## 1. Why Learn the Symbolism in *Principia Mathematica?*



*Principia Mathematica* [PM] was written jointly by Alfred
North Whitehead and Bertrand Russell over several years, and published
in three volumes, which appeared between 1910 and 1913. It presents a
system of symbolic logic and then turns to the foundations of
mathematics to carry out the logicist project of defining mathematical
notions in terms of logical notions and proving the fundamental axioms
of mathematics as theorems of logic. While hugely important in the
development of logic, philosophy of mathematics and more broadly of
"Early Analytic Philosophy", the work itself is no longer
studied for these topics. As a result the very notation of the work
has become alien to contemporary students of logic, and that has
become a barrier to the study of *Principia Mathematica*. We
include a series of definitions of notions such as transfinite
cardinal numbers, well-orderings, rational and real numbers. These are
defined differently in the theory of types of PM than than in
axiomatic set theory. 



This entry is intended to assist the student of PM in reading the
symbolic portion of the work. What follows is a partial translation of
the symbolism into a more contemporary notation, which should be
familiar from other articles in this Encyclopedia, and which is quite
standard in contemporary textbooks of symbolic logic. No complete
algorithm is supplied, rather various suggestions are intended to help
the reader learn the symbolism of PM. Many issues of interpretation
would be prejudged by only using contemporary notation, and many
details that are unique to PM depend on that notation. It will be seen
below, with some of the more contentious aspects of the notation, that
doctrines of substance are built into the notation of PM. Replacing
the notation with a more modern symbolism would drastically alter the
very content of the book.


## 2. Primitive Symbols of Mathematical Logic (Part I)



Below the reader will find, in the order in which they are introduced
in PM, the following symbols, which are briefly described. More detail
is provided in what follows:




|  |  |
| --- | --- |
| [?] | pronounced "star"; indicates a number, or chapter,
as in [?]1, or [?]20.  |
| * | a centered dot (an old British decimal point); indicates a
numbered sentence in the order by first digit (all the 0s preceding
all the 1s etc.), then second digit, and so on. The first definitions
and propositions of [?]1 illustrate this
"lexicographical" ordering: 1*01, 1*1,
1*11, 1*2, 1*3, 1*4, 1*5,
1*6, 1*7, 1*71, 1*72.  |
| \(\vdash\) | the assertion-sign; precedes an *assertion*, either an
*axiom* (i.e., a *primitive proposition*, which are also
annotated "\(\Pp\)") or a *theorem*. |
| \(\Df\) | the definition sign; follows a *definition*. |
| \(.\),   \(:\),   \(:.\),  
\(::\),  etc. | are dots used for delimiting punctuation; in contemporary logic,
we use ( ), [ ], \(\{\ \}\), etc.  |
| \(p, q, r\), etc. | are *propositional variables*. |
| \(\lor\), \(\supset\), \(\osim\), \(\equiv\) and . , :, :., etc.
 | are the familiar sentential *connectives*, corresponding
to "*or*", "*if-then*",
"*not*", "*if and only if* , and
"*and*" respectively. (The dual use of dots for
conjunction and punctuation will be explained below.) In the Second
Edition of PM, 1925-27, the Sheffer Stroke
"\(\mid\)" is the one primitive connective. It means
"*not both* ... *and* \_\_\_".  |
| \(x, y, z\), etc. | are *individual variables*, which are to be read with
"typical ambiguity", i.e., with their *logical
types* to be filled in (see below). |
| \(a, b, c\), etc. | are *individual constants*, and stand for individuals (of
the lowest type). These occur only in the Introduction to PM, and not
in the official system. |
| \(xRy, aRb, R(x)\), etc. | are *atomic predications*, in which the objects named by
the variables or constants stand in the relation \(R\) or have the
property \(R\). These occur only in the Introduction.
"\(a\)" and "\(b\)" occur as constants only in
the Second Edition. The predications \(R(x), R(x,y)\), etc., are used
only in the Second Edition. |
| \(\phi\), \(\psi\), \(\chi\), etc.,

and \(f, g\), etc. | are higher-order variables which range over *propositional
functions*, no matter whether those functions are simple or
complex.  |
| \(\phi x\), \(\psi x\), \(\phi(x,y)\), etc. | open atomic formulas in which both "\(x\)" and
"\(\phi\)" are free. [An alternative interpretation is to
view "\(\phi x\)" as a schematic letter standing for a
formula in which the variable "\(x\)" is free.]  |
| \(\hat{\phantom{x}}\) | the circumflex; when placed over a variable in an open formula
(as in "\(\phi \hat{x}\)") results in a term for a
function. [This matter is controversial. See Landini 1998.] When the
circumflected variable precedes a complex variable, the result
indicates a class, as in \(\hat{x}\phi x\) which is the class of \( x
\) which are \( \phi \), \( \{ x \mid \phi x\} \) in modern notation.
 |
| \(\phi\hat{x}, \psi\hat{x}, \phi(\hat{x},\hat{z}),\) etc. | Terms for propositional functions. Here are examples of such
terms which are constants: "\(\hat{x}\) is happy",
"\(\hat{x}\) is bald and \(\hat{x}\) is happy", "\(4
\lt \hat{x} \lt 6\)", etc. If we apply, for example, the
function "\(\hat{x}\) is bald and \(\hat{x}\) is happy" to
the particular individual \(b\), the result is the proposition
"\(b\) is bald and \(b\) is happy".  |
| \(\exists\) and ( ) | are the *quantifiers* "there exists" and
"for all" ("every"), respectively. For
example, where \(\phi x\) is a simple or complex open formula,



|  |  |  |
| --- | --- | --- |
| \((\exists x)\phi x\) | asserts | "there exists an \(x\) such that \(\phi x\)" |
| \((\exists \phi)\phi x\) | asserts | "there exists a propositional function \(\phi\) such that
\(\phi x\)" |
| \((x)\phi x\) | asserts | "every \(x\) is such that \(\phi x\)" |
| \((\phi)\phi x\) | asserts | "every propositional function \(\phi\) is such that \(\phi
x\)" |


[These were used by Peano. More recently, \(\forall\) has been added
for symmetry with \(\exists\). Some scholars see the quantfiers
\((\phi)\) and \((\exists \phi\)) as substitutional.] |
| \(\phi x \supset\_x \psi x\)

\(\phi x \equiv\_x \psi x\) | This is notation that is used to abbreviate universally
quantified variables. In modern notation, these become \(\forall
x(\phi x \supset \psi x)\) and \(\forall x(\phi x \equiv \psi x)\),
respectively. See the definitions for this notation at the end of
 Section 3.2
 below. |
| \(\bang\) | pronounced "shriek"; indicates that a function is
*predicative*, as in \(\phi \bang x\) or \(\phi\bang \hat{x}\).
See
 Section 7. |
| = | the identity symbol; expresses *identity*, which is a
defined notion in PM, not primitive as in contemporary logic. |
| \(\atoi\) | read as "the"; is the inverted *iota* or
*description operator* and is used in expressions for definite
descriptions, such as \((\atoi x)\phi x\) (which is read: the \(x\)
such that \(\phi x\)). |
| [\((\atoi x)\phi x\)] | a definite description in brackets; this is a *scope
indicator* for definite descriptions. |
| \(E\bang\) |  is defined at [?]14*02, in the context \(E\bang
(\atoi x)\phi x\), to mean that the description \((\atoi x)\phi x\) is
*proper*, i.e., there is one and only one thing that is
\(\phi\). |
| \(\exists\bang\) | is defined at [?]24*03, in the context \(\exists
\bang \alpha\), to mean that the class \(\alpha\) is
*non-empty*, i.e., has a member. |


 The evolution of this selection of primitive symbols out of Peano's symbolism is traced in Elkind and Zach (forthcoming). 


## 3. The Use of Dots in PM



An immediate obstacle to reading PM is the unfamiliar use of dots for
punctuation, instead of the more common parentheses and brackets. The
system is precise, and can be learned with just a little practice. The
use of dots for punctuation is not unique to PM. Originating with
Peano, it was later used in works by Alonzo Church, W.V.O. Quine, and
others, but it has now largely disappeared. Alan Turing made a study
of the use of dots from a computational point of view in 1942,
presumably in his spare time after a day's work at Bletchley Park
breaking the codes of the Enigma Machine. Turing suggests that the use
of juxtaposition to indicate conjunction is similar to the use of
juxtaposition arithmetic to indicate multiplication:



> 
> In most systems there is some operation which is described simply by
> juxtaposition, without any special operator. In Church's system this
> is the application of a function to its argument; in Russell's it is
> conjunction and in algebra it is multiplication". (Turing 1942,
> 151)
> 



In his earlier work, such as  *The Principles of Mathematics,*
from 1903, Russell followed Peano's practice of indicating conjunction
by simply juxtaposing formulas. Thus the conjunction of \(p\) and \(q
\) was written \(p q \). Russell began to use the punctuation dot for
conjunction by 1905. The use of dots for punctuation in logic is now
only of historical interest, although some textbooks use a raised dot
\(p \cdot q \) for conjunction. Below we will explain the dual use of
dots for punctuation and conjunction in PM.



The best way to learn to use it is to look at a few samples which are
translated to formulae using parentheses, and thus to get the feel for
it. What follows is an explanation as presented in PM, pages
9-10, followed by a number of examples which illustrate each of
its clauses:



> 
> The use of dots. Dots on the line of the symbols have two uses, one to
> bracket off propositions, the other to indicate the logical product of
> two propositions. Dots immediately preceded or followed by
> "\(\lor\)" or "\(\supset\)" or
> "\(\equiv\)" or "\(\vdash\)", or by
> "\((x)\)", "\((x,y)\)",
> "\((x,y,z)\)" ... or "\((\exists x)\)",
> "\((\exists x,y)\)", "\((\exists x,y,z)\)"
> ... or "\([(\atoi x)(\phi x)]\)" or
> "\([R'y]\)" or analogous expressions, serve to
> bracket off a proposition; dots occurring otherwise serve to mark a
> logical product. The general principle is that a larger number of dots
> indicates an outside bracket, a smaller number indicates an inside
> bracket. The exact rule as to the scope of the bracket indicated by
> dots is arrived at by dividing the occurrences of dots into three
> groups which we will name I, II, and III. Group I consists of dots
> adjoining a sign of implication \((\supset)\) or equivalence
> \((\equiv)\) or of disjunction \(( \lor)\) or of equality by
> definition \((=\Df)\). Group II consists of dots following brackets
> indicative of an apparent variable, such as \((x)\) or \((x,y)\) or
> \((\exists x)\) or \((\exists x,y)\) or \([(\atoi x)(\phi x)]\) or
> analogous expressions. Group III consists of dots which stand between
> propositions in order to indicate a logical product. Group I is of
> greater force than Group II, and Group II than Group III. The scope of
> the bracket indicated by any collection of dots extends backwards or
> forwards beyond any smaller number of dots, or any equal number from a
> group of less force, until we reach either the end of the asserted
> proposition or a greater number of dots or an equal number belonging
> to a group of equal or superior force. Dots indicating a logical
> product have a scope which works both backwards and forwards; other
> dots only work away from the adjacent sign of disjunction,
> implication, or equivalence, or forward from the adjacent symbol of
> one of the other kinds enumerated in Group II. Some examples will
> serve to illustrate the use of dots. (PM, 9-10)
> 


For a deeper discussion of this passage on dot notation, see the
supplement on:



The Use of Dots for Punctuation and For Conjunction.



### 3.1 Some Basic Examples



Consider the following series of extended examples, in which we
examine propositions in PM and then discuss how to translate them step
by step into modern notation. (Symbols below are sometimes used as
names for themselves, thus avoiding some otherwise needed quotation
marks. Russell is often accused of confusing use and mention, so there
may well be some danger in this practice.) 


#### Example 1

 

\[\tag\*{[?]1*2}
{\vdash} \colon p \lor p \ldot {\supset} \ldot p \quad\Pp
\]


This is the second assertion of "star" 1. It is in fact an
axiom or "Primitive Proposition" as indicated by the
'\(\Pp\)'. That this is an assertion (axiom or theorem)
and not a definition is indicated by the use of
"\(\vdash\)". (By contrast, a definition would omit the
assertion sign but conclude with a "\(\Df\)" sign.) Now
the first step in the process of translating [?]1*2 into
modern notation is to note the colon. Recall, from the above quoted
passage, that "a larger number of dots indicates an outside
bracket, a smaller number indicates an inside bracket". Thus,
the colon here (which consists of a larger number of dots than the
single dots occurring on the line in [?]1*2) represents an
outside bracket. The brackets "[" and "]"
represent the colon in [?]1*2. The scope of the colon thus
extends past any smaller number of dots (i.e., one dot) to the end of
the formula. Since formulas are read from left to right the expression
"past" means "to the right of".



So, the first step is to translate [?]1*2 to: 



\[
\vdash[ p \lor p \ldot {\supset} \ldot p]
\]


Next, the dots around the "\(\supset\)" are represented in
modern notation by the parenthesis around the antecedent and
consequent. Recall, in the above passage, we find "... dots
only work away from the adjacent sign of disjunction, implication, or
equivalence ...". Thus, the next step in the translation
process is to move to the formula: 

\[
\vdash [(p \lor p) \supset(p)]
\]

 



Finally, standard modern conventions allow us to delete the outer
brackets and the parentheses around single letters, yielding:



\[
\vdash(p \lor p) \supset p
\]


Our next example involves conjunction:


#### Example 2

 

\[
 \tag\*{[?]3*01}
 p \sdot q \ldot {=} \ldot \osim(\osim p \lor \osim q)
 \quad\Df
\]


The dual use of dots to "indicate" conjunction and
punctuation can be understood through a careful examination of the
details of the paragraph on the use of dots from pages 9 to 11 of PM. Try
reading the dots as punctuation first, and then, if that won't
work, those dots must indicate conjunction.



