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Intuitionistic Logic

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Handbook of Philosophical Logic

Part of the book series: Handbook of Philosophical Logic ((HALO,volume 5))

Abstract

Among these logics that deal with the familiar connectives and quantifiers two stand out as having a solid philosophical-mathematical justification. On the one hand there is a classical logic with its ontological basis and on the other hand intuitionistic logic with its epistemic motivation. The case for other logics is considerably weaker; although one may consider intermediate logics with more or less plausible principles from certain viewpoints none of them is accompanied by a comparably compelling philosophy. For this reason we have mostly paid attention to pure intuitionistic theories.

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Bibliography

  1. P. Aczel. Saturated intuitionistic theories. In [Schmidt, Schlitte and Thiele, 1968, pp. 1–11].

    Google Scholar 

  2. S. Artemov. Explicit provability: the intended semantics for intuitionistic and modal logic. Bull. Symb. Logic, 7, 1–36, 2001.

    Article  Google Scholar 

  3. M. van Atten and D. van Dalen. Arguments for Brouwer’s continutity principle. Forthcoming.

    Google Scholar 

  4. Avigad and S. Feferman. Gödel’s functional (‘Dialectics’) interpretation. In Handbook of Proof Theory, S. R. Buss, ed. pp. 337–406. Elsevier, Amsterdam, 1998.

    Chapter  Google Scholar 

  5. H.Y. Barendregt. The Lambda Calculus. Its Syntax and Semantics. North-Holland, Amsterdam, 1984, 2nd reprint edition in paperback, 1997.

    Google Scholar 

  6. M. Beeson. A theory of constructions and proofs. Preprint No 134. Dept of Maths, Utrecht University, 1979.

    Google Scholar 

  7. M. Beeson. Foundations of Constructive Mathematics. Metarnathematical Studies. Springer Verlag, Berlin, 1985.

    Book  Google Scholar 

  8. E. Bishop and D. Bridges. Constructive Analysis, Springer, Berlin, 1985.

    Book  Google Scholar 

  9. L.E.J. Brouwer. Over de Grondslagen der Wiskunde. Thesis, Amsterdam. Translation ‘On the foundations of mathematics’, in [Brouwer, 1975, pp. 11–101]. New edition in [Brouwer, 1981].

    Google Scholar 

  10. L.E.J. Brouwer. De onbetrouwbaarheid der logische principes. Tijdschrift voor wijsbegeerte, 2, 152–158. Translation ‘The unreliability of the logical principles’ in [1975, pp. 107–111]. Also in [1981].

    Google Scholar 

  11. L.E.J. Brouwer. Begründung der Mengenlehre unabhängig vom logischen Satz vom augsgeschlossenen Dritten. I. Koninklijke Nederlandse Akademie van Wetenschappen Verhandelingen le Sectie 12, no 5, 43 p. Also in [Brouwer, 1975, pp. 150–190].

    Google Scholar 

  12. L.E.J. Brouwer. Collected Works, I. A. Heyting, ed. North Holland, Amsterdam, 1975.

    Google Scholar 

  13. L.E.J. Brouwer. Over de Grondslagen der Wiskunde,Aangevuld met Ongepubliceerde Fragmenten, Correspondentie met D.J. Korteweg. Recensies door G. Mannoury, etc. D. van Dalen, ed. Mathematisch Centrum, varia 1, Amsterdam, 1981.

    Google Scholar 

  14. L.E.J. Brouwer. Brouwer’s Cambridge Lectures on Intuitionism. D. van Dalen, ed. Cambridge University Press, Cambridge, 1981.

    Google Scholar 

  15. L.E.J. Brouwer. Intuitionismus, D. van Dalen, ed. BI-Wissenschaftsverlag, Mannheim, 1992.

    Google Scholar 

  16. J.P. Burgess. The completeness of intuitionistic propositional calculus for its intended interpretation. Notre Dame J. Formal Logic, 22, 17–28, 1981.

    Article  Google Scholar 

  17. C.C. Chang and H.J. Keisler. Model Theory. North-Holland, Amsterdam, 1973.

    Google Scholar 

  18. H.B. Curry and R. Feys. Combinatory Logic I. North-Holland, Amsterdam, 1958.

    Google Scholar 

  19. D. van Dalen. Lectures on intuitionism. In [Mathias and Rogers, 1973, pp. 1–94].

    Google Scholar 

  20. D. van Dalen. A model for HAS. A topological interpretation of the theory of species of natural numbers. Fund Math, 82, 167–174, 1974.

