A generalization is a form of
abstraction
Abstraction in its main sense is a conceptual process wherein general rules and concepts are derived from the usage and classification of specific examples, literal ("real" or "concrete") signifiers, first principles, or other methods.
"An abstr ...
whereby common properties of specific instances are formulated as general concepts or claims.
Generalizations posit the existence of a domain or
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of elements, as well as one or more common characteristics shared by those elements (thus creating a
conceptual model
A conceptual model is a representation of a system. It consists of concepts used to help people knowledge, know, understanding, understand, or simulation, simulate a subject the model represents. In contrast, physical models are physical object su ...
). As such, they are the essential basis of all valid
deductive
Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be false ...
inferences (particularly in
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
, mathematics and science), where the process of
verification
Verify or verification may refer to:
General
* Verification and validation, in engineering or quality management systems, is the act of reviewing, inspecting or testing, in order to establish and document that a product, service or system meets ...
is necessary to determine whether a generalization holds true for any given situation.
Generalization can also be used to refer to the process of identifying the parts of a whole, as belonging to the whole. The parts, which might be unrelated when left on their own, may be brought together as a group, hence belonging to the whole by establishing a common relation between them.
However, the parts cannot be generalized into a whole—until a common relation is established among ''all'' parts. This does not mean that the parts are unrelated, only that no common relation has been established yet for the generalization.
The concept of generalization has broad application in many connected disciplines, and might sometimes have a more specific meaning in a specialized context (e.g. generalization in psychology,
generalization in learning).
In general, given two related concepts ''A'' and ''B,'' ''A'' is a "generalization" of ''B'' (equiv., ''B'' is a
special case
In logic, especially as applied in mathematics, concept is a special case or specialization of concept precisely if every instance of is also an instance of but not vice versa, or equivalently, if is a generalization of . A limiting case is ...
of ''A'') if and only if both of the following hold:
* Every instance of concept ''B'' is also an instance of concept ''A.''
* There are instances of concept ''A'' which are not instances of concept ''B''.
For example, the concept ''animal'' is a generalization of the concept ''bird'', since every bird is an animal, but not all animals are birds (dogs, for instance). For more, see
Specialisation (biology)
A generalist species is able to thrive in a wide variety of environmental conditions and can make use of a variety of different resources (for example, a heterotroph with a varied diet). A specialist species can thrive only in a narrow range of env ...
.
Hypernym and hyponym
The connection of ''generalization'' to ''specialization'' (or ''particularization'') is reflected in the contrasting words
hypernym
In linguistics, semantics, general semantics, and ontologies, hyponymy () is a semantic relation between a hyponym denoting a subtype and a hypernym or hyperonym (sometimes called umbrella term or blanket term) denoting a supertype. In other wor ...
and
hyponym
In linguistics, semantics, general semantics, and ontologies, hyponymy () is a semantic relation between a hyponym denoting a subtype and a hypernym or hyperonym (sometimes called umbrella term or blanket term) denoting a supertype. In other wor ...
. A hypernym as a
generic
Generic or generics may refer to:
In business
* Generic term, a common name used for a range or class of similar things not protected by trademark
* Generic brand, a brand for a product that does not have an associated brand or trademark, other ...
stands for a class or group of equally ranked items, such as the term ''tree'' which stands for equally ranked items such as ''peach'' and ''oak'', and the term ''ship'' which stands for equally ranked items such as ''cruiser'' and ''steamer''. In contrast, a hyponym is one of the items included in the generic, such as ''peach'' and ''oak'' which are included in ''tree'', and ''cruiser'' and ''steamer'' which are included in ''ship''. A hypernym is superordinate to a hyponym, and a hyponym is subordinate to a hypernym.
Examples
Biological generalization
![Generalization process using trees](https://upload.wikimedia.org/wikipedia/commons/f/fd/Generalization_process_using_trees.svg)
An animal is a generalization of a
mammal
Mammals () are a group of vertebrate animals constituting the class Mammalia (), characterized by the presence of mammary glands which in females produce milk for feeding (nursing) their young, a neocortex (a region of the brain), fur or ...
, a bird, a fish, an
amphibian
Amphibians are tetrapod, four-limbed and ectothermic vertebrates of the Class (biology), class Amphibia. All living amphibians belong to the group Lissamphibia. They inhabit a wide variety of habitats, with most species living within terres ...
and a reptile.
Cartographic generalization of geo-spatial data
Generalization has a long history in
cartography
Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an im ...
as an art of creating maps for different scale and purpose.
Cartographic generalization
Cartographic generalization, or map generalization, includes all changes in a map that are made when one derives a smaller-scale map from a larger-scale map or map data. It is a core part of cartographic design. Whether done manually by a cartogra ...
is the process of selecting and representing information of a map in a way that adapts to the scale of the display medium of the map. In this way, every map has, to some extent, been generalized to match the criteria of display. This includes small cartographic scale maps, which cannot convey every detail of the real world. As a result, cartographers must decide and then adjust the content within their maps, to create a suitable and useful map that conveys the
geospatial
Geographic data and information is defined in the ISO/TC 211 series of standards as data and information having an implicit or explicit association with a location relative to Earth (a geographic location or geographic position).
