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In
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
,
philosophy Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...
and related fields, mereology ( (root: , ''mere-'', 'part') and the suffix ''-logy'', 'study, discussion, science') is the study of parts and the wholes they form. Whereas
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
is founded on the membership relation between a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
and its elements, mereology emphasizes the meronomic relation between entities, which—from a set-theoretic perspective—is closer to the concept of
inclusion Inclusion or Include may refer to: Sociology * Social inclusion, aims to create an environment that supports equal opportunity for individuals and groups that form a society. ** Inclusion (disability rights), promotion of people with disabiliti ...
between sets. Mereology has been explored in various ways as applications of
predicate logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
to
formal ontology In philosophy, the term formal ontology is used to refer to an ontology defined by axioms in a formal language with the goal to provide an unbiased (domain- and application-independent) view on reality, which can help the modeler of domain- or a ...
, in each of which mereology is an important part. Each of these fields provides its own axiomatic definition of mereology. A common element of such axiomatizations is the assumption, shared with inclusion, that the part-whole relation
orders Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
its universe, meaning 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 In linguistics, antisymmetry is a syntactic theory presented in Richard S. Kayne's 1994 monograph ''The Antisymmetry of Syntax''. It asserts that grammatical hierarchies in natural language follow a universal order, namely specifier-head-compl ...
), thus forming a
poset In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
. A variant of this axiomatization denies that anything is ever part of itself (irreflexivity) while accepting transitivity, from which antisymmetry follows automatically. Although mereology is an application of
mathematical logic Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...
, what could be argued to be a sort of "proto-geometry", it has been wholly developed by logicians,
ontologists In metaphysics, ontology is the philosophical study of being, as well as related concepts such as existence, becoming, and reality. Ontology addresses questions like how entities are grouped into categories and which of these entities exis ...
, linguists, engineers, and computer scientists, especially those working in
artificial intelligence Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speech re ...
. In particular, mereology is also on the basis for a point-free foundation of geometry (see for example the quoted pioneering paper of Alfred Tarski and the review paper by Gerla 1995). In
general systems theory Systems theory is the interdisciplinary study of systems, i.e. cohesive groups of interrelated, interdependent components that can be natural or human-made. Every system has causal boundaries, is influenced by its context, defined by its structu ...
, mereology refers to formal work on system decomposition and parts, wholes and boundaries (by, e.g.,
Mihajlo D. Mesarovic Mihajlo D. Mesarovic ( Serbian Latin: ''Mihajlo D. Mesarović'', Serbian Cyrillic: Михајло Д. Месаровић; born 2 July 1928) is a Serbian scientist, who is a professor of Systems Engineering and Mathematics at Case Western Reser ...
(1970),
Gabriel Kron Gabriel Kron (1901 – 1968) was a Hungarian American electrical engineer who promoted the use of methods of linear algebra, multilinear algebra, and differential geometry in the field. His method of system decomposition and solution called ...
(1963), or Maurice Jessel (see Bowden (1989, 1998)). A hierarchical version of
Gabriel Kron Gabriel Kron (1901 – 1968) was a Hungarian American electrical engineer who promoted the use of methods of linear algebra, multilinear algebra, and differential geometry in the field. His method of system decomposition and solution called ...
's Network Tearing was published by Keith Bowden (1991), reflecting David Lewis's ideas on gunk. Such ideas appear in theoretical
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
and
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, often in combination with
sheaf theory In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
,
topos In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notio ...
, or
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
. See also the work of Steve Vickers on (parts of) specifications in computer science,
Joseph Goguen __NOTOC__ Joseph Amadee Goguen ( ; June 28, 1941 – July 3, 2006) was an American computer scientist. He was professor of Computer Science at the University of California and University of Oxford, and held research positions at IBM and SRI I ...
on physical systems, and Tom Etter (1996, 1998) on link theory and
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
.


