From a grammatical point of view, a statement is a declarative sentence.
Complex sentences are constructed from expressions denoting certain concepts and logical connectives. The words and phrases NOT, AND, OR, IF... THEN, THEN AND ONLY THEN, EXISTS, ALL and some others are called logical connectives (operators) and denote logical operations with the help of which others are constructed from some sentences.
Sentences without logical connectives are elementary; they cannot be divided into parts so that each part is also a sentence. Elementary statements are also called statements (judgments). Statements contain information about objects, phenomena, and processes.
An elementary statement consists of a subject (logical subject) - what it is about we're talking about in a statement, and a predicate (logical predicate) - that which is affirmed or denied in a statement about the subject.
Thus, a statement is a form of thinking in which a logical connection between concepts acting as subject and predicate is affirmed or denied this statement. The correspondence or inconsistency of this connection with reality makes the statement (judgment) true or false.
The logical connection between the subject and the predicate of a statement is usually expressed in the form of a connective IS or IS NOT, although in the sentence itself this connective may be absent, but only implied. At the same time, the subject of the statement can be expressed not only by the subject of the sentence, just as the predicate can be expressed not only by the predicate (these can also be other members of the sentence). What is considered a subject in a sentence and what is a predicate of a statement is determined by logical accentuation. Logical stress associated with the meaning contained in the sentence for the speaker or listener.
According to the form, statements are divided into simple ones (having logical form « S There is P" or " S do not eat P", Where S– subject, P– predicate) and complex (grammatically expressed in complex sentences).
An example of a simple statement: “All bears love honey,” a complex one: “Some bears love honey and young bamboo shoots.”
Simple sayings allow you to express following types sayings:
· attributive statements – express whether or not a property belongs to an object or class (for example, the Earth is a planet);
· statements about relationships – talk about the existence of a relationship between objects (for example, 3<5 );
· statements of existence (existential statements) – speak about the existence or non-existence of an object or phenomenon.
Operations on a set of statements.
From elementary statements you can compose complex statements using logical operations. Elementary statements that are part of a complex statement are connected by logical operators not by semantic description, but only by their truth values. Consequently, complex statements are functions of the elementary statements included in them. All operations in propositional logic are described only by a truth table.
Operations on a set of statements include:
· Denial. The truth table for it is:
IN natural language it is most often interpreted with the conjunction “and”.
· A disjunction of two elementary statements is true if and only if at least one of the elementary statements is true. It is sometimes called logical addition or logical maximum. The truth table for a disjunction looks like this:
· The XOR operation is given by the following truth table, it is true when only one of the operands is true. This operation is also called strict disjunction or logical inequality.
Mathematical theorems are often formulated in this form. If the theorem is formulated in some other way, then it can be rephrased in the indicated form without losing its essence.
Negation (sign). If A is a statement, then (read: not A) is also a statement; it is true or false depending on whether statement A is false or true. We see that the operation in the theory of statements fully corresponds to the concept of negation in the ordinary sense of the word. The negation operation can be described by the table
Conjunction. As a sign for conjunction, the sign l is used, as well as & (in other words, the conjunction and- And).
If A And IN- statements, then A ˄ IN(reads: A And IN) - a new statement. It is true if and only if A true and IN true.
Unlike the operation of negation, which depends on one elementary statement, conjunction, like all subsequent connectives we give, depends on two elementary statements, which is why they are called two-place connectives, while negation is a one-place connective.
To specify double connectives, it is convenient to write truth matrices in the form of tables with two inputs: the rows correspond to the truth values of one elementary statement, the columns correspond to the values of another elementary statement, and in the cell where the column and row intersect the truth value of the corresponding complex statement is placed.
The truth value of a complex statement A˄ IN is given by the matrix:
As you can see, the definition of the conjunction operation fully corresponds to the ordinary meaning of the conjunction “u”. For example, the problem of protecting automated lines from an accident significantly depends on the reliability of the EA. The influence of vibrations that occur when contacts are closed on the switching wear resistance of the EA is regulated by the ratio of the mechanical and traction characteristics of the electromagnetic drive.
