Fair point. And now that I'm put to it, it is actually quite hard to define. Dictionary definitions do not seem very good so I'll try my own:First, tell me what YOU mean by a continuum.
I think that I can see where Halliday is coming from when he says:A continuum is the set of gradual changes of a qualitative or quantitative construct that exist together no part of which can be distinguished from neighbouring parts except by arbitrary division. From a mathematical viewpoint, this would mean that the change is non-discrete. Cardinal numbers, for instance, therefore do not form a continuum, but positive numbers, which include an infinite number of fractions in between, do. I'm not sure whether the rate of such changes need to be linear or constant (from the viewpoint of the mathematical discipline of differential calculus) but expect they do.
In as much as we do in fact have a continuum of choice between yes and no which is only expressable through modality. That is our dilemma: we have a finite number of modals (in fact quite limitedly finite) to express an infinite number of choices between yes and no. Note also that Haliday did not say that that was the whole picture.Modality refers to the intermediate choices between yes and no.
The trouble with talking about a "continuum of meaning" (or indeed any continuum formed from a qualitative construct) is that such ideas may not be amenable to mathematical analysis. It is as if we would need a sort of non-numerical calculus to do so. We don't have this problem with continuums formed from quantitative constructs because they are amenable to mathematical analysis.
Now that I have defined a continuum, I realise that fitting the modals into one requires that:
1) the modals be ordered into a natural progression of meaning, and
2) the change of meaning between each modal is constant and linear too. In this way the modals are analagous to the cardinal numbers.
This is complicated, however, by the fact that each modal can itself have many different meanings.
Perhaps a clade doesn't need constant rate of change.
Looking forward to your reply.