Friday, 16 January 2015

Bertrand Russell’s Theory of Types

Theory of Types 1

Bertrand Russell’s theory of types seems quite commonsensical at a prima facie level.

That is, different objects, or their many types, can't be treated in the same way within any philosophical or logical system. In Russell’s case, the important thing is that the predicates applicable to different types of objects will differ depending on the type under scrutiny. More specifically, these predicates will be true or false of particular objects depending on the type of object that is predicated.

In Russell’s scheme, objects are arranged in some kind of hierarchy from objects of the least generality to objects of the most extreme generality. In terms of statements, this means that if predicates change according to their types of objects, then statements will likewise change in according to the type of objects they are statements about. So the statements we make about individuals or particulars can't be made about classes. Similarly, the statements we make about classes can't be made about classes of classes. And so on into indefiniteness.

What of different types of individuals? Is this theory true of these types too?Clearly, this theory has relevance to the class of all classes that do not belong to themselves. According to Russell, the theory of types is intended to prevent the paradox discovered in the said class of classes. It was said these classes both are and aren't members of themselves. The original assertion was about classes, not classes of classes. That is, it is said of classes that they either are or aren't members of themselves. This part doesn't result in paradox.

For example, the class of abstract entities is itself an abstract entity. Therefore it's a member of itself. On the other hand, the class of pink shoes is not a pink shoe. Therefore it's not a member of itself. So such classes don't generate any paradoxes.

We introduced another class that is a class of classes. It was, therefore, of a higher logical order than first-order classes. We talked of the class of all classes that aren't members of themselves. This class generated a paradox: it is both a member of itself and not a member of itself. This is the paradox that Russell attempted to solve or prevent. How did he attempt it?

He created his theory of types. And, according to this theory, the predicates and statements we apply to objects depend on the nature of the type of objects being talked about. Clearly, a class of classes is not of the same logical order than a basic class; just as basic classes are of a different logical order than individuals.

We've said that classes can either be members of themselves or not members of themselves. That’s fine. However, we also asked whether or not a class of classes, the class of classes that aren't members of themselves, is a member of itself. It was concluded that it both is and isn't a member of itself. This, of course, is paradoxical. So how does Russell’s theory prevent this paradox from occurring?

Again, different objects have different predicates and statements made about them. A class is a different ‘object’ than a class of classes. It follows that we can't predicate or assert about this class of classes the same kind of things we predicate and assert about first-order classes. We said that the predicate ‘a member of itself’ is either true or false of the classes that either are members of themselves or not. However, our main concern is a class of classes: the class of all classes that aren't members of themselves. So we can't ask of this class the question whether or not it's a member of itself because that kind of question can only be made of classes, not classes of classes. And if these questions can't be asked in the first place, then this class of classes is not in fact paradoxical after all.

A.J. Ayer writes that it is “nonsensical” to even ask if the class of all classes that aren't members of themselves is itself a member or not a member of itself. This question can only be asked of first-order classes, not of second-order classes or classes of classes.

                                                   The Theory of Types 2

It's not only classes, predicates and expressions that differ depending on the objects they're applied to: the theory also applies to ‘numerical expressions’. These expressions, according to Russell and/or Ayer, change their ‘sense’ depending on whether or not they count individuals, classes or classes of classes.

This is an interesting use of the term ‘sense’ if we bear in mind Frege’s notion of sense and reference and all the later theories about reference. Usually proper names, for example, are not supposed to have a sense. So it seems strange, prima facie, that the theory of types claims that numerical expressions have different senses depending on what they are applied to.

However, it may not be numbers themselves that have different senses; but the expressions in which they are constituents. It also seems strange that the way we ‘count’ different types of objects will also be different, regardless of the ‘senses’ of the numbers within the expressions. Again, perhaps I need to know what is meant by ‘numerical expressions’ and whether such expressions have constituents that somehow are relevant to the notion of ‘sense’.

There is a problem with the theory of types. In traditional class theory, the same object can belong to different classes. Though in the theory of types, we must be careful to treat different objects in different ways. This may result in certain classes becoming smaller than they originally were because many classes contain different types of objects as members. This will result in both a proliferation of classes, and also the numerical shrinkage of the classes that are deemed to already exist.

Though, according to set-theory, we define the natural numbers by reference to classes. That is, the class of all two-membered classes is in fact the class we call the number 1. So if class membership, the number of members, defines the natural numbers, then the shrinkage of classes that results from the theory of types may result in their being classes that aren't large enough - in terms of members - to define the higher natural numbers. That is, the higher numbers may not be able to correspond with any classes because there aren't enough classes with correlative higher memberships. The classes needed to account for the higher numbers may simply run out. The may not be enough classes with higher enough memberships to account for the higher natural numbers.

Another way of putting the theory of types - in this class-membership and numbers sense - is that the conditions that allow objects to belong to certain classes becomes more stringent. Classes become better defined, as it were. And if the conditions of class membership are more stringent, then clearly many previous classes will effectively become smaller in terms of their numbers of members. And, again, if this is the case, then certain higher numbers may not find their correlative classes in set-theory. This is clearly a problem for set-theory and the theory of types. Rather than reject a set-theoretic account of numbers, many logicians in Russell’s day and after simply rejected his theory of types. Other theories, of course, were created and adopted after the period in which Russell formulated his theory of types.

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