MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION
Federal Agency for Education
St. Petersburg State University service and economy
Law Institute
Discipline: Logic
on the topic: Complex judgments
Saint Petersburg
The concept of a simple proposition
Judgment- a form of thinking through which something is affirmed or denied about an object (situation) and which has the logical meaning of truth or falsity. This definition characterizes a simple proposition.
The presence of an affirmation or denial of the described situation distinguishes a judgment from concepts.
Characteristic feature a judgment from a logical point of view is that it - if it is logically correct - is always true or false. And this is connected precisely with the presence in the judgment of an affirmation or denial of something. A concept, which, unlike a judgment, contains only a description of objects and situations for the purpose of mentally highlighting them, does not have truth characteristics.
A judgment must also be distinguished from a proposal. Sound shell of judgment – offer. A proposition is always a proposition, but not vice versa. A judgment is expressed in a declarative sentence that asserts, denies, or reports something. Thus, interrogative, imperative and imperative sentences are not judgments. The structures of the sentence and the judgment are not the same. The grammatical structure of the same sentence differs in different languages, while the logical structure of a judgment is always the same among all peoples.
The relationship between judgment and statement should also be noted. Statement is a statement or declarative sentence that can be said to be true or false. In other words, a statement about the falsity or truth of a statement must make sense. A judgment is the content of any statement. Suggestions like "the number n is prime", cannot be considered a statement, since it cannot be said about it whether it is true or false. Depending on what content the variable “n” will have, you can set its logical value. Such expressions are called propositional variables. A statement is denoted by one letter of the Latin alphabet. It is considered as an indecomposable unit. This means that no structural unit is considered as part of it. Such a statement is called atomic (elementary) and corresponds to a simple proposition. From two or more atomic statements, a complex or molecular statement is formed using logical operators (connections). Unlike a statement, a judgment is a concrete unity of subject and object, connected in meaning.
Examples of judgments and statements:
Simple statement - A; simple judgment - "S is (is not) P."
Complex statement – A→B; complex judgment - “if S1 is P1, then S2 is P2.”
Composition of a simple judgment
In traditional logic, a division of judgment into subject, predicate and connective.
The subject is the part of the judgment in which the subject of thought is expressed.
A predicate is a part of a judgment in which something is affirmed or denied about the subject of thought. For example, in a judgment “Earth is a planet of the solar system” the subject is “Earth”, the predicate is “planet” solar system" It is easy to notice that the logical subject and predicate do not coincide with the grammatical ones, that is, with the subject and predicate.
Together the subject and predicate are called in terms of judgment and are denoted by the Latin symbols S and P, respectively.
In addition to terms, a judgment contains a connective. As a rule, the connective is expressed by the words “is”, “essence”, “is”, “to be”. In the example given it is omitted.
Concept of complex judgment
Complex judgment– a judgment formed from simple ones through logical unions of conjunction, disjunction, implication, equivalence.
Logical union- this is a way of combining simple judgments into a complex one, in which the logical value of the latter is established in accordance with the logical values of the simple judgments that comprise it.
The peculiarity of complex judgments is that their logical meaning (truth or falsity) is determined not by the semantic connection of the simple judgments that make up the complex, but by two parameters:
1) the logical meaning of simple judgments included in a complex one;
2) the nature of the logical connective connecting simple propositions;
Modern formal logic abstracts from the meaningful connection between simple judgments and analyzes statements in which this connection may be absent. For example, "If the square of the hypotenuse equal to the sum squares of legs, then higher plants exist on the Sun.”
The logical meaning of a complex proposition is established using truth tables. Truth tables are constructed as follows: at the input, all possible combinations of logical values of simple judgments that make up a complex judgment are written down. The number of these combinations can be calculated using the formula: 2n, where n is the number of simple judgments that make up a complex one. The output is the value of the complex judgment.
Comparability of judgments
Among other things, judgments are divided into comparable having a common subject or predicate and incomparable that have nothing in common with each other. In turn, comparable ones are divided into compatible, fully or partially expressing the same idea and, incompatible, if the truth of one of them necessarily implies the falsity of the other (when comparing such judgments, the law of non-contradiction is violated). The relationship in truth between judgments comparable through subjects is displayed by a logical square.
The logical square underlies all inferences and is a combination of the symbols A, I, E, O, meaning a certain type of categorical statements.
A – General affirmative: All S's are P's.
I – Private affirmative: At least some S are P.
E – General negative: All (none) S are P.
O – Partial Negatives: At least some Ss are not Ps.
Of these, general affirmatives and general negatives are subordinate, and particular affirmative and particular negatives are subordinate.
Judgments A and E are opposed to each other;
Judgments I and O are opposite;
Judgments located diagonally are contradictory.
In no case can contradictory and opposing propositions be simultaneously true. Opposite propositions may or may not be true at the same time, but at least one of them must be true.
The law of transitivity generalizes the logical square, becoming the basis of all immediate inferences and determines that from the truth of subordinate judgments, the truth of the judgments subordinate to them and the falsity of the opposite subordinate judgments logically follow.
Logical connectives. Conjunctive judgment
Conjunctive judgment- a judgment that is true if and only if all the propositions included in it are true.
