The Principles of Mathematics (1903)

§ 81

Can the indefinable element involved in propositional functions be identified with assertion together with the notion of every proposition containing a given assertion, or an assertion made concerning every term? The only alternative, so far as I can see, is to accept the general notion of a propositional function itself as indefinable, and for formal purposes this course is certainly the best; but philosophically, the notion appears at first sight capable of analysis, and we have to examine whether or not this appearance is deceptive.(§ 81 ¶ 1)

We saw in discussing verbs, in Chapter IV, that when a proposition is completely analyzed into its simple constituents, these constituents taken together do not reconstitute it. A less complete analysis of propositions into subject and assertion has also been considered; and this analysis does much less to destroy the proposition. A subject and an assertion, if simply juxtaposed, do not, it is true, constitute a proposition; but as soon as the assertion is actually asserted of the subject, the proposition reappears. The assertion is everything that remains of the proposition when the subject is ommitted: the verb remains an asserted verb, and is not turned into a verbal noun; or at any rate the verb retains that curious indefinable intricate relation to the other terms of the proposition which distinguishes a relating relation from the same relation abstractly considered. It is the scope and legitimacy of this notion of assertion which is now to be examined. Can every proposition be regarded as an assertion concerning any term occurring in it, or are limitations necessary as to the form of the proposition and the way in which the term enters into it?(§ 81 ¶ 2)

In some simple cases, it is obvious that the analysis into subject and assertion is legitimate. In Socrates is a man, we can plainly distinguish Socrates and something that is asserted about him; we should admit unhesitatingly that the same thing may be said about Plato or Aristotle. Thus we can consider a class of propositions containing this assertion, and this will be the class of which a typical number is represented by x is a man. It is to be observed that the assertion must appear as assertion, not as term: thus to be a man is to suffer contains the same assertion, but used as a term, and this proposition does not belong to the class considered. In the case of propositions asserting a fixed relation to a fixed term, the analysis seems equally undeniable. To be more than a yard long, for example, is a perfectly definite assertion, and we may consider the class of propositions in which this assertion is made, which will be represented by the propositional function x is more than a yard long. In such phrases as snakes which are more than a yard long, the assertion appears very plainly; for it is here explicitly referred to a variable subject, not asserted of any one defintie subject. Thus if R be a fixed relation and a a fixed term, ...Ra is a perfectly definite assertion. (I place dots before the R, to indicate the place where the subject must be inserted in order to make a proposition.) It may be doubted whether a relational proposition can be regarded as an assertion concerning the relatum. For my part, I hold that this can be done except in the case of subject-predicate propositions; but this question is better postponed until we have discussed relations[60].(§ 81 ¶ 3)