[PL07] Domain


Notice that when we use the word "everything" in ordinary language, we do not always mean "absolutely everything in the whole universe". Suppose you are going on a trip and I ask, "Have you packed everything into your suitcase?" In such a situation, of course I am not asking whether you have got everything in the universe in your suitcase. Rather, "everything" here probably refers only to those portable items you need for the trip. This is an illustration of what is called restricting the domain of quantification. The domain of quantification is the class of things we are talking about when we use quantifiers such as "every", "all", or "some". To restrict the domain of quantification is to limit the class to a particular group of objects. In the example just given, the domain of quantification includes only the items you need for your trip. Objects such as the moon, or the Eiffel Tower, are therefore not included in the class. Whereas if a physicist proclaims that "everything is made up of elementary particles", the domain of quantification will be larger and presumably includes all the physical objects in the whole universe. Similarly, when a teacher says "everyone is here" in a lecture, it might be that the only people included in the domain are his students.

Restricting the domain of quantification can make formalization easier. For example, we might formalize "everyone is wicked" as " ∀x(Hx→Wx)". But if we restrict the domain of quantification so that we are only talking about human beings, then we can just write down "∀xWx", and leave it as understood that the domain includes all human beings only. But remember : if you do restrict the domain of quantification in formalization, you should define the domain explicitly. For example, consider this simple inference:

Everyone is wicked.
If everyone is wicked, then nobody goes to heaven.
So nobody goes to heaven.

By restricting the domain only to human beings, we can formalize the argument easily :

Translation scheme:

Domain : the set of all human beings
Wx : x is wicked
Gx : x goes to heaven.

Formalized sequent:

∀xWx, (∀xWx→~∃xGx) ⊧ ~∃xGx

What if you do not restrict the domain? In such case the formalization can proceed as follows:

Translation scheme:

Domain : everything
Wx : x is wicked
Px : x is a person
Gx : x goes to heaven

Formalized sequent:

∀x(Px→Wx), (∀x(Px→Wx)→~∃x(Px&Gx)) ⊧ ~∃x(Px&Gx)

So you can see that restricting the domain simplifies the formalization. But do remember that with any argument there should only be a single domain. That implies you should not use different domains to formalize the following argument:

Every human being is an animal.
Every animal can feel pain.
So every human being can feel pain.

The correct way to formalize this argument is not to restrict the domain at all, but to write down something like the following:

Translation scheme :

Hx : x is a human being
Ax : x is an animal
Cx : x can feel pain

Formalized sequent :

∀x(Hx→Ax), ∀x(Ax→Cx) ⊧ ∀x(Hx→Cx)

It would be a mistake to have three domains, one for each premise and another one for the conclusion.

Formalize these English sentences in MPL (let the domain be the set of all human beings) :

  1. Everyone is happy.
  2. If everyone is happy then nobody is sad.
  3. If everyone is happy then someone is not sad.
  4. If someone is sad, then not everyone is happy.
  5. Everybody is sad if someone is sad.
  6. Someone is not sad and not happy.
  7. Nobody is sad and happy.
  8. If nobody is sad and happy, then everyone who is sad is not happy, and everyone who is happy is not sad.
  9. Either Tom is happy, or Jane is sad.
  10. Someone is either happy, or sad, but not both.
  11. If Tom is happy, then everyone is happy.
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