Elements of the theory of names. Logic of names. Types of names General characteristics of a name. Principles of Naming Theory
Logical form- this is a characteristic of thought that does not depend on its specific content, but serves to connect and organize its elements.
In language, the logical form is fixed using propositional, nominal and other variables, as well as logical constants.
Boolean constant is a functor that retains its meaning in any argument. Logical constants are denoted by symbols. Some symbols have already been mentioned, let's name the rest. The functor “and” is denoted by L, “or” by v, “either or” by v, “if, then” – ®, “if and only if when” – “, “it is not true that” – Ø, “it is necessary that” – ÿ, “it is possible that” – à.
It should be noted that questions about the correctness of thinking and the logical consequence of some thoughts from others cannot always be resolved based on “common sense.” Formal logic explores and offers precise ways to solve such questions.
Formal logic is the science of the laws of correct thinking, i.e. such thinking, in which a transition is achieved from previously established positions to new knowledge based on mental patterns formed as a result of repeated repetition in the process of comprehending the truth. Formal logic substantiates its conclusions by creating and using one of the varieties of special, so-called. a formalized language whose sentences consist exclusively of logical constants and variables
Logical law. Correct and incorrect reasoning.
Logical law(or logical truth) is a logical form that generates a true sentence with any substitution of their values (specific content) instead of variables.
The use of forms that are logical laws allows you to remain within the framework of true knowledge and, on the basis of true knowledge, obtain new true knowledge. Reasoning, the form of which is a logical law, is called correct. Deviation from the requirements of logical laws leads to a violation of the correctness of thinking. Correctness is different from the truth of reasoning. Truth characterizes thinking in its relation to reality; if a thought is true, it corresponds to reality. Correctness characterizes reasoning from the point of view of the internal connection between its elements. Correct reasoning can lead to false conclusions. This is possible when the original data is false.
Maintaining correctness with true initial data leads to true results. At the same time, correctness can be defined as a special kind of truth. Logical connections are in accordance with the external world, reflecting the simplest and most universal relations in it. Therefore, logical laws are defined using the concept of truth and are themselves called logical truths. Cognitive errors associated with an incorrect understanding of reality are called meaningful. Errors associated with a violation of correct thinking are called formal or logical. They are divided into paralogisms And sophistry.
Paralogism is an unintentional logical error. As a rule, it is a product of a low logical culture. Sophism- a deliberate violation of the requirements of logic, a technique of intellectual fraud associated with an attempt to pass off a lie as the truth. Etc.: what you haven't lost, you have. - Yes. “You didn’t lose your horns.” Therefore you are horned.
Topic 2. Statement.
1. Propositional logic as the simplest and most fundamental section of formal logic. The concept of utterance. Logical meanings of the statement. Statement, question, command.
2. Statements are simple and complex. Logical conjunctions: conjunction, weak disjunction, strong disjunction, implication, equivalence, negation. Expressing complex statements in symbolic form. Relations between logical forms of statements. Relations of comparability and incomparability. Compatibility relations: following, complete compatibility (equivalence), partial compatibility, cohesion. Relationships of incompatibility: contradiction, opposition.
3. The concept of law in propositional logic. A tabular method for selecting laws in propositional logic. The simplest laws of propositional logic: laws of identity, contradiction, excluded middle. A shortened method for selecting logical laws.
1. Propositional logic as the simplest and most fundamental section of formal logic. The concept of utterance. Logical meanings of the statement. Statement, question, command.
A logical theory that studies the connections between statements, ignoring their internal structure, is called propositional logic or propositional logic. This is the simplest and at the same time fundamental part of formal logic. In it under the statement is understood as a linguistic expression about which only one of two things can be said: whether it is true or false.
Accordingly, truth and falsity act as the logical meanings of a statement.
Should be considered that individual words, when they are not representatives of statements (for example, like “It got cold”), questions, requests And orders statements are not.
2. Statements are simple and complex. Logical conjunctions: conjunction, weak disjunction, strong disjunction, implication, equivalence, negation. Expressing complex statements in symbolic form.
Statements, and their logical forms, can be simple (atomic) or complex (molecular). Simple statements are usually denoted by lowercase letters of the Latin alphabet: p, q, r, ... Capital letters, A, B, C, D, can be used as variables of any statement, simple or complex.
Complex statements are formed with the help of special functors, or logical unions, the most important of which are negation, conjunction, weak and strong disjunction, implication, equivalence. A complex statement is called by the name of the functor with the help of which it is formed.
Let us write down the definitions of these statements and their expressions in symbolic form:
Conjunction(logical product) is a molecular statement, true if and only if all its constituent statements (arguments) are true. Denoted: A Ù B, read: A and B. In colloquial language, conjunctions correspond to the conjunctions “a”, “but”, “yes”, “although”, “however”, etc.
Weak(not exclusive) disjunction is a complex statement that is true if and only if at least one of its arguments is true. (Logical addition). Designated: A Ú B, read: “A or B”; “or” is used in a non-exclusive sense.
Strong(exclusive) disjunction is a complex statement that is true if and only if only one of its arguments is true. Designated: A Ú B, reads: “either A or B.”
Implication is a molecular statement, false if and only if antecedent is true, and consequent false Antecedent or the basis is an expression before operator of implication, and consequent- what comes after. The implication is denoted as “A ® B” and reads: “if A, then B,” or “from A follows B.”
Equivalence- a molecular statement that is true if and only if both argument either truth, either false. That is, when their logical values coincide. It is designated: A «B, it reads: “And if and only if B”, “A if and only if B”.
Here is the truth table for these statements:
Denial of a statement A is a statement that is true if and only if A is false. It is designated A and reads: “not-A”, “it is not true that A”. The definition is expressed using the following table, where “I” stands for “true” and “L” stands for “false”:
| A | A |
| AND | L |
| L | AND |
The listed operations are used both for actions with simple and complex statements, knowing the logical values of the original statements, one can compile truth tables for statements of a more complex form. The order of operations, as in mathematical examples, will be indicated by parentheses. Ex: if I am tired or want to sleep, then I cannot translate this text. This statement is an implication, the antecedent of which is a complex statement - a weak disjunction.
By connecting statements with the help of conjunctions, we can connect their logical forms with them.
3. Relations between logical forms of statements. Relations of comparability and incomparability. Compatibility relations: following, complete compatibility (equivalence), partial compatibility, cohesion. Relationships are incompatible sti: contradiction, opposition.
When discussing practical and scientific issues, various provisions and opinions are compared. They are compared, contrasted, contrasted, and thus enter into various logical relationships with each other. Logical relations between statements are established through the relations of logical forms into which these statements are embodied. Comparable and incomparable forms are distinguished.
The logical forms alpha and beta are comparable, if and only if there is at least one variable contained in both alpha and beta. Etc.: the forms of statements A Ù B and C ® B are comparable, but A Ù B and C ® D are not. That is:
The two statements are comparable if and only if there is at least one simple statement included in the structure of both the first and second statements.
Among comparable logical forms, a distinction is made between compatible and incompatible.
Compatibility of Logical Forms determined by the presence of at least one case when they contain statements that are together as true. Logical forms incompatible in the absence of such a case. Etc.: the forms of statements A Ù B and A Ú B are compatible. Thus, when substituting A and B, true statements are generated that are both true. This can be seen from the table:
| A | IN | A Ù B | A Ú B |
| AND | AND | AND | AND |
Forms A Ú B and A "B are incompatible. Given the same values of A and B, they do not have the same value as “true”.
Compatible Shapes are in a relationship: a) following or obeying, b) , V) partial compatibility, G) clutch.
are in a relationship following or obeying , i.e. from alpha follows beta if and only if whenever the form alpha is transformed into a true statement, the form beta, for the same values of the variables, is also transformed into a true statement.
