Deriving Natural Deduction Rules from Truth Tables

  • Herman Geuvers
  • Tonny Hurkens
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10119)


We develop a general method for deriving natural deduction rules from the truth table for a connective. The method applies to both constructive and classical logic. This implies we can derive “constructively valid” rules for any classical connective. We show this constructive validity by giving a general Kripke semantics, that is shown to be sound and complete for the constructive rules. For the well-known connectives (\(\vee \), \(\wedge \), \(\rightarrow \), \(\lnot \)) the constructive rules we derive are equivalent to the natural deduction rules we know from Gentzen and Prawitz. However, they have a different shape, because we want all our rules to have a standard “format”, to make it easier to define the notions of cut and to study proof reductions. In style they are close to the “general elimination rules” studied by Von Plato [13] and others. The rules also shed some new light on the classical connectives: e.g. the classical rules we derive for \(\rightarrow \) allow to prove Peirce’s law. Our method also allows to derive rules for connectives that are usually not treated in natural deduction textbooks, like the “if-then-else”, whose truth table is clear but whose constructive deduction rules are not. We prove that ”if-then-else”, in combination with \(\bot \) and \(\top \), is functionally complete (all other constructive connectives can be defined from it). We define the notion of cut, generally for any constructive connective and we describe the process of “cut-elimination”.


  1. 1.
    Ariola, Z.M., Herbelin, H.: Minimal classical logic and control operators. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 871–885. Springer, Heidelberg (2003). doi: 10.1007/3-540-45061-0_68 CrossRefGoogle Scholar
  2. 2.
    Curien, P.-L., Herbelin, H.: The duality of computation. In: ICFP, pp. 233–243 (2000)Google Scholar
  3. 3.
    Dyckhoff, R.: Some remarks on proof-theoretic semantics. In: Piecha, T., Schroeder-Heister, P. (eds.) Advances in Proof-Theoretic Semantics, vol. 43, pp. 79–93. Springer, Heidelberg (2016)Google Scholar
  4. 4.
    Francez, N., Dyckhoff, R.: A note on harmony. J. Philos. Logic 41(3), 613–628 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Geuvers, H., Hurkens, T.: Deriving natural deduction rules from truth tables (Extended version). Technical report (2016). herman/PUBS/NatDedTruthTables_Extended.pdf
  6. 6.
    Milne, P.: Inversion principles and introduction rules. In: Dag Prawitz on Proofs and Meaning, Outstanding Contributions to Logic, vol. 7, pp. 189–224 (2015)Google Scholar
  7. 7.
    Negri, S., von Plato, J.: Structural Proof Theory. Cambridge University Press, Cambridge (2001)CrossRefzbMATHGoogle Scholar
  8. 8.
    Parigot, M.: \({\lambda }{\mu }\)-calculus: an algorithmic interpretation of classical natural deduction. In: Voronkov, A. (ed.) LPAR 1992. LNCS, vol. 624, pp. 190–201. Springer, Heidelberg (1992). doi: 10.1007/BFb0013061 CrossRefGoogle Scholar
  9. 9.
    Prawitz, D.: Ideas and results in proof theory. In: Fenstad, J., (ed.) 2nd Scandinavian Logic Symposium, North-Holland, pp. 237–309 (1971)Google Scholar
  10. 10.
    Schroeder-Heister, P.: A natural extension of natural deduction. J. Symb. Log. 49(4), 1284–1300 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Troelstra, A.S., van Dalen, D.: Constructivism in Mathematics, vol. 1. Elsevier, Amsterdam (1988)zbMATHGoogle Scholar
  12. 12.
    van Dalen, D.: Logic and Structure. Universitext, 3rd edn. Springer, London (1994)CrossRefzbMATHGoogle Scholar
  13. 13.
    von Plato, J.: Natural deduction with general elimination rules. Arch. Math. Log. 40(7), 541–567 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Radboud UniversityNijmegenThe Netherlands
  2. 2.Technical University EindhovenEindhovenThe Netherlands

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