Note 31 — Chapter II

Mr Johnson points out to me that in pursuing this line of argument certain restrictions of a somewhat subtle kind are necessary in regard to what may be called our “universe of attributes.” The “universe of objects” which is what we mean by the “universe of discourse,” implies individuality of object and limitation of range of objects ; and if we are to work out a thoroughgoing reciprocity between attributes and objects, we must recognise in our “universe of attributes” restrictions analogous to the above, namely, simplicity of attribute and limitation of range of attributes. By “simplicity of attribute” is meant that the universe of attributes must not contain any attribute which is a logical function of any other attribute or set of attributes. Thus, if A, B are two attributes recognised in our universe, we must not admit such attributes as X (= A and B), or Y (= A or B), or Z (= not-A). We may indeed have a negatively defined attribute, but it must not be the formal contradictory of another or formally involve the contradictory of another. The following example will shew the necessity of this restriction. Let P1, P2, P3, be selected as typical of the whole class P1, P2, P3 P4, P5, P6; and let A1 be an attribute possessed by P1 alone, A2 an attribute possessed by P2 alone, and so on. Then if we recognise A1 or A2 or A3 as a distinct attribute, it is at once clear that P1, P2, P3 will no longer be typical of the whole class; and the same result follows if not-A4 is recognised as a distinct attribute. Similarly, without the restriction in question any selection (short of the whole) would necessarily fail to be typical of the whole class. As a concrete example, suppose that we select from the class of professional men a set of examples that have in common no attribute except those that are common to the whole class. It may turn out that our examples are all barristers or doctors, but none of them solicitors. Now if we recognise as a distinct attribute being “either a barrister or a doctor,” our selected group will thereby have an extra common attribute not possessed by every professional man. The same result will follow if we recognise the attribute “non-solicitor.” Not much need be added as regards the necessity of some limitation in the range of attributes which are recognised. The mere fact of our having selected a certain group would indeed constitute an additional attribute, which would at once cause the selection to fail in its purpose, unless this were excluded as inessential. Similarly, such attributes as position in space or in time &c. must in general be regarded as inessential. For example, I might draw on a sheet of paper a number of triangles sufficiently typical of the whole class of triangles, but for this it would be necessary to reject as inessential the common property which they would possess of all being drawn on a particular sheet of paper.