Abstract We present an axiomatization of the fuzzy bi-Gödel modal logic ${\textbf{K}\textsf{biG}}^{\textsf{f}}$ formulated in the language containing $\triangle $ (Baaz Delta operator) and treating $-\!-\!< $ (co-implication) as the defined connective. We also consider two paraconsistent relatives of ${\textbf{K}\textsf{biG}}^{\textsf{f}}$ — $\textbf{K}\textsf{G}^{2\pm \textsf{f}}$ and $\textsf{G}^{2\pm \textsf{f}}_{\blacksquare ,\blacklozenge }$. These logics are defined on fuzzy frames with two valuations $e_{1}$ and $e_{2}$ standing for the support of truth and falsity, respectively, and equipped with two fuzzy relations $R^{+}$ and $R^{-}$ used to determine supports of truth and falsity of modal formulas. We construct embeddings of $\textbf{K}\textsf{G}^{2\pm \textsf{f}}$ and $\textsf{G}^{2\pm \textsf{f}}_{\blacksquare ,\blacklozenge }$ into ${\textbf{K}\textsf{biG}}^{\textsf{f}}$ and use them to obtain the characterization of $\textbf{K}\textsf{G}^{2}$- and $\textsf{G}^{2}_{\blacksquare ,\blacklozenge }$-definable frames. Moreover, we study the transfer of ${\textbf{K}\textsf{biG}}^{\textsf{f}}$ formulas into $\textbf{K}\textsf{G}^{2\pm \textsf{f}}$, i.e., formulas that are ${\textbf{K}\textsf{biG}}^{\textsf{f}}$-valid on mono-relational frames $\mathfrak{F}$ and $\mathfrak{F}^{\prime}$ iff they are $\textbf{K}\textsf{G}^{2\pm \textsf{f}}$-valid on their bi-relational counterparts. Finally, we establish $\textsf{PSpace}$-completeness of all considered logics.

Abstract This paper is an extended version of Bílková et al. ((2023b). Logic, Language, Information, and Computation. WoLLIC 2023, Lecture Notes in Computer Science, vol. 13923, Cham, Springer Nature Switzerland, 101–117.). We discuss two-layered logics formalising reasoning with probabilities and belief functions that combine the Łukasiewicz $[0,1]$ -valued logic with Baaz $\triangle$ operator and the Belnap–Dunn logic. We consider two probabilistic logics – $\mathsf {Pr}^{{\mathsf {\unicode {x0141}}}^2}_\triangle$ (introduced by Bílková et al. 2023d. Annals of Pure and Applied Logic, 103338.) and $\mathbf {4}\mathsf {Pr}^{{\mathsf {\unicode {x0141}}}_\triangle }$ (from Bílková et al. 2023b. Logic, Language, Information, and Computation. WoLLIC 2023, Lecture Notes in Computer Science, vol. 13923, Cham, Springer Nature Switzerland, 101–117.) – that present two perspectives on the probabilities in the Belnap–Dunn logic. In $\mathsf {Pr}^{{\mathsf {\unicode {x0141}}}^2}_\triangle$ , every event $\phi$ has independent positive and negative measures that denote the likelihoods of $\phi$ and $\neg \phi$ , respectively. In $\mathbf {4}\mathsf {Pr}^{{\mathsf {\unicode {x0141}}}_\triangle }$ , the measures of the events are treated as partitions of the sample into four exhaustive and mutually exclusive parts corresponding to pure belief, pure disbelief, conflict and uncertainty of an agent in $\phi$ . In addition to that, we discuss two logics for the paraconsistent reasoning with belief and plausibility functions from Bílková et al. ((2023d). Annals of Pure and Applied Logic, 103338.) – $\mathsf {Bel}^{{\mathsf {\unicode {x0141}}}^2}_\triangle$ and $\mathsf {Bel}^{\mathsf {N}{\mathsf {\unicode {x0141}}}}$ . Both these logics equip events with two measures (positive and negative) with their main difference being that in $\mathsf {Bel}^{{\mathsf {\unicode {x0141}}}^2}_\triangle$ , the negative measure of $\phi$ is defined as the belief in $\neg \phi$ while in $\mathsf {Bel}^{\mathsf {N}{\mathsf {\unicode {x0141}}}}$ , it is treated independently as the plausibility of $\neg \phi$ . We provide a sound and complete Hilbert-style axiomatisation of $\mathbf {4}\mathsf {Pr}^{{\mathsf {\unicode {x0141}}}_\triangle }$ and establish faithful translations between it and $\mathsf {Pr}^{\mathsf {\unicode {x0141}}^2}_\triangle$ . We also show that the validity problem in all the logics is $\mathsf {coNP}$ -complete.

We present a finitary version of Moss' coalgebraic logic for $T$-coalgebras, where $T$ is a locally monotone endofunctor of the category of posets and monotone maps. The logic uses a single cover modality whose arity is given by the least finitary subfunctor of the dual of the coalgebra functor $T_\omega^\partial$, and the semantics of the modality is given by relation lifting. For the semantics to work, $T$ is required to preserve exact squares. For the finitary setting to work, $T_\omega^\partial$ is required to preserve finite intersections. We develop a notion of a base for subobjects of $T_\omega X$. This in particular allows us to talk about the finite poset of subformulas for a given formula. The notion of a base is introduced generally for a category equipped with a suitable factorisation system. We prove that the resulting logic has the Hennessy-Milner property for the notion of similarity based on the notion of relation lifting. We define a sequent proof system for the logic, and prove its completeness.

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.

In this article, we introduce an epistemic modal operator modelling knowledge over distributive non-associative full Lambek calculus with a negation. Our approach is based on the relational semantics for substructural logics: we interpret the elements of a relational frame as information states consisting of collections of data. The principal epistemic relation between the states is the one of being a reliable source of information, on the basis of which we explicate the notion of knowledge as information confirmed by a reliable source. From this point of view it is natural to define the epistemic operator formally as the backward-looking diamond modality. The framework is a generalization and extension of the system of relevant epistemic logic proposed by Majer and Peliš (2009, college Publications, 123–135) and developed by Bílková et al. (2010, college Publications, 22–38). The system is modular in the sense that the axiomatization of the epistemic operator is sound and complete with respect to a wide class of background logics, which makes the system potentially applicable to a wide class of epistemic contexts. Our system admits a weak form of logical omniscience (the monotonicity rule), but avoids stronger ones (a necessitation rule and a K-axiom) as well as some closure properties discussed in normal epistemic logics (like positive and negative introspection). For these properties we provide characteristic frame conditions, so that they can be present in the system if they are considered to be appropriate for some specific epistemic context. We also prove decidability of the weakest epistemic logic we consider, using a filtration method. Finally, we outline further extensions of our framework to a multiagent system.

We fix a logical connection (Stone ˧ Pred : Setop → BA given by 2 as a schizophrenic object) and study coalgebraic modal logic that is induced by a functor T: Set → Set that is finitary and standard and preserves weak pullbacks and finite sets. We prove that for any such T, the cover modality nabla is a left (and its dual delta is a right) adjoint relative to ω. We then consider monotone unary modalities arising from the logical connection and show that they all are left (or right) adjoints relative to ω.