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Logical Frameworks

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Handbook of Philosophical Logic

Part of the book series: Handbook of Philosophical Logic ((HALO,volume 9))

Abstract

One way to define a logic is to specify a language and a deductive system. For example, the language of first-order logic consists of the syntactic categories of terms and formulae, and its deductive system establishes which formulae are theorems. Typically we have a specific language in mind for a logic, but some flexibility about the kind of deductive system we use; we are able to select from, e.g., a Hilbert calculus, a sequent calculus, or a natural deduction calculus. A logical framework is an abstract characterization of one of these kinds of deductive system that we can use to formalize particular examples. Thus a logical framework for natural deduction should allow us to formalize natural deduction for a wide range of logics from, e.g., propositional logic to intuitionistic type-theories or classical higher-order logic.

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Bibliography

  1. Martín Abadi, Luca Cardelli, Pierre-Louis Curien, and Jean-Jacques Lévy. Explicit substitutions. J. Functional Programming, 1:375–416, 1991.

    Article  Google Scholar 

  2. Peter Aczel. An introduction to inductive definitions. In Jon Barwise, editor, Handbook of Mathematical Logic. North-Holland, Amsterdam, 1977.

    Google Scholar 

  3. Arnon Avron, Furio Honsell, Ian Mason, and Robert Pollack. Using typed lambda calculus to implement formal systems on a machine. J. Auto. Reas., 9:309–352, 1992.

    Article  Google Scholar 

  4. Arnon Avron, Furio Honsell, Marino Miculan, and Cristian Paravano. Encoding modal logics in logical frameworks. Studia Logica, 60(1):161–208, 1998.

    Article  Google Scholar 

  5. Arnon Avron. Gentzenizing Shroeder-Heister’s natural extension of natural deduction. Notre Dame Journal of Formal Logic, 31:127–135, 1990.

    Article  Google Scholar 

  6. Arnon Avron. Simple consequence relations. Inform. and Comput., 92:105–139, 1991.

    Article  Google Scholar 

  7. Arnon Avron. Axiomatic systems, deduction and implication. J. Logic Computat., 2:51–98, 1992.

    Article  Google Scholar 

  8. Henk Barendregt. The Lambda Calculus: Its Syntax and Semantics. North-Holland, Amsterdam, 2nd (revised) edition, 1984.

    Google Scholar 

  9. Henk Barendregt. Introduction to generalized type sytems. J. Functional Programming, 2:125–154, 1991.

    Google Scholar 

  10. Henk Barendregt. Lambda calculi with types. In Samson Abramsky, Dov Gabbay, and Tom S. E. Maibaum, editors, Handbook of Logic in Computer Science, volume 2. Oxford University Press, 1992.

    Google Scholar 

  11. David Basin and Sean Matthews. Structuring metatheory on inductive definitions. Information and Computation, 162(1–2), October/November 2000.

    Google Scholar 

  12. David Basin, Seán Matthews, and Luca Viganò. Labelled propositional modal logics: theory and practice. J. Logic Computat., 7:685–717, 1997.

    Article  Google Scholar 

  13. David Basin, Seán Matthews, and Luca Viganò. A new method for bounding the complexity of modal logics. In Georg Gottlob, Alexander Leitsch, and Daniele Mundici, editors, Proc. Kurt Gödel Colloquium, 1997, pages 89–102. Springer, Berlin, 1997.

    Google Scholar 

  14. David Basin, Seán Matthews, and Luca Viganò. Labelled modal logics: Quantifiers. J. Logic, Language and Information, 7:237–263, 1998.

    Article  Google Scholar 

  15. David Basin, Seán Matthews, and Luca Viganò. Natural deduction for nonclassical logics. Studia Logica, 60(1):119–160, 1998.

