pre-noun adjectival
Formal/literary pre-noun adjectival; used to describe something that can or should exist or happen. Often found in philosophical, legal, or careful writing.
あり得べき未来のシナリオを検討する。
We examine possible future scenarios.
それはあり得べき結果の一つだ。
That is one of the likely outcomes.
pre-noun adjectival
Formal/literary; expresses what ought to be or what is ideal. Less common than the 'possible' sense and often interchangeable with 望ましい in modern Japanese.
あり得べき社会の姿を論じる。
We discuss the desirable form of society.
あり得る is the dictionary form verb meaning 'can exist/happen'; あり得べき is its classical attributive form used as a pre-noun adjectival, now mostly formal/literary.
可能 is a more common, neutral word for 'possible'; あり得べき carries a formal, often philosophical nuance.
望ましい is the standard modern word for 'desirable'; あり得べき in its second sense is more formal and less frequent.
Classical attributive form (連体形) of the verb あり得る (ariuru), from 有り (ari, 'to be') + 得 (u, 'can') + べき (beki, classical auxiliary for possibility/obligation). The form fossilized as a pre-noun adjectival in formal writing.