I used to think of Formal logic as a way to present reasoning in terms of symbols and connectors allowing to do calculus (see Predicate logic).
The term ‘formal’ also conjures the idea of something rigid and uncreative. This is contrasted with other kinds of logics, such as dialectical logic. On the other hand, one might say that this perceived rigidness is a sign of rigor.
Regardless, considerable progress has been made in the field of formal logic in the past decades.