Whole situoids

Before we can proceed with our discussion of the meaning of ``can be comprehended as a whole'', the basic feature of situoids, we will have to draw our attention to the nature of wholes in general. Investigation of part-whole relationships dates back as far as Aristotle. In our investigation of comprehending situoids as a whole we will refer to more recent work, namely rescher. Their investigation of wholes and their parts is gestalt theoretical in origin.

Since gestalt theory presupposes a subject capable of comprehension, it is most suited for our theory. We will regard gestalten with situoids as their instances. Gestalt is a German word and refers to the concept where an entities properties cannot be discovered from the total properties of its parts alone The Internet Community. There are similar but different views on the part-whole relationship, the most influential probably being those of phenomenologicians like Husserl soko1,krecz1, who uses the term ``figural moment'' instead of ``gestalt''. More differences between both approaches can be found in gurwitsch.

Alternative theories of the part-of relation, of parts and wholes can be used in an integrated top-level ontology parallel to our theory of whole situoids and their parts, as there may be other categories in the top-level ontology, that do not behave like situoids, as comprehensible wholes.

The philosophy of Brentano, Meinong and Husserl eventually led to the development of gestalt theory reiser1. The founder of the gestalt theoretical movement is Max Wertheimer, although the term ``gestalt theory'' has first been introduced by Christian von Ehrenfels, a disciple of Brentano. Another root of gestalt theory is Ernst Mach becher, whose work ``Beitraege zur Analyse der Empfindung'' (Contributions to the Analysis of Sensations, 1886) influenced gestalt theorists like Ehrenfels, but also phenomenologists like Husserl.

In rescher, three conditions are stated as prerequisites for a whole entity:

1. The whole must possess some attribute in virtue of its status as a whole, an attribute peculiar to it and characteristic of it as a whole.
2. The parts of the whole must stand in some special and characteristic relation of dependence with one another. They must satisfy some special condition in virtue of their status as parts of a whole.
3. The whole must possess some kind of structure in virtue of which certain specifically structural characteristics pertain to it.

All those three conditions have to be regarded concerning a specific part-of relationship, and therefore a specific decomposition.

Definition 5.15 (Decomposition)   Let $w$ be a specific object and $Pt$ a specific part-of relation, then the class $D$ of $Pt$-parts of $w$ is a decomposition of $w$ if every $Pt$-part of $w$ has some $Pt$-part in common with at least one element of $D$.

There are attributes of a whole, that can be shared or unshared by its parts. An attribute, a quality, the whole possesses and none of its parts do is called unshared, and shared if all parts of the whole possess the attribute.

Definition 5.16 (unshared attributes, shared attributes)   An attribute or quality $Q$ is called a $D$-unshared attribute of a whole $w$ relative to a decomposition $D$ of $w$ into $Pt$-parts, if $Q$ is an attribute of $w$ which is inhered by no $Pt$-part of $w$ belonging to the decomposition $D$. $Q$ is called a $D$-shared attribute of $w$ relative to $D$, if $Q$ is inhered by all $Pt$-parts of $w$ belonging to $D$.

It is clear, that if some attribute is not $D$-unshared, it does not necessarily have to be $D$-shared, and vice versa.

Underivable attributes of a whole are attributes, that are not a logical consequence of some set of attributes of a set of parts of the whole. Underivable attributes, again, have to be regarded relative to a specific set of parts and attributes. An example of some underivable attribute is the weight of a pile of stones. Of course every part of this pile of stones has a weight, but summoning up the weight of the pile by using the weight of the parts requires the additional natural law that weight is additive. An attribute or quality is derivable if it is a logical consequence of the attributes of some set of parts. Rescher and Oppenheim gave the following definition in rescher

Definition 5.17 (underivable attributes, $G$-characterization)   An attribute $Q$ of a whole $w$ is a $D$-$G$-underivable attribute of $w$ relative to a decomposition $D$ of $w$ and to a set $G$ of attributes if '$Q(w)$' is not a logical consequence of the characterization of the elements of $D$ with respect to $G$. By 'characterization of the elements of $D$ with respect to $G$', or briefly '$G$-characterization', is meant a sentence which, for any $n$-adic relation $g$ of $G$, and any $n$ elements $d_1,...,d_n$ of $D$ states whether or not the relation holds between these $n$ elements.

Definition 5.18 (derivable attribute)   An attribute $Q$ of a whole $w$ is a $D$-$G$-derivable attribute of $w$ relative to a decomposition $D$ of $w$ and to a set $G$ of attributes if '$Q(w)$' is a logical consequence of the $G$-characterization of the $D$-parts of $w$.

