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
. 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
and
such
that an agent knows both
and
and yet fails
to know
. In general, from some epistemic statements one cannot
deduce any other epistemic statement. Given the information that an
agent's knowledge includes a set
of sentences, in reality we
can never infer reliably that the agent knows a sentence from the
deductive closure
of
with respect to a deductive
system
(except for those already in
), even if we suppose
to be very weak (but not degenerate in the sense that
.) 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
and
, he should be able to use modus
ponens to infer
''. 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.