Mereology - WikiHQ

Mereology (; from Greek μέρος 'part' (root: μερε-, mere-) and the suffix -logy, 'study, discussion, science') is the philosophical study of part-whole relationships, also called parthood relationships. As a branch of metaphysics, mereology examines the connections between parts and their wholes, exploring how components interact within a system. This theory has roots in ancient philosophy, with significant contributions from Plato, Aristotle, and later, medieval and Renaissance thinkers like Thomas Aquinas and John Duns Scotus. Mereology was formally axiomatized in the 20th century by Polish logician Stanisław Leśniewski, who introduced it as part of a comprehensive framework for logic and mathematics, and coined the word "mereology". use the name Classical Extensional Mereology (CEM) for what is here called General Extensional Mereology (GEM). In this article, following the practice of Varzi (1999) and the SEP, the name Classical Extensional Mereology (CEM) is reserved for a weaker theory, defined in . serves as the general and exhaustive theory of parthood and composition, at least for a large and significant domain of things. Nevertheless, GEM's assumptions are very common in mereological frameworks, due largely to Leśniewski influence as the one to first coin the word and formalize the theory: mereological theories commonly assume that everything is a part of itself (reflexivity), that a part of a part of a whole is itself a part of that whole (transitivity), and that two distinct entities cannot each be a part of the other (antisymmetry), so that the parthood relation is a partial order. An alternative is to assume instead that parthood is irreflexive (nothing is ever a part of itself) but still transitive, in which case antisymmetry follows automatically.

History

Informal part-whole reasoning was consciously invoked in metaphysics and ontology from Plato (in particular, in the second half of the Parmenides) and Aristotle onwards, and more or less unwittingly in 19th-century mathematics until the triumph of set theory around 1910. 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 Leśniewski elaborated this "Polish mereology" over the course of the 20th century. For a selection of the literature on Polish mereology, see Srzednicki and Rickey (1984). For a survey of Polish mereology, see Simons (1987). 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 surveyed in Simons (1987),

{| class="wikitable"

! Operation or term

!SEP!! Others (comma-separated)

| x is a part of y

|<math>Pxy</math>|| <math>x < y</math> || <math>x < y</math> || <math>Pxy</math>

|<math>x \sqsubseteq y</math>|| ⊂, ⊆, ≤, ⪯, ⊑

| x is a proper part of y

|<math>PPxy</math>|| <math>x \ll y</math> || <math>x \ll y</math> ||

|<math>x \sqsubset y</math>|| ⊂, ⊊, <

| x and y overlap

|<math>Oxy</math>|| <math>x \circ y</math> || <math>x \circ y</math> ||

| <math>x \bigcirc y</math>|| 0

|x and y underlap

|<math>Uxy</math>

|<math>x \; \mathsf{Un} \; y</math>

| x and y are disjoint

|<math>Dxy</math>|| <math>x \mid y</math>|| <math>x \mid y</math>||

| <math>x \wr y</math>|| ∣, )(, Z

| the (binary) product of x and y

|<math>x \times y</math>|| <math>x \cdot y</math>|| <math>xy</math>|| <math>x \cap y</math>

| <math>x \sqcap y</math>|| <math>x \land y</math>

| the (binary) sum of x and y

|<math>x + y</math>|| <math>x + y</math>|| <math>x + y</math>|| <math>x \cup y</math>

| <math>x \sqcup y</math>||

| the universal object (top)

|<math>U</math>|| <math>U</math>|| <math>U</math>|| <math>w</math>

| <math>U</math>|| a*, Un

| x is the sum of the set α

| || <math>x\,\mathrm{Su}\,\alpha</math>|| <math>x\,\mathrm{Fu}\,\alpha</math>|| <math>x \Sigma \alpha</math>

| <math>x \; \mathsf{sum} \; \alpha</math>, <math>x \; \mathsf{fu} \; \alpha</math>|| S, F

