## 1. Overview



*Principia Mathematica*, the landmark work in formal logic
written by
 Alfred North Whitehead
 and
 Bertrand Russell,
 was first published in three volumes in 1910, 1912 and 1913. A second
edition appeared in 1925 (Volume I) and 1927 (Volumes II and III). In
1962 an abbreviated issue (containing only the first 56 chapters)
appeared in paperback.



Written as a defense of *logicism* (the thesis that mathematics
is in some significant sense reducible to logic), the book was
instrumental in developing and popularizing modern mathematical logic.
It also served as a major impetus for research in the foundations of
mathematics throughout the twentieth century. Along with
Aristotle's *Organon* and Gottlob Frege's
*Basic Laws of Arithmetic*, it remains one of the most
influential books on logic ever written.



This entry includes a presentation of the main definitions and
theorems used in the development of the logicist project in PM. The
entry indicates a path through the whole work presenting the basic
results proved in *Principia Mathematica* (PM) in a somewhat
more contemporary notation, so as to make it easy to compare the
system of Whitehead and Russell with that of Frege, the other most
prominent advocate of logicism in the foundations of mathematics. The
aim of that program, as described by Russell in the opening lines of
the preface to his 1903 book *The Principles of Mathematics*,
namely to define mathematical notions in terms of logical notions, and
to derive mathematical principles, so defined, from logical principles
alone:



> 
> 
> The present work has two main objects. One of these, the proof that
> all pure mathematics deals exclusively with concepts definable in
> terms of a very small number of fundamental concepts, and that all its
> propositions are deducible from a very small number of fundamental
> logical principles, is undertaken in Parts II-VII of this work,
> and will be established by strict symbolic reasoning in Volume
> II....The other object of this work, which occupies Part I., is
> the explanation of the fundamental concepts which mathematics accepts
> as indefinable. This is a purely philosophical task.... (1903:
> xv)
> 
> 
> 



Though Frege's system was subject to Russell's Paradox,
subsequent examination of his system shows how much of the development
of arithmetic is possible independently of the paradoxical elements of
the system. In particular, recent interest in Frege's system has
led to the isolation of what is called "Frege's
Theorem" as possible in a consistent fragment of Frege's
original system, and from it the goal of deriving arithmetic, as
formalized in Peano's Postulates. See the entry
 Frege's theorem and foundations for arithmetic,
 which presents this aspect of Frege's system in contemporary
notation.



Russell had written *The Principles of Mathematics* (PoM),
which presents the basic elements of his logicist program, before
discovering Frege's similar work in *Foundations of
Arithmetic* and *Basic Laws of Arithmetic*, in
June of 1902. As he describes in the Preface, Russell intended a
formal presentation of his account in a "Volume II" of
PoM. In 1903 he enlisted Alfred North Whitehead to join him in the
writing of this second volume, but soon the project turned into a new
work, *Principia Mathematica*, a massive three volume work,
that was not to be published until 1910 (Volume I), 1912 (Volume II)
and 1913 (Volume III).



The system of PM differed significantly from Frege's system, in
a large part because of the introduction of the *theory of
types* whose purpose was to avoid the paradox that had affected
Frege in a principled fashion. A second important difference from
Frege's system is that PM is based on a logic of relations of
various numbers of arguments, whereas Frege's system was based
on the notions of function and object, with even his distinctively
logical *concepts* being seen as functions (from a number of
objects to the truth values T and F, which are also objects in
Frege's system.) So, it might be said that PM is based on a
theory of ramified types of relations, in contrast to Frege's
second order predicate calculus with concepts. The most important step
is to define set expressions in terms of higher-order functions. Thus
the paradoxical "Russell set", the set of all sets which
are not members of themselves, \(\{ x \mid x \notin x\}\), is defined by
an expression involving functions that will violate the theory of
types. The expression for the offending class is ruled out on the
basis of the theory of types, as is its seeming innocuous complement,
the set of all sets that *are* members of themselves, \(\{ x
\mid x \in x\}\). In contemporary set theory \(\{ x \mid x \notin
x\}\) is the universe of sets, which is not itself a set, and because
no set is an element of itself, \(\{ x \mid x \in x\}\) is just the
empty set. An additional cost of this method is that while for Frege
sets are objects of the lowest types, there will be sets in the PM
theory in a *simple theory* of types, which distinguishes
individuals and sets of individuals and sets of sets of individuals,
etc. Even to derive a hierarchy of sets in the simple theory the
*axiom of reducibility* is needed to guarantee that more
complex "impredicative" definitions pick out sets of the
same simple type. Thus the "least upper bound" of a closed
interval of real numbers will identify a member of that set of a
higher order in the ramified theory. That this least upper bound will
be of the same simple type requires the axiom of reducibily. The cost
of adopting the theory of types to avoid the paradox extends to
difficulties in constructing the natural numbers. While Russell
follows Frege in many important details, in particular in using
Frege's notion of the ancestral of the successor relation to
define the natural numbers, other parts of the construction are
importantly different. Frege was able to define the successor of a
number by using the set of its predecessors. The number 2 is the set
containing 0 and 1, and thus it has two members. They will, however,
be of different types in the hierarchy of simple types, and so the
whole set of natural numbers cannot be defined within the theory of
simple types. Since each step from 0 to 1, to 2, etc, raises the
simple types from 0 to 1 to 2, there will be no simple type of all the
natural numbers, so defined. Instead PM adopts the *axiom of
infinity* which assures the existence of an infinite number of
individuals, allowing for the construction of the natural numbers for
each type above a lower bound of 3 or so (as numbers will be sets of
equinumerous sets of individuals...).



With this turn to the ramified theory of types, along with the extra
axioms of reducibility, and infinity, it is possible for PM to define
a version of Frege's construction of the Natural Numbers so that
the "Peano axioms" can be proved from logic alone. This
takes up to section [?]120, well into Volume II. At
this point the alternative to "Frege's Theorem" is
completed, in the sense that we are presented with a consistent
development of the natural numbers, based on a theory of higher-order
logic with a number of additional axioms. Philosophers soon followed
Ludwig Wittgenstein (1922) and disputed the idea that these additional
axioms, the axioms of reducibility and infinity, are really logical
truths, and so denied that the logicist program of reducing arithmetic
to logic was any more successful than Frege's attempt had
been.



The survey of PM will proceed through the remainder of Volume II and
through Volume III, where the theories of rational and real numbers
are developed. The contrast intended here is not with Frege's
theories of rational and real numbers, which are present in
*Grundgesetze* but are not seen as a natural extension of the
theory of natural numbers. Instead the contemporary account of natural
numbers and real numbers is seen as an elementary extension of the
axiomatic Zermelo-Frankel set theory. A contemporary textbook in
axiomatic set theory, such as Enderton (1977) or Suppes (1960), shows
how to construct rational numbers (and negative integers) as pairs of
natural numbers, thus 3/4 is constructed as the pair with the
operations of addition and multiplication defined as operations on
pairs; thus \(1/2 + 1/3 = 10/12 = 5/6\). These positive rational
numbers are extended to the whole set by adding negative integers, and
then real numbers are defined as Dedekind cuts in the rational
numbers, i.e., the set of partitions of sets of rational numbers. The
arithmetic of real numbers is then defined for these constructions,
and so with sets of real numbers the whole of analysis can be reduced
to arithmetic. PM, however, avoids this "arithmetization"
of analysis, but instead defines rational, real and in fact a huge
class of "relation numbers" as sets of isomorphic sets of
relations. Russell says later that he regrets that the theory of
relation numbers was not picked up by later set theorists, even though
this was some of his most original work in PM. The brief summary of
these later topics that we include below, can therefore be seen as a
summary of the interesting consequences of taking a different route to
the definition of natural numbers based on a logic of relations and
properties, rather than the set theory of contemporary foundations of
mathematics. This entry is thus aimed at an explication of the unusual
order of presentation of these results, in comparison with both Frege
and contemporary set theory, and to illustrate these aspects of the
theory of relations that are not investigated by contemporary
researchers.


## 2. History of and Significance of *Principia Mathematica*


### 2.1 History of *Principia Mathematica*



Logicism is the view that (some or all of) mathematics can be reduced
to (formal) logic. It is often explained as a two-part thesis. First,
it consists of the claim that all mathematical truths can be
translated into logical truths or, in other words, that the vocabulary
of mathematics constitutes a proper subset of the vocabulary of logic.
Second, it consists of the claim that all mathematical proofs can be
recast as logical proofs or, in other words, that the theorems of
mathematics constitute a proper subset of the theorems of logic. As
Russell writes, it is the logicist's goal "to show that
all pure mathematics follows from purely logical premises and uses
only concepts definable in logical terms" (1959: 74).



The logicist thesis appears to have been first advocated in the late
seventeenth century by Gottfried Leibniz. Later, the idea was defended
in much greater detail by Gottlob Frege. During the critical movement
of the 1820s, mathematicians such as Bernard Bolzano, Niels Abel,
Louis Cauchy, and Karl Weierstrass succeeded in eliminating much of
the vagueness and many of the contradictions present in the
mathematics of their day. By the mid- to late-1800s, William Hamilton
had gone on to introduce ordered couples of reals as the first step in
supplying a logical basis for the complex numbers and Karl
Weierstrass, Richard Dedekind, and Georg Cantor had all developed
methods for founding the irrationals in terms of the rationals. Using
work done by H.G. Grassmann and Richard Dedekind, Guiseppe Peano had
then gone on to develop a theory of the rationals based on his now
famous axioms for the natural numbers. By Frege's day, it was
thus generally recognized that large parts of mathematics could be
derived from a relatively small set of primitive notions.



Even so, it was not until 1879, when Frege developed the necessary
logical apparatus, that logicism could finally be said to have become
technically plausible. After another five years' work, Frege
arrived at the definitions necessary for logicising arithmetic and
during the 1890s he worked on many of the essential derivations.
However, with the discovery of paradoxes such as
 Russell's paradox
 at the turn of the century, it appeared that additional resources
would need to be developed if logicism were to succeed.



By 1902, both Whitehead and Russell had reached this same conclusion.
Both men were in the initial stages of preparing second volumes to
their earlier books on related topics: Whitehead's 1898 *A
Treatise on Universal Algebra* and Russell's 1903 *The
Principles of Mathematics.* Since their research overlapped
considerably, they began collaborating on what would eventually become
*Principia Mathematica*. By agreement, Russell worked primarily
on the philosophical parts of the project, including the book's
philosophically rich Introduction, the theory of
 descriptions,
 and the no-class theory (in which set or class terms become
meaningful only when placed in well-defined contexts), all of which
can still be read fruitfully even by non-specialists. The two men then
collaborated on the technical derivations. As Russell writes,



> 
> 
> As for the mathematical problems, Whitehead invented most of the
> notation, except in so far as it was taken over from Peano; I did most
> of the work concerned with series and Whitehead did most of the rest.
> But this only applies to first drafts. Every part was done three times
> over. When one of us had produced a first draft, he would send it to
> the other, who would usually modify it considerably. After which, the
> one who had made the first draft would put it into final form. There
> is hardly a line in all the three volumes which is not a joint
> product. (1959: 74)
> 
> 
> 



Initially, it was thought that the project might take a year to
complete. Unfortunately, after almost a decade of difficult work on
the part of the two men, Cambridge University Press concluded that
publishing *Principia* would result in an estimated loss of 600
pounds. Although the press agreed to assume half this amount and the
Royal Society agreed to donate another 200 pounds, this still left a
100-pound deficit. Only by each contributing 50 pounds were the
authors able to see their work through to publication. (Whitehead,
Russell, & James 1910)



Publication involved the enormous job of type-setting all three
volumes by hand. In 1911, the printing of the second volume was
interrupted when Whitehead discovered a difficulty with the symbolism.
The result was the insertion (on roman numeral pages) of a long
"Prefatory Statement of Symbolic Conventions" at the
beginning of Volume II.



The initial print run of 750 copies of Volume I and 500 copies of each
of Volumes II and II from Cambridge University Press had been sold by
1922 when Rudolf Carnap wrote to Russell asking for a copy. Russell
responded by sending Carnap a 35 page handwritten summary of the
definitions and some important theorems in the work (Linsky 2011:
14-15). As no plates were available for a second printing,
Russell began the work of preparing a second edition that appeared in
1925-27. The first was reset along with a new introduction and
three appendices, and Volume II was reset as well. Volume III was
reproduced by a photographic process, and so the page numbers from the
first edition are the same in this volume. *Principia
Mathematica* is still in print with Cambridge University Press.


As
with many works in mathematics, the later progress of the field of
symbolic logic led to numerous improvements. Work in the
school of logic started by David Hilbert at Gottingen and in the
Polish school of logicians led by S. Lesniewski and his most famous student, Alfred Tarski, began with correcting
what they saw as defects and gaps in PM. See Kahle (2013) and Wolenski (2013). The criticisms were immediate,
begun by Chwistek (1912) soon after the first volume had been
published. A series of important new presentations of mathematical
logic, in particular Hilbert and Ackermann (1928), Hilbert and Bernays
(1934), and Kleene (1952), were adopted as text books by successive
generations of logicians. As pointed out in Urquhart (2013) this lead
to a slow decline in the number of references to PM in technical work
in logic, as well as its gradual replacement by other texts for the
Introduction to Symbolic Logic courses that soon became a staple
offering of university departments of philosophy. By the 1950s PM was
no longer used as a textbook, even in graduate courses. PM's
influence, then, was enormous from 1910 to 1950, with it now having
the status of a recognized classic that is unfamiliar to students of
logic, and even unreadable because of its superseded notation. This
entry, together with the entry on 
 the notation in *Principia Mathematica*,
 are intended to make the contributions of this monumental work
available, and to enable further research on some of the ideas hidden
in those three long volumes.


### 2.2 Significance of *Principia Mathematica*



Achieving *Principia's* main goal proved to be a
challenge. An initial response among mathematicians and logicians in
Germany and Poland was to decry the decline in standards of formal
rigor set by Frege. This complaint was voiced by Frege himself, in a
letter to Philip Jourdain in 1912:



> 
> 
> ...I do not understand the English language well enough to be
> able to say definitely that Russell's theory (*Principia
> Mathematica I*, 54ff) agrees with my theory of functions of the
> first, second, etc. levels. It does seem so. But I do not understand
> all of it. It is not quite clear to me what Russell intends with his
> designation \(\phi \bang \hat{x}\). I never know for sure whether he
> is speaking of a sign or of its content. (Frege 1980: 78)
> 
> 
> 



This claim that the notion of "propositional function" is
subject to use-mention confusions has persisted to this day. This
entry will present a modernized version of the syntax of PM, combined
with an account of the notation for types in the works of Alonzo
Church (1974, 1976). Modern theories of types allow for a coherent
syntax for higher-order languages which many find adequate to meet
these objections. The complaint about the formulation of the syntax of
PM was repeated and a further difficulty was expressed by Godel
(1944) in his influential survey of PM:



> 
> 
> It is to be regretted that this first comprehensive and thorough-going
> presentation of a mathematical logic and the derivation of mathematics
> from it [is] so greatly lacking in formal precision in the foundations
> (contained in [?]1-[?]21 of
> Principia) that it presents in this respect a considerable step
> backwards as compared with Frege. What is missing, above all, is a
> precise statement of the syntax of the formalism. Syntactical
> considerations are omitted even in cases where they are necessary for
> the cogency of the proofs, in particular in connection with the
> "incomplete symbols". These are introduced not by explicit
> definition, but by rules describing how sentences containing them are
> to be translated into sentences not containing them. To be sure,
> however, that (or for what expressions) this translation is possible
> and uniquely determined and that (or to what extent) the rules of
> inference apply to the new kind of expressions, it is necessary to
> have a survey of all possible expressions, and this can be furnished
> only by syntactical considerations. (Godel 1944 [1951: 126])
> 
> 
> 



The issue with respect to defined expressions, including the
"incomplete symbols" for classes and definite descriptions
which are explained below, is still problematic for interpreting PM.
The difficulty is that certain defined expressions such as the
notation for definite descriptions, class abstracts and even the
identity symbol '\(=\)', are not specified in the initial
description of the syntax of the theory, nor are they shown to be
validly used as instances of the axioms with their apparent syntax.
The method of "contextual definition" used in PM is
difficult to formulate rigorously and is not used in contemporary
logical theories. The modern presentation of PM in this entry includes
the symbols for descriptions and classes, thus differing from the
completely rigorous presentations of Church (1976), for example, who
avoids both definite descriptions and class expressions, and takes
identity as an undefined primitive.



