The following definitions are correct

What is a definition?

Define comes from the Latin word for “demarcate”. Like all human actions, defining is primarily purposeful: defining a term or phrase to clarify its use. The definition should result in criteria that can be used to decide whether something falls under the term. So definition is a metalinguistic operation, the purpose of which is to improve understanding. It is often triggered by the perception of an ambiguity or vagueness that would be detrimental to further communication and must therefore be eliminated beforehand.

A difficult question at the beginning is what is actually being defined, words, terms or objects. For example, if I define:

A stove is a device or fireplace where food is cooked
then do I have the stove (object) or stove (Word) or ‘stove’ (term) defined?

The first alternative would be to explicate as a definition of an individual or a set of objects. You cannot define an individual (e.g. you can identify or describe it). A heterogeneous set of objects cannot be defined either; one can only enumerate its elements. A homogeneous set of objects, i.e. a class, seems to be subject to a definition like the following

Sandals are open women's shoes with a higher heel.
It is true that this is formulated as if a class of objects were the subject of definition; but is meant
A sandal is an open women's shoe with a higher heel.
That is, the subject's nominal syntagma is generically determined, as is necessary when one speaks of concepts. Conclusion: Objects are not defined.

Then the question remains whether one defines terms or words (see the section on concept and expression). In fact, both are possible. Since scholasticism, a distinction has been made between two types of definition, the nominal definition and the real definition.

    This means the following: Suppose the expression comes in a lecture hypothesis in front. The word is familiar to the student Erwin, but the lecturer uses it in a way that Erwin does not understand. Erwin therefore asks the lecturer to define ‘hypothesis’. The lecturer then defines:

    • A hypothesis is an assumption about something in an empirical area; it has the status of a hypothesis as long as it is not proven, in particular not deduced from a theory, and also not refuted.
    Here the lecturer has one Real definition given, i.e. he has explained a term.

    On the other hand, suppose scientist W systematizes the lexical relations between terms. He comes across pairs of verbs like open - close, cover - reveal, arm - disarm. He summarizes the phenomenon by formulating: It is a lexical relation between two verbs, one of which means the transfer of an object from state Z1 to Z2 and the other means the transfer of the object from state Z2 to Z1. The only thing missing now is a suitable term for the term so conceived. W therefore defines:

    • Given a meaning B1, which implies the transfer of an object from state Z1 to state Z2, and a meaning B2, which implies the transfer of the object from Z2 to Z1, then B1 and B2 are called reversible to one another.

    W has now assigned a term to the term, i.e. it has a term from ‘reversive’ Nominal definition given.

    Summarized:

    • A Real definition is the explanation of a term, the aim of which is to determine how the term is used.
    • A Nominal definition is the explicit introduction of a term, the aim of which is to establish a term for a term, i.e. to make it manageable by coupling it to a word.

    Nominal definitions occur almost exclusively in science, while real definitions, on the other hand, also occur in daily practice. The latter are therefore far more vital than the former.

    Now we can give a real definition of ‘definition’ (cf. Wikipedia s.v. definition):

      A definition makes the intent of a term explicit. That is, it indicates the set of characteristics that are common to all and only to the entities that form the extension of the term. This definition of definition ’tries to capture what is common to real definition’ and nominal definition ’, while slightly downplaying the latter in favor of the former.

      A definition therefore explains the intention, not the extension of a term. On the one hand, the extension of most of the terms cannot be enumerated. On the other hand, the purpose of the definition would not be achieved by providing a criterion according to which one can decide whether an entity belongs to the extension of a term. Definitions that list what falls under a term are therefore inappropriate.

