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"plainly false" is correct and can be used in written English.
It means something that is obviously or clearly untrue. Example: The politician's promises were plainly false, as he had a history of breaking them.
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The first proposition is plainly false.
The solitarist view of human identity is plainly false, and it can also be dangerous.
So how are we to understand the President's plainly false statement?
He said the contention that he has a grand plan to destroy the hunting media is plainly false.
Brown went on to make another statement that was plainly false: "Britain will always honour its commitment to international justice.
And he made questionable assertions, including one — that the E.P.A. had never found a case of unsafe hydraulic fracturing of natural gas — that was plainly false.
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The state forensic commission in Texas is still finishing its work on Willingham's case, but David Grann's New Yorker article examines the entire case, including the jailhouse informant who plainly gave false testimony and the circumstantial evidence, flimsy in the first place, that was not what it appeared to be to the jury.
That is plainly untrue.
If you look at Clinton's exchange with Steve Kroft in its entirety, I count eight separate times in which she either plainly denied the false claim that Obama was Muslim, labeled that suggestion to be a smear, or expressed sympathy for Obama having to deal with the Muslim innuendo.
"You have grown men making statements that they plainly know to be false reading directly from talking points from a gun lobby that plainly knows them to be false.
Let me say it plainly: This is a false premise.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com