Abduction: To find assumptions from which given conclusions follows; finding hypotheses
or theories to account for facts.
The general form of an abduction
is accordingly: Given facts F for which one wants an explanation, to infer a hypothesis H such that H deductively entails F. An example of C.S.
Peirce makes this clear: Given a sack of beans from which 10 beans
have been sampled, that are all white, to infer by abduction that all
the beans in the bag are white.
1. About the confusions of
abduction, deduction and induction
C.S. Peirce was the
first to stress that abduction is a kind of inference in its own right, though
Aristotle might have been thinking of something like it when he wrote
about "epagoge", which is normally translated as "induction".
Indeed,
abduction and induction have been
much confused, and the usual examples of "inductive generalizations",
of the kind "if suchandsuch holds for a sample
(of a certain kind), then suchandsuch holds for the population", are often better called
abductions, simply because they are not deductively valid, while the
assumption that the population is suchandsuch a kind does explain why
a sample from it would also be of the kind.
Similarly, abduction and deduction
are often confused, if not as often as in the previous case: The sort
of "deductions from the evidence" that
Sherlock Holmes and Hercule Poirot glorify in are nearly always abductions
(explanations) rather than deductions (consequences). See: Theory.
There are two reasons
why Peirce was quite right in calling abduction
an inference: First, because it is
one by definition, although it is also quite true that a good abduction
from given facts to a hypothesis that explains
it deductively if it is true is rarely deductively correct that the hypothesis
deductively follows from from the facts , even though to be a good
deductive explanation the facts must deductively follow from
the hypothesis and such further knowledge one has.
And secondly, because it is especially
abductions that require creative
imagination. The reason is that any given fact (or possibility) may be accounted for
in infinitely many distinct ways, and so to guess a hypothesis that is consistent with
such evidence
as one has may be a great feat of creative imagination.
2. Testing abductions
Since abductions are
essentially creative leaps to explanations,
one important problem is how to test them. There are two tests for
abductions: By deduction and by induction.
The deductive test is
that if T is stated to be an explanation for F, that indeed one can
prove that, given one's other assumptions A, T&A deductively
entails F. This is not sufficient to prove that T is true, but it is
sufficient to prove that (1) T explains F in logical principle if
possibly not in actual fact and that (2) either T or A is false if F is
false. This second logical fact does entail possibilities for testing
T, for it enables one to definitely refute it if one finds that F is
false and if one knows that all one's other assumptions A one used are
true.
The inductive test is
that if T is stated to be an explanation for F, and T is not known to
be false, and one can deduce another statement G from T and such other
assumptions A one uses, and one finds that G is true, then one has
therewith found some evidence that supports T, and that normally makes
T more probable than it was before.
3.
Explaining abduction formally
It is not difficult to
give a general settheoretical outline of
what is involved in abduction. To explain a given fact F, the general
formula comes to this:
 Find a mapping i of
one's domain of theories and facts to a similar fact F* such that there
are a mapping j and a theory T* with the property that T* entails F*
and the value of j(T*) is included in a theory T that has the
properties of entailing and F and to be such that i(T) is included in T*
A diagrammatic picture
of this looks like this:
Thus one starts with F
in need of an explanation, thinks of something similar F* with the help
of a mapping i that relates F to an F* that one also knows and that is
explained by a theory T* that one knows, and then thinks of a new
theory T derived from T* by some mapping j such that T explains F and T
itself maps to a subset of T* by i.
Hence there are several
distinct psychological steps one must go through to find an
explanation, that may be summarized schematically thus:
To explain an F that has
no obvious, clear or satisfactory explanation, try to think of a
restatement of F that one can explain by some theory T*, and then try
to think of some restatement or extension or restriction of T* from
which F would follow.
Note also that, as
stated, and in settheory, what
one ends up with generally satisfies these two inclusions:
j(T*) inc T
i(T) inc T*
and that if these
inclusions are proper, quite a lot may have been added to j(T*) to
obtain a satisfactory T to explain F.
In general, i and j will
be similarities of some kind, and
part of the checking that an abduction is correct is a proof (or at
least some plausible argument) that in fact T* entails F* and in fact T
entails F.
Of course, further
desirable properties for the T one found is that it does not entail
anything one knows to be false, and that it does entail some G that one
can experimentally investigate, and that strenghtens the probability of
T if verified and weakens the probability of T if falsified. See also induction, inference.
