Implication-with-possible-exceptions

We introduce an implication-with-possible-exceptions and define validity of rules-withpossible-exceptions by means of the topological notion of a full subset. Our implication-with-possibleexceptions characterises the preferential consequence relation as axiomatized by Kraus, Lehmann and Magidor [Kraus, Lehmann, and Magidor, 1990]. The resulting inference relation is non-monotonic. On the other hand, modus ponens and the rule of monotony, as well as all other laws of classical propositional logic, are valid-up-to-possible exceptions. As a consequence, the rules of classical propositional logic do not determine the meaning of deducibility and inference as implication-without-exceptions. §0. Introduction. Although non-monotonic logic arose as a subdiscipline of artificial intelligence, the potential of the subject for the field of formal logic should not be underestimated, as non-monotony is connected with the most important difficulties in formalizing practical reasoning. Practical reasoning seems to be governed by general principles that function as rules-with-possible-exceptions. Hence, mathematical axioms, when interpreted in the usual way, will not suffice to describe the underlying calculus. In this paper, the expressive power of the language used to formulate principles of reasoning has been extended to enable the interpretation of formal axioms as valid-up-to-possible-exceptions. The core of the system is the notion of a full subset, a topological generalization of intuitions stemming from Euclidean geometry (a subject that preceded Aristotelean logic). This notion gives rise to a topological semantics of defeasible implication in a natural way. A sound and complete axiomatization is established, relying for the proof on a result of [Kraus, Lehmann, and Magidor, 1990] (in fact, we find the same axiom system). The notion of a full subset is used to define semantics of nested implicational statements. This semantics is associated with two issues. In the first place, we think of an imaginary person that handles (some) rules of inference as rules-withpossible-exceptions. With the semantics defined in this paper, the person turns out to obey-up-to-possible-exceptions every law of classical propositional logic, including modus ponens and the rule of monotony. Thus, it is shown that it is possible for a nonmonotonic formalism to obey-up-to-possible-exceptions the rule of monotony. A much more interesting conclusion is that the rules of classical propositional logic do apparently not determine the usual interpretation of implication (and inference) as implication-without-exceptions. Received July 6, 1999. c © 0000, Association for Symbolic Logic 0022-4812/00/0000-0000/$1.00