Saturday, December 15, 2012

Mereotopology and Lists

Here's a list:

Dog
Cat
Cow

And another list:

Wolf
Dog
Fox

One way we could compare these is by taking 'Dog' as a shared part. The lists overlap at dog. We could call this a mereological approach to lists.

A different way is that we could say, "No, 'Dog' is not a shared part, because they are different lists, not some kind of branching list with 'Dog' at the crossways. But the lists are connected at 'Dog'." We could call this a topological approach to lists.

This bifurcation occurs in lists of all kinds. Two similar pictures, for instance, could be compared in mereological terms (they share features) or in topological terms (their features are corresponding, whether you wish to regard them as actually shared or not). (If it sounds odd to call pictures 'lists', think of a computer screen, which consists of rows of pixels. A picture of this sort would be a list of pixel-patterns according to row.) In practical terms the mereological and the topological approaches are equivalent: when similarity is involved, you can always treat it mereologically as a sort of overlap or part-sharing or topologically as a sort of connection or correspondence. Connection is usually treated as weaker than overlap, which it is logically, since overlapping things always connect, but not vice versa. But in the context of lists, they are closely connected to the same thing, the fact that different lists can have the same items, and therefore are (nonsynonomous) ways of talking about the same thing. The mereological approach just (as a matter of structure) emphasizes the sameness, while the topological approach (again as a matter of structure) de-emphasizes it, and these lists are simple enough that it doesn't matter much which you do.

But there are kinds of lists in which one would want to be able to do both. Take this list.

  • Dog
    1. Collie
    2. Great Dane
  • Cat
    1. Egyptian Mao
    2. Ragdoll
  • Pets Jack Owns
    1. Collie
    2. Goldfish
    3. Egyptian Mao

We have lists within lists, and we want to be able to talk mereologically (the [Collie, Great Dane] list is part of the [Dog, Cat, Pets Jack Owns] list) and topologically (the [Collie, Great Dane] list is connected to the [Collie, Goldfish, Egyptian Mao] list).

The mereotopology of lists has broader implications than just lists in our ordinary sense, because modal logics can all be treated as the logics governing different kinds of relations among different kinds of lists. We can distinguish source lists and target lists (not necessarily mutually exclusive); and modal logic would be an account of what you can determine about target lists from source lists. Suppose that [p] is Box-p (It is necessary that p, It is required that p, It is always true that p, etc.) and <p> is Diamond-p (It is possible that p, It is is permissible that p, It is sometimes true that p, etc.). Then our source list says [p], we can conclude that if there are any target lists, they have p as an item. Likewise, if our source list says <p>, then we can conclude that there is some target list that has p as an item. Given this, however, we can say that, for any source list, [p] indicates that any list connected to the source list (in the relevant way) overlaps with the source list at p, and <p> indicates that some list overlaps with the source list at p and therefore is connected to the source list (in the relevant way).

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