Sabine Frittella

In this paper, we provide a Hilbert-style axiomatization for the crisp bi-Gödel modal logic $\textbf{K}\textsf{biG}$. We prove its completeness w.r.t. crisp Kripke models where formulas at each state are evaluated over the standard bi-Gödel algebra on $[0,1]$. We also consider a paraconsistent expansion of $\textbf{K}\textsf{biG}$ with a De Morgan negation $\neg $, which we dub $\textbf{K}\textsf{G}^{2}$. We devise a Hilbert-style calculus for this logic and, as a consequence of a conservative translation from $\textbf{K}\textsf{biG}$ to $\textbf{K}\textsf{G}^{2}$, prove its completeness w.r.t. crisp Kripke models with two valuations over $[0,1]$ connected via $\neg $. For these two logics, we establish that their decidability and validity are $\textsf{PSPACE}$-complete. We also study the semantical properties of $\textbf{K}\textsf{biG}$ and $\textbf{K}\textsf{G}^{2}$. In particular, we show that Glivenko’s theorem holds only in finitely branching frames. We also explore the classes of formulas that define the same classes of frames both in $\textbf{K}$ (the classical modal logic) and the crisp Gödel modal logic $\mathfrak{G}\mathfrak{K}^{c}$. We show that, among others, all Sahlqvist formulas and all formulas $\phi \rightarrow \chi $ where $\phi $ and $\chi $ are monotone define the same classes of frames in $\textbf{K}$ and $\mathfrak{G}\mathfrak{K}^{c}$.

We discuss two-layered logics formalising reasoning with paraconsistent probabilities that combine the Łukasiewicz [0, 1]-valued logic with Baaz ▵\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\triangle $$\end{document} operator and the Belnap–Dunn logic. The first logic (introduced in [7]) formalises a ‘two-valued’ approach where each event ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} has independent positive and negative measures that stand for, respectively, the likelihoods of ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} and ¬ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lnot \phi $$\end{document}. The second logic that we introduce here corresponds to ‘four-valued’ probabilities. There, ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} is equipped with four measures standing for pure belief, pure disbelief, conflict and uncertainty of an agent in ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document}.We construct faithful embeddings of and into one another and axiomatise using a Hilbert-style calculus. We also establish the decidability of both logics and provide complexity evaluations for them using an expansion of the constraint tableaux calculus for.

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.

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. 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.