| x is the product of the set α

| || <math>x\,\mathrm{Pr}\,\alpha</math>|| <math>x\,\mathrm{Nu}\,\alpha</math>|| <math>x \Pi \alpha</math>

| || P, N

| the complement of x

| <math>\sim x</math>|| <math>\bar x</math> || <math>-x</math> || <math>-x</math>

| <math>-x</math>||

|the (general) sum of the x such that <math>\varphi x</math>

|<math>\sigma x \varphi x</math>

|the (general) product of the x such that <math>\varphi x</math>

|<math>\pi x\, \varphi x</math>

Axioms

The axioms are: 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 Fig. 3.2 in Casati and Varzi (1999: 48).

{| class=wikitable

!Label!!Name!!System!!Included Axioms

|M1||Reflexivity|| ||

|M2||Antisymmetry|| ||

|M3||Transitivity||M||M1, M2, M3

|M4||Weak Supplementation||MM||M, M4

|M5||Strong Supplementation||EM||M, M5

|M5'||Atomistic Supplementation|| ||

|M6||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. Reformulated in terms of Proper Part, MM is Simons's (1987) There is still a concern with parts and wholes, but instead of looking at what parts make up a whole, the emphasis is on what a thing is made of, such as its materials, e.g., the bronze in a bronze statue. Below are two of the main puzzles that philosophers use to discuss constitution.

Ship of Theseus: Briefly, the puzzle goes something like this. There is a ship called the Ship of Theseus. Over time, the boards start to rot, so we remove the boards and place them in a pile. First question, is the ship made of the new boards the same as the ship that had all the old boards? Second, if we reconstruct a ship using all of the old planks, etc. from the Ship of Theseus, and we also have a ship that was built out of new boards (each added one-by-one over time to replace old decaying boards), which ship is the real Ship of Theseus?

Statue and Lump of Clay: Roughly, a sculptor decides to mold a statue out of a lump of clay. At time <math>t_1</math> the sculptor has a lump of clay. After many manipulations at time <math>t_2</math> there is a statue. The question asked is, is the lump of clay and the statue (numerically) identical? If so, how and why?

Constitution typically has implications for views on persistence: how does an object persist over time if any of its parts (materials) change or are removed, as is the case with humans who lose cells, change height, hair color, memories, and yet we are said to be the same person today as we were when we were first born. For example, Ted Sider is the same today as he was when he was born—he just changed. But how can this be if many parts of Ted today did not exist when Ted was just born? Is it possible for things, such as organisms to persist? And if so, how? There are several views that attempt to answer this question. Some of the views are as follows (note, there are several other views):

(a) Constitution view. This view accepts cohabitation. That is, two objects share exactly the same matter. Here, it follows, that there are no temporal parts.

(b) Mereological essentialism, which states that the only objects that exist are quantities of matter, which are things defined by their parts. The object persists if matter is removed (or the form changes); but the object ceases to exist if any matter is destroyed.

(c) Dominant Sorts. This is the view that tracing is determined by which sort is dominant; they reject cohabitation. For example, lump does not equal statue because they're different "sorts".

(d) Nihilism—which makes the claim that no objects exist, except simples, so there is no persistence problem.

(e) 4-dimensionalism or temporal parts (may also go by the names perdurantism or exdurantism), which roughly states that aggregates of temporal parts are intimately related. For example, two roads merging, momentarily and spatially, are still one road, because they share a part.

(f) 3-dimensionalism (may also go by the name endurantism), where the object is wholly present. That is, the persisting object retains numerical identity.

Mereological composition

One question that is addressed by philosophers is which is more fundamental: parts, wholes, or neither? Another pressing question is called the special composition question (SCQ): For any Xs, when is it the case that there is a Y such that the Xs compose Y? This question has caused philosophers to run in three different directions: nihilism, universal composition (UC), or a moderate view (restricted composition). The first two views are considered extreme since the first denies composition, and the second allows any and all non-spatially overlapping objects to compose another object. The moderate view encompasses several theories that try to make sense of SCQ without saying 'no' to composition or 'yes' to unrestricted composition.

