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1ABDUCTIVE AND INDUCTIVE
REASONING: BACKGROUND AND
ISSUES
Peter A. Flach and Antonis C. Kakas
1.1 INTRODUCTION
This collection is devoted to the analysis and application of abductive and inductive
reasoning in a common context, studying their relation and possible ways for integra-
tion. There are several reasons for doing so. One reason is practical, and based on the
expectation that abduction and induction are sufficiently similar to allow for a tight
integration in practical systems, yet sufficiently complementaryfor this integration to
be useful and productive.
Our interest in combining abduction and induction is not purely practical, however.
Conceptually, the relation between abduction and induction is not well understood.
More precisely, there are several, mutually incompatible ways to perceive this relation.
For instance, Josephson writes that ‘it is possible to treat every good (...) inductive
generalisation as an instance of abduction’ (Josephson, 1994, p.19), while Michalski
has it that ‘inductive inference was defined as a process of generatingdescriptions that
imply original facts in the context of background knowledge. Such a general definition
includes inductive generalisation and abduction as special cases’ (Michalski, 1987,
p.188).
One can argue that such incompatible viewpointsindicate that abduction and induc-
tion themselves are not well-defined. Once their definitions have been fixed, studying
their relation becomes a technical rather than a conceptual matter. However, it is not
self-evident why there should exist absolute, Platonic ideals of abduction and induc-
tion, waiting to be discovered and captured once and for all by an appropriate defini-
1
2P.A. FLACH AND A.C. KAKAS
tion. As with most theoretical notions, it is more a matter of pragmatics, of how useful
a particular definition is going to be in a particular context.
A more relativistic viewpoint is often more productive in these matters, looking at
situations where it might be more appropriate to distinguish between abduction and
induction, and also at cases where it seems more useful to unify them. Sometimes
we want to stress that abduction and induction spring froma common root (say hypo-
thetical or non-deductive reasoning), and sometimes we want to take a finer grained
perspective by looking at what distinguishes them (e.g.the way in which the hypothe-
sis extends our knowledge). The following questions will therefore be our guidelines:
When and how will it be usefulto unify, or distinguish, abductionandinduction?
How can abduction and induction be usefully integrated?
Here and elsewhere, by unification we mean considering them as part of a common
framework, while by integration we mean employingthem together,in some mutually
enhancing way, for a practical purpose.
The current state of affairs with regard to these issues is perhaps most adequately
described as an ongoing debate, and the reader should look upon the following chap-
ters as representing a range of possible positionsin this debate. One of our aims in this
introductory chapter is to chart the terrain where the debate is taking place, and to po-
sition the contributions to this volume within the terrain. We will retrace some of the
main issues in this debate to their historical background. We will also attempt a syn-
thesis of some of these issues, primarily motivated by work in artificial intelligence,
sometimes taking positions that may not be shared by every authorin this volume.
The outline of this chapter is as follows. In Section 1.2 we discuss the philosophi-
cal and logical origins of abduction and induction. In Section 1.3 we analyse previous
work on abduction and induction in the context of logic programming and artificial
intelligence, and attempt a (partial) synthesis of this work. Section 1.4 considers the
integration of abduction and induction in artificial intelligence, and Section 1.5 con-
cludes.
Before we embark on this, let us express our sincere thanks for all authors con-
tributing to this volume, without whom we couldn’t have written this introduction –
indeed, some of the viewpoints we’readvocating have been strongly influenced by the
other contributions. Wherever possible we have tried to indicate the original source of
a viewpoint we discuss, but we apologise in advance for any omissions in this respect.
1.2 ABDUCTION AND INDUCTION IN PHILOSOPHY AND LOGIC
In this section we discuss various possible viewpoints on abduction and induction
that can be found in the philosophical and logical literature.The philosophical issue is
mainly one of categorisation (which forms of reasoning exist?), while the logical issue
is one of formalisation.
As far as categorisation is concerned,it seems uncontroversial that deductionshould
be singled out as a separate reasoning form which is fundamentally different from any
other form of reasoning by virtue of its truth-preserving nature. The question, then,
is how non-deductive reasoning should be mapped out. One school of thought holds
ABDUCTIVE AND INDUCTIVE REASONING: BACKGROUND AND ISSUES 3
that no further sub-categorisation is needed: all non-deductive logic is of the same
category, which is called induction. Another school of thought argues for a further
division of non-deductive reasoning into abduction and induction. We will discuss
these two viewpoints in the next two sections. A general analysis of the relationship
between abduction and induction from several different perspectives is also carried out
by Bessant in her contribution to this volume.
1.2.1 Induction as non-deductive reasoning
Let us start by taking a look at a textbook definition of induction.
Arguments can be classified in terms of whether their premisses provide (1) con-
clusive support, (2) partial support, or (3) only the appearance of support (that is,
no real support at all.) When we say that the premisses provide conclusive support
for the conclusion, we mean that if the premisses of the argument were all true,
it would be impossible for the conclusion of the argument to be false. Arguments
that have this characteristic are called deductive arguments. When we say that
the premisses of an argument provide partial support for the conclusion, we mean
that if the premisses were true, they would give us good reasons – but not con-
clusive reasons – to accept the conclusion. That is to say, although the premisses,
if true, provide some evidence to support the conclusion, the conclusion may still
be false. Arguments of this type are called inductive arguments. (Salmon, 1984,
p.32)
This establishes a dichotomy of the set of non-fallacious arguments into either deduc-
tive or inductivearguments, the distinction being based on the way they are supported
or justified: while deductive support is an absolute notion, inductive support must be
expressed in relative (e.g. quantitative) terms.
Salmon further classifies inductive arguments into arguments based on samples,
arguments from analogy, and statistical syllogisms. Arguments based on samples or
inductive generalisations have the following general form:
Xpercent of observed Fs are Gs;
therefore, (approximately) Xpercent of all Fs are Gs.
Arguments from analogy look as follows:
Objects of type Xhave properties F,G,H, ...;
objects of type Yhave properties F,G,H, ..., and also property Z;
therefore, objects of type Xhave property Zas well.
Finally, statistical syllogisms have the following abstract form:
Xpercent of all Fs are Gs;
ais an F;
therefore, ais a G.
Here Xis understood to be a high percentage (i.e. if Xis close to zero, the conclusion
must be changed to ‘ais not a G’).
There are several important things to note. One is that some premisses and conclu-
sions are statistical, talking about relative frequencies (‘Xpercent of’), while others
are categorical. In general, we can obtain a categorical special case from arguments
4P.A. FLACH AND A.C. KAKAS
involving a relative frequency Xby putting X
=
100%. Obviously, the categorical
variant of statistical syllogism is purely deductive. More importantly, categorical in-
ductive generalisation has the following form:
All observed Fs are Gs;
therefore, all Fs are Gs.
As argued in Section 1.2.3, most inductive arguments in artificial intelligence are cat-
egorical, as this facilitates further reasoning with the inductiveconclusion.
Regardless of whether inductive arguments are statistical or categorical, we must
have a way to assess their strength or inductive support, and this is the second way
in which statistics comes into play. Given evidence Ecollected in the premisses of
an inductive argument, we want to know the degree of belief we should attach to
the hypothetical conclusion H. It is widely believed that degrees of belief should be
quantified as (subjective) probabilities – in particular, the degree of belief in Hgiven
Eis usually identified with the conditional probability P
(
H
j
E
)
. The probabilistic
formalisation of inductive support is known as confirmation theory.
It is tempting to consider the degree of confirmation of hypothesis Hby evidence
Eas the degree of validity of the inductive argument ‘E, therefore H’, and treat this
‘inductive validity’ as analogous to deductive validity. Following this line of thought,
several authors speak of confirmationtheory as establishing an ‘inductive logic’:
‘What we call inductive logic is often called the theory of nondemonstrative or
nondeductive inference. Since we use the term ‘inductive’ in the wide sense of
‘nondeductive’, we might call it the theory of inductive inference... However,
it should be noticed that the term ‘inference’ must here, in inductive logic, not
be understood in the same sense as in deductive logic. Deductive and inductive
logic are analogous in one respect: both investigate logical relations between sen-
tences; the first studies the relation of [entailment], the second that of degree of
confirmation which may be regarded as a numerical measure for a partial [entail-
ment]... The term ‘inference’ in its customary use implies a transition from given
sentences to new sentences or an acquisition of a new sentence on the basis of
sentences already possessed. However, only deductive inference is inference in
this sense.’ (Carnap, 1950,
x
44B, pp.205–6)
In other words, confirmation theory by itself does not establish a consequence relation
(a subset of L
L, where Lis the logical language), since any evidence will confirm
any hypothesis to a certain degree. Inductive logic based on confirmation theory does
not have a proof theory in the traditional sense, and therefore does not guide us in
generating possible inductive hypotheses from evidence, but rather evaluates a given
hypothesis against given evidence. The inductive logic arising from confirmation the-
ory is a logic of hypothesis evaluation rather than hypothesis generation. This dis-
tinction between hypothesis generation and hypothesis evaluation is an important one
in the present context, and we will have more to say about the issue in Sections 1.2.3
and 1.3.
