The work that Boole originated was designed to consider matters of truth or validity and falsehood. Hence it can be used to determine the truth value of propositions and so decide if a chain of arguments is in fact valid. In the same way, Boolean algebra can be used to decide whether two statements are logically equivalent or not and so see if a tautology exists. From Shannon's work the same methods can be applied to switching circuits. The fact that Boolean algebra is an algebra often allows the expressions used in these applications to be simplified.
In addition the concepts of AND, OR and their negations are valuable in the work of current search engines. (see https://en.wikipedia.org/wiki/Boolean_search#Boolean_searches).