A spatial logic is a formal language interpreted over any class of structures featuring geometrical entities and relations, broadly construed. In the past decade, spatial logics have attracted much attention in response to developments in such diverse fields as Artificial Intelligence, Database Theory, Physics, and Philosophy. The aim of this handbook is to create, for the first time, a systematic account of the field of spatial logic. The book comprises a general introduction, followed by fourteen (...) chapters by invited authors. Each chapter provides a self-contained overview of its topic, describing the principal results obtained to date, explaining the methods used to obtain them, and listing the most important open problems. Jointly, these contributions constitute a comprehensive survey of this rapidly expanding subject. (shrink)

This is the first work to explicitly target Bourdieu's philosophy of space as a basic organizing force for his social theory. It draws together his work on both social space and physical space, and it applies the logic that binds them together to problems of architecture and urban development.

We focus on a branch of region-based spatial logics dealing with affine geometry. The research on this topic is scarce: only a handful of papers investigate such systems, mostly in the case of the real plane. Our long-term goal is to analyse certain family of affine logics with inclusion and convexity as primitives interpreted over real spaces of increasing dimensionality. In this article we show that logics of different dimensionalities must have different theories, thus justifying further work on different (...) dimensions. We then focus on the three-dimensional case, exploring the expressiveness of this logic and consequently showing that it is possible to construct formulas describing a three-dimensional coordinate frame. The final result, making use of the high expressive power of this logic, is that every region satisfies an affine complete formula, meaning that all regions satisfying it are affine equivalent. (shrink)

Name der Zeitschrift: Semiotica Jahrgang: 2014 Heft: 202 Seiten: 459-480.

Spatial and diagrammatic reasoning is a significant part not only of logical abilities, but also of logical studies. The authors of this paper consider some novel trends in studying this type of reasoning. They show that there are the following two main trends in spatial logic: (i) logical studies of the distribution of various objects in space (logic of geometry, logic of colors, etc.); (ii) logical studies of the space algorithms applied by nature itself ( (...) class='Hi'>logic of swarms, logic of fungi colonies, etc.). (shrink)

Several logical operators are defined as dual pairs, in different types of logics. Such dual pairs of operators also occur in other algebraic theories, such as mathematical morphology. Based on this observation, this paper proposes to define, at the abstract level of institutions, a pair of abstract dual and logical operators as morphological erosion and dilation. Standard quantifiers and modalities are then derived from these two abstract logical operators. These operators are studied both on sets of states and sets of (...) models. To cope with the lack of explicit set of states in institutions, the proposed abstract logical dual operators are defined in an extension of institutions, the stratified institutions, which take into account the notion of open sentences, whose satisfaction is parametrised by sets of states. A hint on the potential interest of the proposed framework for spatial reasoning is also provided. (shrink)

A spatial modal logic is introduced as an extension of the modal logic S4 with the addition of certain spatial operators. A sound and complete Kripke semantics with a natural space interpretation is obtained for SML. The finite model property with respect to the semantics for SML and the cut-elimination theorem for a modified subsystem of SML are also presented.

We propose in this paper to construct modal logics based on mathematical morphology. The contribution of this paper is twofold. First we show that mathematical morphology can be used to define modal operators in the context of normal modal logics. We propose definitions of modal operators as algebraic dilations and erosions, based on the notion of adjunction. We detail the particular case of morphological dilations and erosions, and of there compositions, as opening and closing. An extension to the fuzzy case (...) is also proposed. Then we show how this can be interpreted for spatial reasoning by using qualitative symbolic representations of spatial relationships (topological and metric ones) derived from mathematical morphology. This allows to establish some links between numerical and symbolic representations of spatial knowledge. (shrink)

In this short note we show that any proof of a general spatiality theorem for inductively generated formal topologies requires full classical logic. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

To date, few studies have focused on mapping the mechanisms underlying children’s skills in science. This study investigated to what extent logical reasoning, spatial processing, and working memory, tapped at age 9-10-years, are predictive of physics skills at age 12-13-years. The study used a sample of 81 children (37 girls). Measures of mathematics and reading were also included in the study. Multiple regression analysis showed that spatial processing, and verbal working memory accounted for a similar amount of unique (...) variance (4.5–4.6%) while logical reasoning accounted for 5.7% variance. Physics is a multivariate discipline that draws upon numerous cognitive resources. Logical reasoning ability is a key component in order for children to learn about abstract physics facts, concepts, theories, and applying complex scientific methods. Spatial processing is important as it may sub-serve the assembly of diverse sources of visual-spatial information into a spatial-schematic image. The working memory system provides a flexible and efficient mental workspace that can supervise, coordinate, and execute processes involved in physics problem solving. (shrink)