      The definition of definition ’excludes a number of speech acts or sentence forms, which are also called definition’, and is narrower than the sum of the relevant language usage. The condition that a definition makes the intension of a concept explicit is not fulfilled, for example, by the ostensive definition or definition per ostensionem (through demonstratio ad oculos). It is defined here per ostensionem by Erna's statement in the following dialogue:

      • Erwin: "Tell me, what is a joystick anyway?"
      • Erna, pointing to the joystick of her computer: “That!”
      where Erna's utterance is to be interpreted pragmatically as “anything that has the essential properties of the the Has". Definitions per ostensionem have become an integral part of natural language use and, in particular, language acquisition. They work particularly well for prototypical terms when a focal instance is pointed out. However, they do not meet the above definition condition and are therefore not considered in the following.

      Another type of utterance that looks like a definition is in

      A number word is a numeral.
      Water is H2O.

      This type of equivalence is called reductive definition. The designation correctly reflects the fact that this definition does not explicitly explain the intention of the term, but rather assigns an everyday term to a term that has been explicated (elsewhere) in a science. In a certain sense, this is a nominal definition insofar as it presupposes a scientific understanding of the right equivalent of the equation and assigns the everyday expression to it.

      From a formal point of view, a definition is the association of two expressions with one another. The term that is defined is called that Definiendum, in the examples above hypothesis and reversible. The term that defines the above is called that Definiens.1 The expression definition is ambiguous because it denotes both the operation of defining and defining. (The above definition relates to the former sense.) The term is needed Definiens only in contexts where this ambiguity should be avoided.

      In the assignment, the definiendum is equated with the defining. More precisely, they are declared synonymous. Of course, one cannot apply the narrow distributional synonymy term to definitions, because for purely stylistic reasons the defining cannot be used everywhere for the definiendum. Apart from such points of view, however, it is to be demanded that the Definiendum should state the Definiens in all places where it appears in texts.

      In the following we limit ourselves to real definitions and next ask: how do you define appropriately and correctly? The question has at least two aspects:

      • How do you determine what actually constitutes the intention of a term?
      • How do you formulate a definition?
      The first question relates to semantic analysis, the second to conventions about a type of text.

      Demarcation methodology

      The methods of semantic analysis are the subject of entire textbooks and seminars. In short, the methodology consists of two steps:

      1. Determine in which paradigmatic and syntagmatic relations the definiendum is in the language.
      2. Test your attempt at definition by using the defining in texts for the definiendum.

      Determination of the paradigmatic and syntagmatic relations

      The first step of the methodology consists of the following: When determining the paradigmatic relations, one compares the definiendum with expressions with a similar meaning and differentiates it from these. For example, if you want to define ‘hypothesis’, you have to separate it from theory ’, theorem’, thesis ’, argument’, ‘observation’, ‘proof’ etc. Subsumption under the correct generic term plays an important role. E.g. the generic term for ‘sentence’ ’linguistic unit’, the generic term for order ’is‘ relation ’, etc. When determining the syntagmatic relations, one examines the contexts in which the definiendum is used. For example, when it comes to the term methodology ’, you notice that it occurs in sentences such as Erna has not yet mastered the linguistic methodology, and one concludes from this that there can be a methodology for each discipline.

      Using this method also ensures that you can get a term through its essential and not by any accidental (random) properties defined. The example going back to the Platonists

      ‘Man is a featherless biped’
      shows what is meant in a blatant way, but
      ‘A well is a columnar cavity that extends vertically from the surface of the earth down to at least the water table’
      is no more acceptable. If one examines the paradigmatic relations of human ’, one finds that the term is not in direct contrast to bird’; and if you examine the syntagmatic relations of ‘well’, one finds that it is a place that people go to for water.

      An extension of the requirement to observe the paradigmatic and syntagmatic relations of the Definiendum, which cannot be properly pursued here, is the following: A scientific term can only be defined within the framework of a theory. Laypeople sometimes expect a scientist to explain a term to them and get frustrated when the scientist goes too far. The example of the concept of meaning shows that there are different approaches to its conception in philosophy and in linguistics, all of which make extensive theoretical preconditions. In principle, this is the case with every theoretical term: it cannot be defined without preconditions, but belongs in the framework of a theoretical building in which some terms are elementary, others are derived and where, consequently, terms appear in the definitions of some definitions that in turn are previously defined were.

