The process of determining whether or not a statement, or query, is inferred by a knowledge base is entailment. Classical entailment is regular deduction, which holds that if 1. All humans are mortal 2. Socrates is a human then logically Socrates is mortal. This form of reasoning is, however, completely monotonic, which does not allow for exceptions to stated rules, and therefore also does not allow for the concept of typicality. This inherently limits the expressivity of a given expert system, since reality often contains exceptions. Consider now the addition of a defeasible rule (along with a regular rule) 3. Humans are typically not philosophers 4. Socrates is a philosopher Defeasibility is the introduction of typicality, and the associated concept of exceptionality. A defeasible rule is one that allows itself to be broken, by stating that the rule is typically the case, rather than always the case. However, adding this semantic extension to classical logic raises an important problem regarding reasoning, because a knowledge base with defeasible rules cannot be reasoned about in the same way as a classical knowledge base. Furthermore, whereas classical, monotonic entailment is unique, there are arbitrarily many di erent defeasible entailments, and therefore what a defeasible entailment relation will infer from a given defeasible knowledge base is dependent on which entailment it is. Given the above knowledge base regarding Socrates, the question is can we conclude that Socrates is mortal? The answer is dependent on whether or not we are using a prototypical or presumptive entailment. A prototypical entailment is conservative, and draws fewer inferences, and will therefore conclude that Socrates is an atypical human, and therefore cannot conclude that he is mortal. On the other hand, a presumptive entailment, which draws as many inferences as it reasonably can, will not see any clashes regarding mortality, and infer as much as possible, including that he is mortal.
There have been a number of approaches to characterizing defeasible reasoning.
Circumscription [
Kaliski and Meyer
The defeasible reasoning framework that this paper uses was de ned by Kraus,
Lehmann, and Magidor [
The KLM framework is de ned by three foundational papers, and then further extended by later work. A signi cant barrier of entry to those who wish to either apply or extend this framework is to understand it. The initial KLM papers are both dense and also con ict with one another in certain respects. For example, the meaning of symbols were changed between papers without comment. There is therefore an argument to be made that an overview that compiles the current state of the framework in a single document would enable future research groups to far more easily utilize the KLM framework. 3
A main aim of this paper is how best to extend these properties to isolate exactly
the defeasible entailments that infer in a useful, or understandable way. Work
on this has already been done by Casini et. al. [
An overarching aim of this work is to provide a single point of reference for any project that wishes to extend, constrain, or otherwise modify the framework to de ne a defeasible entailment of interest, or any project that wishes to implement an already de ned defeasible entailment into an application. In particular, the paper details the semantics, properties, and the algorithms associated with various monotonic and nonmonotonic entailments for defeasible knowledge bases, precise de nitions for defeasibility and non-monotonicity, and syntax-sensitive entailments.