Sunday, November 22, 2015

The Interrogative Interpretation of Abductive Inference

Peirce famously suggested that besides the common forms of inference, induction and deduction, there is also a third, as important, which he called abduction. One of his major motivations was an analogy with figures of the syllogism -- there are three original figures of syllogism (the Fourth is a later addition not original to Aristotle's scheme). In this analogy, deduction seemed to correspond to the First Figure and induction to correspond to the Third Figure, which left the question -- what corresponds to the Second Figure? The most famous version of this scheme of figures of reasoning is the one in which Peirce compares the forms of inference in terms of how they handle the Rule, the Case, and the Result:

Deduction
Rule: All the beans in this bag are white.
Case: These beans are from this bag.
Result: These beans are white.

Abduction
Rule: All the beans in this bag are white.
Result: These beans are white.
Case: These beans are from this bag.

Induction
Case: These beans are from this bag.
Result: These beans are white.
Rule: All the beans in this bag are white.

The pattern of terms in the figures of the syllogism are clearly maintained here.

But there have always been puzzles and peculiarities with how abduction actually can work. It doesn't seem to be truth-preserving, for instance. At times Peirce seems to characterize it as the rational form of guessing. Sometimes it sounds like pattern-recognition. At other times he treats it like idea-construction. He often suggested that abduction is hypothesis-making (one then uses deduction to get consequences which are tested by induction). Peirce also seems to have gone back and forth on how exactly to distinguish abduction from induction, and to have stepped away, over time, from the syllogistic analogy.

I mentioned that abduction as Peirce conceives it can't be truth-preserving: given true premises, there is no guarantee that the conclusion is true. But it is very clear that abduction as Peirce conceives has to be possibility-preserving: as long as the premises are possible (or perhaps true), they establish that the conclusion is possible. This is something that I've thought about for quite some time.*

In the recent IEP article on Peirce's Logic, Bellucci and Pietarinen note a recently discovered interpretation of abduction in a letter to Lady Welby that gives a rather different account of abduction than the standard versions.** The contrast is with modus tollens:

Modus Tollens
If A is true, C {is/is not} true.
C {is not/is} true.
Therefore A is not true.

Abduction
If A is true, C {is/is not} true.
C {is/is not} true.
Therefore is A not true?

The conclusion, in other words is in 'interrogative mood', or more precisely, is equivalent to: It is to be inquired whether A is not true.

This is, I believe, closely related to the point I made above about possibility-preservation, with a particular interpretation of possibility, the one that Peirce saw as most relevant. One way to put it: if we take Diamond or weak modality to posit something for investigation or inquiry, then from the premises, an abductive inference gives us a proposition whose truth value is Diamond (interpreted as positing for investigation) rather than True.

The 'positing for investigation' is actually quite substantive, as Peirce understands it; it means that we have reason to invest resources into the inquiry -- which is indeed about what we usually mean when we say something is a possible topic for inquiry, since we don't ever take this kind of possibility to be the bare abstract possibility of being something into which some possible inquirer could possibly inquire under some possible circumstances.

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* It shows up, for instance, in a Dashed Off post in July 2011. Dashed Off posts usually lag behind the original jotting of the notes by six months to a year and a half (although I sometimes clean up the notes when putting them in posts), so this is probably something I started thinking about explicitly by early 2011 at the latest.

** But looking back I see that I was aware of it several years ago; in a Dashed Off post I recorded some jotted-down notes on how abduction might be understood:

abduction as recognition of phenomena as an icon of a symbol (a likeness of a general conception) (Peirce EP 2:287)

abduction leading to conclusions in interrogative mood

abduction as concerned with economy of money, time, thought, and energy (Peirce CP 5.600)

abduction : inference through icon :: induction : inference through index :: deduction : inference through symbol

abduction as divine: NEM 3.206; CP 8.212; CP 6.476-477 MS 843.7
(cp Peirce on agapistic evolution)

abduction as guided by the notion of good

The second note explicitly identifies the interrogative mood interpretation, and the third recognizes the 'worth-the-expense' aspect of Peirce's understanding of the 'interrogative mood'. Again, there's a lag between the original notes and the Dashed Off posts, and the lags have tended to grow longer in the past few years, so this probably goes back to 2012.

This is a reason why it's handy to take notes; I would not have remembered coming across the interrogative interpretation at all. It also provides a reminder that our inquiries, if they are extensive, are often so complex that we cannot trace through everything that has been involved in them. The notes themselves are just quick snapshots; they don't record everything that was going on in my mind, and, indeed, I don't know at all what I thought about the interrogative interpretation at the time. But even so, they show that my bits-and-pieces studies of Peirce has covered ground I don't even remember covering. Peirce, I think, would be pleased at both the example and its moral.

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