But a half-blood would be pretty much the same as a mud-blood (ie muggle).  Pure blood implies that both parents are from wizarding families.  Yes, this is a contradiction that remains improperly resolved within the source material.<br/><br/>The 'best fit' here would be that being Slytherin is an attitude, and the public face of that attitude is to deny that there is such a thing as a mud-blooded Slytherin.  The matter is probably not open for discussion within Slytherin turf.