Disjunction. We will use the sign ˅ as a sign for disjunction. If A and B are statements, then A v B (read: A or B) is a new statement. It is false if A and B are false; in all other cases A v IN true. Thus, the truth matrix for the disjunction operation looks like this:
The disjunction operation corresponds to the usual meaning of conjunction "or". For example, contact wear is monitored by selecting a dip or by weighing the contacts on a scale before and after operation.
Implication. We will use the sign as a sign for implication. If A and B are two statements, then A IN(read: A implies B) - a new statement. It is always true, except when A true, but IN false.
The truth matrix of the implication operation is as follows:
In implication A IN first term A called the antecedent, the second term IN-consequential.
Implication describes to some extent what is expressed in ordinary speech by the words “if A, That IN", "from A should IN», « A- a sufficient condition for IN".
If the increase in resistance in the intercontact gap after the current passes through zero is more intense than the increase in voltage, then re-ignition of the arc will not occur. If the short circuit current significantly exceeds the melting current of the fuse-link, then the fuse-link burns out and the fuse turns off the electrical circuit.
Equivalence. For this operation the sign ⇔ is used. The operation is defined as follows: if A and B- statements, then A ⇔ IN(reads: A equivalent IN) is a new statement that is true if either both statements are true or both are false.
Using the introduced connectives, you can construct complex statements that depend not only on two, but also on any number of elementary statements.
In rated current modes 25...600 A a pair of contacts can perform a dual role: long-term transmission of current in the on position and shutdown, accompanied by the occurrence of an arc. In the first case, the contacts must have low contact resistance; in the second, the requirements for high contact resistance are imposed. In both cases, the same single-stage contact system is used. Both processes affect contact wear.
Note. A non-strict inequality is a disjunction A<В ˅ (А = В).Оно истинно, если истинно по меньшей мере одно из входящих в него простых высказываний. Примерами complex statements encountered in practice are the so-called double inequalities A< В < С(А < В) ˄ (В < С), а, например, означает сложное высказывание (А< В) ˄ ((В Having the truth value of simple statements, it is easy to calculate the truth value of a complex statement based on the definition of connectives. Let a complex statement be given ((B ˅ C) ⇔ (B ˄ A)) and let the elementary statements included in it have the following truth values: A = L, B = I, C =
I. Then B ˅ C = I, B ˄ A = L, so the statement in question ((B ˅ C) ⇔ (B ˄ A)) is false. A complex proposition is one that contains logical connectives and consists of several simple propositions. In the future, we will consider simple propositions as certain indivisible atoms, as elements from the combination of which complex structures arise. We will denote simple propositions by separate Latin letters: a, b, c, d, ... Each such letter represents a certain simple proposition. Where can you see this? Distracting from the complex internal structure of a simple judgment, from its quantity and quality, forgetting that it contains a subject and a predicate, we retain only one property of a judgment - that it can be true or false. Everything else does not interest us here. And when we say that the letter “a” represents a proposition, and not a concept, not a number, not a function, we mean only one thing: that “a” represents truth or falsehood. If by "a" we mean the proposition "Kangaroos live in Australia," we mean the truth; if by “a” we mean the proposition “Kangaroos live in Siberia,” we mean a lie. Thus, our letters "a", "b", "c", etc. – these are variables that can be replaced by true or false. Logical connectives represent formal analogues of conjunctions in our native natural language. Just as complex sentences are built from simple ones with the help of conjunctions “however”, “since”, “or”, etc., so complex propositions are formed from simple ones with the help of logical connectives. Here there is a much greater connection between thought and language, so in what follows, instead of the word “judgment,” which denotes pure thought, we will often use the word “statement,” which denotes thought in its linguistic expression. So, let's get acquainted with the most commonly used logical connectives. Negation. In natural language it corresponds to the expression “It is not true that...”