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 ...
.
:
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.
:
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''.
:
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:
:
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.
:
P.1
:M2,
Antisymmetric: If ''Pxy'' and ''Pyx'' both hold, then ''x'' and ''y'' are the same object.
:
P.2
:M3,
Transitive: If ''Pxy'' and ''Pyz'', then ''Pxz''.
:
P.3
*M4, Weak Supplementation: If ''PPxy'' holds, there exists a ''z'' such that ''Pzy'' holds but ''Ozx'' does not.
:
P.4
*M5, Strong Supplementation: If ''Pyx'' does not hold, there exists a ''z'' such that ''Pzy'' holds but ''Ozx'' does not.
:
P.5
*M5', Atomistic Supplementation: If ''Pxy'' does not hold, then there exists an atom ''z'' such that ''Pzx'' holds but ''Ozy'' does not.
:
P.5'
*Top: There exists a "universal object", designated ''W'', such that ''PxW'' holds for any ''x''.
:
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''.
:
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''.
:
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''.
:
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.
:
_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_...