Despite these reactions to the rigor of the presentation, PM
nevertheless was studied carefully by those interested in the new
symbolic logic including David Hilbert and those in his school in
Gottingen (see Ewald & Sieg 2013: 3 and Chwistek 1912).
Primarily at issue were the kinds of assumptions Whitehead and Russell
needed to complete their project. Although *Principia*
succeeded in providing detailed derivations of many major theorems in
finite and transfinite arithmetic, set theory, and elementary measure
theory, three axioms in particular were arguably non-logical in
character: the axioms of infinity, reducibility and the
"multiplicative axiom" or Axiom of Choice. The axiom of
infinity in effect states that there exists an infinite number of
objects. Arguably it makes the kind of assumption generally thought to
be empirical rather than logical in nature. The multiplicative axiom,
later added to Zermelo's axioms as the Axiom of Choice, asserts
the existence of a certain set containing one element from each member
of a given set. Russell objected that without a rule guiding the
choice, such an axiom was not a logical principle. The axiom of
reducibility was introduced as a means of overcoming the not
completely satisfactory effects of the
 theory of types,
 the mechanism Russell and Whitehead used to restrict the notion of a
well-formed expression, thereby avoiding Russell's paradox.
Although technically feasible, many critics concluded that the axiom
was simply too *ad hoc* to be justified philosophically.
Initially at least, Leon Chwistek (1912) believed that it led to a
contradiction. Kanamori sums up the sentiment of many readers: 



> 
> 
> In traumatic reaction to his paradox Russell had built a complex
> system of orders and types only to collapse it with his Axiom of
> Reducibility, a fearful symmetry imposed by an artful dodger. (2009:
> 411) 
> 
> 
> 



In the minds of many, the issue of whether mathematics could be
reduced to logic, or whether it could be reduced only to set theory,
thus remained open.



In response, Whitehead and Russell argued that both axioms were
defensible on inductive grounds. As they tell us in the Introduction
to the first volume of *Principia*,



> 
> 
> self-evidence is never more than a part of the reason for accepting an
> axiom, and is never indispensable. The reason for accepting an axiom,
> as for accepting any other proposition, is always largely inductive,
> namely that many propositions which are nearly indubitable can be
> deduced from it, and that no equally plausible way is known by which
> these propositions could be true if the axiom were false, and nothing
> which is probably false can be deduced from it. If the axiom is
> apparently self-evident, that only means, practically, that it is
> nearly indubitable; for things have been thought to be self-evident
> and have yet turned out to be false. And if the axiom itself is nearly
> indubitable, that merely adds to the inductive evidence derived from
> the fact that its consequences are nearly indubitable: it does not
> provide new evidence of a radically different kind. Infallibility is
> never attainable, and therefore some element of doubt should always
> attach to every axiom and to all its consequences. In formal logic,
> the element of doubt is less than in most sciences, but it is not
> absent, as appears from the fact that the paradoxes followed from
> premisses which were not previously known to require limitations.
> (1910: 62 [1925: 59])
> 
> 
> 



Whitehead and Russell were also disappointed by the book's
largely indifferent reception on the part of many working
mathematicians. As Russell writes,



> 
> 
> Both Whitehead and I were disappointed that *Principia
> Mathematica* was only viewed from a philosophical standpoint.
> People were interested in what was said about the contradictions and
> in the question whether ordinary mathematics had been validly deduced
> from purely logical premisses, but they were not interested in the
> mathematical techniques developed in the course of the
> work....Even those who were working on exactly the same subjects
> did not think it worth while to find out what *Principia
> Mathematica* had to say on them. I will give two illustrations:
> *Mathematische Annalen* published about ten years after the
> publication of *Principia* a long article giving some of the
> results which (unknown to the author) we had worked out in Part IV of
> our book. This article fell into certain inaccuracies which we had
> avoided, but contained nothing valid which we had not already
> published. The author was obviously totally unaware that he had been
> anticipated. The second example occurred when I was a colleague of
> Reichenbach at the University of California. He told me that he had
> invented an extension of mathematical induction which he called
> 'transfinite induction'. I told him that this subject was
> fully treated in the third volume of the *Principia*. When I
> saw him a week later, he told me that he had verified this. (1959:
> 86)
> 
> 
> 



Despite such concerns, PM proved to be remarkably influential in at
least three ways. First, it popularized modern mathematical logic to
an extent undreamt of by its authors. By using a notation more
accessible than that used by Frege, Whitehead, and Russell managed to
convey the remarkable expressive power of modern predicate logic in a
way that previous writers had been unable to achieve. Second, by
exhibiting so clearly the deductive power of the new logic, Whitehead
and Russell were able to show how powerful the idea of a modern formal
system could be, thus opening up new work in what soon was to be
called metalogic. Third, *Principia Mathematica* re-affirmed
clear and interesting connections between logicism and two of the main
branches of traditional philosophy, namely
 metaphysics 
 and epistemology,
thereby initiating new and interesting work in both of these
areas.



As a result, not only did *Principia* introduce a wide range of
philosophically rich notions (including
 propositional function,
 logical construction, and
 type theory),
 it also set the stage for the discovery of crucial metatheoretic
results (including those of Kurt Godel, Alonzo Church, Alan
Turing and others). Just as importantly, it initiated a tradition of
common technical work in fields as diverse as philosophy, mathematics,
linguistics, economics and computer science.



Today a lack of agreement remains over the ultimate philosophical
contribution of *Principia*, with some authors holding that,
with the appropriate modifications, logicism remains a feasible
project. Others hold that the philosophical and technical
underpinnings of the project remain too weak or too confused to be of
great use to the logicist. (For more detailed discussion, readers
should consult Quine 1963, 1966a, 1966b; Landini 1998, 2011; Linsky
1999, 2011; Hale and Wright 2001; Burgess 2005; Hintikka 2009; and
Gandon 2012.)



There is also lack of agreement over the importance of the second
edition of the book, which appeared in 1925 (Volume I) and (Volumes II
and III were directly reprinted from the first edition in 1927). The
revisions were done by Russell, although Whitehead was given the
opportunity to advise. In addition to the correction of minor errors
throughout the original text, changes to the new edition included a
new Introduction and three new appendices. (The appendices discuss the
theory of quantification, mathematical induction and the axiom of
reducibility, and the principle of extensionality respectively.) The
book itself was reset more compactly, making page references to the
first edition obsolete. Russell continued to make corrections as late
as 1949 for the 1950 printing, the year he and Whitehead's widow
finally began to receive royalties.



Today there is still debate over the ultimate value, or even the
correct interpretation, of some of the revisions, revisions that were
motivated in large part by the work of some of Russell's
brightest students, including Ludwig Wittgenstein and Frank
Ramsey. Appendix B has been notoriously problematic. The appendix
purports to show how mathematical induction can be justified without
use of the axiom of reducibility; but as Alasdair Urquhart
reports,



> 
> 
> The first indication that something was seriously wrong appeared in
> Godel's well known essay of 1944, "Russell's
> Mathematical Logic". There, Godel points out that line (3)
> of the demonstration of Russell's proposition
> [?]89*16 is an elementary logical blunder, while the
> crucial [?]89*12 also appears to be highly questionable.
> It still remained to be seen whether anything of Russell's proof
> could be salvaged, in spite of the errors, but John Myhill provided
> strong evidence of a negative verdict by providing a model-theoretic
> proof in 1974 that no such proof as Russell's can be given in
> the ramified theory of types without the axiom of reducibility.
> (Urquhart 2012)
> 
> 
> 



Linsky (2011) provides a discussion, both of the Appendix itself and
of the suggestion that by 1925 Russell may have been out of touch with
recent developments in the quickly changing field of mathematical
logic. He also addresses the suggestion, made by some commentators,
that Whitehead may have been opposed to the revisions, or at least
indifferent to them, concluding that both charges are likely without
foundation. (Whitehead's own comments, published in 1926 in
*Mind*, shed little light on the issue.)


## 3. Contents of *Principia Mathematica*



*Principia Mathematica* originally appeared in three volumes.








![Title Page, First Edition, Principia Mathematica, Volume 1, see link below for text](pm1.jpg)

Title page of the first edition of *Principia Mathematica*, Volume I (1910)





![Cover of the first edition of Principia Mathematica to *56, see link below for text](pm2.jpg)

Cover of the first paperback issue of *Principia Mathematica to [?]56* (1962)









Together, the three volumes are divided into six parts. The commentary
that follows will go through the sections in order, indicating in
the early parts where a reader can skip ahead to study the unique
features of the development of mathematics in the PM system as
contrasted with that of Frege and contemporary set theory.


### 3.1 Volume I



Volume I is divided into a lengthy Introduction in three
sections, followed by two major Parts I (divided into Sections
A-E) and II (also divided into Sections A-E):


* Preliminary Explanations of Ideas and Notations
* The Theory of Logical Types
* Incomplete Symbols
* Part I: Mathematical Logic
	+ A. The Theory of Deduction
	[?]1-[?]5
	+ B. Theory of Apparent Variables
	[?]9-[?]14
	+ C. Classes and Relations
	[?]20-[?]25
	+ D. Logic of Relations
	[?]30-[?]38
	+ E. Products and Sums of Classes
	[?]40-[?]43
* Part II: Prolegomena to Cardinal Arithmetic
	+ A. Unit Classes and Couples
	[?]50-[?]56
	+ B. Sub-Classes, Sub-Relations, and Relative Types
	[?]60-[?]65
	+ C. One-Many, Many-One and One-One Relations
	[?]70-[?]74
	+ D. Selections [?]80-[?]88
	+ E. Inductive Relations
	[?]90-[?]97


### 3.2 Volume II



Volume II begins with a preliminary section on notational conventions
followed by Parts III (divided into Sections A-C), IV (divided
into Sections A-D), and the first half of Part V (Sections
A-C):


* Prefatory Statement of Symbolic Conventions
* Part III: Cardinal Arithmetic
	+ A. Definition and Logical Properties of Cardinal Numbers
	[?]100-[?]106
	+ B. Addition, Multiplication and Exponentiation
	[?]110-[?]117
	+ C. Finite and Infinite
	[?]118-[?]126
* Part IV: Relation-Arithmetic
	+ A. Ordinal Similarity and Relation-Numbers
	[?]150-[?]155
	+ B. Addition of Relations, and the Product of Two Relations
	[?]160-[?]166
	+ C. The Principle of First Differences, and the Multiplication and
	Exponentiation of Relations
	[?]170-[?]177
	+ D. Arithmetic of
	Relation-Numbers[?]180-[?]186
* Part V: Series
	+ A. General Theory of Series
	[?]200-[?]208
	+ B. On Sections, Segments, Stretches, and Derivatives
	[?]210-[?]217
	+ C. On Convergence, and the Limits of Functions
	[?]230-[?]234


### 3.3 Volume III



Volume III contains the remainder of Part V (Sections D-F) and
concludes with Part VI (divided into Sections A-D):


* Part V: Series (continued)
	+ D. Well-Ordered Series
	[?]250-[?]259
	+ E. Finite and Infinite Series and Ordinals
	[?]260-[?]265
	+ F. Compact Series, Rational Series, and Continuous Series
	[?]270-[?]276
* Part VI: Quantity
	+ A. Generalization of Number
	[?]300-[?]314
	+ B. Vector-Families [?]330-[?]337
	+ C. Measurement [?]350-[?]359
	+ D. Cyclic Families [?]370-[?]375



A fourth volume on geometry was begun but never completed (Russell
1959: 99).



Overall, the three volumes not only represent a major leap forward
with regard to modern logic, they are also rich in early
twentieth-century mathematical developments. To give one example,
Whitehead and Russell were the first to define a series as a set of
terms having the properties of being asymmetrical, transitive and
connected (1912 [1927: 497]). To give another, it is in
*Principia* that we find the first detailed development of a
generalized version of Cantor's transfinite ordinals, which the
authors call "relation-numbers". The resulting
"relation-arithmetic" in turn led to significant
improvements in our understanding of the general notion of structure
(1912: Part IV).



As T.S. Eliot points out, the book also did a great deal to promote
clarity in the use of ordinary language in the early part of the
twentieth century:



> 
> 
> how much the work of logicians has done to make of English a language
> in which it is possible to think clearly and exactly on any subject.
> The *Principia Mathematica* are perhaps a greater contribution
> to our language than they are to mathematics. (1927: 291)
> 
> 
> 



The book is also not without some self-deprecating humour. As
Blackwell points out (2011: 158, 160), the authors twice poke fun at
the length and tedium of the project's many logical derivations.
In Volume I, the authors explain that one cannot list all the
non-intensional functions of \(\phi \bang \hat{z}\) "because
life is too short" (1910 [1925: 73]); and in Volume 3, after
over 1,800 pages of dense symbolism, the authors end Part IV, Section
D, on Cyclic Families, with the comment,



> 
> 
> We have given proofs rather shortly in this Section, particularly in
> the case of purely arithmetical lemmas, of which the proofs are
> perfectly straightforward, but tedious if written out at length. (1913
> [1927: 461])
> 
> 
> 



Evidence that the humour originates more with Russell than with
Whitehead is perhaps found in not dissimilar remarks that appear in
Russell's other writings. Russell's comment when
discussing the axiom of choice, to the effect that given a collection
of sets, it is possible to "pick out a representative
arbitrarily from each of them, as is done in a General Election"
(1959: 92), is perhaps a case in point.



Readers today (i.e., those who have learned logic in the last few
decades of the twentieth century or later) will find the book's
notation somewhat antiquated. Readers wanting assistance are advised
to consult the entry on the
 notation in *Principia Mathematica*.
 Even so, the book remains one of the great scientific documents of
the twentieth century.


## 4. Volume I


### 4.1 Part I: Mathematical Logic


#### 4.1.1 Propositional Logic in PM



The system of *propositional logic* of PM, can be seen as a
system of *sentential logic* consisting of a language, and
rules of inference. PM contains the first presentation of symbolic
logic that deals with propositional logic as a separate theory. Frege
had involved quantification from the beginning, while Peano's
system was interpretable as about propositions and classes with some
different principles holding for each interpretation. The
propositional logic of PM is unusual for modern readers, for various
reasons having to do with its origins in Russell's earlier work
on logic. One is that the axioms of propositional logic are not stated
using only the primitive connectives of the logic, which are \(\lnot\)
and \(\lor\), but instead only use \(\lor\), and \(\supset\) which is
a defined connective.



In this section we will use *A*, *B*, etc as meta-linguistic
variables for formulas. The formulas constructed from atomic
propositions with the connectives are said to express *elementary
propositions* to distinguish them from propositions involving
quantifiers and propositional functions. The system is organized
axiomatically, the axioms, called "primitive propositions"
or "Pp", are presented with the characteristic
'\(\supset\)' of material implication, which is defined
with \(\lnot\) and \(\lor\). The connectives \(\amp\) and \(\equiv\)
are also defined, but not needed in the statements of the axioms. This
peculiarity has its origins in Russell's view from 1903 that




> 
> 
> The propositional calculus is characterized by the fact that all its
> propositions have as hypothesis and as consequent the assertion of a
> material implication. (1903: 13) 
> 
> 
> 



All of the "primitive propositions" of PoM are stated with
only material implication as a primitive connective. The connectives
\(\amp\),\(\lor\) and \(\equiv\), are defined as might be expected.
The notion of negation, expressed by \(\lnot\), is defined using a
notion of quantification over propositions ( \(\lnot A\) means that
*A* implies all propositions). By 1906 Russell had decided to use
\(\lnot\) as a primitive connective, and no longer used propositional
quantifiers, allowing \(\supset\) to be defined, while the primitive
propositions were still stated with \(\supset\) and \(\lor\). That the
system of propositional logic in PM was the result of an evolution of
changes in choices of primitives is mirrored in the choice of theorems
that are proved in the first chapters. While most are proved because
they will be used later in PM, some remain simply as remants of the
earlier systems. In particular PM contains several theorems that were
primitive propositions in earlier systems, though not used in what
follows. In fact one primitive proposition of PoM, known as
"Peirce's Law (\([(p \supset q )\supset p ] \supset p\))
appears to have been proved in an early version of PM as
[?]2*7 but simply deleted (and its number not
reassigned to another theorem) simply to save space (see Linsky
2016).



