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Mereology (; from Ancient 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 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".
Mereological ideas were influential in early , and formal mereology continues to be used by some working on the . Different axiomatizations of mereology have been applied in , used in to analyze "mass terms", used in the cognitive sciences, and developed in . Mereology has been combined with topology, for more on which see the article on mereotopology. Mereology is also used in the foundation of Whitehead's point-free geometry, on which see Tarski 1956 and Gerla 1995. Mereology is used in discussions of entities as varied as musical groups, geographical regions, and abstract concepts, demonstrating its applicability to a wide range of philosophical and scientific discourses.
In metaphysics, mereology is used to formulate the thesis of "composition as identity", the theory that individuals or objects are identical to mereological sums (also called fusions) of their parts. A metaphysical thesis called " mereological monism" suggests that the version of mereology developed by Stanisław Leśniewski and Nelson Goodman (commonly called General Extensional Mereology, or GEM) serves as the general and exhaustive theory of parthood and composition, at least for a large and significant domain of things. This thesis is controversial, since parthood may not seem to be a transitive relation (as claimed by GEM) in some cases, such as the parthood between organisms and their organs. (2026). 9781405106795, Blackwell Pub. ISBN 9781405106795 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.
Ivor Grattan-Guinness (2001) explains part-whole reasoning during the 19th and early 20th centuries, and reviews how Georg 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 1901, in the second volume of Logical Investigations – Third Investigation: "On the Theory of Wholes and Parts" (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 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). Since 1980 or so, however, research on Polish mereology has been almost entirely historical in nature.
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).
In 1930, Henry S. Leonard completed a Harvard PhD dissertation in philosophy, setting out a formal theory of the part-whole relation. This evolved into the "calculus of individuals" of Nelson 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 surveyed in Simons (1987), Casati and Varzi (1999), and Cotnoir and Varzi (2021).
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 , not unlike the case with modal logic.
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.
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:
| x is a part of y | ⊂, ⊆, ≤, ⪯, ⊑ | |||||
| x is a proper part of y | ⊂, ⊊, < | |||||
| x and y overlap | 0 | |||||
| x and y underlap | ||||||
| x and y are disjoint | ∣, )(, Z | |||||
| the (binary) product of x and y | ||||||
| the (binary) sum of x and y | ||||||
| the universal object (top) | a*, Un | |||||
| x is the sum of the set α | S, F | |||||
| x is the product of the set α | P, N | |||||
| the complement of x | ||||||
| the (general) sum of the x such that | ||||||
| the (general) product of the x such that |
The systems in the table below are partial order by 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, then B includes A. The resulting Hasse diagram is similar to Fig. 3.2 in Casati and Varzi (1999: 48).
| M1, M2, M3 |
| M, M4 |
| M, M5 |
| EM, M6, M7 |
| M, M8 |
| EM, M8 |
| EM, M8' |
| M2, M8, M9 |
| M, M5', M8 |
There are two equivalent ways of asserting that the universe is partial order: 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) 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 set and , 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 Unrestricted 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. 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:
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.
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.
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 the sculptor has a lump of clay. After many manipulations at time 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, Theodore 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):In Theodore Sider, John Hawthorne & Dean W. Zimmerman (eds.), Contemporary Debates in Metaphysics. Blackwell Pub. 241—262 (2007).
(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.
(a) Contact—Xs compose a complex Y if and only if the Xs are in contact;
(b) Fastenation—Xs compose a complex Y if and only if the Xs are fastened;
(c) Cohesion—Xs compose a complex Y if and only if the Xs cohere (cannot be pulled apart or moved in relation to each other without breaking);
(d) Fusion—Xs compose a complex Y if and only if the Xs are fused (joined together such that there is no boundary);
(e) Organicism—Xs compose a complex Y if and only if either the activities of the Xs constitute a life or there is only one of the Xs; and
(f) Brutal Composition—"It's just the way things are." There is no true, nontrivial, and finitely long answer.
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.
In set theory, singletons are "atoms" that have no (non-empty) proper parts; set theory where sets cannot be built up from unit sets is a nonstandard type of set theory, called non-well-founded set theory. 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), however, showed how to construct a calculus of individuals lacking "Atomism", i.e., one where every object has a "", so that the universe is infinite.
A detailed comparison between mereology, set theory, and a Semantics "ensemble theory" is presented in chapter 13 of Bunt (1985); when David Lewis wrote his famous , he found that "its main thesis had been anticipated in" Bunt's ensemble theory.
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."
Forrest (2002) revised Lewis's analysis by first formulating a generalization of CEM, called "Heyting mereology", whose sole nonlogical primitive is Proper Part, assumed transitive and antireflexive. According to this theory, 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 lattice) 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.
Mereology may still be valuable to non-nominalists: Eberle (1970) defended the "ontological innocence" of mereology, which is the idea that one can employ mereology regardless of one's ontological stance regarding sets. This innocence results from mereology being formalizable in either of two equivalent ways: quantified variables ranging over a universe of sets, or schematic predicates with a single free variable.
Still, Stanisław Leśniewski and Nelson Goodman, who developed Classical Extensional Mereology, were nominalists, 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. David Lewis, whose attempted to reconstruct set theory using mereology, was also a nominalist.
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 numbers and 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.
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. Tom Etter has proposed recasting aspects of quantum mechanics in explicitly mereological terms, treating quantum "links" or correlations as relations among parts of a distributed process and arguing that the algebraic structure of quantum theory can be understood as arising from systematic constraints on how such parts fit together. 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.
By contrast, typical 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.
The sum-based representation of plural individuals helps to account for the ambiguity between collective and distributive readings in sentences such as:
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.
Krifka, in particular, links the mereological structure of events to that of nominal reference. He shows that the distinction between telicity and atelic verbal predicates parallels the distinction between quantized and cumulative nominal denotations. For example:
On this view, the mereology of nominal arguments (for instance, whether an noun phrase 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).
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 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.
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