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

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. 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)

For an arbitrary similarity type of Boolean Algebras with Operators we define a class of Sahlqvist 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 are canonical, 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 called Henkin's equation is equivalent to a simpler identity that uses only one variable. We give a conceptually simple proof by showing that the firstorder correspondents of these two equations are equivalent over the class of cylindric atom structures. (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)

Although ethical guidelines for doing Internet research are available, most prominently those of the Association of Internet Researchers, ethical decision-making for research on publ...

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)

A special interest in optics among various seventeenth-century painters living in the Dutch city of Delft has intrigued historians, including art historians, for a long time. Equally, the impressive career of the Delft microscopist Antoni van Leeuwenhoek has been studied by many historians of science. However, it has never been investigated who, at that time, had access to the mathematical and optical knowledge necessary for the impressive achievements of these Delft practitioners. We have tried to gain insight into Delft as (...) a ‘node’ of optical knowledge by following the careers of three minor local figures in early seventeenth-century Delft. We argue that through their work, products, discussions in the vernacular and exchange of skills, rather than via learned publications, these practitioners constituted a foundation on which the later scientific and artistic achievements of other Delft citizens were built. Our Delft case demonstrates that these practitioners were not simple and isolated craftsmen; rather they were crucial components in a network of scholars, savants, painters and rich virtuosi. Decades before Vermeer made his masterworks, or Van Leeuwenhoek started his famous microscopic investigations, the intellectual atmosphere and artisanal knowledge in this city centred on optical topics. (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. (shrink)

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 as 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 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.

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

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

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)

In [6] Albert Visser shows that ILP completely axiomatizes all schemata about provability and relative interpretability that are provable in finitely axiomatized theories. In this paper we introduce a system called ILP that completely axiomatizes the arithmetically valid principles of provability in and interpretability over such theories. To prove the arithmetical completeness of ILP we use a suitable kind of tail models; as a byproduct we obtain a somewhat modified proof of Visser's completeness result.

This study offers a re-interpretation of basic elements of Aristotle's semantics and metaphysics on the basis of a meticulous reconstruction of his semantics. By eliminating anachronistic conceptions commonly ascribed to him, many shortcomings or obscurities he is accused of will disappear.

People with cancer, who live better or longer than expected or who recover completely despite a poor medical prognosis, usually go through a profound change and self-development. This paper is an attempt to describe and understand the nature of this transformation by examining how initially unexamined conceptions of oneself, life, illness, etc., become manifest and get developed. One feature of this process is that people leave the present-day medical conception, which is based on the notion of victim of and battle (...) against illness, and discover that they have other resources for dealing with life and illness. It will be argued that at the centre of this transformation lies the discovery and development of the will, which is closely associated with the willingness to examine one's conceptions of oneself, health, illness and life. (shrink)

Le but de cet article est une critique de la philosophie politique de M. Eric Weil. Par opposition à M. Weil, l'auteur définit la „politique" ici comme l'art du réalisable, à préciser comme : l'art d'arriver à court terme à une organisation de la société qu'on estime meilleure, et cela en employant tous les moyens permis. L'homme politique n'est pas, ainsi que l'affirme M. Weil, un „saint de Vtmiversel”, qui, en ce qui concerne l'emploi de la violence, se trouve au-dessus (...) de la loi. Ce „politicien-éducateur” est en vérité l'appui des régimes totalitaires et est d'ailleurs inconcevable en dehors de ceux-ci. En outre la théorie de M. Weil contient un certain nombre de contradictions internes et, si on la mesure à la pratique, elle se trouve être déficiente à beaucoup d'égards. Contre la conception platonicienne de M. Weil l'auteur défend la thèse que la politique n'est pas une infraction, regrettable mais inévitable, à „la morale”. C'est la tâche de l'homme politique de supprimer toutes les absurdités, inconséquences et injustices qui entachent les structures de la société. Ce qui, concrètement, est absurde, inconséquent ou injuste n'est pas déterminé par „la morale”, mais par la conscience du droit et par le sentiment moral d'une époque donnée. Les „certitudes” d'où part l'homme politique ne sont pas des Idées platoniciennes, mais des hypothèses de travail, qui dérivent leur contenu non pas de la contemplation des Idées, mais d'une notion de ce qui est juste et désirable, notion qui est déterminée par les conditions matérielles et spirituelles et qui se trouve dans un changement continuel. (shrink)

This edition of Giraldus Odonis' Logica for the first time gives access to an important and original treatise, which has unduly been neglected since the author's death. It is also important in that it gives evidence of interesting achievements in the field of logic outside the anti-metaphysical circle surrounding Ockham.

This volume not only provides the first critical edition with an English translation of the famous correspondence of Nicholas of Autrecourt , but also an assessment of his views and the views of those to whom the letters were addressed.

The first critical edition of the _Syncategoreumata_ by the thirteenth-century philosopher Peter of Spain, accompanied by a facing-page English translation to make its contents accessible to modern readers. The introduction gives an account of all the manuscripts used for the edition. Extensive indexes have been added to facilitate the reader's orientation in the book.

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)

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)

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 ...

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 ...

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.