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definitiona statement or process by which the meaning of a term is conveyed. Ideally a definition – definiens – will be logically equivalent to the word or term being defined – the definiendum. However, instead of such strict verbal definitions, ostensive definitions may be provided by pointing to examples or by providing general indications of use rather than strict definitions, e.g. in indicating the meaning of colour terms. Numerous further submeanings of the term ‘definition’ should also be noted, including:
- the distinction between ‘descriptive’ and 'S tipulative’ definitions, the former stating a meaning which already has currency, the latter a proposed or reformulated statement of meaning;
- the distinction between ‘nominal’ and ‘real’ definitions, where the intention of the latter is to move beyond a merely conventional statement of meaning to provide a definition of a phenomenon in terms of its underlying or ‘real’ structural determinants (see CONVENTIONALISM and REALISM). See also ANALYTIC AND SYNTHETIC, OPERATIONALISM.
an indication or explanation of the meaning (sense) of a term and/or the content of the concept expressed by the given term; this term (concept) is called the definiendum (abbreviated Dfd), while the set of actions (words) constituting the definition is called the definiens (abbreviated Dfn). The Dfd is always a word (a term, the name of a concept). The Dfn may be either a word or a certain concrete, perfectly real object; in the latter case the definition consists of pointing to the object in a very literal sense, for example, by gesturing or by some other means of “presentation.” Such definitions, which by their very nature convey information only about the content (or even only part of the content) of the concept being defined, are called ostensive definitions. They play an important role in cognition and in everyday practice: it is precisely with their help that our “initial accumulation” of concepts occurs, without which cognition would be impossible.
Insofar as pointing out an object (or class of objects)—a method characteristic of ostensive definition—may be accomplished with words alone (using demonstrative pronouns and descriptions, for example), it is natural to include such linguistic structures in the same class of definitions. However, the overwhelming majority of definitions, in which both the Dfd and the Dfn are verbal, define the meanings of certain expressions (the Dfd) through the meanings of other expressions (the Dfn) that are assumed to be known (within the bounds of the given definition). Such definitions are called verbal; each of them is a sentence in a particular language. (The totality of sentences in a complex definition may always be considered a single complex sentence.) By means of verbal definitions new terms are introduced, or the meanings of terms introduced earlier are clarified; in both cases the definitions are called nominal. If, however, one is concerned not with the term itself but rather with the object or concept denoted by it, the definition is called real. The purpose of such a definition is to establish that the terms of the Dfd and Dfn denote one and the same object. (The classification of definitions as nominal or real is conventional.)
Up to now we have been discussing explicit definitions, which not only permit the Dfd to be introduced into any context as an “abbreviation” for the Dfn but also, in contrast, permit the Dfd to be taken out of an arbitrary context, when necessary, and be “decoded” by means of the Dfn. A classic example of this type of definition may be the definitions examined by Aristotle, per genus et differentiam (”by kind and specific difference”), which affirm the equality of the Dfd and Dfn, in which the Dfd is singled out from some wider class of objects (genus) by indicating one of its specific properties (differentia).
From the contemporary viewpoint, although genus and differentia are often distinguished, they are distinguished only grammatically and not logically. For example, in the definition “a square is a right-angled rhombus” the genus is “rhombus” and the differentia is “right-angled,” whereas in the definition “a square is an equilateral rectangle” the genus is “rectangle” and the differentia is “equilateral.” Expressed with greater precision (which, incidentally, may be considered a specific characteristic of the definition), both of these definitions are equivalent to the definition “a square is simultaneously a rhombus and a rectangle,” in which both terms of the Dfn are of absolutely equal importance.
In scientific usage, implicit definitions are very common. In such definitions the Dfd is not directly given but may be “inferred” from the context. Sometimes implicit definitions may be transformed into explicit, or contextual definitions. (The process of solving a system of equations is an example of such a transformation, inasmuch as it may be regarded, from the outset, as the definition, albeit implicit, of the unknowns.) Cases in which the implicit nature of the definition cannot be removed are of particular importance. This is exactly the case in axiomatic theories, in which axioms implicitly define the initial terms of the theory.
The classification of definitions as ostensive and verbal, real and nominal, corresponds in modern logic to the distinction between semantic and syntactic definitions. In semantic definitions, the Dfd and the Dfn are verbal expressions at different levels of abstraction (the meaning of a term being defined by properties of objects); in syntactic definitions, the Dfd and the Dfn belong to the same semantic level (the meaning of an expression being defined by the meaning of other expressions). Syntactic definitions play an important role in mathematical logic and in its applications to the foundations of mathematics and to the construction of artificial algorithmic languages for programming digital computers. They fulfill the requirement of efficiency in the construction of the Dfd and in the discrimination of the Dfd from objects not satisfying the given definition. These requirements are very much in accord with the mathematically important criterion of constructibility, the measurability of the quantity introduced by the given definition.
