Current Trends in Substructural Logics | Journal of Philosophical Logic Skip to main content
Springer Nature Link
Log in

Current Trends in Substructural Logics

  • Published:
Journal of Philosophical Logic Aims and scope Submit manuscript

Abstract

This paper briefly overviews some of the results and research directions. In the area of substructural logics from the last couple of decades. Substructural logics are understood here to include relevance logics, linear logic, variants of Lambek calculi and some other logics that are motivated by the idea of omitting some structural rules or making other structural changes in LK, the original sequent calculus for classical logic.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. This paper was originally written. In 2012 for the 40th anniversary issue of the Journal of Philosophical Logic; I only had time to make some small revisions and additions. In October 2014.

References

  1. Allwein, G., & Dunn, J.M. (1993). Kripke models for linear logic. Journal of Symbolic Logic, 58, 514–545.

    Article  Google Scholar 

  2. Anderson, A.R., & Belnap, N.D. (1975). Entailment: The logic of relevance and necessity, Vol. I. Princeton: Princeton University Press.

    Google Scholar 

  3. Anderson, A.R., Belnap, N.D., & Dunn, J.M. (1992). Entailment: The logic of relevance and necessity, Vol. II. Princeton: Princeton University Press.

    Google Scholar 

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

    Article  Google Scholar 

  5. Belnap, N.D. (1982). Display logic. Journal of Philosophical Logic, 11, 375–417.

    Google Scholar 

  6. Belnap, N.D. (1990). Linear logic displayed. Notre Dame Journal of Formal Logic, 31, 14–25.

    Article  Google Scholar 

  7. Belnap, N.D., & Wallace, J.R. (1961). A decision procedure for the system \(E_{\overline {I}}\) of entailment with negation. Technical Report 11, Contract No. SAR/609 (16), Office of Naval Research, New Haven.

  8. Bimbó, K. The decidability of the intensional fragment of classical linear logic. (19 pages, submitted for publication).

  9. Bimbó, K. (2001). Semantics for structurally free logics L C+. Logic Journal of the IGPL, 9, 525–539.

    Article  Google Scholar 

  10. Bimbó, K (2005). Admissibility of cut in L C with fixed point combinator. Studia Logica, 81, 399–423.

    Article  Google Scholar 

  11. Bimbó, K. (2007). Relevance logics. In D. Jacquette (Ed.), Philosophy of logic, Handbook of the philosophy of science (D. Gabbay, P. Thagard and J. Woods, eds.) (Vol. 5, pp. 723–789). North-Holland: Elsevier.

  12. Bimbó, K. (2009). Dual gaggle semantics for entailment. Notre Dame Journal of Formal Logic, 50, 23–41.

    Article  Google Scholar 

  13. Bimbó, K. (2012). Combinatory logic: Pure, applied and typed.. Discrete mathematics and its applications. Boca Raton: CRC Press.

  14. Bimbó, K. (2014). Proof theory: Sequent calculi and related formalisms.. Discrete mathematics and its applications. Boca Raton: CRC Press.

  15. Bimbó, K., & Dunn, J.M. (2002). Four-valued logic. Notre Dame Journal of Formal Logic, 42, 171–192.

    Google Scholar 

  16. Bimbó, K., & Dunn, J.M. (2008). Generalized Galois logics: Relational semantics of nonclassical logical calculi, CSLI lecture notes (Vol. 188). Stanford: CSLI Publications.

    Google Scholar 

  17. Bimbó, K., & Dunn, J.M. (2009). Symmetric generalized Galois logics. Logica Universalis, 3, 125–152.

    Article  Google Scholar 

  18. Bimbó, K., & Dunn, J.M. (2010). Calculi for symmetric generalized Galois logics. In J. van Benthem & M. Moortgat (Eds.), Festschrift for Joachim Lambek, Linguistic Analysis (Vol. 36, pp. 307–343). Vashon: Linguistic Analysis.

  19. Bimbó, K., & Dunn, J.M. (2012). New consecution calculi for \(R_{\rightarrow }^{\,t}\). Notre Dame Journal of Formal Logic, 53, 491–509.

  20. Bimbó, K., & Dunn, J.M. (2013). On the decidability of implicational ticket entailment. Journal of Symbolic Logic, 78, 214–236.

    Article  Google Scholar 

  21. Bimbó, K., & Dunn, J.M. (2014). Extracting BB \(^{\prime }\) IW inhabitants of simple types from proofs in the sequent calculus \(LT_{\rightarrow }^{\, t}\) for implicational ticket entailment. Logica Universalis, 8, 141–164.

