Instituto de Investigaciones Filosóficas, Unam
1 INTRODUCTION
The intellectual enterprise of Charles Sanders Peirce, in its broadest sense, was to develop a semiotic theory to give an account for thought and language. With regard to our purposes, the fundamental question Peirce addressed was how synthetical reasoning is possible. 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 judgements a priori
possible?' But antecedently to this comes the question how synthetical judgements
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 Hookway (1992), page 18).
Peirce proposes abduction to be the logic for synthetic reasonig, a method to
acquire new ideas. That is, an ampliative mode of inference. In contrast, deduction
is analitic, an explicative mode of inference, since the information
in the conclusion is already suggested in the premisses. Induction is also considered
by Peirce to be an ampliative mode of inference, though not of a purely synthetic
kind. (CP, 5.170).
The notions of logical inference and of validity that Peirce puts forward however,
go beyond our present understanding of what logic is about. They are linked
to his epistemology, a dynamic view of thought as logical inquiry.
In this paper, our main purpose is to concentrate on the logical form of abduction
as proposed by Peirce. In particular, our interest is on the role played by
the element of surprise. Our aim is to propose abduction as an epistemic
process for the acquisition of knowledge. A natural consequence of this interpretation
is that the logical form of abduction is that of a process instead of an argument,
as it has generally being understood. Our interpretation has natural connections
to theories of epistemic change in artificial intelligence (AI). In particular
to the theory proposed by Gärdenfors (Gärdenfors 1988) in the field
of knowledge representation in artificial intelligence (AI).
The paper is organized as follows: section 2 is dedicated to Peirce; a reconstruction
of his notion of abduction is offered, as well as a description of the epistemic
elements to model thought as a dynamic action. Section 3 presents abduction
in the field of AI and gives the basics of epistemic theories.in AI. In section
4 we present a model for abduction as epistemic change which combines elements
of Peirce and AI. Finally, in section 5 our conclusions are presented.
Charles Sanders Peirce was
the first philosopher to give to abduction a logical form. 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. We will point out a few general
aspects of his theory of inquiry, and later concentrate on some of its more
logical aspects.
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(2),
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).
Later on, Peirce proposed these types of reasoning as the stages composing a
method for logical inquiry, of which hypothesis, now called abduction, is the
beginning:
"From its [abductive] suggestion deduction can draw a prediction which can be tested by induction." (CP 5.171)
The notion of abduction is then enriched by the more general conception of: "the process of forming an explanatory hypothesis" (CP 5.171) and the syllogistic form is replaced by 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.
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, according to the formulation above. 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 Peirce, abductive reasoning is essential for every human inquiry. It plays
a role in perception: "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 claimed by Anderson (Anderson 1986), who suggests a double aspect of abduction, an intuitive and a rational one.
Interpreting Peirce's abduction
The notion of abduction
in Peirce has puzzled scholars all along. Some have concluded that Peirce held
no coherent view on abduction at all (Frankfurt 1958), others have tried to
give a joint account with induction (Reilly 1970) and still others claim it
is a form of inverted modus ponens (Anderson 1986). A more modern view is found
in (Kapitan 1990) 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 (Ayim 1974).
We leave here the reconstruction of Peirce's notion of abduction. A nice concise
account of the development of abduction in Peirce, which clearly distinguishes
three stages in the evolution of his thought is given by F.Fann (Fann 1970).
Another key reference on Peirce's abduction, in its relation to creativity in
art and science is found in (Anderson 1987).
In Peirce's epistemology, thought is a dynamic process, essentially an action between two states of mind: doubt and belief. While the essence of the latter is the "establishment of a habit which determine 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" (Peirce 1955) and it is characterized by being a stormy and unpleasant state from which every human struggles to be freed from:
"The irritation of doubt causes a struggle to attain a state of belief". (Peirce 1955).
Notice that Peirce speaks of belief rather than of knowledge. Thus, the pair 'doubt-belief' is really a cycle between two opposite states. While belief is a habit, doubt is its privation. Doubt, however, Peirce (Peirce 1955) claims, 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 a 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)
By saying this, Peirce is
not only arguing that a genuine doubt is necessary to break up a habit, but
it is also identifying it with surprise. Indeed it seems to be using both terms
interchangeably:
"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).
Moreover, Peirce distinguishes two ways to break up 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, the emphasis is mine)
"Surprise again has
its Active and its Passive variety; -- the former when what one perceives positively
conflicts with expectation, the latter when having no positive expectation but
only the absence of any suspicion of anything out of the common something quite
unexpected occurs, --such as a total eclipse of the sun which one had not anticipated".
(CP 8.315)
Peirce's epistemic model proposes surprise as the trigger for every inquiry,
coming in two varieties, novelty or anomaly. In Aliseda (1997)
I have labeled these two aspects abductive triggers. We will see later
on how these are related to abduction in AI.
