Some relations are clearly internal in the sense that, for example, four would not be four unless it were related to two in the way it is. Some relations are internal to their bearers under one description but not under another, for example, a wife would not be a wife unless suitably related to a husband, but Mary would still be Mary had she not married. Or take the internal relation where Jack is taller than his wife, Joan. Here the relation is internal to both of them together, in symbolic form it can be given as: Jack(R)Joan, where R is the ordered relation of "Taller than".
The doctrine that all relations are internal implies that everything has some relation, however distant, to everything else. Such a doctrine is ascribed by Russell and Moore to certain ideas by Hegel and the American philosopher, Charles Peirce. However neither of these philosophers themselves would describe their own beliefs in this manner, ie, as being doctrinaire. Russell associates it with pragmatism, objective idealism and the absolute idealism of Hegel. It also refers to coherentism, a holist approach to truth.
So for the example given above of Jack(taller than)Joan, Bertrand Russell claims that the ordering of the relation is not internal to Jack and Joan taken together. The order is something external imposed on the couple Jack and Joan. This however leaves the question as to the status of the ordering, since it cannot be non-existent. A further step in the process is needed to get beyond Russell's objection and this is to include the person doing the ordering in the example, so we have Jack is taller than Joan, according to Tom, or in symbolic form (Jack(R)Joan)(R2)Tom. However, here again we have another kind of ordering which is not included in the grouping.
Russell had opposed the doctrine of internal relations in his abandonment of idealism by reverting to the age old doctrine of atomism and a version of Leibnizian monadism, in which the world is conceived as composed of many distinct, independent entities, each of which can be considered in isolation from its relations to other things. The argument itself was first put down on paper in Plato's dialogue Parmenides and is often referred to as the argument of the One and the Many, part of which includes a version of Aristotle's Third Man Argument, which rejects Platonic Forms and introduces objects as being composed of both form and matter.
"This material is in itself the ultimate substance. Evolution, on the materialistic theory, is reduced to the role of being another word for the description of the changes of the external relations between portions of matter. There is nothing to evolve, because one set of external relations is as good as any other set of external relations. There can merely be change, purposeless and unprogressive. But the whole point of the modem doctrine is the evolution of the complex organisms from antecedent states of less complex organisms. The doctrine thus cries aloud for a conception of organism as fundamental for nature. It also requires an underlying activity -- a substantial activity -- expressing itself in achievements of organism.