If A mean B, not B does not mean not A.
A is the core of B. But it is not limited to B.
For example.
If I bought the tickets, then I went to the game.
I did not go to the game. Therefore I did not buy the tickets.
Not necessarily true, as there may have been other reasons for me to have boguht the tickets.
If it said I bought the tickets if and only if I went to the game, than yes, not B would mean not A. Because in this case A is restricted to interrelations only with B. In the primary example, it is not. This is the distinction between just 'if', and if and only if.
Dictators do use logic to give structure to their lectures. They win over the audience not by reasoning, but through an appeal to emotion. However, they would be well advised to structure their appeal to emotion as clearly as possible. That way they know what they are doing. This is what logical reasoning does. Informs us of the structure of our ideas. Inevitably, there is structure to all things, as they could be analyzed. But whether we are aware of that is not clear. For example, we can conceive someone having an emotional outburst, we could clearly pinpoint what has ignited their passions, what they said, and how the two interrelate. Their ignorance of that logical structure of the statements they have made does not preclude such a structure from existing.