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Glossary of Principia Mathematica

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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

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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

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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

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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

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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

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Notes

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  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

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  • 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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