To summarise, one way to categorise arguments is by dividing them into non-
defeasible (i.e. deductive) and defeasible but supported (i.e. inductive) arguments. A
further sub-categorisation can be obtained by looking at the syntactic form of the ar-
gument. Confirmation theory quantifies inductive support in probabilistic terms, and
deals primarily with hypothesis evaluation.
ABDUCTIVE AND INDUCTIVE REASONING: BACKGROUND AND ISSUES 5
1.2.2 Deduction, induction and abduction
After having discussed the view that identifies induction with all non-deductive rea-
soning, we next turn to the trichotomy of deductive, inductive and abductivereasoning
proposed by the American philosopher Charles Sanders Peirce (1839–1914).
Peirce was a very prolific thinker and writer, but only a fraction of his work was
published during his life. His collected works (Peirce, 1958)1therefore reflect, first
and foremost, the evolutionof his thinking, and should be approached with some care.
With respect to abduction and induction Peirce went through a substantial change of
mind during the decade 1890 – 1900 (Fann, 1970). It is perhaps fair to say that many
of the current controversies surroundingabduction seem to be attributable to Peirce’s
mindchange. Below we will briefly discuss both his early, syllogistic theory, which
can be seen as a precursor to the current use of abduction in logic programming and
artificial intelligence, and his later, inferential theory, in which abduction represents
the hypothesis generation part of explanatory reasoning.
Peirce’s syllogistic theory. In Peirce’s days logic was not nearly as well-developed
as it is today, and his first attempt to classify arguments (which he considers ‘the
chief business of the logician’ (2.619), follows Aristotle in employing syllogisms.
The following syllogism is known as Barbara:
All the beans from this bag are white;
these beans are from this bag;
therefore, these beans are white.
The idea is that this valid argument represents a particular instantiation of a reason-
ing scheme, and that any alternative instantiation represents another argument that is
likewise valid. Syllogisms should thus be interpreted as argument schemas.
Two other syllogisms are obtained from Barbara if we exchange the conclusion (or
Result, as Peirce calls it) with either the major premiss (the Rule) or the minor premiss
(the Case):
Case. — These beans are from this bag.
Result. — These beans are white.
Rule. — All the beans from this bag are white.
Rule. — All the beans from this bag are white.
Result. — These beans are white.
Case. — These beans are from this bag.
The first of these two syllogisms (inference of the rule from the case and the result)
can be recognised as what we called previously a categorical inductive generalisation,
generalising from a sample of beans to the population of beans in the bag. The sort
of inference exemplified by the second syllogism (inference of the case from the rule
1References to Peirce’s collected papers take the form X
:
Y, where Xdenotes the volume number and Ythe
paragraph within the volume.
6P.A. FLACH AND A.C. KAKAS
and the result) Peirce calls making a hypothesis or, briefly, hypothesis – the term
‘abduction’ is introduced only in his later theory.2
Peirce thus arrives at the following classification of inference(2.623):
Inference
8
<
:
Deductive or Analytic
Synthetic
Induction
Hypothesis
Comparing this classification with the one obtained in Section 1.2.1, we can point
out the following similarities. That what was called induction previouslycorresponds
to what Peirce calls synthetic inference (another term he uses is ampliative reason-
ing, since it amplifies, or goes beyond, the information contained in the premisses).
Furthermore, what Peirce calls induction corresponds to what we called inductive gen-
eralisation in Section 1.2.1.3
On the other hand, the motivations for these classifications are quite different in
each case. In Section 1.2.1 we were concentrating on the different kinds of support or
confirmation that arguments provide, and we noticed that this is essentially the same
for all non-deductive reasoning. When we concentrate instead on the syllogistic form
of arguments, we find this to correspond more naturally to a trichotomy, separating
non-deductivereasoning into two subcategories. As Horn clause logic is in some sense
a modern upgrade of syllogistic logic, it is perhaps not surprising that the distinction
between abduction and induction in logic programming follows Peirce’s syllogistic
classification to a large extent. This will be further taken up in Section 1.3.
Peirce’s inferential theory. In his later theory of reasoning Peirce abandoned the
idea of a syllogistic classification of reasoning:
‘(...) I was too much taken up in considering syllogistic forms and the doctrine
of logical extension and comprehension, both of which I made more fundamental
than they really are. As long as I held that opinion, my conceptions of Abduc-
tion necessarily confused two different kinds of reasoning.’ (Peirce, 1958, 2.102,
written in 1902)
Instead, he identified the three reasoning forms – abduction, deduction and induction
– with the three stages of scientific inquiry: hypothesis generation, prediction, and
evaluation (Figure 1.1). The underlying model of scientific inquiry runs as follows.
When confronted with a number of observations she seeks to explain, the scientist
comes up with an initial hypothesis; then she investigates what other consequences
this theory, were it true, would have; and finally she evaluates the extent to which
these predicted consequences agree withreality. Peirce calls the first stage, coming up
with a hypothesis to explain the initial observations, abduction; predictions are derived
from a suggested hypothesis by deduction; and the credibility of that hypothesis is
estimated through its predictions by induction. We will now take a closer look at these
stages.
2Peirce also uses the term ‘retroduction’, a translation of the Greek word απαγωγ
´
ηused by Aristotle (trans-
lated by others as ‘reduction’).
3It should be noted that, although the above syllogistic arguments are all categorical, Peirce also considered
statistical versions.
ABDUCTIVE AND INDUCTIVE REASONING: BACKGROUND AND ISSUES 7
observations
predictions
hypothesis
abduction
deduction
induction
Y
T
I
L
A
E
R
Figure 1.1 The three stages of scientific inquiry.
Abduction is defined by Peirce as the process of forming an explanatory hypothe-
sis from an observation requiring explanation. This process is not algorithmic: ‘the
abductive suggestion comes to us like a flash. It is an act of insight, although of ex-
tremely fallible insight’ (Peirce, 1958, 5.181). Elsewhere Peirce describes abduction
as ‘a capacity for ‘guessing’ right’, a ‘mysterious guessing power’ underlying all sci-
entific research (Peirce, 1958, 6.530). Its non-algorithmic character notwithstanding,
abduction
‘is logical inference (...) having a perfectly definite logical form. (...) Namely, the
hypothesis cannot be admitted, even as a hypothesis, unless it be supposed that it
would account for the facts or some of them. The form of inference, therefore, is
this:
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.’ (Peirce, 1958, 5.188–9)
Let us investigate the logical form of abduction given by Peirce a little closer. About
C we know two things: that it is true in the actual world, and that it is surprising. The
latter thing can be modelled in many ways, one of the simplest being the requirement
that C does not follow from our other knowledge about the world. In this volume,
Aliseda models it by an epistemic state of doubt which calls for abductive reasoning
to transform it into a state of belief.
Then, ‘if A were true, C would be a matter of course’ is usually interpreted as ‘A
logically entails C’.4Peirce calls A an explanation of C, or an ‘explanatory hypothe-
sis’. Whether or not this is an appropriate notion of explanation remains an issue of
4Note that interpreting the second premiss as a material implication, as is sometimes done in the literature,
renders it superfluous, since the truth of A
!
C follows from the truth of the observation C.
8P.A. FLACH AND A.C. KAKAS
debate. In this volume, Console and Saitta also propose to identify explanation with
entailment, but Josephson argues against it.
Besides being explanatory, Peirce mentions two more conditions to be fulfilled
by abductive hypotheses: they should be capable of experimental verification, and
they should be ‘economic’. A hypothesis should be experimentally verifiable, since
otherwise it cannot be evaluated inductively. Economic factors include the cost of
verifying the hypothesis, its intrinsic value, and its effect upon other projects (Peirce,
1958, 7.220). In other words, economic factors are taken into account when choosing
the best explanation among the logically possible ones. For this reason, abduction is
often termed ‘inference to the best explanation’ (Lipton,1991).
