Can anyone please explain to me what is meant by the term 'domain of interpretation' in mathematical logic? For example, in the following context:
We shall describe variables which are thought of as representing elements within some particular domain of interpretation
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You know that a statement like $\forall x P(x)$ says 'All objects have property P'. However, what are 'all objects'? We could say that it is 'all objects that exist' ... but that would still not be very clear: all objects in our universe? Right now? At any time? What if I want talk about other kinds of imagined worlds or universes? Maybe we could try the set of all possible objects, i.e. some kind of 'universal set'? But Cantor showed there is no such thing. And finally, what counts as an object anyway?
To solve all these problems in order to evaluate a sentence like $\forall x P(x)$, formal logic demands that we specify a 'universe of discourse' or 'domain of interpretation', which specifies exactly what objects we are talking about. And another nice thing about that is that if you restrict your objects to, say, the natural numbers, then in order to express that every natural number is even or odd, instead of saying $\forall x (NaturalNumber(x) \rightarrow (Even(x) \lor Odd(x))$, we can just say $\forall x (Even(x) \lor Odd(x))$. Indeed, another way of thinking of the domain of interpretation is to say that that specifies exactly what objects the quantifiers quantify over, and what values the variables can possibly take.
It is the "collection" of objects on which the variables and the quantifiers "have meaning".
The formula $∀x \ (x≥0)$ is true when interpreted in the domain of naturals and false in the domain of rationals.