Glossary of Principia Mathematica

This is a list of the notation used in Russell and Whitehead's Principia Mathematica.

The second (but not the first) edition of volume I has a list of notation used at the end.

Glossary

This is a glossary of some of the technical terms in Principia Mathematica that are no longer widely used or whose meaning has changed.

apparent variable

bound variable

atomic proposition

A proposition of the form R(x,y,...) where R is a relation.

Barbara

A mnemonic for a certain syllogism.

class

A subset of the members of some type

codomain

The codomain of a relation R is the class of y such that xRy for some x.

compact

A relation R is called compact if whenever xRz there is a y with xRy and yRz

concordant

A set of real numbers is called concordant if all nonzero members have the same sign

connected

connexity

A relation R is called connected if for any 2 distinct members x, y either xRy or yRx.

continuous

A continuous series is a complete totally ordered set isomorphic to the reals. *275

correlator

bijection

couple

1. A cardinal couple is a class with exactly two elements

2. An ordinal couple is an ordered pair (treated in PM as a special sort of relation)

Dedekindian

complete (relation) *214

defiendum

The symbol being defined

defiens

The meaning of something being defined

derivative

A derivative of a subclass of a series is the class of limits of non-empty subclasses

description

A definition of something as the unique object with a given property

descriptive function

A function taking values that need not be truth values, in other words what is not called just a function.

diversity

The inequality relation

domain

The domain of a relation R is the class of x such that xRy for some y.

elementary proposition

A proposition built from atomic propositions using "or" and "not", but with no bound variables

Epimenides

Epimenides was a legendary Cretan philosopher

existent

non-empty

extensional function

A function whose value does not change if one of its arguments is changed to something equivalent.

field

The field of a relation R is the union of its domain and codomain

first-order

A first-order proposition is allowed to have quantification over individuals but not over things of higher type.

function

This often means a propositional function, in other words a function taking values "true" or "false". If it takes other values it is called a "descriptive function". PM allows two functions to be different even if they take the same values on all arguments.

general proposition

A proposition containing quantifiers

generalization

Quantification over some variables

homogeneous

A relation is called homogeneous if all arguments have the same type.

individual

An element of the lowest type under consideration

inductive

Finite, in the sense that a cardinal is inductive if it can be obtained by repeatedly adding 1 to 0. *120

intensional function

A function that is not extensional.

logical

1. The logical sum of two propositions is their logical disjunction

2. The logical product of two propositions is their logical conjunction

matrix

A function with no bound variables. *12

median

A class is called median for a relation if some element of the class lies strictly between any two terms. *271

member

element (of a class)

molecular proposition

A proposition built from two or more atomic propositions using "or" and "not"; in other words an elementary proposition that is not atomic.

null-class

A class containing no members

predicative

A century of scholarly discussion has not reached a definite consensus on exactly what this means, and Principia Mathematica gives several different explanations of it that are not easy to reconcile. See the introduction and *12. *12 says that a predicative function is one with no apparent (bound) variables, in other words a matrix.

primitive proposition

A proposition assumed without proof

progression

A sequence (indexed by natural numbers)

rational

A rational series is an ordered set isomorphic to the rational numbers

real variable

free variable

referand

The term y in xRy

referent

The term x in xRy

reflexive

infinite in the sense that the class is in one-to-one correspondence with a proper subset of itself (*124)

relation

A propositional function of some variables (usually two). This is similar to the current meaning of "relation".

relative product

The relative product of two relations is their composition

scope

The scope of an expression is the part of a proposition where the expression has some given meaning (chapter III)

Scott

Sir Walter Scott, author of Waverley.

second-order

A second order function is one that may have first-order arguments

section

A section of a total order is a subclass containing all predecessors of its members.

segment

A subclass of a totally ordered set consisting of all the predecessors of the members of some class

selection

A choice function: something that selects one element from each of a collection of classes.

sequent

A sequent of a class α in a totally ordered class is a minimal element of the class of terms coming after all members of α. (*206)

serial relation

A total order on a class. Unfortunately PM insist that this class must be the field of the relation, resulting in the bizarre convention that the class cannot have exactly one element.

significant

well-defined or meaningful

similar

of the same cardinality

stretch

A convex subclass of an ordered class

stroke

The Sheffer stroke. (Only used in the second edition of PM.)

type

As in type theory. All objects belong to one of a number of disjoint types.

typically

Relating to types; for example, "typically ambiguous" means "of ambiguous type".

unit

A unit class is one that contains exactly one element

universal

A universal class is one containing all members of some type

vector

1. Essentially an injective function from a class to itself (for example, a vector in a vector space acting on an affine space)

