be considered as an answer to a certain question.

Therefore, the question can be characterized as a logical form, as if preceding the judgment, representing a kind of "prejudice". Thus, a question is a logical form (construction), which is aimed at obtaining an answer in the form of a certain judgment.

Questions are divided into research and information.

Research questions are aimed at obtaining new knowledge. These are questions that have not yet been answered. For example, the question: How was the universe born?» is exploratory.

Informational questions are aimed at acquiring (transferring from one person to another) already existing knowledge (information). For example, the question: What is the melting point of lead?» is informational.

Questions are also divided into categorical and propositional.

categorical (replenishing, special) questions include interrogative words "who", "what", "where", "when", "why", "how", etc., indicating the direction of the search for answers and, accordingly, the category of objects, properties or phenomena in which to look for the answers you need.

Propositional(from lat. propositio- judgment, suggestion) ( specifying, general) questions, which are also often called, are aimed at confirming or denying some already available information. In these questions, the answer is, as it were, already laid down in the form of a ready-made judgment, which only needs to be confirmed or rejected. For example, the question: Who created periodic system chemical elements? ” is categorical, and the question: “ Is learning mathematics useful?"- propositional.

It is clear that both research and information questions can be both categorical and propositional. One could put it the other way around: both categorical and propositional questions can be both exploratory and informational. For example: " How to create a universal proof of Fermat's Theorem?» – research categorical question:

« Are there planets in the Universe that are inhabited, like the Earth, sentient beings? » is an exploratory propositional question:

« When did logic appear?” – information categorical question: “ Is it true that the number ? What is the ratio of the circumference of a circle to its diameter?” is an informational propositional question.

Any question has a certain structure, which consists of two parts. The first part is some information (expressed, as a rule, by some judgment), and the second part indicates its insufficiency and the need to supplement it with some kind of answer. The first part is called basic (basic)(also sometimes called premise of the question), and the second part desired. For example, in the information categorical question: When was the electromagnetic field theory created?"- the main (basic) part is an affirmative judgment:" The electromagnetic field theory was created, - and the desired part, represented by question word « when”, indicates the insufficiency of the information contained in the basic part of the question, and requires its addition, which should be sought in the field (category) of temporal phenomena. In a research propositional question: " Is it possible for earthlings to fly to other galaxies?", - the main (basic) part is represented by the proposition: " Possible flights of earthlings to other galaxies", - and the desired part, expressed by the particle " whether”, indicates the need to confirm or deny this judgment. In this case, the desired part of the question does not indicate the absence of some information contained in its basic part, but the absence of knowledge about its truth or falsity and requires this knowledge to be obtained.

The most important logical requirement for posing a question is that its main (basic) part be a true proposition. In this case, the question is considered logically correct. If the main part of the question is a false judgment, then the question should be recognized as logically incorrect. Such questions do not require an answer and are subject to rejection.

For example, the question: When was the first trip around the world? "- is logically correct, since its main part is expressed by a true judgment: " The first circumnavigation of the world took place in human history.". Question: " In what year did the famous English scientist Isaac Newton complete his work on the general theory of relativity?"- is logically incorrect, since its main part is represented by a false judgment: " Author general theory relativity is the famous English scientist Isaac Newton».

So, the main (basic part) of the question must be true and must not be false. However, there are logically correct questions, the main parts of which are false judgments. For example, questions: “Is it possible to create a perpetual motion machine?”, “Is there intelligent life on Mars?”, “Will they invent a time machine?”– undoubtedly, should be recognized as logically correct, despite the fact that their basic parts are false judgments: “ . The fact is that the desired parts of these questions are aimed at clarifying the truth values ​​of their main, basic parts, that is, it is required to find out whether the judgments are true or false: “ It is possible to create a perpetual motion machine”, “There is intelligent life on Mars”, “They will invent a time machine”. In this case, the questions are logically correct. If the desired parts of the questions under consideration were not aimed at clarifying the truth of their main parts, but had something else as their goal, these questions would be logically incorrect, for example: Where was the first perpetual motion machine created?”, “When did intelligent life appear on Mars?”, “How much will it cost to travel in a time machine?”. Thus, the main rule for posing a question should be expanded and clarified: the main (basic) part of a correct question should be a true judgment; if it is a false judgment, then its desired part should be aimed at clarifying the truth value of the main part; otherwise the question will be logically incorrect. It is not difficult to guess that the requirement for the main part to be true, mainly applies to categorical questions, and the requirement that the desired part be a finding of the truth of the main part, applies to propositional questions.

It should be noted that correct categorical and propositional questions are similar in that they can always be answered with a true answer (as well as a false one). For example, to a categorical question: When the first one ended World War? "- can be given as a true answer:" In 1918", - and false: " In 1916". To a propositional question: Does the Earth revolve around the Sun?" - can also be given as true: " Yes, it rotates", - and false: " No it doesn't rotate", - answer. Both of these questions are logically correct. So, the fundamental possibility of obtaining true answers is the main feature of correct questions. If it is fundamentally impossible to get true answers to certain questions, then they are incorrect. For example, one cannot get a true answer to a propositional question: Will World War I ever end?” – just as it is impossible to get it for a categorical question: “ How fast does the sun revolve around the stationary earth?».

Any answers to these questions will need to be recognized as unsatisfactory, and the questions themselves - logically incorrect, subject to rejection.

Check yourself:

1. What is a question? What is the relationship between question and judgment?

2. How do research questions differ from information questions? Give five examples of research and information questions each.

3. What are categorical and propositional questions? Give five examples of categorical and propositional questions each.

4. Describe the questions below in terms of whether they belong to research or information, as well as categorical or propositional:

1) When was the law of gravity discovered?

2) Will the inhabitants of the Earth be able to settle on other planets of the solar system?

3) What year was Bonaparte Napoleon born?

4) What is the future of humanity?

5) Is it possible to prevent a third world war?

5. What is the logical structure of the question? Give an example of a categorical research question and highlight the main (basic) and desired parts in it. Do the same with the categorical information question, the propositional research question, and the propositional information question.

6. Which questions are logically correct and which ones are incorrect? Give five examples of logically correct and incorrect questions each. Can a logically correct question have a false body? Is it sufficient to determine the correct question of the requirement that its main part be true?

What unites logically correct categorical and propositional questions?

7. Answer which of the following questions are logically correct and which are incorrect:

1) How many times larger is the planet Jupiter than the sun?

2) What is the area of ​​the Pacific Ocean?

3) In what year did Vladimir Vladimirovich Mayakovsky write the poem "A Cloud in Pants"?

4) How long did the fruitful joint scientific work Isaac Newton and Albert Einstein?

5) What is the length of the equator the globe?

Judgment - this is a form of thinking in which something is affirmed or denied about the connection between an object and its attribute or about the relationship between objects. Main logical characteristic proposition is its truth value - every proposition is either true or false. A proposition is true if and only if the situation described in it actually takes place, otherwise it is false.

