A collection is whatever stands to a general predicate of single subjects in a certain relation sui generis, such that for every such predicate there is a single collection and for every collection there is such a predicate.
A collection is not a thing, but an ens rationis, since its distinctive identity is constituted not only by an arbitrary act, but by the distinctive identities of other things, namely of those that it comprises. At the same time, it so far partakes of the nature of a thing, that it is constituted not at all by anything of a general nature but by individual things, irrespective of their characters. This gives the collection individuality.
A collection is not a thing, and therefore a collection which comprises a single thing and excludes everything else is not identical with that thing. Whatever things may be given, there is just one collection which comprises them all and nothing else. According to this rule, there is a collection which comprises nothing. This collection subsists in every universe.
We may say that a whole is an ens rationis whose being consists in the copulate being of certain other things, either not entia rationis or not so much so as the whole; so that a whole is analogous to a collection, which is, in fact, a special kind of whole. There can be no doubt that the word whole always brings before the mind the image of a collection, and that we interpret the word whole by analogy with collection. The idea of a collection is itself, however, by no means an easy one to analyze. It is an ens rationis, abstraction, or fictitious subject (but the adjective must be understood in a broad sense, to be considered below), which is individual, and by means of which we are enabled to transform universal propositions into singular propositions. Thus, the proposition “all men are mortal,” with a new subject and new predicate, appears as “The collection of men is a collection of mortals”; just as, for other purposes, and by means of other abstractions, we transform the same proposition into “The character of mortality is possessed by every man”; and the members of the collection are regarded as less fictitious than the collection. It very often happens that an object given indirect perception as an individual is, on closer scrutiny, seen to be identifiable with a collection of parts. But it does not seem to be strictly accurate to say that the larger object of perception is identical with that abstraction, the collection of the smaller objects. It is rather something perceived which agrees in its relations with the abstraction so well that, for convenience, it is regarded as the same thing. No doubt the parts of a perceived object are virtually objects of consciousness in the first percept; but it is useless to try to extend logical relations to the sort of thought which antecedes the completion of the percept. By the time we conceive an object as a collection, we conceive that the first reality belongs to the members of the collection and that the collection itself is a mere intellectual aspect, or way of regarding these members, justified, in ordinary cases, by certain facts. We may, therefore, define a collection as a fictitious (thought) individual, whose being consists in the being of certain less fictitious individuals.
The definition […] of a collection is that it is a real individual object whose being consists in the being of whatever may actually exist that possesses a certain character.
A Collection is anything whose being consists in the existence of whatever there may exist that has any one quality; and if such thing or things exist, the collection is a single thing whose existence consists in the existence of all those very things.
According to this definition, a collection is an ens rationis. [—] A collection has essence and may have existence.
A collection is a single object whose being consists in the existence of whatever independent individuals may exist of which a given rhema is true, these independent individuals being called the members of the collection.
In a […] general sense, a collection is simply an individual object whose being consists in the being of whatever objects there may be of a certain general description, these objects being called its members, so that every proposition concerning the collection as subject is equivalent to some relative proposition concerning the members as subjects. But this definition leaves it undetermined what the relation is to be between what is predicated, in any proposition, concerning the collection and the relative predicate concerning the members in the interpreting proposition.
…while numbers may on occasion be, or represent, multitudes, they can never be collections, since collections are not grades of any kind, but are single things.
A collection is a thing to which everything of a certain description peculiar to the individual collection stands in an existential relation essential to the collection. By saying that the relation is existential I mean that anything of that description is by logical necessity in that relation to the collection by virtue of its existence. By saying that the relation is essential to the collection, I mean that by logical necessity if there be nothing of the given description, the collection does not exist, but remains, like the description, in the world of pure ideas; while every object of the given description by being in this relation to the collection imparts to this collection a higher grade of existence than if[,] other things being unchanged[,] that object did not exist.