Complex judgments, their structure and types. Complex judgments
Judgments are divided into simple and complex.
In general, simple and complex judgments are distinguished on the basis of a number of features.
A simple judgment contains only one statement or negation, a complex one - several. A simple judgment contains only one semantic unit, and a complex one contains several such units. A simple judgment can only be decomposed into concepts; from the complex, if necessary, at least two other judgments are distinguished, each of which can be assessed as true or false. These signs can be identified by considering the following judgments.
1) “Democritus is not an idealist” - a simple proposition.
2) “If it is raining, then the roofs are wet” - a difficult judgment.
A judgment is a relatively complete thought reflecting things, phenomena of the real world with their properties and relationships. A judgment has a specific structure. Its elements are subject, predicate, connective, and in some cases quantifier (quantitative) words.
The subject is knowledge about the subject of judgment (logical subject). It is designated by the letter S.
A predicate is knowledge about what is affirmed or denied about the subject of judgment (logical predicate). Designated - R.
A predicate can express both the thought about the existence of an object, about its features, properties, relationships, and the thought about our assessment of it or motives for certain actions, behavior, etc.
Link - establishes that the thinkable in the predicate is inherent or not inherent in the subject of judgment. Sometimes the link is only implied.
The subject and the predicate are called judgment terms.
Each judgment consists of three elements - two terms and a bundle. Each of these members of the judgment is necessarily present or implied in these judgments.
Inclusion judgments include the belonging of an item to a class of items or of one class to another class of items. For example: "CHVVAKUSH is a higher military educational institution."
Composition of a simple judgment
Simple judgment there is a statement about the presence or absence of any signs in any particular object, in part or in all objects of a certain class.
Structure simple judgment contains:
First, one or more subjects of judgment or logical subjects – these are parts representing objects about which something in the judgment is affirmed or denied.
Secondly, judgment predicate or logical predicate – this is partjudgments , expresses what is asserted or denied about objects that representsubjects .
Together subject and predicate are called termsjudgments and are denoted respectively by Latin symbols S and P .
Besides subjects and predicate judgment contains bundle , which, as a rule, is expressed by the words "is", "essence", "is", "to be".
For a clear illustration of the structure of the judgment, we will analyze two examples:
In the judgment "The sun is a red-hot heavenly body" subject one - "Sun", predicate - "red-hot celestial body", and bunch expressed by the word "is".
In the judgment "The earth revolves around the sun" two subject - "Earth" and "Sun", and predicate is the relation "revolves".
MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION
Federal Agency for Education
Saint Petersburg State University of Service and Economics
Institute of Law
Discipline: Logic
on the topic: Complex judgments
Saint Petersburg
Simple judgment concept
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 judgment.
The presence of an affirmation or denial of the described situation distinguishes a judgment from concepts.
A characteristic feature of a judgment from a logical point of view is that it - with its logically correct construction - is always true or false. And this is due precisely to the presence of affirmation or denial of something in the judgment. A concept, which, unlike a judgment, contains only a description of objects and situations with the aim of mentally isolating them, does not have truth characteristics.
A judgment should be distinguished from a proposal as well. Sound shell of judgment - sentence... A judgment is always a proposal, but not vice versa. Judgment is expressed in a declarative sentence in which something is asserted, denied, or communicated. Thus, interrogative, imperative and imperative sentences are not judgments. Sentence and judgment structures are not the same. The grammatical structure of the same sentence differs in different languages, while the logical structure of judgment is always the same for all peoples.
The relationship between judgment and utterance should also be noted. Utterance Is a statement or declarative sentence that can be said to be true or false. In other words, the statement that a statement is false or true must make sense. Judgment is the content of any statement. Suggestions such as "N is prime", cannot be considered a statement, since one cannot say about it whether it is true or false. Depending on what content the variable "n" will have, you can set its boolean value. Such expressions are called propositional variables. The statement is indicated by any one letter of the Latin alphabet. It is regarded as an indecomposable unit. This means that no structural unit is seen in it as a part of it. Such a statement is called atomic (elementary) and corresponds to a simple proposition. From two or more atomic statements by means of logical operators (connectives), a complex or molecular statement is formed. Unlike a statement, a judgment is a concrete unity of a subject and an object, connected in meaning.
Examples of judgments and statements:
Simple statement - A; simple judgment - "S is (is not) P".
Complicated statement - A → B; difficult judgment - “If S1 is P1, then S2 is P2”.
Composition of a simple judgment
In traditional logic, the division of judgment into subject, predicate and connective.
