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We introduce a new topological semantics for belief logics in which the belief modality is interpreted as the interior of the closure of the interior operator. We show that the system wKD45, a weakened version of KD45, is sound and complete with respect to the class of all topological spaces. While generalizing the topological belief semantics proposed in [1, 2] to all spaces, we model conditional beliefs and updates and give complete axiomatizations of the corresponding logics with respect to the class of all topological spaces.

In this paper, we study doxastic attitudes that emerge on the basis of argumentational reasoning. In order for an agent’s beliefs to be called ‘rational’, they ought to be well-grounded in strong arguments that are constructed by combining her available evidence in a specific way. A study of how these rational and grounded beliefs emerge requires a new logical setting. The language of the logical system in this paper serves this purpose: it is expressive enough to reason about concepts such as factive combined evidence, correctly grounded belief, and infallible knowledge, which are the building blocks on which our notions of argument and grounded belief can be defined. Building further on previous work, we use a topological semantics to represent the structure of an agent’s collection of evidence, and we use input from abstract argumentation theory to single out the relevant sets of evidence to construct the agent’s beliefs. Our paper provides a sound and complete axiom system for the presented logical language, which can describe the given models in full detail, and we show how this setting can be used to explore more intricate epistemic notions.

Dependence logic extends the language of first order logic by means of dependence atoms and aims to establish a basic theory of dependence and independence underlying such seemingly unrelated subjects as causality, random variables, bound variables in logic, database theory, the theory of social choice, and even quantum physics. In this work we summarize the setting of dependence logic and recall the main results of this rapidly developing area of research.

In this chapter we overview the main structures of epistemic and doxastic logic. We start by discussing the most celebrated models for epistemic logic, i.e., epistemic Kripke structures. These structures provide a very intuitive interpretation of the accessibility relation, based on the notion of information. This also naturally extends to the multi-agent case. Based on Kripke models, we then look at systems that add a temporal or a computational component, and those that provide a ‘grounded’ semantics for knowledge. We also pay special attention to ‘non-standard semantics’ for knowledge and belief, i.e., semantics that are not based on an underlying relation on the sets of states. In particular, we discuss here neighbourhood semantics and topological semantics. In all of these approaches, we can clearly point at streams of results that are inspired by work by Johan van Benthem. We are extremely pleased and honoured to be part of this book dedicated to his work and influences.

In this chapter we re-evaluate Segerberg’s “full DDL” (Dynamic Doxastic Logic) from the perspective of Dynamic Epistemic Logic (DEL), in its belief-revision-friendly incarnation. We argue that a correct version of full DDL must give up the Success Postulate for dynamic revision. Next, we present (an appropriately generalized and simplified version of) full DDL, showing that it is a generalization of the so-called Topo-logic of Moss and Parikh. We construct AGM-friendly versions of full DDL, corresponding to various revising/contracting operations considered in the Belief Revision literature. We show that DDL can internalize inside one model the “external” doxastic dynamics of DEL, as well as the evidential dynamics investigated by van Benthem and Pacuit. In our Conclusions section, we compare three styles of modeling doxastic dynamics: DDL, DEL and PDL/ETL (the Propositional Dynamic Logic approach, with its Epistemic Temporal Logic variant).

One of van Benthem’s seminal results is the Bisimulation Theorem characterizing modal logic as the bisimulation-invariant fragment of first-order logic. Janin and Walukiewicz extended this theorem to include fixpoint operators, showing that the modal \({\mu }\)-calculus \({\mu }\)ML is the bisimulation-invariant fragment of monadic second-order logic MSO. Their proof uses parity automata that operate on Kripke models, and feature a transition map defined in terms of certain fragments of monadic first-order logic. In this paper we decompose their proof in three parts: (1) two automata-theoretic characterizations, of MSO and \(\mu \)ML respectively, (2) a simple model-theoretic characterization of the identity-free fragment of monadic first-order logic, and (3) an automata-theoretic result, stating that (a strong version of) the second result somehow propagates to the level of full fixpoint logics. Our main contribution shows that the third result is an instance of a more general phenomenon that is essentially coalgebraic in nature. We prove that if one set \(\varLambda \) of predicate liftings (or modalities) for a certain set functor \(T\) uniformly corresponds to the \(T\)-natural fragment of another such set \(\varLambda ^{\prime }\), then the fixpoint logic associated with \(\varLambda \) is the bisimulation-invariant logic of the fixpoint logic associated with \(\varLambda ^{\prime }\).

We present a new topological semantics for doxastic logic, in which the belief modality is interpreted as the closure of the interior operator. We show that this semantics is the most general (extensional) semantics validating Stalnaker’s epistemic-doxastic axioms [22] for “strong belief”, understood as subjective certainty. We prove two completeness results, and we also give a topological semantics for update (dynamic conditioning), i.e. the operation of revising with “hard information” (modeled by restricting the topology to a subspace). Using this, we show that our setting fits well with the defeasibility analysis of knowledge [18]: topological knowledge coincides with undefeated true belief. Finally, we compare our semantics to the older topological interpretation of belief in terms of Cantor derivative [23].

We present a logic for reasoning about the evidence-based knowledge and beliefs and the evidential dynamics of non-logically -omniscient agents. We do this by adapting key tools and techniques from Dynamic Epistemic Logic, Justification Logic, and Belief Revision so as to provide a lightweight, yet fine-grained approach that characterizes well-known epistemic and doxastic attitudes in terms of the evidential reasoning that justifies these attitudes. We then add the dynamic operations of evidence introduction, evidence-based inference, strong acceptance of new evidence (evidential "upgrade"), and irrevocable acceptance of additional evidence (evidential "update"). We exemplify our theory by providing a formal dynamic account of Lehrer’s well-known Gettier-type scenario involving the famous Ferrari and the infamous Messrs. Nogot and Havit.