Classes, sets, groups, systems
The description of the domain (the creation of its ontology) begins with the selection of objects and their classification, which traditionally consists in compiling a tree of subclass classes and assigning individuals to them. In this case, the term “class”, in essence, is used in the meaning of “set”: the assignment of an object to a class is thought of as including it as an element in the corresponding set. The purpose of this text is to show that such a unified approach to the description of the domain structure is a strong simplification and does not allow fixing the diversity of semantic relations of objects.
Let's look at three options for classifying an individual Bug:
The first sequence of co-ordinated entities is unambiguously decided to be described through the assignment of classes and subclasses: A bug is an individual of the class “like”, a class “like” is a subclass of dogs, and that is a subclass of the class “animal”. In this case, the class “animals” is interpreted as the set of all animals, and the class “likes”, as a subset of the set “dogs”. However, such a description, despite the fact that it is sufficiently clear, is essentially tautological, self-referential: we call an individual a Bug a husky, if it is included in a multitude of huskies, and we define the multitude of huskies as the set of all individuals of the huskies - that is duplicates the name. In addition, the description of the class-set is completely exhausted by the description of the individual falling under the concept defining the class. It should also be noted that operating with such classes-sets does not depend on the number of elements in them: like a Bug will be a husky even when it remains the only, last husky on Earth. Moreover, we can even operate with such classes-sets even in the absence of individuals in them: you can build an ontology of already extinct dinosaurs, think of a class in which only in the future will the projected unique device or construct a model of the subject area of ​​mythical animals, heroes of fairy tales, although the power of all classes-sets will be zero.
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So, if we talk about the content side of the analyzed classification (animal - dog - like - Bug), then it (the content side) can not be expressed in any way through the ratio of sets and subsets. In this case, we are dealing with conceptualization - the selection of concepts and the establishment of gender-species relations between them. At the same time, the actual number of elements of the conceptual class, that is, the scope of the concept, does not appear in its definition and is mentioned (and even not meaningfully) only when one concept (“husky”) falls under another (“dog”), that is, when as a kind of genus. Yes, we can state that the scope of the concept “dog” is larger than the scope of the concept “husky”, but the real numerical ratio of these sets has no ontological meaning. Exceeding the volume of a class with the volume of a subclass under generic relations reflects only that, by definition, the genus should include several types - otherwise this classification becomes meaningless. That is, in the genus-species conceptual classification, we are interested in the content of the concepts - how the “dog” type differs from the “cat” type (which also falls under the generic concept “animal”), and not how the volumes of the genus and species sets correlate and the more volumes of species concepts ("dog" and "cat"). And in order to distinguish conceptual classes from truly countable sets, it would be more correct to speak of the individual falling under the concept , and not of including him in a class / set. It is clear that in a formal record the statements “fall under the concept of X” and “is an element of class X” may look the same, but failure to understand the essential difference between these two descriptions can lead to serious errors in the construction of ontology.
In the second variant (service - sled - Bug), we are also not interested in comparing the notion of a “sled” of any set: the semantic content of the statement “Bug - sled” does not depend on whether it is the only sled or there are many. It would seem that here we are dealing with a generic relationship: the concept of “riding” can be viewed as a species relative to the generic concept of “service”. But the connection of the individual "Bug" with the concept of "riding" differs significantly from the connection with the concept of "like": the second is conceptual, the concept is immanent and invariably inherent in the individual, and the first reflects the localization in time specialization . A bug was not born a sled, and perhaps with age it may cease to be her and go into the category of watchdogs, and under old age in general lose all “professions”. That is, speaking of specialization, we can always distinguish the events of acquisition and loss of connection with one or another concept. For example, the Bug could be recognized as an absolute champion of the breed, and then lose this title, which is fundamentally impossible with conceptual concepts: The bug from birth to death, that is, throughout the entire period of its existence as an individual, is a dog and husky. In the same way, a person remains the concept of “person” all his life, but situationally (from event to event) may fall under the special notions “schoolchild”, “student”, “doctor”, “husband”, etc. And as already noted, the connection with these concepts do not mean inclusion in some set (although it may look like this) - attributing a specializing concept is always the result of an individual’s specific relationship with other individuals: entering a school, university, getting a diploma, marriage registration, etc. Therefore, specializing concepts can be called hereafter relational . From the above examples, there is another significant difference between conceptual classification and specialization: an individual may have several specializations (the Bug is a riding one and a champion of a breed, a man is a student and a husband), but cannot simultaneously belong to more than one conceptual hierarchy (A Bug cannot be a dog and a cat).