The subject is a 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 the judgment "Earth is a planet of the solar system" the subject is "Earth", the predicate "planet of the solar system." It is easy to see that the logical subject and the predicate do not coincide with the grammatical ones, that is, with the subject and predicate.
Together, the subject and the predicate are called terms of judgment and are denoted by the Latin symbols S and P, respectively.
In addition to terms, judgment contains a bundle. As a rule, the link is expressed by the words “is”, “essence”, “is”, “to be”. In the example shown, it is omitted.
Complex judgment concept
Difficult judgment- a judgment formed from simple ones by means of logical unions of conjunction, disjunction, implication, equivalence.
Logical union Is a way of connecting simple judgments into a complex one, in which the logical value of the latter is set in accordance with the logical values of the simple judgments that make it up.
The peculiarity of complex judgments is that their logical meaning (truth or falsity) is determined not by the semantic connection of simple judgments that make up the complex, but by two parameters:
1) the logical meaning of simple judgments included in the complex;
2) the nature of the logical connective that connects simple judgments;
Modern formal logic distracts from the meaningful connection between simple judgments and analyzes statements in which this connection may be absent. For instance, "If the square of the hypotenuse is equal to the sum of the squares of the legs, then there are higher plants on the Sun."
The logical value of a complex judgment is established using truth tables. Truth tables are constructed as follows: at the input, all possible combinations of logical values of simple judgments are written out, which make up a complex judgment. The number of these combinations can be calculated by the formula: 2n, where n is the number of simple judgments that make up a complex one. At the output, the value of a complex judgment is written out.
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 by compatible, fully or partially expressing the same thought and, incompatible if the truth of one of them necessarily implies the falsity of the other (when comparing such judgments, the law of consistency is violated). The truth relation between judgments comparable across subjects is displayed by a logical square.
The logical square underlies all inferences and is a combination of symbols A, I, E, O meaning a certain type of categorical statements.
A - Generally affirmative: All S are P.
I - Partly affirmative: At least some S are P.
E - Generally negative: All (none) S are P.
O - Partially negative: At least some S are not P.
Of these, generally affirmative and general negative are subordinate, and partial affirmative and partial negative are subordinate.
Judgments A and E are opposed to each other;
I and O are opposite;
Diagonal judgments are contradictory.
Contradictory and opposed judgments in no case can be simultaneously true. Opposing judgments 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 direct inferences and, determines that, from the truth of the subordinate judgments, it logically follows the truth of the judgments of their subordinates and the falsity of the opposite subordinate judgments.
Logical connectives. Conjunctive judgment
Conjunctive judgment- a judgment that is true if and only if all judgments included in it are true.
It is formed by means of a logical conjunction of a conjunction, expressed by grammatical unions "and", "yes", "but", "however". For instance, "It shines, but it does not warm."
It is symbolically designated as follows: А˄В, where A, B are variables denoting simple judgments, is a symbolic expression of the logical union of a conjunction.
The definition of a conjunction corresponds to a truth table:
A | V | A˄ V |
AND | AND | AND |
AND | L | L |
L | AND | L |
L | L | L |
Disjunctive judgments
There are two types of disjunctive judgments: 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 judgments included in it is true, or "which is false when both statements are false." For instance, "This number is either a multiple or not a multiple of five."
The logical union of disjunction is expressed through the grammatical union "either ... or".
AVB is written symbolically.
The logical value of strict disjunction corresponds to the truth table:
A | V | A˅ V |
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 maybe more) of the simple judgments included in the complex is true. For instance, "Writers can be either poets or prose writers (or both at the same time)".
A loose disjunction is expressed through the grammatical union "or ... or" in a separating-connecting meaning.
Written symbolically ˅ B. A loose disjunction corresponds to a truth table:
A | V | A˅ V |
AND | AND | AND |
AND | L | AND |
L | AND | AND |
L | L | L |
Implicative (conditional) judgments
Implication- a complex judgment that takes on the logical meaning of falsity if and only if the previous judgment ( antecedent) is true, and the subsequent ( consequent) is false.
In natural language, the implication is expressed by the conjunction "if ..., then" in the sense of "probably, that A and not B". For instance, "If a number is divisible by 9, then it is divisible by 3."
