A defense of second-order logic
Axiomathes (forthcoming)
| Abstract | 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 first-order 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, second-order logic provides the benefits of a rich framework without the associated costs. | |||||||||
| Keywords | No keywords specified (fix it) | |||||||||
| Categories | ||||||||||
| Options |
|
|||||||||
Download options| PhilPapers Archive |
Upload a copy of this paper Check publisher's policy on self-archival Papers currently archived: 5,705 |
| External links |
 |
| Through your library | Configure |
Similar books and articlesGregory H. Moore (1980). Beyond First-Order Logic: The Historical Interplay Between Mathematical Logic and Axiomatic Set Theory. History and Philosophy of Logic 1 (1-2):95-137.
S. Shapiro (2012). Higher-Order Logic or Set Theory: A False Dilemma. Philosophia Mathematica 20 (3):305-323.
Stewart Shapiro (1999). Do Not Claim Too Much: Second-Order Logic and First-Order Logic. Philosophia Mathematica 7 (1):42-64.
Matti Eklund & Daniel Kolak (2002). Is Hintikka's Logic First-Order? Synthese 131 (3):371 - 388.
Alexander Paseau (2010). Pure Second-Order Logic with Second-Order Identity. Notre Dame Journal of Formal Logic 51 (3):351-360.
Ignacio Jané (1993). A Critical Appraisal of Second-Order Logic. History and Philosophy of Logic 14 (1):67-86.
Jouko Vaananen (2001). Second-Order Logic and Foundations of Mathematics. Bulletin of Symbolic Logic 7 (4):504-520.
Peter Koellner (2010). Strong Logics of First and Second Order. Bulletin of Symbolic Logic 16 (1):1-36.
Analytics
Monthly downloads |
Added to index2010-06-09Total downloads77 ( #10,399 of 549,198 )Recent downloads (6 months)1 ( #63,397 of 549,198 )How can I increase my downloads? |
My notes
Discussion
