Assume 'S' is a first-order sentence about a subject 'Z'.
When one stipulates a Model for 'S' with a domain 'D' does one always assume that the domain will contain all the objects within the subject 'Z'?
E.g. Let S = 'All lizards like flies'
Then when assigning a model/structure to 'S' do we assume that the domain will contain all the objects which are lizards? Since the sentence is about lizards.
I apologise for the crude example and (I think) poorly structured question, but I hope you understand what I'm getting at.