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Investigation into Combinatory Systems with Dual Combinators

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Abstract

Combinatory logic is known to be related to substructural logics. Algebraic considerations of the latter, in particular, algebraic considerations of two distinct implications (→, ←), led to the introduction of dual combinators in Dunn & Meyer 1997. Dual combinators are "mirror images" of the usual combinators and as such do not constitute an interesting subject of investigation by themselves. However, when combined with the usual combinators (e.g., in order to recover associativity in a sequent calculus), the whole system exhibits new features. A dual combinatory system with weak equality typically lacks the Church-Rosser property, and in general it is inconsistent. In many subsystems terms "unexpectedly" turn out to be weakly equivalent. The paper is a preliminary attempt to investigate some of these issues, as well as, briefly compare function application in symmetric λ-calculus (cf. Barbanera & Berardi 1996) and dual combinatory logic.

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References

  • Barbanera, F., and S. Berardi, ‘A symmetric lambda calculus for classical program extraction’, Information and Computation 125 (1996), 103-117.

    Google Scholar 

  • BimbÓ, K., ‘Dual identity combinators’, (a paper presented at the 20th World Congress of Philosophy, Aug. 1998, Boston (MA)).

  • BimbÓ, K., and J.M. Dunn, ‘Two extensions of the structurally free logic LC’, Logic Journal of IGPL 6 (1998), 403-424.

    Google Scholar 

  • Curry, H.B., ‘Grundlagen der kombinatorischen Logik’, American Journal of Mathematics 52 (1930), 509-536, 789–834.

    Google Scholar 

  • Dunn, J.M., ‘Relevance logic and entailment’, in D. Gabbay, and F. Guenthner (eds.), Handbook of Philosophical Logic. III: Alternatives to classical logic, Reidel, Dordrecht, 1986, pages 117-229.

    Google Scholar 

  • Dunn, J.M., and R.K. Meyer, ‘Combinators and structurally free logic’, Logic Journal of IGPL 5 (1997), 505-537.

    Google Scholar 

  • Meyer, R.K., K. BimbÓ and J.M. Dunn, ‘Dual combinators bite the dust’ (abstract), The Bulletin of Symbolic Logic 4 (1998), 463-464. (Presented by title at the Annual Conference of the Australasian Association for Logic, July 4–6, 1997, University of Auckland, New Zealand.)

    Google Scholar 

  • SchÖnfinkel, M., ‘On the building blocks of mathematical logic’, in J. van Heijenoort (ed.) ,From Frege to Gödel. A source book in mathematical logic, Harvard University Press, Cambridge (MA), 1967, pages 355-366.

    Google Scholar 

  • Smullyan, R.M., Diagonalization and Self-Reference, Clarendon, Oxford, 1994.

    Google Scholar 

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Bimbó, K. Investigation into Combinatory Systems with Dual Combinators. Studia Logica 66, 285–296 (2000). https://doi.org/10.1023/A:1005252431462

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  • DOI: https://doi.org/10.1023/A:1005252431462

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