It is formed through a logical conjunction of conjunction, expressed by the grammatical conjunctions “and”, “yes”, “but”, “however”. For example, “It shines, but it doesn’t warm.”
Symbolically denoted as follows: A˄B, where A, B are variables denoting simple judgments, ˄ is a symbolic expression of the logical conjunction of conjunction.
The definition of a conjunction corresponds to the truth table:
| A | IN | A˄ IN |
| AND | AND | AND |
| AND | L | L |
| L | AND | L |
| L | L | L |
Disjunctive judgments
There are two types of disjunctive propositions: strict (exclusive) disjunction and non-strict (non-exclusive) disjunction.
Strict (exclusive) disjunction- a complex judgment that takes on the logical meaning of truth if and only if only one of the propositions included in it is true or “which is false when both statements are false.” For example, “A given number is either a multiple or not a multiple of five.”
The logical conjunction disjunction is expressed through the grammatical conjunction “either...or.”
A˅B is symbolically written.
The logical value of a strict disjunction corresponds to the truth table:
| A | IN | A˅ IN |
| AND | AND | L |
| AND | L | AND |
| L | AND | AND |
| L | L | L |
Non-strict (non-exclusive) disjunction- a complex judgment that takes on the logical meaning of truth if and only if at least one (but there may be more) of the simple judgments included in the complex is true. For example, “Writers can be either poets or prose writers (or both at the same time)”.
A loose disjunction is expressed through the grammatical conjunction “or...or” in a dividing-conjunctive meaning.
Symbolically written A ˅ B. A non-strict disjunction corresponds to a truth table:
| A | IN | A˅ IN |
| AND | AND | AND |
| AND | L | AND |
| L | AND | AND |
| L | L | L |
Implicative (conditional) propositions
Implication- a complex judgment that takes the logical value of falsity if and only if the previous judgment ( antecedent) is true, and the following ( consequent) is false.
In natural language, implication is expressed by the conjunction “if..., then” in the sense of “it is likely that A and not B.” For example, “If a number is divisible by 9, then it is divisible by 3.”
A complex judgment is a judgment that consists of several simple judgments. Thus, the proposition “Theft is a crime” is simple, it has one subject (“theft”) and one predicate (“crime”). The judgment “The verdict must be legal and justified” - this judgment is formed from two simple ones: “The verdict must be legal” and “The verdict must be justified.”
Complex judgments are formed from simple ones with the help of logical conjunctions: “If... then”, “and” or “and their equivalents.
Complex judgments include conditional, connecting and distributive judgments.
Most legal norms are expressed in the form of complex judgments. For example: “The parties in civil law are the plaintiff and the defendant”, “If the case is initiated without legal grounds, the prosecutor terminates it”, “Incorrect transaction that does not comply with the requirements of the law”, “Assault with the aim of taking possession of state or public property, associated with violence, dangerous for the life or health of the person attacked, or with the threat of such violence (robbery) - is punishable ... ", etc. Let's consider the types of such judgments.
Conditional proposition
A conditional (implicative) proposition is a complex proposition formed from two simple propositions in relation to a cause and a consequence, connected by the logical conjunction “if... then”. Examples of conditional propositions: “If a body is heated, it will expand,” “If the sentence is unfounded, then it is illegal.”
A conditional proposition consists of a reason and a consequence. And the part of the conditional proposition, which expresses the conditions for the existence (non-existence) of any phenomenon, is called the basis, and the part of the conditional proposition, which expresses what is determined by this condition, is called the consequence of the conditional proposition. For example, in the judgment “If a body is heated, then it will expand,” the basis is “if the body is heated,” and the consequence is “then it will expand.”
If the basis of a conditional proposition is denoted by the letter A, and the consequence by the letter I, then the structure of this conditional proposition will be expressed by the formula: if A, then B.
The logical conjunction “if... then” is called implication in mathematical logic, and the conditional proposition is called an implicate proposition. The conjunction “if... then” is denoted by the sign “->”. Using it, you can write down the structure of a conditional proposition in the formula A->B. It reads: “A implies B,” or “If A, then B.”
Not every sentence that contains the conjunction “if... then” is a conditional proposition. Thus, the sentence “If yesterday we did not know that S. would play for the main team of our football team, then today everyone knows this,” although it has the conjunction “if ... then,” is not a conditional proposition, since the conditional consequent he does not express the connection. A conditional proposition can be expressed without the conditional conjunction “if... then”, for example: “He who does not work, does not eat”, “If you hurry, you will make people laugh” and others.
In legal legislation, many conditional propositions are expressed not by the conjunction “if... then”, but by the words “in case”, “when”, etc. The “then” part of the logical conjunction “if... then” is often omitted.
Conditional propositions reflect a variety of conditional dependence of some phenomena on others. The priests reflect the causal relationship between phenomena, the sequence or simultaneity of phenomena in time, the coexistence or impossibility of coexistence of objects and phenomena or their signs is necessary, the connection between means and ends, and the like. Therefore, it is impossible to always consider the basis of a conditional judgment as a cause, and the effect - as the effect of this cause. These concepts are not identical.
A conditional proposition, like any proposition, can be either true or false.