  22. Bimbó, K., & Dunn, J.M. Modalities in lattice-R. (32 pages, ms.)

  23. Birkhoff, G. (1967). Lattice theory, AMS colloquium publications, 3rd edn. (Vol. 25). Providence: American Mathematical Society.

    Google Scholar 

  24. Blok, W.J., & Raftery, J.G. (2004). Fragments of R-mingle. Studia Logica, 78, 59–106.

    Article  Google Scholar 

  25. Brady, R.T. (Ed.) (2003). Relevant logics and their rivals. A continuation of the work of R. Sylvan, R. Meyer, V. Plumwood and R. Brady (Vol. II). Burlington: Ashgate.

  26. Church, A. (1951). The weak theory of implication In A. Menne, A. Wilhelmy & H. Angsil (Eds.), , Kontrolliertes Denken, Untersuchungen zum Logikkalkül und zur Logik der Einzelwissenschaften (pp. 22–37). Karl Alber: Komissions-Verlag.

  27. Cook, S.A. (1971). The complexity of theorem-proving procedures. In: Proceedings of the Third Annual ACM Symposium on the Theory of Computing (pp. 151–158).

  28. Curry, H.B. (1963). Foundations of mathematical logic. New York: McGraw-Hill Book Company. (Dover, New York, NY, 1977).

    Google Scholar 

  29. Došen, K., & Schroeder-Heister, P. (1993). Substructural logics. Oxford: Clarendon.

    Google Scholar 

  30. Dunn, J.M. (1973). A ‘Gentzen system’ for positive relevant implication, (abstract). Journal of Symbolic Logic, 38, 356–357.

    Google Scholar 

  31. Dunn, J.M. (1976). A Kripke-style semantics for R-mingle using a binary accessibility relation. Studia Logica, 35, 163–172.

    Article  Google Scholar 

  32. Dunn, J.M. (1976). Quantification and RM. Studia Logica, 35, 315–322.

    Article  Google Scholar 

  33. Dunn, J.M. (1986). Relevance logic and entailment In D. Gabbay & F. Guenthner (Eds.), , Handbook of philosophical logic, 1st edn. (Vol. 3, pp. 117–224).. Dordrecht: D. Reidel.

  34. Dunn, J.M. (1991). Gaggle theory: An abstraction of Galois connections and residuation with applications to negation, implication, and various logical operators In J. van Eijck (Ed.), , Logics in AI: European workshop JELIA ’90, number 478 in lecture notes in computer science (pp. 31–51). Berlin: Springer.

  35. Dunn, J.M. (1993). Partial-gaggles applied to logics with restricted structural rules In K. Došen & P. Schroeder-Heister (Eds.), , Substructural logics (pp. 63–108). Oxford: Clarendon.

  36. Dunn, J.M. (1995). Gaggle theory applied to intuitionistic, modal and relevance logic In I. Max & W. Stelzner (Eds.), , Logik und Mathematik. Frege-Kolloquium Jena (pp. 335–368). Berlin: W. de Gruyter.

  37. Dunn, J.M., & Hardegree, G.M. (2001). Algebraic methods in philosophical logic, Oxford logic guides (Vol. 41). Oxford: Oxford University Press.

    Google Scholar 

  38. Dunn, J.M., & Meyer, R.K. (1997). Combinators and structurally free logic. Logic Journal of the IGPL, 5, 505–537.

    Article  Google Scholar 

  39. Dunn, J.M., & Restall, G. (2002). Relevance logic In D. Gabbay & F. Guenthner (Eds.), , Handbook of philosophical logic, 2nd edn. (Vol. 6, pp. 1–128).. Amsterdam: Kluwer.

  40. Gentzen, G. (1964). Investigations into logical deduction. American Philosophical Quarterly, 1, 288–306.

    Google Scholar 

  41. Giambrone, S. (1985). T W + and R W + are decidable. Journal of Philosophical Logic, 14, 235–254.

    Article  Google Scholar 

  42. Girard, J.-Y. (1987). Linear logic. Theoretical Computer Science, 50, 1–102.

    Article  Google Scholar 

  43. Goldblatt, R. (2009). Conservativity of Heyting implication over relevant quantification. Review of Symbolic Logic, 2, 310–341.

    Article  Google Scholar 

  44. Goldblatt, R., & Mares, E.D. (2006). An alternative semantics for quantified relevant logic. Journal of Symbolic Logic, 71, 163–187.

