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

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Abstract

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.

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References

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Authors and Affiliations

  1. Departamento de Matemática, Universidade de Lisboa, Campo Grande, Ed. C6, 1749-016, Lisboa, Portugal

    Fernando Ferreira

  2. Queen Mary, University of London, Mile End Road, London, E1 4NS, United Kingdom

    Gilda Ferreira

Authors
  1. Fernando Ferreira
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  2. Gilda Ferreira
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Correspondence to Fernando Ferreira.

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Ferreira, F., Ferreira, G. Commuting Conversions vs. the Standard Conversions of the “Good” Connectives. Stud Logica 92, 63–84 (2009). https://doi.org/10.1007/s11225-009-9186-1

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  • DOI: https://doi.org/10.1007/s11225-009-9186-1

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