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