86:. 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. The theories of Clarke and Whitehead are discussed in Simons (1987: 2.10.2), and Lucas (2000: ch. 10). The entry
187:
Mereotopology is being applied by
Salustri in the domain of digital manufacturing (Salustri, 2002) and by Smith and Varzi to the formalization of basic notions of ecology and environmental biology (Smith and Varzi, 1999, 2002). It has been applied also to deal with vague boundaries in geography
72:(1928) -- see also his (1940). The early historical background of mereotopology is documented in Bélanger and Marquis (2013) and Whitehead's early work is discussed in Kneebone (1963: ch. 13.5) and Simons (1987: 2.9.1). The theory of Whitehead's 1929
63:
in several books and articles he published between 1916 and 1929, drawing in part on the mereogeometry of De Laguna (1922). The first to have proposed the idea of a point-free definition of the concept of topological space in mathematics was
1771:
1635:
233:
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
1993:
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: ch. 5).
1424:
1035:
571:
1352:
253:. 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
2062:
196:
Casati and Varzi (1999: ch.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
798:
704:
1285:
463:
1989:
863:
387:
1230:
1146:
1108:
1192:
336:
176:(RCC). It provides the starting point for the theory of fiat boundaries developed by Smith and Varzi, which grew out of the attempt to distinguish formally between
105:. More advanced treatments of mereotopology include Cohn and Varzi (2003) and, for the mathematically sophisticated, Roeper (1997). For a mathematical treatment of
1666:
1545:
871:
does not necessarily hold. While things that overlap are necessarily connected, connected things do not necessarily overlap. If this were not the case,
2278:
90:
includes two contemporary treatments of
Whitehead's theories, due to Giangiacomo Gerla, each different from the theory set out in the next section.
101:. Lucas (2000: ch. 10) and Casati and Varzi (1999: ch. 4,5) are introductions to mereotopology that can be read by anyone having done a course in
1367:
960:
511:
188:(Smith and Mark, 2003), and in the study of vagueness and granularity (Smith and Brogaard, 2002, Bittner and Smith, 2001, 2001a).
1307:
2063:
Les principes de la connaissance naturelle d’Alfred North
Whitehead — Alfred North Whitehead’s Principles of Natural Knowledge
2085:
Casati & Varzi (1999: ch. 4) and
Biacino & Gerla (1991) have reservations about some aspects of Clarke's formulation.
2313:
2037:
592:
87:
180:
boundaries (in geography, geopolitics, and other domains) which reflect more or less arbitrary human demarcations and
2031:
735:
246:
261:, the subformula to the right of the biconditional is a definition of the atomic formula, whose variables are
2333:
2072:
665:
1247:
409:
1866:
827:
351:
149:
141:
28:
173:
169:
2244:
1201:
1123:
1085:
2180:
2136:(Lecture Notes in Computer Science 988), Berlin/Heidelberg/New York, etc.: Springer, 1995, 475–484.
137:
200:, and follows their exposition closely. The mereological part of GEMTC is the conventional theory
2112:
503:
2231:
2095:
1288:
1153:
580:
242:
1472:
153:
79:
309:
2214:
2197:
266:
235:
2234:”, Proceedings of COSIT Workshop on Spatial Vagueness, Uncertainty and Granularity (2001).
1766:{\displaystyle Cxy\rightarrow \exists z[SCz\land Ozx\land (Pwz\rightarrow (Owx\lor Owy)).}
8:
1854:
1653:
1075:
491:
218:
165:
161:
114:
106:
74:
48:
2163:
1657:
for the mereological sum of two individuals to exist: they must be connected. Formally:
2338:
2013:
2008:
1630:{\displaystyle SCx\leftrightarrow ((Owx\leftrightarrow (Owy\lor Owz))\rightarrow Cyz).}
923:
868:
487:
390:
339:
36:
2146:
2066:, Frankfurt / Paris / Lancaster, ontos verlag, 2007. See also the relevant entries of
93:
Although mereotopology is a mathematical theory, we owe its subsequent development to
2348:
2343:
2023:
1464:
1067:
1047:) denote the mereological sum (fusion) of all individuals in the domain satisfying φ(
250:
211:
102:
1480:
98:
168:, holes, and so on. Mereotopology has been applied also as a tool for qualitative
1648:
603:
281:
157:
145:
118:
110:
83:
60:
20:
2296:
1838:
1052:
611:
262:
254:
183:
boundaries which reflect bona fide physical discontinuities (Smith 1995, 2001).
