Notre Dame Journal of Formal Logic 58 (4):461-506 (2017)

Authors
Hartry Field
New York University
Tore Fjetland Øgaard
University of Bergen
Harvey Lederman
Princeton University
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.
Keywords extensionality axiom   naive comprehension   nonclassical logic  naive class theory
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DOI 10.1215/00294527-2017-0010
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References found in this work BETA

Saving Truth From Paradox.Hartry H. Field - 2008 - Oxford, England: Oxford University Press.
Outline of a Theory of Truth.Saul Kripke - 1975 - Journal of Philosophy 72 (19):690-716.
Can There Be Vague Objects?Gareth Evans - 1978 - Analysis 38 (4):208.
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Relevant Restricted Quantification.J. C. Beall, Ross T. Brady, A. P. Hazen, Graham Priest & Greg Restall - 2006 - Journal of Philosophical Logic 35 (6):587-598.

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Citations of this work BETA

Paths to Triviality.Tore Øgaard - 2016 - Journal of Philosophical Logic 45 (3):237-276.
Model-Theoretic Semantics and Revenge Paradoxes.Lorenzo Rossi - 2019 - Philosophical Studies 176 (4):1035-1054.
Naive Set Theory and Nontransitive Logic.David Ripley - 2015 - Review of Symbolic Logic 8 (3):553-571.

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