The connection between abductive
logic and the epistemic transition between the mental states of doubt and belief
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.
The cognitive process that integrates abductive inference and the epistemic
process may be described as follows: a novel or anomalous experience gives place
to a surprising fact, generating a state of doubt which breaks up a belief
habit. Therefore, abductive reasoning is triggered, and consists on explaining
the surprising fact and calm the state of doubt. Calm the doubt rather than
destroying it since an abductive explanation does not necessarily end in a belief.
An abductive explanation is just a suggestion to be tested before it becomes
a belief.
Research on abductive reasoning
in AI dates back to 1973 (Pople 1973), but it is until fairly recently that
is has attracted great interest. Peirce's abductive formulation has been the
point of departure of recent studies on abductive reasoning, such as logic programming
(Kakas etal 1995), knowledge acquisition (Kakas 90) and natural language processing
(Hobbs etal 1990). However, these approaches have paid little attention to the
elements of this formulation and none to what Peirce said elsewhere in his writings.
Peirce's abductive formulation has been generally interpreted as the following
logical argument:
C
A C
A
Where the status of A is
tentative. The second premise, AC, is assumed to be part of the background
theory BT. The Peircean economic criterion is incorporated as a further
selection process to produce the best explanation, since there might be several
formulae which satisfy the above formulation but are nevertheless not appropriate
as explanations. In AI circles, however, the only thing which is recognized
as Peircean is the abductive formulation and not the additional criteria of
testability and economy.
However intuitive, this formulation certainly captures neither the fact that
C is surprising nor the additional criteria that Peirce proposed. Moreover,
notice that the interpretation of the second premise needs not be committed
to material implication; it could be any other logical implication or even a
computational process in which A is the input and C its output.
It is impossible to give an overview of abduction in AI in this exploding field.
Therefore, we limit ourselves to (i) give a brief description of abduction as
logical inference, (ii) proceed to describe epistemic theories and finally (iii)
establish the connection between them(3)
.
Abduction as logical inference
The general trend in logic based approaches to abduction in AI interprets abduction as "backwards deduction plus additional conditions". What follows is the standard version of abduction deduction via some consistent additional assumption, satisfying certain extra conditions. It combines some common requirements from the literature (Mayer etal 1993, Konolige 1996, Kakas 1995):
Given a theory BT (a set
of formulae) and a formula s (an atomic formula), e is a explanation if:
1.- Bt, e s
2- e is consistent with BT
3.- e is minimal.
4.- e has some restricted syntactical form (usually an atomic formula or a conjunction
of them).
Thus, a formula e is an
explanation of s with respect to a background theory BT if s follows as a logical
conclusion (condition 1. The symbol for denotes the usual notion of logical
consequence) and the explanation is consistent with the theory (condition 2).
Moreover, the best explanation is minimal in the sense that it is the most simple
formula (condition 3), generally interpreted as that which is implied by all
other possible explanations. However, there is no unique characterization of
logical minimality (cf. Aliseda 1997 for a discussion of minimality as a logical
criterion). Finally, condition 4 is a restriction which avoids that trivial
formulas like BTC count as abductive explanations.
To illustrate this definition, here is the most common example in AI:
Example 1:
Let BT be the theory:
The lawn is wet when it rains (rw)
The lawn is wet when the sprinklers are on. (sw)
s: The lawn is wet (w).
When condition 1 is the
only one considered, the following are some of the formulae which satisfy the
condition of logical consequence: ll, a, lla, ll¬ll, BTs. Condition 2 of
consistency is needed since in the Tarskian notion of logical consequence everything
follows from inconsistent premises. Taking into account all conditions we eliminate
inconsistent explanations (ll¬ll) and those of compound form (lla,BTs.),
leaving the first two (ll,a) as possible explanations.
An additional condition not always made explicit is that ¬( BT s) and ¬(
BT ¬s). This says that neither the fact to be explained nor its negation
should follow from the background theory alone. Sometimes, this condition figures
as a precondition for an "abductive problem".
Abduction as logical inference is just one of the multiple interpretations in
AI. Flach (1996) suggests that this multiplicity goes back to two theories of
abduction in Peirce, the early syllogistic one and the later inferential one,
corresponding to the two periods in the development of abduction in Peirce (cf.
.section 2.1).
The main motivation of epistemic
theories in AI is to develop logical and computational mechanisms to incorporate
new information to a scientific theory, data base or set of beliefs. Several
types of change are appropriate in different situations. The pioneer work of
Peter Gärdenfors (Gärdenfors 1988), proposes a normative theory of
epistemic change characterized by the conditions that a rational belief change
operator should satisfy.
