Generalization

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generalization

, generalisation
1. Psychol the evoking of a response learned to one stimulus by a different but similar stimulus
2. Logic the derivation of a general statement from a particular one, formally by prefixing a quantifier and replacing a subject term by a bound variable. If the quantifier is universal (universal generalization) the argument is not in general valid; if it is existential (existential generalization) it is valid
3. Logic any statement ascribing a property to every member of a class (universal generalization) or to one or more members (existential generalization)

Generalization

 

(1) In physiology, generalization is the spread of excitation through the central nervous system of animals and humans. The process of generalization arises under the influence of impulses coming from the periphery (as a result of a strong stimulus, such as food, pain, or a new, undifferentiated stimulus that gives rise to an orientation reaction). Generalization of excitation along the cortex of the cerebral hemispheres occurs in the first stages of the formation of a conditioned reflex.

(2) In pathology, generalization is the conversion of an initially limited infectious or neoplasmic process into an extensive one, with the appearance of foci in other organs. Generalization occurs through the blood channels and lymphatic pathways. Generalization does not include the gradual extension of the territory of a primary focus of affection, if it is not accompanied by the appearance of new foci in other organs.


Generalization

 

a method of increasing knowledge by means of a mental transition from the particular to the general. The transition to a higher level of abstraction also usually corresponds to this method. An example of generalization is the transition from the observation of aggregates of individual objects to the mental classification of these objects into aggregates of equal number, which leads to the concept of natural numbers.

One of the most important means of acquiring scientific knowledge, generalization makes it possible to derive general principles (laws) from the chaos of phenomena that obscures them, and to unify and identify in a single formula sets of different things and events.

The two principal types of generalizations are distinguished by their semantic and epistemological content. The first type generates new semantic units (concepts)—that is, notions, laws, principles, and theories that are not determined by the initial semantic field (primary semantics). The second type does not generate such units and can yield only new variants of old values.

Generalizations of the second type have a comparatively simpler structure than the first type and are often their limiting cases. The second type includes, in particular, extrapolation (for example, extension of the quantum interpretation of Planck’s law of thermal radiation to the field of light phenomena, which made it possible to explain the photoelectric effect). Incomplete induction is also classified as a generalization of the second type (for example, the extension to all substances of the experimentally known property of a number of substances to exist in three aggregate states). In addition, the second type includes the ∀-generalization of pure predicate logic, which is essentially a synonymous transition from A(x) to ∀xA(x), where the condition A(x) is interpreted as universality.

The first type of generalization includes all theoretical generalizations, or generalizations by abstraction, which in the theory of cognition is reflected by the transition from some abstraction of the order n to abstractions of a higher order. In particular, this includes generalization by replacing constants with variables, which is natural for logic and which makes possible the isolation, in “pure form,” of such essences as property and relation. Theoretical generalization also includes generalization based on an idealized experiment that suggests speculative principles, such as the principle of inertia or the principle of relativity. In addition, it includes generalizations through identification in terms of a property, which makes it possible to isolate the common essence of phenomena perceived in different ways. (For example, magnetism, electricity, and light are merely different manifestations of the electromagnetic field.)

The ∀-generalization of applied logic (Locke’s rule) is also a generalization of the first type. It is widely used in mathematical proofs, when, during the transition from a particular value of ξ to all ξ in the identification-abstraction interval, preservation of the truth of the predicate established for a particular value is ensured. This is always possible if the truth of the predicate depends not on the particular value of x, but only on its range of change as determined by the corresponding identification— that is, on the class of abstraction. (In this case the given particular value serves as a generalized representative, or standard, of the class of abstraction.) Here, in contrast to the ∀-generalization of pure logic, a new semantic context of generalization also arises. The original conventional interpretation of the premise is replaced by an interpretation of universality, and the concept of class of abstraction, as related to the content of the particular value, becomes part of the content of a subquantifier variable, making the quantifier “bound.” However, in cases where the class of abstraction coincides with the universal class, the ∀-generalization of applied logic passes into the ∀-generalization of pure logic.

Historically, the development of concepts and theories has been expressed in the increase of knowledge by means of chains of generalizations, in which generalizations of the first or second type serve as links. Chains of generalizations reflect the sequential links of first-order essences with essences of the second, third, and higher orders. These links are different and, depending on their character, are represented by chains of generalizations retaining the semantics of the initial concepts or, on the other hand, by chains of generalizations that alter the primary semantics. The sequential generalization of the concept of number by the construction of systems of natural, integral, rational, real, and complex numbers may serve as an example. Characteristic of this chain, which retains the primary semantics, are expansions of the initial range that satisfy the principle of the constancy of formal laws, according to which the laws of operations defined for elements of the initial domain must be preserved for new elements in all subsequent expansions. However, the chain cannot be extended indefinitely. The arithmetic of transfinite quantitative numbers no longer satisfies this principle, but the resulting transition to the general concept of quantitative number leads to a new understanding of the arithmetic of natural numbers as the arithmetic of the powers of finite sets.

The transition from classical logic to intuitionist logic and the successive transition from classical mechanics to relativistic mechanics and the general theory of relativity may serve as examples of a generalization chain of the second type. In such transitions, the more general theory may have a complete formulation independent of the less general theory, but it must contain the latter as a limiting case. This is the basic meaning of the principle of correspondence for chains of generalizations with changing primary semantics.

REFERENCES

Polya, G. Matemalika i pravdopodobnye rassuzhdeniia. Moscow, 1957. (Translated from English.)
Davydov, V. V. Vidy obobshcheniia ν obuchenii. Moscow, 1972.
Sachkov, Iu. V. “Protsessy obobshcheniia ν sinteze znanii.” In Sintez sovremennogo nauchnogo znaniia. Moscow, 1973. Pages 421–46.
Matiushkin, A. M., and M. M. Novoselov. “Vidy obobshcheniia i problemy psikhologii obucheniia.” Voprosy psikhologii, no. 2, 1974.

F. V. LAZAREV and M. M. NOVOSELOV

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