next up previous
Next: Tense Logics Up: DISCUSSION OF LITERATURE Previous: Modal Logic

Logic of Knowledge


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 tex2html_wrap_inline1391 (read `a knows that'), and its dual tex2html_wrap_inline1393 , defined as tex2html_wrap_inline1395 . The semantics is obtained by the analogous reading of tex2html_wrap_inline1391 as: `it is true in all possible worlds compatible with a's knowledge that'. The propositional logic of tex2html_wrap_inline1391 (similar to tex2html_wrap_inline1542 ) 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'. tex2html_wrap_inline1409 , where p is a person and tex2html_wrap_inline1413 and tex2html_wrap_inline1415 are situations, is true when, if p is in fact in situation tex2html_wrap_inline1415 , then for all he knows he might be in situation tex2html_wrap_inline1413 . That is to say, tex2html_wrap_inline1413 is an epistemic alternative to tex2html_wrap_inline1415 , 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 tex2html_wrap_inline1429 , where q is a proposition of Hintikka's calculus, as tex2html_wrap_inline1433 , where tex2html_wrap_inline1435 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.

next up previous
Next: Tense Logics Up: DISCUSSION OF LITERATURE Previous: Modal Logic

John McCarthy
Mon Apr 29 19:20:41 PDT 1996