#### Logical Atomism

The theory of logical atomism is a crucial tool in Russell’s
philosophical method. Logical atomism contends that, through rigorous and
exacting analysis, language—like physical matter—can be broken down
into smaller constituent parts. When a sentence can be broken down
no further, we are left with its “logical atoms.” By examining the
atoms of a given statement, we expose its underlying assumptions
and can then better judge its truth or validity.

Take, for example, the following sentence: “The King of
America is bald.” Even this deceptively simple sentence can be broken
down into three logical components:

- 1. There exists a King of America.
- 2. There is only one King of America.
- 3. The King of America has no hair.

We know, of course, that there is no King of America.
Thus the first assumption, or atom, is false. The complete statement
“The King of America is bald” is untrue, but it isn’t properly *false* because
the opposite isn’t true either. “The King of America has hair” is
just as untrue as the original statement, because it continues to
assume that there is, in fact, a King of America. If the sentence
is neither true nor false, what kind of claim on the truth can it
make? Philosophers have debated whether the sentence, in fact, has
any meaning at all. What is clear is that applying the concepts
of logical atomism to language reveals the complexity of the concepts
truth and validity.

#### The Theory of Descriptions

The theory of descriptions represents Russell’s most significant
contribution to linguistic theory. Russell believed that everyday
language is too misleading and ambiguous to properly represent the truth.
If philosophy was to rid itself of mistakes and assumptions, a purer,
more rigorous language would be required. This formal, idealized
language would be based on mathematical logic and would look more
like a string of math equations than anything ordinary people might
recognize as a language.

Russell’s theory offers a method for understanding statements that
include definite descriptions. A definite description is a word, name,
or phrase that denotes a particular, individual object. *That chair*, *Bill
Clinton*, and *Malaysia* are all examples
of definite descriptions. The theory of descriptions was created
to deal with sentences such as “The King of America is bald,” where
the object to which the definite description refers is ambiguous
or nonexistent. Russell calls these expressions *incomplete
symbols*. Russell showed how these statements can be broken
down into their logical atoms, as demonstrated in the previous section.
A sentence involving definite descriptions is, in fact, just a shorthand
notation for a *series* of claims. The true, logical
form of the statement is obscured by the grammatical form. Thus,
application of the theory allows philosophers and linguists to expose
the logical structures hidden in ordinary language—and, it is hoped,
to avoid ambiguity and paradox when making claims of their own.

#### Set Theory

The ability to define the world in terms of sets is crucial
to Russell’s project of logicism, or the attempt to reduce all mathematics
to formal logic. A set is defined as a collection of objects, called *members* or *elements*.
We can speak of the set of all teaspoons in the world, the set of
all letters in the alphabet, or the set of all Americans. We can
also define a set negatively, as in “the set of all things that
are *not* teaspoons.” This set would include pencils,
cell phones, kangaroos, China, and anything else that’s not a teaspoon.
Sets can have subsets (e.g., the set of all Californians is a subset
of the set of all Americans) and can be added and subtracted from
one another. In early set theory, any collection of objects could
properly be called a set.

Set theory was invented by Gottlob Frege at the end of
the nineteenth century and has become a major foundation of modern mathematical
thought. The paradox discovered by Bertrand Russell in the early
twentieth century, however, led to a major reconsideration of its
founding principles. Russell’s Paradox showed that allowing any
collection of objects to be termed a set sometimes creates logically
impossible situations—a fact that threatens to undermine Russell’s
greater, logicist project.

#### Russell’s Paradox

Russell’s Paradox, which Russell discovered in 1901, reveals
a problem in set theory as it had existed up to that point. The
paradox in its true form is very abstract and somewhat difficult
to grasp—it concerns the set of all sets that are not members of
themselves. To understand what that refers to, consider the example
of the set containing all the teaspoons that have ever existed.
This set is not a member of itself, because the set of all teaspoons
is not itself a teaspoon. Other sets may, in fact, be members of
themselves. The set of everything that is not a teaspoon does contain
itself because the set is not a teaspoon. The paradox arises if
you try to consider the set of all the sets that are not members
of themselves. This metaset would include the set of all teaspoons,
the set of all forks, the set of all lobsters, and many other sets.
Russell poses the question of whether *that* set
includes itself. Because it is defined as the set of all sets that are
not members of themselves, it must include itself because by definition
it does not include itself. But if it includes itself, by definition it
must not include itself. The definition of this set contradicts
itself.

Many people have found this paradox difficult to fathom,
so in philosophy textbooks it is often taught by analogy with other
paradoxes that are similar but less abstract. One of the most famous
of these is the barber paradox. In a certain town, there is a barber
who shaves the men who do not shave themselves. The paradox arises when
we consider whether the barber shaves himself. On one hand, he can’t
shave himself because he’s the barber, and the barber only shaves
men who don’t shave themselves. But if he doesn’t shave himself,
he must shave himself, because he shaves all the men who don’t shave
themselves. This paradox resembles Russell’s in that the way the
set is defined makes it impossible to say whether a certain thing belongs
to it or not.

Russell’s Paradox is significant because it exposes a
flaw in set theory. If any collection of objects can be called a
set, then certain situations arise that are logically impossible.
Paradoxical situations such as that referred to in the paradox threaten
the entire logicist project. Russell argued for a stricter version
of set theory, in which only certain collections can officially
be called sets. These sets would have to satisfy certain axioms
to avoid impossible or contradictory scenarios. Set theory before
Russell is generally called *naïve set theory*, while
post-Russell set theory is termed *axiomatic set theory*.