A conditional proposition is true if it correctly reflects the conditional dependence of one phenomenon on another. If between the phenomenon referred to in the basis of a conditional proposition and the phenomenon referred to as a consequence of the conditional proposition, there really is a conditional relationship referred to in the conditional proposition, then such a conditional proposition is true, it correctly reflects the connection between phenomena.
If there is no conditional relationship between phenomena and reality, which is discussed in the conditional judgment, then such a conditional judgment is erroneous, it distorts reality. Thus, the judgment “If a body is heated, it will expand” is true, since the conditional relationship between the phenomena (heating of the body and the property of the body to expand), which is discussed in this judgment, really exists. And the proposition “If a body is heated, its volume will decrease” is false, since here we are talking about the presence of such a conditional relationship between phenomena (“heating the body” and “decreasing the volume of the body”), which in fact is absent.
A conditional proposition is true or false, both in the case when it speaks about phenomena that exist in reality, and in the case when it speaks about phenomena whose existence is possible in the future, as well as those that we know about that they do not exist and will not exist. For example, the conditional proposition “If our Earth did not have an atmosphere, then life on it would be impossible” is true; it correctly establishes the existence of a conditional relationship between the existence of an atmosphere and life on Earth.
In mathematical logic, the truth and falsity of the implication A->B is determined by the truth or falsity of simple judgments that make up implicate judgments: grounds and consequences (A and B). An implicative proposition is false only if the reason (A) is true and the consequence (B) is false. In all other cases, namely: when the reason is true and the consequence is true; the reason is erroneous, but the consequence is true; the basis is a delusion and the consequence is false - the implication A->B is true
The truth table of implicative propositions has the following form:
| A | V | A->B |
| ii and XX | andXiX | andXII |
Conditional propositions can be distinguishing or invisible. We have examined conditional invisible judgments. Let us now find out that these are conditional distinguishing judgments, or, as they are called, equivalence judgments.
An emphasizing conditional proposition (equivalence judgment) is a conditional proposition, both parts of which can be both a basis and a consequence.
For example: “If the parts of an object are parts of the same object, then the relief of the individual parts coincides.” If the consequence of this judgment is made the basis, and the basis is the consequence, then the judgment remains true: “If the relief of the individual parts coincides, then these parts are parts of the same object.” The content of the judgment has not changed.
Thus, a conditional proposition will be distinguishing if, when transforming the proposition “If A, then B” into the proposition “If B, then A,” it remains true.
The structure of a distinguishing conditional proposition can be written as follows: A ~ B.
By highlighting a conditional proposition, it is true only in two cases, namely: when the reason and consequence are true and when the reason and consequence are false. In the last two cases, when the basis is true and the consequence is erroneous and when the basis is erroneous and the consequence is true, highlighting the conditional proposition is erroneous.
Here is a truth table for distinguishing conditional propositions:
| A | IN | A ~ B |
| iiX X | andXIX | andX Xi |
2. Establishing the logical meaning of complex judgments using truth tables.
Complex judgments are judgments consisting of several simple judgments interconnected by logical unions. It is by these that the type and logical characteristics and conditions of truth of a complex judgment are determined.
The construction of truth tables goes through the construction of logical functions and has parallels with mathematical functions. That is, a simple judgment is assigned a variable that can take only two values: a logical one (1 - true) or a logical zero (0 - false).
There are five logical conjunctions in total: negation, conjunction, disjunction, implication, equivalence.
Of the listed conjunctions, negation is unary
“not”, “it’s not true that”.
It is symbolically represented by the sign “” and has a truth table:
When compiled through a logical function, the truth table for inversion will look like:
Logic highlights four types complex judgment with binary (paired) conjunctions:
connective union (conjunction)
“and”, “a”, “but”, “yes”, etc. ;
dividing conjunction (disjunction)
“or”, “either”, etc.;
conditional conjunction (implication)
"if.., then";
union equivalence, identity (equivalence)
“if and only if.., then”, “if and only if.”
Connective view (conjunction)
Two or more simple propositions can form a complex one with the help of a connecting conjunction (" A», « But», « Yes», « And", etc.), which is symbolically represented by the sign "&".
For example: “Today is Sunday and we are going out of town.”
This conjunctive judgment can be written as a formula: (S is P) and (S is P), or p& q .
Type of conjunctive judgment:
Judgment with complex subject : S1, S 2, S 3 are P
For example: “Description, comparison, characterization are the main types of implicit definitions”
Judgment with complex predicate : S is P1 and P2
For example: “BSUIR – knowledge and lifestyle”
Judgment with complex subject and predicate : S1, S 2, S 3 are P1 and P2
For example: “Engineers, programmers, economists are graduates of our university and employees of many enterprises”
The conjunction can to express :
▫ Simultaneity“The lecture ended and the bell rang”
▫ Subsequence“The student listened to the lecture, wrote a term paper and defended it”
▫ Transfer"Abstract, course work, diploma - are types of student scientific work"
▫ Location“The BSUIR admissions office building was on the right, and the correspondence department building was on the left”
Since a simple proposition by its nature can be either true or false, the main dependencies of a complex conjunctive proposition will be determined by its logical conjunction. These dependencies are easily detected in the so-called “truth tables” developed by logic for logical unions.
