Definition
Definition
In the development of a logistic system (q. v.) it is usually desirable to introduce new notations, beyond what is afforded by the primitive symbols alone, by means of syntactical definitions or nominal definitions, i.e., conventions which provide that certain symbols or expressions shall stand (as substitutes or abbreviations) for particular formulas of the system. This may be done either by particular definitions, each introducing a symbol or expression to stand for some one formula, or by schemata of definition, providing that any expression of a certain form shall stand for a certain corresponding formula (so condensing many — often infinitely many — particular definitions into a single schema). Such definitions, whether particular definitions or schemata, are indicated, in articles herein by the present writer, by an arrow ?, the new notation introduced (the definiendum) being placed at the left, or base of the arrow, and the formula for which it shall stand (the definiens) being placed at the right, or head, of the arrow. Another sign commonly employed for the same purpose (instead of the arrow) is the equality sign = with the letters Df, or df, appearing either as a subscript or separately after the definiens.
This use of nominal definition (including contextual definition — see the article Incomplete symbol) in connection with a logistic system is extraneous to the system in the sense that it may theoretically be dispensed with, and all formulas written in full. Practically, however, it may be necessary for the sake of brevity or perspicuity, or for facility in formal work.
Such methods of introducing new concepts, functions, etc. as definition by abstraction (q. v.), definition by recursion (q. v.), definition by composition (see Recursiveness) may be dealt with by reducing them to nominal definitions; i.e., by finding a nominal definition such that the definiens (and therefore also the definiendum) turns out, under an intended interpretation of the logistic system, to mean the concept, function, etc. which is to be introduced.
In addition to syntactical or nominal definition we may distinguish another kind of definition, which is applicable only in connection with interpreted logistic systems, and which we shall call semantical definition. This consists in introducing a new symbol or notation by assigning a meaning to it. In an interpreted logistic system, a nominal definition carries with it implicitly a semantical definition, in that it is intended to give to the definiendum the meaning expressed by the definiens; but two different nominal definitions may correspond to the same semantical definition. Consider, for example, the two following schemata of nominal definition in the propositional calculus (Logic, formal, 1)
[A] ? [B] ? ~A ? B.
[A] ? [B] ? ~[A ~B].
As nominal definitions these are inconsistent, since they represent [A] ? [B] as standing for different formulaseither one, but not both, could be used in a development of the propositional calculus. But the corresponding semantical definitions would be identical if — as would be possible — our interpretation of the propositional calculus were such that the two definientia had the same meaning for any particular A and B.
In the formal development of a logistic system, since no reference may be made to an intended interpretation, semantical definitions are precluded, and must be replaced by corresponding nominal definitions.
Of quite a different kind are so-called real definitions, which are not conventions for introducing new symbols or notations — as syntactical and semantical definitions are — but are propositions of equivalence (material, formal, etc.) between two abstract entities (propositions, concepts, etc.) of which one is called the definiendum and the other the definiens. Not all such propositions of equivalence, however, are real definitions, but only those in which the definiens embodies the “essential nature” (essentia, ??s?a) of the definiendum. The notion of a real definition thus has all the vagueness of the quoted phrase, but the following may be given as an example. If all the notations appearing, including ?x, have their usual meanings (regarded as given in advance), the proposition expressed by
(F)(G)[[F(x) ?x G(x)] = (x)[~F(x) ? G(x)]]
is a real definition of formal implication — to be contrasted with the nominal definition of the notation for formal implication which is given in the article Logic, formal, 3. This formula, expressing a real definition of formal implication, might appear, e.g., as a primitive formula in a logistic system.
(A situation often arising in practice is that a word — or symbol or notation — which already has a vague meaning is to be given a new exact meaning, which is vaguely, or as nearly as possible, the same as the old. This is done by a nominal or semantical definition rather than a real definition; nevertheless it is usual in such a case to speak either of defining the word or of defining the associated notion.)
Sometimes, however, the distinction between nominal definitions and real definitions is made on the basis that the latter convey an assertion of existence, of the defimendum, or rather, where the definiendum is a concept, of things falling thereunder (Saccheri, 1697); or the distinction may be made on the basis that real definitions involve the possibility of what is defined (Leibniz, 1684). Ockham makes the distinction rather on the basis that real definitions state the whole nature of a thing and nominal definitions state the meaning of a word or phrase, but adds that non-existents (as chimaera) and such parts of speech as verbs, adverbs, and conjunctions may therefore have only nominal definition. — A.C.