Following Wiggins’ seminal article, On Being in the Same Place at the Same Time, this article presents the first comprehensive account of the relation of material constitution, an asymmetrical, transitive relation which totally orders distinct ‘entities’ (individuals, pluralities or masses of stuff) which ‘spatially coincide.’ Their coincidence in space is explained by a recursive definition of ‘complete-composition’, weaker than strict mereological indiscernibility, which also explains the variety of logically independent similarities in such cases. This account is ‘analytical’, dealing with ‘putative’ (...) cases in general. The main strategy is to rationally reconstruct Aristotle’s notion of material cause. It is developed more fully with a theory of how to justify admitting generable and perishable things into the natural world, in The Kinds of Things: A Theory of Personal Identity Based on Transcendental Argument (Open Court, 1996). (shrink)

A critical survey of the fundamental philosophical issues in the logic and formal ontology of space, with special emphasis on the interplay between mereology (the theory of parthood relations), topology (broadly understood as a theory of qualitative spatial relations such as continuity and contiguity), and the theory of spatial location proper.

It remains controversial whether touch is a truly spatial sense or not. Many philosophers suggest that, if touch is indeed spatial, it is only through its alliances with exploratory movement, and with proprioception. Here we develop the notion that a minimal yet important form of spatial perception may occur in purely passive touch. We do this by showing that the array of tactile receptive fields in the skin, and appropriately relayed to the cortex, may contain the same (...) basic informational building blocks that a creature navigating around its environment uses to build up a perception of space. We illustrate this point with preliminary evidence that perception of spatiotemporal patterns on the human skin shows some of the same features as spatial navigation in animals. We argue (a) that the receptor array defines a ‘tactile field’, (b) that this field exists in a minimal form in ‘skin space’, logically prior to any transformation into bodily or external spatial coordinates, and (c) that this field supports tactile perception without integration of concurrent proprioceptive or motor information. The basic cognitive elements of space perception may begin at lower levels of neural and perceptual organisation than previously thought. (shrink)

Qualitative reasoning about space and time - a reasoning at the human level - promises to become a fundamental aspect of future systems that will accompany us in daily activity. The aim of Spatial and Temporal Reasoning is to give a picture of current research in this area focusing on both representational and computational issues. The picture emphasizes some major lines of development in this multifaceted, constantly growing area. The material in the book also shows some common ground and (...) a novel combination of spatial and temporal aspects of qualitative reasoning. Part I presents the overall scene. The chapter by Laure Vieu is on the state of the art in spatial representation and reasoning, and that by Alfonso Gerevini gives a similar survey on research in temporal reasoning. The specific contributions to these areas are then grouped in the two main parts. In Part II, Roberto Casati and Achille Varzi examine the ontological status of spatial entities; Anthony Cohn, Brandon Bennett, John Gooday, and Nicholas Gotts present a detailed theory of reasoning with qualitative relations about regions; Andrew Frank discusses the spatial needs of geographical information systems; and Annette Herskovits focuses on the linguistic expression of spatial relations. In Part III, James Allen and George Ferguson describe an interval temporal logic for the representation of actions and events; Drew McDermott presents an efficient way of predicting the outcome of plan execution; and Erik Sandewall introduces a semantics based on transitions for assessing theories of action and change. In Part IV, Antony Galton's chapter stands clearly between the two areas of space and time and outlines the main coordinates of an integrated approach. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:IntroductionSpatial, Environmental, and Ecocritical Approaches to Holocaust MemoryEmily-Rose Baker (bio), Michael Holden (bio), Diane Otosaka (bio), Sue Vice (bio), and Dominic Williams (bio)The successful implementation of genocide during the Holocaust depended on the spatial organisation of mass murder. From the concentrated ghettos and camps delimited by walls and barbed wire to the open fields and camouflaged forests where victims were shot en masse, Anne Kelly Knowles et al. (...) argue, specific locations, patterns of migration, deportation and evacuation, and other spatial elements such as scale, were all central to the Nazi regime's creation of a "comprehensive geography of oppression."1 This is hardly surprising given the biopolitical logics and colonial fantasies underpinning Lebensraum ('living space'), the territorial concept guiding the genocidal aims of the Third Reich, which designated zones of life as well as the inverse on the European continent. The Holocaust was also experienced spatially by its victims, who were forced to navigate new (and at times familiar) sites and landscapes as those of genocide, and whose testimonies in the aftermath of the event conjured these very places in detail.In recent years, the spaces of the Holocaust have been variously revisited, reconstructed, and reimagined within Holocaust studies scholarship. As a broader part of the 'spatial turn' in the humanities, Holocaust scholars have increasingly employed spatial perspectives of the event to deepen our understanding of the environments in which the Holocaust took place and the transformation of these spaces over time—particularly in the fields of history and social science.2 While drawing on diverse modes of spatialisation from interdisciplinary perspectives, contributions from these fields have tended to privilege methods we might describe as variously 'forensic,' including mapping, model-making, and investigations of material traces of the past.3 This work has resulted in many significant insights, not only into the [End Page 1] contexts and structures of Nazi genocide and the perpetrator imaginings of space, but also the experiences of victims and survivors.4Another significant body of work has attended to space and culture within Holocaust contexts. From early on in the postwar period, particular locations and buildings became identified with the Holocaust, such as the notorious entrance guard house at Auschwitz-Birkenau, which appears on countless book covers as well as within films and artworks. Engaging with spatial insights and methodologies from the perspective of cultural and memory studies, scholars have attested to the production and mnemonic significance of these and less well-known sites, as well as the symbolic spatial worlds and experiences to which they give rise. These include Paul Jaskot and Gavriel Rosenfeld, who have turned respectively to Holocaust and post-Holocaust architectures as commonly overlooked sites of history and memory. Drawing attention to the construction of Holocaust space, Jaskot shows how reading architecture 'as process' rather than static structures laden with symbolic meaning reveals unique histories of forced labor in which "the goals of the SS intersected with victims' experience."5 Rosenfeld demonstrates the ways in which spaces built in response to Nazi genocide after 1945 have been variously "imbued with Jewish significance" by their users and creators, resulting in the rise to prominence of Jewish-identified architects.6 Along similar lines, Michael Meng's work has explored the ways in which Jews and gentiles interact with ruined synagogues, cemeteries and Jewish districts as important remnants of pre-war Jewish life in the post-war period in German and Polish cities.7Studies such as those noted above have tended to focus on cities and public buildings, as well as state and civic policy. But as the spatial histories and geographies of the Holocaust have helped to show, extermination often took place in rural settings (mostly located in Central and Eastern Europe), such as the killing fields where Jews were shot en masse; the forests where victims hid and were hunted by Germans and local collaborators; or the open spaces of the dissolving Reich through which former concentration camp inmates were forced to march during the final year of the war—sites in which, owing to less contained spatial dynamics, violence played out differently.8These kinds of sites, which are sparsely documented and often simply thought of... (shrink)