For conjunctions The truth table is:
When compiled through a logical function, the truth table for the conjunction will look like:
|
Multiplication function:F= A* B |
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Disjunctive view (disjunction)
Two or more simple propositions can form a complex one with the help of a dividing logical conjunction (“ either...or", "or" and etc). With its help, you can form, for example, such a complex disjunctive judgment: “The forests on the territory of our country are deciduous or coniferous or mixed.” This judgment is written in the form of a formula: (S is P) v (S is P), or pvq .
In logic there is a distinction two meanings dividing (disjunctive) conjunction: dividing-connecting ( weak disjunction ) pvq
For example: “Every student knows the name of the rector of BSUIR or at least the name of his faculty”
Strictly dividing union (strict, or strong disjunction ). pv q
Disjunction can to express :
▫ Choice“It’s either class or a break”
▫ Alternative“Admission to the exam will be either the given test, or testing"
A weak disjunction does not prohibit or exclude the simultaneous truth of simple judgments included in this complex one. Thus, the above proposition “Forests are deciduous or coniferous or mixed” is an example of a weak disjunction: in this case, the conjunction “or” not only separates, but also connects, allowing the presence of the listed three characteristics in the same forest.
But a strong (strict) disjunction excludes the simultaneous truth of simple judgments included in a complex one. Thus, in the judgment “This animal is a wolf or a bear,” the conjunction “or” plays a strictly dividing role; a given animal cannot be both at the same time.
For weak disjunction , the truth table is:
When compiled through a logical function, the truth table for a weak disjunction will look like:
For strong disjunction , the truth table is:
When compiled through a logical function, the truth table for a strong disjunction will look like:
Equivalent view (equivalence)
Two or more simple propositions can form a complex one with the help of a mutually conditioning (identical) conjunction (“ if and only if», « then and only then"), which is symbolically represented by the sign “≡”. This union forms a complex judgment, whose truth characteristics are opposite to the judgment of strict disjunction. The fact is that this union also gives a complex judgment, true only in two cases, when either all simple judgments included in the complex are true, or all are false. For example, “Triangles have equal angles if and only if their sides are equal,” or “If and only if the angles of a triangle are equal, then its sides are also equal.”
This judgment is written as a formula: (S is P) ≡ (S is P), or p≡q .
For example: “You can become a student of BSUIR if and only if....”
Truth table for equivalence :
When compiled through a logical function, the truth table for the equivalence will look like:
Conditional view (implication)
Two or more simple propositions can form a complex one using a conditional conjunction (“ if... then», « when..., then", etc.), which is symbolically represented by the sign "→".
This judgment can be written as a formula: (S is P) → (S is P), or p→q .
For example: “If you completed the test before the bell, you can turn it in early.”
The complex conditional proposition thus formed consists of two elements :
· antecedent (base)(a simple proposition that is concluded between the conjunction “if” and the particle “then”)
· consequent (consequence)(a simple judgment following the particle “that”).
The implication may to express :
▫ Causality“If a lamp is unplugged, it will go out.”
▫ Rationale“Since the conclusion in the laboratory work is not made, the work is not considered valid”
Truth table for implications :
When compiled through a logical function, the truth table for the implication will look like:
Traditional formal logic considers the structure of complex judgments as a mental construction, the elements of which are interconnected in meaning. True, she does not make the relationship between complex judgments the subject of her detailed study. As an exception, we can only talk about the relations and connections between conditional and disjunctive judgments considered by traditional logic, but traditional logic considers them as elements of a more complex form of thought - inference, as a conditional disjunctive syllogism.
The relationships between the four types of complex judgments are the subject of modern formal (mathematical or symbolic) logic. It analyzes and establishes natural dependencies between complex judgments and even has a whole list of so-called equivalence formulas, when complex judgments with one logical conjunction are identical in their truth value to other complex judgments with other logical conjunctions. That is, we are talking about the interchangeability of logical unions. Thus, equivalence can be expressed by implication, implication by disjunction, disjunction by conjunction, and vice versa.
For example: ( p&q) is equivalent to “not-( p→ not- q)" and is equivalent to "not-(not- p v not- q)»;
(p v q) is equivalent to not-(not- p& Not- q);
(p→ q) is equivalent to (not- p v q); (p≡q) is equivalent to ((not- p v q) & (Not- p v q)).
A complex judgment can not only consist of several simple judgments, but also include several logical connectives: (p&q) → p. To establish the truth of such a judgment, it is necessary to establish the main logical conjunction indicating the type of judgment and construct the corresponding truth table.
Complex logical expressions
Complex logical expressions consist of several complex judgments connected using logical operations. When compiling these truth tables, it is necessary to take into account the sequence: 1) inversion 2)conjunction 3)disjunction 4)implication 5)equivalence. To change the specified order, use parentheses!
There is also a certain algorithm for compiling such tables:
Define number of lines , which will be in the table.
2 n + 2 , Where n – number of simple utterances.
Define number of columns , which will be in the table.
To do this, use the function: k + n , Where k – the number of different logical operations included in a complex statement.
Fill in first n columns.
