To believe or accept something as true, especially in archaic or dialectal usage; an old-fashioned form of 'believe'.
Middle English bileven, from Old English belyfan, combining be- (prefix meaning 'make' or 'cause') and leaf/lief (meaning 'to allow' or 'to remain'). The meaning evolved from 'to allow to remain' to 'to accept as true' through Old Norse and Germanic influences.
This is one of those words that shows how 'believe' used to sound quite different—the 'bi-' prefix in bileve reminds us that many English verbs had this 'be-' prefix centuries ago, like 'besmirch' or 'befriend.' Languages are always shedding their old clothes!
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