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.[1] 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".
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.[2] 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.
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. Metaphysical ideas of this era that discuss the concepts of parts and wholes include divine simplicity and the classical conception of beauty.
Ivor Grattan-Guinness (2001) explains part-whole reasoning during the 19th and early 20th centuries, and reviews how 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,[3] 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)[4] 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).[5] 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 Goodman and Leonard (1940).[6] Goodman revised and elaborated this calculus in the three editions of Goodman (1951).[7] 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 basic choice in defining a mereological system, is whether to allow things to be considered parts of themselves (reflexivity of parthood). In naive set theory a similar question arises: whether a set is to be considered a "member" of itself. In both cases, "yes" 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. In set theory, a set is often termed an improper subset of itself. Given such paradoxes, 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: 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.
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\landlnotPyx).
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:
Oxy\leftrightarrow\existsz[Pzx\landPzy].
The parts of z, the "overlap" or "product" of x and y, are precisely those objects that are parts of both x and y.
Uxy\leftrightarrow\existsz[Pxz\landPyz].
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\forallz[Ozx → Ozy].
There have been many notations for mereology. The table below builds on the comparison in Peter Simons, "Parts", page 99.[8]
| Operation or term | SEP | Simons | Leonard–Goodman | Tarski | Pietruszczak[9] | Others (comma-separated) | |
|---|---|---|---|---|---|---|---|
| x is a part of y | Pxy | x<y | x<y | Pxy | x\sqsubseteqy | ⊂, ⊆, ≤, ⪯, ⊑ | |
| x is a proper part of y | PPxy | x\lly | x\lly | x\sqsubsety | ⊂, ⊊, < | ||
| x and y overlap | Oxy | x\circy | x\circy | xcircy | 0 | ||
| x and y underlap | Uxy | x Un y | |||||
| x and y are disjoint | Dxy | x\midy | x\midy | x\wry | ∣,)(Z | ||
| the (binary) product of x and y | x x y | x ⋅ y | xy | x\capy | x\sqcapy | x\landy | |
| the (binary) sum of x and y | x+y | x+y | x+y | x\cupy | x\sqcupy | ||
| the universal object (top) | U | U | U | w | U | a*, Un | |
| x is the sum of the set α | xSu\alpha | xFu\alpha | x\Sigma\alpha | x sum \alpha x fu \alpha | S, F | ||
| x is the product of the set α | xPr\alpha | xNu\alpha | x\Pi\alpha | P, N | |||
| the complement of x | \simx | \barx | -x | -x | -x | ||
| the (general) sum of the x such that \varphix | \sigmax\varphix | \sigmax\varphix | |||||
| the (general) product of the x such that \varphix | \pix\varphix | \pix\varphix |
The axioms are:
M1, Reflexive: An object is a part of itself.
Pxx.
M2, Antisymmetric: If Pxy and Pyx both hold, then x and y are the same object.
(Pxy\landPyx) → x=y.
M3, Transitive: If Pxy and Pyz, then Pxz.
(Pxy\landPyz) → Pxz.
PPxy → \existsz[Pzy\landlnotOzx].
lnotPyx → \existsz[Pzy\landlnotOzx].
lnotPxy → \existsz[Pzx\landlnotOzy\landlnot\existsv[PPvz]].
\existsW\forallx[PxW].
Top is a theorem if M8 holds.
\existsN\forallx[PNx].
Uxy → \existsz\forallv[Ovz\leftrightarrow(Ovx\lorOvy)].
Oxy → \existsz\forallv[Pvz\leftrightarrow(Pvx\landPvy)].
If Oxy does not hold, x and y have no parts in common, and the product of x and y is undefined.
\existsx[\phi(x)]\to\existsz\forally[Oyz\leftrightarrow\existsx[\phi(x)\landOyx]].
M8 is also called "General Sum Principle", "Unrestricted Composition", "Unrestricted Sum", "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.
\existsy[Pyx\land\forallz[lnotPPzy]].
Simons (1987), Casati and Varzi (1999) and Hovda (2008)[10] 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).
| 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) 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 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:
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. 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.
In metaphysics there are many troubling questions pertaining to parts and wholes. One question addresses constitution and persistence, another asks about composition.
In metaphysics, there are several puzzles concerning cases of mereological constitution, that is, what makes up a whole.[11] 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
t1
t2
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):[13]
(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.
One question that is addressed by philosophers is which is more fundamental: parts, wholes, or neither?[14] [15] [16] [17] [18] [19] [20] [21] [22] [23] 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?[24] [25] [26] [27] [28] [29] [30] 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.
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.
Philosophers want to know when some Xs compose something Y. There are several kinds of response:
(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.[32]
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.
Mereology was influential in early set theory, and has been used in work in the foundations of mathematics, especially by nominalists.
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.[34] Mereologically, if "Uniqueness of Composition" is accepted (see), the objects a,b,c determine just the one fusion a+b+c, but collectively, the urelements a,b,c may be used to generate infinitely many sets: {a,b,c}; {a, {b, c}}; {{a}, {b}, {c}};