The notion of truth-functional *semantics* for propositional
logic, using the familiar truth tables, and the notion of
*completeness* of an axiom system, was not developed until soon
after the publication of PM by Bernays (1926). As a result there is no
attempt to find a short list of axioms that will be complete, and so
at later stages of the work there is no simple appeal to
"tautological consequences" which might be easily
justified by semantic considerations.



The language of propositional logic in PM consists of a vocabulary
consisting of:


* Atomic proposition variables: *p*, *q*, *r*,
\(p\_1\), ... (There are no proposition constants.)
* Sentential connectives. Primitive: \(\lnot\) and \(\lor\).
Defined: \(\supset\), \(\amp\), \(\equiv\).
* Punctuation: \((\), \()\), \([\), \(]\), \(\{\), \(\}\), etc.



The well formed formulas (*wffs*) are defined as follows:


* Atomic proposition variables are *wffs*.
* If *A* and *B* are *wffs* then so are: \(\lnot
A\) and \(A \lor B\)



The other familiar connectives are defined:


* Definitions 

\[\begin{align}
\tag\*{[?]1*01} {A \supset B} & \eqdf \lnot {A \lor B}\\
\tag\*{[?]3*01} {A \amp B} & \eqdf {\lnot(\lnot A \lor \lnot B})\\
\tag\*{[?]4*01}({A \equiv B} & \eqdf {(A \supset B) \amp (B \supset A)}\\
 \end{align}
 \]
* Axioms 

\[\begin{align}
\tag\*{Pp [?]1*2} (p \lor p ) & \supset p\\
\tag\*{Pp [?]1*3} q & \supset (p \lor q )\\
\tag\*{Pp [?]1*4} (p \lor q ) & \supset (q \lor p)\\

\tag\*{Pp [?]1*5} [p \lor ( q \lor r )] & \supset [q
\lor ( p \lor r )]\\

\tag\*{Pp [?]1*6} ( q \supset r ) & \supset [ (p \lor q
) \supset (p \lor r ) ]\\ \end{align}\]



In 1926 Paul Bernays showed that this could be reduced by one, as
axiom 4 ([?]1*5) can be proved from the
others.


* Rules of inference:
	+ *Modus ponens* ([?]1*1): From
	\(\vdash A \supset B\) and \(\vdash A\), derive \(\vdash B\)
	+ *Substitution*: From \(\vdash A\) derive \(\vdash A'\)
	where \(A'\) is the result of substituting some formula *B*
	uniformly for any atomic proposition variable that occurs in
	*A*.



There is no explicit statement of a rule of substitution in PM. The
free variables in the propositional logic of PM may be interpreted as
schematic letters, and so the system will require a rule of
substitution of formulas. In this article they are to be interpreted as
real variables ranging over propositions, in which case instances
would be derived by instantiation from generalizations over all
propositions. The announcement in the Introduction that propositions
are not necessary in what follows and so will be avoided suggests the
schematic interpretation of the variables. We follow the variable
interpretation in this article, however, in part to allow our notation
to follow PM, with *p*'s and *q*'s rather than a
new vocabulary of schematic letters *A*, *B*, etc. This
interpretation of the letters as variables will also assist in the
presentation of quantificational logic in PM below.



As is standard for an axiomatic formulation of logic, a
*derivation* of a formula of sentential logic in PM will
consist of an instance of one of the six axioms, the result of a
substitution in a preceding line, or the application of modus ponens
to two preceding lines. Theorems of PM will be proved in order,
allowing the use of (instances of) preceding theorems as lines in
later derivations.



The resulting system is complete, in the sense that all and only
truth-functionally valid sentences are derivable in the system. This
despite the seeming defects of the system by modern standards,
including the redundancy of one of the axioms, the use of defined
symbols in expressions to which the rules of inference apply, and the
use of defined symbols in the axioms. The derivations in
[?]2 to [?]5 are abbreviated, but
with an indication on the side of each line of what justifies it, and
how any abbreviation can be undone. Theorems are proved primarily as
needed in later numbers, but some were axioms, or important theorems
of earlier versions of propositional logic, going back to *The
Principles of Mathematics*. Aside from historical interest in
their actual choices, however, the system of PM can be viewed as based
on any standard system of propositional logic.


#### 4.1.2 The "Ramified" Theory of Types



The theory of types in the initial chapters of PM is
*ramified*, so that within a given type, of propositions, or of
functions of individuals, and functions of functions of individuals,
there will be finer subdivisions. This ramification is necessary for
the application of the logic of PM to what what are called
"epistemological" paradoxes in the Introduction to PM. The
most prominent of these is the (propositional) Liar paradox created by
the proposition that all propositions of a certain sort, say asserted
by Epimenides, are false, when that very proposition is of that sort,
that is the only proposition that Epimenides asserted. The solution in
the ramified theory of types requires that a proposition about a sort
of first level propositions, say that they are all false, will itself
be of the next order.



The paradoxes of the theory of sets are resolved by reducing
assertions about sets to assertions about propositional functions. The
restriction that a function of one type cannot apply to a function of
the same type is enough to block the paradoxes. Thus the distinction
between individuals, functions of individuals, and functions of such
functions, categorized by what came to be called "simple theory
of types" is enough for the purposes of reducing mathematics to
classes, and so to logic. The idea that the full theory of types was
not needed to resolve the mathematical or set theoretical paradoxes
was proposed by Chwistek (1921) and Ramsey (1931), and led to the
later introduction of the terms "ramified theory of types"
and "simple theory of types" that will be used in this
entry.



In the Introduction to PM terminology is introduced for the two ways
that variables may appear in formulas. The "apparent
variables" are *bound variables*, whereas "real
variables" are *free variables*. The proper
interpretation of higher-order variables in PM is the subject of
contemporary dispute among scholars of PM. Landini (1998) and Linsky
(1999) offer two rival accounts. Landini holds that higher-order free
variables should be interpreted as schematic letters, replaceable by
formulas, and that the bound variables are to be interpreted
"substitutionally". The logic of the theory of types in PM
can be seen as an extension of a theory of a standard first order
logic developed in [?]10. Then the more distinctive
notions of PM that depend on the theory of types can be explained.
These include the Axiom of Reducibility, in [?]12
which underlies the so called *ramification* of the theory of
types, the division into *orders* of predicates true of a
single type of argument. The Axiom of Reducibility asserts that for an
arbitrary function of any order there is an equivalent
*predicative* function, that is, one true of exactly the same
range of arguments. This will be explained below. Identity is defined,
in [?]13, with a version of Leibniz' notion of
the identity of indiscernibles that is consistent with the theory of
types. Replacing Leibniz' notion that *x* and *y* are
identical just in case they share the same properties, in PM, *x*
and *y* are identical if and only if they share the same
predicative functions. Then using the notion of identity so defined,
PM presents Russell's theory of definite descriptions, precisely
as it was defined in "On Denoting" (1905). This article
will use the notation for "*r*-types" due to Alonzo
Church 1976, which is explained in the accompanying article "The
Notation of *Principia Mathematica*" in this
Encyclopedia.



Although PM does not single out first order logic from the whole
ramified theory of types, the actual deductive apparatus on the page
looks exactly like a system of first order logic, and the
complications of the logic of higher types can be expressed with an
additional apparatus of type indices. In what follows we will use the
system of *r-types* in Church (1976) for type indices, and the
use of *lambda operators* for propositional functions.





---



**Church's (1976) formulation of the logic of PM with
*r-types*** The language of the higher-order
quantificational logic of PM is called *ramified type theory*,
and the system of types, following Church (1976) will be called
*r*-types. Note that there are two kinds of variables, but they
are all assigned to an r-type. Individual variables behave as a
special case of propositional function variables.


* (argument) variables: \(x\_{\mathbf{\tau}}\),
\(y\_{\mathbf{\tau}}\), \(z\_{\mathbf{\tau}}\), ...for each type
\(\tau\)
* *n*-place propositional function variables:
\(\phi\_{\tau}^{n}, \psi\_{\tau}^{n}\), ...(\(n \geq 1\)), where
\(\tau\) is a type symbol. (\(R^n,S^n, \ldots\) (\(n \geq 1\)) for
*relations in extension*.) \(\chi\) is used for a higher-order
function of functions, as in \(\chi (\phi)\), and \(\Phi\) for the
next order, as in \(\Phi (\chi)\)
* connectives: \(\lnot\), \(\lor\)
* punctuation: \((\), \()\), \([\), \(]\), \(\{\), \(\}\), etc.
* the quantifier symbols: \(\forall\) and \(\exists\).
* the lambda symbol: \(\lambda\)



The system of symbols for *r*-types and the assignment of
*r*-types to variables for different entities (individuals and
functions) is as follows:


* \(\iota\) is the *r*-type for an *individual*.
* Where \(\tau\_1 , \ldots, \tau\_m\) are any *r*-types, then
\((\tau\_1 , \ldots, \tau\_m) / n\) is the *r*-type of a
propositional function of *level n*; this is the *r*-type
of any *m*-ary propositional function of *level* *n*,
which has arguments of *r*-types \(\tau\_1 , \ldots, \tau\_m\),
respectively.



The *order* of an entity is defined as follows:


* the *order* of an individual (of *r*-type \(\iota\))
is 0
* the *order* of a function of *r*-type \( (\tau\_1, \ldots,
\tau\_m ) / n \) is \(n+N\) where *N* is the greatest of the orders of
the arguments \(\tau\_1 \ldots, \tau\_m\)



There are no predicate or individual names in this language. There
are, however complex terms for propositional functions, defined
together with formulas (with the usual notion of bound and free
variables):



Let the expressions \(\phi\_{\tau}\) be variables ranging over propositional
functions of type \(\tau\). We read \(x\_{\tau}\) as a metalinguistic
variable ranging over variables of *r*-type \(\tau\). The
subscript \(\tau\) will be indicated only with the initial quantifier
which governs the variable.



We then can define the *well formed formulas* (wffs) and
*terms* of quantificational logic as follows:


* Variables (for individuals and propositional functions) are
terms.
* If \(\phi\_{\tau}^{n}\) is an *n*-place propositional function
variable of *r*-type \((\tau\_1 \ldots, \tau\_n) /k\) and
\(x\_{i} \) for \( ( 0 \leq i \leq n ) \) are terms of of *r*-types \(\tau\_1 \ldots,
\tau\_n\), respectively, then \(\phi (x\_1, \ldots x\_n)\) is a
*wff*.



(The variables \(x\_n\) are called "argument" variables.
They will include *individual variables* of *r*-type
\(\iota\), but also variables of higher types. The variable \(\phi\)
can occur as a predicate in \(\phi(x)\) and as an argument in \(\Psi
(\phi)\), and cannot be of type \(\iota\) to occur in a
*wff*.)
* If *x* is a variable of type \( \tau \) and *A* is
a *wff* then \(\lambda x A\) is a *term*
of *r*-type \((\tau)/n\) (where *n* is one more than the
highest order of any bound variable in *A* and at least as high
as the order of any free variable in *A*).
* If *x* is an individual variable of type \(\tau\) and *A* is
a wff in which *x* occurs \(free\) then \(\forall x A\) and
\(\exists x A\) are wffs.
* If *A* and *B* are wffs, then so are \(\lnot A\), \(A \amp
B\), \(A \lor B\), \(A \supset B\), and \(A \equiv B\).



The conventional precedence ordering of connectives will allow for
fewer punctuation signs to indicate scope of connectives, thus \(A
\lor B \supset C\) is read as \(( A \lor B) \supset C\)



The comprehension principle for a system of higher-order logic, or set
theory, states which formulas express a property or set. Within a type
theory this allows for what looks like an "unrestricted"
comprehension principle, in that for every well formed expression
*A* with a free variable, *x*, there is a property which is
satisfied by precisely the entities satisfying the formula. It is the
restrictions of types that block the paradoxes, as the problematic
formulas "is not a member of itself" and "does not
apply to itself" are ruled out by the system of types. The
comprehension principle then is characterized by an infinite set of
sentences of the form of:




Comprehension: 

 

\[\exists \phi \forall x\_{\tau} [ \phi (x) \equiv A], \quad (\phi \textrm{ not free in } A)\]


where \(\phi\) is a functional variable of *r*-type \((\tau)/n\)
and *x* is a variable of *r*-type \(\tau\), and the bound
variables of *A* are all of order less than the order of \(\phi\)
and the free variables of *A* are all of order not greater than
the order of \(\phi\).





---




As presented here Church's seemingly straightforward
comprehension principle, with its restrictions on the types of
variables, is for Quine a glaring manifestation of the confusion of
use and mention of language that he sees infecting PM:



> 
> 
> ...there is a characteristic give and take between sign and
> object: the propositional function gets its order from the abstractive
> expression, and the order of the variable is the order of the values.
> Exposition is eased by allowing the word 'order' a double
> sense, attributing orders at once to the notations and, in parallel,
> to their objects. (Quine 1963: 245)
> 
> 
> 



The offense comes from attributing orders (*r*-types) to
propositional functions on the basis of the variables with which they
are defined, but also to the functions themselves, as simply values of
bound higher-order variables. In response, the defender of type theory
must say that any semantic intrepretation of the notion of
propositional function will have to attribute to functions these
distinctions that are marked in linguistic expressions of some of
them, and in particular, the variables involved in their
definition.



What follows in PM up to [?]12 is a presentation of
quantificational logic in the ramified theory of types. The
complications are due to the decision of the authors (surely on
Russell's insistence) to add a new section
[?]9 which allows the earlier theory of
propositional logic to be incorporated directly into a
quantificational logic as is done in contemporary logic. This shows
the extent to which the earlier theory is indeed a theory of
*propositions*, not an account of a fragment of
quantificational logic allowing open sentences containing free
variables.


##### Quantificational Logic in PM



Section [?]10 formulates quantificational logic as
it is currently formulated, namely the axioms and theorems of
propositional logic are assumed to hold for all formulas, and not just
the elementary propositions of [?]1-5. It
appears that Russell became concerned about this assumption, and so a
new section [?]9 was introduced to derive the
principles of quantification theory from elementary propositions
alone. While of interest to scholars of PM, the upshot is the same for
later uses of quantificational logic in PM.



Again, the reader interested in what distinguishes the logicist
project in PM can skip this section, although passing attention may be
paid to the system of higher-order logic that is used, as based here
on the *ramified theory of types*. 



The extension to functions of more than one variable is obvious, and
below, some applications will employ this extension.



The existential quantifier and the other familiar connectives
\(\supset\), \(\amp\) and \(\equiv\) are defined as for propositional
logic. (In what follows *A* is now an arbitrary (possibly
quantificational) formula):

 

\[
\tag\*{[?]10*01} \exists x A \eqdf \lnot \forall x \lnot A
\]

* Axioms of [?]10: All instances of propositional
theorems where wffs are uniformly substituted for propositional
variables.



The system of PM uses a rule of Universal Generalization and an Axiom
which amounts to a rule of Instantiation.

 

\[
\tag\*{[?]10*1} 
\vdash \forall x\_{\tau} A \supset A'
\]


where \(A'\) is like *A* except for having a term *y* of
type \(\tau\) substituted for \(x\_{\tau}\) in *A*.



