mereology

=Mereology=

toc Read here about mereology, the formal term for the theory of parthood relations, and its significance in studying scale, where behavior in inter-scale and intra-scale is compared.

=Definition= Mereology (from the Greek μερος, ‘part’) is the theory of parthood relations: of the relations of part to whole and the relations of part to part within a whole. Its roots can be traced back to the early days of philosophy, beginning with the Presocratics and continuing throughout the writings of Plato, Aristotle and Boethius. (Stanford Encyc of Philosophy)

=Contrast Of Mereology With Set Theory= Mereology is an attempt to lay down the general principles underlying the relationships between an entity and its constituent parts, whatever the nature of the entity, just as set theory is an attempt to lay down the principles underlying the relationships between a set and its members. Unlike set theory, mereology is not committed to the existence of abstracta: the whole can be as concrete as the parts. But mereology carries no nominalistic commitment to concreta either: the parts can be as abstract as the whole. David Lewis's Parts of Classes (1991), which provides a mereological analysis of the set-theoretic universe, is a good illustration of this ontological neutrality of mereology. (It should be noted, however, that this way of conceiving of mereology has sometimes been challenged, the worry being that very different part-whole relations may hold between different kinds of entity. We shall not pursue this option here, but see e.g. Mellor 2006 for a representative example. See also Uzquiano 2006 for some worries concerning the alleged universality and topic-neutrality of mereology.)

=Problem of the Many in Mereology= Peter Unger has defined the Problem of the Many, summarized in the Stanford Encyclopedia of Philosophy as follows. As anyone who has flown out of a cloud knows, the boundaries of a cloud are a lot less sharp up close than they can appear on the ground. Even when it seems clearly true that there is one, sharply bounded, cloud up there, really there are thousands of water droplets that are neither determinately part of the cloud, nor determinately outside it. Consider any object that consists of the core of the cloud, plus an arbitrary selection of these droplets. It will look like a cloud, and circumstances permitting rain like a cloud, and generally has as good a claim to be a cloud as any other object in that part of the sky. But we cannot say every such object is a cloud, else there would be millions of clouds where it seemed like there was one. And what holds for clouds holds for anything whose boundaries look less clear the closer you look at it. And that includes just about every kind of object we normally think about, including humans. Although this seems to be a merely technical puzzle, even a triviality, a surprising range of proposed solutions has emerged, many of them mutually inconsistent. It is not even settled whether a solution should come from metaphysics, or from philosophy of langauge, or from logic.

David Lewis version from his “Many, but Almost One” essay in 1993:

> Think of a cloud—just one cloud, and around it a clear blue sky. Seen from the ground, the cloud may seem to have a sharp boundary. Not so. The cloud is a swarm of water droplets. At the outskirts of the cloud, the density of the droplets falls off. Eventually they are so few and far between that we may hesitate to say that the outlying droplets are still part of the cloud at all; perhaps we might better say only that they are near the cloud. But the transition is gradual. Many surfaces are equally good candidates to be the boundary of the cloud. Therefore many aggregates of droplets, some more inclusive and some less inclusive (and some inclusive in different ways than others), are equally good candidates to be the cloud. Since they have equal claim, how can we say that the cloud is one of these aggregates rather than another? But if all of them count as clouds, then we have many clouds rather than one. And if none of them count, each one being ruled out because of the competition from the others, then we have no cloud. How is it, then, that we have just one cloud? And yet we do.

=Table of Mereological Statements= Source: Stanford Encyclopedia of Philosophy, @http://plato.stanford.edu/entries/mereology/ accessed 6 May 2010 0645 GMT

List of Statements
The following 25 generic statements are used to illustrate and distinguish parthood relationships.
 * The handle is part of the mug.
 * This cap is part of my pen.
 * The left half is your part of the cake.
 * The cutlery is part of the tableware.
 * The contents of this bag is only part of what I bought.
 * That area is part of the living room.
 * The outermost points are part of the perimeter.
 * The first act was the best part of the play.
 * The clay is part of the statue.
 * Gin is part of martini.
 * Writing detailed comments is part of being a good referee.
 * The conclusion is part of the argument.
 * The domain of quantification is part of the model.
 * The suffix is part of the official file name.
 * Rationality is part of personhood.
 * The conclusion is part of the argument.
 * The domain of quantification is part of the model.
 * The suffix is part of the official file name.
 * Rationality is part of personhood.
 * Everything is part of itself.
 * Any part of any part of a thing is itself part of that thing.
 * Two distinct things cannot be part of each other.
 * A biological subunit of a cell is not a part of the organ of which that cell is a part;
 * A handle can be part of a door and the door of a house, though a handle is never part of a house;
 * My fingers are part of me and I am part of the team, yet my fingers are not part of the team.

