This is a question out of model theory. (This preamble is due to the fact that "model" and "theory" are used in different ways in different fields.) Is there any specific relationship between two theories which have the same models? A knee-jerk reaction would be to say that they are isomorphic, but upon reflection this does not seem likely. (This question is not to get mixed up with the fact that two models having the same theory are isomorphic.) Thanks.