Here’s a rewritten four-page chapter from IFL2 on empty domains. Difficult to know how to handle this topic. Many intro texts just skate over the issue. An earlier draft perhaps said a bit too much a bit too confusingly. Hope this strikes a better balance. Last-minute comments always welcome!
(By the way, “sets*” with a star is my usage in the book for when I really mean sets as objects in their own right — the ones that play a starring role in full-blown set* theory — as opposed to when I’m occasionally using lightweight talk of virtual classes which can be translated away.)
9 thoughts on “Empty domains, again”
This is a nice tour of the concerns that lead to free logic. As your text makes clear, the unrelativized quantifiers $\forall$ and $\exists$ behave quite differently from their relativized versions, $\forall x\in D$ and $\exists x\in D$. This difference in behaviour, by the way, underlies to the modern criticism made by logicians of the hidden assumptions of the “subalternation” relation in the Traditional Square of Oppositions. You might want to add a brief comment on that? It is worth noting —or so I claim— that mathematicians hardly ever write sentences that are naturally translated in terms of unrelativized quantifiers. If that is the case, though, wouldn’t it seem worthy for them to use the corresponding natural deduction system, with primitive introduction / elimination rules for $\forall x\in D$ and $\exists x\in D$? The rules for the unrelativized quantifiers may of course easily be derived from the rules for the latter (more natural?) relativized quantifiers.
In IFL, the discussion of empty domains is carried out using concepts from natural deduction while, if I remember rightly, the formal semantics for first-order logic comes much later in the book. That’s fine, but it might be a good idea to add, at that later stage, a short paragraph on the empty domain from the perspective of the semantics. I do not think that it would affect your general conclusion on the matter which, personally, I see as very reasonable, but would perhaps help appreciate it from a different angle. Namely, there are no functions taking the (non-empty) sets of individual constants and individual variables into the empty set, so the very notion of an interpretation in the empty set is ill-defined; nor is it clear how it might profitably be reconstructed without affecting the treatment of non-empty domains or creating endless complications.
Good point: I’ll double check what I’ve written in the formal semantics chapter to signal were the assumption that domains are non-empty is doing some work. Thanks!
Mostly small things (and I hope it’s clear from comments I’ve made before that I’m pretty happy with the approach it takes).
I found it a bit confusing at the start.
You bring in an argument from §32.3 that concludes ∀xGx, and then an argument said to justify it that concludes Fa instead. Well, ok, you say it starts like that. Still, I headed for §32.3 to see what was going on. The first argument in §32.3 concludes Gm. Reading on, I found the intended argument, B′. It’s just a bit harder to find than I’d like.
[BTW, looking earlier in §32, I can’t make sense of the examples of sentences at the bottom of p 300. Why is m in ∃y Mmy not a dummy name? Or n in ∀x(Lx → ∃y(Fy ∧ Rxyn))? Are there special rules for m and n somewhere?]
p 330, “Which is what happens in A. There are no such things as tachyons. So the stipulated domain of quantification is empty.” — Rather than say there are no such things, which seems questionable as a flat out assertion, perhaps say: This is what happens in A when there are no such things as tachyons, because then the stipulated domain of quantification is empty.
I don’t think it’s a good idea to call set theory “set* theory”. It seemed ok when I read your post saying ‘“sets*” with a star is my usage in the book for when I really mean sets as objects in their own right’, but not once I saw how it actually was used.
I think it’s fine to raise the issue of when there’s a commitment to sets existing and when it’s just a way of talking, but saying “set*” whenever you mention set theory seems too heavy-handed and ideological to me — especially since you’ve eliminated most of the unnecessary set-talk that was in the first edition. I don’t think an introductory logic text should be pushing that issue that hard.
Thanks for these comments!
1. I’ve clarified the backreference to §32.3 and also added dots to the first displayed proof in §34.1 to make it doubly clear that it is indeed a proof-fragment.
2. Middle alphabet lower case letters, proper names; early alphabet, dummy names. And ne’er the twain shall meet. Clear I think to a reader coming from working on earlier chapters. But I’m adding a sentence to §32.1(b) which re-emphasizes the point.
3. Fair point about the tachyons.
4. I’ve been wavering about the use of “sets*”! Another worry is introducing any non-standard terminology in an intro book. I agree that the passage you are commenting on is better without.
Excellent chapter. I noticed no substantive problems. Just an extra ‘is’ in the final line of paragraph (c) on page 330.
I’m quite enthusiastic about the fact that you go into this in an introductory text. The fact that you do has persuaded me to buy a copy of IFL2 when it comes out.
Thanks for the encouraging comment (and the typo correction!).
On a first reading, I see no glaring errors (as usual, from my perspective as an intelligent layman; the professionals may disagree!).
“No glaring errors” is a good start :)