LECTURE No. 12. Complex judgments. Formation of complex judgments

1. The concept of complex judgments

The concept of complex judgments is inextricably linked with conjunction, disjunction, implication, equivalence and negation.

These are the so-called logical links. They are used as a unifying link, linking one simple proposition to another. This is how complex sentences are formed. That is complex judgments are judgments created from two simple ones.

The ratio of the truth of judgments is displayed in the tables. These tables reflect all possible cases of truth and falsity of judgments, and each of the simple judgments, which is part of the complex one, is reflected in the "cap" of the table as a letter (for example, a, b). Truth or falsity is reflected in the form of the letters "I" or "L" (true and false, respectively).

Before considering conjunction, disjunction, implication, equivalence and negation, it makes sense to give them a brief description. These logical connectives are called logical constants.

In the literature, you can find their other name - logical constants, but this does not change their essence. In our language, these constants are expressed in certain words. So, the conjunction is expressed by the unions "yes", "but", "although", "but", "and" and others, and the disjunction is expressed by the unions "or", "or", etc. We can talk about the truth of the conjunction if both simple propositions included in it are true. A disjunction is true when only one simple proposition is true. This refers to a strict disjunction, while a non-strict disjunction is true provided that at least one of its constituent simple judgments is true. An implication is always true except in one case.

Let's consider the above in more detail.

conjunction (a^{^}b) - this is a way of linking simple judgments into complex ones, in which the truth of the resulting judgment directly depends on the truth of the composite ones. The truth of such judgments is achieved only when both simple judgments (both a and b) are also true. If at least one of these judgments is false, then the new, complex judgment formed from them should also be recognized as false. For example, in the judgment "This car is of very high quality (a) and has only run ten thousand meters (b)", the truth depends on both its right side and its left side. If both simple propositions are true, then the complex one formed from them is also true. Otherwise (if at least one of the simple propositions is false), it is false. This judgment is a characteristic of a particular car. The falsity of one of the simple propositions, obviously, does not exclude the truth of the other, and this can lead to errors associated with determining the truth of complex propositions formed with the help of a conjunction. Of course, the truth of one simple proposition is not excluded by the falsity of another, but we should not forget that we are characterizing an object, and from this point of view, the falsity of one of the simple propositions is considered from the other side. This is due to the fact that with the falsity of the judgment on one of the points of this characteristic, the characteristic as a whole becomes false (in other words, it leads to the transmission of incorrect information about the machine as a whole).

Disjunction (aVb) is strict and non-strict. The difference between these two types of disjunction is that in a non-strict form its members are not mutually exclusive. An example of a non-strict disjunction might be: "To obtain a workpiece, the part can be finished on the machine (a) or pre-processed with a file (b)". Obviously, here a does not exclude b and vice versa. The truth of such a complex judgment depends on the truth of its members in the following way: if both members are false, the disjunctive judgment formed through them is also considered false. However, if only one simple proposition is false, such a disjunction is recognized as true.

Strict disjunction is characterized by the fact that its members exclude each other (in contrast to non-strict disjunction). The judgment "Today I will do my homework (a) or go for a walk outside (b)" is an example of a strong disjunction. Indeed, you can do only one thing at the moment - do your homework or go for a walk, leaving the lessons for later. Therefore, a strict disjunction is true only when only one of the simple propositions included in it is true. This is the only case in which a strict disjunction is true.

Equivalent It is characterized by the fact that an educated complex proposition is true only in those cases when both simple propositions that make up its composition are true, and false if both of these propositions are false. In literal terms, equivalence looks like a = b.

When negating the proposition, displayed as a, is true when the concept is falsely negated. This is due to the fact that negation and the negated simple proposition not only contradict, but also exclude (deny) each other. Thus, it turns out that when the concept a is true, the concept a is false. Conversely, if a is false, then a negating it is true.

Implication (a - › b) true in all cases except one. In other words, if both simple propositions in the implication are true or false, or if proposition a is false, then the implication is true. However, when the proposition b is false, the implication itself becomes false. This can be seen with an example: "We will throw a working cartridge into the fire (a), it will explode (b)". Obviously, if the first judgment is true, then the second is also true, since the explosion of a cartridge thrown into a fire will inevitably occur. Therefore, considering the first case, we can conclude that if the second proposition is false, then the whole implication is false.