Fill in the remaining columns. In accordance with the truth tables of the corresponding logical operations, and when filling each column, operations are performed on the values of one or two columns located to the left of the one being filled.
Complex judgments are formed from simple ones by combining them in various ways. Typically, the characteristics of simple and complex judgments do not cause difficulties. However, situations are possible when the boundary between simple and complex judgments should be recognized to a certain extent as conditional. This applies to such constructions in which, not without reason, one can identify either one statement (or negation), or two or three. The assessment of a detailed judgment as simple or complex depends to a certain extent on the position of the researcher. Let’s take the judgment: “This person is a police officer and an athlete.” It can also be considered as simple, if we proceed from the fact that the phrase “internal affairs officer and athlete” expresses one concept. On the other hand, we can assume that the person in question is an employee, but has never been involved in sports. It turns out that the construction we are considering, along with true information, also contains false information. This false information cannot be contained in the concept “athlete”, because the concept does not have a truth value. The bearer of truth value is judgment. But can one judgment be the bearer of two truth values? This is only possible when the judgment consists of two judgments, i.e. is complex. Thus, there is reason to interpret this judgment as complex, consisting of two statements: “This person is a police officer” and “This person is an athlete.”
Types of complex judgments according to the nature of the logical conjunction.
1. Conjunctive(or connecting) propositions. They are formed from initial simple judgments through the logical conjunction of the conjunction “and” (symbolically “”) A B, i.e. A and B. In Russian, the logical conjunction of a conjunction is expressed by many grammatical conjunctions: and, a, but, yes, although, and also, despite the fact that. “I will go to college, even though it will be a lot of work.” Sometimes no alliances are required. Here is a statement from one of the American presidents of the early 20th century: “We are facing a new era in which we will obviously rule the world.”
There are 4 possible ways of combining two initial judgments “A” and “B”, depending on their truth and falsity. A conjunction is true in one case if each of the propositions is true. Here is a table of the conjunction.
2. Disjunctive(divisive) judgments.
a) a weak (non-strict) disjunction is formed by the logical conjunction “or”. It is characterized by the fact that the combined judgments do not exclude each other. Formula: A V B (A or B). The conjunctions “or” and “or” are used here in a dividing and connecting sense. Example: “Pontsov is a lawyer or an athlete.” (He may turn out to be both a lawyer and an athlete at the same time). A weak disjunction is true when at least one of the propositions is true.
The semantic boundary between conjunction and weak disjunction is in a certain respect arbitrary.
b) strong (strict) – logical union “either...or”, . Its components (alternatives) exclude each other: A B. (either A or B). It is expressed essentially by the same grammatical means as the weak one: “or”, “either”, but in a different dividing-exclusive meaning. "We will live or die." “Amnesty can be general or partial.” A strict disjunction is true when one of the propositions is true and the other is false.
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AND |
3. Implicative(conditional propositions). They combine judgments based on the logical conjunction “if..., then” and “then..., when” (symbol “→”), (A → B; if A, then B). “If the weather improves, we will find traces of the criminal.” The judgment that comes after the words “if”, “then” is called the antecedent (preceding) or the basis, and the one that comes after “then”, “when” is called the consequent (subsequent) or consequence. The implication is always true, except for the case when the reason is true and the consequence is false. It must be remembered that the conjunction “if ... then” can also be used in a comparative sense (“If gunpowder itself was invented in China in ancient times, then weapons based on the use of the properties of gunpowder appeared in Europe only in the Middle Ages") and, as is easy to see, can express not an implication at all, but a conjunction.
4. Equivalent(equivalent) judgments. They combine judgments with mutual (direct and inverse) dependence. It is formed by the logical union “if and only if..., then”, “if and only if..., when”, “only if”, “only if” symbol “↔” (A ↔ B), if and only if A , then B). “If and only if a citizen has great services to the Russian Federation, he has the right to receive the high award of the Order of Hero of Russia.” The signs “=” and “≡” are also used. An equivalence is true when both propositions are true or both are false.
Equivalence can also be interpreted as a conjunction of two implications, direct and inverse: (р→q) (q → р). Equivalence is sometimes called double implication.
Summarizing what has been said about complex judgments, it should be noted that some also distinguish the so-called counterfactual judgment (the conjunction “if..., then”, the symbol “● →”. This is a sign of counterfactual implication. The meaning is this: the situation described by the anti-incident does not take place, but if it existed, then the state of affairs described by the consequent would exist. For example: “If Pontsov were the mayor of Krasnoyarsk, he would not live in a hotel.”
A complex proposition is a proposition consisting of several simple ones connected by logical connectives. The following types of complex judgments are distinguished: 1) connecting, 2) dividing, 3) conditional | nye, 4) equivalent. The truth of such complex judgments is determined by the truth of their simple constituents.1. Connective (conjunctive) propositions, j
A connective, or conjunctive, is a proposition consisting of several simple ones connected by the logical connective “and”. For example, the proposition “Theft and fraud are intentional crimes” is a connective proposition consisting of two simple ones: “Theft is an intentional crime,” “Fraud refers to intentional crimes.” If the first is denoted by p and the second by q, then it is connective;
a judgment can be expressed symbolically as p l q, where p and q are terms.^ conjunction (or conjunctions), l is the symbol of the conjunction. |
In natural language, the conjunctive connective can also be represented by such expressions as: “a”, “but”, “and also”, “as well”, “although”, “however”, “despite”, “ at the same time" and others. For example: “When the court establishes the amount of compensation to be compensated-| In the case of damage, not only the losses caused must be taken into account! (p), but also the specific situation in which the losses were incurred. repaired (q), as well as the financial situation of the employee (d).” Sim-,| This judgment can be expressed literally as follows: r l q l g.
