ABSTRACT
Relations may be divided into four classes, according as they do or do not possess either of two attributes, transitiveness* and symmetry. Relations such that xRy always implies yRx are called symmetrical; relations such that xRy, yRz together always imply xRz are called transitive. Relations which do not possess the first property I shall call not symmetrical; relations which do possess the opposite property, i.e. for which xRy always excludes yRx, I shall call asymmetrical. Relations which do not possess the second property I shall call not transitive; those which possess the property that xRy, yRz always exclude xRz I shall call intransitive. All these cases may be illustrated from human relationships. The relation brother or sister is symmetrical, and is transitive if we allow that a man may be his own brother, and a woman her own sister. The relation brother is not symmetrical, but is transitive. Half-brother or half-sister is symmetrical but not transitive. Spouse is symmetrical but intransitive; descendant is asymmetrical but transitive. Half-brother is not symmetrical and not transitive;
if third marriages were forbidden, it would be intransitive. Son-in-law is asymmetrical and not transitive; if second marriages were forbidden, it would be intransitive. Brother-in-law is not symmetrical and not transitive. Finally, father is both asymmetrical and intransitive. Of not-transitive but not intransitive relations there is, so far as I know, only one important instance, namely diversity; of not-symmetrical but not asymmetrical relations there seems to be similarly only one important instance, namely implication. In other cases, of the kind that usually occur, relations are either transitive or intransitive, and either symmetrical or asymmetrical.
