  Some presentations of Lewis’s view replace properties with sets of centered worlds (Egan 2006, Ninan 2013). Centered worlds are ordered pairs \(\langle o, w \rangle\) where \(o\) is an object and \(w\) is a world. Suppose, as Lewis does, that properties are sets of possible objects; alternatively, suppose that a property \(F\) is an intension, namely a function whose value at any world \(w\) is the set of objects that have \(F\) at \(w\). Then, as Lewis (1979a) points out, believing a property \(F\) is, in a certain sense, equivalent to believing the set of centered worlds \(\langle o, w \rangle\) such that \(o\) has \(F\) in \(w\).