Failure of Completeness in Proof-Theoretic Semantics
- 267 Downloads
Several proof-theoretic notions of validity have been proposed in the literature, for which completeness of intuitionistic logic has been conjectured. We define validity for intuitionistic propositional logic in a way which is common to many of these notions, emphasizing that an appropriate notion of validity must be closed under substitution. In this definition we consider atomic systems whose rules are not only production rules, but may include rules that allow one to discharge assumptions. Our central result shows that Harrop’s rule is valid under substitution, which refutes the completeness conjecture for intuitionistic logic.
KeywordsProof-theoretic semantics Intuitionistic logic Mints’s rule Harrop’s rule Completeness
Mathematics Subject Classification (2000)03A05 03F03 03F55
This work was supported by the French-German ANR-DFG project “Hypothetical Reasoning – Its Proof-Theoretic Analysis” (HYPOTHESES), DFG grant Schr 275/16-2 to T.P. and P.S.-H. and by grants CNPq 401882/2011-0 and CAPES/DAAD 1110-11-0 to W.d.C.S. We should like to thank the anonymous referees for very valuable detailed comments on earlier versions of this paper. We also thank Grigory Olkhovikov and Tor Sandqvist for helpful comments and suggestions.
- 2.de Campos Sanz,W., & Piecha, T. (2014). A critical remark on the BHK interpretation of implication. In P.E. Bour, G. Heinzmann,W. Hodges, P. Schroeder-Heister (Eds.), 14th CLMPS 2011 Proceedings, Philosophia Scientiae, Vol. 18(3). To appear.Google Scholar
- 4.Dummett, M. (1991). The Logical Basis of Metaphysics. London: Duckworth.Google Scholar
- 5.Goldfarb, W. (2014). On Dummett’s “Proof-theoretic Justifications of Logical Laws”. In T. Piecha & P. Schroeder-Heister (Eds.), Advances in Proof-Theoretic Semantics. Trends in Logic. Dordrecht: Springer. Circulated manuscript, 1998.Google Scholar
- 7.Hallnäs, L. (2006). On the proof-theoretic foundation of general definition theory. In R. Kahle & P. Schroeder-Heister (Eds.), Proof-Theoretic Semantics. Special issue of Synthese (Vol. 148, pp. 589– 602). Berlin: Springer.Google Scholar
- 9.Kleene, S.C. (1971). Introduction to Metamathematics. Wolters-Noordhoff Publishing, Groningen and North-Holland Publishing Company: Amsterdam and London.Google Scholar
- 11.Litland, J. (2012). Topics in Philosophical Logic. Ph.D. thesis, Cambridge: Department of Philosophy, Harvard University.Google Scholar
- 13.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, Studies in Logic and the Foundations of Mathematics (Vol. 63, pp. 179–216). Amsterdam: North-Holland.CrossRefGoogle Scholar
- 14.Mints, G.E. (1976). Derivability of admissible rules. Journal of Mathematical Sciences, 6, 417–421.Google Scholar
- 15.Olkhovikov, G.K., & Schroeder-Heister, P. (2014). Proof-theoretic harmony and the levels of rules: Generalised non-flattening results. In E. Moriconi & L. Tesconi (Eds.), Second Pisa Colloquium in Logic, Language and Epistemology. Pisa: ETS. To appear.Google Scholar
- 17.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
- 21.Sandqvist, T. (2014). Basis-extension semantics for intuitionistic sentential logic. Submitted manuscript.Google Scholar
- 23.Schroeder-Heister, P. (1993). Rules of definitional reflection. In Proceedings of the Eighth Annual IEEE Symposium on Logic in Computer Science (Montreal 1993) (pp. 222–232). Los Alamitos: IEEE Computer Society.Google Scholar
- 24.Schroeder-Heister, P. (2006). Validity concepts in proof-theoretic semantics. In R. Kahle & P. Schroeder-Heister (Eds.), Proof-Theoretic Semantics. Special issue of Synthese (Vol. 148, pp. 525–571). Berlin: Springer.Google Scholar
- 25.Schroeder-Heister, P. (2012). Proof-theoretic semantics. In E. Zalta (Ed.), The Stanford Encyclopedia of Philosophy (Winter 2012 Edition). http://plato.stanford.edu/archives/win2012/entries/proof-theoretic-semantics/.
- 26.Schroeder-Heister, P. (2014). The calculus of higher-level rules, propositional quantification, and the foundational approach to proof-theoretic harmony. In A. Indrzejczak (Ed.), Special issue, commemorating the 80th anniversary of Gentzens and Jaśkowski’s groundbreaking works on assumption based calculi, Studia Logica (Vol. 103). Berlin: Springer. To appear.Google Scholar