The "Boethian Ballads"(generally comprised of Fortune, Trouthe, Gentilesse, and Lak of Steadfastnesse—hereafter LoS) are so called because of their subject matter, generally thought to coincide not only with Chaucer's interest in Boethius's Consolation of Philosophy, but with the time during which he was actively translating the work into his Middle English Boece. However, while some of the poems, such as Trouthe, experiment directly with ideas from the Consolation, LoS's Boethian ideas are more circumspect.
In Chaucer's Boece Book II, Prosa 6, 29–38: "O, ye erthliche bestes, considere ye nat over whiche thyng that it semeth that ye hand power? Now yif thou saye a mows among othere mysz that challanged to hymself-ward ryght and and power over all othere mysz, how gret scorn woldestow han of it! (Glosa. So fareth it by men [that the wikkid men have power over the wikkid men; that is to seye], the body hath power over the body.)" Chaucer uses this rhetoric to great effect in Trouthe, in which he uses a pun on Sir Philip de la Vache to compare the typical foibles of humans with the more "natural" activities of animals. Further, the refrain in this poem, "And trouthe thee shal delivere, it is no drede" focuses on a very typical Boethian view.
In contrast, John Scattergood argues in his article "Social and Political Issues in Chaucer: An Approach to Lak of Stedfastnesse," Chaucer is using the complaint poem genre to address specific political issues occurring at the time of its composition (in the case of LoS, he argues that he is referring to the political upheavals of c. 1387). While there are certainly some aspects of Boethianism in LoS, the refrain "That al is lost for lak of stedfastnesse" seems more focused on individual actions than the mutability of life. In this case, Chaucer's direct address to (purportedly) King Richard II is necessarily reducing the important of the Consolation of this poem—while the undertones of the poem remain focused on the fickleness of life, an invocation to or support of action on the part of the king does not allow for the same sort of inevitability which appears in his other poems.
Why, though, does any of this matter? While the influences of Boethius on Chaucer (and many other medieval writers) are important to consider, it is equally important to not become hamstrung by such associations. It may be useful to think of Chaucer's poems in groupings such as the "Boethian Ballads", but rethinking the different ways in which Chaucer is using Boethian rhetoric creates new readings of his individual poems, as well.