Mereology

Posted by: lizwang on: November 16, 2008

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Mereology is a collection of axiomatic formal systems dealing with parts and their respective wholes and the parts of the parts in a whole. Mereology is an application of predicate logic and a branch of ontology, especially formal ontology.

Before the rise of set theory, part-whole reasoning was casually but unwittingly invoked all over mathematics and metaphysics, including in Aristotle. Ivor Grattan-Guinness (2001) sheds much light on this aspect of the period just before the Cantor-Peano notion of set became canonical. The first to reason consciously and at length about parts and wholes was, apparently, Edmund Husserl in his 1901 Logical Investigations, translated as Husserl (1970). However, the word "mereology" is absent from his writings, and he employed no symbolism even though his doctorate was in mathematics.

Stanisław Leśniewski coined the term in 1927, from the Greek word meros (part). Between 1916 and 1931, he wrote a number of highly technical papers on the subject, translated in Leśniewski (1992). This "Polish mereology" was elaborated over the course of the 20th century by Leśniewski’s students, and by students of his students. For a good selection of this secondary literature, see Srzednicki and Rickey (1984). Simons (1987) reviews Polish mereology at some length. Since then, however, little has been published about it.

A. N. Whitehead was supposed to write a fourth volume of Principia Mathematica on geometry, never published. His 1914 correspondence with Bertrand Russell mentions his working on this volume, and its mereological content. This work appears to have culminated in the mereological systems of Whitehead (1919,1920). His 1929 Process and Reality contains a good deal of informal mereotopology.

Henry Leonard‘s 1930 Harvard Ph.D. dissertation in philosophy set out a formal theory of the part-whole relation which first appeared in print in Goodman and Leonard (1940), who called it "the calculus of individuals." Goodman went on to elaborate this calculus in the three editions of his Structure of Appearance, the last being Goodman (1977). Eberle (1970) clarified the relation between mereology and set theory, and 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.

For some time, philosophers and mathematicians were reluctant to explore mereology, believing that it implied a rejection of set theory, a position known as nominalism. Goodman was indeed a nominalist and his fellow nominalist, Richard Milton Martin employed a version of the calculus of individuals throughout his career, starting in 1941. The calculus of individuals began to come into its own starting only around 1970, when the "ontological innocence" of mereology began to be recognized. One can employ mereology regardless of one’s ontological stance regarding set theory. Quantified variables ranging over a universe of sets, and schematic monadic predicates with a free variable, can be used interchangeably in the formal description of a mereological system. Since that recognition, formal work in ontology and metaphysics has made increasing use of mereology.

Mereology is a form of mathematics, arguably a sort of "proto-geometry," but has been wholly developed by logicians and computer scientists. To date, the only mathematician to write on mereology is Leśniewski’s student Alfred Tarski in the 1920s and 30s (see Tarski 1984). For that matter, mereology is seldom mentioned outside of the literatures on ontology and artificial intelligence. Standard university texts on logic and mathematics are silent about mereology, which has undoubtedly contributed to its undeserved obscurity. Topological notions of boundaries and connection can be married to mereology, resulting in mereotopology; see Casati and Varzi (1999).

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.
Nonetheless, mereology is now largely accepted as a useful tool for formal philosophy, although to date it has received much less attention than set theory.

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 be built up from atoms. Eberle (1970) showed how to devise "atomless" mereologies such that all objects have proper parts and so can be divided at will.

Lewis (1991) showed informally that mereology augmented by a few ontological assumptions and some careful reasoning about singletons, yields a system in which a can be both a member and a subset of b. Lewis’s system is much more than a curiosity; it turns the axioms of Peano arithmetic and of Zermelo-Fraenkel set theory into theorems. On the relation of mereology and ZF, also see Bunt (1985).

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 whose parts are all the objects that are not parts of themselves. Is it a part of itself? (However, every object is an "improper" part of itself.) Hence mereology requires an axiomatic formulation.

The treatment and terminology below follow Casati and Varzi (1999: chpts. 3,4) closely. 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 nonnested axiomatic systems, not unlike the case with modal logic.

