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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
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.
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.
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.
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.
Baldwin, T.: 1979, Interpretations of quantifiers,Mind 88, 215–240.
Beeson, M.: 1982, ‘Recursive models for constructive set theories’, Annals Math. Logic 23, 127–178.
Belnap, N. D.: 1962, ‘Tonk, plonk and phnk’, Analysis 22, 130–134.
Van Benthem, J. F. A. K. and Van Eijck, J.: 1982, ‘The dynamics of interpretation’, J. Semantics 1, 3–20.
Cohen, M. R. and Nagel, E.: 1934, An Introduction to Logic and Scientific Method, Routledge and Kegan Paul, London.
Davidson, D.: 1967, ‘Truth and meaning’, Synthese 17, 304–323.
Davidson, D.: 1984, Inquiries into Truth and Interpretation, Oxford. U. P.
Davies, M. K.: 1981, Meaning, Quantification, Necessity, Routledge and Kegan Paul, London.
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.
Diller, J. and Troelstra, A. S.: 1984, ‘Readability and intuitionistic logic’, Synthese 60, 253–282.
Dummett, M.: 1968–1969, ’The reality of the past’, Proc. Arist. Soc., 69, 239–258.
Dummett, M.: 1973, Frege, Duckworth, London.
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.
Dummett, M.: 1975a, ‘The justification of deduction’, Proc. British Academy LIX, 201–231.
Dummett, M.: 1976, ‘What is a theory of meaning? (II)’, (‘WTM2’) in Evans and McDowell [1976], pp. 67–137.
Dummett, M.: 1977, Elements of Intuitionism, Oxford U.P.
Dummett, M.: 1978, Truth and Other Enigmas(‘TE’), Duckworth, London.
Dummett, M.: 1981, ‘Frege and Wittgenstein’, in I. Block (ed.), Perspectives on the Philosophy of Wittgenstein, Blackwell, Oxford, pp. 31–42.
Dummett, M.: 1982, ‘Realism’, Synthese 52, 55–112.
Evans, G.: 1977, ‘Pronouns, quantification and relative clauses (I)’, Canadian J. Phil., reprinted in Platts [ 1980 ], pp. 255–317.
Evans, G. and McDowell, J. (eds.): 1976, Truth and Meaning, Oxford U.P.
Foster, J. A.: 1976, ‘Meaning and truth theory’, in Evans and McDowell [ 1976 ], pp. 1–32.
Frege, G.: 1893, Grundgesetze der Arithmetik, Jena.
Gentzen, G.: 1934–1935, ‘Untersuchungen Über das logische Schliessen’, Math. Zeitschrift 39,176–210, 405–431.
Hacking, I.: 1979, ‘What is logic?’, J. Philosophy 76, 285–319.
Hallnas, L.: 1983, On Normalization of Proofs in Set Theory, Diss., University of Stock-holm, Preprint No. 1, Dept of Philosophy.
Hart, W. D.: 1982, ‘Prior and Belnap’, Theoria XLVII, 127–138.
Hendry, H. E.: 1981, ‘Does IPC have binary indigenous Sheffer function?’, Notre Dame J. Formal Logic 22, 183–186.
Heyting, A.: 1931, ‘Die intuitionistische Grundlegung der Mathematik’, Erkenntnis 2, 106–115.
Heyting, A.: 1956, Intuitionism, North-Holland, Amsterdam.
Heyting, A.: 1960, ‘Remarques sur le constructivisme’, Logique et Analyse 3, 177–182.
Kleene, S. C.: 1967, Mathematical Logic, John Wiley & Sons, New York.
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.
Loar, B.: 1976, ‘Two theories of meaning’, in Evans and McDowell [1976], pp. 138–161.
McDowell, J.: 1976, ‘Truth conditions, bivalence and verificationism’, in Evans and McDowell [1976], pp. 42–66.
McDowell, J.: 1977, J.: 1977, ‘On the sense and reference of a proper name’, Mind 86,159–185. Also in Platts [ 1980 ].