(Note: The notion of suitable "substitution" is much more
complicated for logic of higher types than it is for first order
logic. In part this is because of the application to an argument of
lambda expressions for a propositional function, e.g., \([\lambda x
\phi (x)] (\nu)\) where \(\nu\) may be a complex term involving
variables and quantifiers in other lambda expressions.)



\[
\tag\*{[?]10*11}
 \textrm{If } \vdash A \textrm{ then } \vdash \forall x\_{\tau} A'
\]


where \(A'\) is like *A* except for having a term *y* of
type \(\tau\) substituted for *x* in *A*



Other quantifier principles, which govern the move of a quantifier
from the inside of a formula to governing the entire formula, so
called "quantifier containment principles" are also
derived as theorems in [?]10. Some that are often used in later
numbers are:

 

\[
\begin{align}

\tag\*{[?]10*12} \forall x\_{\tau} ( A \lor \phi (x ) )
& \supset (A \lor \forall x\_{\tau} \phi (x ))\\

\tag\*{[?]10*21} \forall x\_{\tau} [ A \supset \phi (x ) ]
& \equiv [A \supset \forall x\_{\tau} \phi (x)]\\ 

\end{align}\]


The introduction to [?]10 in PM begins with:



> 
> 
> The chief purpose of the propositions of this number
> [[?]10] is to extend to formal implications
> (*i.e.* to propositions of the form \(\forall x (\phi x \supset
> \psi x)\) as many as possible of the propositions proved previously
> for material implications, *i.e.* for propositions of the form
> \(p \supset q\). (notation updated)
> 
> 
> 



In other words, this section introduces the logic of quantification,
in a way that is familiar to contemporary logic. The propositional
logic of the preceding sections is interpreted as true only of
elementary, first order propositions, and so extended to higher-order
logic by showing how sentences can be presented in "prenex
form", that is with quantifers in intial position preceding a
quantifier free matrix. These theorems are familiar now as
"quantifier confinement" theorems, of the form of:



\[
\tag\*{[?]10*23} \forall x\_{\tau} [ \phi (x ) \supset A ] \equiv [ \exists x\_{\tau} \phi (x)] \supset A 
\]

##### The Axiom of Reducibility



Given that the system of PM contains a ramified theory of types,
however, the move to discussion of classes for the remainder of the
work after [?]20 requires a further axiom, the axiom
of reducibility, in order to allow a simple theory of types of
classes. Consider the fundamental notion from the theory of real
numbers of the *least upper bound* (*l.u.b.*) of a
bounded class of real numbers. Consider the class of all real numbers
whose square is less than or equal to 2, i.e., \(\{ x \mid x^2 \leq
2\}\). A class of reals *S* has an *upper bound* if and
only if \(\exists r \forall s ( s \in S \supset s \leq r)\). If a
bounded class *S* of real numbers has members of some
*r*-type \(\tau\), then the least upper bound must belong to an
*r*-type \(\tau / 1\) because of the quantifier in the definition
ranging over the elements *s* of *S*. We say that the
definition of *S* is "impredicative" because it
involves quantification over a totality to which it is intended to
belong. The theory of real numbers, however, requires that sometimes
the least upper bound of a class is a member of that class, in this
case, the least upper bound of *S*, namely \(\sqrt{2}\), is an
element of *S*.



The resolution of this in the system of PM is to adopt an axiom which
guarantees that any class defined in terms of another class will be of
the same type. Thus impredicative definitions of classes are allowed,
and do not introduce a class of a higher type. This is accomplished by
adopting the Axiom of Reducibility, in [?]12, which
guarantees that for any function \(\phi\), there will be a
co-extensive *predicative* function. More precisely, the Axiom
of Reducibility asserts that for any function of any number of
arguments of an arbitrary level, there is an equivalent function of
level 1, ie. one true of the same entities:




Axiom of Reducibility, 

 

 \[
 \tag\*{Pp [?]12*1}
 \forall \psi \exists \phi \forall x\_{\tau} [\psi (x) \equiv \phi \bang(x) ]
 \]


where \(\phi \bang\) is a *predicative function*.




The exclamation mark "\(\bang\)" is used in PM to indicate
*predicative functions*. In Church's system of
*r*-types this is expressed by saying that the variable *x*
is of *r*-type \(\tau\) and \(\phi\) is of *r*-type \((\tau
)/1\) and \(\psi\) is of *r*-type \((\tau) /n\). In other words,
\(\phi\) is of the lowest order compatible with its arguments. This
notion of predicative functions is taken from the Introduction. In
[?]12 Whitehead and Russell propose a narrower
conception of predicative function, by which \(\phi\) must be a
*matrix*, or function in the definition of which no quantifiers
at all appear. See the accompanying entry on 
 the notation in *Principia Mathematica*.




It has seemed to some, beginning with Chwistek (1912) and continuing
through Copi (1950) that the Axiom of Reducibility is technically
faulty, leading to an inconsistency, or at least redundancy in the
system of PM. Ramsey (1931) early on argued that the supposed
contradiction in fact demonstrated that certain predicative functions
are indefinable. Church (1976), confirms this assessment, and uses the
presentation of *r*-types we describe here to show rigorously the
limitations on what functions are definable in the system of PM.



The interaction of this Axiom with the theory of classes in PM will be
explained below in connection with [?]20 on
classes.


##### Identity in PM



Contemporary logic follows Frege in treating *identity*,
represented by \(=\), as a logical notion. In PM the notion of
identity is defined following Leibniz as indiscernibility, namely
indiscernible objects are identical. That is, \(\forall \phi ( \phi x
\equiv \phi y) \supset x = y\). But since the axiom of reducibility
guarantees that if there is any type of function on which *x* and
*y* differ, they will differ on some predicative function, PM
uses the following definition of identity:

 

\[
\tag\*{[?]13*01}
x\_{\tau} = y\_{\tau} \eqdf \forall \phi [ \phi \bang (x) \supset \phi \bang(y) ],
\]


for \(\phi \bang\) a predicative function.



In contemporary systems of logic an axiom or rule of inference allows
that if \(x = y\), then for any predicate \(\phi\), \(\phi x \equiv
\phi y\). In other words, identicals are indiscernible. The given
definition of identity only suffices if it is not possible that
entities *x* and *y* which share all predicative properties,
cannot be distinguished by some property of a higher order. The axiom
of reducibility guarantees that *x* and *y* sharing
properties of any given higher order will entail sharing predicative
properties, and so by the definition of identity, \(x =
y\).



In the appendix B to the second edition of PM, which was written by
Russell, there is a technical discussion of the consequences of
abandoning the axiom of reducibility. A faulty proof is proposed to
show that the principle of Induction can be derived without using the
axiom of reducibilty in a modified theory of types (see Linsky 2011).
As Russell points out, however, it is not possible to define real numbers using "Dedekindian" classes
of rational numbers without assuming the axiom of reducibility. (The
thesis that every class of reals with an upper bound has a real number
as its least upper bound, discussed above, would not be provable.) As
a result, Russell says "analysis would collapse". In all
of this discussion, however, Russell does not indicate what would
replace the definition of Identity in [?]13, which
so crucially depends on the axiom of reducibility.


##### Definite Descriptions



Russell presented his theory of definite descriptions in "On
Denoting" (1905) and it has probably been the most widely
discussed application of the logic of PM. The role of the theory of
definite desciptions in PM, however, is exhausted by its use in
[?]30 to define what are called "Descriptive
functions". In contemporary logic it is routine to show how the
notion of a "functional relation" can be used to justify
the introduction of function symbols into a language with only
*n*-place predicates. The theory of definite descriptions is
essential for this argument. After [?]30 there are
only a handful of occurrences of description operators in PM. What is
perhaps Russell's most valuable contribution to philosophical
logic and the philosophy of language, is, here, only a device used for
a technical, though programatically important, purpose. The technical
purpose, however, does indicate an important distinction between the
logicism of Frege and Russell. Frege's logic is based on the
notion of concept, which is a case of a function from objects to truth
values. Russell's logic can be seen as further reducing the
mathematical notion of function to his logical notion of
*propositional function*. Some logicians firmly in the
tradition of mathematical logic do not find this to be an advance, but
it does indicate a significant difference between the approaches of
Frege and Russell (see Linsky 2009).



Definite descriptions are expressions of the form "the
\(\phi\)" which occur in the position of terms apparently as the
arguments of functions. Russell's example from "On
Denoting" (1905) is the expression "The present King of
France" which apparently occurs as an argument to the function
"is bald" in the sentence "The present King of
France is bald". In general the expression "the \(\phi\)
is \(\psi\)" is defined as equivalent to the expression
"There is exactly one \(\phi\) and it is \(\psi\)":




Contextual definition of Definite Descriptions

 



\[ 
 \tag\*{[?]14*01}
 \psi ( \imath x \phi (x) ) \eqdf \exists x \forall y \{ [ \phi ( y) \equiv y = x ] \amp \psi (x)\}
\]



The use of the expression \(\eqdf\) which makes it appear that both
flanking expressions are terms, disguises the fact that in this case
of a "contextual definition" what occurs on each side 
are formulas, the right hand side replacing the left hand side, thus
"eliminating" the definite description.



To distinguish the two readings of the expression "The present
King of France is not bald", according to the
"scope" of the description (with respect to negation), PM
uses a "scope indicator" \( [ \imath x \phi ( x )]\) before
the formula from which the description is to be eliminated by the
definition above. Symbolize "The present King of France"
as \(\imath x K(x)\) and "*x* is bald" as \( B(x)\), the two
readings will be symbolized as:

 

\[[\imath x K(x)] \lnot B(\imath x K(x)),\] 


which, eliminating the description by definition, becomes: 



\[ \exists x \forall y \{ [K(x) \equiv y = x ] \amp \lnot B(x) \} \]


which is the reading on which there is exactly one present King of
France and he is not bald, and:

 

\[\lnot [\imath x K(x)]B(\imath x K(x)),\]


which, eliminating the description by definition, becomes:



 \[ \lnot ( \exists x \forall y \{ [K(x) \equiv y = x ] \amp B(x) \} )\]


The latter is the reading on which it is not the case that there is
one and only one present King of France and he is bald. That may be
true if there is not exactly one present King of France, as is
actually the case, as France has no King. In such a case the
description is not "proper", which is expressed with a
special symbol in PM, \(E\bang\), defined as:




*proper description*

 

\[
\tag\*{[?]14*02}
E\bang (\imath x \phi (x)) \eqdf \exists x \forall y [ \phi (y) \equiv y = x ]
\]






In theorem [?]14*3 we find one of the rare
occurrences of bound variables ranging over propositions *p* and
*q* of functions that are not predicative. (Suppose that *p*
and *q* are of some *r*-type \(()/n\) and *f* is a
function of those propositions, *f* might have *r*-type
\((()/n)/m\) for \(m, n > 1\)). Here we also see an occurrence of a
formula \(\imath x \phi (x)\) in subject position expressing a proposition as
an argument of such a function. These expressions do not figure in
theorems later in PM and only occasionally in the introductory
material of some sections. Theorem [?]14*3
asserts that in truth-functional contexts the scope of a (proper)
description does not effect the truth value of a proposition in which
it occurs:

 

\[
\tag\*{[?]14*3}
\begin{align}
\{ \forall p \forall q [ ( p \equiv q ) \supset (\Phi(p) \equiv \Phi(q) )] \amp E\bang ( \imath x \phi(x) ) \} 
\supset \\
\{ \Phi [\imath x \phi (x) ]\chi(\imath x \phi (x) )\equiv [\imath x \phi (x)] \Phi(\chi(\imath x \phi (x ))) \}
\end{align}
\]


This theorem is another indication of the way in which the
philosophical basis of PM, with its propositional functions that are
intensional is left behind as the mathematical content of PM is
introduced with the definition of classes in the next sections.


##### The "No-Classes" Theory of Classes



The theory of sets (*classes*) in PM is based on a number of
contextual definitions, similar in some ways to the theory of
descriptions. In what follows we will occasionally use the expression
"class" for the PM notion, to remind the reader of the
differences between this and an axiomatic theory of sets, such as ZF,
not to indicate that these are "proper classes" in the
sense used in ZF or VGB class theory, to indicate an expression that
does not define a set, such as \(\{ x \mid x = x \}\), which is true
of the *universe* \(\rV\) and so too "large" to be
a set.



The basic definition eliminates terms for classes from contexts in
which they occur, just as the theory of definite descriptions
eliminates descriptions occuring in the positions of terms:




Contextual definition of classes

 

\[
 \phi \{x \mid \psi (x) \} \eqdf \exists \chi 
\left [ 
\begin{split}
\forall x [ \chi \bang (x) \equiv \psi (x) ] \\
{} \amp \phi (\lambda x \chi (x))
\end{split}
\right] 
\tag\*{[?]20*01}
\]


for \(\chi \bang\) a predicative function




In other words, an expression seeming to attribute the property
\(\phi\) to a class \(\{x \mid \psi (x) \}\) is true if and only if
there is some predicative property \(\chi\), which is co-extensive
with \(\psi\), which really has the property \(\phi\).



The notion of membership (\(\in\)) which is the one non-logical
relation symbol of ZF, is defined in the PM system:




Definition of \(\in\) 

 

\[
\tag\*{[?]20*02}
x \in \phi \eqdf \phi \bang (x)
\]


for \(\phi\) a predicative function.




The principal role of this "no-classes" theory of classes,
as it is called, is to show how the theory of types resolves the
paradoxes that had afflicted the naive theory of classes in *The
Principles of Mathematics* and was seen by Russell to afflict
Frege's theory. After these foundational sections, all the
individual variables that appear in PM should be seen as ranging over
classes, (and, as will be explained below, the relation symbols are to
be interpreted as ranging over relations in extension). The paradoxes
appear in different forms, as seen in the Introduction to PM, but the
resolution of the paradox of "the class of all classes which do
not belong to themselves", which appears in Russell's
intial letter to Frege, will be used as our example. This class, which
leads directly to a contradiction, would appear in contemporary
notation as \(\{ x \mid x \notin x \}\). The paradox arises when one
asks whether that class is a member of iteself or not. The expression
that it is a member of itself \(\{ x \mid x \notin x \} \in \{ x \mid
x \notin x \}\) will have two class expressions to be eliminated by
the first definition, and then several uses of the relation symbol
\(\in\) which will also be eliminated. In the end there will be an
expression \(\lnot (\phi\_{\tau} \in \psi\_{\tau})\), which is not
legitimate, since this is not well-formed for any \({\tau}\). A
function must be of a higher order than its arguments.



The effect of these two definitions is to demonstrate that classes
fall into a simple theory of types, and while subject to these type
restrictions, all of the inferences involving class expressions
observe classical quantification theory as stated in
[?]10 above. The definitions of existential and
universal quantification are simple. Note that Russell uses Greek
letters (\(\alpha\), \(\beta\),...) to range over classes:




Definition of quantification over "all classes"



\[
\tag\*{[?]20*07}
\forall \alpha \chi (\alpha) \eqdf \forall \phi \chi ( \{ x \mid \phi \bang(x) \})\]


for \(\phi \bang\) a predicative function.



Definition of quantification over "some classes" 



\[
\tag\*{[?]20*071}
\exists \alpha \chi ( \alpha ) \eqdf \exists \phi \chi \{ x \mid \phi \bang(x) \}\] 


for \(\phi \bang\) a predicative function.




The definition of \(\in\) is extended to classes without change:




Definition of membership of a class in a function

 

\[ 
\tag\*{[?]20*07}
\alpha \in \psi \eqdf \psi \bang ( \alpha)
\]


for \(\psi \bang\) a predicative function.




The remainder of [?]20 consists of theorems proving
that the theorems of quantificational logic developed in
[?]10 apply as well to expressions about classes,
with the "Greek" variables \(\alpha, \beta, \ldots\) in
the place of individual variables \(x, y, \ldots\). Because formulas
with Greek variables look and behave the same as individual variables
with respect to quantificational logic, it is possible to overlook the
interaction of the theory of classes with the theory of types. As
Godel points out in the passage quoted above (Godel 1944
[1951: 126]), the "contextual definitions" of class
variables \(\alpha\), \(\beta\), etc., does not specify the
elimination of class abstracts from all possible contexts, and in
particular those that talk about classes. Linsky (2004) argues that PM
has no notation for classes of propositional functions to distinguish
them from classes of classes, although one could be added. This is
another indication of the turn in PM after the intial sections (up to
[?]21) to an extensional system of classes and
relations.



