Abstract not found

This paper presents a formal theory for reasoning about motion of spatial entities, in a qualitative framework. Taking over a theory intended for spatial entities, we enrich it to achieve a theory whose intended models are spatio-temporal entities, an idea sometimes proposed by philosophers or AI authors but never fully exploited. We show what kind of properties usually assumed as desirable parts of any space-time theory are recovered from our model, thus giving a sound theoretical basis for a natural, qualitative representation of motion.

It is generally accepted that in principle it’s possible to formalize completely almost all of present-day mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In contrast to the QED Manifesto however, we do not offer polemics in support of such a project. We merely try to place the formalization of mathematics in its historical perspective, as well as looking at existing praxis and identifying what we regard as the most interesting issues, theoretical and practical.

This paper deals with the issue of the representation of space and motion, and argues that motion can be taken as a primitive notion on which a theory of space can be built, in which every object is an occurrent and has temporal parts. There has been a lot of discussion around the continuants/ occurrents opposition; while some authors have advocated the use of occurrents only for theories of parts and the geometry of common-sense, the few detailed or convincing work that has been devoted to solving the inherent problems of such an approach has made it easy for its detractors to claim it is a dead-end street. We present here a theory of spatio-temporal entities and show how this theory can be used to define a theory of motion. Thus we define a notion of continuity that is more appropriate than mathematical continuity for characterizing motion, and argue that we have here a basis for a theory of spatio-temporal objects.

1.1 Four assays of quality................................................................ 4 1.2 The structure of appearance.................................................... 11 1.3 Intrinsic versus relational........................................................ 13 1.4 Four refutations....................................................................... 17 2. Qualities and their Places................................................................ 25 2.1 The appearance of space......................................................... 25 2.2 Some brain-mind mysteries..................................................... 26 2.3 Spatial qualia........................................................................... 33 2.4 Appearances partitioned.......................................................... 35 2.5 Ties that bind........................................................................... 38 2.6 Feature-placing introduced...................................................... 43 3 Places Phenomenal and Real............................................................ 47 3.1 Space-time regions.................................................................. 47 3.2 Three varieties of visual field.................................................. 50 3.3 Why I am not an array of impressions..................................... 55 3.4 Why I am not an intentional object......................................... 58 3.5 Sensory identification.............................................................. 61 3.6 Some examples of sensory reference....................................... 66

We explore different intuitive notions of spatio-temporal continuity and give a formal characterization of continuity for space-time histories. We investigate the types of transitions possible for the RCC-8 topological relations under each distinct notion of spatio-temporal continuity and provide a hierarchy of conceptual neighbourhood diagrams.

Continuity from a qualitative perspective is different from both the philosophical and mathematical view of continuity. We explore different intuitive notions of spatio-temporal continuity. We present a general formal framework for continuity and continuous transitions in mereotopology for spatio-temporal histories and thus sketch the correctness of the conceptual neighbourhood for the qualitative spatial representation language RCC-8.

In this paper a model theory for figure ground locations is proposed. Figure ground locations are n-tuples of predicates about relations between regional individuals and a set of regional individuals forming a regional partition of the plane. This language is based on the RCC-theory which is extended by a number of definitions and axioms constraining the models to the set of regular closed sets in a 2-dimensional T4 topological space. In this paper we give two models of figure ground locations: One model in point set topology referring to equivalence classes of regular closed sets. One graph theoretical model which is a set of directed and edgevalued versions of the non-directed non-labeled dual graph of the planar graph representing the regional partition. Both models are shown to be isormorphic with respect to a set of union and intersection operations. 1. INTRODUCTION Much effort has been expended on the problem of constructing formal theories for qualitative spatial reasoning (QS...

Russell had two theories of definite descriptions: one for singular descriptions, another for plural descriptions. We chart its development, in which ‘On Denoting’ plays a part but not the part one might expect, before explaining why it eventually fails. We go on to consider many-valued functions, since they too bring in plural terms—terms such as ‘�4 ’ or the descriptive ‘the inhabitants of London ’ which, like plain plural descriptions, stand for more than one thing. Logicians need to take plural reference seriously if only because mathematicians take many-valued functions seriously. We assess the objection (by Russell, Frege and others) that many-valued functions are illegitimate because the corresponding terms are ambiguous. We also assess the various methods proposed for getting rid of them. Finding the objection ill-founded and the methods ineffective, we introduce a logical framework that admits plural reference, and use it to answer some earlier questions and to raise some more. 1. Russell’s theory of plural descriptions Everybody knows that Russell had a theory of definite descriptions. Not everybody realizes that he had two: one for singular descriptions, another for plural descriptions. The contents of ‘On Denoting ’ have blinkered the popular conception of his agenda. 1.1 The Principles of Mathematics In the Principles class talk is plural talk: ‘so-and-so’s children, or the children of Londoners, afford illustrations ’ of classes; ‘the children of Israel are a class ’ (1903c, pp. 24, 83). Readers brought up on modern set theory must beware. Russell’s plural descriptions each stand for many things, and accordingly his classes are ‘classes as many’: they are many things—the children of Israel are a class—not one. (Unless, of course, they only have a single member. Throughout this paper we use the plural idiom inclusively, to cover the singular as a limiting case. Purists should read ‘the Fs ’ as ‘the F or Fs ’ and adjust the context to suit.) Russell first investigates plural idioms in the chapter on ‘Denoting’. His exposition is complicated, however, by his insistence that distribu-

Many of the spatial attributes in our everyday contexts are vague -- imprecise and indeterminate. Imprecision is intrinsic (such as indefinite boundaries for certain regions) or situational (such as imprecise location information due to ignorance or inherent errors/imprecision in measuring devices). Even though qualitative representations have an intrinsic facility to handle indefinite and imprecise information, specific formalisms have been developed to facilitate representing and reasoning under such vagueness. The notion of an `egg-yolk' structure underlies most of these formalisms. We propose an alternative interpretation for the `eggyolk ' and provide a taxonomic classification of spatial vagueness evolving over time for geographic space.