Volume 1 / Part 1 — *30. DESCRIPTIVE FUNCTIONS.
Reading edition
*30. DESCRIPTIVE FUNCTIONS.
Summary of *30.
The functions hitherto considered, with the exception of a few particular functions such as , have been propositional, i.e. have had propositions for their values. But the ordinary functions of mathematics, such as , , , are not propositional. Functions of this kind always mean "the term having such and such a relation to ." For this reason they may be called descriptive functions, because they describe a certain term by means of its relation to their argument. Thus "" describes the number ; yet propositions in which occurs are not the same as they would be if were substituted for . This appears e.g. from the proposition "," which conveys valuable information, whereas "" is trivial. Descriptive functions, like descriptions in general, have no meaning by themselves, but only as constituents of propositions[55].
The general definition of a descriptive function is:
*30·01.
That is, "" is to mean "the term which has the relation to ." If there are several terms or none having the relation to , all propositions about , i.e. all propositions of the form "," will be false. The apostrophe in "" may be read "of." Thus if is the relation of father to son, "" means "the father of ." If is the relation of son to father, "" means "the son of "; in this case, all propositions of the form "" will be false unless has one son and no more.
All the functions that occur in ordinary mathematics are instances of the above definition; all are obtained in the above manner from some relation. Thus in our notation "" takes the place of what would commonly be "," this latter notation being reserved for propositional functions. We should write "" in place of "," using "" to express the relation of to when .
A definition such as , where the meaning given to the term defined is a description, must be understood to mean that the term defined (in this case ) and the description assigned as its meaning (in this case ( are to be interchangeable in use: the definition is, in a sense, more purely symbolic than other definitions, since the description assigned as the meaning has itself no meaning except in use. It would perhaps be more formally correct to write
But even this definition would not be quite complete, because it omits mention of the scope of the two descriptions and . Thus the complete form would be But it is unnecessary to adopt this form of definition, provided it is understood that the definition *30·01 means that "" may be written for "(" everywhere, i.e. in indications of scope as well as elsewhere. The use of the definition occurs always in accordance with the proposition: which is *30·1, below.
It is to be observed that *30·01 does not necessarily involve For this, by the definition, is equivalent to which, by *14·28, only holds when , i.e. when there is one term, and no more, which has the relation to .
All the conventions as to scope explained in *14 are to be transferred to , i.e., in the absence of any contrary indication, the scope of is to be the smallest proposition, enclosed in dots or other brackets, in which the in question occurs.
We put
*30·02.
This definition serves merely for the avoidance of brackets. It is to be interpreted as meaning In future, we shall often define a new expression as having a descriptive phrase for its meaning; in such a case, the definition is always to be interpreted as above. That is, any proposition in which the new expression occurs is to be the proposition which is obtained by substituting the old expression for the new one wherever the latter occurs.
, in the above, is to be interpreted by first treating as if it were not a descriptive symbol, and applying *30·01 and *14·01 or *14·02 to , and by then applying *30·01 and *14·01 or *14·02 to .
The majority of the propositions of the present number are immediate consequences of the corresponding propositions in *14. Thus *14·31—·34 and *14·113 lead immediately to *30·12—·16, which show that, either always or when exists, the "scope" of or of and makes no difference to the truth-values of such propositions as we are concerned with. We have
*30·18.
so that what holds of everything holds of , provided exists. This results immediately from *14·18, and shows that, provided exists, the fact that "" is an incomplete symbol does not prevent its being substituted as a value of whenever we have (, or an assertion of the propositional function .
One of the most used propositions of this number is:
*30·3.
which results immediately from *14·202. The following analogous proposition results from the above by means of *14·122:
*30·31.
I.e. "" involves, in addition to "," the statement that whatever has the relation to is identical with .
A proposition constantly referred to is:
*30·37.
In the hypothesis, might be replaced by , but one or other of them is essential. For, by *14·21, "" implies and (these are equivalent when ), and therefore cannot be true when and do not exist.
The use of *30·37 is chiefly in cases where or or both are replaced by descriptive functions. Suppose, for example, that is replaced by . By *30·18, we may substitute for if exists. By *14·21, both sides of the implication in *30·37 will become false if does not exist, and therefore the implication will still hold. Hence whether exists or not, we may substitute it for and obtain In like manner, if we replace by , we obtain
A very important proposition is:
*30·4.
This proposition states that, provided exists, to say that is the term which has the relation to is equivalent to saying that a has the relation to . Thus for example " is the occupier of the house " is equivalent to " occupies the house ," " is the writer of Waverley" is equivalent to " wrote Waverley," " is the father of " is equivalent to " begot ." But we cannot argue from "John Smith inhabits London" to "John Smith is the inhabitant of London."
We shall introduce in this and subsequent sections many constant relations for which is always true. When is such that is always true, we have, in virtue of *30·4, for every possible value of . The following proposition is useful in cases where both and are such that and always exist:
*30·41.
Thus if we know that and are always identical, we know not only that and are identical, but also that (and therefore ) always exists.
*30·01.
*30·02.
In interpreting , is to be treated as an ordinary symbol until has been eliminated by *30·01 and *14·01 or *14·02, and then the above definitions are to be applied to .
*30·1.
*30·11.
The following propositions are immediate applications of *14·31 ff., made in accordance with *30·1.
*30·12.
*30·13.
*30·14.
*30·141.
*30·142.
*30·15.
The following two propositions are immediate consequences of *14·113·112.
*30·16.
*30·17.
*30·18.
*30·19.
*30·2.
In proving *30·2, we have to use the definition *30·01, not *30·1, because is not of the form . This appears if we attempt to apply the definition *14·01 to , which leads to an expression containing the meaningless constituent . But by the definition *30·01, every typographical occurrence of the symbol "" means what results when this symbol is replaced by "(" hence "" means "."
*30·21.
*30·22.
Note that we do not necessarily have which is only true when .
*30·3.
*30·31.
*30·32.
*30·33.
*30·34.
*30·341.
Dem.
*30·35.
*30·36.
*30·37.
Dem.
This proposition is very frequently used.
*30·4.
This is a very important proposition, of which the use is constant.
*30·41.
Dem.
*30·42.
The hypothesis ( is fulfilled by a number of important special relations, of which examples will occur in the subsequent numbers of the present section.
*30·5.
Dem.
*30·501.
On the meaning of "," see note to the definition *30·02.
Dem.
*30·51.
*30·52.