Induction is identified by Peirce as the process of testing a hypothesis against reality
through selected predictions. ‘Induction consists in starting from a theory, deducing
from it predictions of phenomena,and observingthose phenomena in orderto see how
nearly they agree with the theory’ (Peirce, 1958, 5.170). Such predictions can be seen
as experiments:
‘When I say that by inductive reasoning I mean a course of experimental inves-
tigation, I do not understand experiment in the narrow sense of an operation by
which one varies the conditions of a phenomenon almost as one pleases. (...) An
experiment (...) is a question put to nature. (...) The question is, Will this be the
result? If Nature replies ‘No!’ the experimenter has gained an important piece
of knowledge. If Nature says ‘Yes,’ the experimenter’s ideas remain just as they
were, only somewhat more deeply engrained.’ (Peirce, 1958, 5.168)
This view of hypothesis testing is essentially what is called the ‘hypothetico-deductive
method’ in philosophy of science (Hempel, 1966). The idea that a verified prediction
provides further support for the hypothesis is very similar to the notion of confirma-
tion as discussed in Section 1.2.1, and also refutation of hypotheses through falsified
predictions can be brought in line with confirmation theory, with a limiting degree of
support of zero.5The main difference from confirmation theory is that in the Peircean
view of induction the hypothesis is, through the predictions, tested against selected
pieces of evidence only. This leads to a restricted form of hypothesis evaluation, for
which we will use the term hypothesis testing.
Peirce’s inferential theory makes two main points. It posits a separation between
hypothesis generation and hypothesis evaluation; and it focuses attention on hypothe-
ses that can explain and predict. Combining the two points, abduction is the process of
generating explanatory hypotheses(be they general‘rules’ or specific ‘cases’, as in the
syllogistic account), and induction corresponds to the hypothetico-deductive method
of hypothesis testing. However, the two points are relatively independent: e.g., we can
perceive generation of non-explanatory hypotheses. We will come back to this point
in the discussion below.
5From a Bayesian perspective P
(
H
j
E
)
is proportional to P
(
E
j
H
)
P
(
H
)
, where P
(
H
)
is the prior probability
of the hypothesis; if Eis contrary to a prediction P
(
E
j
H
)=
0. See Poole’s chapter for further discussion of
the Bayesian perspective.
ABDUCTIVE AND INDUCTIVE REASONING: BACKGROUND AND ISSUES 9
1.2.3 Discussion
In the previous two sections we have considered three philosophical and logical per-
spectives on how non-deductive reasoning may be categorised: the inductivist view,
which holds that no further categorisation is needed since all non-deductivereasoning
must be justified in the same way by means of confirmation theory; the syllogistic
view, which distinguishes between inductive generalisation on the one hand and hy-
pothesis or abduction as inference of specific ‘cases’ on the other; and the inferential
view, which holds that abduction and induction represent the hypothesis generation
and evaluation phases in explanatory reasoning. As we think that none of these view-
points provides a complete picture, there is opportunityto come to a partial synthesis.
Hypothesisgenerationandhypothesisevaluation. The most salient point of Peirce’s
later, inferential theory is the distinction between hypothesis generation and hypothe-
sis evaluation. In most other accounts of non-deductive reasoning the actual hypothe-
sis is already present in the argument under consideration, as can be seen clearly from
the argument forms discussed in Section 1.2.1. For instance, when constructing an
inductive generalisation
Xpercent of observed Fs are Gs;
therefore, (approximately) Xpercent of all Fs are Gs.
our job is first to conjecture possible instantiations of Fand G(hypothesis generation),
and then to see whether the resulting argument has sufficient support (hypothesis eval-
uation).
One may argue that a too rigid distinction between generation and evaluation of
hypotheses is counter-productive,since it would lead to the generation of many, ulti-
mately useless hypotheses. Indeed, Peirce’s ‘economic factors’, to be considered when
constructing possible abductivehypotheses, already blur the distinction to a certain ex-
tent. However, even if a too categorical distinction may have practical disadvantages,
on the conceptual level the dangers of confusing the two processes are much larger.
Furthermore, the distinction will arguably be sharper drawn in artificial reasoning sys-
tems than it is in humans, just as chess playing computers still have no real alternative
to finding useful moves than to consider all possible ones.
In any case, whether tightly integrated or clearly separated, hypothesis generation
and hypothesis evaluation have quite distinct characteristics. Here we would argue that
it is hypothesis generation, being concernedwith possibilities rather than choices, that
is most inherently ‘logical’ in the traditional sense. Deductive logic does not help the
mathematician in selecting theorems, only in distinguishing potential theorems from
fallacious ones. Also, as (Hanson, 1958) notes, if hypothesis evaluation establishes
a logic at all, then this would be a ‘Logic of the Finished Research Report’ rather
than a ‘Logic of Discovery’. An axiomatic formalisation of the logic of hypothesis
generation is suggested by Flach in his chapter in this volume.
We also stress the distinction between generation and evaluation because it provides
a useful heuristic for understanding the various positions of participants in the debate
on abduction and induction. This rule of thumb states that those concentrating on
generating hypotheses tend to distinguish between non-deductive forms of reasoning;
those concentrating on evaluating hypotheses tend not to distinguish between them.
10 P.A. FLACH AND A.C. KAKAS
Not only does the rule apply to the approaches discussed in the previous two sections;
we believe that it can guide the reader, by and large, through the chapters in this
collection.
Inductive generalisation. Turning next to the question ‘What is induction?’, we ex-
pect that any form of consensus will centre around the argument form we called induc-
tive generalisation (see above). In the inductivist approach such sample-to-population
arguments were separated out on syntactic grounds. They also figured in Peirce’s
syllogistic theory as one of the two possible reversals of Barbara.
As we remarked above, hypothesis generation here amounts to instantiating Fand
G. In general the number of possibilities is large, but it can be reduced by constraining
the proportion X. Many artificial intelligence approaches to induction actually choose
Fand Gsuch that Xis (close to) 100%, thereby effectively switching to categorical
inductive generalisations:
All observed Fs are Gs;
therefore, all Fs are Gs.
For instance, instead of observing that 53% of observed humans are female, such
approaches will continue to refine Funtil all observed Fs are female (for instance, F
could be ‘humans wearing a dress’).
The point here is not so much that in artificial intelligence we are only interested
in infallible truths. Often, we have to deal with uncertainties in the form of noisy
data, exceptions to rules, etc. Instead of representing these uncertainties explicitly in
the form of relative frequencies, one deals with them semantically, e.g. by attaching a
degree of confirmation to the inductive conclusion, or by interpretingrules as defaults.
The above formulation of categorical inductive generalisation is still somewhat lim-
iting. The essential step in any inductive generalisation is the extension of the universal
quantifier’s scope from the sample to the population. Although the universally quan-
tified sentence is frequently a material implication, this need not be. A more general
form for categorical inductive generalisation would therefore be:
All objects in the sample satisfy P
(
x
)
;
therefore, all objects in the population satisfy P
(
x
)
.
where P
(
x
)
denotes a formula with free variable x. Possible instantiations of P
(
x
)
can
be found by pretending that there exist no other objects than those in the sample, and
looking for true universal sentences. For instance, we might note that every object in
the sample is either female or male. This approach is further discussed in the chapter
by Lachiche.
Confirmatory and explanatory induction. This more comprehensive formulation
of categorical inductive generalisation also indicates a shortcoming of Peirce’s infer-
ential theory: not all hypotheses are explanatory. For instance, take the inductive gen-
eralisation ‘every object in the population is female or male’. This generalisation does
not, by itself, explain that Maria is female, since it requires the additional knowledge
that Maria is not male. Likewise, an explanation of John being male is only obtained
by adding that John is not female. This phenomenon is not restricted to disjunctive
ABDUCTIVE AND INDUCTIVE REASONING: BACKGROUND AND ISSUES 11
generalisations: the rule ‘every parent of John is a parent of John’s brother’ does not
explain parenthood.
In line with recent developments in inductive logic programming, we would like to
suggest that inductive generalisations like these are not explanatory at all. They sim-
ply are generalisations that are confirmed by the sample. The process of finding such
generalisations has been called confirmatory induction (also descriptive induction).
The difference between the two forms of induction can be understood as follows. A
typical form of explanatory induction is concept learning, where we want to learn a
definition of a given concept Cin terms of other concepts. This means that our induc-
tive hypotheses are required to explain (logically entail) why particular individuals are
Cs, in terms of the properties they have.
However, in the more general case of confirmatory induction we are not given a
fixed concept to be learned. The aim is to learn relationships between any of the
concepts, with no particular concept singled out. The formalisation of confirmatory
hypothesis formation thus cannot be based on logical entailment, as in Peirce’s ab-
duction. Rather, it is a qualitative form of degree of confirmation, which explains its
name. We will have more to say about the issue in Section 1.3.2.
Abduction. Turning next to abduction, it may seem at first that Peirce’s syllogistic
and inferential definitions are not easily reconcilable. However, it is possible to per-
ceive a similarity between the two when we notice that the early syllogistic view of
abduction or hypothesis (p. 5) provides a special form of explanation. The Result (tak-
ing the role of the observation) is explained by the Case in the light of the Rule as a
given theory. The syllogistic form of abduction can thus be seen to meet the explana-
tory requirement of the later inferential view of abduction. Hence we can consider
explanation as a characterising feature of abduction. This will be further discussed in
Section 1.3.2.
