Volume 1 / Part 1 — *37. PLURAL DESCRIPTIVE FUNCTIONS.
Reading edition
*37. PLURAL DESCRIPTIVE FUNCTIONS.
Summary of *37.
In this number, we introduce what may be regarded as the plural of . "" was defined to mean "the term which has the relation to ." We now introduce the notation "" to mean "the terms which have the relation to members of ." Thus if is the class of great men, and is the relation of wife to husband, will mean "wives of great men." If is the class of fractions of the form for integral values of , and is the relation "less than," will be the class of fractions each of which is less than some member of this class of fractions, i.e. will be the class of proper fractions. Generally, is the class of those referents which have relata that are members of .
We require also a notation for the relation of to . This relation we will call . Thus is the relation which holds between two classes and when consists of all terms which have the relation to some member of .
A specially important case arises when always exists if . In this case, is the class of all terms of the form when . We will denote the hypothesis that always exists if by the notation , meaning "the 's of 's exist."
The definitions are as follows:
*37·01.
*37·02.
*37·03.
This definition serves merely for the avoidance of brackets. Without it, "" would be ambiguous as between ( and , which are not equal. In all cases in which a suffix occurs, we shall adopt the same convention, i.e. we shall always put
*37·04.
Thus consists of all classes which have the relation to some member of . is only significant when is a class of classes relatively to members of the converse domain of ; in this case, is a class of classes relatively to members of the domain of .
*37·05.
Here the symbol "" must be treated as a whole, i.e. we must not regard it as making an assertion about . If , we must not suppose that we shall be able to put "," which would be nonsense, just as "" is nonsense even when and .
The notation , introduced in the present number, is extremely useful, and embodies a very important idea. Its use is somewhat different according to the kind of relation concerned. Consider first the kind of relation which leads to a descriptive function, say . If is a class of relations, is the class of the domains of these relations. In this case, is a class each of whose members is of the form , where . Again, let us denote by "" the relation of to ; then if we denote by "" the class of cardinal numbers, will denote all numbers that result from multiplying a cardinal number by , i.e. all multiples of . Thus e.g. will be the class of even numbers. If is a correlation between two classes and , i.e. a relation such that, if , exists and is a member of , while conversely, if , exists and is a member of , then , and we may regard as a transformation applied to each member of and giving rise to a member of . It is by means of such transformations that two classes are shown to be similar, i.e. to have the same (cardinal) number of terms.
In the case of serial relations, the utility of the notation is somewhat different. Suppose, for example, that is the relation of less to greater among real numbers. Then if is any class of real numbers, will be the segment of real numbers determined by , i.e. the class of real numbers which are less than the limit or maximum of . In any series, if is a class contained in the series and is the generating relation of the series, is the segment determined by . If has either a limit or a maximum, say , will be . But if has neither a limit nor a maximum, will be what we may call an "irrational" segment of the series. We shall see at a later stage that the real numbers may be identified with the segments of the series of rationals, i.e. if is the relation of less to greater among rationals, the real numbers will be all classes such as , for different values of . The real numbers which correspond to rationals will be those resulting from a which has a limit or maximum; the irrationals will be those resulting from a which has no limit or maximum.
The present number may be divided into various sections, as follows: (1) First, we have various elementary properties of the terms defined at the beginning of the number; this section ends with *37·29. (2) We have next a set of propositions dealing with relative products, and with such symbols as , , and so on. The central proposition here is
*37·33.
By the definition, . Thus This connects propositions concerning such symbols as with propositions concerning relative products. This second section consists of the propositions from *37·3 to *37·39. (3) We have next a set of propositions on relations with limited domains and converse domains. The chief of these are
*37·401.
*37·412.
*37·41.
These propositions on relations with limited domains and converse domains, together with certain others naturally connected with them, extend from *37·4 to *37·52. (4) We next have a number of very important propositions on the consequences of the hypothesis , i.e. the hypothesis that, for any argument which is a member of , gives rise to a descriptive function . The chief proposition in this section is
*37·6.
Propositions with the hypothesis are applied to the cases of and , in which the hypothesis is verified. This section extends from *37·6 to *37·791. (5) Finally, we have three propositions on the relative product of with other relations. These propositions are useful in relation-arithmetic (Part IV).
The propositions of the present number which are most used in the sequel, apart from those already mentioned, are the following (omitting such as merely embody definitions):
*37·15.
*37·16.
*37·2.
*37·22.
*37·25.
*37·26.
*37·265.
*37·29.
*37·32.
*37·45.
*37·46.
*37·61.
For example, let be the relation of father to son, the class of Etonians, the class of rich men; then "" states "all fathers of Etonians are rich," while "" states "if a boy is an Etonian, his father must be rich." In virtue of the above proposition, these two statements are equivalent.
*37·62.
*37·63.
*37·01.
*37·02.
*37·03.