McDowell, J.: 1978, J.: 1978, ‘Physicalism and primitive denotation: Field on Tarski’, Erkenntnis 13,131–152. Also in Platts [ 1980 ].
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.
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.
McGinn, C.: 1979, ‘Ana priori argument for realism’, J. Philosophy 74, 113–133.
McGinn, C.: 1980, ‘Truth and use’, in Platts [1980], pp. 19–40.
McGinn, C.: 1982, ‘Realist semantics and content ascription’, Synthese 52, 113–134.
Martin, E., Jr.: 1982, ‘Referentiality in Frege’s Grundgesetze’, History and Philosophy of Logic 3, pp. 151–164.
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.
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.
Martin-Löf, P.: 1984, Intuitionistic Type Theory, Notes by Giovanni Sambin of a series of lectures given in Padova, June 1980, Bibliopolis, Naples.
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.
Platts, M. de B.: 1979, Ways of Meaning, Routledge and Kegan Paul, London.
Platts, M. de B. (ed.): 1980, Reference, Truth and Reality, Routledge and Kegan Paul, London.
Prawitz, D.: 1965, Natural Deduction, Dissertation, University of Stockholm.
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.
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.
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.
Prawitz, D.: 1977, ‘Meaning and proofs’, Theoria XLIII, 2–40.
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.
Prawitz, D.: 1980, ‘Intuitionistic logic: a philosophical challenge’, in G. H. von Wright (ed.), Logic and Philosophy, M. Nijhoff, The Hague, pp. 1–10.
Prior, A. N.: 1960, ‘The runabout inference-ticket’, Analysis 21, 38–39.
Schroeder-Heister, P.: 1981, ‘Untersuchungen zur regellogischen Deutung von Aussagen- verknüpfungen’, Dissertation, University of Bonn.
Schroeder-Heister, P.: 1982, ‘Logische Konstanten und Regeln’, Conceptus 16, 45–60.
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.
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.
Smullyan, R. M.: 1968, First Order Logic, Springer-Verlag, Berlin.
Stenlund, S.: 1973, The Logic of Description and Existence, Dept. of Philosophy, University of Uppsala.
Stenlund, S.: 1975, ‘Descriptions in intuitionistic logic’, in S. Kanger (ed.), Proceedings of the Third Scandanavian Logic Symposium, North-Holland, Amsterdam, pp. 197– 212.
Stevenson, J. T.: 1961, ‘Roundabout the runabout inference-ticket’, Analysis 21, 124– 128.
Sundholm, G.: 1981, ‘Hacking’s logic’, J. Philosophy 78, 160–168.
Sundholm, G.: 1983, ‘Constructions, proofs and the meaning of the logical constants’, J. Philosophical Logic 12, 151–172.
Tarski, A.: 1956, Logic, Semantics, Metamathematics, Oxford U.P.
Tennant, N.: 1978, Natural Logic, Edinburgh U.P.
Tennant, N.: 1982, ‘Proof and paradox’, Dialectica 36, 265–296.
Thiel, C.: 1975, ‘Zur Inkonsistenz der Fregeschen Mengenlehre’, in C. Thiel (ed.), Frege und die moderne Grundlagenforschung, A. Hain, Meisenheim, pp. 134–159.
Wagner, S.: 1981, ‘Tonk’, Notre Dame J. Formal Logic 22, 289–300.
Wright, C.: 1976, ‘Truth conditions and criteria’, Proc. Arist. Soc., supp. 50, 217–245.
Wright, C.: 1980, Wittgenstein on the Foundations of Mathematics, Duckworth, London.
Wright, C.: 1980a, ‘Realism, truth-value links, other minds and the past’, Ratio 22, 112–132.
Wright, C.: 1981, ‘Dummett and revisionism’, Philosophical Quarterly 31, 47–67.
Wright, C.: 1982, ‘Strict finitism’, Synthese 51, 203–282.
Zucker, J. and Tragresser, R. S.: 1978, ‘The adequacy problem for inferential logic’, J. Philosophical Logic 7, 501–516.
Editor information
Editors and Affiliations
Rights 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