Page:An Enquiry Concerning the Principles of Natural Knowledge.djvu/118



30. Abstractive Classes. 30.1 A set of events is called an ‘abstractive class’ when (i) of any two of its members one extends over the other, and (ii) there is no event which is extended over by every event of the set.

The properties of an abstractive class secure that its members form a series in which the predecessors extend over their successors, and that the extension of the members of the series (as we pass towards the ‘converging end’ comprising the smaller members) diminishes without limit; so that there is no end to the series in this direction along it and the diminution of the extension finally excludes any assignable event. Thus any property of the individual events which survives throughout members of the series as we pass towards the converging end is a property belonging to an ideal simplicity which is beyond that of any one assignable event. There is no one event which the series marks out, but the series itself is a route of approximation towards an ideal simplicity of ‘content.’ The systematic use of these abstractive classes is the ‘method of extensive abstraction.’ All the spatial and temporal concepts can be defined by means of them.

30.2 One class of events — α, say — is said to ‘cover’ another class of events — β, say — when every member of α extends over some member of β.

If α be an abstractive class and α covers β, then β must have an infinite number of members and there can be no event which is extended over by every member of β. For any member of α, however small, extends over some member of β. The usual case of covering is when both classes, α and β, are abstractive classes; then each member of α, the covering class, extends over the whole