ABDUCTIVE REASONING
SYNTHESE LIBRARY
STUDIES IN EPISTEMOLOGY,
LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE
Editors-in-Chief:
VINCENT F. HENDRICKS, Roskilde University, Roskilde, Denmark
JOHN SYMONS, University of Texas at El Paso, U.S.A.
Honorary Editor:
JAAKKO HINTIKKA, Boston University, U.S.A.
Editors:
DIRK VAN DALEN, University of Utrecht, The Netherlands
THEO A.F. KUIPERS, University of Groningen, The Netherlands
TEDDY SEIDENFELD, Carnegie Mellon University, U.S.A.
PATRICK SUPPES, Stanford University, California, U.S.A.
JAN WOLEŃSKI, Jagiellonian University, Kraków, Poland
VOLUME 330
ABDUCTIVE REASONING
LOGICAL INVESTIGATIONS INTO
DISCOVERY AND EXPLANATION
by
ATOCHA ALISEDA
National Autonomous University of Mexico
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To Rodolfo
Contents
Dedication
Foreword
v
xi
Part I Conceptual Framework
1. LOGICS OF GENERATION AND EVALUATION
1
Introduction
2
Heuristics: A Legacy of the Greeks
3
Is There a Logic of Discovery?
4
Karl Popper and Herbert Simon
5
Logics for Scientific Methodology
6
Discussion and Conclusions
3
3
4
6
12
21
24
2. WHAT IS ABDUCTION?
1
Introduction
2
What is Abduction?
3
The Founding Father: C.S. Peirce
4
Philosophy of Science
5
Artificial Intelligence
6
Further Fields of Application
7
A Taxonomy for Abduction
27
27
28
35
37
39
43
46
viii
Part II
ABDUCTIVE REASONING
Logical Foundations
3. ABDUCTION AS LOGICAL INFERENCE
1
Introduction
2
Logic: The Problem of Demarcation
3
Abductive Explanatory Argument: A Logical Inference
4
Abductive Explanatory Inference: Structural Characterization
5
Discussion and Conclusions
4. ABDUCTION AS COMPUTATION
1
Introduction
2
Semantic Tableaux
3
Abductive Semantic Tableaux
4
Computing Abductions with Tableaux
5
Further Logical and Computational Issues
6
Discussion and Conclusions
Part III
53
53
54
64
75
89
95
95
98
106
110
118
129
Applications
5. SCIENTIFIC EXPLANATION
1
Introduction
2
Scientific Explanation as Abduction
3
Discussion and Conclusions
135
135
135
146
6. EMPIRICAL PROGRESS
1
Introduction
2
Kuipers’ Empirical Progress
3
Empirical Progress in (Abductive) Semantic Tableaux
4
Discussion and Conclusions
153
153
156
160
165
7. PRAGMATISM
1
Introduction
2
Pragmatism
3
Abduction and Epistemology
4
Pragmatism Revisited
5
Discussion and Conclusions
167
167
168
170
174
177
Contents
ix
8. EPISTEMIC CHANGE
1
Introduction
2
Abduction as Epistemic Change
3
Semantic Tableaux Revisited
4
Discussion and Conclusions
179
179
180
186
197
References
203
Author Index
219
Topic Index
223
Foreword
Many types of scientific reasoning have long been identified and recognised
as supplying important methodologies for discovery and explanation in science,
but many questions regarding their logical and computational properties still
remain controversial. These styles of reasoning include induction, abduction,
model-based reasoning, explanation and confirmation, all of them intimately
related to problems of belief revision, theory development and knowledge assimilation. All of these have been addressed both in the philosophy of science
and in the fields of artificial intelligence and cognitive science, but their respective approaches have been in general too far apart to leave room for an integrated
account.
My general concern in this book is scientific discovery and explanation. The
point of departure is to address an old but still unsettled question: does scientific
methodology have a logic? There have been several conflicting stances about
how to pose this question in the first place - not to mention the solutions - each of
which rests on its own assumptions about the scope and limitations of scientific
methodology and also in their attitude to logic. This question has been posed
within several research traditions and for a variety of motivations and purposes,
and that fact has naturally shaped the way it has been tackled. In this respect
the answer is often already implicit in the very posing of the question itself,
which in any case is not a yes-no matter.
Thus a closer look at aspects of this question is in order. While it is clear
to everyone that scientific practice indeed involves both processes of discovery
and explanation, the first point of disagreement is concerned with the proper
scope of the methodology of science. In the predominant view of XXth century philosophy of science, creativity and discovery is simply out of bounds
for philosophical reflection, and thus the above question is focused mainly on
issues of explanation and testing. It is well known that great philosophers and
mathematicians have been brilliant exceptions in the study of discovery in science, but their contributions have set no new paradigms in the methodology of
xii
ABDUCTIVE REASONING
science, and instead have inspired research in cognitive science and artificial
intelligence. There is indeed a rapidly growing research on issues of computational scientific discovery, which is full of computer programs which challenge
the philosophical claim that the so called ‘context of discovery’ does not allow
for any formal treatment. Thus, in this tradition discovery is part of scientific
methodology, on a par with explanation.
As for the place of logic, there are also a variety of instances of what ‘logic’
amounts to in posing the above question. On the one hand, in XXth century
positivist philosophy of science deductive logic was the predominant formal
framework to address issues of explanation and evaluation. From the start it
was clear that inductive logic had too little on offer for a proper logical analysis
of scientific methodology, and this provided the leeway for alternative proposals, such as Popper’s conjectures and refutations logical method. Computer
oriented research, on the other hand, identifies logic with ‘pattern seeking methods’, a notion which fits very well their algorithmic and empirical approach to
the above question. In any case, the use of the term ‘logic’ regarding discovery
has had little to do with providing logical foundations for their programs, either
as conceived in the mathematical logical tradition or as in artificial intelligence
logical research, both of which regard inference as the underlying logical notion. It is clear that classical logic cannot account for any kind of ampliative
reasoning, and so far as the logical tools developed in artificial intelligence are
concerned, not many of them have been applied to issues of discovery in the
philosophy of science.
Aim and Purpose
In this book I offer a logical analysis of a particular type of scientific reasoning, namely abduction, that is, reasoning from an observation to its possible
explanations. This approach naturally leads to connections with theories of
explanation and empirical progress in the philosophy of science, to computationally oriented theories of belief change in artificial intelligence, and to the
philosophical position known as Pragmatism, proposed by Charles Peirce, to
whom the term abduction owes its name. The last part of the book is concerned
with all these applications.
My analysis rests on several general assumptions. First of all, it assumes
that there is no single logical method in scientific practice in general, and with
respect to abduction in particular. In this my view is pluralistic. To be sure,
abduction is not a new form of inference. It is rather a topic-dependent practice
of explanatory reasoning, which can be supported by various notions of inference, classical and otherwise. By this assumption, however, I do not claim it is
possible to provide a logical analysis for all and every part of scientific inquiry.
In this respect, my enterprise is modest and has no pretensions that it can offer
either a logical analysis of great scientific discoveries, or put forward a set of
FOREWORD
xiii
logical systems that would provide general norms to make new discoveries. My
aim is rather to lay down logical foundations in order to explore some of the
formal properties under which new ideas may be generated and evaluated. The
compensation we gain from this very modest approach is that we can gain some
insight into the logical features of some parts of the scientific discovery and explanation processes. This is in line with a well-known view in the philosophy of
science, namely that phenomena take place within traditions, something which
echoes Kuhn’s distinction between normal and revolutionary science. Hence,
another general assumption is that a logical analysis of scientific discovery of
the type I propose is for normal science, not denying there may be a place for
some other kind of logical analysis of revolutionary science, but clearly leaving
it out of the scope of this enterprise.
Another general assumption is that the methodological distinction between
the contexts of discovery and justification is an artificial one. It can be dissolved
if we address abduction as a process rather than as a ready-made product by
itself for us to study. Historical as well as computationally oriented research
of scientific discoveries show very clearly that new ideas do not just come out
of the blue (even though there may be cases of sudden flashes of insight), and
that the process of discovery often involves a lot of explanation, evaluation and
testing on the way, too. So, it may be possible to address the justification part of
science all by itself, as contemporary philosophy has done all along, but once
discovery issues come into play, an integrated account of both is needed.
Content Description
This book is divided into three parts: (I) Conceptual Framework, (II) Logical
Foundations, and (III) Applications, each of which is briefly described in what
follows.
In part I, the setting for the logical approach taken in this book is presented.
One the one hand, chapter 1 offers a general overview of the logics of discovery
enterprise as well as the role of logic in scientific methodology, both in philosophy of science and in the fields of artificial intelligence and cognitive science.
The main argument is that logic should have a place in the normative study in
the methodology of science, on a par with historical and other formal computational approaches. Chapter 2 provides an overview of research on abduction,
showing that while there are general features and in most cases the main inspiration comes from the American pragmatist, Charles S. Peirce, each approach
has taken a different route. To delineate our subject more precisely, and create some order, a general taxonomy for abductive reasoning is then proposed.
Several forms of abduction are obtained by instantiating three parameters: the
kind of reasoning involved (e.g., deductive, statistical), the kind of observation
triggering the abduction (novelty, or anomaly with respect to some background
theory), and the kind of explanations produced (facts, rules, or theories).
xiv
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In part II, the logical foundations of this enterprise are laid down. In chapter
3, abduction is investigated as a notion of logical inference. It is shown that this
type of reasoning can be analyzed within various kinds of logical consequence as
the underlying inference, namely as classical inference (backwards deduction),
statistical or as some type of non-monotonic inference. The logical properties
of these various ‘abductive explanatory kinds’ are then investigated within the
‘logical structural analysis’, as proposed for non-monotonic consequence relations in artificial intelligence and dynamic styles of inference in formal semantics. As a result we can classify forms of abduction by different structural rules.
A computational logic analysis of processes producing abductive inferences is
then presented in chapter 4, using and extending the mathematical framework
of semantic tableaux. I show how to implement various search strategies to generate various forms of abductive explanations. Our eventual conclusion for this
part is that abductive processes should be our primary concern, with abductive
explanatory inferences as their secondary ‘products’.
Part III is a confrontation of the previous analysis and foundations with existing themes in the philosophy of science and artificial intelligence. In particular,
in chapter 5, I analyze the well-known Hempelian models for scientific explanation (the deductive-nomological one, and the inductive-statistical one) as
forms of abductive explanatory arguments, the ultimate products of abductive
reasoning. This then provides them with a structural logical analysis in the
style of chapter 3. In chapter 6, I address the question of the dynamics of empirical progress, both in theory evaluation and in theory improvement. I meet
the challenge made by Theo Kuipers [Kui99], namely to operationalize the
task of ‘instrumentalist abduction’, that is, theory revision aiming at empirical
progress. I offer a reformulation of Kuipers’ account of empirical progress into
the framework of (extended) semantic tableaux, in the style of chapter 4, and
show that this is indeed an appealing method to account for empirical progress
of some specific kind of empirical progress, that of lacunae.
The remaining two chapters have a common argument, namely that abduction
may be viewed as a process of epistemic change for belief revision, an idea which
connects naturally to the notion of abduction in the work of Charles Peirce, and
that of belief revision in the work of Peter Gärdenfors, thus suggesting a direct
link between philosophy and artificial intelligence. In chapter 7, I explore
the connection between abduction and pragmatism, as proposed by Peirce,
showing that the former is conceived as an epistemic procedure for logical
inquiry, and that it is indeed the basis for the latter, conceived as a method
of philosophical reflection with the ultimate goal of generating ‘clear ideas’.
Moreover, I argue that abduction viewed in this way can model dynamics of
belief revision in artificial intelligence. For this purpose, an extended version
of the semantic tableaux of chapter 4 provides a new representation of the
FOREWORD
xv
operations of expansion, and contraction, all of which shapes the content of
chapter 8.
Acknowledgements
Abductive Reasoning presents the synthesis of many pieces that were published in various journals and books. Material of the following earlier publications, recent publications or publications shortly to appear has been used, with
the kind permission of the publishers:
‘Sobre la Lógica del Descubrimiento Cientı́fico de Karl Popper’, Signos
Filosóficos, supplement to number 11, vol. VI, January–June, pp. 115–130.
Universidad Autónoma Metropolitana. México. 2004. (Chapter 1).
‘Logics in Scientific Discovery’, Foundations of Science, Volume 9, Issue 3,
pp. 339–363. Kluwer Academic Publishers. 2004. (Chapter 1).
Seeking Explanations: Abduction in Logic, Philosophy of Science and Artificial Intelligence. PhD Dissertation, Philosophy Department, Stanford University. Published by the Institute for Logic, Language and Computation (ILLC),
University of Amsterdam, 1997. (ILLC Dissertation Series 1997–4). (Chapters
2, 3, 4, 5 and 8).
‘Mathematical Reasoning Vs. Abductive Reasoning: An Structural Approach’, Synthese 134: 25–44. Kluwer Academic Press. 2003. (Chapter 3).
‘Computing Abduction in Semantic Tableaux’, Computación y Sistemas:
Revista Iberoamericana de Computación, volume II, number 1, pp. 5–13.
Centro de Investigación en Computación (CIC), Instituto Politécnico Nacional,
México. 1998. (Chapter 4).
‘Abduction in First Order Semantic Tableaux’, together with A. Nepomuceno. Unpublished Manuscript. 2004. (Chapter 4).
‘Lacunae, Empirical Progress and Semantic Tableaux’, to appear in R. Festa,
A. Aliseda and J. Peijnenburg (eds). Confirmation, Empirical Progress, and
Truth Approximation (Poznan Studies in the Philosophy of the Sciences and the
Humanities, vol. 83), pp. 141–161. Amsterdam/Atlanta, GA: Rodopi. 2005.
(Chapter 6).
‘Abduction as Epistemic Change: A Peircean Model in Artificial Intelligence’. In P. Flach and A. Kakas (eds). Abductive and Inductive Reasoning:
Essays on their Relation and Integration, pp. 45–58. Kluwer Academic Publishers, Applied Logic Series. 2000. (Chapters 7 and 8).
‘Abducción y Pragmati(ci)smo en C.S. Peirce’. In Cabanchik, S., etal. (eds.).
El Giro Pragmático en la Filosofı́a Contemporánea. Gedisa, Argentina. 2003.
(Chapter 7).
The opportunity to conceive this book was provided by a series of research
stays as a postdoc at the Philosophy Department in Groningen University, during
the spring terms of three consecutive years (2000–2002). I thank my home institution, for allowing me to accept the kind invitation by Theo Kuipers. In Gronin-
xvi
ABDUCTIVE REASONING
gen I joined his research group Promotion Club Cognitive Patterns (PCCP) and
had the privilege to discuss my work with its members and guests, notably with
David Atkinson, Alexander van den Bosch, Erik Krabbe, Jeanne Peijenburg,
Menno Rol, Jan Willem Romeyn, Rineke Verbrugge and John Woods. I am
especially grateful to Theo for making me realize of the fertility of logical research on abduction, particularly when it is guided by questions raised in the
philosophy of science.
In the Instituto de Investigaciones Filosóficas at the Universidad Nacional
Autónoma de México, the Seminario de Razonadores (‘Reasoners Seminar’)
has been a wonderful forum over the past six years for the discussion of reasoning themes amongst philosophers, logicians, mathematicians and computer
scientists. I wish to mention the following member colleagues, all of which
have contributed to my thinking about abduction: José Alfredo Amor, Axel
Barceló, Maite Ezcurdia, Claudia Lorena Garcı́a, Francisco Hernández, Raymundo Morado, Ana Bertha Nova, Samir Okasha, Silvio Pinto, Marina Rakova
and Salma Saab.
I also wish to thank the following people for inspiring discussions and generous encouragement towards my abductive research: Hans van Ditmarsch, Peter
Flach, Donald Gillies, Michael Hoffmann, Lorenzo Magnani, Joke Meheus,
Ulianov Montaño, Angel Nepomuceno, Jaime Nubiola, León Olivé, Sami
Paavola, David Pearce, Ana Rosa Pérez Ransanz, Vı́ctor Rodrı́guez, Vı́ctor
Sánchez Valencia†, Matti Sintonen, Ambrosio Velasco, Rodolfo Vergara and
Tom Wasow. In particular, I am grateful to Johan van Benthem, for his guidance during my doctoral research at Stanford. Finally, I thank the anonymous
referee for the many insightful suggestions to improve this book. Of course,
the responsibility of any shortcomings is mine.
Atocha Aliseda,
July 2004,
Tepoztlán, México.
PART I
CONCEPTUAL FRAMEWORK
Chapter 1
LOGICS OF GENERATION AND EVALUATION
1.
Introduction
The general purpose of this chapter is to provide a critical analysis on the
controversial enterprise of ‘logics of discovery’. It is naturally divided into six
parts. After this introduction, in the second part (section 2) we briefly review
the original heuristic methods, namely analysis and synthesis, as conceived
in ancient Greece. In the third part (section 3) we tackle the general question of
whether there is a logic of discovery. We start by analyzing the twofold division
between the contexts of discovery and justification, showing that it may be
not only further divided, but also its boundaries may not be so sharply distinguished. We then provide a background history (from antiquity to the XIXth
century), divided into three periods in time, each of which is characterized by
an epistemological stance (infallibilism or fallibilism) and by the types of logic
worked out (generational, justificatory inductive logics, non-generational and
self-corrective logics). Finally, we motivate the division of this general question into other three questions, namely one of purpose, one of pursuit and one
of achievement, for in general, there is a clear gap between the search and the
findings in the question of a logic of discovery. In the fourth part (section
4), we confront two foremost views on the logic of discovery, namely those
of Karl Popper and Herbert Simon, and show that despite appearances, their
approaches are close together in several respects. They both hold a fallibilist
stance in regard to the well-foundedness of knowledge and view science as a
dynamic activity of problem solving in which the growth of knowledge is the
main aspect to characterize. We claim that both accounts fall under the study
of discovery –when a broad view is endorsed– and the convergence of these
two approaches is found in that neither Simon’s view really accounts for the
4
ABDUCTIVE REASONING
epistemics of creativity at large, nor Popper neglects its study entirely. In the
fifth part (section 5), we advance the claim that logic should have a place in the
methodology of science, on a pair with historical and computational stances,
something that naturally gives place to logical approaches to the logic of discovery, to be cherish in a normative account of the methodology of science.
However, we claim that the label ‘logics of discovery’ should be replaced by
‘logics of generation and evaluation’, for on the one hand ‘discovery’ turns out
to be a misleading term for the processes of generation of new knowledge and
on the other hand, a logic of generation can only be conceived together with
an account of processes for evaluation and justification. In the final part of this
chapter (section 6), we sum up our previous discussion and advance our general
conclusions.
This chapter shows that interest on the topic of the logics of discovery goes
back to antiquity and spans over our present days, as it pervades several disciplines, namely Philosophy of Science, Cognitive Science and Artificial Intelligence. The search of a logic of discovery is not a trivial question, in fact is not
even a single question, and the words ‘logic’ and ‘discovery’ may be interpreted
in several many ways. This chapter will serve to set the ground and philosophical motivation behind our main purpose in this book, namely the logical study
of abductive reasoning, which we motivate in the next chapter (chapter 2).
2.
Heuristics: A Legacy of the Greeks
All roads in the study of discovery lead back to the Greek mathematicians
and philosophers in antiquity. In their study of the processes for solving problems, two were the main heuristic strategies, namely analysis and synthesis.
An extensive description of these methods is found in the writings of Pappus
of Alexandria (300 A.D.), who follows a tradition found in ‘The Treasury of
Analysis’ (analyomenos), a collection of books by earlier mathematicians. The
central part of his description reads as follows (Pappi Alexandrini Collectionis
Quae Supersunt. The translation is from [HR74, Chapter II]):
‘Now analysis is the way from was is sought – as if it were admitted – through its
concomitants (τ α ακoλoυθα, the usual translation reads: consequences) in order to
something admitted in synthesis. For in analysis we suppose that which is sought to be
already done, and we inquire from what it results, and again what is the antecedent of
the latter, until we on our backward way light upon something already known and being
first in order. And we call such a method analysis, as being a solution backwards. In
synthesis, on the other hand, we suppose that which was reached last in the analysis
to be already done, and arranging in their natural order as consequents the former
antecedents and linking them one with another, we in the end arrive at the construction
of the thing sought. And this we call synthesis’.
‘Now analysis is of two kinds. One seeks the truth, being call theoretical. The other
serves to carry out what was desired to do, and this is called problematical. In the
theoretical kind we suppose the thing sought as being and as being true, and then we
Logics of Generation and Evaluation
5
pass through its concomitants (consequences) in order, as though they were true and
existent by hypothesis, to something admitted; then, if that which is admitted to be true,
the thing sought is true, too, and the proof will be the reverse of analysis. But if we come
upon something false to admit, the thing sought will be false, too. In the problematical
kind we suppose the desired thing to be known, and then we pass through its concomitants
(consequences) in order, as though they were true, up to something admitted. If the thing
admitted is possible or can be done, that is, if it is what the mathematicians call given, the
desired thing will also be possible. The proof will again be the reverse of analysis. But
if we come upon something impossible to admit, the problem will also be impossible’.
Analysis and synthesis may be thought of as methods running in reverse
of each another, provided that all steps in analysis are reversed when used for
synthesis. However, this depends on the interpretation given to ‘τ α ακoλoυθα’.
While some scholars regard it as ‘consequences’, suggesting these methods
consist entirely of reversible steps, some others (notably [HR76]), interpret it
as ‘concomitants’, meaning ‘almost any sort of going together. Hence Pappus’
general description of analysis depicts it consistently as a search for premisses,
not as drawing of consequences.’ [HR76, p. 255].
Moreover, the methods of analysis and synthesis do not exist in isolation, but
only make sense in combination. It is not enough to arrive at something known
to be true in order to assert the truth of the initial statement which analysis
takes only as presupposed; the method of synthesis has to provide a proof for it.
Either the proof is constructed by inverting each step of the analysis (under the
‘consequence’ interpretation) or the process of analysis may provide material
to construct a proof, but is no final warrantee for a successful synthesis.
Both types of analysis begin with a hypothetical statement, but while in
the theoretical kind the statement is supposed to be true, in the problematical
kind the statement is supposed to be known. Then, a decisive point is reached
when something is found to be either admittedly true (or possible) or admittedly
false (or impossible). In the first case, the initial statement can find its proof
of being true by way synthesis, if (again) the backward and forward steps of
these methods are indeed reversible. In the second case however, if by way of
analysis we arrive at a statement that is admittedly false (or impossible), this is
enough to assert the falsity (or impossibility) of the initial statement. It comes
at no surprise that this situation ‘is just a special case of the so–called reductio
ad absurdum; because in such a case our starting theorem is, beyond doubt,
false’.[Sza74, p. 126].
It is however a matter of debate to what exactly these methods amounted
to in antiquity, for the text of Pappus ‘does not suffice to reconcile his general
description of analysis with his own mathematical practice, or with the practice
of other ancient mathematicians’ [HR76, pp. 255–256]. An attempt to provide
a modern and formalized account of these heuristic methods has been given by
Hintikka and Remes ([HR74, HR76]), who put forward the thesis that analysis is
6
ABDUCTIVE REASONING
a special case of natural deduction methods. In particular, they propose Beth’s
method of semantic tableaux as a method of analysis1 . As it is well–known,
semantic tableaux are primarily a refutation method (cf. chapter 4); when an
entailment does not hold, open branches of the tableau are records of counter–
examples. Otherwise, the tableau itself provides information to construct a
sequent style proof. Therefore, the method of analysis provides a definite
proof (by reductio) of the falsity of the ‘thing sought’ in the case of refutation
(when counter–examples have been found). As we shall see (cf. chapter 4),
our approach to abduction also takes semantic tableaux as its mathematical
framework for a logical reconstruction of this type of reasoning. Therefore, the
work by Hintikka and Remes may be regarded as a precursor of our own.
3.
Is There a Logic of Discovery?
Contexts of Research
The literature on the subject of scientific discovery is staggeringly confused
by the ambiguity and complexity of the term discovery. A discovery of an idea
leading to a new theory made in science involves a complicated process that
goes from the initial conception of an idea throughout its justification and final
settlement as a new theory. These two aspects are just the two extremes in a
series of intermediate processes including the entertainment of a new idea, its
initial evaluation, which may lead to finer ideas in need of evaluation or may
even be replaced by other ideas, calling for modification of the original one.
Therefore, we must at least acknowledge that scientific discovery is a process
subject to division. This fact naturally confronts us with the problem of how to
provide a proper division.
An attempt to supply a division is given by the contemporary distinction
between the contexts of discovery and the context of justification, originally
proposed by Reichenbach in the 30’s [Rei38]. This division often presupposes
the latter as dealing exclusively with the ‘finished research report’ of a theory,
and thus certainly leaves ample room to the former. In order to bring some order
and clarity into the study of the process of scientific discovery at large, several
authors identify an intermediate step between the two extremes, the conception
and justification of a new idea. While Savary puts forward the phase of ‘working
with ideas’ [Sav95], Laudan introduces the ‘context of pursuit’ as a ‘nether
region’ between the two contexts [Lau80, p. 174]. Another dimension in the
study of the context of discovery is to distinguish between a narrow and a
broad view. While the former view regards issues of discovery as those dealing
1 Although
Beth himself was inspired by the methods of analysis and synthesis to device his method of
semantic tableaux, he ‘neither connected this logical idea in so many words with the ancient geometrical
analysis nor applied his approach to the elucidation of historical problems.’ [HR76, p. 254].
Logics of Generation and Evaluation
7
exclusively with the initial conception of an idea, the latter view is that which
deals with the overall process going from the conception of a new idea to its
settlement as an idea subject for ultimate justification (a distinction introduced
by Laudan in [Lau80]).
It is however also a matter of choice to extend the boundaries of the context of
justification in order to deal with evaluation questions as well, especially when
the truth of a theory is not the sole interest to be searched for. A consequence of
this move is the proposal to rename the ‘context of justification’ as the ‘context
of evaluation’ [Kui00, p.132] or as the ‘context of appraisal’ [Mus89, p. 20].
Under the latter view, the context of discovery has in turn been relabeled the
‘context of invention’, in order to avoid the apparent contradiction that arises
when we speak of the discovery of hypotheses, as discovery is a ‘success word’
which presupposes that what is discovered, must be true.
Therefore, the original distinction between the contexts of discovery and
justification may not only be further divided, but also its boundaries may not be
so sharply distinguished. A separate issue concerned with all these contexts of
research is to inquire whether the ‘context of discovery’ or any other context for
that matter is subject to philosophical reflection and allows for logical analysis.
Background History
In what follows, I will offer an overview of the evolution of the enterprise of
the logic of discovery (based on [Lau80], [Nic80] and [Mus89]), from antiquity
to the XIXth century. This overview will serve us to identify those relevant
aspects that have played a role in the study of discovery in order to place them
in the research agenda nowadays.
Since antiquity the enterprise of a logic of discovery was conceived as
providing an instrument for the generation of genuine new concepts and theories
in science. This enterprise was at the core of Aristotle’s Posterior Analytics as
an attempt to provide an account of the discovery of causes by finding missing
terms in incomplete explanations2 . According to some interpretations ([Lau80],
[Ke89]), the search for a logic of discovery at this period aimed to capture the
‘Eureka moment’, and thus discovery was taken in its narrow interpretation.
Thereafter, the project flourished through concrete proposals for finding good
and effective methods for making causal discoveries, as witnessed in the work
of Bacon, Descartes, Herschel, Boyle, Leibniz and Mill, to name the most
representative ones. The division between contexts in scientific research into
2 On the one hand, Aristotle introduced the notion of
enthymeme to characterize those incomplete arguments
which lack a premise. Once that premise is found, this type of argument is converted into a deductive one.
On the other hand, Aristotle characterized two other types of arguments, namely ‘apagoge’ and ‘epagoge’,
the former later identified with abduction (by Peirce) and the latter generally interpreted as induction.
8
ABDUCTIVE REASONING
discovery and justification was not present at this period and the goal was to
offer a logic of discovery also providing a solution to the problem of theory
justification. The search for such a logic was guided by a strong philosophical
ideal, that of finding a universal system, which would capture the way humans
reason in science. For this purpose Leibniz proposed a ‘characteristica universalis’ consisting of a mathematical language, precise and without ambiguities
(as opposed to natural languages) in which all ideas could be translated to and
by which intellectual arguments were conclusively settled. A consequence of
this ideal was the presumption to decide on the validity of chain of arguments
via a calculus, the ‘calculus ratiocinator’. This naive optimism in what a logic
for science could provide, went to extreme of pretending to have access to all
scientific truths via this calculi3 .
Leibniz’s calculus was a response to Aristotle’s deductive syllogistics, which
did not provided what they were looking for, a logic that would genuinely
generate new sciences. To this end, Bacon proposed the method of ‘eliminative
induction’, a kind of disjunctive syllogism, by which hypotheses are eliminated
in favour of the ‘true’ one. As for Descartes, ‘he gave us the handy hints for
problem solvers of the Regulae, and a few secretive references to the method
of analysis and synthesis proposed by the Greek geometers’ [Mus89, p.18]
synthesis analysis. As it turns out, however, their methods did not provided
rules to generate genuine new ideas, for they are rather directed to working
with already made hypotheses. Bacon’s method is indeed a method of selection
and Descartes method focuses on ways of decomposing a hypothesis into parts
known to be true (analysis into its parts), to later run a synthesis process in order
to prove it.
According to our previous discussion of contexts of research, where to locate
these methods turns out to be a matter of debate. They clearly belong to the
context of justification according to Reichenbach’s distinction and under a narrow view of discovery (as pointed out in [Mus89, p. 18]), but to the context of
discovery under a broad one. And taking a finer view on contexts, they belong
to the context of pursuit or to the context of evaluation or appraisal when the
context of justification is enlarged. In any case, it seems that neither of them
belongs to the context of invention, as these methods fail to give an account of
how is that the hypotheses entertained were initially conceived.
What we find in all these authors is that while their search for a logic of
discovery was guided by Leibniz’s ideal of a universal calculi, the results obtained from its pursuit, did not live up to this search. Clearly, there is a gap
between the ‘search’ and the ‘findings’ in the question of a logic of discovery.
Therefore it seems useful to divide the ambitious question of whether there is
3 Nowadays
we know this attempt is indeed impossible. There is no general mechanical procedure able to
decide on the validity of logical inferences. Predicate logic is undecidable.
Logics of Generation and Evaluation
9
a logic of discovery into three of them, namely its purpose, pursuit, and its
achievement. That is, to distinguish between what is searched for and what
is obtained as a result of that search, in order to evaluate their coherence and
thus give an answer to the question of achievement. But before getting into the
proposed three-question division, we will present Laudan’s [Lau80] critical historical analysis on the question concerning the search of a logic of discovery, in
which he is concerned with discovery in its narrow sense, the view that regards
issues of discovery as those dealing exclusively with the initial conception of
an idea.
Laudan’s Analysis
According to this interpretation, two were the main motivations behind the
enterprise of a logic of discovery, namely the epistemological problem of the
well-foundedness of knowledge and the heuristic and pragmatic problem of
how to ‘accelerate scientific advance’. Regarding the first one, the prevalent
epistemological stance supporting the legitimization of knowledge was an infallibilistic position, one that required an absolute warrant for the newly generated
concepts and theories about the world. The goal was to offer a logic of discovery also providing a solution to the problem of theory justification. There was
not a unique epistemological position in regard to the justification of theories,
but only one of them was compatible with the enterprise of a logic of discovery,
namely the one identified with the generational stance4 . Therefore, infallibilism in turn leads to a generational position, and these two together provide an
absolute warrant of our claims about the world via a truth-preserving logic.
The second motivation regarding the problem of how to ‘accelerate scientific
advance’, was the guiding motor of the ultimate goal for a logic of discovery, that
of providing ‘rules leading to the discovery of useful facts and theories about
nature’ [Lau80, p. 175]. The rules proposed, however, were closely linked to
another aspect in the evolution of the ideas surrounding discovery, namely with
the object of science under examination. Though several characterizations of
what scientific inference amounted to, most methodologists up to mid XVIIIth
century focused their efforts on the characterization of the discovery of empirical laws, of universal statements concerning observable entities, and thus the
4 The
two opposing groups were known as the ‘consequentialists’ and the ‘generators’. The former group
justifies a theory as true through an inspection of (some of) its consequences. In contrast, the latter group
renders a scientific theory as established by showing that it follows logically from observational statements.
Since the Aristotelian era, however, a logical flaw was acknowledged in the consequentialist position, since
it committed the ‘fallacy of affirming the consequent’ (to argue from the truth of a consequence to the truth
of the theory), something which made this view logically inconclusive: ‘if theories were to be demonstrably
true, such demonstration could not come from any (non exhaustive) survey of the truth of their consequences’
[Lau80, p. 177]. The only circumstances in which infallibilism and consequentialist can agree are when it is
possible to (i) provide an exhaustive examination of the consequences of a certain theory and (ii) enumerate
and reject exhaustively all possible theories at hand [Lau80, Note 3, p. 183].
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ABDUCTIVE REASONING
focus was for an inductive logic modelling ‘enumerative induction’. Scientific
research to be analysed had to do with generalizations directly inferred from
observations, with physical laws such as ‘all gases expand when heated’. It was
until the 1750’s that several scientists and philosophers searched for a logic to
discover explanatory theories, those involving theoretical entities as well (such
as found in [Lap04]). In this case, other type of logics were needed, those
which involved a more complex mechanism than that of enumerative induction. Usually, these logics involve a background theory (consisting of laws),
initial conditions and a relevant observation. The goal of these logical apparatus
was to produce a better and truer theory than the background one. The idea
of truth-approximation was behind this conception, and accordingly had been
labeled ‘self-corrective logics’(we shall identify this logics with abduction, cf.
chapter 2). To conclude, up to this point, infallibilism and generational logics
were hand in hand, together providing a truth preserving logic of discovery to
warrant infallible knowledge.
Later on, around mid XIXth century, the enterprise of a logic of discovery
in the terms described above was abandoned and replaced by the search of a
logic of justification, a logic of post hoc evaluation. The turning point of this
move had to do with a major change in view regarding the legitimization of
knowledge, namely by a shift from infallibilism to fallibilism. A consequence
of this change in epistemological stance is that the sources of knowledge were
no longer supported by certain ground, and thus justification could be separated
from proof. Thus, the independence of genesis from the justification of theories
sets in, for there was no longer a need to unfold the generation of theories in
order to characterize their justification, in other words, the enterprise of a logic
of discovery becomes ‘redundant’ from the epistemic point of view, and it
is abandoned. Another consequence is that the consequentialist view for the
justification of theories becomes in fashion again, as it is now compatible with an
fallibilist account, and thus paves the way to the view represented by what later
was known as the ‘hypothetico-deductive method’, which became the standard
in theory justification.
Thus, there are clearly three episodes in the evolution of a logic of discovery
from antiquity up to the mid XIXth century, characterized as follows:
Up to 1750:
Epistemological stance: infallibilism
Logic: generational and justificatory inductive logic.
1750 - 1820:
Epistemological stance: infallibilism
Logic: generational and justificatory inductive logic as well as
self-corrective logics.
Logics of Generation and Evaluation
11
1820 - 1850:
Epistemological stance: fallibilism
Logic: Nongenerational justificatory inductive logic as well as
self-corrective logics.
Underlying Questions
The purpose of a logic of discovery concerns the ultimate goal researchers
engaged in this enterprise wish to achieve in the end, and the response (affirmative or negative) to the question of whether there is such a logic is largely
grounded on philosophical ideals. The pursuit of a logic of discovery regards
the working activities that researchers in the field are engaged in to attain their
goals, and the answer to such a question is given in the form of concrete proposals of logics of discovery. Finally, the achievement of a logic of discovery
provides an evaluation between previous aspects, of whether what is actually
achieved lives up to its purpose. This three-question division allows to evaluate
existing proposals on the logic of discovery as to their coherence, and it also
provides a finer grain distinction in order to compare confronting proposals, for
they may agree on one question while disagree on the other one.
In the first period identified above (from antiquity to mid XIX century), the
search for a logic of discovery was guided by a strong philosophical ideal, that
of finding a universal system which would capture the way humans reason in
science, including the whole spectrum, from the initial conception of new ideas
into their ultimate justification. Following the spirit of Leibniz’s “Characteristica Universalis", this ideal was the motor behind the ultimate purpose of
finding a logic of discovery. As to the question of pursuit, the central method
worked out was the modeling of induction, giving place to proposals such as
Bacon’s ‘eliminative induction’, which as already mentioned, it is in fact a
method for hypothesis selection. Therefore, regarding the question of achievement of a logic of discovery in this period, the products resulting from its pursuit
did captured only to a very small extent what the question of purpose was after,
and therefore the project was not coherent in regard to what was searched for
and what was found in the end. A similar analysis can be given for the second
period. In the third period, the original question of purpose is vanished by
the advancement of fallibilism; for it was then clear that a universal calculi in
which all ideas could be translated to and by which intellectual arguments were
conclusively settled, was an impossible goal to achieve. The question to be
searched for instead is concerned with a logic of justification, and accordingly,
the question of pursuit focused on developing accounts on this matter. There
was complete harmony between the search and the findings, and so the question
of achievement renders this period as coherent. The abandonment of a logic for
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ABDUCTIVE REASONING
the genesis of theories simply deleted the search of such kind of ‘discovery’
rules from the agenda and instead focused exclusively in mechanisms for the
justification of ready–made scientific theories.
However, the aim to find a logic dealing with the conception of new ideas
was not completely discarded, as witnessed later in the work of Charles S.
Peirce (cf. chapter 2 and 7) and others, but this line of research has remained
incoherent as far as the question of achievement is concerned. As absurd as this
position seems to be, this is still the ideal to which the “friends of discovery"
hold to, but as we shall see in the following corresponding section, a closer
analysis into their approach will help to make sense of their stance.
4.
Karl Popper and Herbert Simon
From the philosophy of science point of view, Karl Popper (1902-1994) and
Herbert Simon (1916-2001) were two well-established scholars who reacted
to logical positivism in their own peculiar and apparently opposing views on
the logic of discovery. In this section, we will elucidate some aspects of their
proposals in order to place them in the present discussion of scientific discovery.
On the one hand, when a finer analysis of contexts of research is done, it
seems Popper’s logic may be considered as part of the context of discovery,
and his account on the growth of knowledge by the method of conjectures and
refutations, is in accordance with the “Friends of Discovery" mainstream, of
which Simon is one of the pioneers. On the other hand, we will unfold Simon’s
claim that there is a logic of discovery, in order to show that far from being a
naive aim to unravel the mysteries of scientific creativity, it is a solid proposal
in the direction of a normative theory of discovery processes, grounded on the
view of logic as a pattern seeking activity based on heuristic strategies to meet
its ends.
The Logic of Discovery
It is an unfortunate yet an interesting fact that Popper’s Logik der Forschung
first published in German in 1934, was translated into English and published
twenty five years later as The Logic of Scientific Discovery. An accurate
translation would have been: The Logic of Scientific Research, as found in
translations into other languages, such as Spanish (La Lógica de la Investigación
Cientı́fica).
One reason for its being unfortunate lies in the fact that several renowned
scholars (Simon and Laudan amongst others) have accused Popper of denying
the very subject matter of what the English title of his book suggests, something
along the lines of a logical enterprise into the epistemics of scientific theory
discovery. These complaints are firmly grounded within the accepted view that
Logics of Generation and Evaluation
13
for Popper scientific methodology concerns mainly the testing of theories, and
this approach clearly leaves outside of its scope issues having to do with discovery. Thus, for those philosophers of science interested in discovery processes as
well as in other methods for scientific inquiry outside the realm of justification,
it seems natural to leave Popper out of the picture and take the above complaint
on the title to be just a confusion originated from its English translation.
However, on a closer look into Popper’s philosophy, an additional confusion
stirs up when we find (in a publication shortly after) his position beyond justification issues and concerned with the advancement and discovery in science, as
witnessed by the following quote: ‘Science should be visualized as progressing
from problems to problems - to problems of increasing depth. For a scientific
theory - an explanatory theory - is, if anything, an attempt to solve a scientific
problem, that is to say, a problem concerned with the discovery of an explanation’ [Pop60a, p. 179]. This view is in accord with Simon’s famous slogan
that ‘scientific reasoning is problem solving’ made in research in cognitive
psychology and artificial intelligence (to be later introduced), a claim also put
forward by Laudan in the philosophy of science. Moreover, it seems Popper
was happy with the English title of his book, for being such an obsessive proof
reader of his own work, he made no remarks about it5 .
Additionally, it is also appealing that in the literature of knowledge discovery
in science, especially from a computational point of view, some of Popper’s
fundamental ideas are actually implemented within the simulation of discovery
and testing processes in science6 .
The common view on Popper’s position states that issues of discovery cannot
be studied within the boundaries of methodology, for he explicitly denies the
existence of a logical account for discovery processes, regarding its study a
business of psychology. This position is backed up by the following –much too
quoted– passage:
“Accordingly I shall distinguish sharply between the processes of conceiving a new
idea, and the methods and results of examining it logically. As to the task of the logic
of knowledge -in contradistinction to the psychology of knowledge - I shall proceed on
the assumption that it consists solely in investigating the methods employed in those
systematic tests to which every new idea must be subjected if it is to be seriously entertained. Some might object that it would be more to the purpose to regard it as the
5 Popper
used every opportunity to clarify his claims and terms put forward in his ‘Logik der Forschung’,
as witnessed in a multitude of footnotes and new appendices to ‘The Logic of Discovery’ and in many
remarks found in later publications. In [Pop92, p. 149] he reports (referring to his ‘Postscript: After Twenty
Years’): ‘In this Postscript I reviewed and developed the main problems and solutions discussed in Logik
der Forschung. For example, I stressed that I had rejected all attempts at the justification of theories, and
that I had replaced justification by criticism’.
6 For example, the requirement characterizing a theory as ‘scientific’ when it is subject to refutation, is
translated into the ‘FITness’ criterion, and plays a role in the process of theory generation, for a proposed
theory only survives if it can be refuted within a finite number of instances [SG73].
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ABDUCTIVE REASONING
business of epistemology to produce what has been called a ‘rational reconstruction’ of
the steps that have led the scientist to a discovery -to the finding of some new truth ...
But this reconstruction would not describe these processes as they actually happen: it
can give only a logical skeleton of the procedure of testing. Still, this is perhaps all that
is meant by those who speak of a ’rational reconstruction’ of the ways in which we gain
knowledge.
It so happens that my arguments in this book are quite independent of this problem.
However, my view of the matter, for what is worth, is that there is no such thing as a
logical method of having new ideas, or a logical reconstruction of this process" [Pop34,
P. 134].
First of all, it should come at no surprise that Popper’s position holds a place
in the discussion of discovery issues, as the objects of his analysis are precisely
genuinely new ideas. Moreover, we should emphasize that Popper draws a clear
division between two processes; amidst the conception of a new idea, and the
systematic tests to which a new idea should be subjected to, and in the light of
this division he advances the claim that not the first but only the second one is
amenable to logical examination.
Conjectures and Refutations:
The Growth of Scientific Knowledge
For Popper the growth of scientific knowledge was the most important of
the traditional problems of epistemology [Pop59, P. 22]. His fallibilist position
provided him with the key to reformulate the traditional problem in epistemology, which is focused on the reflection on the sources of our knowledge. Laid
down this way, this question is one of origin and begs for an authoritarian answer [Pop60b, p. 52], regardless of its answer being placed in our observations
or in some fundamental assertions lying at the core of our knowledge. Popper’s
answer to this question is that we do not know or even hope to know about
the sources of our knowledge, since our assertions are only guesses [Pop60b,
p. 53]. Rather, the question and its answer should be the following: “How
can we hope to detect and eliminate error? Is, I believe, By criticizing the
theories or guesses of others" [Pop60b, p. 52]. This is the path to make knowledge grow: “the advance of knowledge consists, mainly, in the modification of
earlier knowledge" [Pop60b, p. 55].
This concern on the growth of knowledge is intimately related to his earlier
mentioned view of science as a problem solving activity, and in this respect he
writes: “Thus, science starts from problems, and not from observations; though
observations may give rise to a problem, specially if they are unexpected"
[Pop60a, p. 179]. Moreover, rather than asking the question ‘How do we
jump from an observation statement to a theory?, the proper question to ask is
the following: “How do we jump to from an observation statement to a good
theory?’ ... By jumping first to any theory and then testing it, to find whether
Logics of Generation and Evaluation
15
it is good or not; i.e. by repeatedly applying the critical method, eliminating
many bad theories, and investing many new ones" [Pop63, p. 55].
Thus, the intention in both the title and content of Popper’s The Logic of
Scientific Discovery was to unfold the epistemics of evaluation and selection
of newly discovered ideas in science, more in particular, of scientific theory
choice. To this end, Popper proposed a rational method for scientific inquiry, the
method of conjectures and refutations, which he refined in later publications
[Pop63]7 . The motivation behind this method was to provide a criterion of
demarcation between science and pseudo-science. Besides its purpose, the
logical method of conjectures and refutations is a norm for progress in science,
for producing new and better theories in a reliable way. For Popper the growth
of scientific knowledge –and even of pre-scientific knowledge– is based on the
learning from our mistakes, which according to him is achieved by the method
of trial and error. He did not provided a precise procedure to perform scientific
progress, leading to better theories, but rather a set of theory evaluation criteria
including a measure for its potential progressiveness (its testability) and the
condition of having a greater ‘empirical content’ than the antecedent theory.
Does Scientific Discovery Have a Logic?
In principle, the pioneering work of Herbert Simon and his team share
the ideal on which the whole enterprise of artificial intelligence was initially
grounded, namely that of constructing intelligent computers behaving like rational beings, something which resembles the philosophical ideal which guided
the search for a logic of discovery in the first period identified above (up to the
XIX century). However, it is important to clarify on what terms is this ideal
inherited in regard to the question of purpose of a logic of discovery, on the
one hand, and put to work with respect to the question of the pursuit of such a
logic, on the other.
In his essay ‘Does scientific discovery have a logic? Simon sets himself the
challenge to refute Popper’s general argument, reconstructed for his purposes
as follows: ‘If ‘There is no such thing as a logical method of having new
ideas, then there is no such thing as a logical method of having small new
ideas” [Sim73a, p. 327]. (My emphasis), and his strategy is precisely to show
that an antecedent in the affirmative does not commit to an assessment of the
consequent, as Popper seems to suggest. Thus, Simon converts the ambitious
aim of searching for a logic of discovery revealing the process of discovery at
large, into an unpretentious goal: ‘Their modesty [of the examples dealt with] as
7 This
method is proposed as an alternative to induction. In fact, it made induction absolutely irrelevant;
there was no problem of induction as there was any induction procedure as a method for scientific inquiry
in the first place, being just an optical illusion to be discarded from the methodology of science.
16
ABDUCTIVE REASONING
instances of discovery will be compensated by their transparency in revealing
the underlying process’ [Sim73a, p. 327].
This humble but brilliant move allows Simon to further draw distinctions on
the type of problems to be analysed and on methods to be used. For Simon and
his followers, scientific discovery is a problem-solving activity. To this end,
a characterization of problems into those that are well structured versus those
that are ill structured is provided, and the claim for a logic of discovery focuses
mainly on the well-structured ones8 . A well structured problem is that for
which there is a definite criterion for testing, and for which there is at least one
problem space in which the initial and the goal state can be represented and all
other intermediate states may be reached with appropriate transitions between
them. An ill-structured problem lacks at least one of the former conditions.
Although there is no precise methodology by which scientific discovery is
achieved, as a form of problem solving, it can be pursued via several methodologies. The key concept in all this is that of heuristics, the guide in scientific
discovery which is neither totally rational nor absolutely blind. Heuristic
methods for discovery are characterized by the use of selective search with
fallible results. That is to say, while they provide no complete guarantee to
reach a solution, the search in the problem space is not blind, but it is selective
according to a predefined strategy. The authors distinguish between ‘weak’ and
‘strong’ methods of discovery. The former is the type of problem solving used
in novel domains. It is characterized by its generality, since it does not require
in-depth knowledge of its particular domain. In contrast, strong methods are
used for cases in which our domain knowledge is rich, and are specially designed for one specific structure. Weak methods include generation, testing,
heuristic methods, and means-ends analysis, to build explanations and solutions
for given problems. These methods have proved useful in artificial intelligence
and cognitive simulation, and are used by several computer programs. Examples are the BACON system which simulates the discovery of quantitative laws
in Physics (such as Kepler’s law and Ohm’s law) and the GLAUBER program,
which simulates the discovery of qualitative laws in chemistry9 .
Heuristics: A Normative Theory of Discovery
Simon proposes an empirical as well as a normative theory for scientific
discovery, both complementing his broad view on this matter:
8 However,
cf. [Sim73b] for a proposal on how to treat ill structured problems and for the claim that there is
no clear boundary between these kinds of problems.
9 It is however a matter of debate whether these computer programs really makes discoveries, since it produces
theories new to the program but not new to the world, and its discoveries seem spoon–fed rather than created.
In the same spirit, Paul Thagard proposes a new field of research, “Computational Philosophy of Science"
[Tha88] and puts forward the computational program PI (Processes of Induction) to model some aspects of
scientific practice, such as concept formation and theory building.
Logics of Generation and Evaluation
17
‘The theory of scientific discovery has both an empirical part and a formal part. As an
empirical theory, it seeks to describe and explain the psychological and sociological
processes that are actually employed to make scientific discoveries. As a formal theory,
it is concerned with the definition and logical nature of discovery, and it seeks to provide
normative advice and prescriptions for anyone who wishes to proceed rationally and
efficiently with a task of scientific discovery.’ [Sim77, P. 265]
Let us first deal with the normative aspect of scientific discovery. When Simon proposes a logic for scientific discovery he is relying on a whole research
program already developed for artificial intelligence and cognitive psychology,
namely the machinery based on heuristic search used for the purposes of devising computer programs to simulate scientific discoveries. For him, a logic of
scientific method does not refer to the pure and formal use of the term ‘logic’,
but rather to ‘a set of normative standards for judging the processes used to
discover or test scientific theories, or the formal structure of the theories themselves. The use of the term ‘logic’ suggests that the norms can be derived from
the goals of the scientific activity. That is to say, a normative theory rests on
contingent propositions like: ‘If process X is to be efficacious for attaining goal
Y, then it should have properties A,B,C’ [Sim73a, p. 328].
The normative theory of scientific discovery is thus a theory for prescribing how actual discoveries may have been made following a rational, though
a fallible set of strategies. This view naturally makes his approach close to
computational and formal approaches. In fact, Simon views a normative theory
of discovery as ‘a branch of the theory of computational complexity’ [Sim73a,
p. 332]. Moreover, he is willing to use the word ‘logic’ in a broad sense, something along the lines of a ‘rational procedure’. Finally, a normative theory of
discovery is one that does not demand a deductive justification for the products
of induction, and thus it liberates it from the ‘justificationist’ positivist view of
science.
Concerning the empirical aspect of scientific discovery, we must admit that
Simon indeed attempts to describe the psychological processes leading to invent new ideas in science, and in fact talks about incubation and unconscious
processes in discovery [Sim77, p. 292], but his position is clearly not aiming
to unravel the mysteries of creativity, precisely because he believes there are
no such mysteries in the first place: ‘new representations, like new problems,
do not spring from the brow of Zeus, but emerge by gradual –and very slow–
stages ... Even in revolutionary science, which creates those paradigms, the
problems and representations are rooted in the past; they are not created out
of whole cloth’ [Sim77, p. 301]. Thus, an empirical theory of discovery is
based on a psychological theory of discovery, one which provides a mechanistic explanation for thinking, which in turn is demonstrated by programming
computers imitating the way humans think. However, on the one hand Simon
claims that the processes to explain scientific discovery are only sufficient –and
not necessary– to account for discoveries that have actually occurred. On the
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ABDUCTIVE REASONING
other hand, he admits there may be forms of making new discoveries in science
which may nevertheless be impossible to characterize: ‘existing informationprocessing theories of thinking undoubtedly fall far short of covering the whole
range of thinking activities in man. It is not just that the present theories are
only approximately correct. Apart from limits on their correctness in explaining the ranges of behaviour to which they apply, much cognitive behaviour still
lies beyond their scope’ [Sim77, p.276]. I leave it as an open question whether
this quote, and in general Simon’s position, implies that there may be some
future time in which an empirical theory of discovery will explain the discovery phenomena at large. I personally believe this is not the case, for he asserts
further that creativity in science is achieved by three factors: luck, persistence
and ‘superior heuristics’ (a heuristic is superior to another one if it is a more
powerful selective heuristics) [Sim77, p. 290], thus making room to human
factors impossible to be characterized in full.
We must note that Simon is not trapped into the discovey–justification dichotomy, but rather views the process of scientific inquiry as a continuum,
which only holds the initial discovery and the final justification of a theory as
the extreme points of a whole spectrum of reasoning processes. In this respect,
his theory is clearly considered as taking a broad view on discovery.
Even though this approach comes from apparently distant disciplines to philosophy of science, namely cognitive psychology and artificial intelligence,
they are proposals which suggest the inclusion of computational tools in the
philosophy of science research methodology and by so doing claim to reincorporate aspects from the context of discovery within its agenda. However, this
approach neither aims nor provides an account of ‘the Eureka’ moment, even
for small ideas.
Although there are several particularities about Simon’s proposal with respect
to all others, in general his claims reflect the spirit of the project on the search
of a logic of discovery for the ‘Friends of discovery’ overall enterprise.
Science as a Problem - Solving Activity:
Popper and Simon Revisited
When presenting the work of these two philosophical giants, we had on focus
their views on inquiry in science, in order to explore here to what extent their
stances are close together. On the one hand, a first conclusion to draw is that
while Popper was genuinely interested in an analysis of new ideas in science,
he rendered the very first process of the conception of an idea to be outside the
boundaries of the methodology of science, and centered his efforts in giving an
account of an ensuing process, that concerned with the methods of analyzing
new ideas logically, and accordingly produced his method of conjectures and
refutations. Whereas for Simon, his aim was to simulate scientific discovery
at large, giving an account both for the generation and evaluation of scientific
Logics of Generation and Evaluation
19
ideas, convinced that the way to go was to give both an empirical and a normative
account of discovery, the former to describe and then represent computationally
the intellectual development of human discoveries made in science. The latter to
provide prescriptive rules, mainly in the form of heuristic strategies to perform
scientific discoveries.
Both authors hold a fallibilist position, one in which there is no certainty of
attaining results and where it is possible to refute already assessed knowledge,
in favour of new one that better explains the world. However, while for Popper
there is one single method for scientific inquiry, the method of conjectures and
refutations, for Simon there are several methods for scientific inquiry, for the
discovery and justification processes respond to several heuristic strategies,
largely based on pattern seeking, the logic of scientific discovery. A further
difference between these approaches is found in the method itself for the advancement of science, in what they regard to be the ‘logic’ for discovery. While
for Popper ideas are generated by the method of blind search, Simon and his
team develop a full theory to support the view that ideas are generated by the
method of ‘selective search’. Clearly the latter account allows for a better
understanding of how theories and ideas may be generated.
Going back to the three-question division concerning the logic of discovery,
namely purpose, pursuit and achievement, it seems useful to analyze these
two approaches in regard to them. Popper’s purpose for a logic of scientific
discovery has no pretensions at all to uncover the epistemics of creativity,
but focuses on the evaluation and selection of new ideas in science. Thus,
his position is clearly akin with the period corresponding to the search for a
logic of justification, as far as the question of purpose is concerned. As to
the question of pursuit, his account provides criteria for theory justification
under his critical rationalism view, by which no theory is finally settled as true,
but is also concerned with the advancement of science, characterizing problem
situations as well as a method for their solution. In fact, for Lakatos, Popper’s
proposal goes beyond its purpose:
‘There is no infallibilist logic of scientific discovery leading infallibly to results, but
there is a fallibilistic logic of discovery which is the logic of scientific progress. But
Popper, who has laid the basis for this logic of discovery was not interested in the metaquestion of what is the nature of this investigation, so he did not realize that it is neither
psychology nor logic, but an independent field, the logic of discovery, heuristics".
[Lak76, p. 167]10 .
Therefore, as to the question of achievement, it seems that Popper actually
found more than what he was looking for, since his account offered not just
10 It is interesting to note that in this book, Lakatos was greatly inspired by the history of mathematics, paying
particular attention to processes that created new concepts - often referring to G. Polya as the founding father
of heuristics in mathematical discovery. (Cf. [Ali00a] for a more detailed discussion of heuristics in the
work of G. Polya).
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ABDUCTIVE REASONING
the justification of theories, but in fact a methodology for producing better and
stronger theories, something which enters the territory of discovery, at least in
so far as the question of the growth of knowledge is concerned.
As for Simon, even though as to the question of purpose he is genuinely
after a logic of scientific discovery, his question is also normative, rather than
just psychologically prescriptive. And his aim is a modest one, to simulate
scientific discoveries for well-structured problems and expressible in some
formal or computational language. As to the question of pursuit, by introducing
both the notion and the methodology of ‘selective search’ in a search space for
a problem, his research program has achieved the implementation of complex
heuristic strategies that simulate well-known scientific discoveries in science.
Thus, judging by the empirical examples provided, it is difficult to refute the
whole enterprise. Finally, as to the question of achievement, I think Simon’s
proposal is a coherent one, at least in its normative dimension and modest
purpose, it provides results accordingly for a logic of discovery.
To conclude, I think Popper’s and Simon’s approaches are close together,
at least in so far as the following basic ideas are concerned: they both hold
a fallibilist stance in regard to the well-foundedness of knowledge and view
science as a dynamic activity of problem solving in which the growth of
knowledge is the main aspect to characterize, as opposed to the view of science
as an static enterprise in search of the assessment of theories as true. But
Popper failed to appreciate the philosophical potential of a normative theory of
discovery and therefore was blinded to the possibility of devising a logic for
the development of knowledge. His view of logic remained static: ‘I am quite
ready to admit that there is a need for a purely logical analysis of theories, for
an analysis which takes no account of how they change and develop. But this
kind of analysis does not elucidate those aspects of the empirical science which
I, for one, so highly prize’ [Pop59, p. 50].
One reason that allows for the convergence of these two accounts, perhaps
obvious by now, is that neither the “Friends of discovery" really account for
the epistemics of creativity at large nor Popper neglects its study entirely. Both
accounts fall naturally under the study of discovery –when a broad view is
endorsed– and neither of them rejects the context of justification, or any other
context for that matter. Therefore, it seems that when the focus is on the
processes of inquiry in science, rather than on the products themselves, any
possible division of contexts of research is doomed to fail sooner or later.
Logics of Generation and Evaluation
21
5.
Logics for Scientific Methodology
A Place for logic in Scientific Methodology
In this section our aim is to advance the following two claims. In the first
place, in our view, the potential for providing a logic for scientific discovery
is found in a normative account in the methodology of science. However, we
have seen that the formal proposals for this purpose have little, if nothing to
do with how a formal logical system looks like. Thus in the second place, our
claim is that logic, as understood in modern systems of logic, has a place in the
study of logics of discovery. Let us deal with each matter in turn.
The two periods identified above (from antiquity to mid XIXth century) had
in mind a logic of discovery of a descriptive nature, one that would capture
and describe the way humans reason in science. As noted, these ‘logics’ had
little success, for they failed to provide such an account of discovery. Thereafter, when the search for a logic of discovery was abandoned, a normative
account prevailed in favour of proposals of logics of justification. Regarding
the approaches of Popper and Simon in this respect, it is clear that while Popper
overlooked the possibility of a normative account of logics of discovery, Simon
centered his efforts in the development of heuristic procedures, to be implemented computationally, but not on logics –per se– of discovery. In fact, some
other proposals which are strictly normative and formal in nature, such as that
found in [Ke97], argue for a computational theory as the foundation of a logic
of discovery, one which studies algorithmic procedures for the advancement of
science.
