Hyperset-based model theory

We will now show how to model situations with hypersets. As Barwise has shown in barsit, there are inherently circular situations, and the best way to model them is by using hypersets.

We will use the same notion as did Barwise in barsit of a state model: A state model is an ordered triple $< R,a,p >$, where $R$ is a relation, $a$ a function and $p \in \{0,1\}$. A situation model is introduced as a set of state models.

Axiom 5.43   There is an operation $M:Entity \mapsto Set$, satisfying the following equations:

• If $b$ is not a situation or infon, then $M(b)=b$.
• If $\phi = \langle\langle R,a;i\rangle\rangle$, then $M(\phi) = <R,b,i>$, where $b$ is the function on $dom(a)$ satisfying $b(x)=M(a(x))$.
• If $s$ is a situation, then $M(s)=\{M(\phi):s\models \phi\}$.

We believe that more research has to be done concerning situation models. For example constraints and the semantic of the two types of constraints we discussed are not reflected by this approach to model situations.

Another approach of modelling situoids has been published by the GOL researchers in herre4.