The inversion principle, and the related notion of harmony, has been extensively discussed in proof-theoretic semantics at least since (Prawitz, 1965). In our paper we make use of multi- conclusion sequent-style calculi to provide a harmonious proof-theoretic analysis of modal logics, and we concentrate on Standard Deontic Logic (SDL) as a running example. Our approach is based on combining display calculi (DSDL) with Dosen's characterization of logicality in purely structural terms. We present a genuinely new double-line version of DSDL, i.e. DdlSDL, and we show that DSDL and DdlSDL are deductively equivalent. This equivalence allows us to show that the rules of DSDL are harmonious: the left introduction rules are inverse of the right one inasmuch as they are deductively equivalent to the bottom-up elimination rule of DdlSDL.

Logicality, Double-line Rules, and Harmony / Gratzl, Norbert; Orlandelli, Eugenio. - ELETTRONICO. - (2016), pp. 81-92. (Intervento presentato al convegno General proof Thoery tenutosi a University of Tuebingen nel 27-29 Novembre 2015) [10.15496/publikation-10394].

### Logicality, Double-line Rules, and Harmony

#### Abstract

The inversion principle, and the related notion of harmony, has been extensively discussed in proof-theoretic semantics at least since (Prawitz, 1965). In our paper we make use of multi- conclusion sequent-style calculi to provide a harmonious proof-theoretic analysis of modal logics, and we concentrate on Standard Deontic Logic (SDL) as a running example. Our approach is based on combining display calculi (DSDL) with Dosen's characterization of logicality in purely structural terms. We present a genuinely new double-line version of DSDL, i.e. DdlSDL, and we show that DSDL and DdlSDL are deductively equivalent. This equivalence allows us to show that the rules of DSDL are harmonious: the left introduction rules are inverse of the right one inasmuch as they are deductively equivalent to the bottom-up elimination rule of DdlSDL.
##### Scheda breve Scheda completa Scheda completa (DC)
2016
General Proof Theory
81
92
Logicality, Double-line Rules, and Harmony / Gratzl, Norbert; Orlandelli, Eugenio. - ELETTRONICO. - (2016), pp. 81-92. (Intervento presentato al convegno General proof Thoery tenutosi a University of Tuebingen nel 27-29 Novembre 2015) [10.15496/publikation-10394].
Gratzl, Norbert; Orlandelli, Eugenio
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/536346
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