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     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
author	Robert McIntyre <rlm@mit.edu>
date	Fri, 28 Oct 2011 00:03:05 -0700
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