Extensionalizing Intensional Second-Order Logic

Neo-Fregean approaches to set theory, following Frege, have it that sets are the extensions of concepts, where concepts are the values of second-order variables. e idea is that, given a second-order entity X, there may be an object εX, which is the extension of X. Other writers have also claimed a similar relationship between second-order logic and set theory, where sets arise from pluralities. is paper considers two interpretations of second-order logic—as being either extensional or intensional—and whether either is more appropriate for this approach to the foundations of set theory. Although there seems to be a case for the extensional interpretation resulting from modal considerations, I show how there is no obstacle to starting with an intensional second order logic. I do so by showing how the ε operator can have the ešect of ‘extensionalising’ intensional second-order entities. It is oŸen thought that there is a close connection between sets and the denotation of the second-order variables under various interpretations of second-order logic. Even if it is denied that second-order entities just are sets (as, for example, is famously claimed by Quine, 1970), it might be thought that sets ‘arise’ from second-order entities in some way. So, for example, Frege’s (inconsistent) set theory had it that sets are extensions of concepts (which are what his second-order variables range over). More recently, attempts to extend the neo-Fregean programme of Bob Hale and CrispinWright (Hale andWright, 2001a) to set theory have followed suit to some extent, albeit with restrictions on which concepts form sets (e.g. Boolos, 1989; Hale, 2000; Shapiro, 2003). In addition, a number of articles which are less explicitly Fregean in motivation have claimed that sets arise from the denotation of second-order variables, where second-order quantication is interpreted as plural quantication (e.g. Burgess, 2004; Linnebo, 2010). Call such an approach to set theory the abstractionist approach1. e idea is that, for a second-order entity X, there may be (though will not always be, on pain of contradiction) an object εX, which is the set of X.2 Central to an abstractionist approach to set theory 1 is is a somewhat wider use of the term ‘abstractionist’ than is common, where it is used to refer to an explicitly Fregean and neo-Fregean approach to mathematics in general. 2 is way of putting things does not do justice to the plural interpretation of second-order logic, nor, arguably, to the interpretation of second-order logic over concepts. Under the plural interpretation, it would not be right to call the denotation of X a single entity. Instead, for plural quantiers, this should read as ‘for any objects xx, there may be a set εxx, which is the set of them.’ And it may not be correct to refer to the values of second-order variables as entities on the concept reading, since