History

Informal part-whole reasoning was consciously invoked in
metaphysics Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of conscio ...
and
ontology In metaphysics, ontology is the philosophical study of being, as well as related concepts such as existence, becoming, and reality. Ontology addresses questions like how entities are grouped into categories and which of these entities exis ...
from
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
(in particular, in the second half of the ''
Parmenides Parmenides of Elea (; grc-gre, Παρμενίδης ὁ Ἐλεάτης; ) was a pre-Socratic Greek philosopher from Elea in Magna Graecia. Parmenides was born in the Greek colony of Elea, from a wealthy and illustrious family. His dates a ...
'') and
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phil ...
onwards, and more or less unwittingly in 19th-century mathematics until the triumph of
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
around 1910. Metaphysical ideas of this era that discuss the concepts of parts and wholes include
divine simplicity In theology, the doctrine of divine simplicity says that God is simple (without parts). The general idea can be stated in this way: The being of God is identical to the "attributes" of God. Characteristics such as omnipresence, goodness, trut ...
and the classical conception of beauty. Ivor Grattan-Guinness (2001) sheds much light on part-whole reasoning during the 19th and early 20th centuries, and reviews how
Cantor A cantor or chanter is a person who leads people in singing or sometimes in prayer. In formal Jewish worship, a cantor is a person who sings solo verses or passages to which the choir or congregation responds. In Judaism, a cantor sings and lead ...
and
Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The sta ...
devised
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
. It appears that the first to reason consciously and at length about parts and wholes was
Edmund Husserl , thesis1_title = Beiträge zur Variationsrechnung (Contributions to the Calculus of Variations) , thesis1_url = https://fedora.phaidra.univie.ac.at/fedora/get/o:58535/bdef:Book/view , thesis1_year = 1883 , thesis2_title ...
, 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 Stanisław Leśniewski (30 March 1886 – 13 May 1939) was a Polish mathematician, philosopher and logician. Life He was born on 28 March 1886 at Serpukhov, near Moscow, to father Izydor, an engineer working on the construction of the Trans-Sibe ...
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 Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...
, in his Appendix E to Woodger (1937) and the paper translated as Tarski (1984), greatly simplified Leśniewski's formalism. Other students (and students of students) of Lesniewski elaborated this "Polish mereology" over the course of the 20th century. For a good selection of the literature on Polish mereology, see Srzednicki and Rickey (1984). For a survey of Polish mereology, see Simons (1987). Since 1980 or so, however, research on Polish mereology has been almost entirely historical in nature. A. N. Whitehead planned a fourth volume of ''
Principia Mathematica The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...
'', on
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, but never wrote it. His 1914 correspondence with
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
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 Ph.D. dissertation in philosophy, setting out a formal theory of the part-whole relation. This evolved into the "calculus of individuals" of
Goodman Goodman or Goodmans may refer to: Businesses and organizations * Goodman Games, American publisher. * Goodman Global, an American HVAC manufacturer. * Goodman Group, an Australian property company. * Goodmans Industries, a British electronic co ...
and Leonard (1940). Goodman revised and elaborated this calculus in the three editions of Goodman (1951). The calculus of individuals is the starting point for the post-1970 revival of mereology among logicians, ontologists, and computer scientists, a revival well-surveyed in Simons (1987), Casati and Varzi (1999), and Cotnoir and Varzi (2021).