A mereological system requires at least one primitive relation e.g., dyadic Parthood, "x is a part of y," written Pxy.

Parthood is nearly always assumed to partially order the universe.

An immediate defined predicate is "x is a proper part of y," written PPxy,

which holds (i.e., is satisfied, comes out true) if Pxy is true and Pyx is false.

An object lacking proper parts is an atom.

The mereological universe consists of all objects we wish to think about, plus all of their proper parts.

Two other common defined predicates are:

Overlap: x and y overlap, written Oxy, if there exists an object z such that Pzx and Pzy both hold. 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.
A number of possible axioms follow. Lower case letters denote variables ranging over objects:

Parthood partially orders the universe:
M1, Reflexive: An object is a part of itself.
M2, Antisymmetric: If Pxy and Pyx both hold, then x and y are the same object.
M3, Transitive: If Pxy and Pyz, then Pxz.
M4, Weak Supplementation: If PPxy holds, there exists a z such that Pzy holds but Ozx does not.
M5, Strong Supplementation: Replace "PPxy holds" in M4 with "Pyx does not hold."
M5′, Atomistic Supplementation: If Pxy does not hold, then there exists an atom z such that Pzx holds but Ozy does not.
Top: There exists a "universal object", designated W, such that PxW holds for any x. Top is a theorem if M8 holds.
Bottom: There exists an atomic "null object", designated N, such that PNx holds for any x.
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.
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. If Oxy does not hold, x and y have no non-empty parts in common, and the product of x and y is defined if and only if Bottom holds.
M8, Unrestricted Fusion: Let φ be a first-order formula φ having one free variable. Then the fusion of all objects satisfying φ exists. Also called "General Sum Principle," "Unrestricted Mereological Composition," or "Universalism." M8 corresponds to the principle of unrestricted comprehension of naive set theory. This principle 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 fusion described in M8 not only exists but is unique.
M9, Atomicity: All objects are either atoms or fusions of atoms.
The above axioms all hold in classical extensional mereology. Other systems of mereology are described in Simons (1987) and Casati and Varzi (1999). There are some analogies between these axioms and those of standard Zermelo-Fraenkel set theory, if "parthood" in mereology is taken as corresponding to subset in set theory.

In the table below, strings of bold letters name mereological systems. These systems 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 Fig. 2, and Fig. 3.2 in Casati and Varzi (1999: 48).

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. 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. 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 [M7] asserts that any two underlapping [overlapping] objects have a unique sum [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 analogues of the universal and null sets, and sum and product are likewise the analogs of set 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, if 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 a:

Boolean algebra lacking a 0;
Join semilattice bounded from above by 1. Binary fusion and W interpret join and 1, respectively.
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.

Label	Name	System	Included Axioms
M1	Parthood is reflexive
M2	Parthood is antisymmetric
M3	Parthood is transitive	M	M1-3
M4	Weak Supplementation	MM	M, M4
M5	Strong Supplementation	EM	M, M5
M5′	Atomistic Supplementation
M6	General Sum Principle	CEM	EM, M6-7
M7	Product	GM	M, M8
M8	Unrestricted Fusion	GEM	EM, M8
M8′	Unique Fusion	GEM	EM, M8′
M9	Atomicity	AGEM	M2, M8, M9
		AGEM	M, M5′, M8

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. Nevertheless, 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.

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

Simons (1987) sees mereology primarily as a tool for doing formal metaphysics. His strengths include: the work of Leśniewski and his descendants; the connections between mereology and a number of continental philosophers, especially Edmund Husserl; the relation between mereology and recent work on formal ontology and metaphysics; mereology and free logic and modal logic; mereology and the study of Boolean algebras and lattice theory.
Casati and Varzi (1999) see mereology primarily as a way of understanding the material world and how humans interact with it. Their strengths include: topology and mereotopology; boundaries and holes; the mereological implications of Alfred North Whitehead‘s Process and Reality and work descended therefrom; mereology as a theory of events; mereology as a "proto-geometry" for physical objects; mereology and theoretical computer science.
Both books include excellent bibliographies.