What does “i promise you” mean?

used as a way of encouraging or warning sb about sth

How is “i promise you” used?

In The Crucible (2014), one line is: “I promise you, Mr. Danforth, I only thought I saw them but I did not.”

The clip at the top of this page is from The Crucible (2014).

I promise you, Mr. Danforth, I only thought I saw them but I did not.

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