And only in the third version of the description of the Bugs - as belonging to a certain kennel and as a member of a specific team that pulls the sleds along the tundra - it is just necessary to mention the set. Only in this case we have the right to say that the individual is an element of a concrete set with a countable number of elements, and does not fall under the concept, which can be represented as an abstract set, conditionally fixing the scope of this concept. And here it is important that the individual is part of another individual, originally defined as a set: a kennel and a team are necessarily a non-empty set of dogs, and the number of elements of this set necessarily enters their definitions as individuals. That is, in this case, we should speak about the part-to-whole relationship: the bug is part of the kennel and part of the team. Moreover, the entry or non-entry of the Bugs into a specific team changes its contents (the teams): if we had a two-piece team, then after removing the Bugs, the team turns into a single one. In such cases, we are dealing not simply with a countable set (dogs in a kennel), but with an individual, whose essence changes as the composition of its elements changes, is determined by this composition, that is, with the system . If a kennel is just an individual group, described through the set of elements included in it, then the team is a system whose essence depends on the number and specificity of its parts.
Consequently, when constructing the ontology of the domain, it is possible to distinguish real objects-sets, defined precisely as the combination of a certain number of individuals. These are: class at school, goods in a box in a warehouse, parts of an electronic device block, etc. And these sets can be subsets of other real counting sets: all students of the school, all goods in the warehouse, all parts of the device. When selecting these sets, it is essential that they (these sets) act as independent individuals (collective, consignment, set of parts), the main attribute of which is precisely the number of elements included in them. Moreover, a change in this attribute can lead to a change in the status of an object, say, with an increase in the number of elements, to turn a quartet into a quintet or a regiment into a brigade. It is also important that the description of these set objects, complex objects is not reduced to the description of individuals within them, although it may include an indication of the allowable type of the latter (string quartet, horse team). And such relationships — not between abstract sets, but between sets, which are individuals, complex objects — are more accurately described as part-whole relations, rather than a subclass class.
So, the traditional classification of individuals through attributing them to one or another class-set cannot be considered homogeneous. It is necessary to distinguish (1) the inclusion of individuals as parts in a complex object (whole), the semantic specificity of which is not reduced to the description of its elements. At the same time (1.1.) An object-whole can be considered only as a named set of individuals (parts in a package, a collection of pictures), for which, in fact, only the number of parts is important. Such objects may be called groups (or collections ). Also (1.2.) An object-integer can meaningfully (and not only quantitatively) be determined by its parts and, as a result, possess attributes that parts do not possess. Such integrity is traditionally called systems , and parts of systems - elements. The second variant of the description of objects by attributing them to classes-subclasses is (2) the falling of individuals under the concept that only formally, tautologically, can be described as the inclusion of individuals in a set whose power is equal to the power of the concept. The conceptual description of individuals, in turn, can be classified into (2.1) a conceptual , globally fixing type of an individual, and (2.2) specializing (relational) , locally in time and space (event-related) connecting the individual with other objects.
The above reasoning, first of all, raises the question of the adequacy and adequacy of the traditional approach to the description of the subject area using a classification based on set theory. And the conclusion is proposed: for fixing in the ontologies of the whole variety of connections of objects, more differentiated classification tools are needed (groups, systems, conceptual and specializing concepts). The formalism of set theory can only be used as a local simplification for the needs of logical inference, and not as the main method of description.