In effect the class variables can be seen as propositional function
variables, restricted to *r*-types in which only predicative
functions appear, in arguments as well, leading to what might be seen
as "hereditarily predicative functions". In other words,
the class variables can be replaced with propositional function
variables in which the *r*-type of the function, and of all the
arguments are of the form \((\beta\_1, \beta\_2, \ldots, \beta\_m )/1\)
and the same applies to \(\beta\_1, \beta\_2, \ldots, \beta\_m\) as well.
This means that variables and terms for classes will obey the
*simple theory of types*. These can be contrasted with
*r*-types by presenting an alternative system of simple types or
"*s*-types".




**Church's (1974) "Simple" Theory of
Types**


* \(\iota\) is the *s*-type for an *individual*.
* Where \(\tau\_1 \ldots, \tau\_m\) are any *s*-types, then
\((\tau\_1 \ldots, \tau\_m)\) is the *s*-type of a propositional
function of a *m*-ary propositional function which has arguments
of *s* types \(\tau\_1 \ldots, \tau\_m\), respectively.



The *order* of an entity in the system of *s*-types is
defined as follows:


* the *order* of an individual (of *r*-type \(\iota\))
is 0
* the *order* of a function of *r*-type \(\tau\_1 \ldots,
\tau\_m\) is \(n+1\) where *n* is the greatest of the order of the
arguments \(\tau\_1 \ldots, \tau\_m\)



Church's notion of "order" is not quite one that is
familiar from talk of "first order logic" and
"second order logic". First order logic will have bound
variables of *s*-type 0, and a logic which quantifies over
variables of *s*-type 1, thus the familiar notion of
"order" is one more than the highest order of any of the
bindable variables in the *s*-type system.




It should be noted that every *s*-type is also an
*r*-type, namely one that is hereditarily predicative. Thus it
might seem that the expressions of the theory of classes are all
simply a special case of formulas of the full system of the ramified
theory of types. This will be true of the assignment of types to
variables, but it must be remembered that the entire formula \(\phi
\{x \mid \psi (x) \}\) about a class is by definition

 

 \[\exists \chi [ \forall x [ \chi (x) \equiv \psi (x)] \amp \phi (\chi)].\]


All we have discussed so far is the relative types of \(\phi\) and
\(\chi\). The Axiom of Reducibility guarantees that there is a
predicative \(\chi\) co-extensive with any \(\psi\) in the defining
condition of a class. To justify use of the class term \(\{x \mid \psi
(x) \}\) one must then just show that there is *some* function
that has the higher-order property \(\phi\). This is the step
comparable to the proof that a definite description is proper, i.e.,
true of exactly one thing, that justifies using that description as a
singular term.


##### Comparison of the Classes of PM with Axiomatic Set Theory



It is widely thought that the system of PM offers a very different
approach to the solution of the paradoxes than that of axiomatic set
theory as formulated in the Zermelo-Fraenkel system ZF. While the
theory of types is thought of as a desperate attempt to save the
logicist program by artifically introducing types in order to resolve
the paradoxes, axiomatic set theory seems to simply postulate sets as
entities and adopts axioms in a first order language with
"\(\in\)" for membership as its one non-logical symbol.
This view has been forcefully expressed by Quine:



> 
> 
> Whatever the inconveniences of type theory, contradictions such as
> [the Russell paradox] show clearly enough that the previous naive
> logic needs reforming....There have been other proposals to the
> same end--one of them coeval with the theory of types. [Quine
> cites Zermelo 1908.] But a striking circumstance is that none of these
> proposals, type theory included, has any intuitive foundation. None
> has the backing of common sense. Common sense is bankrupt, for it
> wound up in contradiction. (Quine 1951: 153)
> 
> 
> 



However, both the view that type theory lacks intuitive support, and
that type theory and axiomatic set theory are based on the same
intuitions dates back to Godel in 1933, referring to set theory
as the "theory of aggregates":



> 
> 
> At least hitherto only one solution which meets these two requirements
> [of avoiding the paradoxes while retaining mathematics and the theory
> of aggregates] has been found....This solution consists in the
> theory of [simple] types....It may seem as if another solution
> were afforded by the system of axioms for the theory of aggregates, as
> presented by Zermelo, Fraenkel and von Neumann; but it turns out that
> this system of axioms is nothing else but a natural generalization of
> the theory of types, or rather, it is what it becomes of the theory of
> types if certain superfluous restrictions are removed. (Godel
> 1933 [1995: 45-46])
> 
> 
> 



The two "restrictions" that Godel intends are the
restriction that types are not *cumulative* and that the levels
of types are limited to the natural numbers 0, 1,...
*n*,.... Godel suggests that one adopt a cumulative
system of types in which a given type includes functions of all lower
types (or *orders*), and the types extend beyond \(\omega\),
\(\omega\) + 1, ...\(\omega^{\omega}\), ..., through all the
ordinals. Such a "natural generalization" of the theory of
types, he asserts, amounts to the same as Zermelo-Fraenkel set theory
(ZF). Godel's claim is spelled out by George Boolos (1971)
as the "iterative conception" of sets, which can be
expressed formally. If one thinks of sets as built up in stages, with
each stage adding all sets of members of the last stage, and the
process extending endlessly, then the axioms of ZF
set theory are indeed provable from the axioms of the theory of the
"iterative conception" of sets. In turn the
"iterative conception" relies on a strong intuition,
contrary to what Quine says. It is the same intuition that underlies
the hierarchy of types.



Following Boolos' presentation of the "iterative
conception of set" it seems that axiomatic set theory and PM do
not differ widely, and express the similar intuitive notions of set
that provide the same solution to the paradoxes.



Strictly as presented in PM, however, the no-classes theory differs
significantly from ZF. The sentences of the PM theory are expressed in
the theory of types, as opposed to the first order theory of ZF. ZF
and PM cannot simply be compared in terms of their theorems. Not only
are there different axioms in the two theories, but the very languages
in which they are expressed differ in logical power. If we follow
Godel and Boolos, however, the two are seen to be based on the
same intuitive basis, and the differences are seen as the same,
barring certain "superfluous restrictions" on the theory
of PM.


##### Relations in PM



[?]21 extends the notion of class which is the
extension of a one place propositional function to the comparable
notion of a "Relation" for functions of two arguments with
the analagous contextual definition.




Contextual definition of a relation in extension 

 

\[
\phi \{x ; y \mid \psi (x,y) \} \eqdf \exists \chi 
\left[
\begin{split}
 \forall x \forall y [
 \chi \bang (x,y) \equiv \psi (x,y)] \\
{} \amp 
\phi ( \lambda x \lambda y \: \chi (x,y))
\end{split}
\right ]
\tag\*{[?]20*01}
\]



(Note: The use of this unusual notation \(\phi \{x; y \mid \psi (x,y)
\}\) in this one definition is meant to avoid the implication that a
relation is interpreted as a set of ordered pairs, that would be
represented by the contemporary notation \(\phi \{\langle x,y \rangle
\mid \psi (x,y) \}\). The PM notation for propositional functions, as
in \(\phi \hat{x}\) uses a caret over the variable where we would
write \(\lambda x \phi(x)\). The PM notation for a class is \(\hat{x}
\phi (x)\). A two-place propositional function is identified with
variables also with carets: \(\phi (\hat{x} \hat{y})\) and the
corresponding relation \(\hat{x} \hat{y} \phi (x,y)\). This notation
does not identify relations as classes of ordered pairs, and that is
how our blend of PM and contemporary notation in \(\phi \{x ; y \mid
\psi (x,y) \}\) is to be taken.)



The introduction of Greek letters for classes in
[?]20 and the use of "Roman letters" R,
S, ...in [?]21 for relations, marks a change in
the notation used in PM. After [?]21 the letters
\(\phi, \psi, \ldots\) rarely appear. As Quine remarks in his study of
the logic of Whitehead and Russell, it would seem that after a certain
point the body of PM makes use of extensional higher-order logic in a
simple theory of types:



> 
> 
> In any case there are no specific attributes [propositional functions]
> that can be proved in *Principia* to be true of just the same
> things and yet to differ from one another. The theory of attributes
> receives no application, therefore, for which the theory of classes
> would not have served. Once classes have been introduced, attributes
> are scarcely mentioned again in the course of the three volumes.
> (Quine 1951: 148)
> 
> 
> 



Quine here hints at the view of PM that is widely shared among
mathematical logicians, who see the ramified theory of types, with its
accompanying Axiom or Reducibility, as a digression taking logic into
a realm of obscure intensional notions, when instead logic, even if
expressed in a theory of types, is extensional and is comparable to
axiomatic set theory presented with a simple hierarchy of sets of
individuals, sets of sets individuals, and so on.



It is certainly true that the the remainder of PM is devoted to the
theory of individuals, classes, and relations (in extension) between
those entities. Thus the ontology of these later portions is a
hierarchy of predicative functions arranged in a simple theory of
types. This has led one interpreter, Gregory Landini (1998), to argue
that only predicative functions are values of bound variables in PM.
What we have interpreted as variables ranging over possibly
non-predicative propositional functions, \(\phi\), \(\psi\),...
are for Landini only schematic letters, and are not bindable
variables. The only bound variables in PM, he asserts, range over
predicative functions. This is a strong version of a view that others
such as Kanamori (2009) have expressed, going back to Ramsey (1931),
namely that the introduction of the Axiom of Reducibility has the
effect of undoing the ramification of the theory of types, at least
for a theory of classes, and so a higher-order logic used for the
foundations of mathematics ought to have only a simple type
structure.



Our interpretation of this change in attention to classes and
relations indicated by the shift in notation is that it indicates the
extent to which the solution to the paradoxes, which required a
ramified theory of (possibly intensional) propositional functions, may
have superseded a logic based on an unproblematic notion of class and
mathematical functions and relations between them, that appeared in
the body of *The Principles of Mathematics* before
Russell's attention was drawn to the paradoxes. In the summary
of the later sections of PM that follows below, it will appear that in
fact the symbolic development follows very closely that of PoM from
ten years earlier. While we do not know much about the order in which
sections of PM were composed, it will appear from this change of
attention from propositional functions to classes and relations, that
the later parts are in fact an earlier stratum in the conceptual
development of the project that started out as a symbolic
"Volume II" to follow PoM.



To remind the reader of the change from talking of propositional
functions to relations in extension, two further notational
alterations are introduced. Greek letters such as \(\alpha\),
\(\beta\), *etc*. , will be used
as variables for ranging over classes as well. The individual
variables which are ambiguous with respect to type, "typically
ambiguous", will now also range over classes. A function
\(\phi\) of two variables *x* and *y* is indicated with the
arguments in parentheses after the function variable: \(\phi(x, y)\).
A two place relation *R* holding between *x* and *y* is
written \(x \relR y\), with the *R* in "infix"
position. The obvious limitation of this notation is that it is not
readily extended to three place relations, adding a third variable,
say *z*. We will follow the practice in PM and write \(x \relR
y\) for binary relations. PM only requires binary relations for most
of the three volumes, although the projected volume IV on geometry
would need a notation for "*x* is between *y* and
*z*", as can be seen from Henry Sheffer's unpublished notes from Russell's lectures on
geometry from 1910 at Cambridge. There he uses the notation
\(y\rels{B}(x,y)\) which blends the two styles.


##### The Algebra of Classes



The notions of the subset relation and the intersection and union of
sets are defined in PM exactly as they are now (albeit with different
terminology). The complement of a set and the universal class \(\rV\)
are not allowed in set theory, and rejected as "proper
classes". In PM, as they only are a set of entities of a given
type \(\tau\), they form a set of the next higher type, \((\tau)/1\).
The complement of a set of a given type is the set of all entities (of
that type) that are not in the set. Each empty set will be the
complement of the universal set (of a given type \(\tau\) ) and so
there will be the empty set of type \(\tau\).

 

\[
\begin{align}
\alpha \subseteq \beta & \eqdf \forall x (x \in \alpha \supset x \in \beta) \tag\*{[?]22*01}\\
\alpha \cap \beta & \eqdf \{x \mid ( x \in \alpha \amp x \in \beta ) \tag\*{[?]22*02}\\
\alpha \cup \beta & \eqdf \{ x \mid (x \in \alpha \lor x \in \beta ) \tag\*{[?]22*03}\\
\end{align}
\]


The type subscript \(\tau\) is added below as a reminder that the
notions of universal set \(\rV\) and complement are each with regard to a given type (and
so an empty set \(\emptyset\) will recur in each type.)



\[
\begin{align}
- \alpha &\eqdf \{ x\_{\tau} \mid \lnot (x \in \alpha )\}\tag\*{[?]22*04}\\
{\alpha - \beta} & \eqdf {\alpha \cap {- \beta}}\tag\*{[?]22*05}\\
\end{align}
\]

##### The Universal Class and the Empty Class

 

\[\tag\*{[?]24*01}
\rV\_{\tau + 1} \eqdf \{ x\_{\tau} \mid (x = x) \}
\]


The subscript on '\(\rV\)' indicates that the universe of
classes of a given (simple) type \(\tau\) will be a member of the next
type. There is no class of all classes of whatever type. This is in
common with axiomatic set theory which holds that there is no set of
all sets.

 

\[\tag\*{[?]24*02}
\emptyset\_{\tau} \eqdf - \rV\_{\tau}
\]

##### Mathematical functions in PM



The logic of PM is based on *propositions*, *propositional
functions* and *relations in extension*, unlike
Frege's which deals with objects, in particular, truth values,
and functions, with the special case of concepts, which are functions
from objects to truth values. PM reduces mathematical functions to
"functional relations" in a way that is familiar from
elementary courses. If there is a binary relation which has a unique
second argument for each first argument, i.e., 

 

 \[\forall x \exists y [x \relR y \amp \forall z (x \relR z \supset z = y)]\] 


then one can introduce a new function symbol \(f\_R\), such that



 \[\forall x \forall y (x\relR y \equiv f\_R(x) = y).\]


Similarly for an \(n+1\) place relation for each \(x\_1\), ...,
\(x\_n\) there is a unique *y* such that \(R(x\_1, \ldots, x\_n,
y)\), then one can introduce an *n*-place function *g*
mapping \(x\_1, \ldots, x\_n\) onto *y*. In PM the expressions for
mathematical functions are definite descriptions, referring to the
last argument of a relation as the "value" of the function
described by that relation. We will use the expression \(f\_R\) to
refer to the functional term referring to the function derived from a
relation *R*. PM uses the explicit definite description "the
*R* of *y*" (written \( R`y \) ) where we would use the functional
expression \(f\_R\). The definition of a *monadic functional
term* then is:

 

\[
\tag\*{[?]30*01} f\_Ry \eqdf (\imath x)(x \relR y)
\]


with the general form for an *n*-place functional term *g*
derived from an \(n+1\) place relation *S* (following
Russell's notation in lectures):

 

 \[g\_S(x\_1, \ldots, x\_n) \eqdf (\imath y )(x\_1 S x\_2, \ldots, x\_n, y)\]


(The diligent reader will find that this presentation does not follow
PM exactly. The example "the father of" based on a
relation *R* expressing "*x* is the father of
*y*" would make "the *R* of
*x*" actually refer to the unique *x* which is the
father of *y*, and so what has been explained above is
appropriate to the *converse* of that relation, \(\relbR\). The
practice of reading the argument of a relational function as the
*x* and the value as the *y* is so well established that we
have taken a liberty with the actual definitions in PM.)



Recall that from this point on in PM, the relations are to be
considered as "relations in extension" and so it is easy
to see how one can treat the relations as ordered \(n+1\)-tuples of
which the last member is unique given the first *n* arguments. In
particular, a monadic function *f* can be seen in the familiar
way as a set of ordered pairs (of \(\langle x, f\_R(x) \rangle\)) for
each argument *x* in the domain of the function.