In this thesis we present two logical systems, $\bf MP$ and $\MP$, for the purpose of reasoning about knowledge and effort. These logical systems will be interpreted in a spatial context and therefore, the abstract concepts of knowledge and effort will be defined by concrete mathematical concepts.

Most relational theories assert both that spatial discourse is reducible to talk about physical objects and their spatial relations, and that the relation of congruence derives from a non-metrical relation which intervals bear or possibly bear to measuring instruments. We have shown that there are serious logical difficulties involved in maintaining both these positions and the thesis of the continuity of space. We have also shown that Grünbaum's motivating argument for the reduction of congruence is unsound, and, moreover, (...) that the prospects of formulating the kind of definition which this reduction requires seem to rest upon a wholly vague notion of physical possibility. Alternatively, if congruence is taken as primitive, it seems plausible that spatial discourse can be reconstructed so that there is no ontological commitment to spatial objects other than physical objects. Whether or not this relational thesis is ultimately defensible requires more elaboration of the relational theory, and investigation of its role in the formalizations of current physical theories. It is odd, considering the wide currency of relational theories, that this work has never been done. (shrink)

Thinking about space is thinking about spatial things. The table is on the carpet; hence the carpet is under the table. The vase is in the box; hence the box is not in the vase. But what does it mean for an object to be somewhere? How are objects tied to the space they occupy? This book is concerned with these and other fundamental issues in the philosophy of spatial representation. Our starting point is an analysis of the (...) interplay between mereology (the study of part/whole relations), topology (the study of spatial continuity and compactness), and the theory of spatial location proper. This leads to a unified framework for spatial representation understood quite broadly as a theory of the representation of spatial entities. The framework is then tested against some classical metaphysical questions such as: Are parts essential to their wholes? Is spatial colocation a sufficient criterion of identity? What (if anything) distinguishes material objects from events and other spatial entities? The concluding chapters deal with applications to topics as diverse as the logical analysis of movement and the semantics of maps. (shrink)

In early modernity, one can find many spatial logic diagrams whose geometric forms share a family resemblance with religious art and symbols. The family resemblance these diagrams bear in form is often based on a vesica piscis or on a cross: Both logic diagrams and spiritual symbols focus on the intersection or conjunction of two or more entities, e.g. subject and predicate, on the one hand, or god and man, on the other. This paper deals with the (...) development and function of logic diagrams, their analogy to religious art and symbols, and their modern application in artificial intelligence. (shrink)

What are the relationships between an entity and the space at which it is located? And between a region of space and the events that take place there? What is the metaphysical structure of localization? What its modal status? This paper addresses some of these questions in an attempt to work out at least the main coordinates of the logical structure of localization. Our task is mostly taxonomic. But we also highlight some of the underlying structural features and we single (...) out the interactions between the notion of localization and nearby notions, such as the notions of part and whole, or of necessity and possibility. A theory of localization—we argue—is needed in order to account for the basic relations between objects and space, and runs afoul a pure part-whole theory. We also provide an axiomatization of the relation of localization and examine cases of localization involving entities different from material objects. (shrink)

We present relational proof systems for the four groups of theories of spatial reasoning: contact relation algebras, Boolean algebras with a contact relation, lattice-based spatial theories, spatial theories based on a proximity relation.