Alpha and beta forms are in a relationship full compatibility or equivalence , if and only if whenever the first corresponds to a true statement, the second also corresponds to a true statement and vice versa. That is, with the same values of the components, the logical meanings of the statements completely coincide. In relation to equivalence there are also statements of the following logical forms:
The relation of equivalence allows, in the process of reasoning, without compromising the meaning, to interchange statements of different forms, as in all of these cases, to eliminate redundant information, as in cases 10, 13, to identify new forms - 12, 15. Formulas that are in a relationship of complete compatibility follow each other from a friend, i.e. are in a relationship mutual succession.
Logical forms alpha and beta are in a relationship partial compatibility , if and only if they correspond to statements that can be both true, but cannot be false together.
Incompatible logical forms are in a relationship: a) contradictions, b) inconsistency.
Logical forms alpha and beta are in a relationship contradictions, if and only if with their help statements are generated that cannot be both true and cannot be false together.
Logical forms alpha and beta are in a relationship disgusting , if and only if they correspond to statements that cannot be both true, but can be false together.
Comparable logical forms alpha and beta are in a concatenation relation if and only if the truth (falsity) of statements of the alpha form does not exclude the falsity (truth) of statements of the beta form, and vice versa.
Establishing relationships between logical forms facilitates meaningful analysis and ensures accuracy and certainty of reasoning.
4. The concept of law in propositional logic. A tabular method for selecting laws in propositional logic. The simplest laws of propositional logic: laws of identity, contradiction, excluded middle. A shortened method of lo selection gical laws.
The laws of logic characterize the correctness of the construction of logical thinking, the process of its flow from the point of view of its certainty, consistency, and consistency of validity. Human practice confirms the adequacy of logical connections to general connections and relationships between things. The laws of formal logic are related to the truth of thinking, but not directly, but indirectly. The correctness of thinking is compatible with both its truth and falsity. Etc.: From the false statements “all fish are mammals” and “a whale is a fish,” the true conclusion “a whale is a mammal” follows.
It should be remembered that from true premises, subject to the laws and rules of logic, it is impossible to obtain a false conclusion - it will necessarily be true.
The law of logic is understood as the necessary connection both between the elements of thought and between thoughts, expressed in a judgment or inference. This connection is expressed in patterns of regular forms that have developed in the process of many centuries of thinking practice. These schemes are expressed in formulas that take the value “AND” for all values of the variables included in them. In propositional logic, these formulas are called identically true. The specificity of the laws of propositional logic is that individual statements act as integral formations as variables included in the structure of logical forms. When substituting any variables into a logical law, the resulting complex statement will always be true.
The number of identically true formulas is unlimited, therefore the number of laws in logic is infinite.
The basic laws of propositional logic are the laws of identity, contradiction, and excluded middle.
Law of identity: every thought in the process of reasoning must be identical to itself. Designated: A « A.
Law of contradiction(non-contradiction): two statements that deny each other cannot be true together; at least one of them is false. Designated: Ø(A ÙØA).
Law of the excluded middle: two statements that negate each other cannot be both false. One of them is necessarily true, the third is excluded. This law applies to contradictory, or so-called. contradictory statements and is denoted: AÚØA.
When a statement is expressed by a formula with a small number of variables, it is convenient to use the tabular method, so abbreviated methods for selecting logical laws are used.
The abbreviated method for selecting logical laws can be found using the example of the form ((A ® B) Ù (B ® C) Ù A) ® C. The train of thought here will be as follows:
1) let's say this form is not a logical law. Then, with some substitution, it will be a false statement.
2) Since this form is an implication, it can turn out to be a false statement only when, with some substitution, its antecedent is true and its consequent is false, that is, when ((A ® B) Ù (B ® C) Ù A) – true and c is false.
3) This antecedent is a conjunction, and for it to be true, it is necessary that both of its terms be true, i.e., (A ® B) Ù (B ® C) and A must be true.
4) Since (A ® B) Ù (B ® C) is a conjunction, if it is true, both terms, A ® B and B ® C must be true.
5) A ® B – true implication; its antecedent A is true according to paragraph 3, B will also be true.
6) Since B ® C is a true implication, and b is true, then C is also true.
7) It turns out that statement C must simultaneously be false, according to clause 2, and true, according to clause 6. this is impossible, since by definition, every statement is either true or false. The resulting contradiction is the result of the assumption in paragraph 1, which will have to be abandoned and admitted that the considered form is a logical law.
It should be borne in mind that the use of the abbreviated method requires good orientation in the definitions of basic logical conjunctions.
Topic 3. Names.
1. The concept of a name. Expressing names in natural language. Volume and content as the main characteristics of a name.
2. The concept of a sign. Types of signs. Characteristics are general (generic) and distinctive (specific). The main and complete content of the name.
3. Name and concept. Names are single, general, empty. The concept of the universe of reasoning and universal names. Names are clear and unclear.
4. Relationships between names. Comparability and incomparability of names. Compatibility and its types - complete compatibility (equal volume), subordination, partial compatibility (intersection). Incompatibility and its types - contradiction, out-of-position, subordination, opposition. Euler circle diagrams and Venn diagrams to depict relationships between names.
5. Operations with volumes of names. Generalization, limitation, expansion, localization, typification. Mental transitions from part to whole and vice versa.
6. Division. Logical division, its goals and structure. Types of logical division - standard and non-standard division, dichotomous and polytomous (by modification of a characteristic).
7. Classification and typology. Classification (typology) natural and artificial. Rules of logical division and errors when they are violated. Analytical division, periodization.
8. Definition, its purpose and structure. Nominal and real definitions. Explicit and implicit definitions. Types of explicit definitions (attributive, genetic, operational). Implicit definitions and their types (through abstraction, contextual, inductive and axiomatic). Specificity of ostensive definitions. Definitions registering, postulating, clarifying. Determination rules and errors when they are violated. Techniques similar to definition (description, characterization, through indicating the opposite, etc.). The meaning of definitions in various spheres of human activity.
1. The concept of a name. Expressing names in natural language. Scope and content as the main characteristics of the name.
Name– a language expression denoting an object or set, a collection of objects. “Object” in this case is understood generally, in the broadest sense. Objects that are mentally combined into a certain set, or class, are called elements of the set (class).
Names denote, name, represent some objects in the language. These items are called name meanings.
The main characteristics of a name are its volume and content.
Name volume is a set, collection, class of objects denoted by a name. Name content- this is a set of attributes of objects conceivable in the name.
2. The concept of a sign. Types of signs. General (generic) characters and distinguishing characteristics nal (species). The main and complete content of the name.
Sign- this is any property, any characteristic of an object. The content of the name captures the characteristics of objects that collectively belong to each object identified by this name, i.e. included in its scope.
The characteristics that make up the content of a name are divided into generic, specific and individual. If we identify a narrower class of objects within a broader class of objects, then the characteristics that distinguish a wider class of objects will be called generic, and those that distinguish a narrower class will be called specific. That is, generic characteristics act as general ones, and specific ones act as distinctive ones.
Birth characteristics- these are signs of the class of objects in which a narrower class (subclass) is distinguished.
Species characteristics– these are the characteristics in accordance with which subclasses within a class are distinguished.
Distinguish basic And complete name content. Basic content of the name– the minimum part of the content of a name from which all its remaining contents are derived (which in this case is called derivative).
3. Name and concept. Names are single, general, empty. The concept of the universe denias and universal names. Names are clear and unclear.
Concept- a form of thinking in which objects of a particular class are identified and generalized according to significant distinctive features. Essential is a sign that determines the qualitative specificity of certain objects and which distinguishes these objects from all others. This feature underlies the selection of objects and their combination into classes. Every concept is characterized by volume and content.
The scope of a concept is a set of objects that have a feature that constitutes the content of the concept. A separate object related to the scope of a particular concept is called an element of the class.
Concepts are expressed in natural language through names- words or phrases. A name consisting of one word is called simple, of the two – complex, expressed by the phrase – descriptive or descriptive.
