This paper formally studies a notion of dialectical argument strength in terms of the number of ways in which an argument can be successfully attacked in expansions of an abstract argumentation framework. The proposed model is abstract but its design is motivated by the wish to avoid overly limiting assumptions that may not hold in particular dialogue contexts or in particular structured accounts of argumentation. It is shown that most principles for gradual argument acceptability proposed in the literature fail to hold for the proposed notion of dialectical strength, which clarifies their rational foundations and highlights the importance of distinguishing between logical, dialectical and rhetorical argument strength.