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Sunday morning thought: I was taking a look at 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, , and .
Deduction strings together, say, is and is to give is .
Induction looks to generalise from is , taking as a sample of , to conclude that is .
Abduction looks to explain why is , having noted that is , by hypothesising that is .
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. And I guess one can ’explain’ a phenomenon, such as a shadow, occurring in a base space by lifting to the total space.
This would be a good insight to put at abductive reasoning.
See also Abduction: A Categorical Characterization, which develops a model for abduction as a functor between categories of structures, and then a topos of “abduction procedures”. I will add this reference to abductive reasoning. Is their framework itself a “lifting” of the syllogistic schema here? :-)
Thanks, Matt. That paper looks quite involved, but it has pointed me to
which I’ve added. Let’s see, first example:
Known Law: If , then
Known Evidence: has occurred
Abduced Conjecture: 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.
Hypothetico-deductivism: given converging arrows, look for a lift to complete triangle. Test lift by postcomposing with 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 know that all beans in that bag are large, so test whether all of these beans are large.
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