Abstract
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 ((a →a) →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.
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References
Avron A., ‘The semantics and proof-theory of linear logic’,Theoretical Computer Science 57, 161–189 (1988).
Anderson, A. R. and Belnap, N. D.,Entailment, Vol. I, Princeton University Press (1975).
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).
Curry, H. B. and Feys, R.,Combinatorial Logic, Vol. I, North-Holland (1958).
Girard, J.-Y., ‘Linear logic’,Theoretical Computer Science,N1, (1987).
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.
Hindley J. R., ‘The principal type-scheme of an object in combinatory logic’,Trans. Amer. Math. Soc. 146, 29–60 (1969).
Hindley, J. R. and Meredith, D., ‘Principal type-schemes and condensed detachment’. Preprint, October (1987).
Jaśkowski S., Über Tautologien in welchen keine Variable mehr als zweimal vorkommt’,Zeitschrift für Math. Logik und Grundlagen der Math 9, 219–228 (1963).
Martins, J. P. and Shapiro, S. C., ‘A model for belief revision’,Non-Monotonic Reasoning Workshop pp. 241–294 (1984).
Moh Shaw-Kwei, ‘The deduction theory and two new logical systems’,Methodos 2, 56–75 (1950).
Robinson J. A., ‘A machine-oriented logic based on the resolution principle’,J. Ass. Computing Machinery 12, 23–4 (1965).
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).
Wos, L., Overbeek, R., Lusk, E., and Boyle, J.,Automated Reasoning, Prentice-Hall (1984).
Zamov N. K., ‘Maslov's inverse method and decidable classes’,Annals of Pure and Applied Logic 42, 165–194 (1989).
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Mints, G., Tammet, T. Condensed detachment is complete for relevance logic: A computer-aided proof. J Autom Reasoning 7, 587–596 (1991). https://doi.org/10.1007/BF01880330
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DOI: https://doi.org/10.1007/BF01880330