A situation example

This example is due to John Perry perry1. The dog Jackie has a broken leg. There is an x-ray taken of Jackie, and this x-ray has a certain pattern, indicating that Jackie has a broken leg.

There are two types of situations of concern. The first is one with an x-ray, $x$, at a time $t$, that has a certain pattern $\Phi$. This pattern is the pattern of a broken leg.

$$T=[s\vert s\models \langle\langle$$   X-ray$$,x,t;1 \rangle\rangle \land \langle\langle$$   $$\mbox{Has-pattern-$\Phi$}$$$$,x,t;1\rangle\rangle ]$$

The second type is one with the x-ray $x$ of some object $y$ and this object has a broken leg, all at a certain time.

$$T'=[s\vert s\models \langle\langle$$   Is-xray-of$$,x,y,t;1\rangle\rangle \land \langle\langle$$   Has-broken-leg$$,y,t;1\rangle\rangle$$

Our constraint will now state the following: If there is an x-ray $x$ and it has a certain pattern $\Phi$, the pattern of a broken leg, then it is an x-ray of some object $y$ and this object has a broken leg.

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

We define a universal $u_s$, the universal of situations with an x-ray of a broken leg, in the following way:

$$u_s=(\Sigma,T_s)$$

$$\Sigma=(\models)$$

$$T_s=[s\vert s \models ,x,t;1\rangle\rangle \land s \models \langle\langle \mbox{Has-pattern-$\Phi$} ,x,t;1\rangle\rangle$$

The situation $s$ is then defined as follows, where $a$ is the x-ray and $t'$ the time the x-ray has been taken.

$$s::u_s$$

$$s \models \langle\langle ,a,t';1\rangle\rangle$$

$$s \models \langle\langle \mbox{Has-pattern-$\Phi$} ,a,t';1\rangle\rangle$$

$$s \models C$$

Now $f$ is an anchor defined on $x$ and $t$, such that $f(x)=a$ and $f(t)=t'$. Now we obtain

$$s \models \phi$$

$$\phi = \langle\langle ,a,t';1\rangle\rangle \land \langle\langle \mbox{Has-pattern-$\Phi$} ,a,t';1\rangle\rangle$$

$$= \langle\langle ,f(x),f(t);1\rangle\rangle \land \langle\langle \mbox{Has-pattern-$\Phi$} ,f(x),f(t);1\rangle\rangle$$

$$= (\langle\langle ,x,t;1 \rangle\rangle \land \langle\langle \mbox{Has-pattern-$\Phi$} ,x,t;1\rangle\rangle )[f]$$

$$= cond(T)[f]$$

Then $P$ is the following proposition:

$$\exists s'(s' \models \exists y(\langle\langle ,x,y,t;1\rangle\rangle \land \langle\langle ,y,t;1\rangle\rangle )[f])$$

$$= \exists s'(s' \models \exists y(\langle\langle ,f(x),y,f(t);1\rangle\rangle \land \langle\langle ,y,f(t);1\rangle\rangle ))$$

$$= \exists s'(s' \models \exists y(\langle\langle ,a,y,t';1\rangle\rangle \land \langle\langle ,y,t';1\rangle\rangle ))$$

This proposition is the pure information carried by the situation $s$, namely the proposition, that in some situation $s'$ there has to be a dog $y$, and $a$ is the x-ray of $y$ at time $t'$.

Let us now see what we will obtain using the second type of constraint. Let the type $T$ be the same as before, $T'=[s\vert s\models \langle\langle$   Has-broken-leg$,y,t;1\rangle\rangle ]$ and $T''=[s\vert s\models \langle\langle$   Is-xray-of$,x,y,t;1\rangle\rangle ]$, and $C'=\langle\langle Involves_R,T,T',T'';1\rangle\rangle$.

The universal $u_s$ is the same as above.

We could state that the described situation is the instance of another universal, a universal of situations with an x-ray of a broken leg of some person. Or we could define an additional universal and say that this situation is also an instance of another universal, the universal of situations with an x-ray of an individual. Again for the sake of simplicity we omit this.

The situation $s_R$ is then defined as follows, where $a$ and $t'$ are as before, and $b$ is assigned to the dog Jackie.

$$s::u_{sR}$$

$$s \models \langle\langle ,a,t';1\rangle\rangle$$

$$s \models \langle\langle \mbox{Has-pattern-$\Phi$} ,a,t';1\rangle\rangle$$

$$s \models \langle\langle ,a,b,t';1\rangle\rangle$$

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

Now $f$ is an anchor defined on $x$ and $t$ as well as $y$, with $f(x)=a$, $f(t)=t'$ and $f(y)=b$, $f$ has to assign $y$ to the dog Jackie, $b$.

Now we obtain the following.

$$\phi = \langle\langle ,a,t';1\rangle\rangle \land \langle\langle \mbox{Has-pattern-$\Phi$} ,a,t';1\rangle\rangle$$

$$= \langle\langle ,f(x),f(t);1\rangle\rangle \land \langle\langle \mbox{Has-pattern-$\Phi$} ,f(x),f(t);1\rangle\rangle$$

$$= \langle\langle ( ,x,t;1 \rangle\rangle \land \langle\langle \mbox{Has-pattern-$\Phi$} ,x,t;1\rangle\rangle )[f]$$

$$= cond(T)[f]$$

$$\phi' = \langle\langle ,a,b,t';1\rangle\rangle$$

$$= \langle\langle ,f(x),f(y),f(t);1\rangle\rangle$$

$$= (\langle\langle ,x,y,t;1\rangle\rangle )[f]$$

$$= cond(T'')[f]$$

Now we will obtain the proposition

$$\exists s''(s'' \models \langle\langle ,b,t';1\rangle\rangle )$$

This is the information we wanted to obtain out of the described situation.