First of all, there is a problem with your question. Boolean logic is an algebra where the values are true or false. Whenever you write a condition (if) or conditional loop (for / while), you make use of this logic. Often you want two conditions to hold at the same time or only one of them, etc. It doesn't matter if you use a bool type or not. You can also express the same logic as ints and whatnot. So if you say "I could write that without using bool", that doesn't mean that your reasoning doesn't involve boolean logic. Having bool types is never necessary but helps prevent confusions. basically the same arguments that go for and against a typesystem apply here, too.
Apart form that, it is really useful to understand boolean logic. It helps to:
1) read (and also write, but please don't) rather unintuitive code, e.g.
Code
status = noError || fixIt()
i.e. the first part gets evaluated and if noError is true, you known the disjunction has to be true as well. thus, fixit() will not run. If there is an error, though, fixIt() runs and its return value determines the value that will be stored in status.
2) bitwise logic: AND/OR/XOR/NOT on a bit level are crucial to understand if you do low level, space-efficient stuff like compressions.
3) boolean logic is "the easy version" of first (and higher) order logic. These have various uses as well: They are the foundation of static type checkers used in compilers, programming languages (e.g. prolog) that work entirely in first order logic exist, there are classification methods, i.e. a certian form of machine learning, that rely on it (FOIL, etc), and so on, and so on
tl;dr:
you NEED to understand boolean logic, especially the very basic stuff. this hasn't much to do with having bool types or not. it's also the foundation for more complicated (but also useful) stuff