A connecting proposition can be either two- or multi-component; in symbolic notation: r l q l g l... l p. Let us give an example of a connecting proposition that includes more than 20 conjuncts:
“The cart rushes through potholes, Booths, women, Boys, shops, lanterns, Palaces, gardens, monasteries, Bukharians, sleighs, vegetable gardens flash past, Merchants, shacks, men, Boulevards, towers, Cossacks, Pharmacies, fashion stores, Balconies, lions on the gates And flocks of jackdaws on the crosses.”
(A.S. Pushkin)
In language, a connecting proposition can be expressed by one of three logical-grammatical structures.
1. The connective connective is represented in a complex subject according to the scheme: Si and S2 are P. For example: “Confiscation of property and deprivation of rank are additional criminal sanctions.”
2) The connective is represented in a complex predicate according to the scheme: S is Pi and Pi. For example: “A crime is a socially dangerous and illegal act.”
3) The connection is represented by a combination of the first two methods according to the scheme: Si and Si are Pi and P2. For example: “Nozdryov was also on friendly terms with the police chief and the prosecutor and treated him in a friendly manner” (N.V. Gogol). p q pAq and I. I and L L l I L l L L
A conjunctive proposition is true if all its constituent conjuncts are true and false if at least one of them is false. The conditions for the truth of the judgment p l q are shown in the table (Fig. 31), where truth is denoted by I and falsity by L. In the first two columns of the table, p and q are taken as independent and therefore accept all possible combinations of values I and L: II, IL , LI, LL. The third column shows the value of the judgment r l q. Of the four line-by-line options, it is true only in the 1st line, when both conjuncts are true: p and q. In all other cases it is false: in the 2nd
and the 3rd lines due to the falsity of one of the terms, and in the 4th due to the falsity of both terms.
2. Separation (disjunctive) judgments.
A disjunctive, or disjunctive, is a proposition that consists of several simple ones connected by the logical connective “or”. For example, the proposition “A sales contract may be concluded orally or in writing” is divided.? a special judgment consisting of two simple ones: “Purchase agreement;
agreement may be concluded orally”; “A purchase and sale agreement? may be concluded in writing."
If the first is denoted by p, and the second by q, then the dividing judgment is symbolic! can be expressed as p v q, where p and q are the terms of the disjunction (disjunctions), v is the symbol of the disjunction.A disjunctive judgment can be either two- or multiple-stated: р v q v ... v p.
In language, a disjunctive judgment can be expressed in one way1| of three logical-grammatical structures. ;
1) The disjunctive connective is presented in a complex subject p2) The disjunctive connective is presented in a complex predicate p3) The disjunctive connective is represented by a combination of the first two methods according to the scheme: Si or S2 is PI or P2. For example: “Link! expulsion can be applied as a main or additional |
no sanctions." |
Lax and strict disjunction. Since the copula “or” is simplified in natural language in two meanings - connective-separation and exclusive-separation, then two types of disjunctive judgments should be distinguished: 1) non-strict (weak) conjunction and 2) strict (strong) disjunction.
1) A loose disjunction is a judgment in which the connective “or” is used in a connecting-disjunctive meaning (si) (vol v). For example: “Melee weapons can be piercing and cutting” symbolically p v q. The connective “or” in this case is F divides, since such types of weapons exist separately, and unites^ because there are weapons that are both piercing and cutting
The truth conditions of a non-strict disjunction are presented in Te face (Fig. 32). The judgment p v q will be true if XG is true of one member of the disjunction (1, 2, 3rd lines - II, IL, L!
P q pvq I I I L I l I and l L l
A disjunction will be false if both its members are false (4th line - LL).
2) Strict disjunction - a judgment in which the connective “or” is used in a disjunctive meaning (symbol?). For example: “An act may be intentional or careless,” symbolically p? q.
The terms of a strict disjunction, called alternatives, cannot both be true. If an act was committed intentionally, then it cannot be considered careless, and, conversely, an act committed through negligence cannot be considered intentional. p q P^q and and L and l I l and I l l l
The truth conditions for a strict disjunction are presented in the table (Fig. 33). Judgment p? q will be true if one member is true and the other is false (2nd and 3rd lines IL, LI); it will be false if both terms are true (1st line - AI) or both are false (4th line - LL). Thus, a judgment of strict disjunction will be true if one alternative is true and false - both if the alternatives are both false and simultaneously true.
