In a Nutshell:
The rule of eating of frogs for most scholars is built upon the rule of slaughtering and killing them. The scholars differ due to the conflicting views relating to evidences that seem to suggest killing of frogs is prohibited.
Evidences
There are a number of ahadith which forbid the killing of frogs.
Juristic Analysis
The Hanafis, Shafi'is, Hanbalis and Zahiris rely on these narrations to prohibit the eating of frogs arguing whatever has been forbidden to kill is also forbidden to eat. If it had been permissible to eat, it would have been permissible to kill.
Malik, Ibn Abi Laila, al-Sha'bi and al-Thawri in one of his two opinions however permit eating frogs based upon the Quranic verses:
"It is permitted for you to catch and eat seafood" [5:96]
and the generality of the well known hadith:
"Its water is purifying and its dead are lawful."
They also argue the ahadith about killing frogs are weak.
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