Wednesday, November 14, 2012

Definitional Mereotopology

Let us start with a basic idea: definitions of concepts can have parts. We can think about definitional parts in two ways, depending on whether we take the whole definition to count as a 'part' of the definition. When we count the definition itself as a 'part' of the definition, let's call this a basic definitional part of the concept defined and give it the abbreviation P. Basic definitional parts have the following basic properties:

(1) xPx
-> definitions are basic definitional parts of themselves

(2) xPy & yPz -> xPz
-> if x is a basic definitional part of y, and y is a basic definitional part of z, then x is also a basic definitional part of z.

(3) xPy & yPx -> x=y
-> if x and y are both basic definitional parts of each other, then they are the same definition, at least for any relevant purposes

So suppose our definition is that human beings are rational animals. Then we can say that rationality is a basic definitional part of the concept of humanity, and so is animality. It's also true (by (1)), that rational-animality is a basic definitional part (it's the basic definitional part that happens to be the whole thing). If part of the definition of 'rationality' is (just for example) 'capable in principle of grasping abstract universal concepts', this is a basic definitional part of rationality, and it is also (by (2)) a basic definitional part of anything of which rationality is a basic definitional part. And if rational-animality is a basic definitional part of animal-rationality and vice versa, then they can (by (3)) be treated as simply the same thing.

But we often use 'part' in a narrower sense, in which wholes are not counted as parts in the relevant sense. Let's call this a strict definitional part of the concept defined, and give it the abbreviation PP. Strict definitional parts are related to basic definitional parts in the following way:

(4) xPPy <-> xPy & ~yPx
-> that is, x is a strict definitional part of y when it is a basic definitional part of y, but y is not a definitional part of it
-> or, in other words: strict definitional parts are basic definitional parts that are not equal to the whole

It's also clear that definitions can overlap. Basic definitional overlap, which we can abbrevaite as O, is easily defined: it occurs when there is something that is a basic definitional part of both definitions. (We could also define a strict definitional overlap using strict definitional parts.) Examples of definitional overlap between the definition of humanity and the definition of doghood would include things like animality, mammality, vertebratehood, and so forth.

We can also see that things can be relevant to definitions in various ways. Let's say we have basic definitional relevance, given the abbreviation C, whenever something is relevant to a definition in a way that meets the following conditions:

(5) xCx
-> definitions are definitionally relevant to themselves.

(6) xCy -> yCx
-> whenever x is definitionally relevant to y, the reverse is also true.

It is clear from this that if x definitionally overlaps with y, they are definitionally relevant to each other. Likewise, it is clear from this that all basic definitional parts and all strict definitional parts are definitionally relevant to the concepts for which they are parts.

We can get more specific by uniting our two concepts. Let something be internally relevant to the definition, abbreviated IP, if it meets the following condition:

(7) xIPy <-> (xPy & (zCx -> zOy))
-> x is internally relevant to the definition of y when x is part of y and when something, call it z,is definitionally relevant to x it is so in such a way that z definitionally overlaps with y.

And we can likewise let something be externally relevant to the definition (EC), when it is definitionally relevant to it but not so as to overlap with it: for instance, if definitions are indirectly relevant to each other without actually sharing any parts. (It's actually quite an interesting question whether this is possible, and, if so, the conditions under which it is.)

All of these, of course, are just some of the simpler possible mereotopological concepts applied to definitions of concepts; thus mereotopology can provide a rigorous vocabulary for talking about definitions.

2 comments:

  1. John Donne6:25 AM

    Thanks for this. I find basic  property #2 especially interesting, for it points me in a direction to think of those phenomena to which it does not or need not apply, and to mull over how such things differ from "definitions".

    I think I'll ponder it for a couple of days, at least.

    ReplyDelete
  2. branemrys6:53 AM

    That's definitely one of the useful things about exercises like this.

    ReplyDelete

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