[?]3*01 defines the use of dots to indicate conjunction.
(That first dot, when read as punctuation would extend until an equal
number of dots, namely the dot before the = sign, yielding the
incoherent expression: "( \(p ( q) =\_\mathit{df} ( \osim (\osim
p \lor \osim q) ) \) ". It must, therefore, indicate a
conjunction.) The dots around a sign of equality by definition
\(=\_\mathit{df} \) are in Group I, and so the parentheses that replace
them extend to the ends of the expression: 

 

\[
(p \: .\: q) =\_\mathit{df} ( \osim (\osim p \lor \osim q) )
\]

 Then, we
delete the outer parentheses on the right and left as unnecessary for
interpreting the formula, so we have: 

\[
p . q =\_\mathit{df} \osim (\osim p \lor \osim q)
\]

 in a modern notation

\[ p \amp q =\_\mathit{df} \osim (\osim p \lor \osim q) \] 


Notice that the scope of the negation sign "\(\osim\)" in
[?]3*01 is not indicated with dots, even in the PM system,
but rather uses parentheses.


#### Example 3

 

\[
\tag\*{[?]9*01}
 \osim \{(x) \sdot \phi x\} \ldot {=} \ldot (\exists x) \sdot \osim \phi x
 \quad\Df
\]


We apply the rule "dots only work away from the adjacent sign of
disjunction, implication, or equivalence, or forward from the adjacent
symbol of one of the other kinds enumerated in Group II" (where
Group II includes "\((\exists x)\)"). In this case the
first dot extends to the punctuation symbol \( \} \) which is allowed
optionally to replace dots. No such punctuation after the quantifier
(or after the negation) occur in the modern equivalent which would be:

\[
\osim (x)\phi x =\_\mathit{df} (\exists x)\osim \phi x
\]

 or 

\[
\osim \forall x\phi x =\_\mathit{df} \exists x\osim \phi x
\]



The ranking of connectives in terms of relative "force",
or *scope*, is a standard convention in contemporary logic. If
there are no explicit parentheses to indicate the scope of a
connective those which have precedence in the ranking are presumed to
be the principal connective, and so on for subformulas. Thus, instead
formulating the following DeMorgan's law as the cumbersome:



\[
[(\osim p) \lor (\osim q)] \equiv[\osim (p \amp q)]
\]


we nowadays write it as:

 

\[
\osim p \lor \osim q \equiv\osim (p \amp q)
\]


This simpler formulation follows from the convention that \(\equiv\)
has wider scope than \(\lor\) and &, and the latter have wider
scope than \(\osim\). Indeed parentheses are often unneeded around
\(\equiv\), given a further convention on which \(\equiv\) has wider
scope than \(\supset\). Thus, the formula \(p \supset q \equiv\osim
p\lor q\) becomes unambiguous. We might represent these conventions by
listing the connectives in groups with those with widest scope at the
top: 

 

\[\begin{array}{c}
\equiv \\
\supset \\
\amp, \lor \\
\osim 
\end{array}\]


For Whitehead and Russell, however, the symbols \(\supset\),
\(\equiv\), \(\lor\) and \(\ldots =\ldots \Df\), in Group I, are of
equal force. Group II consists of the variable binding expressions,
quantifiers and scope indicators for definite descriptions, and Group
III consists of conjunctions. Negation is below all of these. So the
ranking in PM would be:

 

\[\begin{array}{c}
\supset, \equiv, \lor \text{ and } \ldots = \ldots \quad\Df \\
(x), (x,y) \ldots (\exists x), (\exists x,y) \ldots [(\atoi x)\phi x] \\
p \sdot q \\
\osim 
\end{array}\]


This is what Whitehead and Russell seem to mean when they say
"Group I is of greater force than Group II, and Group II than
Group III." Consider the following:


#### Example 4

 

\[
\tag\*{[?]3*12}
 {\vdash} \colon \osim p \ldot {\lor} \ldot \osim q \ldot {\lor} \ldot p \sdot q
\]


This theorem illustrates how to read multiple uses of the same number
of dots within one formula. Grouping "associates to the
left" both for dots and for a series of disjunctions, following
the convention of reading from left to right and the definition:



\[
\tag\*{[?]2*33} p \vee q \vee r \ldot {=} \ldot (p \vee q) \vee r \quad\Df
\]


In [?]3*12, the first two dots around the \(\lor\) simply
"work away" from the connective. The second
"extends" until it meets with the next of the same number
(the third single dot). That third dot, and the fourth "work
away" from the second \(\lor\), and the final dot indicates a
conjunction with least force. The result, formulated with all possible
punctuation for maximum explicitness, is:

 

\[
\{[(\osim p) \lor (\osim q)] \lor (p \amp q)\}
\]


If we employ all the standard conventions for dropping parentheses,
this becomes:

 

\[
(\osim p \lor \osim q) \lor (p \amp q)
\]


This illustrates the passage in the above quotation which says
"The scope of the bracket indicated by any collection of dots
extends backwards or forwards beyond any smaller number of dots, or
any equal number from a group of less force, until we reach either the
end of the asserted proposition or a greater number of dots or an
equal number belonging to a group of equal or superior
force."



Before we look at a wider range of examples, a detailed example
involving quantified variables will prove to be instructive. Whitehead
and Russell follow Peano's practice of expressing universally
quantified conditionals (such as "All \(\phi\)s are
\(\psi\)s") with the bound variable subscripted under the
conditional sign. Similarly with universally quantified biconditionals
("All and only \(\phi\)s are \(\psi\)s"). That is, the
expressions "\(\phi x \supset\_x \psi x\)" and
"\(\phi x \equiv\_x \psi x\)" are defined as follows:



\[
\tag\*{[?]10*02} 
\phi x \supset\_x \psi x \ldot {=} \ldot (x) \ldot \phi x \supset \psi x \quad\Df
\]

 

\[
\tag\*{[?]10*03}
 \phi x \equiv\_x \psi x \ldot {=} \ldot (x) \ldot \phi x \equiv \psi x \quad\Df 
\]


and correspond to the following more modern formulas,
respectively:

 

\[
\forall x(\phi x \supset \psi x)
\] 

 

\[
\forall x(\phi x \equiv \psi x)
\]


As an exercise the reader might be inclined to formulate a rigorous
algorithm for converting PM into a particular contemporary symbolism
(with conventions for dropping parentheses), but the best way to learn
the system is to look over a few more examples of translations, and
then simply begin to read formulae directly.


### 3.2 More Examples



In the examples below, each formula number is followed first by
*Principia* notation and then its modern translation. Notice
that in [?]1*5 parentheses are used for punctuation in
addition to dots. (Primitive Propositions [?]1*2,
[?]1*3, [?]1*4, [?]1*5, and
[?]1*6 together constitute the axioms for propositional
logic in PM.) Proposition [?]1*5 was shown to be redundant
by Paul Bernays in 1926. It can be derived from appropriate instances
of the others and the rule of modus ponens.




|  |  |
| --- | --- |
| [?]1*3 | \({\vdash} \colon q \ldot {\supset} \ldot p \lor q \quad\Pp\)

\(q \supset p \lor q\)  |
| [?]1*4 | \({\vdash} \colon p \lor q \ldot {\supset} \ldot q \lor p
\quad\Pp\)

\(p \lor q \supset q \lor p\)  |
| [?]1*5 | \({\vdash} \colon p \lor (q \lor r ) \ldot {\supset} \ldot q
\lor (p \lor r ) \quad\Pp\)

\(p \lor (q \lor r ) \supset q \lor (p \lor r )\)  |
| [?]1*6 | \({\vdash} \colondot q \supset r \ldot {\supset} \colon p \lor q
\ldot {\supset} \ldot p \lor r \quad\Pp\)

\((q \supset r ) \supset(p \lor q \supset p \lor r )\)  |
| [?]2*03 | \({\vdash} \colon p \supset \osim q \ldot {\supset} \ldot q
\supset\osim p\)

\((p \supset\osim q) \supset(q \supset\osim p)\)  |
| [?]3*3 | \({\vdash} \colondot p \sdot q \ldot {\supset} \ldot r \colon
{\supset} \colon p \ldot {\supset} \ldot q \supset r\)

\([(p \amp q) \supset r] \supset [p \supset(q \supset r)]\)  |
| [?]4*15 | \({\vdash} \colondot p \sdot q \ldot {\supset} \ldot \osim r
\colon {\equiv} \colon q \sdot r \ldot {\supset} \ldot \osim p\)

\([(p \amp q )\supset\osim r ] \equiv [(q \amp r )\supset\osim p ]\)
 |




|  |  |
| --- | --- |
| [?]5*71 | \({\vdash} \colondot q \supset\osim r \ldot {\supset} \colon p
\lor q \sdot r \ldot {\equiv} \ldot p \sdot r\)

\((q \supset\osim r) \supset \{ [(p \lor q) \amp r ] \equiv (p \amp r)
\}\)  |
| [?]9*04 | \(p \ldot {\lor} \ldot (x) \ldot \phi x \colon {=} \ldot (x)
\ldot \phi x \lor p \quad\Df\)

\(p \lor \forall x\phi x =\_\mathit{df} \forall x(\phi x \lor p)\)
 |




|  |  |
| --- | --- |
| [?]9*521 | \({\vdash} \colons (\exists x) \ldot \phi x \ldot {\supset}
\ldot q \colon {\supset} \colondot (\exists x) \ldot \phi x \ldot
{\lor} \ldot r \colon {\supset} \ldot q \lor r\)

[\((\exists x\phi x) \supset q] \supset [((\exists x\phi x) \lor r)
\supset (q \lor r)\)]  |
| [?]10*55 | \({\vdash} \colondot (\exists x) \ldot \phi x \sdot \psi x
\colon \phi x \supset\_x \psi x \colon {\equiv} \colon (\exists x)
\ldot \phi x \colon \phi x \supset\_x \psi x\)

\(\exists x(\phi x \amp \psi x) \amp \forall x(\phi x \supset \psi x)
\equiv \exists x\phi x \amp \forall x(\phi x \supset \psi x)\)
 |



 Notice that there are two uses of double dots ':'
in [?]10*55 to indicate conjunctions.

## 4. Propositional Functions



There are two kinds of functions in PM. Propositional functions such
as "\(\hat{x}\) is a natural number" are to be
distinguished from the more familiar mathematical functions, which are
called "descriptive functions" (PM, Chapter \(\ast\)31).
Descriptive functions are defined using relations and definite
descriptions. Examples of descriptive functions are \(x + y\) and
"the successor of \(n\)".



Focusing on propositional functions, Whitehead and Russell distinguish
between expressions with a free variable (such as "\(x\) is
hurt") and names of functions (such as "\(\hat{x}\) is
hurt") (PM, 14-15). The propositions which result from the
formula by assigning allowable values to the free variable
"\(x\)" are said to be the "ambiguous values"
of the function. Expressions using the circumflex notation, such as
\(\phi \hat{x}\) only occur in the introductory material in the
technical sections of PM and not in the technical sections themselves
(with the exception of the sections on the theory of classes),
prompting some scholars to say that such expressions do not really
occur in the formal system of PM. This issue is distinct from that
surrounding the interpretation of such symbols. Are they
"term-forming operators" which turn an open formula into a
name for a function, or simply a syntactic device, a placeholder, for
indicating the variable for which a substitution can be made in an
open formula? If they are to be treated as term-forming operators, the
modern notation for \(\phi \hat{x}\) would be \(\lambda x\phi x\). The
\(\lambda\)-notation has the advantage of clearly revealing that the
variable \(x\) is *bound* by the term-forming operator
\(\lambda\), which takes a predicate \(\phi\) and yields a term
\(\lambda x\phi x\) (which in some logics is a singular term that can
occur in the subject position of a sentence, while in other logics is
a complex predicative expression). Unlike \(\lambda\)-notation, the PM
notation using the circumflex cannot indicate scope. The function
expression "\(\phi(\hat{x},\hat{y}\))" is ambiguous
between \(\lambda x\lambda y\phi xy\) and \(\lambda y\lambda x\phi
xy\), without some further convention. Indeed, Whitehead and Russell
specified this convention for relations in extension (on p. 200 in the
introductory material of [?]21, in terms of the order of the
variables), but the ambiguity is brought out most clearly by using
\(\lambda\) notation: the first denotes the relation of being an \(x\)
and \(y\) such that \(\phi xy\) and the second denotes the converse
relation of being a \(y\) and \(x\) such that \(\phi xy\).


## 5. The Missing Notation for Types and Orders



This section explains notation that is not in *Principia
Mathematica*. Except for some notation for "relative"
types in \(\ast 63\), and again in early parts of Volume II, there are
famously no symbols for types in *Principia Mathematica*!
Sentences are generally to be taken as "typically
ambiguous" and so standing for expressions of a whole range of
types and so just as there are no individual or predicate constants,
there are no particular functions of any specific type. So not only
does one not see how to symbolize the argument:



All men are mortal
  

Socrates is a man
  

Therefore, Socrates is mortal 



but also there is no indication of the logical type of the function
"\(\hat{x}\) is mortal". The project of PM is to reduce
mathematics to logic, and part of the view of logic behind this
project is that logical truths are all completely general. The
derivation of truths of mathematics from definitions and truths of
logic will thus not involve any particular constants other than those
introduced by definition from purely logical notion. As a result no
notation is included in PM for describing those types. Those of us who
wish to consider PM as a logic which can be applied, must supplement
it with some indication of types.



