The logic of knowledge was first investigated as a modal logic by
Hintikka in his book Knowledge and belief (1962). We shall only
describe the knowledge calculus. He introduces the modal operator
(read `a knows that'), and its dual 
, defined as 
.
The semantics is obtained by the analogous reading of as: `it is
true in all possible worlds compatible with a's knowledge that'.
The propositional logic of (similar to 
) turns out to be
S4, that is M+Ax. 3; but there are some complexities over
quantification. (The last chapter of the book contains another
excellent account of the overall problem of quantification in modal
contexts.) This analysis of knowledge has been criticized in various
ways (Chisholm 1963, Follesdal 1967) and Hintikka has replied in
several important papers (1967b, 1967c, 1972). The last paper
contains a review of the different senses of `know' and the extent to
which they have been adequately formalized. It appears that two
senses have resisted capture. First, the idea of `knowing how',
which appears related to our `can'; and secondly, the concept of
knowing a person (place, etc.) when this means `being acquainted
with' as opposed to simply knowing who a person is.
In order to translate the (propositional) knowledge calculus
into `situation' language, we introduce a three-place predicate into
the situation calculus termed `shrug'. 
, where p
is a person and 
and 
are situations, is true when, if p is
in fact in situation , then for all he knows he might be in
situation . That is to say, is an epistemic alternative
to , as far as the individual p is concerned--this is
Hintikka's term for his alternative worlds (he calls them model-sets).
Then we translate 
, where q is a proposition of
Hintikka's calculus, as 
, where 
is the fluent which translates q. Of course we have to supply
axioms for shrug, and in fact so far as the pure knowledge-calculus
is concerned, the only two necessary are
and
that is, reflexivity and transitivity.
Others of course may be needed when we add tenses and other machinery to the situation calculus, in order to relate knowledge to them.