Many aspects of how humans form and combine concepts are notoriously difficult to capture formally. In this paper, we focus on the representation of three particular such aspects, namely overexten- sion, underextension, and dominance. Inspired in part by the work of Hampton, we consider concepts as given through a prototype view, and by considering the interdependencies between the attributes that define a concept. To approach this formally, we employ a recently introduced family of operators that enrich Description Logic languages. These (...) operators aim to characterise complex concepts by collecting those instances that apply, in a finely controlled way, to ‘enough’ of the concept’s defin- ing attributes. Here, the meaning of ‘enough’ is technically realised by accumulating weights of satisfied attributes and comparing with a given threshold that needs to be met. (shrink)

The main result of the present paper is that the variable-free fragment of logic K*, the logic with a single K-style modality and its “reflexive and transitive closure,” is EXPTIMEcomplete. It is then shown that this immediately gives EXPTIME-completeness of variable-free fragments of a number of known EXPTIME-complete logics. Our proof contains a general idea of how to construct a polynomial-time reduction of a propositional logic to its n-variable—and even, in the cases of K*, PDL, CTL, ATL, and some others, (...) variable-free—fragments. Complexity of countermodels for such fragments is considered. (shrink)

In the paper we consider complexity of intuitionistic propositional logic and its natural fragments such as implicative fragment, finite-variable fragments, and some others. Most facts we mention here are known and obtained by logicians from different countries and in different time since 1920s; we present these results together to see the whole picture.

Like modal logic, temporal logic, and description logic, separation logic has become a popular class of logical formalisms in computer science, conceived as assertion languages for Hoare-style proof systems with the goal to perform automatic program analysis. In a broad sense, separation logic is often understood as a programming language, an assertion language and a family of rules involving Hoare triples. In this survey, we present similarities between separation logic as an assertion language and modal and temporal logics. Moreover, we (...) propose a selection of landmark results about decidability, complexity and expressive power. (shrink)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 BASIC MODAL LOGIC . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.

This book illustrates the program of Logical-Informational Dynamics. Rational agents exploit the information available in the world in delicate ways, adopt a wide range of epistemic attitudes, and in that process, constantly change the world itself. Logical-Informational Dynamics is about logical systems putting such activities at center stage, focusing on the events by which we acquire information and change attitudes. Its contributions show many current logics of information and change at work, often in multi-agent settings where social behavior is essential, (...) and often stressing Johan van Benthem's pioneering work in establishing this program. However, this is not a Festschrift, but a rich tapestry for a field with a wealth of strands of its own. The reader will see the state of the art in such topics as information update, belief change, preference, learning over time, and strategic interaction in games. Moreover, no tight boundary has been enforced, and some chapters add more general mathematical or philosophical foundations or links to current trends in computer science. The theme of this book lies at the interface of many disciplines. Logic is the main methodology, but the various chapters cross easily between mathematics, computer science, philosophy, linguistics, cognitive and social sciences, while also ranging from pure theory to empirical work. Accordingly, the authors of this book represent a wide variety of original thinkers from different research communities. And their interconnected themes challenge at the same time how we think of logic, philosophy and computation. Thus, very much in line with van Benthem's work over many decades, the volume shows how all these disciplines form a natural unity in the perspective of dynamic logicians (broadly conceived) exploring their new themes today. And at the same time, in doing so, it offers a broader conception of logic with a certain grandeur, moving its horizons beyond the traditional study of consequence relations. (shrink)

This book provides a detailed exposition of one of the most practical and popular methods of proving theorems in logic, called Natural Deduction. It is presented both historically and systematically. Also some combinations with other known proof methods are explored. The initial part of the book deals with Classical Logic, whereas the rest is concerned with systems for several forms of Modal Logics, one of the most important branches of modern logic, which has wide applicability.

This chapter presents a new semantics for inductive empirical knowledge. The epistemic agent is represented concretely as a learner who processes new inputs through time and who forms new beliefs from those inputs by means of a concrete, computable learning program. The agent’s belief state is represented hyper-intensionally as a set of time-indexed sentences. Knowledge is interpreted as avoidance of error in the limit and as having converged to true belief from the present time onward. Familiar topics are re-examined within (...) the semantics, such as inductive skepticism, the logic of discovery, Duhem’s problem, the articulation of theories by auxiliary hypotheses, the role of serendipity in scientific knowledge, Fitch’s paradox, deductive closure of knowability, whether one can know inductively that one knows inductively, whether one can know inductively that one does not know inductively, and whether expert instruction can spread common inductive knowledge—as opposed to mere, true belief—through a community of gullible pupils. (shrink)

This is an exploratory paper about combining logics, combining theories and combining structures. Typically when one applies logic to such areas as computer science, artificial intelligence or linguistics, one encounters hybrid ontologies. The aim of this paper is to identify plausible strategies for coping with ontological richness.

