Gabriele SACCO a,1, Loris BOZZATO b and Oliver KUTZ ba- Free University of Bozen-Bolzano, Italyb- Fondazione Bruno Kessler, ItalyORCiD ID: Gabriele Sacco https://orcid.org/0000-0001-5613-5068, Loris Bozzatohttps://orcid.org/0000-0003-1757-9859, Oliver Kutzhttps://orcid.org/0000-0003-1517-7354Abstract. Defeasible reasoning is a kind of reasoning where some generalisations may not be valid in all circumstances, that is general conclusions may fail in some cases. Various formalisms have been developed to model this kind of reasoning, which is characteristic of common-sense contexts. However, it is not easy for a modeller to choose among these systems the one that better fits its domain from an ontological point of view.In this paper we first propose a framework based on the notions of exceptionalism and defeasibility in order to be able to compare formalisms and reveal their ontological commitments. Then, we apply this framework to compare four systems, showing the differences that may occur from an ontological perspective.
4.2. Circumscription The key idea of circumscription is introducing a class abnormality predicates Abi, which is minimised. Therefore, we can represent e -generalisation as x((P(x)Ab(x)) Qx) which can be interpreted as if something is a P and is not abnormal then it is a Q too. So, in this case we have two kinds of generalisation even if they are not distinguished at the object level. That is, it is not introduced a new form of generalisation, but only a
id: 59f4729d73b899f621f188b1b01e2fc9 - page: 9
In terms of Denition 3 above, the justication here is given by being recognised as abnormal. In fact, are considered exceptions those individuals which necessarily are abnormal. This is due to the mechanism of reducing the extension of Ab as much as possible, which represent the assumption that we are always in the most normal possible condition. In fact, in circumscription we order the interpretations according to the extension of the Ab predicates, where the smaller the better. For example, consider the knowledge base containing the propositions x(Bird(x) Ab(x) Flies(x)), Bird(tweety), Bird(chilly), Flies(chilly), (tweety 6= chilly). Chilly is correctly considered abnormal, since it does not y and we can conclude that Tweety ies because we consider the minimal models, and a model where Ab contains only Chilly is more normal than one containing also Tweety. 4.3. Default Logic
id: 693089576a71906fa1e7df336bcd2604 - page: 10
The intuition behind default logic is to add a set of rules that allow to specify which assumptions should be made, given that they are consistent. A rule has the following form: h : /i where is a sentence corresponding to the prerequisite for the application of the rule, is one that correspond to the justication for the application of the rule and is the conclusion of the rule. The idea is that if we know and is consistent with what we know, we can conclude . These rules are the ones that allow to represent defeasible generalisations. In fact, default rules do not necessarily apply to all the individuals, but only to those which is consistent to do so. Therefore, exceptions to a default rule are those individual for which the justication does not apply. Interestingly, this is an inverse perspective with respect to our view above, since in this case the justication is needed for applying the defeasible generalisation, rather than to individuate the exceptions. For instance, co
id: f68d26bd31dae72200125ae969c7b7ee - page: 10
In this case we have only a rule which says that if something is a bird and it is not inconsistent to assume that it ies, then we can conclude that it ies. We can see that this is true only for Tweety, since we know that Chilly does not y and so we have in our expansion that Tweety ies.
id: ae56ea6e9d75daf35994b4b133e8bb71 - page: 10