By a simple judgment called proposition expressing the relationship of two terms. The terms in a simple judgment are called subject and predicate judgments. The subject of judgment (S ) is what is said in the judgment, i.e. subject of thought. Judgment predicate ( R) what is said about the subject, what signs are attributed to him or not are called. In addition to the subject and the predicate, the structure of the judgment includes the quantifier and the connective. The judgment quantifier indicates the amount of judgment, i.e. indicates the total, partial or singular quantity of the subject of judgment (expressed by the words "all", "none", "some", "this"). A copula denotes a relationship between a subject ( S ) and predicate ( R ) judgments, due to which thought takes the form of a judgment. The link indicates the quality of the judgment. (Expressed by the words "is", "is not", "is", "is not").

Unified classification of simple categorical propositions. Depending on the quantity and quality, there are generally affirmative, general negative, particular affirmative and particular negative judgments.

Affirmative ( BUT) called a judgment that is general in quantity and affirmative in quality. Canonical form "All S's are P's" .

General negative ( E) called a judgment that is general in quantity and negative in quality. Canonical form "No S is a P" .

private affirmative (I ) called a judgment that is partial in quantity and affirmative in quality. Canonical form "Some S's are P's" .

private negative ( O) called a judgment that is partial in quantity and negative in quality. Canonical form "Some S's are not P's» .

Distribution of terms in simple categorical judgments. In simple judgments, terms can be distributed ( S+ , R + ), or not distributed ( S- , R - ). A term is called distributed if it is taken in full in the judgment. A term is called undistributed if in the judgment it is taken in terms of volume. The distribution of terms in a judgment is derived from the definition of relations between concepts that express the terms of a judgment. When determining the distribution of terms in simple categorical judgments, one should be guided by the following rules:

a) B general affirmative judgments ( BUT) : subject ( S R ) is always undistributed in the case of a subordination relation between the subject and the predicate of the judgment; subject ( S ) is always distributed and the predicate ( R ) is always distributed in the case of an equivalence relation between the subject and the predicate of the judgment;

b) B general negative judgments ( E): subject ( S ) and predicate ( R ) judgments are always distributed;

c) B private affirmative judgments (I ) : subject ( S ) and predicate ( R ) are undistributed in the case of an intersection relation between the subject and the predicate of the judgment; and subject ( S ) is undistributed, and the predicate ( R) is distributed in the case of a subordination relationship between the predicate and the subject of the judgment;

d) B private negative judgments ( O) : subject of judgment ( S ) is always undistributed, and the judgment predicate ( R ) is always distributed.

Complicated judgment is called a proposition, consisting of several simple, connected by logical connectives. Writing a complex proposition in the symbolic language of logic, in which simple propositions are replaced by symbols p, q, r, s, t ..., and logical unions to the symbols that replace them Ù, v, → , ↔ is called the logical form of a compound proposition. There are five main types of logical connection:

Asserting the presence of several situations at the same time - conjunction (Ù );

Statement of the presence of at least one of several situations - weak disjunction(v);

Asserting the existence of only one of several situations - strong disjunction ();

One situation is a sufficient condition for the occurrence of another situation - implication (→);

One situation is sufficient and necessary condition for the occurrence of another situation - equivalent (↔).

Depending on the type of logical connection, the following complex judgments are distinguished:

- connecting judgments- judgments in which simple judgments are interconnected by a logical connective conjunction ( Ù ). Boolean form: ( R Ù q );

- disjunctive judgments- judgments in which simple judgments are interconnected by a logical connective weak disjunction ( v) or strong disjunction (). Boolean form: ( R v q ); (pq );

- conditional propositions- judgments in which simple judgments are interconnected by a logical link implication ( → ) or the equivalent ( ↔ ). Boolean form: ( R → q ), (R ↔ q ), where R - basis of judgment q - a consequence of judgment. In conditional propositions in the correct logical form, the base always comes first, and the conclusion at the end of the formula.

The truth values ​​of complex judgments depend on the truth values ​​of the constituent judgments and on the type of their connection, which is determined by compiling truth tables:

- conjunction (Ù ) takes the value " True» only in the case of simultaneous truth of all variables; in other cases, the conjunction takes the value " Lie» (See: Fig. 18);

- weak (nonstrict) disjunction(v) takes on the value " Lie» only in the case of simultaneous falsity of all variables; in other cases, the weak disjunction takes the value " True» (See: Fig. 19);

- strong (strict) disjunction() takes the value " Lie» in case of simultaneous truth or falsity of all variables; in other cases, strong disjunction takes the value " True» (See: Fig. 20);

- implication (→ ) takes the value " Lie"only in the case of the truth of the basis of the judgment and the falsity of the consequence of the judgment; in other cases, the implication takes the value " True» (See: Fig. 21);

- equivalent (↔ ) takes the value " Lie"in the case of the truth of the foundation and the falsity of the consequence of the judgment, or vice versa, the falsity of the foundation and the truth of the consequence of the judgment; in other cases, the equivalent takes the value " True» (See: Fig. 22).

negation of judgment- this is an operation consisting in the transformation of the logical content of the negated judgment, the end result of which is the formulation of a new judgment, which is in relation to the contradiction to the original judgment. The negation of a simple attributive judgment is made according to the following equivalences: A = O; E = I; I = E; O = A - where A, E, I, O - types of simple categorical judgments, - a sign of external negation.

The negation of a complex judgment is made according to the following equivalences:

(p Ù q) ↔ (p v q)– 1st De Morgan's law

(p v q) ↔ (p Ù q)– 2nd De Morgan's law

(p q) ↔ (p ↔ q)

(p → q) ↔ (p Ù q)

(p ↔ q) ↔ (p Ù q) v (p Ù q)

We express the above in the form of complex schemes:

Rice. 23-24

Rice. 27.

Typical examples on the topic "Judgment"

Task 6. Bring the statement to the correct logical form, give a unified classification of judgments, give their schemes and the designations A, E, I, O accepted in logic.

To solve the problem, we use sentence reduction algorithm natural language to canonical form categorical judgments and analysis of simple judgments.

1. Determine subject and predicate statements, naming them accordingly S and R (composite S and R emphasize with one solid line).

2. When defining a predicate, keep in mind the following:

If the predicate is expressed noun or phrase with noun, then in this case predicate remains unchanged.

sample 1:

« Some lawyers (S) - lawyers (R) ».

If the predicate is expressed adjective or communion, which can be represented , then in this case .

Sample 2:

« Some roses (S) beautiful (R) ». → « Some roses (S) - beautiful flowers (R) ».

If the predicate is expressed verb, which can be represented one word or phrase, then in this case a generic concept for the subject of the statement should be added to the predicate, a turn the verb into its corresponding participle.

Sample 3:

« Some students of our group (S) handed over today logically (R) ». → "Some students of our group (S) there is students who passed the logic test today (R) ».