All the above examples of conjunction, disjunction, implication consisted of two variables. However, this is not always the case. There may be three or more variables. Considering complex judgments for truth, we get literal formulas. The latter can be characterized as true or false. In this regard, a formula is called identically true if it is true for any combination of its variables. The name identically false has a formula that takes only a false value (the value "false"). The last kind of such formulas is the satisfiable formula. Depending on the combinations of variables included in it, it can take both the value "true" and the value "false".

2. Expression of statements

Sentences are expressed using symbols. - variables and signs denoting logical terms. There are no other symbols for this purpose.

Variable statements are expressed as letters of the Latin alphabet (a, b, c, d, etc.). Such letters are called variable statements, as well as propositional variables. In simple terms, this group of symbols refers to simple judgments that make up a statement. These judgments are expressed in the form of narrative sentences.

Another group of characters, used to express statements in the form of formulas, these are signs. They represent logical terms such as conjunction and disjunction, which can be strict or non-strict, negation, equivalence and implication. A conjunction is displayed as an upward tick (^) and a disjunction as a downward tick (V). For a strict disjunction, a dot is placed above the checkbox. The implication has the sign "-›", negation (-), equivalence (=).

The last type of symbols with which statements are expressed are parentheses.

Symbols denoting logical terms and types of connectives are characterized by different strengths. Thus, the ligament ^ is considered the strongest, that is, it binds stronger than all the others. The V ligament is stronger than the -, which is only important in some cases. Thus, determining the strength of connectives becomes important when writing formulas without using parentheses. If we have a statement expressed by the formula (a^b)Vc, you don’t have to write parentheses, but directly indicate that a^bVc. The same rule applies when using the symbol - ›. However, this rule is not true in all cases. That is, in many cases it is unacceptable to omit parentheses. For example, when the conjunctive connective of the concept a is carried out with two other concepts connected by the relation of implication and separated by parentheses, it is unacceptable to omit the latter (a^(b - c)). This is obvious, since otherwise it would be necessary to first carry out the conjunction and only then the implication. From a school mathematics course we know that it is impossible to omit parentheses in such a case. The following example can illustrate such a situation: 2 X (2 + 3) = 10 и 2 X 2 + 3 = 7. The result is obvious.

In connection with the above, it can be noted that not every symbolic expression of statements is a formula. This requires the presence of certain signs. For example, the formula must be constructed correctly. Examples of such a construction could be: (a^b), (aVb), (a - b), (a = b). This construction is noted as a PPF, i.e. a correctly constructed formula. Examples of incorrectly constructed formulas could be: a^b,aVb, Vb,a - b, (a^b) etc. In the first three cases, the incorrectness of the formula lies in the fact that the concepts united by sheaves must be enclosed in brackets. The last formula has an open bracket, while the third example is characterized by the fact that one simple concept is not combined with another, despite the fact that there is a disjunction symbol.

In our daily life, we often, sometimes without noticing it, use not only simple, but also complex judgments. Such judgments, as already mentioned above, are formed from two or more simple judgments with the help of logical connectives, which are called disjunction, conjunction, implication and negation, as well as equivalence. These links are expressed using signs: ^ for the conjunction V for disjunction, -> for implication. familiar = display equivalence, and the sign a means negation. There are two options for displaying the disjunction. The first one is a simple downward tick for a simple disjunction. For complex, the same checkmark is used, but with a dot on top. The graphic representation of the formulas of complex judgments is very important, as it allows you to more clearly understand their structure, nature and meaning.

Logical connectives unite simple propositions, which are essentially declarative sentences. And there are quite a lot of options here. Sentences can consist of nouns and adjectives, verbs, participles, etc. Some sentences are simple propositions, others are complex. Complex judgments or statements are characterized by the fact that they can be divided into two simple ones, united by a logical constant. However, this is not possible with all complex sentences. When, as a result of dismemberment, a statement changes its meaning, such an operation is unacceptable. For example, when we say “The area was old, and the houses in it had long since fallen into disrepair,” we mean a conjunction where one side, “the area was old,” is united by the conjunction “and” with the second part, “the houses in it have long since fallen into decay.” . The meaning of the statement has not changed, despite the fact that we examined simple propositions in isolation from each other. However, in the statement “There is a beautiful and fast car parked in the parking lot,” an attempt to separate will lead to distortion of the originally transmitted information. So, considering simple propositions separately, we get: “a beautiful (car) is parked in the parking lot” - this is the first proposition combined with the second conjunction “and”. The second proposition is: “(there is a) fast car parked in the parking lot.” As a result, you might think that there were two cars - one beautiful, the other fast.

