David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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Axiomathes 20 (2-3):365-383 (2010)
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||Second-order logic Nonstandard models Semantics Reference Putnam Quine Azzouni|
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References found in this work BETA
W. V. Quine (1986). Philosophy of Logic. Harvard University Press.
Stewart Shapiro (1997). Philosophy of Mathematics: Structure and Ontology. Oxford University Press.
Hilary Putnam (1978). Meaning and the Moral Sciences. Routledge & K. Paul.
Hartry Field (1989). Realism, Mathematics & Modality. Basil Blackwell.
Michael D. Resnik (1997). Mathematics as a Science of Patterns. New York ;Oxford University Press.
Citations of this work BETA
Catarina Dutilh Novaes (forthcoming). Axiomatizations of Arithmetic and the First-Order/Second-Order Divide. Synthese.
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