Fundamentality

There are philosophers who are concerned with the question of fundamentality. That is, which is more ontologically fundamental: the parts or their wholes? There are several responses to this question, though one of the default assumptions is that the parts are more fundamental. That is, the whole is grounded in its parts. This is the mainstream view. Another view, explored by Schaffer (2010), is monism, where the parts are grounded in the whole. Schaffer does not just mean that, say, the parts that make up my body are grounded in my body. Rather, Schaffer argues that the whole cosmos is more fundamental and everything else is a part of the cosmos. Then, there is the identity theory which claims that there is no hierarchy or fundamentality to parts and wholes. Instead wholes are just (or equivalent to) their parts. There can also be a two-object view which says that the wholes are not equal to the parts—they are numerically distinct from one another. Each of these theories has benefits and costs associated with them. This theory, though well explored, has its own problems: it seems to contradict experience and common sense, to be incompatible with atomless gunk, and to be unsupported by space-time physics.

Many more hypotheses continue to be explored. A common problem with these theories is that they are vague. It remains unclear what "fastened" or "life" mean, for example. And there are other problems with the restricted composition responses, many of them which depend on which theory is being discussed.

Mathematics

thumb|right|400px|The 11 exhaustive logical relations. These diagrams are fundamental to mereotopology, as they illustrate the distinction between tangential and non-tangential parts, a key concept in the mathematical formalization of part-whole relations.

Mereology was influential in early set theory, and has been used in work in the foundations of mathematics, especially by nominalists.

Set theory

From the beginnings of set theory, there has been a dispute between conceiving of sets "mereologically", where a set is the mereological sum of its elements, and conceiving of sets "collectively", where a set is an entity defined by intransitive membership, imposing additional structure upon urelements. Mereologically, if "Uniqueness of Composition" is accepted (see ), the objects a,b,c determine just the one fusion a+b+c, but collectively<nowiki>, the urelements a,b,c may be used to generate infinitely many sets: {a,b,c}; {a, {b, c; </nowiki><nowiki></nowiki>; {<nowiki></nowiki>, {b}, {c; etc. The collective conception is now dominant, but some of the earliest set theorists adhered to the mereological conception: Richard Dedekind, in "Was sind und was sollen die Zahlen?" (1888), avoided the empty set and used the same symbol for set membership and set inclusion, which are two signs that he conceived of sets mereologically. also used the mereological conception. who first laid out the difference between collections and mereological sums. is certainly significant for, though it does not fully explain, its current popularity. when David Lewis wrote his famous ', he found that "its main thesis had been anticipated in" Bunt's ensemble theory. axiomatized Zermelo-Fraenkel (ZFC) set theory using only classical mereology, plural quantification, and a primitive singleton-forming operator, governed by axioms that resemble the axioms for "successor" in Peano arithmetic. This contrasts with more usual axiomatizations of ZFC, which use only the primitive notion of membership. Lewis's work is named after his thesis that a class's subclasses are mereological parts of the class (in Lewis's usage, this means that a set's subsets, not counting the empty set, are parts of the set); this thesis has been disputed.