As mentioned, the weight of a pile of stones is not derivable by purely logical means from the weight of the stones alone. However, we could add a theory of some kind, and derive this attribute with the help of this theory. We will call this attribute then $D$-$G$-$T$-derivable, if $T$ is the theory concerned, or $D$-$G$-$T$-underivable, if it is impossible to deduce the attribute of the whole with the means of $T$. We will call $D$-$G$-$T$-underivable attributes simply ``underivable'' attributes.

The existence of an underivable attribute corresponds to the First Ehrenfelscriterion, that has been formulated in Christian von Ehrenfels' first study of Gestalt-theory, ``Ueber Gestaltqualitaten'': ``The whole is more than the sum of its parts.'' Sometimes those attributes, that only come into existence when the parts are assembled, are called emergent attributes.

Considering the second condition for wholes, the dependence of certain characteristics of one part upon those of other parts, we will have to consider configurations. ``An ordered set of objects, $p_1, p_2,...,p_n$, which stand in the relation $R$ to each other, i.e. for which $R(P_1, p_2,...,p_n)$ holds, will be said to form a configuration of kind $R$.''rescher As we can see, Rescher and Oppenheim's configurations are certain kinds of states of affairs, in our terminology, and the kind of the configuration, the kind of the state of affairs is defined by the relation universal.

For the easier part of quantitative attributes, an attribute $f$ of a part $p_1$ is called $\phi$-dependant upon some class $G$ of quantitative attributes of the objects $p_i$, if the $f$-value of $p_1$ in every configuration of kind $R$ is related by $\phi$ to the values of the attributes in $G$ of $p_i$. $\phi$ may be strong or weak, defining the value of $f$ completely, or just statistically. It is possible for the configuration to consist only of one object, or the class $G$ consisting only of the attribute $f$.

For non-quantitative attributes, similar observations can be made. The perceived impression may depend on other parts surrounding or the whole, as can be seen in figure 5.5.

Figure 5.5: Zoellner Illusion, parallel lines

For the requirement that the parts of the whole must stand in some special, characteristic relation of dependence with one another, wholes may form dependence systems, that are defined as follows in rescher:

Definition 5.19 (Dependence system)   A configuration is a $\phi$-dependence-system relative to a set $G$ of attributes if each part of the configuration has some $G$-attribute which is $\phi$-dependent upon the $G$-attributes of the remaining parts.

The third criterion refers to some kind of structure the whole must possess in virtue of its status as a whole. This involves three things:

1. The parts of the whole.
2. A domain of positions in some kind of topological structure.
3. An assignment assigning the parts to positions of the domain.

Often, we are not interested in the specific parts of some whole, but only the types of the parts and the position they occupy in the whole. This leads us to the concept of a complex:

Definition 5.20 (Complex)   A complex is characterized by the following three features:

1. A set $G$ of topologically structured attributes.
2. A topologically structured space $X$, constituting the domain of positions.
3. An assignment $f$ of exactly one $G$-attribute to each $X$-position.

Different complexes may have the same structure, they are isomorphic. For two complexes to be isomorphic, their topological domains must have the same structure, the sets of attributes of the two complexes must have the same structure and the attributes of both complexes have to be assigned to corresponding positions of the domains.

Isomorphic complexes are related by transformations. Different but isomorphic complexes possess certain structural similarities. Consider a piece of music and another, different piece of music, that is a transposition of the first. They are certainly different, but isomorphic, and they share the same melody. Therefore, there are attributes that are shared by all complexes of a group of isomorphic complexes. They are called complexial features, attributes that are invariant under isomorphic transformations. As Rescher and Oppenheim point out, those attributes may even be underivable. The possession of attributes that are invariant under those transformations is stated as the Second Ehrenfels-criterion.

Let us now apply this formalism to situoids. First we examine what kinds of decompositions of situoids exist. We will examine some possible decompositions shortly, without much discussion and motivation, before we concentrate on one specific decomposition which we will examine further.

Since situoids exist in time and space, our first possible decomposition is a spatio-temporal one. We will denote this part-of relation with $\leq _{st}$. For this we project the situoid on parts of its framing chronoid or topoid or both. The attributes we may consider are the duration or spatial size the resulting entities have. Then we do have a derivable attribute, namely the new duration and spatial extent the whole situoid has. The spatio-temporal parts of the situoid, possibly, but not necessarily, situoids on its own, may be cause and result of each other, however, this does not always have to be the case, especially if the spatio-temporal parts are not situoids.