Readers should note that the explanation of types outlined below is
not going to correspond with the statements about types in the text of
PM. Alonzo Church [1976] developed a simple, rational reconstruction
of the notation for both the simple and ramified theory of types as
implied by the text of PM. (There are alternative, equivalent
notations for the theory of types.) The full theory can be seen as a
development of the *simple theory of types*. 


### 5.1 Simple Types



A definition of the simple types can be given as follows:


* \(\iota\) (Greek iota) is the type for an *individual*.
* Where \(\tau\_1,\ldots,\tau\_n\) are any types, then
\(\ulcorner(\tau\_1,\ldots,\tau\_n)\urcorner\) is the type of a
propositional function whose arguments are of types
\(\tau\_1,\ldots,\tau\_n\), respectively.
* \(\ulcorner\)( )\(\urcorner\) is the type of propositions.



Here are some intuitive ways to understand the definition of type.
Suppose that "Socrates" names an individual. (We are here
ignoring Russell's considered opinion that such ordinary
individuals are in fact classes of classes of sense data, and so of a
much higher type.) Then the individual constant "Socrates"
would be of type \(\iota\). A monadic propositional function which
takes individuals as arguments is of type \((\iota)\). Suppose that
"is mortal" is a predicate expressing a property of
individuals. The function "\(\hat{x}\) is mortal" will be
of the type \((\iota)\). A two-place or *binary* relation
between individuals is of type \((\iota,\iota)\). Thus, a relation
expression like "parent of" and the function
"\(\hat{x}\) is a parent of \(\hat{z}\)" will be of type
\((\iota,\iota)\). 



Propositional functions of type \((\iota)\) are often called
"first order"; hence the name "first order
logic" for the familiar logic where the variables only range
over arguments of first order functions. A monadic function of
arguments of type \(\tau\) are of type \((\tau)\) and so functions of
such functions are of type \(((\tau))\). "Second order
logic" will have variables for the arguments of such functions
(as well as variables for individuals). Binary relations between
functions of type \(\tau\) are of type \((\tau,\tau)\), and so on, for
relations of having more than 2 arguments. Mixed types are defined by
the above. A relation between an individual and a proposition (such as
"\(\hat{x}\) believes that \(\hat{P}\)") will be of type
\((\iota\),( )).


### 5.2 Ramified Types



To construct a notation for the full ramified theory of types of PM,
another piece of information must be encoded in the symbols. Church
calls the resulting system one of *r-types*. The key idea of
ramified types is that any function defined using quantification over
functions of some given type has to be of a higher "order"
than those functions. To use Russell's example:



\(\hat{x}\) has all the qualities that great generals have 



is a function true of persons (i.e., individuals), and from the point
of view of *simple* type theory, it has the same simple logical
type as particular qualities of individuals (such as bravery and
decisiveness). However, in ramified type theory, the above function
will be of a higher order than those particular qualities of
individuals, since unlike those particular qualities, it involves a
quantification over those qualities. So, whereas the expression
"\(\hat{x}\) is brave" denotes a function of r-type
\((\iota)/1\), the expression "\(\hat{x}\) has all the qualities
that great generals have" will have r-type \((\iota)/2\). In
these r-types, the number after the "/" indicates the
*level* of the function. The order of the functions will be
defined and computed given the following definitions.



Church defines the r-types as follows:


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



The *order* of an entity is defined as follows (here we no
longer follow Church, for he defines orders for variables, i.e.,
expressions, instead of orders for the things the variables range
over):


* 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 order of the arguments
\(\tau\_1,\ldots,\tau\_m\).



These two definitions are supplemented with a principle which
identifies the levels of particular defined functions, namely, that
the level of a defined function should be one higher than the highest
order entity having a name or variable that appears in the definition
of that function.



To see how these definitions and principles can be used to compute the
order of the function "\(\hat{x}\) has all the qualities that
great generals have", note that the function can be represented
as follows, where "\(x, y\)" are variables ranging over
individuals of r-type \(\iota\) (order 0),
"GreatGeneral\((y)\)" is a predicate denoting a
propositional function of r-type \((\iota)/1\) (and so of order 1),
and "\(\phi\)" is a variable ranging over propositional
functions of r-type \((\iota)/1\) (and so of order 1) such as
*great general*, *bravery*, *leadership*,
*skill*, *foresight*, etc.:

 

\[
(\phi)\{[(y)(\textrm{GreatGeneral}(y) \supset \phi(y)] \supset \phi \hat{x} \}
\]


We first note that given the above principle, the r-type of this
function is \((\iota)/2\); the level is 2 because the level of the
r-type of this function has to be one higher than the highest order of
any entity named (or in the range of a variable used) in the
definition. In this case, the denotation of GreatGeneral, and the
range of the variable "\(\phi\)", is of order 1, and no
other expression names or ranges over an entity of higher order. Thus,
the level of the function named above is defined to be 2. Finally, we
compute the order of the function denoted above as it was defined: the
sum of the level plus the greatest of the orders of the arguments of
the above function. Since the only arguments in the above function are
individuals (of order 0), the order of our function is just 2. 



Quantifying over functions of r-type \((\tau)/n\) of order \(k\) in a
definition of a new function yields a function of r-type
\((\tau)/n+1\), and so a function of order one higher, \(k+1\). Two
kinds of functions, then, can be of the *second order*: (1)
functions of first-order functions of individuals, of r-type
\(((\iota)/1)/1\), and (2) functions of r-type \((\iota)/2\), such as
our example "\(\hat{x}\) has all the qualities that great
generals have". This latter will be a function true of
individuals such as Napoleon, but of a higher order than simple
functions such as "\(\hat{x}\) is brave", which are of
r-type \((\iota)/1\).



Logicians today use a different notion of "order". Today,
first-order logic is a logic with only variables for individuals.
Second order logic is a logic with variables for both individuals and
properties of individuals. Third-order logic is a logic with variables
for individuals, properties of individuals, and properties of
properties of individuals. And so forth. By contrast, Church would
call these logics, respectively, the logic of functions of the types
\((\iota)/1\) and \((\iota,\ldots,\iota)/1\), the logic of functions
of the types \(((\iota)/1)/1\) and
\(((\iota,\ldots,\iota)/1,\ldots,(\iota,\ldots,\iota)/1)/1\), and the
logic of functions of the types \((((\iota)/1)/1)/1\) etc. (i.e., the
level-one functions of the functions of the preceding type). Given
Church's definitions, these are logics of first-, second- and
third-order functions, respectively, thus coinciding with the modern
terminology of "\(n\)th-order logic".


## 6. Variables



As mentioned previously, there are no individual or predicate
constants in the formal system of PM, only variables. The
Introduction, however, makes use of the example "\(a\) standing
in the relation \(R\) to \(b\)" in a discussion of atomic facts
(PM, 43). Although "\(R\)" is later used as a variable
that ranges over relations in extension, and
"\(a,b,c,\ldots\)" are individual variables, let us
temporarily add them to the system as predicate and individual
constants, respectively, in order to discuss the use of variables in
PM.



PM makes special use of the distinction between "real", or
free, variables and "apparent", or bound, variables. Since
"\(x\)" is a variable, "\(xRy\)" will be an
atomic formula in our extended language, with "\(x\)" and
"\(y\)" real variables. When such formulae are combined
with the propositional connectives \(\osim\), \(\lor\), etc., the
result is a *matrix*. For example, "\(aRx \ldot {\lor}
\ldot xRy\)" would be a matrix.



As we saw earlier, there are also variables which range over
functions: "\(\phi\), \(\psi\), \(\ldots,f, g\)", etc. The
expression "\(\phi x\)" thus contains two variables and
stands for a proposition, in particular, the result of applying the
function \(\phi\) to the individual \(x\).



Theorems are stated with real variables, which gives them a special
significance with regard to the theory. For example,

 

\[
\tag\*{[?]10*1}
 \vdash \colon (x) \ldot \phi x \ldot {\supset} \ldot \phi y \quad\Pp
\]


is a fundamental axiom of the quantificational theory of PM. In this
Primitive Proposition the variables "\(\phi\)" and
"\(y\)" are real (free), and the "\(x\)" is
apparent (bound). As there are no constants in the system, this is the
closest that PM comes to a rule of universal instantiation.



Whitehead and Russell interpret "\((x) \sdot \phi x\)" as
"the proposition which asserts *all* the values for
\(\phi \hat{x}\)" (PM 41). The use of the word "all"
has special significance within the theory of types. They present the
*vicious circle principle*, which underlies the theory of
types:



> 
> ... generally, given any set of objects such that, if we suppose
> the set to have a total, it will contain members which presuppose this
> total, then such as set cannot have a total. By saying that the set
> has "no total", we mean, primarily, that no significant
> statement can be made about "all its members". (PM, 37)
> 



Specifically, then, a quantified expression, since it talks about
(presupposes) "all" the members of a totality, must
express a member of a different, higher, logical type than those
members in order to observe the vicious circle principle. Thus, when
interpreting a bound variable, we must assume that it ranges over a
specific type of entity, and so types must be assigned to the other
entities represented by expressions in the formula, in observance with
the theory of types.



A question arises, however, once one realizes that the statements of
primitive propositions and theorems in PM such as [?]10*1
are taken to be "typically ambiguous" (i.e., ambiguous
with respect to type). These statements are actually schematic and
represent all the possible specific assertions which can be derived
from them by interpreting types appropriately. But if statements like
[?]10*1 are schemata and yet have bound variables, how do
we assign types to the entities over which the bound variables range?
The answer is to first decide which type of thing the free variables
in the statement range over. For example, assuming that the variable
\(y\) in [?]10*1 ranges over individuals (of type
\(\iota)\), then the variable \(\phi\) must range over functions of
type \((\iota)/n\), for some \(n\). Then the bound variable \(x\) will
also range over individuals. If, however, we assume that the variable
\(y\) in [?]10*1 ranges over *functions* of type
\((\iota)/1\), then the variable \(\phi\) must range over functions of
type \(((\iota)/1)/m\), for some \(m\). In this case, the bound
variable \(x\) will range over functions of type \((\iota)/1\).



So \(y\) and \(\phi\) are called "real" variables in
[?]10*1 not only because they are free but also because
they can range over any type. Whitehead and Russell frequently say
that real variables are taken to ambiguously denote "any"
of their instances, while bound variables (which also ambiguously
denote) range over "all" of their instances (within a
legitimate totality, i.e. type).


## 7. Predicative Functions and Identity



The exclamation mark "!" following a variable for a
function and preceding the argument, as in "\(f\bang
\hat{x}\)", "\(\phi \bang x\)", "\(\phi\bang
\hat{x}\)", indicates that the function is *predicative*,
that is, of the lowest order which can apply to its arguments. In
Church's notation, this means that predicative functions are all
of the first level, with types of the form \((\ldots)/1\). As a
result, predicative functions will be of order one more than the
highest order of any of their arguments. This analysis is based on
quotations like the following, in the *Introduction* to PM:




> 
> We will define a function of one variable as predicative when it is of
> the next order above that of its argument, i.e., of the lowest order
> compatible with its having that argument. (PM, 53)
> 



Unfortunately in the summary of [?]12, we find "A
predicative function is one which contains no apparent variables,
i.e., is a matrix" [PM, 167]. Reconciling this statement with
that definition in the *Introduction* is a problem for
scholars. 



To see the shriek notation in action, consider the following
definition of identity: 

 

\[
\tag\*{[?]13*01}
 x = y \ldot {=} \colon (\phi) \colon \phi \bang x \ldot {\supset} \ldot \phi \bang y \quad\Df 
\]


That is, \(x\) is identical with \(y\) if and only if \(y\) has every
predicative function \(\phi\) which is possessed by \(x\). (Of course
the second occurrence of "=" indicates a definition, and
does not independently have meaning. It is the first occurrence,
relating individuals \(x\) and \(y\), which is defined.) 



To see how this definition reduces to the more familiar definition of
identity (on which objects are identical iff they share the same
properties), we need the Axiom of Reducibility. The Axiom of
Reducibility states that for any function there is an equivalent
function (i.e., one true of all the same arguments) which is
predicative:



Axiom of Reducibility: 

\[
\tag\*{[?]12*1} \vdash \colon (\exists f) \colon \phi x \ldot {\equiv\_x} \ldot f\bang x \quad\Pp
\]



To see how this axiom implies the more familiar definition of
identity, note that the more familiar definition of identity is:



\[ 
x = y \ldot {=} \colon (\phi) \colon \phi x \ldot {\supset} \ldot
 \phi y \quad\Df
\] 


for \(\phi\) of "any" type. (Note that this differs from
[?]13*01 in that the shriek no longer appears.) Now to
prove this, assume both [?]13*01 and the Axiom of
Reducibility, and suppose, for proof by *reductio*, that \(x =
y\), and \(\phi x\), and not \(\phi y\), for some function \(\phi\) of
arbitrary type. Then, the Axiom of Reducibility [?]12*1
guarantees that there will be a predicative function \(\psi \bang\),
which is coextensive with \(\phi\) such that \(\psi \bang x\) but not
\(\psi \bang y\), which contradicts [?]13*01.