Axioms and primitive notions

Reflexivity: A basic choice in defining a mereological system, is whether to consider things to be parts of themselves. In
naive set theory Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike Set theory#Axiomatic set theory, axiomatic set theories, which are defined using Mathematical_logic#Formal_logical_systems, forma ...
a similar question arises: whether a set is to be considered a "subset" of itself. In both cases, "yes" gives rise to paradoxes analogous to
Russell's paradox In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains a ...
: 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 An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
formulation. A mereological "system" is a
first-order theory First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantif ...
(with
identity Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), ...
) whose
universe of discourse In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range. Overview The doma ...
consists of wholes and their respective parts, collectively called ''objects''. Mereology is a collection of nested and non-nested
axiomatic system In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contai ...
s, not unlike the case with
modal logic Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...
. 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 In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...
( dyadic
predicate Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: **Predicate (mathematical logic) **Propositional function **Finitary relation, o ...
). 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: *An immediate defined
predicate Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: **Predicate (mathematical logic) **Propositional function **Finitary relation, o ...
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. Compared to parthood (which is a
partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
), ProperPart is a
strict partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
. :PPxy \leftrightarrow (Pxy \land \lnot Pyx). 3.3 :An object lacking proper parts is an ''atom''. The mereological
universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. Acc ...
consists of all objects we wish to think about, and all of their proper parts: *Overlap: ''x'' and ''y'' overlap, written ''Oxy'', if there exists an object ''z'' such that ''Pzx'' and ''Pzy'' both hold. :Oxy \leftrightarrow \exists z zx \land Pzy 3.1 :The parts of ''z'', the "overlap" or "product" of ''x'' and ''y'', are precisely those objects that are parts of both ''x'' and ''y''. *Underlap: ''x'' and ''y'' underlap, written ''Uxy'', if there exists an object ''z'' such that ''x'' and ''y'' are both parts of ''z''. :Uxy \leftrightarrow \exists z xz \land Pyz 3.2 Overlap and Underlap are reflexive,
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
, and
intransitive In grammar, an intransitive verb is a verb whose context does not entail a direct object. That lack of transitivity distinguishes intransitive verbs from transitive verbs, which entail one or more objects. Additionally, intransitive verbs are ...
. 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 \forall z zx \rightarrow Ozy 3.31 The axioms are: *Parthood partially orders the
universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. Acc ...
: :M1, Reflexive: An object is a part of itself. :\ Pxx. P.1 :M2, Antisymmetric: If ''Pxy'' and ''Pyx'' both hold, then ''x'' and ''y'' are the same object. :(Pxy \land Pyx) \rightarrow x = y. P.2 :M3, Transitive: If ''Pxy'' and ''Pyz'', then ''Pxz''. :(Pxy \land Pyz) \rightarrow Pxz. P.3 *M4, Weak Supplementation: If ''PPxy'' holds, there exists a ''z'' such that ''Pzy'' holds but ''Ozx'' does not. :PPxy \rightarrow \exists z zy \land \lnot Ozx P.4 *M5, Strong Supplementation: If ''Pyx'' does not hold, there exists a ''z'' such that ''Pzy'' holds but ''Ozx'' does not. :\lnot Pyx \rightarrow \exists z zy \land \lnot Ozx P.5 *M5', Atomistic Supplementation: If ''Pxy'' does not hold, then there exists an atom ''z'' such that ''Pzx'' holds but ''Ozy'' does not. :\lnot Pxy \rightarrow \exists z zx \land \lnot Ozy \land \lnot \exists v [PPvz. P.5' *Top: There exists a "universal object", designated ''W'', such that ''PxW'' holds for any ''x''. :\exists W \forall x [PxW]. 3.20 :Top is a theorem if M8 holds. *Bottom: There exists an atomic "null object", designated ''N'', such that ''PNx'' holds for any ''x''. :\exists N \forall x [PNx]. 3.22 *M6, Sum: If ''Uxy'' holds, there exists a ''z'', called the "sum" or "fusion" of ''x'' and ''y'', such that the objects overlapping of ''z'' are just those objects that overlap either ''x'' or ''y''. :Uxy \rightarrow \exists z \forall v vz \leftrightarrow (Ovx \lor Ovy) P.6 *M7, Product: If ''Oxy'' holds, there exists a ''z'', called the "product" of ''x'' and ''y'', such that the parts of ''z'' are just those objects that are parts of both ''x'' and ''y''. :Oxy \rightarrow \exists z \forall v vz \leftrightarrow (Pvx \land Pvy) P.7 :If ''Oxy'' does not hold, ''x'' and ''y'' have no parts in common, and the product of ''x'' and ''y'' is undefined. *M8, Unrestricted Fusion: Let φ(''x'') be a
first-order In mathematics and other formal sciences, first-order or first order most often means either: * "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of high ...
formula in which ''x'' is a
free variable In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not ...
. Then the fusion of all objects satisfying φ exists. :\exists x
phi(x) Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek: ''fi'' ) is the 21st letter of the Greek alphabet. In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voicele ...
\to \exists z \forall y yz_\leftrightarrow_\exists_x[\phi_(x)_\land_Oyx._P.8 :M8_is_also_called_"General_Sum_Principle",_"Unrestricted_Mereological_Composition",_or_"Universalism"._M8_corresponds_to_the_set_builder_notation.html" ;"title="phi_(x)_\land_Oyx.html" ;"title="yz \leftrightarrow \exists x yz_\leftrightarrow_\exists_x[\phi_(x)_\land_Oyx._P.8 :M8_is_also_called_"General_Sum_Principle",_"Unrestricted_Mereological_Composition",_or_"Universalism"._M8_corresponds_to_the_set_builder_notation">principle_of_unrestricted_comprehension_of_naive_set_theory_ Naive_set_theory_is_any_of_several_theories_of_sets_used_in_the_discussion_of_the_foundations_of_mathematics. Unlike_Set_theory#Axiomatic_set_theory,_axiomatic_set_theories,_which_are_defined_using_Mathematical_logic#Formal_logical_systems,_forma_...
,_which_gives_rise_to_Russell's_paradox_ In_mathematical_logic,_Russell's_paradox_(also_known_as_Russell's_antinomy)_is_a__set-theoretic_paradox_discovered_by_the_British_philosopher_and_mathematician_Bertrand_Russell_in_1901._Russell's_paradox_shows_that_every_set_theory_that_contains_a_...
._There_is_no_mereological_counterpart_to_this_paradox_simply_because_''parthood'',_unlike_set_membership,_is_Reflexive_relation.html" ;"title="phi (x) \land Oyx">yz \leftrightarrow \exists x[\phi (x) \land Oyx. P.8 :M8 is also called "General Sum Principle", "Unrestricted Mereological Composition", or "Universalism". M8 corresponds to the set builder notation">principle of unrestricted comprehension of
naive set theory Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike Set theory#Axiomatic set theory, axiomatic set theories, which are defined using Mathematical_logic#Formal_logical_systems, forma ...
, which gives rise to
Russell's paradox In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains a ...
. There is no mereological counterpart to this paradox simply because ''parthood'', unlike set membership, is Reflexive relation">reflexive. *M8', Unique Fusion: The fusions whose existence M8 asserts are also unique. P.8' *M9, Atomicity: All objects are either atoms or fusions of atoms. : \exists y[Pyx \land \forall z[\lnot PPzy. P.10


