Conjunctive Versus Disjunctive Abduction —A Pragmatic Difference Between Abduction and Inverse Resolution

The philosopher Charles Sanders Peirce (see [1] for further details) distinguished three types of logical inference: deduction, induction and abduction. Deduction is a form of explicative inference; the conclusion follows necessarily from the premises. Induction and abduction, however, are forms of ampliative inference, the results of which do not follow necessarily from the premises. Harman [2] claims that (enumerative) induction is a special case of abduction or “inference to the best explanation”. We agree with Harman and examine one aspect where this claim appears to be borne out. We investigate inverse resolution [3] which is claimed to be inductive. The inverse resolution operators can in fact be shown to satisfy a definition of abduction. However, we claim that the difference between abduction and the inverse resolution V-operators is a pragmatic one: abductions are computed as conjunctions of literals while the V-operators produce disjunctions of literals. Notions of minimality are also important to both though in different ways. These results can also be extended to a first-order language. We also show that, in the propositional case, the inverse resolution W-operators lead to a conservative extension of the original theory. The W-operators introduce a new propositional symbol not present in the language of the original domain theory. However, since they lead to a conservative extension, they do not provide an increase in expressibility. This represents a limitation of this type of learning.