mereology
please note:
- the text and code below is from The Pseudopedia
- it has been imported raw for GetWiki
{{nofootnotes|date=February 2010}}In
philosophy,
mereology (from the Greek μέρος, root: μερε(σ)-, "part" and the suffix -logy "study, discussion, science") is a collection of axiomatic
first-order theories dealing with parts and their respective wholes. In contrast to
set theory, which takes the set–member relationship as fundamental, the core notion of mereology is
meronomic, which means based on part–whole relationships. Mereology is both an application of
predicate logic and a branch of
formal ontology.
History
Informal part-whole reasoning was consciously invoked in
metaphysics and
ontology from
Plato (particularly in the second half of the
Parmenides) and
Aristotle onwards, and more or less unwittingly in 19th century mathematics until the triumph of
set theory around 1910.
Ivor Grattan-Guinness (2001) sheds much light on part-whole reasoning during the 19th and early 20th centuries, and reviews how
Cantor and
Peano devised
set theory. Apparently, the first to reason consciously and at length about parts and wholes was
Edmund Husserl in his 1901
Logical Investigations (Husserl 1970 is the English translation). However, the word "mereology" is absent from his writings, and he employed no
symbolism even though his doctorate was in mathematics.
Stanisław Leśniewski coined "mereology" in 1927, from the Greek word μέρος (
méros, "part"), to refer to a formal theory of part-whole he devised in a series of highly technical papers published between 1916 and 1931, and translated in Leśniewski (1992). Leśniewski's student
Alfred Tarski, in his Appendix E to Woodger (1937) and the paper translated as Tarski (1984), greatly simplified Leśniewski's formalism. Other students (and students of students) of 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, on
geometry, but never wrote it. His 1914 correspondence with
Bertrand Russell reveals that his intended approach to geometry can be seen, with the benefit of hindsight, as mereological in essence. This work culminated in Whitehead (1916) and the mereological systems of Whitehead (1919, 1920).In 1930, Henry 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 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) and Casati and Varzi (1999).Standard university texts on logic and mathematics are silent about mereology, which has undoubtedly contributed to its undeserved obscurity. Although mereology is an application of
mathematical logic, arguably a sort of "proto-geometry", it has been wholly developed by logicians,
ontologists, and computer scientists, especially those working in
artificial intelligence. "Mereology" can also refer to formal work on system decomposition (by, e.g.,
Gabriel Kron or Maurice Jessel), or on parts, wholes and boundaries. Such ideas appear in theoretical
computer science and
physics, often in combination with
Sheaf,
Topos, or
Category Theory. See the work of Steven Vickers on (parts of) specifications,
Joseph Goguen on physical systems, and Tom Etter on
Link Theory.
Axioms and primitive notions
It is possible to formulate a "naive mereology" analogous to
naive set theory. Doing so gives rise to paradoxes analogous to
Russell's paradox. Let there be an object
O such that every object that is not a proper part of itself is a proper part of
O. Is
O a proper part of itself? No, because no object is a proper part of itself; and yes, because it meets the specified requirement for inclusion as a proper part of
O. (Every object is, of course, an
improper part of itself. Another, though differently structured, paradox can be made using
improper part instead of
proper part; and another using
improper or proper part.) Hence mereology requires an
axiomatic formulation. A mereological "system" is a
first-order theory (with
identity) whose
universe of discourse consists of wholes and their respective parts, collectively called
objects. Mereology is a collection of nested and non-nested
axiomatic systems, not unlike the case with
modal logic.The treatment, terminology, and hierarchical organization below follow Casati and Varzi (1999: Ch. 3) closely. For a more recent treatment, correcting certain misconceptions, see Hovda (2008). Lower-case letters denote variables ranging over objects. Following each symbolic axiom or definition is the number of the corresponding formula in Casati and Varzi, written in bold.A mereological system requires at least one primitive
binary relation (
dyadic predicate). The most conventional choice for such a relation is
Parthood (also called "inclusion"), "
x is a
part of
y", written
Pxy. Nearly all systems require that Parthood
partially order the universe. The following defined relations, required for the axioms below, follow immediately from Parthood alone:
PPxy ↔ (Pxy and ˜ Pyx).
3.3
An object lacking proper parts is an
atom. The mereological
universe consists of all objects we wish to think about, and all of their proper parts:
- Overlap: x and y overlap, written Oxy, if there exists an object z such that Pzx and Pzy both hold.
Oxy ↔ eξsts z[Pzx and 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 ↔ eξsts z[Pxz and Pyz ].
3.2
Overlap and Underlap are
reflexive,
symmetric, and in
transitive.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 ↔ ∀ z[Ozx → Ozy].
3.31
The axioms are:
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 and Pyx) → x = y.
P.2
M3,
Transitive: If
Pxy and
Pyz, then
Pxz.
(Pxy and Pyz) → Pxz.
P.3
- M4, Weak Supplementation: If PPxy holds, there exists a z such that Pzy holds but Ozx does not.
PPxy → eξsts z[Pzy and ˜ Ozx].
P.4
- M5, Strong Supplementation: Replace "PPxy holds" in M4 with "Pyx does not hold".
˜ Pyx → eξsts z[Pzy and ˜ 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.
˜ Pxy → eξsts z[Pzx and ˜ Ozy and ˜ eξsts v [PPvz]].
P.5'
- Top: There exists a "universal object", designated W, such that PxW holds for any x.
eξsts W ∀ 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.
- 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 which overlap either x or y.
Uxy → eξsts z ∀ v [Ovz ↔ (Ovx or 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 which are parts of both x and y.
Oxy → eξsts z ∀ v [Pvz ↔ (Pvx and 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 formula in which x is a free variable. 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
principle of unrestricted comprehension of
naive set theory, which gives rise to
Russell's paradox. There is no mereological counterpart to this paradox simply because
Parthood, unlike set membership, is
reflexive.
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
(other abbreviations are explained below). In
hold as axioms or are theorems. M9,
, in the sense that if all the theorems of system A are also theorems of system B, but the converse is not necessarily true, then B
A. The resulting
and Fig. 3.2 in Casati and Varzi (1999: 48).{| class=wikitable
