A Defense of Second-Order Logic

Second-order logic has a number of attractive features, in particular the strong expressive resources it offers, and the possibility of articulating categorical mathematical theories (such as arithmetic and analysis). But it also has its costs. Five major charges have been launched against second-order logic: (1) It is not axiomatizable; as opposed to first-order logic, it is inherently incomplete. (2) It also has several semantics, and there is no criterion to choose between them (Putnam, J Symbol Logic 45:464–482, 1980). Therefore, it is not clear how this logic should be interpreted. (3) Second-order logic also has strong ontological commitments: (a) it is ontologically committed to classes (Resnik, J Phil 85:75–87, 1988), and (b) according to Quine (Philosophy of logic, Prentice-Hall: Englewood Cliffs, 1970), it is nothing more than ‘‘set theory in sheep’s clothing’’. (4) It is also not better than its first-order counterpart, in the following sense: if first-order logic does not characterize adequately mathematical systems, given the existence of non-isomorphic firstorder interpretations, second-order logic does not characterize them either, given the existence of different interpretations of second-order theories (Melia, Analysis 55:127–134, 1995). (5) Finally, as opposed to what is claimed by defenders of second-order logic [such as Shapiro (J Symbol Logic 50:714–742, 1985)], this logic does not solve the problem of referential access to mathematical objects (Azzouni, Metaphysical myths, mathematical practice: the logic and epistemology of the exact sciences, Cambridge University Press, Cambridge, 1994). In this paper, I argue that the second-order theorist can solve each of these difficulties. As a result, secondorder logic provides the benefits of a rich framework without the associated costs. My thanks go to Philip Catton, Steven French, Sarah Kattau, Bernard Linsky, Judy Pelham, Jaroslav Peregrin, Peter Simons, Göran Sundholm, and Peter Woodruff for helpful discussions. O. Bueno (&) Department of Philosophy, University of Miami, Coral Gables, FL 33124, USA e-mail: [email protected] 123 Axiomathes (2010) 20:365–383 DOI 10.1007/s10516-010-9101-4