Given the treatment of "relations" as "relations in
extension" it is no accident that the development of the logic
of relations in [?]30-[?]38
looks familiar to contemporary logicians, with even some of the
notation from PM surviving into contemporary usage. A series of
notions are defined in a way quite familiar to the modern treatment of
relations as sets of *n*-tuples:


##### The Converse of a Relation

 

\[
\tag\*{[?]31*02} {\relbR} = \{\lambda x \lambda y (y \relR x)\}
\]


or, in terms of pairs: 

 

\[{\relbR} = \{ \langle x, y \rangle \mid (y \relR x) \}\]

##### Domains, Ranges and Fields of Relations



The notions of the *domain*, *range*, and *field* of a relation
are also given a contemporary definition (and so also the notions of
the *domain*, *range* and *field* of a function).



\[
\begin{align}
\tag\*{[?]33*11} \Domain \; R &\eqdf \{x \mid \exists y ( x \relR y ) \}\\
\tag\*{[?]33*111} \Range \; R &\eqdf \{y \mid \exists x (x \relR y) \}\\
\tag\*{[?]33*112} Field \; R &\eqdf \{x \mid \exists y (x \relR y \vee y \relR x ) \} \\
\end{align}
\]


Note that it is possible that a relation can have its domain in one type
and range in another. This adds complications in the theory of
cardinal numbers when a relation of similarity (equinumerousity) holds
between classes of different types. (See the discussion of [?]100
below.)


##### The Product of Two Relations



The *composition* of relations *R* and *S* is called
their *relative product* and uses a different symbol \(R\mid
S\) where we write \(R \circ S\):

 

\[
\tag\*{[?]34*01} R \circ S \eqdf \lambda x \lambda z \{ \exists y ( x \relR y \amp y \relS z ) \}
\]

##### Restricted Relations



In the case of the *restriction* of (the  *range*  of) a relation *R* to a
particular class \(\beta\), is given
 this definition, with the symbol now used instead for the restriction of the  *domain*  :

 

\[
\tag\*{[?]35*02} R \upharpoonright \beta \eqdf \lambda x \lambda y (x \relR y \amp y \in \beta)
\]


In his survey of PM, Quine (1951: 155) complains that this last 100
pages of Part I is occupied with proving theorems relating redundant
definitions of the same notions. Thus PM defines the notion of domain
and range and then introduces notions that again define the same
classes, which are proved to be equivalent. PM defines the notation 
of '\(R\pmdq\beta\)'
to be read as "the terms
which have the relation *R* to members of \(\beta\)" and
uses the example:



> 
> 
> If \(\beta\) is the class of great men, and *R* is the relation
> of wife to husband, \(R\pmdq\beta\) will mean "wives of great
> men". (PM, 278)
> 
> 
> 



In contemporary logic with the notation of set theory used above,
there is no need for a special symbol for this notion, as it is
written as:

 

\[
\tag\*{[?]37*01} R\pmsq \pmsq \beta \eqdf \{ x \mid \exists y (y \in \beta) \amp x \relR y \}
\]

##### Products and Sums of Classes of Classes

 

\[
\tag\*{[?]40*01} {} \cap \alpha \eqdf \{ x \mid \forall \beta (\beta \in \alpha \supset x \in \beta) \}
\]


This is the *intersection* of \(\alpha\).

 

\[
\tag\*{[?]40*02} {} \cup \alpha \eqdf \{ x \mid \exists \beta (\beta \in \alpha \amp x \in \beta) \}
\]


is the *union* of \(\alpha\).


### 4.2 Part II: Prolegomena to Cardinal Arithmetic


##### The Cardinal Number 1

 

\[
\tag\*{[?]52*01} 1 \eqdf \{ \alpha \mid \exists x ( \alpha = \{x \} ) \}
\]


So the cardinal number 1 is the class of all singletons. There will be
a different number 1 for each type of *x*. Frege, by contrast,
defines the natural number 1 as the extension of a certain concept,
namely being identical with the number 0, which itself is the
extension of the (empty) concept of not being self identical. In
axiomatic set theory the natural numbers are particular finite
ordinals, in particular the series with 0 as the empty set \(\emptyset\), 1 is \(\{
0 \}\), 2 is \(\{0, 1 \}\), and so on. This construction
is named the *von Neumann ordinals*.


##### Pairs

 

\[
\tag\*{[?]54*02} 2 \eqdf \{ \alpha \mid \exists x \exists y (x \neq y \amp \alpha = \{y \} \cup \{ x \} ) \}
\]


Similarly, the number 2 is the class of all pairs, rather than a
particular pair. In the type theory of PM there will be distinct
couples for the types of *y* and *x*. When they are of the
same type the couple is called "homogenous". Even with
homogenous pairs there will be distinct classes of pairs for each
type, and thus a different number 2 for each type. The same notion
applies to relations.


##### Ordered Pairs



The notion of an ordered pair, called an "*ordinal
couple*" is defined as:

 

\[
\tag\*{[?]55*01} \langle x, y \rangle \eqdf \textrm{ the extension of } \lambda x \lambda y (x \in \{x \} \amp y \in \{y\})
\]


The idea is that the order of the relation \(\lambda x \lambda y (x
\in \{x\} \amp y \in \{y\})\) determines the first and second element
of the ordered pair. It is a relation in extension, which is the
analogue of a property in extension or class. A relation in extension
has a distinction between the first and second elements due to the
order of the defining relation. The closest in contemporary language
would be:

 

\[\phi \langle x, y \rangle \eqdf \exists \psi \forall u \forall v ( \psi (u, v) \equiv \lambda x \lambda y [ x \in \{x\} \amp y \in \{y\} ] (u, v) \amp \phi (\psi) ) \]


Given the definition of extensions of relations this is the version of
the no-classes theory for relations. After attending classes of
Russell the year before, and having several discussions, Norbert
Wiener (1914) proposed the following definition (in modern
notation):

 

\[\langle x, y \rangle \eqdf \{\{\{ x\}, \emptyset \}, \{\{ y\}\}\} \textrm{ where } \emptyset \textrm{ is the empty set.}\]


Wiener's accomplishment was to capture the ordering of the pair
which in PM is captured by the ordering of the arguments of relations
with the unordered notion of set membership.


##### The end of PM to [?]56



The paperback abridged edition of *PM to
[?]56* only goes this far, so the remaining
definitions have only been available to those with access to the full
three volumes of PM.


##### Relative Types



This section presents a discussion of relations between individuals of
distinct types, introducing a notation for types, \(t\pmsq x\) for the
type to which *x* belongs. This section is little used in Volume
I. The special consequences for this notion when dealing with relative
types of cardinal numbers is the topic of the Preface to Volume II,
which was added after the first volume was already in print. The delay
due to working out these details partially explains the three year gap
between the publication of Volume I in 1910, and the remaining volumes
II and III in 1913. Section [?]65 (On the Typical
Definition of Ambiguous Symbols), is a discussion of *typical
ambiguity*, the ambiguity of variables with respect to type.



\[
\tag\*{[?]70*01} f: \alpha \rightarrow \beta \eqdf
\] 


The functions *f* from \(\alpha\) onto \(\beta\), that is, the
\(\Domain \; f = \alpha\) and \(\Range \; f = \beta\)


##### Similarity of Classes

 

\[
\tag\*{[?]73*01} \alpha \approx \beta \eqdf (\exists f) f : \alpha \stackrel{1-1}{\longrightarrow} \beta.
\]


There is a one-one function mapping \(\alpha\) onto \(\beta\)
(*similarity* of \(\alpha\) and \(\beta\)). Contemporary
discussions say that \(\alpha\) and \(\beta\) are
*equinumerous*. Difficulties arise with respect to the
definition of cardinal numbers when the relation of similarity they
involve is one that has a domain and range in different types. See
[?]100 below.



The main theorem in this chapter is a proof of the Cantor-Bernstein
theorem, that if a set \(\alpha\) is similar to a subset *z* of
another set \(\beta\) and \(\beta\) is similar to a subset \(\delta\)
of \(\alpha\) then \(\alpha\) and \(\beta\) are themselves
similar:

 

\[
 \forall \alpha \forall \beta \forall \gamma \forall \delta 
\left[\left (
\begin{split}
\alpha \approx \gamma & {}\amp \beta \approx \delta \\
&{} \amp \gamma \subseteq \beta \\
&{} \amp \delta \subseteq \alpha \\
\end{split}
\right) \supset \alpha \approx \beta \right] 
\tag\*{[?]73*88}
\]


The proof here explicitly follows the proof by Ernst Zermelo from
1908. Whitehead and Russell call this the
"Schroder-Bernstein" theorem.


##### The Axiom of Choice (Multiplicative Axiom)



The Multiplicative Axiom, or "Axiom" of Choice, is not an
axiom of PM, what is termed a "primitive proposition", but
is instead a defined expression that is added as an hypothesis to
theorems for which it is used. This reflects the emerging awareness at
the time of the role of the Axiom of Choice in various proofs, in
particular, Zermelo's proof that every class can be
*well-ordered*.



\[{}\
\begin{aligned}&\textrm{Multiplicative}\\&\textrm{Axiom}\end{aligned} \eqdf 
\forall \alpha \left\{ 
\begin{split}
&\forall \beta (\beta \in \alpha \supset \beta \neq \emptyset ) \amp {}\\
&\;\;\forall \beta \forall \delta \left [
\left(
\begin{split}
\beta & \in \alpha \amp {}\\
 \delta & \in \alpha \amp {}\\
\beta & \neq \delta
\end{split}
\right) 
\supset 
(\beta \cap \delta = \emptyset) 
\right] 
\supset \\
&\;\;\;\; \exists \beta \forall \delta \exists \gamma
\left[
\begin{split}
& \delta \in \alpha \supset \\
& \delta \cap {} \beta = \{\gamma\} 
\end{split}
\right]
\end{split}
\right\}
\tag\*{[?]88*03}
\]


If \(\alpha\) is a class of mutually exclusive, non-empty, classes, then there is
a ("choice") set \(\beta\) such that the intersection of
\(\beta\) with each member of \( \delta \) of \(\alpha\) is a unique (chosen) member of
\(\delta \).


##### \(\Rast\) The Ancestral Relation

 

\[
\quad\Rast \eqdf
\left \{
\begin{split}
& \langle x, y \rangle \mid (\exists u x\relR u \lor \exists u uRx ) \amp {} \\
& \forall \alpha 
\left [ 
\left[
\begin{split}
& x \in \alpha \amp {}\\
&\forall z \forall w (z \in \alpha \amp zRw \supset w \in \alpha ) 
\end{split}
\right] \supset y \in \alpha 
\right] 
\end{split}\right \}
\tag\*{[?]90*01}
\]


This follows Frege's definition, namely, that *y* is in all
the *R*-hereditary classes that contain *x* or (*x* is
in the field of *R* ).


##### The Powers of a Relation



The "powers" of a relation \(R (\textrm{Pot}\pmsq R)\) are
the relations *R*, \(R^2\), \(R^3\), ...where



\[{R^2 \eqdf R \circ R},\quad {R^{n+1} \eqdf R \circ R^n},\quad \ldots\]


These definitions begin with [?]91*03, using the notion of
the ancestral of a higher-order relation between relations defined
beginning with *R*.



The main result of this section is another proof of the
Cantor-Bernstein theorem: "This proof is essentially the same as
Bernstein's published originally by Borel [1898: Note 1, pp.
102-7]" (PM I, 589). In this proof the one-one relation
between the sets \(\alpha\) and \(\beta\) is constructed from the
powers of two relations *R* that maps \(\alpha\) into \(\beta\),
and *S* that maps \(\beta\) into \(\alpha\). The one-one mapping
is constructed in stages. First all of \(\alpha\) is mapped onto
\(\beta\) by *R*. Those elements in \(\beta\) not in the range of
*R* are mapped onto \(\alpha\) by *S*. But some elements in
the range of *S* will have already been mapped by *R*. They
need to be shifted to a new image in \(\beta\), again by *R*.
This process is iterated through all of the powers of *R*, and
then it is shown that the resulting relation is one to one from
\(\alpha\) onto \(\beta\). See Hinkis (2013) for a history of the many
different proofs of this theorem.


## 5. Volume II


### 5.1 Prefatory Statement of Symbolic Conventions



The writing of this preface delayed the publication of the second
volume of PM, as Whitehead and Russell struggled over the
complications it raised. The difficulties arise from the *typical
ambiguity* of terms and formulas of the theory of types. Every
constant, such as those for the numbers \(0,1, \ldots, \aleph\_0\) will
have a definition relative to each type. Without assuming the Axiom of
Infinity for individuals, there is no guarantee that a given constant
designates a non-empty class in a given type. The preface introduces
the notion of "formal numbers", which are to be
interpreted as belonging to a type that makes them not identical with
the \(\emptyset\) for that type. Volume II begins with Part III,
"Cardinal Arithmetic". The notions of cardinal numbers are
developed in full generality, extending to infinite cardinals.
Consequently the theory of natural numbers, which are called
"Inductive Cardinals" in PM, is introduced with a series
of definitions of special cases of notions that are first introduced
in a general form applying to any numbers or classes. For example,
addition of natural numbers, as in the famous proof that \(1 + 1 = 2\)
in [?]110*04 is proved for the special
case of the addition of classes that applies to cardinal numbers,
'\(+\_c\)'. The Summary to section A introduces the notion
of *homogenous cardinals*, which are classes of similar classes
whose members are all of the same type. It is possible to define
similarity between two classes \(\alpha\) and \(\beta\) of distinct
types say \(\tau\) and \(\tau '\), and cardinals are classified as
*descending* and *ascending* as the domain of the
relevant similarity relation is of a higher type than the range, and
when of a lower type, respectively. The theory of cardinal numbers is
straightforward with homogenous cardinals, however the exceptions must
be kept in mind, as is evidenced in [?]100.


### 5.2 Part III: Cardinal Arithmetic


##### Definition of cardinal numbers

 

\[
\tag\*{[?]100*01} 
\rN\_c \eqdf \{ x \mid \forall y (y \in x \leftrightarrow \forall z \forall w (z,w \in y \leftrightarrow z \approx w) \}
\]


*Cardinal Numbers* are classes of equinumerous (similar)
classes. We can add a notion of the *number* of a class to
allow for a direct comparison with Frege: 

 

\[\# \{x \mid \phi ( x) \} \eqdf \{ y \mid y \approx \{x \mid \phi (x) \} \}.\] 


(In set theory of course this is too large to be a set, and so is just
a "proper class".)


##### Hume's Principle in PM



*Hume's Principle* which is described in Frege (1884:
SS63) as asserting that the content of the proposition that
"the number which belongs to the concept F is the same as the
number which belongs to the concept G" is equivalent to
"the concept F is similar to the concept G". In terms of
classes this becomes \(\alpha \approx \beta \equiv \# \alpha = \#
\beta\). Hume's principle is the focus of much of the discussion
of "Neo-Logicism", the doctrine that Frege's
construction of the numbers can be built on a consistent foundation
(see the entry on 
 Frege's theorem).



Only one direction of this equivalence is provable in PM:



\[
\tag\*{[?]100*321}
\alpha \approx \beta \supset \# \alpha = \# \beta
\]


The failure of the other direction, the implication from right to
left, is due to the possibility that \(\alpha\) and \(\beta\) are of
different types, so that any similarity relation between them will
have its domain and range in different types. Suppose there are
\(\aleph\_0\) individuals, and consider two higher types with the
cardinals in them of even larger, but distinct, cardinalities, say
\(\alpha\) in some high type has cardinality \(\aleph\_1\) and that
\(\beta\) in an even higher type and has cardinality \(\aleph\_2\).
There are no sets of individuals similar to \(\alpha\) or \(\beta\),
so no similarity relation with a domain in \(\alpha\) or \(\beta\)
will hold with respect to any set in the type of individuals. Suppose
that \(\#\) is defined in terms of such *descending* relation.
Therefore \(\# \alpha = \{ \Lambda \} = 0\) and \(\# \beta = \{
\Lambda \} = 0\) so \(\# \alpha = \# \beta\), yet \(\alpha \not\approx
\beta\) on any similarity relation of whatever domain and range,
because their cardinalities differ. Whitehead and Russell assert that
the case of \(\alpha\) and \(\beta\) being in different types is the
only way to construct an exception to this direction of Hume's
principle, and offer as a restricted version:

 

\[
\tag\*{[?]100*34}
\exists \gamma [\gamma \in (\alpha \cap \beta) ] \supset ( \alpha \approx \beta \equiv \# \alpha = \# \beta )
\]


The antecedent guarantees that \(\alpha\) and \(\beta\) are of the
same type, and so the cardinal numbers involved are *homogenous
cardinals*. (Landini (2016) argues that this section of PM is
confused.)


