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| In [[philosophy]] and [[mathematical logic]], '''mereology''' (from the Greek μέρος, root: μερε(σ)-, "part" and the suffix -logy "study, discussion, science") treats parts and the wholes they form. Whereas [[set theory]] is founded on the membership relation between a set and its elements, mereology emphasizes the [[meronomy|meronomic]] relation between entities, which from a set theoretic perspective is closer to that of [[Inclusion (set theory)|inclusion]] between [[set (mathematics)|sets]].
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| Mereology has been axiomatized in various ways as applications of [[predicate logic]] to [[formal ontology]], of which mereology is an important part. A common element of such axiomatizations is the assumption, shared with inclusion, that the part-whole relation [[partial order|orders]] its universe, meaning that everything is a part of itself ([[reflexive relation|reflexivity]]), that a part of a part of a whole is itself a part of that whole ([[transitive relation|transitivity]]), and that two distinct entities cannot each be a part of the other ([[antisymmetric relation|antisymmetry]]). A variant of this axiomatization denies that anything is ever part of itself (irreflexive) while accepting transitivity, from which antisymmetry follows automatically.
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| Standard university texts on logic and mathematics are silent about mereology, which has undoubtedly contributed to its obscurity. Although mereology is an application of [[mathematical logic]], what can be argued a sort of "proto-geometry", it has been wholly developed by logicians, [[ontology|ontologists]], linguists, engineers, and computer scientists, especially those working in [[artificial intelligence]].
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| "Mereology" can also refer to formal work in [[General Systems Theory]] on system decomposition and parts, wholes and boundaries (by, e.g., [[Mihajlo D. Mesarovic]] (1970), [[Gabriel Kron]] (1963), or Maurice Jessel (see (Bowden 1989, 1998)). A hierarchical version of [[Gabriel Kron]]'s Network Tearing was published by Keith Bowden (1991), reflecting David Lewis's ideas on [[Gunk (mereology)|Gunk]]. Such ideas appear in theoretical [[computer science]] and [[theoretical physics|physics]], often in combination with [[Sheaf theory|Sheaf]], [[Topos]], or [[Category Theory]]. See also the work of [[Steve Vickers (computer scientist)|Steve Vickers]] on (parts of) specifications in Computer Science, [[Joseph Goguen]] on physical systems, and Tom Etter (1996, 1998) on Link Theory and [[Quantum Mechanics]].
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| The class concept in [[object-oriented programming]] lends a mereological aspect to programming not found in either [[imperative program]]s or [[declarative program]]s. [[Inheritance (object-oriented programming)|Method inheritance]] enriches this application of mereology by providing for passing procedural information down the part-whole relation, thereby making method inheritance a naturally arising aspect of mereology.
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| ==History==
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| Informal part-whole reasoning was consciously invoked in [[metaphysics]] and [[ontology]] from [[Plato]] (in particular, in the second half of the ''[[Parmenides (dialogue)|Parmenides]]'') and [[Aristotle]] onwards, and more or less unwittingly in 19th-century mathematics until the triumph of [[set theory]] around 1910. [[Ivor Grattan-Guinness]] (2001) sheds much light on part-whole reasoning during the 19th and early 20th centuries, and reviews how [[Georg Cantor|Cantor]] and [[Peano]] devised [[set theory]]. It appears that the first to reason consciously and at length about parts and wholes was [[Edmund Husserl]] in his 1901 ''Logical Investigations'' (Husserl 1970 is the English translation). However, the word "mereology" is absent from his writings, and he employed no symbolism even though his doctorate was in mathematics.
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| [[Stanisław Leśniewski]] coined "mereology" in 1927, from the Greek word μέρος (''méros'', "part"), to refer to a formal theory of part-whole he devised in a series of highly technical papers published between 1916 and 1931, and translated in Leśniewski (1992). Leśniewski's student [[Alfred Tarski]], in his Appendix E to Woodger (1937) and the paper translated as Tarski (1984), greatly simplified Leśniewski's formalism. Other students (and students of students) of Lesniewski elaborated this "Polish mereology" over the course of the 20th century. For a good selection of the literature on Polish mereology, see Srzednicki and Rickey (1984). For a survey of Polish mereology, see Simons (1987). Since 1980 or so, however, research on Polish mereology has been almost entirely historical in nature.
