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On the theory of types1

Published online by Cambridge University Press:  12 March 2014

W. V. Quine
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Extract

In this paper the theory of logical types will be examined, and certain departures from it will be suggested. Though the purpose of the paper is not primarily expository, an approach has been possible which presupposes no familiarity with special literature. Matters at variance with such an approach have been confined to appendices and footnotes.

In the early pages the logical paradoxes will be considered—an infinite series of them, of which Russell's paradox is the first. Then Russell's simple theory of types will be formulated, in adaptation to a minimal set of logical primitives: inclusion and abstraction. Two aspects of the theory will be distinguished: an ontological doctrine and a formal restriction. It will be found that by repudiating the former we can avoid certain unnatural effects of the type theory—notably the reduplication of logical constants from type to type, and the apparent dependence of finite arithmetic upon an axiom of infinity. But the formal restriction itself has unnatural effects, which survive, even in an aggravated form, after the type ontology has been dropped. A liberalization of the formal restriction will be proposed which removes the more irksome of these anomalies.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 3 , Issue 4: Including an Update to A Bibliography of Symbolic Logic , December 1938 , pp. 125 - 139
Copyright
Copyright © Association for Symbolic Logic 1938

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Footnotes

1

The main ideas of this paper were presented in an address before the mathematical fraternity Pi Mu Epsilon and the New York University Philosophical Society at their annual joint meeting in New York, February 24, 1938.

References

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