    Google Scholar 

  21. D. van Dalen. The use of Kripke’s Schema as a reduction principle. Journal of Symbolic Logic, 42, 238–240, 1977.

    Article  Google Scholar 

  22. D. van Dalen. An interpretation of intuitonistic analysis. Ann. Math. Logic, 13, 1–43, 1978.

    Article  Google Scholar 

  23. D. van Dalen. How to glue analysis models. Journal of Symbolic Logic, 49, 1339–1349, 1984.

    Article  Google Scholar 

  24. D. van Dalen. Glueing of analysis models in an intuitionistic setting. Studia Logica, 45, 181–186, 1986. To appear.

    Article  Google Scholar 

  25. D. van Dalen. The continuum and first-forder logic. JSL, 57, 1417–1424, 1993.

    Google Scholar 

  26. D. van Dalen. Hermann Weyl’s intuitionistic mathematics. Bull. Symb. Logic, 4, 145–169, 1995.

    Article  Google Scholar 

  27. D. van Dalen. Logic and Structure, 3rd edn. Springer Verlag, Berlin, 1997.

    Google Scholar 

  28. D. van Dalen. From Brouwerian counterexamples to the creating subject. Studia Logica, 62, 305–314, 1999.

    Article  Google Scholar 

  29. D. van Dalen. Mystic, Geometer and Intuitionist. The Life of L.E.J. Brouwer. Volume 1: The Dawning Revolution, Oxford University Press, 1999.

    Google Scholar 

  30. D. van Dalen. Development of Brouwer’s Intuitionism. In Proof Theory. History and Philosophical Significance, V. F. Hendricks, S. A Pedersen and K. F. Jorgensen, eds. pp. 117–152. Kluwer, Dordrecht, 2000.

    Google Scholar 

  31. D. van Dalen and R. Statman. Equality in the presence of apartness. In [Hintikka, Niiniluoto and Saarinen, 1979, pp. 95–118].

    Google Scholar 

  32. M. Davis, ed. The Undecidable. Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions. Raven Press, New York, 1965.

    Google Scholar 

  33. O. Demuth and A. Kucera. Remarks on constructive mathematical analysis. In Logic Colloquium ‘78, M. Boffa, D. van Dalen and K. McAloon, eds. pp. 81–130. North Holland, Amsterdam, 1979.

    Google Scholar 

  34. M. Dummett. The philosophical basis of intuitionistic logic. In Logic Colloquium ‘73, H. E. Rose and J. C. Shepherdson, eds. pp. 5–40, North-Holland, Amsterdam, 1973. Also in M. Dummett, Truth and Other Enigmas, pp. 215–247, Duckworth, London, 1978.

    Google Scholar 

  35. Dummett, 1977] M. Dummett. Elements of Intuitionism. Oxford University Press, Oxford, 1977.

    Google Scholar 

  36. V.H. Dyson and G. Kreisel. Analysis of Beth’s Semantic Construction of Intuitionistic Logic. Technical Report no. 3. Appl. math. and statistics lab. Stanford University, 65 pp. 1961.

    Google Scholar 

  37. S. Feferman. Constructive theories of functions and classes. In Logic Colloquium ‘78, M. Boffa, D. van Dalen and K. McAloon, eds. pp. 159–224. North Holland, Amsterdam, 1979.

    Google Scholar 

  38. J.E. Fenstad, ed. Proceedings of the Second Scandinavian Logic Symposium, North- Holland, Amsterdam, 1971.

    Google Scholar 

  39. K. Fine. An intermediate logic without the finite model property. Unpublished, 1970.

    Google Scholar 

  40. M.C. Fitting. Intuitionistic Logic, Model Theory and Forcing, North-Holland, Amsterdam, 1969.

    Google Scholar 

  41. M. Fourman. Notions of choice sequence. In [Troelstra and van Dalen, 1982, pp. 91–106].

    Google Scholar 

  42. M.P. Fourman and D. S. Scott. Sheaves and logic. In [Four-man, Mulvey and Scott, 1979, pp. 302–401].

    Google Scholar 

  43. M.P. Fourman, C.J. Mulvey and D.S. Scott. Applications of Sheaves. Proceedings Durham 1977. Lecture Notes No 753, Springer Verlag, Berlin, 1979.

