The intuitive idea behind the possible worlds approach is that an agent can build different models of the world using some suitable language. He usually does not know exactly which one of the models is the right model of the world. However, he does not consider all these models equally possible. Some world models are incompatible with his current information state, so he can exclude these incompatible models from the set of his possible world models. Only a subset of the set of all (logically) possible models are considered possible by the agent. For example, an agent possesses the information that he is 30 years old. Then among the models of the world he will not consider possible all those models in which he is not 30 years old. The smaller the set of worlds an agent considers possible, the smaller his uncertainty, and the more he knows.

The set of worlds considered possible by an agent depends on the ``actual world'', or the agent's actual state of information. This dependency can be captured formally by introducing a binary relation, say , on the set of possible worlds (read possible models of the world.) To express the idea that for agent , the world is compatible with her information state when he is in the world , we require that the relation holds between and . One says that is an epistemic alternative to (for agent ). If a sentence is true in all worlds which agent considers possible then we say that this agent knows . Formally, the concept of models is defined as follows:

We can easily check that according to definition 3, if is valid then so is , for all and all natural numbers . These rules can be interpreted as saying that any agent 's knowledge is closed under logical laws: whenever knows all premises of a valid inference rule then he also knows the conclusion.

If we restrict the class of models by imposing appropriate conditions
on the epistemic alternativeness relations 's then we get larger
classes of valid formulae and may obtain characteristic models for
extensions of **K**. The well-known results for modal logic
can be transferred to epistemic logic without any difficulty. The
following theorem summarizes some completeness and decidability
results for modal epistemic logic (cf. [Che80],
[HC96], [Gol87], [HM92], [FHMV95]).

The logic **S5** is considered by many researchers as the
standard logic of rational knowledge, and **KD45** as the
standard belief logic. It is generally accepted that negative
introspection is a more demanding condition than positive
introspection. Therefore many researchers argue that it is more
reasonable to adopt **S4**, rather than **S5**, as
the logic of knowledge.