Behind the images, the actual logical work iscarried out by reasoning-specific operations on these spatial layout models.

Modal logics, originally conceived in philosophy, have recently found many applications in computer science, artificial intelligence, the foundations of mathematics, linguistics and other disciplines. Celebrated for their good computational behaviour, modal logics are used as effective formalisms for talking about time, space, knowledge, beliefs, actions, obligations, provability, etc. However, the nice computational properties can drastically change if we combine some of these formalisms into a many-dimensional system, say, to reason about knowledge bases developing in time or moving objects. To study (...) the computational behaviour of many-dimensional modal logics is the main aim of this book. On the one hand, it is concerned with providing a solid mathematical foundation for this discipline, while on the other hand, it shows that many seemingly different applied many-dimensional systems (e.g., multi-agent systems, description logics with epistemic, temporal and dynamic operators, spatio-temporal logics, etc.) fit in perfectly with this theoretical framework, and so their computational behaviour can be analyzed using the developed machinery. We start with concrete examples of applied one- and many-dimensional modal logics such as temporal, epistemic, dynamic, description, spatial logics, and various combinations of these. Then we develop a mathematical theory for handling a spectrum of 'abstract' combinations of modal logics - fusions and products of modal logics, fragments of first-order modal and temporal logics - focusing on three major problems: decidability, axiomatizability, and computational complexity. Besides the standard methods of modal logic, the technical toolkit includes the method of quasimodels, mosaics, tilings, reductions to monadic second-order logic, algebraic logic techniques. Finally, we apply the developed machinery and obtained results to three case studies from the field of knowledge representation and reasoning: temporal epistemic logics for reasoning about multi-agent systems, modalized description logics for dynamic ontologies, and spatio-temporal logics. The genre of the book can be defined as a research monograph. It brings the reader to the front line of current research in the field by showing both recent achievements and directions of future investigations (in particular, multiple open problems). On the other hand, well-known results from modal and first-order logic are formulated without proofs and supplied with references to accessible sources. The intended audience of this book is logicians as well as those researchers who use logic in computer science and artificial intelligence. More specific application areas are, e.g., knowledge representation and reasoning, in particular, terminological, temporal and spatial reasoning, or reasoning about agents. And we also believe that researchers from certain other disciplines, say, temporal and spatial databases or geographical information systems, will benefit from this book as well. Key Features: Integrated approach to modern modal and temporal logics and their applications in artificial intelligence and computer science Written by internationally leading researchers in the field of pure and applied logic Combines mathematical theory of modal logic and applications in artificial intelligence and computer science Numerous open problems for further research Well illustrated with pictures and tables. (shrink)

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 region-based 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(A ∧ ⟨α⟩ B) where A and B are Boolean terms and α is a relational term. Examining what we can (...) say about dynamic models when we use formulas to describe them, we successively address the axiomatization/completeness issue and the decidability/complexity issue of our dynamic logics of the region-based theory of discrete spaces. (shrink)

This article considers how Spatial Big Data is situated and produced through embodied spatial experiences as data processes appear and act in small moments on mobile phone applications and other digital spatial technologies. Locating Spatial Big Data in the historical and geographical contexts of Sydney and Hong Kong, it traces how situated knowledges mediate and moderate the rising potency of discourses of cartographic reason and data logics as colonial cartographic imaginations expressed in land divisions and urban (...) planning continue on, in a world that increasingly values models of calculability, interoperability and authority. It draws on ethnographic material gathered through walking interviews in both cities, and in doing so, it argues that by using ethnographic ‘moments’, it is possible to decentre the focus on data processes to consider the critical potential of a politics of everyday experiences that produce and reflect the structures of data logics. Through these ethnographic moments, this article examines how mobile technologies are complicit in the production of Spatial Big Data, and the impact this has on the increasing regimentation and surveillance of modes of being and expression via mobile media. At the same time, it will argue that while spatial calculability has expanded from cartographic reason into data logics, the epistemological universality of Spatial Big Data is constantly being resisted – in moments of experimentation, failure, intuition, memory and desire, the ghosts of the incalculable epistemes, experiences and people, forgotten by the emphasis on calculation, continue to speak. (shrink)

