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A natural extension of natural deduction

Published online by Cambridge University Press:  12 March 2014

Peter Schroeder-Heister
Peter Schroeder-Heister*
Affiliation:
Fachgruppe Philosophie, Universität Konstanz, Postfach 5560, 7750 Konstanz, Federal Republic of Germany
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Extract

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].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 49 , Issue 4 , December 1984 , pp. 1284 - 1300
Copyright
Copyright © Association for Symbolic Logic 1984

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References

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