## 8. Definite Descriptions



The inverted Greek letter iota "\(\atoi\)" is used in PM,
always followed by a variable, to begin a definite description.
\((\atoi x) \phi x\) is read as "the \(x\) such that \(x\) is
\(\phi\)", or more simply, as "the \(\phi\)". Such
expressions may occur in subject position, as in \(\psi(\atoi x) \phi
x\), read as "the \(\phi\) is \(\psi\)". The formal part
of Russell's famous "theory of definite
descriptions" consists of a definition of all formulas
"...\(\psi(\atoi x) \phi x\)..." in which a
description occurs. To distinguish the portion \(\psi\) from the rest
of a larger sentence (indicated by the ellipses above) in which the
expression \(\psi(\atoi x) \phi x\) occurs, the *scope* of the
description is indicated by repeating the definite description within
brackets:

 

\[
[(\atoi x) \phi x] \sdot \psi(\atoi x) \phi x
\]


The notion of scope is meant to explain a distinction which Russell
famously discusses in "On Denoting" (1905). Russell says
that the sentence "The present King of France is not bald"
is ambiguous between two readings: (1) the reading where it says of
the present King of France that he is not bald, and (2) the reading
which denies that the present King of France is bald. The former
reading requires that there be a unique King of France on the list of
things that are not bald, whereas the latter simply says that there is
not a unique King of France that appears on the list of bald things.
Russell says the latter, but not the former, can be true in a
circumstance in which there is no King of France. Russell analyzes
this difference as a matter of the scope of the definite description,
though as we shall see, some modern logicians tend to think of this
situation as a matter of the scope of the negation sign. Thus, Russell
introduces a method for indicating the scope of the definite
description.



To see how Russell's method of scope works for this case, we
must understand the definition which introduces definite descriptions
(i.e., the inverted iota operator). Whitehead and Russell define:



\[
\tag\*{[?]14*01}
 [(\atoi x) \phi x] \sdot \psi(\atoi x) \phi x \ldot {=} \colon
(\exists b) \colon \phi x \ldot {\equiv\_x} \ldot x=b \colon \psi b
\quad\Df \]


This kind of definition is called a *contextual definition*,
which are to be contrasted with *explicit* definitions. An
explicit definition of the definition description would have to look
something like the following:

 

\[
(\atoi x)(\phi x) = \colon \ldots \quad\Df
\]


which would allow the definite description to be replaced in any
context by whichever defining expression fills in the ellipsis. By
contrast, [?]14*01 shows how a sentence, in which there is
occurrence of a description \((\atoi x)(\phi x)\) in a context
\(\psi\), can be replaced by some other sentence (involving \(\phi\)
and \(\psi\)) which is equivalent. To develop an instance of this
definition, start with the following example:



Example.
  

The present King of France is bald. 



Using \(PKFx\) to represent the propositional function of being a
present King of France and \(B\) to represent the propositional
function of being bald, Whitehead and Russell would represent the
above claim as:

 

\[
[(\atoi x)(PKFx)] \sdot B(\atoi x)(PKFx)
\]


which by [?]14*01 means:

 

\[
(\exists b) \colon PKFx \ldot {\equiv\_x} \ldot x=b \colon Bb
\]


In words, there is one and only one \(b\) which is a present King of
France and \(b\) is bald. In modern symbols, using \(b\)
non-standardly, as a variable, this becomes: 

 

\[
(\exists b)[\forall x(PKFx \equiv x=b) \amp Bb]
\]


Now we return to the example which shows how the scope of the
description makes a difference:



Example.
  

The present King of France is not bald. 



There are two options for representing this sentence.

 

\[
[(\atoi x)(Kx)] \sdot \osim B(\atoi x)(Kx)
\]


and

 

\[
\osim [(\atoi x)(Kx)] \sdot B(\atoi x)(Kx)
\]


In the first, the description has "wide" scope, and in the
second, the description has "narrow" scope. Russell says
that the description has "primary occurrence" in the
former, and "secondary occurrence" in the latter. Given
the definition [?]14*01, the two PM formulas immediately
above become expanded into primitive notation as:

 

\[
\begin{align}
(\exists b) \colon PKFx \equiv\_x x=b \colon \osim Bb\\
\osim (\exists b) \colon PKFx \equiv\_x x=b \colon Bb
\end{align}
\]


In modern notation these become:

 

\[
\begin{align}
\exists x[\forall y(PKFy \equiv y=x) \amp \osim Bx]\\
\osim \exists x[\forall y(PKFy \equiv y=x) \amp Bx]
\end{align}
\]


The former says that there is one and only one object which is a
present King of France and this object is not bald; i.e., there is
exactly one present King of France and he is not bald. This reading is
false, given that there is no present King of France. The latter says
it is not the case that there is exactly one thing that is a present
King of France and that object is bald. This reading is true because
there is not even one present King of France. 



Although Whitehead and Russell take the descriptions in these examples
to be the expressions which have scope, the above readings in both
expanded PM notation and in modern notation suggest why some modern
logicians take the difference in readings here to be a matter of the
scope of the negation sign. 


## 9. Classes, Relations, and Functions



The circumflex "^" over a variable preceding a
formula is used to indicate a class, thus \(\hat{x} \psi x\) is the
class of things \(x\) which are such that \(\psi x\). In modern
notation we represent this class as \(\{x \mid \psi x\}\), which is
read: the class of \(x\) which are such that \(x\) has \(\psi\).
Recall that "\(\phi \hat{x}\)", with the circumflex over a
variable after the predicate variable, expresses the propositional
function of being an \(x\) such that \(\phi x\). In the type theory of
PM, the class \(\hat{x} \phi x\) has the same logical type as the
function \(\phi \hat{x}\). This makes it appropriate to use the
following contextual definition, which allows one to eliminate the
class term \(\hat{x} \psi x\) from occurrences in the context \(f\):

\[
\tag\*{[?]20*01} 
 f\{ \hat{z}(\psi z)\} \ldot {=} \colon (\exists \phi) \colon \phi \bang x \ldot {\equiv\_x} \ldot \psi x \colon f \{ \phi\bang \hat{z}\} \quad\Df
\]

 or in modern notation: 

\[
f\{z \mid \psi z\} =\_\mathit{df} \exists \phi[\forall x(\phi x \equiv \psi x) \amp f(\lambda x \phi x)]
\]

 where \(\phi\) is a
predicative function of \(x\)



Note that \(f\) has to be interpreted as a higher-order function which
is predicated of the function \(\phi \bang \hat{z}\). In the modern
notation used above, the language has to be a typed language in which
\(\lambda\) expressions are allowed in argument position. As was
pointed out later (Chwistek 1924, Godel 1944, and Carnap 1947)
there should be scope indicators for class expressions just as there
are for definite descriptions. (The possibility of scope ambiguities
in propositions about classes is mentioned in the final sentences of
the *Introduction* (PM I, 84)). Chwistek, for example, proposed
copying the notation for definite descriptions, thus replacing
[?]20*01 with:

 

\[
[\hat{z}(\psi z)] \sdot f\{ \hat{z}(\psi z)\} \ldot {=} \colon (\exists \phi) \colon \phi \bang x \ldot {\equiv\_x} \ldot \psi x \colon f \{ \phi\bang \hat{z} \}
\]


Contemporary formalizations of set theory make use of something like
these contextual definitions, when they require an
"existence" theorem of the form \(\exists x\forall y(y \in
x \equiv \ldots y\ldots)\), in order to justify the introduction of a
singular term \(\{y \mid \ldots y\ldots \}\). See Suppes (1960). Given
the law of extensionality, it follows from \(\exists x\forall y(y \in
x \equiv \ldots y\ldots)\) that there is a unique such set. The
relation of membership in classes \(\in\) is defined in PM by first
defining a similar relationship between objects and propositional
functions: 

\[
\tag\*{[?]20*02}
x \in (\phi\bang \hat{z}) \ldot {=} \ldot \phi \bang x \quad\Df 
\]

 or, in modern notation: 

\[
x \in \lambda z\phi z =\_\mathit{df} \phi x
\]

 



[?]20*01 and [?]20*02 together are then used
to define the more familiar notion of membership in a class. The
formal expression "\(y \in \{ \hat{z}(\phi z)\}\)" can now
been seen as a context in which the class term occurs; it is then
eliminated by the contextual definition [?]20*01.
(Exercise)



In PM there is a class of all classes, Cls, defined as: 

\[
\tag\*{[?]20*03} \Cls = \hat{\alpha} \{ (\exists \phi ). \alpha = \hat{z} (
\phi ! z) \} \quad\Df 
\]





PM also has Greek letters for classes: \(\alpha, \beta, \gamma\), etc.
These will appear as bound (real) variables, apparent (free) variables
and in abstracts for propositional functions true of classes, as in
\(\phi \hat{\alpha}\). Only definitions of the bound Greek variables
appear in the body of the text, the others are informally defined in
the *Introduction*: 

\[
\tag\*{[?]20*07} 
(\alpha) \sdot f \alpha \ldot {=} \ldot (\phi) \sdot f \{ \hat{z}(\phi\bang z)\} \quad\Df
\]

 or, in modern notation,

\[
\forall \alpha\, f\alpha =\_\mathit{df} \forall \phi f\{z\mid\phi z\}
\]

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



Thus universally quantified class variables are defined in terms of
quantifiers ranging over predicative functions. Likewise for
existential quantification: 

\[
\tag\*{[?]20*071} 
 (\exists \alpha) \sdot f \alpha \ldot {=} \ldot (\exists \phi) \sdot f \{ \hat{z}(\phi\bang z)\} \quad\Df
\]

 or, in modern notation,

\[
\exists \alpha\, f\alpha =\_\mathit{df} \exists \phi f\{z\mid\phi z\}
\]

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



Expressions with a Greek variable to the left of \(\in\) are defined:

\[
\tag\*{[?]20*081}
\alpha \in \psi\bang \hat{\alpha} \ldot {=} \ldot \psi \bang \alpha \quad\Df
\]

 



These definitions do not cover all possible occurrences of Greek
variables. In the Introduction to PM, further definitions of \(f
\alpha\) and \(f \hat{\alpha}\) are proposed, but it is remarked that
the definitions are in some way peculiar and they do not appear in the
body of the work. The definition considered for \(f \hat{\alpha}\)
is:

 

\[
f \hat{\alpha} \ldot {=} \ldot (\exists \psi) \sdot \hat{\phi} \bang x \equiv\_x \psi \bang x \sdot f \{ \psi\bang \hat{z} \}
\]


or, in modern notation,

 

\[
\lambda \alpha\, f\alpha =\_\mathit{df} \lambda \phi f\{x \mid \phi x\}
\]


That is, \(f \hat{\alpha}\) is an expression naming the function which
takes a function \(\phi\) to a proposition which asserts \(f\) of the
class of \(\phi\)s. (The modern notation shows that in the proposed
definition of \(f \hat{\alpha}\) in PM notation, we shouldn't
expect \(\alpha\) in the definiens, since it is really a bound
variable in \(f \hat{\alpha}\); similarly, we shouldn't expect
\(\phi\) in the definiendum because it is a bound variable in the
definiens.) One might also expect definitions like
[?]20*07 and [?]20*071 to hold for cases in
which the Roman letter "\(z\)" is replaced by a Greek
letter. The definitions in PM are thus not complete, but it is
possible to guess at how they would be extended to cover all
occurrences of Greek letters. This would complete the project of the
"no-classes" theory of classes by showing how all talk of
classes can be reduced to the theory of propositional functions.


## 10. Concluding Mathematical Logic



Although students of philosophy usually read no further than
[?]20 in PM, this is in fact the point where the
"construction" of mathematics really begins. [?]21
presents the "General Theory of Relations" (the theory of
relations in extension; in contemporary logic these are treated as
sets of ordered pairs, following Wiener). \(\hat{x} \hat{y} \psi(x,
y)\) is the relation between \(x\) and \(y\) which obtains when
\(\psi(x, y)\) is true. In modern notation we represent this as as the
set of ordered pairs \(\{\langle x, y \rangle \mid \psi(x, y)\}\),
which is read: the set of ordered pairs \(\langle x, y \rangle\) which
are such that \(x\) bears the relation \(\psi\) to \(y\). 



The following contextual definition ([?]21*01) allows one
to eliminate the relation term \(\hat{x} \hat{y}\psi (x, y)\) from
occurrences in the context \(f\): 

 

\[
 f \{ \hat{x} \hat{y} \psi ( x, y )\} \ldot {=} \colondot
(\exists \phi) \colon \phi \bang ( x, y ) \ldot {\equiv\_{x,y}} \ldot \psi( x,y ) \colon f \{ \phi\bang (\hat{u}, \hat{v} )\} 
\quad\Df
\]


or in modern notation: 

 

\[
f \{\langle x, y \rangle \mid \psi( x, y )\} =\_\mathit{df} \exists \phi[\forall xy (\phi(x, y) \equiv \psi( x, y) ) \amp
f ( \lambda u \lambda v \phi(u,v))]
\]


where \(\phi\) is a predicative function of \(u\) and \(v\).



*Principia* does not analyze relations (or mathematical
functions) in terms of sets of ordered pairs, but rather takes the
notion of propositional function as primitive and defines relations
and functions in terms of them. The upper case letters \({R}, {S}\)
and \({T}\), etc., are used after [?]21 to stand for these
"relations in extension", and are distinguished from
propositional functions by being written between the arguments. Thus
it is \(\psi(x,y)\) with arguments after the propositional function
symbol, but \(xRy\). From [?]21 functions "\(\phi\) and
\(\psi\)", etc., disappear and only relations in extension,
\({R}\), \({S}\) and \({T}\), etc., appear in the pages of *Principia*. While propositional functions might be
"intensional", that is two functions may be true of the
same objects yet not be identical, no distinct relations in extension
are true of all the same objects. The logic of *Principia* is
thus "extensional", from page 200 in volume I, through to
the end in Volume III. 



