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Prospects for a naive theory of classes. (English) Zbl 1417.03272

Summary: 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 R. T. Brady [Notre Dame J. Formal Logic 24, 431–449 (1983; Zbl 0488.03026); in: Paraconsistent logic. Essays on the inconsistent. München etc.: Philosophia Verlag. 437–471 (1989; Zbl 0691.03038); Universal logic. Stanford, CA: CSLI Publications (2006; Zbl 1234.03010); J. Philos. Log. 43, No. 2–3, 261–281 (2014; Zbl 1343.03003)], 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 A. Bacon [Notre Dame J. Formal Logic 54, No. 1, 87–104 (2013; Zbl 1273.03073)]. 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.

MSC:

03E70 Nonclassical and second-order set theories
03B60 Other nonclassical logic
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