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What does for all mean?

Looking for the meaning or definition of the word for all? Here's what it means.

Preposition
  1. In spite of, despite.
  2. (mathematics) (Indicates that a statement is true no matter what element of a set it is made about.)
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Examples
Still, for all of this, South Carolina is now represented in the U.S. Senate by Tim Scott, a Republican and an african-american.
In return for all this solicitude the lawyer had an unaddressed and ambiguous scrawl in his possession.
Yet for all that, the hospitality of her little home in Ajaccio was lavish and famous.
For this problem of the categories, in all its abstractness, is still a common problem for all of us.
I have come to thank you for all your care of Ada, and your kindness to us all.
There were lights, but for all the sight of people Addington might have been abed.

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