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Chapitre D'ouvrage Année : 2024

A note on schematic validity and completeness in Prawitz's semantics

Résumé

Prawitz’s semantics, an instance of proof-theoretic semantics [1, 13], has come in basically two forms: semantics of valid arguments – SVA, see e.g. [9] – and theory of grounds – ToG, see e.g. [10]. Here, I shall focus mostly on SVA, and I will only occasionally refer to ToG. SVA is based on the notion of valid argument. The latter is inspired by Prawitz’s normalisation results for Gentzen’s Natural Deduction [8], stating that derivations for Γ ⊢ A can be transformed, through suitable reductions, to derivations for Γ∗ ⊆ Γ ⊢ A without detours. A detour is given by a formula which occurs both as conclusion of an introduction, and as a major premise of an elimination. In intuitionistic logic, Prawitz’s theorems imply what Schroeder-Heister called the fundamental corollary [12]: A is a theorem iff there is a closed derivation of A ending by an introduction. This may confirm Gentzen’s claim that introductions fix meaning, while eliminations are unique functions of the introductions [2].

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Philosophie
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hal-04573084 , version 1 (13-05-2024)

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  • HAL Id : hal-04573084 , version 1

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Antonio Piccolomini d'Aragona. A note on schematic validity and completeness in Prawitz's semantics. Francesco Bianchini; Vincenzo Fano; Pierluigi Graziani. Current topics in logic and the philosophy of science. Papers from SILFS 2022 postgraduate conference, 4, Società Italiana di Logica e Filosofia delle Scienze (SILFS), 2024, 978-1-84890-455-2. ⟨hal-04573084⟩

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