We consider some modal languages with a modal operator $D$ whose semantics is based on the relation of inequality. Basic logical properties such as definability, expressive power and completeness are studied. Also, some connections with a number of other recent proposals to extend the standard modal language are pointed at.

In many logics dealing with information one needs to make statements not only about cognitive states, but also about transitions between them. In this paper we analyze a dynamic modal logic that has been designed with this purpose in mind. On top of an abstract information ordering on states it has instructions to move forward or backward along this ordering, to states where a certain assertion holds or fails, while it also allows combinations of such instructions by means of operations (...) from relation algebra. In addition, the logic has devices for expressing whether in a given state a certain instruction can be carried out, and whether that state can be arrived at by carrying out a certain instruction.This paper deals mainly with technical aspects of our dynamic modal logic. It gives an exact description of the expressive power of this language; it also contains results on decidability for the language with arbitrary structures and for the special case with a restricted class of admissible structures. In addition, a complete axiomatization is given. The paper concludes with a remark about the modal algebras appropriate for our dynamic modal logic, and some questions for further work. (shrink)

In many logics dealing with information one needs to make statements not only about cognitive states, but also about transitions between them. In this paper we analyze a dynamic modal logic that has been designed with this purpose in mind. On top of an abstract information ordering on states it has instructions to move forward or backward along this ordering, to states where a certain assertion holds or fails, while it also allows combinations of such instructions by means of operations (...) from relation algebra. In addition, the logic has devices for expressing whether in a given state a certain instruction can be carried out, and whether that state can be arrived at by carrying out a certain instruction. This paper deals mainly with technical aspects of our dynamic modal logic. It gives an exact description of the expressive power of this language; it also contains results on decidability for the language with 'arbitrary' structures and for the special case with a restricted class of admissible structures. In addition, a complete axiomatization is given. The paper concludes with a remark about the modal algebras appropriate for our dynamic modal logic, and some questions for further work. The paper only contains some sketchy examples showing how the logic can be used to capture situations of dynamic interest, far more detailed applications are given in a companion to this paper (De Rijke [33]). (shrink)

Building on the recent scholarship of Bonnie Kent, Christian Trottmann, and especially L.M. de Rijk, this volume gathers together studies by other specialists on Odonis, covering his ideas in economics, logic, metaphysics, ethics, natural ...

This modern, advanced textbook reviews modal logic, a field which caught the attention of computer scientists in the late 1970's.

The strong connection between the two and its development into the Middle Ages form a major subject of this volume.Other themes featuring in this book are Plato ...

Modal Logic, originally conceived as the logic of necessity and possibility, has developed into a powerful mathematical and computational discipline. It is the main source of formal languages aimed at analyzing complex notions such as common knowledge and formal provability. Modal and modal-like languages also provide us with families of restricted description languages for relational and topological structures; they are being used in many disciplines, ranging from artificial intelligence, computer science and mathematics via natural language syntax and semantics to philosophy. (...) This volume presents a broad and up-to-date view of the field, with contributions covering both the foundations of modal logic itself and each of the aforementioned application areas. Complemented with an editorial introduction covering the roots of modal logic, this book is indispensable for any advanced student and researcher in non-classical logic and its applications. (shrink)

We study a propositional bimodal logic consisting of two S4 modalities £ and [a], together with the interaction axiom scheme a £ϕ → £ aϕ. In the intended semantics, the plain £ is given the McKinsey-Tarski interpretation as the interior operator of a topology, while the labelled [a] is given the standard Kripke semantics using a reflexive and transitive binary relation a. The interaction axiom expresses the property that the Ra relation is lower semi-continuous with respect to the topology. The (...) class of topological Kripke frames characterised by the logic includes all frames over Euclidean space where Ra is the positive flow relation of a differential equation. We establish the completeness of the axiomatisation with respect to the intended class of topological Kripke frames, and investigate tableau calculi for the logic, although tableau completeness and decidability are still open questions. (shrink)

We study several modal languages in which some (sets of) generalized quantifiers can be represented; the main language we consider is suitable for defining any first order definable quantifier, but we also consider a sublanguage thereof, as well as a language for dealing with the modal counterparts of some higher order quantifiers. These languages are studied both from a modal logic perspective and from a quantifier perspective. Thus the issues addressed include normal forms, expressive power, completeness both of modal systems (...) and of systems in the quantifier tradition, complexity as well as syntactic characterizations of special semantic constraints. Throughout the paper several techniques current in the theory of generalized quantifiers are used to obtain results in modal logic, and conversely. (shrink)

We define bisimulations for temporal logic with Since and Until. This new notion is compared to existing notions of bisimulations, and then used to develop the basic model theory of temporal logic with Since and Until. Our results concern both invariance and definability. We conclude with a brief discussion of the wider applicability of our ideas.

v. 1. On the twelfth century theories of fallacy.--v. 2. The origin and early development of the theory of supposition.

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.

v. 1. General introduction, the works on logic -- v. 2. The metaphysics, semantics in Aristotle's strategy of argument.

