Constraints and Involvement

Now, after defining universals, or situation types, we can show how to use this formalism in practise.

We will use constraints as in perry1, infons with a pictural state of affairs, whose relation relates amongst situation types.

Definition 5.32 (Constraint)   A constraint is an infon $i=\langle\langle R,a_1,\ldots,a_n;p\rangle\rangle$, such that $a_1,\ldots,a_n$ are assigned to situation types.

We will use two special types of constraints. There are more, the most basic being $\langle\langle \top,a,b;1\rangle\rangle$ ( $\langle\langle \bot,a,b;1\rangle\rangle$), stating that if a situation is of type $a$ than it is also (not) of type $b$. However, some more complicated types of constraints are of concern to us.

We can define what we mean by the involvement of two situations. We distinguish between simple involvement, which is a binary relation, and relative involvement, which is a ternary relation.

Definition 5.33 (Simple and relative involvement)   Simple involvement is a binary relation. If the type of a situation $T$ involves the type of a situation $T'$, then for every situation of type $T$ there is one of type $T'$, noted as

$$\langle\langle Involves,T,T';1\rangle\rangle$$

Relative involvement is a ternary relation. If $T$ involves $T'$ relative to $T''$, then for any pair of situations of type $T$ and $T''$ there is a situation of type $T'$, noted as

$$\langle\langle Involves_R,T,T',T'';1 \rangle\rangle$$

The theory defined by a universal can assert constraints to situations. Let $C$ be a constraint, then $s \models C$ will assert the constraint $C$ to a situation $s$ (which is the free variable in the theory defined by the universal the situation is an instance of).

Now we express the information carried by an infon. Information here is a proposition relative to some constraint of the form $\langle\langle Involves,\ldots;1\rangle\rangle$.

Axiom 5.39   Let $C$ be some constraint. The infon $i$ carries the pure information that $P$ relative to $C$ iff

1. $C=\langle\langle Involves,T,T';1\rangle\rangle$
2. For any assignment $f$ such that $i=cond(T)$, $anchor(i)=f$, $P$ is the proposition $P=\exists s'(s' \models \exists a_1,\ldots,a_n (cond(T')[f]))$.

The assignment function for the conditioning infon of $T$ has to be the same as for the conditioning infon of $T'$, and all other parameters of $cond(T')$ are existentially quantified. Note also, that it is not required, that if $s \models i$ and therefore $s$ is of the type $T$, it itself has to be of type $T'$, but there only has to exist some situation of type $T'$.

We need another axiom for relative involvement.

Axiom 5.40   Let $C$ be some relative constraint. Then the infon $i$ carries the incremental information that $P$ relative to $C$ and the infon $i'$ iff

1. $C=\langle\langle Involves_R,T,T',T'';1\rangle\rangle$
2. For any assignment $f$ such that $i=cond(T)$, $anchor(i)=f$ and $i'=cond(T'')[f]$, $P$ is the proposition $P=\exists s'(s' \models \exists a_1,\ldots,a_n (cond(T')[f]))$.

Now let us assume, that for a universal $u$ with situations as its instances, $u=(T,\Sigma)$, and $T=\{s\models C_1,\ldots,s\models C_n s\models B_1,\ldots,s\models B_m\}$, where $C_1,\ldots,C_n$ are constraints and $B_1,\ldots,B_m$ are basic infons. Of course are constraints only a special type of basic infons.

We will see in an example, how this formalism is applied.