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Bridges from Classical to Nonmonotonic Logic
David Makinson
Department of Computer Science
King’s College London
Contents
Preface
Acknowledgements
1.Introduction
1.1. We are all nonmonotonic
1.2. Recalling some features of classical consequence
1.3. Some misunderstandings and a habit to suspend
1.4. Three ways of getting more conclusions out of your premises
2. Using background assumptions
2.1. From classical consequence to pivotal assumptions
2.2. From pivotal assumptions to default assumptions
2.3. Specializations and generalizations
2.4. Review and explore
3. Restricting the set of valuations
3.1. From classical consequence to pivotal valuations
3.2. From pivotal valuations to default valuations
3.3. Specializations and generalizations
3.4. Review and explore
4. Using additional rules
4.1. From classical consequence to pivotal rules
4.2. From pivotal rules to default rules
4.3. Generalizations and variants
4.4. Review and explore
5. Connections with probabilistic inference
5.1. Basic concepts and axioms for probability
5.2. Probabilistic characterizations of classical consequence
5.3. Supraclassical probabilistic consequence relations
5.4. Bringing probabilistic and qualitative inference closer together
5.5. Review and explore
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6. Some brief comparisons
6.1. Relations with belief revision
6.2. Links with update, counterfactuals, and conditional directives
6.3. Some representation theorems for consequence relations
6.4. Review and explore
Appendix: Proof of Theorem 4.3-1.
Glossary of Special Symbols
Answers to Selected Exercises
References
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Preface
Who are we writing for?
This book is directed to all those who have heard of nonmonotonic reasoning and would like
to get a better idea of what it is all about. What are its driving ideas? In what ways does it
differ from classical logic? How does it relate to probability? More generally, how does it
behave and how can it be used? We will try to answer these questions as clearly as possible,
without undue technicality on the one hand, nor vagueness or hand waving on the other.
It is written for the student in a classroom, the instructor teaching a course, but also for the
solitary reader. The lone reader faces a particularly difficult task, for there is nobody to turn to
when the going gets tough. In sympathy, we try to help as far as possible: explaining all points
fully, saying some things in more than one way to get the point across, providing
recapitulations at the end of each chapter, giving exercises and selected solutions. These will
be useful to all readers, but especially for those working alone.
What does the reader need to know already?
Tackling a subject such as this does require a minimal background, and in honesty we must
make it clear from the beginning what is needed. To those without any grounding in classical
logic at all, who may out of curiosity have picked up the volume, we say: put it down, and go
to square one. Get the rudiments of classical propositional logic (alias truth-functional logic)
under your belt, and then come back. Otherwise you will be like someone trying to learn
about algebra before knowing any arithmetic, counterpoint without sol-fa, or style without the
elements of grammar.
For that purpose, students of computer science, mathematics, philosophy or linguistics should
take the introductory course in logic offered by their department, and read whatever textbook
the instructor recommends. If working alone with an interest in computation or mathematics,
try chapters 6-8 of James Hein Discrete Structures, Logic, and Computability (Boston: Jones
and Bartlett, second edition 2002). Those whose background is in the humanities should try
L.T.F. Gamut Logic, Language and Meaning: Volume I. Introduction to Logic (Chicago
University Press, 1991). And for those lucky enough to find it, there is the author’s out-of-
print Topics in Modern Logic (London: Methuen 1973; chapters 1 and 3).
In the Introduction of this book we do review some features of classical propositional logic,
but with a special purpose. We bring out certain aspects that tend to be passed over in most
presentations, but are essential for unfolding our theme. In particular, we explain some very
general concepts that reveal themselves not only in classical consequence but also in other
contexts. Among these are the general notion of a closure operation or relation, the property of
compactness, and the idea of a Horn rule.
