Which logical connective is designated as. Complex judgments

Complex judgments are judgments formed from simple ones using logical connectives.

The connection between the elements of a complex judgment is carried out using logical unions (logical connectives).

Logical connections:

Their main feature is that logical conjunctions are unambiguous, while grammatical conjunctions have many meanings and shades.

1. CONJUNCTION(from Latin conjunctio – union, connection).

Sign: ˄ or &

And», « A», « But», « Yes», « Although», « which», « but», « however», « at the same time"etc.

Judgment " She likes apple juice and green tea"is a conjunction (connection) of two simple propositions: " she likes apple juice" And " she likes green tea».

Ab or A& b

2. DISJUNCTION(from Latin disjunctio – disunion).

Sign: ˅

In Russian, conjunctions correspond to conjunctions: “ or», « or», « either... or».

Judgment " We'll go to the cinema or to the park" is a disjunction of two simple propositions: " we'll go to the cinema" or "we'll go to the park". This connection is not strict, that is, it does not imply only one choice, since we can go to the cinema or take a walk in the park.

Recording this judgment using logical connectives will look like: Ab

3. Strict disjunction

Sign: .

The conjunction “or” can be used in a strict sense - when the members of the disjunction exclude each other.

Recording this judgment using logical connectives will look like:

4. IMPLICATION( from Latin implico – closely connect)

Sign: → .

In the language, analogues of this connective are conjunctions: “ if... then»; « when..., then»; « as soon as... then"etc.

Usually, with the help of implication, cause-and-effect relationships like: “ If the sun comes out, it will become warm». a → b. The first element of the implication is called basis(antecedent), second – consequence(consequent).

5. EQUIVALENCE( from Late Lat. aequivalens – equivalent; equivalent)

Sign: ↔ or ≡ .

In the language, analogues of this connective are conjunctions: “ if and only if»; « then and only then when...»; « only on the condition that... then».

Judgment: " Only then will the child receive candy when he has finished all the soup" is the equivalent.

Recording this judgment using a logical connective will look like: a↔ b or a≡ b

6 .NEGATION

Sign: ~ or ¬ . are put before judgment~a or¬a ; or a line that is placed over a judgment

In language, negation is expressed by conjunctions and words: “ Not», « wrong"etc.

Judgment: " Car won't start" is written as ~a

Judgment: " Likes or doesn't like"contains strict disjunction and negation.

Exercises: Write your judgments in the form logical form using logical connectives.

Self-test: Write judgments in logical form using logical connectives

To test yourself, highlight the “formula” column and change the font color

A proposition is called complex if it contains logical connectives and consisting of several simple propositions.

In what follows we will consider simple judgments as certain indivisible atoms, as elements from the combination of which arise complex structures. We will denote simple propositions by separate in Latin letters: a, b, c, d, ... Each such letter represents some simple proposition. Where can you see this? Taking a break from the complex internal structure a simple judgment, from its quantity and quality, forgetting that it has a subject and a predicate, we retain only one property of a judgment - that it can be true or false. Everything else doesn't 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 are formal analogues of conjunctions in our native natural language. How complex sentences are constructed from simple ones with the help of conjunctions “however”, “since”, “or”, etc., and complex judgments are formed from simple ones with the help of logical connectives. Here we feel 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 denial externally is 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 meanings: 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 a conductor passes electric current, then 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 something 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, however, 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.

Agreements about which we're talking about, are expressed by truth tables for logical connectives, showing in which cases a statement with one or another connective is considered true, and in which – 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 English poet” is true, so his denial “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. Third case: you did not marry the one to whom you promised, although you remain faithful to him, cherishing the memories of the 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 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 absorb 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, having met local resident, asks him: “Who are you, from what kind of tribe?” “I’m great!” - 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?

Logical multiplication or conjunction is an operation expressed by the connective “and” and denoted by the dot “ ” (or the signs & or  ). Statement A  B is true if and only if both statements A and B are true.

