A new characterization of abduction and induction is proposed, which is based on the idea that the various aspects of the two kinds of inference rest on the essential features of increment of (so to speak, extensionalized) comprehension and, respectively, of extension of the terms involved. These two essential features are in a reciprocal relation of duality, whence the highlighting of the dual aspects of abduction and induction. Remarkably, the increment of comprehension and of extension are dual ways to realize, in the limit, a 'deductivization' of abduction and induction in a similar way as the Closed World Assumption does in the case of the latter.
As is well known, the rst logical analysis of abduction (and the term itself)
has been given in the works of Charles Sanders Peirce. His vast speculation on
abduction has been subdivided by Fann [
What has remained of the rst theory, mainly in the AI and logic programming literature, is the triad of examples relating deduction, induction, and abduction (2.623),3 of which the pertinent one is the following: (Rule) All the beans from this bag are white (Result) These beans are white (Case) These beans are from this bag
On the other hand, the later peircean theory of abduction is summarized by this notorious reasoning schema (5.189), to which a massive literature is dedicated, especially in the eld of philosophy of science:
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.
The would be a matter of course implies something like the fact that A is a cause or at any rate an explanation of C (as this is what generally accounts for C being ‘naturally’ expected, given A),4 whence the matching of abduction with (backward) causal reasoning and/or IBE.
Despite the formal clarity of ( Abd1) and the informal intent of ( Abd2), often abduction is represented by this inference schema:
M ! P P M
Abd0
Now, presumably the ‘rule’ ( Abd0) is simply to be conceived as a forgetful
interpretation of (Abd2), and of this other schema (extracted from ( Abd1)),
which is at the heart of ALP:
3 I’m using the standard quotation format for the Collected Papers of C. S. Peirce [
AbdQ
But the fact is that the schema ( Abd0) ‘forgets too much’: if we want to characterize abduction as IBE, we must operate in a causal/explanatory framework; while if we want to talk of abduction in a purely logical way, e.g. when doing logic programming, we must at least use predicate logic. A formulation in terms of mere propositional logic like that of ( Abd0), besides being vaguely evocative, is not sucient for neither approach; and, what is worst, it leaves the door open to easy criticisms against abduction in general (see note 8 below).
On the other side, it must be said that, often, many useful logical features of reasoning are captured in terms of pure propositional logic; so the problem is: can we characterize abduction within propositional logic in a less trivial and more illuminating way than ( Abd0) does? And can we extract from such a characterization any new (or, up to now, largely overlooked) feature of abductive reasoning? The answer to both questions is I hope positive, as will be shown in the next section. 2
The analysis of abduction and of its relation of duality with induction which is to follow is based on, and is an explication of, the works belonging to the early logical speculation of Peirce, which are framed in a syllogistic (and, later, probabilistic) setting.
Other works insisting on the duality between abduction and induction are [
I shall represent induction and abduction by the following inference schemata,
which are dual to each other:
5 Few remarks about the importance of the duality relation. It is a key concept in
category theory; in fact, category theory itself seems to have been born to solve a
duality problem, with Mac Lane’s 1950 article Duality in Groups (cf. [
S1 _ S1 ! P . .
.
Sj ! P M ! P _ Sj ! M
M ! P1 ^ Ind
S ! P1 . .
.
S ! Pk S ! M
Abd
These denitions arise as a way to connect the schema ( Abd1) above (p. 2), and the analogous schema for induction, to the earliest characterization of inducton and abduction given by Peirce (in (2.511), and also in (2.424425)), in which multiple subject/predicate classes are explicitly inserted.
The natural means (suggested, more or less in clear terms, by Peirce himself in (2.461516)) to move from a multiplicity of objects to a single class is simply to take their union or intersection,6 if they are subjects or, respectively, predicates.
The enlargement of a class by adding members to an union, and the shrinkage of a class by adding members to an intersection, are respectively the extensional counterparts of the concepts of breadth and depth of a term (i.e. a class), which Peirce investigates thoroughly in (2.407 .). They are akin to the concepts of extension and comprehension,7 and to the aforementioned notions of range and content examined in the early hempelian analysis of scientic explanation (see note 5).
The central aspect in the present approach is the idea that induction and
abduction are essentially based upon, respectively, the enlargement and the
shrinkage of classes, all the other features being derived from these. Console
and Saitta [
To see how the inductive and abductive inferences work in this conceptual framework, consider Fig. 1 and Fig. 2, which illustrate what happens in (the set-theoretical versions of) the inferences ( Ind ) and (Abd ) with j = k = 2. Briey, an inductive inference is the more conrmed/plausible the more subjects it examines, up to the ideal point when the subject class has been enlarged so much that it coincides with the middle term class; at which point all the tests Sj ! P are passed, so that M n P = ? (the shaded area, i.e. the potential 6 I take for granted the shifting from predicate logic to set notation and then to propositional logic, in the case of categorical propositions of the form A (universal aermative); so that 8x (A(x) ! B(x)) becomes A B, which in turn can be written as A ! B. Taking !; :; _; ^ instead of ; n; [; \ (as indeed happens in the schemata (Ind ) and (Abd )) is really a way to let propositional logic speak for the relevant logical features of objects otherwise represented by predicate logic. 7 Things are actually more complicated: Peirce makes a distinction between essential, substantial and informed breadth and depth, each subjected to dierent rules; and he is critical precisely of the careless attening of these concepts on those of extension and comprehension.
M
S1
S2 P1
M counterexample space for the conclusion, is empty), and the arrow in the rst premise of (Ind ) can be inverted, so that a deduction is obtained.
Dually, an abductive inference is the more conrmed/plausible the more properties of subjects it considers, up to the ideal point when the predicate class has been shrunk so much that it coincides with the middle term class. That being the case, all the tests S ! Pk are passed, so that S n M = ? (again, the shaded area, i.e. the potential counterexample space, is empty), and the arrow in the rst premise of ( Abd ) can be inverted, so as to get a deduction.
This reversal of arrows in induction and abduction is linked with completion
or circumscription techniques, so basically with the Closed World Assumption .
On this point an interesting analysis has been made by Lachiche [
The present view of induction and abduction is an ‘incremental’ one: we increase the number of tests up to the limiting point in which all of the relevant subjects (induction) or all of the relevant properties (abduction) have been tested; even before reaching this ideal point, we get more and more condent about our conclusion as we proceed further. 8 In the opposite direction, we have as limiting/trivial cases, respectively, single-case induction and ( AbdQ), as in both these inference forms only a single test is taken into consideration.
In this work I have considered implications (i.e. inclusions) as categorical, and tests as yes/no questions; it may be interesting, then, to extend this approach in order to capture probabilistic features as well. This in turn is needed if one wants to dene some measure of conrmation/plausibility and, possibly, to implement the new inference gures in a logic programming system. 8 Thus it can be avoided, or at least minimized, the problem of ( Abd0) committing the fallacy of arming the consequent : an eect can have, e.g., two competing causes/explanations; but it is improbable that for a conjunction of eects (i.e. for a more detailed description of the eect in question or, in the limit, for its ‘maximally detailed’, or ‘complete’, description) both of the same two explanations remain available. I wish to thank a reviewer for having pointed this matter.