The reader should also be familiar with a few simple mathematical tools. These are of two
kinds. First: some ‘working set theory’ – basic operations on sets, such as finite intersection,
union and relative complement, indexed families of sets, infinite intersection and union;
ordered pairs, Cartesian products, the treatment of relations and functions as sets of ordered
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pairs, and the notion of a well-ordering. Second: mathematical induction as a method of proof,
in both its simple and course-of-values forms over countable sets. This is needed in order to
establish routine properties of the inductively defined sets that will be encountered, beginning
with the set of formulae of propositional logic.
These tools are often taught along with elementary logic, and also as part of first courses in
computing and discrete mathematics. For the reader who lacks them, there is no alternative to
going back to an elementary course or to reading the first few chapters of an introductory
book. A good old standby is Seymour Lipschutz Set Theory and Related Topics (New York:
McGraw-Hill Education, second edition 1998), especially the first five chapters.
Alternatively, chapters 1-4 of the book of Hein mentioned above. For those who would like to
do it in elegance and style, the first twelve chapters of Paul Halmos’ Naïve Set Theory (New
York: van Nostrand, 1960) are unbeatable.
The proofs of some results in this book, for example Theorem 2.2–2, appeal to maximality
principles such as Zorn’s Lemma. Familiarity with these principles, as set out say in chapter 9
of Lipschutz or chapters 14-17 of Halmos, is needed for full understanding of those proofs,
and is in any case a premium investment. However, even without them the student may get a
partial understanding: read the theorem for the finite case only, and recall that for a finite set,
if some subset of it has a property then it is always included in at least one maximal subset
with the same property.
In the author’s opinion, all these things should be taught in high school, and in some
privileged places they are. They are fun to learn, and indispensable in the electronic world.
But sadly, there are also students who are allowed to go through an entire undergraduate
education without ever being exposed to them.
So, for those who do not have the necessary minimal background we say goodbye – and hope
to see you again later. For those who are ready, we say: welcome and read on!
Main themes
From the outside, nonmonotonic logic is often seen as a rather mysterious affair. Even from
the inside, it can appear to lack unity, with multiple systems proposed by as many authors
going in different directions. The few available textbooks tend to perpetuate this impression.
Our main purpose is to take some of the mystery out of the subject and show that it is not as
unfamiliar as may at first sight seem. In fact, it is easily accessible to anybody with the
minimal background that we have described in classical propositional logic and basic
mathematical tools - provided certain misunderstandings and a tenacious habit, both signalled
in the first chapter, are put aside.
As we shall show, there are logics that act as natural bridges between classical consequence
and the principal kinds of nonmonotonic logic to be found in the literature. These logics,
which we call paraclassical, are very simple to define and easy to study. They provide three
main ways of getting more out of your premises than is allowed by strict deduction, that is, by
good old classical consequence. In other words, they are principled ways of creeping,
crawling or jumping to conclusions. Like classical logic, they are perfectly monotonic, but
they already display some of the distinctive features of the nonmonotonic systems that they
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prefigure, as well as providing easy conceptual passage to them. They give us, in effect, three
main paths from the classical homeland to nonmonotonic shores.
The book examines the three one by one. We begin with the simplest among them, whose
moving idea is to use additional background assumptions along with current premises. Then
we consider ways of getting the same result by excluding certain classical valuations. And
finally we examine a third means to the same end, by adding rules alongside the premises.
In each case we obtain a monotonic bridge system that may be rendered nonmonotonic in a
natural and meaningful way. In fact, each of the bridge systems leads to quite a range of
nonmonotonic ones, and this is where the diverse systems of the literature show themselves.
But they are no longer a disorderly crowd. They fall into place as variants, natural and indeed
to be expected, of three basic ideas.
The book then turns to the subtle question of the relation between logic and probability –
more specifically, between classical logic, probabilistic inference, and nonmonotonic
reasoning. On the one hand, there are several different ways of characterizing classical
consequence in probabilistic terms. On the other hand, we can also use probability to define
supraclassical consequence operations, both monotonic and nonmonotonic. They turn out to
be quite different in certain important respects from the qualitative operations of earlier
chapters; but as we show there are also ways of bringing them closer together.
