INTRODUCTORY REMARKS

In this chapter we further develop the basic three-valued approach to defeasible inference by introducing a notion of “general extension” for Default Logic that falls in line with the analogous notion for nonmonotonic networks introduced in Chap. 2. To an extent, the present work can be viewed as a vindication of the skeptical approach to defeasible inheritance of Horty et al., without at the same time underestimating its indebtedness to the results of Fitting (1990, 1993), Przymusiński (1991), and Przymusińska and Przymusiński (1994) in logic programming.

As will be clear, such a notion of general extension provides a generalization of the notion of extension for Default Logic that was originally due to Reiter (1980). The two versions differ mainly in the way conflicts among defaults are handled. Characteristically, the present notion of general extension allows defaults not explicitly triggered to preempt other defaults. A consequence of the adoption of such a notion of extension is that the resulting framework turns out to be mathematically well behaved, in that the collection of all the general extensions of a default theory has a nontrivial algebraic structure. This fact has two major technical fallouts: First, it turns out that every default theory has a general extension; second, general extensions allow one to define a well-behaved, skeptical relation of defeasible consequence for default theories, satisfying the principles of Reflexivity, Cut, and Cautious Monotonicity formulated by Gabbay (1985) (see Section 1.2).

This chapter deals with the definition of a relation of defeasible consequence based on the notion of general extension introduced in Chap. 3, as well as some alternative developments. The basic notion of defeasible consequence is introduced in Section 4.1: The main result is that the relation so defined satisfies Gabbay's three desiderata (Theorem 4.1.6). Section 4.2 presents alternative developments, namely: (1) the important case of seminormal default theories (which turn out to have a unique minimal general extension); (2) defeasible consequence as based on extensions that are nonminimal (“optimal” in the sense of Manna and Shamir); and (3) a variant of general extensions that, at the cost of a slight complication, avoids certain somewhat counterintuitive results. In Section 4.3 we draw conclusions and comparisons to other approaches, and in Section 4.4 we sketch how to give a “transfinite” version of the present approach. Proofs of selected theorems can be found in Section 4.5.

DEFEASIBLE CONSEQUENCE

As we set out to define a relation of defeasible consequence, it will be convenient to introduce an abbreviated notation for extensions: We will use boldface uppercase Greek letters to stand for triples of sets of defaults, as in Γ = (Γ+, Γ−, Γ*). Similarly, given sequences of sets of defaults and we write Γn for If Γn is a sequence of sets of defaults (n ≤ 0), we write limGn for the unique Γ = (Γ+, Γ−, Γ*) such that Finally, to simplify notation still further, we write just W + ϕ in place of W ∪ {ϕ}.

Logic is an ancient discipline that, ever since its inception some 2,500 years ago, has been concerned with the analysis of patterns of valid reasoning. The beginnings of such a study can be traced back to Aristotle, who first developed the theory of the syllogism (an argument form involving predicates and quantifiers). The field was further developed by the Stoics, who singled out valid patterns of propositional argumentation (involving sentential connectives), and indeed flourished in ancient times and during the Middle Ages, when logic was regarded, together with grammar and rhetoric (the other two disciplines of the trivium), as the foundation of humanistic education. However, the modern conception of logic is only approximately 150 years old, having been initiated in England and Germany in the latter part of the nineteenth century with the work of George Boole (An Investigation of the Laws of Thought, 1854), Gottlob Frege (Begriffsschrift, 1879), and Richard Dedekind (Was sind und was sollen die Zahlen?, 1888). Thus modern symbolic logic is a relatively young discipline, at least compared with other formal or natural sciences that have a long tradition.

Throughout its long history, logic has always had a prescriptive as well as a descriptive component. As a descriptive discipline, logic aims to capture the arguments accepted as valid in everyday linguistic practice.

FIRST-ORDER LOGIC

It was mentioned that first-order logic (henceforth fol) was originally developed for the representation of mathematical reasoning. Such a representation required the establishment of a high standard of rigor, meant to guarantee that the conclusion follows from the premises with absolute deductive cogency. In this respect, fol turned out to be nothing but a stunning success. The account of deductive reasoning provided by fol enjoys a number of important mathematical properties, which can also be used as a crucial benchmark for the assessment of alternative logical frameworks. (The reader interested in an introduction to the nuts and bolts of fol can consult any of the many excellent introductory texts that are available, such as Enderton, 1972.)

From the point of view of abstract consequence relations, fol provides an implementation of the so-called no-counterexample account: A sentence Φ is a consequence of a set Γ of sentences if and only if one cannot reinterpret the (nonlogical part of the) language in which Γ and Φ are formulated in such a way as to make all sentences in Γ true and Φ false. An inference from premises ψ1, …, ψk to a conclusion Φ is valid if Φ is a consequence of {ψ1,…, ψk}, i.e., if the inference has no counterexample.

BACKGROUND AND MOTIVATION

Defeasible inheritance networks were originally developed to gain a sound mathematical understanding of the way inheritance systems store, gain access to, and manipulate taxonomic information with exceptions (a survey can be found in Thomason, 1992). An inheritance network can be identified with a collection of signed links (positive or negative) over a set of nodes. Such links are of the form n1 → n2 or n1 ↛ n2, respectively, where n1 and n2 are nodes in the net. Such nodes are labeled by lexical items referring to categories of individuals. It is convenient to identify nodes with their labels.When the network is to be interpreted defeasibly, a link n1 → n2 represents the fact that objects of category n1 tend to be of category n2, whereas a link of the form n1 ↛ n2 represents the fact that objects of category n1 tend not to be of category n2. A path over a net Γ is a sequence of links from Γ, at most the last one of which is allowed to be negative. So both n1 → n2 → n3 and n1 → n2 ↛ n3 are paths, whereas n1 ↛ n2 → n3 is not. A path is positive or negative according to whether its last link is positive or negative.