3.2 Simple and complex statements

Simple and complex statements

Saying is a more complicated formation than a name. When decomposing sentences into simpler parts, we always get certain names. Say, the statement “The sun is a star” includes the names “Sun” and “star” as its parts.

A statement is a grammatically correct sentence, taken together with the meaning (content) expressed by it and which is true or false.

The notion of utterance is one of the original, key concepts of modern logic. As such, it does not allow a precise definition, equally applicable in its different sections.

The statement is considered true if the description given by it corresponds to the real situation, and false if it does not correspond to it. "Truth" and "false" are called "truth values ​​of statements."

From individual statements in different ways you can build new statements. For example, from the statements “The wind blows” and “It is raining”, you can form more complex statements “The wind blows and it rains”, “Either the wind blows or it rains”, “If it rains, the wind blows”, etc.

A statement is called simple if it does not include other statements as its parts.

A statement is called complex if it is obtained with the help of logical connectives from other simpler statements.

Consider the most important ways to construct complex statements.

A negative utterance consists of the original utterance and the negation, usually expressed by the words “not”, “wrong, that”. A negative utterance is thus a complex utterance: it includes as its part a different utterance. For example, the negation of the statement “10 is an even number” is the statement “10 is not an even number” (or: “It is not true that 10 is an even number”).

Let us denote the statements by the letters A, B, C, ... The full meaning of the notion of the negation of a statement is given by the condition: if the statement A is true, its negation is false, and if A is false, its negation is true. For example, since the statement “1 is a positive integer” is true, its negation “1 is not a positive integer” is false, and since “1 is a prime number” is false, its negation “1 is not a prime number” - is true.

Combining two statements with the word "and" gives a complex statement called a conjunction. Sayings that are joined in this way are called "conjunction members."

For example, if the statements “Today is hot” and “Yesterday it was cold” were connected in this way, the conjunction “Today is hot and it was cold yesterday”.

A conjunction is true only in the case when both statements are true; if at least one of its members is false, then the whole conjunction is false.

In ordinary language, two statements are connected by the union “and” when they are related to each other in content or meaning. The nature of this connection is not quite clear, but it is clear that we would not consider the conjunction "He went to coat, and I went to university" as an expression that has meaning and can be true or false. Although the statements “2 is a prime number” and “Moscow is a big city” are true, we are not inclined to consider their conjunction “2 is a simple number and Moscow is a big city” to be true, since the components of this statement are not related to each other in meaning. Simplifying the meaning of conjunction and other logical connectives and refusing for this the obscure notion of the “connection of statements in meaning”, logic makes the meaning of these connectives both broader and more definite.

Combining two statements with the word "or" gives the disjunction of these statements. The statements that form the disjunction are called "members of the disjunction."

The word "or" in everyday language has two different meanings. Sometimes it means "one or the other or both," and sometimes "one or the other, but not both together." For example, the statement “This season I want to go to the“ Queen of Spades ”or“ Aida ”allows for the possibility of a double visit to the fans. In the statement “He studies at Moscow University or at Yaroslavl University” it is implied that the person mentioned only studies at one of these universities.

The first meaning of "or" is called non-exclusive. Taken in this sense, the disjunction of two statements means that at least one of these statements is true, regardless of whether they are both true or pet. Taken in the second, the exclusive or the strict sense, the disjunction of two statements asserts that one of the statements is true, and the second is false.

A non-exclusive disjunction is true when at least one of the statements in it is true and false only when both its members are false.

An exclusionary clause is true when only one of its members is true, and it is false, when both its members are true or both are false.

In logic and mathematics the word "or" is almost always used in a non-exclusive sense.

A conditional statement is a complex statement, usually formulated with the help of the bundle “if ..., then ...” and establishing that one event, state, etc. is in one sense or another a ground or condition for another.

For example: "If there is a fire, that is, smoke", "If the number is divisible by 9, it is divisible by 3", etc.

The conditional statement is composed of two simpler statements. The one of them, to which the word “if” is prefaced, is called the ground, or antecedent (previous), the statement after the word “that,” is called the effect, or sequential (next).

When affirming a conditional statement, we first of all mean that it cannot be so that what is said in its basis takes place, and what is said in consequence is absent. In other words, it cannot happen that the antecedent is true, and the consequent is false.

In terms of a conditional utterance, the concepts of a sufficient and necessary condition are usually defined: an antecedent (base) is a sufficient condition for a sequential (effect), and a sequential is a necessary condition for an antecedent. For example, the truth of the conditional statement “If choice is rational, then the best available alternative is chosen” means that rationality is a sufficient basis for choosing the best available opportunity and that choosing such an opportunity is a necessary condition for its rationality.

A typical function of a conditional statement is to justify one statement by referring to another statement. For example, the fact that silver is electrically conductive can be justified by referring to the fact that it is metal: "If silver is metal, it is electrically conductive."

