Skip to main content
Article thumbnail
Location of Repository

Canonicity and Bi-Approximation in Non-Classical Logics

By Tomoyuki Suzuki

Abstract

Non-classical logics, or variants of non-classical logics, have rapidly been developed together with the progress of computer science since the 20th century. Typically, we have found that many variants of non-classical logics are represented as ordered algebraic structures, more precisely as lattice expansions. From this point of view, we can think about the study of ordered algebraic structures, like lattice expansions or more generally poset expansions, as a universal approach to non-classical logics.\ud Towards a general study of non-classical logics, in this dissertation, we discuss canonicity and bi-approximation in non-classical logics, especially in lattice expansions and poset expansions. Canonicity provides us with a connection between logical calculi and space-based semantics, e.g. relational semantics, possible world semantics or topological semantics. Note that these results are traditionally considered over bounded distributive lattice-based logics, because they are based on Stone representation. Today, thanks to the recent generalisation of canonical extensions, we can talk about the canonicity over poset expansions. During our investigation of canonicity over poset expansions, we will find the notion of bi-approximation, and apply it to non-classical logics, especially with resource sensitive logics

Publisher: University of Leicester
Year: 2010
OAI identifier: oai:lra.le.ac.uk:2381/8931

Suggested articles

Citations

  1. (1959). A completeness theorem in modal logic. doi
  2. (1981). A course in universal algebra, doi
  3. (1989). A new proof of Sahlqvist’s theorem on modal definability and completeness. doi
  4. (2007). A relational semantics for distributive substructural logics and the topological characterization of the descriptive frames. CALCO-jnr
  5. (2005). A Sahlqvist theorem for distributive modal logic. doi
  6. (2003). A Sahlqvist theorem for relevant modal logics. Studia Logica, 73:383– 411, doi
  7. (1978). A topological representation theory for lattices. Algebra Universalis, doi
  8. (2000). Algebraic Foundations of Manyvalued Reasoning. Trends in Logic. doi
  9. (1985). Algebraic laws for nondeterminism and concurrency. doi
  10. (2006). Algebras and coalgebras. doi
  11. (1969). An axiomatic basis for computer programming. doi
  12. (2000). An introduction to cylindric set algebras. doi
  13. (1977). An Introduction to Modal Logic. doi
  14. (2008). An Introduction to Non-Classical Logic. doi
  15. (2000). An Introduction to Substructural Logics. doi
  16. (2010). Bi-approximation semantics for substructural logic.
  17. (1951). Boolean algebras with operators I. doi
  18. (1952). Boolean algebras with operators II. doi
  19. (1994). Bounded distributive lattices with operators.
  20. (2001). Bounded lattice expansions. doi
  21. (1998). Canonical completions of lattices and ortholattices.
  22. (2005). Canonical extensions and relational completeness of some substructural logics. doi
  23. (2007). Canonical extensions of double quasioperator algebras: an algebraic perspective on duality for certain algebras with binary operators. doi
  24. (2010). Canonicity results of substructural and lattice-based logics. The Review of Symbolic Logic, doi
  25. (1999). Categorical Logic and Type Theory, doi
  26. (1998). Categories for the Working Mathematician. doi
  27. (1975). Completeness and correspondence in the first and second order semantics for modal logic. doi
  28. (1997). Constructive canonicity in non-classical logics. doi
  29. (1991). Domain theory in logical form. doi
  30. (1996). Duality for algebras of relevant logics. doi
  31. (1997). Duality for lattice-ordered algebras and normal algebraizable logics.
  32. (2000). Dynamic Logic. Foundations of Computing. doi
  33. (2006). Elementary canonical formulae: extending Sahlqvist’s theorem. doi
  34. (1998). First-Order Modal Logic, doi
  35. (2007). First-order modal logic. doi
  36. (1963). Functorial Semantics of Algebraic Theories and Some Algebraic Problems in the Context of Functorial Semantics of Algebraic Theories. doi
  37. (2003). General frames for relevant modal logics. doi
  38. (2006). Generalized Kripke frames. doi
  39. (1994). Handbook of Categorical Algebra 1, volume 50 of Encyclopedia of Mathematics and its Applications. doi
  40. (1994). Handbook of Categorical Algebra 3, volume 52 of Encyclopedia of Mathematics and its Applications. doi
  41. (1985). Heyting Algebra I, Duality Theory.
  42. (1986). Introduction to higher order categorical logic, volume 7 of Cambridge studies in advanced mathematics.
  43. (2002). Introduction to Lattices and Order. doi
  44. (2000). Introduction to Process Algebra. doi
  45. (1997). Kleene algebra with tests. doi
  46. (2007). Kripke completeness of some distributive substructural logics.
  47. (1993). Kripke models for linear logic. doi
  48. (1948). Lattice theory, volume XXV
  49. (2006). Lattices of Intermediate and Cylindric Modal Logics.
  50. (1985). Logics without the contraction rule. doi
  51. (2006). MacNeille completions and canonical extensions.
  52. (2006). Mathematical modal logic: a view of its evolution. doi
  53. (1997). Modal logic, volume 35 of Oxford Logic Guides. doi
  54. (2002). Modal logic, volume 53 of Cambridge Tracts in Theoretical Computer Science. doi
  55. (2002). Non-canonicity of MV-algebras.
  56. On canonicity of poset expansions. Algebra Universalis. doi
  57. (1994). On the canonicity of Sahlqvist identities. doi
  58. (2006). On the representation of Kleene algebras with tests. doi
  59. (1972). Ordered topological spaces and the representation of distributive lattices. doi
  60. (1937). Partially ordered sets. doi
  61. (1960). Polarity and duality. doi
  62. (1954). Polyadic boolean algebras. doi
  63. (1956). Predicates, terms, operations, and equality in polyadic boolean algebras. doi
  64. (1986). Relevance logic and entailment. doi
  65. (1982). Relevant logics and their rivals. Part 1. The basic philosophical and semantical theory. doi
  66. (2003). Relevant logics and their rivals. Volume II. A continuation of the work of doi
  67. (2007). Residuated lattices: an algebraic glimpse at substructural logics, doi
  68. Sahlqvist theorem for substructural logic. doi
  69. (1995). Sahlqvist’s theorem for Boolean algebras with operators with an application to cylindric algebras. Studia Logica, doi
  70. (1974). Semantic analysis of orthologic. doi
  71. (1993). Semantics for substructural logics. doi
  72. (2002). Separation logic: A logic for shared mutable data structures. doi
  73. (1989). Sequent-systems and groupoid models. doi
  74. (1992). Sheaves in Geometry and Logic: A First Introduction to Topos Theory. doi
  75. (1975). Some connections between elementary and modal logic. doi
  76. (1986). sometimes” and “not never” revisited: on branching versus linear time temporal logic. doi
  77. (1997). Stone duality for lattices. Algebra Universalis, doi
  78. (1982). Stone spaces, volume 3 of Cambridge studies in advanced mathematics.
  79. (2003). Substructural logics and residuated lattices - an introduction. doi
  80. Suppose R◦(x1, x2, x), i.e. if R(x1, x2, y′) then x ≤ y′ for every y′ ∈ Y . By assumption, we obtain R(x1, x2, y), which derives x doi
  81. (1972). The semantics of entailment. doi
  82. (1977). The temporal logic of programs. doi
  83. (1936). The theory of representations for Boolean algebras. doi
  84. (2006). Topoi: the Categorial Analysis of Logic. doi
  85. (1937). Topological representations of distributive lattices and Brouwerian logics. Casopis Pest. doi
  86. (1988). Topology and duality in modal logic. doi

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.