# The Principles of Mathematics (1903)

## § 65

To sum up. When a class-concept, preceded by one of the six words all, every, any, a, some, the, occurs in a proposition, the proposition is, as a rule, not about the concept formed of the two words together, but about an object quite different from this, in general not a concept at all, but a term or complex of terms. This may be seen by the fact that propositions in which such concepts occur are in general false concerning the concepts themselves. At the same time, it is possible to consider and make propositions about the concepts themselves, but these are not the natural propositions to make in employing the concepts. Any number is odd or even is a perfectly natural proposition, whereas Any number is a variable conjunction is a proposition only to be made in a logical discussion. In such cases, we say that the concept in question denotes. We decided that denoting is a perfectly definite relation, the same in all six cases, and that it is the nature of the denoted object and the denoting concept which distinguishes the cases. We discussed at some length the nature and the differences of the denoted objects in the five cases in which thoese objects are combinations of terms. In a full discussion, it would be necessary also to discuss the denoting concepts: the actual meanings of these concepts, as opposed to the nature of the objects they denote, have not been discussed above. But I do not know that there would be anything further to say on this topic. Finally, we discussed the, and showed that this notion is essential to what mathematics calls definition, as well as to the possibility of uniquely determining a term by means of concepts; the actual use of identity, though not its meaning, was also found to depend upon this way of denoting a single term. From this point we can advance to the discussion of classes, thereby continuing the development of the topics connected with adjectives.(§ 65 ¶ 1)