Volume 1 / Part 4 — *151. ORDINAL SIMILARITY.
Reading edition
*151. ORDINAL SIMILARITY.
Summary of *151.
In this number, we give the definition of ordinal similarity, and various equivalent forms; we prove that ordinal similarity is reflexive (*151·13), symmetrical (*151·14) and transitive (*151·15), and we give some particular cases of ordinal similarity (*151·6 ff.). Propositions in this number should be compared with those in *73, to which they are analogous.
The class of ordinal correlators of and is written , where "" stands for "similar ordinally." We put
(We might equally well put which is an equivalent but more condensed form of the definition.) We then define " is ordinally similar to " as meaning that there is at least one ordinal correlator of and , i.e.
We shall find that if and generate well-ordered series, they have at most one correlator (*250·6), but this does not hold in general for other series.
After giving the elementary properties of ordinal similarity, we have three important propositions on its connection with cardinal similarity, namely: (*151·18) if is similar to , the field of is similar to the field of (the converse does not hold in general, but holds if and are finite serial relations); (*151·19) if is similar to , there is a relation similar to and having for its field, and vice versa; (*151·191) is an ordinal correlator of and when, and only when, it is a cardinal correlator of and and .
We then have a set of propositions on correlators of the form (*151·2—·243). Most of the correlators with which we shall be concerned are of this form. The most useful proposition here is
*151·22.
A useful consequence of this proposition is
*151·231.
This consequence is useful because the hypothesis is satisfied by most of the relations which occur as correlators.
We have next a number of propositions on the inferribility of or from or , and connected matters (*151·25—·29).
We have
*151·25.
*151·26.
*151·29.
*151·29 is never used, but is inserted in order to show that our definition of "ordinal similarity" agrees with what is commonly understood by that term. If and are regarded as serial, so that "" means " precedes in the -series," and "" means " precedes in the -series," then our proposition states that two series are ordinally similar when their terms can be so correlated that predecessors in either are correlated with predecessors in the other, and successors with successors, i.e. when the two series can be correlated without change of order.
We have next (*151·31—·52) a set of miscellaneous propositions, of which the most useful are
*151·401.
*151·5.
*151·401 will be useful in such cases as the following: Let and be relations between relations, then and will be the corresponding relations of their domains. Suppose , . Then, by *151·401, if is a correlator of and , is a correlator of and .
*151·5 shows that if is a correlator of and , it correlates with , with , with , and with .
Our next set of propositions (*151·53—·59) is concerned with the correlation of powers of and and kindred matters. We show (*151·55) that a correlator of and is also a correlator of and , and therefore if and are similar, so are and (*151·56); we show also (*151·59) that if and are similar, so are and . These propositions are used in the theory of progressions (*263·17).
The remaining propositions (*151·6 to the end) are concerned with applications to particular cases. The most useful of these are
*151·61.
which shows how to raise the type of a relation without changing its relation-number;
*151·64.
*151·65.
We prove also that all members of (i.e. all relations of the form , where ) are similar *151·63), and that all relations of the form are similar (*151·631).
*151·01.
*151·02.
*151·1.
*151·11.
*151·12.
*151·121.
*151·13.
*151·131.
Dem.
*151·14.
*151·141.
Dem.
*151·15.
*151·16.
*151·161.
*151·162.
*151·17.
*151·18.
Dem.
*151·19.
Dem.
*151·191.
Dem.
*151·2.
Dem.
*151·21.
*151·22.
Dem.
*151·23.
The above proposition (*151·23) is very useful. It is the analogue of *73·15. (It should be observed that, in all propositions concerning likeness, plays the same part as plays in propositions concerning similarity.) By means of *151·23, we can establish likeness in all those numerous cases in which a relation which is not usually one-one becomes one-one when confined to a certain converse domain, as for example if we have to deal with , where , or with , where . Thus e.g. by the above proposition, if is any relation whose field is , where , will be an ordinally similar relation whose field is .
*151·231.
*151·232.
*151·24.
*151·241.
*151·242.
*151·243.
*151·25.
Dem.
*151·251.
*151·252.
*151·253.
*151·254.
Dem.
This proposition is the analogue of *72·54. "" means "," not "."
*151·26.
Dem.
*151·261.
*151·262.
*151·263.
*151·264.
Dem.
*151·27.
*151·271.
*151·28.
Dem.
The above proposition shows that ordinal similarity as we have defined it has the properties which are commonly associated with the term "ordinal similarity," namely that and are ordinally similar when their fields can be so correlated that two terms having the relation are always correlated with two terms having the relation , and vice versa.
The hypothesis is redundant in *151·28; this is shown in the following proposition.
*151·281.
Dem.
*151·29.
*151·31.
Dem.
*151·32.
*151·33.
Dem.
*151·4.
Dem.
*151·401.
Dem.
*151·41.
This proposition is the analogue of *73·63.
The following proposition is used frequently both in relation-arithmetic and in the theory of series.
*151·5.
Dem.
*151·51.
Dem.
*151·52.
*151·53.
Dem.
*151·54.
Dem.
*151·55.
*151·56.
*151·56 is used in *263·17.
The two following propositions are lemmas for **151·59, which is used in *263·17.
*151·57.
Dem.
*151·58.
Dem.
*151·59.
The remaining propositions of this number consist of applications to particular cases.
*151·6.
This proposition is only significant when is a relation between relations.
*151·61.
*151·62.
*151·63.
Dem.
The above proposition shows that all ordinal couples (i.e. all members of are ordinally similar. The following proposition shows the same for couples whose referent and relatum are identical.
*151·631.
Dem.
*151·64.
The following proposition is frequently used in relation-arithmetic.
*151·65.