Michael Potter, a creator of Scott–Potter set theory, has criticized Lewis's work for failing to make set theory any more easily comprehensible, since Lewis says of his primitive singleton operator that, given the necessity (perceived by Lewis) of avoiding philosophically motivated mathematical revisionism, "I have to say, gritting my teeth, that somehow, I know not how, we do understand what it means to speak of singletons." Potter says Lewis "could just as easily have said, gritting his teeth, that somehow, he knows not how, we do understand what it means to speak of membership, in which case there would have been no need for the rest of the book." Even after the currently-dominant "collective" conception of sets became prevalent, mereology has sometimes been developed as an alternative foundation, especially by authors who were nominalists and therefore rejected abstract objects such as sets. The advantage of mereology for nominalists is its relative ontological economy compared to set theory, since mereology, when Uniqueness of Composition is accepted, will generate at most one entity from some given entities (namely their sum or fusion), whereas infinitely many sets are generated from just one urelement (e.g. Wikipedia, {Wikipedia}, <nowiki></nowiki>, <nowiki>}</nowiki>, {<nowiki>}</nowiki>} ...). and consciously developed mereology as an alternative to set theory as a foundation of mathematics. Goodman defended the Principle of Nominalism, which states that whenever two entities have the same basic constituents, they are identical. Most mathematicians and philosophers have accepted set theory as a legitimate and valuable foundation for mathematics, effectively rejecting the Principle of Nominalism in favor of some other theory, such as mathematical platonism.

Richard Milton Martin, who was also a nominalist, employed a version of the calculus of individuals throughout his career, starting in 1941. 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.

Burgess and Rosen (1997) provide a survey of attempts to found mathematics without using set theory, such as using mereology.

General systems theory

In general systems theory, mereological notions of part, whole and boundary are used to describe how complex systems can be decomposed and recomposed. Early work by Mihajlo D. Mesarovic and collaborators on multilevel and hierarchical control treated each level of organization as a system with its own internal structure and environment, while at the same time regarding each level as a component of a more inclusive system. Their formalism makes explicit use of system boundaries, interfaces and mappings between subsystems, and is often cited as a paradigmatic application of rigorous part–whole analysis in systems theory.

A complementary engineering tradition originates with Gabriel Kron's Diakoptics, or "method of tearing", in which a large network or field problem is split into subproblems whose solutions are later recombined to obtain the behaviour of the original system. Later authors showed that diakoptics can be understood using algebraic topology, with the interfaces between subsystems represented by shared chains or cochains, so that the overall method operates on a structured mereological decomposition of the network. Building on Kron, Keith Bowden developed "hierarchical tearing", a multilevel variant in which subsystems are recursively partitioned into sub-subsystems, and argued that diakoptics provides the basis for an "ontology of engineering" that takes networks, components and their interconnections as the primary units of analysis. In these approaches, the parts of a system are not merely smaller pieces of the whole but can carry "holographic" information about it, since behaviour at the interfaces encodes constraints coming from the rest of the system.

The same part–whole perspective appears in work that combines mereological ideas with sheaf theory, topos theory and category theory. Joseph Goguen pioneered the use of categories and sheaves in general systems theory and in the semantics of distributed and concurrent systems, treating local behaviours over components or regions as "sections" that can be glued together along their overlaps to produce global behaviour. In theoretical computer science, Steve Vickers has argued that locale theory and topos theory provide natural mathematical settings for modelling specifications and state spaces as systems of "observable parts": basic opens correspond to pieces of information, their overlaps encode compatibility, and their joins represent more complete states. These frameworks make precise how global structures emerge from compatible local data, closely mirroring mereological intuitions about how wholes depend on patterns of overlap among their parts.

Mereological themes also surface when general systems theory is applied to theoretical physics. Bowden has suggested that diakoptic and holographic methods can be interpreted as forms of "physical computation", in which physical processes perform the calculations required to propagate constraints between parts of a system. In such work, the focus is not only on what entities exist but on how they are nested, overlapped and dynamically related, reinforcing the role of mereology as a unifying formal thread within general systems theory.

Linguistic semantics

In formal semantics and cognitive science, mereology has been used extensively to model the meanings of mass nouns, count nouns, plurals, measure phrases, and event predicates. A common assumption is that the domain of individuals (and often the domain of events) forms a lattice or sum structure, equipped with a mereological part-of relation and a sum (or fusion) operation. On this view, entities can combine by sum (fusion) and stand in part–whole relations, and many linguistic phenomena are captured in terms of these operations and relations.

