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, 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.

Abduction

• 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 $1\to E$ through some law, $C \to E$, relating cause to effect via a map $1 \to C$, specifying that the cause is present.

Classifying kinds of abduction: Taking that model of abduction being a lift, we have given a map, $h: A \to C$, and we’re looking to provide an explanation in the form of a $B$ and two maps $f: A \to B$ and $g:B \to C$. I think it’s abductive in at least the following cases:

(1) There’s an existing $g: B \to C$ and we just need to establish $f: A \to B$, such that $g \cdot f = h$.

Example: Patient $X$ has a range of symptoms, $C$. Medical books tell us that people with disease $B$ have symptoms $C$. Hypothesis $X$ has disease $B$.

(2) There are a range of candidates $g_i: B \to C$. We need to find the best $g_i$, such that there’s a suitable $f_i: A \to B$ with $g_i \cdot f_i = h_i$.

Example: Person $Y$ is found dead, murdered in the ballroom. There were 6 people, $Z_i$, in the house that night, any of whom might have done it. We look for the best candidate hypothesis “$Z_k$ murdered $Y$.

Or, there’s a circular shadow on the floor. Is it caused by a sphere, a cyclinder, a disc, …? This makes the lift picture concrete, with the shadow projection $g$.

(3) We need to construct a suitable $B$ in a setting where we don’t even know what type of thing it might be. (Magnani 2001: ‘creative’ rather than ‘selective’.)

Being human, we’re rather too inclined to jump to a $B$ even when there’s very little evidence.

Example: why has my cow’s milk run dry, a plague of rats has rampaged through my barn, and my child has a fever? It must be witchcraft from that old woman over the hill.

Or should we take (2) to be about selection of a $b_i$ from a $B$. So, e.g., $B$ is the type of potential murderers, and we’re working out which term of $B$ is responsible?

But then (1) is rather similar. There’s a map from the type of diseases to the type of sets of symptoms. The choice then is of which particular disease is relevant.

Much depends on how close to being invertible is the map. If symptom clusters are very distinctive, it’s easy to find the disease. In psychiatry however there’s often a great overlap – low mood, lack of energy,…

Abduction in parameter estimation: given parameterized statistical models $P \to O^I$ and data $1 \to O^I$,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.

Induction

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 $A \to B$. Typically we need some extra structure on $A$, e.g., that there be some distance between points. Then nearest neighbor algorithms generate extensions.

What variation here? Seems that the three objects and two arrows are in place. It’s largely about choosing an arrow from a homset.

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 $Lan$, $Ran$, $Lift$, $Rift$, Notes, p. 3.

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.

Sometimes this final check is merely inductive. We can only try out some of the predictions.

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, $\mathbf{H}/A$, $x: A \vdash f: [B(x), C(x)]$ completes a triangle, is it an abduction? Say for $A$ football teams, $B(x)$ supporters of $x$, $C(x)$ players of $x$. Why does person $b: B(a)$ support team $a:A$? Because he likes best of all players $c:C(a)$ who plays for $a$.

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

The Para construction

Abstractly, we can think of $(P, f) : \mathbf{Para}(\mathcal{C})(A, B)$ as a learner which is learning a map of type $A \to B$ in $\mathcal{C}$. Our learning algorithm will search through the parameter space $P$ in order to find a $p : P$ such that the map $f(p, -): A \to B$ is best, according to some criteria.

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?

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)

References

• Matteo Capucci, Induction is induction

• Magnani, L. (2001). Abduction, reason, and science. Dordrecht: Kluwer.