The volume denoted by the name is called denotation, and a separate subject of this volume is designatum name.
There are single, general, and empty names.
Single name denotes one object and is expressed by a proper noun. That is, the scope of a single name includes one element.
Common name denotes more than one thing. That is, the scope of the common name includes more than one element. The scope of common names is the classes (sets) of objects they cover. The class that is the scope of the name, called meaning this name.
Zero ( empty) names are names whose scope does not contain a single element. A class without a single element is called zero or empty.
A special type of common name is universal names. Their name, as you might guess, comes from the word “universum”. Each area of cognition has its own class of objects under study. These can be physical bodies, living organisms, numbers, etc. In logic and methodology of cognition, this kind of class, or set, is called universe corresponding field of knowledge, or, as they also say, universe of reasoning . This means that statements and reasoning in a given area of knowledge relate to these objects. For example, for biology the universe as a whole will be the class of all living beings, for the corresponding section - the class of vertebrates. In logic and methodology of science universe can sometimes be interpreted as extremely wide variety, – a set that includes all objects as its elements. So:
Name called universal, if in the specific part of its content only such features are recorded that are inherent in each element of the class, which is the universe of reasoning. For example, if an object is metal, then it has the property of conducting electric current.
Among the universal names, there are those whose specific content reflects some objective law, and those whose content does not reflect such a law. Names first type- so-called naturally or need universal, their specific content expresses some necessary pattern associated with the laws of an objective nature (logic or nature). Ex: “an object for which it is true that it has property P or does not have property P.” This general formulation corresponds to “a triangle with a sum of angles of 180 degrees” (in the case of Euclidean geometry). Names of the second type can be attributed to accidentally universal. Ex: everyone gathered in the audience put on their hats. Then “the person in the given audience wearing a hat” will accidentally be a universal name.
The name is called clear(precise, definite), if with respect to any object it is possible to accurately and unambiguously decide whether this object is included or not included in the scope of the given name. Otherwise, the name is said to be fuzzy (vague, vague, vague, imprecise) in scope.
4. Relationships between names. Comparability and incomparability of names. Compatibility and its types - complete compatibility (equal volume), subordination, partial compatibility (intersection). Incompatibility and its types - contradiction, out-of-position, subordination, opposition. Euler's circle diagrams and Venn diagrams to depict relationships between names.
Relationships between names are distinguished depending on the specifics of the relationship between their contents and volumes.
Names comparable among themselves if their contents have common characteristics. The names are incomparable, if their contents do not have common features that allow identifying grounds for comparison. Comparable names are divided into conscientious And incompatible.
Names compatible if their volumes at least partially coincide, i.e., they have common elements. Otherwise the names are incompatible.
Compatibility Relationships are divided into: 1) relationships equivalence (equivalence), 2) submission, 3) intersections (crossings).
Names whose volumes completely coincide, are equal in volume (equivalent).
The names are in respect of subordination, if the volume of one is completely included in the volume of the other, but does not coincide with it. The inclusive name is called subordinating, included – subordinates.
The names are intersecting (crossing), if their volumes only partially include each other.
Name incompatibility manifests itself in cases where there are: 1) relationships subordination, 2) contradictions, 3) opposites.
subordinate, if their total volumes constitute part of the volume of a certain subordinate name. The presence of a more general subordinating name is necessary for the subordinating relationship.
Incompatible names are called contradictory, if they completely exhaust the scope of the third, subordinate name, and one of them denotes objects devoid of properties included in the content of the second name. Two such names, exhausting the entire universe in scope, exclude the possibility of a third volume located between them.
Incompatible names are called opposite, if their contents express any extreme characteristics in some ordered series of gradually changing properties. Many pairs of opposite names are vague in scope. Opposite names do not exhaust the scope of the class within which they are compared. Each such name includes in its scope only extreme sets volume elements of this class.
Euler circle diagrams and Venn diagrams are used to depict the relationships between names.
Equivolume ratio(equivalence)
A and B. Etc.: A – square, B – rectangle,
whose diagonals are mutually perpendicular.
Subordination relationship
A – student, B – first year student.
Intersection relationship(crossing)
A – student, B – resident of Minsk.
Subordination relationship
A – resident of Rudenssk, B – resident of Minsk,
S is a citizen of the Republic of Belarus.
Attitude of contradiction
A - student, B - non-student
Opposite attitude
A - the richest citizens
Republic of Belarus, B – the poorest
citizens of the Republic of Belarus
Relationship between incomparable names.
In contrast to subordination, in the case of
incomparability of names no longer specified
a wide class subordinating their volumes.
5. Operations with volumes of names. Generalization, limitation, expansion, localization, typification. Mental transitions from part to whole and vice versa.
The relationships between names by volume allow you to perform logical operations with them, as a result of which new names appear. The most important operations are generalization, restriction, expansion, localization, typification.
Generalization of volume A- this is a logical operation, as a result of which a name is formed with volume B, containing volume A. That is, to generalize the name A means to form another name B (genus), which would subordinate A (species).
Moreover, when generalizing, the name B may still be unknown, the content must be selected, the scope must be established or clarified, and the name itself must be reformulated. The process of generalization is an integral component of scientific knowledge. In the process of cognition, the generalizing name itself can be generalized, etc. the limit of generalization is some universal name. In different sciences these are names that fix fundamental scientific concepts- so-called scientific categories. For example, in mathematics, in geometry - a point, a plane; in logic - a property, a relation: in physics, in mechanics - force, mass, material point.
Limitation is a logical operation inverse to generalization. When limiting, a name with volume B is found, which contains volume A. Volume limitation A is the finding of another such name B (kind), which is in a relationship of subordination to the name A of the genus). The limit of the limitation is names whose volumes are equal to one item, i.e. single names. For example, the limit of the name “capital” is Minsk, Warsaw, etc.
Typing- a special type of restriction.
Type is a name to which homogeneous objects correspond to one degree or another.
If some items make up the volume of name A and among them there are those who undoubtedly, i.e. with a degree equal to one, belong to volume b, and others have this property in some ways, i.e. ., less than one, then the name with volume B represents type. Ex: by limiting the scope of the name “man”, you can get “low man” or “tall man”. “low man” is a type; the other type is the “tall man.” So:
Type is a name with an unclear scope.
The term “type” can also be used in another sense, when typical representatives include only those objects that certainly, with a degree equal to one, belong to the scope of the fuzzy name. In this case, the content of the type in a concentrated form contains the characteristics of related objects. In this sense type- This sample name, a standard for describing and evaluating objects. Remember the typical representatives of the characters of Russian literature of the 19th century, for example, the characters of Gogol or Dostoevsky (“teenager”, “underground man”, etc.).
Volume expansion A– addition to volume A of new objects that are identical to the old ones in some way.
Localization of volume name A- an operation inverse to expansion, the removal from volume A of objects that are identical to the remaining ones according to some characteristics. Ex: in biology, whales were once removed from the class of fish, but the scope and content of the name “fish” remained unchanged.
The fact is that in the case of adding or removing certain objects from the scope of a certain name, the scope or content of the name is not changed. The sign according to which the volume is allocated and recorded remains unchanged.
Mental transitions from part to whole and from whole to part differ from logical operations with volumes of names. In logical operations, relationships between generic and specific characteristics are established. Thus, the generalized name contains the result of the generalization, but not vice versa. The species has everyone characteristics of the genus, but not vice versa, the genus does not possess everyone signs of the species. Unlike the relationship between genus and species, part does not contain c content barely. Therefore, confusing the operation of generalization or limitation with the operation of mental transition from part to whole or from whole to part is unacceptable and leads to misconceptions.
6. Division. Logical division, its goals and structure. Types of logical division - standard and non-standard division, dichotomous and polytomous (according to modification of the characteristic).
Logical division– an operation by which the volume of a name (genus) is distributed among classes (species) in accordance with some characteristic. In this case the genus is called divisible name, kinds - division members, sign – basis of division. Sometimes a feature may also be called a point of view or aspect of consideration.