[?]22 on the "Calculus of Classes" presents the
elementary set theory of intersections, unions and the empty set which
is often all the set theory used in elementary mathematics of other
sorts. The student looking for the set theory of *Principia* to
compare it with, say the Zermelo-Fraenkel system, will have to look at
various numbers later in the text. The Axiom of Choice is defined at
[?]88 as the "Multiplicative Axiom" and a version of
the Axiom of Infinity appears at [?]120 in Volume II as
"Infin ax". The set theory of *Principia* comes
closest to Zermelo's axioms of 1908 among the various familiar
axiom systems, which means that it lacks the Axiom of Foundation and
Axiom of Replacement of the now standard Zermelo-Fraenkel axioms of
set theory. The system of *Principia* differs importantly from
Zermelo's in that it is formulated in the simple theory of
types. As a result, for example, there are no quantifiers ranging over
all sets, and there is a set of all things (for each type). 



[?]30 on "Descriptive Functions" provides Whitehead
and Russell's analysis of mathematical functions in terms of
relations and definite descriptions. Frege had used the notion of
function, in the mathematical sense, as a basic notion in his logical
system. Thus a Fregean "concept" is a function from
objects as arguments to one of the two "truth values" as
its values. A concept yields the value "True" for each
object to which the concept applies, and "False" for all
others. Russell, from well before discovering his theory of
descriptions, had preferred to analyze functions in terms of the
relation between each argument and value, and the notion of
"uniqueness". With modern symbolism, his view would be
expressed as follows. For each function \(\lambda x f(x)\), there will
be some relation (in extension) \(R\), such that the value of the
function for an argument \(a\), that is \(f(a)\), will be the unique
individual which bears the relation \(R\) to \(a\). The result is that
there are no function symbols in *Principia*. As Whitehead and
Russell say, the familiar mathematical expressions such as
"\(\sin \pi/2\)" will be analyzed with a relation and a
definite description, as a "descriptive function". The
"descriptive function", \(R'y\) (the \(R\) of
\(y)\), is defined as follows: 

 

\[
\tag\*{[?]30*01} 
R'y = (\atoi x)xRy \quad\Df
\]


If the relation \(R\) is between persons \( x \) and \( y \) when \( x
\) is the father of \( y \) then the function will take an individual
\(y \) as argument to the value \(x \) as their father. For instance,
if \(xRy\) is the relation ' \(x\) is father of \(y\)',
then R'\(y\) is the function which maps \(y\) to the father of
\(y\) (if he exists). Note that here the left argument \(x\)
corresponds to the value of the function, whereas the right argument
\(y\) of R is the argument, or input, to the function R'\(y\).
Likewise if \(xSy\) is the relation of a number to its successor,
\(n\) to \(n + 1\), then S'\(y\) would be the argument of the
function which maps \(y\) to the number that it succeeds, rather than
expressing the "successor function " which maps a number
to its successor. This is the reverse of the order that is now
commonly used when relating functions and relations. Nowadays we
reduce functions to a binary relation between the argument in the
first place and value in the second place. This may lead to some
confusion in the definitions of notions such as the *domain*
and *range* of a relation below. 



We conclude this section by presenting a number of prominent examples
from these remainder of Volume I, with their intuitive meaning,
location in PM, definition in PM, and a modern version. (Some of these
numbers are theorems rather than definitions.) Note, however, that the
modern formulations will sometimes logically differ from the original
version in PM, such as by treating relations as classes of ordered
pairs, etc. More prominent is the practice in PM of defining notions
as relations, or higher order relations between relations, rather than
as functions determined by those relations. In his account of the
logic of *Principia*, W.V. Quine (1951) objects to the
complexity and even redundancy of much of this symbolism in comparison
with axiomatic set theory. These formulas can be worked out, however,
with a step by step application of the definitions. 



For each formula number, we present the information in the following
format:




|  |  |
| --- | --- |
| PM Symbol | (Intuitive Meaning)    [Location]

PM Definition

Modern Version  |



\(\ast 22 \) *Calculus of Classes* 




|  |  |
| --- | --- |
| \(\alpha \subset \beta\)  | (\(\alpha\) is a *subset* of \(\beta\))   
[[?]22*01]

\(x\in \alpha \ldot {\supset\_x} \ldot x\in \beta\)

\(\alpha \subseteq \beta\)  |
| \(\alpha \cap \beta\)  | (the *intersection* of \(\alpha\) and \(\beta)\)
   [[?]22*02]

\(\hat{x} (x \in \alpha \sdot x \in \beta\))

\(\alpha \cap \beta\)  |
| \(\alpha \cup \beta\)  | (the *union* of \(\alpha\) and \(\beta\))   
[[?]22*03]

\(\hat{x} (x \in \alpha \lor x \in \beta\))

\(\alpha \cup \beta\)  |
| \(-\alpha\)  | (the *complement* of \(\alpha)\)   
[[?]22*04]

\(\hat{x} (x\osim \in \alpha\)) [i.e., \(\hat{x} \osim (x \in
\alpha\)) by [?]20*06]

\(\{x \mid x \not\in \alpha \}\)  |
| \(\alpha - \beta\)  | (\(\alpha\) *minus* \(\beta)\)   
[[?]22*05]

\(\alpha \cap -\beta\)

\(\{x \mid x\in \alpha \amp x\not\in \beta \}\)  |



\(\ast 23 \) *Calculus of Relations* 




|  |  |
| --- | --- |
| \(R \: \subset \! \! \! * \: S\)  | (\(R\) is a *subrelation* \(S\))   
[[?]23*01]  

\(xRy \: . \supset\_{x,y} \: . xSy \)

\(\forall x \forall y (x R y \: \supset \: x S y)\)  |
| \(R \: \dot{\cap} \: S\)  | (the *intersection* of \(R\) and \(S\))   
[[?]23*02]

\(\hat{x}\hat{y} (x R y \: . \: x S y)\)

\(\{\langle x, y \rangle | Rxy \; \amp \; Sxy \}\)  |
| \( \dot{-} R \)  | (the *negation* of \(R\))   
[[?]23*04]

\( \dot{-} R = \hat{x} \hat{y} \{ \sim (xRy) \} \)

\(\{\langle x, y \rangle | \sim Rxy \}\)  |



\(\ast 24 \) *The Existence of Classes* 




|  |  |
| --- | --- |
| \(\mathrm{V}\)  | (the *universal* class)    [[?]
24*01]

\(\hat{x} (x\) = \(x)\)

\(\mathrm{V}\) or \(\{x \mid x = x\}\)  |
| \(\Lambda\)  | (the *empty* class)    [[?]24*02]

\(-\mathrm{V}\)

\(\varnothing\)  |
| \(\exists! \alpha \)  | (the class \( \alpha \) *exists*)   
[[?]24*03]

\((\exists x ). x \in \alpha \)

\(\exists x ( x \in \alpha )\)  |



\(\ast 25 \) *The Existence of Relations* 




|  |  |
| --- | --- |
| \(\dot{\exists}! R\)  | (the relation \(R\) *exists*)   
[[?]25*03]

\((\exists x,y) .\: xRy\)

\( \exists x \exists y Rxy\)  |



\(\ast 30 \) *Descriptive Functions* 




|  |  |
| --- | --- |
| \(R'y\)  | (the \(R\) of \(y)\) (a descriptive function)   
[[?]30*01]

(\(\atoi x)(xRy)\)

\(R'y\) is the (possibly partial) function where \(f\_R (y) = x
\) if \(x\) R\( y\) and this \(x\) is unique, and otherwise is
undefined. |



\(\ast 31 \) *Converses of Relations* 




|  |  |
| --- | --- |
| Cnv  | (the relation between a relation and its *converse*)
   [[?]31*01]

\(\hat{Q} \hat{P} \{ xQy . \equiv\_{x,y} . yPx \}\)

\(\{ \langle Q , P \rangle \: \mid \: \forall x \forall y (Qxy \equiv
Pyx) \: \} \)  |
| \(\breve{R}\)  | (the *converse* of \( R \) )   
[[?]31*02]

\(\hat{x} \hat{z} (zRx)\)

\(\{\langle x,z\rangle \mid Rzx\}\)  |



\(\ast 32 \) *Referents and Relata of a Given Term* 




|  |  |
| --- | --- |
| \(\overrightarrow{R}'y\)  | (the *R-predecessors* of \(y)\)   
[[?]32*01]

\(\hat{x} (xRy)\)

\(\{x \mid Rxy \}\)  |
| \(\overleftarrow{R}'x\)  | (the *R-successors* of \(x)\)   
[[?]32*02]

\(\hat{z} (xRz)\)

\(\{z \mid Rxz \}\)  |



\(\ast 33 \) *Domains and Fields of Relations* 




|  |  |
| --- | --- |
| \( \text{D} 'R\)  | (the *domain* of \(R)\)   
[[?]33*01]

\(\hat{x} \{ (\exists y) \sdot xRy \}\)

\(\{x \mid \exists y Rxy \}\) also: \(\mathcal{D}` R\)  |
| \( \backd 'R\)  | (the *converse domain* (range) of \(R)\)   
[[?]33*02]

\(\hat{z} \{(\exists x) \sdot xR z \}\)

\(\{z \mid \exists x Rxz \}\) also \(\mathcal{R}`R\)  |
| \(C'R\)  | (the *field* of \(R)\)   
[[?]33*03]

\(\hat{x} \{(\exists y): xRy \ldot {\lor} \ldot yRx\}\)

\(\{x \mid \exists y (Rxy \lor Ryx)\}\) also \(\mathcal{F}`R\)
 |



\(\ast 34 \) *The Relative Product of Two Relations* 




|  |  |
| --- | --- |
| \(R\mid S\)  |  (the *relative product* of \(R\) and \(S)\)   
[[?]34*01]

\(\hat{x} \hat{z} \{(\exists y) \sdot xRy \sdot ySz \}\)

\(R \circ S\) or \(\{\langle x,z\rangle \mid \exists y(Rxy \amp
Syz)\}\)  |



\(\ast 35 \) *Limited Domains and Converse Domains* 




|  |  |
| --- | --- |
| \(\alpha \upharpoonleft R\)  | (the *restriction* of the domain of \(R\) to \(\alpha )\)
   [[?]35*01]

\(\hat{x} \hat{y}[ x\in \alpha \sdot xRy]\)

\(\{\langle x,y \rangle \mid x \in \alpha \amp Rxy \}\)  |
| \(R \restriction \beta\)  | (the *restriction* of the range of \(R\) to \(\beta)\)
   [[?]35*02]

\(\hat{x} \hat{y}[xRy \sdot y \in \beta]\)

\(\{\langle x,y \rangle \mid Rxy \amp y \in \beta\}\)  |
| \(\alpha \uparrow \beta\)  | (the relation of members of \(\alpha\) to members of \(\beta)\)
   [[?]35*02].

\(\hat{x} \hat{y}[x \in \alpha \sdot y \in \beta\)]

\(\{\langle x,y\rangle \mid x \in \alpha \amp y\in \beta \}\), the
*Cartesian product* of \(\alpha\) and \(\beta \).  |



\(\ast 36 \) *Relations with Limited Fields* 




|  |  |
| --- | --- |
|  \(P \restriction \! \! \! \downharpoonright \alpha\)  | (the *restriction* of \(R\) to \(\alpha)\)   
[[?]36*01]

\(\alpha \upharpoonleft P \restriction \alpha\)

\(\{\langle x,y \rangle \mid x \in \alpha \amp y \in \alpha \amp Rxy
\}\)  |



\(\ast 37 \) *Plural Descriptive Functions* 




|  |  |
| --- | --- |
| \(R ''\beta\)  | (the terms which have the relation \(R\) to members of
\(\beta\))    [[?]37*01] [?]

