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Saturday, October 09, 2010

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Some Further Thoughts Regarding the Bundling-Theory-of-Individuals,

The present post is focused on three types of questions. The first type concerns the logic of the *bundling operation*. ‘B’, ‘B*’, etc., are bundle variables; ‘U’, ‘U*’, ‘V’, etc., are universal variables; ‘x’, ‘y’, etc., are individual variables.

The second type raises some questions about how Bill’s Bundling Theory of Individuals (BTI) makes sense of several common sense relations such as part-whole: e.g., a table leg is a proper-part of the whole table; a room is a proper-part of the whole house, etc. The third type of questions raises some conceptual issues about BTI; e.g., Bill’s proposal to replace ‘exemplification’ between universals and individuals with a relation of bundled-with between universals or the relation of *ontological part*. Here I shall also raise some quetions about multiple-exemplification and how BTI handles this property of universals.

Section A: The Logic of the Bundling.

1) Bill proposes the following two definitions:
D1. X is complete =df for any predicate P, either x satisfies P or it is not the case that x satisfies P.
D2. Bundle B has the property P-ness =df P-ness is an ontological 'part' of B.

2) D1 is ambiguous between an excluded middle reading and a gappy reading. According to the excluded middle reading, for every predicate either it or its negation is satisfied by an individual. According to the gappy reading, the second clause of the definians means that the individual satisfies neither the predicate nor its negation. The two readings of D1 may give different answers to the question as to the number of universals it takes to constitute an individual according to BTI. Therefore, D1 must be disambiguated. I shall assume the excluded middle reading and hence modify D1 as follows:

D1*. X is complete =df for any predicate P, either x satisfies P or x satisfies ~P (but not both).

2.1) The clause ‘but not both’ indicates that we assume throughout the law of non-contradiction; i.e., no individual satisfies both a predicate and its negation simultaneously. The law of non-contradiction is assumed to hold for all clauses of BTI.

2.2) BTI dispenses with the traditional asymmetric relation of exemplification between universals and individuals and replaces it with a symmetric relation between bundles. Bill says:

“Now on a scheme like this [i.e., BTI] there is no exemplification (EX) strictly speaking. EX is an asymmetrical relation -- or relational tie: If x exemplifies P-ness, then it is not the case that P-ness exemplifies x. Bundling is not exemplification because bundling is symmetrical: if U1 is bundled with U2, then U2 is bundled with U1. So what do we mean when we say of a particular construed as a bundle that is has -- or 'exemplifies' or 'instantiates' using these terms loosely -- a property? We mean that it has the property as a 'part.' Not as a spatial or temporal part, but as an ontological part. “

2.3) Definition D2 is intended to codify this replacement. Unfortunately, there is a systematic confusion here. Suppose U is exemplified by x. Let x be construed as the bundle Bx. Then the former relation of exemplification between U and x is now replaced by some relation between U and Bx (since x just is Bx). What is this relationship? In the above quotation, Bill seems to identify this relation as a symmetric relation between two *universals* ‘U1’ and ‘U2, such that “if U1 is bundled-with U2, then U2 is bundled-with U1.”

But notice that the bundled-with relation between two universals may be quite different from the relation that holds between a universal and a *bundle* (such as Bx, for instance) which constitutes a given individual. Presumably, it is the later that replaces exemplification, not the former. And presumably it is the later that is defined by D2 in terms of the notion of *ontological part*. So even if the relation of *bundled-with* that holds between two universals is symmetric, the relation between a universal and a bundle (i.e., the relation of *ontological-part*) may not be, unless it is explicitly so stipulated.

2.4) Bill may pry apart the bundled-with relation and the ontological-part relation so that the former holds only among universals and the later only between universals and bundles. According to this scheme, the ontological-part relation that holds between universals and bundles replaces exemplification. But, then, the following question needs an answer: Is the relation of ontological-part symmetric or asymmetric?

If it is the later, then no distinction has been as yet drawn between the traditional asymmetric exemplification relation and the ontological-part relation. If it is the former, then no clear distinction between the relation of bundled-with and the relation of ontological-part is drawn.

2.5) Alternatively, Bill might conflate the bundled-with relation with the ontological-part relation and render them both symmetric. But then this must be explicitly incorporated into the definition D2.

2.6) So these issues need to be clarified.

3) Let R stand for whatever relation replaces exemplification in BTI. The following questions about the logic of R need to be determined:
(a) Is R symmetric?
(b) Is R reflexive?
(c) Is R transitive?

3.1) These questions are indispensable in order to determine whether BTI can indeed provide a coherent account of individuals in terms of bundles of universals.