The connection between the substantiating and substantiated (grounds and consequences) expressed by the conditional statement is difficult to characterize in general terms, and only sometimes its nature is relatively clear. This connection may be, firstly, a connection of the logical sequence that takes place between the premises and the conclusion of a correct inference ("If all living multicellular creatures are mortal, and the jellyfish is such a creature, then it is mortal"); secondly, by the law of nature (“If the body is subjected to friction, it will begin to heat up”); thirdly, a causal connection (“If the Moon is in a node of its orbit at a new moon, a solar eclipse occurs”); fourthly, social regularity, rule, tradition, etc. (“If society changes, man changes,” “If advice is intelligent, it must be implemented”).

The connection that is expressed by the conditional utterance usually combines the conviction that a consequence with a certain need “follows” from the foundation and that there is some general law, having managed to formulate that, we could logically derive the consequence from the foundation.

For example, the conditional statement “If bismuth is a plastic metal” seems to imply the general law “Nes metals are plastic”, making the consequent of this statement a logical consequence of its antecedent.

In conventional language and in the language of science, a conditional statement, in addition to the function of substantiation, can also perform a number of other tasks: formulate a condition that is not related to any implied general law or rule (“If I want, I will cut my cloak”); fix any sequence (“If last summer was dry, this year it is rainy”); to express disbelief in a peculiar form (“If you solve this problem, I will prove Fermat's great theorem”); opposition (“If an elder grows in a garden, an uncle lives in Kiev”), etc. The multiplicity and heterogeneity of the functions of the conditional utterance makes its analysis much more difficult.

The use of conditional statements associated with certain psychological factors. Thus, we usually formulate such a statement only if we do not know with certainty whether its antecedent and the consequent are true or not. Otherwise, its use seems unnatural (“If cotton wool is metal, it is electric wire to”).

The conditional statement is very widely used in all areas of reasoning. In logic, it is represented, as a rule, by means of implicative utterance, or implication. In this case, the logic clarifies, systematizes and simplifies the use of “if ..., then ...”, frees it from the influence of psychological factors.

The logic is distracted, in particular, from the fact that the relationship between the basis and the effect characteristic of the conditional utterance, depending on the context, can be expressed with the help of the ns only “if ... then ...”, but also other language means. For example, “Since water is liquid, it transmits pressure uniformly in all directions,” “Although clay is not metal, it is plastic,” “If wood was metal, it would be electrically conductive,” etc. These and similar statements are represented in the language of logic by means of implication, although the use of "if ... then ..." in them would not be completely natural.

Claiming the implication, we argue that it cannot happen that its foundation takes place, and the consequence is absent. In other words, the implication is false only if this basis is true and the consequence is false.

This definition assumes, like the previous definitions of connectives, that every statement is either true or false and that the truth value of a complex sentence depends only on the truth values ​​of its constituent statements and on the way they are connected.

An implication is true when both its base and its consequence are true or false; it is true if its foundation is false, and the consequence is true. Only in the fourth case, when the base is true, and the consequence is false, is the implication false.

Implication does not assume that statements A and B are somehow related to each other in content. In the case of truth B, the statement “if A, then B” is true regardless of whether A is true or false and is connected in meaning with B or not.

For example, the following statements are considered true: “If there is life on the Sun, then two and two are four”, “If the Volga is a lake, then Tokyo is a big village,” etc. The conditional statement is also true when A is false, and at the same time, again, it doesn’t matter whether or not B is true and is connected by content with A or not. The following statements are true: “If the Sun is a cube, then the Earth is a triangle”, “If two or two equals five, then Tokyo is a small city”, etc.

In ordinary reasoning, all these statements are unlikely to be considered as meaningful and even less so as true.

Although implication is useful for many purposes, it is not entirely consistent with the usual understanding of conditional communication. The implication covers many important features of the logical behavior of the conditional utterance, but it is not sufficiently adequate description of it.

In the past half century, energetic attempts have been made to reform the theory of implication. In this case, it was not about abandoning the concept of implication described above, but about introducing along with it another concept that takes into account not only the truth values ​​of statements, but also their connection in content.

Equality is closely associated with implication , sometimes referred to as “double implication”.

Equivalence is a complex statement “L, if and only if B”, formed from Li B's statements and decomposing into two implications: “if A, then B”, and “if B, then A”. For example: "A triangle is equilateral if and only if it is equilateral." The term “equivalence” also refers to a bunch of “..., if and only if ...”, with the help of which this complex statement is formed from two statements. Instead of "if and only if" for this purpose can be used "if and only if", "if and only if, when", etc.

If logical connectives are defined in terms of truth and falsehood, equivalence is true if and only if both of its utterances have the same truth value, i.e. when both are true or both are false. Accordingly, equivalence is false when one of its statements is true and the other is false.