Beyond a doubt this man is honest.
"Beyond a doubt" seems to be a phrase here and I believe it's a noun phrase, but I'm told it's not. Its a adverb phrase how?
What is it?
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In classical parlance, it's a sentence adverbial. It gives the speaker's (here immovable) opinion about the truth of the statement in the (following) matrix sentence. I'd call it a pragmatic marker subclass modality (the word 'modal' here means 'addressing the speaker's view of the truth value of the attached proposition'). Compare 'Allegedly', 'Clearly', 'Arguably', 'Possibly' ....