There are obvious resemblances between some of the ways of generating nonmonotonic
inference relations and constructions that have been used for other purposes – for example, the
logics of belief revision and update, as well as counterfactual conditionals and conditional
directives. The last chapter of the book discusses the links between these different kinds of
‘belief management’ and their residual differences. It also presents sample representation
theorems for some of the principal systems of nonmonotonic logic. We do not attempt
anything like a complete coverage of all the different representations that may be found in the
literature; there are too many, and still growing. Two strategic examples are selected, one with
a complete proof and the other with its central construction merely sketched.
Topics Not Dealt With
We owe it to the reader – and especially to the instructor who may be thinking of using the
book – to say clearly what topics, associated with nonmonotonic reasoning, are not covered.
All inference relations studied are defined over a purely propositional language using only
Boolean connectives. They could be extended to first-order classical logic, but most of the
interesting questions appear to arise already at the propositional level. In this subject, to work
in a first-order language is to increase overheads for little added value in understanding.
We will not be examining logics in which classical connectives are reinterpreted in a non-
Boolean manner. Systems of logic programming with negation using various forms of
‘negation as failure’ are sometimes presented in this way although, as we will suggest in
chapter 4, they are more transparently seen as rules about propositions in a fragment of the
classical language. The study of logic programming has now become a world of its own, with
an emphasis on questions of computational complexity, which are not at the centre of our
concern. For a recent review of logic programming using the ‘answer set semantics’ see
Gelfond and Leone (2002); for an overview of various approaches to logic programming with
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negation see Brewka, Dix and Konolige (1997) chapters 6-7; and for an encyclopaedic
account see the book of Baral (2003).
Nor will we be examining languages that add non-classical connectives to the usual classical
ones. In particular, we will not be considering autoepistemic logics. These are formed by
adding to the Boolean connectives a special kind of modal operator whose ‘introspective’
reading engenders nonmonotonic behaviour. Despite their correspondence with certain
maverick default-rule logics under translations due to Konolige and others, they are quite
different from the mainstream ones, for reasons that we will explain in chapter 4. The reader
interested in studying autoepistemic logic is referred to chapter 4.2 of Brewka, Dix and
Konolige (1997) or, in more detail, the overview of Konolige (1994).
There are also two approaches to nonmonotonic reasoning in which logical connectives have
little role to play. They are the theory of defeasible inheritance nets, and the abstract theory of
argument defeat. In both cases the focus is on the notion of one path (in a net) or argument (in
a discussion) defeating or undercutting another, and on ascertaining the final effect of
complex patterns of iterated defeat. Here too, the issues at stake are rather special, and not
discussed in this volume. For defeasible inheritance nets, the reader is referred to the overview
of Horty (1994). For argument defeat, consult the recent survey of Prakken and Vreeswijk
(2001) and the papers of García and Simari (to appear) and Bench-Capon (to appear).
Thus this book does not pretend to be anything near a comprehensive treatment of all the
investigations that have gone under the name of nonmonotonic logic. But it does present some
very central ideas, structuring them in a coherent perspective, and seeks to explain them as
clearly as possible.
Strategy of Presentation
When the author first began planning this text, he followed engrained habits and sought for
maximal generality. "We have available a multitude of different formal approaches to
nonmonotonic reasoning", he thought. "So in order to put some order into the affair, we need
to find the most general schema possible, under which they all fall. We can then present the
different accounts as so many special cases of the general schema".
However, it quickly became clear that this strategy is hopeless. It does work quite well when
focussing on the kinds of consequence relation that are generated, for these may be classified
according to the sets of regularity conditions that they satisfy. One may thus begin with a
minimal set of conditions, defining a very broad class of consequence relations, and gradually
add to the conditions, defining narrower classes. Indeed, this was the strategy followed in the
author’s paper ‘General theory of cumulative inference’ (1989) and survey ‘General Patterns
in Nonmonotonic Reasoning’ (1994). But if, as in this book, one is primarily interested in the
principal modes of generation of consequence relations, the strategy makes little sense. The
modes differ abruptly from each other; and when one abstracts upon them in an attempt to
obtain a ‘most general mode’, one comes up with formulations that are opaque to intuition,
clumsy to work with, and mathematically close to empty.