Characterization of the parthood relationships

 * part, regardless of attachment [1]
 * detached part
 * cognitively salient part
 * arbitrarily demarcated part
 * self-connected part
 * disconnected part
 * homogeneous part
 * gerrymandered part
 * material part
 * immaterial part
 * extended part
 * unextended part
 * spatial part
 * temporal part
 * relation of material constitution [2,3]
 * constitution: relation of mixture composition [2,4]
 * relation of conceptual inclusion [2,5]
 * difference as material bodies, events, geometric entities, or spatial regions
 * difference as abstract entities such as propositions, sets, types, or properties
 * general relation
 * partial ordering—a reflexive, transitive, antisymmetric relation
 * reflexivity
 * transitive [6]
 * nontransitive [6]
 * asymmetric postulate

Notes on parthood characterizations
[1] any portion of a given entity, regardless of whether the portion itself is attached to the remainder [2] contrversial mereological status [3] Although this constitution relation was included by Aristotle in his threefold taxonomy of parthood (Metaphysics, Δ, 1023b), many contemporary authors would rather construe it as a sui generis, non-mereological relation (see e.g. Wiggins 1980, Rea 1995, and Thomson 1998). [4] This ingredient-mixture relationship is subject to controversy, as the ingredients may involve significant structural connections besides spatial proximity and may therefore fail to retain certain important chemical characteristics they have in isolation (see Sharvy 1983). [5] The term ‘part’ in this case can be argued as appearing only in the surface grammar and disappears at the level of logical form, e.g., paraphrase as “A good referee is one who writes detailed comments.” (For more examples and tentative taxonomies, see Winston et al. 1987, Iris et al. 1988, and Gerstl and Pribbenow 1995.) [6] What counts as a biological subunit of a cell may not count as a subunit, i.e., a distinguished part of the organ, but that is not to say that it is not part of the organ at all. Similarly, if there is a sense of ‘part’ in which a handle is not part of the house to which it belongs, or my fingers not part of my team, it is a restricted sense: the handle is not a functional part of the house, though it is a functional part of the door and the door a functional part of the house; my fingers are not directly part of the team, though they are directly part of me and I am directly part of the team. It is obvious that if the interpretation of ‘part’ is narrowed by additional conditions (e.g., by requiring that parts make a functional or direct contribution to the whole), then transitivity may fail. In general, if x is a φ-part of y and y is a φ-part of z, x need not be a φ-part of z: the predicate modifier ‘φ’ may not distribute over parthood. But that shows the non-transitivity of ‘φ-part’, not of ‘part’. And within a sufficiently general framework this can easily be expressed with the help of explicit predicate modifiers (Varzi 2006a).

Mapping statements to Characterizations

 * The handle is part of the mug: part, regardless of attachment [1]; cognitively salient part; self-connected part; homogeneous part; material part; extended part; spatial part; difference as material bodies, events, geometric entities, or spatial regions; general relation; partial ordering—a reflexive, transitive, antisymmetric relation
 * This cap is part of my pen: detached part; cognitively salient part; self-connected part; homogeneous part; material part; extended part; spatial part; difference as material bodies, events, geometric entities, or spatial regions; general relation; partial ordering—a reflexive, transitive, antisymmetric relation
 * The left half is your part of the cake: arbitrarily demarcated part; self-connected part; homogeneous part; material part; extended part; spatial part; difference as material bodies, events, geometric entities, or spatial regions; general relation; partial ordering—a reflexive, transitive, antisymmetric relation
 * The cutlery is part of the tableware: disconnected part; homogeneous part; material part; extended part; spatial part; difference as material bodies, events, geometric entities, or spatial regions; general relation; partial ordering—a reflexive, transitive, antisymmetric relation
 * The contents of this bag is only part of what I bought: gerrymandered part; material part; extended part; spatial part; difference as material bodies, events, geometric entities, or spatial regions; general relation; partial ordering—a reflexive, transitive, antisymmetric relation
 * That area is part of the living room: immaterial part; extended part; spatial part; difference as material bodies, events, geometric entities, or spatial regions; general relation; partial ordering—a reflexive, transitive, antisymmetric relation
 * The outermost points are part of the perimeter: unextended part; spatial part; difference as material bodies, events, geometric entities, or spatial regions; general relation; partial ordering—a reflexive, transitive, antisymmetric relation
 * The first act was the best part of the play: temporal part; difference as material bodies, events, geometric entities, or spatial regions; general relation; partial ordering—a reflexive, transitive, antisymmetric relation
 * The clay is part of the statue: relation of material constitution [2,3]
 * Gin is part of martini: constitution: relation of mixture composition [2,4]
 * Writing detailed comments is part of being a good referee: relation of conceptual inclusion [2,5]; difference as abstract entities such as propositions, sets, types, or properties
 * The conclusion is part of the argument: general relation; partial ordering—a reflexive, transitive, antisymmetric relation
 * The domain of quantification is part of the model: general relation; partial ordering—a reflexive, transitive, antisymmetric relation
 * The suffix is part of the official file name: general relation; partial ordering—a reflexive, transitive, antisymmetric relation
 * Rationality is part of personhood: general relation; partial ordering—a reflexive, transitive, antisymmetric relation
 * The conclusion is part of the argument: difference as abstract entities such as propositions, sets, types, or properties; reflexivity
 * The domain of quantification is part of the model: difference as abstract entities such as propositions, sets, types, or properties
 * The suffix is part of the official file name: difference as abstract entities such as propositions, sets, types, or properties; transitive [6]
 * Rationality is part of personhood: difference as abstract entities such as propositions, sets, types, or properties
 * Everything is part of itself:
 * Any part of any part of a thing is itself part of that thing:
 * Two distinct things cannot be part of each other: asymmetric postulate
 * A biological subunit of a cell is not a part of the organ of which that cell is a part; : nontransitive [6]
 * A handle can be part of a door and the door of a house, though a handle is never part of a house: nontransitive [6]
 * My fingers are part of me and I am part of the team, yet my fingers are not part of the team: nontransitive [6]