Even if the syllogistic and inferential view of abductioncan thus be reconciled, it is
still possible to distinguish between approaches which are primarily motivatedby one
of the two views. The syllogistic account of abduction has been taken up, by and large,
in logic programming and other work in artificial intelligence addressing tasks such
as that of diagnosis and planning. In this volume, the logic programming perspective
on abduction can be found in the contributions by Christiansen, Console and Saitta,
Inoue and Haneda, Mooney, Poole, Lamma et al., Sakama, and Yamamoto. The logic
programming and artificial intelligence perspective will be more closely examined in
the next section. On the other hand, the chapters by Aliseda, Josephson, and Psillos
are more closely related to the inferential perspective on abduction.
1.3 ABDUCTION AND INDUCTION IN LOGIC PROGRAMMING AND
ARTIFICIAL INTELLIGENCE
In this section, we will examine how abductionand inductionappearin the field of arti-
ficial intelligence (AI) and its specific subfield of logic programming. In Section 1.3.1
we will argue that in these fields abduction and induction are generally perceived as
distinct reasoning forms, mainly because they are used to solve different tasks. Con-
sequently, most of what follows should interpreted from the viewpoint of Peirce’s
12 P.A. FLACH AND A.C. KAKAS
earlier, syllogistic theory. In Section 1.3.2 we argue that abductive hypotheses primar-
ily provide explanations, while inductive hypotheses provide generalisations. We then
further investigate abduction and induction from a logical perspective in Section 1.3.3,
pointing out differences in the way in which they extend incomplete theories. In Sec-
tion 1.3.4 we investigate how more complex reasoning patterns can be viewed as being
built up from simple abductive and inductive inferences. Finally, in Section 1.3.5 we
address the computational characteristics of abduction and induction.
1.3.1 A task-oriented view
In AI the two different terms of abduction and induction exist separately and are used
by different communities of researchers. This gives the impression that two distinct
and irreducible forms of non-deductive reasoning exist. We believe this separation to
be caused by the fact that in AI, irrespective of the level at which we are examining
the problem, we are eventually interested in tackling particular tasks such as planning,
diagnosis, learning, and language understanding. For instance, a prototypical AI ap-
plication of abductive reasoning is the problem of diagnosis. Here abduction is used
to produce a reason, according to some known theory of a system, for the observed
(often faulty) behaviour of the system. A typical inductive task, on the other hand,
is the problem of concept learning from examples. From a collection of observations
which are judged according to some background information to be similar or related
we draw hypotheses that generalise this observed behaviour to other as yet unseen
cases.
What distinguishes this AI view from the philosophical and logical analyses dis-
cussed in the previous section is the morepractical perspective required to tackle these
tasks. Hence in AI it is necessary to study not only the issue of hypothesis evalua-
tion but also the problem of hypothesis generation, taking into account the specific
characteristics of each different task. These tasks require different effects from the
non-deductive reasoningused to address them, resulting in different kinds of hypothe-
ses, generated by different computational methods. As we will argue in Section 1.3.2,
abductive hypotheses are primarily intended to provide explanations and inductive hy-
potheses aim at providing generalisations of the observations.
The point we want to stress here is that in AI hypothesis generation is a real is-
sue, while in philosophy and logic it often seems to be side-stepped since the analysis
usually assumes a given hypothesis. Since abduction and induction produce different
kinds of hypotheses, with different relations to the observations and the background
theory, it seems natural that this increased emphasis on hypothesis generation rein-
forces the distinguishing characteristics of the two reasoning forms. However, despite
this emphasis on hypothesis generation in AI it is not possible to avoid the problem of
hypothesis evaluation and selection amongst several possible alternatives. Returning
to this problem we see that work in AI where the emphasis lies on hypotheses selection
tends to conclude that the two forms of reasoning are not that differentafter all – they
use the same kind of mechanism to arrive at the conclusion. This is seen in Poole’s
work which uses Bayesian probability for the selection of hypotheses and Josephson’s
work where several, more qualitative criteria are used.
ABDUCTIVE AND INDUCTIVE REASONING: BACKGROUND AND ISSUES 13
Peirce revisited. AI’s emphasis on solving practical tasks notwithstanding, most
research is still aimed at providing general solutions in the form of abductive and in-
ductive engines that can be applied to specific problemsby providing the right domain
knowledge and setting the right parameters. In order to understand what these systems
are doing, it is still necessary to use abstract (logical) specifications. Let us examine
this more closely, using the case of logic programming and its two extensions of ab-
ductive and inductive logic programming.
Logic programming assumes a normal form of logical formulae, and therefore has
a strong syllogistic flavour. Consequently, the logic programming perception of ab-
duction and induction essentially follows Peirce’s earlier, syllogistic characterisation.
Here are Peirce’s two reversals of the syllogism Barbara, recast in logic programming
terms:
Case.—from this bag(b).
Result.—white(b).
Rule.—white(X):-from this bag(X).
Rule.—white(X):-from this bag(X).
Result —white(b).
Case —from this bag(b).
The first pattern, inference of a general rule from a case (description) and a result
(observation) of a particular individual, exemplifies the kind of reasoning performed
by inductive logic programming (ILP) systems. The second pattern, inferring a more
complete description of an individual from an observation and a general theory valid
for all such individuals, is the kind of reasoning studied in abductive logic program-
ming (ALP).
The above account describes ILP and ALP by example,and does not providea gen-
eral definition. Interestingly, attempts to providesuch a general definition of abduction
and induction in logic programming typically correspond to Peirce’s later, inferential
characterisation of explanatory hypotheses generation. Thus, in ALP abductive infer-
ence is typically specified as follows:
‘Given a set of sentences T(a theory presentation), and a sentence G(obser-
vation), to a first approximation, the abductive task can be characterised as the
problem of finding a set of sentences Δ(abductive explanation for G) such that:
(1) T
Δ
j
=
G,
(2) T
Δis consistent. ’ (Kakas et al., 1992, p.720)
The following is a specification of induction in ILP:
‘Given a consistent set of examples or observations Oand consistent background
knowledge Bfind an hypothesis Hsuch that: B
H
j
=
O’ (Muggleton and De
Raedt, 1994)
In spite of small terminological differences the two specifications are virtually iden-
tical: they both invert a deductive consequence relation in order to complete an incom-
plete given theory, prompted by some new observations that cannot be deductively
14 P.A. FLACH AND A.C. KAKAS
accounted for by the theory alone.6If our assessment of the distinction between ab-
duction and induction that is usually drawn in AI is correct, we must conclude that
the above specifications are unable to account for this distinction. In the remainder of
Section 1.3 we will try to understand the differencesbetween abduction and induction
as used in AI in modern, non-syllogistic terms. For an account which stays closer to
syllogisms, the reader is referred to the chapter by Wang.
1.3.2 Explanation and generalisation
Let us further analyse the logical processes of abduction and induction from the utility
perspective of AI, and examine to what extent it is possible to distinguish two such
processes on the basis of the function they are intended to perform. We will argue
that such a distinction is indeed possible, since the function of abduction is to provide
explanations, and the function of induction is to provide generalisations. Some of our
views on this matter have been influenced directly by the contribution by Console and
Saitta where more discussion on this possibility of distinction between abduction and
induction can be found.
First, it will be convenient to introduce some furtherterminology.
Observables and abducibles. We will assume a common first-order language for
all knowledge (known, observed, or hypothetical). We assume that the predicates
of this language are separated into observables and non-observables or background
predicates. Domain knowledge or background knowledge is a general theory con-
cerning non-observable predicates only. Foreground knowledge is a general theory
relating observable predicates to background predicates and each other. Instance
knowledge (sometimes called scenario knowledge) consists of formulae containing
non-observable predicates only, possibly drawn from a restricted subset of such pred-
icates. Known instance knowledgecan be part of the background knowledge. Obser-
vations are formulae containing observable predicates, known to hold; predictions are
similar to observations, but their truthvalue is not given.
It will often be useful to employ the notion of an individual to refer to a particu-
lar object or situation in the domain of discourse. For example, instance knowledge
will usually contain descriptions of individuals in terms of non-observable predicates
(thence the name). An unobserved or new individual is one of which the description
becomes known only after the abductive or inductive hypothesis has been formed. As
a consequence, the hypothesis cannot refer to this particular individual; however, the
hypothesis may still be able to provide a prediction for it when its description becomes
available.
Given this terminology, we can specify the aim of induction as inference of fore-
ground knowledge from observations and other known information. Typically, this
information consists of background and instance knowledge, although other known
6Extra elements that are often added to the above definitions are the satisfaction of integrity constraints for
the case of abduction, and the avoidance of negative examples for the case of induction; these can again be
viewed under the same heading, namely as being aimed at exclusion of certain hypotheses.