\(\hat{x} \{(\exists y) \sdot y\in \beta \sdot x Ry\}\)

\(\{x \mid \exists y(y\in \beta \amp Rxy)\}\)  |
|  \( R\_{\in}\)  | (the relation of \(\alpha\) to \(\beta \) when \(\alpha\) is the
class of terms which have \(R \) to members of \(\beta\))   
[[?]37*02]

\(\hat{\alpha} \hat{\beta} ( \alpha = R '' \beta )\)

\(\{\langle \alpha, \beta \rangle \mid x \in \alpha \amp \exists z ( y
\in z \amp z \in \beta \amp Ryz ) \}\)  |



\(\ast 38\) *Double descriptive functions*. PM uses a
metalinguistic variable "\( \venus \)" that can be
replaced by a any of a range of relations between individuals,
classes, or relations, that are treated as *operations* on
their arguments. The operation of intersection can be represented as a
higher order function of its first argument. Thus \(\cap \beta `
\alpha = \alpha \cap \beta\). 




|  |  |
| --- | --- |
|  \(\venus \: y\)  |  (the relation of \( x \: \venus \: y \) to \(x \) for any \( x
\))    [[?]38*02]

\(\hat{u} \hat{x} ( u = x \: \venus\: y)\)

\(\{\langle u,x \rangle \mid u = x \: \venus \: y \}\)  |



This notion will be used later. An example with the notion of relative
product is an instance, thus:




|  |  |
| --- | --- |
|  \(\mid R\)  |  (the relation of one power of \(R\) to the next)   
[[?]38*02]

\(\hat{P}\hat{S}(P = R \mid S )\)

\(\{ \langle P, S \rangle \mid P = R \circ S \}\)  |
|  \(\alpha \: \venus\_{\! \! \! ,,} \: y\)  |  (the class of values of \(x \: \venus \: y \: \) when \(x\) is
an \( \alpha \))    [[?]38*03]

\( \venus \: y \) " \(\alpha \)

\(\{u \mid \exists x ( x \in \alpha \: \amp \: u = x \: \venus y \: )
\: \}\)  |
| \(s ' \kappa\)  | (the *sum* or union of the \(\kappa\)s)   
[[?]40*02]

\(\hat{x} \{ ( \exists \alpha ). \: \alpha \in \kappa \; . \; x \in
\alpha \}\)

\(\cup \kappa\), or \(\{ x \mid \exists \beta ( \beta \in \kappa \amp
x \in \beta ) \}\)  |
| \(\dot{s} ' \lambda \)  | (the *sum* of the relations in \(\lambda\))   
[[?]41*02]

\(\hat{x}\hat{y} \{ ( \exists R ). \: R \in \lambda \; . \; x R y \}\)

\(\{ \langle x, y \rangle \mid \exists R \: (R \in \lambda \; \amp \;
Rxy ) \}\)  |


## 11. Prolegomena to Cardinal Arithmetic (Part II)



Contemporary philosophers would consider the transition to mathematics
to begin with the theory of sets (or proper classes which are too
large to be sets), but in PM that is also a part of Mathematical
Logic. The Prolegomena to Arithmetic thus begins with the definitions
in terms of logic of explicitly arithmetical notions, the cardinal
numbers 1 and 2. 




|  |  |
| --- | --- |
| \(I \) | (the relation of identity)    [[?]50*01]

\( I = \hat{x}\hat{y} (x = y) \)

\(\{ \langle x , y \rangle \mid x = y \}\)  |
| \(J \) | (the relation of diversity)    [[?]50*02]

\( I = \dot{-} I \)

\(\{ \langle x , y \rangle \mid x \neq y \}\)  |
| \(\iota ' x\) | (the *unit class* of \(x\)) as defined by theorem
   [[?]51*1] from definition   
[[?]51*01]

\(\hat{y} (y = x)\)

\(\{ y \mid y = x \}\) (the *singleton* \(x\))  |
| \(\mathbf{1}\) | (the cardinal number 1)    [[?]52*01]

\(\hat{\alpha} \{ (\exists x) \sdot \alpha = \iota'x \}\)

\(\{ \alpha \mid \exists x \; ( \alpha = \{x \} ) \}\) (the class of
all singletons)

The variable \( x \) is typically ambiguous here, so will be a
distinct number 1 for each type that \( x \) can assume.

This applies to 2, as well, and all the natural numbers, as we will
see below.  |
| \(\mathbf{2}\) | (the cardinal number 2)    [[?]54*02]

\(\hat{\alpha} \{ (\exists x,y) \sdot x \neq y \sdot \alpha =
\iota'x \cup \iota'y \}\)

\(\{ \alpha \mid \exists y \exists z( y \neq z \amp \alpha = \{y \}
\cup \{z\} ) \}\) (the class of all pairs)  |
| \(x \downarrow y\) | (the *ordinal couple* of \(x\) and \(y\))   
[[?]55*01]

\(\iota'x \uparrow \iota'y\)

\(\langle x, y \rangle\) (the *ordered pair* \(\langle x,y
\rangle\))  |



The paperback abridged edition of *Principia Mathematica 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.
Russell did not make the decision to end the 1962 abridged version at
this point, but the choice is understandable. It is here that
contemporary set theory begins to look even more different from PM.
Set theory follows Norbert Wiener (1914) by representing relations as
sets of ordered pairs, which themselves defined as sets. (Wiener's
proposal of \( \langle x, y \rangle =\_\mathit{df} \{ \{ \{x \},
\emptyset \}, \{ \{y \} \} \} \) has generally been replaced by
Kuratowski's simpler \(\{ \{x \}, \{ x,y \} \}\)) . The remainder of
PM examines the structure of relations that lead to the mathematics of
natural and real numbers, and the portion of the theory of transfinite
sets that can be carried out in the theory of types. This looks very
different from the development of these notions in axiomatic set
theory. 




|  |  |
| --- | --- |
|  \(\Cl\ ` \alpha\)  | (the class of *subclasses* of \(\alpha\))   
[[?]60*12]

\(\hat{\beta} (\beta \subset \alpha) \}\)

\(\wp{\alpha}\), the *power set* of \(\alpha\), \(\{x | x
\subseteq \alpha\}\)  |
|  \(\Cl\ \ex\ ` \alpha\)  | (the class of *existent subclasses* of \(\alpha\))
   [[?]60*13]

\( \hat{\beta} (\beta \subset \alpha \; . \; \exists ! \beta ) \}\)

\(\{x | x \subseteq \alpha \; \amp \; x \neq \emptyset\}\)  |
|  \(\Rl\ ` P\)  | (the class of *sub relations* of \(P\))   
[[?]61*12]

\(\hat{R} \{ R \subset \! \! \! \! \cdot \; P \}\)

\(\{ R \mid \forall x \forall y ( \langle x, y \rangle \in R \supset
\langle x,y \rangle \: \in P) \}\)  |
|  \(\in\)  | (the *relation of membership in a class*)   
[[?]62*01]

\(\hat{x} \hat{\alpha} ( x \in \alpha)\)

\(\{ \langle x,y \rangle \mid \: x \in y \}\)  |



\(\ast 63\) *Relative Types of Classes.* The theory of types in
PM allows for expressions relating classes of different types. The gap
between set theory and the theory of classes in PM comes from the lack
of a cumulative theory of classes of *any* type. These PM
system allows definitions of relations between, say, individuals and
classes of individuals. These are needed in the account of real
numbers in terms of classes of classes of ratios in Volume III.




|  |  |
| --- | --- |
|  \(t'x\)  | (the type of which \(x\) is a member)   
[[?]63*01]

\(\iota ` x \cup - \iota ` x\)

\(\{ x \} \cup \{ y \mid y \not \in \{ x \} \}\)  |
|  \(t\_0'\alpha\)  | (the type in which \(\alpha\) is contained)   
[[?]63*02]

\(\alpha \cup - \alpha\)

\(\alpha \cup \{ x \mid x \not \in \alpha \}\)  |
|  \(t\_1' \kappa\)  | (the type next below that in which \(\kappa\) is contained)
   [[?]63*03]

\(t\_0 `s`\kappa\)

\(\cup \: \{ \: \cup \{ \alpha \mid \alpha \in \kappa \} \: , \: \{
\beta \mid \beta \not \in \cup \{ \alpha \mid \alpha \in \kappa \} \}
\: \}\)  |
|  \(t\_{11}' \alpha\)  | (the type of pairs of classes of types \(t\_1 ' \alpha\))
   [[?]64*022]

\(t ' ( t\_1 ' \alpha \uparrow t\_1 ' \alpha )\)

The type of a pair of classes of a given type will be the same as that
of classes of those classes. This definition is in order as it stands,
but would be very complex to write in contemporary notation. We leave
it as an open problem for readers to devise a concise formula
formulation.  |
| \(\alpha \rightarrow \beta\) | (The relations with *referents* in \( \alpha \) and
*relata* in \( \beta \) ) (from \( \alpha \) *onto* \(
\beta \))    [[?]70*01]

\(\hat{R} (\overrightarrow{R}"\backd 'R \subset \alpha
\sdot \overleftarrow{R}"D'R \subset \beta )\)

\( \{ R \mid \forall x \forall y \: ( Rxy \supset [ \: \{z \mid Rxz \}
\in \alpha \: \amp \: \{u \mid Rxu \} \in \beta \} \: ]\: \} \)

Since 1 is the class of singleton classes, \((1 \rightarrow 1) \) will
be the class of one to one (surjective) relations.  |
| \(\alpha \mathbin{\overline{\mathrm{sm}}} \beta\) | (the class of similarity relations between \(\alpha\) and
\(\beta\))    [[?]73*03]

\( \{ R \mid R \in 1 \rightarrow 1 \: .\: \alpha = D'R \: .\:
\beta = \backd 'R \} \)

\(\{f \mid f : \alpha \stackrel{1-1}{\longrightarrow} \beta\}\)
 |
| \(\mathrm{sm}\) | (the relation of *similarity*)   
[[?]73*02]

\(\hat{\alpha} \hat{\beta}(\exists! \alpha
\mathbin{\overline{\mathrm{sm}}} \beta)\)

\(\alpha \approx \beta\)  |



\(\ast 80\) *Selections*. A *selection function* for a
class \(\kappa\) is a function \(f\) making each element \(x\) of
\(\kappa\) to a member of \(x\). These are denoted by \( \in\_{\Delta}
`\kappa\). The cardinal number of the product of two classes \(\alpha
X \beta\) is the cardinal number of the class of all pairs of members
selected from \(\alpha\) and \(\beta\), so the guarantee that such
selections exist is called the *Multiplicative Axiom* in PM.
This is now known as the *Axiom of Choice*, which had been
identified as an assumption used in proofs in set theory by Ernst
Zermelo in 1904. In PM it is defined as asserting that if a class
\(\kappa\) is a set of mutually exclusive, non-empty classes, then
there exists a class \(\mu\) which contains exactly one member of each
element of \(\kappa\).




|  |  |
| --- | --- |
| \(\in\_{\Delta}`\kappa\) | (the *selective relations* for \(\in\))   
[[?]80*01]

\((1 \rightarrow Cls) \; \cap \; Rl` \in \; \cap \;
\overleftarrow{\backd}`\kappa\)

\(\{ f \mid \forall \alpha (\alpha \in \kappa \supset f(\alpha) \in
\alpha ) \}\)  |
| \(\Cls ^2 \ \excl\)  | (*class of mutually exclusive classes*)   
[[?]84*01]

\(\hat{\kappa} ( \alpha , \beta \in \kappa \; .\; \alpha \neq \beta \;
. \supset\_{\alpha , \beta} . \; \alpha \cap \beta = \Lambda )\)

\(\{ \kappa \mid \forall \alpha \forall \beta (\alpha , \beta \in
\kappa \; \amp \; \alpha \neq \beta \supset \alpha \cap \beta =
\emptyset)\}\)  |
|  \(\Cls\ \ex ^{2} \ \excl\)  | (*class of mutually exclusive non-empty classes*)
 [[?]84*03]

\(\Cls^2 \; \excl \; - \overleftarrow{\in} ` \Lambda\)

\(\{ \kappa \mid \; \forall \alpha (\alpha \in \kappa \supset \alpha
\neq \emptyset) \; \amp\) \(\; \forall \alpha \forall \beta \: [
\alpha \in \kappa \: \amp \: \beta \in \kappa \supset (\alpha = \beta
\vee \alpha \cap \beta = \emptyset) ] \}\)  |
|  Mult ax  |  (the *Multiplicative Axiom*)   
[[?]88*03 ]

\(\kappa \: \epsilon \: \mathrm{Cls \; ex^2 excl} \: .
\supset\_{\kappa} \: : (\exists \mu) : \alpha \: \epsilon \: \kappa \:
. \supset\_{\alpha} . \: \mu \cap \alpha \: \epsilon \: 1\)

\(\forall \kappa \{ [ \; \forall \alpha (\alpha \in \kappa \supset
\alpha \neq \emptyset ) \; \amp\) \(\; \forall \alpha \forall \beta \:
(\alpha \in \kappa \: \amp \: \beta \in \kappa \supset (\alpha = \beta
\vee \alpha \cap \beta = \emptyset)) \:] \;\supset\) \(\; \exists \mu
\forall \alpha \exists x \: (\alpha \in \kappa \supset \mu \cap \alpha
= \{x \} ) \}\)  |



\(\ast 90\) *Inductive Relations*. The concluding section of
Volume I presents a generalization of the structure of the natural
numbers that underlies the principle of mathematical induction. 




|  |  |
| --- | --- |
| \(R\_\*\)  |  (the *ancestral* of \(R)\)   
[[?]90*01]

\(\hat{x} \hat{y} \{ x \in C' R \colon \breve{R}"\mu
\subset \mu \sdot x \in \mu \ldot {\supset\_{\mu}} \ldot y \in \mu \}\)

\(\{ \langle x, y \rangle \mid x \in \mathcal{F}`R \; \amp\) \(\;
\forall\_{\mu} [ \forall z \forall w [( z \in \mu \; \amp \; Rzw )
\supset w \in \mu ] \supset y \in \mu ] \}\)

 Now written \(R^\*\) this follows Frege's
definition: \(y\) is in all the \(R\)-hereditary classes that contain
\(x\).  |
| \(R\_{\text{ts}}\) | (the relation between \( R \) and the series of its
*powers* \( R^n\) for \(n \gt 0\) , i.e., \( R (= R^1) \),
\(R^2\) \(R^3\), etc. )    [[?]91*02]

\(( \: \mid R)\_{\ast}\)

\(\{ \langle P,S \rangle \mid P = R^n \: \amp \: S = R^{n+1} \} \)
 |
|  Pot \(' R\) | (the *positive powers*, i.e., *Potentia*, of
\(R\))    [[?]91*03]

\(\overrightarrow{R}\_{\text{ts}} ' R\)

\(\{ S \: \mid \; \exists n >0 \: ( S = R^n ) \} \)  |
| \(R\_{\text{po}}\) | (the union of the positive powers of \(R\))   
[[?]91*05]

\(\dot{s}' \text{Pot} ' R\)

