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Abductive reasoning


From Wikipedia, the free encyclopedia

*Abduction* is a method of logical inference introduced by Charles Sanders
Peirce which comes prior to induction and deduction for which the
colloquial name is to have a "hunch". Abductive reasoning starts when an
inquirer considers of a set of seemingly unrelated facts, armed with an
intuition that they are somehow connected. The term /abduction/ is
commonly presumed to mean the same thing as hypothesis; however, an
abduction is actually the process of inference that produces a hypothesis
as its end result^[1]. It is used in both philosophy and computing.


Contents

[hide ]

* 1 Deduction, induction, and abduction

* 2 Logic-based abduction 
* 3 Set-cover abduction 
* 4 History of the concept 
* 5 Applications 
* 6 Abductive validation 
* 7 See also 
* 8 References 
* 9 Notes 
* 10 External links 


[edit ] Deduction, induction, and abduction

Main article: Logical reasoning

Deduction allows deriving /b/ as a consequence of /a/. In other words,
deduction is the process of deriving the consequences of what is assumed.
Given the truth of the assumptions, a valid deduction guarantees the truth
of the conclusion. It is true by definition and is independent of sense
experience. For example, if it is true (given) that the sum of the angles
is 180° in /all/ triangles, and if a certain triangle has angles of 90°
and 30°, then it can be deduced that the third angle is 60°.

Induction allows inferring /a/ entails /b/ from multiple instantiations of
/a/ and /b/ at the same time. Induction is the process of inferring
probable antecedents as a result of observing multiple consequents. An
inductive statement requires empirical evidence for it to be true. For
example, the statement 'it is snowing outside' is unjustified until one
looks or goes outside to see whether it is true or not. Induction requires
sense experience.

Abduction allows inferring /a/ as an explanation of /b/. Because of this,
abduction allows the precondition /a/ to be inferred from the consequence
/b/. Deduction and abduction thus differ in the direction in which a rule
like “/a/ entails /b/” is used for inference. As such abduction is
formally equivalent to the logical fallacy affirming the consequent or
/Post hoc ergo propter hoc /, because there are multiple possible
explanations for /b/.

Unlike deduction and in some sense induction, abduction can produce
results that are incorrect within its formal system. Hence the conclusions
of abduction can only be made valid by separately checking them with a
different method, either by deduction or exhaustive induction. However, it
can still be useful as a heuristic , especially when something is known
about the likelihood of different causes for /b/.


[edit ] Logic-based abduction

In logic , explanation is done from a logical theory /T/ representing a
domain and a set of observations /O/. Abduction is the process of deriving
a set of explanations of /O/ according to /T/ and picking out one of those
explanations. For /E/ to be an explanation of /O/ according to /T/, it
should satisfy two conditions:

* /O/ follows from /E/ and /T/;

* /E/ is consistent with /T/.

In formal logic, /O/ and /E/ are assumed to be sets of literals. The two
conditions for /E/ being an explanation of /O/ according to theory /T/ are
formalized as:

T \cup E \models O; T \cup E is consistent.

Among the possible explanations /E/ satisfying these two conditions, some
other condition of minimality is usually imposed to avoid irrelevant facts
(not contributing to the entailment of /O/) being included in the
explanations. Abduction is then the process that picks out some member of
/E/. Criteria for picking out a member representing "the best" explanation
include the simplicity , the prior probability , or the explanatory power
of the explanation.

A proof theoretical abduction method for first order classical logic based
on the sequent calculus and a dual one, based on semantic tableaux
(analytic tableaux ) have been proposed (Cialdea Mayer & Pirri 1993). The
methods are sound and complete and work for full first order logic,
without requiring any preliminary reduction of formulae into normal forms.
These methods have also been extended to modal logic .

Abductive logic programming is a computational framework that extends
normal logic programming with abduction. It separates the theory /T/ into
two components, one of which is a normal logic program, used to generate
/E/ by means of backward reasoning , the other of which is a set of
integrity constraints, used to filter the set of candidate explanations.


