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The aim of the work is to provide a language to reason about Closed Interactions, i.e. all those situations in which the outcomes of an interaction can be determined by the agents themselves and in which the environment cannot interfere with they are able to determine. We will see that two different interpretations can be given of this restriction, both stemming from Pauly Representation Theorem. We will identify such restrictions and axiomatize their logic. We will apply the formal tools to (...) 

We define a multimodal version of Computation Tree Logic (ctl) by extending the language with path quantifiers E δ and A δ where δ denotes one of finitely many dimensions, interpreted over Kripke structures with one total relation for each dimension. As expected, the logic is axiomatised by taking a copy of a ctl axiomatisation for each dimension. Completeness is proved by employing the completeness result for ctl to obtain a model along each dimension in turn. We also show that (...) 



The general verificationist thesis says that What is true can be known or formally: φ → ◊Kφ VT Fitch's argument trivializes this principle. It uses a weak modal epistemic logic to show that VT collapses truth and knowledge, by taking a clever substitution instance for φ: P ∧ ¬KP → ◊ K(P ∧ ¬KP) Then we have the following chain of three conditionals (a) ◊ K(P ∧ ¬KP) → ◊ (KP ∧ K¬KP) in the minimal modal logic for the knowledge (...) 

Three notions of definability in multimodal logic are considered. Two are analogous to the notions of explicit definability and implicit definability introduced by Beth in the context of firstorder logic. However, while by Beth’s theorem the two types of definability are equivalent for firstorder logic, such an equivalence does not hold for multimodal logics. A third notion of definability, reducibility, is introduced; it is shown that in multimodal logics, explicit definability is equivalent to the combination of implicit definability and reducibility. (...) 

We establish the dichotomy property for stable canonical multiconclusion rules for IPC, K4, and S4. This yields an alternative proof of existence of explicit bases of admissible rules for these logics. 

A recurring issue in any formal model representing agents' (changing) informational attitudes is how to account for the fact that the agents are limited in their access to the available inference steps, possible observations and available messages. This may be because the agents are not logically omniscient and so do not have unlimited reasoning ability. But it can also be because the agents are following a predefined protocol that explicitly limits statements available for observation and/or communication. Within the broad literature (...) 

We consider a simple modal logic whose nonmodal part has conjunction and disjunction as connectives and whose modalities come in adjoint pairs, but are not in general closure operators. Despite absence of negation and implication, and of axioms corresponding to the characteristic axioms of _T_, _S4_, and _S5_, such logics are useful, as shown in previous work by Baltag, Coecke, and the first author, for encoding and reasoning about information and misinformation in multiagent systems. For the propositionalonly fragment of such (...) 

Since it was first proposed by Moses, Shoham, and Tennenholtz, the social laws paradigm has proved to be one of the most compelling approaches to the offline coordination of multiagent systems. In this paper, we make four key contributions to the theory and practice of social laws in multiagent systems. First, we show that the "Alternatingtime Temporal Logic" of Alur, Henzinger, and Kupferman provides an elegant and powerful framework within which to express and understand social laws for multiagent systems. Second, (...) 

The literature on quantum logic emphasizes that the algebraic structures involved with orthodox quantum mechanics are non distributive. In this paper we develop a particular algebraic structure, the quasilattice (Jlattice), which can be modeled by an algebraic structure built in quasiset theory Q. This structure is non distributive and involve indiscernible elements. Thus we show that in taking into account indiscernibility as a primitive concept, the quasilattice that 'naturally' arises is non distributive. 

In this paper, we formulate some approaches to belief fusion and revision using epistemic logic semantics. Fusion operators considered in this paper are majority merging, arbitration, and general merging. Some modalities corresponding to belief fusion and revision operators are incorporated into epistemic logics. The Kripke semantics of these extended logics are presented. While most existing approaches treat belief fusion and revision operators as metalevel constructs, we directly incorporate these operators into our object logic language. By doing so, we both extend (...) 

