Abstract: Even if philosophers can’t agree on whether to locate the source of vagueness in the world, in our language, or in our knowledge, we can agree that at least some ordinary mereological claims are vague. For some candidate vague object, like Edinburgh or Mt Everest,  we may regard some of its parts as being definitely part of it (core parts) and others as being indefinitely part of it (penumbral parts). But it would be a mistake to identify a vague object the sum of all its core parts (minimal fusion) or with the sum of all its core and penumbral parts (maximal fusion). Both the minimal and maximal fusions are precise objects (modulo concerns about higher-order vagueness) and, by definition, a vague fusion is not a determinate entity. Thus, we must find an alternative approach. Among the various accounts of vagueness, supervaluationism has received a great deal of attention, but almost no attention has been paid to mereological theories set within a supervaluational logic. In this paper, we provide a formal semantics for a vague mereology based on the idea of penumbral connections. We develop a model theory based on configurations of simple graphs and investigate a supervaluational modal logic for this semantics.