Possible-worlds semantics for epistemic logic

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 $i$ 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 $R_i$, on the set of possible worlds (read possible models of the world.) To express the idea that for agent $i$, the world $t$ is compatible with her information state when he is in the world $s$, we require that the relation $R_i$ holds between $s$ and $t$. One says that $t$ is an epistemic alternative to $s$ (for agent $i$). If a sentence $\alpha$ is true in all worlds which agent $i$ considers possible then we say that this agent knows $\alpha$. Formally, the concept of models is defined as follows:

We can easily check that according to definition 3, if $\avec{\alpha}{\land}{n}\to \beta$ is valid then so is $\avec{K_i\alpha}{\land}{n}\to K_i\beta$, for all $i\in Agent$ and all natural numbers $n=0,1,2,\ldots$. These rules can be interpreted as saying that any agent $i$'s knowledge is closed under logical laws: whenever $i$ 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 $R_i$'s then we get larger classes of valid formulae and may obtain characteristic models for extensions of K$_N$. 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$_N$ is considered by many researchers as the standard logic of rational knowledge, and KD45$_N$ 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$_N$, rather than S5$_N$, as the logic of knowledge.