Axiom of reducibility

The axiom of reducibility was introduced by Bertrand Russell as part of his ramified theory of types, an attempt to ground mathematics in first-order logic.

History: the problem of impredicativity

With Russell's discovery (1901, 1902) that of a paradox (contradiction) 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 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 "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, 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 his 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 good, impredicative functions had to be disallowed:

"A function whose argument is an individual and whose value is always a first-order proposition will be called a first-order function. A function involving a first-order function or proposition as apparent variable will be called a second-order function, and so on. A function of one variable which is of the order next above that of its argument will be called a predicative function; the same name will be given to a function of several variables [etc] . . .."

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): "A predicative function of x is one whose values are propositions of the type next above that of x, if x is an individual or a proposition, or that of values of x if x is a function. It may be described as one in which the apparent variables, if any, are all of the same type as x or of lower type; and a variable is of lower type than x if it can significantly occur as argument to x, or as argument to an argument to x, and so forth"

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"

Again, they propose the definition of a predicative function as one that doesn't violate the The Theory of Logical Types. Indeed the authors assert such violations are "incapable [to achieve]" and "impossible":

"We are thus lead to the conclusion, both from the vicious-circle principle and from direct inspection, that the functions to which a given object a can be an argument are incapable of being arguments to each other, and that they have no term in common with the functions to which they can be arguments. We are thus led to construct a hierarchy".

Indeed, the authors even use the word impossible:

". . .if we are not mistaken, that not only is it impossible for a function φz^{^} to have itself or anything derived from it as argument, but that, if ψz^{^} is another function such there are arguments a with which both "φa" and "ψa" are significant, then ψz^{^} and anything derived from it cannot significantly be argument to φz^{^}".

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 makes its first appearance in Bertrand Russell's (1908) Mathematical logic as based on the theory of types, but not after some five years of trial and error . In his words:

"Thus a predicative function of an individual is a first-order function; and for higher types of arguments, predicative functions take the place that first-order functions take in respect of individuals. We assume then, that every function is equivalent, for all its values, to some predicative function of the same argument. This assumption seems to be the essence of the usual assumption of classes [modern sets] . . . we will call this assumption the axiom of classes, or the axiom of reducibility.

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 [the same] axiom of reducibility.

In his 1903 he proposed how one would go about evaluating such a 2-place function; indeed he compared the process to double integration: One after another, plug into x definite values a_m (i.e. the particular a_j is "a constant" or a parameter held constant), then evaluate f(a_m,y_n) across all the n instances of possible y_n. In other words: For all y_n evaluate f(a₁, y_n), then for all y_n evaluate f(a₂, y_n), etc until all the x = a_m are exhausted). This would create an m by n matrix of values: TRUE or UNKNOWN. (In this exposition, the use of indices are a modern convenience).

In his 1908 Russell makes 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 we assert that "∀y: f(x, y) is true", then x is the apparent variable because it is unspecified.

Russell now defines a matrix of "individuals" as a first-order matrix, and he follows a similar process to define a second-order matrix, etc. Finally he introduces the definition of a predicative function:

A function is said to be predicative when it is a matrix. It will be observed that, in a hierarchy in which all the variables are individuals or matrices, a matrix is the same thing as an elementary function [cf 1913:127, meaning: the function contains no apparent variables]. ¶ "Matrix" or "predicative function" is a primitive idea"

From this reasoning, he then uses the same wording to propose the same axioms of reducibility as he did in his 1908.

As an aside, Russell in his 1903 considered, and then rejected, "a temptation to regard a relation as definable in extension as a class of couples", i.e. the modern set-theoretic notion of ordered pair. An intuitive version of this notion appeared in Frege's (1879) Begriffsschrift (translated in van Heijenoort 1967:23); Russell's 1903 followed closely the work of Frege (cf Russell 1903:505ff). Russell worried that "it is necessary to give sense to the couple, to distinguish the referent from the relatum: thus a couple becomes essentially distinct from a class of two terms, and must itself be introduced as a primitive idea. It would seem, viewing the idea philosophically, that sense can only be derived from some relational proposition . . . it seems therefore more correct to take an intensional view of relations, and to identify them rather with class-concepts than with classes". As shown below, Norbert Wiener (1914) reduced the notion of relation to class by his definition an ordered pair.

Criticism of the axiom of reducibility

Zermelo 1908

The outright prohibition implied by Russell's axiom of reducibility was roundly criticised by Ernst Zermelo in his 1908 Investigations in the foundations of set theory I, stung as he was by a demand similar to that of Russell that came from Poincaré:

"According to Poincaré (1906, p. 307) a definition is "predicative" and logically admissible only if it excludes all objects that are "dependent" upon the notion defined, that is, that can in any way be determined by it".

