 Possiblity theory is inspired by fuzzy set theory (Zadeh, 1978). The latter is designed to accommodate linguistic phenomena of vagueness (see Égré &amp; Barberousse 2014, Raffman 2014, Williamson 1994, Field 2016, as well as the entry on  vagueness).  Vagueness arises for predicates like “tall” for which there are extreme, paradigmatic and borderline cases. We might represent this phenomenon formally by a membership function \(\mu_{T}: H \rightarrow [0,1]\), where \(\mu_{T}(h)\) is the degree to which person of height \(h \in H\) belongs to the set of tall people \(T\). Then, \(\mu^{-1}(1)\) is the set of all tall heights; \(\mu^{-1}(0)\) is the set of all short heights; and \(\cup_{r\in(0,1)}\mu^{-1}(r)\) is the set of all borderline heights. Since many distinct heights are tall, it is clear why such membership functions should not satisfy Additivity.