Explicit real definitions, in which the Dfd is introduced by describing the means for its construction, formation, production, and realization, are called genetic definitions. In physics and other natural sciences, these requirements are fulfilled by using operational definitions—that is, definitions of physical quantities by a description of the operations by means of which they are measured, and definitions of properties of objects by a description of the reactions of these objects to specific experimental actions. Thus, for example, the length of an object is defined by the results of measuring it, and the concept of the “alkaline solution,” by the phrase “a solution is termed alkaline if it turns litmus paper blue.”
Genetic definitions in the deductive sciences are realized in the form of inductive and recursive definitions. An inductive definition of some function or predicate consists of direct clauses, which specify the meanings of the function or predicate being defined for some objects in its domain of definition, and an indirect clause, which states that no objects outside the domain of the direct clauses of the given definition can satisfy it. A distinction is made between fundamental inductive definitions of certain subject domains and nonfundamental ones, which delimit certain subsets of domains already defined. Thus, the inductive definition of “natural number” (or of “formula” in propositional calculus) is fundamental, but the definition of “even number” (or, correspondingly, of “theorem” in propositional calculus) is nonfundamental. Inductive definitions of both types, which generate the objects defined by them in a certain sequence, justify the application to these objects of proofs by mathematical induction. Cases in which there is a univocal sequence of generation are especially important. Such inductive definitions are called recursive definitions. They take the form of a system of equations or equivalences, some of which are explicit definitions of certain “initial” meanings of the function or predicate being defined, while others describe the means of obtaining new meanings from those already defined with the help of various substitutions and “recursion schemes.” In a certain sense, recursive definitions are the best means of fulfilling the efficiency requirements for definitions, which are of great importance in philosophy and in practical concerns.
Definitions of all kinds (including those considered above) must fulfill a number of general requirements (principles), the violation of which may invalidate a proposition that formally resembles a definition. The rule of interchangeability (eliminability), which constitutes a requirement of equivalence between the Dfd and the Dfn of real definitions, stipulates the possibility of interchanging the Dfd and Dfn of explicit nominal definitions. The rule of univocacy (or definiteness) is the natural requirement of uniqueness of the Dfd for each Dfn—but, of course, not the reverse. (Although it guarantees the absence of homonymy within a particular theory, the rule of univocacy does not preclude synonymy. Furthermore, any explicit definition generates the synonymous pair Dfd≡Dfn, and for the same concept it is possible to have different definitions, among which comparison is often very productive.) Finally, there is the rule excluding the vicious circle: that is, the Dfn of a definition must not depend on the Dfd. The fulfillment of this most natural condition (obviously, when it is violated, the definition “defines nothing”) presents serious difficulties, especially since in the “most precise of the sciences”—mathematics—it is exceedingly inconvenient to refrain completely from impredicative definitions, which violate this rule. It should be noted that inductive and recursive definitions, in whose formulation the Dfn contains mention of the Dfd, nevertheless satisfy this requirement: an analysis of such definitions shows that at every step in the generation of the objects defined by these definitions, the entire Dfd is not used, but only those parts constructed previously (in the preceding steps).
Thus, the fulfillment of the “rules of definition,” such as the principle of efficiency, is by no means a universal, absolute “law,” but presupposes a necessary consideration of the specific characteristics of a particular situation. In unformalized scientific theories, and to an even greater degree in practical matters, where the role of definition is no less important than in the deductive sciences, definitions as a rule lack the precise canonical forms with which the foregoing exposition was primarily concerned. They are most frequently of an implicit and contextual nature; moreover, the complete “explanation” of the concept defined is quite often provided by the context as a whole.
A classic example of a dialectical approach to the problem of definition is K. Marx’ Das Kapital, in which the categories of political economy are not introduced once and for all by formal definitions but are revealed in greater depth in the course of the logical and historical analysis. Although they are very productive, the tendencies toward specifying and refining the types of definitions used in particular fields do not provide a foundation for a single, rigorous, and complete “classification” of definitions. Consequently, one cannot speak of a single “theory of definitions,” although, of course, the use of this term is fully justified within a specific methodological system. Like the concept of proof, which for all its possible refinements denotes in the final analysis “anything that proves something,” the term “definition” applies not only to formal objects of some specific type but also to anything that somehow or other defines something. Definitions at different levels of abstraction, precision, and formality not only constitute the basis for all scientific knowledge but also serve as important tools in the construction of specific scientific disciplines and, more broadly, for the comprehension of any practical activity.
REFERENCESEngels, F. Anti-Dühring. In K. Marx and F. Engels, Soch., 2nd ed., vol. 20.
Aristotle. Analitiki pervaia i vtoraia. Moscow, 1952. (Translated from Greek.)
Tarski, A. Vvedenie v logiku i metodologiiu deduktivnykh nauk. Moscow, 1948. (Translated from English.)
Gorskii, D. P. “O vidakh opredelenii i ikh znachenii v nauke.” In Problemy logiki nauchnogo poznaniia. Moscow, 1964.
Curry, H. B. Osnovaniia matematicheskoi logiki. Moscow, 1969. Chapters 1–3. (Translated from English.)
IU. A. GASTEV