### 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:

2. There is only one King of America.

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*.