The Principles of Mathematics (1903)

§ 82

More difficult questions must now be considered. Is such a proposition as Socrates is a man implies Socrates is a mortal, or Socrates has a wife implies Socrates has a father, an assertion concerning Socrates or not? It is quite certain that, if we replace Socrates by a variable, we obtain a propositional function; in fact, the truth of this function for all values of the variable is what is asserted in the corresponding formal implication, which does not, as might be thought at first sight, assert a relation between two propositional functions. Now it was our intention, if possible, to explain propositional functions by means of assertions; hence, if our intention can be carried out, the above propositions must be assertions concerning Socrates. There is, however, a very great difficulty in so regarding them. An assertion was to be obtained from a proposition by simply omitting one of the terms occurring in the proposition. But when we omit Socrates, we obtain ... is a man implies ... is a mortal. In this formula it is essential that, in restoring the proposition, the same term should be substituted in the two places where dots indicate the necessity of a term. It does not matter what term we choose, but it must be identical in both places. Of this requisite, however, no trace whatever appears in the would-be assertion, and no trace can appear, since all mention of the term to be inserted is necessarily omitted. When an x is inserted to stand for the variable, the identity of the term to be inserted is indicated by the repetition of the letter x; but in the assertional form no such method is available. And yet, at first sight, it seems very hard to deny that the proposition in question tells us a fact about Socrates, and that the same fact is true about Plato or a plum-pudding or the number 2. It is certainly undeniable that Plato is a man implies Plato is a mortal is, in some sense or other, the same function of Plato as our previous proposition is of Socrates. The natural interpretation of this statement would be that the one proposition has to Plato the same relation as the other has to Socrates. But this requires that we should regard the propositional function in question as definable by means of its relation to the variable. Such a view, however, requires a propositional function more complicated than the one we are considering. If we represent x is a man implies x is a mortal by ϕx, the view in question maintains that ϕx is the term having to x the relation R, where R is some definite relation. The formal statement of this view is as follows: For all values of x and y, y is identical with ϕx is equivalent to y has the relation R to x. It is evident that this will not do as an explanation, since it has far greater complexity than what it was to explain. It would seem to follow that propositions may have a certain constancy of form, expressed in the fact that they are instances of a given propositional function, without its being possible to analyze the proposition into a constant and a variable factor. Such a view is curious and difficult: constancy of form, in all other cases, is reducible to constancy of relations, but the constancy involved here is presupposed in the notion of constancy of relation, and cannot therefore be explained in the usual way.(§ 82 ¶ 1)

The same conclusion, I think, will result from the case of two variables. The simplest instance of this case is xRy, where R is to be a constant relation, while x and y are independently variable. It seems evident that this is a propositional function of two independent variables: there is no difficulty in the notion of the class of all propositions of the form xRy. This class is involved--or at least all those members of the class that are true are involved--in the notion of the classes of referents and relata with respect to R, and these classes are unhesitatingly admitted in such words as parents and children, masters and servants, husbands and wives, and innumerable other instances from daily life, as also in logical notions such as premisses and conclusions, causes and effects, and so on. All such notions depend upon the class of propositions typified by xRy, where R is constant while x and y are variable. Yet it is very difficult to regard xRy as analyzable into the assertion R concerning x and y, for the very sufficient reason that this view destroys the sense of the relation, i.e. its direction from x to y, leaving us with some assertion which is symmetrical with respect to x and y, such as the relation R holds between x and y. Given a relation and its terms, in fact, two distinct propositions are possible. Thus if we take R itself to be an assertion, it becomes an ambiguous assertion: in supplying the terms, if we are to avoid ambiguity, we must decide which is referent and which relatum. We may quite legitimately regard ...Ry as an assertion, as was explained before; but here y has become constant. We may then go on to vary y, considering the class of assertions ...Ry for different values of y; but this process does not seem to be identical with that which is indicated by the independent variability of x and y in the propositional function xRy. Moreover, the suggested process requires the variation of an element in an assertion, namely of y in ...Ry, and this is in itself a new and difficult notion.(§ 82 ¶ 2)

A curious point arises, in this connection, from the consideration, often essential in actual Mathematics, of a relation of a term to itself. Consider the propositional function xRx, where R is a constant relation. Such functions are required in considering, e.g., the class of suicides or of self-made men; or again, in considering the values of the variable for which it is equal to a certain function of itself, which may often be necessary in ordinary Mathematics. It seems exceedingly evident, in this case, that the proposition contains an element which is lost when it is analyzed into a term x and a relation R. Thus here again, the propositional function must be admitted as fundamental.(§ 82 ¶ 3)