TARK Conference 2013 Conference Paper
- Katarina Britz
- Ivan Varzinczak
ditionals, but, more fundamentally, it relates to where and how the notion of normality is used in such statements. Indeed, in a KLM defeasible statement α |∼ β, the normality spotlight is somewhat put on α, as though normality was a property of the premise and not of the conclusion. Whether the situations in which β holds are normal or not plays no role in the reasoning that is carried out. In the original KLM framework, normality is also linked to the premise as a whole, rather than its constituents. Technically this meant one could not refer directly to normality of a sentence in the scope of logical operators. This limitation is overcome by taking a (modal) conditional approach à la Boutilier [5] — the resulting conditional logics are sufficiently general to allow for the expression of a number of different forms of defeasible reasoning. However, the emphasis remains on the defeasibility of arguments, or of conditionals. In this paper we investigate a related, but incomparable, notion which we refer to as defeasible modes of inference [11]. 1 These amount to defeasible versions of the traditional notions of actions, obligations, knowledge and beliefs, to name a few, as studied in modal logics. For instance, in an action context, one can say that normally the outcome of a given action a is α. However we may also want to state that the outcome of a is usually (or normally) α, which is different from the former statement. To see why, the first statement says that in the most normal worlds, the result of performing the action a is always α, whereas in the second one it is in the most normal situations resulting from a’s execution that α holds — regardless of whether the situation in which the claim is uttered is normal or not. For a concrete example, assume one arrives at a dark room and wants to toggle the light switch. Exceptionally, the light will not turn on. This can be either because the light bulb is blown (the current situation is abnormal) or because an overcharge was caused while switching the light (the action behaves abnormally). In the former case, the normality of the situation, or state, before the action is assessed, whereas in the latter the relative normality of the situation is assessed against all possible outcomes. Here we are interested in the formalization of the latter type of statement, where it becomes important to shift the notion of normality from the premise of an inference to the effect of an action, and, importantly, use it in the scope of other logical constructors. Our next example concerns obligations and weaker versions thereof. There is a subtle difference between stating Nonmonotonic logics are usually characterized by the presence of some notion of ‘conditional’ that fails monotonicity. Research on nonmonotonic logics is therefore largely concerned with the defeasibility of argument forms and the associated normality (or abnormality) of its constituents. In contrast, defeasible modes of inference aim to formalize the defeasible aspects of modal notions such as actions, obligations and knowledge. In this work we enrich the standard possible worlds semantics with a preference ordering on worlds in Kripke models. The resulting family of modal logics allow for the elegant expression of defeasible modalities. We also propose a tableau calculus which is sound and complete with respect to our preferential semantics.