A non-monotonic logic is a formal logic whose conclusionrelation is not monotonic. In other words, non-monotonic logics are devised to capture and represent defeasible inferences (cf. defeasible reasoning), i.e., a kind of inference in which reasoners draw tentative conclusions, enabling reasoners to retract their conclusion(s) based on further evidence.[1] Most studied formal logics have a monotonic entailment relation, meaning that adding a formula to a theory never produces a pruning of its set of conclusions. Intuitively, monotonicity indicates that learning a new piece of knowledge cannot reduce the set of what is known. A monotonic logic cannot handle various reasoning tasks such as reasoning by default (conclusions may be derived only because of lack of evidence of the contrary), abductive reasoning (conclusions are only deduced as most likely explanations), some important approaches to reasoning about knowledge (the ignorance of a conclusion must be retracted when the conclusion becomes known), and similarly, belief revision (new knowledge may contradict old beliefs).
Abductive reasoning is the process of deriving a sufficient explanation of the known facts. An abductive logic should not be monotonic because the likely explanations are not necessarily correct. For example, the likely explanation for seeing wet grass is that it rained; however, this explanation has to be retracted when learning that the real cause of the grass being wet was a sprinkler. Since the old explanation (it rained) is retracted because of the addition of a piece of knowledge (a sprinkler was active), any logic that models explanations is non-monotonic.
Reasoning about knowledge
If a logic includes formulae that mean that something is not known, this logic should not be monotonic. Indeed, learning something that was previously not known leads to the removal of the formula specifying that this piece of knowledge is not known. This second change (a removal caused by an addition) violates the condition of monotonicity. A logic for reasoning about knowledge is the autoepistemic logic.
Belief revision
Belief revision is the process of changing beliefs to accommodate a new belief that might be inconsistent with the old ones. In the assumption that the new belief is correct, some of the old ones have to be retracted in order to maintain consistency. This retraction in response to an addition of a new belief makes any logic for belief revision to be non-monotonic. The belief revision approach is alternative to paraconsistent logics, which tolerate inconsistency rather than attempting to remove it.
^Strasser, Christian; Antonelli, G. Aldo. "Non-Monotonic Logic". plato.stanford.edu/index.html. Stanford Encyclopedia of Philosophy. Retrieved 19 March 2015.
References
Bidoit, N.; Hull, R. (1989). "Minimalism, justification and non-monotonicity in deductive databases". Journal of Computer and System Sciences. 38 (2): 290–325. doi:10.1016/0022-0000(89)90004-4.
Brewka, G. (1991). Nonmonotonic Reasoning: Logical Foundations of Commonsense. Cambridge University Press. ISBN978-0-521-38394-3.
Cadoli, M.; Schaerf, M. (1993). "A survey of complexity results for non-monotonic logics". Journal of Logic Programming. 17 (2–4): 127–60. doi:10.1016/0743-1066(93)90029-G.
Gabbay, D.M. (1985). "Theoretical foundations for non-monotonic reasoning in expert systems". In Apt, K.R. (ed.). Logics and Models of Concurrent Systems. Vol. 13. NATO ASI Series, Series F: Computer and Systems Sciences: Springer. pp. 439–457. doi:10.1007/978-3-642-82453-1_15. ISBN978-3-642-82453-1.
Ginsberg, M.L., ed. (1987). Readings in Nonmonotonic Reasoning. Morgan Kaufmann. ISBN978-0-934613-45-3.
Horty, J.F. (2001). "Nonmonotonic Logic". In Goble, Lou (ed.). The Blackwell Guide to Philosophical Logic. Wiley. ISBN978-0-631-20692-7.
Łukaszewicz, W. (1990). Non-Monotonic Reasoning. Ellis-Horwood. ISBN978-0-13-624446-2.
Lundberg, C.G. (2000). (PDF). Journal of Economic Psychology. 21 (6): 691–709. doi:10.1016/S0167-4870(00)00027-1. S2CID 11723465. Archived from the original (PDF) on 2017-09-07.
Makinson, D. (2005). Bridges from Classical to Nonmonotonic Logic. College Publications. ISBN9781904987000.