2. A vector-family is a non-empty commuting family of injective functions from some class to itself (VIB)

Symbols introduced in Principia Mathematica volume I

Symbol	Approximate meaning	Reference
✸	Indicates that the following number is a reference to some proposition
α,β,γ,δ,λ,κ, μ	Classes	Chapter I page 5
f,g,θ,φ,χ,ψ	variable functions (though θ is later redefined as the order type of the reals)	Chapter I page 5
a,b,c,w,x,y,z	variables	Chapter I page 5
p,q,r	variable propositions (though the meaning of p changes after section 40).	Chapter I page 5
P,Q,R,S,T,U	Relations	Chapter I page 5
. : :. ::	Dots used to indicate how expressions should be bracketed, and also used for logical "and".	Chapter I, Page 10
$\hat x$	Indicates (roughly) that x is a bound variable used to define a function. Can also mean (roughly) "the set of x such that...".	Chapter I, page 15
!	Indicates that a function preceding it is first order	Chapter II.V
⊦	Assertion: it is true that	*1(3)
~	Not	*1(5)
∨	Or	*1(6)
⊃	(A modification of Peano's symbol Ɔ.) Implies	*1.01
=	Equality	*1.01
Df	Definition	*1.01
Pp	Primitive proposition	*1.1
Dem.	Short for "Demonstration"	*2.01
.	Logical and	*3.01
p⊃q⊃r	p⊃q and q⊃r	*3.02
≡	Is equivalent to	*4.01
p≡q≡r	p≡q and q≡r	*4.02
Hp	Short for "Hypothesis"	*5.71
(x)	For all x This may also be used with several variables as in 11.01.	*9
(∃x)	There exists an x such that . This may also be used with several variables as in 11.03.	9, 10.01
≡_x, ⊃_x	The subscript x is an abbreviation meaning that the equivalence or implication holds for all x. This may also be used with several variables.	10.02, 10.03, *11.05.
=	x=y means x is identical with y in the sense that they have the same properties	*13.01
≠	Not identical	*13.02
x=y=z	x=y and y=z	*13.3
℩	This is an upside-down iota (unicode U+2129). ℩x means roughly "the unique x such that...."	*14
[]		*14.01
E!	There exists a unique...	*14.02
ε	A Greek epsilon, abbreviating the Greek word ἐστί meaning "is". It is used to mean "is a member of" or "is a"	*20.02 and Chapter I page 26
Cls	Short for "Class". The 2-class of all classes	*20.03
,	Abbreviation used when several variables have the same property	20.04, 20.05
~ε	Is not a member of	*20.06
Prop	Short for "Proposition" (usually the proposition that one is trying to prove).	Note before *2.17
Rel	The class of relations	*21.03
⊂ ⪽	Is a subset of (with a dot for relations)	22.01, 23.01
∩ ⩀	Intersection (with a dot for relations). α∩β∩γ is defined to be (α∩β)∩γ and so on.	22.02, 22.53, 23.02, 23.53
∪ ⨄	Union (with a dot for relations) α∪β∪γ is defined to be (α∪β)∪γ and so on.	22.03, 22.71, 23.03, *23.71
− ∸	Complement of a class or difference of two classes (with a dot for relations)	22.04, 22.05, 23.04, 23.05
V ⩒	The universal class (with a dot for relations)	*24.01
Λ ⩑	The null or empty class (with a dot for relations)	24.02
∃!	The following class is non-empty	*24.03
‘	R ‘ y means the unique x such that xRy	*30.01
Cnv	Short for converse. The converse relation between relations	*31.01
Ř	The converse of a relation R	*31.02
$\overrightarrow{ R}$	A relation such that $x\overrightarrow{ R}z$ if x is the set of all y such that $y\overrightarrow{ R}z$	*32.01
$\overleftarrow{ R}$	Similar to $\overrightarrow{ R}$ with the left and right arguments reversed	*32.02
sg	Short for "sagitta" (Latin for arrow). The relation between $\overrightarrow{ R}$ and R.	*32.03
gs	Reversal of sg. The relation between $\overleftarrow{ R}$ and R.	32.04
D	Domain of a relation (αDR means α is the domain of R).	*33.01
D	(Upside down D) Codomain of a relation	*33.02
C	(Initial letter of the word "campus", Latin for "field".) The field of a relation, the union of its domain and codomain	*32.03
F	The relation indicating that something is in the field of a relation	*32.04
$\|$	The composition of two relations. Also used for the Sheffer stroke in *8 appendix A of the second edition.	*34.01
R², R³	Rⁿ is the composition of R with itself n times.	34.02, 34.03
$\upharpoonleft$	$\alpha\upharpoonleft R$ is the relation R with its domain restricted to α	*35.01
$\upharpoonright$	$R\upharpoonright \alpha$ is the relation R with its codomain restricted to α	*35.02
$\uparrow$	Roughly a product of two sets, or rather the corresponding relation	*35.04
⥏	P⥏α means $\alpha\upharpoonleft P \upharpoonright\alpha$ . The symbol is unicode U+294F	*36.01
“	(Double open quotation marks.) R“α is the domain of a relation R restricted to a class α	*37.01
R_ε	αR_εβ means "α is the domain of R restricted to β"	*37.02
‘‘‘	(Triple open quotation marks.) αR‘‘‘κ means "α is the domain of R restricted to some element of κ"	*37.04
E!!	Means roughly that a relation is a function when restricted to a certain class	*37.05
♀	A generic symbol standing for any functional sign or relation	*38
”	Double closing quotation mark placed below a function of 2 variables changes it to a related class-valued function.	*38.03
p	The intersection of the classes in a class. (The meaning of p changes here: before section 40 p is a propositional variable.)	*40.01
s	The union of the classes in a class	*40.02
$\|\|$	$R\|\|S$ applies R to the left and S to the right of a relation	*43.01
I	The equality relation	*50.01
J	The inequality relation	*50.02
ι	Greek iota. Takes a class x to the class whose only element is x.	*51.01
1	The class of classes with one element	*52.01
0	The class whose only element is the empty class. With a subscript r it is the class containing the empty relation.	54.01, 56.03
2	The class of classes with two elements. With a dot over it, it is the class of ordered pairs. With the subscript r it is the class of unequal ordered pairs.	54.02, 56.01, *56.02
$\downarrow$	An ordered pair	*55.01
Cl	Short for "class". The powerset relation	*60.01
Cl ex	The relation saying that one class is the set of non-empty classes of another	*60.02
Cls², Cls³	The class of classes, and the class of classes of classes	60.03, 60.04
Rl	Same as Cl, but for relations rather than classes	61.01, 61.02, 61.03, 61.04
ε	The membership relation	*62.01
t	The type of something, in other words the largest class containing it. t may also have further subscripts and superscripts.	63.01, 64
t₀	The type of the members of something	*63.02
α_x	the elements of α with the same type as x	65.01 65.03
α(x)	The elements of α with they type of the type of x.	65.02 65.04
→	α→β is the class of relations such that the domain of any element is in α and the codomian is in β.	*70.01
sm	Short for "similar". The class of bijections between two classes	*73.01
sm	Similarity: the relation that two classes have a bijection between them	*73.02
P_Δ	λP_Δκ means that λ is a selection function for P restricted to κ	*80.01
excl	Refers to various classes being disjoint	*84
↧	P↧x is the subrelation of P of ordered pairs in P whose second term is x.	*85.5
Rel Mult	The class of multipliable relations	*88.01
Cls² Mult	The multipliable classes of classes	*88.02
Mult ax	The multiplicative axiom, a form of the axiom of choice	*88.03
R_*	The transitive closure of the relation R	*90.01
R_st, R_st	A relations saying one relation is a positive power of R times another	91.01, 91.02
Pot	(Short for the Latin word "potentia" meaning power.) The positive powers of a relation	*91.03
Potid	("Pot" for "potentia" + "id" for "identity".) The positive or zero powers of a relation	*91.04
R_po	The union of the positive power of R	*91.05
B	Stands for "Begins". Something is in the domain but not the range of a relation	*93.01
min, max	used to mean that something is a minimal or maximal element of soeme class with respect to some relation	93.02 93.021
gen	The generations of a relation	*93.03
✸	P✸Q is a relation corresponding to the operation of applying P to the left and Q to the right of a relation. This meaning is only used in 95 and the symbol is defined differently in 257.	*95.01
Dft	Temporary definition (followed by the section it is used in).	*95 footnote
I_R,J_R	Certain subsets of the images of an element under repeatedly applying a function R. Only used in *96.	96.01, 96.02
$\overleftrightarrow{R}$	The class of ancestors and descendants of an element under a relation R	*97.01