\( \{ \langle x, y \rangle \mid \exists S \: \exists n \gt 0 \:( S =
R^n \: \amp \: Sxy ) \} \)  |
| \(xB'P\) | ( \(x\) *begins* the relation \(P\))   
[[?]93*01]

\(x \in D ' P - \backd 'P\)

\(\{ x \mid \exists y \: Pxy \; \amp \sim \exists z Pzx \}\)
 |
| \(x \:\) min\(\_P ` \alpha\) | ( \(x\) is a *minimal* member of \( \alpha \) with
respect to \(P\))    [[?]93*02]

\(x \in \alpha \cap C`P - \breve{P} `` \alpha\)

\(x \in \alpha \; \amp \; x \in \mathcal{F}P \; \amp \; \sim \exists z
\: (Pzx \: \amp \: z \in \alpha )\)  |
| \(\stackrel{\leftrightarrow}{R} `x\) | ( the *family* of \(R\), ancestry and posterity)
   [[?]97*01]

\(\overrightarrow{R}`x \cup (\iota ` x \cap C ` R)\cup
\overleftarrow{R}` x\)

\(\{ y \mid Rxy \; \vee \; ( y = x \: \amp \: x \in \mathcal{F} `R) \;
\vee \; Rxy \} \)  |


## 12. Cardinal Arithmetic (Part III)



With \(\ast 100\) at the start of Volume II, *Principia
Mathematica* finally begins developing the theory of cardinal
numbers with the *Frege-Russell Definition* of numbers as
classes of equinumerous classes.




|  |  |
| --- | --- |
| \(\mathrm{N\_c}\) | (the relation between a class and its cardinal number)
   [[?]100*01]

\(\overrightarrow{\mathrm{sm}}\)

\(\{ \langle \alpha, \beta \rangle \mid \beta = \{ \gamma \mid \gamma
\approx \alpha \} \} \)  |
| \(\mathrm{NC}\) | (the *Cardinal Numbers*)   
[[?]100*02]

\( \mathrm{D} ' \mathrm{N\_c} \)

\( \{ \alpha \mid \exists \beta ( \alpha = \{ \gamma \mid \gamma
\approx \beta \} \: \}\)  |
| \(\mathbf{0}\) | (the cardinal number 0)    [[?]101*01]

\(0 = \mathrm{N\_c}'\Lambda\)

\(\{\varnothing\}\)

 The class of all classes equinumerous with the
empty set is just the singleton containing the empty set. |
| \(\text{N}\_0 \text{c}'\alpha\) | ( the *homogeneous cardinal* of \(\alpha\))   
[[?]103*01]

\(\text{Nc}' \alpha \; \cap \; t'\alpha\)

\(\{ \beta \mid \beta \approx \alpha \}\) for \(\beta\) of the same
type as \(\alpha\)  |
| \(\text{N}\_0\text{C}\)  | (the *Homogeneous Cardinals*)   
[[?]103*02]

\(\text{D}' \text{N}\_0 \text{c} \)

\(\{ \alpha \mid \exists \beta (\alpha \; \text{is the homogenous
cardinal of}\; \beta ) \}\)  |
| \(\alpha + \beta\) | (the *arithmetic sum* of \(\alpha\) and \(\beta\))
   [[?]110*01]

\(\downarrow (\Lambda \cap \beta)"\iota"\alpha \cup
(\Lambda \cap \alpha) \downarrow"\iota"\beta \)

 This is the union of \(\alpha\) and \(\beta\)
after they are made disjoint by pairing each element of \(\beta\) with
\(\{ \alpha \}\) and each element of \(\alpha\) with \(\{ \beta \}\).
The classes \(\alpha\) and \(\beta\) are intersected with the empty
class, \(\Lambda\), to adjust the type of the elements of the
sum.

\(\{ \langle \{ a \} , \emptyset \rangle \mid a \in \alpha \} \cup \{
\emptyset , \{b \} \rangle \mid b \in \beta \}\)  |
| \(\mu +\_c \nu\) | (the *cardinal sum* of \(\mu\) and \(\nu\))   
[[?]110*02]

\(\hat{\xi}\{(\exists \alpha,\beta) \sdot \mu = \mathrm{N\_0
c}'\alpha \sdot \nu = \mathrm{N\_0
c}'\beta\sdot\xi\,\mathrm{sm}(\alpha + \beta)\}\)

 Cardinal addition is the arithmetic sum of
homogeneous cardinals: 

\( \{\gamma \mid \exists \alpha \exists \beta \: [ \gamma \approx (
\alpha + \beta ) ] \: \} \) when \(\alpha \) and \( \beta \) are
homogenous cardinals.  |



The reader can now appreciate why this elementary theorem is not
proved until page 83 of Volume II of *PM*:

 

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


Whitehead and Russell remark that "The above proposition is
occasionally useful. It is used at least three times, in
...". This joke 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.




|  |  |
| --- | --- |
|  \(\beta \times \alpha\)  |  (the *product* of classes)   
[[?]113*02]

\(s` \alpha \downarrow\_{\! \! \!,,} ``\beta\)

\(\{ \langle x , y \rangle \mid x \in \beta \; \amp \; y \in \alpha
\}\)  |
|  \(\mu \times\_{\text{c}} \nu\)  |  (the *product* of homogenous cardinal numbers)
   [[?]113*03]

\(\hat{\xi} \{ (\exists \alpha, \beta) . \: \mu = N\_0\text{c}` \alpha
\: . \: \nu = N\_0\text{c}` \beta \; . \; \xi \: \text{sm} \: (\alpha
\times \beta)\}\)

If \(\mu = \bar{\bar{\alpha}} \; \amp \; \nu = \bar{\bar{\beta}}\)
then \(\mu \times \nu = \{ \beta \mid \beta \approx ( \alpha \times
\beta) \}\)  |
|  \(\alpha\) exp \( \beta \)  |  (the *exponentiation* of classes)   
[[?]116*01]

Prod ' \( \alpha \downarrow\_{\! \! \! ,,} \)'' \(
\beta \)

 \(\{ f \mid \mathcal{D} f = \beta \; \amp \;
\mathcal{R} f \subseteq \alpha \}\)  |
| \(\mu^{\nu}\)  | (the *exponentiation* of cardinal numbers)   
[[?]116*02]

\(\hat{\gamma} \{ (\exists \alpha, \beta) . \: \mu = N\_0\text{c}`
\alpha \: . \: \nu = \mathrm{N}\_0\text{c}` \beta \; . \; \gamma\) sm
\((\alpha\) exp \(\beta)\}\)

\(\{ \gamma \mid \exists \alpha \exists \beta \: ( \mu =
\bar{\bar{\alpha}} \; \amp \; \nu = \bar{\bar{\beta}} \; \amp \;
\gamma \approx \alpha^{\beta} )\}\)  |



The following theorem, that the cardinality of the power set of
\(\alpha\) is 2 raised to the power of the cardinality of \(\alpha\),
\(\; \bar{\bar{ \wp{\alpha} }} = 2^{\bar{\bar{\alpha}}}\), is called
"*Cantor's Proposition*", and is said to be
"very useful" (PM II, 140): 

 

\[\tag\*{[?]116*72} 
\text{Nc}'\text{Cl}'\alpha = 2^{\text{Nc}'\alpha} 
\]


Next the notion of *greater than* arbitrary cardinals, finite
and infinite. The cardinal number of \( \alpha \) is greater than the
cardinal number of \( \beta \) just in case there is a subset of \(
\alpha \) that is equinumerous with \( \beta \), but there is no
subset of \(\beta \) that is equinumerous with \(\alpha \). Cantor's
famous "diagonal argument" shows that cardinal number \(
\aleph\_c \) of the class of real numbers is greater than \( \aleph\_0
\), the cardinal number of the class of natural numbers. 




|  |  |
| --- | --- |
|  \(\mu \gt \nu \)  |  (*greater than*)    [[?]117*01]

\( (\exists \alpha, \beta ) \: . \: \mu = \text {N}\_0\text{c}`
\alpha\: . \: \nu = \mathrm{N}\_0\text{c}` \beta \: . \: \exists ! \:
\text{Cl} ' \alpha \: \cap \: \text{Nc} ' \beta \: . \:
\sim \exists ! \: \text{Cl} ' \beta \: \cap \: \text{Nc} '
\alpha \)

\( \exists \delta (\delta \in \wp{\alpha} \; \amp \; \delta \in
\bar{\bar{\beta}} ) \; \amp \; \sim \exists \gamma (\gamma \in
\wp{\beta} \; \amp \; \gamma \in \bar{\bar{\alpha}} ) \)  |



The more familiar result, *Cantor's Theorem*, proves the power
set of \(\alpha\) is strictly larger, \(2^{\bar{\bar{\alpha}}} \gt
\bar{\bar{ \alpha }}\). 

 

\[\tag\*{[?]117*661} 
 \mu \in \text{N}\_0\text{C} \; . \: \supset \: . \; 2^{\mu} \gt \mu 
\]



|  |  |
| --- | --- |
| NC induct | (the *Inductive Cardinals*)   
[[?]120*01]

\(\hat{\alpha}\{\alpha({+\_c}1)\_\* 0\}\)

\(\{x \mid 0 S^\* x\}\)

 The inductive cardinals are the
"*natural numbers*", that is, 0 and all those
cardinal numbers that are related to 0 by the ancestral of the
"successor relation" \(S\), where \(xSy\) just in case \(y
= x +1\). |
| Infin ax | (the *Axiom of Infinity*)   
[[?]120*03]

\(\alpha \in \text{NC induct}\:\sdot \supset\_{\alpha} \sdot \:
\exists!\alpha\)

\(\forall \alpha (\alpha \in \{x \mid 0S^\* x\} \supset \alpha \neq
\varnothing)\) |



The Axiom of Infinity asserts that all inductive cardinals are
non-empty. (Recall that 0 = \(\{ \varnothing \}\), and so 0 is not
empty.) The Axiom of Infinity 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 it is not an *axiom* of PM as
[[?]120*03] is a *definition*, so this is just
further notation in PM!




|  |  |
| --- | --- |
| Prog | (*Progressions*, or \(\omega\) orderings)   
[[?]122*01]

\((1 \rightarrow 1) \cap \hat{R} (D`R = \overleftarrow{R\_{\ast}}
'' B 'R)\)

\(\{ R \mid R\) is isomorphic to the ancestral of a relation for which
every subset of the domain has a first element. } |



"By a 'progression' we mean a series which is like
the series of the inductive cardinals in order of magnitude (assuming
that all the inductive cardinals exist) *i.e.* a series whose
terms can be called \(1\_R, 2\_R, 3\_R, \ldots \nu\_R, \ldots \). It is
not convenient to *define* a progression as a series which is
ordinally similar to that of the inductive cardinals, both because
this definition only applies if we assume the axiom of infinity, and
because we have in any case to show that (assuming the axiom of
infinity) the series of inductive cardinals has certain properties,
which can be used to afford a direct definition of
progressions." (PM II, 245)




|  |  |
| --- | --- |
|  \(\aleph\_0\) | (the smallest of Cantor's transfinite cardinals)   
[[?]123*01]

\(D''\)Prog

\(\bar{\bar{\omega}}\)  |


## 13. Relation Arithmetic (Part IV)



The notion of *relation number*, is the generalization of the
notion of a well-ordering to an arbitrary relation. Just as a cardinal
number is defined in PM as a class of equinumerous classes, an
arbitrary relation number is a class of *ordinally similar*
relations.




|  |  |
| --- | --- |
|  \(S^{;}Q\)  |  (S is a *correlator* of Q)   
[[?]150*01]

\(S \mid Q \mid \breve{S}\)

\(S \circ Q \circ S^{-1}\)  |
|  \(P \: \overline{\text{smor}} \: Q\)  |  (the class of *similarities of* between \(P\) and \(Q\)
)    [[?]151*01]

\(\hat{S} \{ S \in 1 \rightarrow 1 \; . \; C'Q = \backd 'S
\; . \; P = S^{;} Q \}\)

\(\{ f \mid \mathcal{F} P \stackrel{1-1}{\longrightarrow} \mathcal{F}
Q \; \amp \; \forall x \forall y [ (x \in \mathcal{D} f ) \supset Pxy
\equiv Q f(x) f(y)]\}\)  |
|  \(P \;\) smor \(\:Q\)  |  (\(P\) is *ordinary similar to* \(Q\) )   
[[?]151*02]

\(\{ \langle P, Q \rangle \mid \exists ! \; P \:
\overline{\text{smor}} \: Q \} \)

\(P \cong Q\) ( \(P\) is *isomorphic* to \(Q\)).  |
|  Nr\(`P\)  |  (the *relation number* of \(P\))   
[[?]152*01]

\(\overrightarrow{\text{smor}} ' P\)

\(\{ Q \mid P \cong Q\}\)  |



\(\ast 170\) The relation of *first differences* orders classes
on the basis of an ordering of their members. The method is a
variation on the notion of *lexicographic ordering* of classes
as in the alphabetical ordering of words in a dictionary. See Fraenkel
(1968). PM uses two versions of the notion.




|  |  |
| --- | --- |
|  \(P\_{\text{cl}}\)  |  (ordering of classes by *first differences* of \(P\))
   [[?]170*01]

\(\hat{\alpha}\hat{\beta} \{ \alpha, \beta \in \text{Cl}`C`P \; . \;
\exists ! \; \alpha - \beta - \breve{P}`` ( \beta - \alpha ) \}\)