      Test of definition

      The second step in the methodology is essentially to ensure that the definition is not too narrow or too broad. There are also uses of the word denoting the definiendum which one has not thought of and of which one has to consider whether one wants to include or exclude them. In the case of homonyms, the i.a. no difficulty. Assuming that I have defined 'sentence' as the 'smallest linguistic unit with an illocutive force', and I come across the assertion “the symphony has four movements” in a text, I will at best feel compelled to put a comment in front of my definition, that in our context it is a question of the sentence as a linguistic structure, but does not change the definition. In the case of polysemes, however, the definition may need to be reconsidered. Suppose I have defined "wing" as the "lateral extremity of a bird or flying object" and now come across the expression Wing of a building. Then I may find that my definition is too narrow and change it to something like "lateral extremity of an object".

      The last example shows two things: First, if I change the definition in the manner described, i.e. narrow it or - as here - expand it, then I actually define a different term. Every definition that meets elementary logical requirements defines a term, regardless of the question of whether there is a name of its own for it. However, that is the perspective of a nominal definition. If I intend a real definition, I must first get clarity about the definiendum and then find the best definition for it. However, this may require going through the two steps of the methodology in a spiral.

      Second, the example of the wing - and especially the example of the sentence - shows once again from a different point of view that real definitions actually define concepts, not words. A word can have many different senses. Whenever these cannot be reduced to a common denominator, one must select one of them as a definiendum for the definition. So one can be methodologically guided by the actual use of a term in the manner described, but from a theoretical point of view one defines a term independently of the question of how it is named.

      This is where the scientific definition differs from the lexicographic definition. In any case, the latter is a nominal definition, because it describes the Significatum of an expression of a certain language. The purpose of a dictionary is to indicate its meaning, i.e. all its uses, for every word in the language. For the catchphrase sentence E.g. in a German dictionary you can find ten homonyms, each of which can be polysemous. In this way a comprehensive explanation of the meaning of a word can come about. The lexicographical indication of the meaning of a single disambiguated (i.e. neither homonymous nor polysemic) reading of such a word is most comparable with the scientific definition of a term.

      The classic definition

      So how do you formulate a definition? The scholastics (following Plato and Aristotle) ​​answered this question with the catchy sentence:

      Definitio fit per genus proximum et differentiam specificam. (Scholasticism)

      "The definition [of a species concept] is made by the next higher genus and the species-forming difference."

      (Genus = genus
      Species = species)

      Aristotle's example was:

      Man is the rational living being.

      We can add:

      A white horse is a white horse.
      A bachelor is an unmarried man.

      where living beings ’is the next higher genus, i.e. the immediate hyperonym, and rational’ is the specific difference; and accordingly in the other examples. The terms ‘Genus’ and Species ’are here - unlike in biology - not to be understood as designations of certain levels of a taxonomy, but as relative to one another: If is too hyponym, then there is a species of, and a gender of.

      This procedure therefore presupposes a taxonomy, so that one can easily specify the generic term for any term. This is possible for most nouns, but not trivial for all of them: What is, for example, the generic term for Structure ’, Action’, Intention ’, to name just a few head nuts? Non-noun terms often have no generic term. This applies, for example, to "wet", "old", "love", "sleep" and many others. The classic procedure cannot be applied to such. In any case, where applicable, it is the best.

      Here are some principles for proper definitions:

      1. Theoretically, one could define a term by referring to the subordinate terms, i.e. its hyponyms or the parts of the designated object. One could define 'dog' by listing (representative) dog breeds and 'lamp' as an object, which is composed of a luminous element (“illuminant” in commercial language) and an opaque or transparent screen, so that the screen completely or partially covered and directs its light in a certain direction (bundles or diffuses).