Omar Rivera’s Delimitations of Latin American Philosophy: Beyond Redemption is an important contribution to the interpretation of central figures and questions of the Latin American philosophical tradition, particularly Peruvian Marxist José Carlos Mariátegui and questions of identity and liberation. Rivera establishes productive dialogues between foundational figures such as Simón Bolívar, José Martí, and Mariátegui and decolonial thinkers like María Lugones, Aníbal Quijano, and Gloria Anzaldúa to posit delimitations of Latin American philosophy that might allow it to move beyond redemptive logics (...) that essentialize identities and reinscribe oppressions in narratives of liberation. This review explores the central arguments of the book and proposes a philosophical and a political issue that emerges from them. (shrink)

Data analytics, particularly the current rhetoric around “Big Data”, tend to be presented as new and innovative, emerging ahistorically to revolutionize modern life. In this article, we situate one branch of Big Data analytics, spatial Big Data, through a historical predecessor, geodemographic analysis, to help develop a critical approach to current data analytics. Spatial Big Data promises an epistemic break in marketing, a leap from targeting geodemographic areas to targeting individuals. Yet it inherits characteristics and problems from geodemographics, (...) including a justification through the market, and a process of commodification through the black-boxing of technology. As researchers develop sustained critiques of data analytics and its effects on everyday life, we must so with a grounding in the cultural and historical contexts from which data technologies emerged. This article and others develop a historically situated, critical approach to spatial Big Data. This history illustrates connections to the critical issues of surveillance, redlining, and the production of consumer subjects and geographies. The shared histories and structural logics of spatial Big Data and geodemographics create the space for a continued critique of data analyses’ role in society. (shrink)

It is generally assumed, and usually taken for granted, that reality is fully contained in space. However, when taking a closer look at the strange behavior of the entities of the micro-world, we are forced to abandon such a prejudice and recognize that space is just a temporary crystallization of a small theatre for reality, where the material entities can take a place and meet with each other. More precisely, phenomena like quantum entanglement, quantum interference effects and quantum indistinguishability, when (...) analyzed attentively, tell us that there is much more in our physical reality than what meets our three-dimensional human eyes. But if the building blocks of our physical reality are non-spatial, what does it mean? Can we understand what the nature of a non-spatial entity is? And if so, what are the consequences for our view of the world in which we live and evolve as a species? This article was written having in mind one of the objectives of the Center Leo Apostel for Interdisciplinary Studies, that of a broad dissemination of scientific knowledge. Hence, it addresses a transversal audience of readers, both academic and nonacademic, hoping to stimulate in this way the interdisciplinary dialogue about foundational issues in science. (shrink)

Dynamic Topological Logic ( ) is a modal logic which combines spatial and temporal modalities for reasoning about dynamic topological systems , which are pairs consisting of a topological space X and a continuous function f : X → X . The function f is seen as a change in one unit of time; within one can model the long-term behavior of such systems as f is iterated. One class of dynamic topological systems where the long-term behavior (...) of f is particularly interesting is that of minimal systems ; these are dynamic topological systems which admit no proper, closed, f -invariant subsystems. In such systems the orbit of every point is dense, which within translates into a non-trivial interaction between spatial and temporal modalities. This interaction, however, turns out to make the logic simpler, and while s in general tend to be undecidable, interpreted over minimal systems we obtain decidability, although not in primitive recursive time; this is the main result that we prove in this paper. We also show that interpreted over minimal systems is incomplete for interpretations on relational Kripke frames and hence does not have the finite model property; however it does have a finite non-deterministic quasimodel property. Finally, we give a set of formulas of which characterizes the class of minimal systems within the class of dynamic topological systems, although we do not offer a full axiomatization for the logic. (shrink)

This print edition of Read's account of logical thought includes the original publication's diagrams and tables. In this excellent book, Read commences by offering an overview of past attitudes and definitions of logic. Individual chapters consider the various means by which logical processes are conceived and developed in the mind. Philosophical arguments, spatial reasoning and mathematical forms of logic are discussed in great depth, with illustrations appended where deemed necessary. Read, an academic and philosopher, employs his decades (...) long experience of teaching students about the logical disciplines to write a detailed and comprehensive accounts of logic's aspects. Throughout this work, he attempts to retain a clarity of expression which makes the concepts discussed understandable for readers previously unversed in the types of logical expression. Read also sets out the principles behind setting up a hypothesis, and highlights the common logical fallacies which crop up in human reasoning. Extensive attention is given to the process of syllogism, by which sound conclusions are drawn from the premises offered. Throughout the book, Read is attentive in defining and explaining the esoteric terminology native to logic. As well as allowing the reader swift comprehension of the various subtopics, this explanatory style confers improved flow to the book. At the conclusion we are offered many questions corresponding to each chapter: these are intended to test the reader's understanding, and enforce the mental habits of utilising logical and critical thought. First published in 1898 and subject to several editions with revisions and refinements thereafter, Logic Deductive and Inductive retains a good reputation among academics and logicians. Its successful balance between striking a concise introductory tone with an embrace of the core elements of the subject has served to preserve its popularity into the modern day. (shrink)