The basis for division can be a feature that is inherent only in some objects of a certain class. In this case, objects are divided into those that have this characteristic and those that do not. This division is called dichotomous. Etc.: dividing numbers into even and odd. Division according to a characteristic that all objects of the genus possess and which varies among species is called polytomous. Dichotomous division is simpler and is used, as a rule, at the initial stage of studying subjects, when there is clarity regarding some of the subjects designated by the dividing name.
Logical division can be classical and non-classical. At classic division genus and species are names with clear volume, at non-classical- This fuzzy, vague names or types.
The division operation can be characterized from the point of view
Semantic analysis of natural language made it possible to implement a typology of linguistic expressions in accordance with the types of mental structures of which they are, their properties and relationships. But natural language expressions can be considered as signs that are bearers of names. Taking this into account, all meaningful (meaningful) linguistic expressions in modern logic are considered as names. In the process of cognitive and practical activity, real or conventional things become the subject of human thought. A person cannot do without designating these objects.
In other words, there is a naming relationship between objects (real or imaginary) and the way they are used in the process of exchange of opinions. The naming relationship involves two objects: the denoted and the denoted.
Designating is a product of human mental activity and is subjective in nature.
What is designated can be dependent on the subject of cognition (when we are talking about imaginary objects) and independent (when we are talking about objectively existing objects). Designating can be words, sentences, combinations of sentences.
So, linguistic expressions that have the property of being denoting are called i m e n a m i. The names include individual words (“Shevchenko”, “Dnepr”, “river”) and phrases (“the author of the poem “The Dream””, “the river on the banks of which the capital of Ukraine is located”). Each of the names denotes either an individual object or a collection of objects.
That which the name I refers to is called d e n o t a t o m(designatum, nominee) or the meaning of the name.
The same denotation may have different names. Thus, the names “T. Shevchenko” and “author of the poem “Dream”” indicate the same person. This circumstance makes it necessary to explain what makes it possible to associate (correlate) in each specific case a certain name with the corresponding object (denotation). It turns out that some intermediary is involved in the naming process, without which it is impossible either to use names or to find and distinguish one object from another. The mediator is information, knowledge about the designated object. This information is called the cm and l o m (concept) of the name.
The meaning (concept) and meanings (denotation) make up the content of the name. The bearers of a name can be not only words and phrases, but also some sentences.
The meaning (concept) of a name sentence is the information that the sentence contains (something is affirmed or denied), and the meaning is an abstract object, logical valence ("true" or "false").
Only true names have meaning ("France", "radio inventor", "Kyiv"). Imaginary names only symbolically designate this, since in reality the objects they designate do not exist (such are the names “Pegasus”, “absolutely rigid body”, “v-1”, etc.).
All names have meaning. Identifying the meaning of a name is very important, because it is the meaning that is the link that connects the name with the object. Logic in the theory of names is interested precisely in the explanation of how names are connected with objects of extra-linguistic reality.
Let us consider the need to analyze the theory of names for logic.
Logic makes a name the object of analysis in order to solve, first of all, the following questions:
1) how the name and concepts are related, namely: the meaning of the name and the content of the concept;
2) how the logical meaning of a statement depends on the meanings of the names included in it;
3) what kind of logical means can ensure the invariance of statements during their interaction in the process of inference.
Types of names
Depending on whether the name indicates a separate object or distinguishes an object from many objects, all names are divided into:
Own and
Names designate (individual) things.
Common names distinguish one item from many items.
For example, “state”, “city”, “book”, “natural satellite”.
Comparing names with general names denoting sets, let us pay attention to the fact that common names indicate an indefinite representative from a set of objects - some kind of state, then a city, etc. In fact, common names, unlike proper names, have no meaning or meaning.
For example, if the word "city" is a name for "Kyiv", "Warsaw", then it turns out that it is a name above names, since each object that it names has its own name.
He explained the situation quite convincingly with the correct understanding of the common name B. Russell. He pointed out that the word "man" denotes a few people, and not a specific person.
Therefore, it makes sense to say that a common name does not denote, but represents a specific (arbitrary) object from a set, since a variable (x) in mathematics represents a certain arbitrary number. In this sense, one can interpret common names as unique objective variables, this is, firstly, and secondly, a consequence of this fact is that common names are not names in the proper sense of the word, because objective variables are not names either.
All this allows us to conclude that the class of names does not cover the entire set of linguistic expressions, but coincides only with the category of constant terms. This indicates the diversity of relationships between verbal signs and objects. The naming (designation) relationship is only one of these relationships.
Therefore, when we talk about the meaning, meaning, principles of naming, we mean the nature of the connection of names (constant terms) with the objects that they represent. The procedures for determining the meaning of a name are different in nature. In some cases, a name directly indicates its meaning, in others, additional actions are needed to identify the meaning (special explanations, references to context, etc.).
Names in natural language are expressed not only by a word or phrase ("Shakespeare", "Shakespeare's homeland"), but also by entire sentences using the designated description operator, which is called the iota operator and in natural language is written in the form of the expression "one who.. ". For example, “the one who wrote the poem “Aeneid”,” “The one who first discovered America.” The form of the expression "one who..." does not clearly convey its name in natural language.
Let's take for example a name that sounds like this: “The one who is the author of “Kobzar”. The denotation of this name is a real person named Shevchenko, he was born in 1814 in the village of Morintsy in the Cherkassy region; was a serf in Engelhardt; One of Shevchenko's creative activities was poetry, and contributed to the birth of "Kobzar".
Analyzing this name, you can easily make sure that there is a nuance (aspect, emphasis, shade) inside that can be used to distinguish one name from another with the same denotations. It is this nuance, isolated from the entire array of information about the subject (which we currently possess), that makes up the meaning of the name.
Or take the sentence: “The one who is the author of the painting “Ekaterina”,” which is also a name with the same denotation, but in this case the meaning will be a different shade of information, namely: T. Shevchenko had the talent of an artist, was a friend of Soshenko, who drew attention to the abilities of young Taras, graduated from the St. Petersburg Academy of Arts.
Obviously, there are names whose meaning is quite easy to establish. But the situation becomes more complicated when the name is considered out of context, say the word “Kyiv”. The denotation can be a city, a warship, or a hotel. To unambiguously establish the meaning of the name, additional analysis and explanation is needed.
If the meaning of a name is determined by a specific situation or context, it is called p r o s t i m or not descriptive.
For example, "Jupiter", "Dnepr", "Ukraine".
If the meaning of a name is determined by its construction, it is called with warehouse or descriptive.
For example, “Plato’s student”, “teacher of Alexander the Great”, “capital of France” and others.
Isolating different shades in the array of information about a denotation creates a situation where the same denotation has different names (this is exactly what is typical for descriptive names, where each new name is a new semantic shade).
But quite often the same name indicates different denotations. This is no longer splitting the array of information about the denotation into shades (as is the case with complex names), but finding new arrays of information that makes it possible to clearly separate one denotation from another. This is precisely the property for NON-descriptive
Thus, the procedure for identifying the meaning and denotation of a name presupposes the presence of a context of speech expression.
Under CONTEXT for an arbitrary expression (A) we mean a linguistic expression that includes (A) without violating the syntactic rules of the language used.
It is clear that contexts for A will be parts of a sentence, whole sentences, or fragments of text. For example, take the names “Aristotle’s teacher and founder of the Academy,” “Aristotle’s teacher and friend,” “Aristotle’s teacher and author of the theory of ideas.” All examples have the proper name "Aristotle". Expressions that include the name "Aristotle" without violating the syntactic rules of the given language are called the context for that name.
If the scope of the name includes only one object, then such a name is called single.
Common name is a name whose scope includes more than one element. A class that is the scope of a common name is called meaning this name.