    Google Scholar 

  44. A. Fraenkel, Y. Bar-Hillel, A. Levy and D. van Dalen. Foundations of Set Theory. North-Holland, Amsterdam, 1973.

    Google Scholar 

  45. H. Friedman. The disjunction property implies the numerical existence property. Proc Nat Acad. Sci., 72, 2977–2878, 1975.

    Google Scholar 

  46. H. Friedman. The intuitionistic completeness of intuitionistic logic under Tarskian semantics. Abstract, SUNY at Buffalo, 1977.

    Google Scholar 

  47. H. Friedman. New and old results on completeness of HPC. Abstract, SUNY at Buffalo, 1977.

    Google Scholar 

  48. H. Friedman. Classically and intuitionistically provably recursive functions. In Higher Set Theory, pp. 21–27, Springer Verlag, Berlin, 1977.

    Google Scholar 

  49. D.M. Gabbay. Decidability of the Kreisel-Putnam system. Journal of Symbolic Logic, 35, 431–437, 1970.

    Article  Google Scholar 

  50. D.M. Gabbay. Semantic proof of the Craig Interpolation theorem for intuitionistic logic and extensions I, II. In Logic Colloquium 69, (eds R.O. Gandy and C.M.E. Yates), pp. 391–410. North-Holland, Amsterdam, 1971.

    Chapter  Google Scholar 

  51. D.M. Gabbay. On some new intuitionistic propositional connectives, Studio Logica, 36, 127–139, 1977.

    Article  Google Scholar 

  52. D.M. Gabbay. Semantical Investigations an Heyting’s Intuitionistic Logic. D. Reidel, Dordrecht, 1981.

    Google Scholar 

  53. D.M. Gabbay and D.H. de Jongh. Sequences of decidable finitely axiomatisable intermediate logics with the disjunction property. Journal of Symbolic Logic, 39, 67–79, 1974.

    Article  Google Scholar 

  54. J. Gallier. On the correspondence between proofs and A-terms. In The Curry-Howard Isomorphism, (ed. Ph. de Groote), pp. 55–138. Cahiers de centre de Logique, 8. Academia Louvain-la-Neuve, 1995.

    Google Scholar 

  55. G. Gentzen. Über das Verhältnis zwischen intuitionistischer und klassischer Arithmetik. English translation in [Szabo, 1969, pp. 53–67].

    Google Scholar 

  56. G. Gentzen. Untersuchen über das logische Schliessen I, II. Math. Zeitschrift, 39, 176–210, 405–409, 1935.

    Article  Google Scholar 

  57. J.Y. Girard. Une extension de l’interpretation de Gödel à l’analyse et son application à l’élimination des coupures dans l’analyse et lat théorie des types. In [Fenstad, 1971, pp 63–92].

    Google Scholar 

  58. J.Y. Girard, P. Taylor and Y. Lafont. Proofs and Types. Cambridge University Press, Cambridge, 1989.

    Google Scholar 

  59. V. Glivenko. Sur quelques points de la logique de M. Brouwer, Academie Royale de Belgique, Bull. de la class des science, (5), VI. 15, pp. 183–188, 1929.

    Google Scholar 

  60. C.A. Goad. Monadic infinitary propositional logic: a special operator. Repts Math. Logic, 10, 43–50, 1978.

    Google Scholar 

  61. K. Gödel. Zum intuitionistischen Aussagenkallkiil. Akademie der Wissenschaften in Wien. Math. naturwiss. Klasse. Anzeiger, 69, 65–66, 1932. Also in Ergebnisse eines Math. Koll., 4, 42.

    Google Scholar 

  62. K. Gödel. Zur intuitionistischen Arithmetik und Zahlentheorie. Ergebnisse eines mathematischen Kolloquiums, 4, 34–38, 1933.

    Google Scholar 

  63. Cf. Journal of Symbolic Logic, 31, 484–494, 1966.

    Google Scholar 

  64. K. Gödel. Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes. Dialectica, 12, 280–287. English translation in Journal of Philosophical Logic, 9, 133–142, 1980.

    Google Scholar 

  65. R. Goldblatt. Topoi. The Categorical analysis of Logic, North Holland, Amsterdam, 1979. Revised edition, 1984.

    Google Scholar 

  66. S. Görnemann, A logic stronger than intuitionism, Journal of Symbolic Logic, 36, 249–261, 1971.

    Article  Google Scholar 

  67. R. Grayson. Concepts of general topology in constructive mathematics and in sheaves. Ann. Math. Logic, 20, 1–41, 1981.