. Negation is usually indicated by the sign "" placed before the letter representing some proposition: "a" reads "It is not true that a." Example: “It is not true that the Earth is a sphere.” You should pay attention to one subtle circumstance. Above we talked about simple negative judgments. How to distinguish them from complex judgments with negation? Logic distinguishes two types of negation - internal and external. When the negation is inside a simple proposition before the connective “is,” then in this case we are dealing with a simple negative proposition, for example: “The earth is not a sphere.” If a negation is externally attached to a judgment, for example: “It is not true that the Earth is a ball,” then such a negation is considered as a logical connective that transforms a simple judgment into a complex one. Conjunction. In natural language, this connective corresponds to the conjunctions “and”, “a”, “but”, “however”, etc. Most often, a conjunction is indicated by the “&” symbol. Now this icon is often found in the names of various companies and enterprises. A proposition with such a connective is called conjunctive, or simply conjunction, and looks like this: a&b. Example: “Grandfather’s basket contained boletus and boletus.” This complex judgment is a conjunction of two simple propositions: “There were boletus mushrooms in my grandfather’s basket” and “There were boletus in my grandfather’s basket.” Disjunction. In natural language, this connective corresponds to the conjunction “or”. It is usually denoted by a "v". A judgment with such a connective is called disjunctive, or simply disjunction, and looks like this: a v b. The conjunction “or” in natural language is used in two different senses: loose “or” - when the members of the disjunction do not exclude each other, i.e. can be simultaneously true, and a strict “or” (often replaced by a pair of conjunctions “either... or...") - when the members of the disjunction exclude each other. In accordance with this, two types of disjunction are distinguished - strict and non-strict. Implication. In natural language it corresponds to the conjunction “if... then”. It is indicated by the sign “->”. A proposition with such a connective is called implicative, or simply implication, and looks like this: a -> b. Example: “If an electric current passes through a conductor, the conductor heats up.” The first member of the implication is called the antecedent, or basis; the second is a consequent, or consequence. In everyday language, the conjunction “if... then” usually connects sentences that express the cause-and-effect relationship of phenomena, with the first sentence fixing the cause, and the second the effect. Hence the names of the members of the implication. Representing natural language statements in symbolic form using the above notations means their formalization, which in many cases turns out to be useful. 4) A beautiful island lay in the warm ocean. And everything would be fine, but strangers got into the habit of settling on this island. They come and come from all over the world, and the indigenous people have begun to be squeezed. In order to prevent the invasion of foreigners, the ruler of the island issued a decree: “Every visitor who wants to settle on our blessed island is obliged to make some judgment. If the judgment turns out to be true, the stranger should be shot; if the judgment turns out to be false, he should be hanged.” If you are afraid, then shut up and turn back! The question is: what judgment must be made in order to stay alive and still settle on the island? Truth tables Now we come to a very important and difficult question. A complex proposition is also a thought that affirms or denies something and which therefore turns out to be true or false. The question of the truth of simple judgments lies outside the realm of logic - it is answered by specific sciences, everyday practice or observation. Is the statement “All whales are mammals” true or false? We need to ask a biologist and he will tell us that this proposition is true. Is the statement “Iron sinks in water” true or false? We need to turn to practice: let’s throw some piece of iron into the water and make sure that this judgment is true. In short, the question of the truth or falsity of simple propositions is always ultimately decided by reference to the reality to which they relate. But how to establish the truth or falsity of a complex proposition? Let us have some conjunction “a & b” and we know that the proposition “a” is true and the proposition “b” is false. What can be said about this complex statement as a whole? If in reality there was an object to which the connective “&” refers, then the difficulty would not arise: having discovered this object, we could say: “There is! The conjunction is true!”; having searched around and not found the corresponding object, we would have stated: “The conjunction is false.” But the fact is that in reality nothing corresponds to logical connectives - as well as to the conjunctions of natural language! These are means of connecting thoughts or sentences that we have invented; these are tools of thinking that have no analogues in reality. Therefore, the question of the truth or falsity of statements with logical