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

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

SymbolApproximate meaningReference✸Indicates that the following number is a reference to some propositionα,β,γ,δ,λ,κ, μClassesChapter I page 5f,g,θ,φ,χ,ψVariable functions (though θ is later redefined as the order type of the reals)Chapter I page 5a,b,c,w,x,y,zVariablesChapter I page 5p,q,rVariable propositions (though the meaning of p changes after section 40).Chapter I page 5P,Q,R,S,T,URelationsChapter I page 5. : :. :: Dots used to indicate how expressions should be bracketed, and also used for logical "and". Chapter I, Page 10Indicates (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 *1.01Df Definition *1.01Pp Primitive proposition *1.1Dem.Short for "Demonstration"*2.01. Logical and *3.01p⊃q⊃r p⊃q and q⊃r *3.02≡ Is equivalent to *4.01p≡q≡rp≡q and q≡r*4.02Hp 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*13.02x=y=zx=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.*14.01E! There exists a unique... *14.02εA 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 26Cls Short for "Class". The 2-class of all classes*20.03,Abbreviation used when several variables have the same property*20.04, *20.05~εIs not a member of*20.06Prop Short for "Proposition" (usually the proposition that one is trying to prove).Note before *2.17RelThe 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, *23.05V ⩒The universal class (with a dot for relations)*24.01Λ ⩑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*30.01CnvShort for converse. The converse relation between relations*31.01ŘThe converse of a relation R*31.02A relation such that if x is the set of all y such that *32.01Similar to with the left and right arguments reversed*32.02sgShort for "sagitta" (Latin for arrow). The relation between and R.*32.03gsReversal of sg. The relation between and R.32.04DDomain of a relation (αDR means α is the domain of R).*33.01D(Upside down D) Codomain of a relation*33.02C(Initial letter of the word "campus", Latin for "field".) The field of a relation, the union of its domain and codomain*32.03FThe 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. *34.01R2, R3Rn 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*35.02Roughly 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*37.01RεαRεβ 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"*37.04E!!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. *38.03pThe intersection of the classes in a class. (The meaning of p changes here: before section 40 p is a propositional variable.)*40.01sThe union of the classes in a class*40.02 applies R to the left and S to the right of a relation*43.01IThe equality relation*50.01JThe inequality relation*50.02ιGreek iota. Takes a class x to the class whose only element is x. *51.011The class of classes with one element*52.010The class whose only element is the empty class. With a subscript r it is the class containing the empty relation. *54.01, *56.032The 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, *56.02An ordered pair*55.01ClShort for "class". The powerset relation*60.01Cl exThe relation saying that one class is the set of non-empty classes of another*60.02Cls2, Cls3The class of classes, and the class of classes of classes*60.03, *60.04RlSame as Cl, but for relations rather than classes*61.01, *61.02, *61.03, *61.04εThe membership relation*62.01tThe type of something, in other words the largest class containing it. t may also have further subscripts and superscripts.*63.01, *64t0The type of the members of something*63.02αxthe elements of α with the same type as x*65.01 *65.03α(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. *70.01Short for "similar". The class of bijections between two classes*73.01smSimilarity: the relation that two classes have a bijection between them*73.02PΔλPΔκ means that λ is a selection function for P restricted to κ*80.01exclRefers to various classes being disjoint*84↧P↧x is the subrelation of P of ordered pairs in P whose second term is x.*85.5Rel MultThe class of multipliable relations*88.01Cls2 MultThe multipliable classes of classes*88.02Mult axThe multiplicative axiom, a form of the axiom of choice*88.03R*The transitive closure of the relation R*90.01Rst, RtsRelations saying that one relation is a positive power of R times another*91.01, *91.02Pot(Short for the Latin word "potentia" meaning power.) The positive powers of a relation*91.03Potid("Pot" for "potentia" + "id" for "identity".) The positive or zero powers of a relation*91.04RpoThe union of the positive power of R*91.05BStands for "Begins". Something is in the domain but not the range of a relation*93.01min, maxused to mean that something is a minimal or maximal element of some class with respect to some relation*93.02 *93.021genThe 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.*95.01DftTemporary definition (followed by the section it is used in).*95 footnoteIR,JRCertain subsets of the images of an element under repeatedly applying a function R. Only used in *96.*96.01, *96.02The class of ancestors and descendants of an element under a relation R*97.01

Symbols introduced in Principia Mathematica, Volume II

SymbolApproximate meaningReferenceNcThe cardinal number of a class*100.01,*103.01NCThe class of cardinal numbers*100.02, *102.01, *103.02,*104.02μ(1)For a cardinal μ, this is the same cardinal in the next higher type.*104.03μ(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 +cThe sum of two cardinals*110.02CrpShort for "correspondence". *110.02ς (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

SymbolApproximate meaningReferenceBordAbbreviation of "bene ordinata" (Latin for "well-ordered"), the class of well-founded relations*250.01ΩThe class of well ordered relationsBy convention, PM does not allow well-orderings on a class with 1 element.250.02

Glossary of Principia Mathematica

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