A special type of common name is universal names, or universes . They record all classes of objects, all elements studied in a particular area of cognition. Names belonging to the same universe are called related .
Null (empty) names in the most general form are defined as names whose scope does not contain a single element. A class that does not contain a single element is called null, or empty.
There are also names descriptive And own . Descriptive names identify objects by indicating their corresponding characteristics. Proper names designate objects by direct correlation with them, due to the fact that certain traditions and naming norms have developed in the culture of the human community.
It is important to distinguish between collective and non-collective names. Non-collective is called such a name, each element of the volume of which represents something single, integral. collective is called such a name, each element of which is a collection, collection, union of some objects.
There are positive and negative names. It is based on the fact that objects can be characterized both by the presence and absence of certain properties in objects. Positive is considered to be a name whose contents indicate the properties inherent in objects. Negative is considered to be a name whose contents indicate properties that objects do not have.
Finally, we indicate the division of names into clear and fuzzy. If the name is such that with respect to any object it is possible to accurately, unambiguously decide whether this object is included or not included in the scope of the given name, then this name is called clear (precise, defined) in scope (e.g. rational number, subsistence farming, criminal liability). Otherwise the name is considered fuzzy (vague, vague, blurred, inaccurate) in volume (for example, an expensive product, a young man, a good appearance).
COMPATIBILITY RELATIONSHIP
Names count compatible if their volumes at least partially coincide, i.e. these volumes have common elements.
Types of compatible names:
1) Equal volume (equivalent) names are considered whose volumes completely coincide (Fig. 1). With the relation of equal volume of names A And B every item identified by a name A,may be indicated by a name B, and vice versa.
2) Names are in relation submission , if the volume of one is completely included in the volume of the other, but does not coincide with it. In this case, the inclusive name is called subordinate, or generic, and the included name is called subordinate, or specific. If the name A obeys the name B(Fig. 2), then all signs B inherent in the content of the name A, and each item denoted by a name A, can be denoted by the name B(but not vice versa).
3) Intersecting (crossing) are names whose volumes only partially include each other. Moreover, some objects designated by the name A, can be indicated by the name B, and vice versa. If the names A And B are in relation to intersection (Fig. 3), then objects included simultaneously in the volumes of names A And B, that is, located at the intersection of these volumes, have the same characteristics.
Relationships between related names.
Incompatibility relationship
In case of incompatibility in the content of one of the names, signs are indicated that exclude the signs of the content of the other name.
Types of incompatible names:
1) Contradictory two incompatible names are called, the specific content of one of which (i.e., the totality of its specific characteristics) is a negation of the specific content of the other. Such names completely exhaust the scope of the third name subordinating them (Fig. 4).
2) Subordinates such incompatible names are called, the volumes of which in total constitute part of the volume of some subordinate (generic) name. Because the A And B, being external, are simultaneously subordinated WITH, so they are also called outside the scope relatively WITH(Fig. 5).
3) Opposite they call names whose contents express any extreme characteristics in some ordered series of gradually changing properties (Fig. 6).
Generalization and restriction as operations on names
Generalization of volume A- a logical operation that results in a name with a volume B, containing the volume A. In other words, generalize the name A- means to form such another name B(genus), which would subordinate to itself the name A(view). The limit of generalization in each specific case is a certain universal name.
Limitation- a logical operation inverse to generalization. It consists of finding a name with a volume B, which is contained in the volume A. Limit volume A- means to find such another name B(species), which would be in a relationship of subordination to A(family). The limit of the limitation is names whose volumes are equal to one item (single names).
A special type of constraint is type selection, or typing . Type is a name to which homogeneous objects correspond to one degree or another. If some items make up the volume of the name A and among them there are those that unconditionally (i.e. with degree equal to 1) belong to the volume B, and others have this property to some (less than 1) degree, then a name with volume B represents a type.
Connection to volume A new objects that are identical to old ones according to some criterion is called a logical operation extensions volume A.
The inverse operation of expansion, i.e. removal from volume A objects that are identical to the remaining ones according to some characteristics are called localization name volume A.
Logical operations with volumes of names should not be confused with mental transitions from part to whole and, conversely, from whole to part. The specificity of the latter is most clearly revealed when they are compared with the operations of generalization and limitation.
DIVISION OPERATION
Logical division is a logical operation by which the scope of a name (genus) is distributed among classes (species).
Analytical division - This is an operation associated with the mental isolation of its parts as a whole. These operations should not be mixed.
The division can be classical or non-classical. At classic division of both genus and species - names with a clear volume, with non-classical they are fuzzy, vague names, or types.
Classic logical division consists of finding for a name A such names A 1 , A 2 , ..., A n( n– final number) that:
a) each volume A 1 , A 2 , ... , A n is in a relation of subordination to volume A);
b) sum of volumes A 1 , A 2 , ... , A n is equal to volume A;
c) each pair of volumes A 1 , A 2 , ... , A n is connected by a relation of incompatibility. At the same time the name A called divisible name , A A 1 , A 2 , ... , A n – division members .
It is possible that the basis for division is a feature inherent in only a part of objects of a certain class. In this case, objects are divided into those that possess this feature and those that do not. This division is called dichotomous(Greek dicho - into two parts, tome - section). In contrast, division according to a characteristic that all objects of the genus possess and which varies among species is called polytomous Greek polis – a lot).
The difference between division and dismemberment is based on the different nature of the “whole-part” and “genus-species” relationships.
DIVISION RULES
1. Rule of adequacy.The division must be proportionate. This means that in case of division each of the volumes A 1 , A 2 , ... , A n must be a type of volume A, and the sum of A 1 , A 2 , ... , A n must exhaust the entire volume A;in case of dismemberment mental connection of parts must be equal to the whole. Deviation from this rule leads to errors, the most famous of which are: " division with extra members", when some of the volumes (parts) A 1, A 2, ... , A n is not a species A(not included as part of the whole A); "incomplete division", when not all types (parts) of the divisible genus (whole) are named, and the sum of the volumes of the members of the division is less than the volume of the name being divided.
2. Rule of Distinction. Members of division (dismemberment) must exclude each other, i.e. their volumes should not have common elements in the case of classical division, and parts should not overlap each other in the case of dismemberment.
3. Rule of uniqueness of basis. Division must be done using the same base. When this rule is followed, objects included in the scope of the divisible name are endowed with one single attribute - the one that acts as the basis for the division. Deviation from this rule leads to an error, which is called mixing bases.
Instead of the term "division", the term "classification" is sometimes used as a synonym. Classification in a narrow sense (it is in this sense that we will use this term in the future) - this is a multi-stage, branched division, such that each of the members obtained during this operation becomes the subject of further division.
According to the classical and non-classical division, a distinction should be made between classical and non-classical classification. The last one is called typology .
So far, no simple and unambiguous term has been assigned to the multi-stage and branched division. This operation can be called hierarchization .
Classification and hierarchization are subject to all division rules. In addition, they have their own special rules.
1. Sequence rule . In the case of classification, one should move from the genus to the closest species, and in the case of hierarchization, from the whole to its parts of the same level, without skipping them. If this rule is violated, the permissible error is “ leap in classification (hierarchization) ».
2. Rule of materiality of grounds . Classification (hierarchization) should be made according to essential characteristics. The criterion for the significance of a particular attribute is the ability of the object possessing it to serve as a means of solving the task at hand.
A special case of dismemberment is periodization Its peculiarity is, firstly, an indication of the development of the displayed object over time. Secondly, the members of the division (periods) are distinguished by their measure as a unity of qualitative and quantitative characteristics of an object.
DEFINITION OR DEFINITION (GENERAL CHARACTERISTICS)
In logic, there are primarily two different meanings of the term “definition”. Firstly, under definition is understood as an operation that allows you to select an object among other objects, to clearly distinguish it from them. This is achieved by indicating a feature inherent in this, and only this, object. This feature is called distinctive (specific). What do we do, for example, if we want to extract squares from a class of rectangles? We point out a feature that is inherent in squares and not inherent in other rectangles, – to the equality of their sides.