3. Determine quantifier word ("all", "some", "none", "this").

4. Determine logical link("is", "is not")

5. Record judgment in canonical form: quantifier - subject ( S) - connective - predicate ( R) .

6. Record judgment formula, to determine the quantitative and qualitative characteristics of the judgment.

7. Graphically portray relations between terms of judgment.

8. Determine distribution terms.

Example 1:

"The ancient Greeks made a great contribution to the development of philosophy."

Solution:

1. In this sentence, only the subject is logically defined - "ancient Greeks" ( S ). The predicate is expressed by the phrase "made a great contribution to the development of philosophy" ( R ).

2. Bring the predicate to canonical form. To do this, we select to the subject of the judgment ( "Ancient Greeks") generic concept ( "People"). AT canonical predicate form will be expressed as a phrase "People who have made a great contribution to the development of philosophy".

3. quantifier word in a sentence missing, but from the analysis of the meaning of the sentence it is clear that we are talking only about some of the ancient Greeks. Judgment quantifier - " Some».

4. The proposal states that the subject « Ancient Greeks» ( S Made a great contribution to the development of philosophy» ( R ). Means logical connective affirmative (« there is»).

5. Canonical judgment form: Some ancient Greeks (S) there is people. who made a great contribution to the development of philosophy (R) ».

6. Formula judgments - Some S's are P's . Quantitative-qualitative characteristic of the judgment - private affirmative

7. We graphically depict the relationship between the terms of the judgment. We define the relationship between the concept " Ancient Greeks» ( S ) and the concept " People who made a great contribution to the development of philosophy» ( R ) as a ratio crossing .

8. Define distribution terms: both terms are taken in terms of volume, which means they are undistributed ( S - , R - ) (Fig. 28).

Example 2:

"No one can be held criminally responsible twice for the same crime."

Solution:

1. In this offer subject is not explicitly defined. From an analysis of the meaning of the statement, it is clear that It's about the concept of Human» (S ) . Predicate expressed by the phrase "" ( R ).

2. Bring the predicate to canonical Human"") generic concept (" Creature"). In canonical form predicate will be expressed by the phrase "" ( R ).

3. quantifier word in a sentence missing, but from the analysis of the meaning of the sentence it is clear that it is about the whole volume the concept of "person" S ). quantifier judgments - none».

4. The sentence denies that the subject has “ Human» ( S ) property expressed in the predicate " Can be held criminally responsible twice for the same crime» ( R). (« do not eat»).

5. Write down the judgment in canonical form: " No one human (S) do not eat a living being that can be criminally responsible twice for the same crime (R) ».

6. Recording formula judgments - No S is P general negative (E ).

7. Graphically depict the relationship between the terms of the judgment. We define the relationship between the concept " Human» ( S ) and the concept " A living being that can be criminally responsible twice for the same crime» ( R ) as a ratio incomparability .

8. Define distribution terms: both terms are taken in full, which means they are distributed (S+ , R + ) (Fig. 29).

Example 3:

"Some mushrooms are not edible."

Solution:

1. In this sentence, logically only the subject is defined - " Mushrooms" ( S ) . Predicate expressed by the word edible» ( R ).

2. Bring the predicate to canonical form. To do this, we select the subject of the judgment (“ Mushrooms"") generic concept (" Living organisms"). In canonical form, the predicate will be expressed by the phrase " edible living organisms» ( R ).

3. quantifier the word is present in the sentence we are talking about part of the scope of the concept " Mushrooms» (S ). quantifier judgment word - " Some».

4. Offer denied availability subject « Mushrooms» ( S ) property expressed in predicate « Edible» ( R ). Logical connective is negative (« do not eat»).

5. Write down the judgment in canonical form: " Some mushrooms (S) do not eat edible living organisms (R) ».

6. Recording formula judgments - Some S's are not P's . We determine the quantitative and qualitative characteristics of the judgment - private negative (O ).

7. Graphically depict the relationship between the terms of the judgment. We define the relationship of the relationship between the concept " Mushrooms» ( S ) and the concept " edible living organism» ( R ) as a ratio crossing .

8. Define distribution terms: S taken in terms of volume, a R taken in full, means, distribution theirs is: S - , R + (Fig. 30).

Task 7. Consider complex judgments, express them in symbolic notation. Indicate the antecedent and consequent in implicative judgments.

Example 1:

Their labor rights, freedoms and legitimate interests by all not prohibited means.

Solution:

a) " The worker has the right to protection their labor rights R);

b) "The worker has the right to protection their freedoms by all means not prohibited" - ( q);

in) "The worker has the right to protection their legitimate interests by all means not prohibited" - ( r).

conjunction (Ù );

r u qÙ r

4. p, q, r are conjuncts.

Example 2:

"Humanity can die either from the depletion of earth's resources, or from an environmental catastrophe, or as a result of the third world war."

Solution:

1. We divide this complex judgment into simple ones and express them in the correct notation adopted in Russian, i.e. in the relation of the subject and the predicate and denote these simple judgments in the form adopted in formal logic:

a) "Humanity can die from the depletion of earth's resources" - ( R);

b) "Humanity can die from an ecological catastrophe" - ( q);

in) "Humanity may perish as a result of the third world war" - ( r).

weak disjunction(v);

3. The formula for this complex judgment looks like this:

R v q v r

4. p, q, r are clauses.

Example 3:

“A citizen, due to a physical disability, illness or illiteracy, cannot sign with his own hand, then at his request another citizen can sign the transaction.”

Solution:

1. We divide this complex judgment into simple ones and express them in the correct notation adopted in Russian, i.e. in the relation of the subject and the predicate and denote these simple judgments in the form adopted in formal logic:

a) “A citizen, due to a physical handicap, cannot sign with his own hand” - ( R);

b) “A citizen, due to illness, cannot sign with his own hand” - ( q);

in) “A citizen, due to illiteracy, cannot sign with his own hand” - ( r);

G) “At the request of this citizen, another citizen can sign the transaction” - ( s).

2. In this case, there is a statement of the presence of at least one of several situations, but other situations can also be present at the same time - weak disjunction(v); one of these situations or all of them at the same time is a sufficient condition for the occurrence of another situation - implication(→); thus, we have jointly weak disjunction and implication;

3. The formula for this complex judgment looks like this:

(R v q v r) → s

4. p, q, r are disjuncts; (R v q v r) – antecedent; s is the consequent.

Example 4:

“The marriage is dissolved if the court determines that further living together spouses and the preservation of the family became impossible.

Solution:

1. We divide this complex judgment into simple ones and express them in the correct notation adopted in Russian, i.e. in the relation of the subject and the predicate and denote these simple judgments in the form adopted in formal logic:

a) “The court found that the further joint life of the spouses became impossible” - ( R);

b) “The court found that the preservation of the family became impossible” - ( q);

in) "Marriage is dissolved" - ( r).