For an arbitrary similarity type of Boolean Algebras with Operators we define a class ofSahlqvist identities. Sahlqvist identities have two important properties. First, a Sahlqvist identity is valid in a complex algebra if and only if the underlying relational atom structure satisfies a first-order condition which can be effectively read off from the syntactic form of the identity. Second, and as a consequence of the first property, Sahlqvist identities arecanonical, that is, their validity is preserved under taking canonical embedding algebras. (...) Taken together, these properties imply that results about a Sahlqvist variety V van be obtained by reasoning in the elementary class of canonical structures of algebras in V.We give an example of this strategy in the variety of Cylindric Algebras: we show that an important identity calledHenkin's equation is equivalent to a simpler identity that uses only one variable. We give a conceptually simple proof by showing that the first-order correspondents of these two equations are equivalent over the class of cylindric atom structures. (shrink)

Paperback. This volume is a new interpretation of Plato's earlier and later Theory of Ideas, starting from a detailed analysis of the dialogue, The Sophist.The way in which Plato announces his novel Metaphysics has been puzzling scholars for a long time. Did Plato really introduce Change into the Transcendent World and thus abandon his Theory of Unchangeable Forms?Many of Plato's commentators have claimed that the use of modern techniques of logico-semantical analysis can be a valuable aid in unravelling this problem, (...) as well as other difficulties Plato raised and attempted to solve. In this commentary, Plato's own logico-semantical approach to metaphysical problems is followed. (shrink)

Despite its elusiveness as a concept, ‘artificial intelligence’ (AI) is becoming part of everyday life, and a range of empirical and methodological approaches to social studies of AI now span many disciplines. This article reviews the scope of ethnomethodological and conversation analytic (EM/CA) approaches that treat AI as a phenomenon emerging in and through the situated organization of social interaction. Although this approach has been very influential in the field of computational technology since the 1980s, AI has only recently emerged (...) as such a pervasive part of daily life to warrant a sustained empirical focus in EM/CA. Reviewing over 50 peer-reviewed publications, we find that the studies focus on various social and group activities such as task-oriented situations, semi-experimental setups, play, and everyday interactions. They also involve a range of participant categories including children, older participants, and people with disabilities. Most of the reviewed studies apply CA’s conceptual apparatus, its approach to data analysis, and core topics such as turn-taking and repair. We find that across this corpus, studies center on three key themes: openings and closing the interaction, miscommunication, and non-verbal aspects of interaction. In the discussion, we reflect on EM studies that differ from those in our corpus by focusing on praxeological respecifications of AI-related phenomena. Concurrently, we offer a critical reflection on the work of literature reviewing, and explore the tortuous relationship between EM and CA in the area of research on AI. (shrink)

Professor de Rijk's interest here is in the views on reality put forward by the medieval thinkers from Boethius to William of Ockham, but especially in the 12th-14th centuries, the period from Abelard onwards.Theology was naturally a key influence, but sematic theories - the philosophical theories on how terms signify, or how a name has its meaning and how this is affected by its context - were fundamental as the starting point of ontological speculation. The categories formulated in order (...) to differentiate various types of context and their impact on the semantics of the verb esse, 'to be', and its related forms. De Rijk's aim is to understand how these medieval thinkers interpreted reality according to their own semantic views, and to see how their own particular concerns - for instance William of Ockham's application of the 'principle of parsimony' to ontology - shaped the nature of their thought. (shrink)

We introduce a notion of bisimulation for graded modal logic. Using this notion, the model theory of graded modal logic can be developed in a uniform manner. We illustrate this by establishing the finite model property and proving invariance and definability results.

Peirce algebras combine sets, relations and various operations linking the two in a unifying setting. This paper offers a modal perspective on Peirce algebras. Using modal logic a characterization of the full Peirce algebras is given, as well as a finite axiomatization of their equational theory that uses so-called unorthodox derivation rules. In addition, the expressive power of Peirce algebras is analyzed through their connection with first-order logic, and the fragment of first-order logic corresponding to Peirce algebras is described in (...) terms of bisimulations. (shrink)

We examine the expressive power of probabilistic context free grammars (PCFGs), with a special focus on the use of probabilities as a mechanism for reducing ambiguity by filtering out unwanted parses. Probabilities in PCFGs induce an ordering relation among the set of trees that yield a given input sentence. PCFG parsers return the trees bearing the maximum probability for a given sentence, discarding all other possible trees. This mechanism is naturally viewed as a way of defining a new class of (...) tree languages. We formalize the tree language thus defined, study its expressive power, and show that the latter is beyond context freeness. While the increased expressive power offered by PCFGs helps to reduce ambiguity, we show that, in general, it cannot be decided whether a PCFG removes all ambiguities. (shrink)

This article argues in favour of voluntary active euthanasia in human beings on the grounds that we (society in general) perform euthanasia on valued companion animals when their suffering is considered great. I argue that suffering is a morally relevant criterion that should be considered in all cases (human and animal) of euthanasia. I further argue that human beings possess autonomy, a morally relevant difference to companion animals, that allows them to reason about their futures in a way that animals (...) cannot, which gives us more reason to grant euthanasia, if requested autonomously by terminally ill patients who are experiencing extreme suffering. I consider objections around the sanctity of human life and sufficient palliation. I argue that neither of these objections give sufficient reason to deny competent human beings euthanasia if autonomously requested. (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)

IT is well known that the art of logic (logica or diale(c)tica) knew a remarkable flourishing period during the twelfth century. In the first half of the century its main centres in Paris were: the School of Notre DameI, of St. Victor2, of the Petit Pont3 and of Mont Ste Geneviève4. The present paper aims to offer some new evidence from the manuscripts on the teaching of logic as given in the School of Mont Ste.