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| [[A.N. Whitehead]] planned a fourth volume of ''[[Principia Mathematica]]'', on [[geometry]], but never wrote it. His 1914 correspondence with [[Bertrand Russell]] reveals that his intended approach to geometry can be seen, with the benefit of hindsight, as mereological in essence. This work culminated in Whitehead (1916) and the mereological systems of Whitehead (1919, 1920).
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| In 1930, Henry Leonard completed a Harvard Ph.D. dissertation in philosophy, setting out a formal theory of the part-whole relation. This evolved into the "calculus of individuals" of Goodman and Leonard (1940). Goodman revised and elaborated this calculus in the three editions of Goodman (1951). The calculus of individuals is the starting point for the post-1970 revival of mereology among logicians, ontologists, and computer scientists, a revival well-surveyed in Simons (1987) and Casati and Varzi (1999).
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| ==Axioms and primitive notions==
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| It is possible to formulate a "naive mereology" analogous to [[naive set theory]]. Doing so gives rise to paradoxes analogous to [[Russell's paradox]]. Let there be an object '''O''' such that every object that is not a proper part of itself is a proper part of '''O'''. Is '''O''' a proper part of itself? No, because no object is a proper part of itself; and yes, because it meets the specified requirement for inclusion as a proper part of '''O'''. (Every object is, of course, an ''improper'' part of itself. Another, though differently structured, paradox can be made using ''improper part'' instead of ''proper part''; and another using ''improper or proper part''.) Hence, mereology requires an [[axioms|axiomatic]] formulation.
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| A mereological "system" is a [[first-order logic|first-order theory]] (with [[identity (philosophy)|identity]]) whose [[universe of discourse]] consists of wholes and their respective parts, collectively called ''objects''. Mereology is a collection of nested and non-nested [[axiomatic system]]s, not unlike the case with [[modal logic]].
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| The treatment, terminology, and hierarchical organization below follow Casati and Varzi (1999: Ch. 3) closely. For a more recent treatment, correcting certain misconceptions, see Hovda (2008). Lower-case letters denote variables ranging over objects. Following each symbolic axiom or definition is the number of the corresponding formula in Casati and Varzi, written in bold.
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| A mereological system requires at least one primitive [[binary relation]] ([[dyadic]] [[Predicate (logic)|predicate]]). The most conventional choice for such a relation is '''Parthood''' (also called "inclusion"), "''x'' is a ''part'' of ''y''", written ''Pxy''. Nearly all systems require that Parthood [[partial order|partially order]] the universe. The following defined relations, required for the axioms below, follow immediately from Parthood alone:
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| *An immediate defined [[Predicate (logic)|predicate]] is "x is a '''proper part''' of ''y''", written ''PPxy'', which holds (i.e., is [[Contentment|satisfied]], comes out true) if ''Pxy'' is true and ''Pyx'' is false. If Parthood is a [[partial order]], ProperPart is a [[strict partial order]].
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| :<math>PPxy \leftrightarrow (Pxy \and \lnot Pyx).</math> '''3.3'''
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| :An object lacking proper parts is an ''atom''. The mereological [[universe of discourse|universe]] consists of all objects we wish to think about, and all of their proper parts:
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| *'''Overlap''': ''x'' and ''y'' overlap, written ''Oxy'', if there exists an object ''z'' such that ''Pzx'' and ''Pzy'' both hold.
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| :<math>Oxy \leftrightarrow \exists z[Pzx \and Pzy ].</math> '''3.1'''
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| :The parts of ''z'', the "overlap" or "product" of ''x'' and ''y'', are precisely those objects that are parts of both ''x'' and ''y''.
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| *'''Underlap''': ''x'' and ''y'' underlap, written ''Uxy'', if there exists an object ''z'' such that ''x'' and ''y'' are both parts of ''z''.
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| :<math>Uxy \leftrightarrow \exists z[Pxz \and Pyz ].</math> '''3.2'''
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| Overlap and Underlap are [[reflexive relation|reflexive]], [[symmetric]], and [[Transitive relation|intransitive]].
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| Systems vary in what relations they take as primitive and as defined. For example, in extensional mereologies (defined below), ''Parthood'' can be defined from Overlap as follows:
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| :<math>Pxy \leftrightarrow \forall z[Ozx \rightarrow Ozy].</math> '''3.31'''
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| The axioms are:
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| *'''Parthood''' [[Partial order|partially orders]] the [[universe]]:
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| :M1, '''[[Reflexive relation|Reflexive]]''': An object is a part of itself.