    Article  Google Scholar 

  68. R. Grayson. Heyting-valued semantics. In Logic Colloquium 1982, G. Lolli, G. Longo and A. Marcja, eds. pp. 181–208. North Holland, Amsterdam, 1984.

    Chapter  Google Scholar 

  69. A. Grzegorczyk. A philosophically plausible interpretation of intuitionistic logic, Indagationes Mathematicae, 26, 569–601, 1964.

    Google Scholar 

  70. R. Harrop. On the existence of finite models and decision procedures for propositional calculi. Proc. Carob. Phil. Soc, 54, 1–13, 1958.

    Article  Google Scholar 

  71. W. den Hartog. A Proof Theoretic Study. The Theory of Pseudo-order as a Conservative Extension of the Theory of Apartness. Dept of Math. Rijksuniversiteit Utrecht, Preprint no. 77, 1978.

    Google Scholar 

  72. J. van Heijenoort. From Frege to Gödel, a Source Book in Mathematical Logic 1879- -1931, Harvard University Press, Cambridge, MA, 1967.

    Google Scholar 

  73. A. Heyting. Die formalen Regeln der intuitionistischen Logik. Sitzungsberichte der preussischen Akademie von Wissenschaften, pp. 42–56, 1930. ‘Die formalen Regeln der intuitionistischen Mathematik’, Ibid. pp. 57— 71, 158–169. Also (partly) in Two Decades of Mathematics in the Netherlands, Amsterdam, 1978.

    Google Scholar 

  74. A. Heyting. Mathematische Grundlagenforschung. Intuitionismus Beweistheorie. Springer, Berlin, 1934.

    Google Scholar 

  75. A. Heyting, Intuitionism. An Introduction, North Holland, Amsterdam, 1956.

    Google Scholar 

  76. J. Hintikka, I. Niiniluoto and E. Saarinen, eds. Essays on Mathematical and Philosophical logic, D. Reidel, Dordrecht, 1979.

    Google Scholar 

  77. G. van Hoeven and I. Moerdijk. Sheaf models for choice sequences. Annals of Pure and Applied Logic, 27, 63–107, 1984.

    Article  Google Scholar 

  78. T. Hosoi. On intermediate logics, I, J. Fac. Sci. Univ. of Tokyo, 14, 293–312, 1967.

    Google Scholar 

  79. W. Howard. The formulae-as-types notion of construction. In [Seldin and Hindley, 1980, pp. 479–490].

    Google Scholar 

  80. G.E. Hughes and M.J. Cresswell. An Introduction to Modal Logic. Methuen, London, 1968.

    Google Scholar 

  81. R.C. Hull. Counterexamples in intuitionistic analysis using Kripke’s schema. Z. Math. Logik und Grundlagen der Math, 15, 241–246, 1969.

    Article  Google Scholar 

  82. M. Hyland. The effective topos. In [Troelstra and van Dalen, 1982, pp. 165–216].

    Google Scholar 

  83. M. Hyland, P.T. Johnstone and A.M. Pitts, Tripos theory. Math. Proc. Cambr. Phil. Soc., 88, 205–232, 1980.

    Article  Google Scholar 

  84. V.A. Jankov. Constructing a sequence of strongly independent superintuitionistic propositional calculi. Soviet Math. Dok, 9, 806–807, 1968.

    Google Scholar 

  85. S. Jaskowski. Recherches sur le système de la logique intuitioniste. Actes du Congrès Intern. de Phil. Scientifique. VI. Phil des mathématiques, Act. Sc. et Ind 393. pp. 58–61. Paris, 1936.

    Google Scholar 

  86. I. Johansson, Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus, Cornpositio Math, 4, 119–136, 1936.

    Google Scholar 

  87. P. Johnstone. Stone Spaces. Cambridge University Press, Cambridge, 1982.

    Google Scholar 

  88. D.H. de Jongh. A class of intuitionistic connectives. In The Kleene Symposium, J. Barwise, H. J. Keisler and K. Kunen, eds. pp. 103–112. North Holland, Amsterdam, 1980.

    Chapter  Google Scholar 

  89. D.H. de Jongh and C. Smorynski. Kripke models and the intuitionistic theory of species. Ann. Math. Logic, 9, 157–186, 1976.

    Article  Google Scholar 

  90. A. Kino, J. Myhill and R.E. Vesley. Intuitionism and Proof Theory. Proceedings of the Summer Conference at Buffalo, New York, 1968. North Holland, Amsterdam, 1970.