connectives is not a question of specific sciences or material practice, but a purely logical question. And logic solves it. We agree or accept agreements regarding when statements with one or another logical connective are considered true and when they are false. Of course, these agreements are based on some rational considerations, but it is important to keep in mind that these are our arbitrary agreements, adopted for the purposes of convenience, simplicity, fruitfulness, but not imposed on us by reality. Therefore, we are free to change these agreements and do so whenever we see fit. The agreements in question are expressed by truth tables for logical connectives, showing in which cases a statement with a particular connective is considered true and in which cases it is considered false. In doing so, we rely on the truth or falsity of simple judgments that are components of a complex judgment. "True" ("i") and "false" ("l") are called the "truth values" of a proposition: if a variable represents a true proposition, it takes on the value "true"; if it is false, it takes the value “false”. Each variable can represent either true or false. Negation applies to one proposition. This proposition can be true or false, so the table for negation is as follows: If the original proposition is true, then we agree to consider its negation false; if the original judgment is false, then we consider its negation to be true. This agreement seems to match our intuition. Indeed, the proposition “Byron was an English poet” is true, so its negation “It is not true that Byron was an English poet” is naturally considered false. The proposition “Athens is in Italy” is false, so its negation “It is not true that Athens is in Italy” is naturally considered true. For convenience, we present truth tables for other logical connectives together: All connectives given here connect two propositions. For two propositions there are four possibilities: both can be true; one is true, the other is false; one is false, the other is true; both are false. All these possibilities are taken into account as cases 1-4. A conjunction is true only in one case - when both its terms are true. In all other cases we consider it false. Overall, it seems pretty natural. Let's say you say to your chosen one: “I will marry you and will be faithful to you.” You really married this person and are faithful to him. He is satisfied: you did not deceive him, the conjunction as a whole is true. Second case: you got married, but are not faithful to your husband. He is indignant, believes that you deceived him - the conjunction is false. The third case: you did not marry the one to whom you promised, although you remain faithful to him, cherishing the memories of your first and, alas, only love. Again, he is upset: you deceived him - the conjunction is false. Finally, the fourth option: you didn’t marry him and, naturally, you don’t remain faithful to him. Your admirer is furious: you blatantly deceived him - the conjunction is false. Similar considerations justify the truth table for disjunction. The situation with implication is somewhat more complicated. Consider the proposition “If the sun rose, it became light outside.” Here the implication connects two simple propositions: “The sun has risen” and “It has become light outside.” When both of them are true, then we consider the implication as a whole to be true. Now the second case: the sun has risen, but there is no light outside. If this suddenly happened, we will consider our implication to be false: apparently, we did not take something into account when we formulated such a connection between the two judgments. Third case: the sun did not rise, but it became light outside. Will this disprove our implication? Not at all, this is quite possible: the lights came on on the street, it became light, but this does not contradict the connection between the sunrise and the onset of daylight. The implication can be considered true. Finally, the fourth case: the sun did not rise and there was no light. This is quite natural; our implication remains true. Explaining the truth tables for logical connectives, we tried to show that these tables to some extent correspond to our linguistic intuition, our understanding of the meaning of natural language conjunctions. However, the degree of such correspondence should not be overestimated. Natural language conjunctions are much richer and more subtle in semantic content than logical connectives. The latter grasp only that part of this content that relates to the relations of truth or falsity of simple statements. Logical connectives do not take into account more subtle semantic connections. Therefore, sometimes a rather large discrepancy is possible between logical connectives and conjunctions of natural language. With the help of these connections they create programs for computers, and now you can understand what part of our thinking a computer can assimilate and use. 5) How to divide 7 apples equally among 12 boys without cutting any apples into 12 pieces? (The imposed condition is intended to exclude the simplest solution: cut each apple into 12 parts and give each boy one slice from each apple, or cut 6 apples in half and cut the 7th apple into 12 parts.) 