The basic elements of these theories are the following. Given a consistent theory
BT closed under logical consequence, called the belief state, and a sentence
s, the incoming belief, there are three epistemic attitudes of BT with
respect to s: it is accepted (sBT), rejected (¬sBT), or it is undetermined
(sBT,¬sBT), that is, the theory has no position in regard to s. Given these
attitudes, three are the main operations to incorporate s into BT , thereby
effecting an epistemic change in our currently held beliefs:
. Expansion: (BT
+ s)
An accepted or undetermined sentence is added to .
. Revision: (BT* s)
In order to incorporate a rejected s into BT and maintain consistency in the
resulting belief system, enough sentences in conflict with s are deleted from
BT (in some suitable manner) and only then is s added.
. Contraction (BT - c)
Some sentence c is retracted from BT, together with enough sentences implying
it.
Of these operations, revision is the most complex one. It may indeed be defined as a composition of the other two. First contract those beliefs of BT that are in conflict with s, and then expand the modified theory with sentence s.While expansion can be uniquely defined, this is not so with contraction or revision, as several formulas can be retracted to achieve the desired effect. Let me illustrate this point with the following example:
Example 2:
Let BT be the theory:
The lawn is wet when it rains (r w)
It rains (r)
s: The lawn is not wet (¬w).
In order to incorporate
¬w into BT 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 expand the contracted theory with ¬w consistently. The contraction operation
per se cannot state in purely logical or set-theoretical terms which of these
two options should 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
economy. (4)
In this particular example, it seems unreasonable to delete r, unless it is
considered an "erroneous observational fact".
These are the basics of epistemic theories in AI. In practice, however, full-fledged
systems of belief revision can be quite diverse. They differ in an least three
aspects: (a) belief state representation (sets, bases or possible worlds), (b)
operations of epistemic change (postulates or constructive methods), and (c)
epistemological stance. That is, the epistemic quality to be preserved. While
the "foundationalists" argue that beliefs must be justified, the "coherentists"
consider a priority to maintain the overall coherence of the system, regardless
of having justifications.
Practical connections of
abduction to theories of epistemic change have often been noted. But these in
general use abduction as a means of producing explanations for incoming beliefs
(cf. Aliseda 97 for some references). Our claim is stronger. Abductive reasoning
itself provides a model for epistemic change ( It is clear however, that in
this context, an input sentence is not an observation but a belief for which
a belief is sought). Let me discuss some reasons for this.
In the first place, let us see to what the abductive triggers (cf. 2.2) correspond
in the epistemic model. The "novelty" condition may be formalized
as follows:
¬( Bt s) and ¬( BT ¬s)
s is novel with respect to BT since neither s nor its negation is inferred from the theory. This case corresponds to the condition of epistemic undetermination:
sBT and ¬sBT
Since BT is closed under logical consequence. In other words, the fact s is surprising because the theory has no position in its regard. The abductive trigger of anomaly may be formally expressed as follows:
¬( BT s) and ( BT ¬s)
That is, the theory does not explain the surprising fact but rather its negation. In this case, the corresponding epistemic attitude is rejection:
¬(sBT)
Moreover, the epistemic operations to incorporate beliefs may be modeled with abduction, as it is explained in detailed in the next section.
Our argument in this paper consists in interpreting abduction as epistemic change. For this purpose we combine logical and epistemic elements from Peirce and AI described in previous sections. In Aliseda (1997) I proposed a taxonomy for abduction, within which our present argument is enclosed. Therefore, we briefly present this taxonomy to later concentrate in the essential elements for our claim.
Abduction is a general process of explanation, whose products are specific explanations, with a certain inferential structure. As for the logical schema for abduction, it may be viewed as a threefold relation:
BT, e s
Between an observation or
a belief s (e.g. w), an abduced item e (e.g. r), and a background theory BT
(e.g. rw, sw). (Other parameters are possible here, such as a preference ranking
- but these would rather concern the further selection process.) Against this
background, I have proposed three main parameters that determine types of abduction.
(i) An 'inferential parameter' () sets some suitable logical relationship among
explananda, background theory, and explanandum. This may be classical semantic
entailment, statistical inference, or even any non-standard interpretation of
logical consequence. (ii) Next, 'triggers' determine what kind of abduction
is to be performed: s may be a novel phenomenon, such as the lawn being wet
in example 1, or it may be in conflict with the theory, such as observing that
the lawn is not wet in example 2. (iii) Finally, 'outcomes' e are the various
products of an abductive process: facts, rules, or even new theories. In addition
to these parameters, the extra conditions of consistency and minimality are
taken into account for certain types of abduction (cf. Aliseda 1997 for logical
characterizations of several 'abductive logics' generated by setting these parameters
and additional conditions).
This taxonomy generalizes the standard one of abduction as logical inference
in AI (cf. 3.1). It goes further in that it does not limit the underlying consequence
to be classical, nor the form of 'abducible outcomes' to be facts. More important,
however, is that this schema proposes revision as a case of abductive reasoning.