Truth table of logical multiplication function

Logical addition or disjunction is an operation expressed by the connective “or” (in the non-separating sense of the word) and denoted by “+” (or the sign  ). Statement A  B is false if and only if both statements A and B are false.

Truth table of logical addition function

By implication is an operation expressed by connectives “if..., then”, “from... follows”. Statement A  B is false if and only if A is true and B is false.

Truth table logical function"implication"

In ordinary speech, the connective “if..., then” describes the cause-and-effect relationship between statements. But in logical operations the meaning of statements is not taken into account. Statements A and B forming a compound statement A  B, may be completely unrelated in content. Only their truth or falsity is considered.

Logical equality or equivalent (or double implication ) is an operation expressed by the connectives “if and only then”, “necessary and sufficient”, “... equivalent to...”, and is denoted by the sign  or ~ . The statement AB is true if and only if the values ​​of A and B coincide.

Truth table of the logical function "equivalence"

The implication can be expressed through disjunction and negation:

A  B = Ā  IN.

Equivalence can be expressed through negation, disjunction and conjunction:

A  B = (Ā  IN)  ( A).

Thus, the operations of negation, disjunction and conjunction are sufficient to describe and process logical statements.

For everyone compound statement you can build a truth table that will determine its truth or falsity for various combinations of initial values ​​of simple statements. For example, consider the truth table of a logical expression (A IN) (Ā  )

Truth table

Example . Determine the result of the logical operation F = (A  B)  (C  D) at given values logical variables A, B, C – true, D – false.

Solution .

From the constructed truth table it follows that F=1

There are five widely used logical connectives. These are negation (represented by the sign ¬), conjunction (sign), disjunction (sign v), implication (sign) and equivalence (sign).

Statement ¬ A(reads "not A") means that the statement A false. In other words, ¬ A true when A false, and false when A true.

Statement A B(reads " A And B") means a statement that is true and A, And B. It is true only if both statements are true A And B.

Statement A v B (« A or B") is true if at least one of the statements is true A And B.

Statement A B reads " A entails B" or "if A, That B" It is incorrect if A true, B false, and true in all other cases.

Finally, a statement AB true if the statements A And B either both are true or both are false.

To indicate the structure of connections, parentheses are used, just as it is done in algebra to indicate the order of execution. arithmetic operations. So, for example, the statement ¬ A B means " A wrong, but B true”, and the statement ¬( A B) - “it is not true that A And B both are true." And just as in algebra, to reduce the number of brackets, the order of precedence of connectives is established according to the strength of the connection. Above we have listed the ligaments in order of weakening of the connection. For example, a conjunction connects more strongly than an implication, so the statement A B C understood as A (B C), but not like ( A B) C. This corresponds to what in algebra a + b ? c means a + (b ? c), but not ( a + b) ? c.

Here are some examples of compound statements.

A well-known tongue twister states: “the heron wasted away, the heron was withered, the heron was dead.” This statement can be written in the form: “the heron is stunted” “the heron is dry” “the heron is dead.”

Ratio 0< Z < 1 есть конъюнкция «Z > 0» « Z < 1», a соотношение |Z| > 1 - disjunction " Z> 1" v " Z < -1». Определение логической связки данное выше, можно записать так:

[(A B) (A B) v (¬ A ¬ B)] [(A B) v (¬ A ¬ B) (A B)]

We leave it to the reader to translate into ordinary language the following statement:

“The light is on” “The light bulb is not lit” “There is no electricity” v “The plugs are burnt out” v “The light bulb is burnt out.”

If we assume that statements can only be true or false and, beyond this, nothing can be said about a statement, then the listed connectives are sufficient to express all conceivable constructions from statements. Even two connectives are enough, for example, negation and conjunction or negation and disjunction. This situation occurs, in particular, with regard to mathematical statements. Therefore, other connectives are not used in mathematical logic.

However, natural language reflects more diversity in the evaluation of statements than simply dividing them into true and false. For example, a statement may be viewed as meaningless or as unreliable, although possible (“there must be wolves in this forest”). Special sections of logic, in which other connectives are found, are devoted to these issues. Great value For modern science these sections (unlike the classical mathematical logic) do not have, and we will not touch them.