The aim of this paper is to give new kinds of modal logics suitable for reasoning about regions in discrete spaces. We call them dynamic logics of the regionbased theory of discrete spaces. These modal logics are linguistic restrictions of propositional dynamic logic with the global diamond E. Their formulas are equivalent to Boolean combinations of modal formulas like E where A and B are Boolean terms and α is a relational term. Examining what we can say about dynamic models (...) 

We provide syntactic necessary and sufficient conditions on the formulae reducible by the secondorder quantifier elimination algorithm DLS. It is shown that DLS is compete for all modal Sahlqvist and Inductive formulae, and that all modal formulae in a single propositional variable on which DLS succeeds are canonical. 

We describe BDDbased decision procedures for the modal logic K. Our approach is inspired by the automatatheoretic approach, but we avoid explicit automata construction. Instead, we compute certain fixpoints of a set of types — which can be viewed as an onthefly emptiness of the automaton. We use BDDs to represent and manipulate such type sets, and investigate different kinds of representations as well as a “levelbased” representation scheme. The latter turns out to speed up construction and reduce memory consumption (...) 

This article studies the mathematical properties of two systems that model Aristotle's original syllogistic and the relationship obtaining between them. These systems are Corcoran's natural deduction syllogistic and ?ukasiewicz's axiomatization of the syllogistic. We show that by translating the former into a firstorder theory, which we call T RD, we can establish a precise relationship between the two systems. We prove within the framework of firstorder logic a number of logical properties about T RD that bear upon the same properties (...) 

We study finitely generated free Heyting algebras from a topological and from a model theoretic point of view. We review Bellissima’s representation of the finitely generated free Heyting algebra; we prove that it yields an embedding in the profinite completion, which is also the completion with respect to a naturally defined metric. We give an algebraic interpretation of the Kripke model used by Bellissima as the principal ideal spectrum and show it to be first order interpretable in the Heyting algebra, (...) 

We investigate the expressive power of memory logics. These are modal logics extended with the possibility to store (or remove) the current node of evaluation in (or from) a memory, and to perform membership tests on the current memory. From this perspective, the hybrid logic (↓), for example, can be thought of as a particular case of a memory logic where the memory is an indexed list of elements of the domain. 

In this paper, we extend the canonicity methodology in Ghilardi & Meloni (1997) to arbitrary lattice expansions, and syntactically describe canonical inequalities for lattice expansions consisting of meet preserving operations, multiplicative operations, adjoint pairs, and constants. This approach gives us a uniform account of canonicity for substructural and latticebased logics. Our method not only covers existing results, but also systematically accounts for many canonical inequalities containing nonsmooth additive and multiplicative uniform operations. Furthermore, we compare our technique with the approach in (...) 

In this paper we consider an independencefriendly modal logic, IFML. It follows from results in the literature that qua expressive power, IFML is a fragment of secondorder existential logic, , that cannot be translated into firstorder logic. It is also known that IFML lacks the tree structure property. We show that IFML has the , a weaker version of the tree structure property, and that its satisfiability problem is solvable in 2NEXP. This implies that this paper reveals a new decidable (...) 

We present Arrow Update Logic, a theory of epistemic access elimination that can be used to reason about multiagent belief change. While the beliefchanging of Arrow Update Logic can be transformed into equivalent beliefchanging from the popular Dynamic Epistemic Logic approach, we prove that arrow updates are sometimes exponentially more succinct than action models. Further, since many examples of belief change are naturally thought of from Arrow Update Logicrelativized” common knowledge familiar from the Dynamic Epistemic Logic literature. 

Taking Backward Induction as its running example, this paper explores avenues for a logic of informationdriven social action. We use recent results on limit phenomena in knowledge updating and belief revision, procedural rationality, and a ‘Theory of Play’ analyzing how games are played by different agents. 



