Skip to main content

Proof Theory and Meaning

  • Chapter
Handbook of Philosophical Logic

Part of the book series: Synthese Library ((SYLI,volume 166))

Abstract

The meaning of a sentence determines how the truth of the proposition expressed by the sentence may be proved and hence one would expect proof theory to be influenced by meaning-theoretical considerations. In the present Chapter we consider a proposal that also reverses the above priorities and determines meaning in terms of proof. The proposal originates in the criticism that Michael Dummett has voiced against a realist, truth-theoretical, conception of meaning and has been developed largely by him and Dag Prawitz, whose normalization procedures in technical proof theory constitute the main technical basis of the proposal.

Dedicated to Stig Kanger on the Occasion of his 60th birthday.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Aczel, P.: 1977, ‘The strength of Martin-Löfs type theory with one universe’, in S. Mietinen and J. Väänänen (eds.), Proceedings of the Symposium on Mathematical Logic (Oulo 1974), Report No. 2, Dept. of Philosophy, University of Helsinki, pp. 1–32.

    Google Scholar 

  • Aczel, P.: 1978, ‘The type theoretic interpretation of constructive set theory’, in A. Macintyre et al., (eds.), Logic Colloquium ’77, North-Holland, Amsterdam, pp. 55–66.

    Google Scholar 

  • Aczel, P.: 1980, ‘Frege structures and the notions of proposition, truth and set’, in J. Barwise et al. (eds.), The Kleene Symposium, North-Holland, Amsterdam, pp. 31–59.

    Google Scholar 

  • Aczel, P.: 1982, ‘The type theoretic interpretation of constructive set theory: Choice principles’, in A, S. Troelstra and D. van Dalen, (eds.), The L. E. J. Brouwer Centenary Symposium, North-Holland, Amsterdam, pp. 1–40.

    Chapter  Google Scholar 

  • Baldwin, T.: 1979, Interpretations of quantifiers,Mind 88, 215–240.

    Article  Google Scholar 

  • Beeson, M.: 1982, ‘Recursive models for constructive set theories’, Annals Math. Logic 23, 127–178.

    Article  Google Scholar 

  • Belnap, N. D.: 1962, ‘Tonk, plonk and phnk’, Analysis 22, 130–134.

    Article  Google Scholar 

  • Van Benthem, J. F. A. K. and Van Eijck, J.: 1982, ‘The dynamics of interpretation’, J. Semantics 1, 3–20.

    Article  Google Scholar 

  • Cohen, M. R. and Nagel, E.: 1934, An Introduction to Logic and Scientific Method, Routledge and Kegan Paul, London.

    Google Scholar 

  • Davidson, D.: 1967, ‘Truth and meaning’, Synthese 17, 304–323.

    Article  Google Scholar 

  • Davidson, D.: 1984, Inquiries into Truth and Interpretation, Oxford. U. P.

    Google Scholar 

  • Davies, M. K.: 1981, Meaning, Quantification, Necessity, Routledge and Kegan Paul, London.

    Google Scholar 

  • Diller, J.: 1980, ‘Modified realization and the formulae-as-types notion’, in J. P. Seldin and R. Hindley (eds.), To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, Academic Press, London, pp. 491–502.

    Google Scholar 

  • Diller, J. and Troelstra, A. S.: 1984, ‘Readability and intuitionistic logic’, Synthese 60, 253–282.

    Article  Google Scholar 

  • Dummett, M.: 1968–1969, ’The reality of the past’, Proc. Arist. Soc., 69, 239–258.

    Google Scholar 

  • Dummett, M.: 1973, Frege, Duckworth, London.

    Google Scholar 

  • Dummett, M.: 1975, ‘The philosophical basis of intuitionistic logic’, in H. E. Rose and J. Sheperdson (eds.), Logic Colloquium ’73, North-Holland, Amsterdam, pp. 5–40.

