|
|
| Line 1: |
Line 1: |
| In [[formal ontology]], a branch of [[metaphysics]], and in [[ontology (computer science)|ontological computer science]], '''mereotopology''' is a [[first-order theory]], embodying [[mereology|mereological]] and [[topological]] concepts, of the relations among wholes, parts, parts of parts, and the [[boundary (topology)|boundaries]] between parts.
| | Yesterday I woke up and [http://Www.bbc.co.uk/search/?q=realized+- realized -] At the moment I have been solitary for a little while and [http://lukebryantickets.citizenswebcasting.com luke bryan tour dates for 2014] after much [http://www.ffpjp24.org luke bryan my kind] bullying from friends I now locate myself registered for book luke Bryan ([http://lukebryantickets.omarfoundation.org http://lukebryantickets.Omarfoundation.Org]) on line dating. They guaranteed me that there are plenty of interesting, sweet and regular individuals to meet up, therefore here goes the toss!<br>My buddies and household are magnificent and [http://Www.Britannica.com/search?query=hanging hanging] out with them at tavern gigabytes or dishes is definitely critical. I haven't ever been into dance clubs as I come to realize you can do not own a significant conversation with the sound. I additionally luke bryan concert videos ([http://okkyunglee.com made my day]) have two very adorable and undoubtedly cheeky dogs who are almost always enthusiastic to meet up fresh individuals.<br>I endeavor to maintain as physically fit as possible staying at the fitness center several times weekly. I love my athletics and strive to play or watch since many a potential. I shall often at Hawthorn suits being wintertime. Note: In case that you contemplated purchasing an activity I do not mind, I've observed the carnage of fumbling fits at stocktake revenue.<br><br>Also visit my site; luke bryant tickets ([http://lukebryantickets.hamedanshahr.com http://lukebryantickets.hamedanshahr.com]) |
| | |
| ==History and motivation==
| |
| Mereotopology begins with theories [[A. N. Whitehead]] articulated in several books and articles he published between 1916 and 1929. Whitehead's early work is discussed in Kneebone (1963: chpt. 13.5) and Simons (1987: 2.9.1). The theory of Whitehead's 1929 ''[[Process and Reality]]'' augmented the part-whole relation with topological notions such as [[contact (mathematics)|contiguity]] and [[connected space|connection]]. Despite Whitehead's acumen as a mathematician, his theories were insufficiently formal, even flawed. By showing how Whitehead's theories could be fully formalized and repaired, Clarke (1981, 1985) founded contemporary mereotopology.<ref>Casati & Varzi (1999: chpt. 4) and Biacino & Gerla (1991) have reservations about some aspects of Clarke's formulation.</ref> The theories of Clarke and Whitehead are discussed in Simons (1987: 2.10.2), and Lucas (2000: chpt. 10). The entry [[Whitehead's point-free geometry]] includes two contemporary treatments of Whitehead's theories, due to Giangiacomo Gerla, each different from the theory set out in the next section.
| |
| | |
| Although mereotopology is a mathematical theory, we owe its subsequent development to [[logic]]ians and theoretical [[computer science|computer scientists]]. Lucas (2000: chpt. 10) and Casati and Varzi (1999: chpts. 4,5) are introductions to mereotopology that can be read by anyone having done a course in [[first-order logic]]. More advanced treatments of mereotopology include Cohn and Varzi (2003) and, for the mathematically sophisticated, Roeper (1997). For a mathematical treatment of [[point-free geometry]], see Gerla (1995). [[lattice (order)|Lattice]]-theoretic ([[algebraic structure|algebraic]]) treatments of mereotopology as [[contact algebra]]s have been applied to separate the [[topology|topological]] from the [[mereology|mereological]] structure, see Stell (2000), Düntsch and Winter (2004).
| |
| | |
| [[Barry Smith (ontologist)|Barry Smith]] (1996), Anthony Cohn and his coauthors, and Varzi alone and with others, have all shown that mereotopology can be useful in [[formal ontology]] and [[ontology (computer science)|computer science]], by formalizing relations such as [[contact (mathematics)|contact]], [[connected space|connection]], [[boundary (topology)|boundaries]], [[interior (topology)|interior]]s, holes, and so on. Mereotopology has been most useful as a tool for qualitative [[spatial-temporal reasoning]], with constraint calculi such as the [[Region Connection Calculus]] (RCC).
| |
| | |
| ==Preferred approach of Casati & Varzi==
| |
| Casati and Varzi (1999: chpt.4) set out a variety of mereotopological theories in a consistent notation. This section sets out several nested theories that culminate in their preferred theory '''GEMTC''', and follows their exposition closely. The mereological part of '''GEMTC''' is the conventional theory '''[[mereology|GEM]]'''. Casati and Varzi do not say if the [[model theory|models]] of '''GEMTC''' include any conventional [[topological space]]s.
