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.

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: A cardinal couple is a class with exactly two elements 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: The logical sum of two propositions is their logical disjunction 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

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: Essentially an injective function from a class to itself (for example, a vector in a vector space acting on an affine space) A vector-family is a non-empty commuting family of injective functions from some class to itself (VIB)