6) On one island live two tribes - good fellows who always tell the truth, and liars who always lie. A traveler comes to the island who knows about this, and, meeting a local resident, asks him: “Who are you, from what kind of tribe?” "I'm done!" - the aborigine answers proudly. “That’s good,” the traveler rejoiced, “you will be my guide!” They walk around the island and suddenly see another aborigine in the distance. “Go ask him,” the traveler says to his guide, “what tribe is he from?” The conductor ran back and reported. “He said he was great!” “Aha,” thought the traveler, “now I know exactly what tribe you are from!” How did the traveler guess who his guide was? Let us formulate the basic rules for the formation of new sentences from the original ones using the basic connectives and conjunctions of ordinary spoken language. The rules of the Russian language alone are not enough, since sometimes we put different meanings into the same sentence formulated in Russian. For example, consider the turn of phrase “If, then,” with which we formulate two sentences: Question: Do these sentences say the same thing or is there a situation where one of the sentences is true and the other is false? In other words, the question is whether these sentences are equivalent. Until we clearly define the rules for constructing phrases of this kind, the question cannot be answered unambiguously. On the one hand, when formulating the first sentence, we often mean the second sentence. However, let's look at these proposals from a different perspective. First, let's write down the sentence diagrams. To do this, we denote the sentence “Misha will pass the exam with flying colors” by the letter A, and the sentence “Misha will go to the disco” - with the letter IN. Then these proposals can be schematically written as follows: I) "If A, That IN", 2) “If not A, then not IN". Now let's substitute instead A And IN other predictions. Instead of A take: “The table is made of oak”, instead of IN"The table is wooden." Then we get another pair of sentences: Since these sentences are constructed according to the same schemes as the first two, it means that the equivalence of the first pair of sentences must mean the equivalence of the second pair. However, the first sentence in ordinary speech is obviously a true statement, since oak is a tree, and the second sentence is, in common sense, false, since the table can be made of another tree, such as pine. Thus, in general case sentences constructed according to the “If A, That IN" and "If not A, then no IN" cannot be considered logically identical. So, in order to eliminate ambiguity in the construction of sentences, we need clear rules that allow us to determine the truth or falsity of the resulting sentence depending on the truth or falsity of the original sentences A And IN. Let us give the conjunctions “and”, “or”, as well as the schemes “if, then”, “then and only then”, “it is not true that” an unambiguous logical meaning. Let the letters A and B stand for arbitrary sentences. Let's start with simple situations. 1. Negation sign~| (-i) or. Expression ~li(-L, A) reads: "not A" or “it is not true that A.” Sentence Meanings ~A define by a table from which it is clear that the proposal ~l true exactly when the original sentence A false: When formulating sentences that are simple in structure, the particle “not” can sometimes be “carried inside” the sentence. For example, a sentence “It is not true that the number V6 is an integer” can be formulated as follows: “The number l/6 is not an integer.” Also the sentence “It is not true that straight A And b intersect" formulate: "Direct A And b We won’t ask.” Often an object that does not have some property is called a term with the particle “not”. For example, an integer that is not even is called odd. Therefore, it is equally correct to say “The whole number is odd” and “The whole number is not even.” But without the stipulation that the number is an integer, we have sentences with different meanings. For example, “The number 0.2 is not even” is true, but the sentence “The number 0.2 is odd” is false. Consider the phrase “odd function.” Here we have an independent term and the word “odd” cannot be written and pronounced separately, that is, the sentence “The function is odd” is not a negation of the sentence “The function is even.” Indeed, there is an example of a function in which both sentences are false. For example, the function )t=x+ is neither even nor odd (try to explain this). 2. Conjunction sign l. Expression LlW reads: "A and B". Sometimes conjunction is denoted by &. Sentence Meanings AlV depending on the proposals that make it up A and B defined by the table: So the proposal AlV true only in one case, when both sentences A And IN are true. In other cases this sentence is false. When formulating a proposal AlV Instead of the conjunction “and”, you can use other conjunctions that have the same logical meaning of simultaneously fulfilling each of the sentences: “a”, “but”. Example 1.3.1. Sentence "Number" 111
