Mercurial > dylan
comparison categorical/plausible.org_archive @ 2:b4de894a1e2e
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| author | Robert McIntyre <rlm@mit.edu> |
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| date | Fri, 28 Oct 2011 00:03:05 -0700 |
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| 1 # -*- mode: org -*- | |
| 2 | |
| 3 | |
| 4 Archived entries from file /home/r/aurellem/src/categorical/plausible.org | |
| 5 | |
| 6 * Consistent reasoning as a commutative diagram | |
| 7 :PROPERTIES: | |
| 8 :ARCHIVE_TIME: 2011-07-09 Sat 01:00 | |
| 9 :ARCHIVE_FILE: ~/aurellem/src/categorical/plausible.org | |
| 10 :ARCHIVE_OLPATH: Deductive and inductive posets/Assigning plausibilities to inductive posets | |
| 11 :ARCHIVE_CATEGORY: plausible | |
| 12 :END: | |
| 13 Inductive categories enable the following neat trick: we can interpret | |
| 14 the objects of \(P^*\) as states of given information and interpret | |
| 15 each arrow \(a\rightarrow ab\) in \(P^*\) as an inductive inference: the arrow | |
| 16 \(a\rightarrow ab\) represents an inferential leap from the state of | |
| 17 knowledge where only \(a\) is given to the state of knowledge where | |
| 18 both \(a\) and \(b\) are given\mdash{} in this way, it represents | |
| 19 the process of inferring \(b\) when given \(a\), and we label the | |
| 20 arrow with \((b|a)\). | |
| 21 | |
| 22 This trick has several important features that suggest its usefulness, | |
| 23 namely | |
| 24 - Composition of arrows corresponds to compound inference. | |
| 25 - In the special case of deductive inference, the inferential arrow is an | |
| 26 identity; the source and destination states of knowledge are the same. | |
| 27 - One aspect of the consistency requirement of Jaynes[fn:1] takes the form of a | |
| 28 commutative square: \(x\rightarrow ax \rightarrow abx\) = | |
| 29 \(x\rightarrow bx \rightarrow abx\) is the categorified version of | |
| 30 \((AB|X)=(A|X)\cdot(B|AX)=(B|X)\cdot(A|BX)\). | |
| 31 - We can make plausibility assignments by enriching the inductive | |
| 32 category \(P^*\) over some monoidal category, e.g. the set of real numbers | |
| 33 (considered as a category) with its usual multiplication. /When we do/, | |
| 34 the identity arrows of \(P^*\) \mdash{}corresponding to | |
| 35 deductive inferences\mdash{} are assigned a value of certainty automatically. | |
| 36 | |
| 37 [fn:1] /(IIIa) If a conclusion can be reasoned out in more than one | |
| 38 way, then every possible way must lead to the same result./ | |
| 39 | |
| 40 |