# Philosophical justifications for the assumption of a non-empty domain in classical first-order logic?

Are there any "canonical" (or at least, quite good) papers that attempt to justify the supposition made in the model theory for classical first-order logic that the domain in non-empty?

I know that free logic was developed (at least in part) to avoid this assumption, since "something exists" doesn't seem (to many) to be a logical truth.

Intuitively, I could see the assumption being justified in something like the following manner: you cannot have a logic that quantifies over individuals without there being at least one individual. In this sense, it is just a necessary presupposition, but perhaps not to be regarded as a "logical truth" in the same sense as the law of excluded middle might be.

Is there any literature discussing this issue in the philosophy of logic? In particular, any literature which attempts to defend this assumption?

• In classical first-order logic, the empty domain is just a "single" special case, and it makes sense to exclude it, because this simplifies things. The first time I tried to seriously apply modal logic, this is the place where I got stuck, because handling non-existence and invalidity is quite challenging. But I wonder about classical multi-sorted first order logic, because some domains could be empty, but not all, and hence we are no longer talking about a "single" special case. – Thomas Klimpel Jul 10 '13 at 17:42

One reason is to preserve certain intuitive relationships that we would like. All children like icecream implies Some child likes icecream only if the set of children is never empty. The assumption that universal quantification is a strengthening of existential quantification only holds if the domain is not empty. See this blog post for a discussion of this use within mathematics.

• +1 for a nice mathematical justification. I was hoping for more philosophical justifications, but this is helpful nonetheless. – Dennis Jan 8 '13 at 18:02
• @Dennis The justification in this answer is philosophical, not mathematical. (The blog post is mathematical, though.) – 6005 Feb 17 '15 at 0:37

In mathematics, we might have a statement:

For all x in S, we have P(x)

where S is a set, the domain of quantification that may or may not be empty.

Classical first-order logic is a little weird in that every domain of quantification is assumed to be non-empty. It simply isn't necessary. I guess this assumption makes it possible to introduce free variables by universal specification when required-- something you do not usually do in mathematics. In most mathematical proofs, free variables are introduced either by a premise or by existential specification (instantiation).

Quine's short 1954 paper, Quantification and the Empty Domain (Journal of Symbolic Logic, Vol 19 Number 3 September 1954), addresses this. Quines two reasons are:

1. If some first-order statement is true in every domain larger than a domain D, then it is true in D. But this is only true if D is not the empty set. Philosophically, we want our statements to be true of all sufficiently large domains of discourse, but this theorem says that including finite ("small") domains of discourse does not change things. However, including the empty domain of discourse would change things.

2. It's easy to tell anyway whether a first-order statement is true for the empty domain. So it seems unnecessary to include this in our theory when we can just handle it as a special case.

I would summarize this to say that there are clearly practical reasons to exclude the empty domain; this isn't to say that these reasons are natural, but I would guess Quine is more concerned about what is practical than about what is natural.

In R. M. Martin's 1965 Of Time and the Null Individual, it is actually proposed that we introduce a "null individual" to our logic. As best I can tell, the effect of this is that the empty domain can be discussed, and that essentially the same logical theorems hold true for the empty domain as held in classical first-order logic. Whether you consider this to be justification for leaving the empty domain out in the first place is another matter. Perhaps the "null individual" is little more than an artificial way of making the empty set behave as if it were nonempty. But perhaps it makes sense philosophically to say there must always be a null individual x for which statements such as Px and Qxx can be written, even if we do not assume that this null individual actually exists; in this way we avoid the problems the empty domain could pose for logic, but at the same time we avoid assuming a priori that at least one thing exists (for the whole point of the null individual is that it is not considered to exist).

There is some other discussion in A Note on Truth, Satisfaction and the Empty Domain, Timothy Williamson, 1999.