Goguen has popularized the initial algebra view of semantics via his "no junk, no confusion" slogan. By "no junk", he means that models of a theory presentation should not have unnecessary elements, and "no confusion" that terms should not be mapped to equal values unless they are provably equal. Sometimes, "no junk" is also interpreted as every element in the model is a denotation of a term, while "no confusion" as two different terms denote different elements in the model. [These are classically equivalent statements, but they are not intuinistically equivalent, so I mention both].
My questions are:
What is a 'good' formalization of this slogan? By this I mean an explicit statement of "no junk, no confusion" in the meta-logic (since we're talking about models), where the logical strength of the corresponding statement is well understood.
Are there logics in which these requirements can be internalized?
What would be the corresponding slogan to "no junk, no confusion" for final coalgebras?