*37·04.
*37·05.
*37·1.
*37·101.
*37·102.
*37·103.
*37·104.
*37·105.
*37·106.
Dem.
*37·11.
*37·111.
*37·12.
*37·13.
Dem.
*37·131.
Dem.
*37·14.
Dem.
*37·15.
Dem.
*37·16.
*37·17.
Dem.
*37·171.
Dem.
*37·18.
Dem.
*37·181.
*37·2.
Dem.
The above proposition (*37·2) is one of the forms of asyllogistic inference due to Leibniz's teacher Jungius. The instance given by Jungius is: "Circulus est figura; ergo qui circulum describit, is figuram describit[56]." Here the class of circles is our , the class of figures is our , and the relation of describing is our .
*37·201.
*37·202.
*37·21.
Dem.
*37·211.
*37·212.
*37·22.
This proposition is very frequently used. The fact that here we have identity, while in *37·21 we only have inclusion, is due to the fact that *10·42 states an equivalence, while *10·5 only states an implication.
Dem.
*37·221.
*37·222.
*37·23.
*37·231.
The type of "" here is that type whose members are of the same type as . In the proof, use is made of the convention that a Greek letter always stands for an expression of the form .
Dem.
As appears in the above proof, it is necessary, when a proposition containing "" is to be proved, to abandon the notation with Greek letters, and revert to the explicit functional notation.
*37·24.
Dem.
*37·25.
Dem.
*37·26.
Dem.
*37·261.
*37·262.
*37·263.
*37·264.
Dem.
*37·265.
Dem.
*37·27.
*37·271.
*37·28.
*37·29.
Dem.
*37·3.
Dem.
*37·301.
*37·302.
*37·31.
Dem.
*37·311.
*37·32.
Dem.
*37·321.
*37·322.
*37·323.
*37·33.
Dem.
*37·34.
Dem.
*37·341.
*37·35.
Dem.
*37·351.
*37·352.
*37·353.
Dem.
*37·354.
*37·355.
*37·36.
*37·37.
*37·371.
This definition serves merely for the avoidance of brackets. Like *37·03, this definition will be extended to all suffixes.
*37·38.
*37·39.
*37·4.
Dem.
*37·401.
*37·402.
Dem.
*37·41.
*37·411.
Dem.
*37·412.
Dem.
*37·413.
Dem.
*37·42.
*37·421.
*37·43.
Dem.
*37·431.
*37·44.
*37·441.
*37·45.
*37·451.
*37·46.
*37·461.
*37·462.
*37·47.
Dem.
*37·5.
Dem.
*37·501.
Dem.
*37·502.
*37·51.
Dem.
*37·52.
The following propositions, down to *37·7 exclusive, are concerned with the special properties of which result from the hypothesis , defined in *37·05. The hypothesis is important, because it has many consequences and is satisfied in many cases with which we wish to deal.
*37·6.
This proposition is very important, and is used constantly.
Dem.
*37·601.
Dem.
*37·61.
Dem.
*37·62.
Dem.
The above is the type of inference concerning which Jevons says[57]: "I remember the late Prof. De Morgan remarking that all Aristotle's logic could not prove that 'Because a horse is an animal, the head of a horse is the head of an animal.'" It must be confessed that this was a merit in Aristotle's logic, since the proposed inference is fallacious without the added premiss " the head of the horse in question." E.g. it does not hold for an oyster or a hydra. But with the addition , the above proposition gives an important and common type of asyllogistic inference.
*37·63.
Dem.
This proposition is very frequently used.
*37·64.
Dem.
*37·65.
Dem.
*37·66.
Dem.
*37·67.
Dem.
*37·68.
Dem.
*37·69.
Dem.
A specially important case of is or . This case will be further studied later (in *70); for the present, we shall only give a few preliminary propositions about it. It will be observed that the hypothesis or is always verified, in virtue of *32·12·121. Hence the following applications of *37·6 ff.:
*37·7.
*37·701.
*37·702.
*37·703.
*37·704.
*37·705.
*37·706.
*37·707.
*37·708.
*37·709.
*37·71.
*37·711.
*37·712.
*37·713.
*37·72.
Dem.
*37·721.
*37·73.
*37·731.
Observe that the 's which occur in this proposition will not be all of the same type. E.g. if relates individuals to individuals, the first will be the class of no individuals, while the second and third will be the class of no classes. Thus the ambiguity which attaches to the type of must be differently determined for different occurrences of in this proposition. In general, when this is the case with our ambiguous symbols, we shall adopt a notation which indicates the fact. But when the ambiguous symbol is , it seems hardly worth while.
*37·74.
Dem.
*37·75.
*37·76.
Dem.
*37·761.
*37·77.
*37·771.
*37·772.
*37·773.
*37·78.
*37·781.
*37·79.
*37·791.
*37·8.
Dem.
*37·81.
*37·82.