A formal system is broadly defined as any well-defined system of abstract thought based on the model of mathematics. (Def: http://en.m.wikipedia.org/wiki/Formal_system)

A formal system is broadly defined as any well-defined system of abstract thought based on the model of mathematics. Reference: Wikipedia.

The entailment of the system by its logical foundation is what distinguishes a formal system from others which may have some basis in an abstract model. Often the formal system will be the basis for or even identified with a larger theory or field (e.g. Euclidean geometry) consistent with the usage in modern mathematics such as model theory.