Zermelo countered:

"A definition may very well rely upon notions that are equivalent to the one being defined; indeed in every definition definiens and definiendum are equivalent notions, and the strict observance of Poincaré's demand would make every definition, hence all of science, impossible.".

Wiener 1914

In his 1914 A simplification of the logic of relations, Norbert Wiener removes the need for the axiom of reducibility as applied to relations between two variables x, and y e.g. φ(x,y). He does this by introducing a way to express a relation as a set of ordered pairs: "It will be seen that what we have done is practically to revert to Schröder's treatment of a relation as a class [set] of ordered couples". Indeed van Heijenoort observes that "[b]y giving a definition of the ordered pair of two-elements in terms of class operations, the note reduced the theory of relations to that of classes".But Wiener opined that while he had dispatched Russell and Whitehead's two-variable version of the axiom *12.11, the single-variable version of the axiom of reducibility for (axiom *12.1 in Principia Mathematica) was still necessary.

Skolem 1922

Thoralf Skolem in his 1922 Some remarks on axiomatized set theory took a less than positive attitude toward "Russell and Whitehead" (i.e. their work Principia Mathematica):

"Until now, so far as I know, only one such system of axioms has found rather general acceptance, namely that constructed by Zermelo (1908). Russell and Whitehead, too, constructed a system of logic that provides a foundation for set theory; if I am not mistaken, however, mathematicians have taken but little interest in it

Skolem then observes the problems of what he called "nonpredicative definition" in the set theory of Zermelo:

"...the difficulty is that we have to form some sets whose existence depends upon all sets . . . Poincaré called this kind of definition and regarded it as the real logical weakness of set theory"

While Skolem is mainly addressing a problem with Zermelo's set theory, he does make this observation about the axiom of reducibility:

". . . they [Russell and Whitehead], too, simply content themselves with circumventing the difficulty by introducing a stipulation, the axiom of reducibility. Actually, this axiom decrees that the nonpredicative stipulations will be satisfied. There is no proof of that; besides, so far as I can see, such a proof must be impossible from Russell and Whitehead's point of view as well as from Zermelo's"

Wittgenstein 1918 - (1922 English translation)

Ludwig Wittgenstein, while imprisoned in a prison camp, finished his Tractatus Logico-Philosophicus. His introduction credits "the great works of Frege and the writings of my friend Bertrand Russell". Not a self-effacing intellectual, he pronounced that "the truth of the thoughts communicated here seems to me unassailable and definitive. I am, therefore, of the opinion that the problems have in essentials been finally solved". So given such a attitude it is no surprise that Russell's theory of types comes under criticism:

3.33

In logical syntax the meaning of a sign ought never to play a role; it must admit of being established without mention being thereby made of the meaning of a sign; it ought to presuppose only the description of the expressions.

3.331

From this observation we get a further view -- into Russell's Theory of Types. Russell's error is shown by the fact that in drawing up his symbolic rules he has to speak of the meaning of the signs.

3.332

No proposition can say anything about itself, because the proposition sign cannot be contained in itself (that is the "whole theory of types").

3.333

A function cannot be its own argument, because the functional sign already contains the prototype of its own argument and it cannot contain itself . . .. Herewith Russell's paradox vanishes..

This appears to support the same argument Russell uses to erase his "paradox". But is Wittgenstein committing the error of which he accuses Russell? This "using the signs" to "speak of the signs" Russell criticises in his introduction that preceded the original English translation:

"What causes hesitation is the fact that, after all, Mr Wittgenstein manages to say a good deal about what cannot be said, thus suggesting to the sceptical reader that possibly there may be some loophole through a hierarchy of languages, or by some other exist."

This problem appears later when Wittgenstein arrives at this gentle disavowal of the axiom of reducibility -- one interpretation of the following is that Wittgenstein is saying that Russell has made (what is known today as) a category error; Russell has asserted (inserted into the theory) a "further law of logic" when all the laws (e.g. the unbounded Sheffer stroke adopted by Wittgenstein) have already been asserted:

6.123

It is clear that the laws of logic cannot themselves obey further logical laws. (There is not, as Russell supposed, for every "type" a special law of contradiction; but one is sufficient, since it is not applied to itself.)

6.1231

The mark of logical propositions is not their general validity. To be general is only to be accidentally valid for all things. An ungeneralised proposition can be tautologous just as well as a generalised one.

6.1232

Logical general validity, we could call essential as opposed to accidental general validity, e.g., of the proposition "all men are mortal". Propositions like Russell's "axiom of reducibility" are not logical propositions, and this explains our feeling that, if true, they can only be true by a happy chance.

6.1233

We can imagine a world in which the axiom of reducibility is not valid. But it is clear that logic has nothing to do with the question of whether our world is really of this kind or not.

