One final remark on Russell's theory: He views it as a defect of his logical system that it comes out in that system to be a logical truth that something exists. He says that, but it doesn't fit with most of his presentation, since, after all, he seems to take classes to be logical objects, and so, even if there isn't anything else, there must at least be the empty class.

Russell didn't like this, he did view it as a logical defect, and so his official theory, which he only mentions in a footnote in what we read, is his "no class" theory.

The idea is similar to Frege's context principle: we can introduce class language if we can see how to determine the truth values of sentences using class language. If we can do that without actually using classes, we can act as if there are such things without any actual commitment to them or any other objects.

Here's how: $a\in\{x|\phi (x)\}$ if and only if $\phi (a)$.
$\{x|\phi (x)\}=\{x|\psi (x)\}$ if and only if $\forall x (\phi (x)\leftrightarrow \psi (x))$.

Thus, while Russell relies on classes to introduce all of mathematics, he doesn't actually believe in them.

This procedure, while it is formally analogous to the one Frege uses to introduce directions and numbers has the opposite purpose: Frege is introducing directions and numbers, telling us what they are; while Russell is eliminating classes, telling us there are no such things.

That is one of the most important differences between them: Russell's logicism eliminates mathematics in favor of logic. Frege's logicism introduces mathematics on the basis of logic.

Now, two comments on Frege:

1. Why does he have concepts and their extensions? Why wouldn't the concepts alone do the job? The answer is that he is worried about a traditional question of medieval logic: that of the unity of the proposition. For Frege concepts are sometimes like functions, not like an ordinary object. A concept can be incomplete: it has a hole in it. Thus, when we insert Plato into "x is a philosopher" they join, unify, into a proposition. If both were just objects, and we put them next to each other, we would have no explanation of how they join. Look at Russell's theory:

Socrates,$\{x|x$ is a philosopher$\}$

Nothing happens. They just sit there next to each other. To form a proposition, we need a relation:

Socrates $\in\{x|x$ is a philosopher$\}$

The $\in$ is also a relation or concept, and so it should, according to Russell, also be a class. But then we're stuck. To avoid an obvious infinite regress, Frege has two different kinds: concepts and objects.

But, concepts are not objects, while numbers are objects, Thus, Frege needs a way to associate an object with a concept. What he does is introduce the extension of the concept.

For Frege, because of the context principle, to tell us what an object is, it is enough to say when identity sentences using the concept are true. The fundamental principle is:

Extension($\phi (x)$)= Extension($\psi (x)$) if and only if $\forall x (\phi (x)\leftrightarrow \psi (x))$.

Several of you have complained to me, as Russell complained to Frege, that it is not at all clear what an extension is. For Frege, the question is answered by the principle above. The complaint is the merely psychological one that we don't know what picture to associate with extensions. His reaction is a (longwinded and technical) shrug.

Now, the main criticism of both theories. Russell heard that Cantor thought he had proved that there is no largest set. Russell thought sets were classes, and he was quite sure that Cantor was wrong: the class of everything is obviously the biggest class.

He sat down to find Cantor's mistake, and found the following instead:

Consider the class of all classes that are not members of themselves ($\{x|x\notin x\}$). Call the class $R$. Is $R$ in $R$?

A moment's thought shows that $R\in R\leftrightarrow R\notin R$. This is known as Russell's paradox, and it shows that there is something wrong with the "obvious" principle that every property (propositional function) determines a class. The same defect infects Frege's theory.

Russell discovered this, and spent a year trying to figure out where he went wrong. Finally, he wrote a very diffident letter to Frege, asking where the mistake was. Frege replied, by return mail, "Arithmetic totters."

An adjective is heterological if it does not apply to itself, homological if it does.

Thus, polysyllabic and short are homological, and monosyllabic and long are heterological.

Is "heterological" heterological?

Suppose the barber of Seville shaves only those who do not shave themselves. Who shaves the barber?

Is the sentence "this sentence is false" true or false?

Frege was devastated, and tried to work out a geometrical foundation for mathematics.

Russell invented the (ramified) theory of types: Each class has a level, and a class of classes of level n is of level n+1. Thus, no class can be a member of itself. That restores consistency, but make it impossible to prove that there are infinitely many things, and hence that all the numbers exist, and it makes mathematics unworkable without a new axiom, the axiom of reducibility, that Russell never thought it was reasonable to take to be a logical principle.

-- ProfessorShaughanLavine? - 21 Jan 2005