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| :<math>\ Pxx.</math> '''P.1'''
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| :M2, '''[[Antisymmetric relation|Antisymmetric]]''': If ''Pxy'' and ''Pyx'' both hold, then ''x'' and ''y'' are the same object.
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| :<math>(Pxy \and Pyx) \rightarrow x = y.</math> '''P.2'''
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| :M3, '''[[Transitive relation|Transitive]]''': If ''Pxy'' and ''Pyz'', then ''Pxz''.
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| :<math>(Pxy \and Pyz) \rightarrow Pxz.</math> '''P.3'''
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| *M4, '''Weak Supplementation''': If ''PPxy'' holds, there exists a ''z'' such that ''Pzy'' holds but ''Ozx'' does not.
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| :<math>PPxy \rightarrow \exists z[Pzy \and \lnot Ozx].</math> '''P.4'''
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| *M5, '''Strong Supplementation''': Replace "''PPxy'' holds" in M4 with "''Pyx'' does not hold".
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| :<math>\lnot Pyx \rightarrow \exists z[Pzy \and \lnot Ozx].</math> '''P.5'''
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| *M5', '''Atomistic Supplementation''': If ''Pxy'' does not hold, then there exists an atom ''z'' such that ''Pzx'' holds but ''Ozy'' does not.
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| :<math>\lnot Pxy \rightarrow \exists z[Pzx \and \lnot Ozy \and \lnot \exists v [PPvz]].</math> '''P.5' '''
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| *'''Top''': There exists a "universal object", designated ''W'', such that ''PxW'' holds for any ''x''.
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| :<math>\exists W \forall x [PxW].</math> '''3.20'''
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| :Top is a theorem if M8 holds.
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| *'''Bottom''': There exists an atomic "null object", designated ''N'', such that ''PNx'' holds for any ''x''.
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| :<math>\exists N \forall x [PNx].</math> '''3.22'''
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| *M6, '''Sum''': If ''Uxy'' holds, there exists a ''z'', called the "sum" or "fusion" of ''x'' and ''y'', such that the objects overlapping of ''z'' are just those objects that overlap '''either''' ''x'' or ''y''.
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| :<math>Uxy \rightarrow \exists z \forall v [Ovz \leftrightarrow (Ovx \or Ovy)].</math> '''P.6'''
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| *M7, '''Product''': If ''Oxy'' holds, there exists a ''z'', called the "product" of ''x'' and ''y'', such that the parts of ''z'' are just those objects that are parts of '''both''' ''x'' and ''y''.
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| :<math>Oxy \rightarrow \exists z \forall v [Pvz \leftrightarrow (Pvx \and Pvy)].</math> '''P.7'''
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| :If ''Oxy'' does not hold, ''x'' and ''y'' have no parts in common, and the product of ''x'' and ''y'' is undefined.
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| *M8, '''Unrestricted Fusion''': Let φ(''x'') be a [[first-order logic|first-order]] formula in which ''x'' is a [[free variable]]. Then the fusion of all objects satisfying φ exists.
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| :<math>\exists x [\phi(x)] \to \exists z \forall y [Oyz \leftrightarrow \exists x[\phi (x) \and Oyx]].</math> '''P.8'''
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| :M8 is also called "General Sum Principle", "Unrestricted Mereological Composition", or "Universalism". M8 corresponds to the [[set builder notation|principle of unrestricted comprehension]] of [[naive set theory]], which gives rise to [[Russell's paradox]]. There is no mereological counterpart to this paradox simply because ''Parthood'', unlike set membership, is [[Reflexive relation|reflexive]].
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| *M8', '''Unique Fusion''': The fusions whose existence M8 asserts are also unique. '''P.8' '''
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| *M9, '''Atomicity''': All objects are either atoms or fusions of atoms.
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| :<math> \exists y[Pyx \and \forall z[\lnot PPzy]].</math> '''P.10'''
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| ==Various systems==
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| Simons (1987), Casati and Varzi (1999) and Hovda (2008) describe many mereological systems whose axioms are taken from the above list. We adopt the boldface nomenclature of Casati and Varzi. The best-known such system is the one called ''classical extensional mereology'', hereinafter abbreviated '''CEM''' (other abbreviations are explained below). In '''CEM''', '''P.1''' through '''P.8' ''' hold as axioms or are theorems. M9, ''Top'', and ''Bottom'' are optional.