ABDUCTIVE AND INDUCTIVE REASONING: BACKGROUND AND ISSUES 15
foreground knowledgemay also be used. In some cases it may be empty, for instance
when we are learning the definition of a recursive predicate, when we are learning
the definitions of several mutually dependent predicates, or when we are doing data
mining. The observations specify incomplete (usually extensional) knowledge about
the observables, which we try to generalise into new foreground knowledge.
On the other hand, in abduction we are inferring instance knowledge from ob-
servations and other known information. The latter necessarily contains foreground
information pertaining to the observations at hand. Possible abductive hypotheses are
built from specific non-observablepredicates called abducibles in ALP. The intuition
is that these are the predicates of which the extensions are not completely knownas in-
stance knowledge. Thus, an abductive hypothesis is one which completesthe instance
knowledge about an observed individual. This difference between the effect of abduc-
tion and induction on observable and instance knowledge is studied in the chapter by
Console and Saitta.
Explanation. Non-deductive reasoning as used in AI provides two basic functions
that are generally useful in addressing different problems. These two functions are
(a) finding how a piece of information came about to be true according to a general
theory describing the domain of interest, and (b) constructing theories that can de-
scribe the present and future behaviour of a system. Purely from this utility point of
view, non-deductive reasoning is required to provide these two basic effects of expla-
nation and generalisation. Informally, for the purposes of this chapter it is sufficient
for explanation to mean that the hypothesis reasoned to (or generated) by the non-
deductive reasoning does not refer to observables (i.e. consists of instance knowledge)
and entails a certain formula (an observation), and for generalisation to mean that
the hypothesis can entail additional observable informationon unobserved individuals
(i.e. predictions).
As we have seen before, both abduction and induction can be seen as a form of
reversed deduction in the presence of a background theory, and thus formally qualify
as providing explanationsof some sort. The claim that abduction is explanatory infer-
ence indeed seems undisputed, and we do not find a need to say more about the issue
here (see the chapters by Console and Saitta, Josephson, and Psillos for a discussion of
abduction as explanatory inference). We only point out that if an abductive explana-
tion Δis required to consist of instance knowledge only, then clearly abductionneeds
a given theory Tof foreground knowledge, connecting observables to background
predicates, in order to be able to account forthe observation with Δ. An abductive ex-
planation thus makes sense only relative to this theory Tfrom which it was generated:
it explains the observation according to this particular theory.
However, if induction provides explanations at all, these explanations are of a dif-
ferent kind. For instance, we can say that ‘all the beans from this bag are white’ is an
explanation for why the observed beans from the bag are white. Notice however that
this kind of explanation is universal: ‘observed Xs are Y’ is explained by the hypoth-
esis that ‘all Xs are Y’. This explanation does not depend on a particular theory: it
is not according to a particular model of the ‘world of beans’. It is a general, meta-
level explanation that does not provide any insight to why things are so. As Josephson
16 P.A. FLACH AND A.C. KAKAS
puts it, inductive hypotheses do not explain particular observations, but they explain
the frequencies with which the observations occur (viz. that non-white beans from this
bag are never observed).
Generalisation. We thus find that inductive hypotheses are not explanatory in the
same way as abductive hypotheses are. But we would argue that being explanatoryis
not the primary aim of inductive hypotheses in the first place. Rather, the main goal of
induction is to provide generalisations. In this respect, we find that the ILP definition
of induction (p. 13) is too much focused on the problem of learningclassification rules,
without stressing the aspect of generalisation. An explanatory hypothesis would only
be inductive if it generalises. The essential aspect of induction as applied in AI seems
to be the kind of sample-to-population inference exemplified by categorical inductive
generalisation, reproduced here in its more general form from Section 1.2.3:
All objects in the sample satisfy P
(
x
)
;
therefore, all objects in the population satisfy P
(
x
)
.
As with Peirce’s syllogisms, the problem here is that P
(
x
)
is already assumed to be
given, while in AI a major problem is to generate such hypotheses. The specification
of confirmatory or descriptive induction follows this pattern, but leaves the hypothesis
unspecified:
Given a consistent set of observations Oand a consistent background knowledge
B, find a hypothesis Hsuch that: M
(
B
O
)
j
=
H
(Helft, 1989; De Raedt and Bruynooghe, 1993; Flach, 1995)
Hence the formal requirement now is that any generated hypothesis should be true
in a certain model constructed from the given knowledge and observations (e.g. the
truth-minimal model).
This specification can be seen as sample-to-population inference. For example, in
Peirce’s bean example (p. 5), Bis ‘these beans are from this bag’ (instance knowledge),
Ois ‘these beans are white’ (observation), and H– ‘all the beans from this bag are
white’ – is satisfied by the model containing ‘these beans’ as the only beans in the
universe. Under the assumption that the population is similar to the sample, we achieve
generalisation by restricting attention to formulae true in the sample. Note that the
induced hypothesis is not restricted to one explaining the whiteness of these beans:
we might equally well have induced that ‘all white beans are from this bag’.
Above we defined a hypothesis as generalising if it makes a prediction involving
an observable. We have to qualify this statement somewhat, as the following example
shows (taken from the chapter by Console and Saitta, Example ??,p.??). Let our
background theory contain the following clauses:
measles(X):-brother(X,Y),measles(Y).
red_spots(X):-measles(X).
brother(john,dan).
The observation is red spots(john). A possible explanation for this observa-
tion is measles(john). While this explanation is clearly completing instance
knowledge and thus abductive, adding it to our theory will lead to the prediction
ABDUCTIVE AND INDUCTIVE REASONING: BACKGROUND AND ISSUES 17
red spots(dan). Thus, the hypothesis that John has measles also seems to qualify
as a generalisation. We would argue however that this generalisation effect is already
present in the background theory. On the other hand, an inductive hypothesis produces
agenuinely new generalisation effect, in the sense that we can find new individualsfor
which the addition of the hypothesis to our knowledge is necessary to derive some ob-
servable propertyfor these individuals(usually this propertyis that of the observations
on which the induction was based). With an abductive hypothesis this kind of exten-
sion of the observable property to other new individuals does not necessarily require
the a priori addition of the abductive hypothesis to the theory but depends only on the
properties of this individual and the given background theory: the generalisation, if
any, already exists in the background theory.
We conclude that abductive and inductive hypotheses differ in the degree of gen-
eralisation that each of them produces. With the given background theory Twe im-
plicitly restrict the generalising powerof abduction as we requirethat the basic model
of our domain remains that of T. The existence of this theory separates two levels
of generalisation: (a) that contained in the theory and (b) new generalisations that are
not given by the theory. In abduction we can only have the first level with no in-
terest in genuinely new generalisations, while in induction we do produce such new
generalisations.
1.3.3 Extending incomplete theories
We will now further examine the general logical process that each of abduction and
induction takes. The overall process that sets the two forms of reasoning of abduction
and induction in context is that of theory formation and theory development. In this
we start with a theory T(that may be empty) which describes at a certain level the
problem domain we are interested in. This theory is incomplete in its representation
of the problem domain, as otherwise there is no need for non-deductive ampliative
reasoning. New information given to us by the observationsis to be used to complete
this description. As we argue below, abduction and inductioneach deal with a different
kind of incompleteness of the theory T.
Abductive extensions. In a typical use of abduction, the description of the problem
domain by the theory Tis further assumed to be sufficient, in the sense that it has
reached a stage where we can reason with it. Typically this means that the incom-
pleteness of the theory can be isolated in some of its non-observable predicates,which
are called abducible (or open) predicates. We can then view the theory Tas a repre-
sentation of all of its possible abductive extensions T
Δ, usually denoted T
(
Δ
)
, for
each abducible hypothesis Δ. An enumeration of all such formulae (consistent with
T) gives the set of all possible abductive extensions of T. Abductive entailment with
Tis then defined by deductive entailment in each of its abductive extensions.
Alternatively, we can view each abductive formula Δas supplying the missing in-
stance knowledge for a different possible situation or individualin our domain, which
is then completely described by T
(
Δ
)
. For example, an unobserved individual and its
background properties can be understood via a corresponding abductive formula Δ.
Once we have these background properties, we can derive – using T– other proper-
18 P.A. FLACH AND A.C. KAKAS
ties for this new individual.7Given an abductive theory Tas above, the process of
abduction is to select one of the abductive extensions T
(
Δ
)
of T in which the given
observation to be explained holds, by selecting the corresponding formula Δ. We can
then reason deductively in T
(
Δ
)
to arrive at other conclusions. By selecting Δwe are
essentially enabling one of the possible associations between Δand the observation
among those supplied by the theory T.