    Google Scholar 

  91. S.C. Kleene, Introduction to Meta-mathematics, North Holland, Amsterdam, 1952.

    Google Scholar 

  92. S.C. Kleene. Realisability: A retrospective survey. In [Mathias and Rogers, 1973, pp. 95–112].

    Google Scholar 

  93. S C. Kleene and R.E. Vesley. The Foundations of Intuitionistic Mathematics,especially in Relation to Recursive Functions. North Holland, Amsterdam, 1965.

    Google Scholar 

  94. J.W. Klop. Combinatory Reduction Systems. Thesis, Rijks Universiteit Utrecht. Also MC Tract 127. Math. Centre Amsterdam, 1980.

    Google Scholar 

  95. A.N. Kolmogorov. On the principle of the excluded middle, Russian. Matematiéeski Sbornik, 32, 646–667, 1925. English translation in [Heijenoort, 1967, pp. 414–437].

    Google Scholar 

  96. G. Kreisel. A remark on free choice sequences and the topological completeness proofs. J. Symbolic Logic, 23, 369–388, 1958.

    Article  Google Scholar 

  97. G. Kreisel. Interpretations of analysis by means of constructive function-als of finite type. In Constructivity in Mathematics, A. Heyting, ed. pp. 101–128. North Holland, Amsterdam, 1959.

    Google Scholar 

  98. G. Kreisel, On weak completenes of intuitionistic predicate logic, JSL, 27, 139–158, 1962.

    Google Scholar 

  99. G. Kreisel. Mathematical Logic, In Lectures on Modern Mathematics III, T. L. Saaty, ed. pp. 95–195. Wiley & Sons, New York, 1965.

    Google Scholar 

  100. G. Kreisel. Informal rigour and completeness proofs. In Problems in the Philosophy of Mathematics, I. Lakatos, ed. North Holland, Amsterdam, 1967.

    Google Scholar 

  101. G. Kreisel. Church’s Thesis, a kind of reducibility axiom of constructive mathematics. In [Kino, Myhill and Vesley, 1970, pp. 121–150].

    Google Scholar 

  102. G. Kreisel and H. Putnam. Eine Unableitbarkeitsbeweismethode für den intuitionistischen Aussagenkalkül. Archiv. f. math. Logik, 3, 74–78, 1957.

    Article  Google Scholar 

  103. G. Kreisel and A.S. Troelstra, Formal systems for some branches of intuitonistic analysis, Ann. Math. Logic, 1, 229–387, 1970.

    Article  Google Scholar 

  104. S. Kripke. Semantical analysis of intuitionistic logic I. In Formal Systems and Recursive Functions, J. Crossley and M. Dummett, eds. pp. 92–129. North Holland, Amsterdam, 1965.

    Chapter  Google Scholar 

  105. D. Leivant. Failure of completeness properties of intuitionistic predicate logic for constructive models. Ann. Sc. Univ. Clermont. Sér Math., 13, 93–107, 1976.

    Google Scholar 

  106. D. Leivant, Syntactic translations and provably recursive functions, Journal of Symbolic Logic, 50, 682–688, 1985.

    Article  Google Scholar 

  107. E. Lemmon and D.S. Scott, Intensional logics. Published in 1977 as An Introduction to Modal Logic, K. Segerberg, ed. Blackwell, Oxford, 1966.

    Google Scholar 

  108. V.A. Lifschitz. Problems of decidability for some constructive theories of equalities. In Studies in Constructive Mathematics and Mathematical Logic,A.O. Slisenko, ed. Consultants Bureau, New York, 1969.

    Google Scholar 

  109. S. MacLane and I. Moerdijk. Sheaves in Geometry and Logic, Springer, Berlin, 1992.

    Google Scholar 

  110. D.C. McCarty. Constructive validity is nonarithmetic. JSL, 53, 1036–1041, 1988.

    Google Scholar 

  111. J.C.C. McKinsey, Proof of the independence of the primitive symbols of Heyting’s calculus of propositions, JSL, 4, 155–158, 1939.

    Google Scholar 

  112. C. McLarty, Elementary Categories, Elementary Toposes. Oxford University Press, 1992.

    Google Scholar 

  113. A.A. Markov. Konstruktivnaja logika. Usp. Mat. Nauk, 5, 187–188, 1950.

    Google Scholar 

  114. P. Martin-Löf. Hauptsatz for the theory of species. In [Fenstad, 1971].

    Google Scholar 

  115. P. Martin-Löf. Constructive mathematics and computer programming. In Logic, Methodology and the Philosophy of Science VI, L.J. Cohen, J. Los, H. Pfeiffer and K.P. Podewski, eds. pp. 153–179. North Holland, Amsterdam, 1982.