Section B: BTI and the Part-Whole Relation.

1) I wish to examine how the common sense relation of part-whole is to be construed according to BTI. Examples of the part-whole relation that I have in mind are illustrated by the relation that a table-leg has to the whole table (namely, the former is a proper-part of the later) or a room has to the whole house, etc.

1.1) Clearly, in every case when the part-whole relation holds between two individuals x and y, then one or the other (or both) individuals have at least one property (most likely many more) which the other lacks. Moreover, when x is part of the whole y, then some properties of x are also properties of y (e.g., address: if the room has the property of being at the address 23456 W. Whole Road, Sunshine, AZ), then so does the whole house) and vice versa, whereas other properties of x are not transferable to y (e.g., the size of the room is smaller than he house) and vice versa. Can BTI distinguish between all of these cases?

1.2) Suppose x is a proper-part of y. Then x has the property of ‘being a proper-part of y’ which y lacks (it makes no sense to say that an individual is a proper-part of itself). Similarly, y has the property of ‘having x as a proper-part’, which naturally x lacks.


2) Now, by D1* both x and y are complete: i.e., for every property P, either P is exemplified by x or ~P is exemplified by x; the same holds for y. Since x is a proper-part of y, it follows that x has at least one property that y lacks. Let this property be Q. So x exemplifies Q; but, then, y exemplifies ~Q.

2.1) The following question arises: what does it mean to say that x is a proper-part of y? Let ‘Bx’ be the x-bundle and ‘By’ the y–bundle. Does BTI contain the conceptual resources to articulate the said relationship between x and y?

2.2) Can the part-whole relationship be explicated in terms of the relation of bundled-with? I don’t think so. The bundled-with relation is a relation between universals, whereas the part-whole relation is between individuals and, therefore, it must be construed as a relation between bundles.

2.3) Can the part-whole relation be explicated in terms of the relation of *ontological-part*? This proposal also does not seem to be adequate. The relation of ontological-part is a relation between a universal and a bundle, whereas since the part-whole relation is a relation between individuals, according to BTI, it must be captured by some relation between bundles.

2.4) One proposal is this: If x is a proper-part of y, then the bundle Bx is *included* in the bundle By.

D3. A bundle B is said to be included in a bundle B*=df every ontological part (i.e., universal) of B is an ontological part of B*, but not vice versa.

But now by D3, every property that is an ontological part of Bx is also an ontological part of By. One of these properties is Q. It follows, then, that Q is an ontological part of bundle By, since Q is an ontological part of bundle Bx and bundle Bx is included in bundle By. But now bundle By includes as ontological parts both property Q and property ~Q. Hence it includes contradictory properties.

2.5) Bill might reject the idea that when a bundle B1 includes as an ontological part a property P and bundle B1 is included in B2, then it follows that bundle B2 includes as an ontological part P. But then if Bill rejects this inference, then how does he propose to capture the idea that an individual is a proper-part of a whole as in the example of a room being a proper-part of the whole house?


Section C: Multiple-Exemplification.

1) I foresee some difficulties with Bill’s proposal to replace the asymmetric relation of ‘exemplification’ with the symmetric relation of ‘bundling-with’. One of the distinguishing characteristics of universals is that they can be multiply exemplifiable. Let U be a universal and x and y individuals that exemplify U (according to the standard jargon). Moreover, let V be a universal exemplified by x and not-V a universal exemplified by y. By Bill's proposal we then have the following, where ‘Bx’ represents the x-bundle and ‘By’ the y-bundle:

(i) U is bundled with Bx;
(ii) U is bundled with By;
(iii) Bx is bundled with U; (symmetry)
(iv) Bx is bundled with By; (transitivity of the bundling relation)
(v) V is bundled with Bx;
(vi) not-V is bundled with By;
(vii) V is bundled with By (from (iv), (v), and transitivity);
(viii) By is bundled with V (symmetry);
(ix) V is bundled with not-V. (contradiction)

One way to block this argument is to maintain that bundling-with is not a transitive relation ((iv) above). But this move seems to me to be very counterintuitive, for I cannot think of any other case where bundling-with is not transitive.

" Definition D2 is intended to codify this replacement. Unfortunately, there is a systematic confusion here. Suppose U is exemplified by x." But there is no exemplification on the bundle theory. So I've lost you already.

Bill,

My point in that segment is this. Since there is no exemplification in BTI, some other suitable relation takes its place. This relation is either the relation between universals; e.g., bundled-with, or the relation between a universal and a bundle; i.e., ontological part, or some other relation not yet defined.

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