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Grishin Algebras and Cover Systems for Classical Bilinear Logic

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Abstract

Grishin algebras are a generalisation of Boolean algebras that provide algebraic models for classical bilinear logic with two mutually cancelling negation connectives. We show how to build complete Grishin algebras as algebras of certain subsets (“propositions”) of cover systems that use an orthogonality relation to interpret the negations.

The variety of Grishin algebras is shown to be closed under MacNeille completion, and this is applied to embed an arbitrary Grishin algebra into the algebra of all propositions of some cover system, by a map that preserves all existing joins and meets.

This representation is then used to give a cover system semantics for a version of classical bilinear logic that has first-order quantifiers and infinitary conjunctions and disjunctions.

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References

  1. Bell J.L., Slomson A.B.: Models and Ultraproducts. North-Holland, Amsterdam (1969)

    Google Scholar 

  2. Bell, J. L., ‘Cover schemes, frame-valued sets and their potential uses in spacetime physics’. In A. Reimer (ed.), Spacetime Physics Research Trends, Horizons in World Physics, volume 248. Nova Science Publishers, 2005. Manuscript at http://publish.uwo.ca/~jbell.

  3. Davey B.A., Priestley H.A.: Introduction to Lattices and Order. University Press, Cambridge (1990)

    Google Scholar 

  4. Kosta Došen.: ‘Sequent systems and groupoid models II’. Studia Logica 48(1), 41–65 (1989)

    Article  Google Scholar 

  5. Kosta Došen.: ‘A brief survey of frames for the Lambek calculus’. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 38, 179–187 (1992)

    Article  Google Scholar 

  6. Galatos, Nikolaos, Peter Jipsen, Tomasz Kowalski, and Hiroakira Ono, Residuated Lattices : An Algebraic Glimpse at Substructural Logics, volume 151 of Studies in Logic and the Foundations of Mathematics. Elsevier, 2007.

  7. Jean-Yves Girard.: ‘Linear logic’. Theoretical Computer Science 50, 1–102 (1987)

    Article  Google Scholar 

  8. Robert Goldblatt.: ‘Semantic analysis of orthologic’. Journal of Philosophical Logic 3, 19–35 (1974). doi:Reprinted in 9.

    Article  Google Scholar 

  9. Robert Goldblatt.: Mathematics of Modality. CSLI Lecture Notes No. 43. Stanford University, CSLI Publications (1993)

  10. Robert Goldblatt.: ‘A Kripke-Joyal semantics for noncommutative logic in quantales’. In: G. Governatori, I. Hodkinson, Y. Venema (eds.), Advances in Modal Logic Volume 6, pp. 209–225. College Publications, London (2006) www.aiml.net/volumes/volume6/.

  11. Goldblatt Robert, ‘Cover semantics for quantified lax logic’. Journal of Logic and Computation, 2010. doi:10.1093/logcom/exq029

  12. Goldblatt, Robert, Quantifiers, Propositions and Identity: Admissible Semantics for Quantified Modal and Substructural Logics. Number 38 in Lecture Notes in Logic. Cambridge University Press and the Association for Symbolic Logic, 2011.

  13. Grishin, V. N., ‘On a generalisation of the Ajdukiewicz-Lambek system’. In A. I. Mikhailov (ed.), Studies in Non-Classical Logics and Formal Systems, pp. 315–334. Nauka, Moscow, 1983. English translation in V. M. Abrusci and C. Casadio (eds.), New Perspectives in Logic and Formal Linguistics, Proceedings 5th Roma Workshop, Bulzoni Editore, Rome, 2002. Corrected version available as pp. 1–17 in http://symcg.pbworks.com/f/essllinotesnew.pdf .

  14. Joachim Lambek.: ‘Some lattice models of bilinear logic’. Algebra Universalis 34, 541–550 (1995)

    Article  Google Scholar 

  15. Lambek, Joachim, ‘Bilinear logic and Grishin algebras’. In Ewa Orlowska (ed.), Logic at Work: Essays Dedicated to the Memory of Helena Rasiowa. Physica-Verlag, 1999, pp. 604–612

  16. Mac Lane, Saunders, and Ieke Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Springer-Verlag, 1992.

  17. MacNeille H.M.: ‘Partially ordered sets’. Transactions of the American Mathematical Society 42, 416–460 (1937)

    Article  Google Scholar 

  18. Munkres, James R., Topology. Prentice Hall, 2000.

  19. Niefield. , Susan B., Rosenthal Kimmo I.: ‘Constructing locales from quantales’. Mathematical Proceedings of the Cambridge Philosophical Society 104, 215–234 (1988)

    Article  Google Scholar 

  20. Ono, Hiroakira, ‘Semantics for substructural logics’. In P. Schroeder-Heister and K. Došen (eds.), Substructural Logics. Oxford University Press, 1993, pp. 259–291.

  21. Hiroakira Ono: ‘Closure operators and complete embeddings of residuated lattices’. Studia Logica 74(3), 427–440 (2003)

    Article  Google Scholar 

  22. Hiroakira Ono., Yuichi Komori.: ‘Logics without the contraction rule’. The Journal of Symbolic Logic 50(1), 169–201 (1985)

    Article  Google Scholar 

  23. Rosenthal Kimmo I., Quantales and Their Applications, volume 234 of Pitman Research Notes in Mathematics. Longman Scientific & Technical, 1990.

  24. Mark Theunissen., Yde Venema: ‘MacNeille completions of lattice expansions’. Algebra Universalis 57, 143–193 (2007)

    Article  Google Scholar 

  25. Troelstra A.S.: Lectures on Linear Logic. CSLI Lecture Notes No. 29. CSLI Publications, Stanford, California (1992)

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  1. Victoria University of Wellington, Wellington, New Zealand

    Robert Goldblatt

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  1. Robert Goldblatt
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Correspondence to Robert Goldblatt.

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Dedicated to Ryszard Wójcicki on his 80th birthday

Special issue in honor of Ryszard Wójcicki on the occasion of his 80th birthday Edited by J. Czelakowski, W. Dziobiak, and J. Malinowski

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Goldblatt, R. Grishin Algebras and Cover Systems for Classical Bilinear Logic. Stud Logica 99, 203 (2011). https://doi.org/10.1007/s11225-011-9360-0

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