Semantics and the Liar Paradox

  • Albert Visser
Part of the Synthese Library book series (SYLI, volume 167)

Abstract

The semantical paradoxes are not a scientific subject like Inductive Definitions, Algebraic Geometry or Plasma Physics. At least not yet. On the other hand the paradoxes exert a strong fascination and many a philosopher or logician has spent some thought on them, mostly in relative isolation. The literature on the paradoxes is vast but scattered, repetitive and disconnected. This made it impossible to give a presentation in which all ideas in the literature receive their due.

Keywords

Philosophical Logic Meaningful Sentence Liar Paradox Inductive Definition Liar Sentence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Aczel, P.: 1977, ‘An introduction to inductive definitions’, in J. Barwise (ed.), 1977, pp. 739–782.Google Scholar
  2. Aczel, P.: 1980, ‘Frege structures and the notions of proposition, truth and set’, in J. Barwise et al. (eds.), 1980, pp. 31–59.Google Scholar
  3. Aczel, P. and Feferman, S.: 1980, ‘Consistency of the unrestricted abstraction principle using an intensional equivalence operator’, in J. P. Seldin and J. R. Hindley (eds.) 1980, pp.67–98.Google Scholar
  4. Barwise, J. (ed.): 1977, Handbook of Mathematical Logic, North Holland, 1977.Google Scholar
  5. Barwise, J., Keisler, H. J., and Kunen, K. (eds.): 1910, The Kleene Symposium, North Holland, Amsterdam.Google Scholar
  6. Behmann, H.: 1931, ‘Zu den Widersprüchen der Logik und der Mengenlehre’, Jahresbericht der Deutschen Mathematiker-Vereinigung 40, 37–48.Google Scholar
  7. Belnap, N. D., Jr.: 1982, ‘Gupta’s rule of revision theory of truth’, Journal of Philosophical Logic 11, 103–116.Google Scholar
  8. Blau, U.: 1983, ‘Vom Henker, vom Lügner und von ihrem Ende’, Erkenntnis 19, 27–44.Google Scholar
  9. Blau, U.: 1984, ‘Wahrheit von innen und aussen’, unpublished manuscript, Seminar fUr Philosophie, Logik und Wissenschaftstheorie, Univ. München.Google Scholar
  10. Bochvar, D. A.: 1981, ‘On a three-valued logical calculus and its application to the analysis of the classical extended functional calculus’, English translation in: History and Philosophy of Logic 2, 87–112.Google Scholar
  11. Bunder, M. W.: 1980, ‘The naturalness of illative combinatory logic as a basis for mathematics’, in J. P. Seldin and J. R. Hindley (eds.), 1980, pp. 55–64.Google Scholar
  12. Burge, T.: 1979, ‘Semantical paradox’, Journal of Philosophy 76, 169–198. Reprinted in Martin [1984], pp. 83-118.CrossRefGoogle Scholar
  13. Burge, T.: 1982, ‘The liar paradox: Tangles and chains’, Philosophical Studies 41, 353–366.Google Scholar
  14. Burgess, J. P.: to appear, ‘The truth is never simple’, Department of Philosophy, Princeton University, 1879 Hall, Princeton, New Jersey 08544.Google Scholar
  15. Cantini, A.: 1979, ‘“Tarski extensions” of theories’, preprint, Mathematisches Institut, München.Google Scholar
  16. Cantini, A.: 1980, ‘A note on three-valued logic and Tarski theorem on truth definitions’, Studia Logica 39, 405–414.Google Scholar
  17. Chihara, C. S.: 1973, Ontolgy and the Vicious Circle Principle, Cornell University Press, Ithaca & London.Google Scholar
  18. Chihara, C. S.: 1979, ‘The semantic paradoxes; A diagnostic investigation’, Philosophical Review 88, 590–618.Google Scholar
  19. Chihara, C. S.: 1984, ‘Priest, the Liar, and Gödel’, Journal of Philosophical Logic 13,117–124.Google Scholar
  20. Cowan, D. A.: 1980, Language & Negation, Joseph Publishing Company, San Mateo (Ca.).Google Scholar
  21. Curry, H. B., Feys, R., Craig, W.: 1958, Combinatory Logic, Vol. 1, North Holland, Amsterdam.Google Scholar
  22. Davidson, D.: 1969, ‘On saying that’, in D. Davidson and J. Hintikka, (eds.), Words and Objections, Essays on the Work of W. V. Quine, Reidel, Dordrecht, pp. 158–174.Google Scholar