We could also decompose one situoid in multiple sub-situoids, all of which are part of one situoid. We call this a situoidal decomposition and denote the appropriate part-of relation $\leq _{s}$. The parts form a web of situoids, influencing each other and being cause or result of each other. However, the structure of the whole situoid is generally derivable by the structure of its subsituoids.

Because situoids are described by infons, we can consider an infonic decomposition of situoids, formed by the relation $\leq _{inf}$ or simply $\leq$. The entities that are part of a situoid in this sense behave much like situoids and sometimes are situoids, but they contain less information. They may provide a more abstract view on a situoid, or a more limited. Consider a situoid $w$ of a specific plaza. Then the infonic parts of this situoid may contain a single human, a fountain, a tree, one square meter of concrete plate, and so on. When all those parts are assembled, the situoid $w$ is an instance of the universal $Plaza$, but none of its parts are. The attribute of ``being a plaza'' emerged, when the parts where put together. The second condition however will fail. There is no interdependence between infons in a situoid. Each infon describes some basic information of a situoid by stating a relation, a configuration of objects. They may be informational equivalent, or one is entailing another (logically or with respect to some theory), but many infons obtaining in situoids are independent of each other, and are describing different aspects of the situoid.

We have not discussed all the possible decompositions in detail, as we just wanted to get the idea. There is another decomposition that we will discuss in more detail, namely decomposing the situoid into situations by using the relation $\leq _{sit}$, entities that exist at time points. This can be achieved by projecting the situoid on the boundaries of its framing chronoid. As a result, we get a sequence of situations. We will now examine this decomposition in detail and try to find out, whether situoids fulfil all three conditions of wholes regarding this decomposition.

First we have to examine whether the relation $\leq _{sit}$ defines a decomposition of situoids. A situation is the projection of a situoid on one of the time boundaries of its framing chronoid. Therefore, $sit \leq _{sit} w$, iff $sit$ is a situation, $w$ a situoid, $chr$ the chronoid framing $w$ and $sit$ is the projection of $w$ on one of the time boundaries of $chr$. Let $D$ be the class of all situations of the situoid $w$. We can see, that $\leq _{sit}$ is not reflexive, so we will consider it an irreflexive part-of relation. Then, $D$ is a decomposition of $w$ regarding the part-of relation $\leq _{sit}$ if every $\leq _{sit}$-part of $w$ overlaps with some $\leq _{sit}$-part of $D$. However, our understanding of the relation $\leq _{sit}$ defies the usual understanding of a part-of relation, as it is a relation between two distinct classes of entities, situations and situoids, and cannot be applied to the parts. For the sake of the argument, we will make the relation $\leq _{sit}$ reflexive: $\forall x (x \leq _{sit} x)$.

Now we can show, that $D$, the class of all situations defined by a situoid $w$, is a decomposition of $w$. Let $overlaps(x,y)=\exists z(z \leq _{sit} x \land z \leq _{sit} y)$. Then it can be easily shown, that $\forall x (x \in D \rightarrow \exists y (y \leq _{sit} w \land overlaps(x,y)))$, because $D$ contains all the situational parts of $w$.

The result is pretty trivial, but will become more useful when we start extending the relation $\leq _{sit}$. The class $D$ of all situations of the situoid $w$ is still a decomposition of the relation $\leq _{sit} \cup \leq _{s}$. We can also extend this relation further, by adding an infonic part-of relation on situations, and $D$ will still be a decomposition of $w$.

Obviously, the sum of all situations in chronological order is a sequence of entities, that can be described by states of affairs, just like in situation theory, or situation calculus. Our second condition for wholes is satisfied, because situations may be causes of their following situations, or results of previous situations. Therefore, the infons obtaining in one situation may depend on the infons obtaining in previous situations and with this, the perception of one situation may depend on their neighboring situations and the whole.