    Google Scholar 

  • Dummett, M.: 1975a, ‘The justification of deduction’, Proc. British Academy LIX, 201–231.

    Google Scholar 

  • Dummett, M.: 1976, ‘What is a theory of meaning? (II)’, (‘WTM2’) in Evans and McDowell [1976], pp. 67–137.

    Google Scholar 

  • Dummett, M.: 1977, Elements of Intuitionism, Oxford U.P.

    Google Scholar 

  • Dummett, M.: 1978, Truth and Other Enigmas(‘TE’), Duckworth, London.

    Google Scholar 

  • Dummett, M.: 1981, ‘Frege and Wittgenstein’, in I. Block (ed.), Perspectives on the Philosophy of Wittgenstein, Blackwell, Oxford, pp. 31–42.

    Google Scholar 

  • Dummett, M.: 1982, ‘Realism’, Synthese 52, 55–112.

    Article  Google Scholar 

  • Evans, G.: 1977, ‘Pronouns, quantification and relative clauses (I)’, Canadian J. Phil., reprinted in Platts [ 1980 ], pp. 255–317.

    Google Scholar 

  • Evans, G. and McDowell, J. (eds.): 1976, Truth and Meaning, Oxford U.P.

    Google Scholar 

  • Foster, J. A.: 1976, ‘Meaning and truth theory’, in Evans and McDowell [ 1976 ], pp. 1–32.

    Google Scholar 

  • Frege, G.: 1893, Grundgesetze der Arithmetik, Jena.

    Google Scholar 

  • Gentzen, G.: 1934–1935, ‘Untersuchungen Über das logische Schliessen’, Math. Zeitschrift 39,176–210, 405–431.

    Google Scholar 

  • Hacking, I.: 1979, ‘What is logic?’, J. Philosophy 76, 285–319.

    Article  Google Scholar 

  • Hallnas, L.: 1983, On Normalization of Proofs in Set Theory, Diss., University of Stock-holm, Preprint No. 1, Dept of Philosophy.

    Google Scholar 

  • Hart, W. D.: 1982, ‘Prior and Belnap’, Theoria XLVII, 127–138.

    Google Scholar 

  • Hendry, H. E.: 1981, ‘Does IPC have binary indigenous Sheffer function?’, Notre Dame J. Formal Logic 22, 183–186.

    Article  Google Scholar 

  • Heyting, A.: 1931, ‘Die intuitionistische Grundlegung der Mathematik’, Erkenntnis 2, 106–115.

    Article  Google Scholar 

  • Heyting, A.: 1956, Intuitionism, North-Holland, Amsterdam.

    Google Scholar 

  • Heyting, A.: 1960, ‘Remarques sur le constructivisme’, Logique et Analyse 3, 177–182.

    Google Scholar 

  • Kleene, S. C.: 1967, Mathematical Logic, John Wiley & Sons, New York.

    Google Scholar 

  • Kreisel, G.: 1962, ‘Foundations of intuitionistic logic’, in E. Nagel et al. (eds.), Logic, Methodology and Philosophy of Science, Stanford University Press, pp. 198–210.

    Google Scholar 

  • Loar, B.: 1976, ‘Two theories of meaning’, in Evans and McDowell [1976], pp. 138–161.

    Google Scholar 

  • McDowell, J.: 1976, ‘Truth conditions, bivalence and verificationism’, in Evans and McDowell [1976], pp. 42–66.

    Google Scholar 

  • McDowell, J.: 1977, J.: 1977, ‘On the sense and reference of a proper name’, Mind 86,159–185. Also in Platts [ 1980 ].

    Google Scholar 

  • McDowell, J.: 1978, J.: 1978, ‘Physicalism and primitive denotation: Field on Tarski’, Erkenntnis 13,131–152. Also in Platts [ 1980 ].

    Google Scholar 

  • McDowell, J.: 1978a, ‘On “The reality of the past”,’ in Hookway, C. and P. Pettit (eds.), Action and Interpretation, Cambridge U.P., pp. 127–144.

