This material includes somewhat deeper results such as the existence of disjunctive normal forms for statements. The corresponding normal form is prenex normal form. In first-order languages, the results are striking. The compactness theorem asserts that a model exists for infinite sets of axioms, as long as every finite set of them has a model. For example, since all finite maps are 4-colorable, the same is true of infinite maps.
Proof theory is the study of certain kinds of symbol manipulation.
So, for example, one could make a countable list of all statements which are true for all groups. The topic of being able to recognize membership in a set is the province of Recursion Theory. Here one asks what is calculable in a finite number of steps. Algebraic Logic studies logical systems via associated algebraic structures. Just as ordinary logic may be studies with Boolean algebras one may formalize the calculi with many-valued logics using other algebraic systems.