Incompleteness of Intuitionistic Propositional Logic with Respect to Proof-Theoretic Semantics
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Prawitz proposed certain notions of proof-theoretic validity and conjectured that intuitionistic logic is complete for them [11, 12]. Considering propositional logic, we present a general framework of five abstract conditions which any proof-theoretic semantics should obey. Then we formulate several more specific conditions under which the intuitionistic propositional calculus (IPC) turns out to be semantically incomplete. Here a crucial role is played by the generalized disjunction principle. Turning to concrete semantics, we show that prominent proposals, including Prawitz’s, satisfy at least one of these conditions, thus rendering IPC semantically incomplete for them. Only for Goldfarb’s  proof-theoretic semantics, which deviates from standard approaches, IPC turns out to be complete. Overall, these results show that basic ideas of proof-theoretic semantics for propositional logic are not captured by IPC.
KeywordsGeneral proof theory Proof-theoretic semantics Intuitionistic logic Prawitz’s conjecture Incompleteness Logical constants Kripke semantics
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This work was carried out within the French-German ANR-DFG project “Beyond Logic: Hypothetical Reasoning in Philosophy of Science, Informatics, and Law”, DFG Grant Schr 275/17-1. We are grateful to two referees for their helpful comments and suggestions.
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