We propose a general system that combines the powerful features of modal logic and many-sorted reasoning. Its algebraic semantics leads to a many-sorted generalization of boolean algebras with operators, for which we prove the analogue of the Jónsson-Tarski theorem. Our goal was to deepen the conne ctions between modal logic and program verification, while also testing the expressiveness of our system by defining a small imperative language and its operational semantics.

We extend Łukasiewicz logic obtaining the infinitary logic Infinitary Riesz Logic (⁠|$\mathcal{IR}$|Ł) whose models are algebras C(X, [0, 1]), where X is a basically disconnected compact Hausdorff space. Equivalently, our models are unit intervals in Dedekind |$\sigma $|-complete Riesz spaces with strong unit. The Lindenbaum–Tarski algebra of |$\mathcal{IR}$|Ł is, up to isomorphism, an algebra of [0, 1]-valued Borel functions. Finally, our system enjoys standard completeness with respect to the real interval [0, 1].

In this paper we study Moisil logic, a many-valued system based on the idea of nuancing. We prove a completeness theorem for graded deduction and make a first attempt towards a game semantics for this logic.

The state theory on MV-algebras is a generalization of Boolean probability theory and is a counterpart of the theory of states defined on lattice-ordered groups. We first investigate the metric space naturally associated to an MV-algebra with a state. The metric completion of anMV-algebra is defined and characterized in relation with the geometric properties of the corresponding state. We propose a concept of independent probability MV-algebras, attempting to solve an open problem from Riečan and Mundici.