2. In this case, there is a statement of the simultaneous presence of several situations - conjunction (Ù ); both of these situations are a sufficient condition for the occurrence of another situation - implication(→); thus takes place jointly conjunction and implication;

3. The formula for this complex judgment looks like this:

(r u q) → r

4. р, q – conjuncts; (R v q) – antecedent; r is the consequent.

Task 8. Write down the logical formulas of complex judgments in the language of propositional logic and construct truth tables for them.

To solve the problem, we use the algorithm for analyzing complex statements:

1. Identify and write down all the simple propositions that make up the sentence. Label them with symbols.

2. Determine the logical connection between simple judgments.

3. Write down the formula for a complex judgment. If the judgment is conditional, then it is necessary to determine the reason and the consequence.

4. Compile and fill in the truth table of a complex judgment.

Example 1

"Insult can be inflicted accidentally or intentionally"

Solution:

a) "Insult can be inflicted by accident" - (R)

b) "Insult may be intentional" – (q)

2. Union " or» in the statement asserts the presence of only one of the two situations. The logical connection in this judgment is strong disjunction ().

3. The formula of a complex judgment: p q.

4. We build a truth table for the judgment of this form.

To build a truth table, you need to know the number of columns when entering the table (the number of variables) and the number of rows in the table ( x = 2n , where X - the number of rows in the table, n - the number of variables in the formula). This table has three columns ( R , q, p q) and four lines (2 2 = 4). In the first column we write down all the truth options for R (I and L). In the second column, against each of the values ​​​​of the first column, it fixes the values ​​\u200b\u200bfirst both times as AND, and then both times as L. Under the logical union sign, a strong disjunction () writes the final result, focusing on the truth table placed on page 3, fig. 20. The formula of this judgment is feasible, since it takes both the value of I and the value of L.

R	q	p q
And	And	L
L	And	And
And	L	And
L	L	L

The system for constructing truth tables for any number of propositional ones can be understood from the following considerations:

AT general case number of all possible sets of values n variables is 2n. For example, the number of valid interpretations for a single variable is 2 1 = 1 ; for two variables - 2 2 = 4 ; for three variables - 2 3 = 8; for four variables is 16 , for five - 32 etc.

For example, let the sequence of propositional variables р 1 , р 2 , …p n consists only of one variable ( n= 1). Then there is only two value set:<and > and<l >:

Let the sequence of propositional variables р 1 , р 2 , …p n comprises two variables ( n= 2). In this case, the sets of specified values ​​will be such pairs (there are four):

<and , and >, <l , and >, <and , l >, <l , l >.

If this sequence contains three variables, then the sets of such values ​​will be such combinations ( eight triplets):

<и, и, и>, <л, и, и>, <и, л, и>, <л, л, и>,

<и, и, л>, <л, и, л>, <и, л, л>, <л, л, л>

Formal logic uses the following propositional connectives: , ^, v, →, ↔, where

Symbol denial(additions);

^ - character conjunctions(associations);

v - symbol non-strict disjunction(separation-unification);

- symbol strict disjunction(separation-exclusions);

→ - character implications(logical consequence).

↔ - symbol equivalences(logical identity).

When denial(additions) statement ( BUT) takes the value "true" only if BUT false. And vice versa, if BUT true, then ( BUT)- false.

Example 2

"Turning your back on the most intriguing events of history, it is impossible to understand the logic of this story."

Solution:

1. Define and write down simple judgments:

a) "Man has turned his back on the most intriguing events in history" - R (base)

b) “A person cannot understand the logic of this story” - q (consequence)

2. Union " if, ... then ..." means that the situation expressed by the base ( "man has turned his back on the most intriguing events in history") is sufficient condition for the occurrence of the situation expressed by the consequence ( “a person cannot understand the logic of this story”). The logical connection in this judgment is implication (→ )

3. Judgment formula: p → q

4. We build a truth table for a judgment of this form (see p. 4, Fig. 21).

Under the sign of the logical union, the implication ( → ) we write down its truth values. The formula of this judgment is feasible, since it takes both the value of I and the value of L.

R	q	p → q
And	And	And
L	And	And
And	L	And
L	L	And

Example 3

“If a student is in this faculty, then he is capable or very diligent.”

Solution:

1. Define and write down simple judgments:

a) "The student is studying at this faculty" - R(base)

b) "This student is capable" - q(consequence)

in) "This student is diligent" - r(consequence)

2. Union " if..then.." means that the situation expressed by the reason ("the person studies at this faculty") is a sufficient condition for the occurrence of the situation expressed by the consequence ("he is capable or very diligent"). The logical connection in the judgment is the implication ( → ). As a result, there is a union “or” between the judgments, which means the statement of the presence of at least one of the two situations. Logical connection - weak disjunction (v).

3. Judgment formula: p → (q v r)

4. We build a truth table for the judgment of this form. The number of columns in the input to the table is three (the variables in the formula are 3), and the number of rows in the table is 8. In order to determine the truth values ​​of this formula, it is necessary to determine the procedure. The first step is to find the truth value of the weak disjunction (v), and then the truth value of the implication ( → ).

The truth values ​​of the implication ( → ) are the truth values ​​of the given formula. The formula of this judgment is feasible, since it takes both the value of I and the value of L.

Task 9. Determine the modality of the judgment, write judgments using modal operators:

Modality(from lat. modus - measure, method) is explicitly or implicitly expressed in the judgment judgment characteristic, additional information about the logical and actual status of the judgment, about its regulatory, evaluative, temporal and other characteristics, about the degree of its validity.

initial information in judgment express, as we already know, subject, predicate, quantifier word and mode of expression this information is the formula (S-P) .

Concerning additional information, it can be very different. So, for example, the logician of the middle of the XIII century. William Sherwood counted six species modal forms: true, false, Maybe, impossible, by chance and necessary. AT contemporary In logical thinking, modalities that appear under the names are used more often than others. alethic, deontic and epistemic.

The concept of "alethic"(from Greek aletheia - truth) means "true". Alethic modality in this sense is a relation to basic requirement of logic- to express criteria true and false statements.

Alethic modality is expressed in judgments and terms necessity-accident or possibilities-impossibility information about the features of the logical or actual determinism of judgments.

Asserting the existence of something, as true to reality , denoted symbolically as p.

Likewise, affirmation of the non-existence of something, as a negative reality , denoted by -ÿ ù p.

Example:

"Availability causation between the actions committed by this person and the resulting socially dangerous consequences ( p) is an indispensable condition for bringing him to criminal responsibility ( q)».

ÿ (p ® q).

As opposed to "necessity", "chance" is not associated with inevitability, but fixes only private events in their arbitrary occurrence and existence.

Example:

p) sometimes contributes to the occurrence of cardiovascular diseases ( q)».

In terms of alethic modality, this statement looks like this:

ù ÿ (p ® q).

As for the "possibility" of something, then she is always bound with the compatibility of the phenomenon under consideration with other phenomena, constituting for this phenomenon environment of his existence.