  23. Dowden, B. H.: 1979, ‘The Liar Paradox and Tarski’s Undefinability theorem', dissertation, Stanford University.Google Scholar
  24. Dowden, B. H.: 1984, ‘Accepting inconsistencies from the paradoxes’, Journal of Philosophical Logic 13, 125–130.Google Scholar
  25. Feferman, S.: 1967, ‘Set-theoretical foundations of category theory’, Reports of the Midwest Category Seminar. III, Lecture Notes in Mathematics, vol. 106, Springer, Berlin, pp. 207–246.Google Scholar
  26. Feferman, S.: 1975a, ‘A language and axioms for explicit mathematics’, Algebra and Logic, Lecture Notes in Mathematics, vol. 450, Springer, Berlin, pp. 87–139.Google Scholar
  27. Feferman, S.: 1975b, ‘Investigative logic for theories of partial functions and relations. I and II’, unpublished notes, Stanford University, Stanford, California.Google Scholar
  28. Feferman, S.: 1975c, ‘Non-extensional type free theories of partial operations and classifications. I’ ⊧ ISILC Proof Theorv Symposion. Kiel. 1974, Lecture Notes in Mathematics. vol. 500, Springer, Berlin, pp. 73–118.Google Scholar
  29. Feferman, S.: 1976, ‘Comparison of some type-free semantic and mathematical theories’, unpublished notes, Stanford University, Stanford, California.Google Scholar
  30. Feferman, S.: 1977, ‘Categorical foundations and foundations of category theory’, in R. Butts and J. Hintikka (eds.), 1977, Logic, Foundations of Mathematics and Computability Theory, D. Reidel, Dordrecht, pp. 149–169.Google Scholar
  31. Feferman, S.: 1979, ‘Constructive theories of functions and classes’, in M. Boffa, D. van Dalen and K. McAloon (eds.), Logic Colloquium’ 78, North Holland, Amsterdam, pp. 159–224.Google Scholar
  32. Feferman, S.: 1984, ‘Toward useful type-free theories. I’, Journal of Symbolic Logic 49, 75–111. Reprinted in Martin [1984], pp. 237-288.CrossRefGoogle Scholar
  33. Finsler, P.: 1975, Aufsätze zur Mengenlehre, Wissenschaftliche Buchgesellschaft, Darmstadt.Google Scholar
  34. Fitch, F. B.: 1948, ‘An extension of basic logic’, Journal of Symbolic Logic 13, 95–106.Google Scholar
  35. Fitch, F. B.: 1963, ‘The system Cd of combinatory logic’, Journal of Symbolic Logic 28, 87–97.Google Scholar
  36. Fitch, F. B.: 1966, ‘A consistent modal set theory’, abstract, Journal of Symholic Logic 31, 701.Google Scholar
  37. Fitch, F. B.: 1974, Elements of Combinatory Logic, Yale U. P., New Haven & London.Google Scholar
  38. Fitch, F. B.: 1980, ‘A consistent combinatory logic with an inverse to equality’, Journal of Symbolic Logic 45, 529–543.Google Scholar
  39. Frege, G.: 1975, Funktion, Begriff, Bedeutung, Vandenhoeck & Ruprecht, Güttingen.Google Scholar
  40. Gilmore, P. C.: 1974, ‘The consistency of partial set theory without extensionality’, In Axiomatic Set Theory, Proc. of Symposia in Pure Mathematics, vol. 13, Part II, AMS, Providence R.I., pp. 147–153.Google Scholar
  41. Gilmore, P. C.: 1980, ‘Combining unrestricted abstraction with universal quantification’, in J. P. Seldin and J. R. Hindley (eds.), 1980, pp. 99–123.Google Scholar
  42. Giidel, K.: 1944, ‘Russell’s mathematical logic’, in P. A. Schilpp (ed.), The Philosophy of Benrand Russell. Tudor, New York, pp. 123–153. Reprinted in P. Benacerraf and H. Putnam, (eds.), 1964, Philosophy of Mathematics, Prentice Hall, Englewood Cliffs, New Jersey, pp. 211-232.Google Scholar
  43. Grover, D.: 1977, ‘Inheritors and paradox’, The Journal of Philosophy 74,590–604.Google Scholar
  44. Gupta, A.: 1982, ‘Truth and paradox’, Journal of Philosophical Logic 11. 1–60. Reprinted in Martin [1984], pp. 175-236.CrossRefGoogle Scholar
  45. Gupta, A. and Martin, R. L.: 1984, ‘A fixed point theorem for the weak Kleene valuation scheme’, Journal of Philosophical Logic 13, 131–135.Google Scholar
  46. Gupta, A.: 1987, ‘The meaning of truth’, in E. LePore, New Directions in Semantics, Academic Press, London, pp. 453–480.Google Scholar
