# 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

| 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 |
| R$_ε$ | αR$_ε$β 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 |
| t$_0$ | The type of the members of something | *63.02 |
| α$_x$ | 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 |
| P$_Δ$ | λP$_Δ$κ 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 |
| R$_{st}$, R$_{ts}$ | 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 |
| R$_{po}$ | 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 |
| I$_R$,J$_R$ | 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 | 250.02 |