Therefore, while the way to go seems to be normative, there is nothing in what
we have analysed which resembles a logical theory, as understood in the logical
tradition. The way the “Friends of Discovery” rightly argue in this respect is to
say that the word ‘logic’ should not at all be interpreted as logic in the traditional
sense, and rather think of it as some kind of ‘rationality’ or ‘pattern seeking’.
While we believe the approach taken by Simon and his followers has been very
fruitful for devising computer programs that simulate scientific discoveries, it
has more or less dismissed the place of logic in its endeavors.
Now let us look at the situation nowadays. In spite of few exceptions, logic
(classical or otherwise) in philosophy of science is, to put it simply, out of
fashion. In fact, although classical logic is part of the curricula in philosophy
of science graduate programs, students soon learn that a whole generation of
philosophers regarded logical positivism as a failed attempt (though not just for
the logic), claiming that scientific practice does not follow logical patterns of
reasoning, many of which favoured studies of science based on historical cases.
So, why bother about the place of logic in scientific research?
On the one hand, the present situation in logical research has gone far beyond
the formal developments that deductive logic reached last century, and new
22
ABDUCTIVE REASONING
research includes the formalization of several other types or reasoning, like
induction and abduction. On the other hand, we claim for a balanced philosophy
of science, one in which both methods, the formal and the historical may be
complementary, together providing a pluralistic view of science, in which no
method is the predominant one. Research nowadays has finally overcome the
view that a single treatment in scientific methodology is enough to give us an
understanding of scientific practice. A pluralistic view of science was already
fostered by Pat Suppes [Sup69] in the sixties.
We know at present that logical models (classical or otherwise) are insufficient to completely characterize notions like explanation, confirmation or falsification in philosophy of science, but this fact does not exclude that some
problems in the history of science may be tackled from a formal point of view.
For instance, ‘some claims about scientific revolutions, seem to require statistical and quantitative data analysis, if there is some serious pretension to regard
them with the same status as other claims about social or natural phenomena’
[Sup69, p. 97].
In fact, ‘Computational Philosophy of Science’ may be regarded as a successful marriage between historical and formal approaches. It is argued that
although several heuristic rules have been derived from historical reconstructions in science, they are proposed to be used for future research [MN99]. This
is not to say however, that historical analysis of scientific practice could be
done in a formal fashion, or that logical treatment should care for some kind
of ‘historic parameter’ in its methodology, but we claim instead that these two
views should share their insights and findings in order to complement each
other. After all, recall that part of Hempel’s inspiration to device a ‘logic of
explanation’ was his inquiry on the logic for ‘drawing documents from history’.
Regarding the study of the context of discovery, our position is that it allows
for a precise formal treatment. The dominant trend in contemporary philosophy
of science neglected the study of the processes of theory discovery in science,
partly because other questions were on focus, still we knew nothing about
the canons of justification, and the historical approach to scientific practice
was making its way into methodological issues. It was research on scientific
discovery as problem solving by computational means that really brought back
issues of discovery to the philosophy of science agenda. And the field was ready
for it, as philosophy of science questions had also evolved. Issues of theory
building, concept formation, theory evaluation and scientific progress were at
issue. It is now time to unravel these questions with the hand of logical analysis.
And here again, we get inspiration, or rather the techniques, from work in logics
for common sense reasoning and scientific reasoning developed for artificial
intelligence.
Our claim then is that we must bring back logic into scientific methodology, on a par with computational and historical approaches. This move would
Logics of Generation and Evaluation
23
complement the existing historical account, and as far as its relation to computational approaches in scientific discovery is concerned, it may bring benefits
in regard to giving a foundation to the heuristics, to the logic(s) behind the
(human) processes of discovery in science. But then, how to integrate a logic
in the algorithmic design for scientific discovery? To this end, we must deal
with a logic that besides (or even aside from) its semantics and proof theory,
gives an account of the search strategy tied to a discovery system. That is, we
must provide the automatic procedures to operate a logic, its control strategy,
and its procedures to acquire new information without disturbing its coherence
and hopefully, achieve some learning in the end.
Logics of Generation and Evaluation
In our view, scientific methodology concerns an inquiry into the methods to
account for the dynamics of knowledge. Therefore, the earlier discovery vs.
justification dichotomy is transformed into the complementary study of the generation and evaluation of scientific theories. Thus, the pretentious terminology
of logics of discovery is best labeled as logics for development (as argued in
[Gut80, p.221]) or logics of generation and evaluation , in order to cover both
processes of generation and evaluation of theories, and thus vanishing the older
dichotomy.
By putting forward a logic for scientific discovery we claim no lack of rigour.
But what is clear is that standard deductive logic cannot account for abductive
or inductive types of reasoning. And the field is now ready to use new logics for
this purpose. An impressive body of results has been obtained about deduction
- and a general goal could be the study the wider field of human reasoning while
hanging on to these standards of rigour and clarity.
There are still many challenges ahead for the formal study of reasoning in
scientific discovery, such as giving an integrated account of deductive, inductive,
abductive and analogical styles of inference, the use of diagrams by logical
means, and in general the device of logical operations for theory building and
change. Already new logical research is moving into these directions. In Burger
and Heidema [BH02] degrees of ‘abductive boldness’ are proposed as spectrum
for inferential strength, ranging from cases with poor background information
to those with (almost) complete information. Systems dealing with several
notions of derivability all at once have also been proposed. A formula may be
‘unconditionally derived’ or ‘conditionally derived’, the latter case occurring
when a line in a proof asserts a formula which depends on hypotheses which
may be later falsified, thus pointing to a notion of proof which allows for
addition of lines which are non-deductively derived as well as for deletion of
them when falsifying instances occur. This account is found in the framework
24
ABDUCTIVE REASONING
of (ampliative) adaptive logics, a natural home for abductive inference [Meh05]
and for (enumerative) induction inference [Bat05] alike.
The formalization of analogical reasoning is still a growing area of research,
without a precise idea of what exactly an analogy amounts to. Perhaps research
on mathematical analogy [Pol54], work on analogy in cognitive science [Hel88]
or investigations into analogical argumentation theory recently proposed for
abduction [GW05], may serve to guide research in this direction. Finally, the
study of ‘diagrammatic reasoning’ is a research field on its own right [BE95],
showing that the logical language is not restricted to the two-dimensional left
to right syntactic representation, but its agenda still needs to be expanded on
research for non-deductive logics.
All the above suggests that the use of non-standard logics to model processes
in scientific practice, such as confirmation, falsification, explanation building
and theory improvement is, after all, a feasible project. Nevertheless, this claim
requires a broad conception of what logic is about.
6.
Discussion and Conclusions
In this chapter we claimed that the question of a logic of discovery is better
understood by dividing it into three questions, one on its purpose, another one
on its pursuit, and finally one on its achievement, for in many proposals there
is a clear incongruence between the search and the findings. The search for a
logic of discovery was generally (from antiquity to XIXth century) guided by
the aim of finding a formal characterization of the ways in which humans reason
in scientific discovery, but it has been full of shortcomings, as the products of
this search did not succeeded in describing the discovery processes involved
in human thought, and were rather directed to the selection and evaluation of
already newly discovered ideas in science. Amongst other things, the purely
descriptive nature of this enterprise gave place to the position that issues of
discovery should be studied psychologically and not philosophically, a view
endorsed by Karl Popper under a narrow view of discovery (one which concerns
exclusively the initial act of conceiving an idea). However, we showed that
under a broad view of discovery (one which conceives processes of discovery
at large), Popper’s position does include some aspects of discovery having to
do with the development of science. In contrast, the view endorsed by Herbert
Simon and his followers, claims that there is a normative dimension in the
treatment of logics of discovery and taking a finer perspective on the process
of discovery as a whole, there is indeed place for proposing several heuristic
strategies which characterize and model the generation and evaluation of new
ideas in science, at least for well-structured kind of problems.
Therefore, just as we have to acknowledge discovery as a process subject to
division, we have also to acknowledge an inescapable part in this process, and
accordingly agree with Popper’s motto: ‘admittedly, creative action can never
Logics of Generation and Evaluation
25
be fully explained’ [Pop75, p.179]. But this sentence naturally implies that
creative action may be partially explained (as argued in [Sav95]). The question
then, is to identify a proper line drawing a convenient division in order to carry
out an analysis of the generation of new ideas in science.
Our approach in this book in regard to logics of discovery is logical. In regard
to the question of purpose we hold a normative account in the methodology of
science, one that aims to describe the ways in which a logic of discovery should
behave in order to generate new knowledge. As to question of pursuit, we put
forward the claim that these logics must find their expression in modern logical
systems, as those devised in artificial intelligence.
Chapter 2
WHAT IS ABDUCTION?
OVERVIEW AND
PROPOSAL FOR INVESTIGATION
1.
Introduction
The general purpose of this chapter is to give an overview of the field of
abduction in order to provide the conceptual framework of our overall study
of abductive reasoning and its relation to explanatory reasoning in subsequent
chapters. It is naturally divided into seven parts. After this brief introduction, in
the second part (section 2) we motivate our study via several examples that show
that this type of reasoning pervades common sense reasoning as well as scientific
inquiry. Moreover, abduction may be studied from several perspectives; as a
product or as a process, the latter in turn leading to either the process of
hypotheses construction or of hypotheses selection and finally, abduction makes
sense in connection with its sibling induction, but there are several confusions
arising from this relation. In the third part (section 3), we turn to the founding
father of abduction, the American pragmatist Charles S. Peirce and present
very briefly his theory of abduction. In the fourth part (section 4), we review
abduction in the philosophy of science, as it is related with the central topic of
scientific explanation, existing both in the received view as well as in neglected
ones in this field. In the fifth part (section 5), we present abduction in the field
of artificial intelligence and show that it holds a place as a logical inference, as
a computational process as well as in theories of belief revision. In the sixth
part (section 6), we give an overview of two other fields in which abduction
is found, namely in linguistics and in mathematics (neither of which is further
pursued in this book). Finally, in the seventh part of this chapter (section 7), we
tie up our previous overview by proposing a general taxonomy for abduction,
one that allows two different abductive triggers (novelty and anomaly) , which
in turn lead to different abductive procedures; and one that allows for several
outcomes: facts, rules, or even whole new theories. On our view, abduction is
28
ABDUCTIVE REASONING
not a new notion of inference. It is rather a topic-dependent practice of scientific
reasoning, which can be supported by various types of logical inferences or
computational processes.
Each and every part of this overview chapter is further elaborated in subsequent chapters. Thus, the purpose here is just to motivate our study and set the
ground for the rest of the book.
2.
What is Abduction?
A central theme in the study of human reasoning is the construction of explanations that give us an understanding of the world we live in. Broadly speaking,
abduction is a reasoning process invoked to explain a puzzling observation. A
typical example is a practical competence like medical diagnosis. When a doctor observes a symptom in a patient, she hypothesizes about its possible causes,
based on her knowledge of the causal relations between diseases and symptoms.
This is a practical setting. Abduction also occurs in more theoretical scientific
contexts. For instance, it has been claimed [Han61],[CP, 2.623] that when Kepler discovered that Mars had an elliptical orbit, his reasoning was abductive.
But, abduction may also be found in our day-to-day common sense reasoning.
If we wake up, and the lawn is wet, we might explain this observation by assuming that it must have rained, or by assuming that the sprinklers had been
on. Abduction is thinking from evidence to explanation, a type of reasoning
characteristic of many different situations with incomplete information.
The history of this type of reasoning goes back to Antiquity. It has been compared with Aristotle’s apagoge [CP, 2.776,5.144] which intended to capture a
non-strictly deductive type of reasoning whose conclusions are not necessary,
but merely possible (not to be confused with epagoge, the Aristotelian term for
induction). Later on, abduction as reasoning from effects to causes is extensively discussed in Laplace’s famous memoirs [Lap04, Sup96] as an important
methodology in the sciences. In the modern age, this reasoning was put on the
intellectual agenda under the name ‘abduction’ by C.S. Peirce [CP, 5.189].
To study a type of reasoning that occurs in contexts as varied as scientific
discovery, medical diagnosis, and common sense, suitably broad features must
be provided, that cover a lot of cases, and yet leave some significant substance to
the notion of abduction. The purpose of this preliminary chapter is to introduce
these, which will lead to the more specific questions treated in subsequent
chapters. But before we start with a more general analysis, let us expand our
stock of examples.
What is Abduction?
29
Examples
The term ‘abduction’ is used in the literature for a variety of reasoning processes.
We list a few, partly to show what we must cover, and partly, to show what we
will leave aside.
1. Common Sense: Explaining observations with simple facts.
All you know is that the lawn gets wet either when it rains, or when the
sprinklers are on. You wake up in the morning and notice that the lawn is
wet. Therefore you hypothesize that it rained during the night or that the
sprinklers had been on.
2. Common Sense: Laying causal connections between facts.
You observe that a certain type of clouds (nimbostratus) usually precede
rainfall. You see those clouds from your window at night. Next morning
you see that the lawn is wet. Therefore, you infer a causal connection
between the nimbostratus at night, and the lawn being wet.
3. Common Sense: Facing a Contradiction.
You know that rain causes the lawn to get wet, and that it is indeed raining.
However, you observe that the lawn is not wet. How could you explain this
anomaly?
4. Statistical Reasoning: Medical Diagnosis1 .
Jane Jones recovered quite rapidly from a streptococci infection after she
was given a dose of penicillin. Almost all streptococcus infections clear
up quickly upon administration of penicillin, unless they are penicillinresistant, in which case the probability of quick recovery is rather small.
The doctor knew that Jane’s infection is of the penicillin-resistant type,
and is completely puzzled by her recovery. Jane Jones then confesses that
her grandmother had given her Belladonna, a homeopathic medicine that
stimulates the immune system by strengthening the physiological resources
of the patient to fight infectious diseases.
The examples so far are fairly typical of what our later analysis can deal
with. But actual abductive reasoning can be more complicated than this. For
instance, even in common sense settings, there may be various options, which
are considered in some sequence, depending on your memory and ‘computation
strategy’.
5. Common Sense: When something does not work.
1 This
is an adaptation of Hempel’s famous illustration of his Inductive-Statistical model of explanation as
presented in [Sal92]. The part about homeopathy is entirely mine, however.
30
ABDUCTIVE REASONING
You come into your house late at night, and notice that the light in your
room, which is always left on, is off. It has being raining very heavily, and
so you think some power line went down, but the lights in the rest of the
house work fine. Then, you wonder if you left both heaters on, something
which usually causes the breakers to cut off, so you check them: but they
are OK. Finally, a simpler explanation crosses your mind. Maybe the light
bulb of your lamp which you last saw working well, is worn out, and needs
replacing.
So, abduction involves computation over various candidates, depending on
your background knowledge. In a scientific setting, this means that abductions
will depend on the relevant background theory, as well as one’s methodological
‘working habits’. We mention one often-cited example; even though we should
state clearly at this point that it goes far beyond what we shall eventually deal
with in our analysis.
6. Scientific Reasoning: Kepler’s discovery2 .
One of Johannes Kepler’s great discoveries was that the orbit of the planets
is elliptical rather than circular. What initially led to this discovery was his
observation that the longitudes of Mars did not fit circular orbits. However,
before even dreaming that the best explanation involved ellipses instead of
circles, he tried several other forms. Moreover, Kepler had to make several
other assumptions about the planetary system, without which his discovery
does not work. His heliocentric view allowed him to think that the sun, so
near to the center of the planetary system, and so large, must somehow cause
the planets to move as they do. In addition to this strong conjecture, he also
had to generalize his findings for Mars to all planets, by assuming that the
same physical conditions obtained throughout the solar system. This whole
process of explanation took many years.
It will be clear that the Kepler example has a loose end, so to speak. How we
construct the abductive explanation depends on what we take to be his scientific background theory. This is a general feature of abductions: an abductive
explanation is always an explanation with respect to some body of beliefs. But
even this is not the only parameter that plays a role. One could multiply the
above examples, and find still further complicating factors. Sometimes, no single obvious explanation is available, but rather several competing ones - and we
have to select. Sometimes, the explanation involves not just advancing facts or
rules in our current conceptual frame, but rather the creation of new concepts,
that allow for new description of the relevant phenomena. Evidently, we must
draw a line somewhere in our present study.
2 This
example is a simplification of one in [Han61].
What is Abduction?
31
All our examples were instances of reasoning in which an abductive explanation is needed to account for a certain phenomenon. Is there more unity
than this? At first glance, the only clear common feature is that these are not
cases of ordinary deductive reasoning, and this for a number of reasons. In
particular, the abductive explanations produced might be defeated. Maybe the
lawn is wet because children have been playing with water. Co-occurrence of
clouds and the lawn being wet does not necessarily link them in a causal way.
Jane’s recovery might after all be due to a normal process of the body. What we
learn subsequently can invalidate an earlier abductive conclusion. Moreover,
the reasoning involved in these examples seems to go in reverse to ordinary
deduction (just as analysis runs in reverse to synthesis, cf. chapter 1), as all
these cases run from evidence to hypothesis, and not from data to conclusion,
as it is usual in deductive patterns. Finally, describing the way in which an
explanation is found, does not seem to follow specific rules. Indeed, the precise
nature of Kepler’s ‘discovery’ remains under intensive debate3 .
What we would like to do is the following. Standard deductive logic cannot
account for the above types of reasoning. Last century, under the influence of
foundational research in mathematics, there has been a contraction of concerns
in logic to this deductive core. The result was a loss in breadth, but also a
clear gain in depth. By now, an impressive body of results has been obtained
about deduction - and we would like to study the wider field of abduction while
hanging on to these standards of rigor and clarity. Of course, we cannot do
everything at once, and achieve the whole agenda of logic in its traditional
open-ended form. We want to find some features of abduction that allow for
concrete logical analysis, thereby extending the scope of standard methods. In
the next section, we discuss three main features, that occur across all of the
above examples (properly viewed), that will be important in our investigation.
Three Faces of Abduction
We shall now introduce three broad oppositions that help in clarifying what
abduction is about. At the end, we indicate how these will be dealt with in this
book.
3 For
Peirce, Kepler’s reasoning was a prime piece of abduction [CP, 1.71,2.96], whereas for Mill it was
merely a description of the facts [Mill 58, Bk III, ch II. 3], [CP, 1.71–4]. Even nowadays one finds very
different reconstructions. While Hanson presents Kepler’s heliocentric view as an essential assumption
[Han61], Thagard thinks he could make the discovery assuming instead that the earth was stationary and
the sun moves around it [Tha92]. Still a different account of how this discovery can be made is given in
[SLB81, LSB87].
32
ABDUCTIVE REASONING
Abduction: Product or Process?
The logical key words of judgment and proof are nouns that denote either an
activity, indicated by their corresponding verb, or the result of that activity. In
just the same way, the word abduction may be used both to refer to a finished
product, the abductive explanation, or to an activity, the abductive process
that led to that abductive explanation. These two uses are closely related. An
abductive process produces an abductive explanation as its product, but the two
are not the same. Note that we can make the same distinction with regard to
the notion of explanation. We may refer to a finished product, the explanatory
argument, or to the process of constructing such explanation, the explanatory
process.
One can relate these distinctions to more traditional ones. An example is the
earlier opposition of ‘context of discovery’ versus ‘context of justification’ (cf.
chapter 1). Kepler’s abductive explanation–product “the orbit of the planets is
elliptical”, which justifies the observed facts, does not include the abductive–
process of how he came to make this discovery. The context of discovery has
often been taken to be purely psychological, but this does not preclude its exact
study. Cognitive psychologists study mental patterns of discovery, learning
theorists in AI study formal hypothesis formation, and one can even work with
concrete computational algorithms that produce abductive explanations. To
be sure, it is a matter of debate whether Kepler’s reasoning may be modeled
by a computer. (For a computer program that claims to model this particular
discovery, cf. [SLB81].) However this may be, one can certainly write simple
programs that produce common sense explanations of ‘why the lawn is wet’,
as we will show later on.
Moreover, once produced, abductive explanations are public objects of “justification", which can be checked and tested by independent logical criteria. An
abductive explanation is an element in an explanatory argument, which is in
turn the product of an explanatory process. So, just as abduction, explanation
has also its product and its process sides. The overall procedure as to how
these two reasoning processes can be distinguished and how they are connected
to each other can be exemplified as follows: We may distinguish between the
process of the discovery of a geometrical proof, which produces the required
postulates to be used as premisses and possibly some auxiliary constructions,
and between the actual process of proving the desired theorem, which in turn
produces a proof as a logical argument.
The product–process distinction has been recognized by logicians [Bet59,
vBe93], in the context of deductive reasoning, as well as by philosophers of
science [Rub90, Sal90] in the context of scientific explanation. Both lead to
interesting questions by themselves, and so does their interplay. Likewise, these
two faces of abduction are both relevant for our study. On the product side, our
focus will be on conditions that give a piece of information explanatory force,
What is Abduction?
33
and on the process side, we will be concerned with the design of algorithms
that produce abductive explanations.
Abduction: Construction or Selection?
Given a fact to be explained, there are often several possible abductive explanations, but only one (or a few) that counts as the best one. Pending subsequent
testing, in our common sense example of light failure, several abductive explanations account for the unexpected darkness of the room (power line down,
breakers cut off, bulb worn out). But only one may be considered as ‘best’
explaining the event, namely the one that really happened. But other preference criteria may be appropriate, too, especially when we have no direct test
available.
Thus, abduction is connected to both hypothesis construction and hypothesis selection. Some authors consider these processes as two separate steps,
construction dealing with what counts as a possible abductive explanation, and
selection with applying some preference criterion over possible abductive explanations to select the best one. Other authors regard abduction as a single
process by which a single best explanation is constructed. Our position is
an intermediate one. We will split abduction into a first phase of hypothesis
construction, but also acknowledge a next phase of hypothesis selection. We
shall mainly focus on a characterization of possible abductive explanations. We
will argue that the notion of a ‘best abductive explanation’ necessarily involves
contextual aspects, varying from application to application. There is at least a
new parameter of preference ranking here. Although there exist both a philosophical tradition on the logic of preference [Wri63], and logical systems in
AI for handling preferences that may be used to single out best explanations
[Sho88, DP91b], the resulting study would take us too far afield.
Abduction vs. Induction
Once beyond deductive logic, diverse terminologies are being used. Perhaps
the most widely used term is inductive reasoning [Mill 58, Sal90, HHN86,
Tha88, Fla95, Mic94]. Abduction is another focus, and it is important, at
least, to clarify its relationship to induction. For C.S. Peirce, as we shall see,
‘deduction’, ‘induction’ and ‘abduction’ formed a natural triangle – but the
literature in general shows many overlaps, and even confusions.
Since the time of John Stuart Mill (1806-1873), the technical name given
to all kinds of non-deductive reasoning has been ‘induction’, though several
methods for discovery and demonstration of causal relationships [Mill 58]
were recognized. These included generalizing from a sample to a general property, and reasoning from data to a causal hypothesis (the latter further divided
into methods of agreement, difference, residues, and concomitant variation). A
more refined and modern terminology is ‘enumerative induction’ and ‘explana-
34
ABDUCTIVE REASONING
tory induction’, of which ‘inductive generalization’, ‘inductive projection’,
‘statistical syllogism’, ‘concept formation’ are some instances. Such a broad
connotation of the term induction continues to the present day. For instance,
in the computational philosophy of science, induction is understood “in the
broad sense of any kind of inference that expands knowledge in the face of
uncertainty” [Tha88].
Another ‘heavy term’ for non-deductive reasoning is statistical reasoning,
introducing a probabilistic flavour, like our example of medical diagnosis, in
which possible explanations are not certain but only probable. Statistical reasoning exhibits the same diversity as abduction. First of all, just as the latter is
strongly identified with backwards deduction (as we shall see later on in this
chapter), the former finds its ‘reverse notion’ in probability4 . Both abduction
and statistical reasoning are closely linked with notions like confirmation (the
testing of hypothesis) and likelihood (a measure for alternative hypotheses).
On the other hand, some authors take induction as an instance of abduction.
Abduction as inference to the best explanation is considered by Harman [Har65]
as the basic form of non-deductive inference, which includes (enumerative)
induction as a special case.
This confusion returns in artificial intelligence. ‘Induction’ is used for the
process of learning from examples – but also for creating a theory to explain
the observed facts [Sha91], thus making abduction an instance of induction.
Abduction is usually restricted to producing abductive explanations in the form
of facts. When the explanations are rules, it is regarded as part of induction.
The relationship between abduction and induction (properly defined) has been
the topic for workshops in AI conferences [ECAI96] and edited books [FK00].
To clear up all these conflicts, one might want to coin new terminology altogether. Many authors write as if there were pre-ordained, reasonably clear
notions of abduction and its rivals, which we only have to analyze to get a clear
picture. But these technical terms may be irretrievably confused in their full
generality, burdened with the debris of defunct philosophical theories. Therefore, I have argued for a new term of “explanatory reasoning” in [Ali96a],
trying to describe its fundamental aspects without having to decide if they are
instances of either abduction or induction. In this broader perspective, we can
also capture explanation for more than one instance or for generalizations, –
which we have not mentioned at all – and introduce further fine–structure. For
example, given two observed events, in order to find an explanation that accounts for them, it must be decided whether they are causally connected (eg.
entering the copier room and the lights going on), correlated with a common
cause (e.g. observing both the barometric pressure and the temperature drop4 The
problem in probability is: given an stochastic model, what can we say about the outcomes? The
problem in statistics is the reverse: given a set of outcomes, what can we say about the model?.
What is Abduction?
35
ping at the same time), or just coincidental without any link (you reading this
paragraph in place A while I revise it somewhere in place B). But in this book,
we shall concentrate on explanatory reasoning from simple facts, giving us
enough variety for now. Hopefully, this case study of abduction will lead to
broader clarity of definition as well.
More precisely, we shall understand abduction as reasoning from a single
observation to its abductive explanations, and induction as enumerative induction from samples to general statements. While induction explains a set of
observations, abduction explains a single one. Induction makes a prediction for
further observations, abduction does not (directly) account for later observations. While induction needs no background theory per se, abduction relies on
a background theory to construct and test its abductive explanations. (Note that
this abductive formulation does not commit ourselves to any specific logical
inference, kind of observation, or form of explanation.)
As for their similarities, induction and abduction are both non-monotonic
types of inference (to be defined in chapter 3), and both run in opposite direction
to standard deduction. In non-monotonic inference, new premises may invalidate a previous valid argument. In the terminology of philosophers of science,
non-monotonic inferences are not erosion proof [Sal92]. Qua direction, induction and abduction both run from evidence to explanation. In logical terms, this
may be viewed as going from a conclusion to (part of) its premises, in reverse
of ordinary deduction. We will return to these issues in much more detail in
our logical chapter 3.
3.
The Founding Father: C.S. Peirce
The literature on abduction is so vast, that we cannot undertake a complete
survey here. What we shall do is survey some highlights, starting with the
historical sources of the modern use of the term. In this field, all roads from
XXth century onwards lead back to the work of C.S. Peirce. Together with
the other sources to be discussed, the coming sections will lead up to further
parameters for the general taxonomy of abduction that we propose toward the
end of this chapter.
Understanding Peirce’s Abduction
Charles Sanders Peirce (1839-1914), the founder of American pragmatism was
the first philosopher to give to abduction a logical form, and hence his relevance
to our study. However, his notion of abduction is a difficult one to unravel. On
the one hand, it is entangled with many other aspects of his philosophy, and
on the other hand, several different conceptions of abduction evolved in his
thought. The notions of logical inference and of validity that Peirce puts
forward go beyond our present understanding of what logic is about. They
36
ABDUCTIVE REASONING
are linked to his epistemology, a dynamic view of thought as logical inquiry,
and correspond to a deep philosophical concern, that of studying the nature of
synthetic reasoning.
We will point out a few general aspects of his later theory of abduction, and
then concentrate on some of its more logical aspects. For a more elaborate
analysis of abduction in connection to Peirce’s epistemology and pragmatism,
cf. chapter 7.
The Key Features of Peircean Abduction
For Peirce, three aspects determine whether a hypothesis is promising: it must
be explanatory, testable, and economic. A hypothesis is an explanation if it
accounts for the facts. Its status is that of a suggestion until it is verified, which
explains the need for the testability criterion.
Finally, the motivation for the economic criterion is twofold: a response to
the practical problem of having innumerable explanatory hypotheses to test,
as well as the need for a criterion to select the best explanation amongst the
testable ones.
For the explanatory aspect, Pierce gave the following often-quoted logical
formulation [CP, 5.189]:
The surprising fact, C, is observed.
But if A were true, C would be a matter of course.
Hence, there is reason to suspect that A is true.
This formulation has played a key role in Peirce scholarship, and it has
been the point of departure of many classic studies on abductive reasoning
in artificial intelligence [FK00], such as in logic programming [KKT95],
knowledge acquisition [KM90] and natural language processing [HSAM90].
Nevertheless, these approaches have paid little attention to the elements of
this formulation and none to what Peirce said elsewhere in his writings. This
situation may be due to the fact that his philosophy is very complex and not easy
to be implemented in the computational realm. The notions of logical inference
and of validity that Peirce puts forward go beyond logical formulations. In our
view, however, there are several aspects of Peirce’s abduction which are tractable
and may be implemented using machinery of AI, such as that found in theories
of belief revision (cf. chapter 8).
Our own understanding of abductive reasoning reflects this Peircean diversity
in part, taking abduction as a style of logical reasoning that occurs at different
levels and in several degrees. These will be reflected in our proposal for a
taxonomy with several ‘parameters’ for abductive reasoning.
In AI circles, this formulation has been generally interpreted as the following
logical argument–schema:
What is Abduction?
37
C
A→C
A
Where the status of A is tentative (it does not follow as a logical consequence
from the premises).
However intuitive, this interpretation certainly captures neither the fact that
C is surprising nor the additional criteria Peirce proposed. Moreover, the interpretation of the second premise is not committed to material implication. In
fact, some have argued (cf. [FK00]) that this is a vacuous interpretation and
favour one of classical logical entailment (A |= C). But other interpretations
are possible; any other nonstandard form of logical entailment or even a computational process in which A is the input and C the output, are all feasible
interpretations for “if C were true, A would be a matter of course".
The additional Peircean requirements of testability and economy are not
recognized as such in AI, but are nevertheless incorporated. The latter criterion
is implemented as a further selection process to produce the best explanation,
since there might be several formulae that satisfy the above formulation but are
not appropriate as explanations. As for the testability requirement, when the
second premise is interpreted as logical entailment this requirement is trivialized, since given that C is true, in the simplest sense of ‘testable’, A will always
be testable.
We leave here the reconstruction of Peirce’s notion of abduction. For further
aspects, we refer to chapter 7. The additional criteria of testability and economy
are not part of our general framework for abduction. Testability as understood
by Peirce is an extra-logical empirical criterion, while economy concerns the
selection of explanations, which we already put aside as a further stage of
abduction requiring a separate study.
4.
Philosophy of Science
Peirce’s work stands at the crossroads of many traditions, including logic,
epistemology, and philosophy of science. Especially, the latter field has continued many of his central concerns. Abduction is clearly akin to core notions
of modern methodology, such as explanation, induction, discovery, and heuristics. We have already discussed a connection between process-product aspects
of both abduction and explanation and the well-known division between contexts of discovery and justification. We shall discuss several further points of
contact in chapters 5 and 6 below. But for the moment, we start with a first
foray.
38
ABDUCTIVE REASONING
The ‘Received View’
The dominant trend in philosophy has focused on abduction and explanation as
products rather than a process, just as it has done for other epistemic notions.
Aristotle, Mill, and in this century, the influential philosopher of science Carl
Hempel, all based their accounts of explanation on proposing criteria to characterize its products. These accounts generally classify into argumentative and
non-argumentative types of explanation [Rub90, Sal90, Nag79]. Of particular
importance to us is the ‘argumentative’ Hempelian tradition. Its followers aim
to model empirical why-questions, whose answers are scientific explanations
in the form of arguments. In these arguments, the ‘explanandum’ (the fact
to be explained) is derived (deductively or inductively) from the ‘explananda’
(that which does the explaining) supplemented with relevant ‘laws’ (general
or statistical) and ‘initial conditions’. For instance, the fact that an explosion
occurred may be explained by my lighting the match, given the laws of physics,
and initial conditions to the effect that oxygen was present, the match was not
wet, etcetera.
In its deductive version, the Hempelian account, found in many standard texts
on the philosophy of science [Nag79, Sal90] is called deductive-nomological,
for obvious reasons. But its engine is not just standard deduction. Additional
restrictions must be imposed on the relation between explananda and explanandum, as neither deduction nor induction is a sufficient condition for genuine
explanation. To mention a simple example, every formula is derivable from
itself (ϕ ⊢ ϕ), but it seems counterintuitive, or at least very uninformative, to
explain anything by itself.
Other, non-deductive approaches to explanation exist in the literature. For
instance, [Rub90] points at these two:
[Sal77, p.159] takes them to be: “an assemblage of factors that are statistically relevant".
While [vFr80, p.134] makes them simply: “an answer".
For Salmon, the question is not how probable the explanans renders the
explanandum, but rather whether the facts adduced make a difference to the
probability of the explanandum. Moreover, this relationship need not be in
the form of an argument. For van Fraassen, a representative of pragmatic
approaches to explanation, the explanandum is a contrastive why-question.
Thus, rather than asking “why ϕ?", one asks “why ϕ rather than γ?". The
pragmatic view seems closer to abduction as a process, and indeed, the focus
on questions introduces some dynamics of explaining. Still, it does not tell us
how to produce explanations.
There are also alternative deductive approaches. An example is the work
of Rescher [Res78], which introduces a direction of thought. Interestingly,
this establishes a temporal distinction between ‘prediction’ and ‘retroduction’
(Rescher’s term for abduction), by marking the precedence of the explanandum
What is Abduction?
39
over the hypothesis in the latter case. Another, and quite famous deductivist
tradition is Popper’s logic of scientific discovery [Pop59], which we already
analyzed in chapter 1. Its method of conjectures and refutations proposes the
testing of hypotheses, by attempting to refute them.
What is common to all these approaches in the philosophy of science is the
importance of a hidden parameter in abduction. Whether with Hempel, Salmon,
or Popper, scientific explanation never takes place in isolation, but always in the
context of some background theory. This additional parameter will become
part of our general scheme to be proposed below.
The ‘Neglected View’
Much more marginal in the philosophy of science are accounts of abduction and
explanation that focus on the processes as such. One early author emphasizing
explanation as a process of discovery is Hanson ([Han61]), who gave an account
of patterns of discovery, recognizing a central role for abduction (which he calls
‘retroduction’). Also relevant here is the work by Lakatos ([Lak76]), a critical
response to Popper’s logic of scientific discovery. As already noted in chapter
1, For Lakatos, there is only a fallibilistic logic of discovery, which is neither
psychology nor logic, but an independent discipline, the logic of heuristics. He
pays particular attention to processes that created new concepts in mathematics
– often referring to Polya ([Pol45]) as the founding father of heuristics in
mathematical discovery5 . We will come back to this issue later in the chapter,
when presenting further fields of application.
What these examples reveal is that in science, explanation involves the invention of new concepts, just as much as the positing of new statements (in
some fixed conceptual framework). So far, this has not led to extensive formal
studies of concept formation, similar to what is known about deductive logic.
(Exceptions that prove the rule are occasional uses of Beth’s Definability Theorem in the philosophical literature. A similar lacuna vis-a-vis concept revision
exists in the current theory of belief revision in AI, cf. chapter 8.). We are
certainly in sympathy with the demand for conceptual change in explanation –
but this topic will remain beyond the technical scope of this book (however, cf.
chapter 4 for a brief discussion).
5.
Artificial Intelligence
Our next area of comparison is artificial intelligence. The transition with
the previous section is less abrupt than it may be seem. It has often been
noted, by looking at the respective research agendas, that artificial intelligence
5 In
fact Polya contrasts two types of arguments. A demonstrative syllogism in which from A → B, and B
false, ¬A is concluded, and a heuristic syllogism in which from A → B, and B true, it follows that A is
more credible. The latter, of course, recalls Peirce’s abductive formulation.
40
ABDUCTIVE REASONING
is philosophy of science, pursued by other means (cf. [Tan92]). Research on
abductive reasoning in AI dates back to 1973 [Pop73], but it is only fairly
recently that it has attracted great interest, in areas like logic programming
[KKT95], knowledge assimilation [KM90], and diagnosis [PG87], to name
just a few. Abduction is also coming up in the context of data bases and
knowledge bases, that is, in mainstream computer science.
In this setting, the product-process distinction has a natural counterpart,
namely, in logic-based vs computational-based approaches to abduction.
While the former focuses on giving a semantics to the logic of abduction,
usually defined as ‘backwards deduction plus additional conditions’, the latter
is concerned with providing algorithms to produce abductions.
It is impossible to give an overview here of this exploding field. Therefore,
we limit ourselves to (1) a brief description of abduction as logical inference,
(2) a presentation of abduction in logic programming, and (3) a sketch of the
relevance of abductive thinking in knowledge representation. There is much
more in this field of potential philosophical interest, however. For abduction in
bayesian networks, connectionism, and many other angles, the reader is advised
to consult [Jos94, PR90, Kon96, Pau93, AAA90, FK00].
Abduction as Logical Inference
The general trend in logic based approaches to abduction in AI interprets abduction as backwards deduction plus additional conditions. The idea of abduction as deduction in reverse plus additional conditions brings it very close to
deductive-nomological explanation in the Hempel style (cf. Chapter 5), witness
the following format. What follows is the ‘standard version’ of abduction as
deduction via some consistent additional assumption, satisfying certain extra
conditions. It combines some common requirements from the literature (cf.
[Kon90, KKT95, MP93] and chapter 3 for further motivation). Note however
that given the distinction (product vs. process) and the terminology introduced
so far (abductive explanation, explanatory argument), the following definition
is one which characterizes the conditions for an abductive explanation to be part
of an explanatory argument. Thus this is a characterization of the forward inference from theory and abduced hypothesis to evidence (rather than the backward
inference –which is indeed what mimics the abductive process– triggered by an
evidence taking place with respect to a background theory and producing an
abductive explanation), and we will referred to it as an (abductive) explanatory
argument, in order to highlight that the argument is explanatory, while keeping
in mind it aims to characterize the conditions for an abductive explanation:
(Abductive) Explanatory Argument:
Given a theory Θ (a set of formulae) and a formula ϕ (an atomic formula), α is an
abductive explanation if
1 Θ ∪ α |= ϕ
2 α is consistent with Θ
What is Abduction?
41
3 α is ‘minimal’ (there are several ways to characterize minimality, to be discussed
in chapter 3).
4 α has some restricted syntactical form (usually an atomic formula or a conjunction
of them).
An additional condition not always made explicit is that Θ |= ϕ. This says
that the fact to be explained should not already follow from the background
theory alone. Sometimes, the latter condition figures as a precondition for an
abductive problem.
What can one say in general about the properties of such an ‘enriched’ notion
of consequence? As we have mentioned before, a new logical trend in AI studies
variations of classical consequence via their ‘structural rules’, which govern
the combination of basic inferences, without referring to any special logical
connectives. (Cf. the analysis of non-monotonic consequence relations in
AI of [Gab96], [KLM90], and the analysis of dynamic styles of inference in
linguistics and cognition in [vBe90].) Perhaps the first example of this approach
in abduction is the work in [Fla95] – and indeed our analysis in chapter 3 will
follow this same pattern.
Abduction in Logic Programming
Logic Programming [LLo87, Kow79] was introduced by Bob Kowalski and
Alan Colmerauer in 1974, and is implemented as (amongst others) the programming language Prolog. It is inspired by first-order logic, and it consists
of logic programs, queries, and a underlying inferential mechanism known as
resolution6 .
Abduction emerges naturally in logic programming as a ‘repair mechanism’,
completing a program with the facts needed for a query to succeed. This may
be illustrated by our rain example (1) from the introduction in Prolog:
Program P :
lawn-wet ← rain.
lawn-wet ← sprinklers-on.
Query q: lawn-wet.
6 Roughly speaking, a Prolog program P is an ordered set of rules and facts. Rules are restricted to hornclause form A ← L1 , . . . , Ln in which each Li is an atom Ai . A query q (theorem) is posed to program
P to be solved (proved). If the query follows from the program, a positive answer is produced, and so
the query is said to be successful. Otherwise, a negative answer is produced, indicating that the query
has failed. However, the interpretation of negation is ‘by failure’. That is, ‘no’ means ‘it is not derivable
from the available information in P ’ – without implying that the negation of the query ¬q is derivable
instead. Resolution is an inferential mechanism based on refutation working backwards: from the negation
of the query to the data in the program. In the course of this process, valuable by-products appear: the
so-called ‘computed answer substitutions’, which give more detailed information on the objects satisfying
given queries.
42
ABDUCTIVE REASONING
Given program P , query q does not succeed because it is not derivable from
the program. For q to succeed, either one (or all) of the facts ‘rain’, ‘sprinklerson’, ‘lawn-wet’ would have to be added to the program. Abduction is the process
by which these additional facts are produced. This is done via an extension of the
resolution mechanism that comes into play when the backtracking mechanism
fails. In our example above, instead of declaring failure when either of the
above facts is not found in the program, they are marked as ‘hypothesis’ (our
abductive explanations), and proposed as those formulas which, if added to the
program, would make the query succeed.
In actual Abductive Logic Programming [KKT95], for these facts to be
counted as abductions, they have to belong to a pre-defined set of ‘abducibles’,
and to be verified by additional conditions (so-called ‘integrity constraints’), in
order to prevent a combinatorial explosion of possible explanations.
In logic programming, the procedure for constructing explanations is left entirely to the resolution mechanism, which affects not only the order in which the
possible explanations are produced, but also restricts the form of explanations.
Notice that rules cannot occur as abducibles, since explanations are produced
out of sub-goal literals that fail during the backtracking mechanism. Therefore,
our common sense example (2) in which a causal connection is abduced to
explain why the lawn is wet, cannot be implemented in logic programming7 .
The additional restrictions select the best hypothesis. Thus, processes of both
construction and selection of explanations are clearly marked in logic programming8 .
Logic programming does not use blind deduction. Different control mechanisms for proof search determine how queries are processed. This additional
degree of freedom is crucial to the efficiency of the enterprise. Hence, different
control policies will vary in the abductions produced, their form and the order
in which they appear. To us, this variety suggests that the procedural notion of
abduction is intensional, and must be identified with different practices, rather
than with one deterministic fixed procedure.
Abduction and Theories of Epistemic Change
Most of the logic-based and computation-based approaches to abduction reviewed in the preceding sections assume that neither the explanandum nor its
negation is derivable from the background theory (Θ |= ϕ, Θ |= ¬ϕ). This
leaves no room to represent problems like our common sense light example (5)
in which the theory expects the contrary of our observation. (Namely, that the
7 At least, this is how the implementation of abduction in logic programming stands as of now.
It is of course
possible to write extended programs that produce these type of explanations.
8 Another relevant connection here is research in ‘inductive logic programming’ ([Mic94], [FK00]) which
integrates abduction and induction.
What is Abduction?
43
light in my room is on.) These are cases where the theory needs to be revised in
order to account for the observation. Such cases arise in practical settings like
diagnostic reasoning [PR90], belief revision in databases [AD94] and theory
refinement in machine learning [SL90, Gin88].
When importing revision into abductive reasoning, an obvious related territory is theories of belief change in AI. Mostly inspired by the work of
Gärdenfors [Gar88] (a work whose roots lie in the philosophy of science), these
theories describe how to incorporate a new piece of information into a database,
a scientific theory, or a set of common sense beliefs. The three main types of
belief change are operations of ‘expansion’, ‘contraction’, and ‘revision’. A
theory may be expanded by adding new formulas, contracted by deleting existing formulas, or revised by first being contracted and then expanded. These
operations are defined in such a way as to ensure that the theory or belief system
remains consistent and suitably ‘closed’ when incorporating new information.
Our earlier cases of abduction may be described now as expansions, where
the background theory gets extended to account for a new fact. What is added
are cases where the surprising observation (in Peirce’s sense) calls for revision.
Either way, this perspective highlights the essential role of the background
theory in explanation. Belief revision theories provide an explicit calculus of
modification for both cases. It must be clarified however, that changes occur
only in the theory, as the situation or world to be modeled is supposed to be
static, only new information is coming in. Another important type of epistemic
change studied in AI is that of update, the process of keeping beliefs up–to–
date as the world changes. We will not analyze this second process here – even
though we are confident that it can be done in the same style proposed here.
Evidently, all this ties in very neatly with our earlier findings. (For instance,
the theories involved in abductive belief revision might be structured like those
provided by our discussion, or by cues from the philosophy of science.) We
will explore this connection in more detail in chapter 8.
6.
Further Fields of Application
The above survey is by no means exhaustive. Abduction occurs in many other
research areas, of which we will mention two: linguistics and mathematics.
Although we will not pursue these lines elsewhere in this book, they do provide
a better perspective on the broader interest of our topic.
Abduction in Linguistics
In linguistics, abduction has been proposed as a process for natural language
interpretation [HSAM90], where our ‘observations’ are the utterances that we
hear (or read). More precisely, interpreting a sentence in discourse is viewed as
providing a best explanation of why the sentence would be true. For instance, a
44
ABDUCTIVE REASONING
listener or reader abduces assumptions in order to resolve references for definite
descriptions (“the cat is on the mat” invites you to assume that there is a cat and
a mat), and dynamically accommodates them as presupposed information for
the sentence being heard or read.
Abduction also finds a place in theories of language acquisition. Most prominently, Chomsky proposed that learning a language is a process of theory construction. A child ‘abduces’ the rules of grammar guided by her innate knowledge of language universals. Indeed in [Cho72], he refers to Peirce’s justification for the logic of abduction, – based on the human capacity for ‘guessing the
right hypotheses’, to reinforce his claim that language acquisition from highly
restricted data is possible.
Abduction has also been used in the semantics of questions. Questions are
then the input to abductive procedures generating answers to them. Some work
has been done in this direction in natural language as a mechanism for dealing
with indirect replies to yes-no questions [GJK94]. Of course, the most obvious
case where abduction is explicitly called for are “Why" questions, inviting the
other person to provide a reason or cause.
Abduction also connects up with linguistic presuppositions, which are progressively accommodated during a discourse. The phenomenon of accommodation is a non-monotonic process, in which presuppositions are not direct
updates for explicit utterances, but rather abductions that can be refuted because
of later information. Accommodation can be described as a repair strategy in
which the presuppositions to be accommodated are not part of the background.
In fact, the linguistic literature has finer views of types of accommodation (cf.
the ‘local’/‘global’ distinction in [Hei83]), which might correspond to the two
abductive ‘triggers’ proposed in the next section. A broader study on presuppositions which considers abductive mechanisms and uses the framework of
semantic tableaux to represent the context of discourse, is found in [Ger95].
More generally, we feel that the taxonomy proposed later in this chapter
might correlate with the linguistic diversity of presupposition (triggered by
definite descriptions, verbs, adverbs, et cetera) – but we must leave this as an
open question.
Abduction in Mathematics
Abduction in mathematics is usually identified with notions like discovery and
heuristics. A key reference in this area is the work by the earlier mentioned G.
Polya [Pol45, Pol54, Pol62]. In the context of number theory, for example, a
general property may be guessed by observing some relation as in:
3 + 7 = 10,
3 + 17 = 20,
13 + 17 = 30
Notice that the numbers 3,7,13,17 are all odd primes and that the sum of any of
two of them is an even number. An initial observation of this kind eventually led
45
What is Abduction?
Goldbach (with the help of Euler) to formulate his famous conjecture: ‘Every
even number greater than two is the sum of two odd primes’.
Another example is found in a configuration of numbers, such as in the
well–known Pascal’s triangle [Pol62]:
1
1
1
1
1
2
3
1
3
=
=
=
1 =
1
2
4
8
There are several ‘hidden’ properties in this triangle, which the reader may or
may not discover depending on her previous training and mathematical knowledge. A simple one is that any number different from 1 is the sum of two other
numbers in the array, namely of its northwest and northeast neighbors (e.g. 3
= 1 + 2). A more complex relationship is this: the numbers in any row sum to
a power of 2. More precisely,
n
0
+
...
+
n
n
= 2n
See [Pol62] for more details on ‘abducing’ laws about the binomial coefficients
in the Pascal Triangle.
The next step is to ask why these properties hold, and then proceed to prove
them. Goldbach’s conjecture remains unsolved (it has not been possible to
prove it or to refute it); it has only been verified for a large number of cases (the
latest news is that it is true for all integers less than 4.1011 , cf. [Rib96]). The
results regarding Pascal’s triangle on the other hand, have many different proofs,
depending one’s particular taste and knowledge of geometrical, recursive, and
computational methods. (Cf. [Pol62] for a detailed discussion of alternative
proofs.)
According to Polya, a mathematician discovers just as a naturalist does, by
observing the collection of his specimens (numbers or birds) and then guessing
their connection and relationship [Pol54, p.47]. However, the two differ in that
verification by observation for the naturalist is enough, whereas the mathematician requires a proof to accept her findings. This points to a unique feature
of mathematics: once a theorem finds a proof, it cannot be defeated. Thus,
mathematical truths are eternal, with possibly many ways of being explained.
On the other hand, some findings may remain unexplained forever. Abduction
in mathematics shows very well that observing goes beyond visual perception,
as familiarity with the field is required to find ‘surprising facts’. Moreover, the
relationship between observation and proof need not be causal, it is just pure
mathematical structure that links them together.
Much more complex cases of mathematical discovery can be studied, in
which concept formationis involved. An interesting approach along these lines
is found in [Vis97], which proposes a catalogue of procedures for creating concepts when solving problems. These include ‘redescription’, ‘substitution’, and
‘transposition’, which are explicitly related to Peirce’s treatment of abduction.
46
7.
ABDUCTIVE REASONING
A Taxonomy for Abduction
What we have seen so far may be summarized as follows. Abduction is
a process whose products are specific abductive explanations, with a certain
inferential structure, making an (abductive) explanatory argument. We consider
the two aspects of abduction of equal importance. Moreover, on the process
side, we distinguished between constructing possible abductive explanations
and selecting the best one amongst these. This book is mainly concerned
with the characterization of abductive explanations as products –as an essential
ingredient for explanatory arguments– and with the processes for constructing
these.
As for the logical form of abduction –referring to the inference corresponding
to the abductive process that takes a background theory (Θ) and a given observation (ϕ) as inputs, and produces an abductive explanation (α) as its output–
we have found that at a very general level, it may be viewed as a threefold
relation:
Θ, ϕ ⇒ α
Other parameters are possible here, such as a preference ranking - but these
would rather concern the further selection process. This characterization aims
to capture the direction (from evidence to abductive explanation) of this type of
reasoning. In the end however, what we want is to characterize an (abductive)
explanatory argument, in its deductive forward fashion, that is, an inference
from theory (Θ) and abductive explanation (α) to evidence (ϕ) as follows:
Θ, α ⇒ ϕ
Against this background, we propose three main parameters that determine
types of explanatory arguments. (i) An ‘inferential parameter’ (⇒) sets some
suitable logical relationship among explananda, background theory, and explanandum. (ii) Next, ‘triggers’ determine what kind of abductive process is
to be performed: ϕ may be a novel phenomenon, or it may be in conflict with
the theory Θ. (iii) Finally, ‘outcomes’ (α) are the various products (abductive
explanations) of an abductive process: facts, rules, or even new theories.
Abductive Parameters
Varieties of Inference
In the above schema, the notion of explanatory inference ⇒ is not fixed. It
can be classical derivability ⊢ or semantic entailment |=, but it does not have
to be. Instead, we regard it as a parameter which can be set independently.
It ranges over such diverse values as probable inference (Θ, α ⇒probable ϕ),
in which the explanans renders the explanandum only highly probable, or as
the inferential mechanism of logic programming (Θ, α ⇒prolog ϕ). Further
What is Abduction?
47
interpretations include dynamic inference (Θ, α ⇒dynamic ϕ, cf. [vBe96a]),
replacing truth by information change potential along the lines of belief update
or revision. Our point here is that abduction is not one specific non-standard
logical inference mechanism, but rather a way of using any one of these.
Different Triggers
According to Peirce, as we saw, abductive reasoning is triggered by a surprising phenomenon. The notion of surprise, however, is a relative one, for a
fact ϕ is surprising only with respect to some background theory Θ providing
‘expectations’. What is surprising to me (eg. that the lights go on as I enter
the copier room) might not be surprising to you. We interpret a surprising fact
as one which needs an explanation. From a logical point of view, this assumes
that the fact is not already explained by the background theory Θ: Θ ⇒ ϕ.
Moreover, our claim is that one also needs to consider the status of the
negation of ϕ. Does the theory explain the negation of observation instead
(Θ ⇒ ¬ϕ)? Thus, we identify at least two triggers for abduction: novelty and
anomaly:
Abductive Novelty: Θ ⇒ ϕ, Θ ⇒ ¬ϕ
ϕ is novel. It cannot be explained (Θ ⇒ ϕ), but it is consistent with the
theory (Θ ⇒ ¬ϕ).
Abductive Anomaly: Θ ⇒ ϕ, Θ ⇒ ¬ϕ.
ϕ is anomalous. The theory explains rather its negation (Θ ⇒ ¬ϕ).
In the computational literature on abduction, novelty is the condition for an
abductive, problemabductive problem [KKT95]. My suggestion is to incorporate anomaly as a second basic type.
Of course, non-surprising facts (where Θ ⇒ ϕ) should not be candidates for
abductive explanations. Even so, one might speculate if facts which are merely
probable on the basis of Θ might still need an abductive explanation of some
sort to further cement their status.
Different Outcomes
Abductive explanations themselves come in various forms: facts, rules, or even
theories. Sometimes one simple fact suffices to explain a surprising phenomenon, such as rain explaining why the lawn is wet. In other cases, a rule
establishing a causal connection might serve as an abductive explanation, as in
our case connecting cloud types with rainfall. And many cases of abduction in
science provide new theories to explain surprising facts. These different options
may sometimes exist for the same observation, depending on how seriously we
want to take it. In this book, we shall mainly consider abductive explanations
48
ABDUCTIVE REASONING
in the forms of atomic facts, conjunctions of them and simple conditionals, but
we do make occasional excursions to more complex kinds of statements.
Moreover, we are aware of the fact that genuine abductive explanations sometimes introduce new concepts, over and above the given vocabulary. (For
instance, the eventual explanation of planetary motion was not Kepler’s, but
Newton’s, who introduced a new notion of ‘force’ – and then derived elliptic
motion via the Law of Gravity.) Except for passing references in subsequent
chapters, abduction via new concepts will be outside the scope of our analysis.
Abductive Processes
Once the above parameters get set, several kinds of abductive processes arise.
For example, abduction triggered by novelty with an underlying deductive inference, calls for a process by which the theory is expanded with an abductive
explanation. The fact to be explained is consistent with the theory, so an abductive explanation added to the theory accounts deductively for the fact. However,
when the underlying inference is statistical, in a case of novelty, theory expansion might not be enough. The added statement might lead to a ‘marginally
consistent’ theory with low probability, which would not yield a strong explanation for the observed fact. In such a case, theory revision is needed (i.e.
removing some data from the theory) to account for the observed fact with high
probability. (For a specific example of this latter case cf. chapter 8.)
Our aim is not to classify abductive processes, but rather to point out that several kinds of these are used for different combinations of the above parameters.
In the coming chapters we explore in detail some procedures for computing
different types of outcomes in a deductive format; those triggered by novelty
(chapter 4) and those triggered by anomaly (chapter 8).
Examples Revisited
Given our taxonomy for abductive reasoning, we can now see more patterns
across our earlier list of examples. Varying the inferential parameter, we cover
not only cases of deduction but also statistical inferences. Thus, Hempel’s
statistical model of explanation [Hem65] becomes a case of an (abductive)
explanatory argument. Our example (4) of medical diagnosis (Jane’s quick
recovery) was an instance. Logic programming inference seems more appropriate to an example like (1) (whose overall structure is similar to (4)). As for
triggers, novelty drives both rain examples (1) and (2), as well as the medical
diagnosis one (4). A trigger by anomaly occurs in example (3), where the
theory predicts the contrary of our observation, the lights-off example (5), and
the Kepler example (6), since his initial observation of the longitudes of Mars
contradicted the previous rule of circular orbits of the planets. As for different
outcomes, examples (1), (3), (4) and (5) abduce facts, examples (2) and (6)
What is Abduction?
49
produce rules as their forms of explanantia. Different forms of outcomes will
play a role in different types of procedures for producing abductive explanations. In computer science jargon, triggers and outcomes are, respectively,
preconditions and outputs of abductive devices, whether these be computational
procedures or inferential ones.
This taxonomy gives us the big picture of abductive reasoning. In the remainder of this book, we are going to investigate several of its aspects, which
give rise to more specific logical and computational questions. (Indeed, more
than we have been able to answer!) Before embarking upon this course, however, we need to discuss one more general strategic issue, which explains the
separation of concerns between chapters 3 and 4 that are to follow.
Abductive Logic: Inference + Search Strategy
Classical logical systems have two components: a semantics and a proof theory.
The former aims at characterizing what it is for a formula to be true in a model,
and it is based on the notions of truth and interpretation. The latter characterizes
what counts as a valid proof for a formula, by providing the inference rules of
the system; having for its main notions proof and derivability. These two
formats can be studied independently, but they are closely connected. At least
in classical (first-order) logic, the completeness theorem tells us that all valid
formulas have a proof, and vice versa. Many logical systems have been proposed
that follow this pattern: propositional logic, predicate logic, modal logic, and
various typed logics.
From a modern perspective, however, there is much more to reasoning than
this. Computer science has posed a new challenge to logic; that of providing
automatic procedures to operate logical systems. This requires a further finestructure of reasoning. In fact, recent studies in AI give precedence to a control
strategy for a logic over its complete proof theory. In particular, the heart of
logic programming lies in its control strategies, which lead to much greater
sensitivity as to the order in which premises are given, the avoidance of search
loops, or the possibility to cut proof searches (using the extra-logical operator
!) when a solution has been found. These features are extralogical from a
classical perspective, but they do have a clear formal structure, which can be
brought out, and has independent interest as a formal model for broader patterns
of argumentation (cf. [vBe92, Kal95, Kow91]).
Several contemporary authors stress the importance of control strategies,
and a more finely-structured algorithmic description of logics. This concern is
found both in the logical tradition ([Gab94a, vBe90]), and in the philosophical
tradition ([Gil96]), the latter arguing for a conception of logic as: inference +
control. (Note the shift here away from Kowalski’s famous dictum “Algorithm =
Logic + Control".) In line with this philosophy, we wish to approach abduction
with two things in mind. First, there is the inference parameter, already dis-
50
ABDUCTIVE REASONING
cussed, which may have several interpretations. But given any specific choice,
there is still a significant issue of a suitable search strategy over this inference,
which models some particular abductive practice. The former parameter may
be defined in semantic or proof-theoretic terms. The search procedure, on the
other hand, deals with concrete mechanisms for producing valid inferences. It
is then possible to control which kinds of outcome are produced with a certain
efficiency. In particular, in abduction, we may want to produce only ‘useful’ or
‘interesting’ formulas, preferably even just some ‘minimal set’ of these.
