David Corfield deduction, induction, abduction

General idea

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, MM, PP and SS.

  • Deduction strings together, say, MM is PP and PP is SS to give MM is SS.

  • Induction looks to generalise from MM is SS, taking MM as a sample of PP, to conclude that PP is SS.

  • Abduction looks to explain why MM is SS, having noted that PP is SS, by hypothesising that MM is PP.

Peirce gives related examples:

  • Deduction

    • All beans in that bag are white.
    • These beans are from that bag.
    • Therefore, these beans are white.
  • Induction

    • These beans are from that bag.
    • These beans are white.
    • Therefore, all beans in that bag are white.
  • Abduction

    • These beans are white.
    • All beans in that bag are white.
    • Therefore, these beans are from that bag.

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.

  • Gerhard Schurz, 2008, Patterns of abduction, Synthese 164:201–234,

First example:

Known Law: If CxC x, then ExE x

Known Evidence: EaE a has occurred

Abduced Conjecture: CaC a could be the reason.

So perhaps we could see this as lifting a map from the unit type to some type of effects 1E1\to E through some law, CEC \to E, relating cause to effect via a map 1C1 \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×BA \times B, which may as well be a span between AA and BB, or ACBA \leftarrow C \to B. So induction looks for an extension ABA \to B.

Abduction in parameter estimation: there is given statistical models PO IP \to O^I and data 1O I1 \to O^I, and you look to lift to 1P1 \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.

Extremal liftings and extensions

There may be many candidates, e.g., many explanations. Which is the ‘best’?

In the setting of 2-categories, there are the constructions LanLan, RanRan, LiftLift, RiftRift, 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, H/A\mathbf{H}/A, x:Af:[B(x),C(x)]x: A \vdash f: [B(x), C(x)] is an abduction? Why does bb support tt, because he likes cc who plays for tt.

  • Forming a prequantum bundle is lifting XΩ n+1X \to \Omega^{n+1} to XB nU(1) connX \to B^n U(1)_{conn}.

Database and lifting

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 March 26, 2022 at 03:44:34. See the history of this page for a list of all contributions to it.