Bertrand Russell referred to several different definitions and philosophical
analyses as providingld "logical constructions" of certain entities

and expressions. Examples he cited were the Frege/Russell definition
of numbers as classes of equinumerous classes, the theory of definite descriptions,

the construction of matter from sense data, and several others. Generally
expressions for such entities are called "incomplete symbols" and

the entities themselves "logical fictions". The notion originates with
Russell's logicist program of reducing mathematics to logic, was widely

used by Russell, and led to the later Logical Positivist notion of
construction and ultimately the widespread use of set theoretic models
in philosophy.

1. Honest Toil

2.
Definite Descriptions and Classes

3. Other
Constructions

Bibliography

Russell's most specific formulation of logical construction as a method in Philosophy comes from his essay "Logical Atomism":

One very important heuristic maxim which Dr.Whitehead
and I found, by experience, to be applicable in mathematical logic,and
have since

applied to various other fields, is a form
of Occam's Razor. When some set of supposed entities has neat logical properties,
it turns out, in a

great many instances, that the supposed entities
can be replaced by purely logical structures composed of entities which
have not such neat

properties. In that case, in interpreting
a body of propositions hitherto believed to be about the supposed entities,
we can substitute the

logical structures without altering any of
the detail of the body of propositions in question.This is an economy,
because entities with neat

logical properties are always inferred, and
if the propositions in which they occur can be interpreted without making
this inference, the ground

for the inference fails, and our body of propositions
is secured against the need of a doubtful step. The principle may be stated
in the form:

"Whenever possible, substitute constructions
out of known entities for inferences to unknown entities"(1924, p.160)

Russell was speaking of logical constructions in this memorable passage
from his Introduction to Mathematical Philosophy: "The method of

"postulating"what we want has many advantages; they are the same as
the advantages of theft over honest toil. Let us leave them to others

and proceed with our honest toil." (1919, p. 71)

The notion of logical construction appears frequently with the idea
that what is defined is a "logical fiction", and an "incomplete

symbol". The latter term derives from the use of contextual definitions,
providing an analysis of each sentence in which a defined symbol may occur

without, however, giving an explicit definition, an equation or universal
statement giving necessary and sufficient conditions for the application
of the term in

isolation. The terms "fiction" and "incomplete symbol" apply with differing
aptness to various constructions.

Russell's first use of construction, and the model for later constructions,
is the Frege/Russell definition of numbers as classes. This follows the
kind of

definitions used in the arithmetization of analysis of the preceding
century, in particular, Dedekind's earlier construction of real numbers
as bounded classes

in the rational numbers. Russell's logicist program could not rest
content with postulates for the fundamental objects of mathematics such
as the Peano

Axioms for the natural numbers. Instead numbers were to be defined
as classes of equinumerous classes. Russell also refers to this method
as

"abstraction", now known as the abstraction of an equivalence class.
The definition of equinumerosity, or of the existence of a one to one

mapping between two classes, also called "similarity", is solely in
terms of logical notions of quantifiers and identity. With the numbers

defined, for example, two as the class of all two membered sets, or
pairs, the properties of numbers could be derived by logical means alone.

2. Definite Descriptions and Classes

The most influential of Russell's constructions was the theory of descriptions
from his paper "On Denoting" in 1905. Russell's theory provides an

analysis of sentences of the form "The F is G"where "The F"is called
a definite decription. The analysis proposes that

"The F is G"is equivalent to "There is one and only one F and it is
G";. With this analysis, the logical properties of

descriptions can now be deduced using just the logic of quantifiers
and identity. Among the theorems in *14 of Principia Mathematica are those
showing

that, (1) if there is just one F then "The F is F"is true, and if there
is not, then "The F is G"is always false and, crucially for

the logical manipulation of descriptions, (2) if the F = the G, and
the F is H, then the G is H. In other words, proper (uniquely referring)
descriptions

behave like singular terms. Some of these results are contentious---Strawson
noted that "The present king of france is bald"should be truth

valueless since there is no present king of France, rather than "plainly
false", as Russell's theory predicts.

The theory of descriptions introduces Russell's notion of incomplete
symbol. Definite descriptions "The F"do not show up in the formal

analysis of sentences in which they occur, thus "The F is H"becomes:

(Ex) [(y)(Fy & y=x) & Hx]

of which no subformula, or continuous segment, can be identified as
the analysis of "The F". Much as talk about "the average

family" as in "The average family has 2.2 children" becomes "The number
of children in families divided by the number of

families = 2.2", there is no portion of that analysis that corresponds
with "the average family". Instead we have a formula for

eliminating such expressions from contexts in which they occur, hence
the notion of "incomplete symbol" and the related "contextual

definition". It is standard to see in this the origins of the distinction
between between surface grammatical form and logical form, and thus the
origin

of linguistic analysis as a method in philosophy which operates by
seeing past superficial linguistic form to underlying philosophical analysis.
The theory of

descriptions has been criticized by some linguists who see descriptions
and other noun phrases as full fledged constituents of sentences, and who
see the

sharp distinction between grammatical and logical form as a mistake.

The theory of descriptions is often described as a model for avoiding
ontological commitment to objects such as Meinongian subsistent entities,
and so

logical constructions in general are often seen as being chiefly aimed
at ontological goals. In fact, that goal is at most peripheral to most
constructions.

Rather the goal is to allow the proof of propositions that would otherwise
have to be assumed as axioms or hypotheses. Nor need the ontological goal
be

always elimination of problematic entities. Other constructions should
be seen more as reductions of one class of entity to another, or replacements
of one

notion by a more precise, mathematical, substitute.

Russell's "No-Class" theory of classes from *20 of Principia Mathematica
provides a contextual definition like the theory of descriptions.

One of Russell's early diagnoses of the paradoxes was that they showed
that classes could not be objects. Indeed he seems to have come across
his

paradox of the class of all classes that are not members of themselves
by applying Cantor's argument to show that there are more classes of objects
than

objects. Hence, he concluded, classes could not be objects. Inspired
by the theory of descriptions, Russell proposed that to say something G
of the class

of Fs, G{x: Fx}, is to say that there is some property H coextensive
with (true of the same things as) F such that H is G. Extensionality of
sets is thus

derivable, rather than postulated. If F and H are coextensive then
anything true of {x: Fx} will be true of {x: Hx}. Features of sets then
follow from the

features of the logic of properties, the "ramified theory of types".
Because classes would seem to be individuals of some sort, but on

analysis are found not to be, Russell speaks of them as "logical fictions",
an expression which echoes Jeremy Bentham's notion of a

"legal fiction". Because statements attributing a property to particular
classes are analyzed by existential sentences saying that there is some

propositional function having that property, this construction should
not be seen as avoiding ontological commitment entirely, but rather of
reducing classes

to propositional functions. The properties of classes are really properties
of propositional functions and for every class said to have a property
there really

is some propositional function having that property.

For other constructions such as propositions a contextual definition
is not provided. In any case, constructions do not appear as the referents
of logically

proper names, and so by that account are not part of the fundamental
"furniture" of the world. (Early critical discussions of constructions,

such as Wisdom's, stressed the contrast between logically proper names,
which do refer, and constructions, which were thus seen as ontologically

innocent.)

Beginning with The Problems of Philosophy in 1912, Russell turned repeatedly
to the problem of matter. Part of the problem is to find a refution of

Berkeleyan idealism, of showing how the existence and real nature of
matter can be proved. In Problems Russell argues that matter is a well
supported

hypothesis that explains our experiences. Matter is known only indirectly,
"by description", as the cause, whatever it may be, of our sense

data, which we know "by acquaintance". This is the notion of hypothesis
which Russell contrasts with construction in the passage above.

Russell saw an analogy between the case of simply hypothesizing the
existence of numbers with certain properties, those described by axioms,
and

hypothesizing the existence of matter. While we distinguish the certain
knowledge we may have of mathematical entities from the contingent knowledge
of

material objects, Russell says that there are certain "neat" features
of matter which are just too tidy to have turned out by accident.