    Article  Google Scholar 

  16. George Boolos. The Logic of Provability. Cambridge University Press, 1993.

    Google Scholar 

  17. Robert Boyer and J. Strother Moore. A Computational Logic. Academic Press, New York, 1981.

    Google Scholar 

  18. Robert Bull and Krister Segerberg. Basic modal logic. In Gabbay and Guenthner [1983–89], chapter II.1.

    Google Scholar 

  19. Iliano Cervesato and Frank Pfenning. A linear logical framework. In 11th Ann. Symp. Logic in Comp. Sci. IEEE Computer Society Press, 1996.

    Google Scholar 

  20. Alonzo Church. A formulation of the simple theory of types. J. Symbolic Logic, 5:56–68, 1940.

    Article  Google Scholar 

  21. Martin Davis. Emil Post’s contributions to computer science. In Proc. 4th IEEE Ann. Symp. Logic in Comp. Sci. IEEE Computer Society Press, 1989.

    Google Scholar 

  22. Nicolas G. de Bruijn. Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser theorem. Indagationes Mathematicae, 34:381–392, 1972.

    Google Scholar 

  23. Nicolas G. de Bruijn. A survey of the project Automath. In J. R. Hindley and J. P. Seldin, editors, To H. B. Curry: Essays in Combinatory Logic, Lambda Calculus and Formalism, pages 579–606. Academic Press, New York, 1980.

    Google Scholar 

  24. Joëlle Despeyroux, Frank Pfenning, and Carsten Schürmann. Primitive recursion for higher-order abstract syntax. Technical Report CMU-CS-96–172, Dept. of Computer Science, Carnegie Mellon University, September 1996.

    Google Scholar 

  25. Michael Dummett. The philosophical basis of intuitionistic logic. In Truth and other enigmas, pages 215–247. Duckworth, London, 1978.

    Google Scholar 

  26. Ronald Fagin, Joseph Y. Halpern, and Moshe Y. Vardi. What is an inference rule? J. Symbolic Logic, 57:1018–1045, 1992.

    Article  Google Scholar 

  27. Solomon Feferman. Finitary inductive systems. In Logic Colloquium ’88, pages 191–220. North-Holland, Amsterdam, 1990.

    Google Scholar 

  28. Amy Felty. Specifying and Implementing Theorem Provers in a Higher Order Programming Language. PhD thesis, University of Pennsylvania, 1989.

    Google Scholar 

  29. Amy Felty. Encoding dependent types in an intuitionistic logic. In Huet and Plotkin [1991].

    Google Scholar 

  30. Dov Gabbay and Franz Guenthner, editors. Handbook of Philosophical Logic, vol. I–IV. Reidel, Dordrecht, 1983–89.

    Google Scholar 

  31. Dov Gabbay. Labelled Deductive Systems, vol. 1. Clarendon Press, Oxford, 1996.

    Google Scholar 

  32. Philippa Gardner. Equivalences between logics and their representing type theories. Math. Struct. in Comp. Science, 5:323–349, 1995.

    Google Scholar 

  33. Gerhard Gentzen. Untersuchen über das logische Schließen. Math. Z., 39:179–210, 405–431, 1934.

    Google Scholar 

  34. Kurt Gödel. Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatsh. Math., 38:173–198, 1931.

    Article  Google Scholar 

  35. Joseph A. Goguen and Rod M. Burstall. Institutions: Abstract model theory for specifications and programming. J. Assoc. Comput. Mach., pages 95–146, 1992.

    Google Scholar 

  36. Michael J. Gordon and Tom Melham. Introduction to HOL: A Theorem Proving Environment for Higher Order Logic. Cambridge University Press, Cambridge, 1993.

    Google Scholar 

  37. Michael J. Gordon, Robin Milner, and Christopher P. Wadsworth. Edinburgh LCF: A Mechanized Logic of Computation. Springer, Berlin, 1979.

    Book  Google Scholar 

  38. Ralph E. Griswold. A history of the Snobol programming languages. In Wexelblat [1981], pages 601–660.

    Google Scholar 

  39. Ian Hacking. What is logic? J. Philosophy, 76(6), 1979.

    Google Scholar 

  40. Robert Harper, Furio Honsell, and Gordon Plotkin. A framework for defining logics. J. Assoc. Comput. Mach., 40:143–184, 1993.

