The Scope of a Rule

To speak, we must (a) attribute rule following and (b) attribute belief in natural laws (the 'rules' which the world follows).

Some of these rules have an indefinte scope. This can make it sound as though their scope precedes and exceeds the scope of language - of what we can talk about. We can only imagine such a thing by considering the possibility of a world that we cannot (in principle) talk about, but which, nevertheless, follows the rules. Whatever the value of this as a metaphysical speculation, we can't test it.

A specific case is the rules we presume to underpin the possibility of language itself. We imagine that logical and mathatical laws have a wider scope than simply guiding how we speak. This belief is associated with the kinds of things we say about them - among which are statements like 'these laws have indefinite scope'. Such a statement is true with respect to any testable case, since such a case must be articulable. Also, to qualify it would suggest that there was some articulable case in which they did not hold.

But its generality and lack of qualification do not need to extend to 'things we cannot talk about'. And there can be no rule that defines the boundary between what we can talk about and what we cannot.

When we explore this boundary, we generate paradoxes. This does not mean that the exploration is invalid or absurd. It means we must be careful about the conclusions we draw from these explorations - and about what we mean when we talk about them.

All the objects in the metaphysical world beyond the reach of speculation are paradoxical.

Conclusions drawn from the indefinite scope of our rules of intelligibility will also be paradoxical, since they depend upon speculations about these objects.

Meaning sneaks into the incompleteness and inconsistency proofs at the level of the rules themselves.