Parts of situoids

Now we can introduce formally what we mean by one situoid being part of another.

To capture formally what is meant by a situoid being a part of the world, we introduce the binary relation $\leq \subseteq Situoid \times Situoid$. $s_1 \leq s_2$ if and only if all infons that are facts of $s_1$ are facts of $s_2$. By using the operator $S:Situoid \mapsto Set$ satisfying $S(s)=\{\phi \vert Infon(\phi) \land s \models \phi\}$ we can define this particular part-of relation as $s_1 \leq s_2 \iff S(s_1)\subseteq S(s_2)$.

This is the most basic part-of-relation for situoids. We will mention two other cases of a part-of relation. Let $s$ be a situoid, and $c=chron(s)$ with $prt(s,c)$ its framing chronoid. The projection $s_1$ of $s$ on $c_1 \leq c$, $s_1 = prt_f(s,c_1)$ is called a temporal part of $s$. $s_1 \leq _t s$ if and only if $s_1 = prt_f(s,c_1)$ and $c_1 \leq c$.

The spatial part-of-relation is similar. Let $t$ be the topoid framing the situoid $s$ with $top(s,t)$. $s_1 \leq _s s$ if and only if $s_1 = prs_f(s,t_1)$ and $t_1 \leq t$.

A question that arises is, if $s_1 \leq _t s \rightarrow s_1 \leq s$ and $s_1 \leq _s s \rightarrow s_1 \leq s$, this is to say, if one situoid being a temporal or spatial part of another situoid implies it being a part-of in the sense of the infons being a fact in it.

While this may seem plausible at first, there are some severe problems involved. Consider the following example: A railroad track over some fixed period of time viewed as a situoid $s$. Imagine, there is no train present in the fixed part of the world. Then $s \models \langle\langle Present,Train;0\rangle\rangle$. But at times, there are trains traveling through this part of the world, and if we extend our spatial region, there may also be trains present. Let $s \leq _t s'$, then $s' \models \langle\langle Present, Train;1\rangle\rangle$ may be true.

But not only negative infons will be reverted in extensions of situoids. Consider the above example again. Then $s \models \langle\langle$   Number-of$,Train,0;1\rangle\rangle$ is true, but the extension $s'$ does not support this information, but rather the inverse.

Generally we will deny a relationship between the different part-of relations.

Other variants of part-of relations are discussed in more detail in section 5.2.9.