LOGICAL CONNECTIONS– symbols logical languages, used for education complex statements(formulas) from elementary ones. Logical connectives are also called natural language conjunctions corresponding to these symbols. Typically used logical connectives such as conjunction (the conjunction “and”, symbolic notations: &, ∧ and dot in the form of a multiplication sign, which are often omitted when writing conjunction A And IN How AB), disjunction (a loose conjunction “or”, denoted as “∨”), implication (“if..., then”, denoted by the sign “⊃” and various kinds of arrows), negation (“it is not true that... ", denoted by: , ~ or a line over the negated expression). Of the above, negation is a unary connective. Others are double (binary). In principle, logical connectives can be as local as desired, but in practice, more than binary connectives are used very rarely. In classical logic ( Logics , Propositional logic ) any multi-place logical connectives are expressible in terms of the ones listed. Some practical meaning is given by the use of a ternary logical connective, called a conditional disjunction, connecting three statements A, B And WITH and meaning that " A in case IN, And WITH in case of non- B"or formally: ( B⊃A)&(B⊃C) (Sidorenko E.A. Propositional calculus with conditional disjunction. – In the book: Methods logical analysis. M., 1977).

Classical logic considers logical connectives extensionally (ignoring the substantive meaning of the statements they connect) as truth functions determined by the truth values ​​of the statements they connect. Given two truth values ​​1 (true) and 0 (false) in this logic, statements A And IN can have four possible sets of ordered truth values:<1,1>, <1,0>, <0,1>, <0,0>. The 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&IN value 1 only if A, so IN true, i.e. both have the value 1, in other cases the value A&IN equals 0. Disjunction Α ∨ IN, on the contrary, it is false only in one case, when both are false A, so IN. Implication A⊃ IN is false only if the antecedent is true A and false (consequent) IN. In other cases A⊃ IN takes the value 1. Of the four one-place functions, only negation is of interest, changing the meaning of the statement to the opposite: when A– true, A – 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 equivalences ( A&IN)≡(A∨IN) and (A∨B)≡( A&B), called de Morgan's laws, and also: (Α⊃Β)≡( Α ∨IN), (A&IN)≡(A⊃B), ( Α ∨IN)≡((A⊃IN)⊃A). Any equivalence of the form A≡ IN is valid only when the conjunction ( A⊃IN)&(IN⊃A).

The functions antidisjunction and anticonjunction, defined respectively as ( A∨IN) And ( A&IN), each individually also represents a functionally complete system of connectives. This last circumstance was already known Ch. Pierce (work unpublished during his lifetime, 1880) and was rediscovered by H.M. Sheffer. Using antidisjunction as the only logical connective, Schaeffer in 1913 constructed full calculation statements. Anti-disjunction is denoted by A∣IN and is called the Schaeffer stroke, reading this expression as "not- A and not- B" J. G. P. Nicod used the same notation for anticonjunction (“It is not true that at the same time A And B") and with the help of only this connective in 1917 he formulated a complete propositional calculus with one (total!) axiom and one inference rule. Thus, the Schaeffer 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. Metalogic ). 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 like “If A, That B"even when between statements A And IN(and, accordingly, the events that are discussed in them) there is no real connection. Enough to A was false or IN– true. Therefore, from two sentences: “If A, That IN» and "If IN, That A", at least one thing has to be recognized as true, which does not fit well with the usual use of the conditional copula. 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 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 , e.g. relevant (see Relevant logic ), in the language of which, instead of a material implication (or along with it), other implications are introduced, which are understood intensionally (substantively) and the truth of which cannot be justified truth-functionally. Other logical connectives can also be interpreted intensively.

Literature:

1. Church A. Introduction to mathematical logic, vol. 1. M., 1960;

2. Curry H. Foundations of mathematical logic. M., 1969.

E.A.Sidorenko