An Approach to General Proof Theory and a Conjecture of a Kind of Completeness of Intuitionistic Logic Revisited
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Thirty years ago I formulated a conjecture about a kind of completeness of intuitionistic logic. The framework in which the conjecture was formulated had the form of a semantic approach to a general proof theory (presented at the 4th World Congress of Logic, Methodology and Philosophy of Science at Bucharest 1971 ). In the present chapter, I shall reconsider this 30-year old conjecture, which still remains unsettled, but which I continue to think of as a plausible and important supposition. Reconsidering the conjecture, I shall also reconsider and revise the semantic approach in which the conjecture was formulated.
KeywordsInference Rule Predicate Logic Intuitionistic Logic Natural Deduction Logical Constant
Work on this chapter was done within the project Interpretation and Meaning, funded by Bank of Sweden Tercentenary Foundation.
- 1.Dummett, M. (1991). The Logical Basis of Metaphysics. London: Duckworth.Google Scholar
- 2.Howard, W. (1980). The formulae-as-types notion of construction. In J. R. Hindley & J. Seldin (Eds.), To H. B. Curry: Essays on combinatory logic, lambda calculus, and formalism (pp. 479–490). San Diego: Academic Press.Google Scholar
- 3.Martin-Löf, P. (1971). Hauptsatz for the intuitionistic theory of iterated inductive definitions. In J. E. Fenstad (Ed.), Proceedings of the Second Scandinavian Logic Symposium (pp. 179–216). Amsterdam: North-Holland.Google Scholar
- 4.Prawitz, D. (1965). Natural deduction: a proof-theoretical study. Stockholm: Almqvist & Wicksell. (Reprinted 2006. Mineola, New York: Dover Publications.)Google Scholar
- 5.Prawitz, D. (1970). Constructive semantics. In Proceedings of the First Scandinavian Logic Symposium, Åbo 1968, Filosofiska studier 8 (pp. 96–114) Uppsala: Filosofiska Föreningen och Filosofiska institutionen vid Uppsala Universitet.Google Scholar
- 6.Prawitz, D. (1971). Ideas and results in proof theory. In J. E. Fenstad (Ed.), Proceedings of the Second Scandinavian Logic Symposium (pp. 225–250). Amsterdam: North-Holland.Google Scholar
- 7.Prawitz, D. (1973). Towards a foundation of a general proof theory. In P. Suppes, et al. (Eds.), Logic, Methodology and Philosophy of Science IV (pp. 225–250). Amsterdam: North Holland.Google Scholar
- 8.Prawitz, D. (2005). Logical consequence from a constructivist point of view. In S. Shapiro (Ed.), The Oxford Handbook of Philosophy of Mathematics and Logic (pp. 671-695). Oxford: Oxford University Press.Google Scholar
- 9.Prawitz, D. (2006). Meaning approached via proofs. Synthese, 148, 507–524.Google Scholar
- 10.Schroeder-Heister, P. (2006). Validity concepts in proof-theoretic semantics. Synthese, 148, 525–571.Google Scholar
- 11.Tait, W. (1967). Intentional interpretation of functionals of finite type I. The Journal of Symbolic Logic, 32, 198–212.Google Scholar