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| The systems in the table below are [[partial order|partially ordered]] by [[Inclusion (set theory)|inclusion]], in the sense that, if all the theorems of system A are also theorems of system B, but the converse is not [[logical truth|necessarily true]], then B ''includes'' A. The resulting [[Hasse diagram]] is similar to that in [http://plato.stanford.edu/entries/mereology/#4.2 Fig. 2], and Fig. 3.2 in Casati and Varzi (1999: 48).
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| {| class=wikitable
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| |-
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| !Label!!Name!!System!!Included Axioms
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| |-
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| |- style="border-top:1px solid #999;"
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| |-
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| |M1-M3||'''Parthood''' is a partial order||'''M'''||M1–M3
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| |-
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| |M4||'''Weak Supplementation'''||'''MM'''||'''M''', M4
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| |-
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| |M5||'''Strong Supplementation'''||'''EM'''||'''M''', M5
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| |-
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| |M5'||'''Atomistic Supplementation'''|| ||
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| |-
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| |M6||'''General Sum Principle''' (Sum)|| ||
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| |-
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| |M7||'''Product'''||'''CEM'''||'''EM''', M6–M7
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| |-
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| |M8||'''Unrestricted Fusion'''||'''GM'''||'''M''', M8
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| |-
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| | || ||'''GEM'''||'''EM''', M8
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| |-
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| |M8'||'''Unique Fusion'''||'''GEM'''||'''EM''', M8'
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| |-
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| |M9||'''Atomicity'''||'''AGEM'''||M2, M8, M9
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| |-
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| | || ||'''AGEM'''||'''M''', M5', M8
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| |}
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| There are two equivalent ways of asserting that the [[universe]] is [[partial order|partially ordered]]: Assume either M1–M3, or that Proper ''Parthood'' is [[Transitive relation|transitive]] and [[Asymmetric relation|asymmetric]], hence a [[strict partial order]]. Either axiomatization results in the system '''M'''. M2 rules out closed loops formed using ''Parthood'', so that the part relation is [[well-founded]]. Sets are well-founded if the [[axiom of Regularity]] is assumed. The literature contains occasional philosophical and common-sense objections to the transitivity of ''Parthood''.
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| M4 and M5 are two ways of asserting ''supplementation'', the mereological analog of set [[complement (set theory)|complement]]ation, with M5 being stronger because M4 is derivable from M5. '''M''' and M4 yield ''minimal'' mereology, '''MM'''. '''MM''', reformulated in terms of Proper Part, is Simons's (1987) preferred minimal system.
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| In any system in which M5 or M5' are assumed or can be derived, then it can be proved that two objects having the same proper parts are identical. This property is known as ''[[Extensionality]]'', a term borrowed from set theory, for which [[Axiom of Extensionality|extensionality]] is the defining axiom. Mereological systems in which Extensionality holds are termed ''extensional'', a fact denoted by including the letter '''E''' in their symbolic names.
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| M6 asserts that any two underlapping objects have a unique sum; M7 asserts that any two overlapping objects have a unique product. If the universe is finite or if ''Top'' is assumed, then the universe is closed under ''sum''. Universal closure of ''Product'' and of supplementation relative to ''W'' requires ''Bottom''. ''W'' and ''N'' are, evidently, the mereological analog of the [[universal set|universal]] and [[empty set]]s, and ''Sum'' and ''Product'' are, likewise, the analogs of set-theoretical ''[[Union (set theory)|union]]'' and ''[[Intersection (set theory)|intersection]]''. If M6 and M7 are either assumed or derivable, the result is a mereology with ''closure''.
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| Because ''Sum'' and ''Product'' are binary operations, M6 and M7 admit the sum and product of only a finite number of objects. The ''fusion'' axiom, M8, enables taking the sum of infinitely many objects. The same holds for ''Product'', when defined. At this point, mereology often invokes [[set theory]], but any recourse to set theory is eliminable by replacing a formula with a [[quantification|quantified]] variable ranging over a universe of sets by a schematic formula with one [[free variable]]. The formula comes out true (is satisfied) whenever the name of an object that would be a [[Element (mathematics)|member]] of the set (if it existed) replaces the free variable. Hence any axiom with sets can be replaced by an [[axiom schema]] with monadic atomic subformulae. M8 and M8' are schemas of just this sort. The [[syntax]] of a [[first-order theory]] can describe only a [[denumerable]] number of sets; hence, only denumerably many sets may be eliminated in this fashion, but this limitation is not binding for the sort of mathematics contemplated here.
