Finite Model Theory/FO Discrimination

The Property of Elementary Equivalence

FMT is about discrimination of structures. So the basic question is "Can I discriminate a structure from all different structures?", where different means not equal up to isomorphy. This can be done for finite structures (but not for infinite ones). E.g. ...

Properties for a finite Number of Structures

So being of a certain structure can be seen as an elementary property, that discriminates a class of isomorphic structures from all others. Now we can think of properties that discriminate 2 different (nonisomorphic) structures from the rest. They can also be discriminated by simply connecting the properties, like e.g. ...

This can be extended to properties for a finite number of structures, like e.g. ...

Properties for an infinite Number of Structures

Also properties that hold for an infinite number of structures can be thought of. But these are not discriminatable in the above way. E.g. ...

There are Logics that allow this sort of infinite disjunction, like ...

So, are there other ways to discriminate these structure in FO? Sometimes, like in the case of ...

So we need a decision tool (method) that decides if structures with a certain property can be discriminated or not, i.e. responds either "yes" or "no" for all possible properties, i.e. is sound and complete.