Example:

"Pollution environment (p) may contribute to the occurrence of cardiovascular and pulmonary diseases ( q)».

In terms of alethic modality, this statement looks like this:

à (p ® q).

In turn, the "impossibility" of something always tied With the incompatibility of a given phenomenon with others that are its environment for it.

2. Svintsov V.I. Logics. Elementary course for humanitarian specialties. - M.: Skorina, All world, 1998. - 351 p.

3. Oseledchik M.B. Logics. Program, seminar plans, assignments for control works, guidelines. For all specialties. - M.: Publishing House of MGUP, 2007. - 108 p.

Additional

1. Bryushinkin V.N. Logic: Textbook. 3rd ed., add. and correct. - M.: Gardariki, 2001. -334 p.

2. Getmanova A.D. Logic textbook. With a collection of tasks. - 7th ed., Sr. - M.: KNORUS, 2008. - 368 p.

3. Gorsky D.P. Definition. - M.: Thought, 1974.

4. Kirillov V.I., Orlov G.A., Fokina N.I. Exercises in Logic / Ed. V.I. Kirillov. - 4th ed., revised. and additional - M.: MTSUPL, 1999. - 160 p.

5. Malakhov V.P. formal logic. - Textbook. - M.: Academic Project, 2001. - 384 p.

6. Modern vocabulary by logic. - Mn .: " modern word", 1999. - 768 p.

7. Chueshov V.I. Fundamentals of Modern Logic: Study Guide / V.I. Chueshov. - Minsk: New knowledge, 2003. - 207 p.

1. Judgment - a form of thinking in which the connection between an object and its attribute or the relationship between objects is affirmed or denied, and which has the property of expressing either truth or falsehood. For example: "All pines are trees", "Some animals are not predators." If these judgments correspond to reality, then they are true, and if they do not correspond, then they are false.

It should be noted that any judgment is expressed in the form of a sentence, but not every sentence can express a judgment. Unlike declarative sentences, interrogative and exclamatory sentences do not affirm or deny anything, therefore they cannot express a judgment. Exceptions are rhetorical questions and exclamations, because in terms of meaning they affirm or deny something. For example, the famous saying: “And what Russian does not like to drive fast?” - is a rhetorical interrogative sentence (rhetorical question), since it states in the form of a question that every Russian loves fast driving.

As a more complex form of thinking (compared to a concept), a judgment has a specific structure in which four elements can be distinguished:

Subject (S) - what is being discussed in the judgment;

Predicate (P) - what is said about the subject;

A bunch (the words "is", "is") - that which connects the subject and the predicate;

The quantifier (the words "all", "some", "none") is a pointer to the volume of the subject.

Both the subject and the predicate in a judgment can be expressed in more than one word. The division of a judgment into S and P does not coincide with the division of a sentence into a subject and a predicate, since in logic we single out the elements of thought, and in grammar - the elements of its linguistic expression. In addition, grammar speaks of minor members sentences (addition, definition, circumstance), and logic is abstracted from all this.

The structure of thought is always simpler than the structure of the sentence that expresses it, because thoughts are approximately the same in their structure among all peoples, and their languages ​​differ greatly.

Depending on what is affirmed or denied in the judgment - the attribute belonging to the object or the relationship between objects, or the fact of the existence of objects - judgments are divided into three types:

Attribute judgments- these are judgments in which the predicate is some essential, integral feature of the subject. For example, the proposition: "All sparrows are birds" is attributive, because its predicate (to be a bird) is the main sign of a sparrow, its attribute.

Existential judgments- these are judgments in which the predicate indicates the existence or non-existence of the subject. For example, the proposition: “Perpetual motion machines do not exist” is existential, since its predicate (“does not exist”) indicates the non-existence of the subject (perpetual motion machine).

Relative judgments- these are judgments in which the predicate expresses some kind of relation to the subject. For example, the proposition: "Moscow was founded before St. Petersburg" is relative, because its predicate ("founded before St. Petersburg") indicates the age relationship between cities.

2. Simple judgment is a judgment with one subject and one predicate; a judgment in which there is only one semantic unit that has independent value truth, and which is divided only into concepts.

It is necessary to understand that all simple judgments in terms of the volume of the subject and the quality of the bundle are divided into four types. The volume of the subject can be general (“all”) and particular (“some”), and the connective can be affirmative (“is”) and negative (“is not”):

Each type of simple proposition has its own name and symbol:

- generally affirmative judgments(denoted by the Latin letter A) - these are judgments with the total volume of the subject and an affirmative link. His formula: "All S are P." For example: "All the students in our group study logic."

- private affirmative judgments (I)- these are judgments with a particular volume of the subject and an affirmative link: "Some S are P". For example: "Some students are excellent students."

- general negative judgments (E)- these are judgments with the general volume of the subject and a negative link: “All S is not P (or “Not a single S is P”). For example: "All planets are not stars" ("No planet is a star").

- partial negative judgments (O)- these are judgments with a particular volume of the subject and a negative link: "Some S are not P". For example: "Some mushrooms are not edible."

Please note that judgments in which the subject is a single concept are considered general (general affirmative or generally negative) judgments, since they are talking about the entire scope of the subject. For example: "The sun is a celestial body" or "Antarctica is one of the continents of the Earth."

In the future, we will talk about the types of simple judgments, without using their long names, with the help of conventional symbols - Latin letters A, I, E, O.

There is also an additional classification of judgments:

Emphasizing judgments, in which the belonging or absence of a feature is expressed only in a given object. For example, "Only witnesses, and only they, appear in the people's court on a summons." Such judgments can be single, private and general.

Exclusive judgments, in which the belonging or absence of a sign is expressed for all objects, with the exception of their part. For example, "All citizens have legal capacity and legal capacity, except as provided by law."

Modal judgments- these are judgments in which additional information is given about the type of dependence between the subject and the predicate.

Modality is expressed in terms of: perhaps, accidental, necessary, provable, refutable, problematic, obligatory, soluble, forbidden, good, better, bad, worse; believe that; I know that; it will be so that; it has always been like that, etc. Modality is also derived from the context or guessed intuitively.

The subject and predicate of any judgment are called terms of judgment. They always represent some kind of concepts, the volumes of which, as we already know, can be in various relationships with each other and depicted using Euler circles.

If the judgment refers to all objects included in the scope of the term (that is, the subject or predicate), then this term is called distributed (taken in full). A distributed term is denoted by a “+” sign, and on Euler diagrams it is depicted as a full circle (a circle that does not contain another circle and does not intersect with another circle).

The term is called undistributed(not taken in full) if the judgment is not about all the objects included in the scope of this term. An undistributed term is denoted by a “-” sign, and on Euler diagrams it is depicted as an incomplete circle (a circle that contains another circle or intersects with another circle). For example, in the judgment "All sharks (S) are predators (P)" we are talking about all sharks, which means that the subject of this judgment is distributed. However, in this judgment, we are not talking about all predators, but only about a part of predators (namely, those that are sharks), therefore, the predicate of this judgment is undistributed. Draw the relationship between the volumes of the subject and the predicate with circles and you will see that the distributed term (the subject "sharks") corresponds to a full circle, and the undistributed one (the predicate "predators") corresponds to an incomplete one (the circle of the subject falling into it, as it were, cuts out some part of it) .