In this light, the aim of an abductive search procedure is not necessarily
completeness with respect to some semantics. A procedure that generates all
possible abductive explanations might be of no practical use, and might also
miss important features of human abductive activity. In chapter 4, we are
going to experiment with semantic tableaux as a vehicle for attractive abductive
strategies that can be controlled in various ways.
PART II
LOGICAL FOUNDATIONS
Chapter 3
ABDUCTION AS LOGICAL INFERENCE
1.
Introduction
In the preceding overview chapter, we have seen how the notion of abduction
arose in the last century out of philosophical reflection on the nature of human
reasoning, as it interacts with patterns of explanation and discovery. Our analysis brought out a number of salient aspects to the abductive process, which we
shall elaborate in a number of successive chapters. For a start, abduction may
be viewed as a kind of logical inference and that is how we will approach it in
the analysis to follow here. Evidently, though, as we have already pointed out,
it is not standard logical inference, and that for a number of reasons. Intuitively,
abduction runs in a backward direction, rather than the forward one of standard
inference, and moreover, being subject to revision, it exhibits non-standard nonmonotonic features (abductive conclusions may have to be retracted in the light
of further evidence), that are more familiar from the literature on non-standard
forms of reasoning in artificial intelligence. Therefore, we will discuss abduction as a broader notion of consequence in the latter sense, using some general
methods that have been developed already for non-monotonic and dynamic
logics, such as systematic classification in terms of structural rules. This is not
a mere technical convenience. Describing abduction in an abstract general way
makes it comparable to better-studied styles of inference, thereby increasing
our understanding of its contribution to the wider area of what may be called
‘natural reasoning’. To be sure, in this chapter we propose a logical characterization of what we have called an (abductive) explanatory argument, in order
to make explicit that the inference is explanatory (and thus forward chained),
while keeping in mind it aims to characterize the conditions for an abductive
explanation to be part of this inference (cf. chapter 2).
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ABDUCTIVE REASONING
The outcomes that we obtain in this first systematic chapter, naturally divided into five parts, are as follows. After this introduction, in the second part
(section 2), we discuss the problem of demarcation in logic, in order to set the
ground for our analysis of abduction as a logical inference. Placing abduction in a broader universe of logics and stances, make natural to consider it as
a logical inference of its own kind. In the third part (section 3), we discuss
in detail some aspects of abductive inference, such as its direction, format of
premisses and conclusion as well as its inferential strength. We propose a general logical format for abduction, involving more parameters than in standard
inference, allowing for genuinely different roles of premisses. We find a number of natural styles of abductive explanatory inference, rather than one single
candidate. These (abductive) explanatory versions are classified by different
structural rules of inference, and this issue occupies the fourth part (section
4). As a contribution to the logical literature in the field, we give a complete
characterization of one simple style of abductive explanatory inference, which
may also be viewed as the first structural characterization of a natural style of
explanation in the philosophy of science. We then analyze some other abductive explanatory versions (explanatory, minimal and preferential) with respect
to their structural behaviour, giving place to more sophisticated structural rules
with interest of their own. Finally, we turn to discuss further logical issues such
as how those representations are related to more familiar completeness theorems, and finally, we show how abduction tends to involve higher complexity
than classical logic: we stand to gain more explanatory power than what is provided by standard inference, but this surplus comes at a price. In the fifth and
final part of this chapter (section 5),we offer an analysis of previous sections
centering the discussion on abduction as an enriched form of logical inference
with an structure of its own. We then put forward our conclusions and present
related work within the study of non-monotonic reasoning.
Despite these useful insights, pure logical analysis does not exhaust all there
is to abduction. In particular, it’s more dynamic process aspects, and its interaction with broader conceptual change must be left for subsequent chapters, that
will deal with computational aspects, as well as further connections with the
philosophy of science and artificial intelligence. What we do claim, however,
is that our logical analysis provides a systematic framework for studying the
intuitive notion of abductive explanatory inference, which gives us a view of
its variety and complexity, and which allows us to raise some interesting new
questions.
2.
Logic: The Problem of Demarcation
One of the main questions for logic is the problem of demarcation, that is,
to distinguish between logical and non-logical systems. This question is at
Abduction as Logical Inference
55
the core of the philosophy of logic, and has a central place in the philosophy
of mathematics, in the philosophy of science as well as in the foundations of
artificial intelligence.
Some questions in need for an answer for this problem concern the following
ones: what is a logic?, which is the scope of logic?, which formal systems
qualify as logics?, all of these leading to metaphysical questions concerning
the notion of correctness of a logical system: does it make sense to speak of a
logical system as correct or incorrect?, could there be several logical systems
which are equally correct?, is there just one correct logical system? These
questions in turn lead to epistemological questions of the following kind: how
does one recognise a truth of logic? could one be mistaken in what one takes
to be such truths?
There are however, several proposals and positions in the literature in regard
to all these questions. Our strategy to describe the problem of demarcation of
logic will be the following. Our point of departure is Peirce’s distinction of
three types of reasoning, namely deduction, induction, and abduction. We will
compare them according to their certainty level, something that in turn gives
place to different areas of application, mainly in mathematics, philosophy of
science and artificial intelligence. Next, we will introduce an standard approach
in philosophy of logic based on the relationship between informal arguments
and their counterparts in formal logic, namely the view endorsed by Haack
[Haa78]. Her classification of kinds of logics will be presented, that is, the
well-known distinction amongst extensions and deviations of classical logics,
and inductive logics. Moreover, we take up on Haack’s discussion on the
several positions with respect to the legitimization (and proliferation) of logics,
namely instrumentalism, monism and pluralism. Finally, we will introduce a
much less-known approach –but still standard within its field – coming from
artificial intelligence, namely the logical structural approach devised for the
study of non-monotonic reasoning.
Our overall discussion in this section will serve two purposes. On the one
hand, it aims to show that even under a broad view of logic, there is neither a
unique nor a definite answer to the problem of demarcation, not to mention to
each of the former questions. On the other hand, it will set the ground for the
main purpose of this chapter, that is, an analysis of abduction as a specific kind
of logical inference, in order to show, that abduction holds a natural place to be
considered a logical inference of its own kind.
Types of Reasoning: Deduction, Induction, Abduction
From a logical perspective, mathematical reasoning may be identified with
classical, deductive inference. Two aspects are characteristic of this type of
reasoning, namely its certainty and its monotonicity. The first of these is exemplified by the fact that the relationship between premisses and conclusion is that
56
ABDUCTIVE REASONING
of necessity; a conclusion drawn from a set of premisses, necessarily follows
from them. The second aspect states that conclusions reached via deductive
reasoning are non-defeasible. That is, once a theorem has been proved, there is
no doubt of its validity regardless of further addition of axioms and theorems
to the system.
There are however, several other types of formal non-classical reasoning,
which albeit their lack of complete certainty and monotonicity, are nevertheless
rigorous forms of reasoning with logical properties of their own. Such is the
case of inductive and abductive reasoning. As a first approximation, Charles S.
Peirce distinction seems useful. According to him, there are three basic types
of logical reasoning: deduction, induction and abduction. Concerning their
certainty level: ‘Deduction proves that something must be; Induction shows that
something actually is operative; Abduction merely suggests that something may
be’ [CP, 5.171]. Therefore, while deductive reasoning is completely certain,
inductive and abductive reasoning are not. ‘Deduction is the only necessary
reasoning. It is the reasoning of mathematics’ [CP, 4.145]. Induction must
be validated empirically with tests and experiments, therefore it is defeasible;
and abductive reasoning can only offer hypotheses that may be refuted with
additional information. For example, a generalization reached by induction
(e.g. all birds fly), remains no longer valid after the addition of a premisse,
which refutes the conclusion (e.g. penguins are birds). As for abduction, a
hypothesis (e.g. it rained last night) which explains an observation (e.g. the
lawn is wet), may be refuted when additional information is incorporated into
our knowledge base (e.g. it is a drought period).
Deductive reasoning has been the paradigm of mathematical reasoning, and
its logic is clearly identified with Tarski’s notion of logical inference. In contrast,
inductive and abductive types of reasoning are paradigmatic types of reasoning
in areas like philosophy of science, and more recently, artificial intelligence.
Regarding the former, contemporary research indicates that many questions
regarding their logic remain controversial. As it is well known, Carnap’s proposal for an inductive logic[Car55] found ample criticisms. As for abduction,
while some scholars argue that the process of forming an explanatory hypothesis (our abductive process) cannot be logically reconstructed [Pop59, Hem65],
and have instead proposed each a logical characterization of explanation (our
(abductive) explanatory argument)1 ; others have tried to formally characterize
1 As
for the roots and similarities of these two models of explanation, Niiniluoto[Nii00, p. 140] rightly
observes: “After Hempel’s (1942) paper about the deductive–nomological pattern of historical explanation, Karl Popper complained that Hempel had only reproduce his theory of causal explanation, originally
presented in ‘Logik der Forschung’ (1935, see Popper 1945, chap 25, n. 7; Popper 1957, p. 144). With
his charming politeness, Hempel pointed out that his account of D–N explanation is ‘by no means novel’
but ‘merely summarizes and states explicitly some fundamental points which have been recognized by many
scientists and methodologists”.
Abduction as Logical Inference
57
‘retroduction’ (another term for abduction), as a form of inversed deduction
[Han61], but no acceptable formulation has been found. Regarding the latter, recent logico-computational oriented research has focus on studying non-standard
forms of reasoning, in order to build computer programs modeling human reasoning, which being subject to revision, is uncertain and exhibits non-standard
non-monotonic features. Several contemporary authors propose a more finely
structured algorithmic description of logics. This concern is found both in the
logical tradition ([Gab94a, vBe90]), as well as in work in philosophy ([Gil96]).
Logics: Extensions, Deviations, Inductive
Haack[Haa78] takes as primitive an intuitive notion of a formal system, and
from there it hints at the characterization of what is to be a logical system, as
follows:
“The claim for a formal system to be a logic depends, I think, upon its having an
interpretation according to which it can be seen as aspiring to embody canons of valid
argument." [Haa78, page 3].
The next problem to face is that of deciding what counts as valid argumentation. But before we get into her own answer to this question, here are other
criteria aiming to characterize what counts as a logical system. On the one
hand, according to Kneale, logical systems are those that are purely formal,
for him, those that are complete (in which all universal valid formula are theorems). According to Dummet, on the other hand, logical systems are those
which characterize precise notions. Following the first characterization, many
formal systems are left out, such as second order logic. If we follow the second
one, then proposals such as Hintikka’s system of epistemic logic is left out as
well, for the notions of knowledge and belief characterize pretty vague epistemic concepts [Haa78, page 7]. Both these characterizations provide purely
formal criteria for logical demarcation. For Haack, however ‘the prospects for
a well-motivated formal criterion are not very promising’[Haa78, page 7], for
it has the drawback of limiting the scope of logic to the point of even discarding
well accepted formal systems (e.g. predicate logic) on the basis of being in
absence of other metalogical properties (e.g. decidability). Moreover, many
logical systems are indeed undecidable, incomplete, but nevertheless have interesting applications and have proved useful in areas like computer science
and linguistics.
Haack takes a broad view of logic, considering that ‘the demarcation is not
based on any very profound ideas about ‘the essential nature of logic’[Haa78,
page 4], and follows ‘the benefit of the doubt policy’, according to which,
arguments may be assessed by different standards of validity, and thus accepts
several formal systems as logical. For her, the question we should be asking
is whether a system is good and useful rather than ‘logical’, which after all is
not a well-defined concept. Her approach however, is not wholly arbitrary, for
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ABDUCTIVE REASONING
it does not give up the requirement of being rigorous, and takes classical logic
as its reference point, building up a classification of systems of logic based on
analogies to the classical system, as follows:
Extensions (e)
Modal, Epistemic, Eroretic, . . .
Deviations (d)
Intuitionistic, Quantum, Many-valued, . . .
Inductive (i)
Inductive probability logic
Extensions (e) are formal logical systems, which extend the system of classical logic (Lc ) in three respects: their language, axioms and rules of inference
(Lc ⊆ Le , Ac ⊆ Ae , Rc ⊆ Re ). These systems preserve all valid formula of
the classical system, and therefore all previous valid formula remain valid as
well (∀ϕ(Σ |=c ϕ ⇒ Σ |=e ϕ); ϕ ∈ Lc ). So, for instance, modal logic extends
classical system by the modal operators of necessity and possibility together
with axioms and rules for them.
Deviations (d) are formal systems that share the language with the system
of classical logic (Lc ) but that deviate in axioms and rules (Lc = Ld , Ac =
Ad , Rc = Rd ). Therefore, some formulae, which are valid in the classical
system, are no longer valid in the deviant one (∃ϕ(Σ |=c ϕ∧Σ |=d ϕ); ϕ ∈ Lc ).
Such is the case of intuitionistic logic, in which the classical axiom A ∨ ¬A is
no longer valid.
Inductive systems (i) are formal systems that share the language with the
system of classical logic (Lc = Li ), but in which no formula which is valid
by means of the inductive system is valid in the classical one (∀ϕ(Σ |=i ϕ ⇒
Σ |=c ϕ); ϕ ∈ Li ). Here the basis is the notion of ‘inductive strength’, and
the idea is that ‘an argument is inductively strong if its premisses give a certain
degree of support, even if less than conclusive support, to its conclusion: if,
that is, it is improbable that its premisses (Σ) should be true and its conclusion
(ϕ) false’ (not prob(Σ ∧ ¬ϕ)) [Haa78, page 17].
In each of these logical systems there is an underlying notion of logical consequence (or of derivability), which settles the validity of an argument within
the system. While the first two categories pertain to formal systems which are
deductive in nature, the third one concerns inductive ones. But still there may be
several characterizations for both deductive and inductive kinds. For example,
one deviant system, that of relevance logic renders the notion of classical consequence insufficient and asks for more: an argument in relevance logic must
meet the requirement that the premisses be ‘relevant’ to its conclusion. As for
inductive systems, another way of characterizing them is that for which ‘it is
Abduction as Logical Inference
59
improbable, given that the premisses (Σ) are true, that the conclusion is false
(¬ϕ)’ [Haa78, page 17]. We may interpret this statement in terms of conditional
probability as follows2 : (not prob(Σ/¬ϕ)). Notice that deductive validity
is a limiting case of inductive strength, where the probability of the premisses
being true and the conclusion false is cero, for the first characterization, and
where it is certain that the conclusion is true when the premisses are, for this
second one.
In the overall, under this approach arguments may be assessed by deductive
or inductive standards, and thus there may be deductively valid, inductively
strong or neither. As we shall see later in this chapter (cf. section 5.1), this
classification does not explicitly take into account abductive logic, but it can
nevertheless be accommodated within.
Positions: Instrumentalism, Monism, Pluralism
The position taken with respect to the demarcation of logic largely depends
upon the answers given to metaphysical questions concerning the notion of correctness of a logical system, which in turn depend on the distinction between
system-relative and extra-systematic validity/logical truth. Roughly speaking,
a logical system is correct if the formal arguments (and formula) which are
valid (logically true) in that system correspond to informal arguments (statements), which are valid (logically true) in the extra-systematic sense([Haa78,
page 222]). Three positions are characterized by Haack, each of which is
characterized by the answers (affirmative or negative) given to the following
questions:
Questions:
A: Does it make sense to speak of a logical system as correct or incorrect?
B: Are there extra-systematic conceptions of validity/logical truth by means of
which to characterize what is it for a logic to be correct?
C: Is there one correct system?
D: Must a logical system aspire to global application, i.e. to represent reasoning
irrespective of subject-matter, or may a logic be locally correct. i.e. correct within
a limited area of discourse?
2I
thank the anonymous referee for this particular suggestion.
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ABDUCTIVE REASONING
A
B
NO
YES
Instrumentalism
C
NO
YES
Pluralism
Monism
D
Local Pluralism
Global Pluralism
Thus, on the one hand, the instrumentalist position answers the first two
questions negatively. It is based on the idea that the notion of ‘correctness’ for a
system is inappropriate, and that one should rather be asking for its being more
fruitful, useful, convenient... etc than another one. ‘An instrumentalist will
only allow the ‘internal’ question, whether a logical system is sound, whether,
that is, all and only the theorems/syntactically valid arguments of the system
are logically true/valid in the system’[Haa78, page 224]. On the other hand,
both the monist and the pluralist answer these questions in the affirmative,
the difference being that while the monist recognizes one and only one system
of logic, the pluralist accepts a variety of them. Thus, they answer the third
question opposite. Note that the distinction in these questions is only relevant
for the classical logic vs. deviant logic dichotomy. The reason being that for
a monist classical logic and its extensions are fragments of a ‘correct system’,
and for a pluralist classical logic and its extensions are both ‘correct’.
Likewise, while for an instrumentalist there are not extra-systematic conceptions of validity/logical truth by means of which to characterize what is to be a
logic to be correct, for the monist as well as for the pluralist there are, either in
the unitary fashion or in the pluralistic one. A further distinction made by the
pluralist concerns the scope of application for a certain logical system. While a
global pluralist endorses the view that a logical system must aspire to represent
reasoning irrespective of subject-matter, a local pluralist supports the view that
a logical system is only locally correct within a limited area of discourse.
The next question to analyze is the position taken by each of these stances
with regard to whether deviant logics rival classical logic. In order to answer
this question we have the following diagram:
61
Abduction as Logical Inference
(i) Formal argument which is
represents
(iii) Informal argument
(ii) Valid in L
corresponding
to (iii)’s being
(iv) extra-systematically valid
On the one hand, the monist answers this question in the affirmative and supports the view that (i) aspires to represent (iii) in such a way that (ii) and (iv) do
correspond in the ‘correct logic’. On the other hand, the local pluralist answers
this question in the negative by relativizing (iv) to specific areas of discourse
and the global pluralist either fragments the relation between (i) and (iii) (that
is, denies that the formal arguments of a deviant system represent the same
informal arguments as those of classical logic) or fragments the relationship
between (ii) and (iv) (denies that validity in the deviant logic is intended to correspond to extra-systematic validity as that to which validity in classical logic
is intended to correspond). Finally, the instrumentalist rejects (iv) altogether.
Structural Logical Approach
This type of analysis (started in [Sco71]) was inspired in the works of logical
consequence by Tarski [Tar83] and those of natural deduction by Gentzen
[SD93, Pao02]. It describes a style of inference at a very abstract structural
level, giving its pure combinatorics. It has proved very successful in artificial
intelligence for studying different types of plausible reasoning ([KLM90]), and
indeed as a general framework for non-monotonic consequence relations (
[Gab85]). Another area where it has proved itself is dynamic semantics, where
not one but many new notions of dynamic consequences are to be analyzed (
[vBe96a]). The basic idea of logical structural analysis is the following:
A notion of logical inference can be completely characterized by its basic combinatorial
properties, expressed by structural rules.
Structural rules are instructions which tell us, e.g., that a valid inference
remains valid when we insert additional premisses (‘monotonicity’), or that we
may safely chain valid inferences (‘transitivity’ or ‘cut’). To understand this
perspective in more detail, one must understand how it characterizes classical
inference. In what follows we use logical sequents with a finite sequence of
premisses to the left, and one conclusion to the right of the sequent arrow
(Σ ⇒ C). While X, Y and Z are finite sets of formulae, A, B and C are single
formula.
Classical Inference
The structural rules for classical inference are the following:
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ABDUCTIVE REASONING
C⇒C
Reflexivity:
Contraction:
X, A, Y, A, Z ⇒ C
X, A, Y, Z ⇒ C
Permutation:
X, A, B, Y ⇒ C
X, B, A, Y ⇒ C
Monotonicity:
X, Y ⇒ C
X, A, Y ⇒ C
Cut Rule:
X, A, Y ⇒ C Z ⇒ A
X, Z, Y ⇒ C
These rules state the following properties of classical consequence. Any premisse implies itself (reflexivity), deleting repeated premisses causes no trouble (contraction); premisses may be permuted without altering validity (permutation), adding new information does not invalidate previous conclusions
(monotonicity), and premisses may be replaced by sequences of premisses implying them (cut). In all, these rules allow us to treat the premisses as a mere
set of data without further relevant structure. This plays an important role in
classical logic, witness what introductory textbooks have to say about “simple properties of the notion of consequence"3 . Structural rules are also used
extensively in completeness proofs4 .
These rules are structural in that they mention no specific symbols of the
logical language. In particular, no connectives or quantifiers are involved. This
makes the structural rules different from inference rules like, say, Conjunction
of Consequents or Disjunction of Antecedents, which also fix the meaning of
3 In
[Men64, Page 30] the following simple properties of classical logic are introduced:
If Γ ⊆ ∆ and Γ ⊢ φ, then ∆ ⊢ φ.
Γ ⊢ φ iff there is a finite subset ∆ of Γ such that ∆ ⊢ φ.
If Γ ⊢ xi (for all i) and x1 , . . . , xn ⊢ φ then Γ ⊢ φ.
Notice that the first is a form of Monotonicity, and the third one of Cut.
4 As noted in [Gro95, page46]: “In the Henkin construction for first-order logic, or propositional modal
logic, the notion of maximal consistent set plays a major part, but it needs the classical structural rules. For
example, Permutation, Contraction and Expansion enable you to think of the premisses of an argument as a
set; Reflexivity is needed to show that for maximal consistent sets, membership and derivability coincide”.
63
Abduction as Logical Inference
conjunction and disjunction. Under this approach, Haack’s previous classification of extensions of logics is subsumed, for one rule may fit classical logic as
well as extensions: propositional, first-order, modal, type-theoretic, etc. Each
rule in the above list reflects a property of the set-theoretic definition of classical
consequence ([Gro95]), which – with some abuse of notation – calls for inclusion of the intersection of the (models for the) premisses in the (models for the)
conclusion:
P1 , . . . , Pn ⇒ C iff P1 ∩ . . . ∩ Pn ⊆ C.
Now, in order to prove that a set of structural rules completely characterizes
a style of reasoning, representation theorems exist. For classical logic, one
version was proved by van Benthem in [vBe91]:
Proposition 1 Monotonicity, Contraction, Reflexivity, and Cut completely determine the structural properties of classical consequence.
Proof. Let R be any abstract relation between finite sequences of objects and single
objects satisfying the classical structural rules. Now, define:
a* = {A | A is a finite sequence of objects such that ARa}.
Then, it is easy to show the following two assertions:
1 If a1 , . . . , ak Rb, then a∗1 ∩ . . . ∩ a∗k ⊆ b∗ ,
using Cut and Contraction.
2 If a∗1 ∩ . . . ∩ a∗k ⊆ b∗ , then a1 , . . . , ak Rb,
using Reflexivity and Monotonicity.
⊣
Permutation is omitted in this theorem. And indeed, it turns out to be derivable from Monotonicity and Contraction.
We have thus shown that classical deductive inference, observes easy forms of
reflexivity, contraction, permutation, monotonicity and cut. The representation
theorem shows that these rules completely characterize this type of reasoning.
Non-Classical Inference
For non-classical consequences, classical structural rules may fail. Well-known
examples are the ubiquitous ‘non-monotonic logics’. However, this is not to
say that no structural rules hold for them. The point is rather to find appropriate
reformulations of classical principles (or even entirely new structural rules) that
fit other styles of consequence. For example, many non-monotonic types of
inference do satisfy a weaker form of monotonicity. Additions to the premisses
are allowed only when these premisses imply them:
Cautious Monotonicity:
X⇒C
X⇒A
X, A ⇒ C
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ABDUCTIVE REASONING
Dynamic inference is non-monotonic (inserting arbitrary new processes into
a premisse sequence can disrupt earlier effects). But it also quarrels with other
classical structural rules, such as Cut. But again, representation theorems exist.
Thus, the ‘update-to-test’ dynamic style (once in which a process cannot be
disrupted) and is characterized by the following restricted forms of monotonicity
and cut, in which additions and omissions are licensed only to the left side:
Left Monotonicity:
X⇒C
A, X ⇒ C
Left Cut:
X⇒C
X, C, Y ⇒ D
X, Y ⇒ D
For a broader survey and analysis of dynamic styles, see [Gro95, vBe96a].
For sophisticated representation theorems in the broader field of non-classical
inference in artificial intelligence see [Mak93, KLM90]. Yet other uses of nonclassical structural rules occur in relevance logic, linear logic, and categorial
logics (cf. [DH93, vBe91]. [Gab94b]).
Characterizing a notion of inference in this way, determines its basic repertoire for handling arguments. Although this does not provide a more ambitious
semantics, or even a full proof theory, it can at least provide valuable hints. The
suggestive Gentzen style format of the structural rules turns into a sequent calculus, if appropriately extended with introduction rules for connectives. However,
it is not always clear how to do so in a natural manner, as we will discuss later
in connection with abduction.
The structural analysis of a logical inference is a metalevel explication which
is based on structural rules and not on language, as it does not take into account
logical connectives or constants, and in this respect differs from Haack’s approach.
3.
Abductive Explanatory Argument: A Logical Inference
Here are some preliminary remarks about the logical nature of abductive
inference, which set the scene for our subsequent discussion. The standard
textbook pattern of logical inference is this: Conclusion C follows from a set
of premisses P . This format has its roots in the axiomatic tradition in mathematics that follows the deductive method, inherited from Euclid’s Elements, in
which from a certain set of basic axioms, all geometrical truths of elementary
geometry are derived. This work is not only the first logical system of its kind,
but it has been the model to follow in mathematics as well as in other formal
scientific enterprises. Each proposition is linked, via proofs, to previous axioms, definitions and propositions. This method is forward chained, picturing
Abduction as Logical Inference
65
a reasoning from a finite set of premisses to a conclusion, and it is completely
certain and monotonic5 .
Moreover, there are at least two ways of thinking about validity in this setting,
one semantic, based on the notions of model and interpretation (every model in
which P is true makes C true), the other syntactic, based on a proof-theoretic
derivation of C from P . Both explications suggest forward chaining from premisses to conclusions: P ⇒ C and the conclusions generated are undefeasible.
We briefly recall some features that make abduction a form of inference that
does not fit easily into this format. All of them emerged in the course of our preceding chapter. Most prominently, in abduction, the conclusion is the given and
the premisses (or part of them) are the output of the inferential process: P ⇐ C.
Moreover, the abduced premisse has to be consistent with the background theory of the inference, as it has to be explanatory. And such explanations may
undergo change as we modify our background theory. Finally, when different
sets of premisses can be abduced as explanations, we need a notion of preference between them, allowing us to choose a best or minimal one. These various
features, though non-standard when compared with classical logic, are familiar
from neighbouring areas. For instance, there are links with classical accounts
of explanation in the philosophy of science [Car55, Hem65], as well as recent
research in artificial intelligence on various notions of common sense reasoning
[McC80, Sho88, Gab96]. It has been claimed that this is an appropriate broader
setting for general logic as well [vBe90], gaping back to the original program by
Bernard Bolzano (1781–1848), in his “Wissenschaftslehre" [Bol73]. Indeed,
our discussion of abduction in Peirce in the preceding chapter reflected a typical
feature of pre-Fregean logic: boundaries between logic and general methodology were still rather fluid. In our view, current post-Fregean logical research is
slowly moving back towards this same broader agenda. More concretely, we
shall review the mentioned features of abduction in some further detail now,
making a few strategic references to this broader literature.
Directions in Reasoning: Forward and Backward
Intuitively, a valid inference from, say, premisses P1 , P2 to a conclusion C
allows for various directions of thought. In a forward direction, given the premisses, we may want to draw some strongest, or rather, some most appropriate
conclusion. (Notice incidentally, that the latter notion already introduces a certain dependence on context, and good sense: the strongest conclusion is simply
5 This
is not say however –as Hilbert would have liked to claim– that all mathematical reasoning may be
reduced to axiomatics. As it is well know by the incompleteness results of Gödel, there are clear limitations
to reasoning in mathematics through the axiomatic method. Moreover, the view of mathematics as an
experimental, empirical science, found in philosophy[Lak76] as well as in recent work in computer science
[Cha97], shows that axiomatics cannot exhaust all there is to mathematical reasoning.
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ABDUCTIVE REASONING
P1 ∧ P2 , but this will often be unsuited.) Classical logic also has a backward
direction of thought, when engaged in refutation. If we know that C is false,
then at least one of the premisses must be false. And if we know more, say the
truth of P1 and the falsity of the conclusion, we may even refute the specific
premisse P2 . Thus, in classical logic, there is a duality between forward proof
and backward refutation. This duality has been noted by many authors. It
has even been exploited systematically by Beth when developing his refutation
method of semantic tableaux [Bet69]. Read in one direction, closed tableaux
are eventually failed analyses of possible counterexamples to an inference, read
in another they can be arranged to generate a Gentzen-style sequent derivations
of the inference (we shall be using tableaux in our next chapter, on computing
abduction.) Beth’s tableaux can be taken as a formal model of the historical
opposition between methods of ‘analysis’ and ‘synthesis’ in the development
of scientific argument (cf. chapter 1). Methodologically, the directions are
different sides of the same coin, namely, some appropriate notion of inference.
Likewise, in abduction, we see an interplay of different directions. This time,
though, the backward direction is not meant to provide refutations, but rather
confirmations. We are looking for suitable premisses that would support the
conclusion6 .
Our view of the matter is the following. In the final analysis, the distinction
between directions is a relative one. What matters is not the direction of abduction, but rather an interplay of two things. As we have argued in chapter 2,
one should distinguish between the choice of an underlying notion of inference
⇒, and the independent issue as to the search strategy that we use over this.
Forward reasoning is a bottom up use of ⇒ , while backward reasoning is a
top-down use of ⇒. In line with this, in this chapter, we shall concentrate
on notions of inference ⇒ leaving further search procedures to the next, more
computational chapter 4. In this chapter the intuitively backward direction of
abduction is not crucial to us, except as a pleasant manner of speaking. Instead, we concentrate on appropriate underlying notions of consequence for
abduction.
Formats of Inference: Premisses and Background Theory
The standard format of logical inference is essentially binary, giving a transition from premisses to a conclusion:
P1 , . . . , P n
C
6 In this case, a corresponding refutation would rather be a forward process:
if the abduced premisse turns out
false, it is discarded and an alternative hypothesis must be proposed. Interestingly, [Tij97] (a recent practical
account of abduction in diagnostic reasoning) mixes both ‘positive’ confirmation of the observation to be
explained with ‘refutation’ of alternatives.
Abduction as Logical Inference
67
These are ‘local steps’, which take place in the context of some, implicit or
explicit, background theory (as we have seen in chapter 2). In this standard format, the background theory is either omitted, or lumped together with the other
premisses. Often this is quite appropriate, especially when the background
theory is understood. But sometimes, we do want to distinguish between different roles for different types of premisse, and then a richer format becomes
appropriate. The latter have been proposed, not so much in classical logic,
but in the philosophy of science, artificial intelligence, and informal studies
on argumentation theory. These often make a distinction between explicit premisses and implicit background assumptions. More drastically, premisse sets,
and even background theories themselves often have a hierarchical structure,
which results in different ‘access’ for propositions in inference. This is a realistic picture, witness the work of cognitive psychologists like [Joh83].
In Hempel’s account of scientific explanation (cf. chapter 5) premisses play
the role of either scientific laws, or of initial conditions, or of specific explanatory items, suggesting the following format:
Scientific laws + initial conditions + explanatory facts
⇓
Observation
Further examples are found on the borderline of the philosophy of science
and philosophical logic, in the study of conditionals. The famous ‘Ramsey
Test’ presupposes revision of explicit beliefs in the background assumptions
[Sos75, vBe94], which again have to be suitably structured. More elaborate
hierarchical views of theories have been proposed in artificial intelligence and
computer science. [Rya92] defines ‘ordered theory presentations’, which can
be arbitrary rankings of principles involved in some reasoning practice. (Other
implementations of similar ideas use labels for formulas, as in the labelled
deductive systems of [Gab96].) While in Hempel’s account, structuring the
premisses makes sure that scientific explanation involves an interplay of laws
and facts, Ryan’s motivation is resolution of conflicts between premisses in
reasoning, where some sentences are more resistant than others to revision.
(This motivation is close to that of the Gärdenfors theory, to be discussed in
chapter 8. A working version of these ideas is found in a study of abduction
in diagnosis ([Tij97], which can be viewed as a version of our later account
in this chapter with some preference structure added.) More structured views
of premisses and theories can also be found in situation semantics, with its
different types of ‘constraints’ that govern inference (cf. [PB83]).
In all these proposals, the theory over which inference takes place is not just a
bag into which formulas are thrown indiscriminately, but an organized structure
in which premisses have a place in a hierarchy, and play specific different roles.
These additional features need to be captured in richer inferential formats for
more complicated reasoning tasks. Intuitive ‘validity’ may be partly based on
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ABDUCTIVE REASONING
the type and status of the premisses that occur. We cite one more example, to
elaborate what we have in mind.
In argumentation theory, an interesting proposal was made in [Tou58]. Toulmin’s general notion of consequence was inspired on the kind of reasoning
done by lawyers, whose claims need to be defended according to juridical procedures, which are richer than pure mathematical proof. Toulmin’s format of
reasoning contains the necessary tags for these procedures:
Qualifier
Data
✲
Claim
Rebuttal
Warrant
|
Backing
Every claim is defended from certain relevant data, by citing (if pressed) the
background assumptions (one’s ‘warrant’) that support this transition. (There
is a dynamic process here. If the warrant itself is questioned, then one has to
produce one’s further ‘backing’.) Moreover, indicating the purported strength
of the inference is part of making any claim (whence the ‘qualifier’), with a
‘rebuttal’ listing some main types of possible exception (rebuttal) that would
invalidate the claim. [vBe94] relates this format to issues in artificial intelligence, as it seems to describe common sense reasoning rather well. Toulmin’s
model has also been proposed as a mechanism for intelligent systems performing explanation ([Ant89]).
Thus, once again, to model reasoning outside of mathematics, a richer format
is needed. Notice that the above proposals are syntactic. It may be much
harder to find purely semantic correlates to some of the above distinctions: as
they seem to involve a reasoning procedure rather than propositional content.
(For instance, even the distinction between individual facts and universal laws
is not as straightforward as it might seem.) Various aspects of the Toulmin
schema will return in what follows. For Toulmin, inferential strength is a
parameter, to be set in accordance with the subject matter under discussion.
(Interestingly, content-dependence of reasoning is also a recurrent finding of
cognitive psychologists: cf. the earlier-mentioned [Joh83].) In chapter 2, we
have already defended exactly the same strategy for abduction. Moreover, the
procedural flavor of the Toulmin schema fits well with our product-process
distinction.
As for the basic building blocks of abductive explanatory inference, in the
remainder of this book, we will confine ourselves to a ternary format:
Θ|α⇒ϕ
Abduction as Logical Inference
69
This modest step already enables us to demonstrate a number of interesting
departures from standard logical systems. Let us recall some considerations
from chapter 2 motivating this move. The theory Θ needs to be explicit for
a number of reasons. Validity of an abductive inference is closely related to
the background theory, as the presence of some other explanation β in Θ may
actually disqualify α as an explanation. Moreover, what we called ‘triggers’
of explanation are specific conditions on a theory Θ and an observation ϕ.
A fact may need explanation with respect to one theory, but not with respect
to another. Making a distinction between Θ and α allows us to highlight the
specific explanation (which we did not have before), and control different forms
of explanation (facts, rules, or even new theories). But certainly, our accounts
would become yet more sensitive if we worked with some of the above richer
formats.
Inferential Strength: A Parameter
At first glance, once we have Tarski’s notion of truth, logical consequence
seems an obvious defined notion. A conclusion follows if it is true in all models
where the premisses are true. But the contemporary philosophical and computational traditions have shown that natural notions of inference may need more
than truth in the above sense, or may even hinge on different properties altogether. For example, among the candidates that revolve around truth, statistical
inference requires not total inclusion of premisse models in conclusion models,
but only a significant overlap, resulting in a high degree of certainty. Other
approaches introduce new semantic primitives. Notably, Shoham’s notion of
causal and default reasoning ([Sho88]) introduces a preference order on models,
requiring only that the most preferred models of Σ be included in the models
of ϕ.
More radically, dynamic semantics replaces the notion of truth by that of
information change, aiming to model the flow of information. This move leads
to a redesign for Tarski semantics, with e.g. quantifiers becoming actions on
assignments ([vBC94]). This logical paradigm has room for many different
inferential notions ([Gro95, vBe96a]). An example is the earlier mentioned
update-to-test-consequence:
“process the successive premisses in Σ, thereby absorbing their informational content into the initial information state. At the end, check if the resulting
state is rich enough to satisfy the conclusion ϕ”.
Informational content rather than truth is also the key semantic property
in situation theory ([PB83]). In addition to truth-based and information-based
approaches, there are, of course, also various proof-theoretic variations on stan-
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ABDUCTIVE REASONING
dard consequence. Examples are default reasoning: “ϕ is provable unless and
until ϕ is disproved” ([Rei80]), and indeed Hempel’s hypothetico-deductive
model of scientific inference itself.
All these alternatives agree with our analysis of abductive explanatory inference. On our view, abduction is not a new notion of inference. It is rather
a topic-dependent practice of scientific reasoning, which can be supported by
various members of the above family. In fact, it is appealing to think of abductive inference in several respects, as inference involving preservation of
both truth and explanatory power. In fact, appropriately defined, both might
turn out equivalent. It has also been argued that since abduction is a form
of reversed deduction, just as deduction is truth-preserving, abduction must
be falsity-preserving ([Mic94]). However, [Fla95] gives convincing arguments
against this particular move. Moreover, as we have already discussed intuitively,
abduction is not just deduction in reverse.
Our choice here is to study abductive inference in more depth as a strengthened form of classical inference. This is relevant, it offers nice connections
with artificial intelligence and the philosophy of science, and it gives a useful
simple start for a broader systematic study of abductive inference. One can
place this choice in a historical context, namely the work of Bernard Bolzano,
a nineteenth century philosopher and mathematician (and theologian) engaged
in the study of different varieties of inference. We provide a brief excursion,
providing some perspective for our later technical considerations.
Bolzanos’s Program
Bolzano’s notion of deducibility (Ableitbarkeit) has long been recognized as
a predecessor of Tarski’s notion of logical consequence ([Cor75]). However,
the two differ in several respects, and in our broader view of logic, they even
appear radically different. These differences have been studied both from a
philosophical ([Tho81]) and from a logical point of view ([vBe84a]).
One of Bolzano’s goals in his theory of science ([Bol73]), was to show why
the claims of science form a theory as opposed to an arbitrary set of propositions.
For this purpose, he defines his notion of deducibility as a logical relationship
extracting conclusions from premisses forming compatible propositions, those
for which some set of ideas make all propositions true when uniformly substituted throughout. In addition, compatible propositions must share common
ideas. Bolzano’s use of ‘substitutions’ is of interest by itself, but for our purposes here, we will identify these (somewhat roughly) with the standard use of
‘models’. Thompson attributes the difference between Bolzano’s consequence
and Tarski’s to the fact that the former notion is epistemic while the latter is
ontological. These differences have strong technical effects. With Bolzano,
the premisses must be consistent (sharing at least one model), with Tarski, they
Abduction as Logical Inference
71
need not. Therefore, from a contradiction, everything follows for Tarski, and
nothing for Bolzano.
Restated for our ternary format, then, Bolzano’s notion of deducibility reads
as follows (cf. [vBe84a]):
Θ | α ⇒ ϕ if
(1) The conjunction of Θ and α is consistent.
(2) Every model for Θ plus α verifies ϕ.
Therefore, Bolzano’s notion may be seen (anachronistically) as Tarski’s consequence plus the additional condition of consistency. Bolzano does not stop
here. A finer grain to deducibility occurs in his notion of exact deducibility,
which imposes greater requirements of ‘relevance’. A modern version, involving inclusion-minimality for sets of abducibles, may be transcribed (again, with
some historical injustice) as:
Θ | α ⇒+ ϕ if
(1) Θ | α ⇒ ϕ
(2) There is no proper subset of α, α’, such that Θ | α’ ⇒ ϕ.
That is, in addition to consistency with the background theory, the premisse
set α must be ‘fully explanatory’ in that no subpart of it would do the derivation.
Notice that this leads to non-monotonicity. Here is an example:
Θ | a → b, a ⇒+ b
Θ | a → b, a, b → c ⇒+ b
Bolzano’s agenda for logic is relevant to our study of abductive reasoning
(and the study of general non-monotonic consequence relations) for several
reasons. It suggests the methodological point that what we need is not so
much proliferation of different logics as a better grasp of different styles of
consequence. Moreover, his work reinforces an earlier claim, that truth is not all
there is to understanding explanatory reasoning. More specifically, his notions
still have interest. For example, exact deducibility has striking similarities to
explanation in philosophy of science (cf. chapter 5).
Abductive Explanatory Inference as Deduction in Reverse
In this section we define abductive explanatory inference as a strengthened
form of classical inference. Our proposal will be in line with abduction in
artificial intelligence, as well as with the Hempelian account of explanation. We
will motivate our requirements with our very simple rain example, presented
here in classical propositional logic:
Θ : r → w, s → w
ϕ:w
The first condition for a formula α to count as an explanation for ϕ with
respect to Θ is the inference requirement. Many formulas would satisfy this
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ABDUCTIVE REASONING
condition. In addition to earlier-mentioned obvious explanations (r: rain, s:
sprinklers-on), one might take their conjunction with any other formula, even
if the latter is inconsistent with Θ (e.g. r ∧ ¬w). One can take the fact itself
(w), or, one can introduce entirely new facts and rules (say, there are children
playing with water, and this causes the lawn to get wet).
Inference: Θ, α |= ϕ
α’s: r, s, r ∧ s, r ∧ z, r ∧ ¬w, s ∧ ¬w, w, [c, c → w], Θ → w.
Some of these ‘abductive explanations’ must be ruled out from the start. We
therefore impose a consistency requirement on the left hand side, leaving only
the following as possible abductive explanations:
Consistency: Θ, α is consistent.
α’s: r, s, r ∧ s, r ∧ z, w, [c, c → w], Θ → w.
An abductive explanation α is only necessary, if ϕ is not already entailed by
Θ. Otherwise, any consistent formula will count as an abductive explanation.
Thus we repeat an earlier trigger for abduction: Θ |= ϕ. By itself, this does
not rule out any potential abducibles on the above list (as it does not involve the
argument α.) But also, in order to avoid what we may call external explanations –those that do not use the background theory at all (like the explanation
involving children in our example) –, it must be required that α be insufficient
for explaining ϕ by itself (α |= ϕ). In particular this condition avoids the trivial
reflexive explanation ϕ ⇒ ϕ. Then only the following explanations are left in
our list of examples:
Explanation Θ |= ϕ, α |= ϕ
α’s: r, s, r ∧ s, r ∧ z, Θ → w.
Now both Θ and α contribute to explaining ϕ. However, we are still left
with some formulas that do not seem to be genuine explanations (r ∧ z, Θ →
w). Therefore, we explore a more sensitive criterion, admitting only ‘the best
explanation’.
Selecting the Best Explanation
Intuitively, a reasonable ground for choosing a statement as the best explanation,
is its simplicity. It should be minimal, i.e. as weak as possible in performing
its job. This would lead us to prefer r over r ∧ z in the preceding example.
As Peirce puts it, we want the explanation that “adds least to what has been
observed” (cf. [CP, 6.479]). The criterion of simplicity has been extensively
considered both in the philosophy of science and in artificial intelligence. But
its precise formulation remains controversial, as measuring simplicity can be a
tricky matter. One attempt to capture simplicity in a logical way is as follows:
Weakest Abductive Explanation:
α is the weakest abductive explanation for ϕ with respect to Θ iff
Abduction as Logical Inference
73
(i) Θ, α |= ϕ
(ii) For all other formulas β such that Θ, β |= ϕ, |= β → α.
This definition makes the explanations r and s almost the weakest in the
above example, just as we want. Almost, but not quite. For, the explanation
Θ → w, a trivial solution, turns out to be the minimal one. The following is a
folklore observation to this effect:
Fact 1 Given any theory Θ and observation ϕ to be explained from it, α = Θ → ϕ
is the weakest abductive explanation.
Proof. Obviously, we have (i) Θ, Θ → ϕ |= ϕ. Moreover, let α’ be any other explanation. This says that Θ, α′ |= ϕ. But then we also have (by conditionalizing) that
⊣
α′ |= Θ → ϕ, and hence |= α′ → (Θ → ϕ)
That Θ → ϕ is a solution that will always count as an explanation in a
deductive format was noticed by several philosophers of science ([Car55]). It
has been used as an argument to show how the issue would impose restrictions
on the syntactic form of abducibles. Surely, in this case, the explanation seems
too complex to count. We will therefore reject this proposal, noting also that it
fails to recognize (let alone compare) intuitively ‘minimal’ explanations like r
and s in our running example.
Other criteria of minimality exist in the literature. One of them is based on
preference orderings. The best explanation is the most preferred one, given an
explicit ordering of available assertions. In our example, we could define an
order in which inconsistent explanations are the least preferred, and the simplest
the most. These preference approaches are quite flexible, and can accommodate
various working intuitions. However, they may still depend on many factors,
including the background theory. This seems to fall outside a logical framework,
referring rather to further ‘economic’ decision criteria like utilities. A case in
point is Peirce’s ‘economy of research’ in selecting a most promising hypothesis.
What makes a hypothesis good or best has no easy answer. One may appeal
to criteria of simplicity, likelihood, or predictive power. To complicate matters
even further, we often do not compare (locally) quality of explanations given
a fixed theory, but rather (globally) whole packages of ‘theory + explanation’.
This perspective gives a much greater space of options. As we have not been
able to shed a new light from logic upon these matters, we will ignore these
dimensions here.
Further study would require more refined views of theory structure and reasoning practice, in line with some of the earlier references7 , or even more
ambitiously, following current approaches to ‘verisimilitude’ in the philosophy
of science (cf. [Kui87]).
7 Preferences
over models (though not over statements) will be mentioned briefly as providing one possible
inference mechanism for abduction.
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ABDUCTIVE REASONING
We conclude with one final observation. perhaps one reason why the notion of ‘minimality’ has proved so elusive is again our earlier product-process
distinction. Philosophers have tried to define minimality in terms of intrinsic properties of statements and inferences as products. But it may rather be
a process-feature, having to do with computational effort in some particular
procedure performing abduction. Thus, one and the same statement might be
minimal in one abduction, and non-minimal in another.
Abductive Explanatory Characterization Styles
Following our presentation of various requirements for an abductive explanation, we make things more concrete for further reference. We consider five
versions of abductive explanations making up the following styles: plain, consistent, explanatory, minimal and preferential, defined as follows:
Abductive Explanatory Styles
Given Θ (a set of formulae) and ϕ (a sentence), α is an abductive explanation if:
Plain :
(i) Θ, α |= ϕ.
Consistent :
(i) Θ, α |= ϕ,
(ii) Θ, α consistent.
Explanatory :
(i) Θ, α |= ϕ,
(ii) Θ |= ϕ,
(iii) α |= ϕ.
Minimal :
(i) Θ, α |= ϕ,
(ii) α is the weakest such abductive explanation (not equal to Θ → ϕ).
Preferential :
(i) Θ, α |= ϕ,
(ii) α is the best abductive explanation according to some given preferential ordering.
We can form other combinations, of course, but these will already exhibit
many characteristic phenomena. Note that these requirements do not depend
on classical consequence. For instance, in Chapter 5, the consistency and
the explanatory requirements work just as well for statistical inference. The
former then also concerns the explanandum ϕ. (For, in probabilistic reasoning
it is possible to infer two contradictory conclusions even when the premisses
are consistent.) The latter helps capture when an explanation helps raise the
probability of the explanandum.
A full version of abduction would make the formula to be abduced part of the
derivation, consistent, explanatory, and the best possible one. However, instead
Abduction as Logical Inference
75
of incorporating all these conditions at once, we shall consider them one by
one. Doing so clarifies the kind of restriction each requirement adds to the
notion of plain abduction. Our standard versions will base these requirements
on classical consequence underneath. But we also look briefly toward the end at
versions involving other notions of consequence. We will find that our various
notions of abduction have advantages, but also drawbacks, such as an increase
of complexity for explanatory reasoning as compared with classical inference.
Up to now, we can say that abductive inference may be characterized by
reversed deduction plus additional conditions. However, is this all we can say
about the logic of abduction? This definition does not really capture the rationality principles behind this type of reasoning, like its non-monotonic feature
we have talked about.
In what follows, our aim is to present the characterization of abductive inference from the structural perspective we introduced earlier in this chapter.
This approach has become popular across a range of non-standard logics. Our
systematic analysis will explore different abductive styles from this perspective.
4.
Abductive Explanatory Inference:
Structural Characterization
Consistent Abductive Explanatory Inference
We recall the definition:
Θ | α ⇒ ϕ iff
(i) Θ, α |= ϕ
(ii) Θ, α are consistent
The first thing to notice is that the two items to the left behave symmetrically:
Θ|α⇒ϕ
iff
α|Θ⇒ϕ
Indeed, in this case, we may technically simplify matters to a binary format
after all: X ⇒ C , in which X stands for the conjunction of Θ and α, and C
for ϕ. To bring these in line with the earlier-mentioned structural analysis of
nonclassical logics, we view X as a finite sequence X1 . . . , Xk of formulas and
C as a single conclusion.
Classical Structural Rules
Of the structural rules for classical consequence, contraction and permutation
hold for consistent abduction. But reflexivity, monotonicity and cut fail, witness
by the following counterexamples:
Reflexivity: p ∧ ¬p ⇒ p ∧ ¬p
Monotonicity: p ⇒ p, but p, ¬p ⇒ p
Cut: p, ¬q ⇒ p, and p, q ⇒ q, but p, ¬q, q ⇒ q
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ABDUCTIVE REASONING
New Structural Rules
Here are some restricted versions of the above failed rules, and some others that
are valid for consistent abduction:
1 Conditional Reflexivity
(CR)
X⇒B
X ⇒ Xi
2 Simultaneous Cut
1≤i≤k
(SC)
A1 , . . . , Ak ⇒ B
U ⇒ A1 . . . U ⇒ Ak
U ⇒B
3 Conclusion Consistency
(CC)
U ⇒ A1 . . . U ⇒ Ak
A1 , . . . , Ak ⇒ Ai
1≤i≤k
These rules state the following. Conditional Reflexivity requires that the
sequence X derive something else (X ⇒ B), as this ensures consistency.
Simultaneous Cut is a combination of Cut and Contraction in which the sequent
A1 , . . . , Ak may be omitted in the conclusion when each of its elements Ai
is consistently derived by U and this one in its turn consistently derives B.
Conclusion Consistency says that a sequent A1 , . . . , Ak implies its elements
if each of these are implied consistently by something (U arbitrary), which is
another form of reflexivity.
Proposition 2 These rules are sound for consistent abduction.
Proof. In each of these three cases, it is easy to check by simple set-theoretic reasoning that the corresponding classical consequence holds. Therefore, the only thing to
be checked is that the premisses mentioned in the conclusions of these rules must be
consistent. For Conditional Reflexivity, this is because X already consistently implied
something. For Simultaneous Cut, this is because U already consistently implied something. Finally, for Conclusion Consistency, the reason is that U must be consistent, and
⊣
it is contained in the intersection of all the Ai , which is therefore consistent, too.
A Representation Theorem
The given structural rules in fact characterize consistent abduction:
Proposition 3 A consequence relation satisfies structural rules 1 (CR), 2 (SC),
3 (CC) iff it is representable in the form of consistent abduction.
Proof. Soundness of the rules was proved above. Now consider the completeness
direction. Let ⇒ be any abstract relation satisfying 1, 2, 3. Define for any proposition
A,
A∗ = {X | X ⇒ A}
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Abduction as Logical Inference
We now show the following statement of adequacy for this representation:
Claim. A1 , . . . , Ak ⇒ B
iff
∅ ⊂ A∗1 ∩ . . . ∩ A∗k ⊆ B ∗ .
Proof. ‘Only if’. Since A1 , . . . , Ak ⇒ B, by Rule 1 (CR) we have A1 , . . . , Ak ⇒ Ai
(1 ≤ i ≤ k). Therefore, A1 , . . . , Ak ∈ A∗i , for each i with 1 ≤ i ≤ k, which gives the
proper inclusion. Next, let U be any sequence in the intersection of all A∗i , for 1, . . . , k.
That is, U ⇒ A1 , . . . , U ⇒ Ak . By Rule 2 (SC), U ⇒ B, i.e. U ∈ B ∗ , and we have
shown the second inclusion.
‘If’. Using the assumption of non-emptiness, let, say, U ∈ A∗i , for 1, . . . , k. i.e.
(1 ≤ i ≤ k). By
U ⇒ A1 , . . . , U ⇒ Ak . By Rule 3 (CC), A1 , . . . , Ak ⇒ Ai
the second inclusion then, A1 , . . . , Ak ∈ B ∗ . By the definition of the function *, this
⊣
means that A1 , . . . , Ak ⇒ B.
More Familiar Structural Rules
The above principles characterize consistent abduction. Even so, there are
more familiar structural rules that are valid as well, including modified forms
of Monotonicity and Cut. For instance, it is easy to see that ⇒ satisfies a form
of modified monotonicity: B may be added as a premisse if this addition does
not endanger consistency. And the latter may be shown by their ‘implying’ any
conclusion:
Modified Monotonicity:
X⇒A
X, B ⇒ C
X, B ⇒ A
As this was not part of the above list, we expect some derivation from the
above principles. And indeed there exists one:
Modified Monotonicity Derivation:
X, B ⇒ C
1
X⇒A
X, B ⇒ Xi′ s
X, B ⇒ A
2
These derivations also help in seeing how one can reason perfectly well with
non-classical structural rules. Another example is the following valid form of
Modified Cut:
Modified Cut
X⇒A
U, A, V ⇒ B
U, X, V ⇒ C
U, X, V ⇒ B
This may be derived as follows:
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ABDUCTIVE REASONING
Modified Cut Derivation
U, X, V ⇒ C
U, X, V ⇒ U ′ s, V ′ s
U, X, V ⇒ C
U, X, V ⇒ Xi′ s
1
1
X⇒A
2
U, X, V ⇒ A
U, X, V ⇒ B
U, A, V ⇒ B
2
Finally, we check some classically structural rules that do remain valid as
they stand, showing the power of Rule (3):
Permutation
X, A, B, Y ⇒ X
X, B, A, Y ⇒ X
X, A, B, Y ⇒ C
X, A, B, Y ⇒ A X, A, B, Y ⇒ B
X, B, A, Y ⇒ B X, B, A, Y ⇒ A
X, A, B, Y ⇒ Y
X, B, A, Y ⇒ Y
1
3
X, A, B, Y ⇒ C
X, B, A, Y ⇒ C
2
Contraction (one sample case)
X, A, A, Y ⇒ B
1
X, A, A, Y ⇒ Xi′ s, A, Yi′ s
3
X, A, A, Y ⇒ B
X, A, Y ⇒ Xi′ s, A, Yi′ s
X, A, Y ⇒ B
2
Thus, consistent abductive inference defined as classical consequence plus
the consistency requirement has appropriate forms of reflexivity, monotonicity,
and cut for which it is assured that the premisses remain consistent. Permutation
and contraction are not affected by the consistency requirement, therefore the
classical forms remain valid. More generally, the preceding examples show
simple ways of modifying all classical structural principles by putting in one
extra premisse ensuring consistency.
Simple as it is, our characterization of this notion of inference does provide a
complete structural description of Bolzano’s notion of deducibility introduced
earlier in this chapter (section 3.3).
Explanatory Abductive (Explanatory) Inference
Explanatory abductive explanatory inference (explanatory abduction, for
short)was defined as plain abduction (Θ, α |= ϕ) plus two conditions of necessity (Θ |= ϕ) and insufficiency (α |= ϕ). However, we will consider a weaker
version (which only considers the former condition) and analyze its structural
rules. This is actually somewhat easier from a technical viewpoint. The full
version remains of general interest though, as it describes the ‘necessary collaboration’ of two premisses set to achieve a conclusion. It will be analyzed further
79
Abduction as Logical Inference
in chapter 5 in connection with philosophical models of scientific explanation.
We rephrase our notion as:
Weak Explanatory Abduction:
Θ | α ⇒ ϕ iff
(i) Θ, α |= ϕ
(ii) Θ |= ϕ
The first thing to notice is that we must leave the binary format of premisses
and conclusion. This notion is non-symmetric, as Θ and α have different roles.
Given such a ternary format, we need a more finely grained view of structural
rules. For instance, there are now two kinds of monotonicity, one when a
formula is added to the explanations and the other one when it is added to the
theory:
Monotonicity for Abductive Explanations:
Θ|α⇒ϕ
Θ | α, A ⇒ ϕ
Monotonicity for Theories:
Θ|α⇒ϕ
Θ, A | α ⇒ ϕ
The former is valid, but the latter is not. (A counterexample is: p | q, r ⇒ q
but p, q | q, r ⇒ q). Monotonicity for explanations states that an explanation
for a fact does not get invalidated when we strengthen it, as long as the theory
is not modified.
Here are some valid principles for weak explanatory abduction.
Weak Explanatory Reflexivity
Θ|α⇒ϕ
Θ|ϕ⇒ϕ
Weak Explanatory Cut
Θ | α, β ⇒ ϕ
Θ|α⇒β
Θ|α⇒ϕ
In addition, the classical forms of contraction and permutation are valid
on each side of the bar. Of course, one should not permute elements of the
theory with those in the explanation slot, or vice versa. We conjecture that
the given principles completely characterize the weak explanatory abduction
notion, when used together with the above valid form of monotonicity.
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ABDUCTIVE REASONING
Structural Rules with Connectives
Pure structural rules involve no logical connectives. Nevertheless, there are
natural connectives that may be used in the setting of abductive consequence.
For instance, all Boolean operations can be used in their standard meaning.
These, too, will give rise to valid principles of inference. In particular, the
following well-known classical laws hold for all notions of abductive inference
studied so far:
Disjunction of Θ-antecedents:
Θ1 | A ⇒ ϕ
Θ2 | A ⇒ ϕ
Θ1 ∨ Θ2 | A ⇒ ϕ
Conjunction of Consequents
Θ | A ⇒ ϕ1
Θ | A ⇒ ϕ2
Θ | A ⇒ ϕ1 ∧ ϕ 2
These rules will play a role in our proposed calculus for abduction, as we
will show later on.
Another way of expressing monotonicity with the aid of negation and classical derivability is as follows:
Monotonicity:
Θ|α⇒ϕ
Θ | α ⊢ ¬β
Θ | α, β ⇒ ϕ
We conclude a few brief points on the other versions of abduction on our list.
We have not undertaken to characterize these in any technical sense.
Minimal and Preferential Abductive Explanatory Inference
Consider our versions of ‘minimal’ abduction. One said that Θ, α |= ϕ and
α is the weakest such explanation. By contrast, preferential abduction said that
Θ, α |= ϕ and α is the best explanation according to some given preferential
ordering. For the former, with the exception of the above disjunction rule for
antecedents, no other rule that we have seen is valid. But it does satisfy the
following form of transitivity:
Transitivity for Minimal Abduction:
Θ|α⇒ϕ
Θ|β⇒α
Θ|β⇒ϕ
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Abduction as Logical Inference
For preferential abduction, on the other hand, no structural rule formulated so
far is valid. The reason is that the relevant preference order amongst formulas,
in itself needs to be captured in the formulation of our inference rules. A valid
formulation of monotonicity would then be something along the following lines:
Monotonicity for Preferential Abduction:
Θ|α⇒ϕ
α, β < α
Θ | α, β ⇒ ϕ
In our opinion, this is no longer a structural rule, since it adds a mathematical
relation (< for a preferential order) that cannot in general be expressed in terms
of the consequence itself. This is a point of debate, however, and its solution
depends on what each logic artisan is willing to represent in a logic. In any
case, this format is beyond what we will study in this book.
