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Philosophy
Logic

A Cointuitionistic Adjoint Logic

Profile image of Gianluigi BellinGianluigi Bellin

2017, arXiv (Cornell University)

https://doi.org/10.2168/LMCS
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54 pages

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Abstract

Bi-intuitionistic logic (BINT) is a conservative extension of intuitionistic logic to include the duals of each logical connective. One leading question with respect to BINT is, what does BINT look like across the three arcs -logic, typed λ-calculi, and category theory -of the Curry-Howard-Lambek correspondence? Categorically, BINT can be seen as a mixing of two worlds: the first being intuitionistic logic (IL), which is modeled by a cartesian closed category, and the second being the dual to intuitionistic logic called cointuitionistic logic (coIL), which is modeled by a cocartesian coclosed category. Crolard showed that combining these two categories into the same category results in it degenerating to a poset. However, this degeneration does not occur when both logics are linear. We propose that IL and coIL need to be separated, and then mixed in a controlled way using the modalities from linear logic. This separation can be ultimately achieved by an adjoint formalization of bi-intuitionistic logic. This formalization consists of three worlds instead of two: the first is intuitionistic logic, the second is linear bi-intuitionistic (Bi-ILL), and the third is cointuitionistic logic. They are then related via two adjunctions. The adjunction between IL and ILL is known as a Linear/Non-linear model (LNL model) of ILL, and is due to Benton [4]. However, the dual to LNL models which would amount to the adjunction between coILL and coIL has yet to appear in the literature. In this paper we fill this gap by studying the dual to LNL models which we call dual LNL models. We conduct a similar analysis to that of Benton for dual LNL models by showing that dual LNL models correspond to dual linear categories, the dual to Bierman's [5] linear categories proposed by Bellin [3]. Following this we give the definition of bi-LNL models by combining our dual LNL models with Benton's LNL models to obtain a categorical model of bi-intuitionistic logic, but we leave its analysis and corresponding logic to a future paper. Finally, we give a corresponding sequent calculus, natural deduction, and term assignment for dual LNL models.

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References (28)

