What Do Logical Operators In a Grammar Mean?
A grammar generates a language, it doesn't accept it. Automata accept languages.

The whole point of gramars is that they are easily able to generate quite complex languages. Even regular languages (accepted by finite automata or denoted by regular expressions) can be very hard to describe in simple terms.

In this case, the grammar generates (a subset of) logical formulas, with connectives and ($\wedge$), or ($\vee$), and not ($\thicksim$).