Skip to main content
×
Home

A natural extension of natural deduction

  • Peter Schroeder-Heister (a1)
Abstract

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].

Copyright
References
Hide All
[1]Belnap N. D., Tonk, plonk and plink, Analysis, vol. 22 (19611962), pp. 130134.
[2]Došen K., Logical constants. An essay in proof theory, D. Phil. Thesis, Oxford, 1980.
[3]Hendry H. E., Does IPC have a binary indigenous Sheffer function?, Notre Dame Journal of Formal Logic, vol. 22 (1981), pp. 183186.
[4]Kreisel G., Constructivist approaches to logic, Modern logic—a survey (Agazzi E., editor), Reidel, Dordrecht, 1981, pp. 6791.
[5]Kutschera F. v., Die Vollständigkeit des Operatorensystems. {¬,∧,∨, ⊃ für die intuitionistische Aussagenlogik im Rahmen der Gentzensemantik, Archiv für Mathematische Logik und Grundlagenforschung, vol. 11 (1968), pp. 316.
[6]McCullough D. P., Logical connectives for intuitionistic propositional logic, this Journal, vol. 36 (1971), pp. 1520.
[7]Prawitz D., Natural deduction. A proof-theoretical study, Almqvist & Wiksell, Stockholm, 1965.
[8]Prawitz D., Proofs and the meaning and completeness of the logical constants, Essays on mathematical and philosophical logic (Hintikka J.et al., editors), Reidel, Dordrecht, 1979, pp. 2540; slightly revised German translation in Conceptus, vol. 16 (1982), no. 38, pp. 31–44.
[9]Schroeder-Heister P., Untersuchungen zur regellogischen Deutung von Aussagenverknüpfungen, Dissertation, Bonn, 1981.
[10]Schroeder-Heister P., Logische Konstanten und Regeln. Zur Deutung von Aussagenoperatoren, Conceptus, vol. 16 (1982), no. 38, pp. 4559.
[11]Schroeder-Heister P., Generalized rules for quantifiers and the completeness of the intuitionistic operators &, ∨,⊃,⋏,∀,∃Logic Colloquium '83 (Richter M. M.et al., editors), Lecture Notes in Mathematics, Springer-Verlag, Berlin (to appear).
[12]Tennant N., Natural logic, Edinburgh University Press, Edinburgh, 1978.
[13]Zucker J. I. and Tragesser R. S., The adequacy problem for inferential logic, Journal of Philosophical Logic, vol. 7 (1978), pp. 501516.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 4 *
Loading metrics...

Abstract views

Total abstract views: 96 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 20th November 2017. This data will be updated every 24 hours.