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Do you think that doctors have a right to know of the malpractice lawsuits of their patients?
Since its public information anyway, yes.
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Do You think they should be able to turn down patients due to malpractice lawsuits?
This is a trickier question - I understand the very real economic reasons for doing so, but then the altruistic "I want to help people" side of me cries out that's ridiculous and no doctor should ever use such base reasons for refusing to care for a patient in need. Also, the few malpractice suits I have heard of outside of obstetrics usually sound like the case cited - they have very real merits. This is my limited experience of course, but just knowing about cases like this tells me just because someone is involved in a malpractice suit does not mean the person is just waiting for me to slip up in their care so they can sue me. I bet in the cases with real merit (especially those in which there was permanent physical damage), the patient will probably volunteer they were involved in a suit and probably would more or less interview the doctor to discover if s/he was competent. Anytime a doctor does something that seriously breeches the doctor-patient trust, the patient is bound to be hesitant about any future interactions with doctors.
In the end, I doubt most doctors would seriously look at a so called "blacklist" just because of the sheer number of not only those on the list, but the amount of patients - and few doctors outside of general family practice have the luxury of knowing who their patient will be before they show up.
Of course, the doctor does have the right to refuse care to any patient who s/he believes will reasonably be able to find care elsewhere, or if the doctor feels s/he would end up in physical danger by treating the patient. Thus, a dcotor is not obligated to run into the middle of a gun-fire fight to start treating the injured or rush into a burning building to help burn victims.