### Citations

380 |
The second naive physics manifesto
- Hayes
- 1985
(Show Context)
Citation Context ... Cantor's topological ideas, not least in his writings on the general theory of (extensive and intensive) magnitudes which make up one preliminary stage on the road to the formal ontology of the Logical Investigations. (See [8], pp. 83f, 95, 413; [l], "Prolegomena",§§ 22 and 70.) In what follows we shall outline the basic concepts of formal ontology as Husserl conceived it, concentrating our remarks on the specific case of the world of everyday human experience. We shall thus cover some of the same ground that is covered by Patrick Hayes and others working in the territory of 'na'ive physics' [9]. Most recently, formal ontology has been used as a tool of knowledge representation [10], in ways which draw on the insight that the categories of formal ontology, because they are fundamental to a wide variety of different domains, can be fruitfully employed in providing frameworks for translating between knowledge systems constructed on divergent bases. 2 . Mereology vs Set Theory When modern-day philosophers and those working in the artificial intelligence field turn their attentions to ontology, they standardly begin not with mereology or topology but with set-theoretic tools of the sort ... |

231 | Formal ontology in conceptual analysis and knowledge representation
- Guarino, Poli
- 1995
(Show Context)
Citation Context ...ve and intensive) magnitudes which make up one preliminary stage on the road to the formal ontology of the Logical Investigations. (See [8], pp. 83f, 95, 413; [l], "Prolegomena",§§ 22 and 70.) In what follows we shall outline the basic concepts of formal ontology as Husserl conceived it, concentrating our remarks on the specific case of the world of everyday human experience. We shall thus cover some of the same ground that is covered by Patrick Hayes and others working in the territory of 'na'ive physics' [9]. Most recently, formal ontology has been used as a tool of knowledge representation [10], in ways which draw on the insight that the categories of formal ontology, because they are fundamental to a wide variety of different domains, can be fruitfully employed in providing frameworks for translating between knowledge systems constructed on divergent bases. 2 . Mereology vs Set Theory When modern-day philosophers and those working in the artificial intelligence field turn their attentions to ontology, they standardly begin not with mereology or topology but with set-theoretic tools of the sort that are employed in standard model-theoretic semantics. The rationale for insisting on a... |

112 | Mereotopology: A Theory of Parts and Boundaries. - Smith - 1996 |

92 | Parts, wholes, and part-whole relations: The prospects of mereotopology.
- Varzi
- 1996
(Show Context)
Citation Context ...mereology) and is necessarily such that (for the theory of dependence). 'xis part ofy', which we shall symbolize by means of 'P(x,y)', is to be understood as including the limit case where x and y are identical. 'PP(x,y)' shall stand for 'xis a proper part of y' and we shall use 'x + y' to signify the mereological sum of two objects x and y and 'x x y' to indicate their mereological product or intersection. If we define overlap as the sharing of common parts: DMl O(x,y) := 3z(P(z,x) A P(z,y)) overlap then the axioms for standard (non-tensed) mereology can be formulated as follows ([3], [ 18], [20]): AMI P(x,x) reflexivity AM2 P(x,y) " P(y,x) ~ x = y antisymmetry AM3 P(x,y) " P(y,z) ~ P(x,z) transitivity AM4 Vz(P(z,x) ~ O(z,y)) ~ P(x,y) extensionality AMS 3x(<j>x) ~ 3y\7'z (O(y,z) H 3x (<j>x A O(x,z))) fusion (Here and in the sequel initial universal quantifiers are to be taken as understood.) Parthood is a reflexive, antisymmetric, and transitive relation, a partial ordering. In addition, AM4 ensures that parthood is extensional (that no two distinct things have the same parts), where the schema AMS guarantees that for every satisfied property or condition <!> there exists an object, t... |