It is important here to emphasise that the restriction of the hypothesis of abduction
to abducible predicates is not incidental or computational,but has a deeper representa-
tional reason. It reflects the relative comprehensiveness of knowledge of the problem
domain contained in T. The abducible predicates and the allowed abductive formu-
lae take the role of ‘answer-holders’ for the problem goals that we want to set to our
theory. In this respect they take the place of the logical variable as the answer-holder
when deductive reasoning is used for problem solving. As a result this means that the
form of the abductive hypothesis depends heavily on the particular theory Tat hand,
and the way that we have chosen to represent in this our problem domain.
Typically, the allowed abducible formulae are further restricted to simple logical
forms such as ground or existentially quantified conjunctions of abducible literals.
Although these further restrictions may be partly motivated by computational consid-
erations, it is again important to point out that they are only made possible by the
relative comprehensivenessof the particular representation of our problem domain in
the theory T. Thus, the case of simple abduction – where the abducible hypothesis
are ground facts – occurs exactly because the representationof the problem domain in
Tis sufficiently complete to allow this. Furthermore, this restriction is not significant
for the purposes of comparison of abduction and induction: our analysis here is inde-
pendent of the particular form of abducible formulae. The important elements are the
existence of an enumeration of the abductive formulae, and the fact that these do not
involve observable predicates.
Inductive extensions. Let us now turn to the case of induction and analyse this
process to facilitate comparison with the process of abduction as described above.
Again, we have a collection of possible inductive hypothesesfrom which one must be
selected. The main difference now is the fact that these hypotheses are not limited to
a particular subset of predicates that are incompletely specified in the representation
of our problem domain by the theory T, but are restricted only by the language of T.
In practice, there may be a restriction on the form of the hypothesis, called language
bias, but this is usually motivated either by computationalconsiderations, or by other
information external to the theory Tthat guides us to an inductive solution.
Another essential characteristic of the process of induction concerns the role of the
selected inductive hypothesis H. The role of His to extend the existing theory Tto a
new theory T
0
=
T
H, rather than reason with Tunder the set of assumptions Has is
the case for abduction. Hence Tis replaced by T
0
to become a new theory with which
we can subsequently reason, either deductively of abductively, to extract information
7Note that this type of abductive (or open) reasoning with a theory Tcollapses to deduction, when and if
the theory becomes fully complete.
ABDUCTIVE AND INDUCTIVE REASONING: BACKGROUND AND ISSUES 19
from it. The hypothesis Hchanges Tby requiring extra conditions on the observable
predicates that drive the induction, unlike abduction where the extra conditions do not
involve the observable predicates. In effect, Hprovides the link between observables
and non-observables that was missing or incomplete in the original theory T.
Analogously to the concept of abductive extension, we can define inductive ex-
tensions as follows. Consider a common given theory Twith which we are able to
perform abduction and induction. That is, Thas a number of abductive extensions
T
(
Δ
)
. Choosing an inductive hypothesis Has a new part of the theory Thas the ef-
fect of further conditioning each of the abductive extensions T
(
Δ
)
. Hence, while in
abduction we select an abductive extension of T, with induction we extend each of
the abductive extensions with H. The effect of induction is thus ‘universal’ on all the
abductive extensions.
If we now consider the new abductive theory T
0
=
T
H, constructed by induction,
we can view induction as a process of selecting a collection of abductive extensions,
namely those of the new theory T
0
. Hence an inductive extension can be viewed as a
set of abductive extensions of the original theory T that are further (uniformly) condi-
tioned by the common statement of the inductive hypothesis H. This idea of an induc-
tive extension consisting of a set of abductive extensions was used in (Denecker et al.,
1996) to obtain a formalisation of abduction and induction as selection processes in
a space of possible world models over the given theory in each case. In this way the
process of induction can be seen to have a more general form than abduction, able to
select a set of extensions rather than a single one. Note that this does not necessar-
ily mean that induction will yield a more general syntactic form of hypotheses than
abduction.
Analysis. Comparing the possible inductive and abductive extensions of a given the-
ory Twe have an essential difference. In the case of abductionsome of the predicates
in the theory, namely the observables, cannot be arbitrarily defined in an extension.
The freedom of choice of abduction is restricted to constrain directly (via Δ) only the
abducibles of the theory. The observable predicates cannotbe affected except through
the theory: the observables must be grounded in the existing theory Tby the choice of
the abductive conditions on the abducible part of the extension. Hence in an abductive
extension the extent to which the observables can become true is limited by the theory
Tand the particular conditions Δon the rest of the predicates.
In induction this restriction is lifted, and indeed we can have inductive extensions
of the given theory T, the truthvalue of which on the observable predicates need not
be attributed via Tto a choice on the abducibles. The inductive extensions ‘induce’
a more general change (from the point of view of the observables) on the existing
theory T, and – as we will see below – this will allow induction to genuinely gener-
alise the given observations to other cases not derivable from the original theory T.
The generalising effect of abduction, if at all present, is much more limited. The se-
lected abductive hypothesis Δmay produce in T
(
Δ
)
further information on abducible
or other predicates, as in the measles example from the previous section. Assuming
that abducibles and observables are disjoint, any information on an observable derived
in T
(
Δ
)
is a generalisation already contained in T.
20 P.A. FLACH AND A.C. KAKAS
What cannot happen is that the chosen abductive hypothesis Δalone (without T)
predicts a new observation, as Δdoes not affect directly the value of the observable
predicates. Every prediction on an observable derived in T
(
Δ
)
, not previously true in
T(including the observation that drives the abductive process), corresponds to some
further instance knowledge Δ
0
, which is a consequence of T
(
Δ
)
, and describes the
new situation (or individual) at hand. Such consequences are already known to be
possible in the theory T, as we know that one of its possible extensions is T
(
Δ
0
)
.
In the measles example (p. 16), the observation red spots(john) gives rise to the
hypothesis Δ=measles(john). Adopting this hypothesis leads to a new prediction
red spots(dan), corresponding to the instance knowledge Δ
0
=measles(dan),
which is a consequence of T
(
Δ
)
. This new prediction could be obtained directly from
T
(
measles(dan)
)
without the need of Δ=measles(john).
Similarly, if we consider a previously unobserved situation (not derivable from
T
(
Δ
)
) described by Δnew with T
(
Δ
)
Δnew deriving a new observation, this is also
already known to be possible as T
(
Δ
Δnew
)
is one of the possible extensions of T.
For example, if Δnew =measles(mary), then T
(
Δ
)
Δnew, and in fact T
Δnew
derives red spots(mary), which is again not a genuine generalisation.
In short, abduction is meant to select some further conditions Δunder which we
should reason with T. It concerns only this particular situation described by Δand
hence, if Δcannot impose directly any conditions on the observable predicates, the
only generalisations that we can get on the observables are those contained in Tunder
the particular restrictions Δ. In this sense we say that the generalisation is not genuine
but already contained in T. Hence, as argued in the chapter by Console and Saitta,
abduction increases the intension of known individuals (abducible properties are now
made true for these individuals), but does not have a genuine generalisation effect on
the observables (it does not increase the extension of the observables with previously
unobserved individuals for which the theory Talone could not produce this extension
when it is given the instance knowledge that describesthese individuals).
On the other hand, the universal conditioning of the theory Tby the inductive
hypothesis Hproduces a genuine generalisation on the observables of induction. The
extra conditions in Hon the observables introduce new information on the relation of
these predicates to non-observable predicates in the theory T, and from this we get
new observable consequences. We can now find cases where from Halone together
with a (non-observable) part of T, describing this case, we can derive a prediction not
previously derivable in T.
The new generalisation effect of induction shows up more when we consider as
above the case where the given theory for induction has some of its predicates as
abducible (different fromthe observables). It is now possible to have a new individual
described by the extra abducible information Δnew, such that in the new theory T
0
=
T
Hproduced by induction a new observation holds which was not known to be
possible in the old theory T(i.e. it is not a consequence of T
Δnew). Note that we
cannot (as in the case of abduction) combine Hwith Δnew to a set Δ
0
new of instance
knowledge, under which the observation would hold from the old theory T. We can
also have that a new observation holds alone from the hypothesis Hand Δnew for such
previously unobservedsituations not described in the given theory T. These are cases
ABDUCTIVE AND INDUCTIVE REASONING: BACKGROUND AND ISSUES 21
of genuine generalisation not previously known to be possible from the initial theory
T.Summarising this subsection, induction – seen as a selection of a set of extensions
defined by the new theory T
H– has a stronger and genuinely new generalising
effect on the observable predicates than abduction. The purpose of abduction is to
select an extension and reason with it, thus enabling the generalising potential of the
given theory T. In induction the purpose is to extend the given theory to a new theory,
the abductive extensionsof which can provide new possible observable consequences.