Complex judgments are formed from simple ones by various kinds of their connection. Typically, the characteristics of simple and complex judgments are straightforward. However, situations are possible when the border between simple and complex judgments should be recognized to a certain extent conditional. This refers to constructions in which, not without reason, one can identify 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 be considered as simple, if we proceed from the fact that the phrase "police 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 contains, along with true information, false information. This false information cannot be contained in the concept of "athlete", because the concept does not have a truth value. The carriers of the truth value are judgments. But can one judgment be the bearer of two truth values? This is possible only when the judgment consists of two judgments, i.e. is tricky. Thus, there is a reason to interpret this judgment as a complex one, consisting of two statements: "This person is a police officer" and "This person is an athlete."
Types of complex judgments by the nature of the logical union.
1. Conjunctive(or connecting) judgments. They are formed from the initial simple judgments through the logical union 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 unions: and, a, but, yes, although, and also, despite the fact that. "I will go to college, despite the fact that I have to work hard." Sometimes no unions are required. Here is a quote 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 to combine the 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 conjunction table.
2. Disjunctive(dividing) judgments.
a) a weak (non-strict) disjunction is formed by the logical union "or". It is characterized by the fact that the combined judgments are not mutually exclusive. Formula: A V B (A or B). The conjunctions "or", "or" are used here in a separating-connecting meaning. Example: "Ponzov is a lawyer or an athlete." (He can be both a lawyer and an athlete at the same time.) A weak disjunction is true when at least one of the statements is true.
The semantic boundary between conjunction and weak disjunction is, in a certain sense, conditional.
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", "or", but in a different separative - exclusive meaning. "We will survive or die." "Amnesty can be general or partial." A strong disjunction is true when one of the statements is true and the other is false.
AND |
3. Implicative(conditional judgments). They combine judgments based on the logical union "if ..., then", and "then ... when" (symbol "→"), (A → B; if A, then B). "If the weather improves, then we will find traces of the culprit." The judgment after the words “if”, “then” is called the antecedent (previous) or the basis, and the one after “then”, “when” is called the consequent (subsequent) or the effect. The implication is always true, except for the case when the basis is true, and the consequence is false. It must be remembered that the union "if ..., then" can also be used in a comparative sense ("If gunpowder itself was invented in China in ancient times, then the weapon is based on the use of the properties of gunpowder appeared in Europe only in the Middle Ages ") and, as you can easily see, can express not implication at all, but conjunction.
4. Equivalent(equivalent) judgments. They combine judgments with mutual (direct and inverse) dependence. It is formed by a logical union "if and only if ... then", "if and only if ... when", "only under condition", "only in case" the 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 the Hero of Russia. The signs "=", "≡" are also used. Equivalence is true when both statements are true, or both are false.
Equivalence can also be interpreted as a conjunction of two implications, direct and inverse: (p → q) (q → p). Equivalence is sometimes called double implication.
Summing up 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 there would be a state of affairs described by the consequent, for example: "If Pontsov were the mayor of Krasnoyarsk, he would not have lived in the dormitory."
Affirming or denying anything about the existence of objects, about the connections between them and their properties, as well as about the relationship between objects.
Examples of judgments: “The Volga flows into the Caspian Sea”, “A.S. Pushkin wrote the poem "The Bronze Horseman", "The Ussuri Tiger is listed in the Red Book", etc.
Judgment structure
A judgment includes the following elements: subject, predicate, connective, and quantifier.
- Subject (Latin subjektum - "underlying") - what is said in this judgment, its subject ("S").
- A predicate (lat. Praedicatum - "spoken") is a reflection of the attribute of an object, what is said about the subject of judgment ("P").
- A link is a relationship between a subject ("S") and a predicate ("P"). Determines whether the subject has / does not have any property expressed in the predicate. It can be either implied or denoted by the "dash" sign or by the words "is" ("is not"), "is", "is", "essence", etc.
- The quantifier (quantifier word) defines the scope of the concept to which the subject of judgment belongs. Stands before the subject, but may also be absent from judgment. It is indicated by such words as "all", "many", "some", "none", "none", etc.
True and False Judgments
A judgment is true when the presence of signs, properties and relations of objects, asserted / denied in the judgment, corresponds to reality. For example: "All swallows are birds", "9 is more than 2", etc.
If the statement contained in the judgment does not correspond to reality, we are dealing with a false judgment: "The sun revolves around the Earth", "A kilogram of iron is heavier than a kilogram of cotton wool", etc. Correct judgments form the basis of correct conclusions.
