Fractional-valued modal logic and soft bilateralism

In a recent paper, under the auspices of an unorthodox variety of bilateralism, we introduced a new kind of proof-theoretic semantics for the base modal logic K, whose values lie in the closed interval [0, 1] of rational numbers. In this paper, after clarifying our conception of bilateralism – dubbed “soft bilateralism” – we generalize the fractional method to encompass extensions and weakenings of K. Specifically, we introduce well-behaved hypersequent calculi for the deontic logic D and the non-normal modal logics E and M and thoroughly investigate their structural properties.