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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.

International audience

This article proposes a demonstration implementing the data minimization privacy principle, focusing on reducing data collected by government administrations through forms. Data minimization is defined in many privacy regulations worldwide, but has not seen extensive real-world application. We propose a model based on logic and game theory and show that it is possible to create a practical and efficient solution for a real French welfare benefit case.

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 paper, we generalize the basic notions and results of Dempster-Shafer theory from predicates to formal concepts. Results include the representation of conceptual belief functions as inner measures of suitable probability functions, and a Dempster-Shafer rule of combination on belief functions on formal concepts.

The present article contributes to the development of the mathematical theory of epistemic updates using the tools of duality theory. Here, we focus on Probabilistic Dynamic Epistemic Logic (PDEL). We dually characterize the product update construction of PDEL-models as a certain construction transforming the complex algebras associated with the given model into the complex algebra associated with the updated model. Thanks to this construction, an interpretation of the language of PDEL can be defined on algebraic models based on Heyting algebras. This justifies our proposal for the axiomatization of the intuitionistic counterpart of PDEL.

We establish a formal connection between algorithmic correspondence theory and certain dual characterization results for finite lattices, similar to Nation's characterization of a hierarchy of pseudovarieties of finite lattices, progressively generalizing finite distributive lattices. This formal connection is mediated through monotone modal logic. Indeed, we adapt the correspondence algorithm ALBA to the setting of monotone modal logic, and we use a certain duality-induced encoding of finite lattices as monotone neighbourhood frames to translate lattice terms into formulas in monotone modal logic.

We introduce a multi-type display calculus for Propositional Dynamic Logic (PDL). This calculus is complete w.r.t. PDL, and enjoys Belnap-style cut-elimination and subformula property.

The present article provides an analysis of the existing proof systems for dynamic epistemic logic from the viewpoint of proof-theoretic semantics. Dynamic epistemic logic is one of the best known members of a family of logical systems that have been successfully applied to diverse scientific disciplines, but the proof-theoretic treatment of which presents many difficulties. After an illustration of the proof-theoretic semantic principles most relevant to the treatment of logical connectives, we turn to illustrating the main features of display calculi, a proof-theoretic paradigm that has been successfully employed to give a proof-theoretic semantic account of modal and substructural logics. Then, we review some of the most significant proposals of proof systems for dynamic epistemic logics, and we critically reflect on them in the light of the previously introduced proof-theoretic semantic principles. The contributions of the present article include a generalization of Belnap's cut-elimination metatheorem for display calculi, and a revised version of the display-style calculus D.EAK [30]. We verify that the revised version satisfies the previously mentioned proof-theoretic semantic principles, and show that it enjoys cut-elimination as a consequence of the generalized metatheorem.

In the present article, we introduce a multi-type display calculus for dynamic epistemic logic, which we refer to as Dynamic Calculus. The display approach is suitable to modularly chart the space of dynamic epistemic logics on weaker-than-classical propositional base. The presence of types endows the language of the Dynamic Calculus with additional expressivity, allows for a smooth proof-theoretic treatment, and paves the way towards a general methodology for the design of proof systems for the generality of dynamic logics, and certainly beyond dynamic epistemic logic. We prove that the Dynamic Calculus adequately captures Baltag–Moss–Solecki's dynamic epistemic logic, and enjoys Belnap-style cut elimination.