is not divisible by 2, but is divisible by 3" - symbolically you can write 1 AlV, Where A= "111 is divisible by 2", B = " 111 is divisible by 3." 3. Disjunction sign v. Expressions AvB reads: "A or B." Sentence Meanings AvB defined by the table: From the table it is clear that the offer "A or IN" true in those cases when at least one of the sentences A or IN true, and in the case when both sentences A And IN false, sentence AvB takes a false value. Sometimes from the content of sentences A And IN it follows that sentences cannot be simultaneously true. In this case, the sentence is formulated using the conjunction “or”. For example, the sentence "A number is either positive or negative" also has the form "A or IN”, but at the same time it has such an implication that a number cannot be both positive and negative at the same time. The rules formulated above, apparently, do not raise any questions. Let’s move on to the diagram discussed at the beginning of the paragraph “If A, That IN". 4. Sign of implication-Expression A->B reads: “If A, then B.” Sometimes another arrow symbol => is used to denote this connective, as well as a z> sign. Along with the phrase “If A, That IN" others similar to it use: "B when A», “A only when B.” We motivate the definition of sentence meanings A->B. The main difficulty that arises here is to assign a meaning to the sentence L-»# for those cases when A false. To intelligently determine the meanings, remember the correct sentence discussed above: “If the table is oak, then it is wooden.” Here A= “Oak table”, B ="Wooden table." Let the table be made of pine. Then A false, IN true. Let the table be iron. Then A false and IN false. In both cases the offer A is false, and the resulting sentence “If A, That IN" true. Moreover, both of these cases are really possible. Of course, it is possible that we have oak table, Then Aw B simultaneously true. Here is an example of a true sentence A->B, When A=u>B=l, does not exist. Thus, cases when A=u, B=i, or A=l y B=i, or A=l, V=l, must determine a true sentence And only one case, when which A=u, V-l, means that the offer A->B false. So, in mathematical logic the values of the T-sentence are given by the table below: In what follows, throughout the phrase “If A, That IN" will be understood this way. Here's a suggestion A called by parcel, or condition, A In conclusion. Example 13.2. The parents promised their son Petya: if he successfully graduates from the university, they will buy him a car. It is known that the son did not graduate from university, but his parents still bought him a car. Is it possible to say that what the parents said was a lie? To answer the question, consider the proposals: A= “My son is graduating from university”, B ="They're buying him a car." Wherein A=l, B=i. The parents' promise looks like A^>B. By definition, this is a proposal given values A And IN true (third row of table). Therefore, from a logical point of view, the parents’ words are correct. But if their son graduated from college, but they didn’t buy him a car, in this case (and in no other) the promise would not be fulfilled. Now let's look at another logical connective that is often meant when the words “if, then” are said. For example, if in the conditions of example 1.3.2 the parents assumed that if their son Petya did not graduate from college, they would not buy him a car, it would be correct to say: “The car will be bought if and only if Petya graduates.” institute." 5. Equivalence sign or. Expression And it reads: “And if and only if B.” Other formulations are possible: “And if and only if B», “A exactly when B” and so on. Sentence Meanings AB are given by the table: In cases where A And IN accept same values, offer AB true, otherwise the sentence is false. It is easy to see that the phrase "A then and only when IN" consists of two phrases: "A then when IN" And "A only when IN". The first sentence is written B->A, and the second A^>B. These two sentences are simultaneously true in two cases: A=u, B=u, and A=l, B=l. So, we have defined five signs: l (conjunction), v (disjunction), ->
