Satire – the poetry of the fable

Boileau, in his "L'Art poétique", enumerating poetic genres, mentioned the fable, taking away the right to be counted among poetry. But perhaps he did not intend to deny the fable as poetry, but simply forgot it or considered that it belonged more to prose. Boileau may have been wrong, but are all our and foreign textbooks, dictionaries and reference books wrong, which define the fable only as some kind of didactic form and nothing more? One dictionary says that the fable is "a short moral story or poem" that emphasizes the rules of morality. Others define the fable as "a story that aims to give some moral maxim", or which "gives an allegorical interpretation of certain rules of life or morality", etc.