Situoid theory and modality

Jon Barwise showed in barwise1 how to introduce modality into situation theory, while keeping a strictly realistic point of view. We will show how this is done and how to incorporate his approach in our framework.

Definition 6.1 (Classification, Boolean Classification)   A classification $A=(S,\Sigma,\models)$ is a set of situoids or situations $S$, a set of infons $\Sigma$ and the relation $\models \subseteq S \times \Sigma$.

A Boolean classification $A=(S,\Sigma,\models,\land,\neg)$ is a classification together with $\land$ and $\neg$, defined on infons, and satisfying the following:

• $s \models \phi_1 \land \phi_2$ iff $s \models \phi_1$ and $s \models \phi_2$
• $s \models \neg \phi$ iff $s \not\models \phi$

Barwise used Gentzen-type sequents to model information.

Definition 6.2 (Sequent)   A sequent $I$ is a pair $I=\Gamma \Rightarrow \Delta$, where $\Gamma$ and $\Delta$ are sets of infons.

A sequent $I=\Gamma \Rightarrow \Delta$ holds of a situation (or situoid) $s$, $s \models \Gamma \Rightarrow \Delta$, if the following is true: If $s \models \phi$ for all $\phi \in \Gamma$, then $s \models \rho$ for some $\rho \in \Delta$.

A sequent $I$ is information about a set of situoids or situations $S$ if $s \models I$ for all $s \in S$.

The following propositions about sequents hold, just as for classical sequent calculus, such as LK gentzen1.

Theorem 6.1   Let $s$ be any situation or situoid.

1. $s \models \{A\} \Rightarrow \{A\}$ (Identity)
2. If $s \models \Gamma \Rightarrow \Delta$ then $s \models \Gamma \cup \Gamma' \Rightarrow \Delta \cup \Delta'$ (Weakening)
3. If $s \models \Gamma \cup \Sigma_0 \Rightarrow \Delta \cup \Sigma_1$ for each partition $(\Sigma_0,\Sigma_1)$ of some set of infons $\Sigma'$, then $s \models \Gamma \Rightarrow \Delta$ (Global Cut)
4. If $s \models \Gamma \cup \{A\} \Rightarrow \Delta$, then $s \models \Gamma \cup \{A\land B\} \Rightarrow \Delta$ and also $s \models \Gamma \cup \{B\land A\} \Rightarrow \Delta$ ($\land L$)
5. If $s \models \Gamma \Rightarrow \Delta \cup \{A\}$ and $s \models \Gamma' \Rightarrow \Delta' \cup \{B\}$ then $s \models \Gamma \cup \Gamma' \Rightarrow \Delta \cup \Delta' \cup \{A \land B\}$ ($\land R$)
6. If $s \models \Gamma \Rightarrow \Delta \cup \{A\}$ then $s \models \Gamma \cup \{\neg A\} \Rightarrow \Delta$ ($\neg L$)
7. If $s \models \Gamma \cup \{A\} \Rightarrow \Delta$ then $s \models \Gamma \Rightarrow \Delta \cup \{\neg A\}$ ($\neg R$)

Proof.

1. To show is that if $s \models A$ then $s \models A$, that is $s \models A \rightarrow A$, which is a tautology.
2. For all $\phi \in \Gamma$, $s \models \phi$, and for some $\phi' \in \Delta$, $s \models \phi'$. To show is, if for all $\delta \in \Gamma \cup \Gamma'$ $s \models \delta$, there is some $\delta' \in \Delta \cup \Delta'$ with $s \models \delta'$. If this was not the case, there would be no such $\delta' \in \Delta \cup \Delta'$, but a $\phi' \in \Delta \subseteq \Delta \cup \Delta'$, which is a contradiction.
3. Proof can be found in barwise1.
4. To show is, that if $s \models \phi$ for all $\phi \in \Gamma \cup \{A \land B\}$ then there is some $\phi' \in \Delta$ so that $s \models \phi'$. If this was not the case, then $s \not\models \Gamma \cup \{A\} \Rightarrow \Delta$ which is a contradiction. The proof is similar for the other direction.
5. Given is $s \models \Gamma \Rightarrow \Delta \cup \{A\}$ and $s \models \Gamma' \Rightarrow \Delta' \cup \{B\}$. If $s \not\models \Gamma \cup \Gamma' \Rightarrow \Delta \cup \Delta' \cup \{A \land B\}$, then for all $\phi \in \Gamma \cup \Gamma'$ but for no $\phi' \in \Delta \cup \Delta' \cup \{A \land B\}$ $s \models \phi'$. There is some $\delta \in \Delta \cup \{A\}$ and some $\delta' \in \Delta' \cup \{B\}$ with $s \models \delta$ and $s \models \delta'$. If $\delta \in \Delta$ then there is a $\phi' \in \Delta \cup \Delta' \cup \{A \land B\}$ with $s \models \phi'$ which is a contradiction. The same holds for $\delta' \in \Delta'$. So $\delta=A$ and $\delta'=B$ and $s \models A$ and $s \models B$. But then $s \models A \land B$, which is again a contradiction.
6. Given is $s \models \Gamma \Rightarrow \Delta \cup \{A\}$. Assume $s \not\models \Gamma \cup \{\neg A\} \Rightarrow \Delta$, then for all $\phi \in \Gamma \cup \{\neg A\}$ $s \models \phi$, but for no $\phi' \in \Delta$ $s \models \phi'$, while there is a $\delta \in \Delta \cup \{A\}$ with $s \models \delta$. If $\delta \in \Delta$ we have a contradiction, so $\delta=A$ and $s \models A$, while at the same time $s \models \neg A$, which is a contradiction.
7. The proof is similar to the previous.

