 One could then try the following intensional identity conditions: two properties are identical iff they are co-intensional, i.e., necessarily co-extensional, where the necessity in question is logical necessity. This guarantees that spherical and weighing 2 kilos are different even if they happen to be co-extensional. Following this line, one may take properties to be intensions, understood as, roughly, functions that assign extensions (sets of objects) to predicates at logically possible worlds. Thus, for instance, the predicates “has a heart” and “has kidneys” stand for different intensions, for even if they have the same extension in the actual world they have different extensions at worlds where there are creatures with heart and no kidneys or vice versa. This approach is followed by Montague (1974) in his pioneering work in natural language semantics, and in a similar way by Lewis (1986b), who reduces properties to sets of possible objects in his modal realism, explicitly committed to possible worlds and mere possibilia inhabiting them. Most philosophers find this commitment unappealing. Moreover, one may wonder how properties can do their causal work if they are conceived of in this way (for further criticisms see Bealer 1982: 13–19, and 1998: §4; Egan 2004). However, the criterion of co-intensionality may be accepted without also buying the reduction of properties to sets of possibilia (Bealer views this as the identity condition for his conception 1 properties; see below). Still, co-intensionality must face two challenges coming from opposite fronts.