The Domain of Quantification

The domain of quantification (sometimes called the domain of discourse) is the set of things that may be named in place of variables. When the domain is not specified, it is usually assumed to include absolutely everything that exists. That is why the universal quantifier ‘(∀x)’ ordinarily means anything that exists may be named in place of the variable x, and why the existential quantifier ‘(∃x)’ ordinarily means at least one thing that exists may be named in place of the variable x.

However, it is often convenient to specific a different domain of quantification. To see why, consider the following argument:

Anyone who doesn’t have free will doesn’t have moral responsibilities. But some people do have moral responsibilities. Therefore, at least some people have free will.

Using ‘Px’ to mean x is a person, ‘Fx’ to mean x has free will, and ‘Mx’ to mean x has moral responsibilities, we can symbolize this argument using the ordinary domain of quantification, as follows:

(∀x)((Px • ~Fx) ⊃ ~Mx)
(∃x)(Px • Mx)
∴ (∃x)(Px • Fx)

That symbolization looks rather messy. However, we can simplify the symbolization by including only people in the domain, rather than including everything that exists. When the domain of quantification is restricted to include only people, the quantifier ‘(∀x)’ means that anyone (any person, not just anything) may be named in place of x. Similarly, the quantifier ‘(∃x)’ means that someone (at least one person) may be named in place of x. When we restrict the domain of quantification in this way, the predicate ‘Px’ (which stands for x is a person) is no longer needed, since the quantifiers already refer only to people. So, we can eliminate the predicate ‘Px’ and symbolize the above argument like this instead:

(∀x)(~Fx ⊃ ~Mx)
(∃x)Mx
∴ (∃x)Fx

This simplified representation of the argument is much easier to read and understand! As this example illustrates, restricting the domain of quantification (i.e., narrowing it to include only a subset of things that exist) often makes it easier to symbolize an argument and understand its logical form.

The same domain of quantification must be used for all of the propositions in an argument. (We can’t use a different domain when symbolizing the premises than we use when symbolizing the conclusion, for example.) Otherwise, we risk committing the fallacy of equivocation.

On the other hand, sometimes it is useful to extend the domain of quantification to include things that don’t really exist. Philosophers often discuss arguments about things that are possible but not actual, or about things that could exist but perhaps don’t. Consider the following claims, which appear in Anselm of Canterbury’s famous ontological argument for the existence of God:

Something exists in the understanding, at least, than which nothing greater can be conceived. … And assuredly that, than which nothing greater can be conceived, cannot exist in the understanding alone. For, suppose it exists in the understanding alone: then it can be conceived to exist in reality; which is greater.Anselm’s Proslogion contains more than one version of his famous ontological argument for God’s existence. This quotation comes from the following passage in Chapter 2: “And indeed, we believe that you are a being than which nothing greater can be conceived. Or is there no such nature, since the fool has said in his heart, there is no God? But, at any rate, this very fool … is convinced that something exists in the understanding, at least, than which nothing greater can be conceived. For, when he hears of this, he understands it. And whatever is understood, exists in the understanding. And assuredly that, than which nothing greater can be conceived, cannot exist in the understanding alone. For, suppose it exists in the understanding alone: then it can be conceived to exist in reality; which is greater.”

Some things that “exist in the understanding” do not exist in reality, according to Anselm. Nevertheless, we can still discuss these things and formulate arguments about them. We might even discover something about reality: Anselm contends that by thinking carefully about what sorts of things could exist, we can learn that God actually does exist.

In order to symbolize Anselm’s argument, we have to extend the domain of quantification to include things that “exist in the understanding”—i.e., things are conceptually possible: things that may or may not really exist, but which we can imagine or conceive in our minds. We’ll see how to symbolize the first premise of Anselm’s argument on the next page, after the notion of a relational predicate is introduced.