    Article  Google Scholar 

  45. Goré, R. (1998). Substructural logics on display. Logic Journal of the IGPL, 6, 451–504.

    Article  Google Scholar 

  46. Hartonas, C., & Dunn, J.M. (1997). Stone duality for lattices. Algebra Universalis, 37, 391–401.

    Article  Google Scholar 

  47. Henkin, L. (1949). The completeness of the first-order functional calculus. Journal of Symbolic Logic, 14, 159–166.

    Article  Google Scholar 

  48. Jónsson, B., & Tarski, A. (1951). Boolean algebras with operators, I. American Journal of Mathematics, 73, 891–939.

    Article  Google Scholar 

  49. Jónsson, B., & Tarski, A. (1952). Boolean algebras with operators, II. American Journal of Mathematics, 74, 127–162.

    Article  Google Scholar 

  50. Kamide, N. (2002). Kripke semantics for modal substructural logics. Journal of Logic, Language and Computation, 11, 453–470.

    Article  Google Scholar 

  51. Kamide, N. (2003). Normal modal substructural logics with strong negation. Journal of Philosophical Logic, 32, 589–612.

    Article  Google Scholar 

  52. Kripke, S.A. (1959). The problem of entailment, (abstract). Journal of Symbolic Logic, 24, 324.

    Article  Google Scholar 

  53. Kripke, S.A. (1963). Semantical analysis of modal logic I. Normal modal propositional calculi. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 67–96.

  54. Kripke, S.A. (1965). Semantical analysis of intuitionistic logic I. In J.N. Crossley & M.A.E. Dummett (Eds.), Formal Systems and Recursive Functions, Proceedings of the Eighth Logic Colloquium (pp. 92–130). Amsterdam.

  55. Kripke, S.A. (1965). Semantical analysis of modal logic II. Non-normal propositional calculi In J.W. Addison, L. Henkin & A. Tarski (Eds.), The theory of models (pp. 206–220). North-Holland, Amsterdam.

  56. Lambek, J. (1961). On the calculus of syntactic types In R. Jacobson (Ed.), Structure of language and its mathematical aspects (pp. 166–178). Providence: American Mathematical Society.

  57. Lemmon, E.J., Meredith, C.A., Meredith, D., Prior, A.N., & Thomas, I. (1969). Calculi of pure strict implication In J.W. Davis, D.J. Hockney & W.K. Wilson (Eds.), Philosophical logic (pp. 215–250). Dordrecht: D. Reidel.

  58. Lincoln, P., Mitchell, J., Scedrov, A., & Shankar, N. (1992). Decision problems for linear logic. Annals of Pure and Applied Logic, 56, 239–311.

    Article  Google Scholar 

  59. Mares, E.D. (2000). CE is not a conservative extension of E. Journal of Philosophical Logic, 29, 263–275.

    Article  Google Scholar 

  60. Mares, E.D. (2004). Relevant logic: A philosophical interpretation. Cambridge, UK: Cambridge University Press.

  61. Mares, E.D., & Meyer, R.K. (2001). Relevant logics In L. Goble (Ed.), The Blackwell guide to philosophical logic, Blackwell philosophy guides (pp. 280–308). Oxford: Blackwell Publishers.

  62. Meyer, R.K. (1966). Topics in modal and many-valued logic. PhD thesis, University of Pittsburgh, Ann Arbor.

  63. Meyer, R.K. (2004). Ternary relations and relevant semantics. Annals of Pure and Applied Logic, 127, 195–217.

    Article  Google Scholar 

  64. Meyer, R.K., & Routley, R. (1972). Algebraic analysis of entailment I. Logique et Analyse, 15, 407–428.

    Google Scholar 

  65. Meyer, R.K., & Routley, R. (1973). Classical relevant logics I. Studia Logica, 32, 51–66.

    Article  Google Scholar 

  66. Meyer, R.K., & Routley, R. (1974). Classical relevant logics II. Studia Logica, 33, 183–194.

    Article  Google Scholar 

  67. Moorgat, M. (2010). Symmetric categorial grammar: residuation and Galois connections In J. van Benthem & M. Moortgat (Eds.), Festschrift for Joachim Lambek, Linguistic Analysis (Vol. 36, pp. 143–166).. Vashon: Linguistic Analysis.

  68. Morrill, G., & Valentín, O. (2010). Displacement calculus In J. van Benthem & M. Moortgat (Eds.), Festschrift for Joachim Lambek, Linguistic Analysis (Vol. 36, pp. 167–192). Vashon: Linguistic Analysis.

  69. Ono, H. (2003). Substructural logics and residuated lattices — an introduction In V.F. Hendricks & J. Malinowski (Eds.), 50 years of Studia Logica, Trends in Logic (Vol. 21, pp. 21193–228).. Amsterdam: Kluwer.