Thought experiments are widely used in the informal explanation of Relativity Theories; however, they are not present explicitly in formalized versions of Relativity Theory. In this paper, we present an axiom system of Special Relativity which is able to grasp thought experiments formally and explicitly. Moreover, using these thought experiments, we can provide an explicit definition of relativistic mass based only on kinematical concepts and we can geometrically prove the Mass Increase Formula in a natural way, without postulates of conservation (...) 

Public announcement logic is an extension of multiagent epistemic logic with dynamic operators to model the informational consequences of announcements to the entire group of agents. We propose an extension of public announcement logic with a dynamic modal operator that expresses what is true after any announcement: after which , does it hold that Kφ? We give various semantic results and show completeness for a Hilbertstyle axiomatization of this logic. There is a natural generalization to a logic for arbitrary events. 

We explore the finite model theory of the characterisation theorems for modal and guarded fragments of firstorder logic over transition systems and relational structures of width two. A new construction of locally acyclic bisimilar covers provides a useful analogue of the well known treelike unravellings that can be used for the purposes of finite model theory. Together with various other finitary bisimulation respecting model transformations, and Ehrenfeucht–Fraïssé game arguments, these covers allow us to upgrade finite approximations for full bisimulation equivalence (...) 

Qualitative coalitional games were introduced as abstract formal models of goaloriented cooperative systems. A QCG is a game in which each agent is assumed to have some goal to achieve, and in which agents must typically cooperate with others in order to satisfy their goals. In this paper, we show how it is possible to reason about QCGs using Coalition Logic, a formalism intended to facilitate reasoning about coalitional powers in gamelike multiagent systems. We introduce a correspondence relation between QCGs (...) 

We define a multimodal version of Computation Tree Logic (CTL) by extending the language with path quantifiers $E^\delta $ and $E^\delta $ where δ denotes one of finitely many dimensions, interpreted over Kripke structures with one total relation for each dimension. As expected, the logic is axiomatised by taking a copy of a CTL axiomatisation for each dimension. Completeness is proved by employing the completeness result for CTL to obtain a model along each dimension in turn. We also show that (...) 

Logic is breaking out of the confines of the singleagent static paradigm that has been implicit in all formal systems until recent times. We sketch some recent developments that take logic as an account of informationdriven interaction. These two features, the dynamic and the social, throw fresh light on many issues within logic and its connections with other areas, such as epistemology and game theory. 

We generalize stochastic Kripke models and Markov transition systems to stochastic right coalgebras. These are coalgebras for a functor with as an endofunctor on the category of analytic spaces, and is the subprobability functor. The modal operators are generalized through predicate liftings which are setvalued natural transformations involving the functor. Two states are equivalent iff they cannot be separated by a formula. This equivalence relation is used to construct a cospan for logical equivalent coalgebras under a separation condition for the (...) 

The paper develops an interface between syntaxbased logical models of awareness and dynamic epistemic logic. The framework is shown to be able to accommodate a variety of notions of awareness and knowledge, as well as their dynamics. This, it is argued, offers a natural formal environment for the analysis of epistemic phenomena typical of multiagent information exchange, such as how agents become aware of relevant details, how they perform inferences and how they share their information within a group. Technically, the (...) 

The probabilization of a logic system consists of enriching the language (the formulas) and the semantics (the models) with probabilistic features. Such an operation is said to be exogenous if the enrichment is done on top, without internal changes to the structure, and is called endogenous otherwise. These two different enrichments can be applied simultaneously to the language and semantics of a same logic. We address the problem of studying the transference of metaproperties, such as completeness and decidability, to the (...) 

Dynamic epistemic logic, broadly conceived, is the study of logics of information change. This is the first paper in a twopart series introducing this research area. In this paper, I introduce the basic logical systems for reasoning about the knowledge and beliefs of a group of agents. 



Supervaluationism is a well known theory of vagueness. Subvaluationism is a less well known theory of vagueness. But these theories cannot be taken apart, for they are in a relation of duality that can be made precise. This paper provides an introduction to the subvaluationist theory of vagueness in connection to its dual, supervaluationism. A survey on the supervaluationist theory can be found in the Compass paper of Keefe (2008); our presentation of the theory in this paper will be short (...) 

