§ 5. Definitions

The introduction of new terms as a function of known terms is called a definition. It is important to realize that, because of its reference to terms, a definition belongs to the metalanguage, at least in its original meaning. If we say, for instance, 'a submarine is a ship which can go under water' we define, not a submarine, but the term 'submarine'. We can build a submarine, but not define it. The correct form of the definition therefore would be: 'by 'submarine' we denote a ship which can go under water'; or: 'the term 'submarine' is to have the same meaning as the term 'ship capable of going under water' '.

It is customary, however, to write definitions as sentences of the object language, as in the example given. This usage is without danger if we recognize it as a simplified procedure which for purposes of logical inquiry has to be translated into sentences of the metalanguage. In symbolic language we use here the sign ' =Df' meaning 'equal by definition'. Thus we may write

submarine =Df ship capable of going under water
We shall say that the sign ' =Df' is not a proper object term. Sentences in which this sign occurs are improper object sentences; they are introduced by a shifting of the level of language.1

In accordance with the nomenclature of traditional logic, we call the term on the left-hand side of the definition the definiendum, the term on the right-hand side the definiens.

The relation of equality by definition can be considered a special case of the relation of equisignificance, i.e., of. having the same meaning. It constitutes the case where the equality of meaning is not derived from other statements but is introduced by a volitional decision with reference to the introduction of a new sign. The question whether an equisignificance is demonstrable or a matter of definition will therefore be answered differently according as the system of language is constructed. It is possible to construct the same system in different ways, varying with the starting points; what is a definition in one construction may be a demonstrable equisignificance in another.

The relation of equisignificance between symbols can be reduced to the relation of equisignificance between tokens. Two symbols are equisignificant if every token of one symbol is equisignificant to every token of the other symbol.2 This reduction explains the possibility of making definitions. Every definition will occur once for the first time; it then may be interpreted as the convention that every token similar to the token of the definiendum is to be considered equisignificant to every token similar to the token of the definiens.

The example cited is a definition of the special form considered in traditional logic, a definition by determination of genus proximum and differentia specifica. Modern logic has recognized that this is a very limited form of definition and that in scientific as well as in conversational language we ordinarily use other forms of definitions. In general the sign ' =Df' stands between sentences, not between words or phrases. Thus we define the sentence 'the metabolism of a person is normal' by a set of sentences about the percentage of certain substances contained in his blood. The general form of such a definitionjs therefore

a =Df [b, c, . . .]
where the brackets indicate a certain combination of sentences 'b', 'c', . . . . In this combination the sentences may be combined by various logical operations, such as expressed through the words 'and', 'or', 'implies'. We call this a definition by coordination of propositions.

If we introduce new terms in this way we do not define them explicitly but implicitly, i.e., in combination with other terms. Thus we define not the term 'metabolism' but statements of the kind: 'the metabolism is normal', 'the metabolism is subnormal'. Such procedure is permissible because actually we do not use the isolated term 'metabolism' but only certain sentences in which the term occurs; to know the meaning of these sentences is sufficient for all practical purposes. On account of their reference to usage such definitions are often called definitions in use.

The importance of these definitions lies in the fact that they allow us to define an abstract term by reference to concrete terms, whereas the scholastic definition through higher genus and specific difference determines an abstract term by reference to more abstract terms. This is the reason that the traditional form of definitions cannot represent the actual procedure of knowledge, in which a verification of abstract statements is always given by verification of statements about directly observable things, such as thermometers, gauges, or small particles seen in a microscope.