    Google Scholar 

  116. P. Martin-Löf. Intuitionistic Type Theory. Notes by G. Sambin of a series of lectures given in Padova, June 1982. Bibliopolis, Naples, 1984.

    Google Scholar 

  117. A.R.D. Matias and H. Rogers, Jr. Cambridge Summer School in Mathematical Logic, Springer Lecture Notes, Vol. 337, Berlin, 1973.

    Google Scholar 

  118. L.L. Maximova, Craig interpolation theorem and amalgamable varieties. Soviet Math. Dok., 18, 1550–1553, 1977.

    Google Scholar 

  119. G. Minc. Skolem’s method of elimination of positive quantifiers in sequential calculi. Dokl. Akad. Nauk SSR, 169, 861–864, 1966.

    Google Scholar 

  120. G. Minc. On E-theorems. (In Russian). Investigation in constructive mathematics VI. Zapisky Nauk. Sem. Leningrad Steklov Inst., 40, 110–118, 1974.

    Google Scholar 

  121. J. Moschovakis. Disjunction and existence in formalised intuitionistic analysis. In Sets, Models and Recursion Theory, J. Crossley, ed. pp. 309–31. North Holland, Amsterdam, 1967.

    Google Scholar 

  122. J. Moschovakis. A topological interpretation for second-order intuitionistic arithmetic. Comp. Math., 26, 261–276, 1973.

    Google Scholar 

  123. J. Moschovakis. A disjunctive decomposition theorem for classical theories. In Constructive Mathematics, F. Richman, ed. pp. 250–259. Springer Verlag, Berlin, 1980.

    Google Scholar 

  124. S. Negri and J. von Plato. Structural Proof Theory, Cambridge University Press, Cambridge, 2001.

    Google Scholar 

  125. D. Nelson. Constructible falsity. Journal of Symbolic Logic, 14, 16–26, 1949.

    Article  Google Scholar 

  126. T. Nishimura. On formulas of one variable in intuitionistic propositional calculus. Journal of Symbolic Logic, 25, 327–331, 1966.

    Article  Google Scholar 

  127. H. Ono. Some results on intermediate logics. Publ. RIMS, pp. 117 130, Kyoto University 9, 1972.

    Google Scholar 

  128. H. Ono. A study of intermediate predicate logics. Publ. RIMS, pp. 619–649. Kyoto University 8, 1973

    Google Scholar 

  129. H. Ono. Semantical analysis of predicate logics without the contraction rule. Studia Logica, 44, 187–196, 1985.

    Article  Google Scholar 

  130. H. Ono and Y. Komori. Logics without the contraction rule. Journal of Symbolic Logic, 50, 169–201, 1985.

    Article  Google Scholar 

  131. C.J. Posy. Varieties of indeterminacy in the theory of general choice sequences. J. Phil. Logic, 5, 91–132, 1976.

    Article  Google Scholar 

  132. C.J. Posy. The theory of empirical sequences. J. Phil. Logic, 6, 47–81, 1977.

    Article  Google Scholar 

  133. C.J. Posy. On Brouwer’s definition of unextendable order. History and Phil. Logic, 1, 139–149, 1980.

    Article  Google Scholar 

  134. G. Pottinger. A new way of normalising intuitionist propositional logic. Studia Logica, 35, 387–408, 1976.

    Article  Google Scholar 

  135. D. Prawitz. Natural Deduction. A Proof Theoretical Study. Alqvist & Wiksell, Stockholm, 1965.

    Google Scholar 

  136. D. Prawitz. Some results for intuitionistic logic with second-order quantification rules. In [Kino, Myhill and Vesley, 1970, pp. 259–270].

    Google Scholar 

  137. D. Prawitz. Ideas and results in proof theory. In [Fenstad, 1971]. [Prawitz, 1977] D. Prawitz. Meanings and proofs: on the conflict between classical and intuitionistic logic. Theoria, 43, 2–40, 1977.

    Article  Google Scholar 

  138. D. Prawitz. Proofs and the meaning and the completeness of the logical constants. In [Hintikka, Niiniluoto and Saarinen, 1979, pp. 25–40].