Finally, we point out a duality between abduction and induction (first studied in
(Dimopoulos and Kakas, 1996)) as a result of this analysis. In abduction the theory T
is fixed and we vary the instance knowledge to capture (via T) the observable knowl-
edge. On the other hand, in induction the instance knowledge is fixed as part of the
background knowledge B, and we vary the general theory so that if the selected the-
ory Tis taken as our abductive theory then the instance knowledge in Bwill form
an abductive solution for the observations that drove the induction. Conversely, if
we perform abduction with Tand we consider the abductive hypothesis Δexplaining
the observations as instance knowledge, the original theory Tforms a valid inductive
hypothesis.
1.3.4 Interaction between abduction and induction
In the preceding sections we analysed basic patterns of abduction and induction. In
practice hybrid forms of ampliative reasoning occur, requiring an interaction between
these basic patterns. Such interaction is the subject of this section.
Let us consider a simple example originating from (Michalski,1993). We have the
observation that:
O: all bananas in this shop are yellow,
and we want to explain this given a theory Tcontaining the statement:
T: all bananas from Barbados are yellow.
An explanation for this is given by the hypothesis:
H: all bananas in this shop are from Barbados.
Is this a form of abduction or a form of induction, or perhaps a hybrid form? As we
will show,this strongly depends on the choice of observables and abducibles.
Suppose, first, that we choose ‘yellow’ as observable and the other predicates as ab-
ducibles.8The hypothesis Hselects amongst all the possible abductive extensions of
the theory T(corresponding to the different abducible statements of instance knowl-
edge consistent with T) a particular one. In this selected extension the observation is
entailed and therefore the hypothesis explains according to the abductivetheory Tthe
observation. Note that this hypothesis Hdoes not generalise the given observations: it
does not enlarge the extension of the observable predicate ‘yellow’ over that provided
8We can if we wish consider only the predicate ‘from Barbados’ as abducible.
22 P.A. FLACH AND A.C. KAKAS
by the statement of the observation O. In fact, we can replace the universalquantifica-
tion in ‘all bananas from this shop’ by a typical representative through skolemisation.
More importantly, the link of the observation Owith the extra information of His
known a priori as one of the possible ways of reasoning with the theory Tto derive
new observable information.
There is a second way in which to view this reasoning and the hypothesisHabove.
We can consider the predicate ‘from Barbados’ as the observable predicate with a set
of observations that each of the observed bananas in the shop is from Barbados. We
then have a prototypical inductive problem (like the white bean example of Peirce)
where we generate the same statement Has above, but now as an inductive hypothe-
sis. From this point of view the hypothesis now has a genuine generalising effect over
the observations on the predicate ‘from Barbados’. But where did the observations on
Barbados come from? These can be obtained from the theory Tas separate abductive
explanations for each of the original observations (or a typical one) on the predicate
‘yellow’. We can thus understandthis example as a hybrid processof first using (sim-
ple) abduction to translate separately each given observation as an observation on the
abducibles, and then use induction to generalise the latter set of observations, thus
arriving at a general statement on the abducibles.
Essentially, in this latter view we are identifying, by changing within the same
problem the observableand abducible predicates, simple basic formsof abduction and
induction on which we can build more complex forms of non-deductive reasoning.
Referring back to our earlier discussion in Section 1.3, these basic forms are: pure
abduction for explanation with no generalisationeffect (over what already exists in the
theory T); and pure induction of simple generalisations from sample to population.
This identification of basic distinct forms of reasoning has important computational
consequences. It means that we can consider two basic computational models for the
separate tasks of abduction and induction. The emphasis then shifts to the question
of how these basic forms of reasoning and computation can be integrated together to
solve more complex problems by suitably breaking down these problems into simpler
ones.
It is interesting to note here that in the recent framework of inverse entailment as
used by the ILP system Progol (Muggleton, 1995) where we can learn from general
clauses as observations, an analysis of its computation as done in the chapter by Ya-
mamoto reveals that this can be understood as a mixture of abduction and induction.
As described in the above example, the Progol computation can be separated into first
abductively explaining according to the background theory a skolemised, typical ob-
servation, and then inductively generalising over this abductive explanation. The use-
fulness of explicitly separating out abduction and induction is also evident in several
works of theory formation or revision. Basic computational forms of abduction and
induction are used together to address these complexproblems. This will be described
further in Section 1.4 on the integration of abduction and induction in AI.
1.3.5 Computational characteristics
We will close this section by discussing further the computational distinction that the
basic forms of abduction and induction havein their practice in AI and logic program-
ABDUCTIVE AND INDUCTIVE REASONING: BACKGROUND AND ISSUES 23
ming. Indeed, when we examine the computational models used for abduction and
induction in AI, we notice that theyare very different. Their differenceis so wide that
it is difficult, if not impossible, to use the computational framework of one form of
reasoning in order to compute the other form of reasoning. Systems developed in AI
for abduction cannot be used for induction (and learning), and vice versa, inductive AI
systems cannot be used to solve abductive problems.9In the chapter by Christiansen a
system is described where the computation of both forms of reasoning can be unified
at a meta-level, but where the actual computation followed by the system is different
for the separate forms of reasoning.
We will describe here the main characteristics of the computational models of the
basic forms of abduction and induction, discussed above, as theyare found in practical
AI approaches. According to these basic forms, abduction extracts an explanation
for an observation from a given theory T, and induction generalises a set of atomic
observations. For abduction the computation has the following basic form: extract
from the given theory Ta hypothesis Δand check this for consistency. The search
for a hypothesis is done via some form of enhanced deduction method e.g. resolution
with residues (Cox and Pietrzykowski, 1986; Eshghi and Kowalski, 1989; Kakas and
Mancarella, 1990; Denecker and de Schreye, 1992; Inoue, 1992; Kakas and Michael,
1995), or unfolding of the theory T(Console et al., 1991; Fung and Kowalski, 1997).
The important thing to note is that the abductive computation is primarily based on
the computation of deductive consequences from the theory T. The proofs are now
generalised so that they can be successfully terminated ‘early’ with an abductive for-
mula. To check consistency of the found hypothesis, abductive systems employ stan-
dard deductive methods (these may sometimes be specially simplified and adapted to
the particular form that the abductive formulae are restricted to take). If a hypothesis
(or part of a hypothesis) is found inconsistent then it is rejected and another one is
sought. Note that systems that compute constructive abduction (e.g. SLDNFA (De-
necker and de Schreye, 1998) , IFF (Fung and Kowalski, 1997), ACLP (Kakas and
Michael, 1995)), where the hypothesis may not be ground but can be an existentially
quantified conjunction (with arithmetic constraints on these variables) or even a uni-
versally quantified formula, have the same computational characteristics. They arrive
at these more complex hypotheses by extending the proof methods for entailment to
account for the (isolated) incompleteness on the abducible predicates.
On the other hand, the computational model for the basic form of induction in AI
takes a rather different form. It constructs a hypothesis and then refines this under
consistency and other criteria. The construction of the hypothesis is based on methods
for inverting entailment proofs (or satisfaction proofs in the case of confirmatory in-
duction) so that we can obtain a new theory that would then entail (or be satisfied by)
the observations. Thus, unlike the abductive case, the computation cannot be based
on proof methods for entailment, and new methods such as inverse resolution, clause
generalisation and specialisation are used. In induction the hypothesis is generated
from the language of the problem domain (rather than a given theory of the domain),
9With the possible exception of Cigol (Muggleton and Buntine, 1988), a system designed for doing unre-
stricted reversed deduction.
24 P.A. FLACH AND A.C. KAKAS
in a process of iteratively improving a hypothesis to meet the various requirements
posed by the problem. Furthermore, in induction the comparison of the different pos-
sible hypotheses plays a prominent and dynamic role in the actual process of hypothe-
sis generation, whereas in abduction evaluation of the different alternative hypothesis
may be done after these have been generated.
It should be noted, however, that the observed computational differences between
generating abductive hypotheses and generating inductive hypotheses are likely to be-
come smaller once more complex abductive hypotheses are allowed. Much of the
computational effort of ILP systems is spent on efficiently searching and pruning the
space of possible hypotheses, while ALP systems typically enumerate all possible
abductive explanations. The latter approach becomes clearly infeasible when the ab-
ductive hypothesis space grows. In this respect, we again mention the system Cigol
which seems to be the only system employing a unified computational method (inverse
resolution) to generate both abductive and inductive hypotheses.
