“…Propositions 2,10,14,17); (c) based on these semantic relationships, introduce multi-type normal logics into which the original non-normal logics can be embedded via suitable translations (cf. Section 4) following a methodology which was successful in several other cases [18,19,20,21,29,9,28,30,33,48]; (d) retrieve wellknown dual characterization results for axiomatic extensions of monotone modal logic and conditional logics as instances of general algorithmic correspondence theory for normal (multi-type) LE-logics applied to the translated axioms (cf. Section B); (e) extract analytic structural rules from the computations of the first-order correspondents of the translated axioms, so that, again by general results on proper display calculi [31] (which, as discussed in [2], can be applied also to multi-type logical frameworks) the resulting calculi are sound, complete, conservative and enjoy cut elimination and subformula property.…”