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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeApr 9th 2017
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexSunday morning thought: I was taking a look at the [lectures](http://www.hup.harvard.edu/catalog.php?isbn=9780674749672) 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](https://ncatlab.org/nlab/show/abductive+reasoning) (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 $P$. 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.

   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.

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeApr 9th 2017
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   Author: TobyBartels
   Format: MarkdownItexThis would be a good insight to put at [[abductive reasoning]].

   This would be a good insight to put at abductive reasoning.

3. • CommentRowNumber3.
   • CommentAuthorMatt Earnshaw
   • CommentTimeApr 9th 2017
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   Author: Matt Earnshaw
   Format: MarkdownItexSee also [Abduction: A Categorical Characterization](https://www.dropbox.com/s/n7v39plp9828um8/abduction-categorial-characterization.pdf?dl=0), 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? :-)

   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? :-)

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeApr 10th 2017
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   Author: David_Corfield
   Format: MarkdownItexThanks, Matt. That paper looks quite involved, but it has pointed me to * Gerhard Schurz, 2008, Patterns of abduction, Synthese 164:201–234, which I've added. Let's see, 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.

   Thanks, Matt. That paper looks quite involved, but it has pointed me to

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

   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.

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 29th 2017
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   Author: David_Corfield
   Format: MarkdownItexHypothetico-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.

   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.