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 and implicitly knows
, then he knows
explicitly. The notion of awareness
is left unspecified. Some possible interpretations of ``agent
is
aware of
'' are: ``
is familiar with all the propositions
mentioned in
'', ``
is able to figure out the truth of
'', or ``
is able to compute the truth of
within
time
.''
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
are now interpreted as explicit knowledge and
will be evaluated accordingly in the definition of models.
Intuitively,
is the set of formulae that agent
is aware of at state
, and the relations
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
and the relation
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
-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
and
is a
subformula of
then
. 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
follows
logically from
and
is a subformula of
one of
, then
follows from
, for any agent
.
Another possible closure property for awareness is that agent might be
aware of only a subset of the atomic formulae. In this case one
could assume that
consists of exactly those
formulae that are built up from the atomic formulae in
. 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
.