48 |
On the boundary between mereology and topology’.
- Varzi
- 1993
(Show Context)
Citation Context ...ependent upon me, and thus, by the definition of specific dependence it is not a part of me. 5.4 The Concept of Closure The two approaches to topology sketched above may be unified into a single system by means of the notion of closure, which we can think of as an operation of such a sort that, when applied to an object x it results in a whole which comprehends both x and its boundaries. We employ as basis of our definition of closure the notions of mereology outlined above, which we use to provide an axiomatization of topology that is the mereological counterpart of classical topology ([18], [19], [20]). . An operation of closure (c) is defined in such a way as to satisfy the following axioms: ACl P(x, c(x)) expansiveness (each object is a part of its closure) AC2 P(c(c(x)), c(x)) idempotence (the closure of the closure adds nothing to the closure of an object) AC3 c(x + y) = c(x) + c(y) additivity (the closure of the sum of two objects is equal to the sum of their closures) These axioms, the so-called Kuratowski axioms, define a well-known kind of structure, that of a closure algebra, which is the algebraic equivalent of the simplest kind of topological space. (See [21]) 5.5 The Conc... |

47 |
An essay in formal ontology.
- Smith
- 1978
(Show Context)
Citation Context ...s with the interconnections of truths (or of propositional meanings in general) - with inference relations, consistency, proof and validity. Formal ontology deals with the interconnections of things, with objects and properties, parts and wholes, relations and collectives. As formal logic deals with properties of inferences which are formal in the sense that they apply to inferences in virtue of their form alone, so formal ontology deals with properties of objects which are formal in the sense that they can be exemplified, in principle, by objects in all material spheres or domains of reality [2]. · Husserl's formal ontology is based on mereology, on the theory of dependence, and on topology. The title of his third Logical Investigation is "On the Theory of Wholes and Parts" and it divides into two chapters: "The Difference between Independent and Dependent Objects" and "Thoughts Towards a Theory of the Pure Forms of Wholes and Parts". Unlike more familiar 'extensional' theories of wholes and parts, such as those propounded by Lesniewski (whom Husserl influenced), and by Leonard and Goodman (see [3]), Husserl's theory does not concern itself merely with what we might think of as the v... |

33 |
Boundaries: An Essay in Mereotopology.” In
- Smith
- 1997
(Show Context)
Citation Context ...Dependence As we have seen, substances and accidents may be compounded together mereologically to form larger wholes of different sorts. But substances and accidents are. not themselves related mereologica!ly: a substance is not a whole made up of accidents as parts. Rather, the two are linked together via the formal tie of specific dependence, which might be defined as follows (see also [5], and compare the treatment of 'notional' dependence in [3]): DDl SD(x,y) := -,Q(x,y) "-... (E!x ~ E!y) specific dependence where ·-; signifies the de re necessity operator 'xis necessarily such that' (see [17]) and 'E!' is the predicate of existence defined in the usual way in terms of the existential quantifier: E!x := 3y(x = y). Thus to say that xis specifically dependent on y is to say that x and y do not overlap and that xis necessarily such that if x exists then y exists. 23 r'i' I 24 B. Smith/ Basic Concepts of Formal Ontology 11 Ii! We can now define mutual and one-sided specific dependence, in the obvious way, as follows: DD2 DD3 MSD(x,y) := SD(x,y)" SD(y,x) OSD(x,y) := SD(x,y) "-.MSD(x,y) mutual specific dependence one-sided specific dependence My headache, for example, is one-sidedly spec... |