Logic - this is, of course, an independent science, which has its own conceptual apparatus, tools, information base. Any independent science is separated from others and often radically differs in its approach to a particular subject. This should be kept in mind when we consider the constructions of the Russian language from the point of view of logic. Logic studies such constructions more in isolation. Thus, the time factor is often not taken into account when considering various judgments. In Russian, the time factor, in appropriate cases, is always taken into account. Here it should be said about the commutativity of the conjunction, which is inextricably linked with the above features of the language and logic.

Commutativity - this is the equivalence of judgments (statements), when (a^b) = (b^a). In language, the law of commutative conjunction does not apply, since the time factor is taken into account. Indeed, it is impossible to imagine the equivalence of certain judgments, one of which is earlier in time than the other, and vice versa. For example, the statement "It started to rain and we got wet" would not be equivalent.

(a^b) and "We got wet and it started to rain" (b^a). The same situation can be seen in the statements “A shot rang out, and the beast fell” and “The beast fell, and a shot rang out.” Obviously, the time factor is taken into account here, according to which one event or action, reflected in a complex judgment, precedes another, which determines the meaning of the entire statement.

Logic abstracts from time and evaluates the judgment only from the point of view of its correct construction, as well as truth or falsity. In this regard, the above statements are equivalent, since in each individual case, both parts of them are true.

In this way, Conjunctive statements in logic are commutative, the use of the conjunction “and” in judgments from the point of view of language (in the case when the time factor is taken into account) is non-commutative.

Despite the fact that the prepositions with which the conjunction is formed were indicated above, it cannot be said that in the absence of these prepositions in the judgment, the conjunction is impossible. This is not true. Often in sentences that are complex judgments, different punctuation marks are used as connectives. For example, it can be a comma or a dash, and sometimes a period.

Punctuation marks used in statements are placed between simple judgments and connect them with each other. An example of the use of punctuation marks as logical connectives is the sentence "The clouds parted, the sun came out" or "It was frosty outside, all living creatures hid, icicles formed on the roofs." In general, many scientists have dealt with the issues of linguistic expression of the conjunction. Therefore, this issue is well worked out and covered.

A disjunction (recall that its symbolic designation is V, as well as a similar tick, but with a dot at the top) can be strict or nonstrict. The differences between these two types, as already mentioned, lie in the fact that the terms of a non-strict disjunction exclude each other, while the members of a strict disjunction do not.

The law of commutativity with disjunction is valid regardless of what kind of disjunction is meant. Let's remember that disjunction is expressed by conjunctions, the main ones are definitely “or” and “either”. Let us give examples of strict and non-strict disjunction and use them to illustrate the operation of the law of commutativity. The proposition “I will drink sparkling water or still water” is an example of a weak disjunction, while the proposition “I will go to university or stay at home” is a strict one. The difference between them is that in the first case the action will still be performed, regardless of the selected type of water. In the second case, the action (I will go to university) is excluded if you choose the second option and stay at home. In many cases, the conjunction "or" can simply be replaced with the conjunction "or". For example, in the sentence “Either I ski down the mountain or fall along the way,” you can use the conjunction “or” without any changes. However, there is a conjunction that is used independently and is also a disjunctive connective. This is an “either or either” conjunction. It is quite often used when constructing sentences “Today either an auditor or an auditor came”; “He lives either on Moskovskaya or Komsomolskaya street,” etc.

As mentioned above, the law of commutativity in disjunctive statements operates regardless of the type of disjunction. Take for example the following proposition: “I will drink water with or without gas” and “I will drink water without or with gas.” Obviously, there is no difference between them, the meaning remains the same. You can also check other examples, say, “I will go to university or stay at home” and “I will stay at home or go to university.” The content and scope of a complex judgment formed using a disjunction do not change from rearranging its members. That is why we talk about universal commutativity.

