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Deduction, induction, abduction

Take a basic syllogism:

All the balls in that urn are red.
This ball is picked from that urn.

This ball is red.

We can rewrite it like this:

\((\forall x)(FromUrn(x) \rightarrow Red(x))\)
\(FromUrn(y)\)

\(Red(y)\)

This is deduction. Given the two premises, the conclusion must follow. (Technically, modus ponens is being used here.)

If we switch the two assumptions, we naturally get the same conclusion; it’s still a deductive inference.

Now, let’s reorganize a little bit differently:

This ball is picked from that urn.
This ball is red.

?

What is the conclusion. Well, nothing follows deductively, but let’s be a little more loose about the logic. What if we pick 100 balls and they’re all red. What can we say?

We can say,

This ball is picked from that urn.
This ball is red.

All the balls in that urn are red.

That’s induction. Given one (usually many) instances of a phenomenon (the ball being red after being picked from that urn) can give us evidence (strong or weak depending on how many tests we perform) that some general law is in effect. The general law here is, “All the balls in that urn are red.”

Of course, we can test our inductive conclusion (the general law) by performing more experiments, and ensuring that after each trial, the law plus the fact that the ball we picked from that urn implies the ball is red (that is, we test our predictions given by the general law).

Again, if we reorder the premises, we still have induction.

There is one last reordering:

All the balls in that urn are red.
This ball is red.

?

The missing information now is, where did this ball come from? The only statement not mentioned is:

All the balls in that urn are red.
This ball is red.

This ball is picked from that urn.

Now we are performing abduction (a term coined by Charles Sanders Peirce in the early 1900s; note, his name is pronounced like “purse”). The conclusion is an explanation that gives the cause behind what we have observed (this ball is red).

There is a strong similarity between induction and abduction. Both are weaker forms of reasoning than deduction (because their conclusions may be false). However, induction typically requires many experiments to be a strong conclusion, whereas abduction may have a strong conclusion from just one instance because abduction typically describes causal relationships. “This ball is picked from that urn” causes the ball to be red (with background knowledge that they are all red in that urn). On the other hand, induction says “they are all rved in that urn,” which does not cause the picked ball to be red. This is a subtle point, so we’ll move on (for more information, read the paper “Smart inductive generalizations are abductions” by Josephson).

Inductive/abductive logic programming

We learned that Prolog…

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