Let us consider an example to illustrate this idea. Imagine a 100 meter race as a situoid. Two people are running, $A$ and $B$. So we will have a situoid $s$ with a topoid $t$ and $top(s,t)$. The runners start at the location $x_{start}$ and finish at $x_{finish}$. The race lasts 20 seconds, so there is a chronoid $c_{20}$ with $prt(s,c_{20})$. $A$ needs 10 seconds to reach the finish line, therefore running during the chronoid $c_{10}$, $B$ needs 20 seconds. We will express this with the timed infons $\phi_A = \langle\langle Runs,A,x_{start},x_{finish};1;c_{10}\rangle\rangle$ and $\phi_B = \langle\langle Runs,B,x_{start},x_{finish};1;c_{20}\rangle\rangle$ and $s \models \phi_A$ and $s \models \phi_B$. Projecting $s$ on the right boundary of $c_10$ we will obtain a situation $s_A$ where $A$ is finishing the entire race, $s$: $s_A \models \langle\langle Finishes,A,s;1\rangle\rangle$. The second situation will be the projection of $s$ on the right boundary of $c_{20}$, $s_B$. A similar infon obtains here, $s_B \models \langle\langle Finishes,B,s;1\rangle\rangle$, but due to the position of $s_A$ in $s$ we know, that he did not win the race: $s_B \models \langle\langle Wins,B,s;0\rangle\rangle$. However, a detailed investigation of causality in the framework of GOL, and therefore also in the framework of situoids and situations is still an open research topic.

The situoid is structured into situations as the chronoid is in time boundaries, because we are only projecting the situoid on its framing chronoids time boundaries. For the third condition - that the situoid must possess some kind of structure, in virtue of which certain structural characteristics pertain to it - to be satisfied we will need to identify the parts, a domain of positions the parts may occupy and an assignment function assigning each part to a position. The parts are the situations, the positions the time boundaries and the assignment function is our projection of the situoid on the time boundaries. As Rescher and Oppenheim point out in rescher, we usually do not care about the individual parts, but rather about the types of the parts. Doing so, they call the entity of concern a ``complex'', while we will talk about ``universal'' in a more general approach.

We assumed that every situoid is an instance of some universal. Universals define the structure whose possession is required of wholes in rescher. Universals may require the existence of certain situations in a situoid. These are the parts that are assigned to spatio-temporal locations inside the situoid by the universal. Then, a ``transposition'' in the sense of rescher is nothing more than another instance of the same universal. And usually, there are features of a situoid which are solely due to its being an instance of a specific universal.

For example, the universal ``100 meter race'' will require the existence of certain situations, at least a starting and an ending situation, which are assigned to the left respectively the right boundary of the chronoid framing the situoid $w$. The other parts (situations) required by an universal which $w$ is an instance of, if any, behave as well and are accordingly assigned to certain spatiotemporal locations. It is this structure - the assignment of situations to proper space-time locations - which makes some situoid a 100 meter race and which makes it an instance of the universal ``100 meter race''.

Now the question is, whether a situoid is more than just a mere sequence of situations. This is the case, because situoids are continuous. Just like chronoids cannot be understood as sets of time boundaries, but are inherently more, situoids are inherently more than collections of situations. There are even processes, that can never be fully understood in the domain of situations, but only in a continuous structure like situoids.

Figure 5.6: Flying bird example

Let us consider an example, illustrated in figure 5.6. There are two people, $A$ and $B$ 100 meter apart. Both are walking towards each other in a straight line with a velocity of one meter per second. A bird flies between $A$ and $B$, starting from $A$, with 100 meters per second. When the bird reaches $A$ or $B$, it turns around in no time, and flies the other way. There are three concurrent processes, one depending on two others. These processes can be embedded in a situoid, but cannot be understood in the domain of situations. In the domain of situations this example would be modelled by a sequence of situations, at least one for every occurring change. So whenever the bird reaches $A$ or $B$, a situation is needed to state that the bird is changing direction. But when we would ask which direction the bird would be facing when $A$ and $B$ meet, we cannot get an answer if these processes are modelled as sequences of situations. We will not get an answer either in the domain of situoids, but there we cannot ask the question. There are just three processes, one depending on two others, and the changes of heading are of no concern.

There are more Zeno-like paradoxa which can be used to argue in favor of situoids being more than a mere sequence of situations. Imagine Zenos arrow, which is not moving at any point in time, but still changing position. In the domain of situoids, this arrow is indeed moving in a straight line with a velocity $v$, while in the domain of situations this arrow may be changing positions while its velocity is zero. However, more research has to be done with respect to examples such as the above flying-bird example, as the explanation of why it is impossible to determine the direction in which the bird is flying at the end is somewhat awkward.

But still we believe the first condition for wholes satisfied. The whole situoid is more than the sum of its $\leq _{sit}$-parts.

We have found a decomposition and discussed a set of attributes, that fulfills all our conditions for wholes, and can therefore regard situoids as ``whole'' at least with regard to the $\leq _{sit}$ relation.