What is the theory of logical types?

Bertrand Russell, answered from the texts and cited to the page.

Bertrand Russell answers

The theory of types begins from a very simple observation about what goes wrong when we try to speak of "everything." If there were a class containing all objects of whatever sort, it would have to contain itself as a member — since it is itself something. But normally a class is not a member of itself; mankind is not a man.¹ From this springs the contradiction I first encountered in 1901, which I have since called Russell's paradox: form the class of all classes that are not members of themselves, and ask whether it is a member of itself.

If it is, it is not; if it is not, it is.² Each hypothesis implies its contradictory, and we have a flat contradiction at the heart of what seemed like innocent logical machinery. The solution the theory of types provides is this: the supposition that a class either is or is not a member of itself is not false — it is meaningless.³ Classes are logical fictions, and a statement nominally about a class is significant only if it can be translated into a form making no mention of the class.

When the translation is attempted for self-membership, no such form exists; the symbols cease to symbolise anything at all. From this it follows that classes must be arranged in a strict hierarchy — individuals at the bottom, classes of individuals above them, classes of classes above those — and one cannot form a new class mixing members from different levels of that hierarchy without producing symbols that are, strictly, devoid of meaning.⁴

What this means practically is that the formal uses mathematics makes of classes — all of them — can be obtained without ever supposing that classes are genuine constituents of propositions.⁵ When a proposition is correctly analysed, the class-symbol disappears, much as a definite description disappears on analysis into its constituent quantified claims.

This dissolves several difficulties beyond the central paradox: the null class, which is awkward on a purely extensional view, and the unit class, which if identified with its sole member forces one to say that something both has one member and has many.⁶ I should be candid about the theory's status. The need for some doctrine of types is, I think, beyond serious doubt; the precise form that doctrine ought to take is another matter entirely, and much of what appears in Principia Mathematica — the joint work with Whitehead on the logical foundations of mathematics — remains, as I have said elsewhere, inchoate and not fully settled.⁷

My own three criteria for a satisfying solution were: that the contradictions should vanish, that as much of mathematics as possible should survive intact, and that the solution should in the end appeal to what I can only call logical common sense — that it should seem, on reflection, just what one ought to have expected all along.⁸ The first condition is universally accepted.

The third is the one on which I find myself most at odds with those who, like Quine, have produced technically impressive systems that strike me as constructed purely to avoid the paradoxes rather than illuminating why they arise.⁹

Sources

1. Mankind, for example, is not a man. Form now the assemblage of all classes which are not members of themselves.

   Introduction to Mathematical Philosophy

2. If it is, it is one of those classes that are not members of themselves, i.e. it is not a member of itself. If it is not, it is not one of those classes that are not members of themselves, i.e. it is a member of itself. Thus of the two hypotheses—that it is, and that it is not, a member of itself—each implies its contradictory.

   Introduction to Mathematical Philosophy

3. The supposition that a class is, or that it is not, a member of itself is meaningless in just this way.

   Introduction to Mathematical Philosophy

4. to construct symbolically any class whose members are not all of the same grade in the logical hierarchy is to use symbols in a way which makes them no longer symbolise anything.

   Introduction to Mathematical Philosophy

5. all the formal properties that you desire of classes, all their formal uses in mathematics, can be obtained without supposing for a moment that there are such things as classes

   Part II, p. 278

6. if you do not, you will find that the class is a member of itself, which is objectionable, as we saw earlier in this lecture... This is one reason why you must distinguish a unit-class from its only member.

   Part II, p. 278

7. the theory of types emphatically does not belong to the finished and certain part of our subject: much of this theory is still inchoate, confused, and obscure. But the need of some doctrine of types is less doubtful than the precise form the doctrine should take

   Introduction to Mathematical Philosophy

8. the solution should, on reflection, appeal to what may be called 'logical common sense'—i.e. that it should seem, in the end, just what one ought to have expected all along.

   My Philosophical Development, pp. 80–81

9. Professor Quine, for example, has produced systems which I admire greatly on account of their skill, but which I cannot feel to be satisfactory because they seem to be created ad hoc and not to be such as even the cleverest logician would have thought of if he had not known of the contradictions.

   My Philosophical Development, pp. 80–81