Mereology

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In philosophy, mereology (from the Greek μερος meros part and the ending -logy study, discussion, science) is a collection of axiomatic first-order theories dealing with parts and their respective wholes. Mereology is both an application of predicate logic and a branch of formal ontology.

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[edit] History

Informal part-whole reasoning was consciously invoked in metaphysics and ontology from 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 Cantor and Peano devised set theory. Apparently, 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.

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 of a historical nature.

A.N. Whitehead planned but never published a fourth volume of Principia Mathematica on geometry. 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).

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).

Standard university texts on logic and mathematics are silent about mereology, which has undoubtedly contributed to its undeserved obscurity. Although mereology is an application of mathematical logic, arguably a sort of "proto-geometry," it has been wholly developed by logicians, ontologists, and computer scientists, especially those working in artificial intelligence.

"Mereology" can also refer to formal work on system decomposition (by, e.g., Gabriel Kron or Maurice Jessel), or on parts, wholes and boundaries. Such ideas appear in theoretical computer science and physics, often in combination with Sheaf, Topos, or Category Theory. See the work of Steven Vickers on (parts of) specifications, Joseph Goguen on physical systems, and Tom Etter on Link Theory.

[edit] Axioms and primitive notions

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 axiomatic formulation.

A mereological "system" is a first-order theory (with 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 systems, not unlike the case with modal logic.

The treatment, terminology, and hierarchical organization below follow Casati and Varzi (1999: chpt. 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.

A mereological system requires at least one primitive binary relation (dyadic 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 partially order the universe. The following defined relations, required for the axioms below, follow immediately from Parthood alone:

PPxy \leftrightarrow (Pxy \and \lnot Pyx). 3.3
An object lacking proper parts is an atom. The mereological universe consists of all objects we wish to think about, and all of their proper parts:
  • Overlap: x and y overlap, written Oxy, if there exists an object z such that Pzx and Pzy both hold.
Oxy \leftrightarrow \exists z[Pzx \and Pzy ]. 3.1
The parts of z, the "overlap" or "product" of x and y, are precisely those objects that are parts of both x and y.
  • Underlap: x and y underlap, written Uxy, if there exists an object z such that x and y are both parts of z.
Uxy \leftrightarrow \exists z[Pxz \and Pyz ]. 3.2

Overlap and Underlap are reflexive, symmetric, and intransitive.

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:

Pxy \leftrightarrow (Ozx \rightarrow Ozy). 3.31:

The axioms are:

M1, Reflexive: An object is a part of itself.
\ Pxx. P.1
M2, Antisymmetric: If Pxy and Pyx both hold, then x and y are the same object.
(Pxy \and Pyx) \rightarrow x = y. P.2
M3, Transitive: If Pxy and Pyz, then Pxz.
(Pxy \and Pyz) \rightarrow Pxz. P.3
  • M4, Weak Supplementation: If PPxy holds, there exists a z such that Pzy holds but Ozx does not.
PPxy \rightarrow \exists z[Pzy \and \lnot Ozx]. P.4
  • M5, Strong Supplementation: Replace "PPxy holds" in M4 with "Pyx does not hold."
\lnot Pyx \rightarrow \exists z[Pzy \and \lnot Ozx]. P.5
  • M5', Atomistic Supplementation: If Pxy does not hold, then there exists an atom z such that Pzx holds but Ozy does not.
\lnot Pxy \rightarrow \exists z[Pzx \and \lnot Ozy \and \lnot \exists v [PPvz]]. P.5'
  • Top: There exists a "universal object", designated W, such that PxW holds for any x.
\exists W \forall x [PxW]. 3.20
Top is a theorem if M8 holds.
  • Bottom: There exists an atomic "null object", designated N, such that PNx holds for any x.
\exists N \forall x [PNx]. 3.22
  • M6, Sum: If Uxy holds, there exists a z, called the "sum" or "fusion" of x and y, such that the parts of z are just those objects which are parts of either x or y.
Uxy \rightarrow \exists z \forall v [Ovz \leftrightarrow (Ovx \or Ovy)]. P.6
  • 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 which are parts of both x and y.
Oxy \rightarrow \exists z \forall v [Pvz \leftrightarrow (Pvx \and Pvy)]. P.7
If Oxy does not hold, x and y have no parts in common, and the product of x and y is defined iff Bottom holds.
  • M8, Unrestricted Fusion: Let φ(x) be a first-order formula in which x is a free variable. Then the fusion of all objects satisfying φ exists.
\exists xz [\phi (x) \rightarrow \forall y [Oyz \leftrightarrow (\phi (x) \and Oyx)]]. P.8
M8 is also called "General Sum Principle," "Unrestricted Mereological Composition," or "Universalism." M8 corresponds to the 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.
  • M8', Unique Fusion: The fusions whose existence M8 asserts are also unique. P.8'
  • M9, Atomicity: All objects are either atoms or fusions of atoms.
\exists yz[Pyx \and \lnot PPzy]. P.10

