In the science of hadith, these terms are redefined from their normal linguistic usages. They take on technical definitions that indicate how authentic or reliable a piece of information is based on the strength of the chain of narrators.
Sahih Hadith:
Sahih means authentic and refers to a hadith that fulfils the following five criteria:
1. A continuous (Mu'tasil) chain
2. All narrators are upright (Adl)
3. All narrators have retentiveness (Dhabt)
4. It contains no perverseness or aberrance (Shudhoodh)
5. It contains no defect (Illah)
A continuous chain:
All narrators are upright:
All narrators have maintained retentiveness:
It contains no perverseness or aberrance:
It contains no defects:
Hasan Hadith:
In hadith terminology, Hasan refers to a hadith that has the following 5 components:
1. A continuous chain
2. All narrators are upright
3. All narrators are retentive
4. No aberrance
5. No defects
Da'eef Hadith:
A Da'eef hadith is any hadith that falls short of the criteria of a Hasan hadith. There are a number of different categories and levels.
The above answer is substantial but I wanted to add one more thing.
There is another class of Hadith called the Maudo Hadith i.e. the fabricated hadith.
Any action or saying associated with the Holy Prophet SAW, which he did not perform or say in reality, is termed as Maudo Hadith.
Here are some Maudu hadith:
It causes your sustenance to decrease.
And for those that will occur in the Hereafter, they are:
It brings down the wrath of the Lord.
It makes one's reckoning become difficult.
It causes one to abide in the Hellfire forever.
(Classified fabricated by Ibn Jawzee)
Source: Standup4Islam
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