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| If M8 holds, then ''W'' exists for infinite universes. Hence, ''Top'' need be assumed only if the universe is infinite and M8 does not hold. It is interesting to note that ''Top'' (postulating ''W'') is not controversial, but ''Bottom'' (postulating ''N'') is. Leśniewski rejected ''Bottom'', and most mereological systems follow his example (an exception is the work of [[Richard Milton Martin]]). Hence, while the universe is closed under sum, the product of objects that do not overlap is typically undefined. A system with ''W'' but not ''N'' is isomorphic to:
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| * A [[Boolean algebra (structure)|Boolean algebra]] lacking a 0
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| * A [[Join (mathematics)|join]] [[semilattice]] bounded from above by 1. Binary fusion and ''W'' interpret join and 1, respectively.
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| Postulating ''N'' renders all possible products definable, but also transforms classical extensional mereology into a set-free [[model theory|model]] of [[Boolean algebra (logic)|Boolean algebra]].
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| If sets are admitted, M8 asserts the existence of the fusion of all members of any nonempty set. Any mereological system in which M8 holds is called ''general'', and its name includes '''G'''. In any general mereology, M6 and M7 are provable. Adding M8 to an extensional mereology results in ''general extensional mereology'', abbreviated '''GEM'''; moreover, the extensionality renders the fusion unique. On the converse, however, if the fusion asserted by M8 is assumed unique, so that M8' replaces M8, then - as Tarski (1929) had shown - M3 and M8' suffice to axiomatize '''GEM''', a remarkably economical result. Simons (1987: 38–41) lists a number of '''GEM''' theorems.
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| M2 and a finite universe necessarily imply ''Atomicity'', namely that everything either is an atom or includes atoms among its proper parts. If the universe is infinite, ''Atomicity'' requires M9. Adding M9 to any mereological system, '''X''' results in the atomistic variant thereof, denoted '''AX'''. ''Atomicity'' permits economies, for instance, assuming that M5' implies ''Atomicity'' and extensionality, and yields an alternative axiomatization of '''AGEM'''.
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| ==Set theory==
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| [[Stanisław Leśniewski]] rejected set theory, a stance that has come to be known as [[nominalism]]. For a long time, nearly all philosophers and mathematicians avoided mereology, seeing it as tantamount to a rejection of set theory. Goodman too was a nominalist, and his fellow nominalist [[Richard Milton Martin]] employed a version of the calculus of individuals throughout his career, starting in 1941.
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| Much early work on mereology was motivated by a suspicion that [[set theory]] was [[ontology|ontologically]] suspect, and that [[Occam's Razor]] requires that one minimise the number of posits in one's theory of the world and of mathematics. Mereology replaces talk of "sets" of objects with talk of "sums" of objects, objects being no more than the various things that make up wholes.
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| Many logicians and philosophers reject these motivations, on such grounds as:
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| * They deny that sets are in any way ontologically suspect
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| * [[Occam's Razor]], when applied to [[abstract object]]s like sets, is either a dubious principle or simply false
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| * Mereology itself is guilty of proliferating new and ontologically suspect entities such as fusions.
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| For a survey of attempts to found mathematics without using set theory, see Burgess and Rosen (1997).
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| In the 1970s, thanks in part to Eberle (1970), it gradually came to be understood that one can employ mereology regardless of one's ontological stance regarding sets. This understanding is called the "ontological innocence" of mereology. This innocence stems from mereology being formalizable in either of two equivalent ways:
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| *Quantified variables ranging over a [[universe]] of sets
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| *Schematic [[Predicate (mathematical logic)|predicates]] with a single [[free variable]].
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| Once it became clear that mereology is not tantamount to a denial of set theory, mereology became largely accepted as a useful tool for formal [[ontology]] and [[metaphysics]].