    Google Scholar 

  139. D. Prawitz and P. Malmnäs. A survey of some connections between classical, intuitionistic and minimal logic. In [Schmidt, Schütte and Thiele, 1968, pp. 215–229].

    Google Scholar 

  140. H. Rasiowa, An algebraic Approach to Non-Classical Logics, North Holland, Amsterdam. 1974.

    Google Scholar 

  141. H. Rasiowa and R. Sikorski. The Mathematics of Meta-mathematics, Panstwowe Wydawnictwo Naukowe Warszawa, 1963.

    Google Scholar 

  142. C. Rauszer. An algebraic and Kripke-style approach to a certain extension of intuitionistic logic. Dissertationes Mathernaticae, CLXVII, Warszawa, 67 pp. 1980.

    Google Scholar 

  143. W. Rautenberg. Klasische und nichtklassische Aussangenlogik, Vieweg & Sohn, Braunschweig/Wiesbaden, 1979.

    Google Scholar 

  144. G.R. Renardel de Lavalette. The interpolation theorem in fragments of logic. Indag. Math., 43, 71–86, 1981.

    Google Scholar 

  145. H. A. Schmidt, K. Schütte and H.J. Thiele, eds. Contributions to Mathematical Logic. North Holland, Amsterdam, 1968.

    Google Scholar 

  146. P. Schroeder-Heister. A natural extension of natural deduction. Journal of Symbolic Logic, 49, 1284–1300, 1984.

    Article  Google Scholar 

  147. K. Schütte. Der Interpolationssatz der intuitionistischen Predikatenlogik. Math. Ann., 148, 192–200, 1962.

    Article  Google Scholar 

  148. K. Schütte, Vollständige Systeme modaler und intuitionistischer Logic, Ergebnisse der Mathematik und ihrer Grenzgebiete, 42, Springer Verlag, Berlin, 1968.

    Google Scholar 

  149. D.S. Scott. Extending the topological interpretation to intuitionistic analysis II. In [Kino, Myhill and Vesley, 1970, pp. 235–255].

    Google Scholar 

  150. D.S. Scott. Identity and existence in intuitionistic logic. In [Fourman, Mulvey and Scott, 1979, pp. 660–696].

    Google Scholar 

  151. P. Scowcroft. Some purely topological models for intuitionistic analysis. Ann. Pure Appl. Logic, 98, 173–216, 1999.

    Article  Google Scholar 

  152. K. Segerberg, Propositional logics related to Heyting’s and Johanns-son’s. Theoria, 34, 26–61, 1968.

    Article  Google Scholar 

  153. J.P. Seldin and J.R. Hindley. To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, Academic Press, London, 1980.

    Google Scholar 

  154. J.R. Shoenfield. Mathematical Logic, Addison Wesley, Reading, MA, 1967. Re-issued in paperback by A.K. Peters, 2001.

    Google Scholar 

  155. C. Smorynski. Investigation of Intuitionistic Formal Systems by Means of Kripke Models. Diss. Univ. of Illinois, 1973.

    Google Scholar 

  156. C. Smorynski, Elementary intuitionistic theories, Journal of Symbolic Logic, 38, 102–134, 1973.

    Article  Google Scholar 

  157. C. Smorynski, On axiomatising fragments, Journal of Symbolic Logic, 42, 530–544, 1977.

    Article  Google Scholar 

  158. C. Smorynski. The axiomatisation problem for fragments. Ann. Math. Logic, 14, 193--221, 1978.

    Article  Google Scholar 

  159. C. Smorynski. Non-standard models and constructivity. In [Troelstra and van Dalen, 1982, pp. 459–464].

    Google Scholar 

  160. G. Sommaruga. History and Philosophy of Constructive Type theory. Kluwer, Dordrecht, 2000.

    Google Scholar 

  161. M. Stein. Interpretations of Heyting’s arithmetic. An analysis by means of a language with set symbols. Ann. Math. Logic, 19, 1–31, 1980.

    Article  Google Scholar 

  162. M. Stein. A general theorem on existence Aheorems. Zeit. Math. Logik Grundl. Math, 27, 435–452, 1981.

    Article  Google Scholar 

  163. G. Sundholm. Constructions, proofs and the meanings of the logical constants. J. Phil. Logic, 12, 151–172, 1983.

    Article  Google Scholar 

  164. H.C.M. de Swart. Another intuitionistic completeness proof. Journal of Symbolic Logic, 41, 644–662, 1976.

    Article  Google Scholar 

  165. M.E. Szabo, ed. The Collected Papers of Gerhard Gentzen. North Holland, Amsterdam, 1969.

    Google Scholar 

  166. N. Tait. A realisability interpretation of the theory of species. In Logic Colloquium, R. Parikh, ed. pp. 240–251. Springer Verlag, Berlin, 1975.