30 | Fiat and Bona Fide Boundaries: Towards on Ontology of Spatially Extended Objects",
- Smith, Varzi
- 1997
(Show Context)
Citation Context ...s this is manifested in visual p~rception. Here the boundary of a figure is experienced as a part of the figure, and not simultaneously as the boundary of the ground, which is experienced as running on behind the figure. Something similar applies also in the temporal sphere: the beginning and ending of a race, for example, are not in the same sense boundaries of any complement-entities (of all time prior to the race, and of all time subsequent to the race) as they are boundaries of the race itself. (On the usefulness of such variant, nonclassical topolOgies for formal ontology see [17], [23], [24]) B. Smith I Basic Concepts of Formal Ontology The notion of interior is defined as follows: DB4 i(x) := x -b(x) interior We may define a closed object as an object which is identical with its closure. An open object, si~la~ly, is an object which is identic~l with its interior. The complement of a closed object 1s thus open, that of an open object closed. Some objects will be partly open. and partly closed. (Consider for example the semi-open interval (0, l], which consists ~fall real numbers x which are greater than 0 and less than or equal to 1.) These notions can be used to relate the two a... |

8 |
The Structure of Spatial Location'',
- RobertoCasati, Varzi
- 1996
(Show Context)
Citation Context ...ing Aristotle, we shall call 'substances'. Examples of substances are: animals (including human beings), logs of wood, r~cks, potatoes, and forks. Substances have various properties (qualities, features, .attnbutes) and they undergo various sorts of changes (processes, events), for all of which we shall employ, again following Aristotle, the term 'accident'. Examples of ~ccidents are:. whistles, blushes, speakings, runnings, my knowledge of French, the whiteness of this cheese, and the warmth of this stone. Other sorts of denizens of mesoscop!c reality (for example holes [14], spatial regions [15], the niches into which mesoscopi~ substances fit, cultural and institutional objects) will not be treated here, though objects of these sorts, as well as the microscopic objects which serve in some sense as the core material of mesoscopic reality, would need to be dealt with in a more extended treatment. 21 l h I I I I 22 B. Smith I Basic Concepts of Formal Ontology Accidents, like substances, are individual denizens of reality. My headache, like my lump of cheese, exists here and now, and both will cease to exist at some time in the future. Substances and accidents are nonetheless radically ... |

8 | On Substances, Accidents and Universals: In Defence of a Constituent Ontology", - Smith - 1997 |

4 |
On Drawing Lines on a Map", in Andrew U. Frank und Werner Kuhn (Hrsg.), Spatial lnformation Theory. A Theoretical Basis for
- Smith
- 1995
(Show Context)
Citation Context ...ture as this is manifested in visual p~rception. Here the boundary of a figure is experienced as a part of the figure, and not simultaneously as the boundary of the ground, which is experienced as running on behind the figure. Something similar applies also in the temporal sphere: the beginning and ending of a race, for example, are not in the same sense boundaries of any complement-entities (of all time prior to the race, and of all time subsequent to the race) as they are boundaries of the race itself. (On the usefulness of such variant, nonclassical topolOgies for formal ontology see [17], [23], [24]) B. Smith I Basic Concepts of Formal Ontology The notion of interior is defined as follows: DB4 i(x) := x -b(x) interior We may define a closed object as an object which is identical with its closure. An open object, si~la~ly, is an object which is identic~l with its interior. The complement of a closed object 1s thus open, that of an open object closed. Some objects will be partly open. and partly closed. (Consider for example the semi-open interval (0, l], which consists ~fall real numbers x which are greater than 0 and less than or equal to 1.) These notions can be used to relate the... |

2 |
Part-Whole",
- Fine
- 1995
(Show Context)
Citation Context ...ecessary interdependence). Such parts, for example the individual instances of hue, saturation and brightness involved in a given instance of colour, cannot, as a matter of necessity, exist, except in association with their complementary parts in a whole of the given type. There is a huge variety of such lateral dependence relations, giving rise to a correspondingly huge variety of different types of whole which the more standard approaches of extensional mereology are unable to distinguish [ 4]. The topological background of Husserl's work makes itself felt already in his theory of dependence[5] . It comes most clearly to the fore, however, in his treatment of the notion of fusion: the relation which holds between two adjacent parts of an extended totality when there is no qualitative discontinuity between the two [6]. Adjacent squares on a chess-board array are not fused together in this sense; but if we 19 20 B. Smith I Basic Concepts of Formal Ontology imagine a band of colour that is subject to a gradual transition from red through orange to yellow, then each region of this band is fused with its immediately adjacent regions. In the field of what is experienced perceptually, then... |