Various systems

Simons (1987), Casati and Varzi (1999) and Hovda (2008) describe many mereological systems whose axioms are taken from the above list. We adopt the boldface nomenclature of Casati and Varzi. The best-known such system is the one called ''classical extensional mereology'', hereinafter abbreviated CEM (other abbreviations are explained below). In CEM, P.1 through P.8' hold as axioms or are theorems. M9, ''Top'', and ''Bottom'' are optional. The systems in the table below are
partially ordered In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
by
inclusion Inclusion or Include may refer to: Sociology * Social inclusion, aims to create an environment that supports equal opportunity for individuals and groups that form a society. ** Inclusion (disability rights), promotion of people with disabiliti ...
, in the sense that, if all the theorems of system A are also theorems of system B, but the converse is not
necessarily true Logical truth is one of the most fundamental concepts in logic. Broadly speaking, a logical truth is a statement which is true regardless of the truth or falsity of its constituent propositions. In other words, a logical truth is a statement whic ...
, then B ''includes'' A. The resulting
Hasse diagram In order theory, a Hasse diagram (; ) is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set ''(S, ≤)'' one represents ea ...
is similar to Fig. 3.2 in Casati and Varzi (1999: 48). There are two equivalent ways of asserting that the
universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. Acc ...
is
partially ordered In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
: Assume either M1-M3, or that Proper Parthood is transitive and asymmetric, hence a
strict partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
. Either axiomatization results in the system M. M2 rules out closed loops formed using Parthood, so that the part relation is
well-founded In mathematics, a binary relation ''R'' is called well-founded (or wellfounded) on a class ''X'' if every non-empty subset ''S'' ⊆ ''X'' has a minimal element with respect to ''R'', that is, an element ''m'' not related by ''s& ...
. Sets are well-founded if the
axiom of regularity In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set ''A'' contains an element that is disjoint from ''A''. In first-order logic, the ...
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
complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-clas ...
ation, 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 In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned with whether the internal ...
'', a term borrowed from set theory, for which
extensionality In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned with whether the internal ...
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 Universal is the adjective for universe. Universal may also refer to: Companies * NBCUniversal, a media and entertainment company ** Universal Animation Studios, an American Animation studio, and a subsidiary of NBCUniversal ** Universal TV, a ...
and
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
s, and ''Sum'' and ''Product'' are, likewise, the analogs of set-theoretical
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...
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 Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
, 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 In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not ...
. The formula comes out true (is satisfied) whenever the name of an object that would be a
member Member may refer to: * Military jury, referred to as "Members" in military jargon * Element (mathematics), an object that belongs to a mathematical set * In object-oriented programming, a member of a class ** Field (computer science), entries in ...
of the set (if it existed) replaces the free variable. Hence any axiom with sets can be replaced by an
axiom schema In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom. Formal definition An axiom schema is a formula in the metalanguage of an axiomatic system, in which one or more schematic variables ap ...
with monadic atomic subformulae. M8 and M8' are schemas of just this sort. The
syntax In linguistics, syntax () is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure ( constituency) ...
of a
first-order theory First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantif ...
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 Richard Milton Martin (1916, Cleveland, Ohio – 22 November 1985, Milton, Massachusetts) was an American logician and analytic philosopher. In his Ph.D. thesis written under Frederic Fitch, Martin discovered virtual sets a bit before Quine ...
). 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 In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...
lacking a 0; * a
join Join may refer to: * Join (law), to include additional counts or additional defendants on an indictment *In mathematics: ** Join (mathematics), a least upper bound of sets orders in lattice theory ** Join (topology), an operation combining two top ...
semilattice In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a ...
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 A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...
of
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...
. 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.