Mass–count distinction and measure phrases

One of the earliest and most developed applications of mereology in linguistics concerns the mass–count distinction. Bunt's ensemble-theoretic semantics treats mass terms such as water, sand, or gold as denoting sets of mereological sums of small portions of matter, rather than sets of discrete objects. This allows the semantics to capture characteristic properties of mass nouns, such as cumulative reference: if one quantity is water and another is water, then their mereological sum is also water:

There is water in the glass, and there is water in the jug ⟶ there is water in the glass-and-jug together.

By contrast, typical count nouns like book or apple are modeled as denoting sets of atoms in the mereological structure: minimal, indivisible individuals relative to the context. The sum of two atoms is not itself an atom, which helps explain why two books cannot normally be referred to as a book.

Mereological structures have also been used to analyze measure expressions such as three liters of water or two kilos of rice. In many approaches, a homomorphism maps the mereological domain of quantities of stuff onto a numerical measurement scale, preserving sums: the measure of a sum equals the sum of the measures of its parts, at least when the parts are disjoint. This connection between mereology and measurement is used to explain why sentences such as The water in the two bottles weighs three kilos can be interpreted as talking about the total mass of a mereological sum of quantities of water.

Mereology has also been applied to more complex mass expressions, including so-called object mass nouns such as furniture, luggage, or jewelry, which behave grammatically like mass nouns but seem to refer to collections of discrete objects. These cases put pressure on simple extensional mereological characterizations of the mass–count distinction and have motivated refinements of the theory and alternative proposals.

Plurals, distributivity and collectivity

Mereology also plays a central role in semantic theories of plurals. In Link's influential lattice-theoretic approach, singular individual denotations are atoms in a mereological structure, while plural denotations (e.g. the boys) are sums of such atoms. This allows plural predicates to be defined in terms of their behavior on sums. For example, the cumulative behavior of many plural and mass predicates can be stated in mereological terms:

If a and b are sums of boys that laugh, their sum a+b is also something that laughs.
If a and b are quantities of water, their sum a+b is still water.

The sum-based representation of plural individuals helps to account for the ambiguity between collective and distributive readings in sentences such as:

The boys lifted the piano.

On a collective reading, only the sum of boys is required to stand in the lifting relation to the piano. On a distributive reading, each atomic part of that sum (each boy) must lift the piano individually. In Link-style frameworks, distributive readings can be modeled by operators that distribute predicates over the atomic parts of a plural sum, while collective readings apply the predicate to the sum as a whole.

Mereology-based plural semantics has also been used to model more complex patterns such as cumulative readings (Three boys carried five boxes), where the sentence is true as long as the relevant sums of boys and boxes stand in the carrying relation, without specifying a one-to-one pairing.

Events, aspect and verbal predicates

Beyond the nominal domain, mereology has been applied to the semantics of events and aspect. In many event semantics frameworks, events form a mereological structure parallel to that of individuals: complex events are sums of simpler events, and parthood corresponds to temporal or causal inclusion of subevents.

Krifka, in particular, links the mereological structure of events to that of nominal reference. He shows that the distinction between telic and atelic verbal predicates parallels the distinction between quantized and cumulative nominal denotations. For example:

Mary drank beer for ten minutes. – an atelic predicate whose event denotation is closed under taking proper parts: any proper temporal part of a beer-drinking event is still a beer-drinking event.
Mary drank a glass of beer in ten minutes. – a telic predicate whose event denotation has inherent endpoints: proper subevents in general do not count as completed drink-a-glass-of-beer events.

On this view, the mereology of nominal arguments (for instance, whether an NP denotes a quantized or cumulative set of individuals) can systematically affect the mereological structure of events, and hence the aspectual interpretation of the clause.

