The fact that boundaries are ontologically dependent entities is agreed by Franz Brentano and Roderick Chisholm. This article studies both authors as a single metaphysical account about boundaries. The Brentano-Chisholm theory understands that boundaries and the objects to which they belong hold a mutual relationship of ontological dependence: the existence of a boundary depends upon a continuum of higher spatial dimensionality, but also is a conditio sine qua non for the existence of a continuum. Although the view that ordinary material (...) objects and their boundaries (or surfaces) ontologically depend on each other is correct, it does not grasp their asymmetric relationship: while the existence of a surface rigidly depends upon the existence of the very object it belongs to, the existence of a physical object generically depends upon having some surface. In modal terms, both are two kinds of de re ontological dependence that this article tries to distinguish. (shrink)

We think of a boundary whenever we think of an entity demarcated from its surroundings. There is a boundary (a line) separating Maryland and Pennsylvania. There is a boundary (a circle) isolating the interior of a disc from its exterior. There is a boundary (a surface) enclosing the bulk of this apple. Sometimes the exact location of a boundary is unclear or otherwise controversial (as when you try to trace out the margins of Mount Everest, or even the boundary of (...) your own body). Sometimes the boundary lies skew to any physical discontinuity or qualitative differentiation (as with the border of Wyoming, or the boundary between the upper and lower halves of a homogeneous sphere). But whether sharp or blurry, natural or artificial, for every object there appears to be a boundary that marks it off from the rest of the world. Events, too, have boundaries — at least temporal boundaries. Our lives are bounded by our births and by our deaths; the soccer game began at 3pm sharp and ended with the referee's final whistle at 4:45pm. It is sometimes suggested that even abstract entities, such as concepts or sets, have boundaries of their own, and Wittgenstein could emphatically proclaim that the boundaries of our language are the boundaries of our world. Whether all this boundary talk is coherent, however, and whether it reflects the structure of the world or simply the organizing activity of our mind, are matters of deep philosophical controversy. (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.

There is a basic distinction, in the realm of spatial boundaries, between bona fide boundaries on the one hand, and fiat boundaries on the other. The former are just the physical boundaries of old. The latter are exemplified especially by boundaries induced through human demarcation, for example in the geographic domain. The classical problems connected with the notions of adjacency, contact, separation and division can be resolved in an intuitive way by recognizing this two-sorted ontology of boundaries. Bona fide boundaries (...) yield a notion of contact that is effectively modeled by classical topology; the analogue of contact involving fiat boundaries calls, however, for a different account, based on the intuition that fiat boundaries do not support the open/closed distinction on which classical topology is based. In the presence of this two-sorted ontology it then transpires that mereotopology—topology erected on a mereological basis—is more than a trivial formal variant of classical point-set topology. (shrink)

The paper outlines a model-theoretic framework for investigating and comparing a variety of mereotopological theories. In the first part we consider different ways of characterizing a mereotopology with respect to (i) the intended interpretation of the connection primitive, and (ii) the composition of the admissible domains of quantification (e.g., whether or not they include boundary elements). The second part extends this study by considering two further dimensions along which different patterns of topological connection can be classified - the strength of (...) the connection and its multiplicity. (shrink)

Online collection of papers by Devitt, Dretske, Guarino, Hochberg, Jackson, Petitot, Searle, Tye, Varzi and other leading thinkers on philosophy and the foundations of cognitive Science. Topics dealt with include: Wittgenstein and Cognitive Science, Content and Object, Logic and Foundations, Language and Linguistics, and Ontology and Mereology.

