Failure of Completeness in Proof-Theoretic Semantics
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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.
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