Studia Logica

, Volume 92, Issue 1, pp 63–84 | Cite as

Commuting Conversions vs. the Standard Conversions of the “Good” Connectives

  • Fernando FerreiraEmail author
  • Gilda Ferreira


Commuting conversions were introduced in the natural deduction calculus as ad hoc devices for the purpose of guaranteeing the subformula property in normal proofs. In a well known book, Jean-Yves Girard commented harshly on these conversions, saying that ‘one tends to think that natural deduction should be modified to correct such atrocities.’ We present an embedding of the intuitionistic predicate calculus into a second-order predicative system for which there is no need for commuting conversions. Furthermore, we show that the redex and the conversum of a commuting conversion of the original calculus translate into equivalent derivations by means of a series of bidirectional applications of standard conversions.


Natural deduction commuting conversions predicative quantifiers 


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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade de LisboaLisboaPortugal
  2. 2.Queen MaryUniversity of LondonLondonUnited Kingdom

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