Logicality, Double-line Rules, and Harmony

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.