Russell 1927

In his 1927 Introduction to the second edition of Principia Mathematica Russell criticises his own axiom:

"One point in regard to which improvement is obviously desirable is the axiom of reducibility (*12.1.11). This axiom has a purely pragmatic justification: it leads to the desired results, and to no others. But clearly it is not the sort of axiom with which we can rest content. On this subject, however, it cannot be said that a satisfactory solution is as yet obtainable. . . . There is another course recommended by Wittgenstein† [† Tractatus Logico-Philosophicus, *5.54ff] for philosophical reasons. This is to assume that functions of propositions are always truth-functions, and that a function can only occur as in a proposition through its values. There are difficulties . . . It involves the consequence that all functions of functions are extensional. . . . [But the consequences of his logic are that] the theory of infinite Dedekindian and well-ordering collapses, so that irrationals, and real numbers generally, can no longer be adequately dealt with. Also Cantor's proof that 2ⁿ > n breaks down unless n is finite Perhaps some further axiom, less objectionable than the axiom of reducibility, might give these results, but we have not succeeded in finding such an axiom.

Wittgenstein's 5.54ff has more to do with the notion of function but is worth repeating for context:

5.54

In the general propositional form, propositions occur in a proposition only as bases of the truth-operations.

5.541

At first sight it appears as if there were also a different way in which one proposition could occur in another. ¶ Especially in certain propositional forms of psychology, like "A thinks, that p is the case," or "A thinks p," etc. ¶ Here it appears superficially as if the proposition p stood to the object A in a kind of relation. ¶ (And in modern epistemology [Russell, Moore, etc.] those propositions have been conceived in this way.)

5.542

But it is clear that "A believes that p, "A thinks p", "A says p", are of the form " ' p ' thinks p "; and here we have no coordination of a fact and an object, but a coordination of facts by means of a coordination of their objects.

5.5421 [etc: "A composite soul would not be a soul any longer."]

5.5422

The correct explanation of the form of the proposition "A judges p" must show that it is impossible to judge a nonsense. (Russell's theory does not satisfy this condition).

A possible interpretation of Wittgenstein's stance is that the thinker A i.e. 'p' is identically the thought p, in this way the "soul" remains a unit and not a composite. So to utter "the thought thinks the thought" is utter nonsense, because per 5.542 the utterance does not specify anything.

von Neumann 1925

John von Neumann in his 1925 An axiomatization of set theory wrestled with the same issues as did Russell, Zermelo, Skolem, and Fraenkel. He summarily rejected the effort of Russell:

"Here Russell, J. Konig, Weyl, and Brouwer must be mentioned. They arrived at entirely different results [from the set theorists], but the over-all effect of their activity seems to me outright devastating. In Russell, all of mathematics and set theory seems to rest upon the highly problematic "axiom of reducibility", while Weyl and Brouwer systematically reject the larger part of mathematics and set theory as completely meaningless"

He then notes the work of the set theorists Zermelo, Fraenkel and Schoenflies, in which "one understands by "set" nothing but an object of which one knows no more and wants to know no more than what follows about it from the postulates. The postulates [of set theory] are to be formulated in such a way that all the desired theorems of Cantor's set theory follow from them, but not the antinomies. In these axiomatizations, however, we can never be perfectly sure of the latter point. We see only that the known modes of inference leading to the antinomies fail, but who knows whether there are not others?

While he mentions the efforts of David Hilbert to prove the consistency of his axiomatization of mathematics von Neumann placed him in the same group as Russell. Rather, von Neumann considered his proposal to be "in the spirit of the second group . . . We must, however, avoid forming sets by collecting or separating elements [durch Zusammenfassung oder Aussonderung von Elementen], and so on, as well as eschew the unclear principle of "definiteness" that can still be found in Zermelo. ¶ We prefer, however, to axiomatize not "set" but "function".

van Heijenoort observes that ultimately this axiomatic system of von Neumann's, "was simplified, revised, and expanded . . . and it come to be known as the von Neumann-Bernays-Gödel set theory".

David Hilbert 1927

David Hilbert's axiomatic system (see more at Hilbert system) that he presents in his 1925 The Foundations of Mathematics is the mature expression of a task he set about in the early 1900's but let lapse for a while (cf his 1904 On the foundations of logic and arithmetic). His system is neither set theoretic nor derived directly from Russell and Whitehead. Rather, it invokes 13 axioms of logic -- four axioms of Implication, six axioms of logical AND and logical OR, 2 axioms of logical negation, and 1 ε-axiom ("existence" axiom)-- plus a version of the Peano axioms in 4 axioms including mathematical induction, some definitions that "have the character of axioms, and certain recursion axioms that result from a general recursion schema" plus some formation rules that "govern the use of the axioms".

