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New article: Inductive lattices
author Dylan Holmes <ocsenave@gmail.com>
date Tue, 01 Nov 2011 01:55:26 -0500
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1 # -*- mode: org -*-
4 Archived entries from file /home/r/aurellem/src/categorical/plausible.org
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 \((b|a)\).
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.
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./