Concerning properties in decomposition

Before we can state a consistency axiom for artefact models, we need an answer to the question of whether properties are to be considered in this context. That is, do the properties of a decomposed object relate to the properties of the sub–

objects? The following discussion aims at reaching a clarification in answering this question. We conclude that no such relation can be claimed. Our discus-sion goes as follows (the section can easily be skipped if interest is not on the ontological or epistemological aspects):

We consider the problem of how to understand the relation between proper-ties possessed by an object and the properproper-ties which seem to be derived to

sub–objects in a decomposition. We name this relation thederivation relation assuming that derivation concerns properties. For convenience we define the notion of attribution as follows: An object is said to be attributed with an attribute a if the object is ascribed a propertyp ={a, vs}; wherevs is some property value set.

Our quest here is problematic though our solution is uncontroversial. The quest is problematic in the sense that the derivation relation in question seems not to be in focus either in literature on the metaphysics of properties, neither in literature on part–whole theories. Though, Simon treats a similar question in context of essential parts and defines the notion oflocal predication(see page 135 in [155]). Also, Simons states that predicates over objects as sums (the objects we decompose) are to be cumulative (see page 111 in [155]). If a predicate applies to a part, it must apply to the whole as well. But according to Simons, Quine has argued the absurdity in this perspective: It simply does not hold for mass–predicates. However, if we claim a relation to hold — as a conjecture — we should at least try out some different approaches as we intend to do here.

To start with, we shall dogmatically assume that we need to saysomethingabout that derivation relation. That is, there must indeed be a wellformed condition for properties of objects and properties of sub–objects.

A first approach could be that we allow decompositions in which sub–objects not necessarily derive all attributes from the properties of the object being decomposed. The motivation here is that we often may speak of an object having a property even though we know that the object consists of parts which do not possess the property. As an example, consider a table. We may ascribe the property of having the colour blue to that table even though the legs have the colour grey. Thus, motivation comes from our (perhaps vague) way of using natural language. It is one of the ways we perform abstractions in every day life: “This chair is blue” may be true in our conception even though only main parts are blue. Let us consider it formally and in our context. The approach means that the set of attributes to be derived is a subset of the attributes which are attributed the object being decomposed.

The approach emphasizes the situation where this subset is a proper subset.

The situation, however, opens for the possibility of having the empty set of as a special case. If the derived set of attributes is empty, there is no relation to be claimed. It turns out in our formalisation that this implies that there are not sub–objects then. However, this was not our intention, and if we try to accommodate by choosing another formalisation, we discover that the problem is much more far reaching. If a certain attribute of a decomposed object is not designated as to be derived, we allow a property with this attribute to be ascribed the sub–objects after the decomposition. This latter property could

6.2 Artefact models 157

have a different value set, but more problematic; the attribute could be different, so that the property belongs to a different ontological domain of properties. But this is absurd. A wall which is ascribed the property of being made of concrete can then be decomposed into two parts which all are later ascribed the property of being made of wood. That is, we cannot say anything about the derivation relation. In defence, we shall allow for partial decomposition. It may be that a property, which is not derived in one lattice path, is derived in another. Still, we have not said anything about the derivation relation; only that some sort of consistency should exist..

A second approach is to try to repair the leaks of the first one. Leaving out an attribute of a decomposed object may lead to absurdities. We might require that at least one sub–object derives a property with that attribute. That is, in order for the decomposition to be wellformed with respect to derived properties, the derivation relation should maintain that all properties are derived to sub–

objects in some way. We can divide this approach into two cases: (i) We could require that at least one sub–object derives all the properties of the decomposed object, or (ii) we could require that all properties are at least derived in some way to some sub–object; i.e. distributed. The approach, in either case, introduces the notion of essential parts. In (i) the object which derives all properties, is considered essential and thus more important than the other sub–objects. In (ii) such importance depends on what perspective we take when “considering” the decomposed object. In one perspective, material may be essential; in another, mass may be essential, etc. The difference between (i) and (ii) is thus a matter of context, and we shall treat them together. The second approach (although quite appealing) has a built in problem which is rooted in distinguishing essential from non–essential parts. Take for example a television. We say that this television is black even though some parts of it do not possess this property. Let us now decompose the television object into a black box, electronics inside, glass front and a little red lamp flashing. The essential part, concerning the property mentioned, may be the black box; perhaps because it spatially dominates the outer surfaces of the television. However, for that we cannot be sure. It could as well be the glass front which is dominating. But then we might also say that the red light is the essential part, and so on. If we wish to select the second approach, we should at least have a formal way of distinguishing essential and non–essential parts. This issue is in fact a known problem in mereology and we are here far from a real clarification [155]. Furthermore, since we allowed for partial decomposition it may be that no sub–objects can be designated as essential at the time of decomposition.

A third approach is to require that what is said about an object should also apply for all sub–objects into which it is decomposed. We thus avoid the tricky ways of using language, which were mentioned for the first approach. In decomposing an object, we should make sure to designate all attributes of the decomposed

object as to apply to all sub–objects as well; and similar for value sets. For colour properties and material properties, the approach seems appealing. If a decomposed object is red, its parts should be too, and if a decomposed object is made of wood so should its parts. The trouble appears when considering properties like dimensions. If a beam has a length of2m any proper part of it certainly does not.

A fourth approach is to repair the lacks of the third approach. We maintain the requirement that all attributes of a decomposed object are attributed sub–

objects as well. However, we loosen the restriction on value sets. The derivation relation thus only says something about the kind of properties, not what specific properties. However, this is hardly the way a design practitioner works. If we are not allowed to state — using a top–down approach — that a house is to be made of wood because the chimney is not, we loose the fuzzy way designers abstract. The abstraction process may here be impossible to define explicitly and universally.

A fifth variant could be considered as solution to the problem of the fourth approach. It is as appealing as it is strange, which is partly why it has not been chosen in this paper. Still, if the notion of abstraction — as considered under the fourth approach — is to be defined explicitly, this fifth variant we believe is the best candidate for a foundation. Consider an object with a propertyp.

The object is now decomposed into a number of sub–objects. The rule is now that if a sub–object derives a property, this property does not apply to the decomposed object anymore, if not all sub–objects derive the property. That is, it is removed.

We conclude that there seems not to be any derivation principle which satisfies both the requirement of stating a consistent, universal rule for derivation of pro-perties, and at the same time do not restrict the design process. The seemingly strange solution of the fifth variant needs more clarifications before it can be called a real candidate. Therefore, we have chosen not to claim there to be a relation between properties of an object and properties of the objects into which it is decomposed.

6.2.3 Consistency

In order for an artefact model to be consistent, it must satisfy the following:

(i) All attributes for properties of an object must uniquely identify a corre-sponding value set, and (ii) all attributes for relations must uniquely identify a corresponding value set.