The treatment of epistemic logic as a branch of modal logic brings some advantages. However, there is a high price to pay. The most important objection to the modal approach is that it makes unrealistic assumptions about the reasoning power of the agents. The problem is known as the ``logical omniscience problem'' (LOP) and occurs in several forms. In its strongest form the problem can be stated as follows:

That is, whenever an agent knows all of the formulae in a set and follows logically from , then the agent also knows . In particular, the agent knows all theorems (taking in lemma 9 to be the empty set), and he knows all logical consequences of a sentence that he knows (taking to consist of a single sentence.)

Besides this strong form there are other, generally weaker forms of logical omniscience. The following are listed in [FHMV95]:

- Knowledge of valid formulae: agent knows all logical truths
(rule
**(NEC)**). - Closure under logical implication: if agent knows
and if logically implies (i.e.,
is
valid), then agent knows (rule
**(MON)**). - Closure under logical equivalence: if agent knows
and if and are logically equivalent (i.e.,
is valid), then agent knows
(rule
**(CGR)**). - Closure under material implication: if agent knows
and if agent knows
then agent knows
(axiom
**(K)**). - Closure under conjunction: if agent knows
and if agent knows then agent knows
(axiom
**(C)**).

The list of questionable properties could be extended to include any
other instance of the rule **(RK)** (from
to infer
.) Moreover, the axiom
schemata **(D)**, **(4)** and **(5)** can also be shown
to be too strong for realistic agents. In particular,
under certain circumstances axiom **(5)** suggests that agents
can even decide undecidable problems ([BS92],
[SW94])! In general, there seems to be no genuine epistemic
principle that may claim universal validity^{2.3}.

If epistemic logic is to be interpreted as describing actual knowledge
of realistic (though idealized) agents, then the discussed closure
properties require agents to be very powerful reasoners whose
computational capacities cannot be achieved by real (human or
artificial) agents, who are simply not logically omniscient. Logical
omniscience poses a problem because it contradicts the fact that
agents are limited in their reasoning powers. They are inherently
resource-bounded and therefore cannot handle an unlimited amount of
information. Agents may establish immediately certain logical truths
or simple consequences of what they consciously assented to. However,
there are highly remote dispositional states which could only be
established by complex, time-consuming reasoning. The modal framework
cannot distinguish between a sentence that an agent consciously
assented to and a piece of potential knowledge which could never be
made actual by the agent and is therefore not suited to model
resource-bounded reasoning^{2.4}.