There are various contexts in which it is not pertinent to generate and attend to all the classical consequences of a given premiss—or to trace all the premisses which classically entail a given consequence. Such contexts may involve limited resources of an agent or inferential engine, contextual relevance or irrelevance of certain consequences or premisses, modelling everyday human reasoning, the search for plausible abduced hypotheses or potential causes, etc. In this paper we propose and explicate one formal framework for a (...) 

This paper provides a complete characterization of epistemic models in which distributed knowledge complies with the principle of full communication (van der Hoek et al., 1999; Gerbrandy, 1999). It also introduces an extended notion of bisimulation and corresponding model comparison games that match the expressive power of distributed knowledge operators. 

Biintuitionistic logic is the result of adding the dual of intuitionistic implication to intuitionistic logic. In this note, we characterize the expressive power of this logic by showing that the ﬁrst order formulas equivalent to translations of biintuitionistic propositional formulas are exactly those preserved under biintuitionistic directed bisimulations. The proof technique is originally due to Lindstrom and, in contrast to the most common proofs of this kind of result, it does not use the machinery of neither saturated models nor elementary (...) 

In this paper we propose a method for modeling social influence within the STIT approach to action. Our proposal consists in extending the STIT language with special operators that allow us to represent the consequences of an agent’s choices over the rational choices of another agent. 

In a recent paper, Alexander argues that relaxing the requirement that sound knowers know their own soundness might provide a solution to Fitch’s paradox and introduces a suitable axiomatic system where the paradox is avoided. In this paper an analysis of this solution is proposed according to which the effective move for solving the paradox depends on the axiomatic treatment of the ontic modality rather than the limitations imposed on the epistemic one. It is then shown that, once the ontic (...) 

We present Dynamic Epistemic Temporal Logic, a framework for reasoning about operations on multiagent Kripke models that contain a designated temporal relation. These operations are natural extensions of the wellknown “action models” from Dynamic Epistemic Logic. Our “temporal action models” may be used to define a number of informational actions that can modify the “objective” temporal structure of a model along with the agents’ basic and higherorder knowledge and beliefs about this structure, including their beliefs about the time. In essence, (...) 

We model three examples of beliefs that agents may have about other agents' beliefs, and provide motivation for this conceptualization from the theory of mind literature. We assume a modal logical framework for modelling degrees of belief by partially ordered preference relations. In this setting, we describe that agents believe that other agents do not distinguish among their beliefs ('no preferences'), that agents believe that the beliefs of other agents are in part as their own ('my preferences'), and the special (...) 

Since Montague’s work it is well known that treating a single modality as a predicate may lead to paradox. In their paper “No Future”, Horsten and Leitgeb show that if the two temporal modalities are treated as predicates paradox might arise as well. In our paper we investigate whether paradoxes of multiple modalities, such as the No Future paradox, are genuinely new paradoxes or whether they “reduce” to the paradoxes of single modalities. In order to address this question we develop (...) 



We develop an axiomatic theory of “generalized RoutleyMeyer logics.” These are firstorder logics which are can be characterized by model theories in a certain generalization of RoutleyMeyer semantics. We show that all GRM logics are subclassical, have recursively enumerable consequence relations, satisfy the compactness theorem, and satisfy the standard structural rules and conjunction and disjunction introduction/elimination rules. We also show that the GRM logics include classical logic, intuitionistic logic, LP/K3/FDE, and the relevant logics. 

We make a proposal for formalizing simultaneous games at the abstraction level of player's powers, combining ideas from dynamic logic of sequential games and concurrent dynamic logic. We prove completeness for a new system of 'concurrent game logic' CDGL with respect to finite nondetermined games. We also show how this system raises new mathematical issues, and throws light on branching quantifiers and independencefriendly evaluation games for firstorder logic. 