| |
| | |
| We begin with some [[domain of discourse]], whose elements are called [[individual]]s (a [[synonym]] for [[mereology]] is "the calculus of individuals"). Casati and Varzi prefer limiting the ontology to physical objects, but others freely employ mereotopology to reason about geometric figures and events, and to solve problems posed by research in [[machine intelligence]].
| |
| | |
| An upper case Latin letter denotes both a [[relation (mathematics)|relation]] and the [[Predicate (mathematical logic)|predicate]] letter referring to that relation in [[first-order logic]]. Lower case letters from the end of the alphabet denote variables ranging over the domain; letters from the start of the alphabet are names of arbitrary individuals. If a formula begins with an [[atomic formula]] followed by the [[biconditional]], the subformula to the right of the biconditional is a definition of the atomic formula, whose variables are [[bound variable|unbound]]. Otherwise, variables not explicitly quantified are tacitly [[universal quantifier|universally quantified]]. The axiom '''Cn''' below corresponds to axiom '''C.n''' in Casati and Varzi (1999: chpt. 4).
| |
| | |
| We begin with a topological primitive, a [[binary relation]] called ''connection''; the atomic formula ''Cxy'' denotes that "''x'' is connected to ''y''." Connection is governed, at minimum, by the axioms:
| |
| | |
| '''C1'''. <math>\ Cxx.</math> ([[Reflexive relation|reflexive]])
| |
| | |
| '''C2'''. <math> Cxy \rightarrow Cyx.</math> ([[Symmetric relation|symmetric]])
| |
| | |
| Now posit the binary relation ''E'', defined as:
| |
| | |
| <math>Exy \leftrightarrow [Czx \rightarrow Czy].</math>
| |
| | |
| ''Exy'' is read as "''y'' encloses ''x''" and is also topological in nature. A consequence of '''C1-2''' is that ''E'' is [[Reflexive relation|reflexive]] and [[Transitive relation|transitive]], and hence a [[preorder]]. If ''E'' is also assumed [[axiom of extensionality|extensional]], so that:
| |
| | |
| <math> (Exa \leftrightarrow Exb) \leftrightarrow (a=b),</math>
| |
| | |
| then ''E'' can be proved [[Antisymmetric relation|antisymmetric]] and thus becomes a [[partial order]]. Enclosure, notated ''xKy'', is the single primitive relation of the [[Whitehead's point-free geometry|theories in Whitehead (1919, 1925)]], the starting point of mereotopology.
| |
| | |
| Let ''parthood'' be the defining primitive [[binary relation]] of the underlying [[mereology]], and let the [[atomic formula]] ''Pxy'' denote that "''x'' is part of ''y''". We assume that ''P'' is a [[partial order]]. Call the resulting minimalist mereological theory '''M'''.
| |
| | |
| If ''x'' is part of ''y'', we postulate that ''y'' encloses ''x'':
| |
| | |
| '''C3'''. <math>\ Pxy \rightarrow Exy.</math>
| |
| | |
| '''C3''' nicely connects [[mereology|mereological]] parthood to [[topological]] enclosure.
| |
| | |
| Let ''O'', the binary relation of mereological ''overlap'', be defined as:
| |
| | |
| <math> Oxy \leftrightarrow \exist z[Pzx \and\ Pzy].</math>
| |
| | |
| Let ''Oxy'' denote that "''x'' and ''y'' overlap." With ''O'' in hand, a consequence of '''C3''' is:
| |
| | |
| <math>Oxy \rightarrow Cxy.</math>
| |
| | |
| Note that the [[Conversion (logic)|converse]] does not necessarily hold. While things that overlap are necessarily connected, connected things do not necessarily overlap. If this were not the case, [[topology]] would merely be a model of [[mereology]] (in which "overlap" is always either primitive or defined).
| |
| | |
| Ground mereotopology ('''MT''') is the theory consisting of primitive ''C'' and ''P'', defined ''E'' and ''O'', the axioms '''C1-3''', and axioms assuring that ''P'' is a [[partial order]]. Replacing the '''M''' in '''MT''' with the standard [[axiom of extension|extensional]] mereology '''[[mereology|GEM]]''' results in the theory '''GEMT'''.