Mereological tools have also been used to analyze path expressions and spatial adverbials, for instance in sentences such as The planes flew above and below the clouds. Here, the parts of a complex path or region (segments above vs. below the clouds) can be related to parts of the overall motion event using mereological and often mereotopological relations (parthood plus contact or connection).

Alternative formalisms and limitations

While mereology has provided a powerful set of tools for modeling nominal and event semantics, its application to natural language is not uncontroversial. Nicolas argues that purely mereological (or lattice-theoretic) treatments of mass nouns are too weak to capture certain "intermediate" readings and identity statements involving masses, and advocates using plural logic instead, where mass terms can behave like plural terms that refer to several things at once. Other authors have combined mereology with topological notions (mereotopology) in order to address problems such as the minimal-parts problem and to model notions like connectedness and contact that matter for the interpretation of mass and count expressions.

Moreover, the ordinary-language phrase part of is highly polysemous and context-sensitive. It can express, among other things, spatial inclusion (the handle is part of the door), group membership (She is part of the team), temporal inclusion (that episode is part of the series), and even looser relations of relevance (this is part of the problem). Simons emphasizes that many of these usages do not correspond straightforwardly to a single precise mereological relation, which complicates any attempt to read natural-language part as a simple parthood predicate Pxy.

Because of these difficulties, some authors adopt a cautious stance about the scope of formal mereology in natural language semantics. Casati and Varzi, for example, explicitly restrict their ontology to physical objects and spatial regions, and warn against assuming that all ordinary part–whole talk can be faithfully rendered in terms of a single, global mereological relation. Nonetheless, mereology—often in combination with additional structure such as topology, ordering, or measurement—remains an important component of many contemporary theories of linguistic meaning.

Notes

References

Sources

Bowden, Keith, 1991. Hierarchical Tearing: An Efficient Holographic Algorithm for System Decomposition, Int. J. General Systems, Vol. 24(1), pp 23–38.
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 P., 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, Roberto, and Varzi, Achille C., 1999. Parts and Places: the structures of spatial representation. MIT Press.
Cotnoir, A. J., and Varzi, Achille C., 2021, Mereology, Oxford University Press.
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, "Nonclassical mereology and its application to sets", Notre Dame Journal of Formal Logic 43: 79–94.
Gerla, Giangiacomo, (1995). "Pointless Geometries", in Buekenhout, F., Kantor, W. eds., "Handbook of incidence geometry: buildings and foundations". North-Holland: 1015–31.
Goodman, Nelson, 1977 (1951). The Structure of Appearance. Kluwer.
Goodman, Nelson, and Quine, Willard, 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. A system of geometry based on Lesniewski's mereology, with basic properties of mereological structures.
Hovda, Paul, 2008, "What is classical mereology?" Journal of Philosophical Logic 38(1): 55–82.
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.
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.
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. Ch. 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.
Mesarovic, M.D., Macko, D., and Takahara, Y., 1970, "Theory of Multilevel, Hierarchical Systems". Academic Press.
Nicolas, David, 2008, "Mass nouns and plural logic", Linguistics and Philosophy 31(2): 211–44.
Pietruszczak, Andrzej, 1996, "Mereological sets of distributive classes", Logic and Logical Philosophy 4: 105–22. Constructs, using mereology, mathematical entities from set theoretical classes.
Pietruszczak, Andrzej, 2005, "Pieces of mereology", Logic and Logical Philosophy 14: 211–34. Basic mathematical properties of Lesniewski's mereology.
Pietruszczak, Andrzej, 2018, Metamerology, Nicolaus Copernicus University Scientific Publishing House.
Potter, Michael, 2004. Set Theory and Its Philosophy. Oxford Univ. Press.
Simons, Peter, 1987 (reprinted 2000). 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.

External links

Internet Encyclopedia of Philosophy:
"Material Composition" – David Cornell
Stanford Encyclopedia of Philosophy:
"Mereology" – Achille Varzi
"Boundary" – Achille Varzi