Correspondence of infons and states of affairs

Let us first consider what is meant by the correspondence of states of affairs and pictures of states of affairs. The problem that is still open is, how we account for relations or relators. Given two states of affairs, $\phi=\langle\langle r::R,a_1,\ldots,a_n\rangle\rangle$ and a picture of $\phi$, say $\rho=\langle\langle p::P,b\rangle\rangle$, and $Arg(\rho)=\{x_1,\ldots,x_n\}$. If $\rho$ is a picture of $\phi$, then $\forall i (0 \leq i\land i\leq n \rightarrow b(x_i)=a_i \lor b(x_i)=nil)$. This is to say, that all arguments of the picture $\rho$ have to be either unassigned or assigned to the appropriate objects of $\phi$.

But what are we to do with the relations $R$ and $P$, and the relators $r$ and $p$ for this sake? We could just extend the assignement function to include the relators as well, but this does not appear to be right. The picture is a logical representation of the state of affairs, and its elements behave in logical space as the elements of the state of affairs in reality. This suggests that there is some similarity in the relation of the state of affairs and the picture. We will take the point of view that the relations are identical, and the relators of the state of affairs and its picture are different (but still instances of the same relation).

This may not be satisfying for all philosophers. But we are dealing with a dilemma, GOL still has to face in the future. Relations and relators in GOL are entities, relations universals, relators its instances. GOL appears to be using entities from reality, but it is not. Even GOL cannot state ``what'' some entity is, but only ``how'' it is. Therefore, relations, relators and all other entities in GOL are symbolic structures, names, or pictures of entities in reality. These entities are ``denoted'' by names. GOL takes objects from reality, denotes them by names and describes ``how'' they are in logic. When GOL is talking about relations, then it is a logical description of how certain relations behave. The picture that an intelligent agents forms of a state of affairs is similar, as it is always a specification of the state of affairs in the domain of logic. A picture is a logical representation of its corresponding state of affairs. When GOL talks about some relation $R$ (that does exist in reality, not as a symbolic structure), and uses the name $P$ (which is a symbolic structure) to characterize $R$, then $R$ is characterized by only logical means. Hereby, the symbolic structure $P$ ``corresponds'' to relation $R$, or $R$ is ``denoted'' by $P$. $P$ is therefore the translation of $R$ into logic. This is the reason we are stating that the relations in a state of affairs and its picture are identical: the state of affairs can be described in logic, and therefore its constituting relation-universal, too, and this is what creates the picture. These are the preliminaries of a denotation relation, which is still missing in GOL.

Definition 5.3 (Strong correspondence ($SOA$ and $picSOA$))   Let $s=\langle\langle r::R,x_1,...,x_n\rangle\rangle$ be a state of affairs, $\phi=\langle p::P,arg_1,...,arg_n\rangle$ be a picture of a state of affairs and let $a$ be the assignment function of $\phi$. Then $corr(s,\phi) \iff R=P \land a(arg_1)=x_1\land ... \land a(arg_n)=x_n$. We will then say, that $s$ and $\phi$ correspond strongly.

This is a strong form of correspondence, because all argument places of $\phi$ are assigned an object. But the possibility exists, that only some argument places are filled by appropriate objects, so the picture is incomplete, but still a picture of a state of affairs. We will define what we will call ``weak correspondence'' for this case.

Definition 5.4 (Weak correspondence (SOA and picSOA))   Let $s=\langle\langle r::R,x_1,...,x_n\rangle\rangle$ be a state of affairs, $\phi=\langle p::P,arg_1,...,arg_n\rangle$ be a picture of a state of affairs and let $a$ be the assignment function of $\phi$. Then $wcorr(s,\phi)$, if and only if $R=P$ and there is a non-empty subset $X \subseteq \{arg_1,\ldots,arg_n\}$ such that $a(arg_i)=x_i$ for all $arg_i \in X$ and for all $b \in \{arg_1,\ldots,arg_n\}\ X$ holds $a(b)=nil$.

We will then say, that $s$ and $\phi$ correspond weakly.

A picture of a state of affairs corresponds to a state of affairs, if they have the same structure and the picture elements are assigned to the appropriate objects present in the state of affairs. They correspond weakly, if they have the same structure, and at least one picture element is assigned an appropriate object present in the state of affairs. When we speak about correspondence between pictures and states of affairs, we usually mean weak correspondence, unless stated otherwise.

With this formalism, we can say what is meant by an infon corresponding to a state of affairs. An infon corresponds to a state of affairs, if its pictural state of affairs corresponds to the state of affairs. Of cause, it always corresponds to its picture of a state of affairs itself. Correspondence is our means of accessing the state of affairs in a picture or an infon, and therefore polarity of an infon is of no concern.

Definition 5.5 (Correspondence ($Infon$ and $SOA$))   Let $s=\langle\langle R,x_1,...,x_n\rangle\rangle$ be a state of affairs and $i=\langle\langle P,a;p\rangle\rangle$ be an infon. Then $corr(i,s) \iff corr(s,\langle\langle P,a\rangle\rangle ) \lor \langle\langle P,a \rangle\rangle =s$ and $wcorr(i,s) \iff corr(i,s) \lor wcorr(s,\langle\langle P,a\rangle\rangle )$.

Note, that this form of correspondance is not transitive. However, it will be useful to define a transitive relation based on correspondance. We will introduce the relation $corr^+$ as the transitive correspondance relation.

Axiom 5.3  

$$\forall x \forall y \forall z (corr(x,y)\land corr(y,z) \rightarrow corr^+(x,z))$$

We can only form pictures of possible states of affairs, states of affairs that exist in some possible world. This is asserted in the following axiom.

Axiom 5.4  

$$\forall x (picSOA(x) \rightarrow \exists y (SOA(y) \land corr(x,y)))$$

$$\forall x (Infon(x) \rightarrow \exists y (SOA(y) \land corr(x,y)))$$

Whether the reverse is true or not is an open question. It may be possible that there are states of affairs of which we cannot form a picture in our mind, and no infon either. We will leave it open as an alternative.

Axiom 5.5 (Alternative 1)  

$$\forall x (SOA(x) \rightarrow \exists y (picSOA(y) \land corr(y,x)))$$

Axiom 5.6 (Alternative 2)  

$$\exists x (SOA(x) \land \neg \exists y (picSOA(y) \land corr(y,x)))$$

Some other issue is, whether the correspondance relation is well-founded. Can we construct an unlimited series of pictures of pictures, without ever ending up with a corresponding state of affairs that is not a picture? We will deny this. We will, however, always end up with a state of affairs that is not a picture.

Axiom 5.7   Let $SOA$ be the set of all states of affairs. Every non-empty subset $X\subseteq SOA$ has a $corr^+$-minimal element in $SOA$. No $corr^+$-minimal element in the set $SOA$ is a pictural state of affairs.