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What is the difference between presumption and assumption?
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Presume and assume can mean the same thing, but then they have other meanings that aren't interchangable.

I presume he will introduce the speakers.

I assume he will introduce the speakers.

Basically, that's just 2 ways to say virtually the same thing.

But, one could say, "I will assume the position of Vice President of Marketing on Monday."

One would not say, "I will presume the position."

or, you might say, "I don't presume to know what you are thinking."

You would not say, "I don't assume to know what you are thinking."

I'm sure there are other examples.
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What's the meaning of this sentence:

"I don't presume to know what you are thinking."

Does it mean 'I don't pretend to know...' or ' I don't claim to know...'

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It means both of those, but also connotes the feeling that the speaker realizes it would be presumtuous to assume one knew what the other person was thinking.
Thank you Sam, this was exactly what I was looking for.
Your examples were clear and made it easy to understand (not to mention remember)

-Maya
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AH020387What's the meaning of this sentence: "I don't presume to know what you are thinking."Does it mean 'I don't pretend to know...' or ' I don't claim to know...'?
The sentence above, "I don't presume to know what you are thinking.", means that I don't have enough information to state that I know what you are thinking. A 'presumption' is a conclusion drawn from facts or information already known, which can reasonably lead to a logical conclusion; therefore, the speaker, knowing that there is no factual basis for doing so, is not willing to suggest a knowledge, or conclusion, that cannot reasonably be claimed.