Hilbert states that, with regard to this system, i.e. "Russell and Whitehead's theory of foundations [,] ... the foundation that it provides for mathematics rests, first, upon the axiom of infinity and, then upon what is called the axiom of reducibility, and both of these axioms are genuine contentual assumptions that are not supported by a consistency proof; they are assumptions whose validity in fact remains dubious and that, in any case, my theory does not require . . . reducibility is not presupposed in my theory . . . the execution of the reduction would be required only in case a proof of a contradiction where given, and then, according to my proof theory, this reduction would always be bound to succeed".

It is upon this foundation that modern recursion theory rests.

Ramsey 1925

In 1925 Frank Plumpton Ramsey argued that it is not needed . However in the second edition of Principia Mathematica (1927, page xiv) and in Ramsey's 1926 paper it is stated that certain theorems about real numbers could not be proved using Ramsey's approach. Most later mathematical formalisms (Hilbert's Formalism (mathematics) or Brower's Intuitionism for example) do not use it.

Ramsey showed that you can reformulate the definition of predicative by using the definitions in Wittgenstein's Tractatus Logico-Philosophicus. As a result, all functions of a given order are predicative, irrespective of how they are expressed. He goes on to show that his formulation still avoids the paradoxes. However, the "Tractatus" theory did not appear strong enough to prove some mathematical results.

W. V. Quine in van Heijenoort 1967

In a damning critique that also discusses the pros and cons of Ramsey (1931) Quine calls Russell's formulation of "types" to be "troublesome . . . the confusion persists as he attempts to define "nth order propositions". . . the method is indeed oddly devious . . . the axiom of reducibility is self-effacing", etc..

Like Kleene (see below) Quine observes that Ramsey (1926), (1931) divided the various paradoxes into two varieties (i) "those of pure set theory" and (ii) those derived from "semantic concepts such as falsity and specifiability", and Ramsey believed that the second variety should have been left out of Russell's solution". Quine ends with the damning opinion that "because of the confusion of propositions with sentences, and of attributes with their expressions, Russell's purported solution of the semantic paradoxes was enigmatic anyway".

Stephen Kleene 1952

In his section §12. First inferences from the paradoxes, subchapter "LOGICISM" Kleene 1952 traces the development of Russell's theory of types:

"To adapt the logicistic [sic] construction of mathematics to the situation arising from the discovery of the paradoxes, Russell excluded impredicative definitions by his ramified theory of types (1908, 1910)".

Kleene observes that "to exclude impredicative definitions within a type, the types above type 0 [primary objects or individuals "not subjected to logical analysis"] are further separated into orders. Thus for type 1 [properties of individuals, i.e. logical results of the propositional calculus ], properties defined without mentioning any totality belong to order 0, and properties defined using the totality of properties of a given order below to the next higher order)".

But Kleene parenthetically observes that "the logicistic definition of natural number now becomes predicative when the [property] P in it is specified to range only over properties of a given order; in [this] case the property of being a natural number is of the next higher order" . But this separation into orders makes it impossible to construct the familiar analysis, which [see Kleene’s example at Impredicativity] contains impredicative definitions. To escape this outcome, Russell postulated his axiom of reducibility . . .". But, Kleene wonders, "on what grounds should we believe in the axiom of reducibility?". He observes that, whereas Principia Mathematica is presented as derived from intuitively-derived axioms that "were intended to be believed about the world, or at least to be accepted as plausible hypotheses concerning the world [,] . . . if properties are to be constructed, the matter should be settled on the basis of constructions, not by an axiom". Indeed he quotes Whitehead and Russell 1927 questioning their own axiom:

" '... clearly it is not the sort of axiom with which we can rest content' "(Kleene quoting from Whitehead and Russell's introduction to their 1927 2nd edition of Principia Mathematica.

Kleene references work of Ramsey 1926 but notes that "neither Whitehead and Russell nor Ramsey succeeded in attaining the logicistic goal constructively" and "an interesting proposal . . . by Langford 1927 and Carnap 1931-2, is also not free of difficulties". Kleene ends this discussion with quotes from Weyl 1946 that "the system of Principia Mathematica . . . [is founded on] a sort of logician's paradise . . ." and anyone "who is ready to believe in this 'transcendental world' could also accept the system of axiomatic set theory (Zermelo, Fraenkel, etc), which, for the deduction of mathematics, has the advantage of being simpler in structure".

References

• van Heijenoort, Jean (1967, 3rd printing 1976), From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, Cambridge, MA, ISBN 0-674-32449-8 (pbk)
• Russell, Bertrand (1903) The Principles of Mathematics: Vol. 1, Cambridge at the University Press, Cambridge, UK, republished as a googlebook.
• Whitehead, Alfred North and Russell, Bertrand (1913, 1962 edition), Principia Mathematica to *56, Cambridge at the University Press, London UK, no ISBN or US card catalog number.
• Mario Livio (2009), Is God a Mathematician?, Simon and Schuster, New York, NY, ISBN-13: 978-0-7432-9405-8.