Secondly, the definition is called logical an operation that makes it possible to reveal, clarify or form the meaning of some linguistic expressions with the help of other linguistic expressions. So, if a person does not know what the word “vertshok” means, they explain to him that the vershok – this is an ancient measure of length equal to 4.4 cm. Since a person knows in advance what an “ancient measure of length equal to 4.4 cm” is, the meaning of the word “vertex” becomes clear and understandable to him.
A definition that gives a distinctive characteristic of a certain object is called real. A definition that reveals, clarifies or forms the meaning of some linguistic expressions with the help of others is called nominal.
The method of establishing the meaning of a linguistic expression by directly correlating it with the designated object or its image is called ostensive definition.
IN definition structure there are three parts:
1) a defined name or an expression containing it (indicated by the sign Dfd – abbreviation from Lat. definiendum);
2) an expression that reveals, clarifies or forms the meaning of the defined name (indicated by the sign Dfn - an abbreviation of the Latin definiens);
3) a definitive connective that relates Dfd and Dfn according to their meaning (indicated by the sign º).
Formally, the structure of the definition is represented by the expression: Dfd º Dfn.
RULES OF DEFINITION
1. Rule of proportionality. Dfd and Dfn must be of equal volume.
Deviation from the rule of proportionality leads to errors:
1) "too broad a definition" - volume Dfn is greater than volume Dfd;
2) "too narrow a definition" - volume Dfn is less than volume Dfd;
3) “at the same time too broad and too narrow a definition” - the volumes Dfd and Dfn are in an intersection relationship.
4) definition via empty name- Dfd and Dfn turn out to be incompatible.
2. Anti-vicious circle rule. It is forbidden to define Dfd through Dfn, which, in turn, is defined through Dfd. The violation allowed in this case is called " vicious circle in definition". A special case of the "vicious circle" is tautology – repetition of Dfd and Dfn (even if in a different verbal form) without establishing the meaning of Dfd.
3. Rule of unambiguity. Each Dfn must exactly correspond to one single Dfd, and vice versa. This rule eliminates the phenomena of synonymy and homonymy and prohibits the use of metaphors and artistic images.
4. Rule of simplicity. Dfn must be expressed by a descriptive name that characterizes the defined objects only by their basic characteristics. Otherwise the definition will be redundant. In classical definitions, this rule is satisfied provided that: a) the genus included in Dfn is closest to Dfd, i.e. such that no other name, subordinate to the gender and subordinate to Dfd, has been previously defined; b) in Dfn there are no expressions in the relation of following (subordination).
5. Rule of competence. Dfn can only contain expressions whose values have already been accepted or previously defined. Deviation from this rule is called "defining the unknown through the unknown."
Related information.
3.1. General logical characteristics of a name
The essential characteristic of a person is abstract linguistic thinking. It is based on a person’s ability, distracting from specific objects and phenomena, to turn to their essence. At the same time, both real objects and phenomena (“house”, “morning”) and their properties (“purity”, “harmony”) are designated by names in the language. Consequently, the name is the basic logical and semiotic unit, the elementary form, and the process of thinking is a process of operating with names and establishing special connections between them. A name denotes any object of thought from the point of view of its distinctive features. IN
In language, a name is expressed using words and phrases, which in a sentence are most often used as the subject or nominal part of the predicate. Outside the verbal form, a name does not exist, but a name and a word are not identical: the same name in different languages has a different linguistic form, and many words have several meanings.
3.2. Volume and content of the name
IN Logically, any name has scope and content. The content of a name represents its semantic meaning, that is, the totality of those characteristics of objects and their classes that it denotes.
The volume of a name is represented by the totality of its bearers or designates, which can be either material objects or only imaginable ones.
The volume and content of the name, which characterize it from different sides, are closely related. The study of this connection made it possible to identify a special pattern, which was expressed in the law of the inverse relationship between the content and volume of a name: By increasing the content of the name, we decrease its volume, and vice versa. The content of the name increases due to the inclusion of new features. For example, the name "student". Its scope includes all students of higher educational institutions of all forms of education (full-time, part-time, evening, distance learning, etc.). By adding a new feature to it - “correspondence student”, we enriched the content of the name “student”, but reduced its volume, excluding from it students of all other forms of education. The logical operation in which we move from a name with a larger volume to a name with a smaller volume is called limiting the scope of the name. The limit of the restrictions is
names with minimal volume (single, most often proper).
The reverse operation in logic is called generalization of the scope of the name. It represents a transition from a name with a smaller volume to a name with a larger volume due to the exclusion of certain features from its content. For example, the name “textbook on logic”. By excluding the attribute from its content, we get a name with more volume - “textbook”, but with less content. In this case, the limit of generalization is names with the widest possible scope - categories denoting extremely broad and abstract phenomena, processes and connections (“space”, “good”, “matter”, etc.).
3.3. Types of names
The type of name depends both on the number of its designata and on the characteristics it denotes. By volume, names are divided into single, common and empty (null). In terms of content - on concrete and abstract, positive and negative, relative and non-relative, collective and non-collective.
A single name is a name that has one designation (“first cosmonaut”, “Constitution of the Republic of Belarus”, “Immanuel Kant”). As a rule, proper names also belong to singular ones. Names that have two or more designata are called general (“student”, “law”, “constitution”). Names that do not have designata are called empty (zero). Such names have semantic meaning, but are devoid of subject matter. These include names from the sphere of human fantasy, fairy tales, myths (“mermaid”, “Serpent Gorynych”, “unicorn”), scientific concepts as a result of extreme abstraction (“ideal gas”, “absolutely black body”) and names in the content in which signs are imagined that contradict the nature of the designated objects (“triangular square”, “icy sun”).
Names are divided into abstract and concrete depending on what they mean. If the name denotes real objects and their classes, it is specific (“student”, “house”, “centaur”, “thunderstorm”). Names denoting individual properties of objects and relationships between them are called abstract (“purity”, “love”, “courage”).
Names are divided into positive and negative depending on whether they record the presence of some attribute in the designated object or its absence. A name is called positive that indicates the presence of some attribute in the object (“believer”, “order”).
On the contrary, a name indicating the absence of a characteristic in an object is called negative (“asymmetry”, “inadequacy”). As a rule, negative names are formed using negative particles (not-, without-, a-). If a name without a negative prefix is not used for various reasons (language development, changes in lexical norms), then it is positive (“hatred”, “dissonance”).
Irrelevant are names that designate objects in themselves, regardless of the relationships and connections of these objects with others (“man”, “house”). Relative names are names that designate objects that do not exist independently, but only as members of some relationship (“good” - evil", "day - night").
A name denoting a collection of objects, conceived as a single whole, is called a collective name (“constellation”, “service”). Moreover, the name of integrity does not coincide with the names of the objects that compose it. Thus, the designatum of the name “constellation” is the Constellation Ursa Major and other constellations, and not stars and celestial bodies. Non-collective names are given that denote objects and their classes and are conceived not as independent entities, but existing separately (“planet”, “window”).
Determining the types of name by volume and content, we give it a full logical description: planet - general, specific, positive, irrespective, non-collective. A complete logical description makes it possible to clarify the scope and content of a name, to use its verbal expression more correctly in a text, discussion, etc.
3.4. Relationships between names by volume
The entire set of names can be divided into comparable and incomparable. Names that have at least one common characteristic (“student” and “athlete”) are comparable. Incomparable names do not have common characteristics, therefore, it is impossible to compare them. In logical relations there can only be comparable names. Comparable names, in turn, are divided into compatible and incompatible. Compatible names include names whose volumes completely or partially coincide, and incompatible names include names whose volumes do not coincide either completely or partially. The relationships between names are graphically represented on Euler circles.
