Empty domains #2

Here is a long quotation from Oliver and Smiley in their Plural Logic, motivating their adoption of a universally free logic (i.e. one allowing empty domains and empty terms).

If empty singular terms are outlawed, we are deprived of logic as a tool for assessing arguments involving hypothetical particular things. Scientists argued about the now-discredited planet Vulcan just as they argued about the now-accepted proto-continent Pangaea; intellectuals argued about the merits of Ern Malley, the non-existent poet of the great Australian literary hoax. Sometimes of course we may know that the particular thing exists, but at others we may wish to remain agnostic or we may even have good grounds to believe that it doesn’t exist. Indeed, our reasoning may be designed precisely to supply those grounds: ‘Let N  be the greatest prime number. Then … contradiction. So there is no such thing as N’.

Now consider the effect of ruling out an empty domain. For many choices of domain —people, tables, chairs—the requirement that the domain be necessarily non-empty is frankly absurd. One may retreat to the idea that the non-emptiness of the domain is a background presupposition in any argument about the relevant kind of thing. But then we shall be deprived of logic as a tool for assessing arguments involving hypothetical kinds. We may indeed be certain that there is a human hand or two, but for other kinds this may be a matter of debate: think of the WIMPs and MACHOs of dark matter physics, or atoms of elements in the higher reaches of the Periodic Table. Sometimes we will have or will come to have good grounds to believe that the kind is empty, like the illusory canals on Mars or the sets of naive set theory, and again this may be the intended outcome of our reasoning: ‘Let’s talk about sets. They are governed by a (naive) comprehension principle … contradiction. So there are no such things.’

It may be replied that some domains are necessarily non-empty, say the domain of natural numbers. It follows that the absolutely unrestricted domain of things is necessarily non-empty too. But even if this necessity could be made out, and what’s more made out to be a matter of logical necessity, we would still not want the argument ‘∀xFx , so ∃xFx ’ to be valid. As we see it, the topic neutrality of logic means that it ought to cover both unrestricted and restricted domains, so it ought not to validate a pattern of argument which certainly fails in some restricted cases, even if not in all.

I think there are at least two issues here which we might pause over. First, in just what sense ought logic to be topic neutral? And second (even if we accept some kind of topic neutrality as a constraint on what counts as logic) how far does a defensible form of neutrality require neutrality about e.g. whether our language actually hooks up to the world? But leave those issues to another post. For the moment, what is interesting me is the straight parallelism between the considerations which Oliver and Smiley adduce for allowing empty names and for allowing empty domains. Certainly, I see no hint here (or in the surrounding text) that they conceive the cases as raising significantly different issues.

In the previous post, I noted that Wilfrid Hodges’s very briskly offers similar reasons of topic neutrality to argue for allowing for empty domains, but goes on to present a logical system which doesn’t allow for empty names. And, having in mind similar considerations to Oliver and Smiley’s, I suggested that this half-way-house is not a comfortable resting place: if his (and their) reason for allowing empty domains is a good one, then we should equally allow (as  Oliver and Smiley do) empty names. So the story went.

However, it was suggested that Hodges’s position is much more natural that I made out. Taking parts of two comments, Bruno Whittle puts it this way:

On a plausible way of thinking about things, to give the meaning of a name is to provide it with a referent. So to ban empty names is simply to restrict attention to those which have been given meanings. On the other hand,  …  I would say that to give the meaning of a quantifier—to ‘provide it with a domain’—is to specify zero or more objects for it to range over. To insist that one specify at least one seems barely more natural than insisting that one provide at least two (i.e. that one really does provide objects plural!)

But is it in general supposed to be a plausible way of thinking about singular terms that to give one a meaning is to provide it with a reference? Is this claim is supposed to apply, e.g., to functional terms? If it doesn’t, then coping with ’empty’ terms constructed from partial functions, for example, will still arguably need a free logic (even if we have banned empty unstructured names). While if it is still supposed to apply, then I think I need to know more about the notion of meaning in play here.

Set that aside, however. Let’s think some more about what is meant by “outlawing” or “banning” empty names in logic — given that we can, it seems, (or so Oliver and Smiley insist) find ourselves inadventently using them, and reasoning with them. To be continued.