However, in addition to two-valued logic, in which a judgment can be either true or false, there is also multidimensional logic. According to its terms, the judgment may also be uncertain. This is especially true of future single judgments: "Tomorrow there will / will not be a sea battle" (Aristotle, "On the interpretation"). If we assume that this is a true judgment, then a naval battle cannot fail to happen tomorrow. Therefore, it is necessary for it to happen. Or vice versa: by asserting that a given judgment at the present moment is false, we thereby make it necessary the impossibility of tomorrow
Judgments by the type of utterance
As you know, according to the type of utterance, there are three types of incentive and interrogative. For example, the sentence “I remember a wonderful moment” is of the narrative type. It is appropriate to suggest that such a judgment would also be narrative. It contains certain information, reports on a certain event.
In turn, the interrogative sentence contains a question implying the answer: "What is the coming day for me?" At the same time, it does not state or deny anything. Accordingly, the statement that such a judgment is interrogative is erroneous. An interrogative sentence, in principle, does not contain a judgment, since the question cannot be differentiated according to the principle of truth / falsity.
An incentive type of proposals is formed when there is a certain motivation for action, a request or a prohibition: "Rise, prophet, and see and heed." As for judgments, according to some researchers, they are not contained in sentences of this type. Others believe that we are talking about a kind of modal judgments.
Quality of judgment
In terms of quality, judgments can be either positive (S is P) or negative (S is not P). In the case of an affirmative judgment, a predicate is used to impart a certain property (s) to the subject. For example: "Leonardo da Vinci is an Italian painter, architect, sculptor, scientist, naturalist, as well as an inventor and writer, the largest representative of the Renaissance art."
In a negative judgment, on the contrary, the property is subtracted from the subject: "James Wykeri's theory of the 25th frame has no experimental confirmation."
Quantitative characteristic
Judgments in logic can be general (referring to all subjects of a given class), private (to some of them) and singular (when it comes to a subject that exists in a single copy). For example, it could be argued that a judgment such as "All cats are gray at night" would refer to a generic species as it affects all felines (the subject of the judgment). The statement “Some snakes are not venomous” is an example of a private judgment. In turn, the judgment “The Dnieper is wonderful in calm weather” is an isolated one, since we are talking about one specific river that exists in a single form.
Simple and complex judgments
Depending on the structure, the judgment can be of the simple or complex type. The structure of a simple judgment includes two related concepts (S-P): "A book is a source of knowledge." There are also judgments with one concept - when the second is only implied: "It was getting dark" (P).
A complex appearance is formed by combining several simple judgments.
Classification of simple judgments
Simple judgments in logic can be of the following types: attributive, judgments with relationships, existential, modal.
Attributive (judgments-properties) are aimed at affirming / denying the presence of certain properties (attributes) in an object. These judgments have a categorical form and are not questioned: "The mammalian nervous system consists of the head and outgoing nerve pathways."
In judgments with relationships, certain relationships between objects are considered. They can have a spatio-temporal context, causal, etc. For example: “An old friend is better than two new ones”, “Hydrogen is 22 times lighter than carbon dioxide”.
An existential judgment is a statement of the existence / non-existence of an object (both material and ideal): "There is no prophet in his own country", "The moon is a satellite of the Earth."
Modal judgment is a form of statement that contains a certain modal operator (necessary, good / bad; proven, known / unknown, prohibited, I believe, etc.). For instance:
- “In Russia, it is necessary to carry out educational reform” (the aesthetic modality is the possibility, the need for something).
- “Everyone has the right to personal security” (deontic modality - moral norms of social behavior).
- "A negligent attitude towards state property leads to its loss" (axiological modality - attitude towards material and spiritual values).
- “We believe in your innocence” (epistemic modality - the degree of reliability of knowledge).
Complex judgments and types of logical connectives
As already noted, complex judgments consist of several simple ones. The following techniques act as logical connections between them:
2. Establishing the logical meaning of complex judgments using truth tables.
Complex judgments are judgments consisting of several simple judgments connected by logical unions. It is on them that the type and logical characteristics, the conditions for the 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: logical one (1 - true) or logical zero (0 - false).
There are five logical unions in total: negation, conjunction, disjunction, implication, equivalence.
Of the listed conjunctions, the unary is the negation
"not", "it is not true that".
It is symbolically depicted by the sign "" and has a truth table:
When compiled through a logical function, the truth table for the inversion will look like:
Logic highlights four kinds complex judgment with binary (paired) unions:
connective union (conjunction)
"and", "a", "but", "yes", etc. ;
dividing union (disjunction)
"or", "or" and the like;
conditional union (implication)
"if .. then";
union equivalence, identity (equivalence)
"if and only if .. then", "if and only if".
