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 such-and-such holds for a sample (of a certain kind), then such-and-such holds for the population", are often better called abductions, simply because they are not deductively valid, while the assumption that the population is such-and-such 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 set-theory, 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.