The disjunctive connective in language is usually expressed using the conjunctions “or”, “either”. In order to strengthen the disjunction to an alternative meaning, double conjunctions are often used: instead of the expression “p or q” they use “or p or q”, and together “p or q” - “either p or q”. Since the grammar does not have unambiguous conjunctions for non-strict and strict division, the question of the type of disjunction in legal and other texts must be resolved by a meaningful analysis of the corresponding judgments.
In legal, political and other contexts, disjunction is used to reveal the content and scope of concepts, describe types of offenses or sanctions, describe crimes and civil offenses.
Complete and incomplete disjunction. Among disjunctive judgments, one should distinguish between complete and incomplete disjunction.
A disjunctive judgment is called complete or closed, which lists all the characteristics or all types of a certain kind.
Symbolically, this judgment can be written as follows." For example: “Forests are deciduous, coniferous or mixed.” The completeness of this division (in symbolic notation^ is indicated by the sign<...>) is determined by the fact that there are no other types of forests in addition to those indicated. |
A disjunctive judgment is called incomplete or open, in which not all characteristics or not all types of a certain kind are listed. In symbolic notation, the incompleteness of the disjunction can be! be expressed by ellipsis: р v qv r v... In natural language it is not | the completeness of the disjunction is expressed in words; “etc.,” “and others,” “and that,” like,” “others,” and others.
3. Conditional (implicative) propositions.
Conditional, or implicative, is a proposition consisting of two simple ones connected by the logical connective “if.., then...”, For example: “If the fuse melts, then the electric lamp goes out-| No". The first judgment - “The fuse is melting” is called the antecedent (preceding), the second - “The light bulb goes out” - the consequent (subsequent). If the antecedent is denoted by p, the consequent by q, and the connective “if..., then...” by the sign “->”, then the implicative proposition can be symbolically expressed as p->q:
The conditions for the truth of an implicative judgment are shown in the table (Fig. 34). The implication is true in all cases except one: P q p-»q and I I and L L l I I l l I
if the antecedent is true and the consequent is false (2nd line), the implication will always be false, i The combination of a true antecedent, for example, “The fuse melts,” and a false consequent, “The light bulb does not go out,” is an indicator of the falsity of the implication. I
The truth of the implication is explained as follows. In the 1st line the truth of p implies
the truth of q, or in other words: the truth of the antecedent is sufficient for the truth of the consequent. And indeed, if the fuse melts, then the electric lamp necessarily goes out due to their sequential inclusion in the electrical circuit.
In the 3rd line, if the antecedent is false - “The fuse does not melt”, the consequent is true - “The electric lamp goes out.” The situation is quite acceptable, because the fuse may not melt, but the electric lamp may go out due to other reasons - lack of current in the circuit, burnt out filament in the lamp, short circuit
electrical wiring, etc. Thus, the truth of q when p is false does not disprove the idea that there is a conditional dependence between them, since if p is true, q will always be true.
In the 4th line, if the antecedent - “The fuse does not melt” - is false, the consequent is also false - “The electric lamp does not go out.” Such a situation is possible, but it does not cast doubt on the fact of the conditional dependence of p and q, because if p is true, q will always be true.
In natural language, not only the conjunction “if..., then...” is used to express conditional propositions, but also other conjunctions:
“there... where”, “then..., when...”, “to the extent..., since...”, etc. In the form of conditional propositions in language, such types of objective connections as causal, functional, spatial, temporal, legal, as well as semantic, logical and other dependencies can be presented. An example of a causal proposition is the following statement: “If water is heated at normal atmospheric pressure to 100°C, it will boil.” An example of a semantic dependence: “If a number is divisible by 2 without a remainder, then it is even.”
In legal texts, legal regulations are often recorded in the form of conditions of judgment: permissions, prohibitions, obligations. Grammatical indicators of implication can be, in addition to the conjunction “if..., then...”, such phrases as: “if there is..., it follows”, “in case..., it follows...”, “with condition..., comes..." and others. At the same time, legal implications can be constructed in the law and other texts without special grammatical indicators. For example: “Secret theft of someone else’s property (theft) is punishable...” or “Knowingly false denunciation of a crime is punishable...”, etc. Each of these prescriptions has an implicative formula: “If a certain illegal act is committed, then legal sanction follows.”
Logical dependencies between statements are often expressed in the form of conditional propositions. For example: “If everything that is criminal is punishable, then not everything that is punishable is criminal.” Or another example of reasoning: “If it is true that some birds fly to warm regions in winter, then it is false that not a single bird flies to warm regions.”
In a conditional proposition, the antecedent serves as a factual or logical basis that determines the adoption of the corresponding consequence in the consequent. The dependence between the antecedent-ground and the consequent-consequence is characterized by the property of sufficiency. This means that truth is based
tion determines the truth of the consequence, i.e. if the basis is true, the consequence will always be true (see 1st line in the table, Fig. 34). In this case, the base is not characterized by the property of the necessity of ala. consequence, because if it is false, the consequence can be both true and false (see 3rd and 4th rows in the table, Fig. 34).
4. Equivalent judgments (double implication). Equivalent is a judgment that includes, as components, two judgments connected by a double (direct and inverse) conditional dependence, expressed by the logical connective “if i only if.
