Prospects for a Naive Theory of Classes

Notre Dame Journal of Formal Logic 58 (4):461-506 (2017)
Download Edit this record How to cite View on PhilPapers
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.
PhilPapers/Archive ID
FIEPFA-2
Revision history
First archival date: 2014-10-18
Latest version: 3 (2014-11-23)
View upload history
References found in this work BETA
Universal Logic.Brady, Ross

View all 20 references / Add more references

Citations of this work BETA
Paths to Triviality.Øgaard, Tore Fjetland
Can Modalities Save Naive Set Theory?Fritz, Peter; Lederman, Harvey; Liu, Tiankai & Scott, Dana

Add more citations

Added to PP index
2014-10-18

Total views
215 ( #15,454 of 42,390 )

Recent downloads (6 months)
53 ( #12,134 of 42,390 )

How can I increase my downloads?

Downloads since first upload
This graph includes both downloads from PhilArchive and clicks to external links.