"Culpably negligent" is a common term from American law. It means, very briefly, that someone was negligent in a way that makes them legally liable for the results.
"Culpable ignorance" is not a common legal term, but it means something similar. A person is culpably ignorant if they should have known something but didn't. This can be important in legal cases.
For example, imagine that a doctor gave some medicine to a patient and the patient died from the medication. If the patient's family sued the doctor, the doctor might say, "But I didn't know that this medication was dangerous." If, however, the fact that the medication can kill was commonly known by other doctors - if it was in many books and the company that made the medication knew it and so on - the doctor could be called "culpably ignorant." That is, it may be true that the doctor didn't know the medication was dangerous, but he should have known - it's his job to know, and he could have easily learned.
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