Axioms for situations

Of course, situations are much simpler than situoids. There can be no progress, no change in situations. Only instantaneous states of affairs exist in a situation. Let us state, without much discussion, the first axioms for situations. Most of them should be clear from our previous discussion of situoids.

Let us remember our definition for situations.

Definition 5.24 (Situation (2))   A situation is a comprehensible entity in reality, that exists at a point in time. It is the projection of a situoid on a point of time. The class of all situations is denoted by $Situation$.

Axiom 5.27  

$$\exists x Situation(x)$$

Situations exist at time boundaries. There is a function $tbs:Situation \mapsto TB$, where $TB$ is the set of all time boundaries.

Axiom 5.28  

$$\begin{split}\forall x,b \left( Situation(x) \land tbs(x)=b \... ...y (Situoid(y) \land chrs(y)=c \land tb(c,b))\right) \end{split}$$

We can now define the relation, first mentioned in section 5.2.9, that a situation $s$ is part of a situoid $w$.

Definition 5.25 ( $\leq _{sit}$, situational part-of)  

$$\begin{split}s \leq _{sit} w \iff Situation(s) \land Situoid(... ...\exists t (topoid(t) \land top(s,t) \land top(w,t)) \end{split}$$

We define the supports relation from section 5.2.2 for situations as follows.

Definition 5.26 (Supports relation for situations)   Let $SInfon$ be the class of all infons except timed infons. The relation $\models \subseteq Situation \times SInfon$ is called obtains relation. We use infix notation to denote this relation: $s \models \phi$, which is read as ``The infon $\phi$ obtains in the situation $s$.

Timed infons do not make sense in situations, as situations exist at precisely one point in time, a single time boundary, and the states of affairs that correspond to the obtaining infons have to exist at the same point in time. This does not mean, that the entities participating in the states of affairs have to exist at time-boundaries, too. As an example, consider the state of affairs ``The process $p$'s being earlier than the process $q$''.

As we did for situoids, we can now define another, basic part of relation.

Axiom 5.29  

$$\begin{split}\forall s_1\forall s_2 (Situation(s_1)\land Situ... ...hi (s_1 \models \phi \rightarrow s_2 \models \phi)) \end{split}$$

And also, two situations are identical, if and only if the same infons obtain in them, that is, if they are part of each other. Again, this is insufficient. They also have to exist at the same time boundary, and they have to be a situational part of the same situoid.

Axiom 5.30  

$$\begin{split}\forall s_1\forall s_2 (Situation(s_1) \land Sit... ...} s \land s_2 \leq _{sit} \land tbs(s_1)=tbs(s_2))) \end{split}$$

We extend the function $S(s)$, taking situoids into sets, to accept situations as its arguments.

Axiom 5.31   There is a function $S:Situoid \cup Situation \mapsto Set$, such that

$$x=S(s) \iff \forall \phi (s \models \phi \leftrightarrow \phi \in x)$$

Again, the same as for situoids, we do not want to permit empty situations.

Axiom 5.32  

$$\forall s (Situation(s) \rightarrow \exists \phi( Infon(\phi) \land s \models \phi))$$

Now we said that situations are comprehensible entities. Some of the axioms of section 5.2.8 have to be adapted and applied to situations. The first important condition is, that every situations has to be the instance of at least one universal.

Axiom 5.33  

$$\forall s (Situation(s) \rightarrow \exists u (Universal(u) \land s::u))$$

Again, given a certain situation, we can add more and more information to it, infinitely. This is different to situoids, as they can also be more and more extended in time, while situations cannot.

Axiom 5.34  

$$\forall s (Situation(s) \rightarrow \exists t (Situation(t) \land s \leq t\land s \not= t))$$

The last axiom needed for situations to be comprehensible entities is the following:

Axiom 5.35   The set $S(s)$ of all infons obtaining in a situation $s$ is decidable.