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## Re: [ontolog-forum] Thing and Class

 To: "[ontolog-forum]" "John F. Sowa" Sat, 13 Sep 2008 00:10:43 -0400 <48CB3D43.20904@xxxxxxxxxxx>
 ```Pat and Chris M.,    (01) We usually agree on technical details, but we often interpret their implications differently. But the following is a very clear technical point:    (02) JFS>> So Kripke's semantics is a *proper subset* of Dunn's semantics.    (03) PH> Wrong as stated, since Dunn's semantics has no constructions > corresponding to Kripkean worlds.    (04) CM> It seems that one should be able to convert any Kripke model > into a Dunn model but, off the top of my head, it's not clear to > me that one can do so *uniquely*.    (05) There is an isomorphism from a Kripke model to a Dunn model:    (06) 1. Given a Kripke model (K,R,Phi) and for each world w in K, let M (a Hintikka-style model set) be the set of propositions true in w. Call M the facts of w.    (07) 2. Let L be the set of propositions that are necessarily true in w. Call L the laws of w.    (08) 3. Theorem: If the accessibility relation R is true of any pair of worlds R(w1,w2) then the laws L1 of world w1 are a subset of the facts M2 of world w2.    (09) This is a unique mapping in which every axiom, definition, and proof that is true in Kripke semantics is also true in Dunn's semantics. Just identify each Kripke w with a Dunn pair (M,L). By the way, this mapping holds for Kripke's "normal" modal logics, and for "nonnormal" logics such as deontic logic, in which the laws of a world w are not a subset of the facts of *the same* w.    (010) Pat claimed that Kripke's version is more "usable", but everything that can be done with Kripke's version can be done with Dunn's version. But if the two systems are isomorphic, the next question is how I could claim that Dunn's version is a superset of Kripke's.    (011) CM> You don't get much more from Dunn except a very rarefied notion > of "law".    (012) PH> And I have never seen any useful observation or insight arise > from Dunn's work that was not already clear in the Kripke framework.    (013) My answer is that Dunn's version makes it possible to do things that cannot be done with Kripke's:    (014) 1. Unlike Kripke's "arbitrary" relation R, Dunn's version allows R to be derived from the choice of laws and facts.    (015) 2. Instead of being arbitrary, the choice can be made by principled arguments about which statements are more "entrenched" than others. For example, the ontology is usually considered more fundamental than just an offhand observation or assertion.    (016) 3. Pat made the claim that nobody in AI is using Dunn's semantics. But I claim that the entire Description Logic community is using Dunn's semantics, but they don't know it. For example, they often say that DL's are a version of modal logic, but they don't explain where that modal effect comes from. But with Dunn's semantics, it's obvious:    (017) a) The T-Box of definitions stated in some DL takes priority over the A-Box of simple assertions, since the A-Box must be consistent with the T-Box.    (018) b) That means the T-Box contains laws, and the A-Box contains the ordinary facts: a classic example of Dunn's semantics.    (019) PH> So, apply this 'insight' to temporal descriptions. Possible worlds > are times, here, and the accessibility relation is the time-ordering > relation. What "laws" vary from time to time, and how would someone > reconstruct the time-ordering (past-present-future) from how these > "laws" change? And even if this can be done, what advantages does > all this give over using good old, you know, *times*, things like > [4pm on Friday 12 September 200    (020) To answer the last question first, an explicit time coordinate is very useful, and the choice of using it is independent of whether you choose a Kripke or a Dunn model.    (021) Although temporal logic with the operators Always and Sometimes is very similar to a modal logic of Necessary and Possible, the addition of a temporal Next operator forces an another constraint on accessibility. But in some temporal theories, everything that has happened is considered to be necessarily true for the future. If you choose that version, then all the facts of the past (with their time stamps) become laws for all future times. That causes Dunn's accessibility relation to give you the usual temporal order.    (022) But a very important application is to the semantics of a database (or a knowledge base that includes instances as well as an ontology of types). For the moment, consider an ordinary relational DB:    (023) 1. The tuples in all the DB tables constitute the ground-level facts that describe a possible world.    (024) 2. The DB constraints, which are checked at each update, are the laws L.    (025) 3. The deductive closure of the ground-level facts and the constraints constitute the complete set of facts M.    (026) 4. The accessibility relation R is defined by the permissible updates that are consistent with the constraints.    (027) If the DB constraints are fixed, then you get the modal logic S5. But in most practical DB systems, the database administrator often defines new tables with new constraints and add more constraints on old tables. When the DB administrator adds a new constraint C, it must be true for the current state of the tables, and the DB system will ensure that it remains true in all future updates. But it might not be possible "roll back" the data to a previous state that could violate C.    (028) PH> Who are the 'so many of them' who use S5...    (029) It is common in many theoretical papers about agents. For example,    (030) "Nowadays, the term Dynamic Epistemic Logic... is used to refer to formalisms that add a special class of actions -- epistemic actions -- to the standard logic S5 for knowledge."    (031) From "Multi-Agent systems" by W. van der Hoek & M. Wooldridge, in _Handbook of Knowledge Representation_, Elsevier, Amsterdam, 2008, pp. 887-929.    (032) In a system in which constraints can be changed (which is very common for practical DB systems), the modal logic S5 is too strong, and the appropriate logic is S4.    (033) JFS>> Kripke only developed his semantics for propositional modal logic.    (034) CM> Exsqueeze me?! I direct you to...    (035) I apologize. I was too hasty.    (036) PH> Dunn-STYLE... rejects possible worlds as entities and > re-conceptualizes them in a meta-theory of sentential relationships.    (037) Note the isomorphism above. As Chris M. said, the entities that Kripke called worlds are arbitrary. Identifying each w with the pair (M,L) derived above can hardly be called a "rejection". It preserves every axiom and theorem of Kripke's, and it supports additional options and features that Kripke doesn't support.    (038) PH> Whereas to insist on not referring to possible worlds...    (039) The phrase 'possible world' is a metaphor for arbitrary entities, and neither Dunn nor I have any objection to it as a metaphor. But the above examples show that the pair (M,L) corresponds to some widely used collections of statements in AI and DB theory.    (040) PH> Its like (in fact, it is literally) saying that instead of > using clock times we should instead refer to sets of propositions > that are true at those times.    (041) More precisely, it is like using a temporal database that keeps track of times *and* the data and constraints that are true at those times. There is a profitable industry that does that.    (042) John    (043) _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ Subscribe/Config: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/ Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx Shared Files: http://ontolog.cim3.net/file/ Community Wiki: http://ontolog.cim3.net/wiki/ To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx    (044) ```
 Current Thread Re: [ontolog-forum] Thing and Class, (continued) Re: [ontolog-forum] Thing and Class, Matthew West Re: [ontolog-forum] Thing and Class, Azamat Re: [ontolog-forum] Thing and Class, John F. Sowa Re: [ontolog-forum] Thing and Class, Pat Hayes Re: [ontolog-forum] Thing and Class, Chris Partridge Re: [ontolog-forum] Thing and Class, Pat Hayes Re: [ontolog-forum] Thing and Class, John F. Sowa Re: [ontolog-forum] Thing and Class, Chris Partridge Re: [ontolog-forum] Thing and Class, John F. Sowa Re: [ontolog-forum] Thing and Class, Pat Hayes Re: [ontolog-forum] Thing and Class, John F. Sowa <= Re: [ontolog-forum] Thing and Class, Pat Hayes Re: [ontolog-forum] Thing and Class, John F. Sowa Re: [ontolog-forum] Thing and Class, John F. Sowa Re: [ontolog-forum] Thing and Class, Christopher Menzel Re: [ontolog-forum] Thing and Class, John F. Sowa Re: [ontolog-forum] Thing and Class, Pat Hayes Re: [ontolog-forum] Thing and Class, Christopher Menzel Re: [ontolog-forum] Thing and Class, John F. Sowa Re: [ontolog-forum] Thing and Class, Chris Menzel Re: [ontolog-forum] Thing and Class, Christopher Menzel