    Google Scholar 

  • McDowell, J.: 1981, ‘Anti-realism and the epistemology of understanding’, in H. Parrett and J. Bouveresse (eds.), Meaning and Understanding, de Gruyter, Berlin, pp. 225–248.

    Google Scholar 

  • McGinn, C.: 1979, ‘Ana priori argument for realism’, J. Philosophy 74, 113–133.

    Article  Google Scholar 

  • McGinn, C.: 1980, ‘Truth and use’, in Platts [1980], pp. 19–40.

    Google Scholar 

  • McGinn, C.: 1982, ‘Realist semantics and content ascription’, Synthese 52, 113–134.

    Article  Google Scholar 

  • Martin, E., Jr.: 1982, ‘Referentiality in Frege’s Grundgesetze’, History and Philosophy of Logic 3, pp. 151–164.

    Article  Google Scholar 

  • Martin-Löf, P.: 1975, ‘An intuitionistic theory of types’, in H. E. Rose and J. Sheperdson (eds.), Logic Colloquium ’73, North-Holland, Amsterdam, pp. 73–118.

    Google Scholar 

  • Martin-Löf, P.: 1982, ‘Constructive mathematics and computer programming’, in L. J. Cohen et al., (eds.), Logic, Methodology, and Philosophy of Science VI, North-Holland, Amsterdam, pp. 153–175.

    Google Scholar 

  • Martin-Löf, P.: 1984, Intuitionistic Type Theory, Notes by Giovanni Sambin of a series of lectures given in Padova, June 1980, Bibliopolis, Naples.

    Google Scholar 

  • Peacocke, C. A. B.: 1981, ‘The theory of meaning in analytical philosophy’, in G. Floistad (ed.), Contemporary Philosophy, Vol 1, M. Nijhoff, The Hague, pp. 35–36.

    Google Scholar 

  • Platts, M. de B.: 1979, Ways of Meaning, Routledge and Kegan Paul, London.

    Google Scholar 

  • Platts, M. de B. (ed.): 1980, Reference, Truth and Reality, Routledge and Kegan Paul, London.

    Google Scholar 

  • Prawitz, D.: 1965, Natural Deduction, Dissertation, University of Stockholm.

    Google Scholar 

  • Prawitz, D.: 1971, ‘Ideas and results in proof theory’, in J. -E. Fenstad (ed.), Proceedings of the Second Scandanavian Logic Symposium, North-Holland, Amsterdam, pp. 235–308.

    Chapter  Google Scholar 

  • Prawitz, D.: 1973, ‘Towards a foundation of a general proof theory’, in P. Suppes et al., (eds.), Logic, Methodology and Philosophy of Science IV, North-Holland, Amsterdam, pp.225–250.

    Google Scholar 

  • Prawitz, D.: 1975, ‘Comments on Gentzen-type procedures and the classical notion of truth’, in J. Diller and G. H. Miiller (eds.), Proof Theory Symposion, Lecture Notes in Mathematics 500, Springer, Berlin, pp. 290–319.

    Chapter  Google Scholar 

  • Prawitz, D.: 1977, ‘Meaning and proofs’, Theoria XLIII, 2–40.

    Google Scholar 

  • Prawitz, D.: 1978, ‘Proofs and the meaning and completeness of the logical constants’, in J. Hintikka et al. (eds.), Essays on Mathematical and Philosophical Logic, D. Reidel, Dordrecht, pp. 25–40.

    Google Scholar 

  • Prawitz, D.: 1980, ‘Intuitionistic logic: a philosophical challenge’, in G. H. von Wright (ed.), Logic and Philosophy, M. Nijhoff, The Hague, pp. 1–10.

    Google Scholar 

  • Prior, A. N.: 1960, ‘The runabout inference-ticket’, Analysis 21, 38–39.