Examples include the most general spatiotemporal properties of objects,
that no two can occupy the same place at the same time, and so on. Material

objects are now to be seen as collections of sense data. Influenced
by William James, Russell defended a "neutral monism" by which

matter and minds were both to be constructed from sense data, but in
different ways. Intuitively, the sense data occuring as they do "in" a

mind, are material to construct that mind, the sense data derived from
an object from different points of view to constructthat object. Russell
saw some

support for this in the theory of relativity, and the fundumental importance
of frames of reference in the new physics.

These prominent examples are not the only use of the notion of construction
in Russell's thought. In Principia Mathematica the multiple relation theory

of propositions is introduced by saying that propositions are "incomplete
symbols". Russell's multiple relation theory, that he held from

1910 to 1919 or so, argued that the constitituents of propositions,
say "Desdemona loves Cassio", which is false, are unified in a way that

does not make it the case that they constitute a fact by themselves.
Those constituents occur only in the context of beliefs, say, "Othello
judges that

Desdemona loves Cassio". The real fact consists of a relation of Belief
holding between the constituents Othello, Desdemona and Cassio, thus

B(o,

d, L, c). Because one might also have believed propositions of other
structures, such as B(o, F, a) there need to be many such relations B,
thus the

"multiple" relation theory. Like the construction of numbers, this
construction abstracts out what a number of occurrences of a belief have
in

common, a believer and various objects in a certain order. The analysis
also makes the proposition an incomplete symbol because there is no constituent

in the analysis of "x believes that p"that corresponds to "p".

Russell also suggests that propositional functions are logical constructions
when he says that they are "nothing", but "nonetheless

important for that". (1918, p. 96) Propositional functions are abstracted
from their values, propositions. The propositional function "x is

human"is abstracted from its values "Socrates is human", "Plato is
human", etc. Viewing propositional functions as

constructions from propositions which are in turn constructions by
the multiple relation theory helps to make sense of the theory of types
of propositional

functions in Principia Mathematica. The notion of "incomplete symbol"
does not make as much sense as "construction"

when applied to propositional functions and propositions. This usage
requires a broadening of the notion.

The notion of logical construction had a great impact on the future
course of analytic philosophy. One line of influence was via the notion
of a contextual

definition, or paraphrase, intended to minimize ontological commitment
and to be a model of philosophical analysis. The distinction between the
surface

appearance of definite descriptions, as singular terms, and the fully
analyzed sentences from which they seem to disappear was seen as a model
for making

problematic notions disappear upon analysis. The theory of descriptions
has been viewed as a paradigm of philosophical analysis.

A more technical strand in analytic philosophy was influenced by the
construction of matter. Rudolf Carnap was attempted to carry out the construction
of

matter from sense data, and later Nelson Goodman continued the project.
More generally, however, the use of set theoretic constructions became

widespread among philosophers, and continues in the construction of
set theoretic models, both in the sense of logic where they model formal
theories,

and as objects of interest in their own right.

Carnap, R., The Logical Structure of the World
& Pseudo Problems in Philosophy, trans. R.George, Berkeley: University
of California Press,

1967.

Goodman, N., The Structure of Appearance,
Cambridge Mass: Harvard University Press, 1951.

Russell, B., 1919, Introduction to Mathematical
Philosophy , London: Routledge, reprinted 1993.

Russell, B., 1905, "On Denoting", in Robert
Marsh, Logic and Knowledge: Essays 1901-1950 , London: George Allen and

Unwin, 1956, 39-56.

Russell, B., 1918, "The Philosophy of Logical
Atomism" in The Philosophy of Logical Atomism , D.F.Pears, ed. Lasalle:
Open

Court, 1985, 35-155.

Russell, B., 1924, "Logical Atomism", in The
Philosophy of Logical Atomism , D.F.Pears, ed., Lasalle: Open Court, 1985,

157-181.

Russell, B., 1912, The Problems of Philosophy,
Oxford: Oxford University Press, reprinted 1967.

Whitehead, A.N., and Russell,B.: 1925, Principia
Mathematica Vol.I., second ed., Cambridge: Cambridge University Press,
1925.

Wisdom, J., 1931, "Logical Constructions (I.).",
Mind , XL , April, 188-216.

Copyright © 1996, 2001

Bernard Linsky

bernard.linsky@ualberta.ca