Type-theoretic lexical semantics have gained a lot of interest since the inception of the Generative Lexicon Theory and the development of tools for such approaches. Conceived as an extensions of Montagovian compositional semantics, they represent a satisfying mean to deal with complex lexical phenomena without losing compositionality — and without losing the logical information that a proposition is asserted, refuted or supposed. All formalisms that have been proposed for that purpose rely on a system with many base types and relations between those; we use many-sorted logic, and will thus call those base types lexical sorts. This chapter details why this system of sorts (or base types) is needed and what features it should provide. We propose that turning to classifiers, in languages that possess such a part of speech, provides a linguistically and cognitively motivated starting point for such a system.