2317:
2327:
2297:
Boolean connection algebras: A new approach to the Region-Connection
Calculus
911:
630:
584:
258:
2067:
2057:
207:
1419:{\displaystyle \mathbf {i} (x\times y)=\mathbf {i} x\times \mathbf {i} y,}
2076:, Frankfurt / Lancaster, ontos verlag, Process Thought X1 & X2, 2008.
1355:
716:
65:
44:
24:
47:
concepts, of the relations among wholes, parts, parts of parts, and the
1491:
1460:
222:
2003:
1055:
928:
876:
712:
607:
230:
202:
126:
40:
2129:
2056:
Cf. Peter Simons, "Whitehead and
Mereology", in Guillaume Durand et
1030:{\displaystyle IPxy\leftrightarrow (Pxy\land (Czx\rightarrow Ozy)).}
144:
and their co-authors have shown that mereotopology can be useful in
2027:
2018:
1468:
872:
495:
122:
78:
augmented the part-whole relation with topological notions such as
2181:
Surrounding Space: The
Ontology of Organism-Environment Relations
1526:
results in Casati and Varzi's preferred mereotopological theory,
226:
59:
Mereotopology begins in philosophy with theories articulated by
566:{\displaystyle (Exa\leftrightarrow Exb)\leftrightarrow (a=b),}
265:. Otherwise, variables not explicitly quantified are tacitly
94:
2232:
A unified theory of granularity, vagueness and approximation
879:(in which "overlap" is always either primitive or defined).
191:
2290:
Parts and places: the structures of spatial representation
1347:{\displaystyle \mathbf {i} (\mathbf {i} x)=\mathbf {i} x.}
129:
structure, see Stell (2000), Düntsch and Winter (2004).
2264:
of Casati and Varzi (1999) is irrelevant to this entry.
2134:
Spatial
Information Theory. A Theoretical Basis for GIS
478:" and is also topological in nature. A consequence of
300:." Connection is governed, at minimum, by the axioms:
1869:
1669:
1647:
enables formalizing the necessary condition given in
1548:
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1310:
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1204:
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1088:
963:
830:
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312:
152:, by allowing the formalization of relations such as
2198:
Do
Mountains Exist? Towards an Ontology of Landforms
633:. Call the resulting minimalist mereological theory
1498:are theorems. Likewise, given an axiomatization of
1983:
1765:
1639:Note that the primitive and defined predicates of
1629:
1418:
1346:
1279:
1224:
1186:
1140:
1102:
1029:
857:
792:
698:
565:
457:
381:
330:
2219:Annals of Mathematics and Artificial Intelligence
1643:alone suffice for this definition. The predicate
2325:
2202:Environment and Planning B (Planning and Design)
2247:" in Christopher Welty and Barry Smith (eds.),
2096:Mereotopology: A Theory of Parts and Boundaries
1845:, the axiom that assures the absence of atoms (
2132:”, in Andrew U. Frank and Werner Kuhn (eds.),
1196:Two easy consequences of this definition are:
1110:as the mereological sum of all interior parts
1849:in Casati and Varzi 1999) may be replaced by
793:{\displaystyle Oxy\leftrightarrow \exists z.}
1853:, which postulates that no individual has a
2288:Casati, Roberto, and Varzi, Achille, 1999.
1787:results in what Casati and Varzi call the
591:, is the single primitive relation of the
241:An upper case Latin letter denotes both a
1540:if it satisfies the following predicate:
280:We begin with a topological primitive, a
54:
2320:—by Achille Varzi. With many references.
