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We design an expansion of Belnap--Dunn logic with belief and plausibility functions that allow non-trivial reasoning with inconsistent and incomplete probabilistic information. We also formalise reasoning with non-standard probabilities and belief functions in two ways. First, using a calculus of linear inequalities, akin to the one presented in~\cite{FaginHalpernMegiddo1990}. Second, as a two-layered modal logic wherein reasoning with evidence (the outer layer) utilises paraconsistent expansions of \L{}ukasiewicz logic. The second approach is inspired by~\cite{BaldiCintulaNoguera2020}. We prove completeness for both kinds of calculi and show their equivalence by establishing faithful translations in both directions.

We introduce a~paraconsistent modal logic $\mathbf{K}\mathsf{G}^2$, based on Gödel logic with coimplication (bi-Gödel logic) expanded with a De Morgan negation $\neg$. We use the logic to formalise reasoning with graded, incomplete and inconsistent information. Semantics of $\mathbf{K}\mathsf{G}^2$ is two-dimensional: we interpret $\mathbf{K}\mathsf{G}^2$ on crisp frames with two valuations $v_1$ and $v_2$, connected via $\neg$, that assign to each formula two values from the real-valued interval $[0,1]$. The first (resp., second) valuation encodes the positive (resp., negative) information the state gives to a~statement. We obtain that $\mathbf{K}\mathsf{G}^2$ is strictly more expressive than the classical modal logic $\mathbf{K}$ by proving that finitely branching frames are definable and by establishing a faithful embedding of $\mathbf{K}$ into $\mathbf{K}\mathsf{G}^2$. We also construct a~constraint tableau calculus for $\mathbf{K}\mathsf{G}^2$ over finitely branching frames, establish its decidability and provide a~complexity evaluation.

We introduce two-dimensional logics based on \L{}ukasiewicz and G\"{o}del logics to formalize paraconsistent fuzzy reasoning. The logics are interpreted on matrices, where the common underlying structure is the bi-lattice (twisted) product of the $[0,1]$ interval. The first (resp.\ second) coordinate encodes the positive (resp.\ negative) information one has about a statement. We propose constraint tableaux that provide a modular framework to address their completeness and complexity.

We introduce the logic LRC, designed to describe and reason about agents’ abilities and capabilities in using resources. The proposed framework bridges two—up to now—mutually independent strands of literature: the one on logics of abilities and capabilities, developed within the theory of agency, and the one on logics of resources, motivated by program semantics. The logic LRC is suitable to describe and reason about key aspects of social behaviour in organizations. We prove a number of properties enjoyed by LRC (soundness, completeness, canonicity, and disjunction property) and its associated analytic calculus (conservativity, cut elimination, and subformula property). These results lay at the intersection of the algebraic theory of unified correspondence and the theory of multitype calculi in structural proof theory. Case studies are discussed which showcase several ways in which this framework can be extended and enriched while retaining its basic properties, so as to model an array of issues, both practically and theoretically relevant, spanning from planning problems to the logical foundations of the theory of organizations.