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