Computational distinctions of the two forms of reasoning are amplified when we
consider the different works of trying to integrate abduction and induction in a com-
mon system. In most of these works, each of the two forms of reasoning is computed
separately, and their results are transferred to the other formof reasoning as input. The
integration clearly recognises two different computationalprocesses (for each reason-
ing) which are then suitably linked together. For example, in LAB (Thompson and
Mooney, 1994) or ACL (Kakas and Riguzzi, 1997; Kakas and Riguzzi, 1999) the
overall computation is that of induction as described above, but where now – at the
point of evaluationand improvementof the hypothesis – a specific abductiveproblem
is computed that provides feedback regardingthe suitability of the inductivehypothe-
sis. In other cases, such as RUTH (Ad´e et al., 1994) or Either (Ourston and Mooney,
1994) an abductive process generates new observable input for a subsidiary inductive
process. In all these cases we have well-defined separate problems of simple forms of
abduction and induction each of which is computed along the lines described above.
In other words, the computational viability of the integrated systems depends signifi-
cantly on this separation of the problem and computation into instances of the simple
forms of abduction and induction.
1.4 INTEGRATION OF ABDUCTION AND INDUCTION
The complementarity between abduction and induction, as we have seen it in the pre-
vious section – abduction providing explanations from the theory while induction gen-
eralises to form new parts of the theory – suggests a basis for their integration. Co-
operation between the two forms of reasoning would be useful within the context of
theory development (construction or revision), where a current theory Tis updated to
a new theory T
0
in the light of new observations Oso that T
0
captures O(i.e. T
0
j
=
O).
At the simplest level, abduction and induction simply co-exist and both function as
revision mechanisms that can be used in developing the new theory (Michalski, 1993).
In a slightly more cooperative setting, induction provides new foreground knowledge
in Tfor later use by abduction. At a deeper level of cooperation, abduction and in-
duction can be integrated together within the process of constructing T. There are
several ways in which this can happen within a cycle of development of T, as will be
ABDUCTIVE AND INDUCTIVE REASONING: BACKGROUND AND ISSUES 25
T
T
′
O
′
O
T
∪H
=
O
Abduction
Induction
Figure 1.2 The cycle of abductive and inductive knowledge development.
described below. For further discussion on the integration of abduction and induction
in the context of machine learning see the chapter by Mooney in this volume. Also
the chapter by Sakama studies how abduction can be used to compute induction in an
integrated way.
The cycle of abductive and inductive knowledge development. On the one hand,
abduction can be used to extract from the given theory Tand observations Oabducible
information that would then feed into induction as (additional) training data. One ex-
ample of this is provided by (Ourston and Mooney, 1994), where abduction identifies
points of repair of the original, faulty theory T, i.e. clauses that could be generalised
so that positive observations in Obecome entailed, or clauses that may need to be
specialised or retracted because they are inconsistent with negative observations.
A more active cooperation occurs when, first, through the use of basic abduction,
the original observations are transformed to data on abducible background predicates
in T, becoming training data for induction on these predicates. An example of this was
discussed in Section 1.3.4; another example in (Dimopoulos and Kakas, 1996) shows
that only if, before inductive generalisation takes place, we abductively transform the
observations into other predicates in a uniform way, it is possible to solve the original
inductive learningtask. In this volume, Abe studies this type of integration, employing
an analogy principle to generate suitable data for induction. Similarly, Yamamoto’s
analysis of the ILP system Progol in this volume shows that – at an abstract level – the
computation splits into a first phase of abductively transforming the observations on
one predicate to data on other predicates, followed by a second generalisation phase
to produce the solution.
In the framework of the system RUTH (Ad´e et al., 1994), we see induction feeding
into the original abductivetask. An abductive explanation maylead to a set of required
facts on ‘inducible’ predicates, which are inductively generalised to give a general rule
in the abductive explanation for the original observations, similar to (one analysis of)
the bananas example discussed previously.
These types of integration can be succinctly summarised as follows. Consider a
cycle of knowledge development governed by the ‘equation’ T
H
j
=
O, where Tis
the current theory, Othe observation triggering theory development, and Hthe new
knowledge generated. Then, as shown in Figure 1.2, on one side of this cycle we
26 P.A. FLACH AND A.C. KAKAS
have induction, its output feeding into the theory Tfor later use by abduction, as
shown in the other half of the cycle, where the abductive output in turn feeds into the
observational data Ofor later use by induction, and so on.
Inducing abductive theories. Another way in which induction can feed into abduc-
tion is through the generation of confirmatory (or descriptive) inductive hypotheses
that could act as integrity constraints for the new theory. Here we initially have some
abductive hypotheses regarding the presence or absence of abducible assumptions.
Based on these hypotheses and other data in Twe generate, by means of confirmatory
induction, new sentences Iwhich, when interpreted as integrity constraints on the new
theory T, would support the abducible assumptions (assumptions of presence would
be consistent with I, assumptions of absence would now be inconsistent with I).
This type of cooperation between abductive and inductive reasoning is based on a
deeper level of integration of the two forms of reasoning, where induction is perceived
as hypothesising abductive (rather than deductive) theories. The deductive coverage
relation for learning is replaced by abductive coverage, such that an inductive hypoth-
esis His a valid generalisation if the observations can be abductively explained by
T
0
=
T
H, rather than deductively entailed. A simple example of this is the exten-
sion of Explanation-Based Learning with abduction (Cohen, 1992; O’Rorke, 1994),
such that deductive explanations are allowed to be completed by abductive assump-
tions before they are generalised.
Inducing abductivetheories is particularly useful in cases where the domain theory
is incomplete, and also when performing multiple predicate learning, as also in this
case the background knowledgefor one predicate includes the incomplete data for the
other predicates to be learned. In these cases the given theory Tis essentially an ab-
ductive theory, and hence it is appropriate to use an abductive coverage relation. On
the other hand, it may be that the domain that we are trying to learn is itself inher-
ently abductive or non-monotonic (e.g. containing nested hierarchies of exceptions),
in which case the hypothesis space for learning is a space of abductive theories.
LAB (Thompson and Mooney, 1994) is one of the first learning systems adopting
this point of view (see also Mooney’s contribution to this volume). The class predi-
cates to be learned are the abducible predicates, and the induced theoryHdescribes the
effects of these predicates on other predicates that we can observe directly with rules
of the form observation
class. Then the training examples (each consisting of a set
of properties and its classification) are captured by the induced hypothesis Hwhen the
correct classification of the examples forms a valid abductive explanation, given H, for
their observed properties. Other frameworks for learning abductive theories are given
in (Kakas and Riguzzi, 1997; Kakas and Riguzzi, 1999; Dimopoulos et al., 1997) and
the chapter by Lamma et al. Here, both explanatory and confirmatory induction are
used to generate theories together with integrity constraints. In this volume, Inoue
and Haneda also study the problem of learning abductive logic programs for capturing
non-monotonic theories.
With this type of integration we can perceive abduction as being used to evaluate
the suitability or credibility of the inductive hypothesis. Similarly, abductive expla-
nations that lead to induction can be evaluated by testing the induced generalisation.
ABDUCTIVE AND INDUCTIVE REASONING: BACKGROUND AND ISSUES 27
In this sense, the integration of abduction and induction can help to cross-evaluate the
hypothesis that they generate.
1.5 CONCLUSIONS
The nature of abduction and induction is still hotly debated. In this introductory chap-
ter we have tried to chart the terrain of possible positions in this debate, and also to
provide a roadmap for the contributions to this volume. From a logico-philosophical
perspective, there are broadlyspeaking two positions: either one holds that abduction
provides explanations and induction provides generalisations; or one can hold that ab-
duction is the logic of hypothesis generation and induction is the logic of hypothesis
evaluation. AI approaches tend to adopt the first perspective (although there are ex-
ceptions) – abduction and induction each deal with a different kind of incompleteness
of the given theory, extendingit in different ways.
As stressed in the introduction to this chapter, we do however think that absolute
positions in this debate may be counter-productive. Referring back to the questions
formulated there, we think it will be useful to unify abduction and induction when
concentrating on hypothesis evaluation. On the other hand, when considering hypoth-
esis generation we often perceive a distinction between abduction and induction, in
particular in their computational aspects.
With respect to the second question, abduction and induction can be usefully inte-
grated when trying to solve complex theory development tasks. We have reviewed a
number of AI approaches to such integration. Most of these frameworksof integration
use relatively simple forms of abduction and induction, namely abduction of ground
facts and basic inductive generalisations. Moreover,each of the two is computed sep-
arately and its results transferred to the other, thus clearly recognising two separate
and basic computational problems. From these, they synthesise an integrated form of
reasoning that can produce more complex solutions, following a cyclic pattern with
each form of reasoning feeding into the other.
A central question then arises as to what extent the combinationof such basic forms
of abduction and induction is complete, in the sense that it encapsulates all solutions to
the task. Can they form a generating basis for any method for such theory development
which Peirce describes in his later work as ‘coming up with a new theory’? We hope
that the present collection of papers will contribute towards understanding this issue,
and many other issues pertaining to the relation between abduction and induction.
Acknowledgments
Part of this work was supported by Esprit IV Long Term Research Project 20237 (Inductive
Logic Programming 2).
{
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