# Empty domains: Must do better

In §11.1 of their Plural Logic, Alex Oliver and Timothy Smiley argue — with characteristic vigour — that an acceptable singular logic (as well a plural logic) should allow empty domains. Their two related headline thoughts:

• Logic should be topic-neutral, and one thing it should be neutral about is whether the domain is empty or not (after all, we can want to make deductions about e.g. sets or superstrings even if we doubt that there are such things).
• Standard logic makes the likes of $$\exists x(Fx \vee \neg Fx)$$ a logical theorem:  but a wff of that kind is not in general a logical truth.

Now, you don’t have to buy their defence of free logic to acknowledge that Oliver and Smiley are right about one thing — namely that many of the textbooks are pretty feeble in what they say by way of explanation/defence of the standard quantifier rules and their presumption that domains are populated. They give half a dozen examples. Let’s also take  two further widely adopted textbooks which they don’t mention.

• Bergmann, Moor and Nelson in The Logic Book just tell us that domains are non-empty sets, without comment (unless my eye has skipped over the relevant discussion). They note that the likes of $$\exists x(Fx \vee \neg Fx)$$ are — in their phrase — quantificationally true. But they don’t discuss (again, unless my eye has skipped) the relation here between being quantificationally true and being logically necessary.
• While Barker-Plummer, Barwise and Etchemendy in Language, Proof and Logic offer just this:

In FOL we always assume that the domain of discourse contains at least one object and that every individual constant in the language stands for an object in that domain. (We could give up these idealizations, but it would complicate things considerably without much gain in realism.)

Oliver and Smiley might reasonably protest: Why isn’t being able to argue about some domain while being neutral about whether it is empty not a major gain in realism? And it is just a fib that allowing empty domains, at least, complicates things “considerably”.

Now, I think there should be a strong bias towards sticking to standard rules in an introductory text.  But I do think a text really Must Do Better than is often that case, in at least acknowledging that there are issues here, and it should be frank that debatable choices are being made. So here is my effort at writing something on empty domains for the second edition of IFL. Comments? Thoughts? Suggestions for improvements?

### 14 thoughts on “Empty domains: Must do better”

1. Matthijs Blom

It escapes me how $\exists x (F x \land \lnot F x)$ should be a logical theorem. Did you perhaps mean to write $\exists x (F x \lor \lnot F x)$?

2. I remember being mildly obsessed with this question as an undergrad. I recall an elegant treatment by Rolf Schock in Logics Without Existence Assumptions.

1. I’m currently the undergrad in that position. Would you happen to know where I could find a copy of that? I did some digging but I can find no such thing.

1. Rolf Schock’s book was published in 1968 in Stockholm. It doesn’t seem to have been scanned as a PDF into one of (copyright infringing!) PDF repositories. If your own university library doesn’t have a copy, you will need to use its interlibrary loans system to get a copy. However …

A lot of work on free logics has been done since 1968. So why not start with e.g. this survey article, and the more recent references therein? — https://plato.stanford.edu/entries/logic-free/

3. Typo: on the second page, there is a “between between” in the final indented section.

(I am referring to the linked pages of your book)

4. Matthijs Blom

Coming from the apparently nonstandard side, I still struggle to see the benefits of excluding empty domains. The practice has always seemed weird to me. Like excluding the number 0 from arithmetic: sure, we get to never worry about division by zero ever again, but is that that really worth it? Or, for that matter, taboo the empty set from set theory and see how that goes.

I observe that, allowing empty domains from the outset, we can always go back to considering populated domains only. In contrast, we cannot easily go the other way around, the same way we wouldn’t be able to talk about zero or empty sets without the concepts of, well, zero and emptiness, like many of our ancestors.

@Peter: Is there a way for commenters to make LaTeX work in comments?

1. On the LaTeX issue, there’s a standard WordPress usage for sites where LaTeX is enabled. Insert “latex” immediately after the opening “$” of a math mode expression and it should work. So “$” followed immediately by “latex” followed, after a space, by “\forall xFx$” produces$latex \forall xFx\$.

1. On the empty domain issue. What would you adopt, then, as natural deduction rules for a logic which allows empty domains — at least assuming you don’t want empty names too? I suggest rules in the footnote at the end of an earlier blog post:

But that by my lights isn’t a very natural natural deduction system!

1. Rowsety Moid

(Though your question wasn’t addressed to me, I’m going to reply anyway. The footnote’s system looks pretty good to me, and it seems natural enough if “natural” means, as I think it should, that it’s like what we do in informal reasoning. Your point against it in the footnote doesn’t give any reason to think it isn’t; instead, you seems to be saying it’s not very “natural” in some other sense: that it isn’t enough like Gentzen’s way of using introduction and elimination rules.)