[edit ] Set-cover abduction

A different formalization of abduction is based on inverting the function
that calculates the visible effects of the hypotheses. Formally, we are
given a set of hypotheses /H/ and a set of manifestations /M/; they are
related by the domain knowledge, represented by a function /e/ that takes
as an argument a set of hypotheses and gives as a result the corresponding
set of manifestations. In other words, for every subset of the hypotheses
H' \subseteq H, their effects are known to be /e/(/H/').

Abduction is performed by finding a set H' \subseteq H such that M
\subseteq e(H'). In other words, abduction is performed by finding a set
of hypotheses /H/' such that their effects /e/(/H/') include all
observations /M/.

A common assumption is that the effects of the hypotheses are independent,
that is, for every H' \subseteq H, it holds that e(H') = \bigcup_{h \in
H'} e(\{h\}). If this condition is met, abduction can be seen as a form of
set covering .


[edit ] History of the concept

The philosopher Charles Sanders Peirce introduced abduction into modern
logic. 'The processes by which we form hunches about the world are, in
Peirce's conception, dependent on perceptual judgments, which contain
general elements such that universal propositions may be deduced from
them.'^[2] In his works before 1900, he mostly used the term to mean the
use of a known rule to explain an observation, e.g., “if it rains the
grass is wet” is a known rule used to explain that the grass is wet. In
other words, it would be more technically correct to say, "If the grass is
wet, the most probable explanation is that it recently rained."

Writing in 1910, Peirce admits that he himself, "in almost everything I
printed before the beginning of this century ... more or less mixed up
hypothesis and induction" ^[3] and he traces the confusion of these two
types of reasoning to logicians' too "narrow and formalistic a conception
of inference, as necessarily having formulated judgments from its
premises."^[4] When induction and abduction are presented as principle the
two stages of inference formation are easily collapsed into one
overarching concept – the hypothesis. This is why, in the scientific
method pioneered by Galileo and Bacon, the abductive stage of hypothesis
formation is conceptualized simply as induction. In the twentieth century
this collapse was reinforced by Karl Popper's explication of the
Hypothetico-deductive model , where the hypothesis is considered to be
just “a guess"^[5] (very much in the spirit of Peirce). However, when
the formation of a hypothesis is considered the result of a process it
becomes clear that this "guess" has already been tested and made more
robust in thought as a necessary stage of its acquiring the status of
hypothesis. Indeed many abductions are rejected or heavily modified by
subsequent abductions before they ever reach this stage.

Later Peirce described the process of science as a combination of
abduction, deduction and induction, stressing that new knowledge is only
created by abduction.

Now, that the matter of no new truth can come from induction or from
deduction, we have seen. It can only come from abduction; and abduction
is, after all, nothing but guessing. We are therefore bound to hope that,
although the possible explanations of our facts may be strictly
innumerable, yet our mind will be able, in some finite number of guesses,
to guess the sole true explanation of them. That we are bound to assume,
independently of any evidence that it is true. Animated by that hope, we
are to proceed to the construction of a hypothesis. ^[6]

The step of adopting a hypothesis or a proposition which would lead to the
prediction of what appear to be surprising facts is called abduction.
Abduction produces the hypothesis that is subsequently tested in induction
and applied in deduction (if it is found to be correct). This use is
contrary to the common use of /abduction/ in the social sciences and in
artificial intelligence , where the old meaning (as a rule to explain an
observation) is used. Contrary to this use, Peirce stated that the actual
process of generating a new rule is not “hampered” by logic rules.
Rather, he pointed out that humans have an innate ability to infer
correctly; possessing this ability is explained by the evolutionary
advantage it gives.

Norwood Russell Hanson , a philosopher of science , wanted to grasp a
logic explaining how scientific discoveries take place. He used Peirce's
notion of abduction for this ^[7] .

Further development of the concept can be found in Peter Lipton 's
"Inference to the Best Explanation" (Lipton, 1991).


[edit ] Applications

Applications in artificial intelligence include fault diagnosis , belief
revision , and automated planning . The most direct application of
abduction is that of automatically detecting faults in systems: given a
theory relating faults with their effects and a set of observed effects,
abduction can be used to derive sets of faults that are likely to be the
cause of the problem.

Abduction can also be used to model automated planning ^[8] . Given a
logical theory relating action occurrences with their effects (for
example, a formula of the event calculus ), the problem of finding a plan
for reaching a state can be modeled as the problem of abducting a set of
literals implying that the final state is the goal state.

