Axioms of situation theory

We will give a brief summary of the axioms of one of the later versions of Barwisean situation theory, without much motivation and explanation, mainly to refer to them later. For a complete discussion, see barsit.

Axiom 3.1   Every relation is the major constituent of some basic infon, and everything is a minor constituent of some basic infon.

Axiom 3.2   If $\phi = \langle\langle R,a;i\rangle\rangle$ and $\phi ' = \langle\langle R',a';i'\rangle\rangle$ are basic infons, then $\phi = \phi '$ iff $R=R'$, $i=i'$ and $a_{arg}=a'_{arg}$ for each argument $arg$ of $R$.

Axiom 3.3   Given an $n$-ary relation $R$ and an appropriate sequence $a$ of arguments for $R$, either $R(a)$ or $\neg R(a)$, but not both. $R(a)$ iff $\models \langle\langle R,a;1\rangle\rangle$; $\neg R(a)$ iff $\models \langle\langle R,a;0\rangle\rangle$.

So far we can only talk about basic infons. Axiom 2 states under which circumstances we will consider basic infons identical, namely when they are identical item by item. However, infons are supposed to capture information, so they may be informational equivalent, although they are not identical item by item. Furthermore, given two basic infons, one might be stronger than the other, so that one infon is entailed by the other.

Axiom 3.4   Let $\phi$, $\phi'$, $\phi''$ be infons. Then $\phi \Rightarrow_l \phi$ and if $\phi \Rightarrow_l \phi'$ and $\phi' \Rightarrow_l \phi''$ then $\phi \Rightarrow_l \phi''$. If $\phi \Rightarrow_l \phi'$ and $\phi$ is a fact, then so is $\phi'$.

Axiom 3.5   Let $a$ and $a'$ be appropriate assignments for $R$, and $a$ is a sub-assignment of $a'$. For $i \in \{0,1\}$, let $\phi_i = \langle\langle R,a;i\rangle\rangle$ and let $\phi_i' = \langle\langle R,a';i\rangle\rangle$. Then $\phi_1' \Rightarrow_l \phi_1$ and $\phi_0 \Rightarrow_l \phi_0'$.

Axiom 3.6   Every set of infons $\Sigma$ has a least upper bound (concerning $\Rightarrow_l$) $\bigwedge \Sigma$ and a greatest lower bound $\bigvee \Sigma$. $\bigwedge \Sigma$ is a fact iff each infon in $\Sigma$ is a fact, $\bigvee \Sigma$ is a fact, iff some infon in $\Sigma$ is a fact.

Axiom 3.7   The basic infons $\phi = \langle\langle R,a;i\rangle\rangle$ and $\phi' = \langle\langle R,a';i \rangle\rangle$ are compatible iff $a$ and $a'$ are compatible as functions. Then $\phi \oplus \phi' = \langle\langle R, a \cup a';i\rangle\rangle$.

Now, for the first time, we will consider situations. A situation is ``a part of reality that can be comprehended as a whole in its own right - one that interacts with other things.''barsit They relate to other things of the world in the sense, that they stand in relation to other things, or have properties.

$\models$ is a binary relation, holding between situations and infons. $s \models \phi$ means ``$\phi$ holds in the situation $s$''.

Axiom 3.8   An infon $\phi$ is a fact iff there is a situation $s$ such that $s \models \phi$.

Axiom 3.9   A situation $s_1$ is a part of a situation $s_2$, $s_1 \triangleleft s_2$, if $\{\phi\vert s_1\models \phi$ and $\phi$ is a basic infon $\}\subseteq \{\phi'\vert s_2\models\phi'$ and $\phi'$ is a basic infon$\}$. If $s_1 \triangleleft s_2$ and $s_2 \triangleleft s_1$, then $s_1 = s_2$.

Therefore the part-of relation between situation is a partial ordering.

Barwise takes the axioms now a step further and defined some kind of interpretation for situations by stating the relation between sets and situations.

Axiom 3.10   Every set $F$ of facts determines a smallest situation $s$ such that for every $f \in F: s \models f$.

Axiom 3.11   Every set is the set of constituents of at least one situation.

The last axiom brings in models of situations.

Axiom 3.12   There is a unique operation $M$ taking values in the sets 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 $M(\phi) = <R,b,i>$ (a state model), where $b$ is the function on $dom(a)$ satisfying $b(x)=M(a(x))$.
• if $s$ is a situation, then $M(s)=\{M(\phi):s\models \phi\}$.