Absolute Generality 10: Expanding background domains

Having tried twice and miserably failed to explain to a sullenly sceptical Hungarian girl that you don’t make an espresso macchiato by filling up the cup to the top with hot milk, I was perhaps not in the optimal mood to sit in a cafe and wrestle with the rest of Glanzberg’s paper again. Still, I tried my best.

What I was expecting, after the first three sections, was a story about how “background domains” get contextually set, a story drawing on the thoughts about what goes on in more common-or-garden contextual settings of restricted domains. Though I confess I was sceptical about how this could be pulled off — given that the business of restricting quantifications to some subdomain of everything available to be quantified over in a context and the business of (so to speak) reaching out to everything would seem to be intuitively quite different. But in fact, what Glanzberg gives us is not the whole story about background-domain-fixing (“very little has been said here about how an initial background domain is set” p. 71), but rather a story about how we might go about domain-expansion when we are (supposedly) brought to acknowledge new objects like the Russell class which cannot (on pain of paradox) already be in the domain of objects that we are currently countenancing as all that there are.

Now, Glanzberg says (p. 62) that although domain-expansion isn’t a case of setting a restricted domain, “it is still the setting of a domain of quantification … [so] it should be governed by the principles” discussed in Sec. 3 of the paper. But in fact, as we’ll see, at most one of the principles to do with common-or-garden domain fixing arguably features in the discussion here about domain-expansion (and I’d say not even that one).

What Glanzberg does discuss is the following line of thought (I’ll use a more familiar example, though he prefers to work with Williamson’s variant Russell paradox about interpretations). Suppose I’m cheerfully quantifying over everything, including sets. You then — how? invoking an All in One principle? — get me to countenance that domain as itself a something, an object, and then show me that it is one which on pain of paradox can’t be in the domain I started off with. Ok, so now this new object is a “topic of discourse”, and what is now covered by “(absolutely) everything” should now include that new object — and I suppose we could see this as an application of the same principle about domains including current topics which we mentioned as governing ordinary domain setting. (But equally, as I said before, it in fact isn’t at all clear how we should handle that supposed general principle in the case of ordinary domains. And the thought in the present case just needn’t invoke any wobbly notion of ‘topic’ but comes down to the following more basic point: if we are brought explicitly to acknowledge the existence of an object outside the previous domain of what we counted as “(absolutely) everything”, then that forces an expansion of what we must now — in our new situation — include in “(absolutely) everything”.)

So, to repeat, suppose you bring me to acknowledge e.g. a set-like object beyond those currently covered by “all sets”. I expand my domain of quantification to contain that too. But not just that too. I’ll also need to add … well, what? At least, presumably, all the other objects that I can define in terms of it, using notions that I already have. And so then what? Thinking of all these objects together with the old ones, I can — by the same move as before — take all those together as a domain, and now we have a new object again. And off we go, iterating the procedure. It is, of course, not for nothing that Dummett called this sort of expansion indefinitely extensible!

We are now in very familiar territory, but territory unconnected with the early sections of the paper. We can now ask: just how far along the ordinals should we iterate? Maybe — Glanzberg seems to be saying — it isn’t a matter of indefinite extensibility, but there is a natural limit. But I found the discussion here to be not very clear.

Where does all this leave us then? As I say, the early sections about common-or-garden domain setting in fact drop out as pretty irrelevant. If the paper has does have anything interesting to say, it is in the later sections, particularly in Sec. 6.2 about — so to speak — how indefinitely extensible indefinitely extensible concepts are. So, OK, I’ll return to have another bash at that. (Though I will grumpily add that I think the editors could have taken a firmer line in getting Glanzberg to make his arguments more accessible.)

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