eralreasoningthanstandardlogics,whichdealwithuniversalstatements.Forexample,Nonmonotoniclog.pdf

Nonmonotonic LogicsNonmonotonic logics are used to formalize plausible reasoning. They allow more gen-eral reasoning than standard logics, which deal with universal statements. For example,standard logics can easily represent the argument:All men are mortalSocrates is a man||||||Therefore, Socrates is mortal( (x)(Man(x) ) Mortal(x)); Man(Socrates) / Mortal(Socrates))but cannot represent reasoning such as:Birds typically yTweety is a bird||||||Therefore, Tweety (presumably) ies.Such arguments are characteristic of commonsense reasoning, where default rules andthe absence of complete information are prevalent.The most salient feature of nonmonotonic reasoning is that the conclusion of a nonmo-notonic argument may not be correct. For example, if Tweety is a penguin, it is incorrectto conclude that Tweety ies. Nonmonotonic reasoning often requires jumping to a conclu-sion and subsequently retracting that conclusion as further information becomes available.Thus, as the set of assumptions grows, the set of conclusions (theorems) may shrink. Thisreasoning is called nonmonotonic in contrast to standard logic which is monotonic: as onesset of assumptions grows, ones set of theorems grows as well. (Formally, a system is mo-notonic if for any two theories A and B, whenever A is a subset of B, the theorems of A area subset of the theorems of B.)All systems of nonmonotonic reasoning are fundamentally concerned with the issue ofconsistency: ensuring that conclusions drawn are consistent with one another and with theassumptions.Default Logic (Reiter 1980) introduces a default rule, an inference rule consisting of anassumption, an appeal to the consistency of some formula, and a conclusion. For example,the rule Birds typically y could be written as1 2Bird(x):Fly(x)|||||Fly(x)which reads: if x is a bird, and it is consistent that x ies, then conclude that x ies.Default rules must be applied with care, since con icting default rules could cause in-consistency if used together.