For the last 25 years, since publication of his Logical Studies, Professor Von Wright has steadily explored the field of philosophical logic. The concept of negation, logical paradoxes, the puzzles connected with evidence and probability in confirmation theory, the interrelatedness of the ideas of time and change, and the clarification of the structure of temporal and spatial orderings are among the many areas he has profitably investigated. -- "Philosophical Review".

The Logic of Causation: Definition, Induction and Deduction of Deterministic Causality is a treatise of formal logic and of aetiology. It is an original and wide-ranging investigation of the definition of causation (deterministic causality) in all its forms, and of the deduction and induction of such forms. The work was carried out in three phases over a dozen years (1998-2010), each phase introducing more sophisticated methods than the previous to solve outstanding problems. This study was intended as part (...) of a larger work on causal logic, which additionally treats volition and allied cause-effect relations (2004). The Logic of Causation deals with the main technicalities relating to reasoning about causation. Once all the deductive characteristics of causation in all its forms have been treated, and we have gained an understanding as to how it is induced, we are able to discuss more intelligently its epistemological and ontological status. In this context, past theories of causation are reviewed and evaluated (although some of the issues involved here can only be fully dealt with in a larger perspective, taking volition and other aspects of causality into consideration, as done in Volition and Allied Causal Concepts). Phase I: Macroanalysis. Starting with the paradigm of causation, its most obvious and strongest form, we can by abstraction of its defining components distinguish four genera of causation, or generic determinations, namely: complete, partial, necessary and contingent causation. When these genera and their negations are combined together in every which way, and tested for consistency, it is found that only four species of causation, or specific determinations, remain conceivable. The concept of causation thus gives rise to a number of positive and negative propositional forms, which can be studied in detail with relative ease because they are compounds of conjunctive and conditional propositions whose properties are already well known to logicians. The logical relations (oppositions) between the various determinations (and their negations) are investigated, as well as their respective implications (eductions). Thereafter, their interactions (in syllogistic reasoning) are treated in the most rigorous manner. The main question we try to answer here is: is (or when is) the cause of a cause of something itself a cause of that thing, and if so to what degree? The figures and moods of positive causative syllogism are listed exhaustively; and the resulting arguments validated or invalidated, as the case may be. In this context, a general and sure method of evaluation called ‘matricial analysis’ (macroanalysis) is introduced. Because this (initial) method is cumbersome, it is used as little as possible – the remaining cases being evaluated by means of reduction. Phase II: Microanalysis. Seeing various difficulties encountered in the first phase, and the fact that some issues were left unresolved in it, a more precise method is developed in the second phase, capable of systematically answering most outstanding questions. This improved matricial analysis (microanalysis) is based on tabular prediction of all logically conceivable combinations and permutations of conjunctions between two or more items and their negations (grand matrices). Each such possible combination is called a ‘modus’ and is assigned a permanent number within the framework concerned (for 2, 3, or more items). This allows us to identify each distinct (causative or other, positive or negative) propositional form with a number of alternative moduses. This technique greatly facilitates all work with causative and related forms, allowing us to systematically consider their eductions, oppositions, and syllogistic combinations. In fact, it constitutes a most radical approach not only to causative propositions and their derivatives, but perhaps more importantly to their constituent conditional propositions. Moreover, it is not limited to logical conditioning and causation, but is equally applicable to other modes of modality, including extensional, natural, temporal and spatial conditioning and causation. From the results obtained, we are able to settle with formal certainty most of the historically controversial issues relating to causation. Phase III: Software Assisted Analysis. The approach in the second phase was very ‘manual’ and time consuming; the third phase is intended to ‘mechanize’ much of the work involved by means of spreadsheets (to begin with). This increases reliability of calculations (though no errors were found, in fact) – but also allows for a wider scope. Indeed, we are now able to produce a larger, 4-item grand matrix, and on its basis find the moduses of causative and other forms needed to investigate 4-item syllogism. As well, now each modus can be interpreted with greater precision and causation can be more precisely defined and treated. In this latest phase, the research is brought to a successful finish! Its main ambition, to obtain a complete and reliable listing of all 3-item and 4-item causative syllogisms, being truly fulfilled. This was made technically feasible, in spite of limitations in computer software and hardware, by cutting up problems into smaller pieces. For every mood of the syllogism, it was thus possible to scan for conclusions ‘mechanically’ (using spreadsheets), testing all forms of causative and preventive conclusions. Until now, this job could only be done ‘manually’, and therefore not exhaustively and with certainty. It took over 72’000 pages of spreadsheets to generate the sought for conclusions. This is a historic breakthrough for causal logic and logic in general. Of course, not all conceivable issues are resolved. There is still some work that needs doing, notably with regard to 5-item causative syllogism. But what has been achieved solves the core problem. The method for the resolution of all outstanding issues has definitely now been found and proven. The only obstacle to solving most of them is the amount of labor needed to produce the remaining (less important) tables. As for 5-item syllogism, bigger computer resources are also needed. (shrink)