  47. Haack, S.: 1978, Philosophy of Logic, Cambridge University Press, Cambridge.Google Scholar
  48. Hansson, B.: 1978, ‘Paradoxes in a semantic perspective’, in J. Hintikka, I. Niiniluoto, and E. Saarinen, (ed.), Essays on Mathematical and Philosophical Logic, D. Reidel, Dordrecht, pp.371–385.Google Scholar
  49. Herzberger, H. G.: 1970, ‘Paradoxes of grounding in semantics’, The Journal of Philosophy 67, 145–167.Google Scholar
  50. Herzberger, H. G.: 1980, ‘Notes on periodicity’, unpublished manuscript.Google Scholar
  51. Herzberger, H. G.: 1982a, ‘Naive semantics and the Liar paradox’, The Journal of Philosophy 79, 479–497.Google Scholar
  52. Herzberger, H. G.: 1982b, ‘Notes on naive semantics’, Journal of Philosophical Logic 11, 61–102.Google Scholar
  53. Hughes, G. E.: 1982, John Buridan on Self-Reference, Cambridge U.P., Cambridge.Google Scholar
  54. Kaplan, D.: 1978, ‘On the logic of demonstratives’, Journal of Philosophical Logic 8,81–98.Google Scholar
  55. Kindt, W.: 1976, ‘Über Sprachen mit Wahrheitsprädikat’, in C. Habel and S. Kanngiesser (eds.), 1978, Sprachdynamik und Sprachstruktur, Niemeyer, Tübingen.Google Scholar
  56. Kindt, W.: 1978, ‘The introduction of truth predicates into first-order languages’, in F. Guenthner, and S. J. Schmidt, (eds.), Formal Semantics and Pragmatics for Natural Languages, D. Reidel, Dordrecht, pp. 359–371.Google Scholar
  57. Koyré, A.: 1947, Epimenide Ie Menteur, Hermann et Cie, Paris.Google Scholar
  58. Kripke, S.: 1975, “Outline of a theory of truth”, The Journal of Philosophy 72,690–716.Google Scholar
  59. Lopez-Escobar, E. G. K.: 1979, ‘A formal logic for the study of paradoxes’, Technical Report TR 79-11, University of Maryland, Department of Mathematics, College Park, Maryland.Google Scholar
  60. Manna, Z., and Shamir, A.: 1976, ‘The theoretical aspects of the optimal fixed-point’, Siam Journal of Computing 5, 414–426.Google Scholar
  61. Manna, Z. and Shamir, A.: 1978, ‘The convergence offunctions to fixed-points ofrecursive definitions’, Theoretical Computer Science 6, 109–141.Google Scholar
  62. Martin, R. L. (ed.): 1970, The Paradox of the Liar, Yale University Press, New Haven (Second edition: 1978, Ridgeview Pub., Atascadero (Ca.)).Google Scholar
  63. Martin, R. L. and Woodruff, P. W.: 1975, ‘On representing “true-in-L” in L’, Philosophia 5, 217–221. Reprinted in Martin [1984], pp. 47-52.Google Scholar
  64. Martin, R. L. (ed.): 1984. Recent Essays on Truth and the Liar paradox, Oxford University Press, Oxford.Google Scholar
  65. Martin, R. M.: 1979, ‘The truth about Kripke’s “Truth”’, in R. M. Martin: Pragmatics, Truth and Language, D. Reidel, Dordrecht, pp. 173–180. ai]McGee, V.: to appear, ‘Technical notes on three systems of naive semantics’, Group in Logic and the Methodology of Science U. C. Berkeley, CA 94720.Google Scholar
  66. Moschovakis, Y. N.: 1974, Elementary Induction on Abstract Structures, North Holland, Amsterdam.Google Scholar
  67. Moschovakis, Y. N.: 1977, ‘On the basic notions in the theory of induction’, in R. E. Butts and J. Hintikka (eds.): Logic, Foundations of Mathematics and Computability Theory, D. Reidel, Dordrecht, pp. 207–236.Google Scholar
  68. Myhill, J.: 1984, ‘Paradoxes’, Synthese 60, 129–143.Google Scholar
  69. Nepeĩvoda, N. N.: 1973, ‘A new notion ofpredicative truth and definability’, Mathematičeskie Zumetki 13, 735–745; English translation: Mathematical Notes of the Academy of Sciences of the USSR 13, pp. 439-445.Google Scholar
  70. Parsons, C.: 1974, ‘The liar paradox’, Journal of Philosophical Logic 3. 381–42. Reprinted in Martin [1984], pp. 9-46.Google Scholar
  71. Parsons, C.: 1982, ‘Postscript to “The liar paradox”’, to appear in Mathematics in Philosophy, Cornell University Press, Ithaca, N.Y.Google Scholar
  72. Parsons, T.: 1984, ‘Assertion, denial, and the Liar paradox’, Journal of Philosophical Logic 13. 137–152.Google Scholar
