TITLE: Formalizing "no junk, no confusion"
QUESTION [7 upvotes]: 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?

REPLY [3 votes]: Belatedly, an answer in set-based situations to

What would be the corresponding slogan to "no junk, no confusion" for final coalgebras?

Given an initial algebra, any algebra will have a special subobject which is the image of the initial structure. The subobject may have confused elements (terms) of the initial algebra, and the object may have extra junk. A map between objects will map the first special subobject to the second, possibly confusing more. The initial algebra has no confusion and no junk.
Given a final/terminal coalgebra, the elements of any coalgebra will have a special colouring in terms of images in the elements of the terminal structure. The object may have more than one element with the same colour, and the object may not use all the colours. A map between objects will preserve the colouring, the domain possibly using fewer colours. The terminal coalgebra colours without ambiguity and without redundancy. If two things behave the same way, they are the same; all behaviours are covered.
No junk, no confusion; No redundancy, no ambiguity.