It is also call ...
information within their representation of the world.
Generalization is meant to be context-specific. That is to say, correctly generalized maps are those that emphasize the most important map elements, while still representing the world in the most faithful and recognizable way. The level of detail and importance in what is remaining on the map must outweigh the insignificance of items that were generalized—so as to preserve the distinguishing characteristics of what makes the map useful and important.
Mathematical generalizations
* A
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
is a generalization of a 3-sided
triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...
, a 4-sided
quadrilateral
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
, and so on to
''n'' sides.
* A
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, ...
is a generalization of a 2-dimensional square, a 3-dimensional
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...
, and so on to ''n''
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
s.
* A
quadric
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...
, such as a
hypersphere
In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, cal ...
,
ellipsoid
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a surface that may be defined as the ...
,
paraboloid
In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry.
Every plane ...
, or
hyperboloid
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by defo ...
, is a generalization of a
conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
to higher dimensions.
* A
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
is a generalization of a
MacLaurin series
Maclaurin or MacLaurin is a surname. Notable people with the surname include:
* Colin Maclaurin (1698–1746), Scottish mathematician
* Normand MacLaurin (1835–1914), Australian politician and university administrator
* Henry Normand MacLaurin ( ...
.
* The
binomial formula
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
is a generalization of the formula for
.
See also
{{wikiquote
*
Categorical imperative
The categorical imperative (german: kategorischer Imperativ) is the central philosophical concept in the deontological moral philosophy of Immanuel Kant. Introduced in Kant's 1785 ''Groundwork of the Metaphysic of Morals'', it is a way of evalu ...
(ethical generalization)
* ''
Ceteris paribus
' (also spelled '; () is a Latin phrase, meaning "other things equal"; some other English translations of the phrase are "all other things being equal", "other things held constant", "all else unchanged", and "all else being equal". A statement ...
''
*
Class diagram
In software engineering, a class diagram in the Unified Modeling Language (UML) is a type of static structure diagram that describes the structure of a system by showing the system's classes, their attributes, operations (or methods), and the rela ...
*
External validity
External validity is the validity of applying the conclusions of a scientific study outside the context of that study. In other words, it is the extent to which the results of a study can be generalized to and across other situations, people, stim ...
(scientific studies)
*
Faulty generalization
A faulty generalization is an informal fallacy wherein a conclusion is drawn about all or many instances of a phenomenon on the basis of one or a few instances of that phenomenon. It is similar to a proof by example in mathematics. It is an exampl ...
*
Generic (disambiguation)
*
Critical thinking
Critical thinking is the analysis of available facts, evidence, observations, and arguments to form a judgement. The subject is complex; several different definitions exist, which generally include the rational, skeptical, and unbiased analysis ...
*
Generic antecedent
Generic antecedents are representatives of classes, referred to in ordinary language by another word (most often a pronoun), in a situation in which gender is typically unknown or irrelevant. These mostly arise in generalizations and are particul ...
*
Hasty generalization
A faulty generalization is an informal fallacy wherein a conclusion is drawn about all or many instances of a phenomenon on the basis of one or a few instances of that phenomenon. It is similar to a proof by example in mathematics. It is an examp ...
*
Inheritance (object-oriented programming)
In object-oriented programming, inheritance is the mechanism of basing an object or class upon another object ( prototype-based inheritance) or class ( class-based inheritance), retaining similar implementation. Also defined as deriving new classe ...
,
* ''
Mutatis mutandis
''Mutatis mutandis'' is a Medieval Latin phrase meaning "with things changed that should be changed" or "once the necessary changes have been made". It remains unnaturalized in English and is therefore usually italicized in writing. It is used i ...
''
*
-onym
The suffix ''-onym'' (from grc, ὄνυμα / name) is a bound morpheme, that is attached to the end of a root word, thus forming a new compound word that designates a particular ''class'' of names. In linguistic terminology, compound words ...
*
Ramer–Douglas–Peucker algorithm The Ramer–Douglas–Peucker algorithm, also known as the Douglas–Peucker algorithm and iterative end-point fit algorithm, is an algorithm that decimates a curve composed of line segments to a similar curve with fewer points. It was one of the e ...
*
Semantic compression In natural language processing, semantic compression is a process of compacting a lexicon used to build
a textual document (or a set of documents) by reducing language heterogeneity, while maintaining text semantics.
As a result, the same ideas ca ...
*
Specialization (logic)
Specialization or Specialized may refer to:
Academia
* Academic specialization, may be a course of study or major at an academic institution or may refer to the field in which a specialist practices
* Specialty (medicine), a branch of medical ...
, the opposite process
*
Inventor's paradox
The inventor's paradox is a phenomenon that occurs in seeking a solution to a given problem. Instead of solving a specific type of problem, which would seem intuitively easier, it can be easier to solve a more general problem, which covers the spec ...
References
Generalizations
Critical_thinking
Inductive_reasoning