In this thesis, I investigate the nature of geometric knowledge and its relationship to spatial intuition. My goal is to rehabilitate the Kantian view that Euclid's geometry is a mathematical practice, which is grounded in spatial intuition, yet, nevertheless, yields a type of a priori knowledge about the structure of visual space. I argue for this by showing that Euclid's geometry allows us to derive knowledge from idealized visual objects, i.e., idealized diagrams by means of non-formal logical inferences. (...) By developing such an account of Euclid's geometry, I complete the "standard view" that geometry is either a formal system or an empirical science, which was developed mainly by the logical positivists and which is currently accepted by many mathematicians and philosophers. My thesis is divided into three parts. I use Hans Reichenbach's arguments against Kant and Edmund Husserl's genetic approach to the concept of space as a means of arguing that the "standard view" has to be supplemented by a concept of a geometry whose propositions have genuine spatial content. I then develop a coherent interpretation of Euclid's method by investigating both the subject matter of Euclid's geometry and the nature of geometric inferences. In the final part of this thesis, I modify Husserl's phenomenological analysis of the constitution of visual space in order to define a concept of spatial intuition that allows me not only to explain how Euclid's practice is grounded in visual space, but also to account for the apriority of its results. (shrink)

This book presents the refereed proceedings of the Sixth European Workshop on Logics in Artificial Intelligence, JELIA '96, held in Evora, Portugal in September/October 1996. The 25 revised full papers included together with three invited papers were selected from 57 submissions. Many relevant aspects of AI logics are addressed. The papers are organized in sections on automated reasoning, modal logics, applications, nonmonotonic reasoning, default logics, logic programming, temporal and spatial logics, and belief revision and paraconsistency.

How do people reason about spatial relations? Do people with different cultural backgrounds differ in how they reason about space? The aim of our cross-cultural study on spatial reasoning is to strengthen this link between spatial cognition and culture. We conducted two reasoning experiments, one in Germany and one in Mongolia. Topological relations, such as “A overlaps B” or “B lies within C”, were presented to the participants as premises and they had to find a conclusion that (...) was consistent with the premises. The problem description allowed multiple possible “conclusions”. Our results, however, indicate that the participants had strong preferences: They consistently preferred one of the possible conclusions and neglected other conclusions, although they were also logically consistent with the premises. The preferred and neglected conclusions were similar in Germany and Mongolia. We argue that the preferences are caused by universal cognitive principles that work the same way in the western culture and Mongolia. (shrink)

The logic of 'elsewhere,' i.e., of a sentence operator interpretable as attaching to a formula to yield a formula true at a point in a Kripke model just in case the first formula is true at all other points in the model, has been applied in settings in which the points in question represent spatial positions, as well as in the case in which they represent moments of time. This logic is applied here to the alethic modal (...) case, in which the points are thought of as possible worlds, with the suggestion that its deployment clarifies aspects of a position explored by John Divers under the name 'modal agnosticism.' In particular, it makes available a logic whose Halldén incompleteness explicitly registers the agnostic element of the position - its neutrality as between modal realism and modal anti-realism. (shrink)

Conceptual realism acknowledges the existence of abstract objects: theoretical realism acknowledges the existence of non-observable objects; whereas classical realism acknowledges the existence of observable objects. Similarly, temporal realism accepts the existence of future and past events along with present ones, and spatial realism accepts the events which occur there (else-where) as well as those that occur here. We dealt earlier with the three former kinds of realism and their opposites: nominalism, instrumentalism and (ontological) idealism [2]. This paper contains an (...) examination of the two latter forms of realism: temporal and spatial, and their counterparts: temporal and spatial irrealisms. Analogies and connections between these standpoints will be the focus of the paper. (shrink)

For the last 25 years, since publication of his Logical Studies, Professor Von Wright has steadily explored the field of philosophical logic. The concept of negation, logical paradoxes, the puzzles connected with evidence and probability in confirmation theory, the interrelatedness of the ideas of time and change, and the clarification of the structure of temporal and spatial orderings are among the many areas he has profitably investigated.

