=Paper=
{{Paper
|id=Vol-2540/paper50
|storemode=property
|title=None
|pdfUrl=https://ceur-ws.org/Vol-2540/FAIR2019_paper_35.pdf
|volume=Vol-2540
}}
==None==
https://ceur-ws.org/Vol-2540/FAIR2019_paper_35.pdf
An overview of defeasible entailment
Adam Kaliski1 and Thomas Meyer1
CAIR and University of Cape Town, South Africa
KLSADA002@myuct.ac.za tmeyer@cs.uct.ac.za
1 Introduction and Background
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 ex-
ceptionality. 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 different
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 en-
tailment 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 [9, 10] was one of the earliest attempts at reasoning about typi-
cality and exceptions. Belief revision [1] is another method that allows for a form
of defeasible reasoning. Other approaches to defeasible reasoning include default
logic [11, 12], as well as auto-epistemic logic [8, 4].
Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0)
2 Kaliski and Meyer
2 Problem Statement
The defeasible reasoning framework that this paper uses was defined by Kraus,
Lehmann, and Magidor [5, 7, 6], as well as extensions by Casini et. al. [3, 2]. The
KLM framework is a very strong candidate for defining defeasible entailment
relations, because it defines a number of properties and postulates that isolate a
subset of defeasible entailments as rational. This provides a theoretical bedrock
which further work can use as a foundation to extend or modify the properties
to describe a defeasible entailment useful to them. Another argument in favour
of this framework is that it is computationally well behaved, reducible to classi-
cal entailment. However, there are many defeasible entailments that make little
sense from a human reasoning standpoint that nevertheless satisfy the KLM
properties. This implies that the postulates defining a rational defeasible en-
tailment are too permissive to isolate only those entailments that are useful.
Extending the KLM postulates to isolate only those defeasible entailments that
have meaning would allow for far better analysis of entailments, and allow for a
range of extensions and applications using this framework.
The KLM framework is defined by three foundational papers, and then further
extended by later work. A significant 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 conflict 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 Aims
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. [3], and is included in this paper.
Further extensions in this paper include methods for defining any defeasible en-
tailment relation based on extending or constraining the set of inferences, and
the properties and distinctions between what is termed basic defeasible entail-
ment, and rational defeasible entailment.
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
define a defeasible entailment of interest, or any project that wishes to implement
an already defined defeasible entailment into an application. In particular, the
paper details the semantics, properties, and the algorithms associated with var-
ious monotonic and nonmonotonic entailments for defeasible knowledge bases,
precise definitions for defeasibility and non-monotonicity, and syntax-sensitive
entailments.
An overview of defeasible entailment 3
References
1. Alchourrón, C.E., Gärdenfors, P., Makinson, D.: On the logic of theory change: Par-
tial meet contraction and revision functions. The journal of symbolic logic 50(2),
510–530 (1985)
2. Casini, G., Meyer, T., Moodley, K., Nortjé, R.: Relevant closure: A new form of de-
feasible reasoning for description logics. In: Fermé, Eduardoand Leite, J. (ed.) Log-
ics in Artificial Intelligence. pp. 92–106. Springer International Publishing, Cham
(2014)
3. Casini, G., Meyer, T., Varzinczak, I.: Taking defeasible entailment beyond rational
closure. In: European Conference on Logics in Artificial Intelligence. pp. 182–197.
Springer (2019)
4. Gelfond, M.: On stratified autoepistemic theories. In: AAAI. vol. 87, pp. 207–211
(1987)
5. Kraus, S., Lehmann, D., Magidor, M.: Nonmonotonic reasoning, preferential mod-
els and cumulative logics. Artificial intelligence 44(1-2), 167–207 (1990)
6. Lehmann, D.: Another perspective on default reasoning. Annals of Mathematics
and Artificial Intelligence 15(1), 61–82 (1995)
7. Lehmann, D., Magidor, M.: What does a conditional knowledge base entail? Arti-
ficial intelligence 55(1), 1–60 (1992)
8. Marek, W., Truszczyński, M.: Autoepistemic logic. Journal of the ACM (JACM)
38(3), 587–618 (1991)
9. McCarthy, J.: Circumscription—a form of non-monotonic reasoning. Artificial in-
telligence 13(1-2), 27–39 (1980)
10. McCarthy, J.: Applications of circumscription to formalizing common sense knowl-
edge. Artificial Intelligence 28, 89–116 (1986)
11. Reiter, R.: A logic for default reasoning. Artificial intelligence 13(1-2), 81–132
(1980)
12. Reiter, R., Criscuolo, G.: Some representational issues in default reasoning. Com-
puters & Mathematics with Applications 9(1), 15–27 (1983)