The following sentence came up in a recent conversation with a Polish national:
I don't have a wife and any children.
I corrected his sentence to:
I don't have a wife or any children.
He immediately pointed out that "or" usually means only one of the statements need be true, and that he wanted to convey that both statements were true, i.e.:
I don't have a wife and I don't have any children.
What is the rationale for using "or" in this case?
[As a side question, should we be using "nor" in this case? I'm assuming not, though again I'm unable to explain why.]