Despite the relevance of powerful brands in the present-day market, research on the process of brand name design from a cognitive perspective focuses almost exclusively on the effects of the use of conceptual metaphor, and to a lesser extent, metonymy, overlooking the role played by other cognitive strategies. This paper analyzes the potentiality of mitigation and strengthening cognitive operations as tools for the systematic, risk-free design of new brand names with highly predictable and felicitous connotations. In particular, it focuses on their role in the systematic generation of axiologically positive brands in both Spanish and American wine labels, thus largely reducing the need for the costly and time-consuming cultural checks that branding companies need to run on new brand names before their commercial launching. In so doing, the interaction of the two aforementioned cognitive operations with a number of pragmatic principles and cultural models of social interaction, and their subservience to other cognitive operations, like those of comparison, correlation, and domain expansion and reduction, are also considered. The results of the study offer new insights on the semantics of commercial brand names which should prove useful for branding professionals, as well as data of interest to linguists dealing with inter-linguistic issues and cognitive modeling alike.