  70. Padovani, V. (2013). Ticket Entailment is decidable. Mathematical Structures in Computer Science, 23, 568–607.

    Article  Google Scholar 

  71. Paoli, F. (2002). Substructural logics: A primer, Trends in Logic (Vol. 13). Dordrecht: Kluwer.

    Book  Google Scholar 

  72. Pentus, M. (2006). Lambek calculus is NP-complete. Theoretical Compuer Science, 357, 186–201.

    Article  Google Scholar 

  73. Priestley, H.A. (1970). Representation of distributive lattices by means of ordered Stone spaces. Bulletin of the London Mathematical Society, 2, 186–190.

    Article  Google Scholar 

  74. Restall, G. (1995). Display logic and gaggle theory. Reports on Mathematical Logic, 29, 133–146.

    Google Scholar 

  75. Restall, G. (2000). An introduction to substructural logics. London: Routledge.

    Book  Google Scholar 

  76. Robles, G., & Méndez, J.M. (2006). Converse Ackermann property and constructive negation defined with a negation connective. Logic and Logical Philosophy, 15, 113–130.

    Article  Google Scholar 

  77. Rosenthal, K.I. (1990). Quantales and their applications. Number 234 in Pitman research notes in mathematics. Essex, UK and New York: Longman and J. Wiley.

    Google Scholar 

  78. Routley, R., & Meyer, R.K. (1972). The semantics of entailment – II. Journal of Philosophical Logic, 1, 53–73.

    Article  Google Scholar 

  79. Routley, R., & Meyer, R.K. (1972). The semantics of entailment – III. Journal of Philosophical Logic, 1, 192–208.

    Article  Google Scholar 

  80. Routley, R., & Meyer, R.K. (1973). The semantics of entailment. In H. Leblanc (Ed.), Truth, Syntax and Modality, Proceedings of the Temple University Conference on Alternative Semantics (pp. 194–243). Amsterdam.

  81. Routley, R., Meyer, R.K., Plumwood, V., & Brady, R.T. (1982). Relevant logics and their rivals I. Atascadero: Ridgeview Publishing Company.

    Google Scholar 

  82. Seki, T. (2004). General frames for relevant modal logics. Notre Dame Journal of Formal Logic, 44, 93–109.

    Article  Google Scholar 

  83. Stone, M.H. (1936). The theory of representations for Boolean algebras. Transactions of the American Mathematical Society, 40, 37–111.

    Google Scholar 

  84. Stone, M.H. (1937). Topological representations of distributive lattices and Brouwerian logics. Časopis pro pěstování matematiky a fysiky, Čast matematická, 67, 1–25.

    Google Scholar 

  85. Szabo, M.E. (1969). The Collected Papers of Gerhard Gentzen. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam.

  86. Thistlewaite, P.B., McRobbie, M.A., & & Meyer, R.K. (1988). Automated theorem proving in non-classical logics. London: Pitman.

    Google Scholar 

  87. Troelstra, A.S. (1992). Lectures on linear logic, CSLI lecture notes (Vol. 29). Stanford: CSLI Publications.

    Google Scholar 

  88. Urquhart, A. (1978). A topological representation theorem for lattices. Algebra Universalis, 8, 45–58.

    Article  Google Scholar 

  89. Urquhart, A (1984). The undecidability of entailment and relevant implication. Journal of Symbolic Logic, 49, 1059–1073.

    Article  Google Scholar 

  90. Urquhart, A. (1990). The complexity of decision procedures in relevance logic. In J.M. Dunn & A. Gupta (Eds.), Truth or consequences (pp. 61–76). Amsterdam: Kluwer.

  91. Urquhart, A. (1999). The complexity of decision procedures in relevance logic II. Journal of Symbolic Logic, 64, 1774–1802.

    Article  Google Scholar 

  92. Urquhart, A (2007). Four variables suffice. Australasian Journal of Logic, 5, 66–73.

    Google Scholar 

Download references

Acknowledgments

I am grateful to J. Michael Dunn for helpful comments on this paper. Also, I would like to thank A. Tedder for some English suggestions.

Author information

Authors and Affiliations

  1. Department of Philosophy, University of Alberta, Edmonton, AB, T6G 2E7, Canada

    Katalin Bimbó

Authors
  1. Katalin Bimbó
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Katalin Bimbó.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bimbó, K. Current Trends in Substructural Logics. J Philos Logic 44, 609–624 (2015). https://doi.org/10.1007/s10992-015-9346-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10992-015-9346-x

Keywords