Set theory

The notion of "subset" in set theory is not entirely the same as the notion of "subpart" in mereology.
Stanisław Leśniewski Stanisław Leśniewski (30 March 1886 – 13 May 1939) was a Polish mathematician, philosopher and logician. Life He was born on 28 March 1886 at Serpukhov, near Moscow, to father Izydor, an engineer working on the construction of the Trans-Sibe ...
rejected set theory as related to but not the same as
nominalism In metaphysics, nominalism is the view that universals and abstract objects do not actually exist other than being merely names or labels. There are at least two main versions of nominalism. One version denies the existence of universalsthings t ...
. For a long time, nearly all philosophers and mathematicians avoided mereology, seeing it as tantamount to a rejection of set theory. Goodman too was a nominalist, and his fellow nominalist
Richard Milton Martin Richard Milton Martin (1916, Cleveland, Ohio – 22 November 1985, Milton, Massachusetts) was an American logician and analytic philosopher. In his Ph.D. thesis written under Frederic Fitch, Martin discovered virtual sets a bit before Quine ...
employed a version of the calculus of individuals throughout his career, starting in 1941. Much early work on mereology was motivated by a suspicion that
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
was
ontologically In metaphysics, ontology is the philosophical study of being, as well as related concepts such as existence, becoming, and reality. Ontology addresses questions like how entities are grouped into categories and which of these entities exis ...
suspect, and that
Occam's razor Occam's razor, Ockham's razor, or Ocham's razor ( la, novacula Occami), also known as the principle of parsimony or the law of parsimony ( la, lex parsimoniae), is the problem-solving principle that "entities should not be multiplied beyond neces ...
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 object In metaphysics, the distinction between abstract and concrete refers to a divide between two types of entities. Many philosophers hold that this difference has fundamental metaphysical significance. Examples of concrete objects include plants, hum ...
s like sets, is either a dubious principle or simply false * Mereology itself is guilty of proliferating new and ontologically suspect entities such as fusions. For a survey of attempts to found mathematics without using set theory, see Burgess and Rosen (1997). In the 1970s, thanks in part to Eberle (1970), it gradually came to be understood that one can employ mereology regardless of one's ontological stance regarding sets. This understanding is called the "ontological innocence" of mereology. This innocence stems from mereology being formalizable in either of two equivalent ways: *Quantified variables ranging over a
universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. Acc ...
of sets *Schematic
predicates Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: **Predicate (mathematical logic) **Propositional function **Finitary relation, ...
with a single
free variable In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not ...
. Once it became clear that mereology is not tantamount to a denial of set theory, mereology became largely accepted as a useful tool for formal
ontology In metaphysics, ontology is the philosophical study of being, as well as related concepts such as existence, becoming, and reality. Ontology addresses questions like how entities are grouped into categories and which of these entities exis ...
and
metaphysics Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of conscio ...
. In set theory, singletons are "atoms" that 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 the mereological sum of atoms. Eberle (1970), however, showed how to construct a calculus of individuals lacking "
atoms Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, an ...
", i.e., one where every object has a "proper part" (defined below) so that the
universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. Acc ...
is infinite. There are analogies between the axioms of mereology and those of standard
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as ...
(ZF), if ''Parthood'' is taken as analogous to
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
in set theory. On the relation of mereology and ZF, also see Bunt (1985). One of the very few contemporary set theorists to discuss mereology is Potter (2004).
Lewis Lewis may refer to: Names * Lewis (given name), including a list of people with the given name * Lewis (surname), including a list of people with the surname Music * Lewis (musician), Canadian singer * "Lewis (Mistreated)", a song by Radiohead ...
(1991) went further, showing informally that mereology, augmented by a few
ontological In metaphysics, ontology is the philosophical study of being, as well as related concepts such as existence, becoming, and reality. Ontology addresses questions like how entities are grouped into categories and which of these entities exi ...
assumptions and
plural quantification In mathematics and logic, plural quantification is the theory that an individual variable x may take on ''plural'', as well as singular, values. As well as substituting individual objects such as Alice, the number 1, the tallest building in London ...
, and some novel reasoning about singletons, yields a system in which a given individual can be both a part and a subset of another individual. Various sorts of set theory can be interpreted in the resulting systems. For example, the axioms of ZFC can be proven given some additional mereological assumptions. Forrest (2002) revises Lewis's analysis by first formulating a generalization of CEM, called "Heyting mereology", whose sole nonlogical primitive is ''Proper Part'', assumed transitive and
antireflexive In mathematics, a binary relation ''R'' on a set ''X'' is reflexive if it relates every element of ''X'' to itself. An example of a reflexive relation is the relation " is equal to" on the set of real numbers, since every real number is equal to ...
. There exists a "fictitious" null individual that is a proper part of every individual. Two schemas assert that every
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
join exists (lattices are
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
) 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.


Mathematics

Husserl never claimed that mathematics could or should be grounded in part-whole rather than set theory. Lesniewski consciously derived his mereology as an alternative to set theory as a
foundation of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathe ...
, but did not work out the details. Goodman and Quine (1947) tried to develop the
natural Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are p ...
and
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s 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 Richard Milton Martin (1916, Cleveland, Ohio – 22 November 1985, Milton, Massachusetts) was an American logician and analytic philosopher. In his Ph.D. thesis written under Frederic Fitch, Martin discovered virtual sets a bit before Quine ...
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 In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
. Martin argued that Eberle's (1970) theory of relational individuals solved this problem.
Topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
notions of boundaries and connection can be married to mereology, resulting in
mereotopology In formal ontology, a branch of metaphysics, and in ontological computer science, mereotopology is a first-order theory, embodying mereological and topological concepts, of the relations among wholes, parts, parts of parts, and the boundaries be ...
; see Casati and Varzi (1999: ch. 4,5). Whitehead's 1929 ''
Process and Reality ''Process and Reality'' is a book by Alfred North Whitehead, in which the author propounds a philosophy of organism, also called process philosophy. The book, published in 1929, is a revision of the Gifford Lectures he gave in 1927–28. Whi ...
'' contains a good deal of informal
mereotopology In formal ontology, a branch of metaphysics, and in ontological computer science, mereotopology is a first-order theory, embodying mereological and topological concepts, of the relations among wholes, parts, parts of parts, and the boundaries be ...
.