Types of compatibility:
1. Identity (equivolume).
A – student, B – university student.
the volumes of which completely coincide. Moreover, they have coinciding designata, since they denote the same object, but their content may be different. The relationship between equal names is depicted in Fig. 1.
2. Intersection
A – student, B – chess player, C – chess student
IN In the intersection relation there are names whose volumes partially coincide. In this case, as a result of the intersection of the volumes of names, a new class is formed, formed by designata common to the intersecting names. In Fig. 2 shows the intersection relationship.
3. Submission
A – student, B – student.
In relation to subordination there are names, the scope of one of which is completely included in the scope of the other, but without exhausting it. This relationship is shown in Fig. 3.
Types of incompatibility: 1. Subordination
A – university, B – BNTU, C – BSU.
IN in a relationship of subordination there are two or more species of the same genus. In relation to the generic name they are in a relationship of subordination, and among themselves - subordination, i.e. their volumes do not intersect. The subordination relationship is shown in Fig. 4.
2. Opposite
A – white, B – black, C – color.
IN in the relationship of opposition (contrary) there are names, one of which has certain characteristics, and the other excludes them, replacing them with the opposite. This relationship is shown in Fig. 5.
Since logic studies the forms of thoughts and ways of expressing them in language, logic is also the science of language. In logic, individual aspects of natural languages (languages that arose and develop mainly spontaneously) are studied, and artificial languages are created - special languages of logic. One of these languages is the language predicate logic, widely used in identifying connections between thoughts by their logical forms. The main advantage of this language is that its expressions are unambiguous. There are no homonyms and no unclear expressions. This allows you to strictly record the course of reasoning and accurately resolve the issue of their correctness or incorrectness, as well as a number of other issues.
In logical analysis, language is considered as a system of signs.
Sign - it is a material object used in the process of cognition or communication as a representative of an object.
The following three types of signs can be distinguished: (1) index signs; (2) symbols-images; (3) signs-symbols.
Index signs are connected with the objects they represent materially, for example, as effects with causes. Thus, smoke indicates the presence of fire, increased body temperature indicates a disease, a change in the color of nails indicates a disease of the internal organs, and a change in the height of the mercury column indicates a change in atmospheric pressure.
Imaginative signs are those signs that themselves carry information about the objects they represent (a map of the area, a painting, a drawing), since they are in a relationship of similarity with the designated objects.
Signs-symbols are not materially related and are not similar to the objects they represent.
Logic explores signs of the latter type.
As a rule, signs have objective and semantic meanings. Subject meaning is the object that is represented (or denoted) by the sign. Semantic meaning- a characteristic of an object expressed in language, the representative of which is a sign, which allows one to distinguish the designated object from other objects. Subject meaning is often called simply meaning, and semantic meaning is called meaning.
Some signs have no meaning, e.g. represent non-existent objects (for example, “perpetual motion machine”), and some do not make sense, i.e. designate some objects, but do not carry information about them, at least such that is expressed in language and allows you to unambiguously identify the objects denoted by the sign.
The role of signs in knowledge was studied by Aristotle. Leibniz and other scientists dealt with this problem. The development of the doctrine of signs in the 19th century became especially relevant. in connection with the needs of linguistics and symbolic logic. American philosopher Charles Peirce (1839-1914) laid the foundations of a special science of signs - semiotics. There are three sections in this science - syntax, semantics and pragmatics, which is due to the presence of three aspects of language.
Syntax is a section of semiotics that studies the relationships between material objects that act as signs (rules for constructing and transforming language expressions, etc.). In the process of this research, one is distracted from the meanings and meanings of signs.
Semantics is the branch of semiotics that primarily studies the relationship of signs to the objects they represent, as well as the meanings of signs, since they are one of the means of establishing a connection between signs and their meanings.
Pragmatics studies what individuals and social groups bring to the understanding of signs, a person's relationship to signs, and the relationships between people in the process of sign communication.
One type of sign is names. The doctrine of names, called naming theory
, developed relatively fully by the German scientist Gottlob Frege (1848-1925). A great contribution to the creation of this doctrine was made by the American logicians R. Carnap (1891-1970) and A. Church (1903-1995), as well as the domestic logician E.K. Voishvillo (b. 1913).
The main concept of naming theory is the concept of “name”. Name - This is a word or phrase that denotes an object. Because a name is a sign, it has meaning or meaning (or both). The meaning of a name is the thing denoted by that name. Other names for the meaning of the name - denotation, designatum, nominee.
Meaning (or concept) is information about objects presented in language, which is expressed by a name and which allows one to unambiguously identify objects that are the meanings of a name.
There are two types of names. A name belonging to the first type designates one object. The name of the second type is common to objects of a certain class. Names of the first type are called singular, and of the second - general. Examples of single names: Moon; capital of Russia; author of the novel "War and Peace". Examples of common names: animal having soft earlobes; European state; student. Thus, the meaning of a single name is a single subject. Common name values are items of some class containing more than one element. The class that consists of the objects that are the meanings of a name is called the scope of the name. The volume of a single name is a class consisting of one subject.
mark">universal. Universal names are general names, the scope of which is the entire universe of reasoning, i.e.: the entire subject area about which one is reasoning. For example, “a person who knows some foreign languages or does not know a single foreign language.” Universe of reasoning here is the set of (all) people. The scope of the name is the same set. The name “a person who knows some foreign languages” is not universal, since its scope does not coincide with the set of (all) people is determined by the context in which. name is used.
There can be names with different meanings and the same volume (for example, “the largest city in England” and “the capital of England”), but there cannot be names with the same meaning but different volumes.
Names can denote objects that do not exist in the universe of reasoning. Such names are imaginary. Examples: “mermaid”, “the most distant point of the Universe”. These names are imaginary if the universe of reasoning consists of objects that exist in objective reality. The scope of the imaginary name is the empty set.
mark">valid .
Frege and Church believe that all names have meaning. Voishvillo believes that not all. Arguing his point of view, he divides names into two types according to the type of meaning - into names that have their own meaning, and names that do not have their own meaning. Names that have their own meaning are descriptive names like "the largest river in Europe". The meaning of such names is determined by their structure and the meanings or meanings of the names that make up these descriptive names. If the names included in a complex name do not make sense, then in this case the descriptive name has meaning. This meaning is to indicate the relationship between the meanings of the constituent names. Non-descriptive names like “Volga” have no meaning of their own. If they have any meaning, it is only a given one. Non-descriptive names are given meaning by means of descriptive names that are associated with them. Descriptive names, in turn, include non-descriptive names. They are also given meaning through descriptive language. Obviously, such a process cannot be endless, i.e. some non-descriptive names have meaning but are not meaningful. These names designate objects, but do not carry information about them, which is expressed in language and allows these objects to be distinguished from other objects of the universe. The meanings of such names are determined through the senses or intuition.
In natural language, some expressions, depending on the context, denote various objects, and there are also cases when the meanings of expressions can be these expressions themselves, etc. This situation is unacceptable in the languages of science, which are subject to the following three normative principles: (1) the principle of objectivity; (2) the principle of unambiguity; (3) the principle of interchangeability.
According to the principle of objectivity, statements must affirm or deny something about the meanings of the names included in the sentences, and not about the names themselves. It must, of course, be borne in mind that the meanings of some names are names. Such cases do not contradict the principle of objectivity. For example, in the sentence “Matter is primary, and consciousness is secondary,” “matter” is the name of objective reality, and in the sentence ““Matter” is a philosophical category,” the word “matter,” taken in quotation marks, is the name of the name, the name of the category. Such names are called quotation marks. Sometimes in natural language there are cases where the name of a name is the original name itself. For example, in the sentence “The word table consists of four letters,” the word “table” is the name of the word itself. This use of names is called autonymous. Autonomous use of names is unacceptable in scientific languages, as it leads to misunderstandings. Thus, in the well-known definition of V.I. Lenin: “Matter is a philosophical category to designate objective reality, which is given to a person in his sensations, which is copied, photographed, displayed by our sensations, existing independently of them” - there is an autonomous use of the name “matter”. This caused controversy about what V.I. called. Lenin is matter, objective reality or category, i.e. thought, concept of reality.
