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?