To build on what @Jayne said:
We take the statement: “Holding a gun makes a person powerful,” and rephrase it as, ”if a person is holding a gun, then he feels powerful.” The converse of this, made by switching the antecedent (if…) and consequent (then…) parts of the phrase, is, ”if a person feels powerful, then he is holding a gun.” (The converse of the statement “A implies B” is “B implies A.”)
@grog, what you have there is the contrapositive—the contrapositive of “A implies B” is “not A implies not B.” The contrapositive of “if a person is holding a gun, then he feels powerful” is “if a person is not holding a gun, then he does not feel powerful.”
The truth or falsehood of the original statement does not have any necessary relationship with the truth or falsehood of the converse or the contrapositive. It’s trivial to find an example of this, such as the statement: If it is Sunday, the post office is closed. The converse of this is, if the post office is closed, it is Sunday. But this is not true on national holidays—on Memorial Day, for instance, the post office is closed, and it is not Sunday. The contrapositive is, if it is not Sunday, the post office is not closed. But again, on Memorial Day—a Monday—the post office is closed.
(Logic is much easier to discuss if you avoid emotionally loaded subjects like guns.)