Medieval analyses of molecular propositions include many non-truthfunctional connectives in addition to the standard modern binary connectives (conjunction, disjunction, and conditional). Two types of non-truthfunctional molecular propositions considered by a number of 13th- and 14th-century authors are temporal and local propositions, which combine atomic propositions with `while' and `where'. Despite modern interest in the historical roots of temporal and tense logic, medieval analyses of `while' propositions are rarely discussed in modern literature, and analyses of `where' propositions are almost completely (...) overlooked. In this paper we introduce 13th- and 14th-century views on temporal and local propositions, and connect the medieval theories with modern temporal and spatial counterparts. (shrink)

A fundamental cognitive function found across a wide range of species and necessary for survival is the ability to navigate complex environments. It has been suggested that mobility may play an important role in the development of spatial skills. Despite evolutionary arguments offering logical explanations for why sex/gender differences in spatial abilities and mobility might exist, thus far there has been limited sampling from nonindustrialized and subsistence-based societies. This lack of sampling diversity has left many unanswered questions regarding (...) the effects that environmental variation and cultural norms may have in shaping mobility patterns during childhood and the development of spatial competencies that may be associated with it. Here we examine variation in mobility, performance on large-scale spatial skills, and performance on small-scale spatial skills among Twa forager/pastoralist children whose daily lives have been dramatically altered since settlement and the introduction of government-funded boarding schools. Unlike in previous findings among Twa adults, boys and girls show similar patterns of travel on all measures of mobility. We also find no significant differences in spatial task performance by gender for large- or small-scale spatial skills. Further, children performed as well as adults did on mental rotation, and they outperformed adults on the water-level task. We discuss how children’s early learning environments may influence the development of both large- and small-scale spatial skills. (shrink)

We propose a logic for reasoning about metric spaces with the induced topologies. It combines the 'qualitative' interior and closure operators with 'quantitative' operators 'somewhere in the sphere of radius r.' including or excluding the boundary. We supply the logic with both the intended metric space semantics and a natural relational semantics, and show that the latter (i) provides finite partial representations of (in general) infinite metric models and (ii) reduces the standard '∈-definitions' of closure and interior to (...) simple constraints on relations. These features of the relational semantics suggest a finite axiomatisation of the logic and provide means to prove its EXPTIME-completeness (even if the rational numerical parameters are coded in binary). An extension with metric variables satisfying linear rational (in)equalities is proved to be decidable as well. Our logic can be regarded as a 'well-behaved' common denominator of logical systems constructed in temporal, spatial, and similarity-based quantitative and qualitative representation and reasoning. Interpreted on the real line (with its Euclidean metric), it is a natural fragment of decidable temporal logics for specification and verification of real-time systems. On the real plane, it is closely related to quantitative and qualitative formalisms for spatial representation and reasoning, but this time the logic becomes undecidable. (shrink)

The paramount role of logic in Wittgenstein’s Tractatus is undeniable and must be obvious to anyone even on a cursory reading. Yet, Wittgenstein's formulations often appear metaphorical when he sketches his ideas on logic and its relation to sentence meaning. Sometimes, they seem more apt to invite loose philosophical associations than pinning down rigorously technical details. This impression notwithstanding, the Tractatus still offers one of the deepest philosophical accounts of modern logic and it does so precisely through (...) its suggestive exposition. It is, in particular, the spatial analogies, Wittgenstein’s ‘logical place’ and ‘logical space’, which are crucial in this connection, and while they carry a structure that is specific for Wittgenstein’s own conception of logic, they are sufficiently general to accommodate also alternative conceptions of logic. Thus, even 100 years after its first publication, Wittgenstein’s Tractatus remains a source of inspiration and a highly valuable one for the philosophy of logic. (shrink)

We propose a logic for reasoning about metric spaces with the induced topologies. It combines the ‘qualitative’ interior and closure operators with ‘quantitative’ operators ‘somewhere in the sphere of radiusr’ including or excluding the boundary. We supply the logic with both the intended metric space semantics and a natural relational semantics, and show that the latter (i) provides finite partial representations of (in general) infinite metric models and (ii) reduces the standard ‘ε-definitions’ of closure and interior to simple (...) constraints on relations. These features of the relational semantics suggest a finite axiomatisation of the logic and provide means to prove its EXPTIME-completeness (even if the rational numerical parameters are coded in binary). An extension with metric variables satisfying linear rational (in)equalities is proved to be decidable as well. Our logic can be regarded as a ‘well-behaved’ common denominator of logical systems constructed in temporal, spatial, and similarity-based quantitative and qualitative representation and reasoning. Interpreted on the real line (with its Euclidean metric), it is a natural fragment of decidable temporal logics for specification and verification of real-time systems. On the real plane, it is closely related to quantitative and qualitative formalisms for spatial representation and reasoning, but this time the logic becomes undecidable. (shrink)