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| In set theory, [[Singleton (mathematics)|singletons]] are "atoms" that have no (non-empty) proper parts; many consider set theory useless or incoherent (not "well-founded") if sets cannot be built up from unit sets. The calculus of individuals was thought to require that an object either have no proper parts, in which case it is an "atom", or be the mereological sum of atoms. Eberle (1970) showed how to construct a calculus of individuals lacking "[[Atomism|atoms]]", i.e., one where every object has a "proper part" (defined below) so that the [[universe]] is infinite.
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| There are analogies between the axioms of mereology and those of standard [[Zermelo-Fraenkel set theory]] (ZF), if ''Parthood'' is taken as analogous to [[subset]] in set theory. On the relation of mereology and ZF, also see Bunt (1985). One of the very few contemporary set theorist to discuss mereology is Potter (2004).
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| [[David K. Lewis|Lewis]] (1991) went further, showing informally that mereology, augmented by a few [[ontology|ontological]] assumptions and [[plural quantification]], and some novel reasoning about [[Singleton (mathematics)|singletons]], yields a system in which a given individual can be both a member and a subset of another individual. In the resulting system, the axioms of [[ZFC]] (and of [[Peano arithmetic]]) are theorems.
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| Forrest (2002) revises Lewis's analysis by first formulating a generalization of '''CEM''', called "Heyting mereology", whose sole nonlogical primitive is ''Proper Part'', assumed [[transitive relation|transitive]] and [[antireflexive]]. There exists a "fictitious" null individual that is a proper part of every individual. Two schemas assert that every [[lattice (order)|lattice]] join exists (lattices are [[complete lattice|complete]]) and that meet [[distributive|distributes]] over join. On this Heyting mereology, Forrest erects a theory of ''pseudosets'', adequate for all purposes to which sets have been put.
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| ==Mathematics==
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| Husserl never claimed that mathematics could or should be grounded in part-whole rather than set theory. Lesniewski consciously derived his mereology as an alternative to set theory as a [[foundation of mathematics]], but did not work out the details. Goodman and Quine (1947) tried to develop the [[natural numbers|natural]] and [[real number]]s using the calculus of individuals, but were mostly unsuccessful; Quine did not reprint that article in his ''Selected Logic Papers''. In a series of chapters in the books he published in the last decade of his life, [[Richard Milton Martin]] set out to do what Goodman and Quine had abandoned 30 years prior. A recurring problem with attempts to ground mathematics in mereology is how to build up the theory of [[Relation (mathematics)|relations]] while abstaining from set-theoretic definitions of the [[ordered pair]]. Martin argued that Eberle's (1970) theory of relational individuals solved this problem.
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| To date, the only persons well trained in mathematics to write on mereology have been [[Alfred Tarski]] and Rolf Eberle. Eberle (1970) clarified the relation between mereology and [[Boolean algebra (logic)|Boolean algebra]], and mereology and set theory. He is one of the very few contributors to mereology to prove [[sound]] and [[complete]] each system he describes.
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| [[Topology|Topological]] notions of [[Boundary (topology)|boundaries]] and connection can be married to mereology, resulting in [[mereotopology]]; see Casati and Varzi (1999: chpts. 4,5). Whitehead's 1929 ''[[Process and Reality]]'' contains a good deal of informal [[mereotopology]].
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| ==Mereology and natural language==
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| Bunt (1985), a study of the [[semantics]] of natural language, shows how mereology can help understand such phenomena as the [[mass noun|mass–count distinction]] and [[grammatical aspect|verb aspect]]. But Nicolas (2008) argues that a different logical framework, called [[Plural quantification|plural logic]], should be used for that purpose.
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| Also, [[natural language]] often employs "part of" in ambiguous ways (Simons 1987 discusses this at length). Hence, it is unclear how, if at all, one can translate certain natural language expressions into mereological predicates. Steering clear of such difficulties may require limiting the interpretation of mereology to [[mathematics]] and [[natural science]]. Casati and Varzi (1999), for example, limit the scope of mereology to [[physical object]]s.
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| ==Important surveys==
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| The books by Simons (1987) and Casati and Varzi (1999) differ in their strengths:
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| *Simons (1987) sees mereology primarily as a way of formalizing [[ontology]] and [[metaphysics]]. His strengths include the connections between mereology and:
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| **The work of [[Stanislaw Leśniewski]] and his descendants
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| **Various continental philosophers, especially [[Edmund Husserl]]
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| **Contemporary English-speaking technical philosophers such as [[Kit Fine]] and [[Roderick Chisholm]]
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| **Recent work on [[formal ontology]] and [[metaphysics]], including continuants, occurrents, [[class noun]]s, [[mass noun]]s, and ontological dependence and [[integrity]]
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| **[[Free logic]] as a background logic
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| **Extending mereology with [[tense logic]] and [[modal logic]]
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| **[[Boolean algebra (structure)|Boolean algebra]]s and [[lattice theory]].
