What is the Domain of Discourse when one isn’t specified?

The null set is a set that contains no elements. I think I can express this statement like this:

$$\forall x (x \not\in \emptyset)$$

Here x could be a number, a word, a function, really anything. What is the domain of discourse then? In general if a domain of discourse is specified what is it understood to be. Thank you!

• "Here x could be a number, a word, a function, really anything." Could it? Then it is not the usual set theory (not the usual $\in$)... Namely, the (unrestricted) domain of discourse in the formal sense is simply established by the formal theory you work in. Apr 14 at 9:20
• It depends on the interpretation. Apr 14 at 10:56
• If the formula uses tha language of sets, the domain is that of sets. Apr 14 at 10:56
• Generally speaking, the domain of discourse is the set of all things whenever it is not specificed. Apr 14 at 13:21
• I think OP here is essentially asking for the ontology of sets. The standard answer is the intended model is the von Neumann universe. The set theory book from open logic project has a nice description of it: builds.openlogicproject.org/content/set-theory/story/… Apr 14 at 14:48

1 Answer

Logic statements themselves do not specify any domain. You need an interpretation to specify a domain and to give any of the non-logical symbols any meaning relative to that domain.

Now, usually when you work with logic, you do so in the context of some domain. For example, given that you are using both the $$\in$$ and the $$\emptyset$$ symbols, you presumably intend to work with sets. So your intended domain of course would be sets. That is, the standard interpretation here would be:

Domain: sets

Constant symbol $$\emptyset$$ means the empty set

Relation symbol $$\in$$ means the set-theoretic 'element of' relation

But given a statement like yours you can provide a completely different interpretation, for example:

Domain: fruits

Constant symbol $$\emptyset$$ means bananas

Relation symbol $$\in$$ means the set-theoretic 'tastes better than' relation

And under that interpretation, your statement comes to mean that no fruit tastes better than bananas

• I’m still a bit confused. So when I was reading up on predicate logic they mentioned a domain of discourse that is used. When it’s left out we can usually tell what set to use based on context. So in real analysis we would use the set of real numbers for example. I got confused when you said “ Logic statements themselves do not specify any domain.” because I was unsure what you meant here. Also you said the domain would be sets. But since I’m dealing with the empty set there can be no elements and since an element can be anything is there a domain that would encompass everything? Apr 14 at 19:05
• @Dr.J Logic statements are just strings of symbols ... by themselves they mean nothing. The user, some interpretation, or some context will have to provide that meaning. That's all I meant by that. An empty set has no elements .. so why would that force there to be things other than sets? And it is a set itself. Apr 15 at 1:44
• That makes sense. So are you saying that the domain x would range over would be sets? For the empty set I was trying to use predicate logic to express the idea that the empty set has no elements. But I was unsure what domain to use encompass everything that could be an element if that makes sense. If the domain was the real numbers then the statement would be true. If the domain was the set of soft drink names then it would be true. But I didn’t know if I could find a domain that would contain the numbers, soft drink names and other things that could be elements of a set if that makes sense. Apr 15 at 2:37
• @Dr.J Well, consider this. Given the $\in$ predicate you want to make sense of the formula $x \in y$, and presumably we want to interpret this as '$x$ is an element of $y$. If we do that, then $y$ needs to be a set ... and that means that sets need to be part of the domain. All sets? Is there even such a thing as 'all sets'? But at least some sets. OK, but which sets? In particular, what are the $x$'s that are in the sets? (continued) Apr 15 at 11:03
• If you want those $x$'s to be something other than sets (e.g. numbers), you quickly have a problem, because what happens when the variables or constants denoting those objects are the second argument of the $\in$ relation? For example, given that we are working with sets, we may want to work with the notion of a subset, and introduce an axiom like $\forall x \forall y (x \subseteq y \leftrightarrow \forall z (z \in x \to z \in y))$, which we can then use to prove something like $\forall x \ x \subseteq x$. (cont'd) Apr 15 at 11:07