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## [ontolog-forum] Nonmonotonic Reasoning

 To: "[ontolog-forum]" "John F. Sowa" Fri, 23 Mar 2012 10:59:42 -0500 <4F6C9DEE.2010604@xxxxxxxxxxx>
 ```The topic of nonmonotonic reasoning has come up in many threads on this list. I'd like to make a few comments about it and cite some references.    (01) First, the term 'nonmonotonic reasoning' is broader than the term 'nonmonotonic logic'. It includes methods of belief revision (or theory revision). Those methods keep the usual classical logic, but they revise a classical theory by adding or deleting axioms to create a new classical theory.    (02) Second, the semantics of every version of nonmonotonic reasoning is based on the semantics of a related classical logic. For example, query languages like SQL and rule-based languages like Prolog use a nonmonotonic logic called *negation as failure* (NAF). Proofs in NAF systems can be defined in several ways:    (03) 1. As nonmonotonic logic, add a new rule of inference to the rules for the corresponding classical logic: if an attempted proof of a statement p fails, assume that p is false.    (04) 2. As belief revision, use the Closed World Assumption (CWA) to revise the theory (axioms + facts) by adding the negation of every statement that uses the same ontology, but is not provable. Then you can use the classical rules of inference, but with a much larger set of axioms.    (05) 3. Method #2 makes a huge revision to the axioms in one step. But it's possible to achieve the same effect by incrementally adding axioms one at a time as needed: whenever a NAF step is used in a proof, add the new negated assumption to the set of axioms for the current theory. As a result, the same proof can be carried out from the enlarged set of axioms, but with just classical inference rules.    (06) Method #3 is an incremental method of belief revision that can use exactly the same inference engine as the corresponding nonmonotonic logic. The only difference is that it saves each negated statement and adds it to the list of axioms. It shows that the difference between nonmonotonic logic and belief revision can just be considered a difference in terminology (at least for NAF logics).    (07) For a general discussion of these issues, see the following book on belief revision, which emphasizes the relationships among nonmonotonic logics, classical logics, and belief revision:    (08) Makinson, David (2005) Bridges from Classical to Nomonotonic Logic, King's College Publications, London.    (09) Makinson also happens to be the M in the AGM axioms for belief revision. The following review of his book summarizes some of the issues:    (010) http://www.maths.manchester.ac.uk/~hykel/work/bridges-review/    (011) See the end of this note for a few excerpts from the review.    (012) For related information, see the first page of Makinson's own web site, which has comments about his other articles:    (013) http://sites.google.com/site/davidcmakinson/    (014) For a good review article with 110 references to the literature on belief revision, see    (015) Peppas, Pavlos (2008) Belief revision, in F. van Harmelen, V. Lifschitz, & B. Porter, Handbook of Knowledge Representation, Elsevier, Amsterdam, pp. 317-359.    (016) Note the last paragraph in the following excerpts from the review of Makinson's book. The cases for which the "unwelcome assumption" can be avoided can be found by walking through the lattice of all possible theories that can be stated in the given classical logic.    (017) John _____________________________________________________________________    (018) Source: http://www.maths.manchester.ac.uk/~hykel/work/bridges-review/    (019) The book [develops] the logico-mathematical apparatus required to construct nonmonotonic inference operations out of the classical, monotonic one...    (020) Its key message is that nonmonotonic logics, as in fact many other so-called non-standard logics, are not to be taken as alternatives to the classical one, in the way, that is, in which intuitionistic logic is regarded as opposed to classical logic. Both at the object- and at the meta- level the importance of classical logic is beyond dispute in nonmonotonic logics. Engaging in nonmonotonic logics means aiming at extending classical logic, rather than replacing it tout court...    (021) Chapters 2-4 constitute the core of the book in which the author presents three canonical constructions to obtain nonmonotonic consequence relations out of monotonic ones. The author follows a general pattern for introducing such constructions. The starting point is always the mathematical machinery provided by classical consequence. It is fundamental to notice that the nonmonotonic consequence relations discussed in the book all rest on the very same language as classical propositional logic. The second step consists in introducing a bridge consequence relation. Such a (monotonic) bridge is then used to construct a nonmonotonic consequence relation...    (022) The main idea of Chapter 2 - Using additional background assumptions - is simple and powerful: distinguishing among local and background assumptions. It is reasonable to believe that when making inferences we, intelligent reasoners, do not consider all the information available to us, that is the premisses of our logical argument, as "equally conspicuous". In particular it seems appropriate to distinguish between the information embodying "the current premises" of our argument - the set of local assumptions - from the set of assumptions which we commit to somehow tacitly - the set of background assumptions (or expectations)...    (023) Allowing the set of background assumptions to vary - essentially in a consistency preserving way - with the set of local assumption amounts to enabling nonmonotonic reasoning, as captured by the default assumption consequence relation, fully investigated in Section 2.2...    (024) However, when considering default assumption consequence we face, in the author's terminology, a basic Dilemma. Makinson proves, in fact, that if the set of background assumptions is not closed under classical consequence, then it is not language invariant...    (025) As to the latter unwelcome consequence, a number of solutions are discussed... All these variations allow the set of background assumptions to be closed under classical consequence yet avoiding the collapse of default assumption consequence into the classical one. Section 2.3 (and the corresponding exercises and problems) illustrates this at length...    (026) _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ Config Subscr: 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 join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1J    (027) ```
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