Structural Rules for Nonstandard Inference
All abductive versions so far had classical consequence underneath. In this
section, we briefly explore structural behaviour when the underlying notion of
inference is non standard, as in preferential entailment. Moreover, we throw in
some words about structural rules for abduction in logic programming, and for
induction.
Preferential Reasoning
Interpreting the inferential parameter as preferential entailment means that
Θ, α ⇒ ϕ if (only) the most preferred models of Θ ∪ α are included in the
models of ϕ. This leads to a completely different set of structural rules. Here
are some valid examples, transcribed into our ternary format from [KLM90]:
Θ, α ⇒ α
Reflexivity:
Cautious Monotonicity:
Θ|α⇒β
Θ|α⇒γ
Θ | α, β ⇒ γ
Cut:
Θ | α, β ⇒ γ
α⇒β
Θ|α⇒γ
Disjunction:
Θ|α⇒ϕ
Θ|β⇒ϕ
Θ|α∨β ⇒ϕ
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ABDUCTIVE REASONING
It is interesting to see in greater detail what happens to these rules when
we add our further conditions of ‘consistency’ and ‘explanation’. In all, what
happens is merely that we get structural modifications similar to those found
earlier on for classical consequence. Thus, a choice for a preferential proof
engine, rather than classical consequence, seems orthogonal to the behavior of
abduction.
Structural rules for Prolog Computation
An analysis via structural rules may be also performed for notions of ⇒ with
a more procedural flavor. In particular, the earlier-mentioned case of Prolog
computation obeys clear structural rules (cf. [vBe92, Kal95, Min90]). Their
format is somewhat different from classical ones, as one needs to represent
more of the Prolog program structure for premisses, including information on
rule heads. (Also, Kalsbeek [Kal95] gives a complete calculus of structural
rules for logic programming including such control devices as the cut operator
!). The characteristic expressions of a Gentzen style sequent calculus for these
systems (in the reference above) are sequents of the form [P ] ⇒ ϕ, where P
is a (propositional, Horn clause) program and ϕ is an atom. A failure of a
goal is expressed as [P ] ⇒ ¬ϕ (meaning that ϕ finitely fails). In this case,
valid monotonicity rules must take account of the place in which premisses are
added, as Prolog is sensitive to the order of its program clauses. Thus, of the
following rules, the first one is valid, but the second one is not:
Right Monotonicity
[P ] ⇒ ϕ
[P ; β] ⇒ ϕ
Left Monotonicity
[P ] ⇒ ϕ
[β; P ] ⇒ ϕ
Counterexample: β = ϕ ← ϕ
The question of complete structural calculi for abductive logic programming
will not be addressed in this book, we will just mention that a natural rule for
an ‘abductive update’ is as follows:
Atomic Abductive Update
[P ] ⇒ ¬ϕ
[P ; ϕ] ⇒ ϕ
We will briefly return to structural rules for abduction as a process in the next
chapter.
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Abduction as Logical Inference
Structural Rules For Induction
Unlike abduction, enumerative induction is a type of inference that explains a
set of observations, and makes a prediction for further ones (cf. our discussion
in chapter 2). Our previous rule for conjunction of consequents already suggests
how to give an account for further observations, provided that we interpret the
commas below as conjunction amongst formulae (in the usual Gentzen calculus,
commas to the right are interpreted rather as disjunctions):
α ⇒ ϕ1
α ⇒ ϕ2
α ⇒ ϕ1 , ϕ 2
That is, an inductive explanation α for ϕ1 remains an explanation when a
formula ϕ2 is added, provided that α also accounts for it separately. Note that
this rule is a kind of monotonicity, but this time the increase is on the conclusion
set rather than on the premisse set. More generally, an inductive explanation α
for a set of formulae remains valid for more input data ψ when it explains it:
(Inductive) Monotonicity on Observations
Θ | α ⇒ ϕ1 , . . . , ϕn
Θ|α⇒ψ
Θ | α ⇒ ϕ1 , . . . , ϕn , ψ
In order to put forward a set of rules characterizing inductive explanation,
a further analysis of its properties should be made, and this falls beyond the
scope of this thesis. What we anticipate however, is that a study of enumerative
induction from a structural point of view will bring yet another twist to the
standard structural analysis, that of giving an account of changes in conclusions.
Further Logical Issues
Our analysis so far has only scratched the surface of a broader field. In this
section we discuss a number of more technical logical aspects of abductive styles
of inference. This identifies further issues that seem relevant to understanding
the logical properties of abduction.
Completeness
The usual completeness theorems have the following form:
Θ |= ϕ
Θ⊢ϕ
iff
With our ternary format, we would expect some similar equivalence, with a
possibly different treatment of premisses on different sides of the comma:
Θ, α |= ϕ
iff
Θ, α ⊢ ϕ
Can we get such completeness results for any of the abductive versions we
have described so far? Here are two extremes.
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ABDUCTIVE REASONING
The representation arguments for the above characterizations of abduction
may be reworked into completeness theorems of a very simple kind. (This
works just as in [vBe96a], chapter 7). In particular, for consistent abduction,
our earlier argument essentially shows that Θ, α ⇒ ϕ follows from a set of
ternary sequents Φ iff it can be derived from Φ using only the derivation rules
(CR), (SC), (CC) above.
These representation arguments may be viewed as ‘poor man’s completeness
proofs’, for a language without logical operators. Richer languages arise by
adding operators, and completeness arguments need corresponding ‘upgrading’
of the representations used. (Cf. [Kur95] for an elaborate analysis of this
upward route for the case of categorial and relevance logics. [Gro95] considers
the same issue in detail for dynamic styles of inference.) At some level, no more
completeness theorems are to be expected. The complexity of the desired proof
theoretical notion ⊢ will usually be recursively enumerable (Σ01 ). But, our later
analysis will show that, with a predicate-logical language, the complexity of
semantic abduction |= will become higher than that. The reason is that it mixes
derivability with non-derivability (because of the consistency condition).
So, our best chance for achieving significant completeness is with an intermediate language, like that of propositional logic. In that case, abduction
is still decidable, and we may hope to find simple proof rules for it as well.
(Cf. [Tam94] for the technically similar enterprise of completely axiomatizing
simultaneous ‘proofs’ and ‘fallacies’ in propositional logic.) Can we convert
our representation arguments into full-fledged completeness proofs when we
add propositional operators ¬, ∧, ∨? We have already seen that we do get
natural valid principles like disjunction of antecedents and conjunction of consequents. However, there is no general method that connects a representational
result into more familiar propositional completeness arguments. A case of successful (though non-trivial) transfer is in [Kan93], but essential difficulties are
identified in [Gro95].
Instead of solving the issue of completeness here, we merely propose the
following axioms and rules for a sequent calculus for consistent abduction
(which we label as |=c ) in what follows:
Axiom: p |=c p
Rules for Conjunction:
∧1
Θ |=c ϕ1 ,
Θ |=c ϕ2
Θ |=c ϕ1 ∧ ϕ2
The following are valid provided that α, ψ are formulas with only positive
propositional letters:
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Abduction as Logical Inference
∧2
α |=c α
ψ |=c ψ
α, ψ |=c α
∧3
α, ψ |=c ϕ
α ∧ ψ |=c ϕ
Rules For Disjunction:
∨1
Θ1 |=c ϕ
Θ2 |=c ϕ
Θ1 ∨ Θ2 |=c ϕ
∨2
Θ |=c ϕ
Θ |=c ϕ ∨ ψ
∨3
Θ |=c ϕ
Θ |=c ψ ∨ ϕ
Rules for Negation:
¬1
¬2
Θ, A |=c ϕ
Θ |=c ϕ ∨ ¬A
Θ |=c ϕ ∨ A
Θ ∧ ¬A |=c ψ
Θ ∧ ¬A |=c ϕ
It is easy to see that these rules are sound on the interpretation of |= as
consistent abduction. This calculus is already unlike most usual logical systems,
though. First of all there is no substitution rule, as p |= p is an axiom, whereas
in general ψ |= ψ unless ψ has only positive propositional letters, in which
case it is proved to be consistent. By itself, this is not dramatic (for instance,
several modal logics exist without a valid substitution rule), but it is certainly
uncommon. Moreover, note that the rules which “move things to the left" (¬2 )
are different from their classical counterparts, and others (∧3 ) are familiar but
here a condition to ensure consistency is added. Even so, one can certainly do
practical work with a calculus like this.
For instance, all valid principles of classical propositional logic that do not
involve negations are derivable here. Semantically, this makes sense, as positive
formulas are always consistent without special precautions. On the other hand,
it is easy to check that the calculus provides no proof for a typically invalid
sequent like p ∧ ¬p |= p ∧ ¬p8 .
8 The
reason is that their cut-free classical proofs (satisfying the subformula property) involve only conjunction and disjunction - for which we have the standard rules.
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ABDUCTIVE REASONING
Digression:
A general semantic view of abductive explanatory consequence
Speaking generally, we can view a ternary inference relation Θ | α ⇒ ϕ as a
ternary relation C (T, A, F) between sets of models for, respectively, Θ, α, and
ϕ. What structural rules do is constrain these relations to just a subclass of all
possibilities. (This type of analysis has analogies with the theory of generalized
quantifiers in natural language semantics. It may be found in [vBe84a] on the
model theory of verisimilitude, or in [vBe96b] on general consequence relations
in the philosophy of science.) When enough rules are imposed we may represent
a consequence relation by means of simpler notions, involving only part of the
a priori relevant 23 = 8 “regions" of models induced by our three argument
sets.
In this light, the earlier representation arguments might even be enhanced by
including logical operators. We merely provide an indication. It can be seen
easily that, in the presence of disjunction, our explanatory abduction satisfies
full Boolean ‘Distributivity’ for its abducible argument αi :
Θ|
i
αi ⇒ ϕ
iff
for some i, Θ | αi ⇒ ϕ.
Principles like this can be used to reduce the complexity of a consequence
relation. For instance, the predicate argument A may now be reduced to a point
wise one, as any set A is the union of all singletons {a} with a ∈ A.
Complexity
Our next question addresses the complexity of different versions of abduction.
Non-monotonic logics may be better than classical ones for modelling common
sense reasoning and scientific inquiry. But their gain in expressive power usually
comes at the price of higher complexity, and abduction is no exception. Our
interest is then to briefly compare the complexity of abduction to that of classical
logic. We have no definite results here, but we do have some conjectures. In
particular, we look at consistent abduction, beginning with predicate logic.
Predicate-logical validity is undecidable by Church’s Theorem. Its exact
complexity is Σ01 (the validities are recursively enumerable, but not recursive).
(To understand this outcome, think of the equivalent assertion of derivability:
“there exists a P: P is a proof for ϕ".) More generally, Σ (or Π) notation refers to
the usual prenex forms for definability of notions in the Arithmetical Hierarchy.
Complexity is measured here by looking at the quantifier prenex, followed by a
decidable matrix predicate. A subscript n indicates n quantifier changes in the
prenex. (If a notion is both Σn and Πn , it is called ∆n .) The complementary
notion of satisfiability is also undecidable, being definable in the form Π01 . Now,
abductive consequence moves further up in this hierarchy.
In order to show that consistent abduction is not ∆02 -complete we have the
following. The statement that “Θ, α is consistent” is Π01 , while the statement that
87
Abduction as Logical Inference
“Θ, α |= ϕ” is Σ01 (cf. the above observations). Therefore, their conjunction
may be written, using well-known prenex operations, in either of the following
forms:
∃∀DEC
or
∀∃DEC.
∆02 .
This analysis gives an upper bound
Hence consistent abduction is in
only. But we cannot do better than this. So it is also a lower bound. For the
sake of reductio, suppose that consistent abduction were Σ01 . Then we could
reduce satisfiability of any formula B effectively to the abductive consequence
B, B ⇒ B, and hence we would have that satisfiability is also Σ01 . But then,
Post’s Theorem says that a notion which is both Σ01 and Π01 must be decidable.
This is a contradiction, and hence Θ, α ⇒ ϕ is not Σ01 . Likewise, consistent
abduction cannot be Π01 , because of another reduction: this time from the
validity of any formula B to True, True ⇒ B.
Consistent abduction is not ∆02 -complete. Although it is in ∆02 and is not Π01 ,
the latter is not sufficient to prove its hardness and thereby completeness, for that
we would have to show that every ∆02 predicate may be reduced to consistent
abduction, and we can only prove that it can be written as a conjunction of Σ01
and Π01 , showing that it belongs to a relatively simple part of ∆02 .
By similar arguments we can show that the earlier weak explanatory abduction is in ∆02 – and the same holds for other variants that we considered.
Therefore, our strategy in this chapter of adding amendments to classical consequence is costly, as it increases its complexity. On the other hand, we seem
to pay the price just once. It makes no difference with respect to complexity whether we add one or all of the abductive requirements at once. We do
not have similar results about the cases with minimality and preference, as their
complexity will depend on the complexity of our (unspecified) preference order.
Complexity may be lower in a number of practically important cases. First,
consider poorer languages. In particular, for propositional logic, all our notions
of abduction remain obviously decidable. Nevertheless, their fine-structure will
be different. Propositional satisfiability is NP-complete, while validity is CoNP-complete.
Another direction would restrict attention to useful fragments of predicate
logic. For example, universal clauses without function symbols have a decidable
consequence problem. Therefore we have the following:
Proposition 4 All our notions of abductive explanatory inference are decidable
over universal clauses.
Finally, complexity as measured in the above sense may miss out on some
good features of abductive reasoning, such as possible natural bounds on search
space for abducibles. A very detailed study on the complexity of logic-based
abduction which takes into account different kinds of theories (propositional,
clausal, Horn) as well as several minimality measures are found in [EG95].
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ABDUCTIVE REASONING
The Role of Language
Our notions of abduction all work for arbitrary formulas, and hence they have no
bias toward any special formal language. But in practice, we can often do with
simpler forms. E.g., observations ϕ will often be atoms, and the same holds for
explanations α. Here are a few observations showing what may happen.
Syntactic restrictions may make for ‘special effects’. For instance, our discussion of minimal abduction contained ‘Carnap’s trick’, which shows that the
choice of α = Θ → ϕ will always do for a minimal solution. But notice that
this trivialization no longer works when only atomic explanations are allowed.
Here is another example. Let Θ consist of propositional Horn clauses only.
In that case, we can determine the minimal abduction for an atomic conclusion
directly. A simple example will demonstrate the general method:
Let Θ = {q ∧ r → s, p ∧ s → q, p ∧ t → q} and ϕ = {q}
q ∧ r → s, p ∧ s → q, p ∧ t → q, α? ⇒ q
(i) Θ, α |= ((p ∧ s → q) ∧ (p ∧ t → q)) → q
(ii) Θ, α |= (p ∧ s) ∨ (p ∧ t) ∨ q
That is, first make the conjunction of all formulas in Θ having q for head
and construct the implication to q (i), obtaining a formula which is already
an abductive solution (a slightly simpler form than Θ → ϕ).Then construct
an equivalent simpler formula (ii) of which each disjunct is also an abductive
solution. (Note that one of them is the trivial one). Thus, it is relatively easier
to perform this process over a simple theory rather than having to engage in a
complicated reasoning process to produce abductive explanations.
Finally, we mention another partly linguistic, partly ontological issue that
comes up naturally in abduction. As philosophers of science have observed,
there seems to be a natural distinction between ‘individual facts’ and ‘general
laws’ in explanation. Roughly speaking, the latter belong to the theory Θ, while
the former occur as explananda and explanantia. But intuitively, the logical basis for this distinction does not seem to lie in syntax, but rather in the nature of
things. How could we make such a distinction? ([Fla95] mentions this issue
as one of the major open questions in understanding abduction, and even its
implementations.) Here is what we think has to be the way to go. Explanations
are sought in some specific situation, where we can check specific facts. Moreover, we adduce general laws, not tied to this situation, which involve general
reasoning about the kind of situation that we are in. The latter picture is not
what is given to us by classical logic. We would rather have to think of a mixed
situation (as in, say, the computer program Tarski’s World, cf. [BE93]), where
we have two sources of information. One is direct querying of the current
situation, the other general deduction (provided that it is sound with respect
to this situation.) The proper format for abduction then becomes a mixture of
Abduction as Logical Inference
89
‘theorem proving’ and ‘model checking’ (cf. [SUM96]). Unfortunately, this
would go far beyond the bounds of this book.
5.
Discussion and Conclusions
Is Abductive Explanatory Inference a Logical System?
The notion of abduction as a logical inference goes back to Peirce’s distinction into kinds of logical reasoning, in which abduction plays the role of
hypothetical inference. Therefore, its certainty is low and its non-monotonicity
high. Though these aspects make it difficult to be handled, it is certainly a
logical system of its own kind, which may be classified of the inductive type
within Haack’s approach9 . It shares the language with classical logic and abductive conclusions are not valid by only means of the classical consequence
(but abduction may have the underlying consequence relation of the deductive
type). It can even be somewhat identified with the second characterization of
induction, namely in which it is improbable ‘supposing’/given that the ‘hypotheses’/premisses are true that the conclusion is false, and therefore we can
assert that the conclusion is true in a tentative way. But abduction may also
be classified as a deviant system, such as in the explanatory abductive version,
in which the premisses really contribute for asserting the conclusion, they are
relevant in a way to the conclusion. In other words, the language is the same,
but the consequence relation is more demanding to the conclusion.
The various types of abductive explanatory styles in a larger universe of other
deductive and inductive systems of logic naturally commits us to a pluralistic
and a global view of logic, such as Haack’s own position, in which there is
a variety of logical systems which rather than competing and being rival to
each other, they are complementary in that each of them has a specific notion
of validity corresponding to an extra-systematic one and a rigorous way for
validating arguments, for it makes sense to speak of a logical system as correct
or incorrect, having several of them. And finally, the global view states for
abduction that it must aspire to global application, irrespective of subject-matter,
and thus found in scientific reasoning and in common sense reasoning alike.
Abductive Explanatory Inference as a Structured Logical
Inference
Studying abduction as a kind of logical inference has provided much more
detail to the broad schema in the previous chapters. Different conditions for
a formula to count as a genuine explanation, gave rise to different abductive
styles of inference. Moreover, the latter can be used over different underlying
9 Although Haack does not include explicitly abduction in her classification, she admits her existence [Haa78,
page 12n].
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ABDUCTIVE REASONING
notions of consequence (classical, preferential, statistical). The resulting abductive explanatory logics have links with existing proposals in the philosophy
of science, and even further back in time, with Bolzano’s notion of deducibility.
They tend to be non-monotonic in nature. Further logical analysis of some
key examples revealed many further structural rules. In particular, consistent
abduction was completely characterized. Finally, we have discussed possible
complete systems for special kinds of abduction, as well as the complexity of
abduction in general.
The analysis of abductive explanatory inference at such an abstract metalogical level, has allowed for an outlook from a purely structural perspective.
We have taken its bare bones and study its consequence type with respect to
itself (reflexivity), to the ability to handle new information (monotonicity), to
the loss of repeated information (contraction), to the order in which premisses
appear (permutation) and to the ability of handling chains of arguments (cut).
In short, we tested abductive inference with respect to its ability to react to a
changing world. As it turned out, none of the above properties were really an
issue for plain abduction, for it is ruled by classical consequence, and therefore
observes the same behaviour as that of classical reasoning. It easily allows
reflexivity and new information does not invalidate in any way previous one.
Moreover, in plain abduction premisse order does not affect the outcome of
reasoning. Finally, cut is an easy rule to follow. In contrast, consistent abduction is not even classically reflexive, Still, every formula that in turn explains
consistently (or it is explained consistently by something else) is reflexive, and
thus ensuring that consistency is preserved. Consistent abduction is also very
sensitive to the growth of information, as inconsistent information cannot come
in at all. But if the new data explains something else together with the theory,
then it is possible to add it as new information. Finally, consistent abduction
also handles a somewhat sophisticated kind of cut, giving thus a way to chain
the arguments. The only rules, which these two types of abductive reasoning
share, are contraction of repeated formulae and permutation.
Here is what we consider the main outcomes of our analysis. We can see abductive explanatory inference as a more structured form of consequence, whose
behavior is different from classical logic, but which still has clear inferential
structure. The modifications of classical structural rules, which arise in this
process, may even be of interest by themselves – and we see this whole area as
a new challenge to logicians. Note that we did not locate the ‘logical’ character
of abduction in any specific set of (modified) structural rules. If pressed, we
would say that some modified versions of Reflexivity, Monotonicity and Cut
seem essential – but we have not been able to find a single formulation that
would stand once and for all. (Cf. [Gab94a] and [Gab94b] for a fuller discussion of the latter point.) Another noteworthy point was our ternary format of
inference, which gives different roles to the theory and explanation on the one
Abduction as Logical Inference
91
hand, and to the conclusion on the other. This leads to finer-grained views of
inference rules, whose interest has been demonstrated.
Abductive Explanatory Inference and Geometries
Nevertheless, the structural characterization we have proposed still leaves a
question unanswered, namely, in what sense an structural characterization leads
to a logic, to a full syntactic or semantic characterization. Even more, despite
the technical results presented, some readers may still doubt whether abductive
reasoning can be considered really logical, perhaps it is more appropriate to
render it as a special type of reasoning. After all, by accepting abductive
reasoning as logical we are accepting a system that only produces tentative
conclusions and not certainties as it is the case for classical reasoning. Let me
precise these questions as follows:
1 In what sense the structural characterization of consistent abduction does
lead to its logic?
2 Are non-classical inferences, such as abduction, really logical?
Regarding the first of these questions, its answer concerns a mathematical
technical problem. That is, it implies a reformulation of the representation
theorem into a completeness theorem, for a logical language without operators
(recall that structural rules are pure, they have no connectives). Furthermore,
a syntactic characterization of abduction requires the extension of the logical
language, to include axioms and operators in order to formulate rules with
connectives and so construct an adequate logical abductive calculi. This way
to proceed, which is to obtain a syntax out of an structural characterization, has
been explored with success for other logics, such as dynamic, relevance and
categorial. Regarding a semantics for abduction, there is also some exploratory
work in this direction, using an extended version of semantic tableaux (cf.
next chapter). However, I conjecture that an abductive version such as the one
allowing all conditions at once does produce a logic that is incomplete.
Regarding the second question, its answer concerns a terminological question of what we want to denote by the term logic. Although structural analysis
of consequence has proved very fruitful and has even been proposed as a distinguished enterprise of Descriptive Logic in [Fla95], many logicians remain
doubtful, and withhold the status of bona fide ‘logical inference’ to the products
of non-standard styles.
This situation is somewhat reminiscent of the emergence of non-euclidean
geometries in the nineteenth century. Euclidean geometry was thought of as the
one and only geometry until the fifth postulate (the parallel axiom) was rejected,
giving rise to new geometries. Most prominently, the one by Lobachevsky ,
which admits of more than one parallel, and the one by Riemann admitting
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ABDUCTIVE REASONING
none. The legitimacy of these geometries was initially doubted but their impact
gradually emerged10 . In our context, it is not geometry but styles of reasoning
that occupy the space, and there is not one postulate under critical scrutiny,
but several. Rejecting monotonicity gives rise to the family of non-monotonic
logics, and rejecting permutation leads to styles of dynamic inference. Linear
logics on the other hand, are created by rejecting contraction. All these alternative logics might get their empirical vindication, too – as reflecting different
modes of human reasoning.
Whether non-classical modes of reasoning are really logical is like asking if
non-euclidean geometries are really geometries. The issue is largely terminological, and we might decide – as Quine did on another occasion (cf.[Qui61])
– to just give conservatives the word ‘logic’ for the more narrowly described
variety, using the word ‘reasoning’ or some other suitable substitute for the
wider brands. In any case, an analysis in terms of structural rules does help us
to bring to light interesting features of abduction, logical or not.
Conclusions
Summarizing, we have shown that abduction can be studied with profit as
a purely logical notion of inference. Of course, we have not exhausted this
viewpoint here – but we must leave its full exploration to other logicians. Also,
we do not claim that this analysis exhausts all essential features of abduction, as
discussed in chapter 2. To the contrary, there are clear limitations to what our
present perspective can achieve. While we were successful in characterizing
what an explanation is, and even show how it should behave inferentially under
addition or deletion of information, the generation of abductive explanations
was not discussed at all. The latter procedural enterprise is the topic of our
next chapter. Another clear limitation is our restriction to the case of ‘novelty’,
where there is no conflict between the theory and the observation. For the case
of ‘anomaly’, we need to go into theory revision, as will happen in chapter 8.
That chapter will also resume some threads from the present one, including a full
version of abduction, in which all our cumulative conditions are incorporated.
The latter will be needed for our discussion of Hempel’s deductive-nomological
model of explanation.
Related Work
Abduction has been recognized as a non-monotonic logic but with few exceptions, no study has been made to characterize it as a logical inference. In
[Kon90] a general theory of abduction is defined as classical inference with
10 The
analogy with logic can be carried even further, as these new geometries were sometimes labeled
‘meta-geometries’.
Abduction as Logical Inference
93
the additional conditions of consistency and minimality, and it is proved to be
implied by Reiter’s causal theories [Rei87], in which a diagnosis is a minimal
set of abnormalities that is consistent with the observed behaviour of a system.
Abduction is also proposed as a procedural mechanism in which the input Q
is an abductive stimulus (goal), and we are interested in ∆′ such that ∆ + ∆′
explains Q (with some suitable underlying inference) [Gab94b, p. 199].
Another approach, closer to our own, though developed independently, is
found in Peter Flach’s PhD dissertation “Conjectures: an inquiry concerning
the logic of induction” [Fla95], which we will now briefly describe and compare
to our work (some of what follows is based on a more recent version of his
proposal [Fla96a].)
Flach’s logic of induction
Flach’s thesis is concerned with a logical study of conjectural reasoning, complemented with an application to relational databases. An inductive consequence relation ≺ (≺⊆ LxL, L a propositional language) is a set of formulae;
α ≺ β interpreted as “β is a possible inductive hypothesis that explains α”, or
as: “β is a possible inductive hypothesis confirmed by α”. The main reason
for this distinction is to dissolve the paradoxical situation posed by Hempel’s
adequacy conditions for confirmatory reasoning [Hem43, Hem45], namely that
in which a piece of evidence E could confirm any hypothesis whatsoever11 .
Therefore, two systems are proposed: one for the logic of confirmation and
the other for the logic of explanation, each one provided with an appropriate
representation theorem for its characterization. These two systems share a set
of inductive principles and differ mainly in that explanations may be strengthened without ceasing to be explanations (H5), and confirmed hypotheses may
be weakened without being disconfirmed (H2). To give an idea of the kind of
principles these systems share, we show two of them, the well-known principles
of verification and falsification in Philosophy of Science:
I1 If α ≺ β and |= α ∧ β → γ, then α ∧ γ ≺ β.
I2 If α ≺ β and |= α ∧ β → γ, then α ∧ ¬γ ≺ β.
They state that when a hypothesis β is tentatively concluded on the basis of
evidence α, and a prediction γ drawn from α and β is observed, then β counts
as a hypothesis for both α and γ (I1), and not for α and ¬γ (I2) (a consequence
of the latter is that reflexivity is only valid for consistent formulae).
11 This
situation arises from accepting reflexivity (H1: any observation report is confirmed by itself) and
stating on the one hand that if an observation report confirms a hypothesis, then it also confirms every
consequence of it (H2), and on the other that if an observation report confirms a hypothesis, then it also
confirms every formula logically entailing it (H5).
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ABDUCTIVE REASONING
Comparison to our work
Despite differences in notation and terminology, Flach’s approach is connected
to ours in several ways. Its philosophical motivation is based on Peirce and
Hempel, its methodology is also based on structural rules, and we agree that the
relationship between explananda and explanandum is a logical parameter (rather
than fixed to deduction) and on the need for complementing the logical approach
with a computational perspective. Once we get into the details however, our
proposals present some fundamental differences, from a philosophical as well
as a logical point of view.
Flach departs from Hempel’s work on confirmation [Hem43, Hem45], while
ours is based on later proposals on explanation [HO48, Hem65]. This leads to
a discrepancy in our basic principles. One example is (consistent) reflexivity;
a general inductive principle for Flach but rejected by us for explanatory abduction (since one of Hempel’s explanatory adequacy conditions imply that it
is invalid, cf. chapter 5). Note that this property reflects a more fundamental
difference between confirmation and explanation than H2 and H5: evidence
confirms itself, but it does not explain itself 12 . There are also differences in the
technical setup of our systems. Although Flach’s notion of inductive reasoning
may be viewed as a strengthened form of logical entailment, the representation of the additional conditions is explicit in the rules rather than within the
consequence relation. Nevertheless, there are interesting analogies between
the two approaches, which we must leave to future work. We conclude with
a general remark. A salient point in both our approaches is the importance of
consistency, also crucial in Hempel’s adequacy conditions both for confirmation
and explanation, and in AI approaches to abduction. Thus, Bolzano’s notion
of deducibility comes back as capturing an intrinsic property of conjectural
reasoning in general.
12 Flach
correctly points out that Hempel’s own solution to the paradox was to drop condition (H5) from his
logic of confirmation. Our observation is that the fact that Hempel later developed an independent account
for the logic of explanation [HO48, Hem65], suggests he clearly separated confirmation from explanation.
In fact his logic for the latter differs in more principles than the ones mentioned above.
Chapter 4
ABDUCTION AS COMPUTATION
1.
Introduction
Our logical analysis of abduction in the previous chapter is in a sense, purely
structural. It was possible to state how abductive explanatory logic behaves,
but not how abductive explanations are generated. In this chapter we turn to the
question of abduction as a computational process. There are several frameworks
for computing abductions; two of which are logic programming and semantic
tableaux. The former is a popular one, and it has opened a whole field of
abductive logic programming [KKT95] and [FK00]. The latter has also been
proposed for handling abduction [MP93] and [AN04], and it is our preference
here. Semantic tableaux are a well-motivated standard logical framework. But
over these structures, different search strategies can compute several versions of
abduction with the non-standard behaviour that we observed in the preceding
chapter. Moreover, we can naturally compute various kinds of abducibles:
atoms, conjunctions or even conditionals. This goes beyond the framework of
abductive logic programming, in which abducibles are atoms from a special set
of abducibles.
This chapter is naturally divided into six parts. After this introduction, in
which we give an overview of abduction as computation, in the second part
(section 2) we review the framework of semantic tableaux and propose a characterization of tableaux extensions into open, closed and semiclosed in order to
prepare the ground for the construction of abductive explanations. In the third
part (section 3) we first propose abduction as a process of tableau extension,
with each abductive version corresponding to some appropriate ‘tableau extension’ for the background theory. We then translate our abductive formulations
to the setting of semantic tableaux and propose two strategies for the generation of abductions. In the fourth part (section 4), we put forward algorithms
96
ABDUCTIVE REASONING
to compute these different abductive versions. In particular, explanations with
complex forms are constructed from simpler ones. This allows us to identify
cases without consistent atomic explanations whatsoever. It also suggests that
in practical implementations of abduction, one can implement our views on
different abductive outcomes in chapter 2. The fifth part (section 5) discusses
further logical aspects of tableau abduction. We analyze abductive semantic
tableaux in first order logic, in particular for the case in which the corresponding
formulae (corresponding to the theory and the negation of the observation) are
finitely satisfiable but may generate infinite tableaux. Moreover, we present a
further semantic analysis, validity of structural rules as studied in chapter 3,
plus soundness and completeness of our algorithms. In the sixth and final part
of this chapter (section 6), we offer an analysis of previous sections with respect
to the scope and limitations of the tableaux treatment of abduction. We then put
forward our conclusions and present related work within the study of abduction
in semantic tableaux.
Generally speaking, this chapter shows how to implement abduction, how to
provide procedural counterparts to the abductive versions described in chapter
3. There are still further uses, though which go beyond our analysis so far.
Abduction as revision can also be implemented in semantic tableaux. Upcoming
chapters will demonstrate this, when elaborating a connection with empirical
progress in science (chapter 6) and theories of belief change in AI (chapter 8).
Our formal specification can lead to the development of algorithms which may
be further implemented as we have done elsewhere. (Cf. [Ali97] Appendix A,
for a Prolog code).
Procedural Abduction
Computational Perspectives
There are several options for treating abduction from a procedural perspective.
One is standard proof analysis, as in logical proof theory (cf. [Tro96]) or in
special logical systems that depend very much on proof-theoretic motivations,
such as relevance logic, or linear logic. Proof search via the available rules
would then be the driving force for finding abducibles. Another approach
would program purely computational algorithms to produce the various types
of abduction that we want. An intermediate possibility is logic programming,
which combines proof theory with an algorithmic flavor. The latter is more in
line with our view of abductive logic as inference plus a control strategy (cf.
chapter 2). Although we will eventually choose yet a different route toward the
latter end, we do sketch a few features of this practically important approach.
Abduction as Computation
97
Abducing in Logic Programming
Computation of abductions in logic programming can be formulated as the
following process. We wish to produce literals α1 , . . . , αn which, when added
to the current program P as new facts, make an earlier failed goal ϕ (the
‘surprising fact’) succeed after all via Prolog computation ⇒p :
α is an abductive explanation for query ϕ given program P if:
P ⇒p ϕ
while
α, P ⇒p ϕ
Notice that we insert the abducibles as facts into the program here - as an
aid. It is a feature of the Prolog proof search mechanism, however, that other
positions might give different derivational effects. In this chapter, we merely
state these, and other features of resolution-style theorem proving without further explanation, as our main concerns lie elsewhere. (Cf. [FK00] for several
papers on abductive logic programming).
Two Abductive Computations
Abductions are produced via the PROLOG resolution mechanism and then
checked against a set of ‘potential abducibles’. But as we just noted, there
are several ways to characterize an abductive computation, and several ways
to add a fact to a Prolog program. An approach which distinguishes between
two basic abductive procedures is found in [Sti91], who defines most specific
abduction (MSA) and least specific abduction (LSA). These differ as follows.
Via MSA only pure literals are produced as abductions, and via LSA those that
are not. A ‘pure literal’ is one which may occur both in the body of a clause as
well as in the head, whereas a ‘non-pure literal’ for a program P , is one which
cannot be resolved via any clause in the program and thus may appear only in
the body of a clause. The following example illustrates these two procedures
(it is a combination of our earlier common sense rain examples):
Program P : r ← c, w ← r, w ← s
Query q: w
MSA:
c, s
LSA:
r
This distinction is useful when we want to identify those abductions that
are ‘final causes’ (MSA) from ‘indirect causes’, which may be explained by
something else (LSA).
Structural Rules
This type of framework also lends itself to a study of logical structural rules like
in chapter 3. This time, non-standard effects may reflect computational peculiarities of our proof search procedure. (Cf. [Min90, Kal95, vBe92, Gab94b]
for more on this general phenomenon.) As for Monotonicity, we have already
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ABDUCTIVE REASONING
shown (cf. chapter 3) that right- but not left-insertion of new clauses in a program is valid for Prolog computation. Thus, adding an arbitrary formula at the
beginning of a program may invalidate earlier programs. (With inserting atoms,
we can be more liberal: but cf. [Kal95] for pitfalls even there.) For another
important structural rule, consider Reflexivity. It is valid in the following form:
Reflexivity: α, P ⇒p α
but invalid in the form
Reflexivity: P, α ⇒p α
Moreover, these outcomes reflect the downward computation rule of Prolog.
Other algorithms can have different structural rules1 .
2. Semantic Tableaux
The Classical Method
Tableau Construction
The logical framework of semantic tableaux is a refutation method introduced in
the 50’s independently by Beth [Bet69] and Hintikka [Hin55]. A more modern
version is found in [Smu68] and it is the one presented here, although it will be
referred to as ‘Beth’s Tableaux’. The general idea of semantic tableaux is as
follows:
To test if a formula ϕ follows from a set of premises Θ, a tableau tree for the sentences
in Θ∪{¬ϕ} is constructed, denoted by T (Θ∪{¬ϕ}). The tableau itself is a binary tree
built from its initial set of sentences by using rules for each of the logical connectives
that specify how the tree branches. If the tableau closes (every branch contains an atomic
formula ψ and its negation), the initial set is unsatisfiable and the entailment Θ |= ϕ
holds. Otherwise, if the resulting tableau has open branches, the formula ϕ is not a valid
consequence of Θ.
The rules for constructing the tableau tree are as follows. There are seven
rules, which may be reduced to two general types of transformation rules, one
‘conjunctive’ (α-type) and one ‘disjunctive’ (β-type). The former for a true
conjunction and the latter for a true disjunction suffice if every formula to be
incorporated into the tableau is transformed first into a propositional conjunctive
or a disjunctive normal form. Moreover, the effect of applying an α-type rule
is that of adding α1 followed by α2 in the open branches, while the resulting
operation of an β-type rule is to generate two branches, β1 and β2 in each of
the open branches. Our notation is as follows:
1 In particular, success or failure of Reflexivity may depend on whether a ‘loop clause’ α ← α is present in
the program. Also, Reflexivity is valid under MSA, but not via LSA computation, since a literal α in a rule
α ← α is a pure literal.
99
Abduction as Computation
Rule A:
α1
α2
−→
α
Rule B:
β
β1 | β 2
−→
The seven rules for tableaux construction are the following. Double negations
are suppressed. True conjunctions add both conjuncts, negated conjunctions
branch into two negated conjuncts. True disjunctions branch into two true
disjuncts, while negated disjunctions add both negated disjuncts. Implications
(a → b) are treated as disjunctions (¬a ∨ b).
Negation
¬¬ϕ
−→
ϕ
ϕ∧ψ
−→
ϕ
ψ
Conjunction
¬(ϕ ∧ ψ)
−→
¬ϕ | ¬ψ
ϕ∨ψ
−→
ϕ|ψ
Disjunction
¬(ϕ ∨ ψ)
−→
¬ϕ
¬ψ
Implication
ϕ→ψ
¬(ϕ → ψ)
−→
−→
¬ϕ | ψ
ϕ
¬ψ
A Simple Example
To show how this works, we give an extremely simple example. More elaborate
tableaux will be found in the course of this chapter. Let Θ = {r → w}. Set
ϕ = w. We ask whether Θ |= ϕ. The tableau is as follows.Here, an empty
circle indicates that a branch is open, and a crossed circle
that the branch
is closed. The tableau T (Θ ∪ {¬ϕ}) is as follows:
r→w
¬r
w
¬w
¬w
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ABDUCTIVE REASONING
The resulting tableau is open, showing that Θ |= ϕ. The open branch
indicates a ‘counterexample’, that is, a case in which Θ is true while ϕ is false
(r, w false). More generally, the construction principle is this. A tableau has to
be expanded by applying the construction rules until formulas in the nodes have
no connectives, and have become literals (atoms or their negations). Moreover,
this construction process ensures that each node of the tableau can only carry a
subformula of Θ or ¬ϕ.
Logical Properties
The tableau method as sketched so far has the following general properties.
These can be established by simple analysis of the rules and their motivation,
as providing an exhaustive search for a counter-example. In what follows, we
concentrate on verifying the top formulas, disregarding the initial motivation
of finding counterexamples to consequence problems. This presentation incurs
no loss of generality. Given a tableau for a theory Θ, that is, T (Θ):
If T (Θ) has open branches, Θ is consistent. Each open branch corresponds
to a verifying model.
If T (Θ) has all branches closed, Θ is inconsistent.
Another, more computational feature is that, given some initial verification
problem, the order of rule application in a tableau tree does not affect the result
(in propositional logic). The structure of the tree may be different, but the
outcome as to consistency is the same. Moreover, returning to the formulation
with logical consequence problems, we have that semantic tableaux are a sound
and complete system:
Θ |= ϕ
iff
there is a closed tableau for Θ ∪ {¬ϕ}.
Given the effects of the above rules, which decrease complexity, tableaux
are a decision method for propositional logic. This is different with predicate
logic (partially treated further in the chapter, section 5), where quantifier rules
may lead to unbounded repetitions. In the latter case, the tableau method is
only semi–decidable. For the propositional case we have: if the initial set of
formulas is unsatisfiable, the tableau will close in finitely many steps. But if it
is satisfiable, the tableau may become infinite, without terminating, recording
an infinite model. In this section, we shall only consider the propositional case.
A more combinatorial observation is that there are two faces of tableaux.
When an entailment does not hold, read upside down, open branches are records
of counter-examples. When the entailment does hold, read bottom up, a closed
tableau is easily reconstructed as a Gentzen sequent calculus proof. This is
no accident of the method. In fact, Beth’s tableaux may be seen as a modern
formalization of the Greek methods of analysis and proof synthesis, as we
already observed in chapter 1.
Abduction as Computation
101
Tableaux are widely used in logic, and they have many further interesting
properties. For a more detailed presentation, the reader is invited to consult
[Smu68, Fit90]. For convenience in what follows, we give a quick reference
list of some major notions concerning tableaus.
Closed Branch : A branch of a tableau is closed if it contains some formula
and its negation.
Atomically Closed Branch : A branch is atomically closed if it contains an
atomic formula and its negation.
Open Branch : A branch of a tableau is open if it is not closed.
Complete Branch : A branch Γ of a tableau is complete if (referring to the
earlier-mentioned two main formula types) for every α which occurs in Γ,
both α1 and α2 occur in Γ, and for every β which occurs in Γ, at least one
of β1 , β2 occurs in Γ.
Closed Tableau : A closed tableau is one in which all branches are closed.
Open Tableau : An open tableau is one in which there is at least one open
branch.
Completed Tableau : A tableau T is completed if every branch of T is either
closed or complete.
Proof of X : A proof of a formula X is a closed tableau for ¬X.
Proof of Θ |= ϕ : A proof of Θ |= ϕ is a closed tableau for Θ ∪ {¬ϕ}.
Our Extended Method
Tableaux Extensions and Closures
Here we present our proposal to extend the framework of semantic tableaux
in order to compute abductions. We describe a way to represent a tableau for
a theory as the union of its branches, and operations to extend and characterize tableaux extension types. A propositional language is assumed with the
usual connectives, whose formulas are of three types: literals (atoms or their
negations), α-type (conjunctive form), or β-type (disjunctive form).
Tableaux Representation
Given a theory Θ we represent its corresponding completed tableau T (Θ) by
the union of its branches. Each branch is identified with the set of formulas that
label that branch. That is:
T (Θ) = [Γ1 ] ∪ . . . ∪ [Γk ] where each Γi may be open or atomically closed.
Our treatment of tableaux will be always on completed tableau, so we just
refer to them as tableaux from now on.
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ABDUCTIVE REASONING
Tableau and Branch Extension
A tableau is extended with a formula via the usual expansion rules (explained
in previous subsection). An extension may modify a tableau in several ways.
These depend both on the form of the formula to be added and on the other
formulas in the theory represented in the original tableau. If an atomic formula
is added, the extended tableau is just like the original with this formula appended
at the bottom of its open branches. If the formula has a more complex form, the
extended tableau may look quite different (e.g., disjunctions cause every open
branch to split into two). In total, however, when expanding a tableau with a
formula, the effect on the open branches can only be of three types. Either (i)
the added formula closes no open branch or (ii) it closes all open branches, or
(iii) it may close some open branches while leaving others open. In order to
compute consistent and explanatory abductions, we need to clearly distinguish
these three ways of extending a tableau. We label them as open, closed, and
semi-closed extensions, respectively. In what follows we define these notions
more precisely.
Branch Extension
Given a T (Θ) the addition of a formula {γ} to each of its branches Γi is defined
by the following + operation:
Γi + {γ} = Γi iff Γi closed:
If Γi is a completed open branch:
Case 1 {γ} is a literal.
Γi + {γ} = Γi ∪ {γ}
Case 2 γ is an α-type (γ = α1 ∧ α2 ).
Γi + {γ} = ((Γi ∪ {γ}) + α1 ) + α2
Case 3 γ is a β-type (γ = β1 ∨ β2 ).
Γi + {γ} = {(Γi ∪ {γ}) + β1 ) ∪ ((Γi ∪ {γ}) + β2 }
That is, the addition of a formula γ to a branch is either Γi itself when it is
closed or it is the union of its resulting branches. The operation + is defined
over branches, but it easily generalizes to tableaux as follows:
Tableau Extension:
T (Θ) + {γ} =def
{Γi + {γ} | Γi ∈ T (Θ)}
1≤i≤k
Our notation allows also for embeddings ((Θ+γ)+β). Note that operation +
is just another way of expressing the usual tableau expansion rules. Therefore,
each tableau may be viewed as the result of a suitable series of + extension
steps, starting from the empty tableau.
Abduction as Computation
103
Branch Extension Types
Given an open branch Γi and a formula γ, we have the following possibilities
to extend it:
Open Extension:
Γi + {γ} = δ1 ∪ . . . ∪ δn is open iif each δi is open.
Closed Extension:
Γi + {γ} = δ1 ∪ . . . ∪ δn is closed iff each δi is closed.
Semi-Closed Extension:
Γi + {γ} = δ1 ∪ . . . ∪ δn is semi-closed iff at least one δi is open and at
least one δj is closed.
Extensions can also be defined over whole tableaux by generalizing the above
definitions. A few examples will illustrate the different situations that may
occur.
Examples
Let Θ = {¬a ∨ b, c}.
T (Θ) = [¬a ∨ b, ¬a, c] ∪ [¬a ∨ b, b, c]
Open Extension: T (Θ + {d}) (d closes no branch).
T (Θ) +{d} = [¬a ∨ b, ¬a, c, d] ∪ [¬a ∨ b, b, c, d]
¬a ∨ b
¬a
b
c
c
d
d
Semi-Closed Extension: T (Θ + {a}) (a closes only one branch).
T (Θ) +{a} = [¬a ∨ b, ¬a, c, a] ∪ [¬a ∨ b, b, c, a]
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ABDUCTIVE REASONING
¬a ∨ b
¬a
b
c
c
a
a
Closed Extension: T (Θ + {¬c}) (¬c closes all branches)
T (Θ) +{¬c} = [¬a ∨ b, ¬a, c, ¬c] ∪ [¬a ∨ b, b, c, ¬c]
¬a ∨ b
¬a
b
c
c
¬c
¬c
Finally, to recapitulate an earlier point, these types of extension are related
to consistency in the following way:
Consistent Extension:
If T (Θ) + {γ} is open or semi-closed, then T (Θ) + {γ} is a consistent
extension.
Inconsistent Extension:
If T (Θ) + {γ} is closed, then T (Θ) + {γ} is an inconsistent extension.
Given this characterization, it is easy to calculate for a given tableau, the sets
of literals for each type of extension. In our example above these sets are as
follows:
Open ={x | x = a, x = ¬c, x = ¬b}
Semi-closed ={a, ¬b}
Closed ={¬c}
These constructions will be very useful for the calculations of abductions.
105
Abduction as Computation
Branch and Tableau Closures
As we shall see when computing abductions, given a theory Θ and a formula ϕ,
plain abductive explanations are those formulas which close the open branches
of T (Θ ∪ ¬ϕ). In order to compute these and other abductive kinds, we need to
define ‘total’ and ‘partial closures’ of a tableau. The first is the set of all literals
which close every open branch of the tableau, the second of those literals which
close some but not all open branches. For technical convenience, we define total
and partial closures for both branches and tableaux. We also need an auxiliary
notion. The negation of a literal is either its ordinary negation (if the literal is
an atom), or else the underlying atom (if the literal is negative).
Given T (Θ)= Γ1 ∪ . . . ∪ Γn
(here the Γi are just the open branches of T (Θ)):
Branch Total Closure (BTC) :
The set of literals which close an open branch Γi :
BTC(Γi ) = {x | ¬x ∈ Γi },
where x ranges over literals.
Tableau Total Closure (TTC) :
The set of those literals that close all branches at once, i.e. the intersection
of the BTC’s:
T T C(Θ) =
i=n
BT C(Γi )
i=1
Branch Partial Closure (BPC) :
The set of those literals which close the branch but do not close all the other
open branches:
BP C(Γi ) = BT C(Γi ) − T T C(Θ)
Tableau Partial Closure (TPC) :
The set formed by the union of BPC, i.e. all those literals which partially
close the tableau:
T P C(Θ) =
i=n
BP C(Γi )
i=1
In particular, the definition of BPC may look awkward, as it defines partial
closure in terms of branch and tableau total closures. Its motivation lies in a way
106
ABDUCTIVE REASONING
to compute partial explanations, being formulas which do close some branches
(so they do ‘explain’) without closing all (so they are ‘partial’). We will use the
latter to construct explanations in conjunctive form.
Having defined all we need to exploit the framework of semantic tableau for
our purposes, we proceed to the construction of abductive explanations.
3.
Abductive Semantic Tableaux
In this section we will show the main idea for performing abduction as a kind
of tableau extension. First of all, a tableau itself can represent finite theories.
We show this by a somewhat more elaborate example. To simplify the notation
from now on, we write Θ ∪ ¬ϕ for Θ ∪ {¬ϕ} (that is, we omit the brackets).
Example
Let Θ = {b, c → r, r → w, s → w},
and let ϕ = {w}.
A tableau for Θ, that is T (Θ), is as follows:
b
c→r
r→w
s→w
¬c
¬r
r
¬r
w
¬s
w
¬s
w
w
¬s
w
The result is an open tableau. Therefore, the theory is consistent and each
open branch corresponds to a satisfying model. For example, the second branch
(from left to right) indicates that a model for Θ is given by making c, r false and
b, w true, so we get two possible models out of this branch (one in which s is
true, the other in which it is false). Generally speaking, when constructing the
tableau, the possible valuations for the formulas are depicted by the branches
(either ¬c or r makes the first split, then for each of these either ¬r or w, and
so on).
When formulas are added (thereby extending the tableau), some of these
possible models may disappear, as branches start closing. For instance, when
¬ϕ is added (i.e. ¬w), the result is the following:
107
Abduction as Computation
b
c→r
r→w
s→w
¬c
¬r
r
¬r
w
¬s
w
¬s
w
¬w
¬w
¬w
w
¬s
w
¬w
¬w
¬w
Notice that, although the resulting theory remains consistent, all but one
branch has closed. In particular, most models we had before are no longer
satisfying, as ¬w is now true as well. There is still an open branch, indicating
there is a model satisfying Θ ∪ ¬w (c, r, s, w false, b true), which indicates that
Θ |= w.
Abductive Semantic Tableaux: The Main Ideas
An attractive feature of the tableau method is that when ϕ is not a valid
consequence of Θ, we get all cases in which the consequence fails, graphically
represented by the open branches (as shown above, the latter may be viewed as
descriptions of models for Θ ∪ ¬ϕ.)
This fact suggests that if these counterexamples were ‘corrected by amending the theory’, through adding more premises, we could perhaps make ϕ a
valid consequence of some (minimally) extended theory Θ’. This is indeed the
whole issue of abduction. Accordingly, abduction may be formulated in this
framework as a process of extension, extending a tableau with suitable formulas
that close the open branches.
In our example above, the remaining open branch had the following relevant
(literal) part:
108
ABDUCTIVE REASONING
b
¬c
¬r
¬s
¬w
The following are (some) formulas whose addition to the tableau would close
this branch (and hence, the whole tableau):
{¬b, c, r, s, w, c ∧ r, r ∧ w, s ∧ w, s ∧ ¬w, c ∨ w}
Note that several forms of statement may count here as abductions. In particular, those in disjunctive form (e.g. c ∨ w) create two branches, which then
both close. (We will consider these various cases in detail later on.). Moreover,
notice that most formulae of this set (except ¬b and s ∧ ¬w) is a semi-closed
extension of the original tableau for Θ.
The Generation of Abductions
In principle, we can compute abductions for all our earlier abductive versions
(cf. chapter 3). A direct way of doing so is as follows:
First compute abductions according to the plain version and then eliminate all those that
do not comply with the various additional requirements.
This strategy first translates our abductive formulations to the setting of
semantic tableaux as follows:
Given Θ (a set of formulae) and ϕ (a sentence), α is an abductive explanation if:
Plain :
T ((Θ ∪ ¬ϕ) ∪ α) is closed.
(Θ, α |= ϕ).
Consistent : Plain Abduction +
T (Θ ∪ α) is open
(Θ |= ¬α)
Explanatory : Plain Abduction +
(i) T (Θ ∪ ¬ϕ) is open
(Θ |= ϕ)
(ii) T (α ∪ ¬ϕ) is open
(α |= ϕ)
Abduction as Computation
109
In addition to the ‘abductive conditions’ we must state constraints over our
search space for abducibles, as the set of formulas fulfilling any of the above
conditions is in principle infinite. Therefore, we impose restrictions on the
vocabulary as well as on the form of the abduced formulas:
Restriction on Vocabulary
α is in the vocabulary of the theory and the observation:
α ∈ Voc(Θ ∪ {ϕ}).
Restriction on Form
The syntactic form of α is either a literal, a conjunction of literals (without
repeated conjuncts), or a disjunction of literals (without repeated disjuncts).
Once it is clear what our search space for abducibles is, we continue with
our discussion. Note that while computation of plain abductions involves only
closed branches, the other versions use inspection of both closed and open
branches. For example, an algorithm computing consistent abductions would
proceed in the following two steps:
Generating Consistent Abductions (First Version)
1 Generate all plain abductions, being those formulas α for which T ((Θ∪
¬ϕ) ∪ α) is closed.
2 Take out all those α for which T (Θ ∪ α) is closed.
In particular, an algorithm producing consistent abductions along these lines
must produce all explanations that are inconsistent with Θ. This means many
ways of closing T (Θ), which will then have to be removed in step 2. This
is of course wasteful. Even worse, when there are no consistent explanations
(besides the trivial one), so that we would want to give up, our procedure still
produces the inconsistent ones. The same point holds for our other versions of
abduction.
Of course, there is a preference for procedures that generate abductions in a
reasonably efficient way. We will show how to devise these, making use of the
representation structure of tableaux, in a way which avoids the production of
inconsistent formulae. Here is our idea.
Generating Consistent Abductions (Second Version)
1 Generate all formulas α which do not close all (but some) open branches
of T (Θ).
2 Check which of the formulas α produced are such that
T ((Θ ∪ ¬ϕ) ∪ α) is closed.
110
ABDUCTIVE REASONING
That is, first produce formulas which extend the tableau for the background
theory in a consistent way, and then check which of these are abductive explanations. As we will show later, the consistent formulae produced by the second
procedure are not necessarily wasteful. They might be ‘partial explanations’
(an ingredient for explanations in conjunctive form), or part of explanations in
disjunctive form.
In other words, consistent abductions are those formulas which “if they had
been in the theory before, they would have closed those branches that remain
open after ¬ϕ is incorporated into the tableau”.
In our example above the difference between the two algorithmic versions is
as follows (taking into account only the atomic formulas produced). Version 1
produces a formula (¬b) which is removed for being inconsistent, and version
2 produces a consistent formula (¬w) which is removed for not closing the
corresponding tableau.
4.
Computing Abductions with Tableaux
Our strategy for computing plain abduction in semantic tableaux will be as
follows. We will be using tableaux as an ordinary consequence test, while
being careful about the search space for potential abducibles. The computation
is divided into different forms of explanations. Atomic explanations come first,
followed by conjunctions of literals, to end with those in disjunctive form. Here
we sketch the main ideas for their construction, and give an example for each
kind. There are no detailed algorithms of programs here, but they may be found
in [Ali97] and [VA99]2 .
Plain Abduction
Atomic Plain Abduction
The idea behind the construction of atomic explanations is very simple. One just
computes those atomic formulas which close every open branch of T (Θ ∪ ¬ϕ),
corresponding precisely to its Total Tableaux Closure (TTC(Θ ∪ ¬ϕ)). Here is
an example:
Let Θ = {¬a ∨ b}
ϕ = b.
2 The
implementation of abduction shows that it is not particularly hard to use it in practice (which may
explain its appeal to programmers). It may be a complex notion general, but when well-delimited, it poses
a feasible programming task. In [Ali97] a Prolog Code (written for Arity Prolog) may be found of the main
procedures for implementing plain abduction (atomic, conjunctive and disjunctive explanations altogether).
In [VA99] the essentials for an implementation in the java language is offered for our abductive strategy
(the first version) given in the previous section. This turns out to be a good choice, for the high level of
abstraction to describe objects and classes in object oriented programming, allows for an straightforward
mapping between the formal tableaux and operations and their corresponding functions and procedures. This
is an advantage over implementations in logic programming, in which the process is tied to the resolution
rule.
Abduction as Computation
111
T (Θ ∪ ¬ϕ) is as follows:
¬a ∨ b
¬a
b
¬b
¬b
The two possible atomic plain abductions are {a, b}.
Conjunctive Plain Abduction
Single atomic explanations may not always exist, or they may not be the only
ones of interest. The case of explanations in conjunctive form (α = α1 ∧
. . . ∧ αn ) is similar to the construction of atomic explanations. We look for
literals that close branches, but in this case we want to get the literals that
close some but not all of the open branches. These are the conjuncts of a
‘conjunctive explanation’, and they belong to the tableau partial closure of Θ
(i.e., to TPC(Θ ∪ ¬ϕ)). Each of these partial explanations make the fact ϕ ‘less
surprising’ by closing some (but not all) of the open branches. Together they
constitute an abductive explanation.
As a consequence of this characterization, no partial explanation is an atomic
explanation. That is, a conjunctive explanation must be a conjunction of partial
explanations. The motivation is this. We want to construct explanations which
are non-redundant, in which every literal does some explaining. Moreover, this
condition allows us to bound the production of explanations in our algorithm.
We do not want to create what are intuitively ‘redundant’ combinations. For
example, if p and q are abductive explanations, then p∧q should not be produced
as explanation. Thus we impose the following condition:
Non-Redundancy
Given an abductive explanation α for a theory Θ and a formula ϕ, α is non-redundant if
it is either atomic, or no subformula of α (different from ϕ) is an abductive explanation.
The following example gives an abductive explanation in conjunctive form
which is non-redundant:
Let Θ = {¬a ∨ ¬c ∨ b},
and ϕ = b.
The corresponding tableau T (Θ ∪ ¬ϕ) is as follows:
112
ABDUCTIVE REASONING
{¬a ∨ ¬c ∨ b}
¬a ∨ ¬c
b
¬a
¬c
¬b
¬b
¬b
The only atomic explanation is the trivial one {b}. The conjunctive explanation is {a ∧ c} of which neither conjunct is an atomic explanation.
Disjunctive Plain Abductions
To stay in line with computational practice, we shall sometimes regard abductive explanations in disjunctive form as implications. (This is justified by the
propositional equivalence between ¬αi ∨ αj and αi → αj .) These special explanations close a branch by splitting it first into two. Disjunctive explanations
are constructed from atomic and partial explanations. We will not analyze this
case in full detail, but provide an example of what happens.
Let Θ = {a}
ϕ = b.
The tableau structure for T (Θ ∪ ¬b) is as follows:
a
¬b
Notice first that the possible atomic explanations are {¬a, b} of which the
first is inconsistent and the second is the trivial solution. Moreover, there are
no ‘partial explanations’ as there is only one open branch. An explanation in
disjunctive form is constructed by combining the atomic explanations, that is ,
{¬a ∨ b}. The effect of adding it to the tableau is as follows (T (Θ ∪ {¬b} ∪
{¬a ∨ b})):
a
¬b
¬a
b
Abduction as Computation
113
This examples serves as a representation of our example in chapter 2, in
which a causal connection is found between certain types of clouds (a) and rain
(b), namely that a causes b (a → b).