Symbols introduced in Principia Mathematica volume II

Symbol	Approximate meaning	Reference
Nc	The cardinal number of a class	100.01,103.01
NC	The class of cardinal numbers	100.02, 102.01, 103.02,104.02
μ⁽¹⁾	For a cardinal μ, this is the same cardinal in the next higher type.	*104.03
μ₍₁₎	For a cardinal μ, this is the same cardinal in the next lower type.	*105.03
+	The disjoint union of two classes	*110.01
+_c	The sum of two cardinals	*110.02
Crp	Short for "correspondence".	*110.02
ς	(A Greek sigma used at the end of a word.) The series of segments of a series; essentially the completion of a totally ordered set	*212.01

Symbols introduced in Principia Mathematica volume III

Symbol	Approximate meaning	Reference
Bord	(Abbreviation of "bene ordinata", Latin for well-ordered) The class of well-founded relations	*250.01
Ω	The class of well ordered relations (though for some weird reason PM does not allow well-orderings on a class with 1 element)	250.02

References

Whitehead, Alfred North, and Bertrand Russell. Principia Mathematica, 3 vols, Cambridge University Press, 1910, 1912, and 1913. Second edition, 1925 (Vol. 1), 1927 (Vols 2, 3).

External links

List of notation in Principia Mathematica at the end of volume I
- The Notation in Principia Mathematica—by Bernard Linsky.
Principia Mathematica online (University of Michigan Historical Math Collection):
Proposition ✸54.43 in a more modern notation (Metamath)

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