Logic: can I define a domain containing distinct names referring to the same object?

Question

For example, would a domain {Eric Blair, George Orwell} contain two things, or one thing?

Answer 1

In the "standard" treatment of first-order logic, the domain of the interpretation contains objects; this is the semantical side.

The language contains "names", i.e. individual constants; this is the syntactical side.

When you interpret a language you have to assign "meaning" to the symbols of the language, like assigning a reference to the individual constants. This amounts to assign to every name of the language an object as its reference.

We can have distinct names having the same objcet as reference.

In your example, Eric Blair and George Orwell are two names of the same individual.

Answer 2

You actually have to be really careful. There is a classical separation between intension and extension. Keep this in mind: you have on the one hand the language, on the other hand the world. When you are considering a term of a language, it denotes an object of the world, which is its extension, and it does it in a certain way (with a particular intension). Gottlob Frege called theese two aspects Sinn (Sense, Intension) and Bedeutung (Reference, Extension). So we can say that, in your example, you are denoting the same object with two different intensions.

During the last century, logicians focused on the estensional logic, which is logic that doesn't care about the intension (For example, if I write "a -> b" [a implies b] it is the same than writing "~a OR b" [Not a or b], because theese two propositions actually say the same thing, in different ways though.)

So, if you have a domain in which two terms refers to the same object, it is like if you have two variables, let's say X and Y, and X = Y. The form, the syntax is different, but the estension is the same.

But, you could also want to write the domain which has, as elements, every name of an object of the world.

{Eric Blair, George Orwell, the author of 1984, ...}

Now, you are considering, as objects, the terms of the language. "Eric Blair" doesn't denote the man in the world who has that name, but it denotes itself, "Eric blair" (between quotes), is the object of the world. In this case, since the names are the objects, you can call them with another term of the language. You can say that the X term for Eric Blair (without quotes) is "Eric Blair". And so, X is the element of the language which denotes the object of the world "Eric Blair".