Algorithm for Computing Plain Abductions
The general points of our algorithm for computing plain abductions is displayed
here.
Input:
. A set of propositional formulas separated by commas representing the theory Θ.
. A literal formula ϕ representing the ‘fact to be explained’.
. Preconditions: Θ, ϕ are such that Θ |= ϕ, Θ |= ¬ϕ.
Output:
Produces the set of abductive explanations: α1 , . . . αn such that:
(i) T ((Θ ∪ ¬ϕ) ∪ αi ) is closed.
(ii) αi complies with the vocabulary, form restrictions as well as with the non-redundancy
condition.
Procedure:
. Calculate Θ + ¬ϕ = {Γ1 , . . . , Γk }
. Take those Γi which are open branches: Γ1 , . . . , Γn
. Atomic Plain Explanations
1 Compute TTC(Γ1 , . . . , Γn )= {γ1 , . . . , γm }.
2 {γ1 , . . . , γm } is the set of atomic plain abductions.
. Conjunctive Plain Explanations
1 For each open branch Γi , construct its partial closure: BPC(Γi ).
2 Check if all branches Γi have a partial closure, for otherwise there cannot be a conjunctive solution (in which case, goto END).
3 Each BPC(Γi ) contains those literals which partially close the tableau. Conjunctive
explanations are constructed by taking one literal of each BPC(Γi ) and making their
conjunction. A typical solution is a formula β as follows:
(a1 ∈ BP C(Γ1 ), b1 ∈ BP C(Γ2 ), . . . , z1 ∈ BP C(Γn ))
a1 ∧ b1 ∧ . . . ∧ z1
4 Each β conjunctive solution is reduced (there may be repeated literals). The set of
solutions in conjunctive form is β1 , . . . , βl 3 .
5 END.
. Disjunctive Plain Explanations
1 Construct disjunctive explanations by means of the following combinations: atomic
explanations amongst themselves, conjunctive explanations amongst themselves, conjunctive with atomic, and each of atomic and conjunctive with ϕ. We just show two of
these constructions:
3 There are no redundant solutions. The reason is that each conjunctive solution β is formed by a conjunction
i
of ‘potential explanations’, none of which is itself an abductive solution.
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ABDUCTIVE REASONING
2 Generate pairs from the set of atomic explanations, and construct their disjunctions
(γi ∨ γj ).
3 For each atomic explanation γi construct the disjunction with ϕ as follows: (γi ∨ ϕ).
4 The result of all combinations above is the set of explanations in disjunctive form.
5 Verify the previous set for redundant explanations. Take out all of these. The remaining
are the explanations in disjunctive form.
6 END.
Consistent Abduction
The issue now is to compute abductive explanations with the additional requirements of being consistent. For this purpose we will follow the same
presentation as for plain abductions (atomic, conjunctive and disjunctive), and
will give the key points for their construction. Our algorithm follows version
2 of the strategies sketched earlier (cf. subsection 3.2). That is, it first constructs those consistent extensions on the original tableau for Θ which do some
closing and then checks which of these is in fact an explanation (i.e. closes the
tableau for Θ ∪ ¬ϕ). This way we avoid the production of any inconsistency
whatsoever. It turns out that in the atomic and conjunctive cases explanations
are sometimes necessarily inconsistent, therefore we identify these cases and
prevent our algorithm from doing anything at all (so that we do not produce
formulae which are discarded afterwards).
Atomic Consistent Abductions
When computing plain atomic explanations, we now want to avoid any computation when there are only inconsistent atomic explanations (besides the trivial
one). Here is an observation which helps us get one major problem out of the
way. Atomic explanations are necessarily inconsistent when Θ + ¬ϕ is an
open extension (recall that ϕ is a literal). So, we can prevent our algorithm
from producing anything at all in this case.
Fact 1 Whenever Θ+¬ϕ is an open extension, and α a non-trivial atomic abductive
explanation (i.e. different from ϕ), it follows that Θ, α is inconsistent.
Proof. Let Θ + ¬ϕ be an open extension and α an atomic explanation (α = ϕ). The
latter implies that ((Θ + ¬ϕ) + α) is a closed extension. Therefore, Θ + α must be a
closed extension, too, since ¬ϕ closes no branches. But then, Θ + α is an inconsistent
extension. I.e. Θ, α is inconsistent. ⊣
This result cannot be generalized to more complex forms of abducibles. (We
will see later that for explanations in disjunctive form, open extensions need not
lead to inconsistency.) In case Θ + ¬ϕ is a semi-closed extension, we have to
do real work, however, and follow the strategy sketched above. The key point
in the algorithm is this. Instead of building the tableau for Θ ∪ ϕ directly, and
working with its open branches, we must start with the open branches of Θ.
Abduction as Computation
115
Algorithm
1 Θ, ¬ϕ are given as input and are such that: Θ |= ϕ, Θ |= ¬ϕ.
2 Calculate Θ+¬ϕ. If it is an open extension, then there are no atomic consistent explanations
(by fact 1), go to Step 6.
3 Calculate the set of literals {γ1 , . . . , γn } for which Θ + γi is a semi-closed extension.
4 Select from above set those γi for which (Θ + ¬ϕ) + γi is a closed extension.
5 The set above is the set of Consistent Atomic Explanations.
6 END.
Conjunctive Consistent Explanations
For conjunctive explanations, we can also avoid any computation when there
are only ‘blatant inconsistencies’, essentially by the same observation as before.
Fact 2 Whenever Θ + ¬ϕ is an open extension, and α = α1 ∧ . . . ∧ αn is a
conjunctive abductive explanation, it holds that Θ, α is inconsistent.
The proof is analogous to that for the atomic case.
In order to construct partial explanations, the ingredients of conjunctive explanations, the key idea is to build conjunctions out of those formulas for which
both Θ + γi and (Θ + ¬ϕ) + γi are semiclosed extensions. The reason is as
follows: the former condition assures that these formulas do close at least one
branch from the original tableau. The latter condition discards atomic explanations (for which (Θ + ¬ϕ) + γi is closed) and the trivial solution (γi = ϕ
when (Θ + ¬ϕ) + γi is open), and takes care of non-redundancy as a side effect.
Conjunctions are then constructed out of these formulas and those which induce
a closed extension for Θ + ¬ϕ are the selected conjunctive explanations.
Here is the algorithm sketch:
1 Θ, ¬ϕ are given as input and are such that: Θ |= ϕ, Θ |= ¬ϕ.
2 Calculate Θ + ¬ϕ. If it is an open extension, then there are no conjunctive consistent
explanations (by fact 2), go to Step 8.
3 Calculate the set of literals {γ1 , . . . , γn } for which Θ + γi is a semi-closed extension.
4 Select from above set those γi for which (Θ + ¬ϕ) + γi is a semi-closed extension. This
is the set of Partial Explanations: {γ1 , . . . , γm }, m ≤ n.
5 Construct conjunctions from set of partial explanations starting in length k (number of open
branches) to end in lenght m (number of partial explanations). Label these conjunctions as
follows: ρ1 , . . . , ρs .
6 Select those ρi from above for which (Θ + ¬ϕ) + ρi is a closed extension.
7 The set above is the set of Consistent Conjunctive Explanations.
8 END.
What this construction suggests is that there are always consistent explanations in disjunctive form, provided that the theory is consistent:
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ABDUCTIVE REASONING
Fact 3 Given that Θ ∪ ¬ϕ is consistent, there exists an abductive consistent
explanation in disjunctive form.
The key point to prove this fact is that an explanation may be constructed as
α = ¬X ∨ ϕ, for any X ∈ Θ.
Explanatory Abduction
As for explanatory abductions, recall these are those formulas α which are
only constructed when the theory does not explain the observation already
(Θ |= ϕ) and that cannot do the explaining by themselves (α |= ϕ), but do so
in combination with the theory (Θ, α |= ϕ).
Given our previous algorithmic constructions, it turns out that the first condition is already ‘built-in’, since all our procedures start with the assumption that
the tableau for Θ∪¬ϕ is open4 . As for the second condition, its implementation
actually amounts to preventing the construction of the trivial solution (α = ϕ).
Except for this solution, our algorithms never produce an α such that α |= ϕ,
as proved below:
Fact 4 Given any Θ and ϕ, our algorithm never produces abductive explanations
α with α |= ϕ (except for α = ϕ).
Proof. Recall our situation: Θ |= ϕ, Θ, α |= ϕ, while we have made sure Θ and α are
consistent. Now, first suppose that α is a literal. If α |= ϕ, then α = ϕ which is the
trivial solution. Next, suppose that α is a conjunction produced by our algorithm. If
α |= ϕ, then one of the conjuncts must be ϕ itself. (The only other possibility is that α
is an inconsistent conjunction, but this is ruled out by our consistency test.) But then,
our non-redundancy filter would have produced the relevant conjunct by itself, and then
rejected it for triviality. Finally, suppose that α is a disjunctive explanation. Given the
above conditions tested in our algorithm, we know that Θ is consistent with at least one
disjunct αi . But also, this disjunct by itself will suffice for deriving ϕ in the presence
of Θ, and it will imply ϕ by itself. Therefore, by our redundancy test, we would have
produced this disjunct by itself, rather than the more complex explanation, and we are
in one of the previous cases. ⊣
Therefore, it is easy to modify any of the above algorithms to handle the computation of explanatory abductions. We just need to avoid the trivial solution,
when α = ϕ and this can be done in the module for atomic explanations.
Quality of Abductions
A question to ask at this point is whether our algorithms produce intuitively
good explanations for observed phenomena. One of our examples (cf. disjunctive plain abductions, section 4.1) suggested that abductive explanations for a
4 It
would have been possible to take out this condition for the earlier versions. However, note that in the
case that Θ ∪ ¬ϕ is closed the computation of abductions is trivialized since as the tableau is already closed,
any formula counts as a plain abduction and any consistent formula as a consistent abduction.
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Abduction as Computation
fact and a theory with no causal knowledge (with no formulas in conditional
form) must be in disjunctive form if such a fact is to be explained in a consistent
way. Moreover, Fact 3 stated that consistent explanations in disjunctive form
are always available, provided that the original theory is consistent. However,
producing consistent explanations does not guarantee that these are good or
relevant. These further properties may depend upon the nature of the theory
itself and of whether it makes a good combination with the observation. If the
theory is a bad theory, it will produce bad or weird explanations. The following
example illustrates this point.
A Bad Theory
Let Θ = {r → w, ¬r}
ϕ=w
α = ¬r → w.
The tableau T (Θ ∪ ¬ϕ ∪ α) is depicted as follows:
r→w
¬r
¬r
w
¬w
¬w
r
w
Interpreted in connection with our rain example, our algorithm will produce
the following consistent ‘explanation’ of why the lawn is wet (w), given that
rain causes the lawn to get wet (r → w) and that it is not raining (¬r). One
explanation is that “the absence of rain causes the lawn to get wet” (¬r → w).
But this explanation seems to trivialize the fact of the lawn being wet, as it
seems to be so, regardless of rain!
A better, though more complex, way of explaining this fact would be to
conclude that the theory is not rich enough to explain why the lawn is wet,
and then look for some external facts to the theory (e.g. sprinklers are on, and
they make the lawn wet.) But this would amount to dropping the vocabulary
assumption.
Therefore, producing good or bad explanations is not just a business of
properly defining the underlying notion of consequence, or of giving an adequate
procedure. An inadequate theory like the one above can be the real cause of
bad explanations. In other words, what makes a ‘good explanation’ is not the
abducible itself, but the interplay of the abducible with the background theory
and the relevant observation. Bad theories produce bad explanations. Our
algorithm cannot remedy this, only show it.
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ABDUCTIVE REASONING
Discussion
Having a module in each abductive version that first computes only atomic explanations already gives us some account of minimal explanations (see chapter
3), when minimality is regarded as simplicity. As for conjunctive explanations,
as we have noted before, their construction is one way of computing non-trivial
‘partial explanations’, which make a fact less surprising by closing some, though
not all open branches for its refutation. One might tie up this approach with a
more general issue, namely, weaker notions of ‘approximative’ logical consequence. Finally, explanations in disjunctive form can be constructed in various
ways. E.g., one can combine atomic explanations, or form the conjunction of all
partial explanations, and then construct a conditional with ϕ. This reflects our
view that abductive explanations are built in a compositional fashion: complex
solutions are constructed from simpler ones.
Notice moreover, that we are not constructing all possible formulas which
close the open branches, as we have been taking care not to produce redundant explanations (cf. section 4). Finally, despite these precautions, as we
have noted, bad explanations may slip through when the background theory in
combination with a certain observation are inappropriate.
5.
Further Logical and Computational Issues
Our algorithmic tableau analysis suggests a number of further logical issues,
which we briefly discuss here.
Rules, Soundness and Completeness
The abductive consequences produced by our tableaux can be viewed as
a ternary notion of inference. Its structural properties can be studied in the
same way as we did for the more abstract notions of chapter 3. But the earlier
structural rules lose some of their point in this algorithmic setting. For instance,
it follows from our tableau algorithm that consistent abduction does not allow
monotonicity in either its Θ or its α argument. One substitute which we had
in chapter 3 was as follows. If Θ, α ⇒ ϕ, and Θ, α, β ⇒ γ (where γ is any
conclusion at all), then Θ, α, β ⇒ ϕ. In our algorithm, we have to make a
distinction here. We produce abducibles α, and if we already found α solving
Θ, α ⇒ ϕ, then the algorithm may not produce stronger abducibles than that.
(It might happen, due to the closure patterns of branches in the initial tableau,
that we produce one solution implying another, but this does not have to be.)
As for strengthening the theory Θ, this might result in an initial tableau with
possibly fewer open branches, over which our procedure may then produce
weaker abducibles, invalidating the original choice of α cooperating with Θ to
derive ϕ.
Abduction as Computation
119
More relevant, therefore, is the traditional question whether our algorithms
are sound and complete. Again, we have to make sure what these properties mean in this setting. First, Soundness should mean that any combination
(Θ, α, ϕ) which gets out of the algorithm does indeed present a valid case of
abduction, as defined in chapter 3. For plain abduction, it is easy to see that
we have soundness, as the tableau closure condition guarantees classical consequence (which is all we need). Next, consider consistent abduction. What
we need to make sure of now, is also that all abducibles are consistent with the
background theory Θ. But this is what happened by our use of ‘partial branch
closures’. These are sure (by definition) to leave at least one branch for Θ open,
and hence they are consistent with it. Finally, the conditions of the ‘explanatory’ algorithm ensure likewise that the theory does not explain the fact already
(Θ ⇒ ϕ) and that α could not do the job on its own (α ⇒ ϕ).
Next, we consider completeness. Here, we merely make some relevant
observations, demonstrating the issues (for our motives, cf. the end of this
paragraph). Completeness should mean that any valid abductive consequence
should actually be produced by it. This is trickier. Obviously, we can only
expect completeness within the restricted language employed by our algorithm.
Moreover, the algorithm ‘weeds out’ irrelevant conjuncts, etcetera, which cuts
down outcomes even more. As a more significant source of incompleteness,
however, we can look at the case of disjunctive explanations. The implications
produced always involve one literal as a consequent. This is not enough for
a general abductive conclusion, which might involve more. What we can say,
for instance is this. By simple inspection of the algorithm, one can see that
every consistent atomic explanation that exists for an abductive problem will
be produced by the algorithm. In any case, we feel that completeness is less
of an issue in computational approaches to abduction. What comes first is
whether a given abductive procedure is natural, and simple. Whether its yield
meets some pre-assigned goal is only a secondary concern in this setting.
We can also take another look at issues of soundness and completeness,
relatively independently from our axiom. The following analysis of ‘closure’
on tableaux is inspired by our algorithm - but it produces a less procedural
logical view of what is going on.
An Alternative Semantic Analysis
Our strategy for producing abductions in tableaux worked as follows. One
starts with a tableau for the background theory Θ (i), then adds the negation
¬ϕ of the new observation ϕ to its open branches (ii), and one also closes the
remaining open branches (iii), subject to certain constraints. In particular (for
the explanatory version) one does not allow ϕ itself as a closure atom as it is
regarded as a trivial solution. This operation may be expressed via a kind of
‘closure operation’:
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ABDUCTIVE REASONING
CLOSE (Θ + ¬ϕ) − ϕ.
We now want to take an independent (in a sense, ‘tableau–free’) look at
the situation, in the most general case, allowing disjunctive explanations. (If
Θ is particularly simple, we can make do (as shown before) with atoms, or
their conjunctions.) First, we recall that tableaus may be represented as sets
of open branches. We may assume that all branches are completed, and hence
all relevant information resides in their literals. This leads to the following
observation.
Fact 5 Let Θ be a set of sentences, and let T be a complete tableau for Θ, with
π running over its open branches. Then ∧Θ (the conjunction of all sentences
in Θ) is equivalent to:
π
open in
T
l
a literal l ∈ π.
This fact is easy to show. The conjunctions are the total descriptions of
each open branch, and the disjunction says that any model for Θ must choose
one of them. This amounts to the usual Distributive Normal Form theorem for
propositional logic. Now, we can give a description of our CLOSE operation
in similar terms. In its most general form, our way of closing an open tableau
is really defined by putting:
S
a set of literals
π
open in T ∃l ∈ S : l ∈ π
The inner part of this says that the set of (relevant) literals S ‘closes every
branch’. The disjunction states the weakest combination that will still close the
tableau. Now, we have a surprisingly simple connection:
Fact 6 CLOSE (Θ) is equivalent to ¬(∧Θ) !
Proof. By Fact 1 plus the propositional De Morgan laws, ¬(∧Θ) is equivalent to
l ∈ π. But then, a simple argument, involving choices
π open in T
l a literal
for each open branch, shows that the latter assertion is equivalent to CLOSE (Θ). ⊣
In the case of abduction, we proceeded as follows. There is a theory Θ, and a
surprising fact q (say), which does not follow from it. The latter shows because
we have an open tableau for Θ followed by ¬q. We close up its open branches,
without using the trivial explanation q. What this involves, as said above, is a
modified operation, that we can write as:
CLOSE (Θ) − q
Example
Let Θ = {p → q, r → q}. The tableau T (Θ ∪ ¬q) − q is as follows:
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Abduction as Computation
(Θ ∪ ¬q) − q
¬p
q
¬r
q
¬r
q
¬q
¬q
¬q
¬q
The abductions produced are p or r.
Again, we can analyze what the new operation does in more direct terms.
Fact 7 CLOSE (Θ) − q is equivalent to ¬ [false/q] ∧ Θ.
Proof. From its definition, it is easy to see that CLOSE(Θ) − q is equivalent with
[false/q] CLOSE(Θ). But then, we have that
CLOSE (Θ ∧ ¬q) − q
[false/q] ¬(∧Θ ∧ ¬q)
iff
iff
(by Fact 2)
(by propositional logic)
¬[false/q] ∧ Θ. ⊣
This rule may be checked in the preceding example. Indeed, we have that
[false/q] Θ is equivalent to ¬p ∧ ¬r, whose negation is equivalent to our earlier
outcome p ∨ r.
This analysis suggests abductive variations that we did not consider before.
For instance, we need not forbid all closures involving ¬q, but only those which
involve ¬q in final position (i.e., negated forms of the ‘surprising fact’ to be
explained). There might be other, harmless occurrences of ¬q on a branch
emanating from the background theory Θ itself.
Example
Let Θ = {q → p, p → q}. The tableaux T (Θ) is as follows:
Θ
¬q
p
¬p
q
¬p
q
...
...
¬q
¬q
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ABDUCTIVE REASONING
In this case, our earlier strategy would merely produce an outcome p - as can
be checked by our false-computation rule. The new strategy, however, would
compute an abduction p or q, which may be just as reasonable.
This analysis does not extend to complex conclusions. We leave its possible
extensions open here.
Tableaux and Resolution
In spite of the logical equivalence between the methods of tableau and resolution
[Fit90, Gal92], in actual implementations the generation of abductions turns out
to be very different. The method of resolution used in logic programming does
not handle negation explicitly in the language, and this fact restricts the kind
of abductions to be produced. In addition, in logic programming only atomic
formulas are produced as abductions since they are identified as those literals
which make the computation fail. In semantic tableaux, on the other hand, it is
quite natural to generate abductive formulas in conjunctive or disjunctive form
as we have shown.
As for similarities between these two methods as applied to abduction, both
frameworks have one of the explanatory conditions (Θ ⇒ ϕ) already built in. In
logic programming the abductive mechanism is put to work when a query fails,
and in semantic tableaux abduction is triggered when the tableau for Θ ∪ ¬ϕ
is open.
Abduction in First Order Semantic Tableaux:
The DB-Tableaux Model
The problem of abduction may be considered an impossible task when applied to first order logic [MP93]. The main reason being that the precondition
for an abductive problem (T (Γ ∪ {¬φ} has an open branch) is indeed undecidable. We cannot of course avoid that result but can tackle the problem for
special cases.
In any case, if there is some presumption for modelling explanatory inference
as conceived in the philosophy of science, quantified rules for dealing with
universal laws must be incorporated. To this end, one problem to tackle is
the existence of infinite branches for a given tableaux for a finitely satisfiable
theory. An attractive way to do that would be to explore transformations of
tableaux with infinite branches into others with finished ones; in such a way
that the newly created finite branches would provide information that allows
the construction of abductive explanations.
In this section the main aim is to study the problem of the existence of
infinite branches when semantic tableaux are applied to the systematic search
for a solution to an explanation problem. We propose the modification of the so
called δ-rule (cf. below) into the δ ′ -rule (following previous work in [Dia93];
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Abduction as Computation
[Boo84]; [Nep 02] and [AN04]), in such a way that if a theory is finitely
satisfiable, then the new semantic tableau presents the same advantages than
the standard one to obtain certain explanatory facts.
In addition to the extension rules given for propositional tableaux (cf. subsection 2.1), here are the classical rules for first order tableaux:
γ-rule:
∀xϕ
ϕ(b1 /x)
ϕ(b2 /x)
...
ϕ(bn /x)
b1 , ..., bn are all constants that occur in sentences of the branch, n ≥ 1 (if
no one occurs, then n = 1), the new part of the branch is Φ + ϕ(b1 /x) +
... + ϕ(bn /x) (ϕ(bi /x) is a γ-sentence, for all i ≤ n).
δ-rule
∃xϕ
ϕ(b/x)
where b is a new constant that does not occur in any previous sentence of the
branch, the new part of the branch is Φ + ϕ(b/x) and this is a δ-sentence.
We are interested in those quantified universal formulae of the following
form: ∀x∃yϕ, which are satisfiable in finite domains. As it turns out, even
formulae of this kind may have a tableau with infinite branches. Therefore, it is
appealing to find a way by which it is possible to transform an infinite tableau
into a finite one, in the sense of obtaining finite open branches.
Let us take as an example the following. The standard tableau for ∀x∃yRxy
is as follows:
∀x∃yRxy
∃yRa1 y
Ra1 a2
∃yRa2 y
Ra2 a3
..
.
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ABDUCTIVE REASONING
and so on. The only one branch is infinite, even though the formula is satisfiable
in a unitary domain. Therefore, the classic tableau framework is of no use here.
We will now introduce the DB-tableaux method5 , followed by the presentation
of this same example, showing that a finite model is possible to be found.
DB-tableaux
Definition 1. A DB-tableau is a tableau constructed by means of the usual
extension rules for propositional tableaux, the γ-rule and the following one:
δ ′ -rule:
∃xϕ
ϕ(a1 /x) | . . . | ϕ(an+1 /x)
Where a1 , a2 , ..., an are all constants that occur in previous sentences; an+1 is a
new constant and ϕ(ai /x) is a δ ′ -sentence for every i ≤ n. The branch Φ splits
into the following branches: Φ + ϕ(a1 /x); Φ + ϕ(a2 /x); ...; Φ + ϕ(an /x);
Φ + ϕ(an+1 /x).
The DB-tableau for ∀x∃yRxy is as follows:
∀x∃yRxy
∃yRa1 y
Ra1 a1
Ra1 a2
∃yRa2 y
Ra2 a1
Ra2 a2
Ra2 a3
..
.
As illustrated, the DB-tableau for ∀x∃yRxy has a completed open branch
(in fact, three of them) from which a model M = D, ℑ can be defined
in such a way that M satisfies every formula of the branch, provided that
ℑ(R) = {ℑ(a1 ), ℑ(a1 )}.
5 E.
Dı́az and G. Boolos proposed independently this modification of Beth’s tableaux for the first time in
[Dia93] and [Boo84] respectively.
Abduction as Computation
125
In what follows we will introduce the notions of n-satisfiability and nentailment for DB-tableaux in order to prepare the ground for our coming notion
of an n-abductive problem.
Definition 2. A branch of a DB-tableau has depth n, for n ≥ 1, iff n is the
number of (non repeated) constants occurring in the branch.
Definition 3. A finite set of sentences Γ is n-satisfiable, for n ≧ 1, iff there
exists a domain D of cardinality n in which Γ is satisfiable.
Definition 4. Given a finite set of sentences Γ and a sentence ψ, Γ n-entails
ψ, for n ≧ 1, in symbols: Γ |=n ψ iff Γ ∪ {ψ} is n-satisfiable and there is no
domain of cardinality m, m ≤ n, for which Γ ∪ {¬ψ} is m-satisfiable.
Theorem 1. A finite set of sentences Γ is n-satisfiable, for n ≥ 1 iff the DBtableau of Γ has an open branch that is n in depth.
This result is proved elsewhere (Cf. [Nep99] and [Nep 02]). An straightforward corollary of this theorem (together with definition 4) is the following
theorem:
Theorem 2. Given a set of sentences Γ ⊆ L and a non quantified sentence
ψ ∈ L, Γ |=n ψ, for n ≥ 1, iff the DB-tableau for Γ ∪ {ψ} has a completed
open branch of depth n and Γ ∪ {¬ψ} is not m-satisfiable, for every m ≤ n.
We have thus defined a notion of satisfaction and of entailment which are
relative to domain cardinality. These notions are weaker than the standard ones,
in fact, n-entailment may be labeled \weak entailment for n", given that now
only some models (and not necessarily all) of Γ are models of ψ. As it should
be noted, for every n ≥ 1, if Γ |= ψ then Γ |=n ψ, though Γ |=n ψ does not
imply Γ |= ψ. That is, Tarskian entailment implies n-entailment but not vice
versa.
Moreover, this logical operation is non monotonic. For example, for n = 2,
we have ∀x∃yRxy, ¬Ra2 a1 |=2 Ra2 a2 , but ∀x∃yRxy, ¬Ra2 a1 , ∀x¬Rxx |=2
Ra2 a2 . Here are the corresponding DB-tableaux for illustration:
126
ABDUCTIVE REASONING
∀x∃yRxy
¬Ra2 a1
∃yRa1 y
∃yRa2 y
Ra1 a1
Ra1 a2
Ra2 a1
Ra2 a2
Ra2 a1
Ra2 a2
This tableau is closed by Ra2 a2 , which proves that the first entailment
holds. The next tableau shows that the second entailment does not hold, for the
branches of depth n = 2 close with the premisses alone.
∀x∃yRxy
¬Ra2 a1
∀x¬Rxx
∃yRa1 y
∃yRa2 y
¬Ra1 a1
¬Ra2 a2
Ra1 a1
Ra1 a2
Ra2 a1
Ra2 a2
Abduction in First Order DB-Tableaux
In this section we will propose the use of DB-tableaux to work with some kind
of abductive problems. Our approach so far has the limitation that produces
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Abduction as Computation
only literals as abductive explanations, but the method could be extended to
accommodate more complex forms as well (cf. [AN04]).
Definition 5. Given a set of sentences Γ ⊆ L and a non quantified sentence
ψ ∈ L, for which the DB-tableau for Γ ∪ {¬ψ} has an open branch of depth
n, Γ, ψn is denoted as a n-abductive problem.
Definition 6. Given an n-abductive problem Γ, ψn , 1 ≤ n < ℵ0 , α is a
n-abductive solution iff
1 The DB-tableau of Γ ∪ {¬ψ} ∪ {α} has closed every branch of depth m,
for all m ≤ n.
2 The DB-tableau of Γ ∪ {α} has a completed open branch of depth n.
These conditions correspond to those for the generating an abductive explanation in the classical model, as shown in section 3.2. However, now the two
requirements are not relative to “all branches of the corresponding tableau”, but
rather relative to “all branches of the corresponding DB-tableau which are of
depth n".
Example
Suppose that Γ = {∀x∃yRxy}, and the fact to be explained is ψ = {Ra1 a2 }.
We shall first construct the the DB-tableau for Γ ∪ {¬ψ} :
∀x∃yRxy
¬Ra1 a2
∃yRa1 y
∃yRa2 y
Ra1 a1
Ra1 a2
Ra2 a1
Ra2 a2
Ra2 a3
..
.
Ra1 a3
..
.
According to the first condition, a 2-abductive solution candidate is ¬Ra1 a1 ,
for this formula would close the above open branches of depth 2. Therefore,
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ABDUCTIVE REASONING
all branches of the DB-tableau for
{∀x∃yRxy, ¬Ra1 a2 , ¬Ra1 a1 }
which are m ≤ 2, are closed:
∀x∃yRxy
¬Ra1 a1
¬Ra1 a2
∃yRa1 y
∃yRa2 y
Ra1 a1
Ra1 a2
Ra1 a3
..
.
In fact, in this case there are no branches of smaller depth, since the formula in
Γ has two variables, making it impossible to construct the tableau with branches
of lesser depth. So we can conclude that:
∀x∃yRxy, ¬Ra1 a1 |=2 Ra1 a2
As for condition 2, the DB-tableau for {∀x∃yRxy, ¬Ra1 a1 } has a completed
open branch (in fact two of them) of depth 2:
∀x∃yRxy
¬Ra1 a1
∃yRa1 y
∃yRa2 y
Ra1 a1
Ra1 a2
Ra1 a3
..
.
Ra2 a1
Ra2 a2
Ra2 a3
..
.
Therefore, ¬Ra1 a1 is a 2-abductive solution for the abductive problem
∀x∃yRxy, Ra1 a2 2 .
Abduction as Computation
129
In this section the main aim was to study the problem of the existence of
infinite branches when semantic tableaux are applied to the systematic search
for a solution to an abductive problem. An attractive way to deal with such a
problem is to transform a tableau with infinite branches into another one with
finished ones; in such a way that the newly created finite branches provide
information that allows the construction of abductive explanations, our main
concern. To this end, we proposed the modification of the standard δ-rule for
the existential quantifier into the δ ′ -rule, and accordingly labelled our tableau
method as DB-tableaux, in such a way that if a theory is finitely satisfiable, then
the new semantic tableau presents the same advantages than the standard one
to obtain certain explanatory facts.
We introduced the notions of n-satisfiability and n-entailment, both relative
to a finite domain. As it turned out, one of our main results shows that a
set of sentences is n-satisfiable if and only if its corresponding DB-tableau
has an open branch of depth n. Therefore, we can define the notion of an nabductive problem and its corresponding n-abductive solution for a DB-tableau
of Γ ∪ {¬φ}, as one with an open branch of depth n and α as its solution when
the DB-tableau for Γ ∪ {¬φ} ∪ {α} has closed every branch of depth m (for
all m ≤ n) and for which the DB-tableau for Γ ∪ {α} has a completed open
branch of depth n.
Any theory with postulates that are formalized in this way is supposed to have
an intended interpretation which corresponds to a model with a finite universe.
And when it is n-satisfiable (n ≥ 1), then it is m-satisfiable for every m ≥ n
(pace the upward theorem of Löwenheim-Skolem). However, an advantage of
DB-tableaux in this case is that the method obtains a minimal model, that is, a
model with the least possible cardinality.
Our approach so far seems has the limitation of producing only literals as
abductive explanations, but our method could be extended in a natural way in
order to accommodate more complex forms as well (cf. [AN04]).
6.
Discussion and Conclusions
Abduction in Tableaux
Exploring abduction as a form of computation gave us further insight into
this phenomenon. Our concrete algorithms implement some earlier points from
chapter 3, which did not quite fit the abstract structural framework. Abductions
come in different degrees (atomic, conjunctive, disjunctive-conditional), and
each abductive condition corresponds to new procedural complexity. In practice, though, it turned out easy to modify the algorithms accordingly. Indeed
these findings reflect an intuitive feature of explanation. While it is sometimes difficult to describe what an explanation is in general, it may be easier to
construct a set of explanations for a particular problem.
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As for the computational framework, semantic tableaux are a natural vehicle
for implementing abduction. They allow for a clear formulation of what counts
as an abductive explanation, while being flexible and suggestive as to possible
modifications and extensions. Derivability and consistency, the ingredients of
consistent and explanatory abduction, are indeed a natural blend in tableaux,
because we can manipulate open and closed branches with equal ease. Hence
it is very easy to check if the consistency of a theory is preserved when adding
a formula. (By the same token, this type of conditions on abduction appears
rather natural in this light.)
Even so, our actual algorithms were more than a straight transcription of the
logical formulations in chapter 3 (which might be very inefficient). Our computational strategy provided an algorithm which produces consistent formulas,
selecting those which count as explanations, and this procedure turns out to be
more efficient than the other way around. Nevertheless, abduction in tableaux
has no unique form, as we showed by some alternatives. A final insight emerging from a procedural view of abduction is the importance of the background
theory when computing explanations. Bad theories in combination with certain
observation will produce bad explanations. Sophisticated computation cannot
improve that.
Finally, we explored abduction in first order tableaux. Our approach has
the advantage of simplifying the systematic search for a solution to certain
abductive problems, by means of first order DB-tableaux, when constants occur
in the corresponding formulae and the standard tableaux method is not effective
to find a solution because of the generation of infinite branches. That is, for
theories which are finitely satisfiable but that nevertheless generate infinite
branches under Beth’s tableau method.
Our tableau approach also has clear limitations. It is hard to treat notions
of abduction in which ⇒ is some non-standard consequence. In particular,
with an underlying statistical inference, it is unclear how probabilistic entailments should be represented. Our computation of abductions relies on tableaux
being open or closed, which represent only the two extremes of probable inference. We do have one speculation, though. The computation of what we
called ‘partial explanations’ (which close some but not all open branches) might
provide a notion of partial entailment in which explanations only make a fact
less surprising, without explaining it in full. (Cf. [Tij97] for other approaches
to ‘approximative deduction’ in abductive diagnostic settings.) As for other
possible uses of the tableau framework, the case of abduction as revision was
not addressed here. In a further chapter (8), we shall see that we do get further
mileage in that direction, too.
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Related Work
The framework of semantic tableaux has relatively recently been used beyond its traditional logical purposes, especially in computationally oriented
approaches. One example is found in [Nie96], implementing ‘circumscription’. In connection with abduction, semantic tableaux are used in [Ger95] to
model natural language presuppositions (cf. chapter 2, abduction in linguistics).
A better–known approach is found in [MP93], a source of inspiration for our
work, which we proceed to briefly describe and compare. The following is a
mere sketch, which cannot do full justice to all aspects of their proposal.
Mayer and Pirri’s Tableau Abduction
Mayer and Pirri’s article presents a model for computing ‘minimal abduction’
in propositional and first–order logic. For the propositional case, they propose
two characterizations. The first corresponds to the generation of all consistent
and minimal explanations (where minimality means ‘logically weakest’; cf.
chapter 3). The second generates a single minimal and consistent explanation
by a non-deterministic algorithm. The first–order case constructs abductions by
reversed skolemization, making use of unification and what they call ‘dynamic
herbrandization’ of formulae. To give an idea of their procedures for generating
explanations, we merely note their main steps: (1) construction of ‘minimal
closing sets’, (2) construction of abductive solutions as literals which close all
branches of those sets, (3) elimination of inconsistent solutions. The resulting
systems are presented in two fashions, once as semantic tableaux, and once
as sequent calculi for abductive reasoning. There is an elegant treatment of
the parallelism between these two. Moreover, a predicate–logical extension is
given, which is probably the first significant treatment of first–order abduction
which goes beyond the usual resolution framework. (In subsequent work, the
authors have been able to extend this approach to modal logic, and default
reasoning.)
Comparison to our work
Our work has been inspired by [MP93]. But it has gone in different directions,
both concerning strategy and output. (i) Mayer and Pirri compute explanations
in line with version 1 of the general strategies that we sketched earlier. That
is, they calculate all closures of the relevant tableau to later eliminate the inconsistent cases. We do the opposite, following version 2. That is, we first
compute consistent formulas which close at least one branch of the original
tableau, and then check which of these are explanations. Our reasons for this
had to do with greater computational efficiency. (ii) As for the type of explanations produced, Mayer and Pirri’s propositional algorithms basically produce
minimal atomic explanations or nothing at all, while our approach provides ex-
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planations in conjunctive and disjunctive form as well. (iii) Mayer’s and Pirri’s
approach stays closer to the classical tableau framework, while ours gives it
several new twists. We propose several new distinctions and extensions, e.g.
for the purpose of identifying when there are no consistent explanations at all.
(iv) An interesting point of the Mayer and Pirri presentation is that it shows very
well the contrast between computing propositional and first–order abduction.
While the former is easy to compute, even considering minimality, the latter is
inherently undecidable, but we have shown that even so there are interesting
cases for treating abduction in first order tableaux. (iv) Eventually, we go even
further (cf. chapter 8) and propose semantic tableaux as a vehicle for revision,
which requires a new contraction algorithm.
PART III
APPLICATIONS
Chapter 5
SCIENTIFIC EXPLANATION
1.
Introduction
In the philosophy of science, we confront our logical account of chapter 3
with the notion of scientific explanation, as proposed by Hempel in two of his
models of scientific inference: deductive-nomological and inductive-statistical
[Hem65]. We show that both can be viewed as forms of (abductive) explanatory
arguments, the ultimate products of abductive reasoning. The former with
deductive underlying inference, and the latter with statistical inference.
In this confrontation, we hope to learn something about the scope and limitations of our analysis so far. We find a number of analogies, as well as new
challenges. The notion of statistical inference gives us an opportunity to expand
the logical analysis at a point left open in chapter 3. We will also encounter
further natural desiderata, however, which our current analysis cannot handle.
Our selection of topics in this field is by no means complete. We will cover
some of them in the companion chapter to the philosophy of science (chapter 6),
but many other connections exist relating our proposal to philosophy of science.
Some of these concerns (such as cognitive processing of natural language)
involve abductive traditions beyond the scope of our analysis. (We have already
identified some of these in chapter 2.) Nevertheless, the connections that we
do develop have a clear intention. We view all three fields as naturally related
components in cognitive science, and we hope to show that abduction is one
common theme making for cohesion.
2.
Scientific Explanation as Abduction
At a general level, our discussion in chapters 1 and 2 already showed that
scientific reasoning could be analyzed as an abductive process. This reasoning
comes in different kinds, reflecting (amongst others) various patterns of dis-
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covery, with different ‘triggers’. A discovery may be made to explain a novel
phenomenon which is consistent with our current theory, but it may also point at
an anomaly between the theory and the phenomenon observed. Moreover, the
results of scientific reasoning vary in their degree of novelty and complexity.
Some discoveries are simple empirical generalizations from observed phenomena, others are complex scientific theories introducing sweeping new notions.
We shall concentrate on rather ‘local’ scientific explanations, which can be taken
to be some sort of logical arguments: (abductive) explanatory ones, in our view.
(We are aware of the existence of divergent opinions on this: cf. chapters 1 and
2). That scientific inference can be viewed as some form of abductive inference
should not be surprising. Both Peirce and Bolzano were inspired in their logical
systems by the way reasoning is done in science. Peirce’s abductive formulations may be regarded as precursors of Hempel’s notion of explanation, as will
become clear shortly. Indeed, it has been convincingly claimed that ‘Peirce
should be regarded as the true founder of the theory of inductive-probabilistic
explanation’ ([Nii81, p. 444] and see also [Nii00] for a detailed account).
At the center of our discussion on the logic of explanation lies the proposal
by Hempel and Oppenheim [HO48, Hem65]. Their aim was to model explanations of empirical ‘why-questions’. For this purpose they distinguished
several kinds of explanation, based on the logical relationship between the explanans and explanandum (deductive or inductive), as well as on the form of
the explanandum (singular events or general regularities). These two distinctions generate four models altogether: two deductive-nomological ones (D-N),
and two statistical ones (Inductive-Statistical (I-S), and Deductive-Statistical
(D-S)).
We analyze the two models for singular events, and present them as forms
of abduction, obeying certain structural rules.
The Deductive-Nomological Model
The general schema of the D-N model is the following:
L1 , . . . , Lm
C1 , . . . , Cn
E
L1 , . . . , Lm are general laws which constitute a scientific theory T , and together with suitable antecedent conditions C1 , . . . , Cn constitute a potential
explanation T, C for some observed phenomenon E. The relationship be-
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137
tween explanandum and explananda is deductive, signaled by the horizontal
line in the schema. Additional conditions are then imposed on the explananda:
T, C is a potential explanation of E iff
T is essentially general and C is singular.
E is derivable from T and C jointly, but not from C alone.
The first condition requires T to be a ‘general’ theory (having at least one
universally quantified formula). A ‘singular’ sentence C has no quantifiers or
variables, but just closed atoms and Boolean connectives. The second condition
further constrains the derivability relation. Both T and C are required for the
derivation of E.
Finally, the following empirical requirement is imposed:
T, C is an explanation of E iff
T, C is a potential explanation of E
C is true
The sentences constituting the explananda must be true. This is an empirical condition on the status of the explananda. T, C remains a potential
explanation for E until C is verified.
From our logical perspective of chapter 3, the above conditions define a
form of abduction. In potential explanation, we encounter the derivability
requirement for the plain version (T, C ⊢ E), plus one of the conditions for our
‘explanatory’ abductive style C ⊢ E. The other condition that we had (T ⊢ E)
is not explicitly required above. It is implicit, however, since a significant
singular sentence cannot be derived solely from quantified laws (which are
usually taken to be conditional). An earlier major abductive requirement that
seems absent is consistency (T ⊢ ¬C). Our reading of Hempel is that this
condition is certainly presupposed. Inconsistencies certainly never count as
scientific explanations. Finally, the D-N account does not require minimality
for explanations: it relocates such issues to choices between better or worse
explanations, which fall outside the scope of the model. We have advocated
the same policy for abduction in general (leaving minimal selection to our
algorithms of chapter 4).
There are also differences in the opposite direction. Unlike our (abductive)
explanatory notions of chapter 2, the D-N account crucially involves restrictions
on the form of the explanantia. Also, the truth requirement is a major difference.
Nevertheless, it fits well with our discussion of Peirce in chapter 2: an abducible
has the status of a suggestion until it is verified.
It seems clear that the Hempelian deductive model of explanation is closely
related to our proposal of (abductive) explanatory argument, in that it complies
with most of the logical conditions discussed in chapter 3. If we fit D-N explanation into a deductive format, the first thing to notice is that laws and initial
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conditions play different roles in explanation. Therefore, we need a ternary
format of notation: T | C ⇒HD E. A consequence of this fact is that this
inference is non-symmetric (T | C ⇒HD E ⇔ C | T ⇒HD E). Thus, we
keep T and C separate in our further discussion. Here is the resulting notion
once more:
Hempelian Deductive-Nomological Inference ⇒HD
T | C ⇒HD E iff
(i) T, C ⊢ E
(ii) T, C is consistent
(iii) T ⊢ E, C ⊢ E
(iv) T consists of universally quantified sentences,
C has no quantifiers or variables.
Structural Analysis
We now analyze this notion once more in terms of structural rules. For a
general motivation of this method, see chapter 3. We merely look at a number of
crucial rules discussed earlier, which tell us what kind of explanation patterns are
available, and more importantly, how different explanations may be combined.
Reflexivity
Reflexivity is one form of the classical Law of Identity: every statement implies
itself. This might assume two forms in our ternary format:
E | C ⇒HD E
T | E ⇒HD E
However, Hempelian inference rejects both, as they would amount to ‘irrelevant explanations’. Given condition (iii) above, neither the phenomenon
E should count as an explanation for itself, nor should the theory contain the
observed phenomenon: because no explanation would then be needed in the
first place. (In addition, left reflexivity violates condition (iv), since E is not a
universal but an atomic formula.) Thus, Reflexivity in this way has no place in
a structural account of explanation.
Monotonicity
Monotonic rules in scientific inference provide means for making additions to
the theory or the initial conditions, while keeping scientific arguments valid.
Although deductive inference by itself is monotonic, the additional conditions
on ⇒HD invalidate classical forms of monotonicity, as we have shown in detail
in chapter 3. So, we have to be more careful when ‘preserving explanations’,
adding a further condition. Unfortunately, some of the tricks from chapter
3 do not work in this case, because of our additional language requirements.
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139
Moreover, outcomes can be tricky. Consider the following monotonicity rule,
which looks harmless:
HD Monotonicity on the Theory (invalid):
T, B|D ⇒HD E
T |A ⇒HD E
T, B|A ⇒HD E
This says that, if we have an explanation A for E from a theory, as well as
another explanation D for the same fact E from a strengthened theory, then
the original explanation will still work in the strengthened theory. This sounds
convincing, but it will fail if the strengthened theory is inconsistent with A.
If we add an extra requirement, that is, T, B|A ⇒HD F , then we get a valid
form of monotonicity:
HD Monotonicity on the Theory (valid):
T, B|D ⇒HD E
T, B|A ⇒HD F
T |A ⇒HD E
T, B|A ⇒HD E
This additional condition forces to require that the strengthened theory explain something else (F ), in order to warrantee the consistency of the theory
and explanation with the observation. However, we have not been able to find
any convincing monotonicity rules when only two requirements are set!
What we learn here is a genuine limitation of the approach in chapter 3.
With notions of D-N complexity, pure structural analysis may not be the best
way to go. We might also just bring in the additional conditions explicitly,
rather than encoding them in abductive sequent form. Thus, ‘a theory may
be strengthened in an explanation, provided that this happens consistently, and
without implying the observed phenomenon without further conditions’. It is
important to observe that the complexities of our analysis are no pathologies,
but rather reflect the true state of affairs. They show that there is much more to
the logic of Hempelian explanation than might be thought, and that this can be
brought out in a precise manner. For instance, the failure of monotonicity rules
means that one has to be very careful, as a matter of logic, in ‘lifting’ scientific
explanations to broader settings.
Cut
The classical Cut rule allows us to chain derivations, and replace temporary
assumptions by further premises implying them. Thus, it is essential to standard
reasoning. Can explanations be chained? Again, there are many possible cut
rules, some of which affect the theory, and some the conditions. We consider
one natural version:
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HD Cut (invalid):
T |B ⇒HD E
T |A ⇒HD B
T |A ⇒HD E
Our rain example of chapter 2 gives an illustration of this rule. Nimbostratus
clouds (A) explain rain (B), and rain explains wetness (E), therefore nimbostratus clouds (A) explain wetness (E). But, is this principle generally valid?
It almost looks that way, but again, there is a catch. In case A implies E by
itself, the proposed conclusion does not follow. Again, we would have to add
this constraint separately, in order to have a valid form of cut:
HD Cut (valid):
T |B ⇒HD E
T |A ⇒HD B
T |A ⇒HD E
A⊢E
This rule not only has an additional requirement but it actually involves two
kinds of inference to make it work (⇒HD and ⊢). We will see later on that rules
involving inductive inference do require this kind of combination. Moreover,
there may be reservation to add this extra requirement as predicate logic is
only semi-decidable, and making claims in the negative gives not warrantee to
achieve them. Still, if we meet the requirements set in the premisses we get a
valid form of monotonicity.
Logical Rules
A notion of inference with as many side conditions as ⇒HD has, considerably
restricts the forms of valid structural rules one can get. Indeed, this observation
shows that there are clear limits to the utility of the purely structural rule analysis,
which has become so popular in a broad field of contemporary logic. To get at
least some interesting logical principles that govern explanation, we must bring
in logical connectives. Here are some valid rules.
1 Disjunction of Theories
T2 | C ⇒ E
T1 | C ⇒ E
T 1 ∨ T2 | C ⇒ E
2 Conjunction of two Explananda
T | C ⇒ E2
T | C ⇒ E1
T | C ⇒ E1 ∧ E 2
3 Conjunction of Explanandum and Theory
T | C1 ⇒ E
T | C2 ⇒ F
T | C1 ⇒ F ∧ E
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Scientific Explanation
4 Disjunction of Explanans and Explanandum
T |C⇒E
T |C ∨E ⇒E
5 Weakening Explanans by Theory
T, F | A ⇒ E
T, F | F → A ⇒ E
The first two rules show that well-known classical inferences for disjunction
and conjunction carry over to explanation. The third says that the explanandum
can be strengthened with any consequence of the background theory. However,
it is common to demand that self explanations should be excluded (i.e., T does
not explain T ). Then, it is also required that F is not equal to T . This has
been one of the obstacles to earlier attempts to give necessary and sufficient
conditions for valid explanatory arguments (see [Tuo73]). The fourth shows
how explananda can be weakened ‘up to the explanandum’. The fifth rule
states that explananda may be weakened provided that the background theory
can compensate for this. The last rule actually highlights a flaw in Hempel’s
models, which he himself recognized. It allows for a certain trivialization of
‘minimal explanations’, which might be blocked again by imposing further
syntactic restrictions (see [Hem65, Sal90]). However, note that these rules
do not necessarily obey the earlier syntactic restrictions of the D-N model,
for they involve non-atomic formulae in the explanandum (rules 2 and 3) and
formula, which may contain quantifiers as initial condition (F → A in rule
5). This situation suggests dropping these restrictions if we want to handle
rules with connectives (but then, we would not speak of Hempel’s model for
singular events). Moreover, the complete DN model of explanation involves
both singular statements and generalizations. So, for a further analysis, an
account of structural rules with quantificational principles of predicate logic is
needed.
More generally, it may be said that the D-N model has been under continued
criticism through the decades after its emergence. No generally accepted formal
model of deductive explanation exists. But at least we hope that our style of
analysis has something fresh to offer in this ongoing debate: if only, to bring
simple but essential formal problems into clearer focus.
The Inductive-Statistical Model
Hempel’s I-S model for explaining particular events E has essentially the
same form as the D-N model. The fundamental difference is the status of
the laws. While in the D-N model, laws are universal generalizations, in the
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ABDUCTIVE REASONING
I-S model they are statistical regularities. This difference is reflected in the
outcomes. In the I-S model, the phenomenon E is only derived ‘with high
probability’ [r] relative to the explanatory facts:
L1 , . . . , Lm
C1 , . . . , Cn
[r]
E
In this schema, the double line expresses that the inference is statistical
rather than deductive. This model retains all adequacy conditions of the D-N
model. But it adds a further requirement on the statistical laws, known as maximal specificity (RMS). This requirement responds to a problem which Hempel
recognized as the ambiguity of I-S explanation. As opposed to classical deduction, in statistical inference, it is possible to infer contradictory conclusions
from consistent premises. One of our examples from chapter 2 demonstrates
this.
The Ambiguity of I-S Explanation
Suppose that theory T makes the following statements. “Almost all cases of
streptococcus infection clear up quickly after the administration of penicillin
(L1). Almost no cases of penicillin-resistant streptococcus infection clears up
quickly after the administration of penicillin (L2). Jane Jones had streptococcus
infection (C1). Jane Jones received treatment with penicillin (C2). Jane Jones
had a penicillin-resistant streptococcus infection (C3)." From this theory it is
possible to construct two contradictory arguments, one explaining why Jane
Jones recovered quickly (E), and the other one explaining its negation, why
Jane Jones did not recover quickly (¬E):
Argument 1
Argument 2
L1
L2
C1 , C2
E
[r]
C2 , C3
[r]
¬E
The premises of both arguments are consistent with each other, they could all
be true. However, their conclusions contradict each other, making these arguments rival ones. Hempel hoped to solve this problem by forcing all statistical
laws in an argument to be maximally specific. That is, they should contain all
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143
relevant information with respect to the domain in question. In our example,
then, premise C3 of the second argument invalidates the first argument, since
the law L1 is not maximally specific with respect to all information about Jane
in T . So, theory T can only explain ¬E but not E.
The RMS makes the notion of I-S explanation relative to a knowledge situation, something described by Hempel as ‘epistemic relativity’. This requirement helps, but it is neither a definite nor a complete solution. Therefore, it has
remained controversial1 .
These problems may be understood in logical terms. Conjunction of Consequents was a valid principle for D-N explanation. It also seems a reasonable
principle for explanation generally. But its implementation for I-S explanation
turns out to be highly non-trivial. The RMS may be formulated semi-formally
as follows:
Requirement of Maximal Specificity:
A universal statistical law A B is maximally specific iff for all A′ such that A′ ⊂ A,
A′ B.
We should note however, that while there is consensus of what this requirement means on an intuitive level, there is no agreement as to its precise formalization (cf. [Sal90] for a brief discussion on this). With this caveat, we give a
version of I-S explanation in our earlier format, presupposing some underlying
notion ⇒i of inductive inference.
Hempelian Inductive Statistical Inference ⇒HI
T, C ⇒HI E iff
(i) T, C ⇒i E
(ii) T, C is consistent
(iii) T ⇒i E, C ⇒i E
(iv) T is composed of statistical quantified formulas (which may include forms like
“Most A are B"). C has no quantifiers.
(v) RMS: All laws in T are maximally specific with respect to T,C.
The above formulation complies with our earlier abductive requirements,
but the RMS further complicates matters. Moreover, there is another source of
vagueness. Hempel’s D-N model fixes predicate logic as its language for making form distinctions, and classical consequence as its underlying engine. But
in the I-S model the precise logical nature of these ingredients is left unspecified.
1 One
of its problems is that it is not always possible to identify a maximal specific law given two rival
arguments. Examples are cases where the two laws in conflict have no relation whatsoever, as in the
following example, due to Stegmüller [Ste83]: Most philosophers are not millionaires. Most mine owners
are millionaires. John is both a philosopher and a mine owner. Is he a millionaire?
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Some Interpretations of ⇒i
Statistical inference ⇒i may be understood in a qualitative or a quantitative
fashion. The former might read Θ ⇒i ϕ as: “ϕ is inferred from Θ with
high probability”, while the latter might read it as “most of the Θ models are
ϕ models”. These are different ways of setting up a calculus of inductive
reasoning2 .
In addition, we have similar options as to the language of our background
theories. The general statistical statements A B that may occur in the background theory Θ may be interpreted as either “The probability of B conditioned
on A is close to 1”, or as statements of the form: “most of the A-objects are
B-objects”. We will not pursue these options here, except for noting that the
last interpretation would allow us to use the theory of generalized quantifiers.
Many structural properties have already been investigated for probabilistic generalized quantifiers (cf. [vLa96, vBe84b] for a number of possible approaches).
Finally, the statistical approach to inference might also simplify some features of the D-N model, based on ordinary deduction. In statistical inference,
the D-N notion of non-derivability need not be a negative statement, but rather
a positive one of inference with (admittedly) low probability. This interpretation has some interesting consequences that will be discussed at the end of
this chapter. It might also decrease complexity, since the notion of explanation
becomes more uniformly ‘derivational’ in its formulation.
Structural Analysis, Revisited
Again, we briefly consider some structural properties of I-S explanation. Our
discussion will be informal, since we do not fix any formal explanation of the
key notions discussed before.
As for Reflexivity of ⇒i , this principle fails for much the same reasons as
for D-N explanation. Next, consider Monotonicity. This time, new problems
arise for strengthening theories in explanations, due to the RMS. A law L might
be maximally specific with respect to T |A, but not necessarily so with respect
to T, B|A or to T |A, B. Worse still, adding premises to a statistical theory
may reverse previous conclusions! In the above penicillin example, the theory
without C3 explains perfectly why Jane Jones recovered quickly. But adding
C3 reverses the inference, explaining instead why she did not recover quickly.
If we then add that she actually took some homeopathic medicine with high
chances of recovery (cf. chapter 2), the inference reverses again, and we will
be able to explain once more why she recovered quickly.
2 Recall from chapter 3 that there may be two different characterizations of the qualitative kind, one in which
it is improbable that Θ ∧ ¬ϕ while another one in which given Θ, it is improbable that ¬ϕ.
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145
These examples show once again that inductive statistical explanations are
epistemically relative. There is no guarantee of preserving consistency when
changing premises, therefore making monotonicity rules hopeless. And stating
that the additions must be ‘maximally specific’ seems to beg the question.
Nevertheless, we can salvage some monotonicity, provided that we are willing
to combine deductive and inductive explanations. (There is no logical reason for
sticking to pure formulations.) Here is a principle that we find plausible, modulo
further specification of the precise inductive inference used (recall that ⇒HD
stands for deductive–nomological inference and ⇒HI for inductive–statistical):
Monotonicity on the theory:
T, B|D ⇒HD C
T |A ⇒HI C
T, B|A ⇒HI C
This rule states that statistical arguments are monotonic in their background
theory, at least when what is added explains the relevant conclusion deductively
with some other initial condition. This would indeed be a valid formulation,
provided that we can take care of consistency for the enlarged theory T, B | A.
In particular, all maximal specific laws for T | A remain specific for T, B | A.
For, by inferring C in a deductive and consistent way, there is no place to add
something that would reverse the inference, or alter the maximal specificity of
the rules.
Here is a simple illustration of this rule. If on the one hand, Jane has high
chances of recovering quickly from her infection by taking penicillin, and on
the other she would recover by taking some medicine providing a sure cure (B),
she still has high chances of recovering quickly when the assertion about this
cure is added to the theory.
Not surprisingly, there is no obvious Cut rule in this setting either. Here it is
not the RMS causing the problem, as the theory does not change, but rather the
well-known fact that statistical implications are not transitive. Again, we have
a proposal for a rule combining statistical and deductive explanation, which
might form a substitute: The following is our proposed formulation:
Deductive cut on the explanans:
T |A ⇒HI B
T |B ⇒HD C
T |A ⇒HI C
Again, here is a simple illustration of this rule. If the administration of
penicillin does explain the recovery of Jane with high probability, and this in turn
explains deductively her good mood, penicillin explains with high probability
Jane’s good mood. (This reflects the well-known observation that “most A are
B, all B are C" implies “most A are C". Note that the converse chaining is
invalid.)
Patrick Suppes (p.c.) has asked whether one can formulate a more general representation theorem, of the kind we gave for consistent abduction (cf.
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chapter 3), which would leave room for statistical interpretations. This might
account for the fluid boundary between deduction and induction in common
sense reasoning. Exploring this question however, has no unique and easy
route. As we have seen, it is very hard to formulate a sound monotonic rule
for I-S explanation. But this failure is partly caused by the fact that this type
of inference requires too many conditions (the same problem arose in the H-D
model). So, we could explore a calculus for a simpler notion of probabilistic
consequence, or rather work with a combination of deductive and statistical inferences. Still, we would have to formulate precisely the notion of probabilistic
consequence, and we have suggested there are several (qualitative and quantitative) options for doing it. Thus, characterizing an abductive logical system
that could accommodate statistical reasoning is a question that deserves careful
analysis, beyond the confines of this book.
3.
Discussion and Conclusions
Further Relevant Connections
In this section we briefly present the notion of inductive support, show its
relation to the notion of explanation and attempt to hint at some of its structural
properties. Moreover, we review Salmon’s notion of statistical relevance, as a
sample of more sophisticated post-Hempelian views of explanation. In addition,
we briefly discuss a possible computational implementation of Hempel’s models
more along the lines of our chapter 4, and finally we present Thagard’s notion
of explanation within his account of ‘computational philosophy of science’.