.., That...". For example: “If and only if a person has been awarded orders and medals (p), then he has the right to wear the corresponding order bars (q).”The logical characteristic of this judgment is that the 41 truth of the statement about the award (p) is considered as a necessary and sufficient condition for the truth of the statement about the existence of the right to wear order bars (q). In the same way, the truth of the statement about the existence of the right to wear order plano! (q) is a necessary and sufficient condition for the truth of the statement that the person has been awarded the appropriate order or medal (p). Such mutual dependence can be symbolically expressed by the double implication pt^q, which reads: “If and only if p, then q.” Equivalence is expressed by another sign: p = q.
In natural language, including legal texts, etc. expressions of equivalent judgments use conjunctions: “only when. provided that..., then...”, “if and only if..^ then...”, “only if..., then...” and others. p q p=q and I I and L L l I L l L I
The conditions for the truth of an equivalent judgment are presented in the table (Fig. 35). Judgment p = . true in those cases when both judgments take on the same meaning, being simultaneously either true (1st line) or false (4th line). This means| that the truth of p is sufficient for | recognition of q as true, and vice versa. 1 Fig-35 The relationship between them is characteristic
is also considered necessary: the falsity of p serves as an indicator of the falsity of q, and the falsity of q indicates the falsity of p.
In conclusion, we present a summary table of the conditions for the truth of complex judgments (Fig. 36). P q PAQ pvq P^q P-»q psq I I I I L I I I L I I I
Complex judgments and interpretation of norms.
(^false judgments - connecting, dividing, conditional and equivalent - are used in ordinary reasoning and legal contexts, both independently and in combination, i.e. in various combinations. So, for example, in a connecting judgment, disjunctive judgments can act as conjuncts: (p v q) l (m v p).In a disjunctive judgment, connecting judgments can act as its members, for example: (p nq) v (m l p).The antecedent and consequent of a conditional judgment can also be conjunctively or disjunctively related judgments, for example : (p v q) -> (m l p).
Using a combination of complex judgments, they describe regulatory requirements, define legal concepts, as well as elements of criminal offenses and torts. In the process of interpreting rules of law and various kinds of legal documents (contracts, agreements, etc.), a thorough and accurate logical and grammatical analysis of their structure, identifying the types and sequence of logical connections between the components of a complex judgment is required.
Technical symbols such as brackets play an important role here. In logic, their function is similar to the use of parentheses in the language of mathematics. For example, the arithmetic expression “2 x 3 4=...” cannot be recognized as definite and clear until the sequence of multiplication and addition operations is established. In one case it takes the value “(2 x 3) 4=10”, in the other “2 x (3 4)=14”.
The statement “The crime was committed by A and B or C” is also not distinguished by certainty, since it is not clear which of the two logical connectives - conjunction or disjunction - is the main one. The statement can be interpreted as "A and (B or C)"; it can be interpreted in another way - “(A and B) or C.” In terms of logical significance, these two statements are far from equivalent.
As an example, let us identify the structure, or logical form, of the article providing for liability for fraud, which states: “Taking possession of personal property of citizens or acquiring the right to property by deception or abuse of trust (fraud) is punishable by imprisonment for a term of up to two years with a fine of up to. ..or correctional labor for up to two years.”
In general, this statement, despite the absence of obvious grammatical indicators, is a conditional proposition of the “D-” S type. It contains legally significant actions (D) as an antecedent, and a sanction (S) as a consequent. At the same time, the antecedent and consequent are complex structural formations.
The antecedent (D) lists the actions that together constitute fraud: “Taking possession of personal property of citizens (di) or acquiring the right to
property (d2) by deception (di) or breach of trust (d4).” Grammar This analysis allows us to present the connection between the noted actions in the following form: di or d2 and d3 or d4; symbolically - (di v dz) l (d3 vd4). Of course, in this form the antecedent is not sufficiently definite, since i allows for double reading: the first option (di v dz) n(d3 v d4); second option di v (d2 l ((d3 v d4)).
In this case, the grammatical analysis of the text of the article should be supplemented with a logical one. If we compare the concept of fraud with other property crimes, we can conclude that of the two given, the first interpretation option is correct. In this case, fraud is understood as actions related to the seizure of personal property of citizens or the acquisition of rights to property; Moreover, both the first and the second are carried out through deception) or abuse of trust. This is precisely the meaning represented by the formula (di v d2) l (d3 v d4).
The consequent (S) provides a complex sanction: fraud is “punishable by imprisonment for a term of up to two years (Si) with a fine of up to... ($2) or correctional labor for a term of up to two years (S3).” The connection between the constituent parts of the consequent has the following form: Si and S2 or 8з, or symbolically ((Si l S2) v Sa). A logical analysis of the text shows that such an interpretation is the only possible one.
If the initial conditional proposition D-»S is detailed in accordance with the analysis, then the article about fraud is presented in the following form
((di v d2) l (d3 v d4)) -> ((Si l S2) v S3)
The main sign of this complex judgment is the implication: the antecedent of the judgment is a conjunction, both members of which are the disjunctive of the expression; the consequent of a judgment is a disjunctive expression, one of the members of the mastery of the skills of logical analysis of complex statements using symbolic language to understand the meaning of legal contexts! effective means accurate interpretation and correct application norms (legal process.
