Abstract
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 [6]). 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.
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Notes
- 1.
Prawitz [4]. The elimination rules are there said to be the inverse of the corresponding introduction rules.
- 2.
One may argue that to allow the value to contain new inferences is too liberal, since the justifying operation then produces an argument that goes beyond what was present in the arguments for the premisses. However, this is an angle that is not taken up here.
References
Dummett, M. (1991). The Logical Basis of Metaphysics. London: Duckworth.
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.
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.
Prawitz, D. (1965). Natural deduction: a proof-theoretical study. Stockholm: Almqvist & Wicksell. (Reprinted 2006. Mineola, New York: Dover Publications.)
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.
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.
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.
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.
Prawitz, D. (2006). Meaning approached via proofs. Synthese, 148, 507–524.
Schroeder-Heister, P. (2006). Validity concepts in proof-theoretic semantics. Synthese, 148, 525–571.
Tait, W. (1967). Intentional interpretation of functionals of finite type I. The Journal of Symbolic Logic, 32, 198–212.
Acknowledgments
Work on this chapter was done within the project Interpretation and Meaning, funded by Bank of Sweden Tercentenary Foundation.
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Prawitz, D. (2014). An Approach to General Proof Theory and a Conjecture of a Kind of Completeness of Intuitionistic Logic Revisited. In: Pereira, L., Haeusler, E., de Paiva, V. (eds) Advances in Natural Deduction. Trends in Logic, vol 39. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7548-0_12
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