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     1.2 +++ b/categorical/plausible.org_archive	Fri Oct 28 00:03:05 2011 -0700
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     1.4 +#    -*- mode: org -*-
     1.5 +
     1.6 +
     1.7 +Archived entries from file /home/r/aurellem/src/categorical/plausible.org
     1.8 +
     1.9 +* Consistent reasoning as a commutative diagram
    1.10 +  :PROPERTIES:
    1.11 +  :ARCHIVE_TIME: 2011-07-09 Sat 01:00
    1.12 +  :ARCHIVE_FILE: ~/aurellem/src/categorical/plausible.org
    1.13 +  :ARCHIVE_OLPATH: Deductive and inductive posets/Assigning plausibilities to inductive posets
    1.14 +  :ARCHIVE_CATEGORY: plausible
    1.15 +  :END:
    1.16 +Inductive categories enable the following neat trick: we can interpret
    1.17 +the objects of \(P^*\) as states of given information and interpret
    1.18 +each arrow \(a\rightarrow ab\) in \(P^*\) as an inductive inference: the arrow
    1.19 +\(a\rightarrow ab\) represents an inferential leap from the state of
    1.20 +knowledge where only \(a\) is given to the state of knowledge where
    1.21 +both \(a\) and \(b\) are given\mdash{} in this way, it represents
    1.22 +the process of inferring \(b\) when  given \(a\), and we label the
    1.23 +arrow with \((b|a)\).
    1.24 +
    1.25 +This trick has several important features that suggest its usefulness,
    1.26 +namely
    1.27 + - Composition of arrows corresponds to compound inference.
    1.28 + - In the special case of deductive inference, the inferential arrow is an
    1.29 +   identity; the source and destination states of knowledge are the same.
    1.30 + - One aspect of the consistency requirement of Jaynes[fn:1] takes the form of a
    1.31 +   commutative square: \(x\rightarrow ax \rightarrow abx\) =
    1.32 +   \(x\rightarrow bx \rightarrow abx\) is the categorified version of
    1.33 +   \((AB|X)=(A|X)\cdot(B|AX)=(B|X)\cdot(A|BX)\).
    1.34 + - We can make plausibility assignments by enriching the inductive
    1.35 +   category \(P^*\) over some monoidal category, e.g. the set of real numbers
    1.36 +   (considered as a category) with its usual multiplication. /When we do/,
    1.37 +   the identity arrows of \(P^*\) \mdash{}corresponding to
    1.38 +   deductive inferences\mdash{} are assigned a value of certainty automatically.
    1.39 +
    1.40 +[fn:1] /(IIIa) If a conclusion can be reasoned out in more than one
    1.41 +way, then every possible way must lead to the same result./
    1.42 +
    1.43 +