A number of systems have been proposed which assume still more restricted reasoning capacities of the agents and in this way avoid all forms of logical omniscience. One framework that eliminates logical omniscience completely is the so-called impossible-worlds approach. Logical omniscience can be avoided if one allow ``impossible possible worlds'' in which the valuation of the sentences of the language is arbitrary. In other words, the logical laws do not hold in the ``impossible possible worlds'' ([Cre70], [Cre73], [Hin75], [Ste79], [Ran82], [Wan90]).
The intuition underlying the introduction of impossible worlds is that an agent may regard some models of the (real) world possible, although they are logically impossible. For example, a logical contradiction cannot be true. However, an agent may not have enough resources to determine the truth value of that contradiction and simply assumes it to be true. So he will consider some worlds possible, although logically they are impossible.
Because knowledge is evaluated with respect to all states and the laws of logic do not hold in some states, all forms of logical omniscience are avoided. For instance, the tautology may be false in an impossible world, but an agent may consider that world possible, so does not hold universally. In other words, the necessitation rule is not valid. Similarly, axiom (K) (closure under material implication) fails to hold, because it is possible that in an impossible world both formulae and are true while is false.
The logic determined by the class of all impossible-worlds models is rather uninteresting, because no genuine epistemic statement is universally valid. Epistemic principles can be obtained by imposing appropriate conditions on the models. For example, axiom (K) is valid if for every impossible world, if the value is assigned to both and then it must be assigned to the formula as well.