The ``received view'': modal epistemic logic

Among all approaches to epistemic logic that have been proposed, the modal approach has been the most widely used for modeling knowledge. An important reason for the popularity of that approach is its simplicity: systems of modal logic are given an epistemic interpretation, and the main technical results about epistemic logic can be obtained almost automatically. To interpret modal logic epistemically one reads modal formulae as epistemic statements expressing the attitude of certain agents towards certain sentences, and the semantics for modal logic is also given a new interpretation.

The interpretation of modal axioms as axioms for knowledge are not without difficulties. If we follow the ordinary usage of the word ``knowledge'' then that interpretation is certainly wrong. For example, consider the modal formula $\Box\alpha \land \Box(\alpha \to \beta) \to \Box\beta$. If interpreted epistemically, it says that if an agent knows the two premises of modus ponens then he also knows the conclusion. This is clearly too strong: there may be sentences $\alpha$ and $\beta$ such that an agent knows both $\alpha$ and $\alpha \to \beta$ and yet fails to know $\beta$. In general, from some epistemic statements one cannot deduce any other epistemic statement. Given the information that an agent's knowledge includes a set $\Gamma$ of sentences, in reality we can never infer reliably that the agent knows a sentence from the deductive closure $Cn(\Gamma)$ of $\Gamma$ with respect to a deductive system $Cn$ (except for those already in $\Gamma$), even if we suppose $Cn$ to be very weak (but not degenerate in the sense that $Cn(\Gamma)=\Gamma$.) This point has led many people to raise the question if epistemic logic is possible at all, or do we have to leave the realm of logic when reasoning about knowledge and belief ([Hoc72], [Bar89].)

Given the mentioned difficulty, how can we make epistemic logic possible? The answer is idealization. One restricts attention on the class of rational agents, where rationality is defined by certain postulates: agents have to satisfy at least some conditions to qualify as rational. For example, such a condition may read: ``If an agent is rational then he should know the laws of logic, therefore, if he knows $\alpha$ and $(\alpha\to \beta)$, he should be able to use modus ponens to infer $\beta$''. Those ``rationality postulates'' for knowledge show a striking similarity with the laws of modal logic, so we may attempt to interpret the necessity operator in modal axioms as knowledge operator and try to justify them as axioms for knowledge. A systematic way to justify epistemic axioms is by way of semantics: one tries to find a plausible epistemic interpretation of a semantics for modal logic. Such an interpretation of the possible worlds semantics was proposed by Hintikka ([Hin62]) and adopted widely hence.