In the present paper, we endow the logics of topological quasi Boolean algebras, topological quasi Boolean algebras 5, intermediate algebras of types 1-3, and pre-rough algebras with proper multi-type display calculi which are sound, complete, conservative, and enjoy cut elimination and subformula property. Our proposal builds on an algebraic analysis and applies the principles of the multi-type methodology in the design of display calculi.

Taking an algebraic perspective on the basic structures of Rough Concept Analysis as the starting point, in this paper we introduce some varieties of lattices expanded with normal modal operators which can be regarded as the natural rough algebra counterparts of certain subclasses of rough formal contexts, and introduce proper display calculi for the logics associated with these varieties which are sound, complete, conservative and with uniform cut elimination and subformula property. These calculi modularly extend the multi-type calculi for rough algebras to a ‘nondistributive’ (i.e. general lattice-based) setting.

We introduce proper display calculi for basic monotonic modal logic, the conditional logic CK and a number of their axiomatic extensions. These calculi are sound, complete, conservative and enjoy cut elimination and subformula property. Our proposal applies the multi-type methodology in the design of display calculi, starting from a semantic analysis based on the translation from monotonic modal logic to normal bi-modal logic.

We introduce a proper multi-type display calculus for bilattice logic (with conflation) for which we prove soundness, completeness, conservativity, standard subformula property and cut elimination. Our proposal builds on the product representation of bilattices and applies the guidelines of the multi-type methodology in the design of display calculi.

We present a software tool for reasoning in and about propositional sequent calculi for modal logics of actions. As an example, we implement the display calculus D.EAK of dynamic epistemic logic. The tool generates embeddings of the calculus in the theorem prover Isabelle/HOL for formalising proofs about D.EAK. Integrating propositional reasoning in D.EAK with inductive reasoning in Isabelle/HOL, we verify the solution of the muddy children puzzle for any number of muddy children. There also is a set of meta-tools that allows us to adapt the software for a wide variety of user defined calculi.