    Article  Google Scholar 

  • Schroeder-Heister, P.: 1981, ‘Untersuchungen zur regellogischen Deutung von Aussagen- verknüpfungen’, Dissertation, University of Bonn.

    Google Scholar 

  • Schroeder-Heister, P.: 1982, ‘Logische Konstanten und Regeln’, Conceptus 16, 45–60.

    Google Scholar 

  • Schroeder-Heister, P.: 1983, The completeness of intuitionistic logic with respect to a validity concept based on an inversion principle’, J. Philosophical Logic 12, 359– 377.

    Google Scholar 

  • Schroeder-Heister, P.: 1984, ‘Generalized rules for quantifiers and the completeness of the intuitionistic operators’ in M. Richter et al. (eds.), Computation and Proof Theory, Lecture Notes in Mathematics 1104, Springer, Berlin, pp. 399–426.

    Chapter  Google Scholar 

  • Smullyan, R. M.: 1968, First Order Logic, Springer-Verlag, Berlin.

    Book  Google Scholar 

  • Stenlund, S.: 1973, The Logic of Description and Existence, Dept. of Philosophy, University of Uppsala.

    Google Scholar 

  • Stenlund, S.: 1975, ‘Descriptions in intuitionistic logic’, in S. Kanger (ed.), Proceedings of the Third Scandanavian Logic Symposium, North-Holland, Amsterdam, pp. 197– 212.

    Google Scholar 

  • Stevenson, J. T.: 1961, ‘Roundabout the runabout inference-ticket’, Analysis 21, 124– 128.

    Google Scholar 

  • Sundholm, G.: 1981, ‘Hacking’s logic’, J. Philosophy 78, 160–168.

    Article  Google Scholar 

  • Sundholm, G.: 1983, ‘Constructions, proofs and the meaning of the logical constants’, J. Philosophical Logic 12, 151–172.

    Article  Google Scholar 

  • Tarski, A.: 1956, Logic, Semantics, Metamathematics, Oxford U.P.

    Google Scholar 

  • Tennant, N.: 1978, Natural Logic, Edinburgh U.P.

    Google Scholar 

  • Tennant, N.: 1982, ‘Proof and paradox’, Dialectica 36, 265–296.

    Article  Google Scholar 

  • Thiel, C.: 1975, ‘Zur Inkonsistenz der Fregeschen Mengenlehre’, in C. Thiel (ed.), Frege und die moderne Grundlagenforschung, A. Hain, Meisenheim, pp. 134–159.

    Google Scholar 

  • Wagner, S.: 1981, ‘Tonk’, Notre Dame J. Formal Logic 22, 289–300.

    Article  Google Scholar 

  • Wright, C.: 1976, ‘Truth conditions and criteria’, Proc. Arist. Soc., supp. 50, 217–245.

    Google Scholar 

  • Wright, C.: 1980, Wittgenstein on the Foundations of Mathematics, Duckworth, London.

    Google Scholar 

  • Wright, C.: 1980a, ‘Realism, truth-value links, other minds and the past’, Ratio 22, 112–132.

    Google Scholar 

  • Wright, C.: 1981, ‘Dummett and revisionism’, Philosophical Quarterly 31, 47–67.

    Article  Google Scholar 

  • Wright, C.: 1982, ‘Strict finitism’, Synthese 51, 203–282.

    Article  Google Scholar 

  • Zucker, J. and Tragresser, R. S.: 1978, ‘The adequacy problem for inferential logic’, J. Philosophical Logic 7, 501–516.

    Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1986 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Sundholm, G. (1986). Proof Theory and Meaning. In: Gabbay, D., Guenthner, F. (eds) Handbook of Philosophical Logic. Synthese Library, vol 166. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-5203-4_8

Download citation

  • DOI: https://doi.org/10.1007/978-94-009-5203-4_8

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-8801-5

  • Online ISBN: 978-94-009-5203-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics