The Principles of Mathematics (1903)

§ 33

Before giving any primitive propositions, Peano proceeds to some definitions. (1) If a is a class, x and y are a’s is to mean x is an a and y is an a. (2) If a and b are classes, every a is a b means x is an a implies that x is a b. If we accept formal implication as a primitive notion, this definition seems unobjectionable; but it may well be held that the relation of inclusion between classes is simpler than formal implication, and should not be defined by its means. This is a difficult question, which I reserve for subsequent discussion. A formal implication appears to be the assertion of a whole class of material implications. The complication introduced at this point arises from the nature of the variable, a point which Peano, though he has done very much to show its importance, appears not to have himself sufficiently considered. The notion of one proposition containing a variable implying another such proposition, which he takes as primitive, is complex, and should therefore be separated into its constituents; from this separation arises the necessity of considering the simultaneous affirmation of a whole class of propositions before interpreting such a proposition as x is an a implies that x is a b. (3) We come next to a perfectly worthless definition, which has been since abandoned[30]. This is the definition of such that. The x’s such that x is an a, we are told, are to mean the class a. But this only gives the meaning of such that when placed before a proposition of the type x is an a. Now it is often necessary to consider an x such that some proposition is true of it, where this proposition is not of the form x is an a. Peano holds (though he does not lay it down as an axiom) that every proposition containing only one variable is reducible to the form x is an a[31]. But we shall see (Chap. X) that at least one such proposition is not reducible to this form. And in any case, the only utility of such that is to effect the reduction, which cannot therefore be assumed to be already effected without it. The fact is that such that contains a primitive idea, but one which it is not easy clearly to disengage from other ideas.(§ 33 ¶ 1)

In order to grasp the meaning of such that, it is necessary to observe, first of all, that what Peano and mathematicians generally call one proposition containing a variable is really, if the variable is apparent, the conjunction of a certain class of propositions defined by some constancy of form; while if the variable is real, so that we have a propositional function, there is not a proposition at all, but merely a kind of schematic representation of any proposition of a certain type. The sum of angles of a triangle is two right angles, for example, when stated by means of a variable, becomes: Let x be a triangle; then the sum of the angles of x is two right angles. This expresses the conjunction of all the propositions in which it is said of particular definite entities that if they are triangles, the sum of their angles is two right angles. But a propositional function, where the variable is real, represents any proposition of a certain form, not all such propositions (see §§ 5962). There is, for each propositional function, an indefinable relation between propositions and entities, which may be expressed by saying that all the propositions have the same form, but different entities enter into them. It is this that gives rise to propositional functions. Given, for example, a constant relation and a constant term, there is a one-one correspondence between the propositions asserting that various terms have the said relation to the said term, and the various terms which occur in those propositions. It is this notion which is requisite for the comprehension of such that. Let x be a variable whose values form the class a, and let f(x) be a one-valued function of x which is a true proposition ofr all values of x within the class a, and which is false for all other values of x. Then the terms of a are the class of terms such that f(x) is a true proposition. This gives an explanation of such that. But it must always be remembered that the appearance of having one proposition f(x) satisfied by a number of values of x is fallacious: f(x) is not a proposition at all, but a propositional function. What is fundamental is the relation of various propositions of given form to the various terms entering severally into them as arguments or values of the variable; this relation is equally required for interpreting the propositional function f(x) and the notion such that, but is itself ultimate and inexplicable. (4) We come next to the definition of the logical product, or common part, of two classes. If a and b be two classes, their common part consists of the class of terms x such that x is an a and x is a b. Here already, as Padoa points out (loc. cit.), it is necessary to extend the meaning of such that beyond the case where our proposition asserts membership of a class, since it is only by means of the definition that the common part is shown to be a class.(§ 33 ¶ 2)

§ 33 n. 1. In consequence of the criticisms of Padoa, R. d. M. Vol. VI, p. 112.

§ 33 n. 2. R. d. M. Vol. VII, No. 1, p. 23; F. 1901, p. 21, § 2, Pro. 4. 0, Note.