This is a list of the notation used in Russell and Whitehead's Principia Mathematica.
The second (but not the first) edition of volume I has a list of notation used at the end.
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 elements
2. 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 disjunction
2. 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)