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| *Casati and Varzi (1999) see mereology primarily as a way of understanding the material world and how humans interact with it. Their strengths include the connections between mereology and:
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| ** A "proto-geometry" for physical objects
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| ** [[Topology]] and [[mereotopology]], especially [[Boundary (topology)|boundaries]], regions, and holes
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| ** A formal theory of events
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| ** Theoretical [[computer science]]
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| ** The writings of [[Alfred North Whitehead]], especially his ''[[Process and Reality]]'' and work descended therefrom.
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| Simons devotes considerable effort to elucidating historical notations. The notation of Casati and Varzi is often used. Both books include excellent bibliographies.
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| To these works should be added Hovda (2008), which presents the latest state of the art on the axiomatization of mereology.
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| ==See also==
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| * [[Attitude polarization]]
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| * [[Gunk (mereology)]]
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| * [[Implicate and explicate order according to David Bohm]]
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| * [[Mereological essentialism]]
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| * [[Mereological nihilism]]
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| * [[Mereotopology]]
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| * [[Meronymy]]
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| * [[Monad (Greek philosophy)]]
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| * [[Plural quantification]]
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| * [[Quantifier variance]]
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| * [[Simple (philosophy)]]
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| * [[Whitehead's point-free geometry]]
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| ==References==
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| * Bowden, Keith, 1991. ''Hierarchical Tearing: An Efficient Holographic Algorithm for System Decomposition'', Int. J. General Systems, Vol. 24(1), pp 23–38.
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| * Bowden, Keith, 1998. ''Huygens Principle, Physics and Computers''. Int. J. General Systems, Vol. 27(1-3),pp. 9–32.
| |
| * Bunt, Harry, 1985. ''Mass terms and model-theoretic semantics''. Cambridge Univ. Press.
| |
| * Burgess, John, and Rosen, Gideon, 1997. ''A Subject with No Object''. Oxford Univ. Press.
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| * Burkhardt, H., and Dufour, C.A., 1991, "Part/Whole I: History" in Burkhardt, H., and Smith, B., eds., ''Handbook of Metaphysics and Ontology''. Muenchen: Philosophia Verlag.
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| * Casati, R., and Varzi, A., 1999. ''Parts and Places: the structures of spatial representation''. MIT Press.
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| * Eberle, Rolf, 1970. ''Nominalistic Systems''. Kluwer.
| |
| * Etter, Tom, 1996. ''Quantum Mechanics as a Branch of Mereology'' in Toffoli T., ''et al.'', ''PHYSCOMP96, Proceedings of the Fourth Workshop on Physics and Computation'', New England Complex Systems Institute.
| |
| * Etter, Tom, 1998. ''Process, System, Causality and Quantum Mechanics''. SLAC-PUB-7890, Stanford Linear Accelerator Centre.
| |
| * Forrest, Peter, 2002, "[http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdfview_1&handle=euclid.ndjfl/1071509430 Nonclassical mereology and its application to sets]", ''Notre Dame Journal of Formal Logic 43'': 79-94.
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| * [[Nelson Goodman|Goodman, Nelson]], 1977 (1951). ''The Structure of Appearance''. Kluwer.
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| * -------, and [[Willard Quine]], 1947, "Steps toward a constructive nominalism", ''Journal of Symbolic Logic'' 12: 97-122.
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| *Gruszczynski R., and Pietruszczak A., 2008, "[http://www.math.ucla.edu/~asl/bsl/1404/1404-002.ps Full development of Tarski's geometry of solids]", ''Bulletin of Symbolic Logic'' 14: 481-540. A system of geometry based on Lesniewski's mereology, with basic properties of mereological structures.
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| * Hovda, Paul, 2008, "[http://www.springerlink.com/content/76l18850p2325p16/ What is classical mereology?]" ''Journal of Philosophical Logic'' 38(1): 55-82.