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This article includes a list of general references but it lacks sufficient corresponding inline citations Please help to improve this article by introducing more precise citations June 2008 Learn how and when to remove this template message A non monotonic logic is a formal logic whose conclusion relation is not monotonic In other words non monotonic logics are devised to capture and represent defeasible inferences cf defeasible reasoning i e a kind of inference in which reasoners draw tentative conclusions enabling reasoners to retract their conclusion s based on further evidence 1 Most studied formal logics have a monotonic entailment relation meaning that adding a formula to a theory never produces a pruning of its set of conclusions Intuitively monotonicity indicates that learning a new piece of knowledge cannot reduce the set of what is known A monotonic logic cannot handle various reasoning tasks such as reasoning by default conclusions may be derived only because of lack of evidence of the contrary abductive reasoning conclusions are only deduced as most likely explanations some important approaches to reasoning about knowledge the ignorance of a conclusion must be retracted when the conclusion becomes known and similarly belief revision new knowledge may contradict old beliefs Contents 1 Abductive reasoning 2 Reasoning about knowledge 3 Belief revision 4 See also 5 Notes 6 References 7 External linksAbductive reasoning EditAbductive reasoning is the process of deriving a sufficient explanation of the known facts An abductive logic should not be monotonic because the likely explanations are not necessarily correct For example the likely explanation for seeing wet grass is that it rained however this explanation has to be retracted when learning that the real cause of the grass being wet was a sprinkler Since the old explanation it rained is retracted because of the addition of a piece of knowledge a sprinkler was active any logic that models explanations is non monotonic Reasoning about knowledge EditIf a logic includes formulae that mean that something is not known this logic should not be monotonic Indeed learning something that was previously not known leads to the removal of the formula specifying that this piece of knowledge is not known This second change a removal caused by an addition violates the condition of monotonicity A logic for reasoning about knowledge is the autoepistemic logic Belief revision EditBelief revision is the process of changing beliefs to accommodate a new belief that might be inconsistent with the old ones In the assumption that the new belief is correct some of the old ones have to be retracted in order to maintain consistency This retraction in response to an addition of a new belief makes any logic for belief revision to be non monotonic The belief revision approach is alternative to paraconsistent logics which tolerate inconsistency rather than attempting to remove it See also Edit Philosophy portalLogic programming Negation as failure Stable model semantics Rational consequence relationNotes Edit Strasser Christian Antonelli G Aldo Non Monotonic Logic plato stanford edu index html Stanford Encyclopedia of Philosophy Retrieved 19 March 2015 References EditBidoit N Hull R 1989 Minimalism justification and non monotonicity in deductive databases Journal of Computer and System Sciences 38 2 290 325 doi 10 1016 0022 0000 89 90004 4 Brewka G 1991 Nonmonotonic Reasoning Logical Foundations of Commonsense Cambridge University Press ISBN 978 0 521 38394 3 Brewka G Dix J Konolige K 1997 Nonmonotonic Reasoning An Overview CSLI Lecture Notes Vol 73 Stanford CSLI publications ISBN 9781881526834 Cadoli M Schaerf M 1993 A survey of complexity results for non monotonic logics Journal of Logic Programming 17 2 4 127 60 doi 10 1016 0743 1066 93 90029 G Donini F M Lenzerini M Nardi D Pirri F Schaerf M 1990 Nonmonotonic reasoning Artificial Intelligence Review 4 3 163 210 doi 10 1007 BF00140676 S2CID 23575942 Gabbay D M 1985 Theoretical foundations for non monotonic reasoning in expert systems In Apt K R ed Logics and Models of Concurrent Systems Vol 13 NATO ASI Series Series F Computer and Systems Sciences Springer pp 439 457 doi 10 1007 978 3 642 82453 1 15 ISBN 978 3 642 82453 1 Ginsberg M L ed 1987 Readings in Nonmonotonic Reasoning Morgan Kaufmann ISBN 978 0 934613 45 3 Horty J F 2001 Nonmonotonic Logic In Goble Lou ed The Blackwell Guide to Philosophical Logic Wiley ISBN 978 0 631 20692 7 Lukaszewicz W 1990 Non Monotonic Reasoning Ellis Horwood ISBN 978 0 13 624446 2 Lundberg C G 2000 Made sense and remembered sense Sensemaking through abduction PDF Journal of Economic Psychology 21 6 691 709 doi 10 1016 S0167 4870 00 00027 1 S2CID 11723465 Archived from the original PDF on 2017 09 07 Makinson D 2005 Bridges from Classical to Nonmonotonic Logic College Publications ISBN 9781904987000 Marek W Truszczynski M 1993 Nonmonotonic Logics Context Dependent Reasoning Springer ISBN 978 3 662 02906 0 Abdallah A Nait 1995 The Logic of Partial Information Springer ISBN 978 3 642 78160 5 External links EditAntonelli G Aldo Non monotonic logic In Zalta Edward N ed Stanford Encyclopedia of Philosophy Non monotonic logic at PhilPapers Non monotonic logic at the Indiana Philosophy Ontology Project Retrieved from https en wikipedia org w index php title Non monotonic logic amp oldid 1140643092, wikipedia, wiki, book, books, library,