##### 0 Defined

 

\[
\tag\*{[?]101*1} 0 \eqdf \# \emptyset
\]


The class of all classes equinumerous with the empty set is just the
singleton containing the empty set, so \(0 = \{ \emptyset \}\).


##### The *Arithmetical Sum* of Classes and Cardinals

 

\[
\tag\*{[?]110*01} 
\alpha + \beta \eqdf [\{ \beta \cap \emptyset \} \times \{\alpha \}] \cup [ \{ \alpha \times \{ \beta \} ] 
\] 


(If \( \alpha, \beta \neq \emptyset \) , otherwise \( \alpha +
\emptyset = \alpha, \emptyset + \beta = \beta \) ). This qualification
is hidden in PM by the use of expressions for functional relations
that are sometimes undefined. The *arithmetic sum* of
\(\alpha\) and \(\beta\) is the union of \(\alpha\) and \(\beta\)
after they are made disjoint by summing the pairing the elements of
\(\beta\) with elements of \(\{ \alpha \}\) and the elements of
\(\alpha\) with the elements of \(\{ \beta \}\). (The *Cartesian
product* of \( \gamma \) and \( \delta \), \( \gamma \times \delta
\) is \( \{ \langle x, y \rangle | x \in \gamma \; \amp \; y \in
\delta \} \)). The The classes \(\alpha\) and \(\beta\) are
intersected with the empty class, \(\emptyset\), to adjust the type of
the elements of the sum. It is more recognizable to contemporary set
theory by the equivalent definition (subject to the same exception
when


: 

 

\[\alpha + \beta \eqdf \{ \langle \emptyset, x \rangle | x \in \alpha \} \cup \{ \langle y, \emptyset \rangle | y \in \beta \} \]


The *cardinal sum* of *y* and *x*

 

\[
\tag\*{[?]110*02}
{}\quad y +\_c x = \{ z \mid \exists \alpha \exists \beta [( y = \# \alpha \amp x = \# \beta) \amp z \approx \alpha + \beta ] \} 
\]


\(y +\_c x\) expresses the *cardinal addition* of cardinals
*y* and *x*. It is the arithmetical sum of
"homogeneous cardinals", cardinals of a uniform type, to
which \(\alpha\) and \(\beta\) are related by \(\rN\_0 c\) (itself
defined at [[?]103*01]). The notation indicating that
\(\alpha\) is a homogenous cardinal \(\alpha\) is \(\rN\_0
c\pmsq\alpha\), which we might write as \(\#\_0\) in an extension of
our contemporary notation replacing \(\#\) above.



The reader can now appreciate the notorious fact that \(1 +1=2\), the
most elementary truth of arithmetic, is not proved until page 83 of
Volume II of *Principia Mathematica*, and even then, almost as
an afterthought:

 

\[
\tag\*{[?]110*643} 1 +\_c 1 = 2
\]


Whitehead and Russell remark that "The above proposition is
occasionally useful. It is used at least three times,
in...". This witticism reminds us that the theory of
natural numbers, so central to Frege's works, appears in PM as
only a special case of a general theory of cardinal and ordinal
numbers and even more general classes of isomorphic structures.


##### Exponentiation



Exponentiation for cardinals is defined in such a way that it
coincides with Cantor's notion that the cardinality of the
powerset of a class \(\alpha\) is 2 raised to the power of the
cardinality of \(\alpha\):

 

\[
\tag\*{[?]116*72}
\lvert\lvert \wp \alpha \rvert\rvert = 2^{\lvert\lvert \alpha \rvert\rvert }
\]

##### Greater and Less



Cantor's Theorem:

 

\[
\tag\*{[?]117*661} 2^{y} > y
\]


This is Cantor's theorem that the if a set \(\alpha\) has a
cardinal number *y* then the cardinal number \(2^{y}\) of the
powerset of \(\alpha\) is greater than *y*.


##### The Natural Numbers



The most direct comparison with Frege's development of the
natural numbers comes with the notion of *Inductive Cardinal*
by which PM means the natural numbers 0, 1, 2,..., and the theory
of these numbers including the principle of induction. Although the
numbers 0 and 1, as well as addition of natural numbers \(+\_c\) is has
been defined earlier, they are defined as cardinal numbers and
addition will apply to all cardinal numbers, finite and transfinite.
For the finite natural numbers, special notions need to be defined
first. For the proof of the Peano Postulates it is necessary not only
to define 0, but also the notion of *successor*. For Frege the
notion of (weak) predecessor of a number is defined, thus 0 and 1 are
the predecessors of 1, while 0, 1 and 2 are the predecessors of 2,
etc. The successor of *n* is then defined by counting the
predecessors of a number, in terms of the definition of number, it is
the number of the class of predecessors. This definition would not
work for PM, where each number would be of a higher type, as it is
defined as a set containing that number. There will in fact be natural
numbers for each type, thus a set of all pairs of individuals of type
0, a set of pairs of sets of type 1, etc., for each type. There is no
one type, however, at which there are all of the natural numbers (sets
of equinumerous sets of that type) without an assumption that there
are infinitely many members of some one type.



The solution in PM is to guarantee that for each finite set of
*n* individuals of type 0, there will be some object not in that
set, which can be included in the set defining the successor. That such a
new individual can be found is guaranteed by the Axiom of Infinity,
which in effect asserts the existence of distinct individuals of any
finite number. It is interesting to note the the "Axiom"
of Infinity is not a *primitive proposition* of the logic of
PM. Instead it is an additional hypothesis, to be used as an
antecedent to mathematical assertions upon which it depends. The issue
of whether the system of PM succeeds as *logicism* is thus not
settled by noting that an Axiom of Infinity has to be assumed, but by
determining whether that "Axiom" is derivable from logical
principles alone.



In axiomatic set theory the "Axiom of Infinity" guarantees
the existence of a particular set, the ordinal \(\omega\):



 \[
\exists x [ \emptyset \in x \amp \forall y ( y \in x \supset y \cup \{ y\} \in x ) ]
\]


The *inductive cardinals* (natural numbers) are defined as the
numbers bearing the ancestral of the \(+\_1\) relation to 0. Given that
the \(+\_1\) relation is the PM account of *successor* this is
the same definition as for Frege.



*Inductive Cardinals* N

 

\[
\tag\*{[?]120*01}
\rN \eqdf \{x\mid 0 \relSast x \}
\]


The *inductive cardinals* N are the familiar *natural
numbers*, namely 0 and all those cardinal numbers that are related
to 0 by the ancestral of the "successor relation"
*S*, where \(x\relS y\) just in case \(y = x +1\). 



\[
\tag\*{[?]120*03}
\textrm{Axiom of Infinity} \eqdf \forall y ( y \in \{x\mid 0\relSast x \} \supset y \neq \emptyset )
\]


This Axiom of Infinity asserts that all inductive cardinals are
non-empty. (Recall that \(0 = \{\emptyset \}\), and so 0 is not
empty.) The axiom is not a "primitive proposition" but
instead to be listed as an "hypothesis" where used, that
is as the antecedent of a conditional, where the consequent will be
said to depend on the axiom. Technically is not an *axiom* of
PM as [?]120*03 is a *definition*, so this is just
further defined notation in PM!



Whitehead and Russell do carry out the step of the logicist program of
deriving Peano's Postulates based on the prior definitions of
the notions of Natural number, 0, and successor, as Russell describes
the project later, in (1919). This is in fact what is done in
[?]120 "Inductive Cardinals" of PM, but
is not described as such, either there or in introductory material.
The results are not proved separately, but as they appear in a
development of various results about natural numbers. Indeed some, such as [?]120*31, can only be seen to be versions of a Peano axiom with a bit of work.


1. 0 is a natural number. 

 \[
 \tag\*{[?]120*12} 0 \in \rN
 \]
2. The successor of any number is a number. 

 \[
 \tag\*{[?]120*121} n \in \rN \supset n +\_c 1 \in \rN
 \]
3. No two numbers have the same successor (assuming the axiom of
Infinity). 

 \[
 \tag\*{[?]120*31}

\ \textrm{Axiom of Infinity} \supset (n +\_{c} 1 = m +\_{c} 1 \supset n = m)
 \]
4. Given the way that the successor
operation is defined, it is not a matter of logic that there is an
extra individual to add to a set of size *n* to give one of size
\(n +\_{c} 1\). This is guaranteed by adding the Axiom of Infinity as
an hypothesis to the theorem.
5. 0 is not the successor of any number. 

 \[
 \tag\*{[?]120*124} n +\_{c} 1 \neq 0
 \]
6. Any property \(\phi\) which belongs to 0, and belongs to the
successor of *m* provided that it belongs to *m*, belongs to
all natural numbers *n*. 

 \[
 \tag\*{[?]120*13}
 \forall n \{ [ n \in \rN \; \amp \; \forall m( \phi m \supset \phi (m +\_c 1)) \; \amp \; \phi \: 0] \supset \phi n \}
 \]



In contemporary set theory the notion of the successor of a number is
defined directly for the ordinals as \(s(x) = x \cup \{x \} \)
rather than by adding 1, and addition is defined using the familiar
recursive definition:

 

\[
\begin{align}
x + 0 & = x\\
x + s(y) & = s (x + y)
\end{align}
\]


The use of recursive definitions is justified by a theorem proving
that they describe a unique function. The induction axiom is justified
by showing that any class that contains 0, and for any number *n*
contains \(s(n)\) will contain all of the numbers in \(\omega\). The
existence of \(\omega\) is guaranteed by the ZF axiom of infinity.



At this point, after 225 pages in Volume II, the reader will see how
to compare the logicist reduction of arithmetic in PM with rival
accounts of Frege and of contemporary set theory.



Frege completes his development of the natural numbers at page 68 of the Volume II of his
*Basic Laws of Arithmetic* published in 1903, which follows the
250 pages of Volume I that had been published in 1893. So both Frege
and the authors of PM took great pains to prove more advanced theorems
only after a chain of closely argued lemmas based on their own
formalized symbolic logic.



Frege ends his deductions of the laws of arithemtic with results about
the notions of 0, Successor, and the principle of Induction which
include the Peano Axioms. He does not consider arithmetical functions,
such as addition or multiplication, and thus does not define the
successor of a number *n* as the result of adding 1 to
*n*.



Frege's account is streamlined in other ways as well. He had
always considered the analysis of simple identity sentences to be
important to his logicism, dating back to the *Begriffsschrift*
(1879), on through his *Foundations of Arithmetic* (1884) and
his "On Sense and Reference" (1892), and even in the
appearance of the example "\(2^2 = 2 + 2\)" in the forward
to the *Basic Laws of Arithmetic*. Indeed the analysis of
identity sentences is the starting point of his introduction of the
theory of sense and reference in 1879,
yet Frege does not diverge from
his project enough to show how such an identity would be proved. So
Whitehead and Russell might well have wanted to include their proof of
\(1+1 = 2\) at [?]110*643 as a reminder of
the analysis of mathematical equations in PM using the
"descriptive functions" of [?]30 and
then the account of definite descriptions in
[?]14.



Frege also does not construct the general theory of the arithmetic of
cardinal and ordinal numbers that occupies PM for much of Part III.
Indeed, after the theorems on Arithmetic in SS54 which concludes
Part II of *Basic Laws of Arithmetic*, Frege jumps directly to
the topic of Real Numbers for the remainder of Volume II. Judging
simply on the amount of theorems that had to be proved in leading to
the account of Peano Arithmetic, PM does not differ wildly from
Frege's earlier attempt.



Admittedly the system of PM is an indirect and cumbersome system to
develop if the theory of Arithmetic were the only goal in mind.
Firstly, however, the system of the ramified theory of types is
independently interesting for the foundations of logic that it
provides. After [?]20 the theory of classes, and
development of general notions of arithmetic for relations which
follows, does present the arithmetic of the natural numbers as a
special case which can be generalized to the arithmetic of ordinal and
cardinal numbers all in a logic with a simple hierarcy of types. The
survey below of what follows in Volume II and Volume III shows the
particular way of developing the theory of rational and then real
numbers that PM follows. The results in set theory will seem
primitive, as the results are dated to around 1908, at just the point
when axiomatic set theory began its extraordinary development.
Whitehead and Russell were not active contributors to set theory and
so PM should not be studied for later technical results that may have
been anticipated here. Russell summarized results in PM from the
current state of the study of infinite cardinals and ordinals in a
paper he give in Paris in a paper called "On the Axioms of the
Infinite and Transfinite" (Russell 1911). There is, however, one
result concerning two notions of infinity that appears to originate
with PM.


##### Dedekind Infinity



In PM a class is a *finite* "inductive" class if
and only if it can be put into a one to one correspondence with the
Natural Numbers less than or equal to some natural number \(n\). It will
be *infinite* if and only if it is not inductive.



A class is *Dedekind Infinite* (*Reflexive*) if and only
if it can be put in a one to one correspondence with a proper subset
of itself.



The key theorem in this section is:

 

\[
\tag\*{[?]124*57}
\textrm{If } y \textrm{ is not inductive then } 2^{2^y} \textrm{ is reflexive.}
\]


The Inductive and Reflexive notions of infinity coincide if one
assumes that axiom of Choice. This result does not assume the axiom of
Choice.



George Boolos (1994: 27) describes the details of this argument and
quotes J.R. Littlewood as saying:



> 
> 
> He [Russell] has a secret craving to have proved *some*
> straight mathematical theorem. As a matter of fact there *is*
> one: "\(2^{2^{\alpha}} > \aleph\_0\) if \(\alpha\) is
> infinite". Perfectly good mathematics.
> 
> 
> 



As use of the Axiom of Choice is explicitly indicated, and many
results do not use it, the unique contribution to set theory of PM may
be in its indication of what can be proved without assuming
Choice.


### 5.3 Part IV: Relation-Arithmetic



*Relation Arithmetic* is the study of the generalization of
cardinal and ordinal numbers to classes of similar classes where
similarity is based on an arbitrary relation. A relation *P* is
similar to a relation *Q*, if there is a one to one relation
*S* (a *correlator*) relating the domain of *P* to
the domain of *Q* so that if \(x\relP y\) for some *x* and
*y* then if \(x\relS w\) and \(w\relQ z\) then \(z \relbS y\).
The mapping *S* is an isomorphism between the relations *P*
and *Q*. A *relation number* will then be a class of
relations that are similar to each other. Relation Arithmetic then
generalizes the notions of cardinal arithmetic, such as sum and
product, to arbitrary relation numbers. Russell himself expressed regret that the material in
Part IV was not more carefully studied by his contemporaries (Russell
1959: 86). If a series \( \relP \) is well-ordered, then the class of relations ordinally similar to \( \relP \) will be an ordinal number. The sums of ordinal numbers are are studied in Tarski (1956), but there has been little interest in the more general notion of Relation Arithmetic presented in these sections of PM. See Solomon (1989).


##### Ordinal Similarity


* [?]151*01*P*
and *Q* are *similar ordinally*

\[ \eqdf \exists S [S: \Domain \; P \stackrel{1-1}{\longrightarrow}
\Domain \; Q \amp P = S \circ Q \circ {\relbS} ] \]



*P* and *Q* are *similar ordinally*, written \(P
\smor Q\), just in case there is a one to one mapping \( \relS \) of
the domain of *P* into the domain of *Q* such if \(x\relP
y\), \(x \relS z\), and \( z \relQ w \) then \(w \relbS y\).



