From owner-cse663-fa08-list@LISTSERV.BUFFALO.EDU Mon Sep 15 13:31:09 2008 Date: Mon, 15 Sep 2008 13:30:59 -0400 From: "William J. Rapaport" Subject: 663: Modal Operators vs. Modal Predicates To: CSE663-FA08-LIST@LISTSERV.BUFFALO.EDU ------------------------------------------------------------------------ Subject: Modal Operators vs. Modal Predicates ------------------------------------------------------------------------ Nicolas asked why (some) epistemic logics used a modal operator K subscripted with an agent's name (Ka) instead of a binary predicate K that takes an agent and a proposition as arguments. As I indicated in my answer in class, there are some epistemic logics that do that. Here are some of the differences between such systems (they are more than merely notational variants): 1. Multiple Ka operators allow for each operator to have its own logic. So if one agent's knowledge is best represented logically by one epistemic logic (as we'll see soon, there are lots of them) and a second agent's knowledge is best represented logically by a different one, then using separate operators allows for this. By contrast, a single binary predicate K would have to have the same logic for all agents. 2. "Ka" is an operator, not a predicate. But "K" would have to be a predicate that takes a proposition as an argument; that goes well beyond the facilities of classical logic. However, there is a controversy over whether "Ka" should be considered a predicate or an operator. For some discussion of this, see: Des Rivieres, Jim, & Levesque, Hector J. (1986), "The Consistency of Syntactical Treatments of Knowledge", in Joseph Y. Halpern (ed.), Theoretical Aspects of Reasoning About Knowledge: Proceedings of the 1986 Conference (Los Altos, CA: Morgan Kaufmann): 115-130. and the references cited therein. For my review of this article, see: http://www.jstor.org/stable/info/2274548