For this reason, a totally different principle guides the organization of this book. It is the
principle of theme and variations. For each of the three main ways of ‘going nonmonotonic’ –
namely, by adding background assumptions, restricting the set of valuations, and adding
background rules – we begin by providing a paradigm formulation, and then sketch some of
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the many specializations, variations, and generalizations. A few variations already appear at
the level of the monotonic bridge systems, and they multiply at the nonmonotonic level. They
are most easily understood as an open family of relatives of a central exemplar, rather than as
particular cases of an all-embracing definition.
For the mathematician this may go rather against the grain, but in the present enterprise it is
the only reasonable way to proceed. And after all, even mathematicians are accustomed to this
procedure in certain cases. For example, when giving an overview of different approaches to
axiomatizing set theory, nobody at the present state of play would try to give a most abstract
definition of what in general constitutes a set theory, and then fit in the Zermelo-Fraenkel
version, Quine’s New Foundations, and so on as special cases. The standard procedure is to
delineate a few broad lines of approach in conceptual terms, present exemplars of each, and
finally sketch some of their many variations. That is the way in which we proceed in this
book.
Review and explore sections
Each chapter ends with a ‘Review and explore’ section. It contains three packages to help the
reader review the material covered and to go further into the literature. First, a recapitulation
of the essential message of the chapter. Second, a checklist of the key concepts, both formal
and informal, that were introduced there. The concepts are not explained all over again, but
are simply listed for checking off at review time. The reader can relocate the formal
definitions and informal explanations by looking for the corresponding italicised terms in the
main text. Third, a short selection of texts for further reading. Some of these cover essentially
the same ground as the section itself, albeit from a different angle, in more detail or with
particular attention to aspects that we have not dwelt on. Some go further. In general, these
will be ‘bite size’ – short papers or chapters. Occasionally, a more extended text is mentioned.
What is the best way to read this book?
Do it pencil in hand; scribble in the margins. Take nothing on faith; check out the assertions;
find errors (and communicate them to the author); pose questions.
Notoriously, the best test of whether one has understood a definition is to be able to identify
positive and negative examples. Nor has one understood a theorem very well if unable to
apply it to straightforward cases and recognize some of its more immediate consequences.
Without this one may have the illusion of understanding, but not the real thing.
For this reason, the book includes exercises. Many of them ask readers to verify claims made
in the flow of the text. To help the brave loner maintain the discipline of working out the
exercises, and also to take some of the load off instructors, selected exercises, marked by
asterisks, are given answers at the end of the text.
As well as exercises, there are some problems. They are more demanding. In general they
require more than checking positive and negative instances of a definition or straight
applications with a couple of steps of argument. Their solutions may need perceptive guessing
plus the ability to prove or disprove the guesses. Both the art of making good guesses and the
ability to check them out are acquired skills, and grow with practice.
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Finally, projects are longer-term tasks. Whether to attempt any of them depends on the
interests and time of the reader, and the goals of the instructor. In general, they involve
consulting items from the literature and working on them. When suggesting a project we will
usually indicate a specific text as entry-point.
Those strapped for time and wanting to cover essentials only, could skim or even omit section
2.3 on specializations and generalizations of the basic pattern of that chapter, and likewise
sections 3.3, 4.3 – provided they promise the author to come back some day later! They could
also omit the rather technical section 5.4. Below that, they risk not getting the general picture.
For the instructor
The author’s experience is that in a graduate class, teaching can take place at average of one
section per hour, without counting section 1.1 and the brief ‘review and explore’ sections.
This adds up to about 20 hours of instruction time for the entire book, or about 15 hours for a
minimal version like the one mentioned above.