(implication), (equivalence), 1 (negation), which are called logical seeders. These signs allow from these sentences A And IN receive new offers. In this case, the meaning (true or false) of the new sentence is uniquely determined by the meanings of the sentences A And IN. The rule for obtaining a new sentence from the original sentences is called logical operation. Thus, each of the logical connectives determines logical operation, which has the same name as the corresponding bundle. The operations considered can be used for both statements and predicates. For example, by combining two unary predicates " Number,t more 3" and "Number X negative" with a disjunction sign, we get a one-place predicate: "Number X more than 3 or negative.” The only thing is that in order to connect two predicates with a logical connective, it is necessary that some general area D valid objects that can be substituted into these predicates instead of variables. Let us define two more logical connectives, called kwaitora.mi, which allow us to obtain statements from unary predicates. The term "quantifier" translated from Latin language means "how much". Therefore, these signs are used to answer the question of how many objects satisfy the proposition And- all or at least one. Let's take an arbitrary predicate and select a variable on which its value depends. Let's denote it Oh). 6. General quantifier V. This sign derived from English word AN and is an abbreviation of the following words: “weight”, “every”, “any”, “any”. The expression Vj&4(y) means that the predicate Oh) executed for all valid objects X. It reads: “For all X and from X.” 7. Existential quantifier 3. This sign comes from the English word Exist and is an abbreviation of the following words: “exists”, “there will be”, “at least one”, “some”. The expression 3x4(*) means that the predicate Oh) is executed for at least one of the valid objects.v. It reads: “There is x and from x.” Example 1.3.3. Let the variable X denotes a university student. Let's consider the proposal Oh)= “Student l: has a car.” Then VxA(x) means that all university students have a car. This is a false statement. Offer EhA(x) means that some students have a car, which is a true statement. Thus, initially we had a predicate whose value depended on the value of the variable dg. After performing the operations, statements were obtained whose values no longer depend on the variable X. Let there be a formula L(x), containing a free variable X. Then the statement that the formula Oh) is identically true, we can write it briefly as Vj&4(jc). The operation of obtaining a sentence using quantifiers is called quantification. When using expressions UhA(x) and 3 xA(x) also say: “A quantifier has been added to the variable x” or “The variable x is connected by a quantifier.” Note that quantifier operations are applicable not only to one-place predicates. If a two-place predicate is given A(hu), then you can connect the variable l - a quantifier and form a sentence /xA(xy), the truth of which will depend on only one variable y, and we will have a one-place predicate. In this entry the variable X called associated with a quantifier, and the variable y - free. In the general case, applying a quantifier operation to any of the variables of a /7-place predicate, we end up with an (n-1)-place predicate. Quantifiers can be used to link any number of variables. If we have a two-place predicate A(hu), then formally you can get 8 statements. linking each variable with some quantifier: Vjc fyA(xy), VyVxA(xy), Vx3уА(xy), 3yVxA(xy), 3xVyA(xy), /уЭхА(xy), ЗхЗуА(ху), ЗуЗхА(ху). Some sentences have the same meaning, for example the first and second (predicate A must be true for any values of * and y), as well as the seventh and eighth. The remaining expressions generally give statements of different truth. Example 1.3.4. Let there be only two boys in the class - Petya and Kolya. For independent decision three problems were given, let's denote them by numbers 1, 2, 3. Petya solved problems 1 and 2, and Kolya solved one problem with number 3. Let's introduce the predicate A(hu), which means the boy * solved the problem u. Here the variable X denotes the boy's name, and the variable at- task number. Consider the following statements. Vx3yA(xy)= “Each boy solved at least one problem” - true statement, since Petya solved two problems, and Kolya solved at least one problem. V_yEx,4(;c,y) = “Each problem was solved by at least one student” - true, so problem number 1 was solved by Petya, problem number 2 was also solved by Petya, and problem 3 was solved by Kolya. From the example considered, we can conclude: the order in which quantifiers are written affects the logical meaning of the sentence. Therefore, the clear formulation of the sentence must unambiguously presuppose the order in which the quantifiers of generality and existence occur. Exercise. Analyze the meanings of the statements from Example 1.3.4 on your own, assuming that Petya solved problems numbered 2 and 3. In general, from the predicate Oh) you can get two statements - /xA(x) and 3x4(x). However, the very often written formula Oh) is understood precisely as the statement Vx4(.x), although the general quantifier is omitted when written or formulated. For example, by writing d- 2 >0, they mean that the square of any real number non-negative. Full entry The statement is: Ulg(dg?0). Record (4x + 6y):2, Where*, y - integers, assumes that the specified sum is always divisible by 2, that is, even. To emphasize this, we should write V*Vy((4.x + 6jy):2). Defined in the last two paragraphs mathematical signs and the signs of logical connectives make up the alphabet of the mathematical language. symbols logical languages, used to form complex statements (formulas) from elementary ones. Logical connectives are also called natural language conjunctions corresponding to these symbols. Usually such logical connectives are used as conjunction (the conjunction “and”, symbolic notations: &, l and a dot in the form of a multiplication sign, which are often omitted, writing the conjunction of A and B as AB), disjunction (a loose conjunction “or”, denoted as “v”), implication (“if..., then”, is denoted by the sign, . A propositional truth function assigns to each listed set one of the truth values - 1 or 0. There are 16 such functions in total. The conjunction assigns to the expression A&. B has the value 1 only in the case when both A and B are true, that is, both have the value 1, otherwise the value of A&.B is 0. The disjunction B, on the contrary, is false only in one case, when both A are false , and B. The implication A and B is false only if A is true (antecedent) and B is false (consequent). In other cases, A => B takes the value 1. Of the four unary functions, only negation is of interest, changing the meaning of the statement to the opposite : when A is true, -A is false, and vice versa. All other unary and binary classical functions can be expressed in terms of the presented ones. When the system of logical connectives adopted in the corresponding semantics allows us to define all the others, it is called functionally complete. Complete systems in classical logic include, in particular, conjunction and negation; disjunction and negation; implication and negation. Conjunction and disjunction are definable through each other due to the equivalences (A&B) = -i(-i/4v-i.) and (A v B) a -,(-&-), called de Morgan’s laws, as well as: (A ^B)s(-iA^ B), (A&B) s -,(A e -), (B) = ((A => B) zA). Any equivalence of the form A = B is valid only when the conjunction (A =) B) & (B e A) is generally valid (always true). The functions antidisjunction and anticonjunction, defined respectively as -(B) and -(A&.B), also each separately represent a functionally complete system of connectives. This last circumstance was already known to C. Pierce (unpublished work in 1880 during his lifetime) and was rediscovered by H. M. Shefier. Using antidisjunction as the only logical connective, Schaeffer in 1913 constructed complete calculation statements. Antidisjunction is denoted by A B and is called the Schaefer prime, reading this expression, as “not-D and not-B.” J. G. P. Nicod used the same notation for anticonjunction (“It is not true that both A and B are both”) and, using only this connective, in 1917 he formulated a complete propositional calculus with one (only!) axiom and one rule of inference. Thus, Schaeffer’s stroke is essentially the vertical line itself, which, according to different authors, can mean both anti-disjunction and anti-conjunction. The extensionality of logical connectives gives them uniqueness, simplifies the problem of constructing logical calculi, and makes it possible to solve metatheoretical problems of consistency, decidability, and completeness for the latter (see Metalogics). However, in some cases, the truth-functional interpretation of connectives leads to a significant discrepancy with how they are understood in natural language. Thus, the indicated truth interpretation of the implication forces us to recognize correct sentences of the form “If A, then B” even in the case when there is no real connection. It is enough for A to be false or B to be true. Therefore, of the two sentences: “If A, then B” and “If B, then A,” at least one has to be recognized as true, which does not fit well with the usual use of the conditional connective. Implication in in this case specially called “material”, thereby distinguishing it from a conditional conjunction, which assumes that there is a real connection between the antecedent and the consequent of a true conditional statement. At the same time, the material implication can be perfectly used in many contexts, for example, mathematical ones, when one does not forget about it specific features. In some cases, however, it is the context that does not allow the conditional conjunction to be interpreted as a material implication, suggesting the interconnection of statements. To analyze such contexts, it is necessary to build special non-classical logics, for example, relevant (see Relevant logic), in the language of which instead material implication(or along with it) other implications are introduced that are understood intensionally (substantively) and the truth of which cannot be justified truth-functionally. Other logical connectives can also be interpreted intensively.
