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A natural extension of natural deduction

  • Peter Schroeder-Heister (a1)

One of the main ideas of calculi of natural deduction, as introduced by Jaśkowski and Gentzen, is that assumptions may be discharged in the course of a derivation. As regards sentential logic, this conception will be extended in so far as not only formulas but also rules may serve as assumptions which can be discharged. The resulting calculi and derivations with rules of any finite level are informally introduced in §1, while §§2 and 3 state formal definitions of the concepts involved and basic lemmata. Within this framework, a standard form for introduction and elimination rules for arbitrary n-ary sentential operators is motivated in §4, understood as a contribution to the theory of meaning for logical signs. §5 proves that the set {&, ∨, ⊃, ⋏} of standard intuitionistic connectives is complete, i.e. &, ∨, ⊃, and ⋏ suffice to express each n-ary sentential operator having rules of the standard form given in §4. §6 makes some remarks on related approaches. For an extension of the conception presented here to quantifier logic, see [11].

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[2]Došen K., Logical constants. An essay in proof theory, D. Phil. Thesis, Oxford, 1980.
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[9]Schroeder-Heister P., Untersuchungen zur regellogischen Deutung von Aussagenverknüpfungen, Dissertation, Bonn, 1981.
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[11]Schroeder-Heister P., Generalized rules for quantifiers and the completeness of the intuitionistic operators &, ∨,⊃,⋏,∀,∃Logic Colloquium '83 (Richter M. al., editors), Lecture Notes in Mathematics, Springer-Verlag, Berlin (to appear).
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[13]Zucker J. I. and Tragesser R. S., The adequacy problem for inferential logic, Journal of Philosophical Logic, vol. 7 (1978), pp. 501516.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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