Of course, this will vary with the level and background of the students. In particular, those
whose background in classical logic did not include much exposure to the notion of a
consequence relation/operation or concepts such as compactness may need considerably more
than one hour for the revision in section 1.2. With them in mind, there are particularly many
exercises in that section. On the other hand, the sections entitled ‘specializations and
generalizations’ provide instructors with a starting point for more detailed treatments of their
favourite special topics, with open schedules.
Conventions
Theorems are numbered by their section, followed by a dash and a digit. For example, the first
explicitly displayed theorem in the book occurs in section 1.3 and is numbered Theorem 1.3–
1. However, only major results are displayed and numbered in this way. Many lesser facts are
simply stated in the flow of the text, and the attentive reader should mark them as they are
found. The same applies to definitions. Only a few key ones are given special display; the
others are within the text, with their central terms italicized.
Generally speaking, the notation used is fairly standard. One exception: we use plain brackets
rather than angular ones for ordered pairs – thus (x,y) instead of x,y. In chapter 4 we also
simplify the usual way of writing default rules. Symbols special to nonmonotonic logic are
recalled in a glossary at the end.
In this preface, we have given references in full. Likewise in the suggestions for further
reading at the end of each section. But in the main text, they are called by author name and
date, e.g. Gabbay (1985), and are answered with full details in the reference list at the end of
the book.
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Acknowledgements
As the basic ideas underlying this book were developed, they were articulated in workshop
and conference presentations, journal publications, and the classroom.
The author would like to thank the organisers of the following meetings and occasions, large
and small, for the opportunity to air ideas that went into this book. They include: Workshop
on Cognitive Economics, Porquerolles, September 2001; KR-02 Special Session on Belief
Change, Toulouse, 19-21 April 2002; ASL European Summer Meeting Special Session on
Nonmonotonic Logic, Münster, 4-8 June 2002; Prague International Colloquium on Formal
Epistemology, 10-13 September 2002; King's College London Department of Computer
Science Colloquium, 9 October 2002; Manchester University Mathematics Department
Colloquium, 18 November 2002; University of Leipzig Department of Computer Science, 14
May 2003; Dresden University Department of Computer Science, 19 May 2003; University
of Buenos Aires Department of Computer Science, 17 July 2003; University of Bahía Blanca
Department of Computer Science, 07 August 2003; Augustus de Morgan Conference on
Agents and Knowledge Representation (King’s College, 03 November 2003) ; University of
New South Wales Department of Computer Science Colloquium, 02 April 2004.
The general vision of nonmonotonic logic developed here was outlined, in a form suitable for
professional logicians, in Makinson (2003a). A thumbnail sketch, adapted for an audience of
economists interested in logic, was given in Makinson (2003b). A stripped-down version of
the present text, without such things as exercises, answers and ‘review and explore’ sections,
will appear as the overview Makinson (to appear).
Much of the text was written while the author taught an MSc course on Nonmonotonic
Reasoning in the Department of Computer Science, King's College London in the spring
semesters of 2003 and 2004. Thanks to all the students and auditors for their questions, which
helped make the presentation more transparent; particularly to Audun Stolpe and Robert
Schubert for their perceptive queries, Xavier Parent and Simon Speed for finding bugs in
exercises and providing some model answers; and David Gabelaia for comments and
important observations.
Many other people made helpful comments on particular points during the gestation process.
They include Gerd Brewka, Björn Bjurling, Marc Denecker, Michael Freund, Donald Gillies,
Dov Gabbay, Jörg Hansen, Daniel Lehmann, Joao Marcos, Wlodek Rabinowicz, Hans Rott,
Karl Schlechta, and an anonymous referee for the Logic Journal of the IGPL.
The book is dedicated to the memory of my former co-author Carlos Alchourrón, who died in
January 1996. Despite his fascination with the logic of belief revision he distrusted its
neighbour nonmonotonic logic, and I could never convince him otherwise. While writing this
book, I realized that it is a continuation of our conversations.