Natural language

Bunt (1985), a study of the
semantics Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy Philosophy (f ...
of natural language, shows how mereology can help understand such phenomena as the mass–count distinction and
verb aspect In linguistics, aspect is a grammatical category that expresses how an action, event, or state, as denoted by a verb, extends over time. Perfective aspect is used in referring to an event conceived as bounded and unitary, without reference to ...
. But Nicolas (2008) argues that a different logical framework, called plural logic, should be used for that purpose. Also,
natural language In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languages ...
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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
natural science Natural science is one of the branches of science concerned with the description, understanding and prediction of natural phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer review and repeatab ...
. Casati and Varzi (1999), for example, limit the scope of mereology to
physical object In common usage and classical mechanics, a physical object or physical body (or simply an object or body) is a collection of matter within a defined contiguous boundary in three-dimensional space. The boundary must be defined and identified by t ...
s.


Metaphysics

In
metaphysics Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of conscio ...
there are many troubling questions pertaining to parts and wholes. One question addresses constitution and persistence, another asks about composition.


Mereological constitution

In metaphysics, there are several puzzles concerning cases of mereological constitution, that is, what makes up a whole. 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 The Ship of Theseus is a thought experiment about whether an object that has had all of its original components replaced remains the same object. According to legend, Theseus, the mythical Greek founder-king of Athens, had rescued the children of ...
. 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 the sculptor has a lump of clay. After many manipulations at time t2 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 In philosophy, mereological essentialism is a mereological thesis about the relationship between wholes, their parts, and the conditions of their persistence. According to mereological essentialism, objects have their parts necessarily. If an obj ...
, 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 Nihilism (; ) is a philosophy, or family of views within philosophy, that rejects generally accepted or fundamental aspects of human existence, such as objective truth, knowledge, morality, values, or meaning. The term was popularized by Ivan ...
—which makes the claim that no objects exist, except simples, so there is no persistence problem. (e) 4-dimensionalism or
temporal parts In contemporary metaphysics, temporal parts are the parts of an object that exist in time. A temporal part would be something like "the first year of a person's life", or "all of a table from between 10:00 a.m. on June 21, 1994 to 11:00 p.m. on Ju ...
(may also go by the names
perdurantism Perdurantism or perdurance theory is a philosophical theory of persistence and identity.Temporal parts ...
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.


Special composition question (SCQ)

Philosophers want to know when some Xs compose something Y. There are several kinds of responses: *One response to this question is called ''nihilism''. Nihilism states that there are no mereological complex objects (read: composite objects); there are only simples. Nihilists do not entirely reject composition because they do think that simples compose themselves, but this is a different point. More formally Nihilists would say: Necessarily, for any non-overlapping Xs, there is an object composed of the Xs if and only if there is only one of the Xs. This theory, though well explored, has its own set of problems. Some of which include, but are not limited to: experiences and common sense, incompatible with atomless gunk, and it is unsupported by space-time physics. *Another prominent response is called ''universal composition'' (UC). UC says that so long as the Xs do not spatially overlap, the Xs can compose a complex object. Universal compositionalists are also considered those who support unrestricted composition. More formally: Necessarily, for any non-overlapping Xs, there is a Y such that Y is composed of the Xs. For example, someone's left thumb, the top half of another person's right shoe, and a quark in the center of their galaxy can compose a complex object according to universal composition. Likewise, this theory also has some issues, most of them dealing with our experiences that these randomly chosen parts make up a complex whole and there are far too many objects posited in our ontology. *A third response (perhaps less explored than the previous two) includes a range of ''restricted composition views''. Though there are several views, they all share a common idea: that there is a restriction on what counts as a complex object: some (but not all) Xs come together to compose a complex Y. Some of these theories include: (a) Contact—the Xs compose a complex Y if and only if the Xs are in contact; (b) Fastenation—the Xs compose a complex Y if and only if the Xs are fastened; (c) Cohesion—the 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—the Xs compose a complex Y if and only if the Xs are fused (fusion is when the Xs are joined together such that there is no boundary); (e) Organicism—the 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. This is not an exhaustive list as many more hypotheses continue to be explored. However, a common problem with these theories is that they are vague. It remains unclear what "fastened" or "life" mean, for example. But there are many other issues within the restricted composition responses—though many of them are subject to which theory is being discussed. * A fourth response is called ''deflationism''. Deflationism states that there is variance on how the term "exist" is used, and thus all of the above answers to the SCQ can be correct when indexed to a favorable meaning of "exist." Further, there is no privileged way in which the term "exist" must be used. There is therefore no privileged answer to the SCQ, since there are no privileged conditions for when X composes Y. Instead, the debate is reduced to a mere verbal dispute rather than a genuine ontological debate. In this way, the SCQ is part of a larger debate in general ontological realism and anti-realism. While deflationism successfully avoids the SCQ, it is not devoid of problems. It comes with the cost of ontological anti-realism such that nature has no objective reality at all. For, if there is no privileged way to objectively affirm the existence of objects, nature itself must have no objectivity.


