If a language $\mathcal{L}$ has constants $\{c,d\}$ can an interpretation have that c=d? Or must it keep them as distinct elements?
My definition of interpretation that's been used is seemingly identical to this definition for domain of discourse on wikipedia:
A domain of discourse $D$, usually required to be non-empty).
- For every constant symbol, an element of $D$ as its interpretation.
- For every n-ary function symbol, an n-ary function from $D$ to $D$ as its interpretation (that is, a function $D^n \rightarrow D$).
- For every n-ary predicate symbol, an n-ary relation on D as its interpretation (that is, a subset of $D^n$).
So this definition says every constant needs an interpretation but does that interpretation need to "preserve" the idea that $c,d$ are distinct elements? Or can an interpretation take $c\to c^D$ and $d\to c^D$ in $D$?
There's no reason two different constants can't be interpreted to be the same element. That might even be a theorem in a suitable theory. For example, in the language of group theory you could use one constant in axioms that define it as the group's left identity and the other in axioms that define the second constant as the group's right identity. It would then be a theorem that the two constants are equal.