Continued: Conceptual description of individuals
Let's look at three options for classifying an individual Bug:
- Animal - dog - husky - beetle.
- Service - sled - Bug.
- Kennel - dog sledding - Bug.
The first sequence of co-ordinated entities is unambiguously decided to be described through the assignment of classes and subclasses: A bug is an individual of the class “like”, a class “like” is a subclass of dogs, and that is a subclass of the class “animal”. In this case, the class “animals” is interpreted as the set of all animals, and the class “likes”, as a subset of the set “dogs”. However, such a description, despite the fact that it is sufficiently clear, is essentially tautological, self-referential: we call an individual a Bug a husky, if it is included in a multitude of huskies, and we define the multitude of huskies as the set of all individuals of the huskies - that is duplicates the name. In addition, the description of the class-set is completely exhausted by the description of the individual falling under the concept defining the class. It should also be noted that operating with such classes-sets does not depend on the number of elements in them: like a Bug will be a husky even when it remains the only, last husky on Earth. Moreover, we can even operate with such classes-sets even in the absence of individuals in them: you can build an ontology of already extinct dinosaurs, think of a class in which only in the future will the projected unique device or construct a model of the subject area of ​​mythical animals, heroes of fairy tales, although the power of all classes-sets will be zero.
')
So, if we talk about the content side of the analyzed classification (animal - dog - like - Bug), then it (the content side) can not be expressed in any way through the ratio of sets and subsets. In this case, we are dealing with conceptualization - the selection of concepts and the establishment of gender-species relations between them. At the same time, the actual number of elements of the conceptual class, that is, the scope of the concept, does not appear in its definition and is mentioned (and even not meaningfully) only when one concept (“husky”) falls under another (“dog”), that is, when as a kind of genus. Yes, we can state that the scope of the concept “dog” is larger than the scope of the concept “husky”, but the real numerical ratio of these sets has no ontological meaning. Exceeding the volume of a class with the volume of a subclass under generic relations reflects only that, by definition, the genus should include several types - otherwise this classification becomes meaningless. That is, in the genus-species conceptual classification, we are interested in the content of the concepts - how the “dog” type differs from the “cat” type (which also falls under the generic concept “animal”), and not how the volumes of the genus and species sets correlate and the more volumes of species concepts ("dog" and "cat"). And in order to distinguish conceptual classes from truly countable sets, it would be more correct to speak of the individual falling under the concept , and not of including him in a class / set. It is clear that in a formal record the statements “fall under the concept of X” and “is an element of class X” may look the same, but failure to understand the essential difference between these two descriptions can lead to serious errors in the construction of ontology.
In the second variant (service - sled - Bug), we are also not interested in comparing the notion of a “sled” of any set: the semantic content of the statement “Bug - sled” does not depend on whether it is the only sled or there are many. It would seem that here we are dealing with a generic relationship: the concept of “riding” can be viewed as a species relative to the generic concept of “service”. But the connection of the individual "Bug" with the concept of "riding" differs significantly from the connection with the concept of "like": the second is conceptual, the concept is immanent and invariably inherent in the individual, and the first reflects the localization in time specialization . A bug was not born a sled, and perhaps with age it may cease to be her and go into the category of watchdogs, and under old age in general lose all “professions”. That is, speaking of specialization, we can always distinguish the events of acquisition and loss of connection with one or another concept. For example, the Bug could be recognized as an absolute champion of the breed, and then lose this title, which is fundamentally impossible with conceptual concepts: The bug from birth to death, that is, throughout the entire period of its existence as an individual, is a dog and husky. In the same way, a person remains the concept of “person” all his life, but situationally (from event to event) may fall under the special notions “schoolchild”, “student”, “doctor”, “husband”, etc. And as already noted, the connection with these concepts do not mean inclusion in some set (although it may look like this) - attributing a specializing concept is always the result of an individual’s specific relationship with other individuals: entering a school, university, getting a diploma, marriage registration, etc. Therefore, specializing concepts can be called hereafter relational . From the above examples, there is another significant difference between conceptual classification and specialization: an individual may have several specializations (the Bug is a riding one and a champion of a breed, a man is a student and a husband), but cannot simultaneously belong to more than one conceptual hierarchy (A Bug cannot be a dog and a cat).
