A Four-Valued Dynamic Epistemic Logic

Epistemic logic is usually employed to model two aspects of a situation: the factual and the epistemic aspects. Truth, however, is not always attainable, and in many cases we are forced to reason only with whatever information is available to us. In this paper, we will explore a four-valued epistemic logic designed to deal with these situations, where agents have only knowledge about the available information (or evidence), which can be incomplete or conflicting, but not explicitly about facts. This layer of available information or evidence, which is the object of the agents' knowledge, can be seen as a database. By adopting this sceptical posture in our semantics, we prepare the ground for logics where the notion of knowledge-or more appropriately, belief-is entirely based on evidence. The technical results include a set of reduction axioms for public announcements, correspondence proofs, and a complete tableau system. In summary, our contributions are twofold: on the one hand we present an intuition and possible application for many-valued modal logics, and on the other hand we develop a logic that models the dynamics of evidence in a simple and intuitively clear fashion

A Labelled Sequent Calculus for Public Announcement Logic

Public announcement logic(PAL) is an extension of epistemic logic (EL) with some reduction axioms. In this paper, we propose a cut-free labelled sequent calculus for PAL, which is an extension of that for EL with sequent rules adapted from the reduction axioms. This calculus admits cut and allows terminating proof search

Bilattice logic of epistemic actions and knowledge

International audienceBaltag, Moss, and Solecki proposed an expansion of classical modal logic, called logic of epistemic actions and knowledge (EAK), in which one can reason about knowledge and change of knowledge. Kurz and Palmigiano showed how duality theory provides a flexible framework for modeling such epistemic changes, allowing one to develop dynamic epistemic logics on a weaker propositional basis than classical logic (for example an intuitionistic basis). In this paper we show how the techniques of Kurz and Palmigiano can be further extended to define and axiomatize a bilattice logic of epistemic actions and knowledge (BEAK). Our propositional basis is a modal expansion of the well-known four-valued logic of Belnap and Dunn, which is a system designed for handling inconsistent as well as potentially conflicting information. These features, we believe, make our framework particularly promising from a computer science perspective

Overview of Results Presented in "Trends in Logic XIII"

Book Reviews: Andrzej Indrzejczak, Janusz Kaczmarek, and Michał Zawidzki (editors), “Trends in Logic XIII. Gentzen’s and Jaśkowski’s Heritage. 80 Years of Natural Deduction and Sequent Calculi”, Wydawnictwo UŁ, Łódź (Poland), 2014, 269 pages, ISBN 978-83-7969-161-6

Evidence-Based Beliefs in Many-Valued Modal Logics

Rational agents, humans or otherwise, build their beliefs from evidence – a process which we call consolidation. But how should this process be carried out? In this thesis, we study a multi-agent logic of evidence and the question how agents should form beliefs in this logic. The main contributions of this thesis are twofold. First, we present and study a many-valued modal logic, and show how it can be suitable for modelling multi-agent scenarios where each agent has access to some evidence, which in turn can be processed into beliefs. This is a technical and practical contribution to many-valued modal logics. Second, we open new paths for research in the field of evidence logics: we show a new approach based on many-valued logics, we highlight the concept of consolidations and the importance of looking at their dynamic nature, and build a methodology based on rationality postulates to evaluate them

Notions of Galois connections for Bilattices

Τα διπλέγματα (bilattices) είναι αλγεβρικές δομές προερχόμενες από τα πεδία της αναπαράστασης γνώσης και της μη μονοτονικής λογικής· αποτελούνται από ένα σύνολο εφοδιασμένο με δύο πλέγματα (lattices), όπου το ένα μοντελοποιεί το βαθμό αλήθειας και το δεύτερο την ποσότητα πληροφορίας. Οι αντιστοιχίες Galois είναι πολύ χρήσιμες στα μαθηματικά, διότι αποτελούν μία ενοποιητική αφαίρεση διάφορων αντιστοιχιών μεταξύ διατεταγμένων συνόλων, καθώς και διότι σχετίζονται στενά με τους τελεστές κλειστότητας. Σε αυτή την εργασία, εισάγουμε κάποιες έννοιες δι-αντιστοιχιών Galois, που αποσκοπούν στο να αποτελέσουν το ανάλογο των αντιστοιχιών Galois για διπλέγματα. Η πρώτη διάκριση που κάνουμε είναι ανάμεσα σε διαντιστοιχίες Galois μονής και διπλής κατεύθυνσης. Οι διαντιστοιχίες διπλής κατεύθυνσης αποτελούνται από ένα ζεύγος (συμβατών μεταξύ τους) αντιστοιχιών Galois ανάμεσα στις διατάξεις αλήθειας και πληροφορίας, ενώ οι διαντιστοιχίες μονής κατεύθυνσης είναι αντιστοιχίες Galois εφοδιασμένες με επιπλέον ιδιότητες που επιχειρούν να συλλάβουν τη διπλεγματική δομή. Μια περαιτέρω διάκριση γίνεται μεταξύ συνήθων και ισχυρών διαντιστοιχιών Galois· στις πρώτες, οι συναρτήσεις που παίρνουν μέρος έχουν ισόμορφες εικόνες ως διατάξεις, ενώ στις δεύτερες οι εικόνες είναι ισόμορφα διπλέγματα. Εξετάζουμε τα τέσσερα είδη διαντιστοιχιών Galois που προκύπτον από τις παραπάνω διχοτομήσεις, τόσο σε διπλέγματα με τελεστές άρνησης όσο και σε διπλέγματα χωρίς τέτοιους τελεστές. Διερευνούμε την γενικευσιμότητα των κομψών ιδιοτήτων των αντιστοιχιών Galois (συνθεσιμότητα, αντιστρεψιμότητα, διατήρηση άνω και κάτω φραγμάτων κλπ), καθώς και την συμπεριφορά των εικόνων όσον αφορά ενδιαφέρουσες ιδιότητες των διπλεγμάτων. Τέλος, αναφερόμαστε στους αντίστοιχους τελεστές κλειστότητας που προκύπτουν από τις διαντιστοιχίες και κάνουμε μια νύξη του πώς οι έννοιες που παρουσιάζουμε μπορούν να γενικευτούν σε σύνολα εφοδιασμένα με περισσότερες από δύο διατάξεις.Bilattices are algebraic structures, stemming from the research on knowledge representation and non-monotonic reasoning; they comprise a set equipped with two lattice orders, one modelling degree of truth and one modelling amount of information. Galois connections are very useful throughout mathematics, providing a unifying abstraction for various correspondences between ordered sets, and being in close correspondence with closure operators. We introduce notions of Galois biconnections, intended to be the bilattice analogue of classical Galois connections between lattices. The first distinction we make is between bidirectional and unidirectional Galois biconnections. A bidirectional Galois biconnection is a (compatible) pair of Galois connections between the truth orderings and the knowledge orderings of bilattices, while a unidirectional Galois biconnection is actually a Galois connection equipped with extra properties that seek to capture the bilattice structure. A further distinction is between regular Galois biconnections, which induce order-isomorphic images of the maps, strong Galois biconnections, which furnish bilattice-isomorphic images. We investigate all four species of Galois biconnections on pre-bilattices and on bilattices with negation and conflation. We examine both the survival of elegant properties of Galois connections (composability, invertibility, preservation of joins and meets, etc.) and the preservation of interesting bilattice properties (distributivity, boundedness, interlacing) for the images of the bilattices under the Galois biconnection. Finally, we discuss the naturally emerging biclosure operators on bilattices and hint on the generalisation of these concepts to sets equipped with more than two lattices