Volume 1 / Part 3 — GENERAL NOTE ON CARDINAL CORRELATORS.
Reading edition
The correlators established at various stages throughout Section B present certain analogies to each other, and they or others closely resembling them will be found to be the correlators required in relation-arithmetic (Part IV). We shall therefore here collect together the most important propositions hitherto proved on correlators.
When we have to deal with correlators of two different functions of a single class, as e.g. and , the correlator is usually or or , with a suitable limitation on the converse domain. Sometimes it is or . Thus for example the class , by means of which is defined (*112), has double similarity with if (*112·14); in this case, the double correlator is with its converse domain limited, i.e. In the case of and , the correlator is , i.e. In the case of and , the correlator is , i.e. also correlates with (*85·61) and with (*85·53), and with (*85·27·42) if .
The correlator of with is (*116·131).
Another kind of correlators arises where we are given a correlator of and , and we wish to construct a correlator for some associated classes and where we are given correlators of with and of with , and we wish to construct a correlator of with , where is some double descriptive function in the sense of *38. In this case, the correlator will usually be of the form (with a limited converse domain). Sometimes and will be identical; sometimes will be . Such correlators always depend upon
*55·61.
together with the propositions *74·77 seq. giving cases in which is a one-one relation. It follows from *55·61 that if and are correlators whose converse domains include the domain and converse domain respectively of a relation , then will be a relation holding between and whenever holds between and . Examples of such correlators as are
*112·153.
*113·127.
*113·65.
*114·51.
*116·192.
An exceptionally simple correlator is given by
*115·502.
Another exceptionally simple case is
*73·63.
By means of the above correlators, most correlators that are required can be calculated. Thus it will be seen that *116·192 in the above list is an immediate consequence of *113·127 and *115·502, since
In order to develop the subject, it is almost always necessary, not merely to prove that two classes are similar, but actually to construct a correlator of the two classes. This applies equally to relation-arithmetic, in which analogous correlators are used to prove ordinal similarity.