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| * [[Edmund Husserl|Husserl, Edmund]], 1970. ''Logical Investigations, Vol. 2''. Findlay, J.N., trans. Routledge.
| |
| * Kron, Gabriel, 1963, ''Diakoptics: The Piecewise Solution of Large Scale Systems''. Macdonald, London.
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| * [[David K. Lewis|Lewis, David K.]], 1991. ''Parts of Classes''. Blackwell.
| |
| * Leonard, H.S., and [[Nelson Goodman|Goodman, Nelson]], 1940, "The calculus of individuals and its uses", ''Journal of Symbolic Logic 5'': 45–55.
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| * Mesarovic, M.D., Macko, D., and Takahara, Y., 1970, "Theory of Multilevel, Hierarchical Systems". Academic Press.
| |
| * Nicolas, David, 2008, "[http://d.a.nicolas.free.fr/Nicolas-Mass-nouns-and-plural-logic-Revised-2.pdf Mass nouns and plural logic]", ''Linguistics and Philosophy'' 31(2): 211–44.
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| * [[Stanisław Leśniewski|Leśniewski, Stanisław]], 1992. ''Collected Works''. Surma, S.J., Srzednicki, J.T., Barnett, D.I., and Rickey, V.F., editors and translators. Kluwer.
| |
| *[[John Lucas (philosopher)|Lucas, J. R.]], 2000. ''Conceptual Roots of Mathematics''. Routledge. Chpts. 9.12 and 10 discuss mereology, mereotopology, and the related theories of [[A.N. Whitehead]], all strongly influenced by the unpublished writings of David Bostock.
| |
| *Pietruszczak A., 1996, "[http://www.logika.umk.pl/llp/04/pietrusz.pdf Mereological sets of distributive classes]", ''Logic and Logical Philosophy'' 4: 105-22. Constructs, using mereology, mathematical entities from set theoretical classes.
| |
| *Pietruszczak A., 2005, "[http://www.logika.umk.pl/llp/142/ap.pdf Pieces of mereology]", ''Logic and Logical Philosophy'' 14: 211-34. Basic mathematical properties of Lesniewski's mereology.
| |
| *Potter, Michael, 2004. '' Set Theory and Its Philosophy''. Oxford Univ. Press.
| |
| * Simons, Peter, 1987. ''Parts: A Study in Ontology''. Oxford Univ. Press.
| |
| * Srzednicki, J. T. J., and Rickey, V. F., eds., 1984. ''Lesniewski's Systems: Ontology and Mereology''. Kluwer.
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| * [[Alfred Tarski|Tarski, Alfred]], 1984 (1956), "Foundations of the Geometry of Solids" in his ''Logic, Semantics, Metamathematics: Papers 1923–38''. Woodger, J., and Corcoran, J., eds. and trans. Hackett.
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| * Varzi, Achille C., 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.
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| *[[A.N. Whitehead|Whitehead, A.N.]], 1916, "La Theorie Relationiste de l'Espace", ''Revue de Metaphysique et de Morale 23'': 423-454. Translated as Hurley, P.J., 1979, "The relational theory of space", ''Philosophy Research Archives 5'': 712-741.
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| *------, 1919. ''An Enquiry Concerning the Principles of Natural Knowledge''. Cambridge Univ. Press. 2nd ed., 1925.
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| *------, 1920. ''The Concept of Nature''. Cambridge Univ. Press. 2004 paperback, Prometheus Books. Being the 1919 Tarner Lectures delivered at [[Trinity College, Cambridge]].
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| *------, 1978 (1929). ''[[Process and Reality]]''. Free Press.
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| * Woodger, J. H., 1937. ''The Axiomatic Method in Biology''. Cambridge Univ. Press.
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| ==External links==
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| {{wiktionary}}
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| * [[Stanford Encyclopedia of Philosophy]]:
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| **"[http://plato.stanford.edu/entries/mereology/ Mereology]" – Achille Varzi.
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| **"[http://plato.stanford.edu/entries/boundary/ Boundary]" – Achille Varzi.
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| **"[http://www.wikinfo.org/index.php/Synergy_and_Dysergy_in_Mereologic_Geometries Synergy and Dysergy in Mereologic Geometries]" - Albert Carpenter
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| [[Category:Mereology| ]]
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| [[Category:Mathematical logic]]
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| [[Category:Ontology]]
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| [[Category:Predicate logic]]
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