Important surveys

The books by Simons (1987) and Casati and Varzi (1999) differ in their strengths: *Simons (1987) sees mereology primarily as a way of formalizing
ontology In metaphysics, ontology is the philosophical study of being, as well as related concepts such as existence, becoming, and reality. Ontology addresses questions like how entities are grouped into categories and which of these entities exis ...
and
metaphysics Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of conscio ...
. His strengths include the connections between mereology and: **The work of
Stanisław Leśniewski Stanisław Leśniewski (30 March 1886 – 13 May 1939) was a Polish mathematician, philosopher and logician. Life He was born on 28 March 1886 at Serpukhov, near Moscow, to father Izydor, an engineer working on the construction of the Trans-Sibe ...
and his descendants **Various
continental philosophers Continental philosophy is a term used to describe some philosophers and philosophical traditions that do not fall under the umbrella of analytic philosophy. However, there is no academic consensus on the definition of continental philosophy. Pri ...
, especially
Edmund Husserl , thesis1_title = Beiträge zur Variationsrechnung (Contributions to the Calculus of Variations) , thesis1_url = https://fedora.phaidra.univie.ac.at/fedora/get/o:58535/bdef:Book/view , thesis1_year = 1883 , thesis2_title ...
**Contemporary English-speaking technical philosophers such as
Kit Fine Kit Fine (born 26 March 1946) is a British philosopher, currently university professor and Silver Professor of Philosophy and Mathematics at New York University. Prior to joining the philosophy department of NYU in 1997, he taught at the Uni ...
and
Roderick Chisholm Roderick Milton Chisholm (; November 27, 1916 – January 19, 1999) was an American philosopher known for his work on epistemology, metaphysics, free will, value theory, and the philosophy of perception. The '' Stanford Encyclopedia of Philoso ...
**Recent work on
formal ontology In philosophy, the term formal ontology is used to refer to an ontology defined by axioms in a formal language with the goal to provide an unbiased (domain- and application-independent) view on reality, which can help the modeler of domain- or a ...
and
metaphysics Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of conscio ...
, including continuants, occurrents,
class noun Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differentl ...
s, mass nouns, and ontological dependence and
integrity Integrity is the practice of being honest and showing a consistent and uncompromising adherence to strong moral and ethical principles and values. In ethics, integrity is regarded as the honesty and truthfulness or accuracy of one's actions. Inte ...
** Free logic as a background logic **Extending mereology with
tense logic In logic, temporal logic is any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time (for example, "I am ''always'' hungry", "I will ''eventually'' be hungry", or "I will be hungry ''until'' I ...
and
modal logic Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...
**
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...
s and
lattice theory A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bou ...
. *Casati and Varzi (1999) see mereology primarily as a way of understanding the material world and how humans interact with it. Their strengths include the connections between mereology and: ** A "proto-geometry" for physical objects **
Topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
and
mereotopology In formal ontology, a branch of metaphysics, and in ontological computer science, mereotopology is a first-order theory, embodying mereological and topological concepts, of the relations among wholes, parts, parts of parts, and the boundaries be ...
, especially boundaries, regions, and holes ** A formal theory of events ** Theoretical
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
** The writings of
Alfred North Whitehead Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy, which today has found applicat ...
, especially his ''
Process and Reality ''Process and Reality'' is a book by Alfred North Whitehead, in which the author propounds a philosophy of organism, also called process philosophy. The book, published in 1929, is a revision of the Gifford Lectures he gave in 1927–28. Whi ...
'' and work descended therefrom.Cf. Peter Simons, "Whitehead and Mereology", in Guillaume Durand et
Michel Weber Michel Weber (born 1963) is a Belgian philosopher. He is best known as an interpreter and advocate of the philosophy of Alfred North Whitehead, and has come to prominence as the architect and organizer of an overlapping array of international ...
(éditeurs),
Les principes de la connaissance naturelle d’Alfred North Whitehead — Alfred North Whitehead’s Principles of Natural Knowledge
', Frankfurt / Paris / Lancaster, ontos verlag, 2007. See also the relevant entries of
Michel Weber Michel Weber (born 1963) is a Belgian philosopher. He is best known as an interpreter and advocate of the philosophy of Alfred North Whitehead, and has come to prominence as the architect and organizer of an overlapping array of international ...
and Will Desmond, (eds.),
Handbook of Whiteheadian Process Thought
', Frankfurt / Lancaster, ontos verlag, Process Thought X1 & X2, 2008.
Simons devotes considerable effort to elucidating historical notations. The notation of Casati and Varzi is often used. Both books include excellent bibliographies. To these works should be added Hovda (2008), which presents the latest state of the art on the axiomatization of mereology.


