HADITH TERMINOLOGY
ضعيف
ḍaʿīf
The chain has a problem — a gap in it, or a narrator whose reliability was questioned. It means "not verified", not "proven false".
Daif is a residual category: any report failing one or more conditions of sahih or hasan. Ibn al-Ṣalāḥ defines it exactly that way — "every hadith in which the qualities of the sahih hadith are not gathered, nor the qualities of the hasan".
Crucially, daif is not one grade but a very wide spectrum. Al-Nawawī states that "its weakness varies in degree just as the soundness of the sahih varies". A hadith weak because a narrator's memory was merely average, and a hadith weak because its narrator was a known liar, both receive the word daif at the coarse level.
So daif reports a failure of VERIFICATION, not a finding of falsehood. Ibn al-Ṣalāḥ distinguishes weak reports from fabricated ones precisely on the ground that a weak report may still be true in fact.
Al-Nawawī records that hadith scholars and others permitted leniency in narrating weak hadith — other than the fabricated — and acting on it in matters such as narratives, virtues of deeds, and admonitions, that is, things with no bearing on creed or on lawful-and-prohibited rulings. But that permission is tightly bounded by its own proponents, and some scholars rejected it entirely (Abū Bakr Ibn al-ʿArabī is reported as prohibiting acting on weak hadith altogether). Al-Nawawī also requires a manner of citation: a weak report must not be introduced with the assertive "the Messenger of God said", but with "it is reported that…" or "it has reached us that…".