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Mereology

From Academic Kids

Mereology is the branch of logic, mathematics and metaphysics dealing with part-whole relationships. Stanislaw Lesniewski first formalized the discipline, naming it in 1927 using the Greek root meros (a part). Henry Leonard and Nelson Goodman later greatly elaborated mereology.

Mereology received its motivatation in part from the view that set theory has something ontologically suspicious about it, and from the position, derived from Occam's Razor, that one should minimise the number of entities in one's theory of the world. Mereology replaces talk of "sets" of objects with talk of "sums" of objects, viewed as no more than the various things that make up 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 capacity, although it typically receives less attention than set theory.

Contents

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 one must formulate mereology via axioms.

The standard axioms of mereology form close analogies 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); sometimes this axiom is called Unrestricted Mereological Composition or the General Sum Principle.
  • Extensionality (any two objects with exactly the same parts are the same object)
  • Bottom (there exists an "empty object", an object with no parts and therefore which forms 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 closely resembles set theory. (An important exception is that mereology is often thought to be conceptually all right 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 consider set theory 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 natural language often uses the words "part of" in ambiguous ways. 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.

Mereologists most commonly view the part-of relationship as a partial order, although some have questioned this. The most frequent objection is that the part-of relationship is not necessarily transitive.

A very different link between natural language and mereology is the following: In linguistics, mereology has lately been used in the formal analysis of natural language semantics to understand such phenomena as the mass/count distinction and grammatical aspect.

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