1 | Logische Untersuchu11ge11, !st ed., Halle: Niemeyer, 1900/01, 2nd ed., 1913/21 (both now available in a comparative edition as Husserlia11a XVIII-XIX, The Hague: Nijhoff, - Husserl - 1975 |

1 |
Parts. A Study i11 011tology,
- Simons
- 1987
(Show Context)
Citation Context ...be exemplified, in principle, by objects in all material spheres or domains of reality [2]. · Husserl's formal ontology is based on mereology, on the theory of dependence, and on topology. The title of his third Logical Investigation is "On the Theory of Wholes and Parts" and it divides into two chapters: "The Difference between Independent and Dependent Objects" and "Thoughts Towards a Theory of the Pure Forms of Wholes and Parts". Unlike more familiar 'extensional' theories of wholes and parts, such as those propounded by Lesniewski (whom Husserl influenced), and by Leonard and Goodman (see [3]), Husserl's theory does not concern itself merely with what we might think of as the vertical relations between parts and the wholes which comprehend them on successive levels of comprehensiveness. Rather, his theory is concerned also with the horizontal relations between co-existing parts, relations which serve to give unity or integrity to the wholes in question. To put the matter simply: some parts of a whole t;xist merely side by side, they can be destroyed or removed from the whole without detriment to the residue. A whole, all of whose parts manifest exclusively such side-by-sideness re... |

1 |
Fusion",
- Casali
- 1991
(Show Context)
Citation Context ...plementary parts in a whole of the given type. There is a huge variety of such lateral dependence relations, giving rise to a correspondingly huge variety of different types of whole which the more standard approaches of extensional mereology are unable to distinguish [ 4]. The topological background of Husserl's work makes itself felt already in his theory of dependence[5] . It comes most clearly to the fore, however, in his treatment of the notion of fusion: the relation which holds between two adjacent parts of an extended totality when there is no qualitative discontinuity between the two [6]. Adjacent squares on a chess-board array are not fused together in this sense; but if we 19 20 B. Smith I Basic Concepts of Formal Ontology imagine a band of colour that is subject to a gradual transition from red through orange to yellow, then each region of this band is fused with its immediately adjacent regions. In the field of what is experienced perceptually, then, we can draw a distinction, between intuitively separated contents, contents set in relief from or separated off from adjoining contents, on the one hand, and contents which are fused with adjoining contents, or which flow ove... |

1 | Phenomenology of, Perception, Qualitative Physics and Sheaf Mereology", - Petito - 1994 |

1 |
Studien zur Arithmetik u11d Geometrie. Texte aus dem Nachlass, 1886-1901, The Hague: Nijhoff,
- Husserl
- 1983
(Show Context)
Citation Context ... given that he was a student of the mathematician Weierstrass in Berlin, and that it was Cantor, Husserl's friend and colleague in Halle during the period when the Logical Investigations were being written, who first defined the fundamental topological notions of open, closed, dense, perfect set, boundary of a set, accumulation point, and so on. Husserl consciously employed Cantor's topological ideas, not least in his writings on the general theory of (extensive and intensive) magnitudes which make up one preliminary stage on the road to the formal ontology of the Logical Investigations. (See [8], pp. 83f, 95, 413; [l], "Prolegomena",§§ 22 and 70.) In what follows we shall outline the basic concepts of formal ontology as Husserl conceived it, concentrating our remarks on the specific case of the world of everyday human experience. We shall thus cover some of the same ground that is covered by Patrick Hayes and others working in the territory of 'na'ive physics' [9]. Most recently, formal ontology has been used as a tool of knowledge representation [10], in ways which draw on the insight that the categories of formal ontology, because they are fundamental to a wide variety of different... |