2117:Philosophy and Phenomenological Research
2073:Handbook of Whiteheadian Process Thought
886:) is the theory consisting of primitive
192:Preferred approach of Casati & Varzi
595:, the starting point of mereotopology.
2326:
2249:Formal Ontology in Information Systems
726:, the binary relation of mereological
172:, with constraint calculi such as the
2251:, New York: ACM Press, 2001, 309–321.
699:{\displaystyle \ Pxy\rightarrow Exy.}
249:letter referring to that relation in
206:. Casati and Varzi do not say if the
1799:. Hence the theory whose axioms are
1280:{\displaystyle \ P(\mathbf {i} x)x.}
458:{\displaystyle Exy\leftrightarrow .}
2314:Stanford Encyclopedia of Philosophy
1984:{\displaystyle \forall x\exists y.}
1834:are taken as primitive predicates.
1818:theorem. Hence given the axioms of
1298:has two more axiomatic properties:
858:{\displaystyle Oxy\rightarrow Cxy.}
382:{\displaystyle Cxy\rightarrow Cyx.}
277:in Casati and Varzi (1999: ch. 4).
13:
2283:Notre Dame Journal of Formal Logic
2153:, 20: 2 (September 2001), 131–148.
1927:
1876:
1870:
1682:
751:
593:theories in Whitehead (1919, 1920)
210:of GEMTC include any conventional
121:have been applied to separate the
14:
2360:
2307:
2277:Biacino L., and Gerla G., 1991, "
2245:Granular Partitions and Vagueness
2243:Thomas Bittner and Barry Smith, "
2230:Thomas Bittner and Barry Smith, “
2213:Barry Smith and Berit Brogaard, “
1238:is the universal individual, and
1062:assure that this sum exists if φ(
117:) treatments of mereotopology as
2196:Barry Smith and David M. Mark, “
2179:Barry Smith and Achille Varzi, “
2162:Barry Smith and Achille Varzi, “
2111:Barry Smith and Achille Varzi, “
1406:
1395:
1372:
1334:
1320:
1312:
1261:
1225:{\displaystyle \mathbf {i} W=W,}
1206:
1128:
1090:
2254:
2237:
2224:
2207:
2038:Whitehead's point-free geometry
1837:If the underlying mereology is
1436:is the mereological product of
132:
88:Whitehead's point-free geometry
2190:
2173:
2156:
2139:
2122:
2119:, 60: 2 (March 2000), 401–420.
2105:
2100:Data and Knowledge Engineering
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1141:{\displaystyle \mathbf {i} x=}
1103:{\displaystyle \mathbf {i} x,}
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1:
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2113:Fiat and Bona Fide Boundaries
1486:, can be defined in terms of
221:, whose elements are called
29:ontological computer science
7:
1997:
1510:may be defined in terms of
1074:in hand, we can define the
906:, and axioms assuring that
875:would merely be a model of
273:below corresponds to axiom
10:
2365:
2070:and Will Desmond, (eds.),
1826:is a defined predicate if
1452:distributes over product.
1070:. With σ and the relation
818:in hand, a consequence of
602:be the defining primitive
174:Region Connection Calculus
170:spatial-temporal reasoning
2221:, 35/1–2 (2002), 153–175.
2130:On Drawing Lines on a Map
1775:Given some mereotopology
1518:become theorems. Adding
1187:{\displaystyle \sigma z.}
400:, the binary relation of
2204:, 30(3) (2003), 411–427.
2043:
1455:It can now be seen that
1058:operator. The axioms of
2301:Artificial Intelligence
2170:, 33:2 (1999), 198–222.
947:is an internal part of
2187:, 121 (2002), 139–162.
1985:
1789:Whiteheadian extension
1767:
1631:
1420:
1348:
1281:
1226:
1188:
1142:
1104:
1031:
932:results in the theory
882:Ground mereotopology (
859:
794:
700:
567:
459:
383:
332:
331:{\displaystyle \ Cxx.}
267:universally quantified
55:History and motivation
2279:Connection Structures
2215:Quantum Mereotopology
2185:Theory in Biosciences
2102:, 20 (1996), 287–303.