The expression of logical connectives in the language is very diverse, there are many schemes according to which statements are built. For each of these schemes, you can build a huge number of complex judgments. This is especially characteristic of the Russian language in all its ambiguity. For example, the implication is built according to such schemes as, for example, "A needs B"; "A is enough for B"; "if A, then B", "A, only if B", etc. For example: "In order to know a lot, you need to study a lot"; "To jump from a tower, it is enough to push off with your feet correctly"; "If the car gets stuck, it will have to be pushed"; "You can turn in your session on time only if you start preparing immediately."

A number of formulas exist for equivalence: "A if B, and B if A"; "for A, B is necessary and sufficient"; "And if and only if B", etc. Let us give examples of judgments built on the basis of these schemes. For example: "If a person is engaged in weightlifting, he will become stronger" and "A person will become stronger if he is engaged in weightlifting"; "To enter a university it is necessary and sufficient to pass the entrance exams"; "You have reached the summit when and only when you have set foot on the highest point of the mountain."

In this regard, it is also necessary to mention the ambiguity of conjunctions expressing logical constants (conjunction, disjunction, implication, etc.). For example, the union "if" can often express not an implication, but a conjunction. It depends on the existence of a meaningful connection between judgments. In this regard, it is necessary to consider natural language expressions from the standpoint of their diversity and heterogeneity.

In addition to logical connectives, expressed in the Russian language through conjunctions that are used in the formation of general and particular judgments, there are quantifiers. These are the existential quantifier and the general quantifier.

General quantifier expressed in Russian by the words "each", "any", "all", "none", etc. Usually a formula with a general quantifier is read as "all objects have a certain property".

Existence quantifier expressed by the words "majority", "minority", "some", "many" and "few", "many" and "few", "almost all", etc. This quantifier is expressed as "there are some objects that have a certain property". There is a variant of the existential quantifier in which "there are some objects that are greater than a certain value". In this construction, objects are understood as numbers.

Some judgments constructed using implication are expressed in the subjunctive mood. They have the same formula as other implications (a - › b), but they are usually called counterfactuals. The subjunctive mood makes us understand that the basis and consequence of such judgments are false. However, this falsity is not universal, i.e., under certain circumstances, the truth of such statements is possible. In other words, such judgments can reflect the subject matter correctly and objectively.

Truth is possible if the relationship between reason and effect implies that the truth of the effect follows from the truth of the reason. Otherwise, we can state the falsity of such a judgment.

A statement constructed in the subjunctive mood has the structure "if A, then it would be B". For example, "If you went to all classes in logic, you would successfully pass the exam"; "If the train had not been late, we would have missed the train" and "If the patient had not fallen, his leg would not have hurt."

Counterfactual statements are of great importance for history, philosophy, to a certain extent mathematics and some other sciences. They are used in constructing hypotheses, considering historical and other issues, and determining possible directions for certain processes. For example, discussions on the topic of the Great Patriotic War are still ongoing. As part of this discussion, the question of the possibilities of its alternative course and the results that could have occurred under a different set of circumstances is considered. Also, within the framework of chemistry, physics, and astronomy, counterfactual judgments are often used. For example, practical physics sometimes comes to the conclusion that it is not possible to theoretically determine the exact course of a process. In this case, to achieve the desired result, you have to use the intellectual search method and confirm the results with practice.

The following statement may be an example of a counterfactual statement in physics: "If we pass an electric current through a copper conductor, then the discharge will be stronger." Since the truth of a counterfactual judgment is ambiguous, and by default both its basis and its consequence (and, accordingly, the entire judgment as a whole) are recognized as false, this judgment has to be verified in practice. In this case, the proposition can be either true or false. It depends on which conductor we used earlier. For example, if we took an iron conductor before a copper one, our judgment will be true, since copper gives less resistance when moving along an electric current conductor. However, if we previously used gold as a conductor, the judgment will turn out to be false, again for a reason related to the conductivity of materials - gold has a conductivity much greater than copper. Astronomy calls into question some properties of the orbits of celestial bodies and the features of the movement of the latter, the relative position of planets, stars, systems and galaxies, etc. As a result, counterfactual statements are also used. Sometimes, in order to justify themselves or to smooth over an acute situation, people say: "If this had not happened, then everything would have gone differently." This is also an example of using the subjunctive mood.