[edit] Various systems

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.

The systems in the table below are partially ordered by inclusion, in the sense that if all the theorems of system A are also theorems of system B, but the converse is not necessarily true, then B includes A. The resulting Hasse diagram is similar to that in Fig. 2, and Fig. 3.2 in Casati and Varzi (1999: 48).

Label Name System Included Axioms
M1-M3 Parthood is a partial order M M1–M3
M4 Weak Supplementation MM M, M4
M5 Strong Supplementation EM M, M5
M5' Atomistic Supplementation
M6 General Sum Principle (Sum)
M7 Product CEM EM, M6–M7
M8 Unrestricted Fusion GM M, M8
GEM EM, M8
M8' Unique Fusion GEM EM, M8'
M9 Atomicity AGEM M2, M8, M9
AGEM M, M5', M8

There are two equivalent ways of asserting that the universe is partially ordered: assume either M1–M3, or that Proper Parthood is transitive and 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.

M4 and M5 are two ways of asserting supplementation, the mereological analog of set complementation, 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.

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 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.

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 and empty sets, and Sum and Product are likewise the analogs of set-theoretical union and intersection. If M6 and M7 are either assumed or derivable, the result is a mereology with closure.

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 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 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.

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. Curiously, 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:

Postulating N renders all possible products definable, but also transforms classical extensional mereology into a set-free model of Boolean algebra.

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. Conversely, if the fusion asserted by M8 is assumed unique, so that M8' replaces M8, then Tarski (1929) showed that M3 and M8' suffice to axiomatize GEM, a remarkably economical result. Simons (1987: 38–41) lists a number of GEM theorems.

M2 and a finite universe necessarily imply Atomicity, namely that everything is either 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 M5' implies Atomicity and extensionality, and yields an alternative axiomatization of AGEM.

[edit] Mereology and set theory

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.

Much early work on mereology was motivated by a suspicion that set theory was 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.

Many logicians and philosophers reject these motivations, on such grounds as:

  • They deny that sets are in any way ontologically suspect;
  • Occam's Razor, when applied to abstract objects like sets, is either a dubious principle or simply false;
  • Mereology itself is guilty of proliferating new and ontologically suspect entities such as fusions.

For a survey of attempts to found mathematics without using set theory, see Burgess and Rosen (1997).

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 the mereology being formalizable in either of two equivalent ways:

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.

In set theory, singletons are "atoms" which 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 to be the mereological sum of atoms. Eberle (1970) showed how to construct a calculus of individuals lacking "atoms", i.e., one where every object has a "proper part" (defined below) so that the universe is infinite.

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).

Lewis (1991) went further, showing informally that mereology, augmented by a few ontological assumptions and plural quantification, and some novel reasoning about singletons, yields a system in which a given individual can be both a member and a subset of another individuals. In the resulting system, the axioms of ZFC (and of Peano arithmetic) are theorems.

