The axiom of reducibility was introduced by Bertrand Russell in the early 20th century as part of his ramified theory of types. Russell devised and introduced the axiom in an attempt to manage the contradictions he had discovered in his analysis of set theory.
History
With Russell's discovery (1901, 1902) of a paradox in Gottlob Frege's 1879 Begriffsschrift and Frege's acknowledgment of the same (1902), Russell tentatively introduced his solution as "Appendix B: Doctrine of Types" in his 1903 The Principles of Mathematics. This contradiction can be stated as "the class of all classes that do not contain themselves as elements". At the end of this appendix Russell asserts that his "doctrine" would solve the immediate problem posed by Frege, but "there is at least one closely analogous contradiction which is probably not soluble by this doctrine. The totality of all logical objects, or of all propositions, involves, it would seem a fundamental logical difficulty. What the complete solution of the difficulty may be, I have not succeeded in discovering; but as it affects the very foundations of reasoning..."
By the time of his 1908 Mathematical logic as based on the theory of types Russell had studied "the contradictions" (among them the Epimenides paradox, the Burali-Forti paradox, and Richard's paradox) and concluded that "In all the contradictions there is a common characteristic, which we may describe as self-reference or reflexiveness".
In 1903, Russell defined predicative functions as those whose order is one more than the highest-order function occurring in the expression of the function. While these were fine for the situation, impredicative functions had to be disallowed:
He repeats this definition in a slightly different way later in the paper (together with a subtle prohibition that they would express more clearly in 1913):
This usage carries over to Alfred North Whitehead and Russell's 1913 Principia Mathematica wherein the authors devote an entire subsection of their Chapter II: "The Theory of Logical Types" to subchapter I. The Vicious-Circle Principle: "We will define a function of one variable as predicative when it is of the next order above that of its argument, i.e. of the lowest order compatible with its having that argument. . . A function of several arguments is predicative if there is one of its arguments such that, when the other arguments have values assigned to them, we obtain a predicative function of the one undetermined argument."
They again propose the definition of a predicative function as one that does not violate The Theory of Logical Types. Indeed the authors assert such violations are "incapable [to achieve]" and "impossible":
The authors stress the word impossible:
Russell's axiom of reducibility
The axiom of reducibility states that any truth function (i.e. propositional function) can be expressed by a formally equivalent predicative truth function. It made its first appearance in Bertrand Russell's (1908) Mathematical logic as based on the theory of types, but only after some five years of trial and error. In his words:
For relations (functions of two variables such as "For all x and for all y, those values for which f(x,y) is true" i.e. ∀x∀y: f(x,y)), Russell assumed an axiom of relations, or [the same] axiom of reducibility.
In 1903, he proposed a possible process of evaluating such a 2-place function by comparing the process to double integration: One after another, plug into x definite values am (i.e. the particular aj is "a constant" or a parameter held constant), then evaluate f(am,yn) across all the n instances of possible yn. For all yn evaluate f(a1, yn), then for all yn evaluate f(a2, yn), etc until all the x = am are exhausted). This would create an m by n matrix of values: TRUE or UNKNOWN. (In this exposition, the use of indices is a modern convenience.)
In 1908, Russell made no mention of this matrix of x, y values that render a two-place function (e.g. relation) TRUE, but by 1913 he has introduced a matrix-like concept into "function". In *12 of Principia Mathematica (1913) he defines "a matrix" as "any function, of however many variables, which does not involve any apparent variables. Then any possible function other than a matrix is derived from a matrix by means of generalisation, i.e. by considering the proposition which asserts that the function in question is true with all possible values or with some values of one of the arguments, the other argument or arguments remaining undetermined". For example, if one asserts that "∀y: f(x, y) is true", then x is the apparent variable because it is unspecified.