The Definitional View of Atomic Systems in Proof-Theoretic Semantics

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Dokumentart: Aufsatz
Date: 2017
Source: Pavel Arazim & Tomáš Lávička (eds.), The Logica Yearbook 2016. London: College Publications 2017, pp. 185-200
Language: English
Faculty: 7 Mathematisch-Naturwissenschaftliche Fakultät
Department: Informatik
DDC Classifikation: 004 - Data processing and computer science
100 - Philosophy
Keywords: Logik , Beweistheorie , Definition
Other Keywords: Beweistheoretische Semantik
Definitorische Reflexion
Proof Theory
Proof-Theoretic Semantics
Definitional Reflection
ISBN: 978-1-84890-243-5
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Atomic systems, that is, sets of rules containing only atomic formulas, play an important role in proof-theoretic notions of logical validity. We consider a view of atomic systems as definitions that allows us to discuss a proposal made by Prawitz (2016, DOI: 10.1007/978-3-319-22686-6_2). The implementation of this view in the base case of an inductive definition of validity leads to the problem that derivability of atomic formulas in an atomic system does not coincide with the validity of these formulas. This is due to the fact that, on the definitional view of atomic systems, there are not just production rules, but both introduction and elimination rules for atoms, which may even generate non-normalizable atomic derivations. This shows that the way atomic systems are handled is a fundamental issue of proof-theoretic semantics.

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