There are several approaches which consider this case as abductive (e.g. Aravindan
etal 1994), however, it is not treated from a logical point of view. By incorporating
revision as a case of abductive reasoning, it is recognized that changes in
theories take place not just by accumulation of new events, but also as a consequence
of anomalies. In this way, we approach Peirce's overall notion of abduction,
in which a fact is surprising by being a novel experience or one contrary
to the facts (cf. 2.2).
In this paper, we focused on the parameter concerning the abductive triggers,
therefore we proceed to describe them in detail.
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 is surprising
only with respect to some background theory providing 'expectations'. What is
surprising to me (e.g. that the lights go on as I enter the copier room) might
not be surprising to you.
Thus, we interpret a surprising fact as one which needs an explanation. That
is, a fact which is novel or anomalous. From a logical point of view, it is
expressed as follows:
Abductive Novelty:
s is a novelty. It cannot be explained ¬(BTs), but it is consistent with the theory ¬(BT¬s).
Abductive Anomaly:
s is anomalous. It cannot
be explained ¬(BTe), and the theory rather explains its negation (BT¬s).
Of course, non-surprising facts (where BTs) should not be candidates for explanation.
Even so, one might speculate if facts which are merely probable on the basis
of BT might still need explanation of some sort to further cement their status.
In what follows we present the operations induced by the abductive triggers.
That is, the abductive operations for epistemic change.
Abductive Expansion
Given an abductive novelty (¬(BTs), ¬(BT¬s)), a consistent explanation e for s is computed in such a way that BT, es and then added to BT.
Abductive Revision
Given an abductive anomaly
(¬(BTs), BT¬s), a consistent explanation e is computed as follows: the
theory BT is revised so that it does not explain ¬s. That is, BT´
is obtained so that ¬(BT´¬s) where BT´=BT-{a1, ..., an }(5).
Once BT´ is obtained, a consistent explanation e is calculated in such
a way that BT´,es. Thus, the process of revision involves both contraction
and expansion.
The overall process of abduction as epistemic change may be described as follows:
abductive reasoning is triggered by a surprise, raising a doubt which may be
of two kinds: novelty or anomaly. In the first case the phenomenon to be explained
is completely new and consistent with the theory, so its explanation is incorporated
by the extension operation. In the second case, the anomalous phenomenon calls
for the revision operation. The theory is revised in such a way that its modification
is not in conflict with the theory, and then its explanation is calculated and
incorporated to the modified theory by expansion.
The notions of logical inference
and of abduction that Peirce puts forward go beyond logical formulations. These
notions are linked to his epistemology, a dynamic view of thought as logical
inquiry, and respond to a deep philosophical concern, that of studying the nature
of synthetic reasoning. We have shown, however, that the Peircean notions of
surprise, novelty and anomaly may be expressed as precise logical conditions,
and serve in a model for abductive reasoning.
Our analysis of abduction combines epistemic elements of Peirce and of approaches
in AI and shows that abduction is a very complex phenomenon which requires a
careful analysis. This type of reasoning may have several logical forms, depending
on the selected logical consequence. Moreover, the form of the explanation may
be very diverse. On the other hand, as we have argued in detail in this paper,
there is no unique form for abductive reasoning, there are at least two types
of triggers, novelty and anomaly, and each one induces different operations
to incorporate a new belief and its explanation into the theory.
A natural consequence of this analysis is that the interpretation of Peirce's
abductive formulation goes beyond that of a logical argument, as it has generally
been understood. The abductive formulation takes the form of a process since
the modification of the background theory is involved.
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1.
1 This paper is a slightly modified version (and a translation) of "La abducción como cambio epistémico: C.S. Peirce y las teorías epistémicas en Inteligencia Artificial", published by Analogía Filosófica XII, No.1. 1988. México. pp. 125-144. A short version in English was presented at the 6th Congress of the IASS-AIS. International Association for Semiotic Studies in Guadalajara, México . July 13-18. I thank Michael Hoffman for his useful coments.
2. 2 The different conceptions of abduction go hand in hand with varied terminology. At the beginning Peirce used presumption and hypothesis (CP 2.776,2.623), as shown in the syllogistic form to come, to later use abductionand retroduction (CP 1.68, 2.776, 7.97), as we will see later on.
3.
3 There are many othes apporaches to abduction, such as bayesian networks and connectionism. In these areas the emphasis is in constructing algorithms to generate abuctions.
4.
4.Various ways of dealing with this issue occur in the literature. I mention only that in Gärdenfors (1988). 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 can retract those formulas which are 'least entrenched' first.
5.
5 In many cases, several formulas and not just one must be removed from the theory. E.g., BT={ab, a, b} and s=¬b. In order to have BT, ¬b consistent, either one of the sets {b, a} or {b, ab} must be removed.