    Article  Google Scholar 

  41. J. Roger Hindley and Jonathan P. Seldin. Introduction to Combinators and A-Calculus. Cambridge University Press, Cambridge, 1986.

    Google Scholar 

  42. William Howard. The formulas-as-types notion of construction. In J. P. Seldin and J. R. Hindley, editors, To H. B. Curry: Essays on Combinatory Logic, Lambda-Calculus, and Formalism. Academic Press, New York, 1980.

    Google Scholar 

  43. Gérard Huet and Gordon Plotkin, editors. Logical Frameworks. Cambridge University Press, Cambridge, 1991.

    Google Scholar 

  44. Samin Ishtiaq and David Pym. A relevant analysis of natural deduction. J. Logic Computat., 8:809–838, 1998.

    Article  Google Scholar 

  45. Stephen C. Kleene. Introduction to Metamathematics. North-Holland, Amsterdam, 1952.

    Google Scholar 

  46. Saul A. Kripke. Semantical analysis of modal logic I: normal propositional modal logic. Z. Math. Logik Grundlag. Math, 8:67–96, 1963.

    Google Scholar 

  47. Narciso Martí-Oliet and José Meseguer. Rewriting logic as a logical and semantic framework. In Handbook of Philosophical Logic. second edition, 2002.

    Google Scholar 

  48. Seán Matthews, Alan Smaill, and David Basin. Experience with FS0 as a framework theory. In Gérard Huet and Gordon Plotkin, editors, Logical Environments. Cambridge University Press, Cambridge, 1993.

    Google Scholar 

  49. Seán Matthews. Metatheoretic and Reflexive Reasoning in Mechanical Theorem Proving. PhD thesis, University of Edinburgh, 1992.

    Google Scholar 

  50. Seán Matthews. A theory and its metatheory in FS0. In Dov Gabbay, editor, What is a Logical System? Clarendon Press, Oxford, 1994.

    Google Scholar 

  51. Seán Matthews. Implementing FS0 in Isabelle: adding structure at the metalevel. In Jacques Calmet and Carla Limongelli, editors, Proc. Disco’96. Springer, Berlin, 1996.

    Google Scholar 

  52. Seán Matthews. Extending a logical framework with a modal connective for validity. In Martín Abadi and Takayasu Ito, editors, Proc. TACS’97. Springer, Berlin, 1997.

    Google Scholar 

  53. Seán Matthews. A practical implementation of simple consequence relations using inductive definitions. In William McCune, editor, Proc. CADE-14. Springer, Berlin, 1997.

    Google Scholar 

  54. John McCarthy. History of Lisp. In Wexelblat [1981], pages 173–197.

    Google Scholar 

  55. Raymond McDowell and Dale Miller. A logic for reasoning with higher-order abstract syntax. In Proc. 12th IEEE Ann. Symp. Logic in Comp. Sci., pages 434–416. IEEE Computer Society Press, 1997.

    Google Scholar 

  56. José Meseguer. General logics. In Heinz-Dieter Ebbinghaus, J. Fernandez-Prida, M. Garrido, D. Lascar, and M. Rodriguez Artalejo, editors, Logic Colloquium, ’87, pages 275–329. North-Holland, 1989.

    Google Scholar 

  57. Rob P. Nederpelt, Herman J. Geuvers, and Roel C. de Vrijer, editors. Selected papers on Automath. Elsevier, Amsterdam, 1994.

    Google Scholar 

  58. Hans-Jürgen Ohlbach. Translation methods for non-classical logics: an overview. Bulletin of the IGPL, 1:69–89, 1993.

    Article  Google Scholar 

  59. Sam Owre, John Rushby, Natarajan Shankar, and Friedrich von Henke. Formal verification for fault-tolerant architectures: Prolegomena to the design of PVS. IEEE Trans. Software Eng., 21:107–125, 1995.