      However, this procedure should not be applied to just one concept of a system. Because then one defines ‘Dachshund’ with reference to the genus proximum as a certain type of dog, but Dog ’is defined by listing its sub-terms including dachshunds. In this way one necessarily gets into a circle. One would have to define all concepts of the system by their sub-concepts. But for those who have an infinite number of sub-concepts - e.g. colors - this leads to an infinite regress.

      Although the procedure of a definition by listing sub-concepts or parts cannot work, such definitions are offered again and again. School grammars, for example, define the relative clause as a subordinate clause introduced by a relative pronoun, or the direct object as that element of the sentence that is in the accusative. Such definitions do not correspond to the way people construct concepts and apply them to new objects. The term ‘relative clause’, for example, has been used for centuries on constructions such as in the girl we saw yesterday used where no relative pronoun can be seen; and ditto the concept of the direct object on constructions like we saw a girl, where no accusative can be seen. Such examples show that the constitutive and definitive aspects of such concepts are not their components for us, but the syntagmatic and paradigmatic context in which they stand and the functions which the things referred to fulfill.

      2. The fact that a concept is not defined by listing hyponyms implies that it is not defined by giving alternatives. Assume the following definition:

      To refute something means either to show a contradiction in it or to cite a counterexample to it.
      Such a definition does not express the common denominator of the two alternatives. The definition
      ‘To refute something means to prove that it does not apply’
      avoids this problem. The Scholastic and Aristotelian approach differs from the wrong procedures mentioned in that it first grasps the definiendum in its environment, holistically, so to speak, and only then looks at its internals, analytically, so to speak.

      At this point we have to start by defining the Concept formation distinguish. While the classic definition is a top-down procedure in the sense described, there are both directions in terms of the formation of the term. In other words, you can create a new term by enriching the intention of a more general term with specifics (top-down), or you can create it by discovering commonalities in specific terms, from which one abstracts a generic term (bottom-up ). Both methods of concept formation occur in natural languages ​​and are reflected, for example, in the complexity of the associated names. For example, 'dog-like' is obviously a secondary generic term, while 'sheepdog' is obviously a secondary sub-term.

      3. The Definiendum must not appear again in the Definiens, otherwise the definition is circular. This applies to, for example

      "To accuse someone means to accuse him of an evil of which he is believed guilty".
      While this case is glaring, that remains indirect circularity rather unnoticed. It has the following two definitions:
      • To accuse someone is to accuse them of an evil of which they are believed guilty.
      • To blame someone for something is to accuse them of that thing.

      This type of circularity is common in dictionaries, where, for example, 'ringing' is defined by 'ringing' and 'ringing' is defined by 'ringing'. An example from linguistics is:

      • A noun is a word that inflects after a case.
      • The case is a morphological category that describes the syntactic or semantic relation of a noun to its context.

      In order to avoid circularity, it is therefore necessary to ensure that the terms appearing in the Definiens are (can) themselves defined without reference to the Definiendum.

      4. The freedom from circularity is closely related to the requirement that the concepts appearing in the defining more elementary than must be the definiendum. Take a look at the definition of "direct object" endorsed on this website:

      • The direct object of a verb is that of its dependents, who becomes subject when it is passivated.

      First of all, according to what was said earlier about circularity, it is understood that we cannot now define ‘passivation’ as an operation that makes the direct object the subject. Instead, we define it as follows:

      • The passivation of a sentence is an operation that leads to an otherwise identical sentence in which the subject is demoved by.

      But it now looks as if we had defined a simple term by a complex one. Because in the definition of ‘passivation’ there is ‘subject’, which is actually a simpler term. But direct object ’, which should actually be on the same level as subject’ in terms of conceptual complexity, is in turn defined by ‘passivation’. Of course, one can only decide whether such a definition is really inappropriate if one sees the whole structure of definitions.