A collection of papers presented at the First International Summer Institute in Cognitive Science, University at Buffalo, July 1994, including the following papers: ** Topological Foundations of Cognitive Science, Barry Smith ** The Bounds of Axiomatisation, Graham White ** Rethinking Boundaries, Wojciech Zelaniec ** Sheaf Mereology and Space Cognition, Jean Petitot ** A Mereotopological Definition of 'Point', Carola Eschenbach ** Discreteness, Finiteness, and the Structure of Topological Spaces, Christopher Habel ** Mass Reference and the Geometry of Solids, Almerindo E. Ojeda (...) ** Defining a 'Doughnut' Made Difficult, N .M. Gotts ** A Theory of Spatial Regions with Indeterminate Boundaries, A.G. Cohn and N.M. Gotts ** Mereotopological Construction of Time from Events, Fabio Pianesi and Achille C. Varzi ** Computational Mereology: A Study of Part-of Relations for Multi-media Indexing, Wlodek Zadrozny and Michelle Kim. (shrink)

The perception of spatial bodies is at least in part a perception of bodily boundaries or surfaces. The usual mathematical conception of boundaries as abstract constructions is, however, of little use for cognitive science purposes. The essay therefore seeks a more adequate conception of the ontology of boundaries building on ideas in Aristotle and Brentano on what we may call the coincidence of boundaries. It presents a formal theory of boundaries and of the continua to which they belong, of a (...) sort which allows a resolution of certain Zeno-style paradoxes. The theory proves to be applicable not only in the cognitive science field but also in regard to problems relating to the ontology of geographical and geopolitical boundaries. (shrink)

Can we maintain that purple seems composed of red and blue without giving up the impenetrability of the red and blue parts that compose it? Brentano thinks we can. Purple, according to him, is a chessboard of red and blue tiles which, although individually too small to be perceived, are together indistinctly perceived within the purple. After a presentation of Brentano’s solution, we raise two objections to it. First, Brentano’s solution commits him to unperceivable intentional objects (the chessboard’s tiles). Second, (...) his chessboard account fails in the end to explain the phenomenal spatial continuity of compound colours. We then sketch an alternative account, which, while holding fast to the phenomenal compoundedness of the purple and to the impenetrability of component colours, avoids introducing inaccessible intentional objects and compromising on the continuity of the purple. According to our proposal, instead of being indistinctly perceived spatial parts of the purple, red and blue are distinctly perceived non- spatial parts of it. (shrink)

According to Brentano's theory of boundaries, no boundary can exist without being connected with a continuum. But there is no specifiable part of the continuum, and no point, which is such that we may say that it is the existence of that part or of that point which conditions the boundary. - An adequate theory of the continuum must now recognize that boundaries be boundaries only in certain directions and not in others. This leads to consequences in other areas, too.

Ci imbattiamo in un confine ogni volta che pensiamo a un’entità demarcata rispetto a ciò che la circonda. C’è un confine (una superficie) che delimita l’interno di una sfera dal suo esterno; c’è un confine (una frontiera) che separa il Maryland dalla Pennsylvania. Talvolta la collocazione esatta di un confine non è chiara o è in qualche modo controversa (come quando si cerchi di tracciare i limiti del monte Everest, o il confine del nostro corpo). Talaltra il confine non corrisponde (...) a una discontinuità fisica o a una differenziazione qualitativa (come nel caso dei confini del Wyoming, o della demarcazione tra la metà superiore e la metà inferiore di una sfera omogenea). Ma, che sia netto o confuso, naturale o artificiale, sembra esservi per ogni oggetto un confine che lo separa dal resto del mondo. Anche gli eventi hanno confini, quantomeno confini temporali. Le nostre vite sono delimitate dalle nostre nascite e morti; una partita di calcio comincia alle quindici in punto e termina col fischio finale dell’arbitro alle sedici e quarantacinque. E a volte si dice che anche le entità astratte, come i concetti o gli insiemi, hanno dei propri confini. Se tutto questo parlare di confini abbia davvero un senso, e se rifletta la struttura del mondo o solo l’attività organizzatrice del nostro intelletto, è però oggetto di profonda controversia filosofica. (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 tend to talk about (refer to, quantify over) parts in the same way in which we talk about whole objects. Yet a part is not something to be included in an inventory of the world over and above the whole to which it belongs, and a whole is not something to be included in the inventory over and above its constituent parts. This paper is an attempt to clarify a way of dealing with this tension which may be labeled (...) the Minimalist View: An element in the field of a part-whole relation is to be included in an inventory of the world if and only if it does not overlap any distinct element that is itself included in the inventory. (shrink)