Monday, April 16, 2007

3.331 From this remark we get a comprehensive view of Russell’s “Theory of types”: Russell’s error is shown by his having to speak of the meaning of a sign when putting together his rules for signs.

Black (p. 146) suggests ‘get a comprehensive view of’ for sehen wir in Russell’s ‘Theory of Types’ hinüber while P&McG have ‘turn to’ and Ogden has ‘get a further view – into Russell’s …’ Hinüber’ means ‘over’ and ‘in’ means ‘in,’ so we are seeing into Russell’s theory, getting insight, but also getting an oversight, either looking beyond it or, as Black suggests, looking over the whole thing (but at it, not to something on the other side).

Russell’s system is impure, therefore, in the sense of TLP 3.33.

In Principles of Mathematics (1903) Russell proposed his first, simple version, and in “Mathematical Logic as Based on the Theory of Types” (1908) he proposed the ramified version. The basic idea of the theory of types is that classes are not objects. This makes for a simpler ontology, and means that it is nonsense to talk about a class being a member of a class in the way that an object (a spoon or a barber, say) can be a member of a class. But the simple version does not get rid of all paradoxes. The ramified theory of types is based on the vicious circle principle, i.e.: “Whatever involves all of a collection must not be one of that collection'; or 'If, provided a certain collection had a total, it would have members only definable in terms of that total, then the said collection has no total.” (see Principia Mathematica, 1 (1910), Introduction, ch. 2, p.1). In the simple theory, there are types of objects (real objects, classes of objects, classes of classes, and so on). In the ramified theory there are also types of properties (properties, properties of properties, etc., e.g. shyness is nice or red is a property of objects).

The theory of types is developed in order to avoid certain paradoxes, e.g. those involving infinity (how it seems possible to come up with an infinite number of objects from a finite collection, and thus prove a priori that the world contains an infinite number of objects—i.e. prove the axiom of infinity) and the class of all classes that are not members of themselves. “Now the theory of types emphatically does not belong to the finished and certain part of our subject: much of this theory is till inchoate, confused, and obscure. But the need of some doctrine of types is less doubtful than the precise form the doctrine should take.”[1]

Classes are logical fictions, and if they are treated as being real objects, whose names have real signification, then the sentences in which they are treated this way will be devoid of meaning. “The supposition that a class is, or that it is not, a member of itself is meaningless in just this way.”[2] Since we cannot know whether the axiom of infinity is true, we cannot know whether the world is infinitely or, on the contrary, finitely divisible. If the latter is the case, then logical analysis has a chance of finding the real simples or particulars that make up the world.

F. P. Ramsey, following Wittgenstein, objects to this theory. Propositional functions are symbols, while individuals are objects. So talk of functions of functions is not like talk of functions of individuals. “For the range of values of a function of individuals is definitely fixed by the range of individuals, an objective totality which there is no [getting?] away from. But the range of arguments to a function of functions is a range of symbols, all symbols which become propositions by inserting in them the name of an individual. And this range of symbols, actual or possible, is not objectively fixed, but depends on our methods of constructing them and requires more precise definition.”[3]

Wittgenstein’s criticism in the Tractatus begins (3.331) with the observation that formal logic is supposed to be purely formal, yet Russell has to refer to the Bedeutungen of the signs in the drawing up of his symbolic rules. This shows that something has gone wrong with the introduction of the theory of types. But what? The theory of types amounts to just this: “No proposition can say anything about itself, because the propositional sign cannot be contained in itself.” (3.332, Ogden).

It is, it seems, a bit like Frege’s distinction between concept and object. Talking about concepts and functions makes them sound like objects, but we need to look at their role in the system to understand them properly. The fact that ‘The class of classes’ looks and sounds like ‘The class of spoons’ does not mean that they are the same. In Frege’s terms, ‘The class of classes’ can be analyzed into the function ‘The class of ( )’ and the argument ‘classes’. Functions are not arguments. A good system of symbols will show this distinction. It does not need to be explicitly stated in a seemingly ad hoc way.

Russell’s paradox shows that there are concepts that do not determine a course of values (or value-range). It shows that Basic Law V is false. Neither Russell nor Frege really managed to rescue logicism from this problem though.

Mounce (p. 56) puts Wittgenstein’s objection to Russell’s theory of types this way: “one cannot in a correct symbolism construct a proposition which refers to itself without making it evident that the contained proposition has a different function from the proposition which contains it. But then it will be evident that one cannot construct a proposition which refers to itself. For, given such a misguided attempt, it will be evident that what one has is not one proposition, referring to itself, but different propositions. In short, a theory of types is entirely unnecessary.”

Black (p. 146): ‘It is hard to account for Wittgenstein’s evident animus in this digression. For Wittgenstein’s own programme for ‘logical syntax’ can properly be viewed as an attempt to accomplish what Russell was reaching for in his theory of types. Wittgenstein himself once said that philosophical grammar or logical syntax was ‘a theory of types’ (Phil. Bem. 3, 2). An improved version of Russell’s theory of types might well be a part of logical syntax in Wittgenstein’s conception.”

[1] Introduction to Mathematical Philosophy p. 135.

[2] Ibid., p. 137

[3] Ramsey “Predicative Functions and the Axiom of Reducibility” in Klemke, pp. 355-368, p. 358, originally in Chapter 1 pp. 32-49 of his The Foundations of Mathematics.


N. N. said...
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N. N. said...

Excellent post!

I think I prefer Mounce's simple account.

Concerning Black's comment: it seems to me that Wittgenstein is not objecting to the work that Russell's theory is doing, i.e., there are, as I read Wittgenstein, distinctions between types of objects, and those distinctions must be taken into account. Rather, Wittgenstein is objecting to a theory of types. For a theory of types attempts to say what can only be shown by the symbolism.

DR said...


Yes, Wittgenstein is objecting to a theory of types, and Mounce puts the point well.