Inductive Support
Related to the notion of explanation is the notion of inductive support. Taking
into account some claimed logical principles around this notion we can explore
some of its features under the light of our structural analysis given so far.
Intuitively, ‘e inductively supports h’, referred to as eIh, if h is ‘inducible’
from e, something which presupposes some inductive consequence logical relation between e and h. Some relevant principles are the following (see [NT73]
for motivation and other principles):
(SC) Special Consequence: If eIh and h ⊢ b, then eIb.
(CC) Converse Consequence: If eIh and b ⊢ h, then eIb.
(E) Entailment: If e ⊢ h, then eIh.
Along the lines of chapter 3, we can capture such inductive logical relationship (eIh) in its forward fashion (h ⇒is e), for some suitable notion of ⇒is .
Then a principle like (CC) may be expressed as follows:
(CC) If h ⇒is e and b ⊢ h, then b ⇒is e.
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147
It is easy to see that using this principle we can conclude b, h ⇒is e from
h ⇒is e and b ⊢ h. We have just to appeal to the monotonicty of ⊢ (from
b ⊢ h we can infer b, h ⊢ h). However, it this seems incorrect to infer that
e inductively supports b, h from the fact that e alone inductively supports h,
even if we have that h is deductively inferred from b. Thus, a revision of this
principle is proposed as follows [Bro68]:
(CC*) If h ⇒is e and ‘b explains h’, then b ⇒is e.
This version blocks the previous undesired monotonicity, for the relationship of explanation is not preserved when the explananda is strengthened with
additional formulae (see [NT73, p. 225] for a discussion of the adoption of
this principle). Thus, it seems clear that notions like explanation and confirmation do not obey the classical monotonicty rule and therefore appropriate
formulations have to be put forward. We leave this as an open question for the
case of confirmation. The purpose of this brief presentation was just to show
the possibility of studying this notion at a structural level, as we have done in
chapter 3 for (abductive) explanatory arguments.
Salmon’s Statistical Relevance
Despite the many conditions imposed, the D-N model still allows explanations
irrelevant to the explanandum. The problem of characterizing when an explanans is relevant to an explanandum is a deep one, beyond formal logic. It is
a key issue in the general philosophy of science.
One noteworthy approach regards relevance as causality. W. Salmon first
analyzed explanatory relevance in terms of statistical relevance [Sal71]. For
him, the issue in inductive statistical explanation is not how probable the explanans (T |A) renders the explanandum C, but rather whether the facts cited in
the explanans make a difference to the probability of the explanandum. Thus,
it is not high probability but statistical relevance that makes a set of premises
statistically entail a conclusion.
Now recall that we said that statistical non-derivability (Θ ⇒i ϕ) might be
refined to state that “ϕ follows with low probability from Θ”. With this reinterpretation, one can indeed measure if an added premise changes the probability
of a conclusion, and thereby count as a relevant explanation. This is not all
there is to causality. Salmon himself found problems with his initial proposal,
and later developed a causal theory of explanation [Sal84], which was refined
in [Sal94]. Here, explanandum and explananda are related through a causal
nexus of causal processes and causal interactions. Even this last version is still
controversial (cf. [Hit95]).
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A Computational Account of Hempel’s models?
Hempel treats scientific explanation as a given product, without dealing with
the processes that produce such explanations. But, just as we did in chapter
4, it seems natural to supplement this inferential analysis with a computational
search procedure for producing scientific explanations. Can we modify our
earlier algorithms to do such a job? For the D-N model, indeed, easy modifications to our algorithm would suffice. (But for the full treatment of universal
laws, we would need a predicate-logical version of the tableau algorithm in line
with our proposal of DB-tableaux in chapter 4.) For the inductive statistical
case, however, semantic tableaux seem inappropriate. We would need to find a
way of representing statistical information in tableaux, and then characterize inductive inference inside this framework. Some more promising formalisms for
computing inductive explanations are labeled deductive systems with ‘weights’
by Dov Gabbay [Gab96], and other systems of dependence-driven qualitative
probabilistic reasoning by W. Meyer Viol (cf. [Mey95]) and van Lambalgen &
Alechina ([AL96]).
Thagard’s Computational Explanation
An alternative route toward a procedural account of scientific explanation is
taken by Thagard in [Tha88]. Explanation cannot be captured by a Hempelian
syntactic structure. Instead, it is analyzed as a process of providing understanding, achieved through a mechanism of ‘locating’ and ‘matching’. The model
of explanation is the program PI (‘processes of induction’), which computes
explanations for given phenomena by procedures such as abduction, analogy,
concept formation, and generalization and then accounts for a ‘best explanation’
comparing those which the program is able to construct.
This approach concerns abduction in cognitive science (cf. chapter 2), in
which explanation is regarded as a problem–solving activity modeled by computer programs. (The style of programming here is quite different from ours in
chapter 4, involving complex modules and record structures.) This view gives
yet another point of contact between contemporary philosophy of science and
artificial intelligence.
Explanation in Belief Revision
We conclude with a short discussion that relates abduction with scientific
explanation. Theories of explanation in the philosophy of science mainly concern scientific inquiry. Nevertheless, some ideas by Gärdenfors on explanation
(chapter 8 of his book [Gar88]), turn out illuminating in creating a connection. Moreover, how explanations are computed for incoming beliefs makes a
difference in the type of operation required to incorporate the belief.
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Gärdenfors’ basic idea is that an explanation is, that which makes the explanandum less surprising by raising its probability. The relationship between
explananda and explanandum is relative to an epistemic state, based on a probabilistic model involving a set of possible worlds, a set of probability measures,
and a belief function. Explanations are propositions that effect a special epistemic change, increasing the belief value of the explanandum. Explananda must
also convey information that is ‘relevant’ to the beliefs in the initial state. This
proposal is very similar to Salmon’s views on statistical relevance, which we
discussed in connection with our statistical reinterpretation of non-derivability
as derivability with low probability. The main difference between the two is
this. While Gärdenfors requires that the change is in raising the probability,
Salmon admits just any change in probability. Gärdenfors notion of explanation
is closer to our ‘partial explanations’ of chapter 4 (which closed some but not
all of the available open tableau branches). A natural research topic will be to
see if we can implement the idea of raising probability in a qualitative manner
in tableaux. Gärdenfors’ proposal involves degrees of explanation, suggesting
a measure for explanatory power with respect to an explanandum.
Combining explanation and belief revision (cf. chapter 8) into one logical
endeavor also has some broader attractions. This co-existence (and possible
interaction) seems to reflect actual practice better than Hempel’s models, which
presuppose a view of scientific progress as mere accumulation. This again links
up with philosophical traditions that have traditionally been viewed as a non–
logical, or even anti-logical. These include historic and pragmatic accounts
[Kuh70, vFr80] that focus on analyzing explanations of anomalous instances
as cases of revolutionary scientific change. (“Scientific revolutions are taken
to be those noncumulative developmental episodes in which an older paradigm
is replaced in whole or in part by an incompatible new one" [Kuh70, p.148].)
In our view, the Hempelian and Kuhnian schools of thought, far from being
enemies, emphasize rather two sides of the same coin.
Finally, our analysis of explanation as compassing both scientific inference
shows that the philosophy of science and artificial intelligence share central aims
and goals. Moreover, these can be pursued with tools from logic and computer
science, which help to clarify the phenomena, and show their complexities.
Conclusions
Our analysis has tested the logical tools developed in chapter 3 on Hempel’s
models of scientific explanation. The deductive-nomological model is indeed
a form of (abductive) explanatory inference. Nevertheless, structural rules
provide limited information, especially once we move to inductive-statistical
explanation. In discussing this situation, we found that the proof theory of
combinations of D-N and I-S explanation may actually be better-behaved than
either by itself. Even in the absence of spectacular positive insights, we can
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still observe that the abductive inferential view of explanation does bring it in
line with modern non-monotonic logics.
Our negative structural findings also raise interesting issues by themselves.
From a logical point of view, having an inferential notion like ⇒HD without
reflexivity and monotonicity, challenges a claim made in [Gab85]. Gabbay
argues that reflexivity, ‘restricted monononicity’ and cut are the three minimal
conditions which any consequence relation should satisfy to be a bona fide nonmonotonic logic. (Later in [Gab94a] however, Gabbay admits this is not such
a strong approach after all, as ‘Other systems, such as relevance logic, do not
satisfy even reflexivity’.) We do not consider this failure an argument against
the inferential view of explanation. As we have suggested in chapter 3, there are
no universal logical structural rules that fit every consequence relation. What
counts rather is that such relations ‘fit a logical format’.
Non-monotonic logics have been mainly developed in AI. As a further point
of interest, we mention that the above ambiguity of statistical explanation also
occurs in default reasoning. The famous example of ‘Quakers and Pacifists’
is just another version of the one by Stegmüller cited earlier in this chapter.
More specifically, one of its proposed solutions by Reiter [Rei80] is in fact a
variation of the RMS3 . These cases were central in Stegmüller criticisms of
the positivist view of scientific explanation. He concluded that there is no
satisfactory analysis using logic. But these very same examples are a source of
inspiration to AI researchers developing non-standard logics.
There is a lot more to these connections than what we can cover in this
book. With a grain of salt, contemporary logic-based AI research may be
viewed as logical positivism ‘pursued by other means’. One reason why this
works better nowadays than in the past, is that Hempel and his contemporaries
thought that classical logic was the logic to model and solve their problems. By
present lights, it may have been their logical apparatus more than their research
program what hampered them. Even so, they also grappled with significant
problems, which are as daunting to modern logical systems as to classical ones,
including a variety of pragmatic factors. In all, there seem to be ample reasons
for philosophers of science, AI researchers, and logicians for making common
cause.
Analyzing scientific reasoning in this broader perspective also sheds light on
the limitations of this book. We have not really accounted for the distinction
between laws and individual facts, we have no good account of ‘relevance’,
and we have not really fathomed the depths of probability. Moreover, much of
scientific explanation involves conceptual change, where the theory is modified
with new notions in the process of accounting for new phenomena. So far,
3 In
[Tan92] the author shows this fact in detail, relating current default theories to Hempel’s I-S model.
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151
neither our logical framework of chapter 3, nor our algorithmic one of chapter
4 has anything to offer in this more ambitious realm.
Chapter 6
EMPIRICAL PROGRESS
1.
Introduction
Traditional positivist philosophy of science inherits from logical research
not only its language, but also its focus on the truth question, that is to say,
the purpose of using its methods as means for testing hypotheses or formulae.
As we saw in the previous chapter, Hempelian models of explanation and confirmation seek to establish the conditions under which a theory (composed by
scientific laws) together with initial conditions, explains a certain phenomenon
or whether certain evidence confirms a theory. As for logical research, it has
been characterized by two approaches, namely the syntactic and the semantic.
The former account characterizes the notion of derivability and aims to answer
the following question: given theory H (a set of formulae) and formula E, is E
derivable from H? The latter characterizes the notion of logical consequence
and responds to the following question: is E a logical consequence of H?
(Equivalent to: are the models of H models of E?) Through the truth question
we can only get a “yes–no" answer with regard to the truth or falsity of a given
theory. Aiming solely at this question implies a static view of scientific practice,
one in which there is no place for theory evaluation or change. Notions like
derivation, logical consequence, confirmation, and refutation are designed for
the corroboration –logical or empirical– of theories.
However, a major concern in philosophy of science is also that of theory
evaluation. Since the 1930s, both Carnap and Popper stressed the difference between truth and confirmation, where the last notion concerns theory evaluation
as well. Issues like the internal coherence of a theory as well as its confirming
evidence, refuting anomalies or even its lacunae, are all key considerations to
evaluate a scientific theory with respect to itself and existing others. While there
is no agreement about the precise characterization of theory evaluation, and in
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general about what progress in science amounts to, it is clear that these notions
go much further than the truth question. The responses that are searched for are
answers to the success question, which includes the set of successes, failures
and lacunae of a certain theory. This is interesting and useful beyond testing
and evaluation purposes, since failures and lacunae indicate the problems of
a theory, issues that if solved, would result in an improvement of a theory to
give a better one. Thus, given the success question, the improvement question
follows from it.
For the purposes of this chapter, we take as our basis a concrete proposal
which aims at modeling the three questions about empirical progress set so far,
namely the questions of truth, success and improvement, the approach originally
presented by Theo Kuipers in [Kui00] and in [Kui99]. In particular, we undertake the challenge of the latter publication, namely to operationalize the task of
theory revision aiming at empirical progress, that is, the task of instrumentalist
abduction. Our proposal aims at showing that evaluation and improvement of a
theory can be modeled by (an extension of) the framework of semantic tableaux,
in the style of chapter 4. In particular, we are more interested in providing a
formal characterization of lacunae, namely those phenomena which a scientific
theory cannot explain, but that are consistent with it. This kind of evidence
completes the picture of successes and failures of a theory [Kui99]. The terms
‘neutral instance’ and ‘neutral result’ are also used [Kui00], for the cases in
which a(n) (individual) fact is compatible with the theory, but not derivable
from it. This notion appears in other places in the literature. For example,
Laudan [Lau77] speaks of them as a ‘non-refuting anomalies’, to distinguish
them from the real ‘refuting anomalies’. We also find these kinds of statements
in epistemology. Gärdenfors [Gar88] characterizes three epistemic states of an
agent with respect to a statement as follows: (i) acceptance, (ii) rejection, and
(iii) undetermination, the latter one corresponding to the epistemic state of an
agent in which she neither accepts nor rejects a belief, but simply has no opinion
about it (cf. chapter 8 for a treatment of abduction as epistemic change based
on this proposal). As for myself (cf. chapters 2 and 8), I have characterized
“novelty abduction" as the epistemic operation that is triggered when faced with
a ‘surprising fact’ (which is equivalent to the state of undetermination when the
theory is closed under logical consequence). It amounts to the extension of the
theory into another one that is able to explain the surprise.
These kind of phenomena put in evidence, not the falsity of the theory (as
it is the case with anomalies), but its incompleteness ([Lau77]), its incapability
of solving problems that it should be able to do.
In contemporary philosophy of science there is no place for complete theories, if we take a ‘complete scientific theory’ to mean that it has the capacity
to give explanations –negative or positive– to all phenomena within its area of
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155
competence1 . Besides, in view of the recognition of the need for initial conditions to account for phenomena, the incompleteness of theories is just something
implicit. Therefore, we should accept the property of being “incomplete" as a
virtue of scientific theories, rather than a drawback as in the case of mathematics. A scientific theory will always encounter new phenomena that need to be
accounted for, for which the theory in question is insufficient. However, this is
not to say that scientists or philosophers of science give up when they are faced
with a particular “undecidable phenomenon".
Our suggestion is that the presence of lacunae marks a condition in the
direction of progress of a theory, suggesting an extension (or even a revision) of
the theory in order to “decide" about a certain phenomenon which has not yet
been explained, thus generating a “better" theory, one which solves a problem
that the original one does not. Therefore, lacunae not only play a role in the
evaluation of a theory, but also in its design and generation.
As for the formal framework for modeling the success and the improvement
questions with regard to lacunae, classical methods in logic are no use. While
lacunae correspond to “undecidable statements"2 , classical logic does not provide a method for modifying a logical theory in order to “resolve" undecidable
statements. However, extensions to classical methods may provide suitable
frameworks for representing theory change. We propose an extension to the
method of semantic tableaux (cf. chapter 4), originally designed as a refutation
procedure, but used here beyond its purpose in theorem proving and model
checking, as a way to extend a theory into a new one which entails the lacunae
of the original one. This is a well-motivated standard logical framework, but
over these structures, different search strategies can model empirical progress
in science.
This chapter is naturally divided into four parts. After this introduction, in
the second part (section 2) we describe Kuipers’ empirical progress framework,
namely the characterization of evidence types, the account of theory evaluation
and comparison, as well as the challenge of the instrumentalist abduction task.
In the third part (section 3) we present our approach to empirical progress in
the (extended) semantic tableaux framework. In the final part of this chapter
(section 4), we sum up our previous discussion and offer our general conclusions.
1 It
is clear that a theory in physics gives no account to problems in molecular biology. Explanation here
is only within the field of each theory. Moreover, in our interpretation we are neglecting all well-known
problems that the identification between the notions of explanation and derivability brings about, but this
understanding of a ‘complete scientific theory’ highlights the role of lacunae, as we will see later on.
2 In 1931 Gödel showed that if we want a theory free of contradictions, then there will always be assertions
we cannot decide, as we are tied to incompleteness. We may consider undecidable statements as extreme
cases of lacunae, since Gödel did not limit himself to show that the theory of arithmetic is incomplete, but
in fact that it is “incompletable", by devising a method to generate undecidable statements.
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2.
ABDUCTIVE REASONING
Kuipers’ Empirical Progress
Two basic notions are presented by this approach, which characterize the
relationship between scientific theories and evidence, namely confirmation and
falsification. To this end, Kuipers proposes the Conditional Deductive Confirmation Matrix. For the purposes of this chapter, we are using a simplified and
slightly modified version of Kuipers’ matrix. On the one hand, we give the
characterization by evidence type, that is, successes and failure, rather than by
the notions of confirmation and falsification, as he does. Moreover, we omit
two other notions he characterizes, namely (dis)confirmation and verification.
On the other hand, we do not take account of the distinction between general
successes and individual problems (failures), something that is relevant for his
further characterization of theory improvement. Our version is useful since it
gives lacunae a special place in the characterization, on a par with successes
and failures, and it also permits a quite simple representation in our extended
tableaux framework.
Success and Failure
SUCCESS
Evidence E is a success of a theory H relative to an initial condition C whenever:
H, C |= E
In this case, E confirms H relative to C.
FAILURE
Evidence E is a failure of a theory H relative to an initial condition C whenever:
H, C |= ¬E
In this case E falsifies H relative to C.
This characterization has the classical notion of logical consequence (|=) as
the underlying logical relationship between a theory and its evidence3 . Therefore, successes of a theory (together with initial conditions) are its logical consequences and failures are those formulae for which their negations are logical
consequences. Additional logical assumptions have to be made. H and C must
be logically independent, C and E must be true formulae and neither C nor H
give an account of E for the case of confirmation (or of ¬E for falsification)
by themselves. That is:
1 Logical Independence
H |= C, H |= ¬C
¬H |= C, ¬H |= ¬C
3 But
this need not to be so. Several other notions of semantic consequence (preferential, dynamic) or of
derivability (default, etc.) may capture evidence type within other logical systems (cf. chapter 3 for a logical
characterization of abduction within several notions of consequence).
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157
2 C and E are true formulae.
3 Assumptions for Conditional Confirmation:
H |= E, C |= E
4 Assumptions for Conditional Falsification:
H |= ¬E, C |= ¬E
Notice that the requirement of logical independence assures the consistency
between H and C, a requirement that has sometimes been overlooked (especially by Hempel, cf. chapter 5), but it is clearly necessary when the logical
relationship is that of classical logical entailment. The assumptions of conditional confirmation and conditional falsification have the additional effect of
preventing evidence from being regarded as an initial condition (H, E |= E)4 .
Lacuna
Previous cases of success and failure do not exhaust all possibilities there are
in the relationship amongst H, C and E (¬E). It may very well be the case that
given theory H, evidence E and all previous assumptions, there is no initial
condition C available to account for E (¬E) as a case of success (or failure).
In this case we are faced with a lacuna of the theory. More precisely:
LACUNA
Evidence E is a lacuna of a theory H when the following conditions hold for all available
initial conditions C:
H, C |= E
H, C |= ¬E
In this case, E neither confirms nor falsifies H.
It is clear that the only case in which a theory has no lacuna is given when it
is complete. But as we have seen, there is really no complete scientific theory.
To find lacunae in a theory suggests a condition (at least from a logical point
of view) in the direction of theory improvement. That is, in order to improve a
theory (H1 ) to give a better one (H2 ), we may extend H1 in such a way that its
lacunae become successes of H2 . The existence of lacunae in a theory confront
us with its holes, and their resolution into successes indicates progress in the
theory. By so doing, we are completing the original theory (at least with respect
4 These additional assumptions are of course implicit for the case in which E (¬E) is a singular fact (a literal,
that is an atom or a negation thereof), since a set of universal statements H cannot by itself entail singular
formulae, but we do not want to restrict our analysis to this case, as evidence in conditional form may also
be of interest.
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to certain evidence) and thus constructing a better one, which may include
additional laws or initial conditions.
The above characterization in terms of successes, failures and lacunae makes
it clear that the status of certain evidence is always with respect to some theory
and specific initial condition. Therefore, in principle a theory gives no account
of its evidence as successes or failures, but only with respect to one (or more)
initial condition(s). Lacunae, on the other hand, show that the theory is not
sufficient, that there may not be enough laws to give an account of evidence
as a case of success or failure. But then a question arises: does the characterization of lacunae need a reference to the non-existence of appropriate initial
conditions? In other words, is it possible to characterize lacuna-type evidence
in terms of some condition between H and E alone? In the next subsection we
prove that it is possible to do so in our framework.
Theory Evaluation and Comparison
Kuipers argues for a ‘Context of Evaluation’, as a more appropriate way to
refer to the so-called ‘Context of Justification’ (cf. chapter 1):
‘Unfortunately, the term ‘Context of Justification’, whether or not specified in a falsificationist way, suggests, like the terms ‘confirmation’ and ‘corroboration’, that the
truth or falsity of a theory is the sole interest. Our analysis of the HD-method makes
it clear that it would be much more adequate to speak of the ‘Context of Evaluation’.
The term ‘evaluation’ would refer, in the first place, to the separate and comparative
HD-evaluation of theories in terms of successes and problems’. [Kui00, p.132]
Kuipers proposes to extend Hempel’s HD methodology to account not only
for testing a theory, which only gives an answer to the ‘truth-question’, but also
for evaluation purposes, in which case it allows one to answer the ‘successquestion’, in order to evaluate a theory itself, thus setting the ground for comparing it with others. This approach shows that a further treatment of a theory
may be possible after its falsification, it also allows a record to be made of the
lacunae of a theory and by so doing it leaves open the possibility of improving it.
As Kuipers rightly states, HD-testing leads to successes or problems (failures)
of a certain theory ‘and not to neutral results’ [Kui00, 101].
There are, however, several models for HD-evaluation, including the symmetric and the asymmetric ones (each one involving a macro or a micro argument).
For the purposes of this chapter we present a simplified version (in which we
do not distinguish between individual and general facts) of the symmetric definition, represented by the following comparative evaluation matrix ([Kui00,
117]):
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159
H2 \ H1 F ailures Lacunae Successes
F ailures
0
−
−
Lacunae
+
0
−
Successes
+
+
0
Besides neutral results (0), there are three types of results which are favorable
(+) for H2 relative to H1 , and three types of results which are unfavorable (-)
for H2 relative to H1 .
This characterization results in the following criterion for theory comparison,
which allows one to characterize the conditions under which a theory is more
successful than another: ‘Theory H2 is more successful than theory H1 if there
are, besides neutral results, some favorable results for H2 and no unfavorable
results for H2 ’. Of course, it is not a question of counting the successes (or
failures) of a theory and comparing it with another one, rather it is a matter of
inclusion, which naturally leads to the following characterization: (cf. [Kui00,
112])
Theory Comparison
A theory H2 is (at time t) as successful as (more successful than) theory H1 if and only
if (at time t):
1 The set of failures in H2 is a subset of the set of failures in H1 .
2 The set of successes in H1 is a subset of the set of successes in H2
3 At least in one of the above cases the relevant subset is proper.
This model puts forward a methodology in science, which is formal in its
representation, and it does not limit itself to the truth question. Nevertheless, it is
still a static representation of scientific practice. The characterization of a theory
into its successes, failures and lacunae is given in terms of the logical conditions
that the triple H, C, E should observe. It does not specify, for example, how
a lacuna of a theory is identified, that is, how it is possible to determine that
there are no initial conditions to give an account of a given evidence as a case
of success or failure. Also missing is an explicit way to improve a theory; that
is, a way to revise a theory into a better one. We only get the conditions under
which two – already given and evaluated – theories are compared with respect
to their successes and failures. Kuipers is well aware of the need to introduce
the issues of theory revision, as presented in what follows.
Instrumentalist Abduction
In a recent proposal [Kui99], Kuipers uses the above machinery to define
the task of instrumentalist abduction, namely theory revision aiming at empirical progress. The point of departure is the evaluation report of a theory (its
successes, failures and lacunae), and the task consists of the following:
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Instrumentalist Abduction Task
Search for a revision H2 of a theory H1 such that:
H2 is more successful than H1 , relative to the available data5 .
Kuipers sets an invitation for abduction aiming at empirical progress, namely
the design of instrumentalist abduction along symmetric or asymmetric lines.
More specific, he proposes the following challenge: “to operationalize in
tableau terms the possibility that one theory is ‘more compatible’ with the
data than another" [Kui99, p.320]. In this chapter we take up this challenge
in the following way. We propose to operationalize the above instrumentalist
task for the particular case of successes and lacunae (his Task I). We cast the
proposal in our extended framework of semantic tableaux (cf. chapter 4).
It will be clear that while our proposal remains basically the same as that of
Kuipers’, our representation allows for a more dynamic view of theory evaluation and theory improvement by providing ways to compute appropriate initial
conditions for potential successes, and for determining the conditions under
which an observation is a case of lacuna, both leading to procedures for theory improvement. However, this chapter does not give all the details of these
processes, it only provides the general ideas and necessary proofs.
3.
Empirical Progress
in (Abductive) Semantic Tableaux
Lacuna Evidence Type Characterization
In what follows we aim to characterize lacuna evidence in terms of the
extension type it effects over a certain tableau for a theory when it is expanded
with it. The main motivation for this characterization is to answer the question
raised in the previous section: can we know in general when a certain formula
E qualifies as a case of lacuna with respect to a theory H?
Let me begin by giving a translation of lacuna evidence (cf. subsection 2.2)
into tableaux, denoted here as lacuna* in order to distinguish it from Kuipers’
original usage:
LACUNA*
Given a tableau T (H) for a theory, evidence E is a lacuna* of a theory H, when the
following conditions hold for all available initial conditions C:
T (H) ∪ {¬E} ∪ {C} is an open tableau
5 In
the cited paper this is defined as the first part of Task III (Task III.1). Task III.2 requires a further
evaluation task and consists of the following: “H2 remains more successful than H1 , relative to all future
data". Moreover, there are two previous tasks: Task I, Task II, which are special cases of Task III, the former
amounts to the case of a ‘surprising observation’ (a potential success or a lacuna) and the task is to expand
the theory with some hypothesis (or initial condition) such that the observation becomes a success. This is
called novelty guided abduction. The latter amounts to the case of an ‘anomalous observation’ (a failure),
and the task is to revise the theory into another one which is able to entail the observation. This one is called
anomaly guided abduction. In my own terminology of abduction (cf. chapter 2) I refer to the former as
abductive novelty and to the latter as abductive anomaly.
Empirical Progress
161
T (H) ∪ {E} ∪ {C} is an open tableau
In this case, neither E confirms nor falsifies H.
Although this is a straightforward translation, it does not capture the precise type of extension that evidence E (and its negation) effects on the tableau.
To this end, we recall our proposed distinction between “open", “closed" and
“semiclosed" extensions. While the first two are characterized by those formulae which when added to the tableau do not close any open branch, or close all
of them respectively, the last one are exemplified by those formulae which close
some but not all of the open branches. (For the formal details of the extension
characterization in terms of semantic tableaux see chapter 4).
As we are about to prove that, when a formula E (and its negation) effects an
open extension over a tableau for a certain theory H, it qualifies as a lacuna*.
More precisely:
LACUNA* CLAIM
Given a theory H and evidence E:
IF T (H) + {¬E} is an open extension and T (H) + {E} is also an open extension,
THEN E is a lacuna* of H.
Proof
(i) To be proved: T (H) ∪ {¬E} ∪ {C} is an open tableau for all C.
(ii) To be proved: T (H) ∪ {E} ∪ {C} is an open tableau for all C.
(i) Let T (H) + {¬E} be an open extension (H |= E).
Suppose there is a C (C = E is warranted by the second assumption of
conditional confirmation), such that T (H) + {¬E} + {C} is a closed
extension.
Then T (H) + {C} must be a closed extension, that is, H |= ¬C, but
this contradicts the second assumption for logical independence.
Therefore,
T (H) + {¬E} + {C} is not a closed extension, that is, H, C |= E,
thus concluding that T (H) ∪ {¬E} ∪ {C} is an open tableau.
(ii) Let T (H) + {E} be an open extension (H |= ¬E).
Suppose there is a C (C = ¬E is warranted by the second assumption
of conditional falsification), such that T (H) + {E} + {C} is a closed
extension.
Then T (H) + {C} must be a closed extension, that is, H |= ¬C, but
this contradicts the second assumption of logical independence.
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ABDUCTIVE REASONING
Therefore,
T (H) + {E} + {C} is not a closed extension, that is, H, C |= ¬E,
thus concluding that T (H) ∪ {E} ∪ {C} is an open tableau.
Notice that in proving the above claim we used two assumptions that have
not been stated explicitly for lacunae; one of conditional confirmation in (i)
(C |= E) and the other one of conditional falsification in (ii) (C |= ¬E).
Therefore, we claim these assumptions should also be included in Kuipers’
additional logical assumptions as restrictions on lacunae. This fact gives us
one more reason to highlight the importance of lacunae as an independent and
important case for theory evaluation.
Finally, our characterization of lacuna** evidence type in terms of our proposed tableaux extensions is as follows:
LACUNA**
Given a tableau for a theory T (H), evidence E is a lacuna** of a theory H whenever:
T (H) + {¬E} is an open extension
T (H) + {E} is an open extension
In this case, E neither confirms nor falsifies H.
This account allows one to characterize lacuna type evidence in terms of
theory and evidence alone, without having to consider any potential initial
conditions. This result is therefore attractive from a computational perspective,
since it prevents the search of appropriate initial conditions when there are none.
The proof of the above lacuna claim assures that our characterization implies
that of Kuipers’, that is, open extensions imply open tableaux (lacuna** implies
lacuna*), showing that our characterization of lacuna** is stronger than the
original one of lacuna*. However, the reverse implication (lacuna* implies
lacuna**) is not valid. An open tableau for T (H) ∪ {¬E} ∪ {C} need not
imply that T (H) + {¬E} is an open extension. Here is a counterexample:
Let H = {a → b, a → d, b → c}
E = {b} and C = {¬d}
Notice that T (H) ∪ {¬b} ∪ {¬d} is an open tableau, but both T (H) + {¬b}
and T (H) + {¬d} are semiclosed extensions.
Examples
In what follows we show the fundamentals of our approach in the framework
of Semantic Tableaux (cf. chapter 4) , via an example on a lacuna-type evidence
followed by one of theory improvement.
Lacuna
Let H1 be a theory and E an evidence. H1 = {a → b}, E = {c}
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Empirical Progress
H1 ∪ {¬c}
¬a
b
¬c
¬c
H1 ∪ {c}
¬a
b
c
c
In this case, neither evidence (c) nor its negation (¬c) are valid consequences
of the theory H1 , as shown by the two open tableaux above. Moreover, it is
not possible to produce any (literal) formula acting as initial condition in order
to make the evidence a case of success or failure (unless the trivial case is
considered, that is, to add c or ¬c). For this particular example it is perhaps
easy to be aware of this fact (a and ¬b are the only possibilities and neither one
closes the whole tableau), but the central question to tackle is, in general, how
can we know when a certain formula qualifies as a case of lacuna with respect
to a certain theory?
We are then faced with a genuine case of a lacuna, for which the theory holds
no opinion about the given evidence, either affirmative or negative: it simply
cannot account for it.
As we saw previously, we characterize formulae as lacunae of a theory whenever neither one of them nor their negations close any open branch when the
tableau is extended with them. As we have pointed out, this kind of evidence
not only shows that the theory is insufficient to account for it, but also suggests
a condition for theory improvement, thus indicating the possibility of empirical progress leading to a procedure to perform the instrumentalist task (cf.
[Kui99]).
Theory Improvement
In what follows we give an example of the extension of theory H1 into a theory
H2 , differing in that formula c is a lacuna in the former and a success in the
latter. One way to expand the tableau is by adding the formula a → c to H1
obtaining H2 as follows:
164
ABDUCTIVE REASONING
Let H2 = {a → b, a → c}, E = {c}
H2 ∪ {¬c}
¬a
b
¬a
c
¬a
c
¬c
¬c
¬c
¬c
This tableau has still two open branches, but the addition of a new formula to
H1 (resulting in H2 ), converts evidence c into a candidate for success or failure,
since it has closed an open branch (in fact two), and thus its extension qualifies
as semi–closed. It is now possible to calculate an appropriate initial condition
to make c a success in H2 , namely a. Thus, c is a success of H2 with respect
to a.
But H1 may be extended by means of other formulae, possibly producing
even better theories than H2 . For instance, by extending the original tableau
instead with b → c the following is obtained:
Let H3 = {a → b, b → c}, E = {c}
H3 ∪ {¬c}
¬a
b
¬b
c
¬b
c
¬c
¬c
¬c
As before, c is a success with respect to initial condition a, but in addition, c
is also a success with respect to initial condition b. Formula a as well as b close
the open branch in a non-trivial way. Therefore, in H3 we gained one success
over H2 . Shall we then consider H3 a better theory than both H1 and H2 ?
According to Kuipers’ criterion for theory comparison (cf. section 2.3), both
H2 and H3 are more successful theories than H1 , since the subset of successes in
H1 is a proper subset of both the set of successes in H2 and of H3 . Moreover,
H3 is more successful than H2 , since in the former case c is a success with
respect to two initial conditions and only with respect to one in the latter case.
Empirical Progress
165
We leave here our informal presentation of the extended method of semantic
tableaux applied to empirical progress leading to the instrumentalist task, as
conceived by Kuipers.
4.
Discussion and Conclusions
Success and Failure Characterizacion?
The reader may wonder whether it is also possible to characterize successes
and failures in terms of their extension types. Our focus in this chapter has been
on lacuna, so we limit ourselves to presenting the results we have found so far,
omitting their proofs:
Success IF E is a Success of H THEN
T (H) + {¬E} is a semiclosed extension and there is an initial condition
available C for which T (H) + {C} is a semiclosed extension.
Failure IF E is a failure of H THEN
T (H) + {E} is a semiclosed extension and there is an initial condition C
available for which T (H) + {C} is a semiclosed extension.
For these cases it turns out that the corresponding valid implication is the
reverse of the one with respect to that of lacunae. This means that closed
tableaux imply semiclosed extensions, but not the other way around. In fact, the
above counterexample6 is also useful to illustrate this failure: both T (H)+{¬b}
and T (H) + {¬d} are semiclosed extensions, but E is not a success of H, since
T (H) ∪ {¬b} ∪ {¬d} is an open tableau.
Conclusions
The main goal of this chapter has been to address Kuipers’ challenge in
[Kui99], namely to operationalize the task of instrumentalist abduction. In
particular, the central question raised here concerns the role of lacunae in the
dynamics of empirical progress, both in theory evaluation and in theory improvement. That is, the relevance of lacunae for the success and the improvement questions.
Kuipers’ approach to empirical progress does take account of lacunae for
theory improvement, by including them in the evaluation report of a theory
(together with its successes and failures), but it fails to provide a full characterization of them, in the same way it does for successes and failures (the former
via the notion of confirmation, and the latter via the notion of falsification).
Moreover, this approach does not specify precisely how a lacuna of a theory is
6H
= {a → b, a → d, b → c} and E = {b}.
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ABDUCTIVE REASONING
recognized, that is, how it is possible to identify the absence of initial conditions which otherwise give an account of certain evidence as a case of success
or failure.
Our reformulation of Kuipers’ account of empirical progress in the framework of (extended) semantic tableaux is not just a matter of a translation into a
mathematical framework. There are at least two reasons for this claim. On the
one hand, with tableaux it is possible to determine the conditions under which
a phenomenon is a case of lacuna for a certain theory, and on the other hand,
this characterization leads to a procedure for theory improvement. Regarding
the first issue, we characterize formulae as lacunae of a theory whenever neither one of them nor their negations close any open branch (when the tableau
is extended with them), they are open extensions in our terminology, and we
prove that our formal characterization of lacuna implies that of Kuipers’. As
for the latter assertion, lacunae type evidences not only show that the theory
is insufficient to account for them, but also suggests a condition for theory
improvement, thus indicating possibilities of empirical progress leading to a
procedure for performing the instrumentalist task.
We have suggested that a fresh look into classical methods in logic may be
used not only to represent the corroboration of theories with respect to evidence,
that is, to answer the truth question, but also to evaluate theories with respect
to their evidence as well as to generate better theories than the original ones.
To this end, we have presented (though informally) the initial steps to answer
the success and improvement questions, showing that given a theory and an
evidence, appropriate initial conditions may be computed in order to make
that evidence a success (failure) with respect to a theory; and when this is not
possible, we have proposed a way to extend a theory and thus improve it in
order to account for its lacunae.
Chapter 7
PRAGMATISM
1.
Introduction
In this chapter, I present the philosophical doctrine known as pragmatism, as
proposed by Charles Peirce, namely, as a method of reflexion with the aim at
clarifying ideas and guided at all moments by the ends of the ideas it analyzes.
Pragmatism puts forward an epistemic aim with an experimental solution, and
does so by following the pragmatic maxim, the underlying precept to fix a belief,
and accordingly produce its corresponding habits of action.
As it turns out, abduction has close connections to pragmatism, the former is
indeed the basis for the latter. This connection is however, a complex one. On
the one hand, to link abduction with a method of reflection, we must conceive
it as an epistemic process for logical inquiry of some kind, which transforms
the surprising phenomenon into a settled belief. Furthermore, pragmatism
is centered in the pragmatic maxim, a guide to calculate practical effects of
conceptions. Thus, on the other hand, in order to attend the experimental
aspect of pragmatism in its connection to abduction, we must attend further
aspects of an abductive hypothesis, and these necessarily go beyond its logical
formulation.
This chapter is naturally divided into four parts. After this introduction, in
the second part (section 2), we present Peirce’s notion of pragmatism, its motivation and purposes, and the pragmatic maxim. In the third part (section 3), we
present Peirce’s development of the notion of abduction as well as his epistemic
view. The abductive formulation can thus be seen as a process of belief acquisition by which a surprising fact generates a doubt that is appeased by a belief,
an abductive explanation. In the fourth part (section 4), we present Peirce’s
notion of pragmatism revisited, that is, in connection with his epistemology
and his (later) view on abduction. Pragmatism is a method of reflexion, and
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ABDUCTIVE REASONING
that constitutes its epistemic aspect, but it is also intimately linked with action,
with those habits of conduct involved in a belief, which is what constitutes the
experimental aspect that gives meaning to things. In the fifth and final part
of this chapter (section 5), we put forward our conclusions relating aspects of
Peirce’s pragmatism to abduction and epistemology.
2.
Pragmatism
Pragmatism and Pragmaticism
William James reports it was Charles Peirce who engendered the philosophical doctrine known as Pragmatism. Just as other notions coined by Peirce,
the choice of this term was carefully selected, this time with the intention of
capturing both sides of this concept: an epistemic as well as a practical one:
‘Now quite the most striking feature of the new theory was its recognition of an inseparable connection between rational cognition and rational purpose; and that consideration
it was which determined the preference for the name pragmatism’. [CP, 5.412]
The term itself was taken from Kant’s ‘pragmatisch’, meaning ‘in relation
to some (well) defined human purpose’ and aims to be distinguished from the
other Kantian term ‘praktisch’, which rather belongs to a region of thought
‘where no mind of the experimentalist type can ever make sure of solid ground
under his feet.’ [CP, 5.412]
Later on, however, Peirce preferred to call it pragmaticism, this time to mark
a difference with the notion of pragmatism proposed by W. James. The history
of this term is fairly well described by Peirce in the following passage [CP,
6.882]:
“In 1871, in a Metaphysical Club in Cambridge, Massachusetts, I used to preach this
principle as a sort of logical gospel, representing the unformulated method followed by
Berkeley, and in conversation about it I called it “Pragmatism". In December [November] 1877 and January 1878 I set forth the doctrine in the Popular Science Monthly; and
the two parts of my essay were printed in French in the Revue Philosophique, volumes vi
and vii. Of course, the doctrine attracted no particular attention, for, as I had remarked
in my opening sentence, very few people care for logic. But in 1897 Professor James
remodelled the matter, and transmogrified it into a doctrine of philosophy, some parts
of which I highly approved, while other and more prominent parts I regarded, and still
regard, as opposed to sound logic. About the time Professor Papini discovered, to the
delight of the Pragmatist school, that this doctrine was incapable of definition, which
would certainly seem to distinguish it from every other doctrine in whatever branch of
science, I was coming to the conclusion that my poor little maxim should be called by
another name; and accordingly, in April, 1905 I renamed it Pragmaticism”1 .
The doctrine of pragmatism is proposed as a philosophical method rather
than a mere philosophy of life, thereby marking yet another difference with
James’ pragmatism:
1 Despite
this clarification, for the aims of this chapter, we refer to the Peircean doctrine as pragmatism.
Pragmatism
169
“It will be seen [from the original statement] that pragmatism is not a Weltanschauung
(worldview) but is a method of reflexion having for its purpose to render ideas clear".
[CP, 5.13].
Pragmatism and Meaning
Pragmatism is a methodological principle responding to questions concerning a theory of meaning; it does not pretend to define truth or reality, but rather
to determine the meaning of terms or propositions:
“Suffice it to say once more that pragmatism is, in itself, no doctrine of metaphysics,
no attempt to determine any truth of things. It is merely a method of ascertaining the
meanings of hard words and of abstract concepts... All pragmatists will further agree
that their method of ascertaining the meanings of words and concepts is no other than
that experimental method by which all the successful sciences (in which number nobody
in his senses would include metaphysics) have reached the degrees of certainty that
are severally proper to them today; this experimental method being itself nothing but
a particular application of an older logical rule, "By their fruits ye shall know them."
[CP, 5.464, 5.465]
The notion of ‘the meaning of a concept’ in this pragmatic terms (cf. [CP,
6.481]) is that it is acquired through the following conditions by which a master
of its use is attained. In the first place, it is required to learn to recognize a
concept in whatever of its manifestations, and this is achieved by an extensive
familiarization with its instances. In the second place, it is required to carry out
an abstract logical analysis of the concept, getting to the bottom of its elemental
constitutive parts. But these two requirements are not yet sufficient to know the
nature of a concept in its totality. For this, it is in addition necessary to discover
and recognize those habits of conduct that the belief in the truth of the concept
in question naturally generates, that is, those habits which result in a sufficient
condition for the truth of a concept in any theme or imaginable circumstance.
The Pragmatic Maxim
Pragmatism is centered in its pragmatic maxim, the underlying guiding principle of this doctrine. In its in its original formulation, this maxim reads as
follows:
“Consider what effects that might conceivably have practical bearing you conceive the
object of your conception to have. Then your conception of those effects is the whole of
your conception of the object". (Reveu philosophique VII, [CP, 5.18]).
The core of this principle is that the conception of an object constitutes its
conceivable practical effects, manifest in habits of action, as the following quote
suggests:
‘To develop its meaning, we have, therefore, simply to determine what habits it produces,
for what a thing means is simply what habits it involves. Now, the identity of a habit
depends on how it might lead us to act, not merely under such circumstances as are
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ABDUCTIVE REASONING
likely to arise, but under such as might possibly occur, no matter how improbable they
may be’. [CP, 5.400]
Peirce adds the following in regard to the notion of ‘conduct’:
“It is necessary to understand the word ‘conduct’, here, in the broadest sense. If, for
example, the predication of a given concept were to lead to our admitting that a given
form of reasoning concerning the subject of which it was affirmed was valid, when
it would not otherwise be valid, the recognition of that effect in our reasoning would
decidedly be a habit of conduct". [CP, 6.481].
“For the maxim of pragmatism is that a conception can have no logical effect or import
differing from that of a second conception except so far as, taken in connection with
other conceptions and intentions, it might conceivably modify our practical conduct
differently from that second conception". [CP, 5.196].
The method proposed by this maxim provides a regulative principle for the
evaluation of beliefs and serves as a guide for our actions, and in this respect
it is a normative principle. The pragmatic maxim becomes the rule to achieve
the main aim of the whole doctrine, namely to clarify ideas.
3.
Abduction and Epistemology
Abduction in Peirce
The notion of abduction in the work of Charles Peirce is entangled in many
aspects of his philosophy and is therefore not limited to the conception of
abduction as a logical inference of its own. The notions of logical inference and
of validity that Peirce puts forward go beyond our present understanding of what
logic is about. On the one hand, they are linked to his epistemology, a dynamic
view of thought as logical inquiry, and correspond to a deep philosophical
concern, that of studying the nature of synthetic reasoning.
The intellectual enterprise of Charles Sanders Peirce, in its broadest sense,
was to develop a semiotic theory, in order to provide a framework to give an
account for thought and language. With regard to our purposes, the fundamental
question Peirce addressed was how synthetic reasoning was possible2 . Very
much influenced by the philosophy of Immanuel Kant, Peirce’s aim was to
extend his categories and correct his logic:
“According to Kant, the central question of philosophy is ‘How are synthetical judgments a priori possible?’ But antecedently to this comes the question how synthetical
judgments in general, and still more generally, how synthetical reasoning is possible at
all. When the answer to the general problem has been obtained, the particular one will
be comparatively simple. This is the lock upon the door of philosophy". ([CP, 5.348],
quoted in [Hoo92], page 18).
2 Cf.
[FK00] for Peirce’s classification of inferences into analytic and synthetic.
Pragmatism
171
Peirce proposes abduction to be the logic for synthetic reasoning, that is, a
method to acquire new ideas. As already noted in chapter 2, he was the first
philosopher to give to abduction a logical form.
The development of a logic of inquiry occupied Peirce’s thought since the
beginning of his work. In the early years he thought of a logic composed of
three modes of reasoning: deduction, induction and hypothesis each of which
corresponds to a syllogistic form, illustrated by the following, often quoted
example [CP, 2.623]:
DEDUCTION
Rule.– All the beans from this bag are white.
Case.– These beans are from this bag.
Result.– These beans are white.
INDUCTION
Case.– These beans are from this bag.
Result.– These beans are white.
Rule.– All the beans from this bag are white.
HYPOTHESIS
Rule.– All the beans from this bag are white.
Result.– These beans are white.
Case.– These beans are from this bag.
Of these, deduction is the only reasoning which is completely certain, inferring its ‘Result’ as a necessary conclusion. Induction produces a ‘Rule’
validated only in the ‘long run’ [CP, 5.170], and hypothesis merely suggests
that something may be ‘the Case’ [CP, 5.171]. The evolution of his theory is
also reflected in the varied terminology he used to refer to abduction; beginning
with presumption and hypothesis [CP, 2.776,2.623], then using abduction and
retroduction interchangeably [CP, 1.68,2.776,7.97].
Later on, Peirce proposed these types of reasoning as the stages composing
a method for logical inquiry, of which abduction is the beginning:
“From its [abductive] suggestion deduction can draw a prediction which can be tested
by induction." [CP, 5.171].
Abduction plays a role in direct perceptual judgments, in which:
“The abductive suggestion comes to us as a flash" [CP, 5.181].
As well as in the general process of invention:
“It [abduction] is the only logical operation which introduces any new ideas" [CP,
5.171].
In all this, abduction is both “an act of insight and an inference" as has
been suggested in [And86]. These explications do not fix one unique notion.
Peirce refined his views on abduction throughout his work. He first identified
abduction with the syllogistic form above, to later enrich this idea by the more
general conception of:
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ABDUCTIVE REASONING
“the process of forming an explanatory hypothesis” [CP, 5.171].
And also referring to it as:
“The process of choosing a hypothesis” [CP, 7.219].
Something that suggests that he did not always distinguish clearly between
the construction and the selection of a hypothesis. In any case, this later view
gives place to the logical formulation of abduction ([CP, 5.189]), which we
reproduced again (cf. chapter 2) as follows:
The surprising fact, C, is observed.
But if A were true, C would be a matter of course.
Hence, there is reason to suspect that A is true.
We recall from chapter 2 that in addition to this formulation which makes up
the explanatory constituent of abductive inference, there are two other aspects to
consider for an explanatory hypothesis, namely its being testable and economic.
While the former sets up a requirement in order to give an empirical account of
the facts, the latter is a response to the practical problem of having innumerable
hypotheses to test and points to the need of having a criterion to select the best
explanation amongst the testable ones. We will return to the former additional
criterion later in the chapter when the connection to pragmatism is put forward.
Interpreting Peirce’s Abduction
The notion of abduction has puzzled Peirce scholars all along. Some have concluded that Peirce held no coherent view on abduction at all [Fra58], others
have tried to give a joint account with induction [Rei70] and still others claim it
is a form of inverted modus ponens [And86]. A more modern view is found in
[Kap90] who interprets Peirce’s abduction as a form of heuristics. An account
that tries to make sense of the two extremes of abduction, both as a guessing
instinct and as a rational activity is found in [Ayi74]. This last approach continues to present day. While [Deb97] proposes to reinterpret the concept of
rationality to account for these two aspects, [Gor97] shows abductive inference
in language translation, a process in which the best possible hypothesis is sought
using instinctive as well as rational elements of translation. Thus, abductive
inference is found in a variety of contexts. To explain abduction in perception,
[Roe97] offers a reinterpretation of Peirce’s abductive formulation, whereas
[Wir97] uses the notion of ‘abductive competence’ to account for language
interpretation.
This diversity suggests that Peirce recognized not only different types of
reasoning, but also several degrees within each one, and even merges between
the types. In the context of perception he writes:
Pragmatism
173
“The perceptual judgements, are to be regarded as extreme cases of abductive inferences” [CP, 5.181].
Abductory induction, on the other hand, is suggested when some kind of
guess work is involved in the reasoning [CP, 6.526]3 . Anderson [And87] also
recognizes several degrees in Peirce’s notion of creativity. A nice concise
account of the development of abduction in Peirce, which clearly distinguishes
three stages in the evolution of his thought is given in [Fan70]. Another key
reference on Peirce’s abduction, in its relation to creativity in art and science is
found in [And87]4 .
Peirce’s Epistemology
In Peirce’s epistemology, thought is a dynamic process, essentially an interaction between two states of mind: doubt and belief. While the essence
of the latter is the “establishment of a habit which determines our actions"
[CP, 5.388], with the quality of being a calm and satisfactory state in which
all humans would like to stay, the former “stimulates us to inquiry until it is
destroyed" [CP, 5.373], and it is characterized by being a stormy and unpleasant
state from which every human struggles to be freed:
“The irritation of doubt causes a struggle to attain a state of belief". [CP, 5.374].
Peirce speaks of a state of belief and not of knowledge. Thus, the pair ‘doubtbelief’ is a cycle between two opposite states. While belief is a habit, doubt
is its privation. Doubt, however, is not a state generated at will by raising a
question, just as a sentence does not become interrogative by putting a special
mark on it, there must be a real and genuine doubt:
“Genuine doubt always has an external origin, usually from surprise; and that it is as
impossible for a man to create in himself a genuine doubt by such an act of the will as
would suffice to imagine the condition of a mathematical theorem, as it would be for
him to give himself a genuine surprise by a simple act of the will". ([CP, 5.443]).
Moreover, it is surprise what breaks a habit:
“For belief, while it lasts, is a strong habit, and as such, forces the man to believe until
some surprise breaks up the habit". ([CP, 5.524], my emphasis).
And Peirce distinguishes two ways to break a habit:
“The breaking of a belief can only be due to some novel experience" [CP, 5.524] or
“...until we find ourselves confronted with some experience contrary to those expectations." ([CP, 7.36], my emphasis).
3 Peirce
further distinguishes three kinds of induction [CP, 2.775,7.208], and even two kinds of deduction
[CP, 7.224].
4 This multiplicity returns in artificial intelligence. [Fla96b] suggests that some confusions in modern accounts of abduction in AI can be traced back to Peirce’s two theories of abduction: the earlier syllogistic one
and the later inferential one. As to more general semiotic aspects of Peirce’s philosophy, another proposal
for characterizing abduction in AI is found in [Kru95].
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ABDUCTIVE REASONING
Peirce’s epistemic model proposes two varieties of surprise as the triggers for
every inquiry, which we relate to the previously proposed novelty and anomaly
(cf. chapter 2). Moreover, these are naturally related to the epistemic operations
for belief change in AI (cf. [Ali00b] and chapter 8).
4.
Pragmatism Revisited
A method of inquiry aiming to clarify ideas, by giving meanings to objects
according to those habits of action involved, is what makes pragmatism an active
philosophical method of reflexion. Pragmatism puts forward an epistemic aim
with an experimental solution, precisely providing an ‘unseparable connection
between rational cognition and rational purpose’ [CP, 5.412].
The nature of this method is logical and not metaphysical: “I make pragmatism to be a mere maxim of logic instead of a sublime principle of speculative
philosophy" [CP, 7.220]. Moreover, the method itself is guided by ‘the pragmatic maxim’, the underlying principle by which a belief is fixed; it is a rule to
achieve the highest degree of clarity in the apprehension of ideas. These are the
motivations and aims of pragmatism; being abduction the underlying logic of
this process: “if you carefully consider the question of pragmatism you will see
that it is nothing else than the question of the logic of abduction” [CP, 4.196].
Abduction is then proposed as an epistemic notion as well as a pragmatic
one. How can this be possible? As previously mentioned (cf. chapter 2) for a
hypothesis to be considered an abductive explanation it should be explanatory,
testable and economic. Abduction has indeed a logical formulation, which
shapes its explanatory aspect, characterizing the argumentative format that all
candidates for abductive hypotheses should observe. It is the first of these
aspects what is related to his epistemology. In particular, our previous analysis
was centered on the role played by the element of surprise in this formulation;
and on its connection to the epistemic transition between the states of doubt
and belief. The second of these aspects is what is relevant to the connection to
pragmatism, for this doctrine provides a maxim which precisely characterizes
what is to count as an explanatory hypothesis based on its being subject to
experimental verification.
Therefore, in order to understand the relationship between abduction and
pragmatism we must go beyond the logical form of abduction in two respects.
On the one hand, abduction must be understood as an epistemic process for the
acquisistion of belief and on the other hand, we must take into consideration
the experimental requirement. In what follows we will present the connections
between abduction and epistemology, followed by the connections between
each of these with pragmatism.
Pragmatism
175
Abduction and Epistemology
The connection between abductive logic and the epistemic transition between
the mental states of doubt and belief is clearly seen in the fact that the surprise
is both the trigger of abductive reasoning –as indicated by the first premise of
the logical formulation of abduction– as well as that of the doubt state when an
belief habit has been broken.
The overall cognitive process showing abductive inference as an epistemic
process can be depicted as follows: a novel or an anomalous experience (cf.
chapter 2) gives place to a surprising phenomenon, generating an state of doubt
which breaks up a belief habit and accordingly triggers abductive reasoning.
The goal of this type of reasoning is precisely to explain the surprising fact
and therefore soothe the state of doubt. It is ‘soothe’ rather than ‘destroy’ for
an abductive hypothesis has to be put into test and be economic, attending to
the further criteria Peirce proposed. The abductive explanation is simply a
suggestion that has to be put to test before converting itself into a belief.
Epistemology and Pragmatism
Two texts concerned with inquiry mark the first places in which Peirce
presents the ideas which shape the doctrine of pragmatism, namely ‘The fixation
of belief’ (1877, [CP, 5.358–387]) and ‘How to make our ideas clear’ (1878,
[CP, 5.388–5.410]). The first of these texts, presents the end of inquiry as the
settlement of opinion as well as the methods for fixing a belief. It worships a
logical method over the methods based on an appeal to authority, on tenacity
and on the a priori: ‘the very first lesson that we have a right to demand that
logic shall teach us is, how to make our ideas clear’ [CP, 5.393]. The primary
function of thought is to produce a belief:
‘The action of thought is excited by the irritation of doubt, and ceases when belief is
attained; so that the production of belief is the sole function of thought’.[CP, 5.394].
The second text reiterates the importance of a method of a logical kind and
introduces the pragmatic maxim, as the guiding principle to achieve the highest
degree of clarity in the apprehension of ideas: ‘Our idea of anything is our
idea of its sensible effects’[CP, 5.401]. The effects of our conceptions must
be produced in order to develop their meaning, and they become alive through
habits of action. In fact, a late definition of pragmatism given by Peirce, reads
as follows:
‘What the true definition of Pragmatism may be, I find it very hard to say; but in my
nature it is a sort of instinctive attraction for living facts’.[CP, 5.64]
Abduction and Pragmatism
In regard to the connection between abduction and pragmatism, we find
in Peirce’s writings notes for a conference “Pragmatism – Lecture VII–" (of
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ABDUCTIVE REASONING
which there is evidence it was never delivered). This conference is composed
by four sections5 , of which only the third one, “Pragmatism – The Logic of
Abduction" (cf. [CP, 5.195] – [CP, 5.206]) becomes relevant to our discussion.
In this section of the conference, Peirce states that the question of pragmatism
is nothing else than the logic of abduction, as suggested by the following quote:
“Admitting, then, that the question of Pragmatism is the question of Abduction, let us
consider it under that form. What is good abduction? What should an explanatory
hypothesis be to be worthy to rank as a hypothesis? Of course, it must explain the
facts. But what other conditions ought it to fulfill to be good? The question of the
goodness of anything is whether that thing fulfills its end. What, then, is the end of an
explanatory hypothesis? Its end is, through subjection to the test of experiment, to lead to
the avoidance of all surprise and to the establishment of a habit of positive expectation
that shall not be disappointed. Any hypothesis, therefore, may be admissible, in the
absence of any special reasons to the contrary, provided it be capable of experimental
verification, and only insofar as it is capable of such verification. This is approximately
the doctrine of pragmatism. But just here a broad question opens out before us. What
are we to understand by experimental verification? The answer to that involves the
whole logic of induction". [CP, 5.198].
Peirce puts forward the pragmatic method as that providing a maxim that
completely characterizes the admissibility of explanatory hypotheses. On the
one hand, it is required that every hypothesis is subject to experimental corroboration (verification in Peircean terms) and on the other hand, this corroboration
is manifest in habits of conduct which eliminate the surprise of doubt in question.
Another aspect of equal importance, but apparently outside the boundaries
of the logic of abduction, is concern with the very notion of experimental
corroboration, which according to the preceding text, Peirce poses in the logic
of induction. Moreover, this aspect may go beyond empirical experimentation:
“If pragmatism is the doctrine that every conception is a conception of conceivable
practical effects, it makes conception reach far beyond the practical. It allows for
any flight of imagination, provided this imagination ultimately alights upon a possible
practical effect; and thus many hypothesis may seem at first glance to be excluded by
the pragmatical maxim that are really not excluded". [CP, 5.196].
Besides the suggestion in this quote that experimental corroboration does not
limit itself to merely empirical experimentation, for it gives place to thought
experiments and other manifestations in the realm of ideas, and it also suggests
that the process of conceiving the possible effects of a certain conception already involves the mere action of experimentation, for in the calculation of the
practical effects manifest in the habits of conduct that a certain explanatory hypothesis produces, we are already in the territory of experimental corroboration,
being these found in the practical realm or in the world of ideas.
5 The
section titles are the following ones: “The Three Cotary Propositions", “Abduction and Perceptual
Judgments", “Pragmatism – the Logic of Abduction" and “The Two Functions of Pragmatism".
Pragmatism
5.
