TITLE: In classical predicate logics, why is it usually assumed that at least one object exists?
QUESTION [14 upvotes]: In classical predicate logic it is commonly assumed that the domain of objects is non-empty. This validates inferences such as $$\forall x Fx \models \exists x Fx$$ as well as, if the identity predicate is available, the logical truth of $$\exists x~x=x.$$ What is the motivation for this assumption? In reality it is arguably not a logical contradiction to assume that nothing exists.

REPLY [12 votes]: Contrary to popular belief, many theorems are more elegantly stated if we allow empty structures. For just one example, the following is one of the most useful tests for quantifier elimination.

A theory $T$ has quantifier-elimination if and only if for all models $M\models T$ and $N\models T$, any substructure $A\subseteq M$, and any embedding $f\colon A\to N$, $f$ is a partial elementary map.

The theorem is false if we require the structure $A$ to be non-empty. If you look at Marker's book Model Theory: An Introduction, Theorem 3.1.4, he actually adds the unnatural hypothesis that the language contains a constant symbol in order to get around this issue!
On the other hand, one of the only places that I'm aware of where it really helps to assume all structures are non-empty is the prenex normal form. For example, the sentence $(\forall x\, P(x))\land Q$, where $P$ is a $1$-ary relation symbol and $Q$ is a proposition symbol (a $0$-ary relation symbol)  is not equivalent to any sentence in prenex form. Of course, this can be fixed by changing the theorem: every first-order formula is equivalent over non-empty structures to one in prenex form. But I'll admit it is a little bit unsatisfying.
If anyone has other examples of places where forbidding empty structures genuinely seems to improve the theory, I'd like to hear about them. (Though fair warning: I'm a real partisan on this issue, so I'll probably respond by trying to convince you that there's an elegant solution allowing empty structures.)
In my experience, there are really no problems developing the basics of first-order logic with empty structures - once you fix the proof rules to be sound on empty structures, of course! But again, there are proof systems which are sound and complete on empty structures and just as elegant as traditional proof systems, if not more so. If you want to see the details of the completeness theorem worked out, you can look at Chapter 3 of these lecture notes on model theory, where I give a sequent calculus system that is sound and complete for many-sorted first-order logic in which any or all of the sorts are permitted to be empty.