  1. Gianluigi Bellin. Subnets of proof-nets in multiplicative linear logic with MIX. Mathematical Structures in Computer Science, 7(6):663-699, 1997. URL: http://journals.cambridge.org/action/displayAbstract? aid=44699.
  2. Gianluigi Bellin. Assertions, Hypotheses, Conjectures, Expectations: Rough-Sets Semantics and Proof Theory, pages 193-241.
  3. Springer Netherlands, Dordrecht, 2014. URL: http://dx.doi.org/10.1007/978-94-007-7548-0_ 10, doi:10.1007/978-94-007-7548-0_10.
  4. Gianluigi Bellin. Categorical proof theory of co-intuitionistic linear logic. Logical Methods in Computer Science, 10(3):Paper 16, September 2014.
  5. Nick Benton. A mixed linear and non-linear logic: Proofs, terms and models (preliminary report). Technical Report UCAM-CL-TR-352, University of Cambridge Computer Laboratory, 1994.
  6. G. M. Bierman. On Intuitionistic Linear Logic. PhD thesis, Wolfson College, Cambridge, December 1993.
  7. Gavin M. Bierman. A note on full intuitionistic linear logic. Ann. Pure Appl. Logic, 79(3):281-287, 1996. URL: https://doi.org/10.1016/0168-0072(96)00004-8, doi:10.1016/0168-0072(96)00004-8.
  8. Torben Braüner and Valeria de Paiva. A formulation of linear logic based on dependency-relations. In Computer Science Logic, 11th International Workshop, CSL '97, Annual Conference of the EACSL, Aarhus, Denmark, August 23-29, 1997, Selected Papers, pages 129-148, 1997. URL: https://doi.org/10.1007/BFb0028011, doi:10. 1007/BFb0028011.
  9. Ranald Clouston, Jeremy E. Dawson, Rajeev Goré, and Alwen Tiu. Annotation-free sequent calculi for full intuition- istic linear logic -extended version. CoRR, abs/1307.0289, 2013. URL: http://arxiv.org/abs/1307.0289.
  10. J.R.B. Cockett and R.A.G. Seely. Proof theory for full intuitionistic linear logc, bilinear logic, and mix categories. Theory and Applications of Categories, 3(5):85-131, 1997.
  11. J.R.B. Cockett and R.A.G. Seely. Weakly distributive categories. Journal of Pure and Applied Algebra, 114(2):133 - 173, 1997.
  12. Tristan Crolard. Subtractive logic. Theoretical Computer Science, 254(1-2):151-185, 2001.
  13. Tristan Crolard. A formulae-as-types interpretation of subtractive logic. J. Log. Comput., 14(4):529-570, 2004. URL: https://doi.org/10.1093/logcom/14.4.529.
  14. Carlo Dalla Pozza and Claudio Garola. A pragmatic interpretation of intuitionistic propositional logic. Erkenntnis, 43(1):81-109, 1995.
  15. Luis Estrada-González. Complement-topoi and dual intuitionistic logic. The Australasian Journal of Logic, 9, 2010.
  16. Rajeev Goré. Dual intuitionistic logic revisited. In Automated Reasoning with Analytic Tableaux and Related Methods, International Conference, TABLEAUX 2000, St Andrews, Scotland, UK, July 3-7, 2000, Proceedings, pages 252-267, 2000. URL: https://doi.org/10.1007/10722086_21, doi:10.1007/10722086_21.
  17. Rajeev Goré, Linda Postniece, and Alwen Tiu. Cut-elimination and proof-search for bi-intuitionistic logic using nested sequents. In Advances in Modal Logic 7, papers from the seventh conference on "Advances in Modal Logic," held in Nancy, France, 9-12 September 2008, pages 43-66, 2008. URL: http://www.aiml.net/volumes/volume7/ Gore-Postniece-Tiu.pdf.
  18. Harley Eades III and Valeria de Paiva. Multiple conclusion linear logic: Cut elimination and more. In Logical Foundations of Computer Science -International Symposium, LFCS 2016, Deerfield Beach, FL, USA, January 4-7, 2016. Proceedings, pages 90-105, 2016. URL: https://doi.org/10.1007/978-3-319-27683-0_7, doi: 10.1007/978-3-319-27683-0_7.
  19. J. Lambek and P.J. Scott. Introduction to Higher-Order Categorical Logic. Cambridge Studies in Advanced Mathe- matics. Cambridge University Press, 1988.
  20. Maria Emilia Maietti, Paola Maneggia, Valeria de Paiva, and Eike Ritter. Relating categorical semantics for intu- itionistic linear logic. Applied Categorical Structures, 13(1):1-36, 2005. URL: http://dx.doi.org/10.1007/ s10485-004-3134-z, doi:10.1007/s10485-004-3134-z.
  21. Michel Parigot. Lambda-mu-calculus: An algorithmic interpretation of classical natural deduction. In Andrei Voronkov, editor, Logic Programming and Automated Reasoning, volume 624 of Lecture Notes in Computer Science, pages 190-201. Springer Berlin / Heidelberg, 1992.
  22. Luis Pinto and Tarmo Uustalu. Relating sequent calculi for bi-intuitionistic propositional logic. In Proceedings Third International Workshop on Classical Logic and Computation, CL&C 2010, Brno, Czech Republic, 21-22 August 2010., pages 57-72, 2010. URL: https://doi.org/10.4204/EPTCS.47.7, doi:10.4204/EPTCS.47.7.
  23. Linda Postniece. Deep inference in bi-intuitionistic logic. In Logic, Language, Information and Computation, 16th International Workshop, WoLLIC 2009, Tokyo, Japan, June 21-24, 2009. Proceedings, pages 320-334, 2009. URL: https://doi.org/10.1007/978-3-642-02261-6_26, doi:10.1007/978-3-642-02261-6_26.
  24. Cecylia Rauszer. A formalization of the propositional calculus of hb logic. Studia Logica, 33(1):23-34, 1974.
  25. Cecylia Rauszer. Semi-boolean algebras and their applications to intuitionistic logic with dual operations,. Funda- menta Mathematicae, 83:219-249, 1974.
  26. Cecylia Rauszer. An algebraic and kripke-style approach to a certain extension of intuitionistic logic. In Dissertationes Mathematics, 167:1-67, Institut Mathématique de l'Académie Polonaise des Sciences 1980.
  27. Harold Schellinx. Some syntactical observations on linear logic. Journal of Logic and Computation, 1(4):537-559, 1991.
  28. James Trafford. Structuring co-constructive logic for proofs and refutations. Logica Universalis, 10(1):67-97, 2016.

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Related topics

  • Mathematics
  • Intuitionistic Logic
  • Computer Science
  • Linear Logic
  • Dual (grammatical number)
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