Please note that the distribution of terms in simple judgments may be different depending on the type of judgment. The subject is always distributed in judgments of the form A and E and is always undistributed in judgments of the form I and O, and the predicate is always distributed in judgments of the form E and O, but in judgments of the form A and I it can be either distributed or undistributed, depending on the nature of the relationship between him and the subject in these judgments.

It is not at all necessary to memorize all cases of distribution of terms in a judgment. It is enough to be able to determine the type of relationship between the subject and the predicate in the proposed judgment and depict them with circular diagrams. A full circle, as already mentioned, will correspond to a distributed term, and an incomplete one - to an undistributed one.

3. Relationships can be established between simple judgments. But it must be remembered that simple judgments are divided into comparable and incomparable. Relationships can only be established between comparable concepts.

Comparable judgments have the same subjects and predicates, but may differ in quantifiers and connectives. For example, the judgments: "All mushrooms are edible" and "Some mushrooms are not edible" are comparable judgments, since they have the same subjects and predicates, but the quantifiers and connectives are different.

Incomparable judgments have different subjects and predicates. For example, the judgments: “All mushrooms are edible” and “Some pies are edible” are incomparable, since their subjects do not coincide.

Comparable judgments are, like concepts, compatible and incompatible.

Compatible judgments are propositions that can be true at the same time. For example, the propositions "Some mushrooms are edible" and "Some mushrooms are not edible" are compatible propositions because they can both be true.

Incompatible judgments cannot be both true: the truth of one of them necessarily means the falsity of the other. For example, the judgments “All mushrooms are edible” and “Some mushrooms are not edible” are incompatible, since they cannot be true at the same time: the truth of the first judgment inevitably leads to the falsity of the second.

Compatible judgments can be in the relationship:

Equivalence (this is the relationship between two judgments, in which the subjects, predicates, connectives, and quantifiers are the same);

Subordinations (this is the relationship between two judgments in which predicates and connectives coincide, and the subjects are in relation to species and genus).

Partial coincidence (subcontrarality) is a relationship between two propositions in which subjects and predicates are the same, but the connectives are different. For example, the judgments "Some mushrooms are edible" and "Some mushrooms are not edible" are in a partial match relationship. It should be noted that in this respect there are only private judgments - (I) and (O).

Incompatible judgments can be in the relationship:

Opposites (contraries) is a relationship between two judgments in which subjects and predicates are the same, but the connectives are different. For example, the judgments "All mushrooms are edible" and "All mushrooms are not edible." It is important to emphasize that opposing propositions cannot be both true, but they can be both false.

Contradictions (contradictions) are a relationship between two judgments in which the predicates are the same, the connectives are different, and the subjects differ in their volumes. For example, the judgments "All mushrooms are edible" and "Some mushrooms are not edible." It should be noted that contradictory judgments cannot be both true and cannot be simultaneously false: the truth of one of them necessarily means the falsity of the other, and vice versa, the falsity of one determines the truth of the other.

The considered relations between simple comparable judgments are represented schematically with the help of a logical square. Look at the textbook, what is a logical square. The vertices of the square represent four types of simple propositions (A, I, E, O), while its sides and diagonals represent the relationships between them.

To establish a relationship between two judgments, it is enough to determine what kind each of them belongs to and see what connects them: the diagonal or which side of the square. For example, we need to find out in what relation are the propositions "All people studied logic" and "Some people did not study logic." Having determined that the first judgment is generally affirmative (A), and the second is particular negative (O), we see that they are connected in a square by a diagonal, which means a contradiction relation.

It must also be borne in mind that the truth values ​​of each of the comparable propositions are in some way related to the truth values ​​of the others. Thus, if a proposition of the form A is true or false, then three other comparable propositions (I, E, O) will also be true or false. For example, if the proposition A "All tigers are predators" is true, then the proposition I "Some tigers are predators" is also true, and the proposition E "All tigers are not predators" and the proposition O "Some tigers are not are predators" will be false.

4. Depending on the union with which simple judgments are combined into complex ones, five types of complex judgments are distinguished:

- conjunctive judgment (conjunction). It may consist of two or more simple propositions. For example, the judgment "Lightning flashed, and thunder rumbled, and it began to rain." Its formula is: (), where a, b, c are simple judgments, and the symbol "definition"> disjunctive judgment (disjunction) can be strict and non-strict and consist of two or more simple judgments.

Formula non-strict disjunction: formula" src="http://hi-edu.ru/e-books/xbook912/files/f3.gif" border="0" align="absmiddle" alt="(!LANG:"Denotes divisive unions" or "," either "," or "in a non-exclusive (connective-separative) meaning. An example of such a judgment would be: "He is learning English, or he is learning German." These two simple judgments do not exclude each other, because it is possible to study both English and German at the same time.

Strict disjunction formula: formula"Denotes dividing unions" or "," either "," either "in the exclusive (separating) meaning. An example of such a judgment would be: "He studies English language or he doesn't study English." These two simple judgments exclude each other, because it is impossible to do and not to do the same thing at the same time.

- implicative judgment (implication) always consists of a reason and a consequence following from it. For example, the proposition "If a substance is a metal, then it is electrically conductive"..gif" border="0" align="absmiddle" alt="(!LANG:” denotes conditional conjunctions “if ... then”, “when ... then”. Note that the base and the consequence cannot be interchanged.

- equivalent judgment (equivalence) consists of two equivalent (identical) judgments, therefore, in it, unlike an implication, there can be neither a foundation nor a consequence. For example, the judgment "If a number is even, then it is divisible by 2"..gif" border="0" align="absmiddle" alt="(!LANG:” denotes unions “if and only if ... then”, “when and only when ... then”. It is easy to see that the simple propositions "The number is even" and "The number is divisible by 2 without a remainder" are connected in such a way that the second follows from the first, and the first follows from the second.

- negative judgment (denial) is a complex proposition with the union "it is not true that ...", which is indicated by the symbol "formula" src="http://hi-edu.ru/e-books/xbook912/files/f11.gif" border="0" align ="absmiddle" alt="(!LANG:a, where a is a simple judgment (some kind of statement), and the sign "example"> a simple negative judgment. For example, "The Earth is not a ball." If the negation outwardly joins the judgment (“It is not true that the earth is a sphere”), then such a negation is considered as a logical connective that transforms a simple judgment into a complex one.

Any complex proposition is true or false, depending on the truth or falsity of the simple propositions included in it. Study the textbook truth table of all types of complex propositions, depending on all possible sets of truth values ​​of the two simple ones included in them.

