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Normality, Non-contamination and Logical Depth in Classical Natural Deduction

Abstract

In this paper we provide a detailed proof-theoretical analysis of a natural deduction system for classical propositional logic that (i) represents classical proofs in a more natural way than standard Gentzen-style natural deduction, (ii) admits of a simple normalization procedure such that normal proofs enjoy the Weak Subformula Property, (iii) provides the means to prove a Non-contamination Property of normal proofs that is not satisfied by normal proofs in the Gentzen tradition and is useful for applications, especially in formal argumentation, (iv) naturally leads to defining a notion of depth of a proof, to the effect that, for every fixed natural k, normal k-depth deducibility is a tractable problem and converges to classical deducibility as k tends to infinity.

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Acknowledgements

We wish to thank three anonymous referees for the very close reading, many interesting comments and important corrections.

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Correspondence to Marcello D’Agostino.

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D’Agostino, M., Gabbay, D. & Modgil, S. Normality, Non-contamination and Logical Depth in Classical Natural Deduction. Stud Logica (2019). https://doi.org/10.1007/s11225-019-09847-4

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Keywords

  • Natural deduction
  • Classical propositional logic
  • Normal proofs
  • Non-contamination
  • Tractable reasoning