In the lectures by Charles Peirce of that name, he gives one of those triads he loves so much on different kinds of reasoning: deductive, inductive, and abductive (or retroductive), as filling in different parts of a syllogism. So there are logical relations between 3 concepts, $M$, $P$ and $S$.
Deduction strings together, say, $M$ is $P$ and $P$ is $S$ to give $M$ is $S$.
Induction looks to generalise from $M$ is $S$, taking $M$ as a sample of $P$, to conclude that $P$ is $S$.
Abduction looks to explain why $M$ is $S$, having noted that $P$ is $S$, by hypothesising that $M$ is $P$.
Peirce gives related examples:
Deduction
Induction
Abduction
Seen from the point of view of category theory, these would seem rather like: composition, extension, lifting.
Induction as a kind of extension seems quite reasonable, when we ask whether a sample satisfying a property (so a mono from a small set) may be extended to the whole collection.
Then one can ‘explain’ via abduction a phenomenon, such as a shadow, occurring in a base space by lifting to the total space.
First example:
Known Law: If $C x$, then $E x$
Known Evidence: $E a$ has occurred
Abduced Conjecture: $C a$ could be the reason.
So perhaps we could see this as lifting a map from the unit type to some type of effects $1\to E$ through some law, $C \to E$, relating cause to effect via a map $1 \to C$, specifying that the cause is present.
Check out: can we say that induction in one category is abduction in the opposite, so that they are dual?
slide 20 of this talk by Dan Ghica et al: talk,
Induction starts there from a list of pairs in $A \times B$, which may as well be a span between $A$ and $B$, or $A \leftarrow C \to B$. So induction looks for an extension $A \to B$.
Abduction in parameter estimation: there is given statistical models $P \to O^I$ and data $1 \to O^I$, and you look to lift to $1 \to P$. So parameter selection is abduction? You have an empirical distribution and you realise it as the value of a parameter in a statistical model.
There may be many candidates, e.g., many explanations. Which is the ‘best’?
In the setting of 2-categories, there are the constructions $Lan$, $Ran$, $Lift$, $Rift$, Notes, p. 3.
Given converging arrows, look for a lift to complete the triangle. Test the proposed lift by post-composing with a new arrow.
Extending Peirce’s example:
Abductive lift: All of these beans are white, all beans in that bag are white. Perhaps all these beans come from that bag.
Deductive postcomposition: But we also know that all beans in that bag are large, so test whether all of these beans are large.
Is any extension an induction? Most plausible when along a mono.
Mathematical recursion answers the problem of how to extend from 0 and successor.
If we can see a differential equation as an extension problem, can this be viewed as a form of induction?
Is any lift an abduction? Plausible only for an epi? A map in a context/slice, $\mathbf{H}/A$, $x: A \vdash f: [B(x), C(x)]$ is an abduction? Why does $b$ support $t$, because he likes $c$ who plays for $t$.
Database enquiring as lifting problems, finding a diagonal in a square, in Spivak’s DATABASE QUERIES AND CONSTRAINTS VIA LIFTING PROBLEMS. Combination of abduction and induction?
Authors of Introduction to From Rule to Meaning claim that Brandom’s commitment-preserving, entitlement-preserving and incompatibility relations correspond to deductive, inductive and modal inferences. (p. 7)
Last revised on September 13, 2021 at 02:54:28. See the history of this page for a list of all contributions to it.