Awareness

Another solution to the LOP consists in introducing a new operator of awareness into the language and to require that belief include awareness ([FH88].) The underlying intuition is that agents need to be aware of some concept before they can have beliefs about it: one cannot know something one is completely unaware of. On the other hand, if an agent is aware of a formula $\alpha$ and implicitly knows $\alpha$, then he knows $\alpha$ explicitly. The notion of awareness is left unspecified. Some possible interpretations of ``agent $i$ is aware of $\alpha$'' are: ``$i$ is familiar with all the propositions mentioned in $\alpha$'', ``$i$ is able to figure out the truth of $\alpha$'', or ``$i$ is able to compute the truth of $\alpha$ within time $T$.''

For better comparison with other approaches, my presentation of the awareness framework will not follow the original definition ([FH88]) in details. The main intuitions are retained, however. In particular, there are no modal operators for implicit knowledge and awareness. The knowledge operators of the language $\mathcal{L}_{N}^{K}$ are now interpreted as explicit knowledge and will be evaluated accordingly in the definition of models.

Intuitively, $\mathcal{A}_i(s)$ is the set of formulae that agent $i$ is aware of at state $s$, and the relations $R_1,\ldots,R_N$ are used to model implicit knowledge. The set of formulae that an agent is aware of can be arbitrary and needs not be closed under any law. Moreover, there is no relationship between (implicit) knowledge and awareness at all: the function $\mathcal{A}_i$ and the relation $R_i$ are completely independent. Since explicit knowledge is defined as implicit knowledge plus awareness, it is obvious that if an agent is aware of all formulae of the language then explicit knowledge reduces to implicit knowledge.

Because it is possible that an agent is aware of some sentence but he is not aware of its logical consequences or its equivalent sentences, the theorems and inference rules of modal epistemic systems do not hold in general. So the forms of logical omniscience discussed in chapter 2 are avoided.

That the awareness approach is able to model non-omniscient agents can be seen in another way. We have seen earlier that the impossible-worlds approach avoids all forms of logical omniscience. The following theorem shows that although the intuitions are quite different, the impossible-worlds approach and the awareness approach are equivalent in a precise sense (cf. [Wan90], [Thi93], [FHMV95]).

As an immediate consequence of this theorem, the awareness framework also solves all forms of the LOP: if an undesirable property can be falsified in an impossible-worlds model, then it can also be falsified in an awareness model. In fact, it can be seen easily that the set of $\mathcal{L}_{N}^{K}$-formulae which are valid wrt all awareness models consists of exactly the instances of propositional tautologies. In other words, no genuine epistemic statement is valid with respect to the class of all awareness models.

So far the concept of awareness has been left unspecified, so no meaningful restrictions can be placed on the set of formulae that an agent is aware of. Once a concrete interpretation has been fixed, some closure properties can be added to the awareness function to capture certain types of ``awareness''.

For example, if we consider a computer program that never computes the truth of a formula unless it has computed the truth of all its subformulae, then we may assume that awareness is closed under subformulae, i.e., if $\alpha \in \mathcal{A}_i(s)$ and $\beta$ is a subformula of $\alpha$ then $\beta \in \mathcal{A}_i(s)$. This assumption may seem innocuous at first, but it turns out to have a rather strong impact on the properties of explicit knowledge. It can be shown easily that if awareness is closed under subformulae then an agent's knowledge is closed under material implication, i.e., the schema (K) is valid. In general, whenever $\beta$ follows logically from $\avec{\alpha}{,}{n}$ and $\beta$ is a subformula of one of $\avec{\alpha}{,}{n}$, then $K_i\beta$ follows from $\avec{K_i\alpha}{,}{n}$, for any agent $i$.

Another possible closure property for awareness is that agent might be aware of only a subset $X$ of the atomic formulae. In this case one could assume that $\mathcal{A}_i(s)$ consists of exactly those formulae that are built up from the atomic formulae in $X$. Under this assumption some forms of logical omniscience are avoided, e.g., knowledge of valid formulae or closure under logical implication. However, all forms of the LOP occur again when attention is restricted to the sublanguage generated by $X$.