For \(\prec\) an ordering of individuals, \(\alpha \prec\_{\text{cl}}
\beta\) is

\(\{ \langle \alpha , \beta \rangle \mid \alpha\) , \(\beta \subseteq
\mathcal{F}( \prec ) \; \amp \; \alpha \not\subseteq \beta \; \amp \;
\forall x \forall y (x \in \beta \; \amp \; y \not \in \alpha \;
\supset \; y \prec x )\: \} \)  |

 This is explained in the  *Summary of*  \(\ast 170\):
"\(\alpha\) and \(\beta\) each pick out terms from \(C`P\), and
these terms have an order conferred by \(P\); we suppose that the
earlier terms selected by \(\alpha\) and \(\beta\) are perhaps the
same, but sooner or later, if \(\alpha \neq \beta\), we must come to
terms which belong to one but not the other. We assume that the
earliest terms of this sort belong to \(\alpha\), not to \(\beta\); in
this case, \(\alpha\)has to \( \beta\) the relation \(P\_{\text{cl}}\).
That is, where \(\alpha\) and \(\beta\) begin to differ, it is terms
of \(\alpha\) that we come to, not terms of \(\beta\). We do not
assume that there is a *first* term which belongs to \(\alpha\)
but not \(\beta \), since this would introduce undesirable
restrictions in case \(P\) is not well-ordered." (PM II, 399)



|  |  |
| --- | --- |
|  \(P\_{\text{lc}}\)  |  (*converse* ordering of classes by *first
differences* of \(P\))    [[?]170*02]

Cnv \(` (\breve{P})\_{\text{cl}}\)

\( \{ \langle \alpha, \beta \rangle \mid \alpha \prec\_{cl} \beta \} \)
 |



"Thus \(\alpha P\_{\text{lc}} \beta\) means, roughly speaking,
that \(\beta - \alpha\) goes on longer than \( \alpha - \beta\), just
as \(\alpha P\_{\text{cl}} \beta\) means that \( \alpha - \beta\)
begins sooner. This if \(P\) is the relation of earlier and later in
time, and \(\alpha\) and \(\beta\) are the times when \(A\) and \(B\)
respectively are out of bed, "\(\alpha P\_{\text{cl}}
\beta\)" will mean that \(A\) gets up earlier than \(B\) and
"\(\alpha P\_{\text{lc}} \beta\)" will mean that \(B\) goes
to bed later than \(A\)." (PM II, 401)


## 14. Series (Part V)



"*Series*" in PM are *linear orderings*.
Volume II concludes halfway through this part, with Volume III
beginning at \(\ast250\) and the theory of well-orderings. These
concepts are defined in the now standard way. This section is only
alien to the modern reader because of the notation.




|  |  |
| --- | --- |
|  trans \( P \)  |  (\(P\) is *transitive* relations)   
[[?]201*1]

\(P^2 \subset \! \! \! \! \cdot \;\; P\)

\(\forall x \forall y \forall z (Pxy \: \amp \: Pyz \: \supset \:
Pxz)\)  |
|  connex P  |  (P is *connected*)    [[?]202*1]

\(x \in C` P \; . \supset\_x . \; \stackrel{\leftrightarrow}{P} ` x =
C`P\)

\(\forall x \forall y [(x, y \in \mathcal{F} P ) \: \amp \: x \neq y
\supset Pxy \vee Pyx ]\)  |
| Ser  |  (*series*)    [[?]204*01]

Rl \(\: ' J \cap\) trans \(\cap\) connex

\(\{ P \mid \; \forall x \forall y (Pxy \supset x \neq y) \; \amp \;
P\) is transitive \(\amp \; P\) is connected \(\}\) or \(P\) is a
*linear ordering* |
| sect  |  (*sections*)    [[?]211*01]

sect \( ' P = \hat{\alpha} ( \alpha \subset C ' P \:. \: P
" \alpha \subset \alpha) \)

\(\{ \alpha \mid \alpha \subset \mathcal{F}P \: \amp \: \forall x [
\exists y ( y \in \alpha \: \amp \: Pxy ) \supset x \in \alpha \: ] \:
\} \) |



"The theory of the modes of separation of a series into two
classes, one of which wholly precedes the other, and which together
make up the whole series, is of fundamental importance. \( \ldots \)
Any class which can be the first of such a pair we call a
*section* of our series." (PM II, 603)




|  |  |
| --- | --- |
|  \(\varsigma ' P\) |  (the series of *segments* of \(P\))   
[[?]212*01]

\(P\_{\text{lc}} \restriction \! \! \! \downharpoonright\)
D\(`P\_{\in}\)  |



The *Summary* of \(\ast 211\) explains the definition as
follows: "The members of D\( "P\_{\epsilon} \) are called
the *segments* of the series generated by *P*. In a
series in which every sub-class has a maximum or a sequent [immediate
successor (cf. \( [?]206 ) ] \), \( \text{D} "P\_{\epsilon}
= \overrightarrow{P}"C'P \) \(([?]211*38)\),
i.e. the predecessors of a class are always the predecessors of a
single term, namely the maximum of the class if it exists, or the
sequent if no maximum exists. \( \ldots \) Thus in general the series
of segments will be larger than the original series. For example, if
our original series is of the type of the series of rationals in order
of magnitude, the series of segments is of the type of the series of
real numbers, i.e. the type of the continuum." (PM II, 603)



> 
> We have no need for a special notation for the series of sections,
> since, in virtue of [?]211*13, it is \(\varsigma '
> P\_{\ast} \ldots \). (PM II, 628)
> 



Volume III begins with \(\ast 250\) on *well-orderings*. An
*ordinal number* is then defined as a class of ordinary similar
well-orderings. 




|  |  |
| --- | --- |
| Bord  |  (*well ordered* relations - *Bene ordinata*)
   [[?]250*01]

\(\hat{P} \{\) Cl ex \('C 'P \subset \backd '
\text{min}\_{P} \}\)

\(\{ P \mid \forall \alpha \; [ \: (\alpha \subseteq \mathcal{F}P \:
\amp \: \alpha \neq \emptyset )\supset \exists x (x \in \alpha \amp
\forall z (z \in \alpha \supset \sim Pzx )\: )\: ] \)  |
| \(\Omega\)  |  (the well ordered series)    [[?]250*02]

Ser \(\; \cap \;\) Bord

\(\Omega\) is the class of well ordered linear orderings.  |
|  NO  | (*Ordinal Numbers*)    [[?]251*01]

Nr " \(\Omega\)

The ordinal numbers are classes of isomorphic well-ordered linear
orderings.  |



"Zermelo's Theorem", that the Multiplicative Axiom
(Axiom of Choice) implies that every set can be well-ordered, is
derived in \(\ast 258\). This was first proved in Zermelo (1904). 



\[\tag\*{[?]258*32 } 

\mu \sim \in \: 1 \: . \; \exists ! \in\_{\Delta}'\text{Cl ex} ' \mu \:
. \supset . \: \mu \: \epsilon \: C" \Omega \]

## 15. Quantity (Part VI)



The last section of PM studies the *rational numbers* and
*real* numbers. They are constructed from relations between
entities, such as *being longer than*, or *being heavier
than*, that might be measured with a ruler or balance scale.
Contemporary *measurement theory* studies the relations among
entities in order to determine which *scales*, or systems of
independently characterized numbers, might be assigned to them as
representing the "quantities" of various properties, such
as length or weight, that they possess. Notice that the real numbers
are not constructed as classes of rational numbers, but are of a
uniform type as "Dedekind cuts" in a series of classes of
ratios. In PM, as in contemporary mathematics, because the class
(*segment*) of rational numbers \( \{ r | r^2 \leq 2 \} \) will
have no rational number as a *least upper bound* , that class
itself will be identified with the irrational number \( \sqrt{2}\).
The rational number \( 1/2 \) is identified with its (lower) segment
of rationals, \( \{ r | r \lt 1/ 2 \} \). 




|  |  |
| --- | --- |
|  \(U\)  |  (*greater than* for inductive cardinals)   
[[?]300*01]

\((+\_c 1)\_{\text{po}} \; \upharpoonright \! \! \! \downharpoonright \;
( \text{NC induct } - \iota ` \Lambda)\)

\(\{ \langle n, m \rangle \mid \; n \gt m \}\)  |
|  Prm  |  (*relative primes*)    [[?]302*01]

\(\hat{\rho}\hat{\sigma} \{ \rho , \sigma \: \epsilon \: \text{NC
induct} \; :\) \(\;\rho = \xi \times\_c \tau \: . \: \sigma = \eta
\times\_c \tau . \supset\_{\xi, \eta, \tau} . \tau = 1\}\)

\(r \; \text{and} \; s \; \text{are}\) *relatively prime* iff
\(\forall j \: \forall l \: \forall k \: [ ( r = j \times k \; \amp \;
s = l \times k) \supset k = 1]\)  |
|  \((\rho , \sigma) \text{Prm}\_{\tau} (\mu , \nu )\) |  (\(\rho / \sigma \; \text{is}\; \mu / \nu\) in its *lowest
terms* and \(\tau\) is the highest common factor of \(\mu \;
\text{and} \; \nu\) )    [[?]302*02]

\(\rho \: \text{Prm} \: \sigma \; . \; \tau \in \text{NC induct} -
\iota ` 0 \; . \; \mu = \rho \times\_{\text{c}} \tau \; . \) 
\(\; \nu = \sigma \times\_{\text{c}} \tau\)

The ratio of \(r/s\) is \(m /n\) in its lowest terms with \(k\) as its
highest common factor \(=\_{\text{df}}\) \(r\) and \(s\) are relatively
prime and \( m = r \times k \; \amp \; n = k \times s\)  |
|  \((\rho , \sigma) \text{Prm}(\mu , \nu )\) |  (the ratio \(\rho / \sigma \; \text{is}\; \mu / \nu \; \text{in
its}\) lowest terms)    [[?]302*03]

\((\exists \tau) . \: (\rho, \sigma ) \text{Prm}\_{\tau} (\mu , \nu)\)

The ratio of \(r/s\) is \(m /n\) in its lowest terms \(=\_{\text{df}}
\exists k (\text{The ratio of}\; r/s \; \text{ is} \; m /n \text{in
its lowest terms with}\; k\) as its highest common factor.)  |
|  \(\mu / \nu\)  |  (the *ratio* of relations \( \mu \) and \( \nu \))
   [[?]303*01]

\(\hat{R} \hat{S} \{ (\exists \rho, \sigma) . (\rho, \sigma)
\text{Prm}(\mu, \nu) . \: \dot{\exists} ! \; R^{\sigma} \: \dot{\cap}
\: S^{\rho} \}\)

\(\{ \langle R, S \rangle \mid \exists r \exists s \:( r/s \) is \(
m/n \) in lowest terms and \(\exists x \exists y (R^s xy \amp S^r xy)
\}\)  |



"A distance on a line is a one-one relation whose converse
domain (and its domain too) is the whole line. If we call two such
distances \(R\) and \(S\), we may say that they have the ratio \(\mu /
\nu\) if, starting from some point \(x\) , \(\nu\) repetitions of
\(R\) take us to the same point \(y\) as we reach by \(\mu
\;\)repetitions of \(S\), *i.e.*, if \(xR^{\nu} y \: . \: x
S^{\mu} y\)." (PM III, 260)




|  |  |
| --- | --- |
|  Rat def  |  (*definite ratios*)    [[?]303*05]

\(\hat{X} \{ (\exists \mu, \nu) . \: \mu , \nu \: \epsilon \:
\text{D}`U \: \cap \; \backd `U . \: X = (\mu / \nu) \upharpoonright
\!\!\! \downharpoonright \; t\_{11} ` \mu \}\)

The class of ratios restricted to members of a given type.

Notice that the following definitions will not depend on the Axiom of
Infinity.  |
| \(X \lt\_r Y\)  |  (*less than* between ratios)   
[[?]304*01]

\(( \exists \mu, \nu, \rho , \sigma).\) \(\: \mu, \nu, \rho , \sigma
\: \epsilon \: \text{Nc induct} \: . \sigma \neq 0 .\) \(\; \mu
\times\_c \sigma \lt \nu \times\_c \rho \: . \; X = \mu / \nu \: . \; Y
= \rho / \sigma\)

\(X \lt Y =\_{\text{df}} \exists j \: \exists k \: \exists m \: \exists
n\) \(\: ( j \times m \lt k \times n \; \amp \; X = j/k \; \amp \; Y =
n/m )\)  |
|  \(H\) |  (the relation *less than* between definite ratios)
   [[?]304*02]

\(\hat{X} \hat{Y} \{ X,Y \: \in \: \text{Rat def} \: . \: X \lt\_r Y
\}\)

\(\{ \langle r, s \rangle \mid \; r \: \text{is rational} \; \amp \; s
\: \text{is rational} \; \amp \; r \lt s \}\)

"H" is a capital *eta*
"\(\eta\)", Cantor's symbol for rational numbers.  |
|  \(\Theta\)  |  (*real numbers*)    [[?]310*01]

\((\varsigma ' H ) \upharpoonright \!\!\! \downharpoonright \; (
- \iota ' \Lambda - \iota '\text{D}' H)\)  |



"The series of real numbers other than 0 and infinity" (PM
III, 316) are the series of the segments of rational numbers other
than the empty class and the whole series.


## 16. Conclusion



This summary cites about 110 of the definitions in PM. The last eight
pages (667-674) of Volume I of the second edition (1925)
consists of a complete list of 498 definitions from all three volumes.
Correspondence in the Bertrand Russell Archives confirms that this was
compiled by Dorothy Wrinch. Her list can be used to trace every one of
the other defined expressions of PM back to the notation discussed in
this entry.