Sake means 'for the purpose of' or 'because of' when used in phrases like 'for your sake' or 'for the sake of peace'. As a separate meaning, *sake* is also a Japanese alcoholic drink made from fermented rice.
The English *sake* in 'for the sake of' comes from Old English *sacu* or *sacu*, meaning 'cause, dispute, or matter'. Japanese *sake* (酒), the drink, is unrelated and simply means 'alcohol' or 'rice wine'.
English *sake* and Japanese *sake* sound the same but come from completely different worlds—one about causes and reasons, the other about rice wine. When you say 'for old times’ sake' you’re using a word that once meant 'legal cause' in Old English. The language kept the idea of motivation but lost the courtroom drama.
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