Glossary of Principia Mathematica

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

1.  A cardinal couple is a class with exactly two elements2.  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 disjunction2.  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. Unfortunately PM insist that this class must be the field of the relation, resulting in the bizarre convention that the class cannot have exactly one element.

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)