1 | Mereology as a Theory of Part-Whole", Logique et Analyse, - Boehman - 1990 |

1 |
Continua without Sets", Logic and Logical Philosophy, I
- Asenjo
- 1993
(Show Context)
Citation Context ...ed continuum seems not to be isomorphic to any real-number structure; indeed, standard mathematical oppositions, such as that between a dense and a continuous series, here find no application. Employing set theory as a tool of ontology thus brings new problems of its own, problems which are artefacts of the theory itself and which seem not to reflect features of the intended domain of application. 4., Most set-theoretical constmctions of the continuum are predicated on the highly quest10nable thesis that out of unextended building blocks an extended whole can somehow be constructed (See [12], [13]). The experienced continuum, in contrast, is organi~ed not in such a way that it W?uld ~e built up out of unextended points, but rather m such a way that the wholes, mcludmg the medium of space, come before the parts which these wholes might contain and which might be distinguished on various levels within them. Of course, set theory is a mathematical theory of tremendous power, and none of the above precludes the possibility of recoqstructing topological and other theories adequate for ontological purposes also on a set-theoretic basis. Standard representation theorems indeed imply that for ... |

1 | and Achille Varzi Holes and Other Supe1ficialities, - Casali - 1994 |

1 |
Sur !'operation A d'analysis situs",
- Kuratowski
- 1922
(Show Context)
Citation Context ...pology ([18], [19], [20]). . An operation of closure (c) is defined in such a way as to satisfy the following axioms: ACl P(x, c(x)) expansiveness (each object is a part of its closure) AC2 P(c(c(x)), c(x)) idempotence (the closure of the closure adds nothing to the closure of an object) AC3 c(x + y) = c(x) + c(y) additivity (the closure of the sum of two objects is equal to the sum of their closures) These axioms, the so-called Kuratowski axioms, define a well-known kind of structure, that of a closure algebra, which is the algebraic equivalent of the simplest kind of topological space. (See [21]) 5.5 The Concept of Connectedness , On the basis of the notion of closure we can now define the standard topological notion of (symmetrical) boundary, b(x), as follows: DB2 b(x) := c(x) x c(-x) boundary Note that it is a trivial consequence of the definition of boundary here supplied that the bo~ndary of an object is in every case also the boundary of the complement of that object. It is indeed possible to define in standard topological terms an asymmetrical notion of 'border', as the intersection of an object with the closure of its complement: DB3 b*(x) =xx c(-x) border In_ fa~~· where Kura... |

1 |
Quelques notions fondamentales de I' Analysis Situs aux point du vue de I' Algebre de la Logique'',
- Zarycki
- 1927
(Show Context)
Citation Context ...d topological notion of (symmetrical) boundary, b(x), as follows: DB2 b(x) := c(x) x c(-x) boundary Note that it is a trivial consequence of the definition of boundary here supplied that the bo~ndary of an object is in every case also the boundary of the complement of that object. It is indeed possible to define in standard topological terms an asymmetrical notion of 'border', as the intersection of an object with the closure of its complement: DB3 b*(x) =xx c(-x) border In_ fa~~· where Kuratowski's _axioms were formulated in terms of the single topological pnmitive o~ closure, Zarycki showed [22] that a set of axioms equivalent to those of Kuratowsk1 can be formulated also in terms of the single primitive notion of border, and the same applies, too, in regard to the notions of interior and boundary. A more challenging project is that of devising variant topologies which would recognize boundaries which would be genuinely asymmetric in the sense that seems to be exemplified, for example, in the figure-ground structure as this is manifested in visual p~rception. Here the boundary of a figure is experienced as a part of the figure, and not simultaneously as the boundary of the ground, wh... |

1 | Basic Problems ofMereotopology", in this volume. Formal Ontology in Information Systems N. Guarino (Ed.) - Varzi - 1998 |