18 May is the birthday of philosophical legend Bertrand Russell. Bryan Pickel, lecturer in Philosophy at the University of Edinburgh, explains what any budding philosopher needs to know about Russell’s work.

Bertrand Russell (1872-1970) was a 20th-century philosopher, whose work covered virtually all areas of philosophy. He wrote prolifically on topics including the foundations of mathematics, the structure of language, the nature of thought, the status of science in society, and sexual ethics. His work is a starting point in many areas both within philosophy and in other disciplines such as linguistics. Russell was known for regularly changing his mind, often starting on one side of a debate but winding up on the other just a few years later. One of Russell’s most important contributions was his discovery of the paradox of classes, aka Russell’s paradox.

Classes and logic

Logicians and mathematicians had long been interested in the notion of a “class” as a foundation for mathematics. A class is a group of things. Whenever we can identify a property, there is a class of things with that property. For example, there is a class of cows (containing all the cows in the world, and nothing else), a class of mushrooms, a class of speckled blue eggs, etc.

The founder of modern logic, Gottlob Frege, had thought all mathematical truths were really just logical truths about classes. Logical truths can be proved by reasoning alone. For example, it is a truth of logic that if all cows are mortal and Daisy is a cow, then she is mortal. Frege’s idea was that he could prove any mathematical claim using only logical assumptions about classes.

Russell’s paradox

Russell spotted a problem with this approach. He noted that some classes are members of themselves. For example, the class containing all the classes (the class of cows, the class of mushrooms, the class of all eggs, etc). This itself is also a class (it contains all the classes in the world, and nothing else). So the class of all classes is also a member of itself, because it is a class. (Still following?!). On the other hand, other classes are not members of themselves. The class of all the cows is not itself a cow, and so is not a member of itself.

This is where things get really interesting. Let’s consider the class of all classes that aren’t members of themselves. So this is a class that contains all the classes such as the class of cows and the class of mushrooms, which are not themselves cows or mushrooms; but it does not contain classes (such as the class of all classes) which are also a member of themselves. Let’s call this class R, for “Russell Class”.

The problem is that we can now prove two contradictory claims:

On the one hand, we can prove that R is a member of R. If R were not a member of R, then it would be a member of the class of classes that are not members of themselves. But that class just is R. So, R has to be a member of R.

On the other hand, we could also claim the opposite: that R is not a member of R. If R were a member of R, then it would be one of the classes that were not members of themselves. So, R cannot be a member of R.

You might have to read that twice – I know I did! The main point is, Russell saw that if we say there is such a thing as “R” – the class of classes that are not members of themselves – then we end up proving two contradictory claims: that R is a member of R and that R is not a member of R. These two claims cannot both be true. So the original assumption about classes (that there is a class like R) must have been mistaken.

When Russell wrote to Frege about this paradox, Frege replied that Russell’s discovery “left me thunderstruck, because it has rocked the ground on which I meant to build arithmetic.”

One element of Russell’s solution is that classes are after all not real entities, but are logical fictions. We have expressions for classes such as ‘the class of people interested in mathematical logic’. This expression looks like it describes a particular class just as the expression ‘the biggest cow’ describes a particular cow. But Russell thinks that this is misleading – there are actual things as cows in the world, but there is no such thing as “classes” that actually exist. Instead of being a real thing, a class is just a way of speaking about other things.

We can explain the meaning of an expression for a class such as ‘the class of people interested in mathematical logic’ by explaining what sentences containing this expression mean. So Russell says that the sentence ‘The class of people interested in mathematical logic is not very numerous’ means the same thing as the sentence ‘Not very many people are interested in mathematical logic’.

Notice that the latter sentence does not contain any expression that stands for a class. Russell concludes that for any statement about a class to be meaningful, we should be able to say the same thing without mentioning classes. Russell says that the expression ‘the class of all classes that are not members of themselves’, violates this requirement and so gives rise to paradox.

Russell’s legacy

In keeping with his general interest, Russell applied his idea far beyond mathematical logical. He argued that a host of philosophical problems—including expressions about numbers, material objects, and even people—could be solved by treating other terms as standing for logical fictions.

The next generation of philosophers including Rudolph Carnap, Susan Stebbing, and (in his early work) Ludwig Wittgenstein took Russell’s approach as a starting point for their own contributions. Indeed, his influence continues today. In my own case, my research investigates whether Russell’s notion of a logical fiction can be made to fit into contemporary theories of semantics (meaning). I’m particularly interested in whether Russell’s ideas can be used to solve similar paradoxes concerning the notions of truth and falsity.

And in the meantime, Russell’s paradox is still fertile ground for Philosophical memes, such as this one, from my colleague Brian Rabern.

References and more information

Bryan Pickel (University of Edinburgh profile)

Russell’s Logical Atomism  (Stanford Encyclopedia of Philosophy)

Works by Bertrand Russell (Project Gutenberg)

Gabriel, Gottfried et al (1980), Philosophical and Mathematical Correspondence. University of Chicago Press, Chicago, IL.

Russell, Bertrand (1914/1993), Our Knowledge of the External World. Routledge, London.