  The bilateralist approach to logical consequence maintains that judgments of
different qualities should be taken into account in determining
what-follows-from-what. We argue that such an approach may be actualized by a
two-dimensional notion of entailment induced by semantic structures that also
accommodate non-deterministic and partial interpretations, and propose a
proof-theoretical apparatus to reason over bilateralist judgments using
symmetrical two-dimensional analytical Hilbert-style calculi. We also provide a
proof-search algorithm for finite analytic calculi that runs in at most
exponential time, in general, and in polynomial time when only rules having at
most one formula in the succedent are present in the concerned calculus.