    Article  Google Scholar 

  60. Lawrence C. Paulson. Isabelle: A Generic Theorem Prover. Springer, Berlin, 1994.

    Google Scholar 

  61. Frank Pfenning. The practice of logical frameworks. In Helene Kirchner, editor, Proc. CAAP’96. Springer, Berlin, 1996.

    Google Scholar 

  62. Frank Pfenning. Structural cut elimination I. intuitionistic and classical logic. Information and Computation, 157(1–2), March 2000.

    Article  Google Scholar 

  63. Randy Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

    Google Scholar 

  64. Emil Post. Formal reductions of the general combinatorial decision problem. Amer. J. Math., 65:197–214, 1943.

    Article  Google Scholar 

  65. Dag Prawitz. Natural Deduction. Almqvist and Wiksell, Stockholm, 1965.

    Google Scholar 

  66. Dag Prawitz. Ideas and results in proof theory. In J. E. Fensted, editor, Proc. Second Scandinavian Logic Symp., pages 235–307. North-Holland, Amsterdam, 1971.

    Chapter  Google Scholar 

  67. David J. Pym and Lincoln Wallen. Proof-search in the λΠ-calculus. In Huet and Plotkin [1991], pages 309–340.

    Google Scholar 

  68. Peter Schroeder-Heister. Generalised rules for quantifiers and the completeness of the intuitionistic operators &, ⋁, ⊃, ⊥, ∀, Ǝ. In M. M. Richter et al., editors, Computation and proof theory. Springer, Berlin, 1984.

    Google Scholar 

  69. Peter Schroeder-Heister. A natural extension of natural deduction. J. Symbolic Logic, 49:1284–1300, 1984.

    Article  Google Scholar 

  70. Dana Scott. Rules and derived rules. In S. Stenlund, editor, Logical Theory and Semantical Analysis, pages 147–161. Reidel, Dordrecht, 1974.

    Chapter  Google Scholar 

  71. Alex K. Simpson. Kripke semantics for a logical framework. In Proc. Workshop on Types for Proofs and Programs, Båstad, 1992.

    Google Scholar 

  72. Raymond Smullyan. Theory of Formal Systems. Princeton University Press, 1961.

    Google Scholar 

  73. Guy L. Steele Jr. and Richard P. Gabriel. The evolution of Lisp. In Thomas J. Bergin and Richard G. Gibson, editors, History of Programming Languages, pages 233–330. ACM Press, New York, 1996.

    Chapter  Google Scholar 

  74. Göran Sundholm. Systems of deduction. In Gabbay and Guenthner [1983–89], chapter I.2.

    Google Scholar 

  75. Göran Sundholm. Proof theory and meaning. In Gabbay and Guenthner [1983–89], chapter III.8.

    Google Scholar 

  76. Carolyn Talcott. A theory of binding structures, and applications to rewriting. Theoret. Comp. Sci., 112:99–143, 1993.

    Article  Google Scholar 

  77. A. S. Troelstra. Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Springer, Berlin, 1982.

    Google Scholar 

  78. Johan van Benthem. Correspondence theory. In Gabbay and Guenthner [1983–89], chapter II.4.

    Google Scholar 

  79. Richard L. Wexelblat, editor. History of Programming Languages. Academic Press, New York, 1981.

    Google Scholar 

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Author information

Authors and Affiliations

  1. University of Freiburg, Germany

    David Basin

  2. IBM Unternehmensberatung GmbH, Frankfurt, Germany

    Seán Matthews

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  1. David Basin
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Editors and Affiliations

  1. King’ College, London, UK

    D. M. Gabbay

  2. Centrum für Informations- und Sprachverarbeitung, Ludwig-Maximilians-Universität, München, Germany

    F. Guenthner

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Basin, D., Matthews, S. (2002). Logical Frameworks. In: Gabbay, D.M., Guenthner, F. (eds) Handbook of Philosophical Logic. Handbook of Philosophical Logic, vol 9. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0464-9_2

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  • DOI: https://doi.org/10.1007/978-94-017-0464-9_2

  • Publisher Name: Springer, Dordrecht

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