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     1.4 +#+TITLE: Categorification of Plausible Reasoning
     1.5 +#+AUTHOR: Dylan Holmes
     1.6 +#+MATHJAX: align:"left" mathml:t path:"../MathJax/MathJax.js"
     1.7 +* COMMENT #+OPTIONS: LaTeX:dvipng
     1.8 +
     1.9 +* Deductive and inductive posets
    1.10 +
    1.11 +** Definition
    1.12 +If you have a collection \(P\) of logical propositions, you can order them by
    1.13 +implication: \(a\) precedes \(b\) if and only if \(a\) implies
    1.14 +\(b\). This makes \(P\) into a poset. Since the ordering arose from
    1.15 +deductive implication, we'll call this a /deductive poset/.
    1.16 +
    1.17 +If you have a deductive poset \(P\),  you can create a related poset \(P^*\)
    1.18 +as follows: the underlying set is the same, and for any two
    1.19 +propositions \(a\) and \(b\) in \(P\), \(a\) precedes
    1.20 +\(ab\) in \(P^*\). We'll call this an /inductive poset/.
    1.21 +
    1.22 +** A canonical map from  deductive posets to inductive posets
    1.23 +Each poset corresponds with a poset-category, that is a category with
    1.24 +at most one arrow between any two objects. Considered as categories,
    1.25 +inductive and deuctive posets are related as follows: there is a map
    1.26 +\(\mathscr{F}\) which sends each arrow \(a\rightarrow b\) in \(P\) to
    1.27 +the arrow \(a\rightarrow ab\) in \(P^*\). In fact, since \(a\) implies
    1.28 +\(b\) if and only if \(a = ab\), \(\mathscr{F}\) sends each arrow  in \(P\) to
    1.29 +an identity arrow  in \(P^*\) (specifically, it sends the arrow
    1.30 +\(a\rightarrow b\) to the identity arrow \(a\rightarrow a\)).
    1.31 +
    1.32 +
    1.33 +** Assigning plausibilities to inductive posets
    1.34 +
    1.35 +Inductive posets encode the relative (/qualitative/) plausibilities of its
    1.36 +propositions: there exists an arrow \(x\rightarrow y\) only if \(x\)
    1.37 +is at least as plausible as \(y\).
    1.38 +
    1.39 +*** Consistent reasoning as a commutative diagram
    1.40 +Inductive categories enable the following neat trick: we can interpret
    1.41 +the objects of \(P^*\) as states of given information and interpret
    1.42 +each arrow \(a\rightarrow ab\) in \(P^*\) as an inductive inference: the arrow
    1.43 +\(a\rightarrow ab\) represents an inferential leap from the state of
    1.44 +knowledge where only \(a\) is given to the state of knowledge where
    1.45 +both \(a\) and \(b\) are given\mdash{} in this way, it represents
    1.46 +the process of inferring \(b\) when  given \(a\), and we label the
    1.47 +arrow with \((b|a)\).
    1.48 +
    1.49 +This trick has several important features that suggest its usefulness,
    1.50 +namely
    1.51 + - Composition of arrows corresponds to compound inference.
    1.52 + - In the special case of deductive inference, the inferential arrow is an
    1.53 +   identity; the source and destination states of knowledge are the same.
    1.54 + - One aspect of the consistency requirement of Jaynes[fn:1] takes the form of a
    1.55 +   commutative square: \(x\rightarrow ax \rightarrow abx\) =
    1.56 +   \(x\rightarrow bx \rightarrow abx\) is the categorified version of
    1.57 +   \((AB|X)=(A|X)\cdot(B|AX)=(B|X)\cdot(A|BX)\).
    1.58 + - We can make plausibility assignments by enriching the inductive
    1.59 +   category \(P^*\) over some monoidal category, e.g. the set of real numbers
    1.60 +   (considered as a category) with its usual multiplication. /When we do/,
    1.61 +   the identity arrows of \(P^*\) \mdash{}corresponding to
    1.62 +   deductive inferences\mdash{} are assigned a value of certainty automatically.
    1.63 +
    1.64 +[fn:1] /(IIIa) If a conclusion can be reasoned out in more than one
    1.65 +way, then every possible way must lead to the same result./
    1.66 +
    1.67 +
    1.68 +*** Reciprocal probabilities
    1.69 +The natural numbers have a comparatively concrete origin: they are the
    1.70 +result of decategorifying the category of finite sets[fn:2], or the
    1.71 +coequalizer of the arrows from a one-object category to a two-object
    1.72 +category with a single nonidentity arrow. Extensions of the set of
    1.73 +natural numbers\mdash{} such as
    1.74 +the set of integers or rational numbers or real numbers\mdash{} strike
    1.75 +me as being somewhat more abstract.
    1.76 +
    1.77 +Jaynes points out that our existing choice of scale for probabilities
    1.78 +(i.e., the scale from 0 for impossibility to 1 for
    1.79 +certainty) has a degree of freedom: any monotonic function of
    1.80 +probability encodes the same information that probability does.
    1.81 +
    1.82 +With this in mind, it seems useful to use not /probability/ but
    1.83 +/reciprocal probability/ instead. This scale, which we might
    1.84 +tentatively call freeness, is a scale ranging 1 (certainty) to
    1.85 +positive infinity (impossibility).
    1.86 +
    1.87 +In this way, the ``probability''
    1.88 +associated with choosing one out of \(n\) indistinguishable choices
    1.89 +becomes identified with \(n\).
    1.90 +
    1.91 +The entropy 
    1.92 +
    1.93 +[fn:2] As Baez says.
    1.94 +
    1.95 +
    1.96 +
    1.97 +** self-questions
    1.98 +
    1.99 +What circumstances would make \(\mathscr{F}\) an injection?
   1.100 +
   1.101 +What if \(P=\{\top,\bot\}\)?
   1.102 +
   1.103 +
   1.104 +
   1.105 +** COMMENT 
   1.106 +Inductive and deductive posets are related as follows: there is a monotone
   1.107 +inclusion map \(\mathscr{i}:P^*\hookrightarrow P\) which\mdash{} since \(a\)
   1.108 +implies \(b\) is equivalent to \(a=ab\)\mdash{} sends comparable
   1.109 +propositions in \(P\) to the same proposition in \(P^*\). Conversely,
   1.110 +only comparable propositions in \(P\) are sent to the same proposition
   1.111 +in \(P^*\).
   1.112 +
   1.113 +
   1.114 +
   1.115 +** Inductive posets and plausibility
   1.116 +
   1.117 +* Inverse Probability