Mereology

Literally, ‘theory of parts’. Term introduced by the Polish logician Stanislaw Lesniewski (1886-1939) to cover a theory which used the whole/part relation as a substitute for the class-membership relation to deal with the structure of classes in ways that would avoid various difficulties connected with the vicious CIRCLE PRINCIPLE and the theory of types. (The term also has a technical use within the theory itself.)

The point about the whole/part relation is that, unlike class-membership, it is transitive; that is if a is a part of b, and b is a part of c, then a is a part of c.

The notion has also been used (by N Goodman, The Structure of Appearance (1951)) to deal with problems concerning stuffs (like water) or general qualities (like red): ‘water’ is taken to be a name for the total quantity of water in the universe (so that the Pacific Ocean counts as a part of water); similarly ‘red’, in naming the colour red, names the totality of red things, treated as a single large object split up over space.

Source:
P Simons, Parts (1987), ch. 1

In philosophy and mathematical logic, mereology (from the Greek μέρος meros (root: μερε- mere-, “part”) and the suffix -logy “study, discussion, science”) is the study of parts and the wholes they form. Whereas set theory is founded on the membership relation between a set and its elements, mereology emphasizes the meronomic relation between entities, which—from a set-theoretic perspective—is closer to the concept of inclusion between sets.

Mereology has been explored in various ways as applications of predicate logic to formal ontology, 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 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), thus forming a poset. 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, what could be argued to be a sort of “proto-geometry”, it has been wholly developed by logicians, ontologists, linguists, engineers, and computer scientists, especially those working in artificial intelligence. 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).

“Mereology” can also refer to formal work in general systems theory on system decomposition and parts, wholes and boundaries (by, e.g., Mihajlo D. Mesarovic (1970), Gabriel Kron (1963), or Maurice Jessel (see Bowden (1989, 1998)). A hierarchical version of Gabriel Kron’s Network Tearing was published by Keith Bowden (1991), reflecting David Lewis’s ideas on gunk. Such ideas appear in theoretical computer science and physics, often in combination with sheaf theory, topos, or category theory. See also the work of Steve Vickers on (parts of) specifications in computer science, Joseph Goguen on physical systems, and Tom Etter (1996, 1998) on link theory and quantum mechanics.