      In complex conceptual structures there can be long chains of definitions in which an indirect circularity can easily go undetected. If one wants, firstly, to achieve the goal that concepts become clear through definition, and secondly, to avoid circularity, one comes to the following conclusion: the relation that the concepts appearing in the defining are more elementary than the definiendum is transitive in the sense that Terms that underlie (the definitions of) all others, are the most elementary. These terms are also called primitive or axiomatic Terms (see semantic primitives). In numerous scientific disciplines, basic research is concerned with the question of what their primitive terms are. The question also arises when building a dictionary, because a structure of definitions that is free of circularity is obviously only possible if some terms remain undefined. This is not easily acceptable in a dictionary, but here, as elsewhere, it is the price one pays for freedom from circularity.

      An intuitive understanding of the elementary concepts is assumed; and if they are cleverly chosen, they are rightly so. If you still do not have this intuition, the term becomes clear through its use, i.e. through its appearance in the sentences of a theory or in the definitions of a dictionary. It can be used as a implicitly defined be valid.

      The fact that the terms used in the Definiens are more elementary than the Definiendum is a necessary prerequisite for the operationalization of the Definiendum to be discussed below.

      Definition of relational terms

      Not only nouns, but also adjectival and verbal terms can be defined, as the examples given have already shown. With these, of course, it becomes obvious what often goes unnoticed with substantive terms: If one defines a relational term, one has to define it in explicit reference to its relata. This requirement is violated in the following definitions:

      • A paraphrase is a verbose description.
      • A presupposition is an implicit requirement.
      • A witness is an accomplice.
      • A mother is a woman who has given birth to a child.

      A paraphrase is necessarily a paraphrase of something, and it is also a presupposition. A witness is a witness for something, and a mother is someone's mother. These relata are part of the term that is to be defined. They must therefore be made explicit in the Definiendum and taken up again in the Definiens, e.g. as follows:

      • A Paraphrase of an expression A1 is an expression A2, which is largely synonymous with A1, but more explicit than A1.
      • A proposition P1 is one Presupposition of a proposition P2, iff both P1 and its opposite P2 imply (P1 → P2 and ¬P1 → P2).
      • Given an event E and a person P, then P is witness for E, iff P not centrally involved in E, but was present at E and perceived E, so that P is able to say that E took place (in a certain way).
      • The mother von is the woman who gave birth.

      The last example clearly shows that a definition that does not take relationality into account does not work because it does not explain the use of the definiendum. My mother does not mean “my wife who gave birth to a child”. Rather, the syntactic description must make it clear that in my mother with my referenced person occupies the position marked with ‘’ in the definition; and then the meaning “woman who gave birth to me” is compositionally deducible.

      As you can see, the arguments of a relational concept basically take the form of variables. However, certain conditions or restrictions often apply to such arguments, in particular conditions about the category of the arguments (proposition, event, person, etc.), which must also be made explicit in a definition.

      Relational nouns are mostly abstracts of verbs or adjectives. The latter are semantically and syntactically more elementary than the derived nouns. The definitions of such abstracts are consequently more elegant and easier to understand if you first define the underlying verbs or adjectives. On the basis of this, the definition of the abstract noun is then also very easy. The following examples illustrate the scheme:

      • Presupposition: A proposition P1 presupposes a proposition P2, iff both P1 and its opposite P2 imply (P1 → P2 and ¬P1 → P2).
        P1 is a presupposition of P2, iff P1 is a presupposition of P2.
      • Polysemy: An expression is polysemic iff it has more than one meaning that is related to one another according to more general semantic principles.
        The polysemy of an expression is its property of being polysemous.
      As you can see, the second part of such definitions is so trivial that it can be left out. The second example also shows that the same definition scheme is suitable for all abstractions, not just relational ones.

      For the formulation of definitions it finally follows that the syntactic scheme 'A: B' or 'An A is a B' (where A is the definiendum and B the defining) has advantages because of its simplicity and clarity, but not the only one possible and not always the best. On the one hand, as I said, the Definiendum - and thus also the Definiens - has to be accompanied by its arguments in a schematic form. On the other hand, the prerequisites contained in the definiendum must be made explicit. They must necessarily recur in the definition and are therefore prerequisites for the entire definition. Therefore a definition can easily begin with an introduction to the form

      ‘Given a such that P () and a such that Q (); then A is from iff B ’,
      as just demonstrated with the definition of ‘witness’.