However, it should be remembered that counterfactual propositions consist of false reasons and consequences. Therefore, when using such constructions in science, a certain amount of caution must be observed.

Counterfactual propositions can be expressed using formulas. Such formulas reflect the number of terms of the statement, the type of connective between them and the sign of the implication. The implication in a counterfactual judgment has a certain specificity: it corresponds, among other things, to the conjunction “if... then”. On the left in such a formula are reflected the members of the counterfactual statement corresponding to the conjunction “if”, on the right - the conjunction “then”. The left and right sides are separated by an implication sign, different from that used in classical propositional logic. The difference between these two symbols is that on the back side of the arrow indicating the implication (classic version (-›)), in the counterfactual implication there is a vertical bar (| - ›). Such a sign is not used in classical propositional logic.

3. Denial of complex judgments

Negation of judgment in logic - this is the replacement of an existing bundle within a complex statement with another, opposite to the last one. If we are talking about a formula in which the negation of complex judgments can be expressed, then it should be noted that the negation is graphically expressed as a horizontal line above the negated judgment. Thus, we get two concepts, united by a logical link, over which a horizontal line is drawn. If such a feature already exists, then in order to implement negation it is necessary to remove such a feature.

All of the above applies to operations performed using conjunction and disjunction. However, what has been said above does not mean that the denial of complex judgments is possible only if they contain exclusively conjunctions of conjunction and disjunction. If it is necessary to carry out the operation of negation in relation to a judgment containing an implication, it is necessary to replace this judgment in such a way that, in the absence of any of its changes, the implication is discarded. This means that it is necessary to choose a judgment equivalent to the given one, which would not contain an implication. When we speak of a judgment that is equivalent to one containing an implication, but not containing it, we mean replacing this connective with a conjunction or disjunction. Graphically, this looks like (a - b) = (a V b). Then the operation described above is performed, in which the sign of conjunction is changed to disjunction, and vice versa.

Usually in speech the expression of negation comes down to adding the prefix “not”. Indeed, since the specified prefix is ​​negative, its use to establish the opposite is completely justified.

It is necessary to mention the laws of de Morgan. They are used in the process of negating complex judgments and have a formulaic expression.

There are only four such laws and, accordingly, formulas:

one) _________

a^b = aVb;

2) _____

a ^{^} b = a V b;

one) _________

a Vb = a ^{^} b;

4) _____

a Vb = a ^{^} b.

Having considered the above, it can be noted that the negation of a complex proposition, which contains a conjunction or disjunction, is a "simple" option, in which it is sufficient to carry out the negation operation.

The formula formed using de Morgan's laws is as follows:

(a ^{^} b) V (c ^{^} e) = (a V b) ^{^} (c V e).

Let us give examples of the negation operation. Negation of a complex proposition, in which there is no implication: "I will finish work and go for a walk and go to the store" - "I will finish work, but I will not go for a walk and will not go to the store." The denial of a complex proposition, in which it is necessary to first change the implication to a conjunction or disjunction, can be illustrated by the following example: "If I buy a car, I will go out of town or turn to the dacha" - "I will buy a car, but I will not go out of town and will not turn into dacha". In this example, for convenience, we have omitted the implication elimination step.

It must be said that judgments that negate each other cannot be both true and false at the same time. The situation of contradiction or negation is characterized by the fact that one of the contradictory concepts is always true, while the other is false. There can be no other position in this case.

It is impossible to identify the operation of negation, as a result of which a new judgment is formed, from the negation, which is a part of negative judgments. The negation of judgments can be made both in relation to the entire judgment and its parts and is expressed by the words “is not”, “is not the essence”, “is not”, as well as “wrong”, etc. Based on the foregoing, we can conclude that there are two types of denial - internal and external. As you might guess, the external denies the entire judgment as a whole. For example, "Some soldiers are not paratroopers" is an internal negation, while "It is not true that the Moon is a planet" is an external negation. Thus, the external negation is the negation of the entire judgment as a whole, while the internal one shows the fact of a contradiction or inconsistency between the predicate and the subject.

The following types of negative judgments can be displayed in the form of formulas: "all S are P" and "some S are not P" (these are general judgments); "no S is P" and "some S are P" (private judgments). The last kind of negative propositions is "this S is P" and "this S is not P" (propositions called singular).

Author: Shadrin D.A.

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