Glossary of Principia Mathematica explained

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

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

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
[]	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
R^2, R^3	R^n 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
Rε	αRεβ 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
Cls^2, Cls^3	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
αx	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
	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
PΔ	λPΔκ 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
Cls^2 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
Rst, Rts	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
Rpo	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
IR,JR	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^[1]	250.02

See also

• Glossary of set theory

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):
  • Volume I
  • Volume II
  • Volume III
• Proposition ✸54.43 in a more modern notation (Metamath)

Notes and References

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