This paper sets out a new way of combining Kripke-complete modal logics: lexicographic product. It discusses some basic properties of the lexicographic product construction and proves axiomatization/completeness results.

In this paper we argue that hybrid logic is the deductive setting most natural for Kripke semantics. We do so by investigating hybrid axiomatics for a variety of systems, ranging from the basic hybrid language (a decidable system with the same complexity as orthodox propositional modal logic) to the strong Priorean language (which offers full first-order expressivity).We show that hybrid logic offers a genuinely first-order perspective on Kripke semantics: it is possible to define base logics which extend automatically to a (...) wide variety of frame classes and to prove completeness using the Henkin method. In the weaker languages, this requires the use of non-orthodox rules. We discuss these rules in detail and prove non-eliminability and eliminability results. We also show how another type of rule, which reflects the structure of the strong Priorean language, can be employed to give an even wider coverage of frame classes. We show that this deductive apparatus gets progressively simpler as we work our way up the expressivity hierarchy, and conclude the paper by showing that the approach transfers to first-order hybrid logic. (shrink)

Combining logics has become a rapidly expanding enterprise that is inspired mainly by concerns about modularity and the wish to join together tailor made logical tools into more powerful but still manageable ones. A natural question is whether it offers anything new over and above existing standard languages. By analysing a number of applications where combined logics arise, we argue that combined logics are a potentially valuable tool in applied logic, and that endorsements of standard languages often miss the point. (...) Using the history of quantified modal logic as our main example, we also show that the use of combined structures and logics is a recurring theme in the analysis of existing logical systems. (shrink)

The product of two $\textbf {Alt}$ logics possesses the polynomial product finite model property and its membership problem is $\textbf {coNP}$-complete. Using a reduction from an undecidable domino-tiling problem, we prove that its admissibility problem is undecidable.

I propose a modal logic that enables to reason about self-regarding and otherregarding motivations in strategic games. This logic integrates the concepts of joint action, belief, individual and group payoff. The first part of the article is focused on self-regarding agents. A self-regarding agent decides to perform a certain action only if he believes that this action maximizes his own personal benefit. The second part of the article explores different kinds of other-regarding motivations such as fairness and reciprocity. Differently from (...) self-regarding agents, other-regarding agents also consider the benefits of their choices for the group. Moreover, their decisions can be affected by their beliefs about other agents' willingness to act for the well-being of the group. The analysis also considers team-directed reasoning, i.e. the mode of reasoning that people use when they take themselves to be acting as members of a group or a team. (shrink)

This is an extended version of the lectures given during the 12-thConference on Applications of Logic in Philosophy and in the Foundationsof Mathematics in Szklarska Poręba. It contains a surveyof modal hybrid logic, one of the branches of contemporary modal logic. Inthe first part a variety of hybrid languages and logics is presented with adiscussion of expressivity matters. The second part is devoted to thoroughexposition of proof methods for hybrid logics. The main point is to showthat application of hybrid logics (...) may remarkably improve the situation inmodal proof theory. (shrink)

We study a knowledge logic that assumes that to each set of agents, an indiscernibility relation is associated and the agents decide the membership of objects or states up to this indiscernibility relation. Its language contains a family of relative knowledge operators. We prove the decidability of the satisfiability problem, we show its EXPTIME-completeness and as a side-effect, we define a complete Hilbert-style axiomatization.

We define cut-free display calculi for knowledge logics wherean indiscernibility relation is associated to each set of agents, andwhere agents decide the membership of objects using thisindiscernibility relation. To do so, we first translate the knowledgelogics into polymodal logics axiomatised by primitive axioms and thenuse Kracht's results on properly displayable logics to define thedisplay calculi. Apart from these technical results, we argue thatDisplay Logic is a natural framework to define cut-free calculi for manyother logics with relative accessibility relations.