177
Discussion and Conclusions
In this chapter we analyzed the doctrine of pragmatism in the work of Charles
Peirce, as a philosophical method of reflexion guided by the pragmatic maxim,
the underlying principle by which a belief is fixed, generating habits of action.
The outcome of our analysis shows that pragmatism is closely connected with
Peirce’s abduction. A direct link is found in the requirement regarding experimental corroboration for explanatory hypotheses, those which comply with the
logical formulation of abduction. The experimental corroboration requirement
raises the significance of the pragmatic dimension of the logic of abduction and
gives an answer to the hypotheses selection problem of those that are explanatory. For Peirce, the experimental corroboration of explanatory hypotheses goes
beyond verification, as it requires of a calculation of its possible consequences
of effects; those that produce new habits of conduct, being these epistemic or
practical.
A natural consequence of this analysis for abduction, is that the interpretation
of Peirce’s formulation goes beyond that of a logical argument to become an
epistemic process which may be described as follows: a novel or an anomalous
fact gives place to a surprising fact, generating a state of doubt. Therefore,
abductive reasoning is triggered, which consists on explaining the surprising fact
by providing an explanatory hypothesis that plays the role of a new belief. And
this state will remain as such until another fact is encountered, thus continuing
the epistemic doubt–belief cycle.
It is appealing that this view has close connections with theories of belief
revision in artificial intelligence6 . It also involves extending the traditional
view of abduction in AI, to include cases in which the observation is in conflict
with the theory, as we have suggested in our taxonomy (cf. chapter 2.).
6 In
the case of an abductive novelty, the explanation is assimilated into the theory by the operation of
expansion. In the case of an abductive anomaly, the operation of revision is needed to modify the theory and
incorporate the explanation. Cf. chapter 8 for a detailed analysis of this connection.
Chapter 8
EPISTEMIC CHANGE
1.
Introduction
Notions related to explanation have also emerged in theories of belief change
in AI. One does not just want to incorporate new beliefs, but often also, to
justify them. The main motivation of these theories is to develop logical and
computational mechanisms to incorporate new information to a scientific theory,
data base or set of beliefs. Different types of change are appropriate in different
situations. Indeed, the pioneering work of Carlos Alchourrón, Peter Gärdenfors
and David Makinson (often referred as the AGM approach) [AGM85], proposes
a normative theory of epistemic change characterized by the conditions that a
rational belief change operator should satisfy.
Our discussion of epistemic change is in the same spirit, taking a number of
cues from their analysis. We will concentrate on belief revision, where changes
occur only in the theory. The situation or world to be modelled is supposed
to be static, only new information is coming in. (The other type of epistemic
change in AI which accounts for a changing world is called update, which we
will briefly discuss at the end of the chapter.)
This chapter is naturally divided into four parts. After this introduction, in
which we give a brief introduction of abduction as a process of belief revision, in the second part (section 2) we review theories of belief revision in AI
and then propose abduction as belief revision, relating our previous ‘abductive
triggers’ (chapter 2) with the epistemic attitudes in the belief revision theories
and defining operations for abductive expansion, revision and contraction of
their own. Finally, we show that abduction as a theory of epistemic change
is both committed to the foundationalist and the coherentist epistemological
stances of belief revision theories. In the third part (section 3), we propose an
implementation in tableaux of abduction as a process of belief revision. We
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ABDUCTIVE REASONING
extend our previous analysis of chapter 4 as to include contraction and revision
operations over tableaux. We propose two different strategies for contraction
(without committing ourselves to any one of them) and outline procedures for
computing explanations. In the fourth and final part of this chapter (section 4),
we offer an analysis of previous sections with respect both to the relation of
abduction to belief revision and to the algorithmic sketch we presented previously. We then put forward our conclusions and briefly mention some other
related work in this direction.
Generally speaking, this chapter shows that the notion of abduction goes
beyond that of logical inference and that it may indeed be interpreted as a process
for belief revision, with clear connections with the existing literature in this
field. It also shows that semantic tableaux may have further uses implementing
contraction and revision operations, which go way beyond the standard ones.
It also involves extending the traditional view of abduction in AI, to include
cases in which the observation is in conflict with the theory, as suggested in our
taxonomy in chapter 2.
2.
Abduction as Epistemic Change
Theories of Belief revision in AI
We shall expand on the brief introduction given in chapter 2, highlighting
aspects that distinguish different theories of belief revision. This sets the scene
for approaching abduction as a similar enterprise1 .
The basic elements of this theory are the following. Given a consistent theory
Θ closed under logical consequence, called the belief state, and a sentence ϕ,
the incoming belief, there are three epistemic attitudes for Θ with respect to ϕ:
either ϕ is accepted (ϕ ∈ Θ), ϕ is rejected (¬ϕ ∈ Θ), or ϕ is undetermined
(ϕ ∈ Θ, ¬ϕ ∈ Θ). Given these attitudes, the following operations characterize
the kind of belief change ϕ brings into Θ, thereby effecting an epistemic change
in the agent’s currently held beliefs:
Expansion
A new sentence is added to Θ regardless of the consequences of the larger
set to be formed. The belief system that results from expanding Θ by a
sentence ϕ together with the logical consequences is denoted by Θ + ϕ.
Revision
A new sentence that is (typically) inconsistent with a belief system Θ is
added, but in order that the resulting belief system be consistent, some of
1 The
material of this section is mainly based on [Gar92], with some modifications taken from other approaches. In particular, in our discussion, belief revision operations are not required to handle incoming
beliefs together with all their logical consequences.
Epistemic Change
181
the old sentences in Θ are deleted. The result of revising Θ by a sentence ϕ
is denoted by Θ ∗ ϕ.
Contraction
Some sentence in Θ is retracted without adding any new facts. In order to
guarantee the deductive closure of the resulting system, some other sentences
of Θ may be given up. The result of contracting Θ with respect to sentence
ϕ is denoted by Θ − ϕ.
Of these operations, revision is the most complex one. Indeed the three
belief change operations can be reduced into two of them, since revision and
contraction may be defined in terms of each other. In particular, revision here
is defined as a composition of contraction and expansion: first contract those
beliefs of Θ that are in conflict with ϕ, and then expand the modified theory with
sentence ϕ (known as ‘Levi’s identity’). While expansion can be uniquely and
easily defined (Θ + ϕ = {α | Θ ∨ {ϕ} ⊢ α}), this is not so with contraction or
revision, as several formulas can be retracted to achieve the desired effect. These
operations are intuitively non-deterministic. A simple example (cf. example 3
from chapter 2) to illustrate this point is the following:
Θ: r, r → w.
ϕ: ¬w.
In order to incorporate ϕ into Θ and maintain consistency, the theory must be
revised. But there are two possibilities for doing this: deleting either of r → w
or r allows us to then expand the contracted theory with ¬w consistently. Several
formulas can be retracted to achieve the desired effect, thus it is impossible to
state in purely logical or set-theoretical terms which of these is to be chosen.
Therefore, an additional criterion must be incorporated in order to fix which
formula to retract. Here, the general intuition is that changes on the theory
should be kept ‘minimal’, in some sense of informational economy2 .
Moreover, epistemic theories in this tradition observe certain ‘integrity constraints’, which concern the theory’s preservation of consistency, its deductive
closure and two criteria for the retraction of beliefs: the loss of information
should be kept minimal and the less entrenched beliefs should be removed first.
These are the very basics of the AGM approach. In practice, however, fullfledged systems of belief revision can be quite diverse. They differ in at least
three aspects: (a) belief state representation, (b) characterization of the operations of epistemic change (via postulates or constructively), and (c) epistemological stance.
2 Various
ways of dealing with this issue occur in the literature. I mention only that in [Gar88]. It is based
on the notion of entrenchment, a preferential ordering which lines up the formulas in a belief state according
to their importance. Thus, we may retract those formulas that are the ‘least entrenched’ first. For a more
detailed reference as to how this is done exactly, see [GM88].
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ABDUCTIVE REASONING
Regarding the first aspect (a), we find there are essentially three ways in
which the background knowledge Θ is represented: (i) belief sets, (ii) belief
bases, or (iii) possible world models. A belief set is a set of sentences from a
logical language L closed under logical consequence. In this classical approach,
expanding or contracting a sentence in a theory is not just a matter of addition
and deletion, as the logical consequences of the sentence in question should
also be taken into account. The second approach emerged in reaction to the
first. It represents the theory Θ as a base for a belief set BΘ , where BΘ is
a finite subset of Θ satisfying Cons(BΘ ) = Θ. (That is, the set of logical
consequences of BΘ is the classical belief state). The intuition behind this
is that some of the agent’s beliefs have no independent status, but arise only
as inferences from more basic beliefs. Finally, the more semantic approach
(iii) moves away from syntactic structure, and represents theories as sets WΘ
of possible worlds (i.e., their models). Various equivalences between these
approaches have been established in the literature (cf. [GR95]).
As for the second aspect (b), operations of belief revision can be given
either ‘constructively’ or merely via ‘postulates’. The former approach is more
appropriate for algorithmic models of belief revision, the latter serves as a
logical description of the properties that any such operations should satisfy.
The two can also be combined. An algorithmic contraction procedure may be
checked for correctness according to given postulates. (Say, one which states
that the result of contracting Θ with ϕ should be included in the original state
(Θ − ϕ ⊆ Θ.)).
The last aspect (c) concerns the epistemic quality to be preserved. While
the foundationalists argue that beliefs must be justified (with the exception of a
selected set of ‘basic beliefs’), the coherentists consider it a priority to maintain
the overall coherence of the system and reject the existence of basic beliefs.
Therefore, each theory of epistemic change may be characterized by its
representation of belief states, its description of belief revision operations, and
its stand on the main properties of sets of beliefs one should be looking for.
These choices may be interdependent. Say, a constructive approach might favor
a representation by belief bases, and hence define belief revision operations on
some finite base, rather than the whole background theory. Moreover, the
epistemological stance determines what constitutes rational epistemic change.
The foundationalist accepts only those beliefs that are justified in virtue of other
basic beliefs, thus having an additional challenge of computing the reasons for
an incoming belief. On the other hand, the coherentist must maintain coherence,
and hence make only those minimal changes that do not endanger (at least)
consistency (however, coherence need not be identified with consistency).
In particular, the AGM paradigm represents belief states as sets (in fact, as
theories closed under logical consequence), provides ‘rationality postulates’ to
characterize the belief revision operations (cf. postulates for contraction at the
Epistemic Change
183
end of the chapter), and finally, it advocates a coherentist view. The latter is
based on the empirical claim that people do not keep track of justifications for
their beliefs, as some psychological experiments seem to indicate [Har65].
Abduction as Belief Revision
Abductive reasoning may be seen as an epistemic process for belief revision. In this context an incoming sentence ϕ is not necessarily an observation,
but rather a belief for which an explanation is sought. Existing approaches to
abduction usually do not deal with the issue of incorporating ϕ into the set of
beliefs. Their concern is just how to give an account for ϕ. If the underlying
theory is closed under logical consequence, however, then ϕ should be automatically added once we have added its explanation (which a foundationalist
would then keep tagged as such).
In AI, practical connections of abduction to theories of belief revision have
often been noted. But these in general use abduction to determine explanations
of incoming beliefs or as an aid to perform the epistemic operations for belief revision. Of many references in the literature, we mention [Wil94] (which studies
the relationship between explanations based on abduction and ‘Spohnian reasons’) and [AD94](which uses abductive procedures to realize contractions over
theories with ‘immutability conditions’). Some work has been done relating
abduction to knowledge assimilation [KM90], in which the goal is to assimilate
a series of observations into a theory maintaining its consistency.
Our claim will be stronger. Abduction can function in a model of theory revision as a means to determine explanations for incoming beliefs. But also more
generally, abductive reasoning itself provides a model for epistemic change.
Let us discuss some reasons for this, recalling our architecture of chapter 2.
First, what were called the two ‘triggers’ for abductive reasoning correspond
to the two epistemic attitudes of a formula being undetermined or rejected. We
did not consider accepted beliefs, since these do not call for explanation.
ϕ is a novelty (Θ |= ϕ, Θ |= ¬ϕ): ϕ is undetermined.
ϕ is an anomaly (Θ |= ϕ, Θ |= ¬ϕ: ϕ is rejected.
ϕ is an accepted belief (Θ |= ϕ).
The epistemic attitudes are presented in [Gar88] in terms of membership (e.g.,
a formula ϕ is accepted if ϕ ∈ Θ). We defined them in terms of entailment,
since our theories are not closed under logical consequence.
Our main concern is not the incoming belief ϕ itself. We rather want to
compute and add its explanation α. But since ϕ is a logical consequence of the
revised theory, it could easily be added. Thus, as we shall see, abduction as
epistemic change may be described by two operations: either as an (abductive)
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ABDUCTIVE REASONING
expansion, where the background theory gets extended to account for a novel
fact, or as an (abductive) revision, in which the theory needs to be revised to
account for an anomalous fact. Belief revision theories provide an explicit
calculus of modification for both cases and indeed serve as a guide to define
abductive operations for epistemic change. In AI, characterizing abduction via
belief operators goes back to Levesque [Lev89]. More recently, other work
has been done in this direction ([Pag96], [LU96]). As we shall see, while the
approaches share the same intuition, they all differ in implementation.
In philosophy, the idea of abduction as epistemic change is already present in
Peirce’s philosophical system (cf. chapter 7), in the connection between abduction and the epistemic transition between the mental states of doubt and belief.
It shows itself very clearly in the fact that surprise is both the trigger of abductive reasoning, —as indicated by the first premise of the logical formulation–,
as well as the trigger of the state of doubt, when a belief habit has been broken. Abductive reasoning is a process by which doubts are transformed into
beliefs, since an explanation for a surprising fact is but a suggestion to be tested
thereafter. Moreover, one aspect of abduction is related to the “Ramsey test"
[Ram00]: given a conditional sentence α → ϕ, α is a reason for ϕ iff revising
your current beliefs by α causes ϕ to be believed. This test has been the point
of departure of much epistemological work both in philosophy and in AI.
Next we propose two abductive epistemic operations for the acquisition of
knowledge, following those in the AGM theory while being faithful to our
interpretation of Peirce’s abduction and proposed taxonomy. Then, a discussion
on abductive epistemic theories, in which we sketch ours and compare it to other
work.
Abductive Operations for Epistemic Change
The previously defined abductive novelty and abductive anomaly correspond
respectively, to the AGM epistemic attitudes of undetermination and rejection
(provided that ⇒ is ⊢ and Θ closed under logical consequence).
In our account of abduction, both a novel phenomenon and an anomalous
one induce a change in the original theory. The latter calls for a revision and the
former for expansion. So, the basic operations for abduction are expansion and
revision. Therefore, two epistemic attitudes and changes in them are reflected
in an abductive model.
Here, then, are the abductive operations for epistemic change:
Abductive Expansion
Given an abductive novelty ϕ, a consistent explanation α for ϕ is computed
in such a way that Θ, α ⇒ ϕ, and then added to Θ.
Epistemic Change
185
Abductive Revision
Given an abductive anomaly ϕ, a consistent explanation α is computed as
follows: the theory Θ is revised into Θ′ so that it does not explain ¬ϕ. That
is, Θ′ ⇒ ¬ϕ, where Θ′ = Θ − (β1 , . . . , βl )3 .
Once Θ′ is obtained, a consistent explanation α is calculated in such a way
that Θ′ , α ⇒ ϕ and then added to Θ.
Thus, the process of revision involves both contraction and expansion.
In one respect, these operations are more general than their counterparts in the
AGM theory, since incoming beliefs are incorporated into the theory together
with their explanation (when the theory is closed under logical consequence).
But in the type of sentences to accept, they are more restrictive. Given that in
our model non-surprising facts (where Θ ⇒ ϕ) are not candidates for being
explained, abductive expansion does not apply to already accepted beliefs, and
similarly, revision only accepts rejected facts. Other approaches however, do
not commit themselves to the preconditions of novelty and anomaly that we
have set forward. Pagnucco’s abductive expansion [Pag96] is defined for an
inconsistent input, but in this case the resulting state stays the same. Lobo
and Uzcátegui’s abductive expansion [LU96] is even closer to standard AGM
expansion; it is in fact the same when every atom is “abducible".
Abductive Epistemic Theories
Once we interpret abductive reasoning as a model for epistemic change, in
lines of those proposed in AI, the next question is: what kind of theory is an
abductive epistemic theory?
As we saw previously, there are several choices for representing belief states,
for characterizing the operations for epistemic change and finally, an epistemic
theory adheres to either the foundationalist or the coherentist trend. The predominant line of what we have examined so far, is to stay close to the AGM
approach. That is, to represent belief states as sets (in fact as closed theories), and to characterize the abductive operations of extension and revision
through definitions, rationality postulates and a number of constructions motivated by those for AGM contraction and revision. As for epistemological
stance, Pagnucco is careful to keep his proposal away from being interpreted
as foundationalist; he thinks that having a special set of beliefs like the “abducibles", as found in [LU96], is “against the coherentist spirit of the AGM"
[Pag96, page 174].
3 In
many cases, several formulas and not just one must be removed from the theory. The reason is that sets
of formulas which entail (explain) ϕ should be removed. E.g., given Θ = {α → β, α, β} and ϕ = ¬β, in
order to make Θ, ¬β consistent, one needs to remove either {β, α} or {β, α → β}.
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ABDUCTIVE REASONING
As for epistemological stance, here is our position. The main motivation for
an abductive epistemic theory is to incorporate an incoming belief together with
its explanation, the belief (or set of) that justifies it. This fact places abduction
close to the above foundationalist line, which requires that beliefs are justified
in terms of other basic beliefs. Often, abductive beliefs are used by a (scientific) community, so the earlier claim that individuals do not keep track of the
justifications of their beliefs [Gar88] does not apply. On the other hand, an important feature of abductive reasoning is maintaining consistency of the theory.
Otherwise, explanations would be meaningless (especially if ⇒ is interpreted
as classical logical consequence. C.f. chapter 3). Therefore, abduction is committed to the coherentist approach as well. This is not a case of opportunism.
Abduction rather demonstrates that the earlier philosophical stances are not
incompatible. Indeed, [Haa93] argues for an intermediate stance of ‘foundherentism’. Combinations of foundationalist and coherentist approaches are also
found in the AI literature [Gal92].
In our view, an abductive theory for epistemic change, which aims to model
Peirce’s abduction, naturally calls for a procedural approach. It should produce
explanations for surprising phenomena and thus transform a state of doubt
into one of belief. The AGM postulates describe expansions, contractions and
revisions as epistemic products rather than processes in their own right. Their
concern is with the nature of epistemic states, not with their dynamics. This
gives them a ‘static’ flavour, which may not always be appropriate.
Therefore, in the following, we aim at giving a constructive model in which
abduction is an epistemic activity.
3.
Semantic Tableaux Revisited:
Toward An Abductive Model for Belief Revision
The combination of stances that we just described naturally calls for a procedural approach to abduction as an activity. But then, the same motivations that
we gave in chapter 4 apply. Semantic tableaux provided an attractive constructive representation of theories, and abductive expansion operations that work
over them. So, here is a further challenge for this framework. Can we extend
our abductive tableau procedures to also deal with revision?
What we need for this purpose is an account of contraction on tableaux.
Revision will then be forthcoming through combination with expansion, as has
been mentioned before.
Revision in Tableaux
Our main idea is extremely straightforward. In semantic tableaux, contraction
of a theory Θ, so as to give up some earlier consequences, translates into the
opening of a closed branch of T (Θ). Let us explain this in more detail for the
Epistemic Change
187
case of revision. The latter process starts with Θ, ϕ for which T (Θ∪ϕ) is closed.
In order to revise Θ, the first goal is to stop ¬ϕ from being a consequence of
Θ. This is done by opening a closed branch of T (Θ) not closed by ϕ, thus
transforming it into T (Θ’). This first step solves the problem of retracting
inconsistencies. The next step is (much as in chapter 4) to find an explanatory
formula α for ϕ by extending the modified Θ’ as to make it entail ϕ. Therefore,
revising a theory in the tableau format can be formulated as a combination of
two equally natural moves, namely, opening and closing of branches:
Given Θ, ϕ for which T (Θ ∪ ϕ) is closed, α is an abductive explanation if
1. There is a set of formulas β1 , . . . , βl (βi ∈ Θ) such that
T (Θ ∪ ϕ) − (β1 , . . . , βl ) is open.
Moreover, let Θ1 = Θ − (β1 , . . . , βl ). We also require that
2. T ((Θ1 ∪ ¬ϕ) ∪ α) is closed.
How to implement this technically? To open a tableau, it may be necessary
to retract several formulas β1 , . . . , βl and not just one (cf. previous footnote).
The second item in this formulation is precisely the earlier process of abductive
extension, which has been developed in chapter 4. Therefore, from now on we
concentrate on the first point of the above process, namely, how to contract a
theory in order to restore consistency.
Our discussion will be informal. Through a series of examples, we discover
several key issues of implementing contraction in tableaux. We explore some
complications of the framework itself, as well as several strategies for restoring consistency, and the effects of these in the production of explanations for
anomalous observations.
Contraction in Tableaux
The general case of contraction that we shall need is this. We have a currently
inconsistent theory, of which we want to retain some propositions, and from
which we want to reject some others. In the above case, the observed anomalous phenomenon was to be retained, while the throwaways were not given
in advance, and must be computed by some algorithm. We start by discussing
the less constrained case of any inconsistent theory, seeing how it may be made
consistent through contraction, using its semantic tableau as a guide.
As is well–known, a contraction operation is not uniquely defined, as there
may be several options for removing formulas from a theory Θ so as to restore
consistency. Suppose Θ = {p ∧ q, ¬p}. We can remove either p ∧ q or ¬p – the
choice of which depends, as we have noticed, on preferential criteria aiming at
performing a ‘minimal change’ over Θ.
We start by noting that opening a branch may not suffice for restoring consistency. Consider the following example.
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ABDUCTIVE REASONING
Example 1
Let Θ = {p ∧ q, ¬p, ¬q}
T (Θ)
p∧q
p
q
¬p
By removing ¬p the closed branch is opened. However, note that this is
not sufficient to restore consistency in Θ because ¬q was never incorporated
to the tableau! Thus, even upon removal of ¬p from Θ, we have to ‘recompute’ the tableau, and we will find another closure, this time, because of ¬q.
This phenomenon reflects a certain design decision for tableaux, which seemed
harmless as long as we are merely testing for standard logical consequence.
When constructing a tableau, as soon as a literal ¬l may close a branch (i.e.,
l appears somewhere higher up; or vice versa) it does so, and no formula is
added thereafter. Therefore, when opening a branch we are not sure that all
formulas of the theory are represented on it. Thus, considerable reconfiguration
(or even total reconstruction) may be needed before we can decide that a tableau
has ‘really’ been opened. Of course (for our purposes), we might change the
format of tableaux, and compute closed branches ‘beyond inconsistency’, so as
to make all sources of closure explicit.
‘Recomputation’ is a complication arising from the specific tableau framework that we use, suggesting that we need to do more work in this setting than
in other approaches to abduction. Moreover, it also illustrates that ‘hidden conventions’ concerning tableau construction may have unexpected effects, once
we use tableaux for new purposes, beyond their original motivation. Granting
all this, we feel that such phenomena are of some independent interest, and we
continue with further examples demonstrating what tableaux have to offer for
the study of contraction and restoring consistency.
Global and Local: Strategies for Contraction
Consider the following variant of our preceding example:
189
Epistemic Change
Example 2
Let Θ = {p ∧ q, ¬p}
T (Θ)
p∧q
p
q
¬p
In this tableau, we can open the closed branch by removing either ¬p or p.
However, while ¬p is indeed a formula of Θ, p is not. Here, if we follow standard
accounts of contraction in theories of belief revision, we should trace back the
Θ-source of this subformula (p ∧ q in this case) and remove it. But tableaus
offer another route. Alternatively, we could explore ‘removing subformulas’
from a theory by merely modifying their source formulas, as a more delicate
kind of minimal change. These two alternatives suggest two strategies for
contracting theories, which we label global and local contraction, respectively.
Notice, in this connection, that each occurrence of a formula on a branch has a
unique history leading up to one specific corresponding subformula occurrence
in some formula from the theory Θ being analyzed by the tableau. The following
illustrates what each strategy outputs as the contracted theory in our example:
Global Strategy
Branch–Opening = {¬p, p}
(i) Contract with ¬p:
¬p corresponds to ¬p in Θ
Θ’=Θ − {¬p} = {p ∧ q}
(ii) Contract with p:
p corresponds to p ∧ q in Θ
Θ’ = Θ − {p ∧ q} = {¬p}
Local Strategy
Branch–Opening
= {¬p, p}
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ABDUCTIVE REASONING
(i) Contract with ¬p
Replace in the branch all connected occurrences of ¬p (following its upward
history) by the atom ‘true’: T .
T (Θ)
p∧q
p
q
T
Θ’
= Θ − {¬p} = {p ∧ q, T }
(ii) Contract with p:
Replace in the branch all connected occurrences of p (following its history)
by the atom true: T .
T (Θ)
T ∧q
T
q
¬p
Θ’
=
Θ − {p} = {T ∧ q, ¬p}
Here, we have a case in which the two strategies differ. When contracting
with p, the local strategy gives a revised theory (which is equivalent to {q, ¬p})
with less change than the global one. Indeed, if p is the source of inconsistency,
191
Epistemic Change
why remove the whole formula p ∧ q when we could modify it by T ∧ q?
This simple example shows that ‘removing subformulas’ from branches, and
modifying their source formulas, gives a more minimal change than removing
the latter.
However, the choice is often less clear-cut. Sometimes the local strategy
produces contracted theories that are logically equivalent to their globally contracted counterparts. Consider the following illustration (a variation on example
3 from chapter 2).
Example 3:
Θ = {r → l, r, ¬l}.
T (Θ ∪ ϕ)
r→l
r
¬r
l
¬l
Again, we briefly note the obvious outcomes of both local and global contraction strategies.
Global Strategy
Left Branch–Opening
=
{r, ¬r}
(i) Contract with r:
Θ’
=
Θ − {r} = {r → l, ¬l}
(ii) Contract with ¬r:
Θ’
=
Θ − {r → l} = {r, ¬l}
Right Branch–Opening
= {¬l, l}
(i) Contract with l:
Θ’
=
Θ − {r → l} = {r, ¬l}
(ii) Contract with ¬l:
Θ’
= Θ − {¬l} = {r, r → l}
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ABDUCTIVE REASONING
Local Strategy
Left Branch–Opening
=
{r, ¬r}
(i) Contract with r:
Θ’
=
Θ − {r} = {r → l, ¬l}
(ii) Contract with ¬r:
Θ’
=
{T ∨ l, r, ¬l}.
Right Branch–Opening
=
{r, ¬r}
(i) Contract with l:
Θ’
=
{¬r ∨ T, r, ¬l}
(ii) Contract with ¬l:
Θ’
=
Θ − {¬l} = {r, r → l}
Now, the only deviant case in the local strategy is this. Locally contracting
Θ with ¬r makes the new theory {T ∨ l, r, ¬l}. Given that the first formula is
a tautology, the output is logically equivalent to its global counterpart {r, ¬l}.
Therefore, modifying versus deleting conflicting formulae makes no difference
in this whole example.
A similar indifference shows up in computations with simple disjunctions,
although more complex theories with disjunctions of conjunctions may again
show differences between the two strategies. We refrain from spelling out these
examples here, which can easily be supplied by the reader. Also, we leave the
exact domain of equivalence of the two strategies as an open question. Instead,
we survey a useful practical case, again in the form of an example.
Computing Explanations
Let us now return to abductive explanation for an anomaly. We want to keep
the latter fixed in what follows (it is precisely what needs to be accommodated),
modifying merely the background theory. As it happens, this constraint involves
just an easy modification of our contraction procedure so far.
Example 4
Θ = {p ∧ q → r, p ∧ q},
ϕ = ¬r
There are five possibilities on what to retract (¬r does not count since it is
the anomalous observation). In the following descriptions of ‘local output’,
note that a removed literal ¬l will lead to a substitution of F (the ‘falsum’) for
its source l in a formula of the input theory Θ.
Contracting with p:
Global Strategy: Θ’={p ∧ q → r}
Local Strategy: Θ’={p ∧ q → r, T ∧ q}
Epistemic Change
193
Contracting with ¬p:
Global Strategy: Θ’={p ∧ q}
Local Strategy: Θ’={F ∧ q → r, p ∧ q}
Contracting with q:
Global Strategy: Θ’={p ∧ q → r}
Local Strategy: Θ’={p ∧ q → r, p ∧ T }
Contracting with ¬q:
Global Strategy: Θ’={p ∧ q}
Local Strategy: Θ’={p ∧ F → r, p ∧ q}
Contracting with r:
Global Strategy: Θ’={p ∧ q}
Local Strategy: Θ’={p ∧ q → T, p ∧ q}.
A case in which the revised theories are equivalent is when contracting with
r, so we have several cases in which we can compare the different explanations
produced by our two strategies. To obtain the latter, we need to perform ‘positive’ standard abduction over the contracted theory. Let us look first at the case
when the theory was contracted with p. Following the global strategy, the only
explanation for ϕ = ¬r with respect to the revised theory (Θ’={p ∧ q → r})
is the trivial solution, ¬r itself. On the other hand, following the local strategy,
there is another possible explanation for ¬r with respect to its revised theory
(Θ’={p ∧ q → r, T ∧ q}), namely q → ¬r. Moreover, if we contract with ¬p,
we get the same set of possible explanations in both strategies. Thus, again,
the local strategy seems to allow for more ‘pointed’ explanations of anomalous
observations.
We do not claim that either of our strategies is definitely better than the other
one. We would rather point at the fact that tableaux admit of many plausible
contraction operations, which we take to be a vindication of our framework.
Indeed, tableaux also suggest a slightly more ambitious approach. We outline
yet another strategy to restore consistency. It addresses a point mentioned in
earlier chapters, viz. that explanation often involves changing one’s ‘conceptual
framework’.
Contraction by Revising the Language
Suppose we have Θ = {p ∧ q}, and we observe or learn that ¬p. Following
our global contraction strategy would leave the theory empty, while following
the local one would yield a contracted theory with T ∧ q as its single formula.
But there is another option, equally easy to implement in our tableau algorithms.
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ABDUCTIVE REASONING
After all, in practice, we often resolve contradictions by ‘making a distinction’.
Mark the proposition inside the ‘anomalous formula’ (¬p) by some new proposition letter (say p’), and replace its occurrences (if any) in the theory by the
latter. In this case we obtain a new consistent theory Θ’ consisting of p ∧ q, ¬p’.
And other choice points in the above examples could be marked by suitable new
proposition letters as well.
We may think of the pair p, p′ as two variants of the same proposition, where
some distinction has been made. Here is a simple illustration of this formal
manipulation.
p ∧ q : “Rich and Famous”
¬p: “Materially poor”
¬p’: “Poor in spirit”
In a dialogue, the ‘anomalous statement’ might then be defused as follows.
– A: “X is a rich and famous person, but X is poor.”
– B: Why is X poor?”
Possible answers:
– A: “Because X is poor in spirit”
– A: “Because being rich makes X poor in spirit”
– A: “Because being famous makes X poor in spirit”
Over the new contracted (and reformulated) theories, our abductive algorithms of chapter 3 can easily produce these three consistent explanations (as
¬p′ , p → ¬p′ , q → ¬p′ ). The idea of reinterpreting the language to resolve
inconsistencies suggests that there is more to belief revision and contraction
than removing or modifying given formulas. The language itself may be a
source of the anomaly, and hence it needs revision, too. (For related work in
argumentation theory, cf. [vBe94].) This might be considered as a simple case
of ‘conceptual change’.
What we have shown is that, at least some language changes are easily incorporated into tableau algorithms, and are even suggested by them. Intuitively,
any inconsistent theory can be made consistent by introducing enough distinctions into its vocabulary, and ‘taking apart’ relevant assertions. We must leave
the precise extent, and algorithmic content, of this ‘folklore fact’ to further
research.
Another appealing consequence of accommodating inconsistencies via language change, concerns structural rules of chapter 3. We will only mention
that structural rules would acquire yet another parameter in their notation,
namely the vocabulary over which the formulas are to be interpreted (e.g.
p|V1 , q|V2 ⇒ p∧q|V1 ∪V2 ). Interestingly, this format was also used in Bolzano
[Bol73] (cf. chapter 3). An immediate side effect of this move are refined notions of consistency, in the spirit of those proposed by Hofstadter in [Hof79],
in which consistency is relative to an ‘interpretation’.
Epistemic Change
195
Outline of Contraction Algorithms
Global Strategy
Input: Θ, ϕ for which T (Θ ∪ ϕ) is closed.
Output: Θ’ (Θ contracted) for which T (Θ’ ∪ϕ) is open.
Procedure: CONTRACT(Θ, ¬ϕ, Θ’)
Construct T (Θ ∪ ϕ), and label its closed branches: Γ1 , . . . , Γn .
IF ¬ϕ ∈ Θ
Choose a closed branch Γi (not closed by ϕ, ¬ϕ).
Calculate the literals that open it: Branch–Opening(Γi ) = {γ1 , γ2 }. Choose one of
them, say γ = γ1 .
Find a corresponding formula γ’ for γ in Θ higher up in the branch (γ’ is either γ itself,
or a formula in conjunctive or disjunctive form in which γ occurs.)
Assign Θ’ := Θ − γ’.
ELSE (¬ϕ ∈ Θ)
Assign Θ’ = Θ − ϕ.
IF T (Θ’∪¬ϕ) is open AND all formulas from Θ are represented in the open branch,
then go to END.
ELSE
IF T (Θ’∪¬ϕ) is OPEN
Add remaining formulas to the open branch(es) until there are no more formulas to add
or until the tableau closes.
IF the resulting tableau Θ” is open, reassign Θ’:=Θ” and goto
END.
ELSE CONTRACT(Θ”,¬ϕ, Θ”’).
ELSE CONTRACT(Θ’,¬ϕ, Θ”).
% (This is the earlier-discussed ‘iteration clause’ for tableau recomputation.)
END
% (Θ’ is the contracted theory with respect to ¬ϕ such that T (Θ’ ∪ϕ) is open.)
Local Strategy
Input: Θ, ϕ for which T (Θ ∪ ϕ) is closed.
Output: Θ’ (Θ contracted) for which T (Θ’ ∪ϕ) is open.
Procedure: CONTRACT(Θ, ¬ϕ, Θ’)
Construct T (Θ ∪ ϕ), and label its closed branches: Γ1 , . . . , Γn .
Choose a closed branch Γi (not closed by ϕ, ¬ϕ).
Calculate the literals that open it: Branch–Opening(Γi ) = {γ1 , γ2 }. Choose one of
them, say γ = γ1 .
Replace γ by T together with all its occurrences up in the branch. (γ’ is either γ itself,
or a formula in conjunctive or disjunctive form in which γ occurs.)
Assign Θ’ := [T /γ]Θ.
196
ABDUCTIVE REASONING
IF T (Θ’∪¬ϕ) is open AND all formulas from Θ are represented in the open branch,
then go to END.
ELSE
IF T (Θ’∪¬ϕ) is OPEN
Add remaining formulas to the open branch(es) until there are no more formulas to add
or until the tableau closes.
IF the resulting tableau Θ” is open, reassign Θ’ := Θ” and goto
END.
ELSE CONTRACT(Θ”,¬ϕ, Θ”’).
ELSE CONTRACT(Θ’,¬ϕ, Θ”).
END
% (Θ’ is the contracted (by T substitution) theory with respect to ¬ϕ such that T (Θ’
∪ϕ) is open.)
Rationality Postulates
To conclude our informal discussion of contraction in tableaux, we briefly
discuss the AGM rationality postulates. (We list these postulates at the end of
this chapter.) These are often taken to be the hallmark of any reasonable operation of contraction and revision – and many papers show laborious verifications
to this effect. What do these postulates state in our case, and do they make
sense?
To begin with, we recall that theories of epistemic change differed in the way
their operations were defined (amongst other things). These can be given either
‘constructively’, as we have done, or via ‘postulates’. The former procedures
might then be checked for correctness according to the latter. However, in
our case, this is not as straightforward as it may seem. The AGM postulates
take belief states to be theories closed under logical consequence. But our
tableaux analyze non-deductively closed finite sets of formulas, corresponding
with ‘belief bases’. This will lead to changes in the postulates themselves.
Here is an example. Postulate 3 for contraction says that: “If the formula to
be retracted does not occur in the belief set K, nothing is to be retracted”:
K-3 If ϕ ∈ K, then K − ϕ = K.
In our framework, we cannot just replace belief states by belief bases here. Of
course, the intuition behind the postulate is still correct. If ϕ is not a consequence
of Θ (that we encounter in the tableau), then it will never be used for contraction
by our algorithms. Another point of divergence is that our algorithms do not
put the same emphasis on contracting one specific item from the background
theory as the AGM postulates. This will vitiate further discussion of even more
complex postulates, such as those breaking down contractions for complex
formulas into successive cases.
Epistemic Change
197
One more general reason for this mismatch is the following. Despite their
operational terminology (and ideology), the AGM postulates describe expansions, contractions, and revisions as (in the terminology of chapter 2) epistemic
products, rather than processes in their own right. This gives them a ‘static’
flavor, which may not always be appropriate.
Therefore, we conclude that the AGM postulates as they stand do not seem
to apply to contraction and revision procedures like ours. Evidently, this raises
the issue of which general features are present in our algorithmic approach,
justifying it as a legitimate notion of contraction. There is still a chance that
a relatively slight ‘revision of the revision postulates’ will do the job. (An
alternative more ‘procedural’ approach might be to view these issues rather in
the dynamic logic setting of [dRi94], [vBe96a] .) We must leave this issue to
further investigation.
4.
Discussion and Conclusions
Belief Revision in Explanation
We present here an example which goes beyond our ‘conservative algorithms’
of chapter 4 and that illustrates abduction as a model for epistemic change. This
example relates our medical diagnosis case given in chapter 2. As we have
shown, computing explanations for incoming beliefs gives a richer model than
many theories of belief revision, but this model is necessarily more complex.
After all, it gives a much broader perspective on the processes of expansion and
revision.
In standard belief revision in AI , given an undetermined belief (our case of
novelty) the natural operation for modifying a theory is expansion. The reason
is that the incoming belief is consistent with the theory, so the minimal change
criterion dictates that it is enough to add it to the theory. Once abduction is
considered however, the explanation for the fact has to be incorporated as well,
and simple theory expansion might not be always appropriate. Consider our
previous example of statistical reasoning in medical diagnosis (cf. chapter 2,
and 5.2.4 of chapter 5), concerning the quick recovery of Jane Jones, which we
briefly reproduce as follows:
Θ : L1 , L 2 , L 3 , C1 4
ϕ:E
Given theory Θ, we want to explain why Jane Jones recovered quickly
(ϕ). Clearly, the theory neither claims with high probability that she recov4 Almost
all cases of streptococcus infection clear up quickly after the administration of penicillin (L1).
Almost no cases of penicillin-resistant streptococcus infection clear up quickly after the administration
of penicillin (L2). Almost all cases of streptococcus infection clear up quickly after the administration
of Belladonna, a homeopathic medicine (L3). Jane Jones had streptococcus infection (C1). Jane Jones
recovered quickly (E).
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ABDUCTIVE REASONING
ered quickly (Θ ⇒ ϕ), nor that she did not (Θ ⇒ ¬ϕ). We have a case of
novelty, the observed fact is consistent with the theory. Now suppose a doctor
comes with the following explanation for her quick recovery: “After careful
examination, I have come to the conclusion that Jane Jones recovered quickly
because although she received treatment with penicillin and was resistant, her
grandmother had given her Belladonna”. This is a perfectly sound and consistent explanation. However, note that having the fact that ‘Jane Jones was
resistant to penicillin’ as part of the explanation does lower the probability of
explaining her quick recovery, to the point of statistically implying the contrary. Therefore, in order to make sense of the doctor’s explanation, the theory
needs to be revised as well, deleting the statistical rule L2 and replacing it with
something along the following lines: “Almost no cases of penicillin-resistant
streptococcus infection clear up quickly after the administration of penicillin,
unless they are cured by something else” (L′2 ).
Thus, we have shown with this example that for the case of novelty in
statistical explanation (which we reviewed in chapter 5), theory expansion may
not be the appropriate operation to perform (let alone the minimal one), and
theory revision might be the only way to salvage both consistency and high
probability.
Is Abduction Belief Revision?
In the first part of this chapter we have argued for an epistemic model of
abductive reasoning, in the lines of those proposed for belief revision in AI.
However, this connection does not imply that abduction can be equated to
belief revision. Let me discuss some reasons for this claim.
On the one hand, in its emphasis on explanations, an abductive model for
epistemic change is richer than many theories of belief revision. Admittedly,
though, not all cases of belief revision involve explanation, so the greater richness also reflects a restriction to a special setting. Moreover, in our model, not
all input data is epistemically assimilated, but only that which is surprising;
those facts which have not been explained by the theory, or that are in conflict
with it. (Even so, one might speculate whether facts which are merely probable
on the basis of Θ might still need explanation of some sort to further cement
their status.)
On the other hand, having a model for abduction within the belief revision
framework has imposed some other restrictions, which might not be appropriate
in a broader conception of abductive reasoning. One is that our abduction always
leads to some modification of the background theory to account for a surprising
phenomenon. Therefore, we leave out cases in which a fact may be surprising
(e.g. my computer is not working) even though it has been explained in the
past. This sense of “surprising" seems closer to “unexpected"; there is a need
to search for an explanation, but it does not involve any revision whatsoever.
Epistemic Change
199
Moreover, the type of belief revision accounted for in this model for abductive
reasoning is of a very simple kind. Addition and removal of information are
the basic operations to effect the epistemic changes, and the only criterion for
theory revision is explanation. There is neither room for conceptual change, a
more realistic type of scientific reasoning, nor place for a finer–grain distinction
of kinds of expansion (as in [Lev91]) or of revision. Our approach may also
be extended with a theory revision procedure, in which the revised theory is
more successful than the original one, in the lines proposed in chapter 6 and in
[Kui99].
Abductive Semantic Tableaux as a
Framework for Epistemic Change
The second part of this chapter concerned our brief sketch of a possible use of
semantic tableaux for performing all operations in an abductive theory of belief
revision. Even in this rudimentary state, it presents some interesting features.
For a start, expansion and contraction are not reverses of each other. The latter
is essentially more complex. Expanding a tableau for Θ with formula ϕ merely
hangs the latter to the open branches of T (Θ). But retracting ϕ from Θ often
requires complete reconfiguration of the initial tableau, and the contraction
procedure needs to iterate, as a cascade of formulas may have to be removed.
The advantage of contraction over expansion, however, is that we need not run
any separate consistency checks, as we are merely weakening a theory.
We have not come down in favor of any of the strategies presented. The
local strategy tends to retain more of the original theory, thus suggesting a more
minimal change than the global one. Moreover, its ‘substitution approach’ is
nicely in line with our alternative analysis of tableau abduction in section 4.5.2
of chapter 4, and it may lend itself to similar results. But in many practical
cases, the more standard ‘global abduction’ works just as well. We must leave
a precise comparison to future research.
Regarding different choices of formulas to be contracted, our algorithms are
blatantly non-deterministic. If we want to be more focused, we would have to
exploit the tableau structure itself to represent (say) some entrenchment order.
The more fundamental formulas would lie closer to the root of the tree. In this
way, instead of constructing all openings for each branch, we could construct
only those closest to the leaves of the tree. (These correspond to the less
important formulas, leaving the inner core of the theory intact.)
It would be of interest to have a proof-theoretic analysis of our contraction
procedure. In a sense, the AGM ‘rationality postulates’ may be compared with
the structural rules of chapter 3. And the more general question we have come
up against is this: what are the appropriate logical constraints on a process view
of contraction, and revision by abduction?
200
ABDUCTIVE REASONING
Conclusions
In this chapter we gave an account of abduction as a theory of belief revision.
We proposed semantic tableaux as a logical representation of belief bases over
which the major operations of epistemic change can be performed. The resulting
theory combines the foundationalist with the coherentist stand in belief revision.
We claimed that beliefs need justification, and used our abductive machinery
to construct explanations of incoming beliefs when needed. The result is richer
than standard theories in AI, but it comes at the price of increased complexity.
In fact, it has been claimed in [Doy92] that a realistic workable system for
belief revision must not only trade deductive closed theories for belief bases,
but also drop the consistency requirement. (As we saw in chapter 3, the latter
is undecidable for sufficiently expressive predicate-logical languages. And
it may still be NP–complete for sentences in propositional logic.) Given our
analysis in chapter 3, we claim that any system, which aims at producing genuine
explanations for incoming beliefs, must maintain consistency. What this means
for workable systems of belief revision remains a moot point.
The tableau analysis confirms the intuition that revision is more complex than
expansion, and that it admits of more variation. Several choices are involved,
for which there seem to be various natural options, even in this constrained
logical setting. What we have not explored in full is the way in which tableaux
might generate entrenchment orders that we can profit from computationally. As
things stand, different tableau procedures for revision may output very different
explanations: abductive revision is not one unique procedure, but a family.
Even so, the preceding analysis may have shown that the standard logical tool
of semantic tableaux has more uses than those for which they were originally
designed.
Finally, regarding other epistemic operations in AI and their connection to
abduction, we have briefly mentioned update, the process of keeping beliefs up–
to–date as the world changes. Its connection to abduction points to an interesting
area of research; the changing world might be full of new surprises, or existing
beliefs might have lost their explanations. Thus, appropriate operations would
have to be defined to keep the theory updated with respect to these changes.
Regarding the connection to other work, it would be interesting to compare
our approach with that mentioned which follows the AGM line. Although
this is not as straightforward as it may seem, we could at least check whether
our algorithmic constructions of abductive expansion and revision validate the
abductive postulates found in [Pag96] and [LU96].
AGM Postulates for Contraction
K-1 For any sentence φ and any belief set K, K − φ is a belief set.
K-2 No new beliefs occur in K − φ: K − φ ⊆ K.
Epistemic Change
201
K-3 If the formula to be retracted does not occur in the belief set, nothing is to
be retracted:
If φ ∈ K, then K − φ = K.
K-4 The formula to be retracted is not a logical consequence of the beliefs
retained, unless it is a tautology:
If not ⊢ φ, then φ ∈ K − φ).
K-5 It is possible to undo contractions (Recovery Postulate):
If φ ∈ K, then K ⊆ (K − φ) + φ.
K-6 The result of contracting logically equivalent formulas must be the same:
If ⊢ φ ↔ ψ, then K − φ = K − ψ.
K-7 Two separate contractions may be performed by contracting the relevant
conjunctive formula:
K − φ ∩ K − ψ ⊆ K − φ ∧ ψ.
K-8 If φ ∈ K − φ ∧ ψ, then K − φ ∧ ψ ⊆ K − ψ.
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Author Index
Alchourrón, C., 179, 181
Alechina, N., 148
Aleliunas, 40
Aliseda, A., xv, 19, 34, 95–96, 110, 123, 127, 129,
174
Amor, J., xvi
Anderson, D., 171–173
Antaki, C., 68
Appelt, D., 36, 43
Aravindan, C., 43, 183
Aristotle, 7–8, 28, 38
Atkinson, D., xvi
Ayim, M., 172
Bacon, F., 7–8, 11
Barceló, A., xvi
Barwise, J., 24, 67, 69, 88
Batens, D., 24
Benthem van, J., 32, 41, 47, 49, 57, 61, 63–65,
67–71, 82, 84, 86, 97, 144, 194, 197
Beth, E., 6, 32, 66, 98, 100, 124, 130
Bolzano, B., 65, 70–71, 78, 90, 94, 136, 194
Boolos, G., 122, 124
Bosch van den, A., xvi
Boyle, R., 7
Brody, B., 147
Burger, I., 23
Cabanchik, S., xv
Carnap, R., 56, 65, 73, 88, 153
Cepparello, G., 69
Chaitin, G., 65
Chomsky, N., 44
Church, A., 86
Colmerauer, A., 41
Corcoran, J., 70
Dı́az, E., 122, 124
Debrock, G., 172
Descartes, R., 7–8
Ditmarsch van, H., xvi
Doyle, J., 200
Dubois, D., 33
Dummet, M., 57
Dung, P., 43, 183
Eiter, T., 87
Etchemendy, J., 24, 88
Euclid, 64
Euler, L., 45
Ezcurdia, M., xvi
Fann, K., 173
Festa, R., xv
Fitting, M., 101, 122
Flach, P., xv–xvi, 33–34, 36–37, 40–42, 70, 88, 91,
93–95, 97, 170, 173
Fraassen van, B., 38, 149
Frankfurt, H., 172
Gärdenfors, P., xiv, 43, 67, 148–149, 154, 179–183,
186
Gödel, K., 65
Gabbay, D., 24, 41, 44, 49, 57, 61, 65, 67, 90, 148,
150
Galliers, J., 122, 186
Garcı́a, C., xvi
Gentzen, G., 6, 61, 64, 66, 82–83, 100
Gervás, P., 44, 131
Gillies, D., xvi, 49, 57
Ginsberg, A., 43
Goebel, R., 40
Goldbach,C., 45
Gorlée, D., 172
Gottlob, G., 87
Groen, G., 13
Groeneveld, W., 62–64, 69, 84
Gutting, G., 23
Haack, S., 55, 57–60, 63–64, 89, 186
Hanson, N., 28, 30–31, 39, 57
Harman, G., 34, 183
Heidema, J., 23
Heim, I., 44
Helman, D., 24
220
Hempel, C., 22, 29, 38, 48, 56, 65, 67, 70, 92–94,
135–138, 141–143, 146, 148–150
Hernández, F., xvi
Herschel, J., 7
Hilbert, D., 65
Hintikka, J., 5–6, 57, 98
Hitchcock, C., 147
Hobbs, J., 36, 43
Hoffmann, M., xvi
Hofstadter, D., 194
Holland, J., 33
Holyoak, K., 33
Hookway, C., 170
James, W., 168
Johnson Laird, P., 67–68
Josephson, J., 40
Kakas, A., xv, 34, 36–37, 40, 42, 47, 95, 97, 170,
183
Kalsbeek, M., 49, 82, 97–98
Kanazawa, M., 84
Kant, I., 170
Kapitan, T., 172
Kelly, K., 7, 21
Kempson, R., 44
Kepler, J., 16, 28, 30–32, 48
Kneale, W., 57
Konolige, K., 40, 92
Kowalski, R., 36, 40–42, 47, 49, 95
Krabbe, E., xvi
Kraus, S., 41, 61, 64, 81
Kruijff, G., 173
Kuhn, T., xiii, 149
Kuipers, T., xiv–xv, 7, 154–156, 158–160,
162–166, 199
Kurtonina, N., 84
Lakatos, I., 19, 39, 65
Lambalgen van, M., 144, 148
Langley, P., 31, 43
Laplace, P., 10, 28
Laudan, L., 6–7, 9, 12–13, 154
Lehmann, D., 41, 61, 64, 81
Leibniz, G., 7–8, 11
Levesque, H., 184
Levi, I., 199
Lloyd, J., 41
Lobachevsky, N., 91
Lobo, J., 184–185, 200
Magidor, M., 41, 61, 64, 81
Magnani, L., xvi
Makinson, D., 64, 179
Mancarella, P., 36, 40, 183
Manna, Z., 89
Martin, P., 36, 43
Mayer, M., 40, 95, 122, 131–132
McCarthy, J., 65
Meheus, J., xvi, 22, 24
Mendelson, E., 62
ABDUCTIVE REASONING
Meyer, W., 148
Michalski, R., 33, 42, 70
Mill, J., 7, 31, 33, 38
Mints, G., 82, 97
Montaño, U., xvi
Morado, R., xvi
Musgrave, A., 7–8
Nagel, E., 38
Nepomuceno, A., xv–xvi, 95, 123, 125, 127, 129
Newton, I., 48
Nickles, T., 7, 22
Niemelä, I., 131
Niiniluoto, I., 56, 136, 146–147
Nisbett, R., 33
Nova, A., xvi
Nubiola, J., xvi
Ohm, M., 16
Okasha,S., xvi
Olivé, L., xvi
Oppenheim, P., 136
Pérez Ransanz, A., xvi
Paavola, S., xvi
Pagnucco, M., 184–185, 200
Paoli, F., 61
Pappus of Alexandria, 4–5
Pascal, B., 45
Paul, G., 40
Pearce, D., xvi
Peijenburg, J., xv–xvi
Peirce, C., xii–xiv, 7, 12, 27–28, 31, 33, 35–37, 39,
43–45, 47, 55–56, 65, 72–73, 89, 94,
136–137, 168–171, 173–174, 176, 184, 186
Peng , Y., 40, 43
Perry, J., 67, 69
Pinto, S., xvi
Pirri, F., 40, 95, 122, 131–132
Polya, G., 24, 39, 44–45
Poole, D., 40
Pople, H., 40
Popper, K., xii, 3–4, 12–15, 18–21, 25, 39, 56, 153
Post, E., 87
Prade , H., 33
Quine, W., 92
Rakova, M., xvi
Ramsey, F., 67, 184
Reggia, J., 40, 43
Reichenbach, H., 6, 8
Reilly, F., 172
Reiter, R., 70, 93, 150
Remes, U., 4–6
Rescher, N., 38
Ribenboim, P., 45
Riemann, G., 91
Rijke de, M., 197
Rodrı́guez, V., xvi
Roesler, A., 172
Rol, M., xvi
221
AUTHOR INDEX
Romeyn, J., xvi
Rott, H., 182
Ruben , D-H., 32, 38
Ryan, M., 67
Sánchez Valencia, V., xvi
Saab, S., xvi
Salmon, W., 29, 32–33, 35, 38, 141, 143, 146–147,
149
Savary, C., 6, 25
Scott, D., 61
Shapiro, E., 34
Shoham, Y., 33, 65, 69
Shrager , J., 43
Simon, H., 3, 12–13, 15–18, 20–21, 31–32
Sintonen, M., xvi
Sipma, H., 89
Smullyan, R., 98, 101
Sosa, E., 67
Stegmüller, W., 143, 150
Stickel, M., 36, 43, 97
Suppes, P., 22, 28, 145
Szabó, Á., 5
Tamminga, A., 84
Tan, Y., 40, 150
Tarski, A., 56, 61, 69–71, 88
Teije ten, A., 66–67, 130
Thagard, P., 16, 31, 33–34, 146, 148
Thompson, P., 70
Toni, F., 40, 42, 47, 95
Toulmin, S., 68
Tuomela, R., 141, 146–147
Troelstra, A., 96
Uribe, T., 89
Uzcategui, C., 184–185, 200
Vázquez, A., 110
Velasco, A., xvi
Verbrugge, R., xvi
Vergara, R., xvi
Visser, H., 45
Wasow, T., xvi
Williams, M., 183
Wirth, U., 172
Woods, J., xvi, 24
Wright von, G., 33
Topic Index
BACON, 16
GLAUBER, 16
PI (Processes of Induction), 16
Abduction, xii, 7, 27–29, 31, 33, 35, 37, 39, 55–56,
148, 171, 174–175, 200
abducibles, 87, 97, 109–110, 117, 137
as a process, xiii, 32, 37, 40, 42, 68, 74, 93, 95–96
as a product, 32, 37, 40, 68, 74
as deduction, 40
as epistemic change, 185
computational-based, 40
consistent, 109–110, 114
explanatory, 78, 116
generation, 108, 119
instrumentalist, xiv, 154, 159, 163, 165
linguistics, 131
logic of, 40, 49
minimal, 80, 131
novelty, 154
preferential, 80–81
scientific explanation, 135
semantics, 91
taxonomy, 27, 36, 46, 49
Abductive
anomaly, 27, 47–48, 160, 184
conclusion, 31
consistent (abductive) explanatory inference, 75
expansion, 184
explanation, 30, 32–33, 40, 46, 92, 95, 105, 108,
127
explanatory argument, 48, 53, 64, 70
explanatory inference, 70–71, 74–75, 78, 89–91
explanatory logics, 90
logic programming, 82
novelty, 27, 47–48, 92, 160, 177, 184
outcomes, 46, 48–49
parameters, 46
problem, 41, 127
procedures, 27
process, 32, 39, 46, 48, 135
reasoning, xi, 29, 40, 56, 183, 198
revision, 185, 200
semantic tableaux, 106–107, 131
See also Semantic tableaux
solution, 127
triggers, 27, 46–49, 69, 136, 175
weakest explanation, 72
Analogy, 24, 148
Analysis, 3–5, 8, 31, 66, 100
problematical, 4–5
theoretical, 4
Anomaly, 29, 48, 92, 136, 153, 183–184, 192, 194
Apagoge, 7, 28
Argument
scientific, 66
theory, 67–68
valid, 57–58
See also Explanatory argument
Artificial intelligence, xi, 4, 15, 18, 36, 39, 43, 49,
53, 55–56, 61, 65, 70, 177, 183, 197
Belief revision, xi, xiv, 27, 39, 43, 47, 132, 177,
179–180, 182–184, 186, 197–198, 200
AGM approach, 179, 182, 196–197, 200
Calculus ratiocinator, 8
Characteristica universalis, 8, 11
Cognitive psychology, 18, 32
Cognitive science, xi, 4, 148
Computational philosophy of science, 16, 22, 146
Concept formation, 16, 34, 39, 45, 48, 148
Conceptual change, 194, 199
Confirmation, xi, 34, 66, 153, 156, 158
Conjectures and refutations, 14–15, 39
Context
of appraisal, 7–8
of discovery, xii–xiii, 3, 6–7, 32, 37
of evaluation, 7–8, 158
of invention, 7–8
of justification, xiii, 3, 6–7, 32, 37, 158
224
of pursuit, 6, 8
of research, 6, 8
Corroboration, 158
Creativity, xi–12, 19, 173
See also Discovery
Deduction, 33, 55–56, 171
backwards, 34, 40, 75
blind, 42
inference, 55
inversed, 57
natural, 61
validity, 59
Discovery, xi, 4, 6, 12, 37
as a process, 6
as problem solving, 16
eureka, 7, 18
friends of, 12
guess, 14
initial conception, 6, 9, 13
knowledge, 13
linguistics, 43, 131
logics, 7–8, 19, 21, 23
mathematical, 39, 44
methods, 16
normative theory, 16–17
of causes, 7, 29, 33
patterns, 39, 135
processes, 17
rules, 12
Empirical progress, 154–156, 159–160, 163, 165
Enthymeme, 7
Epagoge, 7, 28
Epistemic change, 180, 182
See also Belief revision
Evaluation, 19
HD, 158
epistemics of, 15
Explanation, xi, 28, 32, 36–39, 72, 89, 137, 192
Hempelian models, xiv, 38, 48, 70–71, 92–93,
135–138, 141–144, 146, 148, 158
abductive, 30
argumentative, 38
best, 30, 72, 80, 148
consistent, 115, 117, 131
construction, 42
irrelevant, 138
minimal, 118
non-argumentative, 38
non-deductive, 38
partial, 115, 118, 130
potential, 136–137
pragmatic, 38
redundant, 111, 118
scientific, xi, 27, 32, 67, 135–136, 148
selection, 42
statistical, 136, 141–142, 145, 198
weakest, 80
ABDUCTIVE REASONING
Explanatory
argument, 32, 40, 46, 48
process, 32
Fallibilism, 3, 10, 14, 19
Falsification, 156
Heuristics, 16, 37, 39, 44, 172
methods, 3, 16
search, 17, 19, 95
strategies, 12
syllogism, 39
Hypothesis, 27, 31, 36, 56, 171
best, 42
construction, 33, 172
economic, 36–37, 73, 172
explanatory, 36
formation, 32
selection, 33, 172, 177
testable, 36–37, 153, 172
Induction, 7, 15, 27, 33–34, 37, 55–56, 171
abductory, 173
eliminative, 8, 11
enumerative, 10, 24, 33, 35, 83
generalization, 34
inductive support, 146
modeling, 11
projection, 34
statistical syllogism, 34
strength, 58–59
systems, 58
Infallibilism, 3, 9–10, 19
Inference, 66
analytic, 170
classical, 61
deductive, 55
dynamic, 47, 64, 92
hypothetical, 89
logical, 35–36, 56, 61, 64, 66
non-classical, 63–64
non-monotonic, 35, 41, 53, 57, 61, 63, 71, 81, 92,
130
probabilistic, 46, 146
scientific, 9, 135–136
statistical, 48, 69, 130, 142, 144
strength, 68–69
synthetic, 170
to the best explanation, 34
update-to-test, 69
Instrumentalism, 55, 59–61
Knowledge
acquisition, 36
assimilation, xi, 40, 183
background, 30
discovery, 13
growth, 20
well-foundedness, 9, 20
Lacunae, xiv, 153–155, 157, 160–163
Logic of discovery, xiii, 3, 9, 12, 15, 39
TOPIC INDEX
descriptive, 21
question of achievement, 9, 11–12, 20
question of purpose, 9, 11, 15, 19–20, 25
question of pursuit, 9, 11, 15, 19–20, 25
Logic programming, 36, 40–42, 48–49, 95–97, 122
abductive, 42, 82
prolog, 41, 82
Logic
adaptive, 24
ampliative, 24
categorial, 64, 91
classical, 58, 60, 63, 66, 71, 89
deductive syllogistics, 8
deductive, 31, 33
See also Deduction
demarcation, 54–55, 57
descriptive, 91
deviations, 55, 58, 60
dynamic, 53, 61, 69, 91, 197
explanation, 22
extensions, 55, 58
for development, 23
generational, 10
inductive, 55–56, 58
intuitionistic, 58
justificatory, 3
linear, 64, 92, 96
modal, 85, 131
natural deduction, 6
of confirmation, 93
of explanation, 93
of generation, 4, 23
of justification, 10, 23
pattern seeking, 12
philosophy, 55
relevance, 58, 64, 91, 96
scientific methodology, 21
second order, 57
self-corrective, 10
sequent calculus, 6, 100
situation theory, 67, 69
Logical
inference, 27
method, xii
procedures, 23, 50
structural analysis, 41, 54, 61–64, 75–78, 81–83,
89, 91, 94, 97, 118, 138, 140, 144, 194
Medical diagnosis, 197
Monism, 55, 59–61
Novelty, 136, 154, 183, 198
Philosophy of science, xi, 22, 55–56, 65, 70, 88, 90,
93, 122, 135, 147, 153
Pluralism, 55, 59–60
Positivism, xii, 17, 153
Pragmatism, xiv, 35, 168, 174–175
pragmatic maxim, 168–169
pragmaticism, 168
Problem solving, 4, 14, 18, 20
225
Problems
ill structured, 16
well structured, 16, 20
Reasoning, 34, 49, 92
ampliative, 24
analogical, 24
backward, 65–66
common sense, 22, 28–29, 42
conjectural, 93
default, 69–70, 131
diagrammatic, 24
fallacy of affirming the consequent, 9
forward, 65–66
human, 57
in diagnosis, 28, 40, 43
inductive, xi, 33
intelligent computers, 15
mathematical, 55
mechanistic explanation, 17
model-based, xi
natural, 53
non-monotonic, 55, 75
preferential, 81
probabilistic, 74
rational method, 15
scientific, xi, 8, 13, 22, 28, 70
statistical, 29, 34
synthetic, 36, 170
theories of, 18
trial and error, 15
universal system, 11
Retroduction, 38, 57, 171
See also Abduction
Scientific practice, 153, 159
Semantic tableaux, xiv, 6, 44, 50, 66, 91, 95, 98,
108, 122, 155, 160, 186, 199
DB-Tableaux, 122–126, 129
abductive, 106–107, 131
closed extension, 102–104
open extension, 102–103
semi-closed extension, 102–103
Statistical relevance, 146–147
Surprise, 43, 47, 97, 160, 173, 175, 177, 184, 198
Synthesis, 3–5, 8, 31, 66, 100
Theory
background, 30, 39–40, 42, 65–67, 72, 198
building, 16
choice, 15
comparison, 159
development, xi
evaluation, 15, 153, 155, 158
improvement, 157, 159, 163
justification, 9
revision, 36, 42–43, 159, 198
See also Belief revision
testing, 13
Universal calculi, 8
Verification, 156, 177
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J. M. Bochénski, A Precis of Mathematical Logic. Translated from French and German by O.
