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Natural Deduction for the Sheffer Stroke and Peirce’s Arrow (and any Other Truth-Functional Connective)

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Abstract

Methods available for the axiomatization of arbitrary finite-valued logics can be applied to obtain sound and complete intelim rules for all truth-functional connectives of classical logic including the Sheffer stroke (nand) and Peirce’s arrow (nor). The restriction to a single conclusion in standard systems of natural deduction requires the introduction of additional rules to make the resulting systems complete; these rules are nevertheless still simple and correspond straightforwardly to the classical absurdity rule. Omitting these rules results in systems for intuitionistic versions of the connectives in question.

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Notes

  1. The general elimination rules coincide with the elimination rules of Parigot’s [12] free deduction; he showed how to obtain the standard elimination rules from them by systematic simplification. A version of the simplified system of classical multiple-conclusion natural deduction was studied by Parigot [13]. The general elimination rules for ∧ and → were studied by von Plato [14].

  2. The use of the special symbol ⊥ is not strictly necessary, as it could be replaced by explicitly contradictory premises (e.g., C|D, C, and D; or more simply C|C and C) when it appears as a premise, and by an arbitrary formula when it appears as the conclusion. This has the advantage that the resulting rules mention no symbols other than |, i.e., are pure. Indeed, this was in part Hazen and Pelletier’s goal and their rules used this approach. This alternative approach can however not be used if no such “explicit contradiction” can be expressed. See also Section 7 below.

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Acknowledgments

I am grateful to Allen Hazen and Jeff Pelletier for helpful comments as well as the question which originally prompted this paper.

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Zach, R. Natural Deduction for the Sheffer Stroke and Peirce’s Arrow (and any Other Truth-Functional Connective). J Philos Logic 45, 183–197 (2016). https://doi.org/10.1007/s10992-015-9370-x

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Keywords

  • Natural deduction
  • Sequent calculus
  • Sheffer stroke