2. Rowsety Moid

That going the other way was difficult historically doesn’t mean it has to be difficult now. It would be useful to know what the problems actually are. If we have a non-empty domain, it can still be empty of Xs, if we want to treat Xs as not existing.

In the 3-page draft for IFL2, the argument for disallowing empty domains seems to come down to this: “we get a particularly simple and natural set of quantifier rules which are truth-preserving so long as we are not talking about nothing.”

I don’t think the rule that goes from (All x)Px to Pa is natural; I think it’s wrong. What’s natural is to go from (Exists x)Px to Pa. So even though I don’t find the Oliver-Smiley argument convincing, I don’t find the reply convincing either.

5. (My earlier comments were about the issue; this is more directly about what should appear in IFL2.)

I think it’s good to have a section that reminds the reader that domains must be nonempty, points out that some inference rules with desirable properties depend on it, mentions that there are other forms of logic that make different choices, mentions Free Logic, and says some people prefer it or at last think it better for some purposes.

One thing I like about the draft page 299 is that it shows that a seemingly innocent move (switching to a domain that contains only the objects of interest) can have an undesirable, unintended consequences.

When you point out that there isn’t One True Logic, that can be connected to this, because it shows that devising a logic can be a somewhat delicate balancing act.

However, I think it’s important not to give the reader the impression that standard QL prohibits empty domains for no good reason, so they they’re being taught this approach only because it’s standard when really it’s rubbish.

Rather than present the Oliver and Smiley argument on page 300, it should be pointed out that we don’t have to allow empty domains before we can be neutral about whether tachyons exist, or can “argue logically about things that we believe don’t exist, precisely in order to try to show that they don’t exist”, or can “argue in an exploratory way, in ignorance of whether what we are talking about exists”.

Of course we don’t! In maths, people regularly reason about things that might not exist and can often even prove they don’t, and in most cases those arguments could be expressed in standard FOL. When someone presents one of the usual proofs that there’s no bijection between the natural numbers and the reals, no one pops up and says they can’t do that in FOL because, if they restricted the domain to bijections between the naturals and the reals, the domain could not be empty.

In most cases, at least, when we want to consider things that don’t (or might not) exist, a suitable larger domain is easily found and is typically what people are working with anyway. (For instance, if the question is whether integers with certain properties exist, the domain can be integers; if tachyons, particles or physical entities.) ‘¬∃xTx’ seems a perfectly good way to say tachyons don’t exist.

I think the three-page draft ends up being too complicated for an introductory text, too ambivalent, too back-and-forth, and too “some would say”.

The “defender of the standard QL rules” spends most of their time fending off criticism rather than making a positive case, and when they finally make a positive claim — “we get a particularly simple and natural set of quantifier rules which are truth-preserving …” — it’s immediately undercut by “Or so the story goes” and a description of what a proponent of free logic would say.

When I get to

“Still, it will be said, this sort of response misses the central worry about adopting quantifier rules that presuppose that domains are populated”

I think: but is it true? And if it misses the central worry, why are we being told about it?

Instead, everything should be set out in a more direct and straightforward fashion.

6. More about books.

When Oliver and Smiley give examples to show that many textbooks are pretty feeble, they use very brief quotes. It’s not possible to tell (without finding the books and reading them) whether they’ve been fairly characterised. I’ve done that in one case.

Their Mendelson quote — “an interpretation with an empty domain has little or no importance in applications of logic” — is from his Introduction to Mathematical Logic, 5th ed. However, that is far from all that book says about empty domains. It has an entire section, 2.16, starting on p 141 and devoted to “Quantification Theory Allowing Empty Domains”.

The quote Oliver and Smiley used comes from that page and section. However, it’s not even all that’s said about empty domains in the same paragraph.

Sider is another author they criticise. I haven’t looked at everything he says about empty domains. However, I did find something interesting about using a universally quantified statement to introduce an object. It’s on p 97 of his Logic for Philosophy, and here it is:

The practice of introducing a name for an object of a certain type is for use with existentially quantified statements of the metalanguage—statements of the form “there exists some object of such and such type”. It’s not for use with universally quantified statements; if you learn that every object is of a certain type, it’s usually not a good idea to say: “call one such object ‘u’.” Instead, wait. Wait until some particular object or objects of interest have emerged in the proof—until, for example, you’ve learned some existentially quantified statements, and have introduced corresponding names. Only then should you use the universally quantified statement—you can now apply it to the objects of interest. For example, if you introduced a name ‘u’, you could use a universally quantified statement ‘everything is of type T ’ to infer that u is of type T.

Scroll to Top