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Rule Separation and Embedding Theorems for Logics Without Weakening

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Abstract

A full separation theorem for the derivable rules of intuitionistic linear logic without bounds, 0 and exponentials is proved. Several structural consequences of this theorem for subreducts of (commutative) residuated lattices are obtained. The theorem is then extended to the logic LR + and its proof is extended to obtain the finite embeddability property for the class of square increasing residuated lattices.

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References

  1. Aglianò, P., ‘Ternary deductive terms in residuated structures’, Acta Sci. Math. (Szeged) 64 (1998), 397-429.

    Google Scholar 

  2. Anderson, A. R., and N. D. Belnap, Jnr., Entailment: The Logic of Relevance and Necessity, Volume 1, Princeton University Press, 1975.

  3. Anderson, A. R., N. D. Belnap, Jnr., and J. M. Dunn, Entailment: The Logic of Relevance and Necessity, Volume 2, Princeton University Press, 1992.

  4. Avron, A., ‘The semantics and proof theory of linear logic’, Theoretical Computer Science 57 (1988), 161-184.

    Google Scholar 

  5. Blok W. J., and D. Pigozzi, ‘Local deduction theorems in algebraic logic’, Colloquia Mathematica Societatis János Bolyai 54 Algebraic Logic, Budapest (Hungary), 1988, pp. 75-109.

    Google Scholar 

  6. Blok W. J., and D. Pigozzi, ‘Algebraizable Logics’, Memoirs of the American Mathematical Society, Number 396, Amer. Math. Soc., Providence, 1989.

    Google Scholar 

  7. Blok W. J., and D. Pigozzi, ‘Algebraic semantics for universal Horn logic without equality’, in J. D. H. SMITH and A. ROMANOWSKA, (eds.), Universal Algebra and Quasigroup Theory, Heldermann Verlag, Berlin, 1992, pp. 1-56.

    Google Scholar 

  8. Blok W. J., and D. Pigozzi, ‘Abstract algebraic logic and the deduction theorem’, The Bulletin of Symbolic Logic, to appear.

  9. Blok W. J., and J. G. Raftery, ‘On assertional logic’, manuscript, 1998.

  10. Blok W. J., and J. G. Raftery, ‘Fragments of R-mingle’, Studia Logica, to appear.

  11. Blok W. J., and C. J. VAN Alten, ‘The finite embeddability property for residuated lattices, pocrims and BCK-algebras’, Algebra Universalis 48 (2002), 253-271.

    Google Scholar 

  12. Czelakowski J., ‘Algebraic aspects of deduction theorems’, Studia Logica 44 (1985), 369-387.

    Google Scholar 

  13. Czelakowski J., ‘Local deduction theorems’, Studia Logica 45 (1986), 377-391.

    Google Scholar 

  14. Dunn J. M., ‘Partial gaggles applied to logics with restricted structural rules’, in P. Schroeder-Heister and K. Došen, (eds.) Substructural Logics, Clarendon Press, Oxford, 1993, pp. 63-108.

    Google Scholar 

  15. Giambrone S., and R. K. Meyer, ‘Completeness and conservative extension results for some Boolean relevant logics’, Studia Logica 48 (1989), 1-14.

    Google Scholar 

  16. Köhler P., and D. Pigozzi, ‘Varieties with equationally definable principal congruences’, Algebra Universalis 11 (1980), 213-219.

    Google Scholar 

  17. Meyer R. K., ‘Conservative extension in relevant implication’, Studia Logica 31 (1972), 39-46.

    Google Scholar 

  18. Meyer R. K., ‘On conserving positive logics’, Notre Dame Journal Formal Logic 14 (1973), 224-236.

    Google Scholar 

  19. Meyer R. K., ‘Intuitionism, entailment, negation’, in H. Leblanc (ed.) Truth, Syntax and Modality, North Holland, Amsterdam, 1973, 168-198.

    Google Scholar 

  20. Meyer R. K., ‘Improved decision procedures for pure relevant logic’, unpublished manuscript, 1973.

  21. Meyer R. K., ‘Improved decision procedures for pure relevant logic’, in C. A. Anderson and M. Zeleny, (eds.) Logic, Meaning and Computation, Kluwer, Dordrecht, 2001, pp. 191-217.

    Google Scholar 

  22. Meyer R. K., and H. Ono, ‘The finite model property for BCK and BCTW’, Studia Logica 53 (1994), 107-118.

    Google Scholar 

  23. Meyer R. K., and R. Routley, ‘Algebraic analysis of entailment, I’, Logique et Analyse N.S. 15 (1972), 407-428.

    Google Scholar 

  24. Nash-Williams C. St. J. A., ‘On well-quasi-ordering finite trees’, Proc. Cambridge Phil. Soc. 59 (1963), 833-835.

    Google Scholar 

  25. Ono H., ‘Semantics for substructural logics’, in P. Schroeder-Heister and K. Došen, (eds.), Substructural Logics, Clarendon Press, Oxford, 1993, pp. 259-291.

    Google Scholar 

  26. Ono H., and Y. Komori, ‘Logics without the contraction rule’, Journal of Symbolic Logic 50 (1985), 169-202.

    Google Scholar 

  27. Okada M., and K. Terui, ‘The finite model property for various fragments of intuitionistic linear logic’, Journal of Symbolic Logic 64 (1999), 790-802.

    Google Scholar 

  28. Raftery J. G., ‘Ideal determined varieties need not be congruence 3-permutable’, Algebra Universalis 31 (1994), 293-297.

    Google Scholar 

  29. Thistlewaite P. B., M. A. McRobbie, and R. K. Meyer, Automated Theorem-Proving in Non-Classical Logics, Pitman, London, 1988.

    Google Scholar 

  30. Troelstra A. S., Lectures on Linear Logic, CSLI Lecture Notes, No 29, 1992.

  31. Wechler W., Universal Algebra for Computer Scientists, Springer-Verlag Berlin Heidelberg, 1992.

    Google Scholar 

  32. Wójcicki R., Theory of Logical Calculi, Kluwer, Dordrecht, 1988.

    Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits, 2050, Johannesburg, South Africa

    Clint J. Van Alten & James G. Raftery

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  1. Clint J. Van Alten
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  2. James G. Raftery
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Van Alten, C.J., Raftery, J.G. Rule Separation and Embedding Theorems for Logics Without Weakening. Studia Logica 76, 241–274 (2004). https://doi.org/10.1023/B:STUD.0000032087.02579.e2

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