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
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.