Journal of Automated Reasoning

, Volume 7, Issue 4, pp 587–596 | Cite as

Condensed detachment is complete for relevance logic: A computer-aided proof

  • Grigori Mints
  • Tanel Tammet


The condensed detachment ruleD is a combination of modus ponens with a minimal amount of substitution. EarlierD has been shown to be complete for intuitionistic and classical implicational logic but incomplete forBCK andBCI logic. We show thatD is complete for the relevance logic. One of the main steps is the proof of the formula ((aa) →a) →a found in interaction with our resolution theorem prover. Various strategies of generating consequences of the axioms and choosing best ones for the next iteration were tried until the proof was found.

Key words

Automated theorem proving relevance logic condensed detachment 


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  1. 1.
    Avron A., ‘The semantics and proof-theory of linear logic’,Theoretical Computer Science 57, 161–189 (1988).CrossRefGoogle Scholar
  2. 2.
    Anderson, A. R. and Belnap, N. D.,Entailment, Vol. I, Princeton University Press (1975).Google Scholar
  3. 3.
    Church, A., ‘The weak theory of implication’, inKontrolliertes Denken, Untersuchungen zum Logikkalkül und zur Logik der Einzelwissenschaften (Festgabe zum 60. Geburtstag von Prof. W. Britzelmayr), A. Menne, A. Wilhelmy, and H. Angstl (eds.), München Kommissionverlag Karl Aber, pp. 22–37 (1951).Google Scholar
  4. 4.
    Curry, H. B. and Feys, R.,Combinatorial Logic, Vol. I, North-Holland (1958).Google Scholar
  5. 5.
    Girard, J.-Y., ‘Linear logic’,Theoretical Computer Science,N1, (1987).Google Scholar
  6. 6.
    Helman, G. H., ‘Restricted lambda abstraction and the interpretation of some non-classical logics’, Ph.D. Dissertation, University of Pittsburgh (1977), University Microfilms, Ann Arbor, iv + 217 pp.Google Scholar
  7. 7.
    Hindley J. R., ‘The principal type-scheme of an object in combinatory logic’,Trans. Amer. Math. Soc. 146, 29–60 (1969).Google Scholar
  8. 8.
    Hindley, J. R. and Meredith, D., ‘Principal type-schemes and condensed detachment’. Preprint, October (1987).Google Scholar
  9. 9.
    Jaśkowski S., Über Tautologien in welchen keine Variable mehr als zweimal vorkommt’,Zeitschrift für Math. Logik und Grundlagen der Math 9, 219–228 (1963).Google Scholar
  10. 10.
    Martins, J. P. and Shapiro, S. C., ‘A model for belief revision’,Non-Monotonic Reasoning Workshop pp. 241–294 (1984).Google Scholar
  11. 11.
    Moh Shaw-Kwei, ‘The deduction theory and two new logical systems’,Methodos 2, 56–75 (1950).Google Scholar
  12. 12.
    Robinson J. A., ‘A machine-oriented logic based on the resolution principle’,J. Ass. Computing Machinery 12, 23–4 (1965).Google Scholar
  13. 13.
    Tammet, T., ‘The resolution program, able to decide some sovable classes’.COLOG-88: International Conference on Computer Logic. Proceedings. LNCS, v. 417, Springer-Verlag, pp. 300–312 (1990).Google Scholar
  14. 14.
    Wos, L., Overbeek, R., Lusk, E., and Boyle, J.,Automated Reasoning, Prentice-Hall (1984).Google Scholar
  15. 15.
    Zamov N. K., ‘Maslov's inverse method and decidable classes’,Annals of Pure and Applied Logic 42, 165–194 (1989).CrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1991

Authors and Affiliations

  • Grigori Mints
    • 1
  • Tanel Tammet
    • 1
  1. 1.Institute of CyberneticsTallinnEstonia

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