Belief revision , the process of adapting beliefs in view of new
information, is another field in which abduction has been applied. The
main problem of belief revision is that the new information may be
inconsistent with the corpus of beliefs, while the result of the
incorporation cannot be inconsistent. This process can be done by the use
of abduction: once an explanation for the observation has been found,
integrating it does not generate inconsistency. This use of abduction is
not straightforward, as adding propositional formulae to other
propositional formulae can only make inconsistencies worse. Instead,
abduction is done at the level of the ordering of preference of the
possible worlds. Preference models use fuzzy logic or utility models.

In the philosophy of science , abduction has been the key inference method
to support scientific realism , and much of the debate about scientific
realism is focused on whether abduction is an acceptable method of
inference.

In historical linguistics , abduction during language acquisition is often
taken to be an essential part of processes of language change such as
reanalysis and analogy ^[9] .

In anthropology , Alfred Gell in his influential book /Art and Agency/
defined abduction, (after Eco^[10] ) as “a case of synthetic inference
'where we find some very curious circumstances, which would be explained
by the supposition that it was a case of some general rule, and thereupon
adopt that supposition”^[11] . Gell criticizes existing
'anthropological' studies of art, for being too preoccupied with aesthetic
value and not preoccupied enough with the central anthropological concern
of uncovering 'social relationships' specifically the social contexts in
which artworks are produced, circulated, and received ^[12] Abduction is
used as the basis of one gets from art to agency in the sense of a theory
of how works of art can inspire a /sensus communis,/ or the commonly-held
views that a characteristic of a given society because they are shared by
everyone in that society. ^[13] The question Gell asks in the book is,
‘how does initially to ‘speak’ to people?’ He answers by saying
that “No reasonable person could suppose that art-like relations between
people and things do not involve at least some form of semiosis .” ^[14]
However, he rejects any intimation that semiosis can be thought of as a
language because then he would have to admit to some pre-established
existence of the /sensus communis/ that he wants to claim only emerges
afterward out of art. Abduction is the answer to this conundrum because
the tentative nature of the abduction concept (Pierce likened it to
guessing) means that not only can it operate outside of any pre-existing
framework, but moreover, it can actually intimate the existence of a
framework. As Gell reasons in his analysis, the physical existence of the
artwork prompts the viewer to perform an abduction that imbues the artwork
with intentionality. A statue of a goddess, for example, in some senses
actually becomes the goddess in the mind of the beholder; and represents
not only the form of the deity but also her intentions (which are adduced
from the feeling of her very presence). Therefore through abduction, Gell
claims that art can have the kind of agency that plants the seeds that
grow into cultural myths. The power of agency is the power to motivate
actions and inspire ultimately the shared understanding that characterizes
any given society.^[15]


[edit ] Abductive validation

Abductive validation is the process of validating a given hypothesis
through abductive reasoning. This can also be called reasoning through
successive approximation. Under this principle, an explanation is valid if
it is the best possible explanation of a set of known data. The best
possible explanation is often defined in terms of simplicity and elegance
(see Occam's razor ). Abductive validation is common practice in
hypothesis formation in science ; moreover, Peirce argues it is a
ubiquitous aspect of thought:

Looking out my window this lovely spring morning I see an azalea in full
bloom. No, no! I do not see that; though that is the only way I can
describe what I see. That is a proposition, a sentence, a fact; but what I
perceive is not proposition, sentence, fact, but only an image, which I
make intelligible in part by means of a statement of fact. This statement
is abstract; but what I see is concrete. I perform an abduction when I so
much as express in a sentence anything I see. The truth is that the whole
fabric of our knowledge is one matted felt of pure hypothesis confirmed
and refined by induction. Not the smallest advance can be made in
knowledge beyond the stage of vacant staring, without making an abduction
at every step.^[16]

It was Peirce's own maxim that "Facts cannot be explained by a hypothesis
more extraordinary than these facts themselves; and of various hypotheses
the least extraordinary must be adopted." ^[17] After obtaining results
from an inference procedure, we may be left with multiple assumptions,
some of which may be contradictory. Abductive validation is a method for
identifying the assumptions that will lead to your goal.


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