The *sum* of series *P* and *Q*, \(P \oplus Q\),
is defined as:

 

\[\begin{align} \tag\*{[?]160*01} &\ P \oplus Q \eqdf
\\ &\quad\{ \langle x, y \rangle | \; x\relP y \lor x \relQ y \lor
[ \exists z (z\relP x \lor x\relP z)\; \amp \; \exists z (z\relQ y
\lor y\relQ z ) ] \} \end{align}\]


As it is put in the headnote to [?]160: 



> 
> 
> ...we may regard the sum of *P* and *Q* as a relation
> which holds between *x* and *y* when either *x*
> precedes *y* in the *P* series, or *x* precedes
> *y* in the *Q* series, or *x* belongs to the
> *P*-series and *y* belongs to the *Q*-series.
> 
> 
> 



The *product* of series *P* and *Q*, \(Q \otimes
P\), relates pairs of members of the  *field*  of *P* to members of the  *field*  of *Q* as follows. ( This should not be confused with the more familiar notion of  *relative product* which was defined in [?]34.) 



As it is put in the headnote to [?]166: 



> 
>  The product \( \relQ \otimes P \) is ... a relation which has for
> its field all the couples that can be formed by choosing a the
> referent in \(C ' P \) and the relatum in \(C ' Q \).
> These couples are arranged by \( \relQ \otimes \relP \) on the
> following principle: If the relatum of the one couple has the relation
> \( \relQ \) to the relatum of the other, we put the one before the
> other, and if the relata of the two couples are equal while the
> referent of the one has the relation \( \relP \) to the referent of
> the other, we put the one before the other. 
> 
> 
> 



\[\begin{align}
\tag\*{[?]166*112}
&\ \langle x, z \rangle Q \otimes P \langle y, w \rangle {}\\
&\quad\equiv [ ( x,y \in Field \; P \; \amp \; z,w \in Field \; Q ) \; \amp \; zQw ] \lor ( z = w \; \amp \; xPy )
\end{align}\]


(Notice that while the \( sum \) of two binary relations is a binary relation, their \( product \) is a relation between pairs. This is the generalization from classes to relations of the fact that the cardinality of the relative product of two classes is the cardinality of the class of ordered pairs of elements taken one from each.) It is possible to prove results that show the differences between products and sums of
relations and of numbers. The product of relations is associative:




\[
\tag\*{[?]166*42} 
(P \otimes Q) \otimes R \text{ is ordinally similar to } P \otimes (Q \otimes R)
\]


The relations distribute in one way:

 

\[
\tag\*{[?]166*45} 
(Q \oplus R) \otimes P = (Q \otimes P) \oplus (R \otimes P)
\]


However, it does not hold in general that:

 

 \[P \otimes (Q \oplus R) = (P \otimes Q) \oplus (P \otimes R).\]





For the purposes of defining rational and real numbers as relations
between relations, it is necessary to define the ordering relation
between individuals related by the relation, that is, in the domain or
range (the *field*) of the relations. This is described in the
summary of [?]170 and a theorem as follows:



> 
> 
> ...\(\alpha\) is said to precede \(\beta\)...if we consider
> the two classes \(\alpha - \beta\) and \(\beta - \alpha\), there are
> members of \(\alpha - \beta\) which are not preceded by any members of
> \(\beta - \alpha\). (Vol. II, 1912: 411 and 1927: 399)
> 
> 
> 


* [?]170*01
\(\alpha \) precedes \( \beta \) in the relation \(P \)

\[
\alpha P\_{lc}\beta \equiv \exists x \{ x \in ( \alpha - \beta) \amp \lnot [ \exists y
(y \in \beta - \alpha \amp y\relP x)] \}
\]



The notions of the sum and product of relation numbers is defined as
the relation number of the sum and product of the relations, with
adjustments made so that the types of the relations are uniform, and
the numbers contain disjoint relations, as was seen in the definition
of sum for cardinal numbers in [?]110 above. If
*x* and *y* are relation numbers, their sum is \(x \dot{+}
y\) and the product is \(x \dot{\times} y\).



It is proved that the operation of sum for relation numbers is associative
and other properties that directly follow from the corresponding
properties of the sums and products of relations. Among the many
theorems is the fact that sum for relation numbers is associative:



\[
\tag\*{[?]180*56}
( y \dot{+} x ) \dot{+} \rho = y \dot{+} ( x \dot{+} \rho)
\]


The distribution of product of relation numbers over sum for relation
numbers holds in one form:

 

\[
\tag\*{[?]184*42} 
( x \dot{+} \rho) \dot{\times} y = (x \dot{\times} y) \dot{+} ( \rho \dot{\times} y )
\]

### 5.4 Part V: Series



A *series* (*linear ordering*) is defined as a relation
that is *irreflexive* \(\forall x (\lnot x\relR x)\),
*transitive* \(\forall x \forall y \forall z (x\relR y \amp
y\relR z \supset x\relR z)\), and *connected* \(\forall x
\forall y (x\relR y \lor y\relR x)\).
([?]204*01) (These properties are restricted
to a specific domain for each relation. Thus a connected relation will
hold between any two members of a given domain.) This is now called a
*linear ordering* of a given set.


##### Sequents



> 
> 
> Thus sequents of \(\alpha\) are its immediate successors. If
> \(\alpha\) has a maximum, the sequents are the immediate successors of
> the maximum; but if \(\alpha\) has no maximum, there will be no one
> term of \(\alpha\) which is immediately succeeded by a sequent of
> \(\alpha\); in this case, if \(\alpha\) has a single sequent, the
> sequent is the "upper limit" of \(\alpha\). (PM Vol. II,
> "Summary of [?]205", 1912: 577 or 1927: 559)
> 
> 
> 


##### Dedekindian Relations



> 
> 
> We call a relation "Dedekindian" when it is such that
> every class [bounded from above] has either a maximum or a sequent with respect to it. (PM
> Vol. II, "Summary of [?]214", 1912: 684 or 1927: 659)
> 
> 
> 
> 



In other words, when the relation \(R\) such as \( \pmlt \) is *Dedekindian* when every class \(\alpha \) has either a maximum or a sequent with respect to \( R \). This is the standard definition by which every segment with an upper bound
has a least upper bound. That least upper bound will either be the
maximum of the set or the least individual greater than all members of
the set.


## 6. Volume III


### 6.1 Part V: Series (continued)


##### Elementary properties of well ordered series.



At [?]250*51 we find a proof that the Axiom of Choice
follows from the Well-Ordering Principle, that is, that every set can
be well-ordered.


##### The series of Ordinals



The "Burali-Forti" paradox is described in the
Introduction to PM as one of the contradictions that can be resolved
by the theory of types:



> 
> 
> It can be shown that every well-ordered series has an ordinal
> number...and that the series of all ordinals (in order of
> magnitude) is well-ordered. It follows that the series of all ordinals
> has an ordinal number, \(\Omega\) say. But in that case the series of
> all ordinals including \(\Omega\) has the ordinal number \(\Omega +
> 1\), which must be greater than \(\Omega\), Hence \(\Omega\) is not
> the ordinal number of all ordinals. (PM Vol. I, 1925: 61 and 1910:
> 63)
> 
> 
> 



In [?]256 we find the resolution of the Burali-Forti
contradiction in observing the relative types of classes of ordinals.
Ordinal numbers, as classes of isomorphic series, will be of a higher
type than their members. The purported "ordinal number of all
ordinal numbers" \(\Omega\) will be restricted to a type above
the type of the ordinals that are its members. Just as there is no
class of all classes (of some type or other) there is also no ordinal
of all ordinal numbers.



Theorem [?]256*56 demonstrates that "in
higher types there are greater ordinals than any to be found in lower
types". (PM Vol.III, 75)


##### Zermelo's Theorem


* [?]258*326
Assuming the Axiom of Choice, every set can be
well-ordered.



This proof of Zermelo's theorem follows Zermelo's
"new proof" of Zermelo (1908). Together with
[?]250*51 this shows that the Axiom of Choice
is equivalent to the Well Ordering principle.


##### The Transfinite Ancestral Relation



The discussion of "transfinitely hereditary" properties in
[?]257 constitutes the discussion of
"transfinite induction" that Russell pointed out to
Reichenbach in the discussion reported above from Russell (1959:
86),


##### Finite Ordinals



It is shown in [?]262 that every infinite well
ordered series consists of a series (well ordered set) of
progressions. (\(\omega\) orderings).


##### The series of Alephs



A result of Hausdorff in (1906),
that \(\omega\_1\), the first
uncountable ordinal) is not the limit of a progression of smaller
ordinals is shown to follow if one assumes the Axiom of Choice
([?]265*49). It is then conjectured that this
cannot be shown without relying on the Axiom of Choice. See
Grattan-Guinness (2000: 403) for a discussion of Hausdorff's
influence on the content of PM.



Dorothy Wrinch, who had been a member of Russell's circle of
unofficial students during the war, published in 1919 an article on
Dedekindian series of ordinal numbers. She describes the result as
investigating "necessary and sufficient conditions that \(P^Q\)
should be Dedekindian or semi-Dedekindian when *P* and *Q*
are well ordered series" (Wrinch 1919: 219). This study follows
up the result in [?]124 that is described in Boolos
(1994), as an investigation of the arithmetic of ordinals without
assuming the Axiom of Choice. Wrinch's paper follows not only
the notation of PM, but also makes use of theorems up
to the end of section V on Series, with numbers following the dot, and
so could easily be added as [?]277. Russell intended
to found a school of "mathematical philosophy", and of
course succeeded in attracting Ludwig Wittgenstein to the foundational
issues in PM, but there is no other indication of logicians attempting
to set their results in the framework of PM.


### 6.2 Part VI: Quantity


 The later portions of PM
should thus be studied for a hint of how the real numbers and the use
of mathematics in measurement can be developed with this rival
foundation on the theory of relations. Gandon (2008, 2012) argues that
the application of mathematics is better explained using this logicist
account than by rivals.



In [?]300-[?]314, positive
and negative integers, ratios and real numbers all defined. The goal
of the section is to begin the study of how these numbers are used in
the measurement of quantities in geometry and physics.


##### Ratios

 

\[ R n/m S \eqdf \forall x \forall y ( x\rels{R^n} y \supset
x\rels{S^m}y) \]


for *n*, *m* relatively prime.



Relations *R* and *S* stand in the *ratio* of
*n* to *m* when \(x\rels{R^n} y\) then \(x\rels{S^m} y\),
where \(R^n\) is the *n*-th power of *R* and \(S^m\) is the
*m*-th power of *S* (see [?]91). The ratio
of two relations is represented by numbers *n* and *m* that
are *relatively prime*, namely, if

 

 \[\forall j \forall k \forall l [(n = j \times l \amp m = k \times l
 )\supset l = 1].\]


*Ratios* thus are relations between relations. *Rational
Numbers*, in keeping with the generalized notions of all numbers
in PM will be classes of similar ratios, and thus the work of
developing the theory of rational, and then real numbers, is carried
out in terms of ratios and relations between ratios.


##### Real Numbers



The real numbers \(\Theta\) are defined as "the series of
segments of the series of ratios", or the set of Dedekind cuts
of the sequence of rational numbers, in their standard ordering.
Technically \(\Theta\) is a relation in extension. The individuals
that are related in the ratios, and thus at the basis of the series of
rational numbers must be infinite in number for the PM version of the
real numbers to have the structure we expect. In the introductory
material to the section they point out that while the construction of
the reals thus requires the Axiom of Infinity, they add it explicitly
to the theorems where needed and try as much as possible to derive the
results upon which it does not rely without making that
assumption.



The theory of real numbers in PM is closer to that of Frege than the
"arithmetizing" accounts of Dedekind or Cantor. Dedekind
postulated that irrational numbers are to be "created" to
fill out the gaps in the series of rational numbers that are marked by
Dedekind cuts (Dedekind 1872 [1901: 15], while Cantor (1883: SS9, para. 8
[1996: 899]) identified real numbers with the limits of sequences of rational
numbers.



Frege, and PM, however, see real numbers as abstracted from the
similiarity of relations with a certain structure. See Gandon (2012)
for a fine grained comparison of the Frege and Russell
constructions.



It is interesting to note that the overall structure and contents of
sections in PM and *Basic Laws* are similar. They both share a
first section on the symbolic logic that they will use, then a series
of definitions and theorems about the notions of number and concepts
used in arithmetic, with a concluding section on real numbers. While
the range of the mathematics that is to be reduced to logic in the two
works is the same, Frege restricts his work more directly to natural
and real numbers, while PM includes the theory of classes and
arithmetic of relations and relations and infinite sets. While all of
these topics are handled in the initial chapters of a contemporary
textbook in axiomatic set theory, as Urquhart (2013) suggests, this
may be the inevitable fate of even the most groundbreaking
mathematical works.



The addition and multiplication of real numbers is defined in the next
sections as they are for other numbers, but taking disjoint instances
of each number similarity class, and performing the corresponding
operation on them. Many results assume the Axiom of Infinity as an
hypothesis. The operations are symbolized as \(+\_s\) and
\(\times\_s\).



> 
> 
> Measurement--*i.e.* the application of ratios and real
> numbers to magnitudes--will be dealt with in Section C; for the
> present, we shall confine ourselves to those properties of magnitude
> which are presupposed by measurement....
> 
> 
> 
> We conceive of a magnitude as a vector, *i.e.* as an operation,
> *i.e.* as a descriptive function in the sense of
> [?]30. Thus for example, we shall so define our
> terms that 1 gramme would not be a magnitude, but the difference
> between 2 grammes and 1 gramme would be a magnitude, *i.e.* the
> relation "+1 gramme" would be a magnitude. On the other
> hand a centimetre and a second would both be magnitudes according to
> our definition, because distances in space and time are
> vectors....
> 
> 
> 
> We demand of a vector (1) that it shall be a one-one relation, (2)
> that it be capable of indefinite repetition, *i.e.* that if the
> vector takes us from *a* to *b*, there shall always be a
> point *c* such that the vector takes us from *b* to
> *c*. (PM, Vol. III, "Summary of Section B", 1913: 339
> and 1927: 339)
> 
> 
> 



The kinds of quantity addressed in this section and the next are
all



> 
> 
> *vector families* that is, classes of one-one relations all
> having the same converse domain, and all having the domain contained
> in the converse domain. (PM Vol. III, 350)
> 
> 
> 


#### 6.2.1 Measurement



Measurement in PM is based on the relations between objects that are
the basis for the operation of measurement, that one object is heavier
than another, or longer than another. Quantities are then equivalence
classes of objects which have the same relations to others. Operations
are defined on quantities, so that:



> 
> 
> ...that is to say, two-thirds of a pound of cheese ought to be
> \((2/3 \times\_s 1/2)\) of a pound of cheese, and similarly in every
> other case. (PM, Volume III, 407)
> 
> 
> 



PM concludes with a seemingly dangling excursion into the special case
of the measurement of "Cyclic Families". For "such
cases as the angles at a point, or the elliptic straight line, we
require a theory of measurement applicable to families which are not
open" (PM, Volume III, 457). The angles at a point will be
measured from 0 to 360 degrees, and then start over again at 0 to
measure an object rotating around a point. The many ratios that
represent the rotations are represented by a "principal
ratio", which is used to assign the measurement of degrees.


#### 6.2.2 There is no "Conclusion" at the end of PM



PM ends abruptly with the proof of a theorem
([?]375*32) concerning cyclic families,
without any concluding remarks or hint of what is to come later. The
thought is that further mathematics, including the Volume IV that
Whitehead was to write on geometry, would have to be developed
piecemeal. First the notions of a given branch of mathematics would
have to be defined in terms of earlier notions, such as classes of
relations with a given structure, and then the important basic results
in that field would be proved one by one, in the style of the work so
far. Establishing logicism would be an ongoing project, as open-ended
as mathematics itself.