A New Definition of Non-monotonic Inference

Nicholas Asher & Yi Mao

In this paper, we introduce two notions of nonmonotonic reasoning: the syntactic notion of Reasonable Consequence |∼ and the semantic notion of Reasonable Entailment |≈. We prove that one is sound and complete relative to the other. There are two layers involved in the definition of |∼: at the bottom is a propositional modal conditional logic which allows unlimited nesting of defaults and embedding into any context analyzable in possible-worlds semantics. The notion of derivibility ⊢ for this level is the classical one. |∼ is built on the top of this system via transforming as many formulas of the form (A > B) as possible into those of the form □(A → B), and ¬(A > B) into (□A → ¬□B) while preserving consistency relative to ⊢. Γ |∼ ϕ iff Γ ⊢ ϕ for all maximal transformations of Γ.

This approach of defining |∼ offers four advantages: First, it nicely handles negated defaults of the form ¬(A > B), to which researchers have not paid much attention but which are important for analyzing various pragmatic phenomena like lexical blocking. Second, thus defined |∼ captures nonmonotonic argument patterns familiar from the artificial intelligence literature such as Defeasible Modus Ponens, Nixon Diamond and all those satisfied by Asher & Morreau (1991). In addition, |∼ is Cumulative. Third, |∼, unlike the inference notion in Default Logic and many other systems of nonmonotonic inference, is insensitive to the order in which premises are considered; although each maximal transformation is order sensitive, the consideration of all possible maximal transformations cancels out the order sensitivity of |∼. There is no extraneous ordering imposed in determining whether |∼ holds between a set of premises and a conclusion. Fourth, compared with the Commensense Entailment notion discussed in Asher & Morreau [1991], the present account is technically simpler. So it provides a better foundation for analyses which depend on nonmonotonic reasoning such as generics (Pelletier & Asher [1997]), common sense obligation (Asher & Bonevac [1997]), and discourse interpretation (Lascarides & Asher [1993]).

Asher, N. and Bonevac, D.: 1997, “Common sense obligation”, in D. Nute(ed.), Defeasible Denotic Logic, Kluwer Academic Publishers.

Asher, N. and Morreau, M.: 1991, “Commonsense entailment: A modal theory of nonmonotonic reasoning”, in J.Mylopoulos and R.Reiter(eds.), Proceedings of the Twelfth International Joint Conference on Artificial Intelligence.

Lascarides, A. and Asher, N,: 1993, “Temporal interpretation, discourse relations and commonsense entailment”, Linguistics and Philosophy 16.

Pelletier, F. J. and Asher, N.: 1997, “Generics and defaults”, in J.van Benthem and A. ter Meulen(eds.), Handbook of Logic and Language, North Holland.

Addresses Nicholas Asher: Department of Philosophy, University of Texas at Austin, USA

E-mail: nasher@mail.la.utexas.edu

Yi Mao : Department of Philosophy, University of Texas at Austin, USA

E-mail: maoyi@mail.utexas.edu