1986:
1768:
1632:
1502:that is analogous to
1421:
1349:
1282:
1227:
1189:
1143:
1105:
1032:
860:
795:
701:
587:. Enclosure, notated
568:
460:
384:
333:
288:; the atomic formula
109:, see Gerla (1995).
16:Branch of metaphysics
2295:Stell J. G., 2000, "
1867:
1855:topological boundary
1667:
1546:
1368:
1308:
1248:
1202:
1154:
1124:
1086:
961:
828:
736:
666:
648:, we postulate that
512:
410:
352:
310:
236:machine intelligence
2334:Mathematical axioms
1654:Process and Reality
1444:, not defined when
1068:first-order formula
583:and thus becomes a
219:domain of discourse
217:We begin with some
107:point-free geometry
99:computer scientists
75:Process and Reality
2014:Point-set topology
2009:Pointless topology
1981:
1763:
1627:
1479:, the topological
1416:
1344:
1277:
1222:
1184:
1138:
1100:
1027:
922:with the standard
855:
790:
696:
625:". We assume that
606:of the underlying
563:
455:
379:
328:
212:topological spaces
37:first-order theory
2024:Topological space
1465:interior operator
1253:
774:
730:, be defined as:
671:
404:, be defined as:
315:
251:first-order logic
103:first-order logic
70:Dimensionstheorie
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1990:
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1982:
1841:and weaker than
1810:The converse of
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1633:
1628:
1481:closure operator
1425:
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1053:variable binding
1036:
1034:
1033:
1028:
914:. Replacing the
864:
862:
861:
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799:
797:
796:
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772:
711:nicely connects
705:
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697:
669:
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569:
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502:is also assumed
464:
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386:
385:
380:
337:
335:
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296:is connected to
257:followed by the
150:computer science
140:, Anthony Cohn,
119:contact algebras
97:and theoretical
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2026:(with links to
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958:
955:is defined as:
829:
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814:overlap." With
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734:
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604:binary relation
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146:formal ontology
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61:A. N. Whitehead
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51:between parts.
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2308:External links
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2145:Barry Smith, “
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612:atomic formula
610:, and let the
579:can be proved
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292:denotes that "
255:atomic formula
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23:, a branch of
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9:
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133:Applications
127:mereological
113:-theoretic (
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1356:Idempotence
924:extensional
719:enclosure.
717:topological
644:is part of
621:is part of
506:, so that:
504:extensional
223:individuals
138:Barry Smith
123:topological
66:Karl Menger
45:topological
25:metaphysics
2328:Categories
2272:References
2260:The axiom
1795:, denoted
1492:Kuratowski
1461:isomorphic
1448:is false.
1051:). σ is a
926:mereology
894:, defined
492:transitive
286:connection
162:boundaries
158:connection
84:connection
80:contiguity
49:boundaries
2339:Mereology
2164:The Niche
2004:Mereology
1958:→
1943:∧
1928:¬
1925:∧
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1895:∧
1877:∃
1871:∀
1779:, adding
1743:∨
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1680:→
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1577:↔
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1403:×
1383:×
1289:Inclusion
1158:σ
1007:→
992:∧
977:↔
877:mereology
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770:∧
752:∃
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652:encloses
608:mereology
543:↔
528:↔
488:reflexive
474:encloses
438:→
423:↔
402:enclosure
391:symmetric
365:→
340:reflexive
247:predicate
231:mereology
166:interiors
125:from the
115:algebraic
95:logicians
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2349:Topology
2344:Ontology
2318:Boundary
2030:through
2019:Topology
1998:See also
1839:atomless
1469:topology
1076:interior
873:topology
869:converse
600:parthood
496:preorder
482:is that
245:and the
243:relation
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728:overlap
284:called
263:unbound
227:synonym
154:contact
111:Lattice
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1428:where
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1056:prefix
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2151:Topoi
2044:Notes
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1814:is a
1528:GEMTC
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910:is a
629:is a
575:then
498:. If
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2168:Nous
1830:and
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898:and
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810:and
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