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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

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

{{term|apparent variable}}

{{defn|bound variable}}

{{term|atomic proposition}}

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

{{term|Barbara}}

{{defn|A mnemonic for a certain syllogism.}}

{{term|class}}

{{defn|A subset of the members of some type}}

{{term|codomain}}

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

{{term|compact}}

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

{{term|concordant}}

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

{{term|connected}}

{{term|connexity}}

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

{{term|continuous}}

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

{{term|correlator}}

{{defn|bijection}}

{{term|couple}}

{{defn|no=1|A cardinal couple is a class with exactly two elements}}

{{defn|no=2|An ordinal couple is an ordered pair (treated in PM as a special sort of relation)}}

{{term|Dedekindian}}

{{defn|complete (relation) *214}}

{{term|definiendum}}

{{defn|The symbol being defined}}

{{term|definiens}}

{{defn|The meaning of something being defined}}

{{term|derivative}}

{{defn|A derivative of a subclass of a series is the class of limits of non-empty subclasses}}

{{term|description}}

{{defn|A definition of something as the unique object with a given property}}

{{term|descriptive function}}

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

{{term|diversity}}

{{defn|The inequality relation}}

{{term|domain}}

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

{{term|elementary proposition}}

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

{{term|Epimenides}}

{{defn|Epimenides was a legendary Cretan philosopher}}

{{term|existent}}

{{defn|non-empty}}

{{term|extensional function}}

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

{{term|field}}

{{defn|The field of a relation R is the union of its domain and codomain}}

{{term|first-order}}

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

{{term|function}}

{{defn|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.}}

{{term|general proposition}}

{{defn|A proposition containing quantifiers}}

{{term|generalization}}

{{defn|Quantification over some variables}}

{{term|homogeneous}}

{{defn|A relation is called homogeneous if all arguments have the same type.}}

{{term|individual}}

{{defn|An element of the lowest type under consideration}}

{{term|inductive}}

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

{{term|intensional function}}

{{defn|A function that is not extensional.}}

{{term|logical}}

{{defn|no=1|The logical sum of two propositions is their logical disjunction}}

{{defn|no=2|The logical product of two propositions is their logical conjunction}}

{{term|matrix}}

{{defn|A function with no bound variables. *12}}

{{term|median}}

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

{{term|member}}

{{defn|element (of a class)}}

{{term|molecular proposition}}

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

{{term|null-class}}

{{defn|A class containing no members}}

{{term|predicative}}

{{defn|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.}}

{{term|primitive proposition}}

{{defn|A proposition assumed without proof}}

{{term|progression}}

{{defn|A sequence (indexed by natural numbers)}}

{{term|rational}}

{{defn|A rational series is an ordered set isomorphic to the rational numbers}}

{{term|real variable}}

{{defn|free variable}}

{{term|referent}}

{{defn|The term x in xRy}}

{{term|reflexive}}

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

{{term|relation}}

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

{{term|relative product}}

{{defn|The relative product of two relations is their composition}}

{{term|relatum}}

{{defn|The term y in xRy}}

{{term|scope}}

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

{{term|Scott}}

{{defn|Sir Walter Scott, author of Waverley.}}

{{term|second-order}}

{{defn|A second order function is one that may have first-order arguments}}

{{term|section}}

{{defn|A section of a total order is a subclass containing all predecessors of its members.}}

{{term|segment}}

{{defn|A subclass of a totally ordered set consisting of all the predecessors of the members of some class}}

{{term|selection}}

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

{{term|sequent}}

{{defn|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)}}

{{term|serial relation}}

{{defn|A total order on a classPM insists that this class must be the field of the relation, resulting in the bizarre convention that the class cannot have exactly one element.}}

{{term|significant}}

{{defn|well-defined or meaningful}}

{{term|similar}}

{{defn|of the same cardinality}}

{{term|stretch}}

{{defn|A convex subclass of an ordered class}}

{{term|stroke}}

{{defn|The Sheffer stroke (only used in the second edition of PM)}}

{{term|type}}

{{defn|As in type theory. All objects belong to one of a number of disjoint types.}}

{{term|typically}}

{{defn|Relating to types; for example, "typically ambiguous" means "of ambiguous type".}}

{{term|unit}}

{{defn|A unit class is one that contains exactly one element}}

{{term|universal}}

{{defn|A universal class is one containing all members of some type}}

{{term|vector}}

{{defn|no=1|Essentially an injective function from a class to itself (for example, a vector in a vector space acting on an affine space)}}

{{defn|no=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

class="wikitable"

!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

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

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

{{overline|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

\overleftrightarrow{R}

|The class of ancestors and descendants of an element under a relation R

|*97.01

Symbols introduced in ''Principia Mathematica'', Volume II

class="wikitable"

!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

class="wikitable"

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

|250.02

See also

Notes

{{reflist}}

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

  • [https://archive.org/stream/PrincipiaMathematicaVolumeI/WhiteheadRussell-PrincipiaMathematicaVolumeI#page/n709/mode/2up List of notation in Principia Mathematica at the end of Volume I]
  • "[http://plato.stanford.edu/entries/pm-notation/ The Notation in Principia Mathematica]" by Bernard Linsky.
  • Principia Mathematica online (University of Michigan Historical Math Collection):
  • [http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=AAT3201.0001.001 Volume I]
  • [http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=AAT3201.0002.001 Volume II]
  • [http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=AAT3201.0003.001 Volume III]
  • [http://us.metamath.org/mpegif/pm54.43.html Proposition ✸54.43] in a more modern notation (Metamath)

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