Cripples, Bastards, and Broken Things - S1-E4
Corrected entry: When Jon talks to Sam about almost having sex with Ros, he says that their child, in case of a pregnancy, would have been a bastard named Snow as well. This is impossible, because the bastard surnames (like Snow in the North) are only for acknowledged children of noblemen. Jon's children wouldn't have the right to carry a surname, as he's not a nobleman.
Correction: Any child born out of wedlock is a bastard. It's not just a name for noblemen, even peasants have to get married to legitimize their children. Snow is also not a surname, so if Jon and Ros would get married and have a child that child will not be called Snow like his/her father.
Bastard surnames are indeed only for high-born because smallfolk in Westeros do not have or use surnames. Children of unmarried smallfolk may be bastards, but they do not get a bastard surname. Trueborn children of high-born bastards may take the bastard surname or change it if they like.
Correction: Actually, it is just for noblemen. Bastards of smallfolk (the peasantry) in Westeros do not receive a bastard surname, only those of noble birth do. But Jon is not smallfolk. Despite being a bastard, he is indeed of noble birth and children of noble bastards are allowed to take the surname for themselves and most do.
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