Booth, Richard (Mahasarakham University) | Casini, Giovanni (University of Pretoria and University of Luxembourg) | Meyer, Thomas Andreas (University of Cape Town) | Varzinczak, Ivan José (Universidade Federal de Rio de Janeiro)
Propositional Typicality Logic (PTL) is a recently proposed logic, obtained by enriching classical propositional logic with a typicality operator. In spite of the non-monotonic features introduced by the semantics adopted for the typicality operator, the obvious Tarskian definition of entailment for PTL remains monotonic and is therefore not appropriate. We investigate different (semantic) versions of entailment for PTL, based on the notion of Rational Closure as defined by Lehmann and Magidor for KLM-style conditionals, and constructed using minimality. Our first important result is an impossibility theorem showing that a set of proposed postulates that at first all seem appropriate for a notion of entailment with regard to typicality cannot be satis- fied simultaneously. Closer inspection reveals that this result is best interpreted as an argument for advocating the development of more than one type of PTL entailment.
We continue recent investigations into the problem of reasoning about typicality. We do so in the framework of Propositional Typicality Logic (PTL), which is obtained by enriching classical propositional logic with a typicality operator and characterized by a preferential semantics à la KLM. In this paper we study different notions of entailment for PTL. We take as a starting point the notion of Rational Closure defined for KLM-style conditionals. We show that the additional expressivity of PTL results in different versions of Rational Closure for PTL — versions that are equivalent with respect to the conditional language originally proposed by KLM.
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We propose a rigorous definition of the notion of typicality, making use of the strict partial order naturally induced among the objects at hand by a given concept. This perspective enables us to transpose in the framework of object determination logic some well-known problems like contextual typicality or contextual inference. Then these problems can be treated with the help of the tools developed in the study of non monotonic logics. We show that it is possible to work in a logical formalism, using a new determination connector: the concept f g, when it exists, denotes the determination of the concept f by the concept g. Relatively to this connector, the relation of typical inference between concepts turns out to satisfy the important property of rational monotony.