| |
| | |
| Let ''IPxy'' denote that "''x'' is an internal part of ''y''." ''IP'' is defined as:
| |
| | |
| <math>IPxy \leftrightarrow (Pxy \and (Czx \rightarrow Ozy)).</math>
| |
| | |
| Let σ''x'' φ(''x'') denote the mereological sum (fusion) of all individuals in the domain satisfying φ(''x''). σ is a [[bound variable|variable binding]] [[Substring|prefix]] operator. The axioms of '''GEM''' assure that this sum exists if φ(''x'') is a [[first-order logic|first-order formula]]. With σ and the relation ''IP'' in hand, we can define the [[interior (topology)|interior]] of ''x'', <math>\mathbf{i}x,</math> as the mereological sum of all interior parts ''z'' of ''x'', or:
| |
| | |
| <math>\mathbf{i}x \leftrightarrow \sigma z[IPzx].</math>
| |
| | |
| Two easy consequences of this definition are:
| |
| | |
| <math>\mathbf{i}W \leftrightarrow W,</math>
| |
| | |
| where ''W'' is the universal individual, and
| |
| | |
| '''C5'''.<ref>The axiom '''C4''' of Casati and Varzi (1999) is irrelevant to this entry.</ref> <math>\ P(\mathbf{i}x)x.</math> ([[inclusion (set theory)|Inclusion]])
| |
| | |
| The operator '''i''' has two more axiomatic properties:
| |
| | |
| '''C6'''. <math>\mathbf{i}(\mathbf{i}x) \leftrightarrow \mathbf{i}x.</math> ([[Idempotence]])
| |
| | |
| '''C7'''. <math>\mathbf{i}(x \times y) \leftrightarrow \mathbf{i}x \times \mathbf{i}y,</math>
| |
| | |
| where ''a''×''b'' is the mereological product of ''a'' and ''b'', not defined when ''Oab'' is false. '''i''' distributes over product.
| |
| | |
| It can now be seen that '''i''' is [[isomorphic]] to the [[interior operator]] of [[topology]]. Hence the [[Duality (mathematics)|dual]] of '''i''', the topological [[closure operator]] '''c''', can be defined in terms of '''i''', and [[Kuratowski]]'s axioms for '''c''' are theorems. Likewise, given an axiomatization of '''c''' that is analogous to '''C5-7''', '''i''' may be defined in terms of '''c''', and '''C5-7''' become theorems. Adding '''C5-7''' to '''GEMT''' results in Casati and Varzi's preferred mereotopological theory, '''GEMTC'''.
| |
| | |
| ''x'' is ''self-connected'' if it satisfies the following predicate:
| |
| | |
| <math> SCx \leftrightarrow ((Owx \leftrightarrow (Owy \or Owz)) \rightarrow Cyz).</math>
| |
| | |
| Note that the primitive and defined predicates of '''MT''' alone suffice for this definition. The predicate ''SC'' enables formalizing the necessary condition given in [[A. N. Whitehead|Whitehead]]'s ''[[Process and Reality]]'' for the mereological sum of two individuals to exist: they must be connected. Formally:
| |
| | |
| '''C8'''. <math> Cxy \rightarrow \exist z[SCz \and Ozx \and (Pwz \rightarrow (Owx \or Owy)).</math>
| |
| | |
| Given some mereotopology '''X''', adding '''C8''' to '''X''' results in what Casati and Varzi call the ''Whiteheadian extension'' of '''X''', denoted '''WX'''. Hence the theory whose axioms are '''C1-8''' is '''WGEMTC'''.
| |
| | |
| The converse of '''C8''' is a '''GEMTC''' theorem. Hence given the axioms of '''GEMTC''', ''C'' is a defined predicate if ''O'' and ''SC'' are taken as primitive predicates.
| |
| | |
| If the underlying mereology is [[Atomic formula|atom]]less and weaker than '''GEM''', the axiom that assures the absence of atoms ('''P9''' in Casati and Varzi 1999) may be replaced by '''C9''', which postulates that no individual has a [[boundary (topology)|topological boundary]]:
| |
| | |
| '''C9'''. <math> \forall x \exist y[Pyx \and (Czy \rightarrow Ozx) \and \lnot (Pxy \and (Czx \rightarrow Ozy))].</math>
| |
| | |
| When the domain consists of geometric figures, the boundaries can be points, curves, and surfaces. What boundaries could mean, given other ontologies, is not an easy matter and is discussed in Casati and Varzi (1999: chpt. 5).
| |
| | |
| ==See also==
| |
| *[[Mereology]]
| |
| *[[Pointless topology]]
| |
| *[[Point-set topology]]
| |
| *[[Topology]]
| |
| *[[Topological space]] (with links to [[T0 space|T0]] through [[Perfectly normal Hausdorff space|T6]])
| |
| *[[Whitehead's point-free geometry]]
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| *Biacino L., and Gerla G., 1991, "[http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.ndjfl/1093635748 Connection Structures,]" ''Notre Dame Journal of Formal Logic'' 32: 242-47.