See also

*
Finitist set theory Finitist set theory (FST) is a collection theory designed for modeling finite nested structures of individuals and a variety of transitive and antitransitive chains of relations between individuals. Unlike classical set theories such as ZFC and ...
*
Gunk (mereology) In mereology, an area of philosophical logic, the term gunk applies to any whole whose parts all have further proper parts. That is, a gunky object is not made of indivisible ''atoms'' or '' simples''. Because parthood is transitive, any part ...
* Implicate and explicate order according to David Bohm * ''
Laws of Form ''Laws of Form'' (hereinafter ''LoF'') is a book by G. Spencer-Brown, published in 1969, that straddles the boundary between mathematics and philosophy. ''LoF'' describes three distinct logical systems: * The "primary arithmetic" (described in C ...
'' by
G. Spencer-Brown George Spencer-Brown (2 April 1923 – 25 August 2016) was an English polymath best known as the author of '' Laws of Form''. He described himself as a "mathematician, consulting engineer, psychologist, educational consultant and practitioner, co ...
*
Mereological essentialism In philosophy, mereological essentialism is a mereological thesis about the relationship between wholes, their parts, and the conditions of their persistence. According to mereological essentialism, objects have their parts necessarily. If an obj ...
*
Mereological nihilism In philosophy, mereological nihilism (also called compositional nihilism) is the metaphysical thesis that there are no objects with proper parts. Equivalently, mereological nihilism says that mereological simples, or objects without any proper ...
*
Mereotopology In formal ontology, a branch of metaphysics, and in ontological computer science, mereotopology is a first-order theory, embodying mereological and topological concepts, of the relations among wholes, parts, parts of parts, and the boundaries be ...
*
Meronomy A meronomy or partonomy is a type of hierarchy that deals with part–whole relationships, in contrast to a taxonomy whose categorisation is based on discrete sets. Accordingly, the unit of meronomical classification is meron, while the unit of ...
*
Meronymy In linguistics, meronymy () is a semantic relation between a meronym denoting a part and a holonym denoting a whole. In simpler terms, a meronym is in a ''part-of'' relationship with its holonym. For example, ''finger'' is a meronym of ''hand' ...
*
Monad (philosophy) The term ''monad'' () is used in some cosmic philosophy and cosmogony to refer to a most basic or original substance. As originally conceived by the Pythagoreans, the Monad is the Supreme Being, divinity or the totality of all things. In the p ...
*
Plural quantification In mathematics and logic, plural quantification is the theory that an individual variable x may take on ''plural'', as well as singular, values. As well as substituting individual objects such as Alice, the number 1, the tallest building in London ...
*
Quantifier variance The term quantifier variance refers to claims that there is no uniquely best ontological language with which to describe the world. The term "quantifier variance" rests upon the philosophical term 'quantifier', more precisely existential quantifier. ...
*
Simple (philosophy) In contemporary mereology, a simple is any thing that has no proper parts. Sometimes the term "atom" is used, although in recent years the term "simple" has become the standard. Simples are to be contrasted with atomless gunk (where something i ...
* Whitehead's point-free geometry *
Composition (objects) Compositional objects are wholes instantiated by collections of parts. If an ontology wishes to permit the inclusion of compositional objects it must define which collections of objects are to be considered parts composing a whole. Mereology, the ...


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 Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy, which today has found applica ...
, 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 Trinity College is a constituent college of the University of Cambridge. Founded in 1546 by Henry VIII, King Henry VIII, Trinity is one of the largest Cambridge colleges, with the largest financial endowment of any college at either Cambridge ...
. *------, 1978 (1929). ''
Process and Reality ''Process and Reality'' is a book by Alfred North Whitehead, in which the author propounds a philosophy of organism, also called process philosophy. The book, published in 1929, is a revision of the Gifford Lectures he gave in 1927–28. Whi ...
''. Free Press. * Woodger, J. H., 1937. ''The Axiomatic Method in Biology''. Cambridge Univ. Press.


External links

* * *
Internet Encyclopedia of Philosophy The ''Internet Encyclopedia of Philosophy'' (''IEP'') is a scholarly online encyclopedia, dealing with philosophy, philosophical topics, and philosophers. The IEP combines open access publication with peer reviewed publication of original pape ...
: *
Material Composition
– David Cornell *
Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. Eac ...
: *
Mereology
– Achille Varzi *
Boundary
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