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The Simple Consistency of Naive Set Theory using Metavaluations

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Abstract

The main aim is to extend the range of logics which solve the set-theoretic paradoxes, over and above what was achieved by earlier work in the area. In doing this, the paper also provides a link between metacomplete logics and those that solve the paradoxes, by finally establishing that all M1-metacomplete logics can be used as a basis for naive set theory. In doing so, we manage to reach logics that are very close in their axiomatization to that of the logic R of relevant implication. A further aim is the use of metavaluations in a new context, expanding the range of application of this novel technique, already used in the context of negation and arithmetic, thus providing an alternative to traditional model theoretic approaches.

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Authors and Affiliations

  1. Philosophy Program, La Trobe University, Melbourne, Vic, 3086, Australia

    Ross T. Brady

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  1. Ross T. Brady
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Correspondence to Ross T. Brady.

Additional information

Dedicated to Richard Sylvan, on the occasion of the 15th anniversary of his passing.

This project was initiated after a discussion with Zach Weber of the University of Melbourne concerning rules and their contraposed rule-forms, which are in essence the components ‘If \( \mathrm{v}\left( \mathrm{A} \right)=\mathrm{T} \)then \( {\rm{v}}\left( {\rm{B}} \right) = {\rm{T}} \)’ and ‘If \( {{{\rm{v}}}^{*}}\left( {\rm{A}} \right) = {\rm{T}}\;{\rm{then}}\;{{{\rm{v}}}^{*}}\left( {\rm{B}} \right) = {\rm{T}} \)’ of the metavaluation \( \mathrm{v}\left( {\mathrm{A}\to \mathrm{B}} \right) \).

We also acknowledge useful discussion of this paper, especially from Greg Restall and Walter Carnielli, when recently presented to the Routley Memorial Conference ‘Beyond the Possible’, held at the University of Melbourne, 27–29th July, 2011. We also thank Dave Ripley for pointing out an error in detail. We also thank the two referees of this journal for their hard work and for their corrections and clarifications.

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Brady, R.T. The Simple Consistency of Naive Set Theory using Metavaluations. J Philos Logic 43, 261–281 (2014). https://doi.org/10.1007/s10992-012-9262-2

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  • DOI: https://doi.org/10.1007/s10992-012-9262-2

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