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These rules taken as inference rules are complete barwise1. Barwise introduces information contexts, also called local logics.

Definition 6.3 (Information context, Boolean information context)   An information context $C=(A,\Rightarrow,N)$ is a classification $A$, a binary relation $\Rightarrow$ relating sets of infons and a set $N \subseteq S$ of situations called normal situations. $N$ is closed under identity, weakening and global cut, and for all $s \in N$ and all $I \in \Rightarrow$: $s \models I$.

A boolean information context is one where $N$ is also closed under the rules ($\land L$), ($\land R$), ($\neg L$) and ($\neg R$).

The idea is that $\Sigma$, the set of infons from $A$, represents the relevant issues, $\Rightarrow$ the information present, while $N$ are the situations supporting this information. They may, but do not have to be all the situations.

Barwise continues by introducing possible worlds, which he calls ``states'' to distinguish them from Lewisian possible worlds in lewis1. We will call them possible worlds anyway.

Definition 6.4 (Possible and impossible worlds, realization of worlds, possible and impossible situations)   Let $C$ be an information context.

1. A possible world is a binary partition $(\Gamma,\Delta)$ of the infons in $C$, $\Omega$ the set of all worlds.
2. The possible world of a situation $s$ is the partition $(\Gamma_s,\Delta_s)$ where $\Gamma_s = \{\phi \in \Sigma\vert s \models \phi\}$ and $\Delta_s = \Sigma - \Gamma_s$. The world is denoted $world(s)$.
3. A possible world $w \in \Omega$ is realized by the situation $s$ if $world(s)=w$.
4. A world $w = (\Gamma,\Delta)$ is impossible, if $\Gamma \Rightarrow \Delta$, otherwise possible. $\Omega_C$ is the set of all possible worlds in the context $C$.
5. A situation $s$ is possible iff $world(s)$ is possible.

Now it can be shown, that every normal situation is possible, and if the information context is sound, every situation is possible as well as if the information context is complete, then every possible world is realized by some normal situation.

Before we can introduce modalities, we need one more definition, defining what is meant by one information context being less informative than another.

Definition 6.5 (Less and more informative information contexts)   Let $A=(S,\Sigma,\models)$ be a fixed classification, and $C_1=(A,\Rightarrow_{C_1},N_1)$ and $C_2=(A,\Rightarrow_{C_2},N_2)$ be information contexts. Then $C_1 \sqsubseteq C_2$ iff

1. for all sets of infons $\Gamma$ and $\Delta$, if $\Gamma \Rightarrow_{C_1} \Delta$, then $\Gamma \Rightarrow_{C_2} \Delta$ and
2. for all $s \in S$, if $s \in N_2$ then $s \in N_1$.

Now modalities can be introduced.

Definition 6.6 (Informational modal framework)   An informational modal framework $M$ is a classification $A=(S,\Sigma,\models)$, and an information context $C_s=(A,\Rightarrow_s,N_s)$ for each situation $s \in S$, satisfying the condition: If $world(s)=world(s')$ then $C_s \sqsubseteq C_{s'}$ and $C_{s'} \sqsubseteq C_s$. All $t \in N_s$ are called $s$-normal. All worlds possible relative to $C_s$ are called $s$-possible.

Now propositions are taken as sets of possible worlds. It follows the final definition in barwise1, modified to fit our terminology.

Definition 6.7 (Accessibility of worlds, $\mathbf{\diamond p}$, $\mathbf{\square p}$)   Let $M$ be a fixed information frame.

1. A possible world $w'$ is accessible from the world $w$ provided that for some situation $s$ with $world(s)=w$, $w'$ is $s$-possible. (It follows that $w'$ is $s$-possible for every $s$ with $world(s)=w$.)
2. Given a proposition $p$, let
   $$\diamond p = \{w \in \Omega\vert$$some state $$w' \in p$$    is accessible from w$$\}$$
   and
   $$\square p = \neg \diamond \neg p$$

With these definition, Barwise shows in barwise1 several theorems known from modal logics, and we will name the more important ones.

A proposition is valid in the informational modal framework $M$, written $M \models p$ iff $s \models p$ for all situations $s$ in $M$. Then for any modal information frame $M$ and any propositions $p$ and $q$, $\square(p \rightarrow q) \rightarrow (\square p \rightarrow \square q)$, $p \rightarrow \diamond p$ and $\square p \rightarrow p$ are valid.

Several other theorems are proven in barwise1, and all of them can be adapted to the framework of situoids and situation of this thesis. Therefore, this may be an elegant way of introducing modalities in the ontology of GOL.