| |
| * Casati, R., and Varzi, A. C., 1999. ''Parts and places: the structures of spatial representation''. MIT Press.
| |
| * Clarke, Bowman, 1981, "[http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.ndjfl/1093883455 A calculus of individuals based on 'connection',]" ''Notre Dame Journal of Formal Logic 22'': 204-18.
| |
| * ------, 1985, "[http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.ndjfl/1093870761 Individuals and Points,]" ''Notre Dame Journal of Formal Logic 26'': 61-75.
| |
| *Cohn, A. G., and Varzi, A. C., 2003, "[http://www.columbia.edu/~av72/papers/Jpl_2003.pdf Mereotopological Connection,]" ''Journal of Philosophical Logic 32'': 357-90.
| |
| *Düntsch, I., and Winter, M., 2004, [http://www.cosc.brocku.ca/Faculty/Winter/JoRMiCS/Vol1/PDF/v1n7.pdf Algebraization and representation of mereotopological structures] ''Journal of Relational Methods in Computer Science 1'': 161-180.
| |
| *Forrest, Peter, 1996, "From Ontology to Topology in the Theory of Regions," ''The Monist 79'': 34-50.
| |
| * Gerla, G., 1995, "[http://www.dmi.unisa.it/people/gerla/www/Down/point-free.pdf Pointless Geometries,]" in Buekenhout, F., Kantor, W. (eds.), ''Handbook of incidence geometry: buildings and foundations''. North-Holland: 1015-31.
| |
| * Kneebone, Geoffrey, 1963. ''Mathematical Logic and the Foundation of Mathematics''. Dover reprint, 2001.
| |
| *[[John Lucas (philosopher)|Lucas, J. R.]], 2000. ''Conceptual Roots of Mathematics''. Routledge. The "prototopology" of chpt. 10 is mereotopology. Strongly informed by the unpublished writings of David Bostock.
| |
| * Randell, D. A., Cui, Z. and Cohn, A. G.: A spatial logic based on regions and connection, Proc. 3rd Int. Conf. on Knowledge Representation and Reasoning, Morgan Kaufmann, San Mateo, pp. 165–176, 1992.
| |
| * Roeper, Peter, 1997, "Region-Based Topology," ''Journal of Philosophical Logic 26'': 251-309.
| |
| * Simons, Peter, 1987 ''Parts: A Study in Ontology''. Oxford University Press
| |
| * Smith, Barry, 1996, "[http://ontology.buffalo.edu/smith/articles/mereotopology.htm Mereotopology: A Theory of Parts and Boundaries]," ''Data and Knowledge Engineering 20 '': 287-303.
| |
| * ------, 1997, "[http://ontology.buffalo.edu/smith/articles/chisholm/chisholm.html Boundaries: An Essay in Mereotopology]" in Hahn, L. (ed.) ''The Philosophy of Roderick Chisholm''. Open Court: 534-61.
| |
| * Stell, John G., 2000, "[http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.25.7820 Boolean connection algebras: A new approach to the Region-Connection-Calculus]", ''Artificial Intelligence 122'': 111-136.
| |
| * Vakarelov, D., 2007, "[http://www.springerlink.com/content/m12516t84tv12256/ Region-Based Theory of Space: Algebras of Regions, Representation Theory, and Logics]" in Gabbay, D., Goncharov, S., Zakharyaschev, M. (eds.), ''Mathematical Problems from Applied Logic II''. Springer: 267-348.
| |
| * Varzi, A. C., 1996, "[http://www.columbia.edu/~av72/papers/Dke_1996.pdf Parts, wholes, and part-whole relations: the prospects of mereotopology,]" ''Data and Knowledge Engineering, 20'': 259-286.
| |
| * ------, 1998, "[http://www.columbia.edu/~av72/papers/Fois_1998.pdf Basic Problems of Mereotopology,]" in Guarino, N., ed., ''Formal Ontology in Information Systems''. Amsterdam: IOS Press, 29-38.
| |
| * ------, 2007, "[http://www.columbia.edu/~av72/papers/Space_2007.pdf Spatial Reasoning and Ontology: Parts, Wholes, and Locations]" in Aiello, M. et al., eds., ''Handbook of Spatial Logics''. Springer-Verlag: 945-1038.
| |
| | |
| ==External links==
| |
| * [[Stanford Encyclopedia of Philosophy]]: [http://plato.stanford.edu/entries/boundary/ Boundary] -- by Achille Varzi. With many references.
| |
| | |
| [[Category:Mathematical axioms]]
| |
| [[Category:Mereology]]
| |
| [[Category:Ontology]]
| |
| [[Category:Topology]]
| |