Assume and presume are often used interchangeably. However, there’s a slight difference:
▪To assume means to take for granted as the basis of argument or action. There doesn’t have to be any proof for the assumption; it can be arbitrary. You can replace the word assume with suppose. ▪To presume means to assert that something is true without yet having complete evidence (hence the prefix “pre…”). However, there’s a belief that at least some evidence exists. So there’s a stronger element of postulation or hypothesis in assume than in presume.
The nouns are presumption and assumption.
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