Glossary of Principia Mathematica

This is a list of the notation used in Alfred North Whitehead and Bertrand Russell's Principia Mathematica (1910–13).

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
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
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
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
A relation such that if x is the set of all y such that *32.01
Similar to with the left and right arguments reversed *32.02
sg Short for "sagitta" (Latin for arrow). The relation between and R. *32.03
gs Reversal of sg. The relation between 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
is the relation R with its domain restricted to α *35.01
is the relation R with its codomain restricted to α *35.02
Roughly a product of two sets, or rather the corresponding relation *35.04
⥏ P⥏α means . 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
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
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
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
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
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
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
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
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[2] 250.02

See also

Notes
  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.

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).
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