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Mereology is the branch of logic, mathematics, and metaphysics dealing with part-whole relationships. It was first formalized by Stanislaw Lesniewski, and greatly elaborated later by Henry Leonard and Nelson Goodman.
Mereology was in part motivated by the view that there is something ontologically suspicious about set theory, and the position, derived from Occam's Razor, that it is best to have the minimum number of entities in one's theory of the world. It replaces talk of sets of objects with talk of "sums" of objects, viewed as no more than the various things that are their parts.
Many modern logicians and philosophers reject these motivations, on such grounds as: (1) There is nothing "suspicious" about sets; (2) Occam's Razor is, particularly as applied to the sort of "objects" logic deals with, either a dubious principle or an outright false one; (3) Mereology does--in the logical sense--introduce new entities. Nonetheless mereology itself is accepted as an extremely useful tool and perfectly good in that right, although it typically receives less attention than set theory.
Mereology and set theory
It is possible to formulate a "naive mereology" analogous to naive set theory, and possible to generate paradoxes analogous to Russell's Paradox. (There is an object whose parts are all the objects that are not parts of themselves. Is it a part of itself?). Hence mereology must be formulated via axioms.
The standard axioms of mereology are closely analogous to those of standard Zermelo-Fraenkel set theory. They include: Universalism (for any two objects x and y there is a third whose parts comprise x and y alone); Extensionality (Any two objects with exactly the same parts are the same object); Bottom (There is an "empty object", an object with no parts and therefore which is a part of everything); Everything is a part of itself (to which we add the definition "proper part: a proper part of x is a part of x which is not identical to x itself); if x is a part of y and y is a part of z, then x is a part of z.
Give a complete list. See the Mereology article in the Stanford Encyclopedia of Philosophy (http://plato.stanford.edu/entries/mereology/) for a good listing of a dozen, in descending order of popularity, with some of the important implication relationships
If the "parthood" relation is taken as corresponding not to the membership relation in set theory, but to the subset relation, then merelogy is very similar to set theory. (An important exception is that mereology is often though to be conceptually alright if it contains no "atoms": if you can keep breaking things into parts forever. In set theory, unit sets are "atoms" which have no proper parts; many people think set theory is useless or incoherent if sets do not eventually "bottom out" into individuals.)
Mereology and natural language
A problem in the field of mereology is that the words "part of" are often used in ambiguous ways in natural language. If one treats the theory as a device for adding nuances to logical reasoning, it need not lead to any problems; but it is doubtful where, if ever, the mereological predicates can exactly translate correlate expressions in natural language.
The part-of relationship is most commonly viewed as a partial order, although some have questioned this. The most frequent objection is that the part-of relationship is not necessarily transitive.
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