Suggestions

Use up and down arrows to review and enter to select.Please wait while we process your payment

If you don't see it, please check your spam folder. Sometimes it can end up there.

If you don't see it, please check your spam folder. Sometimes it can end up there.

Please wait while we process your payment

By signing up you agree to our terms and privacy policy.

Don’t have an account? Subscribe now

Create Your Account

Sign up for your FREE 7-day trial

Already have an account? Log in

Your Email

Choose Your Plan

BEST VALUE

**Save over 50%** with a SparkNotes PLUS Annual Plan!

Purchasing SparkNotes PLUS for a group?

Get Annual Plans at a discount when you buy 2 or more!

Price

$24.99 $18.74 /subscription + tax

Subtotal $37.48 + tax

Save 25% on 2-49 accounts

Save 30% on 50-99 accounts

Want 100 or more? Contact us for a customized plan.

Your Plan

Payment Details

Payment Details

Payment Summary

SparkNotes Plus

You'll be billed after your free trial ends.

7-Day Free Trial

Not Applicable

Renews March 28, 2023 March 21, 2023

Discounts (applied to next billing)

DUE NOW

US $0.00

SNPLUSROCKS20 | 20% Discount

This is not a valid promo code.

Discount Code (one code per order)

SparkNotes Plus subscription is $4.99/month or $24.99/year as selected above. The free trial period is the first 7 days of your subscription. **TO CANCEL YOUR SUBSCRIPTION AND AVOID BEING CHARGED, YOU MUST CANCEL BEFORE THE END OF THE FREE TRIAL PERIOD.** You may cancel your subscription on your Subscription and Billing page or contact Customer Support at custserv@bn.com. Your subscription will continue automatically once the free trial period is over. Free trial is available to new customers only.

Choose Your Plan

Payment Details

Payment Summary

Your Free Trial Starts Now!

For the next 7 days, you'll have access to awesome PLUS stuff like AP English test prep, No Fear Shakespeare translations and audio, a note-taking tool, personalized dashboard, & much more!

Thank You!

You’ve successfully purchased a group discount. Your group members can use the joining link below to redeem their group membership. You'll also receive an email with the link.

https://www.sparknotes.com/group‑discounts/join/Spark123

Members will be prompted to log in or create an account to redeem their group membership.

Thanks for creating a SparkNotes account! Continue to start your free trial.

Please wait while we process your payment

Your PLUS subscription has expired

- We’d love to have you back! Renew your subscription to regain access to all of our exclusive, ad-free study tools.

Please wait while we process your payment

- Looking for exclusive,
**AD-FREE**study tools? Look no further!

- Study Guide

Please wait while we process your payment

Sign up and get instant access to bookmarks.

- Ad-Free experience
- Easy-to-access study notes
- Flashcards & Quizzes
- AP® English test prep
- Plus much more

Themes, Arguments, and Ideas

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 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.

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, 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*.

Did you know you can highlight text to take a note?
x

Please wait while we process your payment

Sign up and get instant access to creating and saving your own notes as you read.

- Ad-Free experience
- Easy-to-access study notes
- Flashcards & Quizzes
- AP® English test prep
- Plus much more