Summary: We study aggregation functions defined as convex combinations of the linguistically quantified propositions “at least \(k\) criteria are satisfied”. Our approach is similar to the TOWA function in spirit, but instead of using a maxitive measure, we propose to rely on a real-valued finitely additive set function. This assumption leads to a new framework. However, it is combinatorial by nature and, in general, it produces functions with high computational costs. Therefore, we analyze some particular settings and introduce new aggregation functions which can alleviate the combinatorial burden thanks to several combinatorial identities. These methods have interesting features and in particular, some of them make it possible to set different types of relationships between criteria by allowing the use of different t-norms. The interest of our proposals is illustrated with a famous example which cannot be modeled by classical aggregation functions such as the Choquet integral.