In order to determine the truth of a complex proposition using a truth table, it must be formalized. This means discarding its content and leaving only its logical form, expressing it with the help of the already known conventional symbols of conjunction, non-strict and strict disjunction, implication, equivalence and negation.

For example, in order to formalize the following statement: "V.V. Mayakovsky was born in 1891 or in 1893. However, it is known that he was not born in 1891. Therefore, he was born in 1893," one must first select those included in him simple judgments and establish the kind of logical connection between them. The above statement includes two simple judgments: “V.V. Mayakovsky was born in 1891”, “V.V. Mayakovsky was born in 1893.gif" border="0" align="absmiddle" alt="(!LANG:. And, finally, the statement of the second simple proposition follows from this conjunction ("he was born in 1893"), and the implication is obtained: ">

- identically true formulas, which are true for all sets of truth values ​​of their simple propositions. Any identically true formula is a logical law.

- identical false formulas, which are false for all sets of truth values ​​of the variables included in them (simple propositions). They represent a violation of logical laws.

- satisfiable (neutral) formulas for different sets of truth values ​​of the variables included in them are either true or false.

If as a result of the formalization of any reasoning, an identically true formula is obtained, then such reasoning is logically flawless. If the result of formalization is an identically false formula, then the reasoning should be recognized as logically incorrect (erroneous). A feasible (neutral) formula testifies to the logical correctness of the reasoning, of which it is a formalization.

Now let's make a truth table for the formula defined "> 2n, where n is the number of variables (simple statements) in the formula. Since there are only two variables in our formula, the table should have four rows. The number of columns in the table is equal to the sum of the number of variables and the number of logical unions included in the formula..gif" border="0" align="absmiddle" alt="(!LANG:a. The fifth column is the truth values ​​of the conjunction consisting of the above strict disjunction and negation, and finally the sixth column is the truth values ​​of the entire formula or implication. The formula under consideration here takes on the value “true” for all sets of truth values ​​of the variables included in it, therefore, it is identically true, and the complex judgment, the formalization of which it acts, is logically flawless.

To perform exercises on the topic "Judgment", you should use the following algorithm:

1) Determine the type of the analyzed language expression, whether it is an interrogative, imperative or declarative sentence.

2) If the sentence is narrative or is a rhetorical question, exclamation, then it contains a judgment. Determine if the proposition is simple or complex.

3) If the proposition is simple, determine whether it is existential, relational, or attributive.

4) If the judgment is attributive, determine its type according to the combined classification by quality and quantity (particularly affirmative, partly negative, generally affirmative, general negative).

5) Indicate whether it is selective or exclusive.

6) Determine the modality of judgment.

7) Select the terms (subject and predicate) of the judgment and determine their distribution in the judgment.

8) If the judgment is complex, determine the simple judgments included in it and the types of logical connectives connecting them.

9) identify the logical form of the judgment, writing it in the form of an appropriate formula.

10) Check the logical correctness of a complex judgment by constructing a truth table.

1. Determine which of the following sentences are judgments:

1) “How I want to sleep!”; 2) “Sleep!”; 3) "I want to sleep"; 4) "What time is it?"; 5) "The Universe is infinite"; 6) "It will never happen!"; 7) When will this day come?

2. Determine the quality and quantity of the following judgments. Bring these judgments to one of four forms - A, E, I or O.

1) Proper names are capitalized.

2) Words can be divided into syllables.

3) The remaining syllables are called unstressed.

4) Some contemporaries of dinosaurs have not died out so far.

5) Nobody understood him.

6) In Russian, not all words have stress.

7) Alone in the field is not a warrior.

3. Establish the distribution of the subject and the predicate in the following judgments and depict the relationship between them using Euler circles.

1) A cloud covers the sky with darkness.

2) Not all students are excellent students.

3) Not a single ostrich flies.

4) Many people don't speak English.

5) Every hunter wants to know where the pheasant is sitting.

4. Determine the relationship between the following judgments:

1) All whales breathe with lungs. Some whales do not breathe with lungs.

2) Some animals are invertebrates. Some animals are not vertebrates.

3) No man is immortal. Some people are not immortal.

4) Some people like to dance. Some people love to sing.

5) Everyone wants to be happy. Some people don't want to be happy.

5. Write down the following complex judgments in the language of propositional logic:

1) If a given geometric figure has all right angles and equal sides, then it is a square.

2) This year there are a lot of mushrooms in the forest: aspen mushrooms, russula, porcini, saffron mushrooms.

3) When the political process develops in the direction of satisfying the interests of either one or the other group, or increasing the welfare of both of them together, then eventually the limits of the possible are reached.

6. Indicate in which examples the union “or” is given the meaning of a weak disjunction, and in which strict.

1) Petrov is an athlete or a student.

2) Petrov is guilty or innocent.

3) This dish is tasty or sweet.

4) He will listen to music or dance.

5) He will work or rest.

Lesson summary

Topic: "Biospheric level of wildlife organization"

Biology

Grade 10

Foundation Level Program for Educational Institutions

Textbook Ponomareva I.N., Kornilova O.A., Loshchilina T.E., Izhevsky P.V. General biology

Teacher Sudneva T.Yu.

Lesson - a generalization of the studied material.

Target: summarize information about the global ecosystem of the Earth - the biosphere, the features of the biospheric level of organization of living matter and its role in ensuring life on Earth;

Tasks:

Check the ability to apply the acquired knowledge about the biospheric level of the organization to substantiate situations.

Continue the development of general educational skills (highlight the main thing, establish cause-and-effect relationships, work with diagrams, establish the correctness of the judgments made and the sequence of objects and phenomena);

To form a cognitive interest in the subject, develop communication skills and the ability to work in groups;

Equipment: table "Biosphere and its boundaries", tasks on cards for each table out of four, answer sheets, hours, table numbering.

During the classes:

Organizing time.

Mark missing, determine the objectives of the lesson.

Generalization and systematization of knowledge

Conversation on:

Name the levels of organization of living matter, starting with the smallest.

What level have we studied?

What is the biosphere?

Where are the boundaries of the biosphere and how are they determined?

Prove that the biosphere is a biosystem.

What important global processes take place at the biospheric level?

What is the significance of the biological diversity of its living matter for the biosphere?

What is the basic strategy of life at the biospheric level of organization?

Student responses:

Levels of organization of living matter: molecular, cellular, organismal, population-species, biogeocenotic, biospheric.

The biosphere occupies the lower layers of the atmosphere 15 km (to the ozone screen), the entire hydrosphere and the upper layers of the lithosphere 3-4 km.

The biosphere consists of biogeocenoses in which living organisms are interconnected with each other and with the surrounding gray area.

At the biospheric level, very important global processes take place that ensure the possibility of the existence of life on Earth: the formation of oxygen, the absorption and conversion of solar energy, the maintenance of a constant gas composition, the implementation of biochemical cycles and energy flow, the development of biological diversity of species and ecosystems.

The diversity of life forms on Earth ensures the stability of the biosphere, its integrity and unity.

