[FOM] basic model theory question

[Jim Farrugia]

Hi,

I am new to the FOM list, and I have some basic questions about model theory
that I am hoping some of you can help me with.

First, does model theory deal with how different truth values of sentences
are due to different interpretations of constant, function, and relation
symbols?

For instance, consider the following remark from David Marker's introductory 
paper in MSRI Publications Vol. 39, which was mentioned recently on this list. 

In one L-structure with domain Z a particular function symbol ("times hat") 
can be interpreted as "plus"; whereas in another L-structure with domain Z, 
this same symbol could be interpreted as "times."

Given these different interpretations, a given sentence may well have different 
truth values in the different structures. And of course there may also be 
(in this case, there will also be) sentences that have identical truth 
values in the two structures.

What I am wondering is whether the study of how sentences differ in their
truth values according to the interpretation of nonlogical vocabulary is
dealt with in model theory, and if so, where and how. If not, why not?

It seems from what I have read so far that a chief focus of model theory is
on sentences that are true under any interpretation (i.e., in all models),
rather than on how different interpretations affect the truth values of 
sentences.

I would appreciate any light you can shed.

Thank you,

Jim Farrugia
(grad student)