# 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, $M$, $P$ 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

• 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, this 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) be extended to the whole collection.

Then one can ‘explain’ 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 $C x$, 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 through some law relating cause to effect.

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 $A \times B$, 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 there is given data $P \to O^I$ and $1 \to O^I$, and looks 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.

### Hypothetico-deductivism

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.

### Examples

#### Induction

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?

#### Abduction

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 $t$.

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

#### Brandom

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 January 12, 2020 at 11:32:30. See the history of this page for a list of all contributions to it.