According to the principle of unambiguity, an expression used as a name must be the name of only one object, if it is a single name, and if it is a general name, then this expression must be a name common to objects of the same class. In natural language, this principle is not always observed. Its observance is necessary when constructing artificial languages, for example, the language of predicate logic.
The principle of interchangeability: if in a complex name you replace the part, which in turn is a name, with another name with the same meaning, then the meaning of the complex name obtained as a result of such replacement must be the same as the meaning of the original complex name. Let the sentence “The Earth revolves around the Sun” be given (we will assume that sentences are also names and the meaning of the sentence is true or false). Let us replace the name “Sun” in the above sentence with the name “central body of the Solar System”. Obviously, the meanings of these names are the same. As a result of such a replacement, we obtain a true sentence from a true sentence.
The principle of interchangeability seems natural, but there are examples of name substitution that contradict it. Consider the sentence: “Ptolemy believed that the Sun revolved around the Earth.” It is true. Let us replace the name “Sun” with the name “central body of the Solar System”, which has the same meaning. We will receive a false offer.
Such inconsistencies with the principle of interchangeability are called antinomies of the naming relation.
It is necessary to distinguish between two ways of using names. The first is that the name simply identifies the item(s). The second is that objects denoted by a name are considered in a certain aspect. If a name is used in the second sense, then it can be replaced by another name with the same meaning, if only in the second name objects are considered in the same aspect. The above substitution could have been made if Ptolemy had believed that the meanings of the names “Sun” and “central body of the solar system” were the same. Then the meaning of the sentence “Ptolemy believed that the Sun revolved around the Earth” would be “false.” The sentence resulting from the replacement would also be false: “Ptolemy believed that the central body of the solar system revolved around the Earth.”
Language expressions are divided into classes depending on the types of meanings they express, as well as on the types of objects that they denote or represent. These classes are called semantic categories.
First of all, sentences are distinguished, as well as parts of sentences that play an independent role in the composition of sentences.
Sentences are divided into classes depending on whether they express judgments, questions, norms, etc. Sentences expressing judgments are called statements.
Among the expressions included in sentences and playing an independent role in them, descriptive and logical terms.
Descriptive terms include: 1) singular names; 2) common names; 3) signs of properties and relationships; 4) signs of signs; 5) signs of objective functions.
Single and common names are described above.
Properties are how objects and phenomena differ from each other. If we compare people, we can say that one is tall and the other is short, one is black-eyed and the other is blue-eyed, etc. By relating a property to an object in our thoughts, we obtain a true or false sentence.
An attitude differs from a property in that in order to obtain a true or false sentence, it (the attitude) must be attributed in thoughts to a pair or three, etc. items. Examples of relations: “greater than”, “located between”, etc.
In modern logic, property signs and relation signs are included in one semantic category - the category of signs representing the characteristics of sequences of objects. In this case, properties are considered as characteristics of sequences consisting of one object, and relations - as characteristics of sequences consisting of several objects (double relations - characteristics of pairs of objects, three-place relations - characteristics of triplets of objects, etc.).
The relation “greater than” is two-place, since in order to obtain a true or false sentence it must be related in thoughts to a pair of objects. The relation “located between” is a three-place relation, it must be related to a triple of objects in order to obtain a true or false sentence.
The sign of “an object is the presence or absence of this or that property or relationship to other objects.” The sign of an n (pair, triple, etc. of objects) is the presence or absence of any relationship between its elements. Words or phrases expressing characteristics of sequences of n objects defined by predicates.
The sentence “This table is yellow” asserts that this table is yellow. The phrase “is yellow” is a sign of a characteristic, and the word “yellow” is a sign of a property. In the sentence “Moscow is larger than Arkhangelsk,” “more” is a sign of the attribute of a pair of objects (Moscow, Arkhangelsk). The content of this sentence can be expressed differently: “Moscow is larger than Arkhangelsk.” Here “there is greater than” (“more”) is a sign of attribute, and “greater than” is a sign of relation.
It is not always easy to distinguish between common names, on the one hand, and signs of properties and relations, on the other. Out of context, for example, the word "red" can be considered both a property sign and a general name. In the latter case, it is the general name for red items.
When constructing the language of predicate logic, we will understand properties as general names of objects, and n-ary relations as general names of n-ary objects. In this broad interpretation we will call common names predicators.
Subject function signs, or functional signs, or subject functors, represent subject functions.
A function is a correspondence by virtue of which objects (an object, a pair, a triple of objects, etc.) from a certain set, called the domain of definition of the function, are correlated with objects from another or the same set, called the values of the function.
An object function is a function whose values are objects. Examples of subject functions: sin, log, +, mass. Applying the functional sign “mass” to the singular name “Earth,” we obtain as a value the singular name “mass of the Earth,” denoting a certain quantity, i.e. item. Thus, this function compares objects (material objects with mass) with other objects (mass values).
Main logical terms The Russian language includes the following words and phrases: “is” (“essence”), “and”, “or”, “if..., then...”, “not”, “it is not true that...”, “everyone” (“everyone”), “all”, “some”, “that... which...”, “therefore”. Some of these terms express relations of reality. For example, “and” expresses the coexistence of two states of affairs or situations, and “if..., then...” - the connection of two situations when, in the presence of the first, the second always takes place. Such relations are called logical in contrast to non-logical relations, i.e. relations represented by descriptive terms.
Consider the sentence: “If not a single member of the Ivanov family is an honest person, and Stepan is a member of the Ivanov family, then Stepan is not an honest person” and determine which semantic categories the expressions that are its parts belong to. In this sentence, “if..., then...” is a logical term, “none” (“all”) is a logical term, “member of the Ivanov family” is a predicator (common name), “not” is a logical term, “is” (“is”) is a logical term, “an honest man” is a predicator (general name), “and” is a logical term, “Stepan” is a singular name.
Let us explain (once again) what part of the meaning of descriptive terms is preserved when the logical form of thought is identified.
Let's analyze two arguments.
(1) All participants in this crime have been identified by the victim. None of the Petrov family members have been identified by the victim. None of the persons who participated in the commission of this crime have been brought to criminal responsibility for its commission. Consequently, not a single member of the Petrov family has been brought to criminal responsibility for committing this crime.
(2) Anyone who is of sound mind can understand logic. Neither of Crox's sons can understand logic. Crazy people are not allowed to vote. Consequently, none of Crox's sons are allowed to vote.
Let us replace the descriptive predicator terms found in each of these arguments with the variables P, Q, R, S in the order in which they appear in the argument.
The first argument includes four descriptive predicator terms, the order of their entry into the argument is as follows: the first is the term “participant in this crime” (P), the second is “identified by the victim” (Q), the third is “a member of the Petrov family” (R), the fourth - “brought to criminal responsibility for committing this crime” (S). It should be noted that the term “not participating in the commission of this crime” can be considered as obtained as a result of applying the logical negation operation “not” to the term “participant in this crime” and denoted “not-P”.
In the second argument, four descriptive terms appear in the following order: “of sound mind” (P), “able to understand logic” (Q), “son of Crox” (R), “qualified to vote” (S). The term "crazy" corresponds to the term "not of sound mind" and is designated "not-P".
Let us rewrite both of the arguments under consideration, substituting the corresponding variables instead of descriptive terms. We will replace the word “therefore” with a line separating the last sentences of the argument from the preceding sentences:
mark">logical form .
When identifying a logical form, information is stored about which semantic category the descriptive term, replaced by a variable, belongs to. Therefore, when constructing formalized languages of logic, in particular the language of predicate logic, different symbols are introduced for expressions of different categories. In addition, when identifying a logical form, different occurrences of the same term in the context are replaced by the same symbol and different terms by different symbols.