    Chapter  Google Scholar 

  167. M. Takahashi. Cut-elimination theorem and Brouwerian-valued models for intuitionistic type theory. Comment. Math. Univ. St Pauli, 19, 55–72, 1970.

    Google Scholar 

  168. R. H. Thomason. A semantical study of onstructible falsity. Z. Math. Logik und Grundlagen der Mathematik, 15, 247–257, 1969.

    Article  Google Scholar 

  169. A.S. Troelstra. Principles of Intuitionism. Springer Verlag, Berlin, 1969.

    Google Scholar 

  170. A.S. Troelstra, Metamathernatical Investigation of Intuitionistic Arithmetic and Analysis. Springer Verlag, 1973.

    Google Scholar 

  171. A.S. Troelstra. Notes on intuitionistic second-order arithmetic. In [Mathias and Rogers, 1973, pp. 171–205].

    Google Scholar 

  172. A.S. Troelstra, Choice Sequences. A Chapter of lntuitionistic Mathematics. Oxford University Press, Oxford, 1977.

    Google Scholar 

  173. A.S. Troelstra. Commentary on Heyting [1930]. In Two Decades of Mathematics in the Netherlands, Part 5, Math. Centre, Amsterdam, 1978.

    Google Scholar 

  174. A.S. Troelstra. A Supplement to Choice Sequences. Report 79–04 of the Math Institute University of Amsterdam. 22 pp. 1979.

    Google Scholar 

  175. A.S. Troelstra. The interplay between logic and mathematics: intuitionism. In Modern Logic-A Survey, E. Agazzai, ed. pp. 197–221. D. Reidel, Dordrecht, 1980.

    Google Scholar 

  176. A.S. Troelstra. On a second-order propositional operator in intuitionistic logic. Studia Logics, 40, 113–140, 1981.

    Article  Google Scholar 

  177. A.S. Troelstra. Arend Heyting and his contribution to intuitionism. Nieuw Archief voor Wiskunde, 24, pp. 1–23, 1981.

    Google Scholar 

  178. A.S. Troelstra. Analysing choice sequences. Journal of Philosophical Logic, 12, 197–260, 1983.

    Article  Google Scholar 

  179. A.S. Troelstra. Realizability. In Handbook of Proof Theory, (ed. S.R. Buss). pp. 407–474. Elsevier, Amsterdam, 1998.

    Chapter  Google Scholar 

  180. A. Troelstra and D. van Dalen, eds. The L.E.J. Brouwer Centenary Symposium, North Holland, Amsterdam, 1982.

    Google Scholar 

  181. A. Troelstra and D. van Dalen. Constructivism in Mathematics, I, II, Elsevier, Amsterdam, 1988.

    Google Scholar 

  182. A. Troelstra and H. Schwichtenberg. Basic proof Theory. Cambridge University Press, Cambridge, 1996.

    Google Scholar 

  183. W. Veldman. An intuitionistic completeness theorem for intuitionistic predicate logic. J. Symbolic Logic, 41, 159–166, 1976.

    Article  Google Scholar 

  184. W. Veldman and M. Janssen. Some observations on intuitionistically elementary properites of linear orderings. Arch. Math. Logic, 29, 171–187, 1990.

    Article  Google Scholar 

  185. W. Veldman and F. Waaldijk. Some elementary results in intuitionistic model theory. JSL, 61, 745–767, 1996.

    Google Scholar 

  186. A. Visser. On the completeness principle: a study of provability in Heyting’s Arithmetic. Annals Math. Logic, 22, 263–295, 1982.

    Article  Google Scholar 

  187. J. T. Zucker and R. S. Tragesser. The adequacy problem for inferential logic. J. Philosophical Logic, 7, 501–516, 1978.

    Google Scholar 

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Authors and Affiliations

  1. University of Utrecht, The Netherlands

    Dirk van Dalen

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  1. King’s College, London, UK

    Dov M. Gabbay

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

    F. Guenthner

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van Dalen, D. (2002). Intuitionistic Logic. In: Gabbay, D.M., Guenthner, F. (eds) Handbook of Philosophical Logic. Handbook of Philosophical Logic, vol 5. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0458-8_1

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