Bird. 1959
ISBN 90-277-0073-7
P. Guiraud, Problèmes et méthodes de la statistique linguistique. 1959 ISBN 90-277-0025-7
H. Freudenthal (ed.), The Concept and the Role of the Model in Mathematics and Natural and
Social Sciences. 1961
ISBN 90-277-0017-6
E. W. Beth, Formal Methods. An Introduction to Symbolic Logic and to the Study of Effective
Operations in Arithmetic and Logic. 1962
ISBN 90-277-0069-9
B. H. Kazemier and D. Vuysje (eds.), Logic and Language. Studies dedicated to Professor
Rudolf Carnap on the Occasion of His 70th Birthday. 1962
ISBN 90-277-0019-2
M. W. Wartofsky (ed.), Proceedings of the Boston Colloquium for the Philosophy of Science,
1961–1962. [Boston Studies in the Philosophy of Science, Vol. I] 1963 ISBN 90-277-0021-4
A. A. Zinov’ev, Philosophical Problems of Many-valued Logic. A revised edition, edited and
translated (from Russian) by G. Küng and D.D. Comey. 1963
ISBN 90-277-0091-5
G. Gurvitch, The Spectrum of Social Time. Translated from French and edited by M. Korenbaum
and P. Bosserman. 1964
ISBN 90-277-0006-0
P. Lorenzen, Formal Logic. Translated from German by F.J. Crosson. 1965
ISBN 90-277-0080-X
R. S. Cohen and M. W. Wartofsky (eds.), Proceedings of the Boston Colloquium for the Philosophy of Science, 1962–1964. In Honor of Philipp Frank. [Boston Studies in the Philosophy
of Science, Vol. II] 1965
ISBN 90-277-9004-0
E. W. Beth, Mathematical Thought. An Introduction to the Philosophy of Mathematics. 1965
ISBN 90-277-0070-2
E. W. Beth and J. Piaget, Mathematical Epistemology and Psychology. Translated from French
by W. Mays. 1966
ISBN 90-277-0071-0
G. Küng, Ontology and the Logistic Analysis of Language. An Enquiry into the Contemporary
Views on Universals. Revised ed., translated from German. 1967
ISBN 90-277-0028-1
R. S. Cohen and M. W. Wartofsky (eds.), Proceedings of the Boston Colloquium for the
Philosophy of Sciences, 1964–1966. In Memory of Norwood Russell Hanson. [Boston Studies
in the Philosophy of Science, Vol. III] 1967
ISBN 90-277-0013-3
C. D. Broad, Induction, Probability, and Causation. Selected Papers. 1968
ISBN 90-277-0012-5
G. Patzig, Aristotle’s Theory of the Syllogism. A Logical-philosophical Study of Book A of the
Prior Analytics. Translated from German by J. Barnes. 1968
ISBN 90-277-0030-3
N. Rescher, Topics in Philosophical Logic. 1968
ISBN 90-277-0084-2
R. S. Cohen and M. W. Wartofsky (eds.), Proceedings of the Boston Colloquium for the
Philosophy of Science, 1966–1968, Part I. [Boston Studies in the Philosophy of Science,
Vol. IV] 1969
ISBN 90-277-0014-1
R. S. Cohen and M. W. Wartofsky (eds.), Proceedings of the Boston Colloquium for the
Philosophy of Science, 1966–1968, Part II. [Boston Studies in the Philosophy of Science,
Vol. V] 1969
ISBN 90-277-0015-X
J. W. Davis, D. J. Hockney and W. K. Wilson (eds.), Philosophical Logic. 1969
ISBN 90-277-0075-3
D. Davidson and J. Hintikka (eds.), Words and Objections. Essays on the Work of W. V. Quine.
1969, rev. ed. 1975
ISBN 90-277-0074-5; Pb 90-277-0602-6
P. Suppes, Studies in the Methodology and Foundations of Science. Selected Papers from 1951
to 1969. 1969
ISBN 90-277-0020-6
J. Hintikka, Models for Modalities. Selected Essays. 1969
ISBN 90-277-0078-8; Pb 90-277-0598-4
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N. Rescher et al. (eds.), Essays in Honor of Carl G. Hempel. A Tribute on the Occasion of His
65th Birthday. 1969
ISBN 90-277-0085-0
P. V. Tavanec (ed.), Problems of the Logic of Scientific Knowledge. Translated from Russian.
1970
ISBN 90-277-0087-7
M. Swain (ed.), Induction, Acceptance, and Rational Belief. 1970
ISBN 90-277-0086-9
R. S. Cohen and R. J. Seeger (eds.), Ernst Mach: Physicist and Philosopher. [Boston Studies
in the Philosophy of Science, Vol. VI]. 1970
ISBN 90-277-0016-8
J. Hintikka and P. Suppes, Information and Inference. 1970
ISBN 90-277-0155-5
K. Lambert, Philosophical Problems in Logic. Some Recent Developments. 1970
ISBN 90-277-0079-6
R. A. Eberle, Nominalistic Systems. 1970
ISBN 90-277-0161-X
P. Weingartner and G. Zecha (eds.), Induction, Physics, and Ethics. 1970 ISBN 90-277-0158-X
E. W. Beth, Aspects of Modern Logic. Translated from Dutch. 1970
ISBN 90-277-0173-3
R. Hilpinen (ed.), Deontic Logic. Introductory and Systematic Readings. 1971
See also No. 152.
ISBN Pb (1981 rev.) 90-277-1302-2
J.-L. Krivine, Introduction to Axiomatic Set Theory. Translated from French. 1971
ISBN 90-277-0169-5; Pb 90-277-0411-2
J. D. Sneed, The Logical Structure of Mathematical Physics. 2nd rev. ed., 1979
ISBN 90-277-1056-2; Pb 90-277-1059-7
C. R. Kordig, The Justification of Scientific Change. 1971
ISBN 90-277-0181-4; Pb 90-277-0475-9
M. Čapek, Bergson and Modern Physics. A Reinterpretation and Re-evaluation. [Boston
Studies in the Philosophy of Science, Vol. VII] 1971
ISBN 90-277-0186-5
N. R. Hanson, What I Do Not Believe, and Other Essays. Ed. by S. Toulmin and H. Woolf.
1971
ISBN 90-277-0191-1
R. C. Buck and R. S. Cohen (eds.), PSA 1970. Proceedings of the Second Biennial Meeting
of the Philosophy of Science Association, Boston, Fall 1970. In Memory of Rudolf Carnap.
[Boston Studies in the Philosophy of Science, Vol. VIII] 1971
ISBN 90-277-0187-3; Pb 90-277-0309-4
D. Davidson and G. Harman (eds.), Semantics of Natural Language. 1972
ISBN 90-277-0304-3; Pb 90-277-0310-8
Y. Bar-Hillel (ed.), Pragmatics of Natural Languages. 1971
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S. Stenlund, Combinators, γ Terms and Proof Theory. 1972
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M. Strauss, Modern Physics and Its Philosophy. Selected Paper in the Logic, History, and
Philosophy of Science. 1972
ISBN 90-277-0230-6
M. Bunge, Method, Model and Matter. 1973
ISBN 90-277-0252-7
M. Bunge, Philosophy of Physics. 1973
ISBN 90-277-0253-5
A. A. Zinov’ev, Foundations of the Logical Theory of Scientific Knowledge (Complex Logic).
Revised and enlarged English edition with an appendix by G. A. Smirnov, E. A. Sidorenka, A.
M. Fedina and L. A. Bobrova. [Boston Studies in the Philosophy of Science, Vol. IX] 1973
ISBN 90-277-0193-8; Pb 90-277-0324-8
L. Tondl, Scientific Procedures. A Contribution concerning the Methodological Problems of
Scientific Concepts and Scientific Explanation. Translated from Czech by D. Short. Edited by
R.S. Cohen and M.W. Wartofsky. [Boston Studies in the Philosophy of Science, Vol. X] 1973
ISBN 90-277-0147-4; Pb 90-277-0323-X
N. R. Hanson, Constellations and Conjectures. 1973
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K. J. J. Hintikka, J. M. E. Moravcsik and P. Suppes (eds.), Approaches to Natural Language.
1973
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M. Bunge (ed.), Exact Philosophy. Problems, Tools and Goals. 1973 ISBN 90-277-0251-9
R. J. Bogdan and I. Niiniluoto (eds.), Logic, Language and Probability. 1973
ISBN 90-277-0312-4
G. Pearce and P. Maynard (eds.), Conceptual Change. 1973
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I. Niiniluoto and R. Tuomela, Theoretical Concepts and Hypothetico-inductive Inference. 1973
ISBN 90-277-0343-4
R. Fraissé, Course of Mathematical Logic – Volume 1: Relation and Logical Formula. Translated from French. 1973
ISBN 90-277-0268-3; Pb 90-277-0403-1
(For Volume 2 see under No. 69).
A. Grünbaum, Philosophical Problems of Space and Time. Edited by R.S. Cohen and M.W.
Wartofsky. 2nd enlarged ed. [Boston Studies in the Philosophy of Science, Vol. XII] 1973
ISBN 90-277-0357-4; Pb 90-277-0358-2
P. Suppes (ed.), Space, Time and Geometry. 1973 ISBN 90-277-0386-8; Pb 90-277-0442-2
H. Kelsen, Essays in Legal and Moral Philosophy. Selected and introduced by O. Weinberger.
Translated from German by P. Heath. 1973
ISBN 90-277-0388-4
R. J. Seeger and R. S. Cohen (eds.), Philosophical Foundations of Science. [Boston Studies in
the Philosophy of Science, Vol. XI] 1974
ISBN 90-277-0390-6; Pb 90-277-0376-0
R. S. Cohen and M. W. Wartofsky (eds.), Logical and Epistemological Studies in Contemporary
Physics. [Boston Studies in the Philosophy of Science, Vol. XIII] 1973
ISBN 90-277-0391-4; Pb 90-277-0377-9
R. S. Cohen and M. W. Wartofsky (eds.), Methodological and Historical Essays in the Natural
and Social Sciences. Proceedings of the Boston Colloquium for the Philosophy of Science,
1969–1972. [Boston Studies in the Philosophy of Science, Vol. XIV] 1974
ISBN 90-277-0392-2; Pb 90-277-0378-7
R. S. Cohen, J. J. Stachel and M. W. Wartofsky (eds.), For Dirk Struik. Scientific, Historical
and Political Essays. [Boston Studies in the Philosophy of Science, Vol. XV] 1974
ISBN 90-277-0393-0; Pb 90-277-0379-5
K. Ajdukiewicz, Pragmatic Logic. Translated from Polish by O. Wojtasiewicz. 1974
ISBN 90-277-0326-4
S. Stenlund (ed.), Logical Theory and Semantic Analysis. Essays dedicated to Stig Kanger on
His 50th Birthday. 1974
ISBN 90-277-0438-4
K. F. Schaffner and R. S. Cohen (eds.), PSA 1972. Proceedings of the Third Biennial Meeting of
the Philosophy of Science Association. [Boston Studies in the Philosophy of Science, Vol. XX]
1974
ISBN 90-277-0408-2; Pb 90-277-0409-0
H. E. Kyburg, Jr., The Logical Foundations of Statistical Inference. 1974
ISBN 90-277-0330-2; Pb 90-277-0430-9
M. Grene, The Understanding of Nature. Essays in the Philosophy of Biology. [Boston Studies
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ISBN 90-277-0462-7; Pb 90-277-0463-5
J. M. Broekman, Structuralism: Moscow, Prague, Paris. Translated from German. 1974
ISBN 90-277-0478-3
N. Geschwind, Selected Papers on Language and the Brain. [Boston Studies in the Philosophy
of Science, Vol. XVI] 1974
ISBN 90-277-0262-4; Pb 90-277-0263-2
R. Fraissé, Course of Mathematical Logic – Volume 2: Model Theory. Translated from French.
1974
ISBN 90-277-0269-1; Pb 90-277-0510-0
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A. Grzegorczyk, An Outline of Mathematical Logic. Fundamental Results and Notions explained
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ISBN 90-277-0359-0; Pb 90-277-0447-3
F. von Kutschera, Philosophy of Language. 1975
ISBN 90-277-0591-7
J. Manninen and R. Tuomela (eds.), Essays on Explanation and Understanding. Studies in the
Foundations of Humanities and Social Sciences. 1976
ISBN 90-277-0592-5
J. Hintikka (ed.), Rudolf Carnap, Logical Empiricist. Materials and Perspectives. 1975
ISBN 90-277-0583-6
M. Čapek (ed.), The Concepts of Space and Time. Their Structure and Their Development.
[Boston Studies in the Philosophy of Science, Vol. XXII] 1976
ISBN 90-277-0355-8; Pb 90-277-0375-2
J. Hintikka and U. Remes, The Method of Analysis. Its Geometrical Origin and Its General
Significance. [Boston Studies in the Philosophy of Science, Vol. XXV] 1974
ISBN 90-277-0532-1; Pb 90-277-0543-7
J. E. Murdoch and E. D. Sylla (eds.), The Cultural Context of Medieval Learning. [Boston
Studies in the Philosophy of Science, Vol. XXVI] 1975
ISBN 90-277-0560-7; Pb 90-277-0587-9
S. Amsterdamski, Between Experience and Metaphysics. Philosophical Problems of the Evolution of Science. [Boston Studies in the Philosophy of Science, Vol. XXXV] 1975
ISBN 90-277-0568-2; Pb 90-277-0580-1
P. Suppes (ed.), Logic and Probability in Quantum Mechanics. 1976
ISBN 90-277-0570-4; Pb 90-277-1200-X
H. von Helmholtz: Epistemological Writings. The Paul Hertz / Moritz Schlick Centenary
Edition of 1921 with Notes and Commentary by the Editors. Newly translated from German
by M. F. Lowe. Edited, with an Introduction and Bibliography, by R. S. Cohen and Y. Elkana.
[Boston Studies in the Philosophy of Science, Vol. XXXVII] 1975
ISBN 90-277-0290-X; Pb 90-277-0582-8
J. Agassi, Science in Flux. [Boston Studies in the Philosophy of Science, Vol. XXVIII] 1975
ISBN 90-277-0584-4; Pb 90-277-0612-2
S. G. Harding (ed.), Can Theories Be Refuted? Essays on the Duhem-Quine Thesis. 1976
ISBN 90-277-0629-8; Pb 90-277-0630-1
S. Nowak, Methodology of Sociological Research. General Problems. 1977
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J. Piaget, J.-B. Grize, A. Szeminśska and V. Bang, Epistemology and Psychology of Functions.
Translated from French. 1977
ISBN 90-277-0804-5
M. Grene and E. Mendelsohn (eds.), Topics in the Philosophy of Biology. [Boston Studies in
the Philosophy of Science, Vol. XXVII] 1976
ISBN 90-277-0595-X; Pb 90-277-0596-8
E. Fischbein, The Intuitive Sources of Probabilistic Thinking in Children. 1975
ISBN 90-277-0626-3; Pb 90-277-1190-9
E. W. Adams, The Logic of Conditionals. An Application of Probability to Deductive Logic.
1975
ISBN 90-277-0631-X
M. Przełeçki and R. Wójcicki (eds.), Twenty-Five Years of Logical Methodology in Poland.
Translated from Polish. 1976
ISBN 90-277-0601-8
J. Topolski, The Methodology of History. Translated from Polish by O. Wojtasiewicz. 1976
ISBN 90-277-0550-X
A. Kasher (ed.), Language in Focus: Foundations, Methods and Systems. Essays dedicated to
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ISBN 90-277-0644-1; Pb 90-277-0645-X
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J. Hintikka, The Intentions of Intentionality and Other New Models for Modalities. 1975
ISBN 90-277-0633-6; Pb 90-277-0634-4
W. Stegmüller, Collected Papers on Epistemology, Philosophy of Science and History of
Philosophy. 2 Volumes. 1977
Set ISBN 90-277-0767-7
D. M. Gabbay, Investigations in Modal and Tense Logics with Applications to Problems in
Philosophy and Linguistics. 1976
ISBN 90-277-0656-5
R. J. Bogdan, Local Induction. 1976
ISBN 90-277-0649-2
S. Nowak, Understanding and Prediction. Essays in the Methodology of Social and Behavioral
Theories. 1976
ISBN 90-277-0558-5; Pb 90-277-1199-2
P. Mittelstaedt, Philosophical Problems of Modern Physics. [Boston Studies in the Philosophy
of Science, Vol. XVIII] 1976
ISBN 90-277-0285-3; Pb 90-277-0506-2
G. Holton and W. A. Blanpied (eds.), Science and Its Public: The Changing Relationship.
[Boston Studies in the Philosophy of Science, Vol. XXXIII] 1976
ISBN 90-277-0657-3; Pb 90-277-0658-1
M. Brand and D. Walton (eds.), Action Theory. 1976
ISBN 90-277-0671-9
P. Gochet, Outline of a Nominalist Theory of Propositions. An Essay in the Theory of Meaning
and in the Philosophy of Logic. 1980
ISBN 90-277-1031-7
R. S. Cohen, P. K. Feyerabend, and M. W. Wartofsky (eds.), Essays in Memory of Imre Lakatos.
[Boston Studies in the Philosophy of Science, Vol. XXXIX] 1976
ISBN 90-277-0654-9; Pb 90-277-0655-7
R. S. Cohen and J. J. Stachel (eds.), Selected Papers of Léon Rosenfield. [Boston Studies in
the Philosophy of Science, Vol. XXI] 1979
ISBN 90-277-0651-4; Pb 90-277-0652-2
R. S. Cohen, C. A. Hooker, A. C. Michalos and J. W. van Evra (eds.), PSA 1974. Proceedings
of the 1974 Biennial Meeting of the Philosophy of Science Association. [Boston Studies in the
Philosophy of Science, Vol. XXXII] 1976
ISBN 90-277-0647-6; Pb 90-277-0648-4
Y. Fried and J. Agassi, Paranoia. A Study in Diagnosis. [Boston Studies in the Philosophy of
Science, Vol. L] 1976
ISBN 90-277-0704-9; Pb 90-277-0705-7
M. Przełeçki, K. Szaniawski and R. Wójcicki (eds.), Formal Methods in the Methodology of
Empirical Sciences. 1976
ISBN 90-277-0698-0
J. M. Vickers, Belief and Probability. 1976
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K. H. Wolff, Surrender and Catch. Experience and Inquiry Today. [Boston Studies in the
Philosophy of Science, Vol. LI] 1976
ISBN 90-277-0758-8; Pb 90-277-0765-0
K. Kosı́k, Dialectics of the Concrete. A Study on Problems of Man and World. [Boston Studies
in the Philosophy of Science, Vol. LII] 1976
ISBN 90-277-0761-8; Pb 90-277-0764-2
N. Goodman, The Structure of Appearance. 3rd ed. with an Introduction by G. Hellman.
[Boston Studies in the Philosophy of Science, Vol. LIII] 1977
ISBN 90-277-0773-1; Pb 90-277-0774-X
K. Ajdukiewicz, The Scientific World-Perspective and Other Essays, 1931-1963. Translated
from Polish. Edited and with an Introduction by J. Giedymin. 1978
ISBN 90-277-0527-5
R. L. Causey, Unity of Science. 1977
ISBN 90-277-0779-0
R. E. Grandy, Advanced Logic for Applications. 1977
ISBN 90-277-0781-2
R. P. McArthur, Tense Logic. 1976
ISBN 90-277-0697-2
L. Lindahl, Position and Change. A Study in Law and Logic. Translated from Swedish by P.
Needham. 1977
ISBN 90-277-0787-1
R. Tuomela, Dispositions. 1978
ISBN 90-277-0810-X
H. A. Simon, Models of Discovery and Other Topics in the Methods of Science. [Boston Studies
in the Philosophy of Science, Vol. LIV] 1977
ISBN 90-277-0812-6; Pb 90-277-0858-4
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ISBN 90-277-0817-7; Pb
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116. R. Tuomela, Human Action and Its Explanation. A Study on the Philosophical Foundations of
Psychology. 1977
ISBN 90-277-0824-X
117. M. Lazerowitz, The Language of Philosophy. Freud and Wittgenstein. [Boston Studies in the
Philosophy of Science, Vol. LV] 1977
ISBN 90-277-0826-6; Pb 90-277-0862-2
118. Not published 119. J. Pelc (ed.), Semiotics in Poland, 1894–1969. Translated from Polish.
1979
ISBN 90-277-0811-8
120. I. Pörn, Action Theory and Social Science. Some Formal Models. 1977 ISBN 90-277-0846-0
121. J. Margolis, Persons and Mind. The Prospects of Nonreductive Materialism. [Boston Studies
in the Philosophy of Science, Vol. LVII] 1977
ISBN 90-277-0854-1; Pb 90-277-0863-0
122. J. Hintikka, I. Niiniluoto, and E. Saarinen (eds.), Essays on Mathematical and Philosophical
Logic. 1979
ISBN 90-277-0879-7
123. T. A. F. Kuipers, Studies in Inductive Probability and Rational Expectation. 1978
ISBN 90-277-0882-7
124. E. Saarinen, R. Hilpinen, I. Niiniluoto and M. P. Hintikka (eds.), Essays in Honour of Jaakko
Hintikka on the Occasion of His 50th Birthday. 1979
ISBN 90-277-0916-5
125. G. Radnitzky and G. Andersson (eds.), Progress and Rationality in Science. [Boston Studies
in the Philosophy of Science, Vol. LVIII] 1978
ISBN 90-277-0921-1; Pb 90-277-0922-X
126. P. Mittelstaedt, Quantum Logic. 1978
ISBN 90-277-0925-4
127. K. A. Bowen, Model Theory for Modal Logic. Kripke Models for Modal Predicate Calculi.
1979
ISBN 90-277-0929-7
128. H. A. Bursen, Dismantling the Memory Machine. A Philosophical Investigation of Machine
Theories of Memory. 1978
ISBN 90-277-0933-5
129. M. W. Wartofsky, Models. Representation and the Scientific Understanding. [Boston Studies
in the Philosophy of Science, Vol. XLVIII] 1979
ISBN 90-277-0736-7; Pb 90-277-0947-5
130. D. Ihde, Technics and Praxis. A Philosophy of Technology. [Boston Studies in the Philosophy
of Science, Vol. XXIV] 1979
ISBN 90-277-0953-X; Pb 90-277-0954-8
131. J. J. Wiatr (ed.), Polish Essays in the Methodology of the Social Sciences. [Boston Studies in
the Philosophy of Science, Vol. XXIX] 1979
ISBN 90-277-0723-5; Pb 90-277-0956-4
132. W. C. Salmon (ed.), Hans Reichenbach: Logical Empiricist. 1979
ISBN 90-277-0958-0
133. P. Bieri, R.-P. Horstmann and L. Krüger (eds.), Transcendental Arguments in Science. Essays
in Epistemology. 1979
ISBN 90-277-0963-7; Pb 90-277-0964-5
134. M. Marković and G. Petrović (eds.), Praxis. Yugoslav Essays in the Philosophy and Methodology of the Social Sciences. [Boston Studies in the Philosophy of Science, Vol. XXXVI] 1979
ISBN 90-277-0727-8; Pb 90-277-0968-8
135. R. Wójcicki, Topics in the Formal Methodology of Empirical Sciences. Translated from Polish.
1979
ISBN 90-277-1004-X
136. G. Radnitzky and G. Andersson (eds.), The Structure and Development of Science. [Boston
Studies in the Philosophy of Science, Vol. LIX] 1979
ISBN 90-277-0994-7; Pb 90-277-0995-5
137. J. C. Webb, Mechanism, Mentalism and Metamathematics. An Essay on Finitism. 1980
ISBN 90-277-1046-5
138. D. F. Gustafson and B. L. Tapscott (eds.), Body, Mind and Method. Essays in Honor of Virgil
C. Aldrich. 1979
ISBN 90-277-1013-9
139. L. Nowak, The Structure of Idealization. Towards a Systematic Interpretation of the Marxian
Idea of Science. 1980
ISBN 90-277-1014-7
SYNTHESE LIBRARY
140. C. Perelman, The New Rhetoric and the Humanities. Essays on Rhetoric and Its Applications.
Translated from French and German. With an Introduction by H. Zyskind. 1979
ISBN 90-277-1018-X; Pb 90-277-1019-8
141. W. Rabinowicz, Universalizability. A Study in Morals and Metaphysics. 1979
ISBN 90-277-1020-2
142. C. Perelman, Justice, Law and Argument. Essays on Moral and Legal Reasoning. Translated
from French and German. With an Introduction by H.J. Berman. 1980
ISBN 90-277-1089-9; Pb 90-277-1090-2
143. S. Kanger and S. Öhman (eds.), Philosophy and Grammar. Papers on the Occasion of the
Quincentennial of Uppsala University. 1981
ISBN 90-277-1091-0
144. T. Pawlowski, Concept Formation in the Humanities and the Social Sciences. 1980
ISBN 90-277-1096-1
145. J. Hintikka, D. Gruender and E. Agazzi (eds.), Theory Change, Ancient Axiomatics and
Galileo’s Methodology. Proceedings of the 1978 Pisa Conference on the History and Philosophy
of Science, Volume I. 1981
ISBN 90-277-1126-7
146. J. Hintikka, D. Gruender and E. Agazzi (eds.), Probabilistic Thinking, Thermodynamics,
and the Interaction of the History and Philosophy of Science. Proceedings of the 1978 Pisa
Conference on the History and Philosophy of Science, Volume II. 1981 ISBN 90-277-1127-5
147. U. Mönnich (ed.), Aspects of Philosophical Logic. Some Logical Forays into Central Notions
of Linguistics and Philosophy. 1981
ISBN 90-277-1201-8
148. D. M. Gabbay, Semantical Investigations in Heyting’s Intuitionistic Logic. 1981
ISBN 90-277-1202-6
149. E. Agazzi (ed.), Modern Logic – A Survey. Historical, Philosophical, and Mathematical Aspects
of Modern Logic and Its Applications. 1981
ISBN 90-277-1137-2
150. A. F. Parker-Rhodes, The Theory of Indistinguishables. A Search for Explanatory Principles
below the Level of Physics. 1981
ISBN 90-277-1214-X
151. J. C. Pitt, Pictures, Images, and Conceptual Change. An Analysis of Wilfrid Sellars’ Philosophy
of Science. 1981
ISBN 90-277-1276-X; Pb 90-277-1277-8
152. R. Hilpinen (ed.), New Studies in Deontic Logic. Norms, Actions, and the Foundations of
Ethics. 1981
ISBN 90-277-1278-6; Pb 90-277-1346-4
153. C. Dilworth, Scientific Progress. A Study Concerning the Nature of the Relation between
Successive Scientific Theories. 3rd rev. ed., 1994 ISBN 0-7923-2487-0; Pb 0-7923-2488-9
154. D. Woodruff Smith and R. McIntyre, Husserl and Intentionality. A Study of Mind, Meaning,
and Language. 1982
ISBN 90-277-1392-8; Pb 90-277-1730-3
155. R. J. Nelson, The Logic of Mind. 2nd. ed., 1989
ISBN 90-277-2819-4; Pb 90-277-2822-4
156. J. F. A. K. van Benthem, The Logic of Time. A Model-Theoretic Investigation into the Varieties
of Temporal Ontology, and Temporal Discourse. 1983; 2nd ed., 1991 ISBN 0-7923-1081-0
157. R. Swinburne (ed.), Space, Time and Causality. 1983
ISBN 90-277-1437-1
158. E. T. Jaynes, Papers on Probability, Statistics and Statistical Physics. Ed. by R. D. Rozenkrantz.
1983
ISBN 90-277-1448-7; Pb (1989) 0-7923-0213-3
159. T. Chapman, Time: A Philosophical Analysis. 1982
ISBN 90-277-1465-7
160. E. N. Zalta, Abstract Objects. An Introduction to Axiomatic Metaphysics. 1983
ISBN 90-277-1474-6
161. S. Harding and M. B. Hintikka (eds.), Discovering Reality. Feminist Perspectives on Epistemology, Metaphysics, Methodology, and Philosophy of Science. 1983
ISBN 90-277-1496-7; Pb 90-277-1538-6
162. M. A. Stewart (ed.), Law, Morality and Rights. 1983
ISBN 90-277-1519-X
SYNTHESE LIBRARY
163. D. Mayr and G. Süssmann (eds.), Space, Time, and Mechanics. Basic Structures of a Physical
Theory. 1983
ISBN 90-277-1525-4
164. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. I: Elements of
Classical Logic. 1983
ISBN 90-277-1542-4
165. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. II: Extensions of
Classical Logic. 1984
ISBN 90-277-1604-8
166. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. III: Alternative to
Classical Logic. 1986
ISBN 90-277-1605-6
167. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. IV: Topics in the
Philosophy of Language. 1989
ISBN 90-277-1606-4
168. A. J. I. Jones, Communication and Meaning. An Essay in Applied Modal Logic. 1983
ISBN 90-277-1543-2
169. M. Fitting, Proof Methods for Modal and Intuitionistic Logics. 1983
ISBN 90-277-1573-4
170. J. Margolis, Culture and Cultural Entities. Toward a New Unity of Science. 1984
ISBN 90-277-1574-2
171. R. Tuomela, A Theory of Social Action. 1984
ISBN 90-277-1703-6
172. J. J. E. Gracia, E. Rabossi, E. Villanueva and M. Dascal (eds.), Philosophical Analysis in Latin
America. 1984
ISBN 90-277-1749-4
173. P. Ziff, Epistemic Analysis. A Coherence Theory of Knowledge. 1984
ISBN 90-277-1751-7
174. P. Ziff, Antiaesthetics. An Appreciation of the Cow with the Subtile Nose. 1984
ISBN 90-277-1773-7
175. W. Balzer, D. A. Pearce, and H.-J. Schmidt (eds.), Reduction in Science. Structure, Examples,
Philosophical Problems. 1984
ISBN 90-277-1811-3
176. A. Peczenik, L. Lindahl and B. van Roermund (eds.), Theory of Legal Science. Proceedings of
the Conference on Legal Theory and Philosophy of Science (Lund, Sweden, December 1983).
1984
ISBN 90-277-1834-2
177. I. Niiniluoto, Is Science Progressive? 1984
ISBN 90-277-1835-0
178. B. K. Matilal and J. L. Shaw (eds.), Analytical Philosophy in Comparative Perspective.
Exploratory Essays in Current Theories and Classical Indian Theories of Meaning and Reference. 1985
ISBN 90-277-1870-9
179. P. Kroes, Time: Its Structure and Role in Physical Theories. 1985
ISBN 90-277-1894-6
180. J. H. Fetzer, Sociobiology and Epistemology. 1985 ISBN 90-277-2005-3; Pb 90-277-2006-1
181. L. Haaparanta and J. Hintikka (eds.), Frege Synthesized. Essays on the Philosophical and
Foundational Work of Gottlob Frege. 1986
ISBN 90-277-2126-2
182. M. Detlefsen, Hilbert’s Program. An Essay on Mathematical Instrumentalism. 1986
ISBN 90-277-2151-3
183. J. L. Golden and J. J. Pilotta (eds.), Practical Reasoning in Human Affairs. Studies in Honor
of Chaim Perelman. 1986
ISBN 90-277-2255-2
184. H. Zandvoort, Models of Scientific Development and the Case of Nuclear Magnetic Resonance.
1986
ISBN 90-277-2351-6
185. I. Niiniluoto, Truthlikeness. 1987
ISBN 90-277-2354-0
186. W. Balzer, C. U. Moulines and J. D. Sneed, An Architectonic for Science. The Structuralist
Program. 1987
ISBN 90-277-2403-2
187. D. Pearce, Roads to Commensurability. 1987
ISBN 90-277-2414-8
188. L. M. Vaina (ed.), Matters of Intelligence. Conceptual Structures in Cognitive Neuroscience.
1987
ISBN 90-277-2460-1
SYNTHESE LIBRARY
189. H. Siegel, Relativism Refuted. A Critique of Contemporary Epistemological Relativism. 1987
ISBN 90-277-2469-5
190. W. Callebaut and R. Pinxten, Evolutionary Epistemology. A Multiparadigm Program, with a
Complete Evolutionary Epistemology Bibliograph. 1987
ISBN 90-277-2582-9
191. J. Kmita, Problems in Historical Epistemology. 1988
ISBN 90-277-2199-8
192. J. H. Fetzer (ed.), Probability and Causality. Essays in Honor of Wesley C. Salmon, with an
Annotated Bibliography. 1988
ISBN 90-277-2607-8; Pb 1-5560-8052-2
193. A. Donovan, L. Laudan and R. Laudan (eds.), Scrutinizing Science. Empirical Studies of
Scientific Change. 1988
ISBN 90-277-2608-6
194. H.R. Otto and J.A. Tuedio (eds.), Perspectives on Mind. 1988
ISBN 90-277-2640-X
195. D. Batens and J.P. van Bendegem (eds.), Theory and Experiment. Recent Insights and New
Perspectives on Their Relation. 1988
ISBN 90-277-2645-0
196. J. Österberg, Self and Others. A Study of Ethical Egoism. 1988
ISBN 90-277-2648-5
197. D.H. Helman (ed.), Analogical Reasoning. Perspectives of Artificial Intelligence, Cognitive
Science, and Philosophy. 1988
ISBN 90-277-2711-2
198. J. Wolenśki, Logic and Philosophy in the Lvov-Warsaw School. 1989 ISBN 90-277-2749-X
199. R. Wójcicki, Theory of Logical Calculi. Basic Theory of Consequence Operations. 1988
ISBN 90-277-2785-6
200. J. Hintikka and M.B. Hintikka, The Logic of Epistemology and the Epistemology of Logic.
Selected Essays. 1989
ISBN 0-7923-0040-8; Pb 0-7923-0041-6
201. E. Agazzi (ed.), Probability in the Sciences. 1988
ISBN 90-277-2808-9
202. M. Meyer (ed.), From Metaphysics to Rhetoric. 1989
ISBN 90-277-2814-3
203. R.L. Tieszen, Mathematical Intuition. Phenomenology and Mathematical Knowledge. 1989
ISBN 0-7923-0131-5
204. A. Melnick, Space, Time, and Thought in Kant. 1989
ISBN 0-7923-0135-8
205. D.W. Smith, The Circle of Acquaintance. Perception, Consciousness, and Empathy. 1989
ISBN 0-7923-0252-4
206. M.H. Salmon (ed.), The Philosophy of Logical Mechanism. Essays in Honor of Arthur W.
Burks. With his Responses, and with a Bibliography of Burk’s Work. 1990
ISBN 0-7923-0325-3
207. M. Kusch, Language as Calculus vs. Language as Universal Medium. A Study in Husserl,
Heidegger, and Gadamer. 1989
ISBN 0-7923-0333-4
208. T.C. Meyering, Historical Roots of Cognitive Science. The Rise of a Cognitive Theory of
Perception from Antiquity to the Nineteenth Century. 1989
ISBN 0-7923-0349-0
209. P. Kosso, Observability and Observation in Physical Science. 1989
ISBN 0-7923-0389-X
210. J. Kmita, Essays on the Theory of Scientific Cognition. 1990
ISBN 0-7923-0441-1
211. W. Sieg (ed.), Acting and Reflecting. The Interdisciplinary Turn in Philosophy. 1990
ISBN 0-7923-0512-4
212. J. Karpinśki, Causality in Sociological Research. 1990
ISBN 0-7923-0546-9
213. H.A. Lewis (ed.), Peter Geach: Philosophical Encounters. 1991
ISBN 0-7923-0823-9
214. M. Ter Hark, Beyond the Inner and the Outer. Wittgenstein’s Philosophy of Psychology. 1990
ISBN 0-7923-0850-6
215. M. Gosselin, Nominalism and Contemporary Nominalism. Ontological and Epistemological
Implications of the Work of W.V.O. Quine and of N. Goodman. 1990 ISBN 0-7923-0904-9
216. J.H. Fetzer, D. Shatz and G. Schlesinger (eds.), Definitions and Definability. Philosophical
Perspectives. 1991
ISBN 0-7923-1046-2
217. E. Agazzi and A. Cordero (eds.), Philosophy and the Origin and Evolution of the Universe.
1991
ISBN 0-7923-1322-4
SYNTHESE LIBRARY
218. M. Kusch, Foucault’s Strata and Fields. An Investigation into Archaeological and Genealogical
Science Studies. 1991
ISBN 0-7923-1462-X
219. C.J. Posy, Kant’s Philosophy of Mathematics. Modern Essays. 1992
ISBN 0-7923-1495-6
220. G. Van de Vijver, New Perspectives on Cybernetics. Self-Organization, Autonomy and Connectionism. 1992
ISBN 0-7923-1519-7
221. J.C. Nyı́ri, Tradition and Individuality. Essays. 1992
ISBN 0-7923-1566-9
222. R. Howell, Kant’s Transcendental Deduction. An Analysis of Main Themes in His Critical
Philosophy. 1992
ISBN 0-7923-1571-5
223. A. Garcı́a de la Sienra, The Logical Foundations of the Marxian Theory of Value. 1992
ISBN 0-7923-1778-5
224. D.S. Shwayder, Statement and Referent. An Inquiry into the Foundations of Our Conceptual
Order. 1992
ISBN 0-7923-1803-X
225. M. Rosen, Problems of the Hegelian Dialectic. Dialectic Reconstructed as a Logic of Human
Reality. 1993
ISBN 0-7923-2047-6
226. P. Suppes, Models and Methods in the Philosophy of Science: Selected Essays. 1993
ISBN 0-7923-2211-8
227. R. M. Dancy (ed.), Kant and Critique: New Essays in Honor of W. H. Werkmeister. 1993
ISBN 0-7923-2244-4
228. J. Woleński (ed.), Philosophical Logic in Poland. 1993
ISBN 0-7923-2293-2
229. M. De Rijke (ed.), Diamonds and Defaults. Studies in Pure and Applied Intensional Logic.
1993
ISBN 0-7923-2342-4
230. B.K. Matilal and A. Chakrabarti (eds.), Knowing from Words. Western and Indian Philosophical
Analysis of Understanding and Testimony. 1994
ISBN 0-7923-2345-9
231. S.A. Kleiner, The Logic of Discovery. A Theory of the Rationality of Scientific Research. 1993
ISBN 0-7923-2371-8
232. R. Festa, Optimum Inductive Methods. A Study in Inductive Probability, Bayesian Statistics,
and Verisimilitude. 1993
ISBN 0-7923-2460-9
233. P. Humphreys (ed.), Patrick Suppes: Scientific Philosopher. Vol. 1: Probability and Probabilistic
Causality. 1994
ISBN 0-7923-2552-4
234. P. Humphreys (ed.), Patrick Suppes: Scientific Philosopher. Vol. 2: Philosophy of Physics,
Theory Structure, and Measurement Theory. 1994
ISBN 0-7923-2553-2
235. P. Humphreys (ed.), Patrick Suppes: Scientific Philosopher. Vol. 3: Language, Logic, and
Psychology. 1994
ISBN 0-7923-2862-0
Set ISBN (Vols 233–235) 0-7923-2554-0
236. D. Prawitz and D. Westerståhl (eds.), Logic and Philosophy of Science in Uppsala. Papers
from the 9th International Congress of Logic, Methodology, and Philosophy of Science. 1994
ISBN 0-7923-2702-0
237. L. Haaparanta (ed.), Mind, Meaning and Mathematics. Essays on the Philosophical Views of
Husserl and Frege. 1994
ISBN 0-7923-2703-9
238. J. Hintikka (ed.), Aspects of Metaphor. 1994
ISBN 0-7923-2786-1
239. B. McGuinness and G. Oliveri (eds.), The Philosophy of Michael Dummett. With Replies from
Michael Dummett. 1994
ISBN 0-7923-2804-3
240. D. Jamieson (ed.), Language, Mind, and Art. Essays in Appreciation and Analysis, In Honor
of Paul Ziff. 1994
ISBN 0-7923-2810-8
241. G. Preyer, F. Siebelt and A. Ulfig (eds.), Language, Mind and Epistemology. On Donald
Davidson’s Philosophy. 1994
ISBN 0-7923-2811-6
242. P. Ehrlich (ed.), Real Numbers, Generalizations of the Reals, and Theories of Continua. 1994
ISBN 0-7923-2689-X
SYNTHESE LIBRARY
243. G. Debrock and M. Hulswit (eds.), Living Doubt. Essays concerning the epistemology of
Charles Sanders Peirce. 1994
ISBN 0-7923-2898-1
244. J. Srzednicki, To Know or Not to Know. Beyond Realism and Anti-Realism. 1994
ISBN 0-7923-2909-0
245. R. Egidi (ed.), Wittgenstein: Mind and Language. 1995
ISBN 0-7923-3171-0
246. A. Hyslop, Other Minds. 1995
ISBN 0-7923-3245-8
247. L. Pólos and M. Masuch (eds.), Applied Logic: How, What and Why. Logical Approaches to
Natural Language. 1995
ISBN 0-7923-3432-9
248. M. Krynicki, M. Mostowski and L.M. Szczerba (eds.), Quantifiers: Logics, Models and Computation. Volume One: Surveys. 1995
ISBN 0-7923-3448-5
249. M. Krynicki, M. Mostowski and L.M. Szczerba (eds.), Quantifiers: Logics, Models and Computation. Volume Two: Contributions. 1995
ISBN 0-7923-3449-3
Set ISBN (Vols 248 + 249) 0-7923-3450-7
250. R.A. Watson, Representational Ideas from Plato to Patricia Churchland. 1995
ISBN 0-7923-3453-1
251. J. Hintikka (ed.), From Dedekind to Gödel. Essays on the Development of the Foundations of
Mathematics. 1995
ISBN 0-7923-3484-1
252. A. Wiśniewski, The Posing of Questions. Logical Foundations of Erotetic Inferences. 1995
ISBN 0-7923-3637-2
253. J. Peregrin, Doing Worlds with Words. Formal Semantics without Formal Metaphysics. 1995
ISBN 0-7923-3742-5
254. I.A. Kieseppä, Truthlikeness for Multidimensional, Quantitative Cognitive Problems. 1996
ISBN 0-7923-4005-1
255. P. Hugly and C. Sayward: Intensionality and Truth. An Essay on the Philosophy of A.N. Prior.
1996
ISBN 0-7923-4119-8
256. L. Hankinson Nelson and J. Nelson (eds.): Feminism, Science, and the Philosophy of Science.
1997
ISBN 0-7923-4162-7
257. P.I. Bystrov and V.N. Sadovsky (eds.): Philosophical Logic and Logical Philosophy. Essays in
Honour of Vladimir A. Smirnov. 1996
ISBN 0-7923-4270-4
258. Å.E. Andersson and N-E. Sahlin (eds.): The Complexity of Creativity. 1996
ISBN 0-7923-4346-8
259. M.L. Dalla Chiara, K. Doets, D. Mundici and J. van Benthem (eds.): Logic and Scientific Methods. Volume One of the Tenth International Congress of Logic, Methodology and Philosophy
of Science, Florence, August 1995. 1997
ISBN 0-7923-4383-2
260. M.L. Dalla Chiara, K. Doets, D. Mundici and J. van Benthem (eds.): Structures and Norms
in Science. Volume Two of the Tenth International Congress of Logic, Methodology and
Philosophy of Science, Florence, August 1995. 1997
ISBN 0-7923-4384-0
Set ISBN (Vols 259 + 260) 0-7923-4385-9
261. A. Chakrabarti: Denying Existence. The Logic, Epistemology and Pragmatics of Negative
Existentials and Fictional Discourse. 1997
ISBN 0-7923-4388-3
262. A. Biletzki: Talking Wolves. Thomas Hobbes on the Language of Politics and the Politics of
Language. 1997
ISBN 0-7923-4425-1
263. D. Nute (ed.): Defeasible Deontic Logic. 1997
ISBN 0-7923-4630-0
264. U. Meixner: Axiomatic Formal Ontology. 1997
ISBN 0-7923-4747-X
265. I. Brinck: The Indexical ‘I’. The First Person in Thought and Language. 1997
ISBN 0-7923-4741-2
266. G. Hölmström-Hintikka and R. Tuomela (eds.): Contemporary Action Theory. Volume 1:
Individual Action. 1997
ISBN 0-7923-4753-6; Set: 0-7923-4754-4
SYNTHESE LIBRARY
267. G. Hölmström-Hintikka and R. Tuomela (eds.): Contemporary Action Theory. Volume 2:
Social Action. 1997
ISBN 0-7923-4752-8; Set: 0-7923-4754-4
268. B.-C. Park: Phenomenological Aspects of Wittgenstein’s Philosophy. 1998
ISBN 0-7923-4813-3
269. J. Paśniczek: The Logic of Intentional Objects. A Meinongian Version of Classical Logic. 1998
Hb ISBN 0-7923-4880-X; Pb ISBN 0-7923-5578-4
270. P.W. Humphreys and J.H. Fetzer (eds.): The New Theory of Reference. Kripke, Marcus, and
Its Origins. 1998
ISBN 0-7923-4898-2
271. K. Szaniawski, A. Chmielewski and J. Wolenśki (eds.): On Science, Inference, Information
and Decision Making. Selected Essays in the Philosophy of Science. 1998
ISBN 0-7923-4922-9
272. G.H. von Wright: In the Shadow of Descartes. Essays in the Philosophy of Mind. 1998
ISBN 0-7923-4992-X
273. K. Kijania-Placek and J. Wolenśki (eds.): The Lvov–Warsaw School and Contemporary Philosophy. 1998
ISBN 0-7923-5105-3
274. D. Dedrick: Naming the Rainbow. Colour Language, Colour Science, and Culture. 1998
ISBN 0-7923-5239-4
275. L. Albertazzi (ed.): Shapes of Forms. From Gestalt Psychology and Phenomenology to Ontology and Mathematics. 1999
ISBN 0-7923-5246-7
276. P. Fletcher: Truth, Proof and Infinity. A Theory of Constructions and Constructive Reasoning.
1998
ISBN 0-7923-5262-9
277. M. Fitting and R.L. Mendelsohn (eds.): First-Order Modal Logic. 1998
Hb ISBN 0-7923-5334-X; Pb ISBN 0-7923-5335-8
278. J.N. Mohanty: Logic, Truth and the Modalities from a Phenomenological Perspective. 1999
ISBN 0-7923-5550-4
279. T. Placek: Mathematical Intiutionism and Intersubjectivity. A Critical Exposition of Arguments
for Intuitionism. 1999
ISBN 0-7923-5630-6
280. A. Cantini, E. Casari and P. Minari (eds.): Logic and Foundations of Mathematics. 1999
ISBN 0-7923-5659-4
set ISBN 0-7923-5867-8
281. M.L. Dalla Chiara, R. Giuntini and F. Laudisa (eds.): Language, Quantum, Music. 1999
ISBN 0-7923-5727-2; set ISBN 0-7923-5867-8
282. R. Egidi (ed.): In Search of a New Humanism. The Philosophy of Georg Hendrik von Wright.
1999
ISBN 0-7923-5810-4
283. F. Vollmer: Agent Causality. 1999
ISBN 0-7923-5848-1
284. J. Peregrin (ed.): Truth and Its Nature (if Any). 1999
ISBN 0-7923-5865-1
285. M. De Caro (ed.): Interpretations and Causes. New Perspectives on Donald Davidson’s Philosophy. 1999
ISBN 0-7923-5869-4
286. R. Murawski: Recursive Functions and Metamathematics. Problems of Completeness and
Decidability, Gödel’s Theorems. 1999
ISBN 0-7923-5904-6
287. T.A.F. Kuipers: From Instrumentalism to Constructive Realism. On Some Relations between
Confirmation, Empirical Progress, and Truth Approximation. 2000
ISBN 0-7923-6086-9
288. G. Holmström-Hintikka (ed.): Medieval Philosophy and Modern Times. 2000
ISBN 0-7923-6102-4
289. E. Grosholz and H. Breger (eds.): The Growth of Mathematical Knowledge. 2000
ISBN 0-7923-6151-2
SYNTHESE LIBRARY
290. G. Sommaruga: History and Philosophy of Constructive Type Theory. 2000
ISBN 0-7923-6180-6
291. J. Gasser (ed.): A Boole Anthology. Recent and Classical Studies in the Logic of George Boole.
2000
ISBN 0-7923-6380-9
292. V.F. Hendricks, S.A. Pedersen and K.F. Jørgensen (eds.): Proof Theory. History and Philosophical Significance. 2000
ISBN 0-7923-6544-5
293. W.L. Craig: The Tensed Theory of Time. A Critical Examination. 2000 ISBN 0-7923-6634-4
294. W.L. Craig: The Tenseless Theory of Time. A Critical Examination. 2000
ISBN 0-7923-6635-2
295. L. Albertazzi (ed.): The Dawn of Cognitive Science. Early European Contributors. 2001
ISBN 0-7923-6799-5
296. G. Forrai: Reference, Truth and Conceptual Schemes. A Defense of Internal Realism. 2001
ISBN 0-7923-6885-1
297. V.F. Hendricks, S.A. Pedersen and K.F. Jørgensen (eds.): Probability Theory. Philosophy,
Recent History and Relations to Science. 2001
ISBN 0-7923-6952-1
298. M. Esfeld: Holism in Philosophy of Mind and Philosophy of Physics. 2001
ISBN 0-7923-7003-1
299. E.C. Steinhart: The Logic of Metaphor. Analogous Parts of Possible Worlds. 2001
ISBN 0-7923-7004-X
300. P. Gärdenfors: The Dynamics of Thought. 2005
ISBN 1-4020-3398-2
301. T.A.F. Kuipers: Structures in Science Heuristic Patterns Based on Cognitive Structures. An
Advanced Textbook in Neo-Classical Philosophy of Science. 2001
ISBN 0-7923-7117-8
302. G. Hon and S.S. Rakover (eds.): Explanation. Theoretical Approaches and Applications. 2001
ISBN 1-4020-0017-0
303. G. Holmström-Hintikka, S. Lindström and R. Sliwinski (eds.): Collected Papers of Stig Kanger
with Essays on his Life and Work. Vol. I. 2001
ISBN 1-4020-0021-9; Pb ISBN 1-4020-0022-7
304. G. Holmström-Hintikka, S. Lindström and R. Sliwinski (eds.): Collected Papers of Stig Kanger
with Essays on his Life and Work. Vol. II. 2001
ISBN 1-4020-0111-8; Pb ISBN 1-4020-0112-6
305. C.A. Anderson and M. Zelëny (eds.): Logic, Meaning and Computation. Essays in Memory
of Alonzo Church. 2001
ISBN 1-4020-0141-X
306. P. Schuster, U. Berger and H. Osswald (eds.): Reuniting the Antipodes – Constructive and
Nonstandard Views of the Continuum. 2001
ISBN 1-4020-0152-5
307. S.D. Zwart: Refined Verisimilitude. 2001
ISBN 1-4020-0268-8
308. A.-S. Maurin: If Tropes. 2002
ISBN 1-4020-0656-X
309. H. Eilstein (ed.): A Collection of Polish Works on Philosophical Problems of Time and Spacetime. 2002
ISBN 1-4020-0670-5
310. Y. Gauthier: Internal Logic. Foundations of Mathematics from Kronecker to Hilbert. 2002
ISBN 1-4020-0689-6
311. E. Ruttkamp: A Model-Theoretic Realist Interpretation of Science. 2002
ISBN 1-4020-0729-9
312. V. Rantala: Explanatory Translation. Beyond the Kuhnian Model of Conceptual Change. 2002
ISBN 1-4020-0827-9
313. L. Decock: Trading Ontology for Ideology. 2002
ISBN 1-4020-0865-1
SYNTHESE LIBRARY
314. O. Ezra: The Withdrawal of Rights. Rights from a Different Perspective. 2002
ISBN 1-4020-0886-4
315. P. Gärdenfors, J. Woleński and K. Kijania-Placek: In the Scope of Logic, Methodology and
Philosophy of Science. Volume One of the 11th International Congress of Logic, Methodology
and Philosophy of Science, Cracow, August 1999. 2002
ISBN 1-4020-0929-1; Pb 1-4020-0931-3
316. P. Gärdenfors, J. Woleński and K. Kijania-Placek: In the Scope of Logic, Methodology and
Philosophy of Science. Volume Two of the 11th International Congress of Logic, Methodology
and Philosophy of Science, Cracow, August 1999. 2002
ISBN 1-4020-0930-5; Pb 1-4020-0931-3
317. M.A. Changizi: The Brain from 25,000 Feet. High Level Explorations of Brain Complexity,
Perception, Induction and Vagueness. 2003
ISBN 1-4020-1176-8
318. D.O. Dahlstrom (ed.): Husserl’s Logical Investigations. 2003
ISBN 1-4020-1325-6
319. A. Biletzki: (Over)Interpreting Wittgenstein. 2003
ISBN Hb 1-4020-1326-4; Pb 1-4020-1327-2
320. A. Rojszczak, J. Cachro and G. Kurczewski (eds.): Philosophical Dimensions of Logic and
Science. Selected Contributed Papers from the 11th International Congress of Logic, Methodology, and Philosophy of Science, Kraków, 1999. 2003
ISBN 1-4020-1645-X
321. M. Sintonen, P. Ylikoski and K. Miller (eds.): Realism in Action. Essays in the Philosophy of
the Social Sciences. 2003
ISBN 1-4020-1667-0
322. V.F. Hendricks, K.F. Jørgensen and S.A. Pedersen (eds.): Knowledge Contributors. 2003
ISBN Hb 1-4020-1747-2; Pb 1-4020-1748-0
323. J. Hintikka, T. Czarnecki, K. Kijania-Placek, T. Placek and A. Rojszczak † (eds.): Philosophy
and Logic In Search of the Polish Tradition. Essays in Honour of Jan Woleński on the Occasion
of his 60th Birthday. 2003
ISBN 1-4020-1721-9
324. L.M. Vaina, S.A. Beardsley and S.K. Rushton (eds.): Optic Flow and Beyond. 2004
ISBN 1-4020-2091-0
325. D. Kolak (ed.): I Am You. The Metaphysical Foundations of Global Ethics. 2004
ISBN 1-4020-2999-3
326. V. Stepin: Theoretical Knowledge. 2005
ISBN 1-4020-3045-2
327. P. Mancosu, K.F. Jørgensen and S.A. Pedersen (eds.): Visualization, Explanation and Reasoning Styles in Mathematics. 2005
ISBN 1-4020-3334-6
328. A. Rojszczak (author) and J. Wolenski (ed.): From the Act of Judging to the Sentence. The
Problem of Truth Bearers from Bolzano to Tarski. 2005
ISBN 1-4020-3396-6
329. A-V. Pietarinen: Signs of Logic. Peircean Themes on the Philosophy of Language, Games,
and Communication. 2006
ISBN-10 1-4020-3728-7
330. A. Aliseda: Abductive Reasoning. Logical Investigations into Discovery and Explanation. 2006
ISBN-10 1-4020-3906-9
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