Glossary of Principia Mathematica

This is a list of the notation used in Alfred North Whitehead and Bertrand Russell's Principia Mathematica (1910–1913).

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

Glossary[edit]

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

definiendum

The symbol being defined

definiens

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

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

relatum

The term y in xRy

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^[1]

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[edit]

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
[]	The scope indicator for definite descriptions.	*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 the 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 codomain 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_ts	Relations saying that 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 some 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[edit]

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[edit]

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^[2]	250.02

Notes[edit]

^ PM insists that this class must be the field of the relation, resulting in the bizarre convention that the class cannot have exactly one element.
^ Note that by convention PM does not allow well-orderings on a class with 1 element.

References[edit]

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[edit]

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)

[1] PM insists that this class must be the field of the relation, resulting in the bizarre convention that the class cannot have exactly one element.

[2] Note that by convention PM does not allow well-orderings on a class with 1 element.

[1]

[2]

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