  73. Pollock, J. L.: 1977, ‘The liar strikes back’, The Journal of Philosophy 74,604–606.Google Scholar
  74. Popper, Sir K. R.: 1954, ‘Self reference and meaning in ordinary language’, Mind 63, 162–169. Reprinted in Sir K. R. Popper: 1972, Conjectures and Refutations, Routledge and Kegan Paul, London, pp. 304-311.Google Scholar
  75. Prawitz, D.: 1965. Natural Deduction: A Proof theoretical Study, Almqvist and Wiksell. Stockholm.Google Scholar
  76. Priest, G.: 1979, ‘The logic of paradox’, Journal of Philosophical Logic 8,219–241.Google Scholar
  77. Priest, G.: 1984, ‘Logic of paradox revisited’, Journal of Philosophical Logic 13. 153–179.Google Scholar
  78. Quine, W. V. O.: 1962, ‘Paradox’, Scientific American of April 1962, 84–96.Google Scholar
  79. Ramsey, F. P.: 1925. ‘The foundations of mathematics’, Proceedings of the London Mathematical Society. Nr. 2, Vol. 25, pp. 338–384, also in F. P. Ramsey: 1978. Foundations, Routledge & Kegan Paul, London, pp. 152-212.Google Scholar
  80. Scott, D. S.: 1960, ‘Combinators and classes’, in λ-Calculus and Computer Sciencc Theory, Lecture Notes in Computer Science, vol. 37, Springer Verlag, Berlin, pp. 1–26.Google Scholar
  81. Seldin, J. P. and Hindley, J. R. (eds.): 1980, ‘To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism. Academic Press, New York.Google Scholar
  82. Skyrms, B.: 1970, ‘Return of the Liar, three valued logic and the concept of truth’. American Philosophical Quarterly 7, 153–161.Google Scholar
  83. Skyrms, B: 1984, ‘Intensional aspects of semantical self-reference’, in Martin [1984]. pp. 119–132.Google Scholar
  84. Smoryński, C.: 1977, ‘The incompleteness theorems’, in Barwise [1977]. pp. 821–865.Google Scholar
  85. Smullyan, R. M.: 1957, ‘Languages in which self-reference is possible’, Journal of Symbolic Logic 22, 55–67. Reprinted in J. Hintikka (ed.), 1969. The Philosophy of Mathematics. Oxford University Press, Oxford, pp. 64-77.Google Scholar
  86. Smullyan, R. M.: 1981, What Is the Name of This Book?, Penguin Books, Harmondsworth.Google Scholar
  87. Smullyan, R. M.: 1984, ‘Chameleonic Languages’, Synthese 60,201–224.Google Scholar
  88. Tarski, A.: 1944. ‘The semantic conception of truth’. Philosophy and Phenomenological Research 4, 341–375.Google Scholar
  89. Tarski, A.: 1956, ‘The concept of truth in formalized languages’. in A. Tarski: Logic, Semantics, Metamathematics, Oxford University Press, Oxford. 152–278.Google Scholar
  90. Tarski, A.: 1969. ‘Truth and proof’, Scientific American of June 1969, pp. 63–70. 75-77.Google Scholar
  91. Thomason, R. H.: 1969. ‘A scmantical study or constructive falsity’. Zeitschrift für mathe-matische Logik und Grundlagen der Mathematik. 15. 247–257.CrossRefGoogle Scholar
  92. Thomason, R. H.: 1975. ‘Necessity, quotation and truth: An indexical theory’. Philosophia 5,219–241.CrossRefGoogle Scholar
  93. Van Fraassen, B. C.: 1968, ‘Presupposition, implication and self-reference’. Journal of Philosophy 65, 135–152.Google Scholar
  94. Van Fraassen, B. C.: 1972, ‘Inference and selfreference’, in D. Davidson and G. Harman (eds.), Semantics of Natural Language, D. Reidel. Dordrecht, pp. 695–708.Google Scholar
  95. Visser, A.: 1984, ‘Four valued semantics and the Liar’, Journal of Philosophical Logic 13, 181–212.CrossRefGoogle Scholar
  96. Woodruff, P. W.: 1984, ‘Paradox, truth and logic, Part 1 ’,Journal of Philosophical Logic 13, 213–232.CrossRefGoogle Scholar
  97. Yablo, S.: 1982, ‘Grounding, dependence and paradox’, Journal of Philosophical Logic 11, 117–137.CrossRefGoogle Scholar
  98. Yablo, S.: 1985, ‘Truth and reflection’, Journal of Philosophical Logic 14, 297–349.CrossRefGoogle Scholar

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© D. Reidel Publishing Company 1989

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  • Albert Visser

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