Connective view (conjunction)
Two or more simple judgments can form a complex with the help of a connecting union (" 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 in the form of the formula: (S is P) and (S is P), or p& q .
A kind of conjunctive judgment:
Judgment with complex subject : S1, S 2, S 3 is P
For example: "Description, comparison, characteristic 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"
Conjunction may to express :
▫ Simultaneity"The lecture ended and the bell rang."
▫ Sequence"The student listened to the lecture, wrote a term paper and defended it"
▫ Enumeration"Abstract, term paper, diploma - are the types of student research papers"
▫ Location"The building of the admissions office of BSUIR was on the right, and the building of the correspondence department was on the left."
Since a simple judgment by its nature can be either true or false, the main dependencies of a complex conjunctive judgment will be determined by its logical union. These dependencies are easily found in the so-called "truth tables" for logical unions developed by logic.
For conjunctions the truth table is as follows:
When compiled through a logical function, the truth table for the conjunction will look like:
Multiplication function:F= A* B |
|||
Disconnecting view (disjunction)
Two or more simple judgments can form a complex one and with the help of a dividing logical union (" either ... either "," or " and etc). With its help, it is possible to form, for example, such a complex dividing judgment: "Forests in our country are deciduous or coniferous or mixed." This judgment is written in the form of the formula: (S is P) v (S is P), or pvq .
In logic, distinguish two meanings dividing (disjunctive) union: dividing-connecting ( weak disjunction ) pvq
For example: "Each 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"Either classes, or a break"
▫ An alternative"Admission to the exam will be either a given test or testing"
A weak disjunction does not prohibit, does not exclude the simultaneous truth of simple judgments included in this complex. Thus, the above judgment "Forests are deciduous or coniferous or mixed" is an example of weak disjunction: in this case, the union "or" not only separates, but also connects, allowing the presence of the above three features in the same forest.
But a strong (strict) disjunction excludes the simultaneous truth of simple judgments included in a complex one. So, in the judgment "This animal is a wolf or a bear" union "or" performs a strictly dividing role; a given animal cannot be both at the same time.
For weak disjunction , the truth table is as follows:
When compiled through a logical function, the truth table for a weak disjunction will look like:
For strong disjunction , the truth table is as follows:
When compiled through a logical function, the truth table for a strong disjunction will look like:
Equivalent view (equivalent)
Two or more simple judgments can form a complex one with the help of a mutually conditioning (identical) union (" if and only if», « then and only then "), which is symbolically represented by the sign "≡". This union forms a complex judgment, in its truth characteristics the opposite of a strict disjunction judgment. The fact is that this union also gives a complex judgment, true only in two cases, when either all of the 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 in the form of the 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 equivalents :
When compiled through a logical function, the truth table for the equivalent will look like:
Conditional view (implication)
Two or more simple judgments can form a complex with the help of a conditional union (" if ... then», « when ... then», Etc.), which is symbolically represented by the" → "sign.
This judgment can be written in the form of the formula: (S is P) → (S is P), or p→q .
For example: "If you completed the test before the call, then you can submit it earlier."
The complex conditional proposition formed in this way consists of two elements :
· antecedent (base)(a simple proposition that is concluded between the conjunction "if" and the particle "then")
· consequent (consequence)(simple judgment following the particle "that").
Implication can to express :
▫ Causal relationship"If you turn off the lamp from the network, it will go out"
▫ Justification"Since the conclusion in the laboratory work is not made, insofar as the work is not considered credited"
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 such a mental structure, the elements of which are interconnected in meaning. True, she does not make the relationship between complex judgments the subject of her detailed research. As an exception, we can only talk about the relations and connections between conditional and dividing judgments considered by traditional logic, but traditional logic considers them as elements of a more complex form of thought - inference, as a conditionally dividing syllogism.
The relationship between the four types of complex judgments is the subject of modern formal (mathematical, or symbolic) logic. It analyzes and establishes regular dependencies between complex judgments and even has a whole list of so-called equivalence formulas, when complex judgments with one logical union in their truth value are identical to other complex judgments with other logical unions. 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 instance: ( p&q) is equivalent to “non- ( 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 union, indicating the type of judgment, and build a corresponding truth table.
Complex logical expressions
Complex logical expressions are made up of several complex judgments associated with 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 – the number of simple statements.
Define number of columns , which will be in the table.
For this, the function is used: k + n , where k – the number of different logical operations included in a complex statement.
Fill in the first n columns.
Fill in the rest of the 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 filled one.