The main strategy of life at the biospheric level is the preservation of the diversity of forms of living matter and the infinity of life, ensuring the dynamic stability of the biosphere.

Knowledge control.

Students are invited to test their knowledge and skills in this section in the form of a game - "turntables". Students are divided into five working groups at five desktops. On the desktops there are tasks for: determining the correctness of the proposed judgments, determining compliance, defining concepts, determining the correct sequence and establishing cause-and-effect relationships. Tasks are divided into 4-5 options. The time for completing tasks is 5 minutes. At the end of the next task, the students change the desktop and choose a new task of a certain variant, indicating it in the answer sheet, which the teacher distributes to the students in advance, in accordance with the sequence of work. During the lesson, students must complete multi-level tasks for five different options (see Appendix).

At the end of the work, students hand over the answer sheets to the teacher.

Application:

I Task: write down the numbers of correct judgments

Option 1

1. 10% of energy goes to each subsequent food level

2. The relief refers to abiotic environmental factors

3. The exhaustible resources of the biosphere include atmospheric air

4. The living matter of the biosphere includes the remains of organisms at different stages of decomposition

5. The study of the laws of life is engaged in general biology

6. Consumers of the second order are herbivores

7. Plants need solar energy to form organic matter.

8. The signal for seasonal changes for plants is temperature.

9. Nitrogen-fixing bacteria are chemosynthetic organisms

10. The biosphere is the shell of the Earth inhabited by living organisms

Option 2

Decide if the sentences given are correct.

1. The ozone shield protects the biosphere from the harmful ultraviolet radiation of the Sun

2. The founder of the doctrine of the biosphere is V.I. Vernadsky

3. The inexhaustible resources of the biosphere include the energy of ebbs and flows

4. Second-order consumers include herbivores

5. The length of the food chain is limited by the loss of energy at each trophic level.

6. Biotic factors include competition

7. Temperature is the limiting factor in the desert.

8. Consumers decompose organic residues to inorganic compounds

9. The biosphere is the part of the Earth where life exists

10. Boron belongs to the universal biogenic elements of the biosphere.

Option 3

Decide if the sentences given are correct.

1. Air refers to biotic environmental factors

2. The sustainability of the biosphere is ensured by human economic activity

3. The inexhaustible resources of the biosphere include flora and fauna

4. Producers include plants that carry out photosynthesis

5. The true decomposers of the biosphere are fungi and bacteria

6. The energy coming from the Sun is spent on the synthesis of organic substances

7. Biological evolution is an important stage in the chemical evolution of the planet

8. The outer hard shell of the globe, bordering the biosphere, is called the mantle

9. Reproduction of organisms determines the pressure and density of life

10. The noosphere is the "intelligent shell" of the Earth

Option 4

Decide if the sentences given are correct.

1. The length of daylight plays a leading role in seasonal changes in plants and animals

2. Plants need thermal energy to form organic matter.

3. Pollination of plants by insects - there is a biotic factor

4. The stability of the biosphere is determined by the constancy of the influx of solar energy

5. Fungi and microorganisms are consumers

6. Nutrients make a continuous cycle in the biosphere

7. The stability of the biosphere is associated with the diversity of living matter

8. The biosphere is one of the global ecosystems

9. The term "biosphere" was introduced into science by V.I. Vernadsky

10. The appearance of oxygen was the most important step in the evolution of the biosphere

II Task: match.

Option 1

Distribute aromorphoses

1. The emergence of a flower and fruit A. Psilophyta

2. The appearance of integumentary, conductive and B. Moss mechanical tissue

3. The emergence of seeds V. Ferns

4. The appearance of the root system G. Conifers

5. The appearance of the stem and leaves D. Flowering

Option 2

Distribute aromorphoses

1. The appearance of protective shells in the egg and an increase in the supply of nutrients A. Lancelet

2. Occurrence of pulmonary respiration B. Pisces

3. The appearance of the chord V. Amphibians

4. Appearance of horny covers G. Reptiles

5. Appearance of bone jaws E. Birds

Option 3

Distribute aromorphoses

1. The emergence of a five-fingered limb A. Worms

2. Appearance of internal fertilization B. Amphibians

3. Closed circulatory system B. Chordates

4. Live birth D. Mammals

5. The emergence of the internal skeleton of D. Reptiles

Option 4

Distribute aromorphoses

1. The appearance of the spine and skull A. Birds

2. The appearance of warm-blooded B. Amphibians

3. The emergence of a three-chambered heart V. Pisces

4. Dismemberment of the body into segments G. Worms

5. The emergence of a chitinous cover D. Insects

III task: express your opinion.

Option 1

Your opinion

Explain how you understand the meaning of V.I. Vernadsky: "On the earth's surface there is no chemical force more constantly acting, and therefore more powerful in its final consequences, than living organisms taken together"

Option 2

Your opinion

How do you understand the meaning of the words of V.I. Vernadsky: “Man and his activity on the planet today have become a powerful geological force, therefore, must be considered in the biological aspect”

Option 3

Your opinion

There is a famous expression: “We did not inherit the Earth from our parents. We borrowed it from our children." What do these words mean?

Option 4

Your opinion

Is Vernadsky's statement correct: “Life is not an external, random phenomenon on the earth's surface. It is closely connected with the structure of the earth's crust, enters into its mechanism and performs the most important functions, without which it could not exist"?

IV Task: define the concepts

Option 1

Define concepts

Biotic factors, living matter, abiogenesis

Option 2

Define concepts

Heterotrophs, biosphere, energy flow

Option 3

Define concepts

Prokaryotes, chemical evolution, bioinert substance

Option 4

Define concepts

Biogenesis, cycling, eukaryotes

V Exercise:

Option 1

Determine the correct sequence

A) Fish → Reptiles → Birds → Mammals

B) Fish→Amphibians→Reptiles→Birds→Mammals

C) Fish → Amphibians → Reptiles → Mammals

Option 2

Determine the correct sequence

A) Mosses → Algae → Ferns → Angiosperms

B) Algae → Ferns → Mosses → Angiosperms → Gymnosperms

Option 3

Determine the correct sequence

A) Paleozoic → Proterozoic → Mesozoic → Archean → Cenozoic

B) Archean → Proterozoic → Paleozoic → Mesozoic → Cenozoic

C) Cenozoic → Mesozoic → Archean → Paleozoic → Proterozoic

Option 4

Determine the correct sequence

A) Carbon dioxide → Plants → Phytophages → Predators → Decomposers

B) Plants → Phytophages → Predators → Carbon dioxide → Decomposers

C) Carbon dioxide → Predators → Plants → Decomposers → Phytophages

Answer sheet.

FI _____________________

№ option _______________

I .Exercise

Numbers of correct judgments: _______________________________

II. Exercise

III . Exercise

Your opinion:

IV . Exercise

Write down the definition of the concepts:

V . Exercise

Write down the number of the correct sequence ____________