And only in the third version of the description of the Bugs - as belonging to a certain kennel and as a member of a specific team that pulls the sleds along the tundra - it is just necessary to mention the set. Only in this case we have the right to say that the individual is an element of a concrete set with a countable number of elements, and does not fall under the concept, which can be represented as an abstract set, conditionally fixing the scope of this concept. And here it is important that the individual is part of another individual, originally defined as a set: a kennel and a team are necessarily a non-empty set of dogs, and the number of elements of this set necessarily enters their definitions as individuals. That is, in this case, we should speak about the part-to-whole relationship: the bug is part of the kennel and part of the team. Moreover, the entry or non-entry of the Bugs into a specific team changes its contents (the teams): if we had a two-piece team, then after removing the Bugs, the team turns into a single one. In such cases, we are dealing not simply with a countable set (dogs in a kennel), but with an individual, whose essence changes as the composition of its elements changes, is determined by this composition, that is, with the system . If a kennel is just an individual group, described through the set of elements included in it, then the team is a system whose essence depends on the number and specificity of its parts.
Consequently, when constructing the ontology of the domain, it is possible to distinguish real objects-sets, defined precisely as the combination of a certain number of individuals. These are: class at school, goods in a box in a warehouse, parts of an electronic device block, etc. And these sets can be subsets of other real counting sets: all students of the school, all goods in the warehouse, all parts of the device. When selecting these sets, it is essential that they (these sets) act as independent individuals (collective, consignment, set of parts), the main attribute of which is precisely the number of elements included in them. Moreover, a change in this attribute can lead to a change in the status of an object, say, with an increase in the number of elements, to turn a quartet into a quintet or a regiment into a brigade. It is also important that the description of these set objects, complex objects is not reduced to the description of individuals within them, although it may include an indication of the allowable type of the latter (string quartet, horse team). And such relationships — not between abstract sets, but between sets, which are individuals, complex objects — are more accurately described as part-whole relations, rather than a subclass class.
So, the traditional classification of individuals through attributing them to one or another class-set cannot be considered homogeneous. It is necessary to distinguish (1) the inclusion of individuals as parts in a complex object (whole), the semantic specificity of which is not reduced to the description of its elements. At the same time (1.1.) An object-whole can be considered only as a named set of individuals (parts in a package, a collection of pictures), for which, in fact, only the number of parts is important. Such objects may be called groups (or collections ). Also (1.2.) An object-integer can meaningfully (and not only quantitatively) be determined by its parts and, as a result, possess attributes that parts do not possess. Such integrity is traditionally called systems , and parts of systems - elements. The second variant of the description of objects by attributing them to classes-subclasses is (2) the falling of individuals under the concept that only formally, tautologically, can be described as the inclusion of individuals in a set whose power is equal to the power of the concept. The conceptual description of individuals, in turn, can be classified into (2.1) a conceptual , globally fixing type of an individual, and (2.2) specializing (relational) , locally in time and space (event-related) connecting the individual with other objects.
The above reasoning, first of all, raises the question of the adequacy and adequacy of the traditional approach to the description of the subject area using a classification based on set theory. And the conclusion is proposed: for fixing in the ontologies of the whole variety of connections of objects, more differentiated classification tools are needed (groups, systems, conceptual and specializing concepts). The formalism of set theory can only be used as a local simplification for the needs of logical inference, and not as the main method of description.
Continued: Conceptual description of individuals
Source: https://habr.com/ru/post/275865/
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