      The case that a definiendum involves arguments or assumptions that have to be made explicit is only one manifestation of the more general situation mentioned at the end of §2.1.1, that people do not use terms in isolation and consequently a definition cannot succeed either pretend it is.

      We saw at the beginning that a definition should make it possible to clarify whether something falls under a term or not. However, firstly, definitions are often quite abstract and, secondly, they have a place within the framework of theoretical buildings that make numerous requirements that the user cannot easily control. The user needs concrete criteria on the basis of which he can decide whether a phenomenon falls under a given term and whether a statement using this term applies.

      2 The characteristics that make up the intension of the term are therefore to be assigned to elementary phenomena that are observable or otherwise clearly intersubjectively ascertainable. Operationalization belongs to the complete definition of terms in an empirical science. It plays a significant methodological role in the empirical departments of such sciences as sociology and psychology, because their methods and the results obtained with them can only be valid if they have correctly operationalized their terms.

      As a first example, consider the notion of presupposition defined above, which plays a certain role in theories of linguistic semantics. The operationalization of this term results directly from its definition:

      To determine whether a given sentence S1 presupposes a sentence S2, proceed as follows:
      1. Determine if S1 implies S2.
      2. Make the negative version of S1 ¬S1.
      3. Determine if ¬S1 implies S2.
      4. If both implications are true, S1 presupposes S2, otherwise not.

      This is illustrated by the following sentence examples:

      B1.Erna regrets that she divorced Erwin.
      B2.Erna divorced Erwin.
      B3.Erna is remorseful.
      B4.It is not the case that Erna regrets that she divorced Erwin.
      B4 '.Erna doesn't regret having divorced Erwin.

      The question is whether B2 is a presupposition of B1. In fact, B1 implies B2. Next you negate B1. To be sure in propositional logic, this can be done in an explicit way, as in B4; But you can also do it in a natural-linguistic way in order to be able to control the meaning more reliably, as in B4 '. In fact, B4 and B4 'also imply B2. Hence B2 is a presupposition of B1. Another question is whether B3 is also a presupposition of B1. Indeed, B1 implies B3. B4 or B4 ', on the other hand, do not imply B3. Consequently, B3 is not a presupposition of B1 (but a different meaning component of it).

      Another simple example of operationalization is offered by the term hyponymy, which can be defined as follows:

      • A term B1 is hyponymous to a term B2, if applicable the intention of B1 includes that of B2.

      As we know, the intention of B1 includes that of B2, if necessary the extension of B2 includes that of B1. This results in the following operationalization of the concept of hyponomy:

      To determine whether B1 is a hyponym for B2, proceed as follows:
      1. Choose any one such that B1 ().
      2. Determine if B2 ().
      3. In the negative, the hypothesis of hyponymy is falsified. If it is positive, go back to step 1 until you are satisfied.3

      Most of the cases of operationalization of concepts that occur in empirical sciences are far less trivial. For example, whether a sentence has the syntactic structure that is ascribed to it, or whether a language is genetically related to another, can only be determined if the concepts of syntactic structure and genetic relationship are operationalized. So far, this has only partially succeeded.4 In sociology and psychology, the operationalization of terms usually involves a measurement, and this often requires surveys in sociology, tests and experiments in psychology. In this sense, experiment design is the operationalization of a term or a thesis.

      Exercises

      Analysis task


      1 These are, for example, nominal definitions.

      2 The operationalization of a hypothesis is then also defined on the basis of this.

      3 Procedures of this kind are particularly used in the dialog logic of Kamlah & Lorenzen 1967, where a proponent and an opponent share the work.

      4 In order to operationalize the genetic relationship, it has been seriously proposed - from a non-linguistic point of view - that two languages ​​are genetically related if a sufficiently large number of linguists, when asked, state that they are genetically related.