Conditional Objects as Nonmonotonic Consequence Relationships

This paper is an investigation of the relationship between conditional objects obtained as a qualitative counterpart to conditional probabilities, and nonmonotonic reasoning. Roughly speaking, a conditional object can be seen as a generic rule which allows us to get a conclusion provided that the available information exactly corresponds to the "context" part of the conditional object. This gives freedom for possibly retracting previous conclusions when the available information becomes more specific. Viewed as an inference rule expressing a contextual belief, the conditional object is shown to possess all properties of a well-behaved nonmonotonic consequence relation when a suitable choice of connectives and deduction operation is made. Using previous results from Adams' conditional probabilistic logic, a logic of conditional objects is proposed. Its axioms and inference rules are those of preferential reasoning logic of Lehmann and colleagues. But the semantics relies on a three-valued truth valuation first suggested by De Finetti. It is more elementary and intuitive than the preferential semantics of Lehmann and colleagues and does not require probabilistic semantics. The analysis of a notion of consistency of a set of conditional objects is studied in the light of such a three-valued semantics and higher level counterparts of deduction theorem, modus ponens, resolution and refutation are suggested. Limitations of this logic are discussed. The rest of the paper is devoted to studying what remains of the logic in the setting of numerical probability theory.