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2017 Prospects for a Naive Theory of Classes
Hartry Field, Harvey Lederman, Tore Fjetland Øgaard
Notre Dame J. Formal Logic 58(4): 461-506 (2017). DOI: 10.1215/00294527-2017-0010

Abstract

The naive theory of properties states that for every condition there is a property instantiated by exactly the things which satisfy that condition. The naive theory of properties is inconsistent in classical logic, but there are many ways to obtain consistent naive theories of properties in nonclassical logics. The naive theory of classes adds to the naive theory of properties an extensionality rule or axiom, which states roughly that if two classes have exactly the same members, they are identical. In this paper we examine the prospects for obtaining a satisfactory naive theory of classes. We start from a result by Ross Brady, which demonstrates the consistency of something resembling a naive theory of classes. We generalize Brady’s result somewhat and extend it to a recent system developed by Andrew Bacon. All of the theories we prove consistent contain an extensionality rule or axiom. But we argue that given the background logics, the relevant extensionality principles are too weak. For example, in some of these theories, there are universal classes which are not declared coextensive. We elucidate some very modest demands on extensionality, designed to rule out this kind of pathology. But we close by proving that even these modest demands cannot be jointly satisfied. In light of this new impossibility result, the prospects for a naive theory of classes are bleak.

Citation

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Hartry Field. Harvey Lederman. Tore Fjetland Øgaard. "Prospects for a Naive Theory of Classes." Notre Dame J. Formal Logic 58 (4) 461 - 506, 2017. https://doi.org/10.1215/00294527-2017-0010

Information

Received: 13 April 2014; Accepted: 7 October 2014; Published: 2017
First available in Project Euclid: 6 June 2017

zbMATH: 06803184
MathSciNet: MR3707648
Digital Object Identifier: 10.1215/00294527-2017-0010

Subjects:
Primary: 3E70

Rights: Copyright © 2017 University of Notre Dame

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Vol.58 • No. 4 • 2017
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