Being part of reality

It may be puzzling to raise the question of what is meant by something being a part of reality, since it appears so obvious. However, in order to specify what situoids are and what they are not, we pursue this path.

``Being a part of reality'' suggests, that reality can be comprehended as something whole. According to Wittgenstein in tractatus, the world, reality or all there is, consists of all the facts and their being all the facts. Situoids, however, are parts of reality, that can be comprehended as wholes. Therefore, their internal structure should correspond to the one of reality, namely being a set of facts. We will, as Barwise did in barsit, use (hyper-)sets of infons for modelling situoids and situations.

We can ask the question whether reality, all there is, can be considered a situoid. Reality would have to be a maximal situoid, with all states of affairs settled. However, due to our axiom 5.21, this cannot be the case. We could always objectify one situoid, and embed it in another, even larger situoid, a situoid with more information. Therefore, there is no largest situoid and reality is not a situoid. Reality cannot be comprehended as a whole.

We assume that there exists a world and reality independent of our mind and beliefs. This reality can be decomposed in situoids, that can be comprehended as totalities. We will sometimes refer to this kind of situoids as ``real'' or ``actual'' situoids. Then there is a second class of situoids, that we will call ``intentional''. They exist in a different mode of reality. They exist as entities in mind. As an example of a situoid of this kind, consider a situoid, where a unicorn stands in a forest.

The question is, whether intentional situoids are a part of reality, or in what sense they exist. To clarify this, we will have to state the following definition. Remember the beginning of this chapter, where we defined a partial part-of order for situoids, regarding the infons that obtain in those situoids.

Our intention is, that all of reality can be structured in two classes, situoids and worlds. We will simply state the existence of a class of all parts of reality, and define a partial order on this class. The order relation is indeed the infonic part-of we have been using throughout this chapter, although this is not explicitly stated here. Most things should become clear after axiom 5.23.

Definition 5.21 (World)   Let $S$ be the set of all situoids. Let $T \supseteq S$ be the set of all parts of reality. The relation $\leq$ defines a partial order on $T$. Every maximal element of the $\leq$-order on $T$ is called a world.

The idea behind this definition is perhaps not obvious. Of concern to us are situoids, either real or imagined. These situoids can be informationally extended in various ways. Usually, we can add either an infon or its complement as information. This gives rise to various different ways the situoid behaves, and therefore to various different ways of how reality is. What we obtain is a partial order. Now, if this partial order has some kind of maximal element, we will call it a world. However, the idea behind a world was that all issues are settled. So we need another axiom to ensure this.

We can then state the following axiom:

Axiom 5.23   Let $S$ be the set of all situoids and $T \supseteq S$ the set of all parts of reality. $partof$ defines a partial order on $T$. Every totally ordered subset $A$ of $T$ has a unique upper bound $m$ in $T$, such that $\forall x (x \in A \land x \neq m \rightarrow x \in S)$.

What worlds are and how we can model them will become clear with the following theorems.

Theorem 5.1   There is a world.

Proof. Because of axiom 5.23, every totally ordered subset of the set of all parts of reality $T$ has an upper bound. Therefore, according to Zorn's lemma, the partial order $(T,\leq )$ has a maximal element. <1.5em - height0.75em width0.5em depth0.25em

We do not only know, that worlds exist, we can say what they look like, because we defined the $\leq$-Relation as referring to sets of states of affairs. For this, we need to extent the relation $\models$ to all elements of the set $T$ of all parts of reality.

Definition 5.22 (Supports relation for worlds)   Let $w$ be a world, and $i$ be a state of affairs. Then $w \models i \iff \exists s (Situoid(s)\land s \leq w \land s \models i$.

Then every chain of situoids in the partial order of parts of reality defines a world, namely an entity, in which every state of affairs that obtains in any situoid of this chain obtains. With this, we are again back to Wittgenstein, where a world consists of all the facts, and their being all the facts tractatus.

We also know, that worlds are not situoids, they are always more, according to axiom 5.21.

This view on worlds and their correspondence to situoids gives us the liberty of choosing, whether we want to accept the existence of multiple, possible worlds or only a single, real world. This choice has to be formulated in another axiom. We therefore have to choose between the following two axioms:

Axiom 5.24   There is only one world.

Axiom 5.25   There is more than one world.

There is another view on worlds. Let $\Sigma$ be a finite set of infons. Then a world $w$ is a binary partition of $\Sigma$, $(\Gamma,\Delta)$. The idea is, that $\Gamma$ is the set of infons that holds of the world $w$, and $\Delta$ is the set of infons that does not hold: $\forall \phi (\phi \in \Gamma \rightarrow w \models \phi)$ and $\forall \phi (\phi \in Delta \rightarrow w \not\models \phi)$. This is a useful limitation of our view, because in many applications only a finite set of infons is of concern. Let us see if it converges to the previous view on worlds.

Definition 5.23 (Consistence of sets of infons)   A set of infons $\Sigma$ is consistent, if the following is true: If some infon $\phi \in \Sigma$ with $\phi=\langle\langle R,a;p \rangle\rangle$, then $\bar{\phi} \not\in \Sigma$ with $\bar{\phi} = \langle\langle R,a;1-p \rangle\rangle$.

We need one more axiom in order to obtain our desired result.

Axiom 5.26   Every consistent, finite set of infons $\Sigma$ defines a situoid $s$, such that $\forall \phi (\phi \in \Sigma \rightarrow s \models \phi)$.

Please note that the situoid may support more infons, for example due to certain constraints.

Theorem 5.2   Let $\Sigma$ be the set of all infons. A partition $(\Gamma,\Delta)$ of $\Sigma$, $\Gamma$ and $\Delta$ consistent, defines a world $w$ in the way that $\forall \phi (\phi \in \Gamma \rightarrow w \models \phi)$ and $\forall \phi (\phi \in \Delta \rightarrow \neg w \models \phi)$.

Proof. First we prove that $w$ is maximal coherent. No $w'$ with $w \leq w'$ and $w \not= w'$ is coherent. Let us assume, there was such a $w'$. Let us assume that $w' \models \phi$ and $w \not\models \phi$. Since $(\Gamma,\Delta)$ is a partition of the set of all infons, $\phi \in \Delta$. The complement of $\phi$, $\bar{\phi}$, is in $\Gamma$, $\bar{\phi}\in\Gamma$, because $\Delta$ is consistent. But $w' \models \phi$ and, because $w \leq w'$, also $w' \models \bar{\phi}$, so $w'$ is incoherent.

Every finite subset $\Gamma'$ of $\Gamma$ defines a situoid $s$ such that $\forall \phi (\phi \in \Gamma' \rightarrow s \models \phi)$. There is a situoid $s_0$ defined by a finite set $\Gamma' \subset \Gamma$ such that $S(s_0)\subseteq \Gamma$. If this was not the case, then there would be a $\phi \in \Delta$ and $s_0 \models \phi$ for all $s_0$ defined by $\Gamma'$. But $\bar{\phi}\in\Gamma$, and $\Gamma' \cup \{\bar{\phi}\}$ defines a situoid, so there is a situoid $s_1$ defined by $\Gamma'$ and $s_1 \models \bar{\phi}$, which is a contradiction.

There is a situoid $s_0$ with $s_0 \leq w$. $w$ is the maximal element in the part-of order, $s_0$ is part of reality, therefore $w$ is a world. <1.5em - height0.75em width0.5em depth0.25em