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 and antireflexive. There exists a "fictitious" null individual that is a proper part of every individual. Two schemas assert that every lattice join exists (lattices are complete) and that meet distributes over join. On this Heyting mereology Forrest erects a theory of pseudosets, adequate for all purposes to which sets have been put.

[edit] Mereology and mathematics

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 and real numbers 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 relations while abstaining from set-theoretic definitions of the ordered pair. Martin argued that Eberle's (1970) theory of relational individuals solved this problem.

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, and mereology and set theory. He is one of the very few contributors to mereology to prove sound and complete each system he describes.

Topological notions of 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.

[edit] Mereology and natural language

Bunt (1985), a study of the semantics of natural language, shows how mereology can help understand such phenomena as the mass–count distinction and verb aspect. But Nicolas (2008) argues that a different logical framework, called plural logic, should be used for that purpose. 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 objects.

[edit] Important surveys

The books Simons (1987) and Casati and Varzi (1999) differ in their strengths:

Simons devotes considerable effort to elucidating historical notations. The notation of Casati and Varzi is often used. Both books include excellent bibliographies. To these works should be added Hovda (2008), which presents the latest state of the art on the axiomatization of mereology.

[edit] See also

[edit] References

  • 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.
  • 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.
  • Casati, R., and Varzi, A., 1999. Parts and Places: the structures of spatial representation. MIT Press.
  • Eberle, Rolf, 1970. Nominalistic Systems. Kluwer.
  • Forrest, Peter, 2002, "Nonclassical mereology and its application to sets," Notre Dame Journal of Formal Logic 43: 79-94.
  • Hovda, Paul, 2008, "What is classical mereology?", Journal of Philosophical Logic, http://www.springerlink.com/content/76l18850p2325p16/
  • Goodman, Nelson, 1977 (1951). The Structure of Appearance. Kluwer.
  • -------, and Willard Quine, 1947, "Steps toward a constructive nominalism," Journal of Symbolic Logic 12: 97-122.
  • Gruszczynski R., and Pietruszczak A., 2008, "Full development of Tarski's geometry of solids," Bulletin of Symbolic Logic 14:481-540. The paper presents a system of geometry based on Lesniewski's mereology. Basic properties of mereological structures are also given.
  • Husserl, Edmund, 1970. Logical Investigations, Vol. 2. Findlay, J.N., trans. Routledge.
  • Lewis, David K., 1991. Parts of Classes. Blackwell.
  • Leonard, H.S., and Goodman, Nelson, 1940, "The calculus of individuals and its uses," Journal of Symbolic Logic 5: 45–55.
  • Nicolas, David "Mass nouns and plural logic", "Linguistics and Philosophy" 31.2, pp.211-244, 2008, http://d.a.nicolas.free.fr/Nicolas-Mass-nouns-and-plural-logic-Revised-2.pdf
  • Leśniewski, Stanisław, 1992. Collected Works. Surma, S.J., Srzednicki, J.T., Barnett, D.I., and Rickey, V.F., editors and translators. Kluwer.
  • 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, "Mereological sets of distributive classes," Logic and Logical Philosophy 4:105-122. The paper shows how one can, by means of mereology, construct mathematical entities which are built from set theoretical classes.
  • Pietruszczak A., 2005, "Pieces of mereology," Logic and Logical Philosophy 14:211-234. The paper presents 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.
  • 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.
  • Varzi, Achille C., 2007, "Spatial Reasoning and Ontology: Parts, Wholes, and Locations" in Aiello, M. et al, eds., Handbook of Spatial Logics. Springer-Verlag: 945-1038.
  • 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.
  • ------, 1919. An Enquiry Concerning the Principles of Natural Knowledge. Cambridge Univ. Press. 2nd ed., 1925.
  • ------, 1920. The Concept of Nature. Cambridge Univ. Press. 2004 paperback, Prometheus Books. Being the 1919 Tarner Lectures delivered at Trinity College, Cambridge.
  • ------, 1978 (1929). Process and Reality. Free Press.
  • Woodger, J. H., 1937. The Axiomatic Method in Biology. Cambridge Univ. Press.

[edit] External links

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